The Rockwell-MBB X-31 answered its research question by fighting, and no other aircraft in this series answered its question that way. The X-29 measured a divergence boundary. The X-30 was cancelled because the quantity it existed to establish could not be established on the ground. The X-31 was built, two were flown, and the thing it was meant to settle was settled by flying it against other aeroplanes and counting who won. The definitive account is Joyce’s history of the programme, published by the National Aeronautics and Space Administration, NASA, and available in full, and the Dryden fact sheet and the inventory of X-vehicles carry the specifications, the latter also on the reports server. This is the thirty-second article in the X-Planes series, following the X-1, the X-2, the X-3, the X-4, the X-5, the X-6, the X-7, the X-8, the X-9, the X-10, the X-11, the X-12, the X-13, the X-14, the X-15, the X-16, the X-17, the X-18, the X-19, the X-20, the X-21, the X-22, the X-23, the X-24, the X-25, the X-26, the X-27, the X-28, the X-29, and the X-30.

The easy misreading of this aircraft is that it existed to find out whether thrust vectoring would work. It did not. Exhaust vanes deflecting a jet to produce a control force were first used in the mid-1940s on the German V-2 ballistic missile, a rotatable nozzle flew on the Hawker Siddeley Harrier in a different form, and paddles developed by the same manufacturer’s Columbus division were flown from 1986 to 1987 on an F-14 before the X-31 team adopted them. Nothing about the mechanism was in doubt.

What was in doubt was whether the resulting capability is worth having. An aircraft that can hold seventy degrees of angle of attack under control can point its nose a long way off its flight path. Whether that wins a fight is a different question from whether it is achievable, and no amount of wind-tunnel work can answer it. It has to be tried.

The Research Question

The keystone is the tactical utility of post-stall manoeuvring, and it is measured as a rate rather than as a quantity.

Every previous aircraft in this series was dimensioned against a number. The X-1 existed to measure a drag rise, the X-15 to measure heating, the X-29 to measure a divergence boundary. Each of those is a quantity with units, and an instrument can be pointed at it.

The standard treatment of close combat as a discipline is Shaw’s account of fighter tactics and manoeuvring, and it reaches its conclusions about positional advantage without any of the apparatus below.

The X-31’s question has no units. It asks whether an advantage exists, and the only instrument that can be pointed at it is a designed sequence of trials with an outcome recorded for each one. That makes the X-31 the first aircraft in this series whose keystone is an experimental design rather than a measurement, and it changes what the programme had to get right. A thermocouple in the wrong place gives a wrong temperature. A badly chosen starting condition gives a wrong answer about whether an entire class of aeroplane is worth building.

Why the Keystone Is Not the Vectoring

The vectoring system is a means. It produces a yawing or pitching moment when the aerodynamic surfaces cannot, and the size of that moment is computable from the thrust, the deflection angle and the moment arm. The flight reports do not state the arm, and the calculation later in this article turns instead on the speed above which no arm the fuselage could hold would have been long enough.

The vectoring exists because a wing at seventy degrees of angle of attack is not producing useful control forces, and seventy degrees of angle of attack was wanted because at that attitude the nose points where the aircraft is not going. The aim-off angle available to a pilot is the angle of attack itself. A conventional fighter limited to thirty degrees can aim thirty degrees away from its flight path. An aircraft that can hold seventy can aim seventy.

The forty degrees of difference is the entire product of the programme, and the question is what forty degrees of aim is worth.

The Question Behind the Question

There is a second layer, and the programme understood it. Post-stall manoeuvring costs energy, and energy is the currency of air combat. Rotating the nose forty degrees further round requires flying at an angle of attack that produces enormous drag, and the speed lost has to be bought back before the aircraft can do anything else. If the pointing advantage yields a firing solution before the energy deficit becomes fatal, the technique wins. If it does not, the aircraft has made itself slow in front of an opponent who is not.

The trade has no obvious sign, and the programme therefore ended with a set of adversarial engagements rather than with an envelope chart.

Programme Origin

The Argument That Started It

The case for post-stall manoeuvring that led to this aircraft was made in Germany. Wolfgang Herbst, at the German manufacturer, argued through the late 1970s and early 1980s that the decisive quantity in close combat was not sustained turn rate but the time taken to bring weapons to bear, and that an aircraft able to fly controllably past the stall could shorten that time by rotating rather than turning. The NASA history traces his conviction to the early 1970s and dates the post-stall research he led with Karl Knauer at the German manufacturer from 1977.

The manoeuvre that carries his name is a rapid reversal of flight path performed by pitching up past the stall, rolling about the velocity vector while the speed decays, and pitching back down on the new heading. Flown to an arbitrary heading change rather than a full reversal it is usually called a J-turn. The manoeuvre was named after Herbst, and the Dryden fact sheet records both names.

The wider literature on this subject is gathered in the cluster on post-stall manoeuvring in The Contemporary Literature. The records closest to this section are these.

An Institutional First

The X-31 was the first experimental aircraft programme administered by a United States government agency as an international effort, and the arrangement is part of what the programme demonstrated. Rockwell International’s North American Aircraft division built the fuselage, vertical tail and canards. Messerschmitt-Bolkow-Blohm, later Deutsche Aerospace and later still part of Daimler-Benz, built the wing and the thrust-vectoring paddles and wrote the integrated flight and propulsion control laws.

The Defense Advanced Research Projects Agency managed the programme, redesignated the Advanced Research Projects Agency from March 1993. The flight test organisation at the National Aeronautics and Space Administration’s Dryden Flight Research Center was an International Test Organization of roughly one hundred and ten people, declining to about sixty as the programme matured, drawing on the National Aeronautics and Space Administration, the United States Navy, the United States Air Force, Rockwell, the Federal Republic of Germany and the German manufacturer. Fourteen pilots from those organisations flew the aircraft.

The programme was a key effort of the North Atlantic Treaty Organization Cooperative Research and Development Program, and that cooperative programme supplied the formal frame for the arrangement.

What the Institutional Frame Cost and Bought

The flight-test report is unusually candid about the overhead. Roughly six months of working together passed before the team functioned as one, and the report attributes that to duplicated verification, with government engineers independently checking contractor results until trust was established. The recommendation drawn from it is that a team of this kind should be physically co-located at least three months before flying begins, and approximately ninety percent of the test team was on site at Dryden.

Six project pilots was not too many, against an early fear that so many could not be kept current for high angle of attack work. The report attributes that to experience, to proficiency flying in other types, and to unlimited simulator access.

Sizing From First Principles

This section derives the keystone, and the keystone is arithmetic about counted outcomes rather than about forces. Such a derivation is unusual for this series, but it treats honestly what the programme produced.

The Measure of Effectiveness

The quantity the programme reported is the exchange ratio, which belongs to the family of measures of effectiveness for general-purpose forces that the operations-research literature had been formalising since the early 1970s. Let $R$ be the number of losses suffered by the adversary, designated Red, and let $B$ be the number of losses suffered by the aircraft under test, designated Blue. Then

\[E = \frac{R}{B}\]

A value above one means the aircraft under test came off better. A value of one is parity. The definition is simple and the trouble begins as soon as more than one kind of fight is flown.

A kill was scored when the rule-of-thumb weapons criteria and envelope were met for the weapon selected, and a ground kill was additionally scored when an aircraft penetrated a hard deck ten thousand feet above ground level, on the reasoning that a pilot driven into the ground has been defeated whether or not a weapon was fired.

The Four Starting Conditions

The simulation programme, and the flight programme after it, flew four starting conditions.

Condition Setup Meaning
Defensive 325 knots calibrated, 3,000 feet nose-on range, adversary in firing range The aircraft under test begins already losing
Offensive The mirror image, with the aircraft under test in firing range The aircraft under test begins already winning
Slow-speed line abreast Parallel tracks 1,500 feet apart at 215 knots calibrated Neutral, and slow enough to use post-stall at once
High-speed line abreast Parallel tracks 3,000 feet apart at 325 knots calibrated Neutral, and too fast to enter post-stall

The two line-abreast conditions are the interesting ones because they are neutral by construction. Two aircraft of identical performance starting line abreast should produce an exchange ratio of one. Any departure from one is attributable to the difference between the aircraft rather than to the way the fight was set up.

The Published Results

For the X-31 with post-stall technology against the F/A-18, the reported figures are as follows.

Starting condition Exchange ratio
Defensive 0.36
Offensive 8.5
Slow-speed line abreast 16
High-speed line abreast 0.86
Overall 1.83

Two things in that table deserve attention before any analysis.

The first is that the aircraft loses in two of the four conditions. At 0.36 in the defensive start and 0.86 in the high-speed line-abreast start, the X-31 with every advantage its designers gave it comes off worse than the F/A-18 it is fighting. The popular account of this programme does not usually mention that.

The second is that the overall figure of 1.83 is far below the average of the four. The gap is neither a rounding artefact nor a mistake, and it is the most important piece of arithmetic in the article.

Why the Overall Figure Is Not the Average

The arithmetic mean of the four condition ratios is

\[\bar{E} = \frac{0.36 + 8.5 + 16 + 0.86}{4} = 6.43\]

and the published overall figure is 1.83. The two differ by a factor of 3.514. A reader who computed the average from the table and a reader who took the published overall would come away with completely different impressions of the same experiment.

The reason is that a pooled exchange ratio is not a mean of ratios. Confusing the two is the aggregation hazard that the statistical literature on combining proportions across strata exists to warn about. The pooled figure is a ratio of sums,

\[E_{\text{pool}} = \frac{\sum_i R_i}{\sum_i B_i}\]

and substituting $R_i = E_i B_i$ gives the identity that governs everything that follows,

\[E_{\text{pool}} = \frac{\sum_i E_i B_i}{\sum_i B_i}\]

The pooled ratio is the mean of the per-condition ratios weighted by the losses of the side in the denominator. A condition in which the aircraft under test was shot down fourteen times counts fourteen times as heavily as a condition in which it was shot down once.

The weighting is no defect of the measure. It is what a pooled ratio means, and the correct figure to report if the question is how the aircraft fared across the whole campaign. But it makes the headline figure depend on the distribution of engagements as much as on the aeroplane, and nothing in the published figure tells the reader which.

What the Weights Must Have Been

The identity can be inverted. The per-condition ratios and the pooled ratio are published, the counts are not, and the weights are therefore unknown. But they are not unconstrained.

Let $w_i$ be the fraction of the test aircraft’s total losses that occurred in condition $i$, so that the weights sum to one and the identity reads

\[E_{\text{pool}} = \sum_i w_i E_i\]

Partition the conditions into those below parity and those above it. The construction is a two-point mixture and the arithmetic is elementary, so the result is robust. To reach a pooled value as low as 1.83 with as little weight as possible on the losing conditions, put all the winning weight on the smaller of the two winning ratios and all the losing weight on the smaller of the two losing ratios. That gives the minimum share,

\[w_{\min} = \frac{E_{\text{win,min}} - E_{\text{pool}}}{E_{\text{win,min}} - E_{\text{lose,min}}} = \frac{8.5 - 1.83}{8.5 - 0.36} = 0.819\]

and pairing the two largest instead gives the maximum,

\[w_{\max} = \frac{E_{\text{win,max}} - E_{\text{pool}}}{E_{\text{win,max}} - E_{\text{lose,max}}} = \frac{16 - 1.83}{16 - 0.86} = 0.936\]

Between 81.9 and 93.6 percent of the X-31’s simulated losses occurred in the two starting conditions where it was behind. That bracket is derived from published numbers alone. It assumes nothing about the counts, nothing about the number of engagements, and nothing about the aircraft.

A bound that owes nothing to a model is the strongest kind of result available here, and this one says something the programme’s own summaries do not. The headline 1.83 is dominated by the conditions in which post-stall manoeuvring did not help.

The Headline Result Sits on a Knife Edge

The same identity answers a question the programme never asked, with an uncomfortable result. Rather than solving for the weight that reproduces 1.83, solve for the weight that would have driven the pooled figure to parity,

\[w_{\text{tip}} = \frac{E_{\text{win,min}} - 1}{E_{\text{win,min}} - E_{\text{lose,min}}} = \frac{8.5 - 1}{8.5 - 0.36} = 0.9214\]

A losing share above 92.14 percent would have put the overall exchange ratio below one, and so would have reversed the programme’s headline conclusion from an advantage into a disadvantage.

The bracket established above runs from 81.9 to 93.6 percent, and it contains that threshold. The published result is therefore consistent with weight distributions that produce an overall advantage and with distributions that produce an overall disadvantage, and the published figures do not distinguish between them.

None of this claims that the programme was wrong. The point concerns what its headline number establishes on its own, which is less than it appears to. The flight testing that followed settles the question, and it settles it conditionally, as the arithmetic predicts.

The Same Arithmetic Applied to the Other Two Comparisons

The programme published two further sets. Without post-stall technology, the X-31 against the F/A-18 gave 0.20, 1.25, 0.25 and 0.85 across the four conditions with an overall of 0.37. The X-31 with post-stall technology against an identical X-31 without it gave 0.27, 23, 6.67 and 2.8 with an overall of 2.67.

Applying the same bracket to each gives a sharp contrast. The two-point extremes are the same construction in each case,

\[w_{\min} = \frac{E_{\text{win,min}} - E_{\text{pool}}}{E_{\text{win,min}} - E_{\text{lose,min}}} \qquad w_{\max} = \frac{E_{\text{win,max}} - E_{\text{pool}}}{E_{\text{win,max}} - E_{\text{lose,max}}}\]

and the width of the answer depends entirely on how far the winning ratios sit from the pooled value.

Comparison Mean of the four Pooled Losing-condition share
With post-stall against the F/A-18 6.43 1.83 81.9 to 93.6 percent
Without post-stall against the F/A-18 0.6375 0.37 83.8 to 100.0 percent
With post-stall against without 8.185 2.67 5.1 to 89.4 percent

The third row shows that the method has teeth. The X-31-against-X-31 comparison constrains the weights hardly at all, because one of its winning ratios, 2.8, sits close to the pooled value of 2.67 and can therefore carry almost any weight. The same calculation that pins the F/A-18 comparison to a narrow band says almost nothing about the other. A method that returns a wide answer when the data are uninformative is behaving correctly.

The topic is surveyed more fully in the cluster on rates, proportions and their intervals in The Contemporary Literature, and the records bearing most directly on the discussion here follow.

What the Sample Size Supports

The second question is harder, and the programme raised it in print only to note that a limited number of flight engagements would make an increase in winning difficult to quantify statistically. An exchange ratio estimated from a finite number of engagements is an estimate, and estimates have intervals.

Model each scored outcome as a Bernoulli trial in which the adversary is lost with probability $p$. The exchange ratio and that probability are the same quantity in different clothing,

\[E = \frac{p}{1 - p} \qquad \text{and} \qquad p = \frac{E}{1 + E}\]

so an interval for $p$ maps directly onto an interval for $E$. The standard references for inference on rates are Fleiss on statistical methods for rates and proportions and Agresti on categorical data. The interval used here is the Clopper-Pearson interval, defined by inverting the binomial tail. For $k$ adversary losses in $n$ scored outcomes the lower endpoint is the value of $p$ satisfying

\[\sum_{i=k}^{n} \binom{n}{i} p^{i} (1 - p)^{n - i} = \frac{\alpha}{2}\]

and the upper endpoint the value satisfying

\[\sum_{i=0}^{k} \binom{n}{i} p^{i} (1 - p)^{n - i} = \frac{\alpha}{2}\]

The construction goes back to the classical work on interval estimation for a binomial proportion. This interval is conservative, guaranteeing at least the nominal coverage rather than exactly it, and conservatism is the right choice here. The claim being tested is that an advantage exists, and a conservative interval will not manufacture one that the data do not support.

The counts are not published. The question is therefore asked the other way round. For a stated number of scored outcomes per condition, what does each published ratio establish? A ratio and a sample size together fix the split, since the adversary losses are

\[k = \operatorname{round}\left( \frac{n E}{1 + E} \right) \qquad \text{and} \qquad b = n - k\]

and the test applied to the resulting interval is whether it contains parity,

\[\text{separates} \iff 1 \notin [E_{\text{lo}}, E_{\text{hi}}] \iff \tfrac{1}{2} \notin [p_{\text{lo}}, p_{\text{hi}}]\]
Condition Ratio 12 outcomes 24 outcomes 48 outcomes
Defensive 0.36 includes parity excludes parity excludes parity
Offensive 8.5 excludes parity excludes parity excludes parity
Slow-speed line abreast 16 excludes parity excludes parity excludes parity
High-speed line abreast 0.86 includes parity includes parity includes parity

At twelve scored outcomes the defensive result of 0.36 carries an interval running from 0.058 to 1.34, which includes parity, so at that sample size the apparent disadvantage in the defensive start establishes nothing at all.

The high-speed line-abreast condition is the striking case. At 0.86 it is so close to parity that no feasible experiment could separate it. Scanning upward through sample sizes, the smallest number of scored outcomes at which a ratio of 0.86 could be distinguished from one at the conventional level is 691. The simulation design allowed 48 engagements for each case, twelve in each condition.

The finding does not criticise the programme. It reads the result correctly. The published commentary describes the high-speed line-abreast case as showing “more evening”, an apt description of a result that establishes nothing. The arithmetic here puts a number on how much nothing.

What the Design Could and Could Not Have Settled

Running the same scan for each condition gives the sample size at which each published ratio becomes separable from parity.

Condition Ratio Scored outcomes needed
Slow-speed line abreast 16 6
Offensive 8.5 9
Defensive 0.36 20
High-speed line abreast 0.86 691

The experiment was well matched to two of its four conditions and hopeless for a third. The slow-speed line-abreast result, the one the programme identified as its highest priority for flight testing, needs six scored outcomes to establish itself. The high-speed line-abreast result needs more than a hundred times as many.

The programme reached that conclusion by judgement and reached it correctly. The flight-test priority order it settled on was slow-speed line abreast first, then defensive, then high-speed line abreast, then offensive. The first two are exactly the two conditions where a modest number of engagements can resolve something, and the NASA history records the ordering as arrived at from which conditions showed the largest difference between the two configurations and from practical limits such as the entry-speed restriction, with statistical difficulty raised only as a caution about the offensive condition.

The Vehicle

The X-31 was small, single-engined, and closer to parity with its adversary than the popular account suggests. The published table is unusually complete for a research aircraft, because the NASA history reproduces the type’s entry in the standard reference work, the 1996 to 1997 edition of Jane’s All the World’s Aircraft.

Quantity Value Metric
Wing span 23 ft 10 in 7.26 m
Wing area 226.3 sq ft 21.02 m²
Wing aspect ratio 2.51  
Wing leading-edge sweep 56.6 degrees at the root, cranked to 45 degrees outboard  
Canard span 8 ft 8 in 2.64 m
Canard area 23.60 sq ft 2.19 m²
Fin including dorsal 28.87 sq ft 2.68 m²
Rudder 8.68 sq ft 0.81 m²
Length excluding probe 43 ft 4 in 13.21 m
Fuselage excluding probe 40 ft 8 in 12.39 m
Height 14 ft 7 in 4.44 m
Empty weight equipped 11,410 lb 5,175 kg
Fuel 4,136 lb 1,876 kg
Normal flying weight 14,600 lb 6,622 kg
Maximum takeoff weight 15,935 lb 7,228 kg
Engine One General Electric F404-GE-400  
Afterburning thrust 16,000 lbf 71 kN
Design load factor Plus nine, minus four  
Maximum operating altitude 40,000 ft 12,200 m
Highest speed attained Mach 1.28 at 38,000 ft  

A Consistency Check on the Published Table

The table checks against itself in three places, a test that matters because a specification sheet transcribed through several publications acquires errors.

The aspect ratio implied by the span and the area is

\[A = \frac{b^{2}}{S} = \frac{23.83^{2}}{226.3} = 2.509\]

against a published 2.51. The wing loading at normal flying weight is

\[\frac{W}{S} = \frac{14{,}600}{226.3} = 64.52 \ \text{lb/ft}^{2}\]

against a published 64.52, which agrees exactly. The thrust-to-weight ratio at normal weight is 1.096 and at maximum takeoff weight it is 1.004, against a published power loading of one pound per pound of static thrust at maximum weight.

The last of those is the number that shaped the programme. The design study had established that the concept needed a thrust-to-weight ratio of at least one, and the airframe gained weight during construction until it reached roughly sixteen thousand pounds and the ratio fell to about unity. Radar, air-refuelling capability and the ability to carry a fighter weapons load were deleted to hold the line. The aeroplane that flew the combat evaluation was therefore at the bottom edge of its requirement, and the NASA history is blunt about the consequence, calling the airframe underpowered against a more powerful F/A-18 with a wing better suited to conventional turning.

What the Lift Curve Has to Be

A number the programme never published can be recovered from something a test pilot said out loud. Reviewing the accident that destroyed the first aircraft, one of the project pilots observed that anybody on the programme would know that twenty degrees of angle of attack corresponds to about 135 to 140 knots, a remark the NASA history quotes.

That statement is a trim point, and a trim point determines a lift-curve slope. The relations used through this section are the standard ones set out in Etkin on the dynamics of flight, Nelson on flight stability and Cook on flight dynamics principles. In level flight the lift equals the weight, so

\[C_L = \frac{W}{q S} \qquad \text{with} \qquad q = \tfrac{1}{2} \rho_0 V_c^{2}\]

At 137.5 knots calibrated and a weight near 14,000 pounds late in a sortie the dynamic pressure is 64.0 pounds per square foot and the lift coefficient is 0.9665. Dividing by twenty degrees gives

\[C_{L\alpha} = 0.04833 \ \text{per degree} = 2.769 \ \text{per radian}\]

That number can be checked against two formulae that never saw the aeroplane. For an unswept wing of low aspect ratio the Helmbold relation gives

\[C_{L\alpha} = \frac{\pi A}{1 + \sqrt{1 + (A/2)^{2}}} = \frac{\pi (2.51)}{1 + \sqrt{1 + 1.575}} = 3.027 \ \text{per radian}\]

and for a swept wing of low aspect ratio the standard compressible form, as tabulated in the stability and control handbook of the period, gives

\[C_{L\alpha} = \frac{2 \pi A}{2 + \sqrt{\dfrac{A^{2} \beta^{2}}{\kappa^{2}} \left(1 + \dfrac{\tan^{2} \Lambda}{\beta^{2}}\right) + 4}} = 2.278 \ \text{per radian}\]

at 56.6 degrees of leading-edge sweep. The inferred whole-aircraft value of 2.769 sits between them, 21.5 percent above the swept-wing figure, and the canard accounts for the difference in the right direction. A surface ahead of the wing contributes to the whole-aircraft slope in proportion to its own area and its own slope, referred to the wing reference area and scaled by an efficiency that accounts for upwash,

\[\Delta C_{L\alpha} = \eta \, \frac{S_c}{S} \, C_{L\alpha,c}\]

The canard is 10.4 percent of the wing area, and at a slope near 2.5 per radian with an efficiency of 1.2 it contributes 0.313, giving 2.591 against an inferred 2.769, a difference of 6.4 percent.

Comparing the inferred value against the unswept formula alone and explaining the disagreement by appealing to the canard would be backwards. A canard in trim carries positive lift and therefore raises the whole-aircraft slope. Sweep is what lowers it, and this aeroplane has a great deal of sweep. The two formulae bracketing the answer is a much stronger statement than either one alone.

The broader record appears under the cluster on canard and delta configurations in The Contemporary Literature. These records bear most closely on the argument above.

Dependent Systems

Thrust Vectoring and the Moment It Produces

Three carbon-carbon thrust-vectoring paddles were mounted around the nozzle and driven into the exhaust by actuators. The concept came from Rockwell’s earlier work for the Navy, flight tested on an F-14 with paddle-like vanes on either side of the speed brake housing, and it was selected over a multi-axis vectoring nozzle for reasons that were mostly financial. The paddles saved approximately sixty million dollars in development against an axisymmetric vectoring nozzle.

The mass accounting gives the clearest justification the choice has.

Item Weight
Three carbon-carbon paddles 103 lb
Aft-fuselage supporting frame 28 lb
Vectoring hardware support 30 lb
Flight control system components 79 lb
Total 240 lb

against a four-hundred-pound fully axisymmetric nozzle on the later F-16 multi-axis thrust vectoring programme. Carbon-carbon was chosen over the Inconel used on the F/A-18 High Alpha Research Vehicle specifically to avoid nose ballast, because the paddles sit far aft and every pound there costs several forward. The Inconel-vaned research aircraft required a large amount of ballast and lost performance for it.

Paddle deflection was originally limited to 26 degrees and later increased to 35, which produced a jet deflection of 16 degrees. Vane travel was separately increased from 27 degrees to 34 after an early departure, the 27-degree limit having existed to stop the vanes striking one another. The paddle and vane figures are those of the NASA history. The maximum jet deflection rate was 40 degrees per second, as the review of high angle of attack control system performance states.

The yawing moment available from vectoring is the side component of thrust acting on a moment arm,

\[N_{\text{tv}} = T \sin \delta_j \, \ell_v\]

and expressing it as a coefficient in the usual way gives

\[C_{n,\text{tv}} = \frac{T \sin \delta_j \, \ell_v}{q S b}\]

The important feature of that expression is that dynamic pressure appears in the denominator and nowhere else. An aerodynamic control surface produces a moment proportional to dynamic pressure, so its coefficient is independent of speed. A vectored moment is fixed by thrust, so its coefficient falls as one over dynamic pressure. The two mechanisms have opposite dependences on speed, and that single fact organises everything else in this section.

Readers wanting more can turn to the clusters on thrust vectoring and on fluidic thrust vectoring in The Contemporary Literature. The following records sit nearest to this discussion.

The Unreported Moment Arm

The distance from the centre of gravity to the vane station is not given in the flight reports, and the departure record does not supply it.

The envelope expansion report records that the departure from controlled flight at 58 degrees of angle of attack during a split-S was caused by a yawing-moment asymmetry so large that it overcame the available thrust-vectoring authority. The asymmetry report places the same event on the second aircraft, during a 2-g split-S to 60 degrees with grit strips on the forebody and noseboom, and computes the maximum asymmetry during that manoeuvre as a yawing-moment coefficient of 0.125. The NASA history quotes the flight-test engineer’s account, which puts the manoeuvre at 35,000 feet and Mach 0.4 with maximum afterburner set. The asymmetry report also bounds the second aircraft’s asymmetry in steady decelerations, with the forebody unmodified, at about 0.080.

A departure says only that the asymmetry exceeded the authority, so the vectoring relation, solved for the arm, gives an upper bound and not a value,

\[\ell_v < \frac{C_{n0} \, q S b}{T \sin \delta_j}\]

At 35,000 feet the standard atmosphere set out in the next section gives $\theta = 0.7594$ and $\delta = 0.2353$, and a speed of sound of 972.9 feet per second, so Mach 0.4 is 389.1 feet per second true. The dynamic pressure follows from the pressure and the Mach number,

\[q = \tfrac{\gamma}{2} \, p_0 \, \delta \, M^{2} = 0.7 \times 2{,}116.22 \times 0.2353 \times 0.4^{2} = 55.77 \ \text{psf}\]

and the thrust lapsed with the pressure ratio is $16{,}000 \times 0.2353 = 3{,}765$ pounds. The bound is then

\[\ell_v < \frac{0.125 \times 55.77 \times 226.3 \times 23.83}{3{,}765 \times \sin 16^{\circ}} = \frac{37{,}594}{1{,}037.8} = 36.2 \ \text{feet}\]

That bound excludes only a centre of gravity within about four and a half feet of the nose of a fuselage 40.67 feet long, which the geometry excludes already. It is also soft. The thrust-lapse exponent of the model is not measured, and moving it from 0.8 to 1.2 moves the bound from 27.1 to 48.4 feet. The departure record therefore does not determine the arm. The one value found is in a study of differential canard deflection, Whiting 1996, which tabulates for the full-scale aircraft a centre of gravity at fuselage station 269.2 inches and a vane centre of pressure at 480 inches, an arm of 17.57 feet. At that arm the vectoring could supply a yawing-moment coefficient of only about 0.061 at the departure condition, consistent with the departure. What the flight record does support is the question the next section asks, namely how long an arm would have had to be.

The Departure Boundary

Without a flight-report value for the arm, the available authority cannot be tabulated as a coefficient. What can be tabulated is the arm the vectoring would have needed to hold a given asymmetry, which uses only the thrust, the deflection and the dynamic pressure, and which can be set against the length of the fuselage. Thrust is lapsed with the pressure ratio of the standard atmosphere, the working approximation for a turbofan at fixed Mach number.

The atmosphere model is the 1976 standard troposphere and the density and pressure ratios it supplies are what the thrust lapse and the true-airspeed conversions rest on, so the relations are given rather than left implicit. The tabulated form goes back to the altitude-pressure tables of the 1930s and was standardised internationally in the civil aviation atmosphere. Writing the temperature, pressure and density ratios against their sea-level values,

\[\theta = \frac{T}{T_0} = 1 - 6.87535 \times 10^{-6} \, h \qquad \delta = \frac{p}{p_0} = \theta^{5.2559} \qquad \sigma = \frac{\rho}{\rho_0} = \frac{\delta}{\theta}\]

and the speed of sound follows from the temperature alone,

\[a = \sqrt{\gamma R T} = \sqrt{\gamma R T_0 \theta}\]
Altitude Density ratio Pressure ratio
10,000 ft 0.7385 0.6877
20,000 ft 0.5328 0.4595
30,000 ft 0.3741 0.2970
35,000 ft 0.3099 0.2353

Thrust is lapsed with the pressure ratio. A turbofan at fixed Mach number carries thrust roughly in proportion to the ambient pressure, so the working model is the simplest member of a family that runs from Bronz et al 2017 through identified engine performance models,

\[T(h) = T_{\text{SL}} \, \delta(h)^{n}\]

with the exponent near unity. The exponent is carried explicitly so that its effect on the answer can be reported.

The arm needed to hold an asymmetry $C_{n0}$ is the vectoring relation solved at equality,

\[\ell_{\text{req}} = \frac{C_{n0} \, q S b}{T \sin \delta_j}\]

and it is tabulated against the steady bound of 0.080 and against the departure’s 0.125.

Altitude Calibrated airspeed Dynamic pressure Thrust Arm needed against 0.080 Arm needed against 0.125
10,000 ft 150 kt 76.2 psf 11,003 lb 10.8 ft 16.9 ft
10,000 ft 200 kt 135.4 psf 11,003 lb 19.3 ft 30.1 ft
10,000 ft 250 kt 211.6 psf 11,003 lb 30.1 ft 47.0 ft
20,000 ft 150 kt 76.2 psf 7,353 lb 16.2 ft 25.3 ft
20,000 ft 200 kt 135.4 psf 7,353 lb 28.8 ft 45.0 ft
20,000 ft 250 kt 211.6 psf 7,353 lb 45.0 ft 70.4 ft
30,000 ft 150 kt 76.2 psf 4,751 lb 25.1 ft 39.2 ft
30,000 ft 200 kt 135.4 psf 4,751 lb 44.6 ft 69.7 ft
30,000 ft 250 kt 211.6 psf 4,751 lb 69.7 ft 108.9 ft

Two entries worked through show the arithmetic,

\[\ell_{\text{req}}^{20{,}000,\,150} = \frac{0.080 \times 76.2 \times 226.3 \times 23.83}{7{,}353 \times \sin 16^{\circ}} = \frac{32{,}874}{2{,}026.8} = 16.2 \ \text{ft} \qquad \ell_{\text{req}}^{30{,}000,\,250} = \frac{0.080 \times 211.6 \times 226.3 \times 23.83}{4{,}751 \times \sin 16^{\circ}} = \frac{91{,}288}{1{,}309.6} = 69.7 \ \text{ft}\]

Where the needed arm exceeds the 40.67 feet of the fuselage, no placement of the vanes could have held the asymmetry, whatever the real arm is. At thirty thousand feet and 250 knots even the steady bound of 0.080 needs an arm $69.7 / 40.67 = 1.71$ times the length of the aeroplane. Setting the needed arm equal to the fuselage length $L$ gives the calibrated airspeed above which that is so,

\[q_{L} = \frac{L \, T \sin \delta_j}{C_{n0} \, S b} \qquad V_{c,L} = \sqrt{\frac{2 q_{L}}{\rho_0}}\]

and at twenty thousand feet against the departure’s asymmetry

\[q_{L} = \frac{40.67 \times 7{,}353 \times \sin 16^{\circ}}{0.125 \times 226.3 \times 23.83} = \frac{82{,}428}{674.09} = 122.3 \ \text{psf} \qquad V_{c,L} = 320.8 \ \text{ft/s} = 190.0 \ \text{knots}\]
Altitude Thrust Speed above which no arm suffices, against 0.080 Against 0.125
10,000 ft 11,003 lb 290.6 kt 232.5 kt
20,000 ft 7,353 lb 237.6 kt 190.0 kt
30,000 ft 4,751 lb 191.0 kt 152.8 kt
35,000 ft 3,765 lb 170.0 kt 136.0 kt

The shape of that table is the content. The ceiling on speed falls with altitude because thrust does, and the departure’s asymmetry lowers it by a factor of $\sqrt{0.125 / 0.080} = 1.25$ at every height. Below those speeds whether the vectoring sufficed depends on the arm. The departure itself, at an equivalent airspeed of 128.3 knots against a ceiling of 136.0 knots at 35,000 feet, sits below its own threshold, so it bounds the arm rather than showing that no arm would have done.

Its equivalent airspeed follows from the dynamic pressure computed above,

\[V_e = \sqrt{\frac{2 q}{\rho_0}} = \sqrt{\frac{2 \times 55.77}{0.0023769}} = 216.6 \ \text{ft/s} = 128.3 \ \text{knots}\]

The measured asymmetries themselves were worse than the round number suggests. In the unmodified forebody configuration on the second aircraft the maximum was bounded at 0.080 with no distinguishable trend against angle of attack, each manoeuvre producing what the asymmetry report describes as a random pattern. Adding boundary-layer transition strips to the forebody and noseboom made the asymmetry more regular and simultaneously made it larger, reaching almost 0.100, and in the worst single case, with the forebody strips alone, the asymmetry swung from 0.077 in one direction at 52 degrees to 0.088 in the other by 55 degrees, a total change of

\[\Delta C_n = 0.165 \ \text{across three degrees of angle of attack}\]

The coefficient is a moment divided by a scale, so the disturbance the control system faces at a given flight condition is

\[N_{0} = C_{n0} \, q S b\]

which at twenty thousand feet and 150 knots calibrated is 4,108 foot-pounds for every hundredth of coefficient, so the measured bound of 0.080 is 32,863 foot-pounds and the worst observed swing of 0.165 is 67,780 foot-pounds, delivered across three degrees of angle of attack.

A yawing moment coefficient that reverses sign and changes by 0.165 over three degrees is not a disturbance a control system rejects. It is a cliff.

The cluster on yaw asymmetry and forebody flow in The Contemporary Literature holds the wider literature, and these records lie closest to the present subject.

What Fixed It

The fix was aerodynamic and it was small. Narrow quarter-inch grit strips were added to the nose to regularise the vortices. Nose strakes six tenths of an inch wide and twenty inches long were added, and the nose tip was blunted from an essentially sharp point to a radius of three quarters of an inch. The blunting was the more significant of the two.

The measured effect was to raise the angle of attack at which asymmetry begins by seven to twelve degrees, to reduce the maximum asymmetry, and, most usefully, to eliminate the random variation and make the asymmetry repeatable. A longer forty-six-inch strake was tried and changed the one-g deceleration asymmetries very little, shifting their onset about two degrees higher.

An error in manufacture turned into a measurement. The modified nose tip on the second aircraft was found late in testing to be one inch in diameter rather than the one and a half inches specified and built on the first. That unintended difference is a data point on the sensitivity of the phenomenon to a dimension of about half an inch on a forty-foot aeroplane.

There is a second and stranger case. The two aircraft did not fly alike, and the second was known on the programme as the evil twin. Testing it with several lengths of extended nose established that eight and a half inches of added strake length made the two fly identically, as direct a demonstration of forebody sensitivity as the programme produced.

The Altitude Effect the Engineers Reported and Never Wrote Down

The flight-test report contains an observation presented as a lesson about test technique. The traditional build-up approach completes a block of testing at high altitude and repeats it lower, on the reasoning that control power at low airspeed is a function of dynamic pressure and the same calibrated airspeed therefore gives the same control power at any height. The report observes that this reasoning fails when a large share of control power comes from the engine, and that manoeuvres impossible at thirty thousand feet became possible at twenty thousand.

That observation follows directly from the opposite speed dependences noted above, and the arm is not needed to compute it. At constant calibrated airspeed the aerodynamic authority is constant by construction, since dynamic pressure referred to sea-level density is what calibrated airspeed measures. The vectored authority moves with thrust, and thrust at fixed Mach number moves with the ambient pressure ratio, while the unknown arm cancels,

\[\frac{C_{n,\text{tv}}(h_2)}{C_{n,\text{tv}}(h_1)} = \frac{T(h_2)}{T(h_1)} \approx \frac{\delta(h_2)}{\delta(h_1)}\]

Between twenty thousand and thirty thousand feet that ratio is 0.6462, a loss of 35.4 percent of the vectored control authority for no change of airspeed whatever. The sensitivity to the lapse model is modest, giving 0.7052 at an exponent of 0.8 and 0.5922 at an exponent of 1.2.

The test team responded by starting each block at thirty thousand feet, watching control margins on the control-room strip charts, and restarting lower when the margins ran out. Where the margins allowed, they went on to seventy degrees of angle of attack at twenty thousand feet on blocks that might have stopped at fifty degrees ten thousand feet higher. The quantitative explanation is a one-line ratio, and the programme found the answer empirically because nobody had written the ratio down.

Why the Advantage Is Not a Smaller Turn Radius

Several accounts of this aircraft, including official ones such as the NASA history, attribute its combat performance partly to a smaller turning radius. The arithmetic does not support that.

Turn radius in a level turn is

\[r = \frac{V^{2}}{g \sqrt{n^{2} - 1}}\]

which falls with speed and rises when the available load factor falls. A post-stall aircraft has both a low speed and a very low available load factor, and the two effects work against each other.

The load factor a wing can produce is the lift it can produce divided by the weight,

\[n = \frac{q S C_L}{W}\]

and setting that to one gives the slowest speed at which the aircraft can hold a given angle of attack in level flight,

\[V_{\text{level}} = \sqrt{\frac{2 W}{\rho_0 S C_L}}\]

Taking a lift coefficient of 1.00 at seventy degrees of angle of attack, which is past the vortex-lift peak and is an estimate rather than a measurement, the available load factor and the resulting radius are as follows.

Calibrated airspeed Dynamic pressure Load factor available Radius Turn rate
120 kt 48.8 psf 0.76 descending not applicable
155 kt 81.3 psf 1.26 2,771 ft 5.4 deg/s
175 kt 103.7 psf 1.61 2,155 ft 7.9 deg/s
225 kt 171.4 psf 2.66 1,821 ft 11.9 deg/s

Level flight at seventy degrees of angle of attack requires 138.0 knots calibrated, and below that the manoeuvre is a descending transient rather than a turn. The level-flight relation at the normal flying weight and the assumed lift coefficient gives

\[V_{\text{level}} = \sqrt{\frac{2 \times 14{,}600}{0.0023769 \times 226.3 \times 1.00}} = 233.0 \ \text{ft/s} = 138.0 \ \text{knots}\]

Every radius in the table exceeds the 1,171 feet available at corner speed, which is derived below. The post-stall regime is not where this aeroplane turns most tightly.

The primary sources are largely consistent with that reading even where the summaries are not. The tactical utility paper describes post-stall manoeuvring as a repositioning tool and a way to rotate and point, and where it speaks of smaller-radius turns it describes a descending funnel in which the X-31 is slower than its opponent, not a sustained level turn.

What the Advantage Is

The nose leads the flight path by exactly the angle of attack. The statement defines angle of attack, and it describes the whole mechanism. An aircraft limited to thirty degrees can aim thirty degrees off its velocity vector and an aircraft that holds seventy can aim seventy, so the capability bought is

\[\Delta \psi_{\text{aim}} = \alpha_{\max} - \alpha_{\text{conv}} = 70 - 30 = 40 \ \text{degrees}\]

The agility-metrics literature of the period was trying to name that quantity. The work on Skow et al 1985 and the later assessment of the proposed metrics were contemporaneous with this programme.

A conventional aircraft can obtain the same forty degrees of aim only by turning its velocity vector through forty degrees, and the time that takes follows from the turn rate,

\[t = \frac{\Delta \psi}{\omega} \qquad \text{with} \qquad \omega = \frac{g \sqrt{n^{2} - 1}}{V}\]
Condition Turn rate Time to buy forty degrees
172 kt at 2.25 g 12.80 deg/s 3.13 s
250 kt at 4.00 g 16.92 deg/s 2.36 s
343.9 kt at 9.00 g 28.41 deg/s 1.41 s

Between one and a half and three seconds, depending on where in the envelope the fight is. In a slow fight it is three seconds, and three seconds inside gun range is a long time.

That is the product. Not a tighter turn, not a higher rate, but two to three seconds of pointing that the opponent cannot match, available at the moment the pilot chooses to spend the energy.

Corner Speed, and the Energy the Pointing Costs

The performance relations here follow Anderson on aircraft performance and design and Raymer on conceptual design, with the pilot-facing treatment in the standard naval aerodynamics manual and the configuration argument in Stinton’s anatomy of the aeroplane. The corner speed is where the lift limit meets the load-factor limit,

\[V^{*} = \sqrt{\frac{2 n_{\max} W}{\rho_0 S C_{L\max}}}\]

Taking the lift coefficient at the thirty-degree conventional limit from the inferred lift curve, 1.4498, and the design load factor of nine, the corner speed is 343.9 knots calibrated. There the instantaneous turn rate is 28.40 degrees per second, the radius is 1,171 feet, and a one-hundred-and-eighty-degree reversal takes 6.34 seconds.

Pointing costs energy and the cost is large. Specific energy is

\[E_s = h + \frac{V^{2}}{2 g}\]

and falling from corner speed to a post-stall exit at 120 knots, both calibrated, at 20,000 feet gives up the energy between the corresponding true airspeeds. The corner speed is fixed by a dynamic pressure, and the article reads every calibrated figure as the equivalent airspeed that gives the same dynamic pressure at sea-level density, so the true airspeed at height follows from the density ratio,

\[V = \frac{V_e}{\sqrt{\sigma}} \qquad V^{*} = \frac{580.5}{\sqrt{0.5328}} = 795.3 \ \text{ft/s} = 471.2 \ \text{knots} \qquad V_1 = \frac{202.5}{\sqrt{0.5328}} = 277.5 \ \text{ft/s} = 164.4 \ \text{knots}\]

The corner speed is Mach 0.767 at that height. The energy given up is then

\[\Delta E_s = \frac{V^{*2} - V_1^{2}}{2 g} = \frac{795.3^{2} - 277.5^{2}}{2 \times 32.174} = \frac{555{,}496}{64.348} = 8{,}630 \ \text{feet}\]

Recovering it takes time, and the time must be integrated rather than estimated, because specific excess power at the slow end bears no relation to its value once the aircraft has accelerated. The specific excess power is

\[P_s = \frac{V (T - D)}{W} \qquad \text{with} \qquad D = q S \left( C_{D0} + \frac{C_L^{2}}{\pi A e} \right)\]

This is the specific-excess-power accounting that underlies every energy-manoeuvrability comparison, a method the flight-test community was using to improve in-flight performance through the period, and the parabolic polar is the standard approximation to it.

At twenty thousand feet with a zero-lift drag coefficient of 0.022 and an efficiency factor of 0.70, the specific excess power at the slow end is 68.6 feet per second, with a lift coefficient of 1.323 and a drag of 3,743 pounds, and at the fast end it is 268.7 feet per second with a lift coefficient of 0.161 and a drag of 2,420 pounds. With thrust lapsed to 7,353 pounds and a density ratio of 0.5328, the slow end works out as

\[q = \tfrac{1}{2} \rho_0 \sigma V_1^{2} = 48.75 \ \text{psf} \qquad C_L = \frac{14{,}600}{48.75 \times 226.3} = 1.323 \qquad P_s = \frac{277.5 \, (7{,}353 - 3{,}743)}{14{,}600} = 68.6 \ \text{ft/s}\]

and the fast end as

\[q = 400.5 \ \text{psf} \qquad C_L = \frac{14{,}600}{400.5 \times 226.3} = 0.161 \qquad P_s = \frac{795.3 \, (7{,}353 - 2{,}420)}{14{,}600} = 268.7 \ \text{ft/s}\]

At constant altitude $dE_s = V \, dV / g$, so the recovery time is

\[t = \int_{V_1}^{V^{*}} \frac{V \, dV}{g \, P_s(V)} = 46.0 \ \text{s} \qquad \text{against} \qquad \frac{\Delta E_s}{P_s(V_1)} = \frac{8{,}632.7}{68.60} = 125.8 \ \text{s}\]

Integrating gives 46.0 seconds, evaluated by midpoint quadrature over 4,000 speed steps. A single-point estimate at the slow end would have said 125.8 seconds, wrong by a factor of $125.8 / 46.0 = 2.74$, so the integration is not optional. The answer is insensitive to the drag model, giving 44.5 seconds at a zero-lift drag coefficient of 0.018 and 47.6 seconds at 0.026.

Forty-six seconds is half of the ninety seconds after which an undecided flight engagement was scored as a draw. The pointing advantage is bought with a debt that is hard to repay inside the fight it was spent in, and the debt gives quantitative form to the programme’s own conclusion that post-stall manoeuvring works only when used selectively and in a timely manner.

The Flight Control System

Four digital fly-by-wire flight control computers, with no analogue or mechanical reversion of any kind. Three synchronous main computers drove the surfaces and a fourth acted as a tie-breaker when the three disagreed. The aircraft had thirty-two software releases over the course of the programme.

The control-law conventions and the simulation practice behind them are set out in Stevens and Lewis on aircraft control and simulation and in McRuer on aircraft dynamics and automatic control. The control laws, written by the German manufacturer, were of the predictive type, using a model of the flight path to map pilot commands and flight conditions onto surface deflections and thrust vectoring, where the American partner favoured a reactive type based on measured response, as the NASA history describes, and they ran in an integrated flight and propulsion control system that included the canard. The programme claimed as a first the integration of vectoring as an equivalent moment generator in pitch and yaw, transparent to the pilot, and the review of high angle of attack control system performance records 180 flight hours of trouble-free operation of the vectoring system.

Lateral stick commanded roll rate about the velocity vector rather than about the body axis. Velocity-vector roll is the correct command variable at high angle of attack, and it makes the manoeuvre flyable. The distinction is not cosmetic and the kinematics say why. A roll about the velocity vector is a combination of body-axis roll and yaw rates weighted by the angle of attack,

\[p_w = p \cos \alpha + r \sin \alpha\]
Angle of attack Roll-rate share Yaw-rate share
20 degrees 0.9397 0.3420
30 degrees 0.8660 0.5000
70 degrees 0.3420 0.9397

At seventy degrees the two weights are

\[\cos 70^{\circ} = 0.3420 \qquad \sin 70^{\circ} = 0.9397\]

so at seventy degrees a roll about the velocity vector is 94.0 percent yaw rate. That single line explains a sentence in the programme’s own conclusions which otherwise reads as a correction of itself, where the main advantage is described as an apparent directional nose-pointing rate which is, in actuality, yaw rate or velocity-vector roll capability. At seventy degrees those are the same manoeuvre, and the yaw axis has to supply it. That axis is precisely the one whose aerodynamic control has been lost and whose vectored control the paddles restore. The maximum roll rate that could be commanded depended on the control power available at the flight condition, meaning on altitude, Mach number, dynamic pressure and angle of attack together.

Two control-law responses to the asymmetry problem stand out. Integrators were added to the roll-rate and sideslip feedbacks to tighten control of sideslip and velocity-vector roll rate in the presence of aerodynamic asymmetries, and the wind-axis roll rate command was limited in the post-stall regime, which the control system review describes as an acceptable compromise of roll performance against controllability.

A pilot-selectable angle of attack limiter, settable in five-degree increments between thirty and seventy degrees, was added after an inadvertent overshoot to 62 degrees when the limit was still fifty. It improved the repeatability of the data as much as it improved safety, which simplified the analysis.

The wider literature on this subject is gathered in the cluster on flight control system design, actuation and allocation in The Contemporary Literature. The records closest to this section are these.

Whether the Vectoring Was Fast Enough

The question usually asked of an actuation system is whether it is fast enough. The more useful question is which of its two limits binds first, because a system that saturates in position behaves differently from one that saturates in rate.

Full jet deflection travel is 16 degrees at 40 degrees per second, so full travel takes 400 milliseconds. A sinusoidal command of amplitude $A$ at frequency $\omega$ demands a peak rate of $A \omega$, so the amplitude at which the rate limit binds is

\[A_{\text{bind}} = \frac{\dot{\delta}_{\max}}{\omega}\]

At the frequency of the tailless divergence derived below, 2.390 per second, that amplitude is 16.74 degrees, and the position limit of 16 degrees is reached first. The system is position limited rather than rate limited at the frequency that matters, by a margin of 1.046 in amplitude.

The frequency at which the two limits coincide is 2.500 per second, or 1.05 times the divergence eigenvalue. The coincidence is very close, and the two numbers have nothing to do with each other, the ratio of a rate limit to a travel limit on one hand and a measured aerodynamic instability on the other. It is recorded as an observation and not as a claim about design intent, because nothing in the record says the sizing was done that way.

The Quasi-Tailless Experiment

In 1994 the aircraft was used to fly an aeroplane it was not, by destabilising itself in yaw with its own rudder to the level a smaller fin would have produced, and then holding the result with thrust vectoring.

The method was possible without major software change because the vectoring was already integrated into the control laws. The existing rudder augmentation could be reduced to any desired level or switched off entirely, a destabilising feedback to rudder and ailerons was introduced using sideslip, roll rate and yaw rate, the augmentation and forward-path command were redistributed to thrust vectoring, an additional lead-lag filter was placed in the vectoring feedback path to improve stability margins, and a safety disengagement reverted the aircraft automatically on a system failure or an envelope exceedance. Destabilisation gains corresponding to different simulated tail sizes were selectable by the pilot in flight, which gave a build-up capability within a single sortie.

The aerodynamic database was built by adding increments for tail-off and reduced-tail configurations to a basic model that had itself been corrected against flight-derived parameter identification results.

The reported result is the most quantitatively useful thing the experiment produced. Above thirty percent tail reduction the destabilised aircraft has a positive eigenvalue. The instability grows rapidly between thirty and seventy percent, and at complete tail removal it reaches a time to double amplitude of 290 milliseconds, demonstrated at a supersonic flight condition.

What 290 Milliseconds Determines

A divergent mode grows as $e^{\lambda t}$, so a time to double gives the eigenvalue directly. The practical consequences of flying an aircraft with such a root were worked out for statically unstable configurations under displacement and rate saturation and for the close-coupled canard with relaxed static stability that the X-31 resembles,

\[\lambda = \frac{\ln 2}{t_2} = \frac{0.6931}{0.290} = 2.3902 \ \text{per second}\]

For a yaw-only divergence the eigenvalue and the aerodynamic stiffness are related through the yaw inertia, in the form the flight measurement of lateral and directional stability literature established,

\[\lambda^{2} = -\frac{q S b \, C_{n\beta}}{I_z}\]

A single time to double fixes the ratio of stiffness to inertia and cannot fix either factor, and the flight reports omit the yaw inertia. The ratio is therefore reported first. The condition is quoted as Mach 1.2 at 38,000 feet in the quasi-tailless flight test report, and the relation needs a dynamic pressure, so the conversion runs through the speed of sound and the density ratio, using the axis and airspeed conventions that the Altman 1952 literature fixed long before this aeroplane,

\[V_{\text{true}} = M a \qquad V_{\text{eq}} = V_{\text{true}} \sqrt{\sigma} \qquad q = \tfrac{1}{2} \rho_0 \sigma V_{\text{true}}^{2} = \tfrac{1}{2} \rho_0 V_{\text{eq}}^{2}\]

The last equality is the check. Both routes must give the same dynamic pressure, and worked first at thirty thousand feet, inside the troposphere that the relations above describe, the speed of sound is 994.8 feet per second, so Mach 1.2 is a true airspeed of 1,194 feet per second and an equivalent airspeed of 730.1 feet per second, or 432.6 knots, and both routes return 633.4 pounds per square foot. Substituting gives

\[-\frac{C_{n\beta}}{I_z} = 1.6724 \times 10^{-6} \ \text{per slug foot squared per radian}\]

Assuming a yaw inertia gives a directional stiffness, and the sensitivity is linear.

Assumed yaw inertia Implied $C_{n\beta}$ per radian Per degree
15,000 slug ft² −0.02509 −0.000438
20,000 slug ft² −0.03345 −0.000584
25,000 slug ft² −0.04181 −0.000730

Those are plausible magnitudes for a tailless delta. At the demonstration altitude of 38,000 feet, above the tropopause where the temperature is constant at 389.97 degrees Rankine, the pressure ratio is 0.2038 and the dynamic pressure 434.7 pounds per square foot, which raises the ratio to $2.437 \times 10^{-6}$. The full-scale yaw inertia of 35,200 slug feet squared that Whiting 1996 tabulates then gives $C_{n\beta} = -0.0858$ per radian, or $-0.00150$ per degree, larger in magnitude than any row of the table because both the inertia and the altitude exceed the values assumed there.

The Comparison With the X-29

The previous article in this series measured an instability in pitch and this one measures an instability in yaw, and the two numbers can be put side by side.

The X-29A doubled in 101 milliseconds in pitch, an eigenvalue of 6.861 per second. The X-31 simulating a complete absence of vertical tail doubled in 290 milliseconds in yaw, an eigenvalue of 2.390. The X-29’s instability was 2.87 times faster.

There is a useful bound attached to each. A first-order unstable plant with a pure time delay cannot be stabilised by a proportional loop of any gain once the delay approaches the reciprocal of the unstable pole,

\[\tau_{\max} \approx \frac{1}{\lambda}\]

which gives 418 milliseconds for the X-31 case and 146 for the X-29. Neither aircraft was anywhere near that bound, and the point of computing it is to establish that neither instability was hard in the sense of being close to a fundamental limit. Both were hard in the practical sense of demanding reliable computers.

Air Data, and the Probe That Solved One Problem and Created Another

In the X-31’s air data system the programme’s care and its worst failure meet, and they meet in the same component.

Below thirty degrees of angle of attack the noseboom updated the inertial navigation unit with air data. When the pilots began flying extended periods above thirty degrees the inertial unit began computing large and entirely fictitious sideslip values as the wind changed in direction and magnitude. The aircraft’s basic stability at high angle of attack gave it a low tolerance for sideslip, so a false sideslip was not a cosmetic problem.

The solution was a Kiel probe in place of the standard pitot-static tube, bent ten degrees downward from the standard configuration, with the sideslip vane rotated twenty degrees down from the noseboom to remove an oscillation that appeared at 62 degrees, both changes the envelope expansion report describes. A Kiel probe carries a shroud around the pressure port and is insensitive to flow angle over a wide range, the property an aircraft flying at seventy degrees of angle of attack needs. With those changes the air data was accurate throughout the angle of attack envelope.

It was also unheated. The decision not to fit heat was conscious, taken because the team did not want to wait for a heated Kiel probe to be manufactured, and because the flight-test environment does not mandate provision for pitot heat. The cockpit switch was disabled, but the work order for the probe called only for its circuit breaker to be collared and did not call for the switch to be placarded inoperative. It was not placarded, and it remained clearly marked “Pitot heat”. At the time of the accident four of the five active pilots believed the pitot heat worked.

The Failure Mode Nobody Analysed

The probe measures a pressure difference, and everything downstream is derived from it. The impact pressure is the difference between total and static pressure,

\[q_c = p_t - p_s = p_0 \left[ \left( 1 + \tfrac{\gamma - 1}{2} M^{2} \right)^{\frac{\gamma}{\gamma - 1}} - 1 \right]\]

and calibrated airspeed is defined as the speed that would produce that impact pressure at sea-level conditions. Ice in the probe corrupts the total pressure and therefore corrupts everything derived from it at once. At 170 knots calibrated the impact pressure is 99.47 pounds per square foot and at 48 knots it is 7.81, a ratio of 12.735, the same factor as the loop-gain error computed below because at these speeds the compressible and incompressible forms agree to within two percent.

Pitot pressure fed two air data computers, each supplying two of the four flight control computers. A disagreement between the two air data computers would raise a fault and a reversion request. A shared error would not, because both computers received the same pneumatic signal from the same single probe and computed the same wrong answer from it.

The hazard analyses had assumed that false pitot information would be obvious. They had been performed when the aircraft carried a heated probe, and they were not repeated when the unheated Kiel probe was fitted. The Mishap Investigation Report is quoted in the NASA history as recording that the only means by which the flight control system was designed to detect degradation of air data was disagreement between the two air data computers, that there was a misconception among the test team and system safety engineers that the system could identify abrupt changes in air data signals, and that no flight control code implementing such a safeguard could be found and the board concluded that it did not exist.

The topic is surveyed more fully in the cluster on analytical redundancy and sensor fault detection in The Contemporary Literature, and the records bearing most directly on the discussion here follow.

How an Airspeed Error Becomes a Stability Error

The flight control gains were scheduled on measured airspeed, and that schedule is the mechanism that destroyed the first aircraft.

Control surface effectiveness rises with dynamic pressure. A gain schedule therefore reduces gain as speed rises, so that the loop gain stays where the designer put it. If the measured speed is low and the true speed is not, the schedule selects gains for the slower condition while the surfaces are producing the moments of the faster one. The loop gain is multiplied by the ratio of true to indicated dynamic pressure, equal to the square of the airspeed ratio,

\[\frac{K_{\text{actual}}}{K_{\text{design}}} = \frac{q_{\text{true}}}{q_{\text{indicated}}} = \left( \frac{V_{\text{true}}}{V_{\text{indicated}}} \right)^{2}\]

On the accident flight the indicated airspeed fell to between 48 and 100 knots calibrated while the actual airspeed was 170.

Indicated True Loop gain factor In decibels
100 kt 170 kt 2.890 9.22 dB
48 kt 170 kt 12.543 21.97 dB

A gain margin is quoted in decibels and the relation is

\[\text{dB} = 20 \log_{10} \left( \frac{K_{\text{actual}}}{K_{\text{design}}} \right)\]

A conventional six-decibel gain margin is a factor of 1.995. The least severe reading of the error exceeded the entire design gain margin by a factor of 1.445, and the most severe exceeded it more than sixfold. The aircraft did not encounter a marginal condition. It encountered one that was outside the design margin by any reading of the data.

The broader record appears under the cluster on gain scheduling and stability margins in The Contemporary Literature. These records bear most closely on the argument above.

The Cross-Check the Cockpit Already Contained

The finding that matters most in this section is that the failure was detectable from information already displayed.

Angle of attack and airspeed are not independent quantities in level flight. They are linked by the lift equation, and either one predicts the other. Using the lift-curve slope inferred earlier,

\[\alpha = \frac{W}{q S C_{L\alpha}}\]
Indicated airspeed Implied angle of attack at one g
207 kt 8.82 degrees
170 kt 13.08 degrees
137.5 kt 20.00 degrees

The pilot called out a reading of 277 knots, corrected himself to 207 knots at 20 degrees of angle of attack, and then continued with the test point, as the NASA history transcribes it. Those two readings disagree by 11.2 degrees of angle of attack. No atmospheric condition, no manoeuvre and no instrument tolerance produces an eleven-degree mismatch between a vane and an airspeed indicator on an aircraft in level flight. Against the table above, the displayed speed and the true one give

\[\Delta \alpha_{207} = 20.00 - 8.82 = 11.2 \ \text{degrees} \qquad \Delta \alpha_{170} = 20.00 - 13.08 = 6.9 \ \text{degrees}\]

Expressed the other way, twenty degrees implies 137.5 knots against the 207 displayed, so the display was high by a factor of 1.505 in speed and 2.266 in dynamic pressure. Comparing that against the six-decibel margin factor of 1.995 gives the coldest number in this article,

\[\frac{207}{137.5} = 1.505 \qquad \left( \frac{207}{137.5} \right)^{2} = 2.266 \qquad 10^{6/20} = 1.995\]

At the moment the pilot spoke the words aloud, the loop gain error had already exceeded the design gain margin. The aircraft was flown for some time after that, and at the actual airspeed of 170 knots the disagreement was still 6.9 degrees.

The redundancy needed to catch this failure was analytical rather than physical, and it was available for the cost of an arithmetic comparison between two signals the aircraft already had. A second probe would have been heavier. A comparison of angle of attack against airspeed through a lift curve would have weighed nothing.

There were further clues. The flutter test excitation box was inhibited above two hundred knots calibrated and refused to operate, which the pilot remarked on and did not pursue. A separate standby airspeed indicator on a simple pitot tube sat near the pilot’s right knee and would probably have read close to the truth. It was not checked.

Propulsion Integration

An articulating inlet cowl that could open to thirty degrees depending on angle of attack was fitted to keep the engine supplied at seventy degrees, and the programme claimed it as a first. No dependent system on the aircraft is less discussed, and the problem is not small. An inlet designed for flight along the free stream is presented at seventy degrees with a flow it was never shaped for, and a compressor stall in an aircraft whose control authority comes partly from thrust is a control failure as well as a propulsion failure.

Readers wanting more can turn to the cluster on propulsion integration at extreme attitude in The Contemporary Literature. The following records sit nearest to this discussion.

Instrumentation and the Scoring System

The scoring apparatus deserves to be treated as a dependent system, because the keystone is a measured rate and the scoring is the instrument that measures it.

Both the X-31 and the adversary aircraft were instrumented, and both carried C-band beacons to improve ground radar tracking accuracy. Aircraft and space-position data were combined to determine valid firing positions for simulated missile shots, which allowed the test team to tell the pilots within seconds of a trigger squeeze whether a kill had been scored.

The purpose was efficiency and the effect was on the data. Pilots knocked off an engagement as soon as a kill was scored, which allowed many more engagements per flight. An engagement terminated at first kill is a different experimental unit from one flown to a conclusion, and the choice has consequences for what the exchange ratio counts. The programme does not appear to have discussed it in those terms.

Pseudo three-dimensional visualisation tools showed what both aircraft were doing during close-in combat, used in real time to improve situational awareness in the control room and afterwards for debriefing.

The cluster on weapons, displays and the off-boresight problem in The Contemporary Literature holds the wider literature, and these records lie closest to the present subject.

The Flight Test Record

The programme flew 580 flights, an X-plane record, comprising 559 research missions and 21 in Europe for the 1995 Paris Air Show, with fourteen pilots.

The headline totals are published. The distributions are not, and they can be recovered by counting the per-flight log in the appendix of the NASA history. That log carries a flight identifier, a date, a pilot and a purpose for every sortie of both aircraft, and parsing it returns quantities nobody tabulated.

Quantity Value
Flight records parsed 661 across both programmes
Records in the 1990 to 1995 programme 578
Published total for that programme 580
Highest sequence, first aircraft 292
Highest sequence, second aircraft 372
First flight parsed 11 October 1990
Last flight parsed 29 April 2003
Distinct flying days 349
Days carrying more than one flight 187
Busiest day 4 flights

The parse recovers 578 of a published 580, and the three sequence numbers it could not recover account for the difference. That is 99.66 percent of the published total, which is close enough to treat the extraction as sound and far enough from exact to say so.

The tempo follows from the counts and the span. The computation is bookkeeping rather than flight-test technique but is reported with the same care,

\[\dot{N} = \frac{N}{\Delta t} \qquad \text{and} \qquad \frac{N}{D} = \text{flights per flying day}\]

which over the 1,712 days from first flight to the end of the first programme gives 10.28 flights per month and 1.66 flights per distinct flying day.

The annual distribution shows the shape of the programme.

Year Flights
1990 9
1991 98
1992 64
1993 160
1994 181
1995 66
2001 10
2002 41
2003 32

The 1992 dip is the move from Palmdale to Dryden and the reorganisation under the International Test Organization. The 1993 and 1994 peaks are envelope expansion and the tactical utility testing. The gap from 1995 to 2001 is the six years between the end of the first programme and the start of the second.

The longest intervals with no flight by either aircraft are similarly informative.

Gap From To What it is
2,077 days 19 June 1995 24 February 2001 Between the two programmes
406 days 6 April 2001 17 May 2002 Within the second programme
94 days 20 January 1992 23 April 1992 The move to Dryden
84 days 19 January 1995 13 April 1995 The stand-down after the accident
59 days 21 November 1990 19 January 1991 Before the second aircraft flew

The eighty-four-day stand-down falls out of the arithmetic rather than out of the narrative, and it dates the programme’s recovery from the loss of the first aircraft to within a day.

The Milestones

Date Event
11 October 1990 First flight of the first aircraft, 38 minutes, 10,000 feet, from Palmdale
19 January 1991 First flight of the second aircraft
April 1992 First flight from Dryden under the International Test Organization
6 November 1992 Controlled flight at 70 degrees angle of attack, and a velocity-vector roll at 70 degrees
29 April 1993 Minimum-radius 180-degree turn using a post-stall manoeuvre, the Herbst manoeuvre
June 1993 First tactical manoeuvre against a cooperative F/A-18
November 1993 Start of unchoreographed close-in combat testing
1994 Quasi-tailless demonstration
19 January 1995 Loss of the first aircraft
June 1995 Paris Air Show, and the end of the programme

Before the move to Dryden the aircraft flew 108 test missions at Palmdale, achieving thrust vectoring in flight and expanding the post-stall envelope to forty degrees of angle of attack.

The wider literature on this subject is gathered in the cluster on high angle of attack aerodynamics in The Contemporary Literature. The records closest to this section are these.

The Tactical Utility Flight Test

The flight programme flew 87 sorties in the tactical utility phase, of which 80 were scored. Those produced 407 engagements, of which 325 were scored. Pinball II had recommended that each of the six pilots of the International Test Organization fly every starting condition. Reducing those counts to rates,

\[\frac{80}{87} = 92.0 \ \text{percent of sorties scored} \qquad \frac{325}{407} = 79.9 \ \text{percent of engagements scored}\] \[\frac{407}{80} = 5.09 \ \text{engagements per scored sortie} \qquad \frac{325}{80} = 4.06 \ \text{scored engagements per scored sortie}\]

Four scored engagements per sortie is a high rate, the direct product of the real-time kill determination described above. It is also the reason the flight programme reached a few hundred engagements rather than a few dozen.

The flight programme could not reproduce the simulation exactly, and the most consequential difference is an envelope restriction. The aircraft was limited to a post-stall entry speed of 225 knots calibrated maximum, against 325 knots in the simulation, and the programme’s own analysis put the effect of that restriction alone at a thirty percent reduction in post-stall use, according to the NASA history. Only the slow-speed line-abreast condition was substantially unaffected.

A later series raised the entry speed. The flight-test conclusion was that 265 knots calibrated was slightly better than 225 and that 300 knots would be desirable for a future fighter, a finding that the demonstrator was constrained below the useful value of its own capability. The NASA history lists that finding among the flight-test conclusions.

What the Flight Testing Concluded

The conclusions of the tactical utility flight test, as the NASA history sets them out, are quoted here in structure because they are more careful than the summaries built on them.

  • Post-stall close-in combat at seventy degrees of angle of attack had overwhelming success.
  • Flying the aircraft conventionally, limited to thirty degrees, showed that the benefits of post-stall technology were greater than expected.
  • Operational guest pilots with minimal training could fight effectively using post-stall technology.
  • The aircraft was very successful against the then front-line Navy fighters, the F-14D and the F/A-18C.
  • Against opponents with superior thrust-to-weight ratios it was far less successful. The Air Force F-15C and F-16C would not enter the close-in arena and escaped to an altitude sanctuary.
  • Limiting the aircraft to 45 degrees of angle of attack was insufficient and 70 should be used.
  • Advanced short-range missiles with greater angle of attack and off-boresight launch capability are highly desirable for an aircraft with post-stall technology.

The fifth of those is the most important sentence the programme produced, yet it is almost never quoted. A manoeuvre advantage that applies inside a particular arena is worth nothing against an opponent who declines to enter that arena. The F-15C and F-16C pilots of the 422nd Test and Evaluation Squadron exploited their thrust-to-weight advantage to stay above the fight.

The result exposes no limitation of post-stall technology as implemented. It shows what kind of advantage the technology confers. An exchange ratio describes an engagement more than an aeroplane, and the adversary gets a vote on which engagement occurs.

The topic is surveyed more fully in the cluster on air combat effectiveness, simulation and measurement in The Contemporary Literature, and the records bearing most directly on the discussion here follow.

What the Pilots Said

The pilots’ own summary is more restrained than the fact sheet built on it, and the divergence between the two is documented.

The flight-test paper delivered in July 1994 records that all the pilots on the programme had strong military backgrounds and that one point they agreed on is that the aircraft does not necessarily represent a total revolution in air combat. They would not advocate trading away other important fighter characteristics to acquire the capability. What they concluded was that enhanced pitch pointing and velocity-vector manoeuvring opened new options in close-in combat, as a repositioning tool to drive the fight or as a way to rotate and point for a weapons employment the opponent cannot counter, and that it must be used selectively and in a timely manner to succeed.

The NASA fact sheet on the aircraft opens the corresponding passage by stating that the X-31 constituted a revolution in air combat in the post-stall region, and then reproduces the pilots’ qualifications immediately afterwards. The two statements sit in consecutive sentences and the first inverts the source of the second. No accusation of bad faith is intended. The inversion is ordinary and very common when a careful conclusion is summarised, and it is recorded here because the summary is far more widely read than the paper.

There is a second observation from the same paper that has aged well. Carefree handling removed the pilot’s cues about energy state. A conventional fighter warns its pilot through buffet and degrading handling that the energy is going. An aircraft that remains docile throughout its envelope gives no such warning, and the report names restoring those cues as a challenge for future designs.

The broader record appears under the clusters on handling qualities of highly augmented aircraft and on agility as a measured quantity in The Contemporary Literature. These records bear most closely on the argument above.

Comparison With Ground Prediction

The X-31 is unusually well placed for this comparison because it ran the same experiment twice, once in a simulator and once in the air, and then ran the simulator experiment twice as well.

The Simulation Campaigns

Two manned simulation campaigns, called Pinball I and Pinball II, preceded the flight testing. The design was systematic. Four pilots flew each week, one flying the adversary and the other three flying the X-31 with post-stall technology, which gives four groups of three pairings and therefore twelve pilot combinations per week. Twelve combinations across four starting conditions gives 48 engagements for each of six cases, and 240 engagements were flown with the baseline configuration against 192 with a modified high-lift configuration.

Where the Two Simulations Disagreed With Each Other

The two campaigns disagreed sharply, and that disagreement is data about the method rather than about the aeroplane.

Starting condition Pinball I Pinball II
Defensive 1.2 0.47
Offensive 22 23
Slow-speed line abreast 20 6.67
High-speed line abreast 1.5 2.8

Taking the ratio of one campaign to the other condition by condition puts a number on the disagreement, the comparison that the measures-of-effectiveness literature would call a repeatability check,

\[\rho_i = \frac{E_i^{\text{I}}}{E_i^{\text{II}}}\]
Condition Pinball I Pinball II Ratio
Defensive 1.2 0.47 2.553
Offensive 22 23 0.957
Slow-speed line abreast 20 6.67 2.999
High-speed line abreast 1.5 2.8 0.536

The offensive condition reproduced to within 4.3 percent and the slow-speed line-abreast condition differed by a factor of 2.999. Writing $\bar{E}$ for the mean of a campaign’s four conditions, the comparisons are

\[1 - \frac{22}{23} = 0.043 \qquad \frac{20}{6.67} = 2.999 \qquad \frac{2.8}{1.5} = 1.867 \qquad \frac{\bar{E}^{\text{I}}}{\bar{E}^{\text{II}}} = \frac{(1.2 + 22 + 20 + 1.5)/4}{(0.47 + 23 + 6.67 + 2.8)/4} = \frac{11.175}{8.235} = 1.357\]

The defensive condition differed by a factor of 2.553 in one direction and the high-speed line-abreast by 1.867 in the other. The means of the two campaigns are 11.175 and 8.235, a ratio of 1.357, so the campaigns agree far better in aggregate than they do anywhere in particular. That pattern is the signature of sampling variation rather than of a systematic difference.

The programme’s explanation for the defensive disagreement is that the use of post-stall technology on the initial move differed. In Pinball I a standard initial move was used that usually neutralised the attacker’s advantage and turned a defensive start into an offensive situation. In Pinball II that move was used only occasionally.

That explanation is almost certainly correct, and for the measure it is devastating. It means that the exchange ratio in the defensive condition is a measurement of pilot technique rather than of aircraft capability, and that a change in the tactics the pilots chose moved the reported number by a factor of two and a half with the same aeroplane and the same simulator.

Set that beside the interval arithmetic above. At twelve scored outcomes the defensive condition cannot separate 0.36 from parity anyway. The two campaigns disagreeing by a factor of two and a half is therefore not evidence that one of them is wrong. It is what a quantity with that much sampling variation looks like when measured twice.

The Simulation Also Disagreed With the Flight Data in a Way That Mattered

The single most important disagreement is that the simulation did not predict the departure. The flight-test report states it plainly. The fidelity of the simulation was described as quite good and it always lagged behind the flight data, and the report says the failure to predict the departure ahead of time is the most dramatic illustration of that lag.

The lag has a nameable mechanism. The aerodynamic model was built from wind tunnel data and corrected by parameter identification from flight, so the simulation could only know about an aerodynamic effect after the aircraft had flown into it. For a phenomenon like forebody vortex asymmetry, which is sensitive to a half-inch of nose radius and to Reynolds number, that is a structural limitation rather than a modelling shortfall.

The Wind Tunnel Did Not Give the Complete Picture

The flight-test report’s first technical lesson is exactly that sentence, and the report specifies where the tunnel fell short. Above fifty degrees of angle of attack the aircraft encountered lateral kicks the pilots called lurches, which had not appeared on the small-scale model. The asymmetry data showed no distinguishable trend against flight condition across a Reynolds number range of 1.5 to 4.0 million based on a forebody diameter of about 3.2 feet, as the asymmetry report gives it. The range is narrow, and its narrowness is one reason the tunnel could not have settled the question.

The comparison of flight against wind tunnel and water tunnel yawing-moment asymmetries became a research subject in its own right, with reports published in 1994 and revisited in 2001 and again in 2003.

A Negative Result on High Lift

A modification that should have helped did not, and the programme cancelled it on the evidence. Flight researchers recognised that a high-lift configuration would give the aircraft dramatically more lift below thirty degrees of angle of attack. Without post-stall technology the high-lift aircraft outperformed the baseline against the F/A-18, as expected.

With post-stall technology the high-lift aircraft was slightly less effective, because pilots gave up post-stall opportunities trying to fly in the high-lift mode. The pattern was that a pilot able to exploit high lift would fly post-stall until the fight went low and then drop below thirty degrees to use high lift and avoid the hard deck, which turned the fight conventional. Since the aircraft’s advantage was in post-stall flight, the international test community decided against implementing high lift.

A capability that improves the aircraft in isolation and degrades it in combination is a real finding, of a kind that only a manned simulation with free tactics can produce.

Readers wanting more can turn to the cluster on flight test, parameter identification and aerodynamic modelling in The Contemporary Literature. The following records sit nearest to this discussion.

What the Data Changed

The Capability Was Adopted and the Aircraft Was Not

Thrust vectoring entered production, and it did so on aircraft that did not need the X-31 to prove it was possible. The F-22 carries two-dimensional vectoring nozzles. The F-16 multi-axis thrust vectoring programme demonstrated three-dimensional vectoring on a production airframe during the X-31’s own flight programme, and the X-31 flight-test report cites it as evidence that incorporation is achievable without significant trade-offs.

What the X-31 contributed was not the mechanism but the evidence about employment, and the evidence is more qualified than the reputation. Post-stall manoeuvring supplements conventional performance and does not replace it. It wins where the fight is slow and close and loses where the opponent can decline that fight.

The Quasi-Tailless Result Outlived Everything Else

The quasi-tailless demonstration is the part of this programme with the longest reach, and it was almost an afterthought. It established that an aircraft with no vertical tail at all could be held at supersonic speed by thrust vectoring, at a divergence rate of 290 milliseconds to double.

A smaller or absent vertical tail also reduces radar cross-section, an advantage the quasi-tailless flight test report lists beside lower drag, weight and structural complexity. The question of how to control yaw without a fin is now the central configuration question of unmanned combat aircraft design, and the answers in current use are split effectors, differential drag devices and vectoring. The X-31’s contribution was to establish that the vectoring answer works at a measured level of instability rather than an assumed one.

There is a second and less-noticed result from the same programme. Differential canard deflection was studied as a yaw generator with and without the vertical tail, a redundant-effector question of the kind that modern control allocation exists to solve.

The cluster on unmanned combat aircraft and the tailless problem in The Contemporary Literature holds the wider literature, and these records lie closest to the present subject.

The Accident Changed the Air Data System, Eventually

The aircraft that was destroyed by a probe returned with redundant air data. The second aircraft flew again from 2001 to 2003 in the Vectoring Extremely Short Takeoff and Landing Control Tailless Operations Research programme, which used thrust vectoring to fly approaches at high angle of attack. The NASA history records that it first flew with the original Kiel probe, this time heated, and a second heated probe feeding an additional air data computer that monitored the first, and that it tested a flush air data system of pressure ports in the nose with no protruding probe to ice.

The arc is satisfying, but it should not be overstated. The flush system was a separate experiment by the European manufacturer flown in parallel, and the X-31’s second life was a separate programme with its own objectives. But the pairing is real, and the added monitor addresses the exact failure mode that destroyed the first aircraft, because a second, independent pneumatic path can reveal a blockage that a single probe shares with every computer downstream.

The Mishap Board’s recommendations were institutional rather than technical for the most part. It recommended that the system safety analyses be redone with a focus on single-point failures, that a mitigation for the loss of raw pitot-static data be found and verified in simulation, that temporary operating procedures be reviewed and controlled, that all cockpit switches and displays be audited for correct function and labelling, that programme personnel be trained in the basics of system safety, and that the parachute be replaced with one of slower descent rate.

The last of those is no control-system finding, but it belongs in the record. The pilot descended under a seventeen-foot canopy at approximately 28.3 feet per second, a rate typical for naval aircraft that operate mostly over water, and was injured on the hard playa surface.

The wider literature on this subject is gathered in the cluster on air data, icing and flush air data sensing in The Contemporary Literature. The records closest to this section are these.

What Some Test Team Members Thought the Report Missed

Not everyone accepted the Mishap Report’s emphasis, and the disagreement is instructive. Team members felt that large airspeed errors and suspected icing should have mandated stopping the test until the discrepancy was understood, and that the report placed too little weight on the test team’s own methodology. One is quoted saying that the test pilot and flight control engineers ought to understand the importance of the right gains, and another that seventy-knot errors and icing conditions are not acceptable for good flight tests.

The lesson the programme’s pilots drew was procedural rather than technical. Rogers Smith put it to the Society of Experimental Test Pilots in 1996 as an instruction to speak up, stop the tests and think when something does not make sense, and Ken Szalai, the Dryden director at the time, put it as calling a halt for anything that does not sound, feel or smell right.

Both formulations are about a failure of attention rather than a failure of engineering, and the arithmetic in this article supports that reading. The information needed to detect the failure was displayed, spoken aloud, and independently available on a second instrument.

The Contemporary Literature

Cluster Records
Air combat effectiveness, simulation and measurement 287
Air combat decision making by learned policies 125
Rates, proportions and their intervals 349
Uncertainty quantification 53
Pursuit, evasion and differential games 166
Agility as a measured quantity 76
Post-stall manoeuvring 103
High angle of attack aerodynamics 552
Yaw asymmetry and forebody flow 91
Canard and delta configurations 288
Thrust vectoring 209
Fluidic thrust vectoring 137
Control power and effector sizing 60
Engine thrust with altitude 13
The standard atmosphere 10
Propulsion integration at extreme attitude 326
Axis transformations and the velocity-vector roll 43
Flight control system design, actuation and allocation 491
Gain scheduling and stability margins 97
Unstable dynamics 65
Handling qualities of highly augmented aircraft 245
Envelope protection and loss of control 86
Unmanned combat aircraft and the tailless problem 289
Machine learning in flight control and aircraft modelling 203
Clearing a learned controller for flight 37
Air data, icing and flush air data sensing 391
Airspeed systems as a measurement discipline 21
Analytical redundancy and sensor fault detection 231
Flight test, parameter identification and aerodynamic modelling 480
Structures, materials and the airframe 88
Weapons, displays and the off-boresight problem 76
Missile guidance and interception 95
Missile autopilots and missile flight control 20
Rotorcraft flight dynamics and control 56
Multirotor flight control 38
The vehicle’s own literature 36
Other aerodynamic and flight mechanics literature 341
The general control literature 410
Total 6,684

The X-31 flew before three of the fields that now own its questions existed. This section surveys the modern literature on each, written from current knowledge rather than from the editorial date of the article.

Air combat effectiveness, simulation and measurement

The keystone’s own literature. Measures of effectiveness, combat models, piloted and constructive air combat simulation, and the evaluation of fighters in engagements.

The measure the programme reported has a literature of its own, and very little of it names the measure. Measures of effectiveness and combat effectiveness appear in 38 of the 287 titles, and the measures of effectiveness for general-purpose forces that the exchange ratio belongs to are part of it. Beyond-visual-range combat appears in 21 titles, while the exchange ratio, the kill ratio and the loss exchange together appear in five, so the quantity the keystone section analyses is almost never the subject of a title.

287 records. 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[Calculation of the exchange ratio for the Adaptive Maneuvering Logic program 1985] [Carpenter and Falco 1975] [Cetin and Kandemir 2017] [Chae and Choi 2018] [Chan and Vogel 1988] [Chang et al 2022] [Chao Feng and Peng Yao 2016] [Chaoran et al 2021] [Chen et al 2017] [Chen et al 2022] [Chen et al 2024] [Chen et al 2025] [Ciavarelli and Brictson 1978] [Connelly and Myers 1992] [Hague 1983] [Costes 1988] [Covino and Healy 1991] [Coward and Rupp 1982] [Cowdale 2014] [Cox and Roy 1988] [Dantas et al 2021] [Dees and Cornett 1989] [Dees and Cornett 1990] [Development Of Air Combat HDEVS Model Implemented In HDEVSim++ Environment 2017] [Dewangga and Sasongko 2019] [Dixon et al 1989] [Dong et al 2015] [Dong et al 2025] [Dou et al 2023] [Drenth et al 2024] [Dupuy 1986] [Mitchell] [Endsley 1993] [Engel 1963] [Exchange Ratios] [Fenghua He and Yu Yao 2010] [Flynn 1987] [Fu and Wang 2011] [Fu et al 2015] [Fu Li et al 2008] [Poropudas and Virtanen 2009] [Gao et al 2022] [Gao et al 2023] [Gao et al 2025] [Geiselman and Osgood 1995] [Goodrich et al 1992] [Goodrich et al 1995] [Grimm and Well 1987] [Gyue-jeong et al 2025] [Ha et al 2015] [Ha et al 2018] [Hague and Hague 1987] [Haichao et al 2023] [Han et al 2019] [Hankins 1976] [Hankins 1979] [Haoyu et al 2018] [Hara 1964] [Harris et al 1985] [Hayslip et al 1990] [Hayward 1968] [Head 1981] [Heng et al 2023] [Herrick 1985] [Herrick 1988] [Herrick 1988] [Herrick 1990] [Hooks and Huff 1968] [Houck et al 1989] [Hu et al 2018] [Huang et al 2020] [Hutchins and Jones 1975] [Hutchins 1978] [Huynh et al 1987] [Lane 1986] [İşci and Günel 2021] [Jarmark 1976] [Ja¨rmark 1989] [Jefferies and Pinney 1990] [Jia et al 2023] [Jia et al 2025] [Jian et al 2024] [Jiang et al 2023] [Jiang et al 2024] [Jiaqi et al 2023] [Johnson 1984] [Kaplan et al 1994] [Katz 1994] [Kelley and Lefton 1977] [Kelley and Lefton 1979] [Kelly 1988] [Kelly et al 1979] [Ketchel 1977] [Kim and Lee 2022] [Kitowski 1992] [Krause 1997] [Kruk and Runnings 1989] [Kung and Chiang 2015] [Lafferty et al 1991] [Leatham and Linch 1974] [Lee et al 2022] [Lewis 1986] [Lewis and Yeo 1985] [Lewis and Aiken 1985] [Lewis et al 1987] [Lewis et al 1987] [Li and Fan 2018] [Li and wang 2021] [Li et al 2020] [Li et al 2021] [Li et al 2022] [Li et al 2022] [Li et al 2022] [Li et al 2022] [Li et al 2022] [Li et al 2022] [Li et al 2022] [Li et al 2022] [Lim and Park 2012] [Liqun et al 2017] [Liu et al 2022] [Liu et al 2022] [Liu et al 2023] [Liu et al 2025] [Luhn 1988] [Luo et al 2025] [Ma et al 2020] [Macone 1996] [Macone 2000] [Malone et al 1987] [Mansikka et al 2023] [Marchi and Sweeney 1992] [McGuinness et al 1981] [McKee 1998] [Measures of Effectiveness and Measures of Performance 1995] [Meng et al 2022] [Mentz et al 2006] [Merz 1989] [Miller and Hadley 1983] [Minglang et al 2018] [Moroney et al 1979] [Stark 1976] [Muhl 1989] [Neuman 1987] [Nichols 1998] [Oran 1999] [Pan et al 2017] [Panec 1991] [Park et al 2016] [Parrott 1978] [Pennington and Meintel 1980] [Persing et al 2005] [Petty and Barbosa 2016] [Pohlmann and Reed 1978] [Prilliman et al 1969] [Pugh 1973] [Pugh and Mayberry 1973] [Ramteke et al 2022] [Ramírez López and Żbikowski 2018] [Rao and Fu 1989] [Austin et al 1991] [Real‐Time Knowledge Based System to Control Major Air‐Combat Simulator 1991] [Reinisch et al 2022] [Ren et al 2023] [Rogers et al 1973] [Ruther et al 2022] [Saied et al 2018] [Scukins et al 2023] [Scukins et al 2024] [Hoffren and Vilenius 1998] [Shakir 1996] [Shi et al 2018] [Shi et al 2018] [Shinar 1986] [Shinar and Davidovitz 1987] [Shinar et al 1989] [Shull and Griffin 1990] [Simkins and Breyer 1989] [Simpson 1976] [Howells et al 1997] [Skowronski 1988] [Smullen and Nowak 1994] [Sobel 2015] [Sovelius et al 2019] [Sternberg et al 1994] [Sun et al 2020] [Suseno and Sasongko 2016] [Glaerum] [Tan et al 2022] [Tan et al 2025] [Lazarus] [Thomas 1999] [Hague 1982] [Portrey et al] [Topbaş et al 2024] [Torrance et al 1957] [UML-based combat effectiveness simulation system modeling within MDE 2018] [Ummah et al 2020] [Viken and Burley II 1992] [Warner et al 1993] [Waag and Bell 1997] [Waag and Bell 1997] [Wallace and Bowers 1982] [Wang and Tian 2025] [Wang et al 2017] [Wang et al 2019] [Wang et al 2020] [Wang et al 2022] [Wang et al 2023] [Wang et al 2024] [Wang et al 2025] [Wang et al 2025] [Washington Univ Seattle Dept Of Statistics 2007] [Weckl 1987] [Weilin et al 2018] [Wenham 1970] [Wentao et al 2022] [White and Olmstead 1997] [Wilson 1989] [Woodruff and Oliver 1971] [Wooldridge et al 1982] [Wu et al 2015] [Wu et al 2018] [Xi et al 2023] [Xi et al 2024] [Xie et al 2025] [Xu et al 2018] [Yagci and Nikbay 2022] [Yang et al 2020] [Yang et al 2022] [Yang et al 2024] [Yang et al 2025] [Yang et al 2025] [Yoo et al 2021] [Zhang and Shang 2023] [Zhang et al 2015] [Zhang et al 2022] [Zhang et al 2023] [Zhao and Huang 2018] [Zhao et al 2017] [Zhao et al 2024] [Zhen-ning et al 2020] [Zheng and Feiguo 2017] [Zhong et al 2022] [Zhou et al 2026] [Zhu et al 2020] [Zoujie and Wuwei 2016]

Air combat decision making by learned policies

Air combat flown by software. Manoeuvre decision making, situation assessment and engagement tactics produced by reinforcement learning, neural networks and other learned agents.

The question the X-31 answered with six test pilots and a scoring radar is now answered with learned policies. Of these 125 records, 77 name reinforcement learning or one of its algorithms, and 101 were published from 2022 onward. Beyond-visual-range engagements are named in 25 titles and within-visual-range, close-range or one-to-one engagements in 19. An agent trained against one adversary policy and one distribution of starting conditions learns an advantage conditional on both, and the aggregate win rate it reports is the same weighted mean this article has spent its keystone section unpacking.

The modern work has one advantage the X-31 did not. It can fly ten thousand engagements rather than four hundred, which moves the high-speed line-abreast case from unresolvable to routine.

125 records. [Jiang et al 2025] [Wu et al 2025] [Smith and Dike] [Crumpacker et al 2022] [Jun Fang et al 2016] [Kaneshige and Krishnakumar 2007] [Zhang et al 2025] [Bae et al 2023] [Baykal and Baspinar 2023] [Cao et al 2023] [Chen et al 2023] [Chen et al 2025] [Costa et al 2025] [Dantas et al 2022] [Dantas et al 2022] [Dantas et al 2023] [Dantas et al 2025] [Eilbert and Hayslip 1989] [Ernest and Carroll 2016] [Salhi et al 2026] [Fan et al 2022] [Floyd et al 2017] [Fu and Yang 2026] [Gao et al 2024] [Gong et al 2022] [Gu et al 2025] [Guo et al 2022] [Guo et al 2026] [Guo et al 2026] [Haci et al 2026] [Han et al 2025] [Hao et al 2026] [He et al 2025] [Selmonaj et al 2023] [Hong et al 2022] [Hu et al 2021] [Hu et al 2022] [Hu et al 2025] [Huo et al 2025] [Zhang et al 2023] [Isci and Koyuncu 2022] [Jia et al 2023] [Jiandong et al 2021] [Jiang et al 2022] [Jing et al 2024] [Keles et al 2025] [Kong et al 2020] [Kong et al 2020] [Kong et al 2023] [Kuroswiski et al 2023] [Kuroswiski et al 2025] [Lei et al 2022] [Li et al 2020] [Li et al 2022] [Li et al 2022] [Li et al 2024] [Li et al 2024] [Li et al 2025] [Li et al 2025] [Li et al 2025] [Liang et al 2024] [Lima Filho et al 2021] [Liu et al 2025] [Liu et al 2026] [Liu et al 2026] [Mu et al 2026] [Jiang et al 2025] [Gunsch et al] [Park et al 2021] [Peng et al 2024] [Pengzhuang and Hanwen 2023] [Piao et al 2020] [Qiu et al 2020] [Quan et al 2026] [Zhang et al 2023] [Rodin and Massoud Amin 1992] [Ruan et al 2022] [Saldiran et al 2023] [Saldiran et al 2024] [Schosser et al 2026] [Scukins et al 2023] [Shalumov 2020] [Sheeba et al 2025] [Shin et al 2026] [Sun et al 2025] [Tan et al 2024] [Tasbas and Aydinli 2021] [Chen et al 2024] [Wang and Wang 2024] [Wang et al 2020] [Wang et al 2021] [Wang et al 2023] [Wang et al 2024] [Wang et al 2024] [Wang et al 2024] [Wang et al 2024] [Wang et al 2024] [Wang et al 2025] [Wang et al 2025] [Wu et al 2025] [Xie et al 2020] [Yang et al 2019] [Yang et al 2020] [Yang et al 2022] [Yang et al 2024] [Yang et al 2025] [Yang et al 2025] [Yang et al 2025] [Yao et al 2026] [Yin et al 2022] [Zhang et al 2022] [Zhang et al 2023] [Zhang et al 2024] [Zhang et al 2024] [Zhang et al 2026] [Zhang et al 2026] [Zhao et al 2023] [Zhao et al 2024] [Zhao et al 2024] [Zheng and Duan 2023] [Zheng et al 2024] [Zheng et al 2024] [Zhou et al 2024] [Zhu et al 2024] [Zhu et al 2024]

Rates, proportions and their intervals

The arithmetic behind the keystone’s second question. Interval estimation and hypothesis tests for a binomial proportion, for two proportions and for rates, together with the choice of sample size.

Much of this literature was written for medicine. Of the 349 records, 199 name a binomial or a proportion, and 90 appeared in medical, epidemiological or biometric venues, which is one of the places where the inference on small counts that an engagement campaign needs has been worked out most carefully.

349 records. [Wani and Lo 1975] [Pigeot 1990] [Singh 2012] [Khare et al 2021] [Hauck 1984] [Eberhardt and Fligner 1977] [Gray and Lewis 1967] [Bedrick 1987] [Mulekar and Young 1993] [Mickey and Elashoff 1985] [Guan 2012] [Turpin et al 2023] [Zhou and Qin 2005] [Phillips and Holland 1986] [Pervaiz et al 2021] [Subramani 2016] [Flanders 1985] [Wallenstein 1997] [Holland 1989] [Kim et al 2022] [Gupta et al 2024] [Gupta 2024] [Ryu 2009] [Donald and Donner 1987] [Ramanana-Rahary et al 2009] [Agresti and Coull 1998] [Hill 1970] [Almendra-Arao et al 2023] [Mantel and Hauck 1986] [Enang et al 2014] [Raja and Maqbool 2020] [Misra et al 2021] [Reiczigel et al 2008] [Plante 1991] [Andersson 2022] [Andersson 2024] [Angus 1987] [Khamkong 2012] [Ting and Graybill 1991] [Aranishi and Ikeda 2015] [Asrabadi 1985] [Bae et al 2017] [Bahrampour et al 2022] [Bandyopadhyay et al 2017] [Bandyopadhyay et al 2024] [Basu et al 2026] [Shi and Bai 2008] [Joseph and Bélisle 2019] [Rahardja et al 2021] [Bayesian Inference for Binomial Proportion 2004] [Rahardja and Zhao 2012] [Joseph et al 2008] [Sambucini 2017] [Beal 1987] [Berkachy and Donzé 2019] [Liu 2026] [Nam 1993] [Binomial parameter R estimation in Clopper-Pearson method 2012] [Boiculese and Dimitriu 2015] [Braun and Pregibon 1984] [Breebaart 2015] [Brenner and Quan 1990] [Burstein 1973] [Böhning 1988] [Calvez 2022] [Cao et al 2024] [Cao et al 2026] [Thakkar 2025] [Katakam 2025] [Kadhim 2020] [Chan and Zhang 1999] [Chandra et al 2019] [Chang et al 2022] [Chen 1977] [Chen 1984] [Chen 1997] [Chen et al 2015] [Chen et al 2026] [Mantel 1963] [Bu and Sun 2024] [Berg 2001] [Goldstein and Langholz 2007] [Jaech 1970] [Kabaila 2007] [Comparison of Accuracy Properties for Confidence Intervals of the Cross-Product Ratio of Binomial Proportions under Direct-Direct Sampling Scheme 2021] [Jin and Lu 2009] [Sakakibara 2014] [Kaya et al 2021] [Comparison of Two Means, Two Variances or Two Proportions 2012] [Andersson 2024] [Altman 2000] [Confidence Interval for Survival Model based on Partly Interval-Censored data 2015] [Chen et al 2004] [Pradhan and Banerjee 2008] [Fishman 1986] [Price and Bonett 2008] [Confidence Intervals for a Simple Binomial Proportion 2012] [Graybill and Wang 1979] [Li et al 2013] [Confidence Intervals For Signal To Noise Ratio Of A Poisson Distribution 2012] [Lee 2010] [Koopman 1984] [Kharrati-Kopaei and Dorosti-Motlagh 2019] [Confidence intervals for the ratio of two poisson means and poisson predictor intervals 1971] [Kahle et al 2016] [Lloyd 1990] [Sato 1990] [Nam 1995] [Conlon and Thomas 1990] [Almendra-Arao et al 2016] [Yamauchi] [Copas 1992] [Pradhan et al 2013] [Ding 2024] [Cothran 1979] [Wang and Rousseau 2021] [Crane 1974] [Daly 1992] [Dang and Krishnamoorthy 2021] [David Sotres-Ramos and Jesus Rivera-Cortez 2020] [de Paula Castro et al 2021] [Dean and Pagano 2015] [Deep et al 2022] [Determining Sample Sizes Needed to Detect a Difference between Two Proportions 2003] [Anderson and Burstein 1970] [Disney 2016] [Dong and Glynn 2019] [Dozzi and Riedwyl 1984] [Duan et al 2016] [Edwardes and Vollset 1994] [Nazman and Olmuş 2020] [Ekpenyong 2014] [Qin 1999] [Böhning et al 2022] [Hauck 1987] [Donner and Hauck 1988] [Pal and Singh 2019] [Rao 1981] [Phillips and Holland 1987] [Frey and Pérez 2012] [Somerville and Brown 2013] [Xiao et al 2017] [Jäntschi and Bolboacă 2010] [Storer and Kim 1990] [Shao et al 2024] [Ironyt and Pereirat 1986] [Liang 1987] [Krishnan and Suresh 2025] [Fagan 1996] [Fagan 1999] [Fairweather 1972] [Feng et al 2022] [Fujino 1980] [Garthwaite et al 2023] [Gelfand and Thomas 1974] [Puza and O’neill 2006] [Davis 1985] [Singh et al 2016] [Greenland 1989] [Subramani 2013] [Gillett 1989] [Goldman 1963] [Gould 1983] [Graybill 1958] [Graybill and Morrison 1960] [Green 1982] [Halperin 1980] [Hao et al 2016] [Hasan and Krishnamoorthy 2023] [Hauck and Anderson 1986] [Hawila and Berg 2022] [Herrmann and Szatrowski 1982] [Herrmann and Szatrowski 1984] [Liu and Pierce 1993] [Triveni and Danish 2023] [Deely and Gupta 1988] [Hirose et al 2023] [Hohls 1998] [Holladay and Schilling 2024] [Yanagawa and Fujii 1990] [Percus and Percus 2010] [Hung 1997] [Nijimbere 2017] [A Bayesian analysis of 1998] [Belhaj 2018] [Lin and Yang 2006] [Tryon and Lewis 2007] [Hong 2002] [Pigeot and Strugholtz 1994] [Janze 2017] [Jȩdrzejczak et al 2022] [Jia 2020] [Jin et al 2017] [Reed 2004] [Joseph et al 1995] [Jäntschi 2021] [Jäntschi 2025] [Khandeparkar and Sabnis 2025] [Kikuchi et al 2015] [Kokaew et al 2021] [Kokaew et al 2022] [Krippendorff 1970] [Kudo 1956] [Lawson and Ahlstrom 2021] [Lloyd and Kabaila 2010] [Liang 1989] [Lin 1995] [Liu et al 2018] [Locke 1984] [Malinowski 2005] [L’Ecuyer et al 2023] [Malekzadeh and Mahmoudi 2020] [Zhang and Boos 1997] [Rahardja and Yang 2015] [McGrath and Burke 2024] [McGrath and Irving 1973] [McHugh 1961] [McNickle et al 1996] [Rao 2002] [Honglei and Sigui 2019] [Habtzghi et al 2014] [Fosgate 2005] [Müller M.Sc. and Bertsche 2019] [Nawa 2022] [Kang 2008] [Zhou et al 2004] [Vallejo et al 2013] [Ngamkham et al 2016] [Niwitpong 2015] [Coolen and Coolen-Schrijner 2007] [Berger and Gold 1973] [Banerjee and Biswas 2014] [Klein and Linton 2013] [Banerjee and Biswas 2012] [Wang 2010] [Gupta and Liang 1988] [Turpin and Jens 2018] [Buhrman 1973] [Soulakova and Roy 2012] [Berengut and Petkau 1979] [Frishman 1971] [Wallenstein and Berger 1981] [García-Pérez 2005] [Guilbaud and Hauck 1983] [Moore and Nickens 1968] [Orawo 2021] [Ouellette 2025] [Ožbot and Škrjanc 2024] [Fujii and Yanagimoto 2005] [Park 2019] [Park and Leemis 2019] [Parker 1996] [Peer and Azriel 2024] [Peng et al 2019] [Peskun 1993] [Nebenzahl and Sobel 1972] [Emily and Friguet 2016] [Abdullahi et al 2023] [Yanagawa and Fujii 1995] [Yanagawa and Fujii 1995] [Mantel 1985] [Lee 1984] [Rahardja 2019] [Rahardja and Young 2021] [Rahardja et al 2016] [Kadilar et al 2007] [Marsaglia 1965] [Reeve 2016] [Remund and Kitto 2011] [Roy 1993] [Bourke 2001] [Afshartous 2008] [Katsis 2005] [Magaret and Stanaway 2011] [Gonçalves et al 2012] [Guenther 1974] [Samuels and Lu 1992] [Sandelius 1952] [Santner 1998] [Sargent et al 1992] [Schader and Schmid 1990] [Scheffe 1944] [Khare et al 2022] [Udomprasertgul 2001] [Cho 2007] [Shao and Ye 2026] [Shen 1998] [Shiffler and Adams 1987] [Shukla and Subrahmanyam 1999] [Smithpreecha et al 2016] [Sobel and Groll 1966] [Krishnamoorthy and Peng 2007] [Halls 2004] [Nadarajah and Kotz 2006] [Li and Ma 2022] [Statistical Inference for Two Proportions] [Nelson 1972] [Steiger and Wilson 1999] [Yu et al 2013] [Sungboonchoo and Paichit 2023] [Sungboonchoo et al 2021] [Sungboonchoo et al 2021] [Tableman 1987] [Tables for Use with Binomial Samples-Contingency Tests, Confidence Limits and Sample Size Estimates 1957] [Faust 1990] [Tanaka 1961] [Tang 2020] [Tang et al 2015] [Tango 1998] [Consonni and La Rocca 2008] [Thangjai et al 2023] [Hauck and Donner 1988] [The Comparison of Proportions from Several Independent Samples 2003] [Bettin 2021] [Thulin 2014] [Somes 1986] [Hauck 1979] [Donner and Hauck 1986] [The Logrank and Mantel-Haenszel Tests 1998] [Brumback and He 2011] [Basilevsky 1980] [Vitaliano 1979] [Zieliński 2009] [Zieliński 2022] [Breslow and Liang 1982] [Andersson 2023] [Witkovský and Wimmer 2007] [The Weighted Polya Posterior Confidence Interval For the Difference Between Two Independent Proportions 2006] [Pigeot and Strugholtz 1998] [Easterling 2015] [Thong and Shan 2015] [Martín Andrés and Álvarez Hernández 2012] [Martín Andrés and Herranz Tejedor 2003] [El-Neweihi and Proschan 1979] [Buchanan 1961] [Vollset 1993] [Olivier and May 2006] [Pradhan et al 2014] [Weinberger and Yemini 2022] [Wiley et al 2021] [Wilkinson 1961] [Williams 1988] [Wu and Shan 2024] [Xiang et al 2018] [Xue 2015] [Yan and Xu 2016] [Yarandi 1994] [Zielińska-Kolasińska and Zieliński 2023] [Zieliński 2018]

Uncertainty quantification

Intervals on computed quantities. Uncertainty quantification and sensitivity analysis in aerodynamic, structural, propulsion and simulation models.

A result with an interval attached is a modern expectation and it was not a 1993 one. The X-31 published point estimates of exchange ratios with no uncertainty statement, and the arithmetic in this article’s keystone section is an attempt to supply retrospectively what the discipline now supplies as a matter of course.

53 records. [Aircraft Turbofan Engine Linear Model Additive Static Unstructured and Structured Uncertainty Norm Estimation 2023] [Asouti et al 2023] [Barthelemy and Bergen 1989] [Bellosta et al 2021] [Bhardwaj and Kapania 1995] [Campbell et al 2026] [Capiez-Lernout and Soize 2017] [Covella and Fantuzzi 2023] [Dalle et al 2018] [Denham et al 2018] [Eldredge and Mousavi 2026] [Wang and Tang 2004] [Ezvan et al 2016] [Favaregh et al 2016] [Fravolini et al 2017] [Freitas 2020] [Gießler et al 2025] [Giunta 1999] [Głębocki and Jacewicz 2020] [Holifield and Tufts 2024] [Jiang and Li 2018] [Jung and Baeder 2019] [Jung and Baeder 2023] [Khurana et al 2023] [Kraft 2018] [Li et al 2018] [Liu et al 2017] [Liu et al 2026] [Maes et al 2015] [Magrini et al 2021] [Misra and Bai 2019] [Nakamura-Zimmerer et al 2023] [Nelson 2018] [Chen et al 2004] [Papoutsis-Kiachagias et al 2021] [Phillips and West 2018] [Qiao et al 2026] [Rosić and Diekmann 2015] [Schaefer et al 2015] [Schaefer et al 2019] [Schaefer et al 2024] [Stoldt et al 2021] [Suga and Yamazaki 2015] [Tang et al 2015] [Thomas and Dowell 2025] [van den Brandt and de Visser 2018] [Wignall and Houlden 2020] [Willier et al 2024] [Xiao Jiaping and Shen Zuojun 2016] [Yang and Yee 2023] [Ye et al 2022] [Yuan et al 2022] [Zang et al 2015]

Pursuit, evasion and differential games

The formal theory of the engagement. Differential games, pursuit and evasion, capture and the value of a game, for aircraft, missiles, spacecraft and generic agents.

The mathematical treatment of one-versus-one air combat as a differential game predates the X-31 by decades, with Isaacs on differential games as its founding text, and the programme’s reports do not appear to draw on it. Of the 166 records, 54 predate 1990, the year the X-31 first flew, and they are the part of the literature the programme could have read. That literature poses the engagement as a two-sided optimisation with a value function, and the value at a given starting state is precisely the quantity the exchange ratio estimates by sampling.

A formal treatment would have said in advance that the value depends on the starting state, which is the article’s central finding arrived at by arithmetic on published outcomes. The two literatures were solving the same problem in different buildings.

166 records. [Abbas Badakaya et al 2022] [Adams et al 2025] [Al-Talabi 2017] [Al-Talabi 2017] [Al-Talabi and Schwartz 2016] [Alexopoulos et al 2017] [Altaher et al 2020] [Ardema and Rajan 1987] [Ardema et al 1987] [Asgharzadeh-Bonab et al 2026] [Azamov and Ibaydullayev 2020] [Azamov and Ibaydullayev 2021] [Badakaya et al 2022] [Barron and Jensen 1995] [Becerra et al 2015] [Ben-Asher 1994] [Ben-Asher 1996] [Berkovitz 1985] [Berkovitz 1986] [Bernhard 1987] [Bernhard and Pourtallier 1994] [Bertram and Wei 2021] [Bopardikar et al 2008] [Bugnon and Mohler 1988] [Calise and Yu 1985] [Camci and Kayacan 2016] [Carr et al 2018] [Chen and Guo 2025] [Chen et al 2017] [Chentsov 2020] [Chentsov 2021] [Chentsov and Khachai 2020] [Chentsov and Khachay 2019] [Chentsov and Khachay 2020] [Chukanov et al 2024] [Cline and Vincent 1988] [Cui et al 2020] [de Villiers et al 1987] [De Villiers et al 1988] [Dong Chyung 1970] [An et al 2011] [Fang et al 2020] [Fisac and Sastry 2015] [Friedman 1973] [Fuchs et al 2018] [Galperin and Skowronski 1987] [Galperin and Zheng 1989] [Galperin and Zheng 1989] [Garcia et al 2021] [Gard 2023] [Guelman 1993] [Guelman and Shinar 1988] [Guelman et al 1990] [Hao Chen et al 2015] [Hayoun et al 2015] [Hayoun et al 2016] [Henareh and Norouzi 2016] [Hexner et al 2019] [Ho et al 1965] [Hsueh and Ling 2017] [Hu and Zhou 2017] [Hu et al 2024] [Ibragimov et al 2023] [Ibragimov et al 2026] [Imado 1990] [Imado 1993] [Imado and Ishihara 1993] [Ivanov 2021] [Ja. Rubinovich 2018] [King and Peterson 2019] [Lagunov 1977] [Leitmann 1969] [Leitmann 1969] [Liang et al 2022] [Liao et al 2021] [Medveď 1978] [Meier 1969] [Menon 1989] [Menon 1991] [Menon and Chatterji 1996] [Menon and Cheng 1990] [Menon and Duke 1990] [Menon and Duke 1992] [Merz 1975] [Merz and Hague 1977] [Meschler et al 1967] [Miloh 1982] [Moritz et al 1987] [Nikoukhah and Delebecque 1992] [Nonlinear differential game “pursuit-evasion”: information aspect 2021] [Olsder and Papavassilopoulos 1986] [Lee et al 2017] [Duan et al 2024] [Pachter and Yavin 1981] [Pachter and Yavin 1983] [Pachter and Yavin 1986] [Pachter and Yavin 1986] [Pan et al 2026] [Pashkov and Sinitsyn 1995] [Pham 2021] [Platzer 2012] [Pontryagin and Mishchenko 1988] [Prasad and Rajan 1987] [Rao and Shaiju 2023] [Wang et al 2023] [Sun et al 2009] [Xie et al 2014] [Ji et al 2018] [Rodin 1990] [Sahin and Kumbasar 2020] [Segal and Miloh 1995] [Segal and Miloh 1996] [Segal and Miloh 1999] [Lin et al 2007] [Sharifi et al 2022] [Shinar and Glizer 1992] [Shinar and Glizer 1999] [Shinar et al 1989] [Shishkina 1987] [Sinar and Farber 1982] [Skowronski and Stonier 1987] [Skowronski and Stonier 1987] [Sun and Tsiotras 2015] [Sun et al 2016] [Sun et al 2017] [Sun et al 2017] [Sun et al 2019] [Sweriduk and Calise 1994] [Szots and Harmati 2019] [Tamer and Max 1975] [Tang et al 2023] [Turetsky and Glizer 2022] [Turetsky and Shima 2016] [Turetsky and Shima 2016] [Umar et al 2024] [Wang and Liu 2025] [Wang et al 2019] [Wang et al 2026] [Wei and Yang 2018] [Weintraub et al 2020] [Williams et al 1989] [Xi and Cai 2024] [Xi et al 2023] [Xie and Ma 2025] [Xie and Ma 2026] [Xu et al 2022] [Yan 1995] [Yang et al 2022] [Yang et al 2024] [Yang et al 2024] [Yavin 1985] [Yavin 1986] [Yavin 1986] [Yavin 1986] [Yavin 1987] [Yavin 1987] [Yavin 1987] [Yavin 1988] [Yavin 1991] [Yavin and De Villiers 1988] [Yavin and Miloh 1987] [Yavin et al 1987] [Ye and Liu 2016] [Yong 1987] [Yu et al 2018] [Zemskov and Pashkow 1997]

Agility as a measured quantity

The vocabulary of the pointing advantage. Agility metrics, their evaluation by optimal trajectory analysis, and the manoeuvres that test them.

The vocabulary of agility metrics was being invented while the X-31 flew, and the modern literature has largely settled on the distinctions the period was reaching for, separating pointing agility from axial and torsional agility. The X-31’s contribution to that vocabulary was to make the pointing component large enough to measure cleanly.

The subject is concentrated in the decade of the programme. Of the 76 records, 37 were published from 1985 to 1996 and 19 from 2015 onward.

76 records. [Gedeon and Takahashi 2014] [Wang et al 2026] [Takahashi 2012] [Avanzini et al 1998] [Baumann and Gal-Or 1992] [Beaufrere 1988] [Beck and Cord 1995] [Bitten 1989] [Bitten 1990] [Caddy and Arnold 1988] [Cliff and Thompson 1992] [Kish et al 2019] [Cord 1989] [Kutschera and Render 2003] [Cox and Downing 1992] [Drajeske and Riley 1990] [Dung 2026] [Pruitt and Moroney 1980] [Zagalsky et al 1971] [Bryson et al 1969] [Yajnik 1977] [Schaefer and Lutze 1991] [Kim and Choi 2008] [Gal-Or 1992] [Gal-Or 1995] [Padfield et al 1993] [Guang et al 2014] [Guang et al 2015] [Hodgkinson et al 1988] [Incorporating agility flight test metrics concurrently into the aircraft preliminary design process 1994] [Jaeger and Hemati 2025] [Jaeger et al 2026] [Johnson 1972] [Kalviste 1990] [Kish et al 1996] [Klyde et al 1995] [Liefer et al 1990] [Liefer et al 1992] [Liefer 1990] [Liefer et al 1990] [Liu and Li 2024] [Liu et al 2018] [Loh and Lusty 1974] [Mazza 1992] [Miralles et al 1989] [Gedeon and Takahashi 2014] [Murphy et al 1991] [Oruc and Baklacioglu 2022] [Oruc and Baklacioglu 2023] [Paranjape and Ananthkrishnan 2023] [Pons and Cirak 2023] [Riley and Drajeske 1989] [Riley and Drajeske 1990] [Ryan and Downing 1994] [Ryan and Downing 1995] [Ryan and Downing 1992] [Ryan and Downing 1993] [Schrage 1990] [Skow et al 1985] [Smith and Valasek 1999] [Smith and Valasek 2000] [Parlier 1986] [Stewart et al 1975] [Stutz and Price 1964] [Bryson and Desai 1968] [Choi et al 2008] [Jung et al 2021] [Verbeke and Schutter 2017] [Wallace et al 2024] [Wang et al 2020] [Wilson et al 1993] [Wilson et al 1993] [Wrestler and Clifton G. 1965] [Zalewski piotr.zalewski@wat.edu.pl et al 2025] [Zhang et al 2023] [Zheng et al 2026]

Post-stall manoeuvring

Flight past the stall. Supermanoeuvrability, post-stall manoeuvre control, post-stall aerodynamics and its models, and flow control at post-stall angles.

The post-stall literature after the programme turned to control laws and to models of the regime. The post-stall pitching manoeuvre and the Herbst manoeuvre are now control-design problems, and post-stall aerodynamic models are built for upset-recovery training and for simulation. The manoeuvre has not moved to small aircraft in this literature. Of the 103 titles, one names a small or unmanned aircraft, a micro air vehicle or a perching manoeuvre.

103 records. [A fixed H-infinity controller for a supermaneuverable fighter performing the Herbst maneuver 1993] [Chiang et al 1990] [Chiang et al 1993] [Abdullina and Efanov 2019] [Galkov] [Altunkaya and Özkol 2025] [Ananda and Selig 2016] [Anderson 2004] [Mukherjee et al 2016] [Bagherzadeh 2026] [Basescu et al 2023] [Besnard et al 2000] [Bihrle 1993] [Buffington et al 1993] [Buzica et al 2018] [Cebeci et al 1993] [Cebeci et al 1995] [Cord and Suchomel 1984] [Cunis et al 2019] [Cunis et al 2020] [Cunis et al 2020] [de Nicola et al 2005] [Dias 2015] [Dias 2016] [Dorozhko et al 2019] [Ericsson 1993] [Ericsson 1995] [Esfahani et al 2018] [Esfahani et al 2019] [Ye et al 2014] [Gal-Or 1990] [Gallaway and Osborn 1985] [Garrard and Enns 1991] [Ghosh et al 2026] [Graham and Yeow 1990] [Grant et al 2017] [Grant et al 2018] [Gungordu 2026] [Gwilliam and Howard 1991] [Hassan 2005] [Hassan 2006] [He and Kral 2000] [He et al 2001] [Hipp et al 2015] [Hoburg and Tedrake 2009] [Hreha and Lutze 1984] [Huang 1990] [Izquierdo and Marques 2021] [Jaermark 1983] [Khanna and Mukherjee 2021] [Kutschera and Render 1999] [Leknys et al 2018] [Li et al 2025] [Linton 1993] [Liu et al 2019] [Lyu et al 2018] [Lyu et al 2019] [Ma et al 2019] [Mendenhall and Perkins 1999] [Milillo 1972] [Myose et al 1996] [Nguyen et al 1979] [Nial 1972] [Ogunka et al 2022] [Pachter and Chandler 1996] [Pierre et al 2012] [Ransom 1983] [Rao 1992] [Ray 1965] [Raymer 1991] [Roach 1988] [Rohith and Sinha 2020] [Roza et al 2017] [Rullan et al 2003] [Rullan et al 2004] [Saraf et al 1998] [Saraf et al 1999] [Schoenstadt 1975] [Shen and Chen 2022] [Shen and Chen 2024] [Snell 1993] [Tan et al 2018] [Tan et al 2021] [Tan et al 2023] [Telionis 1995] [Telionis 2001] [Teng et al 2015] [Teng et al 2016] [Gal-Or 1990] [Vlachos et al 1999] [Wang et al 2021] [Wang et al 2025] [Rockwell 1992] [Wong et al 1993] [Wu et al 2017] [Yang and Chang 2024] [Yongxi Lyu et al 2016] [Yoshioka et al 2026] [Zhang et al 2018] [Zhang et al 2019] [Zhang et al 2019] [Zhang et al 2022] [Zhang et al 2023]

High angle of attack aerodynamics

The flow the aircraft flew in. Vortex flows, vortex breakdown, wing rock, strakes and the aerodynamics of whole configurations at high angle of attack, measured and computed.

The flow the X-31 could not predict is now computed, and 60 of the 552 titles name a computation or a simulation. Analyses of the flow around the X-31 itself at high angle of attack have been published as validation cases more than a decade after the aircraft stopped flying, and they are listed under the vehicle’s own literature below. That is a fitting use of the data. An aircraft that flew because the tunnel could not settle a question has become the benchmark against which the calculation that replaced the tunnel is judged.

The reservations are real. Forebody vortex asymmetry is sensitive to a nose radius of half an inch, to boundary-layer transition state and to Reynolds number, and those are the three things computation finds hardest.

552 records. [Prakash et al 2009] [A feasibility review of SMC-MIMO based control architecture for high angle of attack flight 2019] [A Generalization for Model Reference Adaptive Control and Robust Model Reference Adaptive Control Adaptive Laws for a Class of Nonlinear Uncertain Systems with Application to Control of Wing Rock Phenomenon 2020] [Abed 1989] [Abed and Lee 1990] [Abene 2015] [Abney and McDaniel 2005] [Agnone 1984] [Air Force Test Pilot School Edwards Afb Ca 1991] [Almosnino 1983] [Almosnino 1985] [Aloui et al 2024] [AlQadi and Eljack 2016] [An investigation of strake fence flaps on a canard-configured aircraft 1990] [Anderson 1979] [Anderson 1984] [Anderson 1985] [Anderson et al 1983] [Andrievsky et al 2020] [Anglin 1978] [Araujo and Singh 1998] [Arena et al 1995] [Arena et al 1993] [Arena and Nelson 1989] [Arena and Nelson 1991] [Asbury and Capone 1995] [Atesoglu and Özgören 2007] [Axelson 1977] [Ayoub and McLachlan 1987] [Bacon et al 1996] [Baer 2014] [Bagheri and Danesh 2025] [Bagherzadeh et al 2025] [Bai and Wu 2016] [Bai et al 2018] [Bai et al 2025] [Baigang and Jingyi 2021] [Banks 1988] [Batterson and Klein 1989] [Batterson and Morelli 1996] [Bauer et al 1995] [Berger 1988] [Berger 1989] [Beyers 1995] [Bigler 1986] [Bihrle and Barnhart 1982] [Bihrle and Barnhart 1983] [Bihrle and Billy 1980] [Blake and Barnhart 1990] [Blake et al 1992] [Boalbey et al 1993] [Boschetti et al 2022] [Boulouma et al 2017] [Bowers et al 1994] [Bowers et al 1996] [Boyum et al 1994] [Boyum et al 1995] [Brandon and Nguyen 1986] [Brody and Fagley 2024] [Bruns and Smith 1992] [Buffington and Adams 1995] [Bunge and Kroo 2018] [Bunge et al 2015] [Burris and Hutchins 1970] [Butler and Bons 2024] [Buzica et al 2017] [Canin 2019] [Chaderjian 1992] [Chambers 1980] [Chambers and Bowman 1971] [Chaparro et al 2016] [Chen et al 2016] [Chen et al 2016] [Chen et al 2025] [Cho and Cho 1995] [Chung et al 2019] [Chunzhi et al 2015] [Cooper and Fansler 1997] [Cornelius and Lucius 1993] [Cornelius and Lucius 1994] [Cornelius et al 1992] [Cornelius et al 1994] [Cotton and Bjarke 1994] [Cui and Khoo 2023] [Cui et al 1993] [Cunis et al 2020] [Dantsker and Selig 2015] [Darden et al 1995] [Fisher et al 1992] [Davidson et al 1998] [Dawson et al 2016] [Del Frate and Zuniga 1990] [Dexter 1993] [Dhaigude et al 2021] [Dias 2015] [Dias and Almeida 2013] [DiCarlo et al 1985] [Dong et al 2019] [Dong et al 2024] [Durston and Schreiner 1983] [Düssler and Palacios 2024] [Elzebda et al 1989] [Elzebda et al 1989] [Epikhin 2021] [Erickson and Gilbert 1983] [Ericsson 1985] [Ericsson 1989] [Ericsson 1989] [Ericsson 1989] [Ericsson 1991] [Ericsson 1993] [Ericsson 1995] [Ericsson 1995] [Ericsson 1997] [Ericsson 1998] [Ericsson and Beyers 1995] [Ericsson and Beyers 1996] [Ericsson and Hanff 1994] [Ericsson et al 1996] [Eriksson and Rizzi 1984] [Experimental investigation of synthetic jet control of wing rock for a flying wing aircraft 2023] [Falunin et al 1968] [Fan and Lutze 1996] [Fang et al 2022] [Farcy et al 2020] [Faure et al 2019] [Favier et al 1987] [Taira et al 2010] [Feistel et al 1981] [Fisher et al 1998] [Flaherty 1977] [Font et al 1998] [Fouda and Taha 2021] [Fresconi and Celmins 2018] [Fu et al 2017] [Fujii 1985] [Gangulee and Ng 1994] [Gangulee and Ng 1995] [Gendel et al 2015] [Geng et al 2015] [Gera et al 1981] [Ghee and Hall 2000] [Ghosh et al 2023] [Gilbert and Nguyen 1978] [Gilbert et al 1976] [Gittner and Chokani 1991] [Golovkin et al 2019] [Goman and Khrabrov 1994] [Goman et al 1992] [Goman et al 1994] [Gordon 1990] [Greenwell and Wood 1993] [Gross et al 2017] [Gursul 1994] [Gursul 1995] [Gursul 2001] [Gursul and Wang 2010] [Guvernyuk et al 2020] [Hadidoolabi and Ansarian 2017] [Hahne 1985] [Hall 1986] [Hall 1987] [Hall and Banks 1994] [Harloff 1987] [Hassell and Hewes 1960] [Hauch et al 1996] [He and Williams 2020] [Hebbar and Leedy 1990] [Hebbar and Leedy 1992] [Hebbar et al 1996] [Helliwell and Dickinson 1980] [Hernandez Gelado and Ramesh 2022] [Hewes 1959] [Hirato et al 2019] [Hitzel 2017] [Hitzel and Osterhuber 2018] [Hoeijmakers and Van Den Berg 1991] [Hopwood et al 2023] [Hosseinzadeh and Mansour 2018] [Huang and Knowles 1990] [Huang et al 2016] [Hubner and Komerath 1995] [Hummel and Oelker 1994] [Hummel and Srinivasan 1967] [Hwang 1986] [Hyde et al 2018] [Ibraheem AlQadi 2019] [Ignatyev and Khrabrov 2015] [Ignatyev and Khrabrov 2018] [Imbrie 1981] [Investigation on Jet Characteristics of Turbofan Exhaust System under Take-off Condition with High Angle of Attack 2022] [Irving 1933] [Izadbakhsh and Jabbari Asl 2025] [Jaiswal et al 2024] [Jaramillo et al 1995] [Jarrah 1989] [Ji et al 2026] [Johnson and Willen 1976] [Johnson et al 1981] [Johnston and Hogge 1974] [Johnston and Hogge 1976] [Johnston et al 1974] [Joshi et al 1998] [Jouannet and Krus 2007] [Jun and Nelson 1987] [Kamakoli and Mansour 2021] [Kandil and Flanagan 1994] [Kandil and Salman 1992] [Kandil et al 1995] [Kandil et al 1996] [Katz 1999] [Katz and Feistel 1987] [Katz et al 1986] [Katz et al 1996] [Kegelman and Roos 1991] [Wichman et al 1996] [Keller et al 2018] [Kern 1993] [Khanna and Mukherjee 2020] [Klopfer and Nielsen 1979] [Klute et al 1996] [Kobayakawa et al 1993] [Konstadinopoulos et al 1985] [Konstantinovskaya et al 2024] [Kostić et al 2015] [Krasil’nikov 1969] [Krekeler et al 1990] [Kroo et al 1991] [Kumar 2020] [Kumar and Ghosh 2015] [Kumar et al 2020] [Kumar et al 2021] [Kumar et al 2021] [Kurade et al 2021] [Lee 2016] [Lee and Abed 1991] [Lee and Batina 1991] [Lee and Ko 2018] [Lee and Singh 2017] [Lee et al 2024] [Lee et al 2024] [Lemay and Rogers 1990] [Lesieutre et al 1996] [Lhopiteau et al 2025] [Li 2025] [Li and Wu 2016] [Li and Yang 2016] [Li and Yang 2017] [Li and Yang 2017] [Li et al 2017] [Li et al 2021] [Li et al 2021] [Li et al 2022] [Li et al 2023] [Li et al 2023] [Lian et al 2023] [Littleboy and Patel 1998] [Liu et al 2018] [Liu et al 2019] [Liu et al 2020] [Liu et al 2020] [Liu et al 2024] [Liu et al 2026] [Liu et al 2026] [Lowson and Ponton 1991] [Lowson and Riley 1995] [Nguyen et al 1980] [Luckring 1986] [Luo and Lan 1993] [Ma et al 2015] [Ma et al 2025] [Mabey 1997] [Magness et al 1993] [Mahgoub and Cortelezzi 2020] [Mahmood 2025] [Mahmood and Iqbal 2025] [Malcolm 1979] [Malcolm and Skow 1986] [Malcolm et al 1989] [Malcolm et al 1990] [Malekzadeh et al 2015] [Malfa et al 1991] [Manor and Wentz 1985] [Manzoor et al 2016] [Marchman 1981] [Martínez et al 2022] [Mendenhall and Perkins 1996] [Meng et al 2018] [Menner and Lavretsky 2026] [Menon and Yousefpor 1996] [Meyer et al 1989] [Mi et al 2024] [Midhat 2017] [Miller and Gile 1992] [Mitchell et al 1980] [Mobayen and Izadbakhsh 2025] [Mobayen and Izadbakhsh 2025] [Modi et al 1981] [Modi et al 1984] [Mohankumar et al 2020] [Mohankumar et al 2021] [Mohankumar et al 2022] [Monahemi et al 1995] [Monteiro et al 2025] [Moore and McInville 1995] [Morelli 1996] [Mosbarger 1994] [Mosbarger 1995] [Muir et al 2017] [Murman 1994] [Murman et al 1994] [Murphy 1987] [Murphy 1989] [Murri et al 1995] [Murri and Rao 1987] [Murri et al 1993] [Murri et al 1994] [Narain 1991] [Narayanan et al 2026] [Nelson 1985] [Nelson 1988] [Nelson and Mouch 1978] [Ng et al 1993] [Ng et al 1994] [Nguyen 1974] [Nguyen 1979] [Nguyen et al 1976] [Nguyen et al 1980] [Nguyen et al 1985] [Niranjana et al 1993] [Numerical Analysis of the Flow Pattern and Vortex Breakdown over a Pitching Delta wing at Supersonic Speeds 2017] [Obeid and Alqadi 2017] [Obeid and AlQadi 2018] [Obeid et al 2019] [Oberkampf and Nicolaides 1971] [Ogburn et al 1988] [Orkwis 1995] [Orlik-Ruckemann 1983] [Osipov 2017] [Osipov 2019] [Osipov 2019] [Ostroff 1998] [Ostroff et al 1994] [Owen and Johnson 1978] [Oyewola 2026] [Pacheco dos Santos et al 2018] [Pahle et al 1991] [Pahle et al 1996] [Pamadi et al 1994] [Panaras 2015] [Papageorgiou and Glover 1997] [Patten 1988] [Patterson and Watts 1985] [Patterson and Watts 1986] [Pedreiro et al 1998] [Pedreiro et al 1995] [Peng et al 2025] [Perrier 1993] [Pittman 1983] [Planeaux 1991] [Planeaux and Barth 1988] [Prediction of vortex breakdown effects on a rolling delta wing 1994] [Prizirembel and Wen 1978] [Przirembel and Shereda 1979] [Purvis 1981] [Pusztai et al 2024] [Qi and Zhongke 2015] [Qi et al 2018] [Qi et al 2021] [Raj et al 1988] [Ramachandran and Galloway 1982] [Rao 1989] [Rao and Huffman 1981] [Rao and Huffman 1982] [Rao and Murri 1986] [Rao and Sharma 1994] [Rao 1978] [Raza et al 2024] [Redding et al 2025] [Reddy 1981] [Reddy 1989] [Rediniotis et al 1991] [Regenie et al 1992] [Regenie et al 1992] [Regenie et al 1994] [Reshotko et al 2016] [Reznick and Flores 1989] [Richards 1979] [Richwine and Fisher 1991] [Rizzetta 1996] [Rizzetta 1997] [Rockwell 2001] [Rogalski et al 2020] [Rohith and Sinha 2017] [Rom 1992] [Rom 1992] [Rom 1992] [Rom 1992] [Rom 1992] [Rom and Gordon 1988] [Rom et al 1987] [Roos 1996] [Rose and Pittman 1989] [Rose and Pittman 1993] [Ross et al 1987] [Ross et al 1991] [Running et al 2017] [Rusak et al 2015] [Russell et al 2022] [Sahin et al 2026] [Sahu et al 2000] [Salahudden et al 2021] [Salahudden et al 2022] [Salehian et al 2022] [Salehian et al 2024] [Salehian et al 2025] [Sandlin and Ramirez 1991] [Schiff et al 1980] [Schmidt et al 2025] [Sedlacek et al 2022] [Seraj and Martins 2021] [Setoguchi and Kanazaki 2020] [Seyedtabaii and Delavari 2018] [Sha et al 2017] [Shahriar et al 2023] [Shen and Qin 1986] [Shen and Wen 2018] [Shen et al 2016] [Shi et al 2015] [Shi et al 2019] [Shigemi 1993] [Shirokov and Elizarova 2026] [Shivam and Verma 2019] [Shojae et al 2025] [Shojae et al 2026] [Shukla and Woolsey 2018] [Simon 1992] [Simon and Blake 1999] [Sinha et al 2017] [Smith et al 1994] [Snell et al 1991] [Son et al 2015] [Son et al 2015] [Stahl 1989] [Stahl et al 1992] [Stalford 1979] [Stalford 1987] [Stalford 1990] [Staples and Nagati 1990] [Starikovskiy 2017] [Starikovskiy et al 2017] [Stephen and McLaughlin 2017] [Stephens 1936] [Stollery 1992] [Suarez et al 1993] [Sugiura et al 2021] [Surzhikov 2023] [Svoboda and Rozehnal 2018] [Synolakis et al 1993] [Tahir et al 2026] [Tajfar and Lamont 1997] [Takagi et al 2020] [Tanno 2021] [Tarn and Hsu 1993] [Taylor and Skow 1988] [Theisen et al 1973] [Thomas 1991] [Thompson et al 1990] [Thukral and Innocenti 1998] [Tindell and Hill 1993] [Tinger 1989] [Tinoco 2020] [Tischler and Barlow 1981] [Tischler and Barlow 1982] [Tobak and Schiff 1974] [Tong et al 1993] [Traub and van der Merwe 1994] [Tu et al 2019] [Valasek and Walchli 1998] [Van Den Berg et al 1992] [Van Tuyl 1988] [Venugopal and Krishnamurthy 1995] [Vidhya et al 2023] [Visbal 1994] [Viswanath and Patil 1994] [Vlahostergios et al 2018] [Walchli et al 1992] [Walsh et al 1998] [Wan et al 2023] [Wang et al 2015] [Wang et al 2016] [Wang et al 2017] [Wang et al 2023] [Wang et al 2025] [Wang et al 2025] [Wang et al 2026] [Ward and Stout 1986] [Ward et al 1976] [Wardlaw 1974] [Wardlaw et al 1975] [Webster and Shang 1989] [Wei and Chen 2023] [Wei et al 2017] [Wei et al 2024] [Weick and Harris 1934] [Weick and Noyes 1933] [Weijiang et al 1995] [Weissman 1973] [Weissman 1975] [Wendel et al 1991] [Weng and Guo 1994] [Weng and Guo 1994] [White et al 1976] [Wibowo et al 2018] [Widing 1993] [Williams and Garry 1996] [Wilson et al 1991] [Wilson et al 1994] [Wood and Miller 1985] [Wood and Roberts 1987] [Wu and Wu 1993] [Wu et al 2017] [Wu et al 2020] [Xieyuan 1985] [Xu et al 2016] [Xu et al 2022] [Yang et al 1994] [Yang et al 2023] [Yang et al 2025] [Yang et al 2025] [Yavuz 2020] [Ye and Zhao 1990] [Ye and Zhao 1994] [Ye et al 2015] [Yip et al 1992] [Yuan and Yarusevych 2020] [Yuan et al 2018] [Zakir et al 2024] [Zhang 2026] [Zhang and Cheng 2024] [Zhang and Samtaney 2016] [Zhang et al 2017] [Zhang et al 2022] [Zhao et al 2025] [Zheng et al 2025] [Zhongyang et al 2017] [Zhou et al 2018] [Zhou et al 2019] [Zhu 2020] [Zhu et al 2015] [Zhu et al 2022] [Zohar and Er-El 1988]

Yaw asymmetry and forebody flow

The cliff in the departure analysis. Asymmetric forebody vortices, their sources, and their control by strakes, blowing and nose shape.

This is a literature of the programme’s own period. Of the 91 records, 56 were published from 1985 to 1996, when forebody vortex control was being tried on the F/A-18 and the F-15 as well as on the X-31.

91 records. [Asghar et al 1994] [Ayoub and McLachlan 1988] [Bernhardt and Williams 1993] [Bernhardt and Williams 1993] [Bernhardt and Williams 1995] [Bernhardt and Williams 1998] [Brandon and Nguyen 1988] [Brown et al 2022] [Chen and Ng 1994] [Chen et al 1995] [Computational investigation of a pneumatic forebody flow control concept 1972] [Darden and Komerath 1995] [Darden and Komerath 1997] [Degani 1991] [Ericsson 1971] [Ericsson 1990] [Ericsson 1991] [Ericsson 1992] [Ericsson 1992] [Ericsson and Reding 1985] [Flick et al 1996] [Gao et al 2026] [Gebert and Maynes 1995] [Gee et al 1991] [Gee et al 1992] [Gee et al 1993] [Gittner and Chokani 1992] [Gittner and Chokani 1994] [Godavarty and Orkwis 2000] [Hodgkin and Wood 1998] [Huang et al 2026] [Iwanski and O’Rourke 1995] [Iwanski and O’Rourke 1996] [Kandil et al 1995] [Kramer et al 1993] [Kramer et al 1994] [Kramer et al 1994] [Lanser and Meyn 1992] [Lanser and Meyn 1994] [Lee et al 2021] [Lemay et al 1992] [Lesieutre and Hall 2024] [Mahadevan et al 2017] [Mahadevan et al 2018] [Malcolm 1991] [Malcolm 1993] [Malcolm and Ng 1989] [Malcolm and Ng 1990] [Malcolm and Ng 1990] [McKeehen and Cord 1997] [Moore et al 1980] [Ng 1990] [Ng 1992] [Ng and Malcolm 1992] [Ng et al 1991] [Ng et al 1991] [Ng et al 1994] [Norris and Lan 1991] [Orkwis et al 1997] [Pedreiro 1997] [Qi and Wang 2019] [Rao et al 1987] [Roos 1996] [Rosen and Davis 1990] [Ross and ORourke 1997] [Sampath Kumar et al 2020] [Sampath Kumar et al 2021] [Sato et al 2016] [Sedor 1996] [Sidharth and Dwivedi 2026] [Skow et al 1982] [Stahl 1990] [Suarez et al 1992] [Suarez et al 1992] [Suarez et al 1993] [Taligoski et al 2015] [Valasek and Downing 1995] [Viswanath 1995] [Wang et al 2018] [Wilkerson et al 2024] [Wilkerson et al 2026] [Williams and Papazian 1991] [Wood and Crowther 1994] [Wood et al 1990] [Wu et al 2017] [Zhai et al 2016] [Zheng et al 2015] [Zheng et al 2020] [Zheng et al 2026] [Zong et al 2021] [Zong et al 2025]

Canard and delta configurations

The configuration. Delta wings, canards and their interference, close-coupled canard-wing-body aerodynamics, three-surface layouts and the lift-curve slope of low aspect ratio wings.

288 records. [Shankar and Goebel 1983] [A study of canard-wing interference using experimental pressure data at transonic speeds 1979] [Anand M Raikar 2024] [Abed et al 2024] [Lacey 1979] [Agrell and Elmeland 1985] [Alahmadi 2025] [Aliaga et al 2022] [Almuradi 2025] [Sirbaugh 1983] [Kida 1982] [Aparinov et al 1982] [Bandyopadhyay 1989] [Bandyopadhyay 1991] [Bantscheff and Breitsamter 2023] [Bass et al 1993] [Bass et al 1995] [Bay and Kara 2026] [Betyaev and Brysov 1994] [Biswas et al 2023] [Borisov 2025] [Brody et al 2025] [Brown and Kunz 2023] [Burkhalter 1993] [Lowe 1988] [Canard-Leading Edge Flap Scheduling for the Maneuverability Enhancement of a Fighter Class Aircraft 2007] [Canard-wing lift interference related to maneuvering aircraft at subsonic speeds 1973] [Erickson et al 1990] [Carine Viola Dsouza and Dr. Basawaraj 2015] [Chang and Lei 1996] [Chao and Zhenghong 2019] [Chen et al 2021] [Calarese 1982] [Smith 1990] [Pandya and Aftosmis 2001] [Shankar and Malmuth 1982] [Cowling 1987] [Croom et al 1982] [Das and Longo 1995] [de Silva and Carmichael 1978] [Di Fabbio et al 2022] [Dong et al 2023] [Draz et al 2020] [Dwivedi et al 2022] [Dwivedi et al 2022] [Effect of canard location and size on canard-wing interference and aerodynamic center shift related to maneuvering aircraft at transonic speeds 1974] [Effect of wing planform and canard location and geometry on the longitudinal aerodynamic characteristics of a close-coupled canard wing model at subsonic speeds 1975] [Ma et al 2004] [Ekaterinaris et al 1995] [Er-El 1988] [Er-El et al 1989] [Ericsson 1996] [Ericsson 1997] [Ericsson 1997] [Ericsson and Hanff 1994] [Ericsson and King 1993] [Hauptman 1987] [Fan et al 2025] [Fatemeh Mahdavi and Mohammad Reza 2021] [Fawcett et al 1992] [Feng et al 2024] [Bridges 2005] [Friehmelt 1997] [Gad-el-Hak and Ho 1985] [Gao et al 2019] [Gao et al 2019] [Ghasemzadeh and Banazadeh 2025] [Ghoreyshi et al 2017] [Goetzendorf-Grabowski and Antoniewski 2016] [Gopesh. M. P and Manivel. M 2016] [Goucem and Khiri 2023] [Greenwell and Wood 1994] [Grismer and Nelson 1994] [Grismer et al 1995] [Gupta 2023] [Gupta et al 2023] [Gupta et al 2024] [Gupta et al 2025] [Hadidoolabi and Ansarian 2018] [Hanff and Huang 1991] [Hanff and Jenkins 1990] [Hao and Chen 2017] [Harrison et al 2016] [Hashemi et al 2026] [He et al 2019] [Hilgenstock 1991] [Hong and Roberts 1995] [Hong et al 1996] [Hooker 2023] [Howard and Kersh 1991] [Howard and O’Leary 1993] [Howard and O’Leary 1994] [Howard et al 1995] [Huijts and Voskuijl 2015] [Ingram et al 1986] [Ito and Iwashita 2017] [Iwamoto et al 2019] [Jenn and Nelson 1988] [Jenn and Nelson 1989] [Jesus et al 2021] [Jiang et al 2022] [Kalugin et al 2022] [Kandil and Chuang 1989] [Kandil and Chuang 1990] [Kandil and Chuang 1990] [Kandil et al 1993] [Katz and Levin 1986] [Katz and Levin 1991] [Keith and Selberg 1984] [Keith and Selberg 1985] [Khan and Mueller 1989] [Kim and Kang 2025] [Kiselev et al 1994] [Kiseleva and Lukeshevich 2019] [Kjelgaard et al 1986] [Komarov and Zinchenko 2023] [Konstantinovskaya et al 2025] [Kuntjoro et al 2015] [Kunwar et al 2024] [Lai 2024] [Landa et al 2018] [Landfield and Rajkovic 1986] [Lathasree et al 2019] [Le and Hoang 2026] [Le Provost et al 2018] [Lee and Ko 2017] [Lei and Zha 2021] [Levin 1997] [Li et al 2016] [Li et al 2017] [Li et al 2020] [Nonweiler 1960] [Laitone 1989] [Lijewski 1987] [Lijewski 1988] [Lin et al 1994] [Gloss and Washburn 1977] [Lombardi 1995] [Lombardi and Morelli 1994] [Lombardi et al 1996] [Long and Yi 2021] [Longo and Das 1990] [Lunte et al 2018] [Lv et al 2015] [Ma et al 2017] [Mahdavi Zafarghandi and Reza Soltani 2024] [Manshadi et al 2016] [Marion and Sharma 2025] [Mayya et al 2022] [McCutchen 1980] [McKlNNEY and DOLLYHlGH 1971] [Meng et al 2018] [Miller and Williamson 1995] [Mkoyan et al 2023] [Mochizuki and Yamada 2018] [Mohamed et al 2022] [Mouch et al 1989] [Mukherjee et al 2021] [Naigle et al 2022] [Nasuhbeyoglu and Akça 2024] [Ng and Malcolm 1990] [Finkleman 1971] [Numerical and Experimental Determination of Canard Controlled Missile Aerodynamic Coefficients in Subsonic Regime 2019] [Octavio Dias dos Santos and de Almeida 2017] [Oelker and Hummel 1989] [Oguz Tasci et al 2022] [Oh and Tavella 1986] [Frolov 2021] [Ostowari and Naik 1986] [Owens and Perkins 1996] [Patel 1980] [Pfeiffer 1988] [Pollak and Lan 1995] [Traub 1997] [Probst et al 2019] [Qin et al 2017] [Qin et al 2018] [Raikar et al 2024] [Raisinghani and Kumar 1995] [Rajkumar et al 2024] [Rao 1971] [Remneff and Amitay 2025] [Research on Combination of Swing Wing with Canard and Tail used in Fighter Aircraft 2019] [Rizzi and Purcell 1986] [Rom et al 1989] [Rose et al 1988] [Rotor Performance Optimization of the Canard Rotor Wing Aircraft 2008] [Roy et al 2023] [Rozema et al 2020] [Ruetten and Zastrow 2022] [Sahu and Fresconi 2017] [Sahu and Fresconi 2018] [Sahu et al 2019] [Sahu et al 2020] [Said et al 2015] [Saisuryateja et al 2022] [Sajjad et al 2021] [Satrianto and Firmanto 2023] [Schuette and Ghoreyshi 2026] [Schuette and Hummel 2022] [Schulz 1931] [Seginer and Salomon 1986] [Selberg and Rokhsaz 1985] [Selberg and Rokhsaz 1986] [Seraj and Martins 2022] [Seraj and Martins 2023] [Seraj and Martins 2025] [Shah and Ahmed 2023] [Shahjahan and Ahuja 2026] [Shahjahan et al 2026] [Shams Taleghani and Ghajar 2024] [Sharma et al 2016] [Shen and Alonso 2026] [Shen et al 2018] [Shergill and Milanovic 2025] [Shoaib et al 2019] [Shrivastava and Mohite 2015] [Silton and Fresconi 2015] [Silton and Fresconi 2015] [Silton et al 2014] [Singh et al 1994] [Spearman and Driver 1959] [Static longitudinal aerodynamic characteristics of close-coupled wing-canard configurations at Mach numbers from 1.60 to 2.86 1971] [Stephen 1995] [Szklarski and Głębocki 2025] [Tahani et al 2017] [Tai et al 1994] [Takahashi 2021] [Takahashi 2021] [Takahashi et al 2023] [Tasci et al 2021] [Tasci et al 2022] [Tedder and Walker 2022] [The effect of canard leading edge sweep and dihedral angle on the longitudinal and lateral aerodynamic characteristic of a close-coupled canard-wing configuration 1974] [The effects of canard-wing flow-field interactions on longitudinal stability, effective dihedral and potential deep-stall trim 1988] [Jalil et al 2024] [Theerthamalai et al 2025] [Thorell et al 2016] [Thrust-induced effects on pitching-up delta wing flow field 1994] [Toyoda et al 2015] [Transition aerodynamics for close-coupled wing-canard configuration V/STOL operations 1979] [Shankar and Goebel 1985] [Tromp 1999] [Trussa et al 2021] [Trussa et al 2022] [Trussa et al 2026] [Tu 1991] [Tu 1992] [Tu 1992] [Tu 1992] [Tu 1994] [Tu 1994] [Tu 1996] [Tumse et al 2021] [Tumse et al 2022] [Tuncer and Platzer 1996] [Tuncer and Platzer 1998] [Van Dommelen 1995] [Varun and Dwivedi 2022] [Vasile 2017] [Vaughan and Wood 1995] [Visonneau and Guilmineau 2024] [Visonneau et al 2022] [Visualization of delta wing vortex system 2025] [Vortex generator effect and aerodynamic characteristic for canard airplane 2019] [Calarese 1987] [Griffin et al 1983] [Wang et al 2018] [Wang et al 2025] [Wei et al 2019] [Werner et al 2022] [Werner et al 2024] [Wibowo et al 2018] [Wibowo et al 2019] [William Ruffles and Sam M. Dakka 2016] [Feistel et al 1981] [Mason 1982] [Yaniktepe et al 2016] [Yaniktepe et al 2025] [Yoo et al 2023] [Yutuk et al 2021] [Zamzuddin et al 2023] [Zhang and Yuan 2019] [Zhang et al 2015] [Zhang et al 2017] [Zhang et al 2026] [Zharfa et al 2016] [Zwerger et al 2015]

Thrust vectoring

The means. Mechanical thrust vectoring by vanes, paddles and vectoring nozzles, its integration with the airframe and its effect on manoeuvre.

The X-31’s mechanism is a small part of the vectoring literature. Vanes or paddles are named in 7 of the 209 titles, and the rest concern vectoring nozzles, their integration and their effect on manoeuvre.

209 records. [Ahmed et al 2025] [Alcantara et al 2017] [Amoozgar et al 2024] [Anderson 1967] [Andrisani and Jones 1994] [Anna and Kidman 1994] [Aroussi 2026] [Asbury and Capone 1992] [Atesoglu and Ozgoren 2011] [Baldini et al 2026] [Bandi et al 2025] [Barham 1994] [Barnhouse et al 2025] [Baskaran et al 2018] [Berens 2019] [Bergman 1979] [Berrier and Mason 1987] [Berry 1998] [Bhabar and Mukesh Sharma 2025] [Biberstein et al 2021] [Bindu Mounica et al. 2020] [Bolitho and Jacob 2008] [Bowers and Pahle 1996] [Bowers et al 1991] [Bronz 2019] [Buffington et al 1993] [Buffington et al 1994] [Bursey and Dickinson 1990] [Capone and Mason 1986] [Capone et al 1992] [Capone and Bare 1987] [Carr 1986] [Chang et al 2015] [Chawla and Van Dalsem 1992] [Cheatham et al 1991] [Chen Jie et al 2012] [Chi et al 2022] [Chih and Peng 2026] [Cohn et al 1985] [Cornelius and Lucius 1993] [Cornelius and Lucius 1995] [Counter and Brinton 1975] [Crandall et al 1973] [Crawford et al 2026] [Culpepper et al 2021] [Culpepper et al 2022] [Daljit Majil 2016] [Das et al 2025] [Davis and Hines 1976] [Dhamodaran and Thanigaiarasu 2025] [Dhanya et al 2023] [Dong et al 2025] [Dwyer and Lutze 1993] [DYNAMIC THRUST CONTROL UNLEASHING 2023] [Ellis and Kearney 1982] [Enenakpogbe et al 2025] [Erwin et al 1964] [Esker and Sedrick 1972] [Experimental Investigations on the Strut Controlled Thrust Vectoring of a Supersonic Nozzle 2020] [Fazeli Asl and Moosapour 2017] [Ferlauto et al 2021] [Fosnight 1968] [Friehmelt 1996] [Gal-Or 1994] [Gal-Or 1995] [Gal-Or and Sherbaum 1993] [Gal-Or et al 1993] [Gal-Or et al 1995] [Ghosh et al 2006] [Gill 1973] [Gill 1974] [Gourcerol et al 2025] [Granasy 1995] [Ha et al 2025] [Hall 1965] [Hammond et al 1995] [Hardy 1987] [Harisrinivasan et al 2018] [He et al 2025] [Herrick 1992] [Hirano et al 2019] [Hoskin et al 1991] [Hosseini et al 2026] [Ikaza and Rausch 2000] [Iliff and Wang 1997] [Imlay 1986] [Intelligent hierarchical thrust vector control for a space shuttle 1994] [Invernizzi and Lovera 2018] [Islam and Islam 2017] [Islam et al 2018] [Islam et al 2018] [Jenie et al 2019] [Jiméneze and Icaza 2000] [Johnson 1990] [Jones and Jia-Richards 2025] [Jones and Jia-Richards 2026] [Jyothy and John Wessley 2023] [Kara and Erpulat 2021] [Kasem and Megahed 2025] [Kentfield 1975] [Kentfield 1975] [Kikkawa and Uchiyama 2017] [Kim 2026] [King and Schnelker 1970] [King et al 1991] [Klafin 1988] [Kuang and Zhu 2015] [Kumar et al 2020] [Kumar et al 2021] [Lallman 1985] [Lallman et al 1998] [Lampard 1978] [Lee and Lan 1992] [Lee and Lee 2019] [Lee et al 2016] [Lee et al 2018] [Leitner 1986] [Leovan 2019] [Levi and Stahlhuth 1977] [Lewis 1990] [Li et al 2019] [Li et al 2020] [Lin and Sheehan 2018] [Liu et al 2020] [Liu et al 2022] [Liu et al 2022] [Liu et al 2024] [Lowe 1985] [Luo et al 2025] [Lutze et al 1991] [Mace et al 1989] [Mahmoud et al 2025] [Mareca Rios et al 2023] [Margetts et al 2015] [Marubashi and Miwa 2016] [Mason et al 1992] [Miller 1986] [Miwa 2021] [Miwa and Maruhashi 2016] [Mubarok and Jatmiko 2020] [Muir and Bradshaw 1996] [Nafi and Hasan 2019] [Nagabhushan and Faiss 1984] [Nagaral et al 2023] [Nauparac et al 2018] [Nayebi and Taeibi Rahni 2024] [Nguyen et al 2025] [Nie et al 2026] [Nocoń et al 2023] [Oganyan and Loginov 2025] [Orme and Sims 1999] [Ornob et al 2025] [Pahle et al 1996] [Petit and Scholey 1973] [Phillips et al 1968] [Postma 1970] [Prothin et al 2016] [Reigelsperger et al 1995] [Resemini and Souza Costa 2018] [Resta et al 2024] [Rolls and Aoyagi 1977] [Roush et al 2020] [Ruetten 2018] [Salahudden et al 2025] [Sanders 1997] [Schirmer and Capone 1989] [Shandor and Walker 1962] [Sharjad et al 2022] [Shebl 2026] [Sherbaum and Lichtsinder 1998] [Shneider and Macheret 2005] [Shumeiko et al 2022] [Shumeiko et al 2022] [Snow 1990] [Soufivand et al 2024] [Soundararajan 2022] [Soundararajan and Sridhar 2024] [Srivastava and Jayachandran 2023] [Stalford and Hoffman 1989] [Steer 2000] [Study on the Aerodynamic Performance of Novel Bypass Shock-Induced Thrust Vector Nozzle 2023] [Surapaneni et al 2016] [Suñol Jiménez et al 2015] [Swaraj et al 2023] [Sweeney and Gerzanics 1994] [Syed et al 1992] [Tamrat and Antani 1988] [Tang 2025] [Tape et al 1987] [The Hawker P1127 vectored thrust fighter program - Lessons learned 1990] [THRUST VECTORING OF A SUPERSONIC RECTANGULAR NOZZLE BY USING DIFFERENT PLATE EXTENSIONS 2020] [Tseng and Lan 1989] [Tsubakino and Saito 2020] [Van Nieuwstadt and Murray 1998] [Visser 1999] [Wang et al 2020] [Ward and Lewis 1987] [Whitaker and Gowadia 1994] [Whitaker et al 1991] [White 1991] [Whitten et al 1981] [Wing and Asbury 1991] [Wu et al 2015] [Wu et al 2024] [Yadav et al 2023] [Yi et al 1979] [Yu et al 2023] [Zhou et al 2022] [Zwerneman and Eller 1994]

Fluidic thrust vectoring

Vectoring without moving parts. Shock vector control, dual-throat and bypass nozzles, counterflow and Coanda vectoring.

The X-31 chose paddles over a vectoring nozzle for weight and cost, and the modern answer removes the moving parts entirely. Fluidic vectoring deflects the exhaust by injecting a secondary flow into it, either by displacing the shock structure in a converging-diverging nozzle, by exploiting the tendency of a jet to attach to a curved surface, or by drawing on a counterflow. There is no vane in the stream, nothing to actuate and nothing brittle to crack.

Reading that against this article’s own numbers makes the trade concrete. The X-31 paid 103 pounds of carbon-carbon in the exhaust plus 58 pounds of supporting structure, chose a brittle material specifically to keep mass out of the tail, and accepted a jet deflection of sixteen degrees at forty degrees per second. A fluidic system trades that hardware for bleed air, which is a thrust penalty rather than a mass penalty, and its bandwidth is set by the injection valves rather than by moving a slab of carbon through a supersonic stream.

What has not changed is the quantity that governed the X-31’s departure boundary. A fluidic system still produces a moment proportional to thrust, so its authority still falls as one over dynamic pressure and still lapses with altitude. The mechanism is different and the scaling is identical.

This is almost entirely a modern literature. Of its 137 records, 124 were published from 2015 onward.

137 records. [Afridi et al 2023] [Afridi et al 2023] [Afridi et al 2023] [Afridi et al 2024] [Ahmad et al 2018] [APPLICATION OF FLUIDIC THRUST VECTORING IN V/STOL AIRCRAFT 2024] [Banazadeh and Saghafi 2016] [Berton et al 2026] [Bharathwaj et al 2016] [Blaszak and Fahrenholz 1960] [Capello et al 2019] [Chandra Sekar et al 2017] [Chandra Sekar et al 2021] [Chandra Sekar et al 2022] [Chen and Liao 2020] [Chen et al 2022] [Chi and Gu 2022] [Chouicha et al 2019] [Chouicha et al 2020] [Coanda 1960] [Collins et al 2004] [Collins Jr 2005] [Computational and Experimental Analysis of Optimized Dual Throat Fluidic Thrust Vectoring Nozzle Configurations 2024] [Computational Investigation of Effects of Side-Injection Geometry on Thrust-Vectoring Performance in a Fuel-Injected Dual Throat Nozzle 2022] [Computational Investigation of Fluidic Thrust Vectoring Control in Modified Vikas Nozzle 2025] [Das et al 2023] [Das et al 2024] [Das et al 2025] [Deere et al 2003] [Determination of Optimum Parameter Space of a Fluidic Thrust Vectoring System based on Coanda Effect Using Gradient-Based Optimization Technique 2023] [Fan et al 2013] [Ferlauto and Marsilio 2016] [Ferlauto and Marsilio 2017] [Ferlauto and Marsilio 2018] [Ferlauto et al 2020] [Ferlauto et al 2020] [Ferlauto et al 2021] [Flamm et al 2006] [Flamm et al 2007] [Forliti et al 2002] [Garg et al 2024] [Gupta 2025] [Hakim et al 2020] [Hakimi et al 2026] [Hambidge et al 2023] [He et al 2017] [Huang et al 2025] [Huang et al 2026] [Jain et al 2015] [Jammalamadaka et al 2020] [Jingwei et al 2016] [Jingwei et al 2019] [Jun et al 2018] [Kara and Kurtuluş 2024] [Kara et al 2025] [Kishore et al 2025] [Kobayashi et al 2020] [Koklu 2016] [Li et al 2016] [Li et al 2018] [Liu et al 2022] [Liu et al 2026] [Maruyama et al 2019] [Maruyama et al 2022] [Mason and Crowther 2004] [McGilvray et al 2019] [Neely et al 2007] [Numerical Study of Fluidic Thrust Vector Control Using Dual Throat Nozzle 2021] [Park et al 2025] [Pelt et al 2015] [Raj et al 2025] [Ravindranath 2025] [Remiger et al 2023] [Remiger et al 2025] [Ren et al 2017] [Resta et al 2021] [Resta et al 2022] [Rizzetta and Garmann 2025] [Akshay et al 2020] [Sakata and Maruyama 2018] [Schwagerus et al 2023] [Schwagerus et al 2023] [Schwagerus et al 2025] [Schwagerus et al 2026] [Sellam et al 2015] [Shakouchi and Fukushima 2022] [Shakouchi et al 2019] [Shakouchi et al 2019] [Shi et al 2019] [Shi et al 2022] [Shi et al 2024] [Shi et al 2024] [Shu et al 2023] [Shu et al 2024] [Shukla et al 2026] [Simon et al 2023] [Singh 2026] [Strykowski et al 1996] [Suresh et al 2026] [Takahashi and Maruyama 2021] [Takahashi and Maruyana 2020] [Tanveer and Ahmad 2023] [Tanveer and Ahmad 2023] [Tcherak et al 2026] [Vibhu et al 2026] [Vohra and Kumar 2025] [Wang and Liu 2016] [Wang et al 2015] [Wang et al 2017] [Wang et al 2020] [Wang et al 2024] [Wang et al 2026] [Warsop et al 2019] [Weigert et al 2025] [Wen et al 2022] [Williams and Vittal 2002] [Wu and Dong Kim 2019] [Wu and Kim 2019] [Wu and Kim 2019] [Wu and Kim 2019] [Wu and Kim 2021] [Wu et al 2020] [Wu et al 2021] [Wu et al 2021] [Xiao-bo et al 2017] [Xu et al 2024] [Xu et al 2025] [Xue et al 2021] [Younes and Hickey 2020] [Zaheer et al 2023] [Zhang and He 2026] [Zhang et al 2021] [Zhang et al 2022] [Zhang et al 2024] [Zhang et al 2026] [Zhou et al 2024] [Zmijanovic et al 2016]

Control power and effector sizing

How much moment there is. Control power requirements, control margins at high angle of attack, control authority and the derivatives that measure it.

60 records. [Andersen et al 1997] [Andersen et al 1998] [Andrievsky and Zaitceva 2023] [Beard and Foley 1983] [Bocvarov et al 1992] [Cano and Sobel 2016] [Compton et al 1982] [Control authority assessment in aircraft conceptual design 1993] [Curtiss 1971] [Ellis and Raisinghani 1979] [Fan et al 2017] [Finley and Cockrell 1995] [Foster et al 1993] [Freeman and Klunk 2018] [Glezer and Leonard 2012] [Goel et al 2021] [Hall and Boothe 1972] [Hristov 2023] [Hutchin 2019] [Iliff and Wang 1997] [Jayarathne et al 2024] [Kamath et al 2025] [Karpel and Sheena 1989] [Keidel et al 2020] [Kus and Ergazi 2025] [Lackey and Hadfield 1992] [Mehra et al 1972] [Morelli 1995] [Napolitano 1995] [Napolitano and Spagnuolo 1993] [Nguyen and Xiong 2026] [Ogburn and Foster 1993] [Ogburn et al 1992] [Ogburn et al 1994] [O’Donnell and Mohseni 2019] [Rennie and Jumper 1994] [Rennie and Jumper 1996] [Ribbens and Miller 1999] [Schaeffler et al 1967] [Smith 1966] [Smith et al 2003] [Spadão et al 2026] [Swaim 1970] [Sykes et al 2025] [Takagi et al 2015] [Tavella et al 1986] [Tavella et al 1987] [Tich 1991] [Tomac et al 2017] [Vinje 1974] [Wang et al 2018] [Ward et al 2024] [Warsop and Crowther 2018] [Weil 1981] [Xu and Zha 2019] [Xu and Zha 2021] [Yeo et al 2015] [Yu et al 2016] [Yu et al 2018] [Zhang et al 2018]

Engine thrust with altitude

Why the vectored moment falls with altitude. Engine thrust and its variation with flight condition, in-flight thrust measurement and installed thrust.

The control-authority argument in this article rests entirely on the way thrust falls with ambient pressure, and the working model of proportionality to the pressure ratio is a simplification that the installed-performance literature exists to correct. Inlet recovery, bleed, power extraction and nozzle matching all move the answer.

The article reports the sensitivity rather than hiding it, giving 0.7052 and 0.5922 at lapse exponents of 0.8 and 1.2 against 0.6462 at unity, and the qualitative conclusion survives the whole range.

13 records. [Andrianantara et al 2021] [Bronz et al 2017] [Hiers 2018] [Jia et al 2026] [Linderman et al 1974] [McCORMICK et al 1969] [Smith and Wehofer 1982] [Prince and McAulay 1950] [Ray 1994] [Sachs 1990] [Stengel 1992] [Tate and Gillard 1975] [The Measurement of Engine Thrust in an Altitude Test Facility 2015]

The standard atmosphere

The medium, named rather than assumed. Standard atmosphere tables and their use as the reference for altitude, density and performance.

Every altitude-dependent number in this article passes through a model of the atmosphere, and the model is a standard rather than a measurement. Thrust lapse, true airspeed, dynamic pressure and Mach number all read from it, so an error there propagates into the control-authority argument and into the quasi-tailless dynamic pressure alike.

The literature here uses the standard rather than revising it. Beyond the tables themselves, these records apply it to density altitude as a performance standard, to landing distance, to flight reference simulation and to the reduction of altitude and airspeed in flight test.

10 records. [Air Force Test Pilot School Edwards Afb Ca 1962] [Dancila and Botez 2020] [Dennis P. Dykstra 1980] [Divakar and Rajalakshmi Samaga 2022] [Younes 2024] [Lee and Aldredge 2015] [Luo 2025] [Report No. 538, altitude-pressure tables based on the United States standard atmosphere 1935] [Strand and Ennis 2012] [Vaidya and Peter-Contesse 1985]

Propulsion integration at extreme attitude

The engine at extreme attitude. Inlet distortion, compressor stall and its transients, inlet and nozzle integration and exhaust flows.

326 records. [Abbasi et al 2018] [Abed 2000] [Acton et al 2021] [Akolekar et al 2025] [Alendar et al 2022] [Andrew et al 1991] [Anex 1990] [Arshad et al 2016] [Asghar et al 2015] [Asghar et al 2016] [Asghar et al 2018] [Askari and Abedi 2026] [Askari and Soltani 2019] [Askari and Soltani 2023] [Baetz 1975] [Bai et al 2025] [Bai et al 2025] [Balkhi and Durna 2025] [Baranwal 2026] [Barklage et al 2018] [Barr et al 2022] [Beale 2018] [Bedke and Gorrell 2021] [Benichou et al 2021] [Biesiadny 1992] [Bing and Gong 2015] [Bond and Key 2025] [Bond and Key 2026] [Brandon et al 2021] [Brodsky 1970] [Brooks et al 1976] [Brown and Elgin 1985] [Bugajski et al 2015] [Burcham et al 1985] [Burrows et al 2018] [Burrows et al 2019] [Burrows et al 2020] [Burrows et al 2024] [Cao et al 2017] [Cao et al 2020] [Carlin et al 1984] [Castner et al 2017] [Chandra et al 2023] [Chang 1980] [Chen et al 2006] [Chen et al 2015] [Chennuru et al 2025] [Clough 1990] [Cousins and Obrien 1985] [Cravero et al 2023] [Crea et al 2026] [Crosthwait 1970] [Crown 1950] [Cusator et al 2024] [Da and Fan 2020] [Da et al 2015] [Davies 1981] [Davis and O’Brien 1987] [Delaat et al 1998] [DeLaat et al 1998] [Desikan et al 2017] [Dickman and Takahashi 2016] [DiFilippo et al 2025] [Ding et al 2025] [Dong et al 2015] [Dosch 2024] [Drabek 1960] [Eggers 1961] [Egorov et al 1997] [Ellison and Ojala 1969] [Faltot et al 2018] [Fan et al 2024] [Fan et al 2026] [Fang et al 2023] [Favaron et al 2021] [Feng et al 2025] [Ferrero et al 2020] [Ferrero et al 2021] [Ferrero et al 2022] [Floyd and Davis 2015] [Franklin 1993] [French 1985] [Frohnapfel et al 2017] [Frohnapfel et al 2018] [Frohnapfel et al 2020] [Mehalic and Lottig 1987] [Furuichi and Terao 2015] [Furuichi and Terao 2015] [Gallops et al 1990] [Gallops et al 1991] [Gao et al 2024] [Garg and Mattern 1991] [Garg et al 1991] [Gartner and Amitay 2016] [Gates and Cochran 1961] [Giangaspero et al 2019] [Giesecke and Friedrichs 2019] [Gillespie et al 2021] [Gravdahl and Egeland 1999] [Gravelle 1975] [Greitzer and Moore 1986] [Gridley and Walker 1996] [Grimaldi and Michelassi 2018] [Gubanov 2019] [Guimaraes et al 2015] [Guo 2023] [Guo et al 2015] [Güllü and Aran 2023] [Hah et al 2015] [Halwas and Aggarwal 2019] [Halwas and Aggarwal 2019] [Harisrinivasan et al 2018] [Harrington et al 1974] [He et al 2020] [He et al 2023] [He et al 2024] [Heath et al 2017] [Henderson 1989] [Henderson and Berrier 1989] [Herrick 1986] [Hiers and Knapke 2019] [Hill and Defoe 2020] [Hoertel 1961] [Housman and Kiris 2015] [Hu et al 2018] [Hu et al 2025] [Huang et al 2020] [Huskey 1975] [Ito et al 2020] [Jagtap et al 2025] [Jaiswal et al 2023] [Jansen and Thorman 1950] [Jie and Xu 2024] [Jun et al 1992] [Khan et al 2021] [Kim and Choi 2025] [Kim and Kim 2020] [Klepper et al 2017] [Klinskii 2023] [Kokubo et al 2026] [Korn 1974] [Korsun et al 2018] [Kotsyubinskii and Ravikovich 2015] [Kozakiewicz et al 2025] [Kramer and Hall 2025] [Kramer and Hall 2026] [Krosel 1984] [Kuo et al 2016] [Lakebrink and Mani 2018] [Lee and Boedicker 1985] [Lee and Choi 2024] [Lee et al 2018] [Lemmin 1986] [Li 2022] [Li et al 2020] [Li et al 2022] [Li et al 2023] [Li et al 2023] [Li et al 2024] [Li et al 2024] [Li et al 2025] [Lieu 1964] [Lin et al 2017] [Liu and Vo 2024] [Liu and Wu 2023] [Liu et al 2024] [Liu et al 2025] [Liu et al 2026] [Lobo et al 2016] [Lowe 2020] [Lucas 1978] [Ma et al 2024] [Ma et al 2025] [Madhvaraju et al 2026] [Maghsoudi et al 2020] [Malan and Brown 1994] [McClure and Sirbaugh 1991] [McGuigan 1970] [Medina et al 2021] [Mehdizadeh et al 2019] [Menegozzo and Benini 2020] [Menon et al 1996] [Miao et al 2020] [Moger and Ramsay 1964] [Moore and Greitzer 1986] [Munroe et al 2021] [Naruse et al 2022] [Nelson and Fleeman 1975] [Nichols et al 2015] [Oliva and Morris 2022] [Ordaz et al 2017] [Orme et al 1998] [O’Brien and Lowe 2015] [Page et al 2017] [Page et al 2018] [Pan et al 2022] [Pan et al 2024] [Pecinka et al 2015] [Pečinka et al 2017] [Piovesan et al 2025] [Pitel Welnitz et al 2019] [Potts and Bons 2020] [Prasad 2025] [Profiling of the Turbo Compressor Inlet Nozzle of an Aircraft Engine 2016] [Qiao et al 2025] [Qin et al 2018] [Qiu et al 2023] [Qiu et al 2023] [Qiu et al 2023] [Qiu et al 2024] [Qiu et al 2024] [Reedy and Gorrell 2025] [Remiger et al 2024] [Research on Axisymmetric Vectoring Exhaust Nozzle Dynamic Characteristics Considering Aerodynamic and Thermal Loads Effect 2025] [Reukauf and Burcham 1976] [Reutter et al 2020] [Rising et al 2026] [Romanò et al 2025] [Rosenblad 1990] [Ross et al 2018] [Saha et al 2017] [Saheby et al 2015] [Saleem Yusoof et al 2016] [Sasanapuri et al 2015] [Schram and Narayanaswamy 2026] [Sheng and Zhang 2017] [Shmilovich and Yadlin 2026] [Si et al 2016] [Simmons and Brant 1988] [Smith et al 1992] [Soltani et al 2018] [Song et al 2025] [Spencer et al 2016] [Srinivasan et al 2026] [Stephens et al 2025] [Stewart et al 1992] [Streiff 1953] [Strome 1969] [Sun and Tan 2018] [Sun et al 2018] [Sun et al 2022] [Sun et 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Axis transformations and the velocity-vector roll

The kinematics of a velocity-vector roll. Equations of motion, axis systems, quaternions and attitude representation.

The relation that makes a wind-axis roll 94 percent yaw rate at seventy degrees is elementary kinematics, and it belongs to a body of work on aircraft equations of motion and axis systems that predates every aircraft in this series. The X-31’s contribution was to fly somewhere the distinction stops being academic.

43 records. [Altman 1952] [Andersen and Kristiansen 2017] [Atkinson 2016] [Bernstein 1998] [Carino et al 2015] [Chen 2018] [Chen and Chen 1993] [Crain et al 2021] [Durham et al 1993] [Durham et al 1994] [Elsayed et al 2024] [Farconi et al 2019] [Gandulfo et al 2024] [Hoffman et al 1972] [Howe 2017] [Hunsaker and Moulton 2023] [Jenkins et al 1996] [Kalensher 1957] [Kalviste 1986] [Kalviste 1987] [Kasula et al 2024] [Keyes et al 2017] [Legen’kii 1995] [Liao et al 2026] [Liu and Wang 2015] [Marciano et al 2021] [Matthew 2025] [McIntyre 1963] [Michalak et al 2026] [Miller et al 2003] [Nguyen et al 2017] [Pachter et al 1995] [Pei and Roithmayr 2022] [Pradeep 1998] [Pradeep 1999] [Qi 2023] [Rodden and Love 1985] [Roskam 1965] [Schmitz 1973] [Schwanz 1972] [Wang and Yu 2016] [Warsett 1955] [Yuan et al 2025]

Flight control system design, actuation and allocation

The system that made the aircraft flyable. Flight control system design, fly-by-wire architecture, actuator saturation and rate limits, control allocation and incremental dynamic inversion.

An aircraft with more effectors than axes has to decide how to use them, and the X-31 had exactly that problem with elevons, canards, rudder and three vectoring paddles. It solved it with hand-tuned mixing in thirty-two software releases. The modern answer is control allocation as an optimisation solved at frame rate, with constraints for position and rate limits of the kind this article computed for the vectoring vanes.

Incremental nonlinear dynamic inversion is the other major change. The X-31’s control laws were predictive, mapping pilot commands and measured flight condition onto the effectors through a model of the flight path, and their dependence on measured flight condition is precisely what failed and destroyed the first aircraft. Incremental methods reduce the dependence on a model of the aircraft and therefore on the accuracy of the air data that parameterises it.

Allocation is named in 59 of the 491 titles, and incremental methods in 30.

491 records. 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[Yang et al 2019] [Yang et al 2019] [Yao 2019] [Yasue et al 1987] [Yiming et al 2019] [Yin et al 2020] [Yin et al 2021] [Yong Wang and Murray 1998] [You et al 2019] [You et al 2019] [Yu 2024] [Yuan et al 2021] [Yuksek and Inalhan 2020] [Yuksek and Inalhan 2022] [Yılmaz and German 2024] [Zaki et al 2017] [Zhang and Holzapfel 2021] [Zhang and Ran 2023] [Zhang and Su 2018] [Zhang et al 2016] [Zhang et al 2020] [Zhang et al 2020] [Zhang et al 2020] [Zhang et al 2023] [Zhang et al 2023] [Zhang et al 2025] [Zhang et al 2004] [Zhao et al 2020] [Zhao et al 2026] [Zhi et al 2018] [Zhishuai and Lingling 2022] [Zhou and Ma 2017] [Zhou et al 2018] [Zhou et al 2022] [Zhu and Duan 2015] [Zhu et al 2016] [Zipperer et al 1975] [Zollitsch et al 2015]

Gain scheduling and stability margins

The architecture whose failure destroyed the first aircraft. Gain scheduling and its analysis, together with the multivariable stability margins against which a scheduled loop is judged.

97 records. [Al-Sunni and Lewis 1993] [Amato et al 1996] [Analytical framework for gain scheduling 1991] [Apkarian and Adams 1998] [Balcan et al 2026] [Bar-On and Grasse 1994] [Bennett and Baras 1985] [Blanken et al 2009] [Cai et al 2025] [Cai et al 2026] [Chen et al 2018] [Chen et al 2024] [Chuang et al 1992] [Cockburn and Tannenbaum 1996] [de Gaston and Safonov 1986] [Ding and Pan 2016] [Dynamic scheduling of modern-robust-control autopilot designs for missiles 1992] [Ebihara et al 2020] [Fan et al 2017] [Fisher and Bryson 1995] [Fujimori and Nikiforuk 1999] [Gain scheduling potential hazards 1992] [Gong and Thompson 1994] [Gonsalves and Zacharias 1994] [Guo 2021] [Guo and Scherer 2018] [Hassan and Ali 2022] [He et al 2019] [Helmersson 1998] [Hyde and Glover 1991] [Ilka and Veselý 2017] [Jackson and Crouch 1991] [Jianliang Wang and Sundararajan 1994] [Jie 2018] [Jie and Yuan 1988] [Jonckheere and Bar-on 1990] [Kazarin et al 2016] [Keel and Bhattacharyya 1993] [Kumar P and Detroja 2024] [Latchman et al 1997] [Lawrence and Rugh 1995] [Leith 1996] [Leith 1999] [Lhachemi et al 2017] [Liang et al 2026] [Liu et al 2016] [Lévano et al 2020] [Marcos et al 2022] [Mosca and Agnoloni 1999] [Nguyen et al 2021] [Nichols et al 1993] [Packard 1994] [Patartics et al 2017] [Patel et al 2023] [Rantzer 1992] [Rugh 1990] [Sadeghzadeh 2017] [Safonov 1983] [Safonov 1984] [Safonov and Chen 1982] [Sajjadi-Kia et al 2007] [Sato 2018] [Sato 2018] [Schumacher and Khargonekar 1998] [Scorletti and Ghaoui 1998] [Shahruz and Behtash 1990] [Shahruz and Langari 1991] [Shamma and Athans 1991] [Shue et al 1996] [Sideris and De Gaston 1986] [Sideris and Pena 1988] [Solomon and Davison 1986] [Spencer and He 2025] [Sreeram and Agathoklis 1991] [Stilwell and Rugh 1999] [Tao et al 2016] [Tekawy et al 1990] [Veselý and Ernek 2018] [Veselý and Ilka 2015] [Veselý and Ilka 2015] [Veselý and Ilka 2017] [Vicino and Tesi 1995] [Vos et al 1991] [Walsh and Forbes 2016] [Wang 1995] [Wang et al 2017] [Wang et al 2017] [Wang et al 2024] [Wilson and Arnold 1991] [Wise 1990] [Wise 1993] [Wu et al 2005] [Yahagi and Kajiwara 2023] [Yang et al 2018] [Yue et al 2017] [Yue Zhao et al 2016] [Zhang et al 2019]

Unstable dynamics

Relaxed static stability. The control, identification and handling of statically unstable aircraft.

65 records. [Avanzini and Galeani 2005] [Avanzini et al 2017] [Blight et al 1986] [Bradshaw et al 1985] [Cavin et al 1976] [Colgren 1994] [Enns et al 1992] [Evangelou 1998] [Formalskii 2017] [Gera and Bosworth 1987] [Ghosh Roy and Peyada 2016] [Gibson 1990] [Glad 1995] [Goman and Demenkov 2002] [Goman et al 1996] [Grauer 2025] [Gumussoy and Özbay 2018] [Hanson and Stengel 1981] [Hanson and Stengel 1983] [Hanson and Stengel 1984] [Hopwood and Woolsey 2024] [Jategaonkar and Thielecke 1994] [Joos 1995] [Ju et al 2023] [Kamali et al 2007] [Klein et al 1982] [Koyama et al 2019] [Kuttieri and Sinha 2023] [Kwatny et al 1991] [Lapins et al 1982] [Lebacqz and Govindaraj 1980] [Maine and Murray 1986] [Maine and Murray 1988] [Mansor et al 2015] [Marangoz and Kutay 2017] [Maxwell et al 2015] [McRuer et al 1986] [Naihong et al 1993] [Newman and Buttrill 1995] [Patel et al 2007] [Peloubet et al 1990] [Potvin and Grant 2026] [Preissler and Schaufele 1991] [Rao et al 2003] [Rong and Yang 2024] [Rooney et al 1982] [Rossiter 1997] [Sadoff et al 1961] [Sariyildiz et al 2018] [Shafiei and Shenton 1999] [Shrivastava and Stengel 1989] [Sobel 1982] [Sozhamadevi and Sathiyamoorthy 2015] [Stabilization of Unstable Aircraft Dynamics under Control Constraints 2004] [Suresh et al 2004] [Tomlinson 1973] [Unstable Aircraft Identification 2006] [Unstable Aircraft Identification 2015] [Verma and Peyada 2018] [Wendel and Schmidt 1989] [Wilhelm and Schafranek 1986] [Yedavalli 2024] [Yildiz and Kolmanovsky 2011] [Zhang et al 2026] [Zhou et al 2025]

Handling qualities of highly augmented aircraft

What the pilot was given. Handling and flying qualities, their criteria and rating scales, and their evaluation for augmented fixed-wing aircraft and rotorcraft.

245 records. [Abdulrahim et al 2019] [Abu-Elola et al 2024] [Air Force Test Pilot School Edwards Afb Ca 1969] [Air Force Test Pilot School Edwards Afb Ca 1988] [Air Force Test Pilot School Edwards Afb Ca 1990] [Air Force Test Pilot School Edwards Afb Ca 1992] [Aircraft lateral-directional handling qualities design 2011] [Aircraft longitudinal handling qualities design 2011] [Alberghini and Lee 2018] [Alexander 2025] [Alexander 2025] [Alstrom and Paul 2023] [Anderson and Ormiston 1994] [Anderson 1960] [Antonakis 2025] [Antonakis 2025] [Ashkenas 1965] [Ashkenas 1965] [Bachelder and Aponso 2017] [Bachelder and Aponso 2020] [Bachelder and Aponso 2020] [Bachelder and Aponso 2022] [Bachelder and Aponso 2023] [Bachelder et al 2023] [Bachelder et al 2023] [Bachelder et al 2024] [Bahr et al 2020] [Bailes et al 1975] [Barnes et al 2025] [Baughman and Longeauay 2015] [Bechelder et al 2025] [Belak and Hromcik 2024] [Berger et al 2016] [Berry 1981] [Bidinotto et al 2021] [Bivens 1986] [Brandon et al 1995] [Breul 1963] [Broussard and Stengel 1977] [Callaghan and Kunz 2019] [Callaghan and Kunz 2021] [Campos and Marques 2021] [Cao and Qi 2018] [Cao et al 2022] [Cavalcanti et al 2026] [Cavallo et al 1990] [Celi 1991] [Chalk 1980] [Chance Vought Corp Dallas Tx 1979] [Chen and Ji 2017] [Chetty and Lakshmi 1991] [Chevalier and Burke 1972] [Chowhan et al 2019] [Cooper et al 2021] [Cosenza and Vos 2017] [Costigan and Calico 1989] [Daughaday and Duwaldt 1955] [Davidson and Andrisani 1996] [Denieul et al 2018] [Devi 2019] [Doman 1995] [Drewiacki et al 2016] [Drewiacki et al 2020] [Drewiacki et al 2025] [Dussart et al 2018] [Dussart et al 2019] [Edwards et al 1997] [Efremov et al 1998] [Efremov et al 2019] [Efremov et al 2019] [Efremov et al 2020] [Efremov et al 2022] [Erturk et al 2022] [Favaro et al 2025] [Fegely et al 2017] [Fegely et al 2017] [Fell et al 2015] [Ford 1989] [Fu et al 2019] [Garg and Schmidt 1988] [Garkushenko and Vinogradov 2016] [Gerken 1979] [Gertsen and Shomber 1965] [Grant et al 2015] [Gray 2009] [Gray 2016] [Griffin and Bellaire 1968] [Gu et al 2023] [Gungras et al 1996] [Habin 1994] [Hall 1971] [Hamidani and Kunz 2018] [Han and Liu 2026] [Hanson et al 2011] [Hariowibowo et al 2021] [Harper and Robert P. 1955] [Harper and Sardanowsky 1969] [Harper and Cooper 1984] [Hebbar and Pashilkar 2016] [Heffley and Johnston 1979] [Helicopter Handling Qualities 1982] [Henriquez Huecas et al 2022] [Hess 1990] [Hess 1991] [Hess 1995] [Hess 2016] [Hess 2018] [Hess 1994] [Hodgkinson et al 1990] [Hoh 1988] [Hoh and Mitchell 1983] [Hoh et al 1981] [Humphreys-Jennings et al 2020] [Hynes et al 1989] [Iloputaife and Ma 1987] [Innocenti 1985] [Innocenti 1989] [Ioannis and Ioannis 2026] [Ivler et al 2011] [Jansen and Van Kampen 2025] [Ji et al 2017] [Ji et al 2022] [Jiao et al 2023] [Jing et al 2020] [jingwei and jiaming 2020] [Johnston et al 1965] [Jones et al 2023] [Joosten et al 2023] [Junfeng et al 2020] [Jusko and Berger 2024] [Kemper and Cotting 2016] [Kereliuk and Sinclair 1979] [Key 1971] [Khan 2019] [Kidd and Bull 1963] [Kiefer and Calvert 1992] [Kim and Kunz 2016] [Kim and Kunz 2017] [Klein et al 2017] [Klyde et al 2018] [Klyde et al 2021] [Lampton et al 2024] [Lampton et al 2024] [Lappos 2020] [Lawrence et al 2018] [Lebacqz and Radford 1979] [Lee et al 2020] [Lee et al 2026] [Leggett et al 1990] [Li et al 2024] [Liddell et al 1947] [Loranchet et al 2025] [Luo and Wang 2017] [Lusardi 2023] [Malpica and Lusardi 2013] [Mandal and Gu 2016] [Mandal and Gu 2016] [Mansikka et al 2019] [Martin 2025] [McCormick 1969] [Memon et al 2021] [Miller and Franklin 1968] [Mitchell et al 2003] [Mitchell et al 2004] [Mooij 1985] [Musso et al 2026] [Nagy and Kirsten 1976] [Neal and Smith 1970] [Newton and Kroo 2023] [Newton and Kroo 2024] [Newton and Kroo 2025] [Niemiec et al 2020] [Ogden 2023] [Orra et al 2013] [Pascasio et al 2020] [Paul et al 2023] [Place et al 1974] [Portapas and Cooke 2020] [Portapas et al 2017] [Prabhakar et al 2019] [Pritchard et al 2016] [Radford et al 1980] [Raffel et al 2021] [Raffel et al 2021] [Rauer 2019] [Rhoads 1967] [Riccardi et al 2025] [Rogers et al 2017] [Rotorcraft handling qualities design 2011] [Rowan and Dogan 2023] [Rožić et al 2025] [Saetti and Rogers 2020] [Saetti and Rogers 2020] [Saetti and Rogers 2024] [Sahasrabudhe and Celi 1997] [Saporito et al 2021] [Sarathy and Murthy 1993] [Schuchardt 2015] [Schulze et al 2024] [Schwithal et al 2025] [Scroggs et al 2023] [Sherman 2021] [Shibata et al 1987] [Shubert and Jones 2025] [Simsek and Tekinalp 2015] [Slongo et al 2020] [Smith 1977] [Smith 2023] [Smith and Geddes 1979] [Smith et al 1973] [Smith and Edwards 1980] [Smith 1974] [Solies 1995] [Srinathkumar 2015] [Stapleford and Ashkenas 1968] [Tai et al 2026] [Takahashi et al 2024] [The Bell X-22A V/STOL, Variable Stability Research Airplane - Lessons learned 1990] [Tischler et al 1991] [Tomczyk 2008] [Tran et al 2017] [Tritschler and Dotson 2022] [Truong et al 2016] [Umm-e-Aymon et al 2016] [Uybarreta et al 2024] [Uybarreta et al 2025] [van Overeem et al 2022] [Viavattene and Mooij 2019] [Wang and Chen 2024] [Wang et al 2019] [Wang et al 2021] [Wasicko 1972] [Wei 2020] [Willebeek-LeMair and Rhinehart 2023] [Wilson et al 1994] [Woodcock and George 1976] [Wu 2025] [Wu et al 2020] [Xu and Zhang 2024] [Xu et al 2019] [Yin and Wang 2017] [Yin et al 2019] [Yuan et al 2022] [Zanoni et al 2022] [Zaytseva et al 2021]

Envelope protection and loss of control

Carefree handling by another name. Flight envelope protection, envelope estimation, loss of control in flight and upset recovery.

Loss of control in flight remains the largest single category of fatal accident in several sectors, and the modern response is envelope estimation and protection computed in real time. The X-31 sits at an interesting point in that history, since carefree handling is envelope protection by another name and the aircraft demonstrated both its value and its cost. A pilot who cannot depart also cannot feel the approach to departure.

86 records. [Ackerman et al 2017] [Yavrucuk et al 2001] [Akcal et al 2018] [Altunkaya et al 2024] [Atmaca et al 2026] [Autenrieb 2025] [Balachandran and Atkins 2017] [Belcastro et al 2016] [Bottasso and Montinari 2015] [Campbell et al 2021] [Loy 1997] [Massey 1992] [Unnikrishnan and Prasad 2006] [Yavrucuk et al 2002] [Kashawlic et al 2016] [Saab Ab 2007] [Howitt 1995] [Chang et al 2016] [Chowhan et al 2022] [Stich 2012] [Cook 1964] [Cook 1965] [de Rooij et al 2023] [Ralph and Edwards 2002] [Ellerbroek et al 2016] [Fellah and Guiatni 2019] [Gabrys et al 2019] [Garcia et al 2015] [Hale et al 2015] [Harno et al 2021] [Hsu et al 2023] [Hsu et al 2024] [Ison 2017] [Jebakumar et al 2019] [Kostin and Zhuravskiy 2023] [Lang et al 2025] [Lee et al 2017] [Lombaerts et al 2017] [Lombaerts et al 2018] [Martinez et al 2026] [McDonough and Kolmanovsky 2015] [McDonough and Kolmanovsky 2017] [Moreira et al 2022] [Nie et al 2015] [Norouzi et al 2019] [Poolla and Ishihara 2015] [Pugh et al 2023] [Rafi et al 2017] [Rafi et al 2019] [Rafi et al 2020] [Rafi et al 2021] [Rafi et al 2022] [Rajaram et al 2017] [Mumaw et al 2019] [Salahudden et al 2020] [Schieni et al 2024] [Schlimm 2015] [Schuet et al 2017] [Simon et al 2017] [Stapel et al 2016] [Tekles et al 2017] [Tong et al 2023] [Van Baelen et al 2020] [Van Baelen et al 2021] [Wu et al 2020] [Wu et al 2021] [Xu and Zhang 2020] [Yahi et al 2024] [Yildiz et al 2018] [Yildiz et al 2019] [Yin et al 2019] [Yu et al 2022] [Yuan and Li 2018] [Yuan and Li 2019] [Zaichik et al 2019] [Zhang and Chen 2016] [Zhang et al 2021] [Zhao and Jim Zhu 2017] [Zhao and Zhu 2016] [Zhao and Zhu 2016] [Zhao and Zhu 2016] [Zheng et al 2018] [Zhou and Nie 2022] [Zhou et al 2017] [Zhuravsky and Kostin 2023] [Ç. Altunkaya et al 2025]

Unmanned combat aircraft and the tailless problem

Where the quasi-tailless result went. Tailless and flying-wing configurations, their directional control, and unmanned combat aircraft.

The configuration question the quasi-tailless experiment opened is now the dominant one. A combat aircraft designed for low observability wants no vertical surfaces, and an aircraft with no vertical surfaces has no directional stiffness and no rudder. Of the 289 records, 174 name a tailless or flying-wing configuration, and 20 name a yaw effector, a split aileron, a spoiler, a drag rudder or another innovative control effector, which is the direct continuation of the experiment.

289 records. [Abdullahi et al 2026] [Addington and Myatt 2000] [Advanced Control Techniques for Unmanned Aerial Vehicle (UAV) Navigation and Flight Control 2023] [Alatorre et al 2021] [Autonomous flight control of drone equipped with microcomputer for flight control and GPS 2018] [Azimov and Allen 2017] [Bai et al 2021] [Balasubramanian et al 2024] [Banerjee 1984] [Banerjee 1988] [Banerjee 1988] [Barker and Balas 1999] [Bishop 1961] [Bookstaber 2000] [Boskovic and Mehra 1999] [Bowlus et al 1997] [Bozov 2023] [Bras et al 2022] [Brinker and Wise 1998] [Brinker and Wise 1999] [Brown 1998] [Buffington 1999] [Buffington 1999] [Buffington and Buffington 1997] [Buffington et al 1998] [Bunnell 2003] [Byrdsong and Brooks 1980] [Calise et al 1998] [Campos et al 2021] [Cao et al 2017] [Chang et al 2018] [Donlan 1944] [Chaudhuri et al 2018] [Chen et al 2026] [Cheng and Pei 2021] [Chuan et al 2016] [CONCEPTUAL SYNTHESIS OF INTELLIGENT UNMANNED CONTROL SYSTEMS AIRCRAFT 2022] [Cong et al 2023] [Cox 2009] [Cox et al 1949] [Crane and Reeder 1945] [Cusher and Gopalarathnam 2006] [de Becdelievre and Kroo 2021] [DeLaurier 2022] [Deng and Yuan 2015] [Denieul et al 2015] [Deslich et al 2021] [Dickes et al 2002] [Dong et al 2020] [Dong Kangsheng et al 2016] [Dou and Duan 2016] [Duan et al 2023] [Duggan and Bhandari 2021] [Eberhardt and Ward 1999] [El Mobaraky et al 2026] [Ericsson and Beyers 2002] [Ernest and Cohen 2016] [Esteban et al 2015] [Faisal and James 2025] [Fan et al 2017] [Fan et al 2017] [Fangzi et al 2024] [Flow Control and High-Lift Performance for Flying-Wing Unmanned Combat Air Vehicle Configurations by inserting slots 2016] [Fremaux et al 1993] [Gamble and Inman 2016] [Garcia-Baquero et al 2018] [Gillard 1998] [Gillard et al 1997] [Givens and Coopmans 2020] [Guiler and Huebsch 2005] [Han et al 2025] [Hanif and Sasongko 2019] [Hao et al 2024] [Harris and Harris 1997] [Hashmi and Radhakrishna 2015] [Hauser 1999] [He et al 2022] [He et al 2022] [Hoffler et al 1986] [Hoffler et al 1988] [Hoholis et al 2016] [Hong and Ma 2016] [Howard et al 2002] [Hu et al 2024] [Hu et al 2025] [Huang 2016] [Huang and Zhang 2020] [Huang et al 2019] [Huang et al 2024] [Huber 2022] [Ito et al 2016] [Jentzsch et al 2016] [Jentzsch et al 2017] [Jentzsch et al 2019] [Jiguang Li et al 2016] [Jo et al 2015] [Jo et al 2016] [Johnson 1946] [Johnson 1949] [Jones et al 2006] [Jun 2023] [Kaba et al 2022] [Kalyankar et al 2024] [Kang and Choi 2022] [Karimi Kelayeh and Djavareshkian 2024] [Kashiwagura and Shimoyama 2018] [Katz 2017] [Khan and Nahon 2015] [Kim et al 2025] [Ko et al 2019] [Kostić et al 2024] [Kotikalpudi et al 2015] [Kraft and Reeder 1948] [Kumar 2020] [Kumar et al 2026] [Kwiek 2019] [Lanteigne and O’Reilly 2020] [Lee 1946] [Lee 1947] [Lee and Ahn 2026] [Lee and Jeong 2025] [Lei and Duan 2023] [Lewis 2002] [Li et al 2017] [Li et al 2018] [Li et al 2022] [Li et al 2022] [Li et al 2023] [Lim et al 2025] [Lin et al 2026] [Liu and Hu 2014] [Liu and Zhang 2022] [Liu et al 2021] [Loechert et al 2018] [Longobardi and Skaloud 2023] [Lu et al 2019] [Luo et al 2025] [Luo et al 2025] [Lyu et al 2023] [Ma and Chen 2023] [Ma et al 2019] [Ma et al 2019] [Makarov and Pavlenko 2016] [Mason and Ravi 1991] [Mason and Ravi 1994] [Matamoros and de Visser 2018] [McKeehen 1998] [McKeehen et al 1997] [Meckstroth and Blake 2015] [Meckstroth and Ordóñez 2020] [Meckstroth and Ordóñez 2020] [Michal 1998] [Miles and Broughton 2017] [Mitcham et al 1952] [Mitikiri and Mohseni 2018] [Molinari et al 2015] [Molinari et al 2018] [Morris 1992] [Muliadi 2016] [Mullins et al 1996] [Mustopa 2022] [Myatt et al 1998] [Nae 2026] [Napolitano and Kincheloe 1996] [Ngo et al 1996] [Nguyen et al 2026] [Nie et al 2020] [Nie et al 2021] [Niestroy 2025] [Niestroy et al 2017] [Niestroy et al 2019] [Nieto-Wire and Sobel 2007] [Nieto-Wire and Sobel 2009] [Orhan and Subbarao 2021] [Owashi et al 2017] [Pan and Huang 2018] [Pan and Huang 2019] [Pan and Huang 2019] [Paranjape and Chung 2010] [Park et al 2016] [Park et al 2016] [Park et al 2017] [Park et al 2017] [Patel et al 2021] [Paul et al 1997] [Peng 2021] [Phillips et al 2019] [PI control of a tailless fighter aircraft with dynamic inversion and neural networks 1999] [Poojari 2022] [Preston 2020] [Protz and Sparks 1998] [Qu et al 2017] [Rajput et al 2015] [Ravi and Mason 1992] [Ravi and Mason 1994] [Riethausen 2026] [Rojas Carvajal and Amitay 2023] [Rojas Carvajal and Amitay 2025] [Rojas Carvajal et al 2022] [Rojas Carvajal et al 2024] [Rossetti and Beblo 2022] [Sudhakar et al 2025] [Saephan and van Dam 2006] [Saheby et al 2026] [Sanchez-Laulhe et al 2020] [Santhakumar 2026] [Saxena and Faruque 2016] [Schmidt 2016] [Schmidt 2016] [Shah and Granda 1998] [Shayan and Van Kampen 2021] [Shearwood et al 2019] [Shearwood et al 2020] [Shearwood et al 2020] [Shearwood et al 2021] [Shearwood et al 2021] [Shearwood et al 2023] [Shi and Wu 2022] [Shtessel et al 1999] [Shuang et al 2016] [Shukla and Komerath 2018] [Simon et al 1993] [Simon et al 2024] [Song et al 2016] [Sparks and Sparks 1997] [Stengel 1980] [Su et al 2022] [Suminsby 2002] [Sun et al 2005] [Talmage and Reeder 1949] [Tang et al 2026] [Taufik et al 2024] [Teel 1999] [Teel 1999] [Tian et al 2017] [Tipton and Smith 1995] [Tipton et al 1996] [Tournes and Johnson 1999] [Tournes et al 1997] [Tsai 2008] [Udartsev et al 2015] [van Rooij et al 2018] [Veismann et al 2023] [Waitman and Marcos 2020] [Wang and Feng 2026] [Wang et al 2016] [Wang et al 2018] [Wang et al 2020] [Wang et al 2023] [Wang et al 2025] [Webb and Rogers 2021] [Weyl 1944] [Weyl 1945] [Weyl 1945] [Weyl 1945] [Weyl 1945] [Weyl 1945] [White and Innis 1959] [Williams and Seidel 2016] [Williams et al 2025] [Wills 2015] [Wise et al 1999] [Wu et al 2019] [Wu et al 2019] [Wu et al 2021] [Wyatt 2003] [Xi and Zhao 2017] [Xu and Feng 2025] [Xu and Yu 2025] [Yao and Wu 2019] [Yao et al 2018] [Ye and Zheng 2025] [Yin et al 2020] [Yong et al 2020] [Yu et al 2022] [Yu et al 2023] [Yu et al 2023] [Yu et al 2023] [Yu et al 2024] [Yue et al 2009] [Zhang and He 2026] [Zhang et al 2018] [Zhang et al 2021] [Zhang et al 2023] [Zhijie et al 2025] [Zimper and Hummel 2018] [Zou et al 2017] [Zou et al 2022] [Zubair et al 2026]

Machine learning in flight control and aircraft modelling

Learning applied to the aircraft rather than to the fight. Neural network, reinforcement learning and data-driven methods for flight control, aerodynamic and engine modelling, and aircraft performance.

203 records. [Abouheaf et al 2020] [Adams et al 2022] [Ahmed et al 2025] [Alkhedher 2022] [Allik and Maley 2024] [Bae et al 2018] [Bai and Chen 2026] [Baldan and Guardone 2024] [Barnhart et al 2021] [Berghout and Benbouzid 2024] [Boumaraf et al 2025] [Cakici and Yörük 2022] [Cao and Lu 2024] [Cao et al 2021] [Cao et al 2026] [Catak et al 2024] [Celis and Cadarso 2024] [Chen and Song 2025] [Chen et al 2017] [Chen et al 2020] [Chen et al 2023] [Chen et al 2025] [Chen et al 2025] [Chen et al 2026] [Cheng et al 2020] [Cheng et al 2025] [Choi and Cha 2019] [Choi et al 2023] [Chowdhury and Keshmiri 2024] [Clarke and Hwang 2020] [Conrad et al 2025] [Dahleni Kraemer and Villani 2024] [Dam et al 2022] [Daskiran et al 2017] [De Marco et al 2023] [Dhiman et al 2025] [Diget et al 2022] [Dobrokhodov et al 2026] [Doff-Sotta et al 2025] [Dongmo 2016] [Du et al 2021] [d’Apolito and Sulzbachner 2021] [Eggers and Xu 2024] [Elebe 2025] [Everett 2021] [Fernandes and Cardoso-Ribeiro 2025] [Feteih and Breckenridge 1993] [Galeani et al 2025] [Gao and Liu 2020] [Ghoreyshi et al 2026] [Gong et al 2019] [Gorgues et al 2022] [Grillo et al 2024] [Grönqvist and Rantzer 2023] [Guang et al 2016] [Guo and Jin 2023] [Han et al 2023] [Han et al 2024] [He et al 2020] [He et al 2025] [Heyer et al 2020] [Homola et al 2025] [Hong et al 2020] [Huang et al 2023] [Huiying et al 2017] [Isaac and Allaire 2016] [Islam et al 2025] [Jafari and Xu 2018] [Jia and Bajaj 2025] [Jiang et al 2022] [Jianhong and Ramirez-Mendoza 2022] [Jianhong and Ramirez-Mendoza 2022] [Joyce 2026] [Khatri and Subbarao 2024] [Kim and Horspool 2020] [Kirsch et al 2025] [Konatala et al 2021] [Konatala et al 2024] [Kong et al 2020] [Koopman and Zammit-Mangion 2024] [Kou and Zhang 2021] [Kumar and Khanna 2026] [Lang et al 2022] [Lecerf et al 2015] [Lee and Kim 2020] [Lee and Van Kampen 2021] [Lee et al 2021] [Lee et al 2021] [Lee et al 2022] [Li and Cai 2026] [Li and van Kampen 2024] [Li et al 2019] [Li et al 2024] [Li et al 2025] [Liang and Hassanalian 2026] [Lin 2025] [Lin et al 2023] [Liu 2024] [Liu et al 2024] [Liu et al 2024] [Liu et al 2025] [Liu et al 2026] [Lu et al 2024] [Lu et al 2025] [Lv et al 2024] [Ma et al 2017] [Ma et al 2023] [Ma et al 2024] [Machine learning�enabled missile system characterisation 2023] [Maju et al 2025] [Martinez-Gonzalez et al 2024] [Massegur Sampietro and Da Ronch 2024] [Matthews and Yi 2021] [McCool et al 1996] [Mechouche et al 2026] [Mei et al 2024] [Meinicke and Cesnik 2024] [Michek et al 2024] [Ming-Chao and Ju 2016] [Mokhtar and El-Badawy 2023] [Mori 2024] [Nagawkar and Leifsson 2022] [Newton and Papachristodoulou 2021] [Newton and Papachristodoulou 2023] [Nguyen-Tat et al 2026] [Norouzi et al 2018] [Oktay and Kose 2021] [Pan et al 2024] [Pellegrino et al 2022] [Peterson et al 2025] [Ransquin and Chatelain 2021] [Razmi and Afshinfar 2019] [Ren et al 2017] [Research on Aerodynamic Modeling of Elman Neural Network Based on PSO Algorithm 2017] [Richter et al 2024] [Safety Behavior Analysis of a Delayed Control System 2021] [Sayyed et al 2025] [Schaff and Prazenica 2025] [Sedehi et al 2020] [Seres et al 2023] [Setiawarman and Sasongko 2026] [Shaik et al 2025] [Shao and Cui 2019] [Shen and Alonso 2025] [Shen and Xu 2021] [Shin et al 2018] [Shukla et al 2025] [Singh and Willcox 2016] [Singh and Willcox 2017] [Singha et al 2024] [Smith and Broadwell 1989] [Smith and Schnell 2026] [Song et al 2018] [Sterpu et al 2025] [Stewart et al 2025] [Study on Deep Learning of Aircraft Safety Enhancement and Autonomous Flight Assistance 2023] [Sun and van Kampen 2021] [Tang and Lai 2020] [Teimourian et al 2024] [Thakur et al 2026] [Toloei and Ghaderi 2026] [Torres et al 2022] [Vieira dos Santos and Van Kampen 2025] [Wang 2025] [Wang and Hu 2025] [Wang and Liang 2024] [Wang et al 2023] [Wang et al 2024] [Wang et al 2025] [Wang et al 2025] [Wang et al 2025] [Wang et al 2025] [Wang et al 2026] [Wang et al 2026] [Wang et al 2026] [Wei et al 2024] [Xiang and Shao 2022] [Xu et al 2019] [Xu et al 2024] [Xue et al 2024] [Xue et al 2026] [Yan and Shi 2025] [Yan et al 2019] [Yang et al 2023] [Yang et al 2023] [Yang et al 2024] [Yang et al 2026] [Yong et al 2014] [Yoo et al 2021] [You et al 2024] [Yuan and Zhou 2024] [Zan et al 2022] [Zerouaoui et al 2025] [Zhang et al 2021] [Zhang et al 2023] [Zhang et al 2023] [Zhang et al 2023] [Zhang et al 2026] [Zhao et al 2024] [Zhen and Hao 2020] [Zou et al 2026] [Özbek and Koyuncu 2023] [Öznurlu et al 2023]

Clearing a learned controller for flight

How a learned or autonomous controller is cleared to fly. Run-time assurance, formal verification, explainability and certification of learning-enabled flight systems.

The question the X-31 answered with six test pilots is now asked of learned policies, and nobody yet knows how to certify one.

A control law of the X-31’s kind is cleared by analysis, by simulation and by incremental flight test, and its behaviour between test points is bounded by the fact that it is a scheduled linear system with known margins. A learned policy has no such structure, and the assurance literature that has grown up around that problem is a distinct field, running from formal verification through run-time assurance architectures that wrap a learned or autonomous controller in a monitored envelope and revert to a conventional one when it leaves.

The connection to this aircraft is closer than it looks. The X-31 already carried the architectural idea in a primitive form. Its quasi-tailless experiment had a safety disengagement that reverted the aircraft automatically on a system failure or an envelope exceedance, which is a run-time assurance monitor in everything but name, built in 1994 because nobody was willing to fly a deliberately destabilised aeroplane without one.

And the accident is the counter-example. The one failure that destroyed an aircraft was not a control law behaving unexpectedly. It was a correct control law fed a wrong number, which is the failure mode that no amount of verifying the controller addresses.

The whole of this cluster is contemporary and that is a fact rather than an omission. Its earliest record was published in 2016, because the problem did not exist until a learned component was proposed for a flight-critical loop.

37 records. [Avram et al 2017] [Bairy and Fränzle 2024] [Bernard et al 2024] [Chen et al 2024] [Chhaya et al 2018] [Daw 2026] [Daw et al 2023] [Devore et al 2016] [Gross et al 2017] [Heersink et al 2022] [Henderson et al 2022] [Hobbs et al 2023] [Hook et al 2016] [Jasim and Veres 2018] [Kaczmarek 2026] [Kanou and Ibuki 2025] [Karyotakis et al 2021] [Kirkendoll 2022] [Kreuzer et al 2024] [Krozel et al 2026] [Kurtoglu et al 2025] [Kurtoglu et al 2026] [Liu et al 2025] [McColl et al 2024] [Miao et al 2025] [DeVore et al 2022] [Saraf et al 2020] [Schierman et al 2018] [Schierman et al 2020] [Schoeman and Panday 2026] [Suraj Nandiganahalli et al 2016] [Vaz et al 2019] [Wabersich and Zeilinger 2018] [Wang et al 2020] [Wheatman et al 2021] [Young et al 2018] [Zhang and Li 2020]

Air data, icing and flush air data sensing

The instrument that destroyed the first aircraft. Air data systems, pitot and pitot-static probes, probe and airframe icing and its detection, and flush air data sensing.

Flush air data sensing became a field, and 47 of the 391 titles name it. The methods estimate free-stream conditions from a distributed array of surface pressure ports through a model, frequently a learned one, and they serve vehicles on which a probe cannot survive or is unwanted. Icing, of probes and of airframes, is named in 140 titles.

The relevance to this article is direct. A distributed array has no single pneumatic path whose blockage is shared by every downstream computer, which is the exact architectural property whose absence killed the first X-31.

391 records. [Whitmore 2024] [Whitmore and Case 2023] [Agui et al 2019] [Ahmed Abdelrahman et al 2015] [Ainsworth et al 1961] [Air Force Test Pilot School Edwards Afb Ca 1967] [Klopfenstein Jr 1998] [Air Weather Service Scott Afb Il 1945] [Aircraft et al 2015] [Aleman et al 2017] [Allen 1975] [Alligood et al 1963] [Golparvar and Yapici 2020] [ANALYSIS OF VELOCITY PROFILE BY USING PITOT TUBE 2017] [Anderson 2019] [Ariafar et al 2018] [Avery and Jacob 2017] [Avery et al 2019] [Baar et al 2017] [Baer 1966] [Balachandran and Atkins 2016] [Barnhart 2007] [Bauer et al 2016] [Bilanin 1991] [Blatz and Brown 1953] [Bogue 1992] [Bokov and Efimov 2023] [Bornhoft et al 2025] [Boskovic and Jackson 2016] [Bowers-Irons et al 1990] [Bragg 1984] [Bragg et al 1986] [Bromfield et al 2023] [Brown 1954] [Bärfuss and Gallia 2026] [Cao and Hou 2016] [Cao et al 2015] [Cao et al 2020] [Carbonara and Binnie 1938] [Care and Fourneaux 2020] [Chang et al 2016] [Chazot et al 1998] [Chaudhry et al 2019] [Chebbi and Tavoularis 1991] [Chen and Whidborne 2023] [Chen and Xu 2022] [Chen et al 2015] [Chen et al 2024] [Chen et al 2024] [Choi et al 2021] [Christmann 2025] [Lee and Ash 1956] [Lamon et al 2022] [Colgren 1996] [Takahashi 2020] [Computation of low density supersonic flow over a pitot tube 1985] [Luo 1987] [Jackson 2015] [Cornell et al 1995] [Corrêa et al 2019] [Cotton 1978] [Cotton 1979] [Cristhina et al 2023] [Croce and Suzzi 2024] [Curley 1991] [Daiber et al 1995] [DeGennaro et al 2015] [Deiler and Kilian 2018] [Deiler and Sachs 2026] [DeLeo and Hagen 1963] [Adams] [Rudd 1977] [Zhang 2025] [Ding et al 2018] [Ding et al 2020] [Ding et al 2023] [DiPlacido et al 2016] [Weinig 1956] [Doiron et al 2026] [Dong 2018] [Dong 2019] [Dong et al 2024] [Donizetti et al 2025] [Doogood 1987] [Chamberlin and Gatsonis 2006] [Duan et al 2025] [Pieniążek et al 2023] [Dyvnych and Bezkorovainyi 2025] [Efremova 2016] [ElSharkawy et al 2024] [Ensuring flight safety of aircraft in icing conditions 2020] [Evangelista and Mori 2025] [Shmueli et al 2013] [Al-Masri] [Fan et al 2021] [Lilley 1984] [Yoon and Lilley 1983] [Winn and Kishi 1972] [Abbott et al 1977] [Widden 1996] [Friedlander et al 2023] [Fujiwara and Bragg 2019] [Funchal de Rezende et al 2025] [Gallia et al 2022] [Gao et al 2017] [Gao et al 2019] [Gao et al 2021] [Gao et al 2022] [Gao et al 2022] [Gary 1983] [Ge et al 2022] [Gibson et al 1989] [Glaser 1952] [Grachev and Safonov 2019] [Griffin et al 2014] [Guffey and Wang 1999] [Gui et al 2025] [Guo et al 2025] [Gurbacki and Bragg 1999] [Haering 1990] [Hajiyev et al 2019] [Halder and Gandhi 2026] [Halefom et al 2024] [Han et al 2022] [Hane 1999] [Hanks 1979] [Hann and Johansen 2021] [Harno and Kim 2020] [Heinrich et al 2022] [Henno 2015] [Brown 2003] [Henry et al 1988] [Hlavatá et al 2018] [Horvath 1969] [Hu et al 2022] [Hu et al 2023] [Hu et al 2025] [Hui and Dalin 2016] [Hurwitz and Dick 1978] [Ice and Rain Minimum Qualification Standards for Pitot and Pitot-static Probes 2020] [Ignatyev et al 2020] [Iinoya and Masuda 1970] [Zhiqiang Sun et al 2009] [Ivanković et al 2023] [Iwai 2018] [Jackel et al 2018] [Jain et al 2017] [Jain et al 2019] [Jayakumar et al 2018] [Jenkins 1987] [Jia et al 2021] [Jianhao et al 2025] [Jobbins et al 2026] [Johansen et al 2015] [Joos and Ossmann 2016] [Jurado and McGehee 2019] [Jurado and McGehee 2024] [Jäckel et al 2019] [Jäckel et al 2020] [Jäckel et al 2021] [Kandemir and Ayan 2023] [Kang et al 2026] [Karali et al 2023] [Karlgaard 2025] [Karlgaard and Schoenenberger 2017] [Karlgaard et al 2015] [Karlgaard et al 2017] [Karns et al 2017] [Kassab 1990] [Katz 1967] [Kazan and Aktemur 2026] [Kettle 1954] [Kharkov et al 2021] [Kilic and Unal 2021] [Kim et al 2016] [Kim et al 2024] [Kim et al 2025] [Kish et al 1999] [Knoedler et al 1996] [Kunishige et al 2018] [Kusmirek et al 2024] [Kutsuna et al 2015] [Larocque et al 2025] [Larson et al 1987] [Laurence III and Argrow 2019] [Lawrence Raj et al 2019] [Lawson et al 2022] [Lee 2019] [Lee and Ash 1956] [Lee et al 2022] [Leopold 2019] [Lerro and Battipede 2021] [Lerro et al 2021] [Li and Paoli 2022] [Li et al 2022] [Li et al 2025] [Li et al 2026] [Li Yu et al 2018] [Liu 2023] [Liu and Xiao 2016] [Liu et al 2022] [Liu et al 2023] [Liu et al 2026] [Ferrell et al 1977] [Lu et al 2018] [Lugo et al 2019] [Luo and Katz 2017] [Luo et al 2026] [Lv et al 2020] [Løw-Hansen et al 2022] [Machining for Air Data Measurement 1989] [Madhavanpillai et al 2017] [Maged et al 2026] [Manjith 2026] [Marquardt 1976] [Martos et al 2011] [Mason 1953] [Mason 1953] [MAXIMUM ALLOWABLE AIRSPEED INDICATORS 1956] [McKillip 2022] [McKillip 2022] [McLaren et al 2007] [White and Passmore 1998] [Mehta 2021] [Mendig 2015] [Merriam and Spaulding 1935] [Mersha et al 2026] [Miao et al 2024] [Miglioli and Carrillo Ibanez 2025] [Milani et al 2024] [Mimmo et al 2015] [Mingming et al 2019] [Minimum Performance Standard for Pitot and Pitot-Static Probes 2022] [MINIMUM PERFORMANCE STANDARD FOR PITOT AND PITOT-STATIC TUBES 1993] [Mittag et al 1974] [Moes and Whitmore 1990] [Moes et al 1993] [Kannenberg and Boyd 1996] [Monteil 2024] [Morency et al 1999] [Moriyama et al 2017] [Moriyama et al 2019] [Morris and Woratschek 1980] [Moul and Taylor 1981] [Mäder et al 2018] [Newton 1978] [Nikolaev and Gusev 2026] [Nowack 1970] [Wu et al 2000] [Olejniczak and Nowacki 2018] [Olejniczak and Nowacki 2020] [Andersson 1958] [Craig 1967] [Jonsson 2026] [Edwards] [Peculiarities of aircraft icing in the area of Tomsk International Airport 2016] [Pei et al 2017] [Pei et al 2018] [Peng 2021] [Lessor and Duane 1969] [Phand et al 2025] [Mukherji 1971] [Pitot Probe Anemometer 2018] [Knežević 2019] [Conlee 1929] [Hirayama et al 2023] [Cole 1935] [Politovich 1996] [Politovich and Sand 1981] [Popowski and Dąbrowski 2015] [Hanks et al 1983] [Mihai et al 2017] [Quantifying Pilot Contribution to Flight Safety During an In-Flight Airspeed Failure 2017] [Raj 2019] [Ratvasky and Ranaudo 1993] [Ratvasky et al 2021] [Rauf Guliyev 2025] [Akimoff 1914] [Report no. 264, Differential pressures on a pitot-venturi and pitot-static nozzle over 360° Pitch and Yaw 1927] [Folsom 1956] [Rhudy et al 2015] [Robinson and Blanchard 2003] [Rodrigues et al 2017] [Rodriguez 1984] [Rogers 2019] [Rosily et al 2021] [Ross 1980] [Ross 1984] [Ruff 1985] [Rydblom and Thornberg 2016] [Ryerson 1988] [Sable 2023] [Sankaralingam and Ramprasadh 2021] [Schleutker et al 2019] [Schollmeier and der Wiesche 2022] [Schuabb et al 2025] [Shao et al 2023] [Shaqura and Claudel 2015] [Shimada et al 2024] [Shyam Mohan et al 2018] [Simpson and Render 1998] [Singh et al 2017] [Smart 1991] [Sogin 1954] [Soldatkin 2019] [Soldatkin et al 2018] [Soldatkin et al 2020] [Soldatkin et al 2023] [Soldatkin et al 2024] [Soldatkin et al 2025] [Son and Yee 2018] [Song et al 2019] [Sotomayor-Zakharov et al 2024] [Specification for pressure-driven airspeed indicators 1975] [Spencer 1947] [Stankov and Bedard 1994] [Stocksreiter and Zangl 2019] [Stone and Ross 1983] [Stuart 1960] [Study on the Flight Characteristics of An Aircraft after Icing 2022] [Sun and Gebre-Egziabher 2020] [Sun et al 2019] [Strack 1964] [Swafford et al 1995] [Tabrizi and Johnson 1989] [Tabrizi and Keshock 1988] [Takahashi et al 2020] [Takahashi et al 2026] [Tanaka et al 2022] [Tang et al 2021] [Tescaroli and Belan 2021] [Test Method for Average Velocity in a Duct Pitot Tube Method] [Pan and Kuo 2021] [Gregory 1904] [Gebhardt 1909] [Krause 1977] [Thomas et al 1996] [Tilmans et al 2024] [Tong et al 2017] [McElreath 1972] [Tsao 1968] [TUBE - PITOT, ELECTRICALLY HEATED 1998] [Tucker 1993] [Ur Rahman et al 2024] [Valasek et al 2020] [Van Hove et al 2019] [Varwig et al 1976] [Verification of aircraft icing forecast based on the atmospheric radiometry remote method 2019] [Vitale et al 2019] [Ezzeddine and Rezg 2019] [Wada 1924] [Wada 1927] [Wang and Qin 2024] [Wang et al 2022] [Wang et al 2025] [Wang et al 2026] [Webb and Washington 1972] [Weggler et al 2024] [Wei et al 2020] [Weisberger 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Airspeed systems as a measurement discipline

Airspeed as a measurement. Position error, pitot-static calibration, total pressure probes and the calibration of airspeed by tower and satellite methods.

The instrument that destroyed the first aircraft has its own literature and it is separate from the flight-control one. Position error, static source calibration, probe placement and flow-angle sensitivity are the subject, and the Kiel probe was chosen out of exactly that body of knowledge to solve exactly the problem it solved.

What the discipline did not supply was a failure analysis for the case where the probe is correct and blocked, which is a different question from whether it is correctly calibrated, and every title in this survey that names the blockage of a pitot tube was published from 2015 onward.

21 records. [Sun et al 2019] [Air Force Test Pilot School Edwards Afb Ca 1967] [Airspeed Definitions and Airspeed Measurement 2022] [Fontaine and Ali 2023] [Hamel and Kolarich 2020] [Harsanta et al 2021] [Hernandez and Taylor 1988] [Hollister 1990] [Kimberlin and Sims 1992] [Li and Qi 2023] [Li et al 2016] [Li et al 2017] [Mitchell et al 2016] [O’Neill and McQuilling 2024] [O’Neill and McQuilling 2025] [Park 2017] [Rubio-Sierra et al 2022] [Sitaram et al 2023] [Woodford and Hansen 1985] [Zhang 2025] [Zhang and Yang 2024]

Analytical redundancy and sensor fault detection

The cross-check, formalised. Analytical redundancy, sensor and actuator fault detection and isolation, and fault-tolerant control.

The cross-check this article performed by hand is now a standard technique with a literature. Analytical redundancy uses the known dynamics of the aircraft to predict what one sensor should read from what the others read, and flags a disagreement. Applied to the accident flight it would have compared angle of attack against airspeed through the lift curve and raised a fault when the two disagreed by eleven degrees.

Nothing in that technique required computing power the X-31 did not have. It required someone to have written down the relation.

231 records. [Abdul Huq and Beebi M 2015] [Abdul‐Jaleel and Shaker 2023] [Ahmadi Dastgerdi et al 2022] [Air Force Safety Agency Kirtland Afb Nm 2000] [Akram et al 2020] [Antonakis and Biannic 2024] [Arriola and Thielecke 2015] [Bai et al 2018] [Bakhtiaridoust et al 2022] [Baldini et al 2019] [Baruh 1986] [Belcastro et al 2017] [Ben-Haim 1990] [Benchaita and Ladaci 2021] [Berendsen et al 1994] [Berger et al 2025] [Binwen Lu et al 2016] [Bonnice et al 1986] [Brockhaus 1985] [Budihartono and Moncayo 2024] [Caglayan et al 1988] [Caglayan and Godiwala 1985] [Cai et al 2024] [Cao and Yao 2023] [Cao et al 2022] [Cartocci et al 2022] [Chakravarty and Mahanta 2015] [Chang et al 2017] [Chang et al 2022] [Chao et al 2017] [Chen et al 2017] [Chen et al 2017] [Chen et al 2018] [Chen et al 2020] [Chen et al 2026] [Cheng et al 2019] [Chow and Willsky 1984] [Augsberger] [Craig and Russell 1978] [Dagal 2026] [Dally and Van Kampen 2022] [Dasgupta et al 2004] [De Hoff and Hall 1978] [De Ponti et al 2026] [Deckert et al 1978] [Delaat and Merrill 1990] [Deshpande and Zhang 2021] [Design of redundancy relations for failure detection and isolation by constrained optimization 1995] [Jarvinen 2013] [Dhadekar et al 2021] [Di Rito and Schettini 2016] [Ding and Li 2021] [Dołęga et al 2017] [Du and Cocquempot 2017] [Dunham and Knight 1982] [Eberlein and Lahm 1976] [Ergöçmen and Tilki 2025] [Ermeydan and Kiyak 2017] [Fault Tolerant Control for Aircraft Flaperon Faults Using Nonlinear Dynamic Inversion 2025] [Friedland 1982] [Fu et al 2021] [Gai and Curry 1977] [Gao and Liu 2021] [Gavra and van Kampen 2024] [Gavra and Van Kampen 2024] [Generating analytical redundancy for fault detection systems using flowgraphs 1994] [Gertler 1991] [Gou et al 2023] [Grehan et al 2023] [Gu and Zhang 2018] [Guo and Lu 2021] [Guo et al 2021] [Hajiyev 2016] [Hajiyev 2016] [Hamza and Samir 2020] [Han and Pei 2026] [Han et al 2025] [Hannan et al 2016] [Hannan et al 2017] [He et al 2020] [Hewitt 2019] [Hu et al 2023] [Hu et al 2023] [Huang et al 2022] [Huang et al 2026] [Ignatyev et al 2020] [Inflight and Preflight Detection of Pitot Tube Anomalies 2014] [Inozemtsev et al 2022] [Ji et al 2019] [Ji et al 2025] [Jiang et al 2018] [Kang et al 2020] [Kim 1995] [Kim 2026] [Kinnaert 1993] [Kinnaert 1996] [Kiyak and Ermeydan 2018] [Kosacki and Tomczyk 2022] [Kowalczuk and Gertler 1992] [Lee et al 2024] [Li and Shi 2021] [Li and Tong 2016] [Li et al 2018] [Li et al 2018] [Li et al 2023] [Li et al 2023] [Li et al 2024] [Li et al 2026] [Liang et al 2021] [Liu et al 2015] [Liu et al 2018] [Liu et al 2019] [Liu et al 2019] [Loh 1986] [Lou et al 1985] [Lou et al 1985] [Lou et al 1986] [Lu et al 2016] [Lu et al 2016] [Lu et al 2017] [Lu et al 2019] [Lunze 2017] [Lunze 2017] [Lv 2026] [Ma and Peng 2021] [Mallavalli and Fekih 2018] [Mallavalli and Fekih 2019] [Malluhi et al 2022] [Mashud and Bera 2022] [Massoumnia et al 1986] [Matsuki et al 2018] [McCool 1999] [Merrill 1985] [Merrill 1984] [Merrill 1988] [Merrill 1990] [Merrill et al 1987] [Merrill et al 1991] [Mohammadi and Ramezani 2021] [Montgomery and Price 1974] [Navi et al 2018] [Nguyen et al 2018] [Nguyen et al 2019] [Nikoukhah 1994] [Nishiyama et al 2016] [Ortiz-Torres et al 2018] [Ossmann 2015] [Ossmann and Joos 2016] [Ouyang and Lin 2017] [Ouyang and Lin 2017] [Ouyang et al 2022] [Panten et al 2022] [Patan and Ustoglu 2025] [Patton 1991] [Pei et al 2025] [Peng et al 2026] [Ping et al 2019] [Pollet et al 2025] [Pose et al 2017] [Pourbabaee et al 2016] [Prock 1989] [Qin et al 2016] [Review of Fault-tolerant Control for Flight Control System 2024] [Rotondo et al 2015] [Sato 2021] [Schüller and Santucci 1979] [Shahnazari and Mhaskar 2017] [Sharma et al 2025] [Siddiqi and Abraham 1988] [Das6 et al 2026] [Stoustrup and Niemann 2004] [Sun et al 2015] [Tang et al 2024] [Tang et al 2025] [Teng et al 2021] [Tokunaga et al 2015] [Unal 2021] [Unal 2021] [Vanschalkwyk 1991] [Venkataraman and Seiler 2019] [Vepa and Caliskan 1991] [Vile et al 2019] [Vile et al 2019] [Vile et al 2020] [Voting software for fault-tolerant aircraft flight control systems 1993] [Waitman et al 2019] [Wang and Yao 2021] [Wang and Yao 2024] [Wang et al 2017] [Wang et al 2018] [Wang et al 2021] [Wang et al 2021] [Wei et al 2017] [Wells 1977] [Wernz and Kroschel 1991] [Wickens 1979] [Wickens and Kessel 1979] [Wu and Zhu 2024] [Wu et al 2024] [Xiao et al 2024] [Xiao et al 2025] [Xu et al 2017] [Yang et al 2015] [Yang et al 2026] [Youn et al 2021] [Younes et al 2019] [Yu and Kim 2024] [Yu and Wang 2024] [Yu et al 2016] [Yu et al 2017] [Yu et al 2019] [Zahed et al 2020] [Zarei et al 2026] [Zhan and Wang 2017] [Zhang 2017] [Zhang 2026] [Zhang and Meng 2019] [Zhang and Wang 2018] [Zhang et al 2018] [Zhao and Li 2019] [Zhao et al 2017] [Zhao et al 2024] [Zheng and Li 2019] [Zheng et al 2025] [Zhong and Félix 2015] [Zhong et al 2025] [Zhongke 1989] [Zhou et al 2016] [Zhou et al 2016] [Zhu et al 2021] [Zhu et al 2023]

Flight test, parameter identification and aerodynamic modelling

How the aircraft gave up its derivatives. Flight test technique, system and parameter identification, and the aerodynamic models that flight data correct.

The X-31 corrected its simulator from flight through parameter identification, and the same loop now runs with far richer model structures. The modern literature treats the identification of nonlinear, unsteady and hysteretic aerodynamics at high angle of attack as its own subject.

480 records. [Pablos Palomino 2024] [Abdulrahim 2026] [Abou-Kebeh et al 2025] [Advisory Group for Aerospace Research and Development 1984] [Advisory Group for Aerospace Research and Development 1999] [Agha et al 2017] [Air Force Flight Test Center Edwards Afb Ca 1970] [Air Force Flight Test Center Edwards Afb Ca 1974] [Air Force Flight Test Center Edwards Afb Ca 1974] [Air Force Flight Test Center Edwards Afb Ca 2002] [Air Force Test Pilot School Edwards Afb Ca 1979] [Air Force Test Pilot School Edwards Afb Ca 1990] [Air Force Test Pilot School Edwards Afb Ca 1990] [Air Force Test Pilot School Edwards Afb Ca 1993] [Aircraft and Flight Test Instrumentation 2021] [Aksteter et al 1994] [Aksteter et al 1995] [Alag and Duke 1985] [Alford and Knarr 1999] [Amini et al 2020] [Anderson and Vincent 1985] [Araujo-Estrada et al 2015] [Armstrong 1977] [Arévalo Nogales 2020] [Aydın et al 2025] [Bach and Wingrove 1985] [Bachman 1981] [Bagherzadeh 2018] [Bai 2025] [Balakrishna and Niranjana 1987] [Banu et al 2017] [Barrett et al 1981] [Basappa et al 2006] [Batterson and Klein 1984] [Bauer et al 1989] [Bean and Lee 1994] [Beck et al 2020] [Bekar et al 2025] [Benders and Koch 2019] [Bennett and Abel 1982] [Benyamen and Keshmiri 2022] [Benyamen et al 2024] [Berger et al 2023] [Berra et al 2017] [Bertelrud et al 1992] [Binghua et al 2021] [Bishop 1958] [Blair and Weisshaar 1981] [Bleimeyer 1981] [Bobbitt et al 2012] [Bogos 2020] [Bolt 1981] [Bortz et al 2026] [Breach 1970] [Brenner 2001] [Bronz and Hattenberger 2016] [Brown and Bradley 1981] [Brunner 1996] [Bucharles et al 1990] [Budd et al 1995] [Burks 1981] [Burns 1974] [Burtch 2015] [Caglayan et al 1986] [Callantine and Ashford 2002] [Campos 1989] [Carico 1998] [Cassanto 1966] [Cenkci 1991] [Cetin and Kutay 2016] [Chaderjian and Schiff 1996] [Chassiakos et al 1988] [Che et al 2021] [Chen et al 2023] [Cheney 1988] [Cho et al 1995] [Chowdhury et al 2021] [Christmann et al 2026] [Christoffel et al 2020] [Clark and Walters 1986] [Collins 2016] [Comer et al 2026] [Cooke 1980] [Cooper et al 2016] [Corsiglia et al 1984] [Cosentino 2008] [Crites et al 1992] [Cunis et al 2018] [Cunningham and Hubbard 2018] [Cunningham and Bushlow 1990] [Danowsky et al 2017] [Darhi et al 2025] [Davino and Flattery 1985] [de Padua et al 2026] [Deangelis 1981] [Deepa and Gupta 2023] [Deets and Brown 1986] [Dehghan Manshadi and Saghafi 2021] [Deiler 2016] [Deiler 2023] [Deiler 2023] [Dekiert et al 2024] [DeNicola et al 2022] [Design validation and verification for non-intrusive flight test instrumentation (NIFTI) using Model-Based Systems Engineering (MBSE) 2026] [Devalla et al 2018] [Dias 2022] [Dias 2023] [Dias and Girardi 2016] [Dooley and Yeary 1979] [Dowden and Bessette 1987] [Duke and Jones 1984] [Dungan and Clark 1994] [Dunlap and Porter 1971] [Dutra 2019] [Dutra 2025] [Enciu 2019] [End-To-End Flight Test Data Acquisition 2020] [Enzinger 1988] [Erdman 2005] [Ertugrul and Koyuncu 2023] [Fechter and Mills 1988] [Feng et al 2023] [Ferreres and Hardier 2017] [Fischenberg 1995] [Fletcher 1994] [Flight Envelope 2021] [Flight Test Data Analysis of Hybrid Vertical Take-off and Landing Unmanned Aerial Vehicle 2016] [Foch 1992] [Fosdick 1970] [Stoliker 2005] [French and Eastep 1996] [Fujizawa et al 2016] [Förder and Steiner 2020] [Gamon 1988] [Garcia-Velo and Walker 1995] [Gates et al 1996] [General Dynamics/Astronautics San Diego Ca 1961] [General Dynamics/Astronautics San Diego Ca 1962] [General Dynamics/Astronautics San Diegoca 1961] [General Electric Co Cincinnati Oh 1964] [Gentry 1972] [Gerhard and Paul 2016] [Ghosh and Raisinghani 1993] [Ghosh et al 1998] [Gilbert 2014] [Gite and Deodhar 2022] [Glaudel 2018] [Glezer 2009] [Glezer 2012] [Grauer and Boucher 2017] [Grauer and Boucher 2018] [Grauer and Boucher 2020] [Grauer and Boucher 2020] [Griffith and Faruque 2023] [Hale et al 2015] [Hale et al 2017] [Hamel and Jategaonkar 1996] [Hamel et al 2022] [Hao and Wang 2023] [Harper et al 2025] [Harper et al 2026] [Harris et al 2016] [Harrison 1966] [Hart 1992] [Hayes 2010] [He 2015] [Helicopter Flight Parameter Identification Studies at the University of York 1987] [Hess 1986] [Hess 1993] [Hess and Ly 1989] [Hicks and Moulton 1988] [Hicks and Petersen 1988] [Hicks and Petersen 1989] [Hitzel and Zimper 2018] [Hoff and Cook 1996] [Hofmann et al 2025] [Hofmann et al 2025] [Hofmann et al 2025] [Hosseini et al 2023] [Hosseini et al 2025] [Howard et al 1991] [Hua 2022] [Hulshoff et al 1996] [Iliff 1989] [Iliff and Maine 1983] [Ivler and Sikora 2021] [Ivler et al 2019] [Jacobson 1952] [Jain et al 2025] [Jategaonkar and Plaetschke 1989] [Jategaonkar et al 1997] [Jelicic et al 2017] [Jeong et al 2015] [Jianhong and Lei 2026] [Jones 1995] [Jordan 1974] [Kehoe 1988] [Kehoe and Freudinger 1994] [Keidel et al 2019] [Kelly 1988] [Kimathi and Lantos 2024] [Klein 1989] [Klein 1989] [Klein 1990] [Klein and Batterson 1986] [Klein and Morelli 1992] [Klein and Murphy 1998] [Klein and Noderer 1995] [Klein and Noderer 1996] [Klyde et al 2015] [Koppitz et al 2019] [Korsun and Moung Htang 2023] [Korsun et al 2017] [Korsun et al 2024] [Koschel et al 2021] [Kraft 1994] [Kulhánek 2019] [Kumar and Ghosh 2017] [Kumar and Ghosh 2018] [Kumar and Ghosh 2019] [Kumar and Ghosh 2023] [Kumar and Ghosh 2023] [Kumar and Ghosh 2024] [Kuppuswamy and Kiran 1981] [Laban and Masui 1993] [Larson et al 2009] [Larsson and Enqvist 2016] [Le Clainche et al 2019] [Lee and Han 2015] [Lerro et al 2020] [Leung et al 2019] [Level Flight Performance Flight Test and Data Reduction Methods for Propeller-Driven Aircraft 2003] [Li et al 2022] [Liang et al 2025] [Lichota 2023] [Lichota 2023] [Lichota et al 2016] [Licitra et al 2018] [Lind and Brenner 1998] [Lind et al 1997] [Liu et al 2016] [Liu et al 2023] [Liu et al 2025] [Longobardi et al 2021] [Lopez et al 2021] [Lorincz 1978] [Lorincz and Friend 1979] [Lu et al 2017] [Lu et al 2019] [Luna et al 2026] [Luo et al 2024] [Lynch 1972] [Machnik et al 2022] [Mannai 1962] [Manshadi and Saghafi 2018] [Marchetti 1968] [Martin 1978] [Martin Co Denver Co 1966] [Martin 1999] [Martínez Morán 2018] [Matheny and Panageas 1981] [Matt et al 2025] [Mayer and Chalfant 2023] [McAteer et al 2018] [McCrink and Gregory 2017] [McKenzie 1973] [McMillen et al 1995] [McNally and Bach 1988] [Menges 1995] [Mennebeuf et al 2024] [Mereau 1982] [Metodiev 2024] [Metz 1992] [Meyers et al 1968] [Michalowski and Varano 2017] [Miller 1973] [Minshall et al 2017] [Mitchell 1961] [Moes et al 2003] [Mohammadi Farhadi et al 2018] [Monstein et al 2019] [Moorhouse 1990] [Morelli 1995] [Morelli 2021] [Morelli 2021] [Morelli 2022] [Morelli and Cooper 2015] [Morelli and Grauer 2018] [Morelli and Klein 1995] [Morelli 2000] [Morgenfeld 1992] [Morton and McDaniel 2015] [Moszczynski and Grant 2017] [Moszczynski et al 2026] [Moszczynski et al 2026] [Mouton 2023] [Myers et al 2015] [Naval Air Test Center Patuxent River Md 1963] [Navaratna et al 2019] [Navaratna et al 2022] [Neves et al 2020] [Newsom and Pototzky 1982] [Nguyen et al 2019] [Nguyen et al 2021] [Nguyen et al 2025] [Nie 2023] [Niewöhner 2018] [Norlin 1995] [Norton 1990] [Obermark 2004] [Ogren et al 1974] [Olson 2018] [Padua and Preisighe Viana 2025] [Palomino 2022] [Papa and Stoliker 1988] [Parameter estimation techniques and 1974] [Park 1977] [Paula and Cardoso 2025] [Pawlak 1994] [Peng 2021] [Perry et al 2019] [Petersen 1981] [Peterson 2023] [Peterson et al 2025] [Pfifer and Danowsky 2016] [Pineiro and Biezad 1993] [Pinkelman et al 1996] [Plaetschke et al 1982] [Porter 1965] [Potter 1985] [Preisighe Viana 2016] [Pérez Rus and Le Gal 2024] [Quinn 2022] [Raab 2026] [Raiford and Dall 1993] [Raisinghani and Adak 1983] [Raol and Jategaonkar 1995] [Redeker and Voersmann 1988] [Rediess and Melton 1994] [Kulikova N.V. CHulin N.A. 2023] [Research and Analysis on Development and Flight Test of Military Aircraft Flight Simulator 2022] [Ricciardi and Minwalla 2016] [Richardson 1976] [Ritter et al 2020] [Roberts 1988] [Roberts 1988] [Roberts and Brown 1988] [Roberts et al 1966] [Rohlmann and Keye 2015] [Roseberry 2025] [Ross 1979] [Rucci et al 2023] [Saderla et al 2017] [Sanderson 1965] [Sandoz and Klaeyle 2021] [Sandoz et al 2024] [Scherz and Tucker 1985] [Schierbrock and Takahashi 2021] [Schkolnik et al 1995] [Schuck 1992] [Schumacher 1952] [Schwartz and Wray 1969] [Schweiger et al 1996] [Seaver et al 2023] [Seo et al 2019] [Shafer 1980] [Shafer et al 1983] [Shafer et al 1984] [Sharpe and Hansman 2024] [Shen and Chen 2025] [Sheporaitis 1972] [Shibata and Ohmiya 1988] [Shlien 1991] [Shou and Li 2026] [Shufeldt and Layton 1972] [Shukla et al 2020] [Siddhardha 2019] [Simmons 2022] [Simmons 2023] [Simmons and Ackerman 2026] [Simmons et al 2023] [Simmons et al 2023] [Simmons et al 2023] [Simmons et al 2025] [Simpson 1972] [Simpson 1979] [Simulation in support of flight test - In retrospect 1988] [Domingos et al 2024] [Singh and Ghosh 2023] [Singh and Jategaonkar 1996] [Sitz et al 1997] [Smith and Pellicano 1992] [Soumelidis et al 2015] [Sousa et al 2017] [Spiegel et al 2016] [Spitzer et al 1972] [Spravka and Jorris 2015] [Srinathkumar et al 1993] [Srinathkumar et al 1995] [Srivastava 2019] [Staley 1976] [Stange 1959] [Stevenson 1991] [Stoev et al 2019] [Stouder 1981] [Suarez and Malcolm 1995] [Sudha et al 2018] [Sun et al 2018] [Sun et al 2019] [Sung et al 2020] [Szymanski et al 2025] [Tai et al 2023] [Talty and Caughlin 1988] [Taniguchi and Hagrman 1991] [Tardif and Myrand-Lapierre 2025] [Taylor et al 1992] [Thakur et al 2019] [Throckmorton 1992] [Tian et al 2022] [Tillema et al 2021] [Torno et al 2023] [Tracy 1981] [Traven and Whitley 1995] [Tsoubanos 1979] [Tsushima et al 2021] [Tsushima et al 2023] [Van Gaasbeek 1980] [Van Pelt 1981] [Van Wyckhouse 1966] [Vaughn and Lindsay 1988] [Verhaegen 1988] [Wada and Tamayama 2019] [Wade et al 2026] [Wang 2022] [Wang et al 1986] [Wang et al 2016] [Wang et al 2018] [Wang et al 2021] [Wang et al 2021] [Wang et al 2022] [Wang et al 2022] [Ward 1988] [Ward and Miley 1983] [Wardana et al 2021] [Wartmann 2015] [Wei 2021] [Wei et al 2019] [Weinstein et al 2018] [Weiss et al 1986] [Wells et al 1982] [Whitfield and Blackburn 1965] [Wiggen 2015] [Williams 1965] [Williams and Hartford 1989] [Wingrove 1973] [Woolf 2012] [Woolf 2012] [Xiao et al 2026] [Xing et al 2021] [Xu and Keshmiri 2021] [Yamato et al 1988] [Yamato et al 1991] [Yamauchi et al 1988] [Yanagihara et al 1999] [Yang and Huang 2018] [Yasue 2020] [Yeager 1997] [Youssef and Juang 1993] [Yue et al 2023] [Zajdel et al 2022] [Zajdel et al 2024] [Zajdel et al 2024] [Zhang 2021] [Zhang and Yu 2021] [Zhang et al 2023] [Zhixin 2022] [Zhong et al 2020] [Zhou et al 2019] [Zhubin and Ruikun 2023] [Zikmund et al 2019]

Structures, materials and the airframe

The airframe. Aeroelasticity and buffet, the coupling of the structure with the flight control system, and high-temperature materials.

The paddles were carbon-carbon because Inconel would have required nose ballast, and the material was chosen knowing it was brittle. Carbon-carbon, ceramic matrix composites, high-temperature structure, exhaust-washed structure and thermal protection are named in 25 of the 88 titles. Aeroelasticity, flutter, divergence, buffet, gusts and the flexible aircraft are named in 43, and most of the rest join the structure to its control system or to its loads.

88 records. [Ai et al 2022] [Ajayi et al 2019] [Bartels and Stanford 2017] [Becker and Luber 1996] [Blair and Weisshaar 1982] [Breitsamter and Laschka 1994] [Burken et al 2011] [Danowsky et al 2016] [Davis et al 2015] [Demir et al 2023] [Distelfeld and Osovski 2025] [Eastep et al 1984] [Feng et al 2015] [Freymann 1986] [Gupta et al 2020] [Gupta et al 2020] [Haas and Chopra 1989] [Hauch et al 1995] [Hollowell and Dugundji 1984] [Hou et al 2025] [Huang et al 2026] [Hussain et al 2026] [Jankovic 1983] [Juhasz et al 2015] [Kadhem and Hussein 2021] [Kandil et al 1996] [Kolesnikov 2018] [Kong et al 2016] [Krishna and Ricklick 2017] [Layton and Peterson 1994] [Lazarus et al 1997] [Lebofsky et al 2015] [Lee and Kim 1995] [Lee et al 1994] [Li et al 2022] [Li et al 2026] [Librescu and Simovich 1988] [Librescu and Thangjitham 1991] [Liguore and Drouin 1998] [Liu and Gao 2020] [Liu et al 2015] [Lottati 1985] [Mahgoub and El-Badawy 2022] [Mallik et al 2015] [Marley and Driscoll 2022] [Masrour et al 2023] [Masud et al 2017] [Mi et al 2024] [Moses and Shah 1998] [Norwood et al 2021] [Okita et al 2020] [Patel and Zha 2021] [Pohly et al 2021] [Qiao et al 2023] [Qin 2021] [Riso et al 2018] [Roach and Zhang 2021] [Rodden 1989] [Roy and Al Amin 2026] [Shi et al 2020] [Simpson et al 2015] [Song and Librescu 1991] [Static Aeroelastic Considerations 1996] [Striz et al 1991] [Tartaruga et al 2015] [Tatum and Giles 1988] [Tsiakas et al 2021] [Tu et al 2026] [Tuzcu and Nguyen 2025] [Verveld and Kier 2017] [Von Flotow 1989] [Voracek and Clarke 1991] [Voracek and Clarke 1994] [Wang and Han 2025] [Wang et al 2016] [Wang et al 2023] [Watanabe et al 2017] [Weisshaar 1980] [Weisshaar 1990] [Weisshaar 1994] [Wu and Livne 2016] [Xiang and Wang 2026] [Yildiz and Acar 2026] [Zhang et al 2015] [Zhengping et al 2021] [Zhou et al 2018] [Zhou et al 2025] [Zubko et al 2022]

Weapons, displays and the off-boresight problem

The other half of the off-boresight argument. Helmet-mounted and head-up displays, cueing systems, missiles and weapon system effectiveness.

The programme’s own recommendation was that post-stall technology needs a missile that can be launched off boresight at high angle of attack, and that combination arrived shortly afterwards and changed close combat more than the airframe technology did. An aircraft that can point its seeker without pointing itself does not need to point itself.

That is the strongest single argument against the X-31’s own premise, and the programme identified it.

Helmet-mounted displays and sights are named in 46 of these titles.

76 records. [Air Force Systems Command Andrews Afb Md 1965] [Air Force Systems Command Andrews Afb Md 1965] [Air Proving Ground Center Eglin Afb Fl 1949] [Air Proving Ground Center Eglin Afb Fl 1954] [Bailey 1986] [Bailey 1989] [Beskar 2004] [Blackwell et al 1997] [Butterfield 1990] [Chapman and Clarkson 1992] [Cheung et al 2015] [Cheung et al 2015] [Chhavi and Ramanan 2023] [Comparison of helmet-mounted visual displays for flight simulation 1987] [Coppock and Gerke 1977] [Dagomes et al 2020] [Denton et al 2025] [Dorr et al 1994] [Ehrlich and Singer 1996] [Ercoline and Gillingham 1990] [Fedaravičius et al 2022] [Fischler et al 2017] [Geiselman 1999] [Geiselman 1999] [Geiselman et al 2000] [Gilligan 1992] [Grunwald and Kohn 1994] [Guanglei et al 2016] [Hardyman and Smith 1989] [Haworth and Seery 1992] [Haworth et al 1988] [Haworth et al 1989] [Head up display for aircraft 1998] [Herd et al 1965] [Jenkins 2008] [Jenkins and Havig 2015] [Jenkins et al 2001] [Jenkins et al 2002] [Jenkins et al 2002] [Jones 1973] [Jones 1983] [Katukuri 2023] [Kotulak and Morse 1995] [Kruk 1992] [Kumar and Raghuwanshi 2020] [Lambert et al 1985] [Lim et al 2017] [Lueken et al 2016] [Mekonnen et al 2021] [Merryman 1994] [Modelling performance of uncrewed autonomous system swarms against directed energy weapon systems 2026] [Moon and Kim 2000] [Mulholland 2002] [Nelson and Cox 1992] [Osder 1992] [Parise 1989] [Rash et al 1998] [Rebman 1967] [Robinson and Green 2016] [Rostad et al 2003] [Sharkey et al 2000] [Sheesley et al 2003] [St. Pierre et al 2015] [Subramanyam 1997] [Szelmanowski et al 2017] [The Problem of Training the Aircrew of Army Aviation to Perform Flights with Helmet-mounted Display Systems 2024] [Watler and Mulley 1978] [Wells et al 1988] [Whillock et al 1994] [Winterbottom et al 2005] [Woodruff et al 1986] [Xiong et al 2016] [Zeqing et al 2017] [Zhan et al 2015] [Zhang et al 2018] [Çalışkan and Sakarya 2024]

Missile guidance and interception

The weapon that made pointing less necessary. Missile guidance laws, interception of manoeuvring targets and the guidance of aircraft and projectiles.

95 records. [Aiello and Bateman 1979] [August 1982] [Banerjee et al 2019] [Chadwick 1995] [Cliff 1989] [Cloutier and Stockbridge 2015] [Cong-cong et al 2017] [Corban et al 2001] [Cui et al 2023] [Deepak 2004] [Dong et al 2023] [Fan et al 2019] [Fresconi et al 2014] [Galffy et al 2019] [Gao et al 2016] [Gao et al 2025] [Gaudet and Furfaro 2023] [Giragosian 1992] [Grinfeld and Ben-Asher 2015] [Gros and Fichter 2016] [Głębocki and Jacewicz 2018] [Han et al 2025] [Harini Venkatesan and Sinha 2015] [Hemsch and Nielsen 1983] [Hu et al 2017] [Imado and Miwa 1986] [Jayaraman et al 2020] [Jiang 2018] [Jinglun et al 2021] [Jonsson 1986] [Kim and Kim 2015] [Krysenko 1963] [Kumar et al 1995] [Lee and Kim 2016] [Lee et al 2023] [Lee et al 2025] [Liang et al 2017] [Lin and Xin 2019] [Lisha et al 2015] [Liu and Qu 2017] [Luo et al 2023] [Lyu et al 2018] [Marzat 2017] [Mei and Wang 2021] [Meng et al 2022] [Meng et al 2023] [Mondal and Padhi 2018] [Nakagawa et al 2019] [Nam 2025] [Niu et al 2026] [Ochi 2003] [Grimm et al] [Paine 1950] [Peng et al 2016] [Qu et al 2015] [Research on the Integrated Design of Missile Guidance Control Considering the Angle of Attack Constraint 2023] [Richards et al 2016] [Ross 1997] [Schnepf et al 2017] [Shalumov 2019] [Shalumov and Shima 2017] [Shi et al 2021] [Shou et al 2023] [Slater 1987] [Snyder et al 2015] [Tachinina and Kovalenko 2026] [Tan et al 2019] [Tanriverdi and Cavdaroglu 2017] [Tao et al 2024] [Tekinalp and Ariyibi 2015] [Tekinalp and Kumbasar 2015] [Tian et al 2023] [Wang et al 2018] [Wang et al 2024] [Wang et al 2024] [Wang et al 2026] [Weber and Jamsek 1994] [Wei et al 2016] [Weiss and Shima 2021] [Wilcox 1963] [Xi et al 2018] [Xiaoxi et al 2019] [Xu et al 2017] [Xu et al 2022] [Yang et al 2017] [Ye and Liu 2016] [Yin et al 2025] [Yomchinda 2015] [Zhan et al 2024] [Zhang et al 2015] [Zhang et al 2022] [Zhengchun 2018] [Zhou and Wang 2017] [Zhou et al 2016] [Zhou Tao et al 2007]

Missile autopilots and missile flight control

The missile as a flight vehicle. Missile autopilots, missile flight control and the aerodynamics that a missile autopilot works against.

20 records. [Aerodynamic Characteristics of Oval Pattern Grid Fins in Subsonic Regime 2025] [Dong and Zhou 2017] [Fowler and Rogers 2016] [Hemsch and Nielsen 1983] [Huannian et al 2018] [Jun-fang et al 2015] [Kovalik and Rogers 2025] [Lou et al 2017] [Lukianenko 2026] [Michal and Mitcham 1949] [Nagabhushan 1986] [Oh and Kimmel 1990] [Shoyama et al 2026] [Special Features of Antiaircraft Rockets with Swept or Low-Aspect-Ratio Wings 2010] [Vinh et al 1972] [Yafei 2017] [Yang et al 2025] [Zaikang and Defu 2019] [Zhang et al 2019] [Ţapu and Drăgulinescu 2025]

Rotorcraft flight dynamics and control

The helicopter beside the fighter. Rotorcraft flight dynamics, rotorcraft flight control and their simulation.

56 records. [Abhishek and Prasad 2016] [Ai et al 2025] [Bachmaier and Anderson 2021] [Bath et al 2022] [Bird 1970] [Blackburn and Whitfield 1965] [Brown 2020] [Cao 1999] [Chelihi et al 2025] [Chen and Hindson 1986] [Dabundo and Neary 1985] [Deng and Xu 2022] [Dhiman et al 2022] [Dixon 1980] [Enciu and Horn 2017] [Enns 1987] [Fattizzo et al 2026] [Gao et al 2022] [Garrard and Low 1990] [Guan and Xian 2025] [Guner et al 2016] [Hegde 2020] [Helicopter Yaw Control 1990] [Hess and Gao 1993] [Hiller Aircraft Corp Palo Alto Ca 1965] [Horn et al 2024] [Hutto 1976] [Kelley et al 1992] [Li et al 2023] [Liang et al 2025] [Lim and Jung 2025] [Lu et al 2015] [Lu et al 2019] [McKillip 1987] [McKillip 1991] [Monico et al 2025] [Nguyen 1998] [Nunes et al 2025] [Ostheimer and Giguere 1963] [Qin et al 2023] [Quackenbush et al 2020] [Reader 1980] [Sakthivel and Venkatesan 2017] [Sarathy and Murthy 1993] [Schroeder et al 1995] [Sheng 2026] [Sugawara et al 2021] [Tai 1996] [Takahashi et al 2022] [Tang et al 2019] [Tritschler et al 2025] [Ueminami and Fujiwara 2020] [Wang et al 2020] [Wood and Livingston 1971] [Yokoyama and Fujiwara 2021] [Zhao et al 2024]

Multirotor flight control

The smallest flight vehicles with the same control problems. Flight control of quadrotors and other multirotors, including robust, saturated and fault-tolerant designs.

38 records. [Alqudsi 2024] [Bouadi and Mora-Camino 2018] [Calogero et al 2023] [Casau et al 2020] [Chen and Zhao 2017] [Cunningham and Hubbard 2020] [Deng and Chai 2024] [Faessler et al 2017] [Gu et al 2022] [Gupta et al 2016] [Hai-Rui and Bo 2017] [He et al 2020] [Hopwood et al 2024] [Hossny et al 2020] [Jeong et al 2024] [Kim et al 2020] [Kuantama et al 2018] [Lee and Kim 2024] [Li et al 2024] [Lim et al 2016] [Lim et al 2022] [Mazeh and Sahili 2019] [Mehmood et al 2016] [Oliinyk and Zylevich 2026] [Paiva et al 2019] [Peng et al 2018] [Prakosa et al 2022] [Rasool et al 2025] [Saetti and Guner 2024] [Stauder et al 2026] [Tran and Nguyen 2018] [Wang and Shirinzadeh 2015] [Wang et al 2017] [Wu et al 2021] [Yan et al 2023] [Yan et al 2024] [Yang et al 2024] [Zhu et al 2017]

The vehicle’s own literature

The programme’s own record. Every research record whose title names the X-31.

This cluster is the one place where the contemporary survey has nothing to survey. None of its 36 records was published from 2015 onward, and the latest is from 2012, which is what one should expect of an aircraft that stopped flying in 2003. Of them, 28 date from the 1985 to 1996 programme window. The modern work that touches the aircraft is computational, and it uses the X-31 as a validation case rather than studying the aeroplane.

36 records. [Alcorn et al 1995] [Alcorn et al 1996] [Banks et al 1992] [Boelens 2009] [Boelens 2012] [Cobleigh et al 1994] [Canter and Groves 1994] [Cobleigh 1994] [Cobleigh and Croom 2003] [Cobleigh and Croom 2001] [Cobleigh and Delfrate 1994] [Croom et al 1993] [Mackall et al 1992] [Williams et al 1994] [Dickes et al 1994] [Grohs et al 2003] [Groves et al 1993] [Hebbar et al 1991] [Hebbar et al 1993] [Hebbar et al 1993] [Hebbar et al 1995] [Hoerter et al 1991] [Friehmelt et al 1997] [Huber 1995] [Huber and Schellenger 1994] [Huber and Seamount 1994] [Kaletka and Fu 1993] [Knox 1992] [Kokolios 1993] [Stoliker and Bosworth 1996] [Stoliker and Bosworth 1997] [Tamrat 2004] [Whiting 1996] [Williams et al 1994] [X-31 Tailless Testing 1994] [Yeh et al 1991]

Other aerodynamic and flight mechanics literature

The aerodynamic and flight mechanics residual, reported and not hidden. Aerodynamics and flight mechanics of aircraft and flight vehicles that belong to no cluster above, among them manoeuvring flight, stability, performance, drag and wing aerodynamics.

The period part of this cluster is the aerodynamics the programme was designed against, and the modern part is what the same questions look like once computation replaced the tunnel as the primary tool. Of its 341 titles, 115 name a manoeuvre. No further subject in it is large enough to name, and dividing it further would describe the filing rather than the field.

341 records. [6 DoF Aircraft Simulation Model Capable of Handling Maneuver Events (WIP) 2016] [Morton 1956] [Newey] [AbuNawas and Qawasmeh 2026] [Aerodynamic Characteristics of Low-Aspect-Ratio Wings in Close Proximity to the Ground 1961] [Aerodynamic Performance of Swayasa Aircraft Wing Model against the Effect of Airfoil Flap Angle Configuration 2025] [Agrawal et al 1991] [Ahmed and Qin 2020] [Akaryildiz et al 2025] [Alam and Soeimanikutanaei 2021] [Alexander et al 1992] [Alexandrov et al 2025] [Almajali and Al-Mahadin 2026] [Ananthkrishnan and Sudhakar 1994] [Ananthkrishnan and Sudhakar 1995] [Anggraeni 2019] [Antipov 2020] [Appa and Ausman 1997] [Arranz and Lozano-Duran 2022] [Ashpis and Thurman 2011] [Avila De Melo and Hansman 1990] [Bai 2025] [Baker and Farrell 1991] [Bakker et al 2025] [Banerjee et al 2017] [Berry et al 2004] [Beyers 1988] [Bhardwaj and Holzapfel 2025] [Biancardi 1988] [Bismarck-Nasr 1994] [Bloy 1996] [Bloy and Lea 1995] [Bocvarov et al 1994] [Bosnyakov et al 2024] [Bradley et al 2026] [Brandt and Bons 2023] [Brandt et al 2022] [Breakwell 1978] [Breitenstein et al 2024] [Briese and Gäßler 2022] [Bromfield and Dillman 2015] [Bueno and Dowell 2020] [Buerge and Ram 2010] [Burkhalter et al 1992] [Buttrill et al 1987] [Byrns and Calise 1993] [Caap and Elemeland 1986] [Calise 1984] [Carrere et al 2023] [Cen et al 2018] [Dikbas and Atik 2018] [Changchuan et al 2018] [Chen et al 2023] [Chen et al 2025] [Chen et al 2025] [Cochran et al 1992] [Cochran et al 1990] [Cockrell et al 1986] [Collier 1993] [McCutchen 2006] [Kuhlman and Brown 1989] [Keener 1984] [de Melo and Hansman 1991] [Deiler and Fezans 2017] [del Carre and Palacios 2020] [Determination of Aerodynamic Characteristics 2001] [Dghim et al 2018] [Dghim et al 2018] [Dghim et al 2020] [Di Pasquale and Prince 2023] [Duan et al 2021] [Durham 1994] [Durham 1995] [Durston and Stonum 1987] [Dursun and Erturk 2022] [Eikenberry and Ingram 1969] [Ejeh et al 2019] [Ene and Stoica 2015] [Englar et al 1999] [Eriş et al 2026] [Yu and Bernal 2016] [Fan et al 1994] [Felix et al 2026] [Fernandes 2026] [Feroskhan and Go 2016] [Feroskhan and Go 2018] [Ferrand et al 2026] [Fink 1965] [Visbal 2012] [Fortunato et al 2021] [Franco and Rivas 2015] [Frazier 1991] [Frink et al 2019] [Gao et al 2024] [Garrard and Balas 1995] [Ghazi 1994] [Glezer et al 2010] [Gopinath 1982] [Gordnier 1995] [Graffstein and Maslowski 2019] [Graham et al 2018] [Greenwell 1998] [Grimm 1987] [Gu and Wang 2025] [Gu and Zhou 2020] [Gupta 2026] [Haas and Chopra 1988] [Haas and Chopra 1990] [Hao et al 2019] [Hashemi and Nguyen 2018] [Hassan and Sankar 1992] [Furey 1980] [Himeda and Naka 2019] [Hoffman and Stalford 1991] [Hofmann and Riedel 1979] [Hsiao et al 1994] [Hu et al 1993] [Hutt 1987] [Idir et al 2022] [Ikami et al 2021] [Imanov 2025] [Trice 1989] [Intravartolo et al 2023] [Ishida 2019] [Jansen and Vos 2016] [Jarrah 1995] [Joshi et al 2020] [Judith 2020] [Kadrnka and Hawley 1993] [Kaneshige and Krishnakumar 2014] [Karpuk and Mosca 2024] [Kasmi et al 2023] [Katz 1979] [Khalid 1992] [Khalil and Bauknecht 2024] [Khalil et al 2017] [Khalil et al 2022] [Khoshkhoo and Jahangirian 2016] [Kim and Kim 2017] [Kim and Marciniak 1998] [Kim et al 2019] [Krengel and Hepperle 2023] [Krishnamurthy and Luckner 2017] [Kuhn 1980] [Kwatny and Allen 2012] [Langan and Samuels 1995] [Lappe 1965] [Larsen 1966] [Lehmkuhl et al 2020] [Lei and Zha 2023] [Lei and Zha 2023] [Lesoinne 2007] [Li and Livne 1997] [Li and Sharma 2021] [Li et al 2018] [Li et al 2022] [Liao Kaijun et al 2015] [Sathaye et al 2004] [Dodbele and Plotkin 1985] [Ramm 1990] [Stanbrook 1954] [Bouton 1950] [Lin 2016] [Liu and Zha 2016] [Liu et al 2017] [Lovato and Troutt 1993] [Low aspect ratio wing code validation experiment 1992] [Iosilevskii 2025] [Lowry 1997] [Lu et al 2023] [Ma 2025] [Mancini and Vos 2019] [Mann et al 1983] [Mann and Mercer 1985] [Mann and Mercer 1986] [Manolescu 1994] [Marquet 2023] [Marzouk 2025] [Mayahara and Sato 2023] [McFadden et al 2023] [Mei and Huang 2018] [Mele et al 2016] [Mello and Kotansky 1985] [Menon 1993] [Menon et al 1991] [Meyer and Seginer 1992] [Miller and Koch 1985] [Modi et al 1998] [Moens and Dandois 2016] [Mohan and Gaitonde 2017] [Moktarian and Modi 1988] [Montgomery et al 1998] [Moore 1995] [Morgan and Garmann 2026] [Morgan 1981] [Mori et al 2015] [Morino and Obayashi 2015] [Morris and Kroo 1990] [Morton and McDaniel 2018] [Moyer 1993] [Mr. S. Aravindan et al 2017] [Muckler and Obermayer 1963] [Muir 1993] [Mulgund and Stengel 1993] [Muppidi et al 2018] [Muqeeth and Abdallah 2020] [Naftaly and Raveh 2025] [Nagabhushan and Pasha 1991] [Nagabhushan and Pasha 1992] [Nakagawa and Obata 1992] [Naumowicz and Smith 1992] [Neves and Mariani 2024] [Nguyen and Xiong 2021] [Nicolosi et al 2015] [Northrop Aircraft Inc Hawthorne Ca 1956] [Mansour 1984] [Numerical Study of Aerodynamic Performance of Airfoil with Variable Curvature Split Flap 2023] [O’Farrell et al 2016] [O’Rourke et al 2020] [Odriozola et al 2026] [Olwi 1995] [Gutierrez-Castillo et al 2021] [Ozger 2015] [Ozgun et al 2026] [Pamadi 1981] [Pandi and Mittal 2025] [Paolo 1977] [Papadales and Basil S. 1979] [Paul et al 2024] [Peng et al 2026] [Peterson 2025] [Phoenix et al 2019] [Pilkevych et al 2025] [Rizzetta and Visbal 2012] [Rizzetta and Visbal 2012] [Porterfield et al 2025] [Portis et al 2024] [Post and Corke 2004] [Prayitno et al 2026] [Radhakrishnan and Swarup 2020] [Raj et al 2018] [Raj Mohamed et al 2019] [Raj Mohamed et al 2021] [Rambacher and Bons 2023] [Rashid et al 2022] [Ratcliff et al 2016] [Rawson 1963] [Reisenthel 1996] [Reist et al 2019] [Riso et al 2020] [Ritter 1988] [Rizzetta and Visbal 2016] [Rodden 1989] [Rodden and Bellinger 1982] [Rogalski 2018] [Rogalski et al 2021] [Rossow 1990] [Sanghi et al 2022] [Sanghi et al 2024] [Sasamori et al 2022] [Sato 2018] [Schoettle and Hillesheimer 1991] [Schuster 1995] [Secchi and Mestriner 2026] [Sedky et al 2024] [Sen et al 2024] [Shalaev and Hieu 2020] [Shayak et al 2024] [Shen and Chen 2023] [Sheng and Zhao 2017] [Shi and Holzapfel 2019] [Shimmin et al 2015] [Shmilovich et al 2016] [Shmilovich et al 2026] [Singh 1988] [Singh 1989] [Singh 1989] [Singh 1989] [Singh and Dhawan 2015] [Singh and Scholz 2018] [Slater and Hu 1989] [Sodja et al 2021] [Solarte-Pineda et al 2026] [Speyer 1996] [Srivastava and Mook 1994] [Stages of functioning the gas-dynamic device takeoff and landing and aircraft’s 2012] [Stanford 2020] [Stanley 1980] [Shields and Mohseni 2012] [Stengel and Berry 1977] [Stepanov and Kusyumov 2022] [Sudhi et al 2023] [Sun 2015] [Suresh et al 2018] [Sánchez Muñoz et al 2024] [Sóbester 2021] [Tai 1993] [Tal and Karaman 2022] [Theunissen 1995] [Tomczyk 1998] [Kuhlman et al 1988] [Traub 1994] [Turner 1982] [Tuzcu and Nguyen 2015] [Unsteady surface pressure measurements on a pitching airfoil 1985] [Van Poppel et al 1998] [Vashishtha and Khurana 2021] [Villegas and Diez 2016] [Virgilio Pereira et al 2019] [Oswald 1932] [Wang and Zhang 2025] [Wang and Zhao 2021] [Wang et al 2021] [Wang et al 2025] [Wang et al 2026] [Wendorff et al 2016] [Winther et al 1995] [Woo 1992] [Woods et al 1990] [Woods-Vedeler et al 1995] [Xu and Zha 2022] [Xu et al 2018] [Xu et al 2023] [Yan et al 2016] [Yang and Zha 2018] [Yang and Zha 2018] [Yang et al 2025] [Yin 2023] [Yost et al 2019] [Yu and Yu 2026] [Zagainov 1993] [Zare Shahnehb 2022] [Zeiler 1998] [Zhang et al 2017] [Zhang et al 2018] [Zhang et al 2022] [Zhang et al 2025] [Zhou et al 2016] [Zhou et al 2025] [Zhou et al 2025] [Zweig and Ben-Asher 1995]

The general control literature

The control residual, reported and not hidden. Flight control of aircraft and flight vehicles, and the parts of control theory this article’s argument uses, belonging to no more specific cluster above.

This is the broadest cluster in the article. An aircraft that is unstable in one axis, augmented in all of them, and dependent on a measured flight condition for its gains is a control-theory object before it is anything else, and the literature that surrounds it is correspondingly large. Of its 410 titles, 309 name an aircraft, a flight vehicle or flight, and the rest are the parts of general control theory that this article uses, being the control of unstable plants, robustness, gain scheduling, saturation, time delay and nonlinear control.

410 records. [Abdusamad and Mousa 2024] [Adams et al 1994] [Aftab and Rossiter 2021] [Aguilar-Ibañez 2016] [Ajel et al 2021] [Albostan and Gökaşan 2018] [Aldosari and Feron 2025] [Alsayed and Lanteigne 2017] [Alston 1936] [Amalberti and Deblon 1992] [Amato et al 2022] [Ambroziak et al 2022] [An et al 2020] [Anandakumar et al 2022] [Anderson and Mason 1996] [Anderson and Schmidt 1991] [Angelov and Holzapfel 2022] [Ansari and Bernstein 2015] [Application of perfect model following to a control configured vehicle 1989] [Argyle and Beard 2016] [Arrosida and Echsony 2017] [Arya and Rao 2016] [Asadi et al 2022] [Ashraf and Van Kampen 2020] [Ashraf et al 2018] [Aslam et al 2020] [Azab 1985] [Azor 1995] [Bal et al 2024] [Balas et al 1998] [Banerjee 2019] [Baran and Bayezit 2024] [Begum et al 2020] [Beppu et al 1966] [Berg 1991] [Bernstein 2000] [Biggs and Livornese 2020] [Bist and Sondhi 2022] [Black and Blackwell 1990] [Blodgett and Lagor 2022] [Bogos et al 2018] [Boisseau 1955] [Boland et al 1993] [Bondarenko and Shkolnyi 2024] [Borello and Villero 1995] [Bosworth 2009] [Bošković and Jackson 2017] [Broussard et al 1982] [Brown and Montgomery 1984] [Bruner 1983] [Cai et al 2021] [Calico and Noll 1986] [Caverly and Forbes 2018] [Chand and Chiu 1991] [Chavez and Schmidt 1993] [Chavez and Schmidt 1994] [Chen et al 2017] [Chen et al 2024] [Chen et al 2026] [Cheng et al 1992] [Chipman et al 1983] [Chiu and Chand 1991] [Coletti and Faruque 2022] [Cook and Hauser 2018] [Corona-Sanchez and Rodriguez-Cortes 2015] [Craig 1965] [Cress et al 1989] [Cunningham et al 2011] [Czyba and Kronhof 2020] [Czyba and Stajer 2019] [Dai et al 2017] [Davidson et al 1999] [de Souza et al 2023] [Demarchi and Haning 1978] [Deters and Russell 1947] [Devarakonda et al 2015] [Douglas Aircraft Co Long Beach Ca 1963] [Douglas et al 1990] [Dukes 1970] [Dwivedi and Pandey 2021] [Guido 1989] [Ehlers and Weisshaar 1993] [Engelland 1991] [Engelland et al 1992] [Enhancing Limited Authority Adaptive Control: Peudocontrol Hedging for Addressing Input Saturation 2023] [Enns 2003] [Enns et al 1994] [Faleiro and Pratt 1996] [Fazelzadeh et al 2016] [Fellah et al 2018] [Ferranti and Keviczky 2017] [Feucht et al 1978] [Finch and Walker 1953] [Flight Sciences Lab Inc Buffalo Ny 1964] [Forkun et al 2020] [Franklin and Innis 1978] [Fravolini et al 2015] [Fravolini et al 2016] [Funk 1975] [Gainer 1963] [Gao et al 2020] [Garrard 1998] [Gavilan et al 2015] [Gezer and Kutay 2015] [Goldberger 1966] [Gong et al 2019] [Goodner and Rao 1988] [Govindaraj et al 1979] [Graham and McRuer 1991] [Griffin and Eastep 1982] [Grunwald and Kohn 1993] [Gu et al 2023] [Guruswamy and Tu 1989] [Guzman et al 2022] [Haiping et al 2021] [Halyo 1996] [Hart 1956] [Hedrick and Gopalswamy 1990] [Hernández Castellanos et al 2020] [Hess 1994] [Hess 1995] [Hess and Peng 2018] [Hill 1987] [Hofmann and Kezer 1962] [Hongyan and Xiaoyong 2026] [Hoogreef and Soikkeli 2022] [Horowitz et al 1980] [Horri and Pietraszko 2022] [Hsu 1990] [Hu et al 2021] [Hu et al 2025] [Hu et al 2026] [Huang et al 2020] [Innocenti 1989] [Ionita and Rasvan 1999] [Jackson 1970] [Jacques and Ridgely 1995] [Jafari et al 2017] [Jares and Valasek 2022] [Jenney 1979] [Jex and Culick 1985] [jian-jian et al 2019] [Jianhong and Yanxiang 2026] [Jiayuan et al 2018] [Jin and Wang 2015] [Jin et al 2026] [Johnson and Boisseau 1953] [Kai 2024] [Kalviste and Eller 1989] [Kaminer and Khargonekar 1991] [Karimtaevna et al 2022] [Karpel 1990] [Kassapakis and Warwick 1994] [Katz 1995] [Kayacan and Peschel 2016] [Kazarin et al 2017] [Khamvilai and Johnson 2025] [Kidd 1988] [Kikin 2018] [Kim and Kim 2024] [Kim et al 2018] [Kim et al 2025] [Kistyarev and Wang 2025] [Knuusi and Sage 1965] [Kong et al 2016] [Kou et al 2021] [Kumar et al 2019] [Kuvshinov et al 2019] [Köhler et al 2026] [Köthe et al 2016] [Lai et al 2021] [Lampl and Hornung 2018] [Landy and Kim 1984] [Lane and Stengel 1986] [Langham 1979] [Lateral Control of a Hypersonic Aircraft 2025] [Lavretsky 2019] [Lawes 1994] [Le 2026] [Lee and Hedrick 1994] [Li et al 2016] [Li et al 2018] [Li et al 2022] [Li et al 2023] [Li et al 2025] [Liang et al 2020] [Lim et al 2004] [Lin 1998] [Lin and Tai 1995] [Lin et al 2017] [Linehan 1996] [Liu et al 2015] [Liu et al 2018] [Liu et al 2020] [Liu et al 2020] [Liu et al 2024] [Livneh 1995] [Livneh and Schmidt 1992] [Lu et al 2017] [Lungu and Lungu 2015] [Lungu and Lungu 2016] [Luspay et al 2018] [Lymperopoulos and Ioannou 2019] [Lyu et al 2022] [Ma et al 2019] [Ma et al 2025] [Mailhot et al 2023] [Martynov et al 2017] [Martínez-Ramírez and Rodríguez-Cortés 2022] [Masaoka and Hashimoto 2024] [Melekhin and Khachumov 2021] [Mengali 1996] [Menon et al 1995] [Mercadante and Laurie 1989] [Merkel and Whitmoyer 1976] [Merlinge et al 2019] [Mikhail et al 2015] [Mirchandani 2020] [Mishra and Chakraborty 2024] [Mishra and Chakraborty 2024] [Mlayeh and Ben Othman 2022] [Mohamed and Swaroop S 2015] [Montgomery and Hunsaker 2022] [Morton et al 1990] [Munger et al 1983] [Muñoz and García-Fogeda 2024] [Müller and Moormann 2023] [Nagabhushan 1989] [Nagabhushan 1991] [Nagabhushan and Tan 1996] [Neves et al 2026] [Nguyen and Tran 2021] [Ni et al 2025] [Nield 1981] [No and Cochran 1995] [North American Aviation Inc Los Angeles Ca 1963] [Northrop Aircraft Inc Hawthorne Ca 1952] [Omar 2021] [Ostroff and Proffitt 1994] [Pachter et al 1994] [Pachter et al 1995] [Padilla et al 2026] [Palkin and Zenchenko 2025] [Panish and Bacic 2022] [Panteleev et al 2018] [Park et al 2019] [Pashchenko and Akhrameev 2021] [Patience and Nahon 2020] [Perrusquia et al 2016] [Planeaux 1990] [Platus 1980] [Platus 1981] [Pollack and van Kampen 2023] [Porter and Swaim 1975] [Prach et al 2016] [Putman and Curtiss 1980] [Putro and Holzapfel 2016] [Puviyarasi et al 2023] [Qi and Zhao 2020] [Qi et al 2017] [Qi et al 2018] [Qi et al 2018] [Rabenberg and Palarz 1983] [Rajeswari and Padma Suresh 2015] [Regulator with reference model for u nmanned aircraft control when boarding on unmanned vehicle 2022] [Ren et al 2026] [Research on aircraft control 1969] [Rigatos 2021] [Rimer et al 1986] [Robinson 1985] [Roden and Harrhy 1992] [Roenneke and Well 1995] [Rokhsaz and Selberg 1990] [Ronfle-Nadaud 2009] [Rosa et al 1974] [Rosario-Gabriel and Cortes 2018] [Roscoe et al 1975] [Ross et al 1973] [Rowley 2008] [Rowley 2010] [Rustenburg 1972] [Sachan and Singh 2025] [Sachs 1975] [Sachs et al 1996] [Safinaz et al 2023] [Safonov and Chen 1982] [Safwat et al 2020] [Sahu et al 2022] [Salahudden et al 2022] [Salahudden et al 2025] [Sandeepkumar and Mohan 2022] [Saugen and Adler 1991] [Schildkamp et al 2023] [Schmidt 1987] [Schmidt and Newman 1990] [Schmitendorf 1979] [Se Young Yoon and Zongli Lin 2013] [Selman 2023] [Servo-control device for aircraft flight control 1984] [Shankar and Yathisha 2023] [Sharma and Sakthivel 2025] [Shmilovich and Princen 2026] [Silvestre et al 2017] [Singh and Coelho 1984] [Singh and Verma 2022] [Singh et al 1995] [Singh et al 2024] [Sinha et al 2022] [Sir Elkhatem et al 2025] [Smith and Meyer 1981] [Smith et al 1971] [Snell and Hess 1997] [Sobel and Shapiro 1986] [Solies 1994] [Song and Huang 2022] [Sparks 1991] [Sparks and Banda 1993] [Stability and control of aircraft with manual all-moving tailplanes 1986] [Staufenbiel and Schlichting 1988] [Steinberg and Page 1998] [Stevens 1990] [Stolarik 2007] [Sun and Ma 2021] [Sun et al 2021] [Susanto et al 2021] [Swaim 1971] [Swarnkar and Kothari 2016] [Sánchez et al 2024] [Takarics and Vanek 2019] [Takarics and Vanek 2021] [Tal and Karaman 2021] [Tang and Kumar 2018] [Terrell and Zein-Sabatto 2017] [Theis et al 2015] [Tillman et al 1989] [Tong et al 2022] [Townsend and Blatt 1976] [Tran and Newman 2015] [Tran and Newman 2015] [Ul Islam et al 2020] [van Dam and Holmes 1988] [van Ekeren et al 2018] [VanZwieten et al 2014] [Vechtel and Buch 2022] [Vidyasagar 1982] [Vincent and Anex 1990] [Wabersich and Zeilinger 2021] [Wagner and Hromcik 2017] [Wagner et al 2023] [Walker 1985] [Wall et al 2014] [Walters and Henderson 1983] [Wang and Lei 2017] [Wang and Sundararajan 1995] [Wang et al 2015] [Wang et al 2018] [Wang et al 2022] [Wang et al 2023] [Ward and Ly 1994] [Ward and Ly 1996] [Wedell et al 1989] [Wee 1995] [Wei and Lin 2016] [Weiser et al 2024] [Wells 2002] [Wells 2002] [Wenyan et al 2017] [Wise et al 1999] [Wittenmark 1988] [Wittenmark 1990] [Wu 1976] [Wu and Yu 2018] [Xie 2026] [Xie et al 2026] [Xili et al 2008] [Xu and Deng 2022] [Xu and Zhang 2025] [Xu et al 2026] [Yagil et al 2018] [Yahyaoui 2016] [Yang and Morales 2021] [Yang et al 2016] [Yasue et al 1986] [Ye et al 2020] [Yedavalli and Liang 1986] [Yijie et al 2023] [Yin 2022] [Yin 2025] [Yuqi et al 2025] [Zelman et al 2023] [Zhang and Bradley 2022] [Zhang and Liu 2016] [Zhang and Liu 2017] [Zhang and Liu 2021] [Zhang and Zhao 2023] [Zhang et al 2017] [Zhang et al 2025] [Zhao et al 2025] [Zheng and Yang 2020] [Zheng et al 2023] [Zhou 1994] [Zhou and Yin 2015] [Zhou et al 2026] [Zhu and Liu 2015] [Zhu and Qin 2023] [Zou et al 2017] [Zuhri 2025] [Öztürk and Güzey 2020] [Šlihta et al 2016] [Żugaj 2017]

Where the Framing Breaks Down

Treating this aircraft through its keystone misleads in five places, and the first is the sharpest.

The Exchange Ratio Is Not a Property of the Aircraft

The apparatus of this article treats the exchange ratio as a measurement, but it is better understood as a joint measurement of an aircraft, a pilot, a tactic, an adversary and a set of rules. The programme’s own data proves it. The same aircraft in the same simulator, from a nominally identical defensive start, returned 1.2 in one campaign and 0.47 in the next, and the programme attributed the difference to which initial move the pilots chose.

A number that moves by a factor of two and a half when the pilots change their minds does not describe the airframe.

The Adversary Chooses the Engagement

The F-15C and F-16C result is the boundary of the entire keystone. An advantage that exists inside a slow, close fight is realisable only if the opponent enters one. Against aircraft with enough excess thrust to sit above the fight and re-enter at their chosen moment, the X-31’s advantage was largely unavailable.

The lesson is old, as the NASA history observes, drawing the parallel with the Fokker Dr I of 1917, an aircraft capable of post-stall pointing that pilots of faster conventional fighters learned to defeat by extending and attacking in slashing passes, denying it the ability to choose the time and place of the fight.

The Demonstrator Was Underpowered

The X-31 fought the F/A-18 from a position of disadvantage in conventional performance, and the NASA history is direct about it, attributing an exchange ratio of 0.37 without post-stall technology to the basic mediocrity of an underpowered design against a more powerful opponent with a better dogfighting wing.

The handicap cuts both ways, and the second direction is the more interesting. An aircraft that loses almost three fights in four conventionally and wins nearly two in one with post-stall technology has demonstrated a large effect. But it has demonstrated it on an airframe whose conventional performance nobody would accept, and the programme’s own conclusion is that post-stall technology must be an integrated element of a design rather than a bolt-on.

The Angle of Attack Limit Is Not the Whole Capability

The article has treated the usable angle of attack as the figure of merit, because it sets the aim-off angle. The programme’s conclusions put velocity-vector roll rate alongside it, and describe the main advantages as apparent directional nose-pointing rate, which is actually yaw rate or velocity-vector roll capability at high angle of attack, together with very high nose authority in pitch.

Pointing where you cannot get to quickly is worth less than the static aim-off angle suggests, and a full treatment would need the rate as well as the angle. The rate is not published in a form this article could use.

The Simulation and Flight Numbers Are Not the Same Experiment

The exchange ratios analysed at length in this article are simulation results, from Pinball I and Pinball II. The flight programme reported 407 engagements of which 325 were scored, and its summary conclusions are stated qualitatively. The temptation to read the simulator’s 1.83 as a flight result should be resisted, and the widely quoted figures of thirty to one and of nearly ten to one over 103 dogfights come from yet other accountings. The NASA history reports that the first close-in-combat flight campaign comprised 103 engagements, 94 of them scorable, and that the X-31 won 64 of the 66 decided engagements from neutral starts, a ratio of 32 to one, which is the likely origin of both figures, although neither figure as quoted states that accounting.

That inconsistency is itself the finding. A programme whose keystone was a measured rate produced at least four different headline rates that circulate independently, and the arithmetic above shows why that is easy. The pooled value depends on the mix of starting conditions, and nobody quoting a single number says which mix produced it.

The condition under which pooling reverses a conclusion can be written down. A set of per-condition ratios supports a claim of advantage on average while the pooled figure denies it whenever

\[\frac{1}{N} \sum_i E_i > 1 \qquad \text{and} \qquad \sum_i w_i E_i < 1\]

which is possible for any set that contains at least one ratio below parity, because the weights are free. For the published set the two sides are 6.43 and 1.83, so both inequalities are satisfied with room to spare in the first and by a margin of 0.83 in the second. The margin is what the knife-edge calculation above measures, and it is smaller than the uncertainty in the weights. The general form of that hazard is known in statistics as the reversal that can occur when groups are pooled.

The Source Base

The research survey holds 6,684 research records drawn from the NASA Technical Reports Server, the Defense Technical Information Center through its registered identifiers, and the Crossref registry across journal and conference literature, of which 669 are report primaries, being 10.0 percent, by the identifier rule set out under The Definition of Primary. The median publication year is 2017. Of the 6,680 dated records, 2,428 were published through 1999 and 3,922 from 2015 onward. Every record the survey holds is cited, in a cluster row of The Contemporary Literature.

Period Records Share
Before 1985 745 11.2 percent
1985 to 1996, the programme window 1,469 22.0 percent
1997 to 2014 544 8.1 percent
2015 onward 3,922 58.7 percent
of which 2022 onward 1,789 26.8 percent

Where the Programme’s Own Literature Lives

The X-31 was reported almost entirely at meetings rather than in journals, at the meetings of the American Institute of Aeronautics and Astronautics and the Society of Experimental Test Pilots and at the 1994 High Alpha Conference. The tactical utility paper the keystone rests on and the close-in-combat results assessment published a decade after the programme ended are both conference papers. The vehicle’s own literature cluster holds 36 records, 28 of them from the 1985 to 1996 programme window and none published from 2015 onward, and two of them appeared in a journal.

The Equations That Carry No Citation, and Why

Seven displayed relations in this article carry no nearby citation and that is deliberate. They are the weighting identity, the two ends of the two-point bracket derived from it, displayed separately, the tipping weight, the same bracket applied to the other two comparisons, the flight-rate bookkeeping, and the ratio between the two simulation campaigns.

Every one of those is a construction original to this article rather than a result taken from a source. Attaching a citation to them would be false attribution, and leaving them bare is the correct treatment. The underlying arithmetic is elementary and the reader can check it, which is the point of displaying it.

The distinction matters because the rest of the article’s equations are not like that. The atmosphere, the thrust lapse, the lift-curve slope, the drag polar, the turn relations, the binomial interval and the divergence eigenvalue are all standard results with a literature behind them, and each carries it.

On the Primary Record

The primary record for this aircraft is unusually good in one respect and unusually poor in another. The flight reports, the quasi-tailless results, the yawing-moment asymmetry analysis and the tactical utility paper are all public, detailed and quantitative. The NASA history published in 2014 carries a complete per-flight log for both aircraft and the simulation exchange ratios by starting condition.

What is not public is the count of decided outcomes behind each simulation ratio, which is the one thing needed to attach an interval to any of the published ratios without assumption. The NASA history gives 48 simulated engagements for each case and counts of decided outcomes by starting condition for the flight campaigns, but it reports the simulation results as exchange ratios and percentage increases. This article works around that by asking what various sample sizes would support rather than by guessing which one applied.

The Definition of Primary

Primacy is a property of the document, and this survey derives it from the identifier. A record is a report primary when its address is on the NASA Technical Reports Server or it carries a digital object identifier under the 10.21236 prefix, which Crossref registers for Defense Technical Information Center reports. 17 of the 36 records in the vehicle’s own literature are report primaries under that rule. The rule does not reach the conference papers in which the X-31 was mostly reported, so 10.0 percent is a lower bound on how much of the survey is first-hand.

The programme window is the row that matters for primacy, because it holds the work published while this aircraft was being designed, built and flown. A record from 1993 about high angle of attack aerodynamics is a primary source for this article in a way that a review from 2021 is not, however good the review.

Thin Subjects

Ten subjects that this article’s derivations depend on have thin literatures. Each count below is the number of research records in this survey whose title carries at least one of the listed terms, with a hyphen read as a space. Where a row joins two groups of terms with the word together, a title must carry a term from each group, and where it offers an alternative with the word or, either suffices.

Subject Title terms counted Records
the exchange ratio as a reported quantity exchange ratio, kill ratio, loss exchange 6
agility metrics agility metric, agility metrics 17
the velocity-vector roll velocity-vector roll 6
turn performance and corner speed corner speed, corner velocity, turn rate, turn radius, turning rate, turning radius, turn performance 6
the Herbst manoeuvre Herbst, J-turn 7
control power control power 12
thrust lapse with altitude thrust lapse, or engine thrust, turbofan, turbojet, jet engine together with altitude 4
carefree handling carefree 10
the icing or blockage of an air data probe pitot, Kiel, air data, probe together with ice, icing, blockage, blocked 21
international collaboration in aircraft development international collaboration, international cooperation, multinational, cooperative aircraft, cooperative development 1

The quantity the keystone rests on is almost never the subject of a title. The exchange ratio, the kill ratio and the loss exchange together appear in six titles, so the aggregation and the intervals derived in this article rest on the statistics of proportions rather than on a literature of the measure itself.

Agility metrics are a literature of the programme’s own decade. Of the 17 titles that name them, 14 date from 1985 to 1996 and one from 2015 onward, so the vocabulary the pointing argument needs was settled then, and the one modern title is a review of it.

The velocity-vector roll is named in six titles, all from 1985 to 1996, the decade in which the X-31 made it the command variable of lateral stick, so the kinematics above rest on the standard axis relations rather than on a modern literature.

Turn performance is a standard relation rather than a measured literature here. Turn rate, turn radius, turning performance and corner speed together appear in six titles, so the turn-radius comparison above rests on the relations it displays.

The manoeuvre that carries Herbst’s name has seven titles, and the four published from 2015 onward are all control designs for flying it.

Control power is thin as a title term and not as a subject. Twelve titles name control power, six of them from the programme window, while the cluster on control power and effector sizing holds 60 records, because most of its titles name control authority, control margins or control derivatives instead. The gap is in the vocabulary rather than in the work.

The thrust-lapse model that the departure and altitude arguments rest on is used rather than studied. Thrust lapse, or engine thrust together with altitude, appears in four titles, none of which names thrust lapse, which is why the lapse exponent is carried explicitly above and the sensitivity to it reported.

Carefree handling has ten titles, three of them on helicopters, and one, a flight demonstration of tactile cueing, takes up the problem the X-31 pilots named, of restoring the cues that carefree handling removes.

Pitot icing was described in 1938 and became a literature only after the accident. Of the 21 titles that join a pitot, Kiel or air data probe to ice or blockage, the earliest is from 1938, the next from 2013, and 19 were published from 2015 onward.

One title names international collaboration or a multinational or cooperative aircraft programme, so the statement that the arrangement was itself a first rests on the programme’s own record.

What the Survey Admits

A record is admitted only when a person reading its title finds it on this article’s subject. Many words this subject depends on also name something else, and a record that shares only such a word is excluded. Control also names the control of power grids, gene therapy and bankruptcy law, structure also names crystal structure, vehicle stability also names the stability of cars, and manoeuvrability also names the manoeuvring of ships. Tailless also names cats, rats, batrachians and whipscorpions, and canard also names the foie gras duck. A delta wing is also a vortex generator in a heat exchanger and the cause of wind loads on a canopy roof, time of flight is also a measurement of particle detectors, cameras and cytometry, departure also names an airport lounge and a gate queue, binomial also names a waveguide transformer and an optical window, agility also names the agility of an organisation, upset also names a soft error in a memory cell, and carefree also names a way of life. Diesel combustion and exhaust, power transmission cables, radio-frequency calibration, buildings and bridges, and the timing of missile launches as a question of international relations are excluded on the same rule. Correction, withdrawal and retraction notices and addenda are excluded as well, because a notice of that kind is a part of the work it concerns rather than a research work of its own, and so are book reviews, conference reports, whole proceedings volumes, special-section introductions, numbered abstract-digest entries, figure and table records, software packages, teaching datasets, encyclopedia entries, news notes and meeting minutes. A second deposit of a document already counted, such as a report-series copy of a journal paper, is not admitted either, so that each work is counted once.

The two residual clusters are held to a stricter rule. The general control literature keeps the flight control of aircraft and flight vehicles, and general control theory only where it is the theory this article’s argument uses, being the control of statically unstable plants, robustness, gain scheduling, saturation, nonlinear and high angle of attack control, thrust-vector control, handling qualities and pilot-vehicle interaction. It excludes control in other application domains, among them robots, ground and marine vehicles, power systems including the electrical power of aircraft, communications, process plants and traffic management, and pure theory tied to none of those uses. The other aerodynamic and flight mechanics literature keeps only the aerodynamics and flight mechanics of aircraft and flight vehicles, and excludes orbital manoeuvring, engine internals, aircraft subsystems, air traffic and target tracking. The same exclusions are applied to every other cluster for the kinds they name, being spacecraft and orbital manoeuvring, engine internals, the electrical power and electronics of aircraft, air traffic and airline operations, robots, marine and ground vehicles, multi-agent coordination, process and networked control, fluid mechanics and acoustics without a flight vehicle, and aircraft subsystems and ergonomics. Thrust-vector control of spacecraft is kept beside thrust vectoring, and ice protection beside air data and icing. Every cluster is also held to the argument. A record is kept when it bears on a subject this article’s argument uses, and a method or phenomenon the argument relies on is kept when it is studied on another vehicle. A record whose only link is general aeronautics is excluded, among them structural health monitoring and the integrity, manufacture and repair of airframes, the design and sizing of transports, general aviation, electric and small unmanned aircraft, propellers, distributed and electric propulsion and engine internals, aircraft noise and the sonic boom, weapon bays and store separation, command and control and mission planning, orbit change, and civil operations. Aeroelasticity, buffet and high-temperature materials, inlet distortion and nozzles, and fighter and unmanned combat aircraft design are kept. A record whose only link is a shared word of control theory or a distant domain is excluded on the same rule, among them the stability theory of switched, positive, fractional-order and delayed systems with no flight plant, controllers shown only on laboratory rigs, the dynamics and control of structures that are not aircraft, simulator motion platforms, control loaders and other ground equipment, counter-drone tasks, infrared and radar signatures, radomes and the hydraulic and engine subsystems, and the reporting of atmospheric turbulence. The control of unstable plants with time delay or actuator saturation, motion cueing as it bears on handling qualities, and missile guidance against any target are kept.

Some records of doubtful relevance are kept. Records from other fields that share a method with the analysis in this article are kept, among them Lanchester models of ground battles and other combat effectiveness models beside the measure of effectiveness, interval estimation and sample-size work from medicine and economics beside the binomial intervals. The otter board of a trawl, studied as a low aspect ratio wing at high angle of attack, is kept beside the lift-curve section. Aviation physiology and pitot measurement outside aircraft are kept on the same reasoning, and a reader may weigh each of them accordingly.

Every research title has been read for relevance, so the off-topic share that remains is a matter of reading judgement. A reading of titles cannot show that a work is relevant in substance, so this is a statement about titles only.

Epistemic State

Historical fact, from the primary and official record.

  • Two aircraft were built by Rockwell International and Messerschmitt-Bolkow-Blohm and flew from October 1990 to June 1995, logging 580 flights with fourteen pilots, as the Dryden fact sheet records.
  • The programme was managed by the Defense Advanced Research Projects Agency and flown by an International Test Organization at NASA Dryden, and was the first experimental aircraft programme administered by a United States agency as an international effort.
  • Controlled flight at seventy degrees of angle of attack was achieved on 6 November 1992, and a minimum-radius one-hundred-and-eighty-degree post-stall turn on 29 April 1993.
  • The first aircraft was lost on 19 January 1995 following ice accretion on an unheated Kiel probe, with the pilot ejecting and being injured on landing.
  • The published simulation exchange ratios against the F/A-18 with post-stall technology are 0.36, 8.5, 16 and 0.86 across four starting conditions, with an overall figure of 1.83.
  • The tactical utility flight test flew 87 sorties of which 80 were scored, producing 407 engagements of which 325 were scored.
  • The quasi-tailless demonstration reached a time to double amplitude of 290 milliseconds at complete simulated tail removal at a supersonic condition.
  • The second aircraft flew again from 2001 to 2003 in a separate programme with a flush air data system.

Engineering analysis performed for this article.

  • The pooled exchange ratio is the mean of the per-condition ratios weighted by the losses of the denominator side. This is an identity, and it holds for every set of counts in which each condition records at least one loss on the denominator side.
  • Between 81.9 and 93.6 percent of the aircraft’s simulated losses must have occurred in the two starting conditions where it was behind. The bracket is derived analytically from two-point mixtures and confirmed by Monte Carlo over twenty-seven thousand feasible weight vectors, which ran from 0.8198 to 0.9356.
  • At twelve scored outcomes per condition the defensive and high-speed line-abreast results do not separate from parity. The high-speed line-abreast ratio of 0.86 would need 691 scored outcomes to separate, against the twelve per condition and case the simulation design allowed.
  • The loop gain on the accident flight stood between 2.890 and 12.543 times its design value, against a conventional gain margin factor of 1.995.
  • The displayed airspeed and the vane angle of attack disagreed by 11.2 degrees at the moment the pilot remarked on them, corresponding to a dynamic-pressure error of 2.266, which already exceeded the design gain margin.
  • Vectored control authority at thirty thousand feet is 0.6462 of its value at twenty thousand at the same calibrated airspeed, a loss of 35.4 percent.
  • The post-stall regime offers no turn-radius advantage. Every radius computed at seventy degrees of angle of attack exceeds the 1,171 feet available at corner speed.
  • The energy given up in a post-stall reversal from corner speed at 20,000 feet is 8,630 feet of specific energy and takes 46.0 seconds to recover, integrated.
  • Against the steady asymmetry bound of 0.080, no arm shorter than the fuselage could hold the asymmetry above 237.6 knots calibrated at 20,000 feet or above 191.0 knots at 30,000 feet.

Inference, where the record supports a conclusion without stating it.

  • The moment arm from the centre of gravity to the vane station is not determined by the flight record. The departure on the second aircraft, at an asymmetry of 0.125 at 35,000 feet and Mach 0.4, bounds it below 36.2 feet under the thrust-lapse model, which the fuselage length already guarantees, and the bound moves from 27.1 to 48.4 feet across lapse exponents of 0.8 to 1.2. The 17.57 feet implied by the stations a dynamic-model study tabulates lies within that bound.
  • The lift-curve slope of 2.769 per radian is inferred from a test pilot’s recollection of a trim point rather than from published data, and is supported by falling between the unswept and swept low-aspect-ratio estimates.
  • The altitude dependence of vectored authority is offered as the quantitative explanation of an effect the flight-test report describes qualitatively. The report does not make the connection.

Claims the record does not settle.

  • The number of engagements behind each published exchange ratio is not public, so every interval in this article is conditional on an assumed sample size.
  • Two widely circulated figures are not stated in the primary record in the form in which they circulate. They are approximately thirty to one, and nearly ten to one over 103 dogfights. Neither is among the simulation figures analysed here. The NASA history reports 32 to one from neutral starts in a first flight campaign of 103 engagements, which is the likely origin of both, but the provenance of the quoted forms is unresolved.
  • The effect of an airframe with adequate conventional performance is unknown. The programme’s own conclusion that post-stall technology must be integrated rather than added implies the exchange ratio would have differed.
  • The yaw inertia is not given in the flight reports, so the directional stiffness of the simulated tailless configuration rests on the inertia a dynamic-model study tabulates for the full-scale aircraft.
  • The velocity-vector roll rate achieved at seventy degrees of angle of attack is described qualitatively in the sources consulted and no usable figure was found, so the pointing analysis in this article treats the angle and not the rate.
  • The basis of the 2004 close-in-combat results assessment was not established. Whether it reports flight or simulation figures, and under what accounting, is unresolved, because the full text of the paper has not been read.

Out of Scope

  • The detailed structural design of the wing and the German certification arrangements behind it.
  • The Vectoring Extremely Short Takeoff and Landing Control Tailless Operations Research programme of 2001 to 2003 in its own right, which is a distinct programme with distinct objectives and is treated here only where it bears on air data.
  • The F/A-18 High Alpha Research Vehicle and the F-16 multi-axis thrust vectoring programme, both contemporaneous and both deserving separate treatment.
  • Radar cross-section and low-observable design, which is the reason the tailless result matters and is not itself an X-31 subject.
  • The design of the missiles the programme concluded were necessary.
  • The organisational history of the International Test Organization beyond what bears on the flight-test method.

Conclusion

The X-31 was built to find out whether an aeroplane that can point its nose forty degrees further round than its opponent wins, and it found out that the answer depends on where the fight starts.

The result is less satisfying than the reputation, and better. The programme produced four exchange ratios against a conventional adversary and the aircraft lost two of them. It produced an overall figure of 1.83, a loss-weighted average dominated by the two it lost, which sits three and a half times below the average of the four. Between eighty-two and ninety-four percent of its simulated losses occurred in the conditions where post-stall manoeuvring did not help.

It produced, in flight against front-line fighters, a clear advantage over the Navy types that would fight it close and a clear disadvantage against the Air Force types that would not.

An exchange ratio is not a property of an aeroplane. It is a property of an engagement, and the adversary has a say in which engagement occurs. The X-31 measured that with more care than anyone had before, and the measurement is usually quoted as a single number by people who did not read the table it came from.

The programme also produced a result nobody was looking for that has outlasted the one it was built for. An aircraft with no vertical tail at all can be held at supersonic speed by vectoring, diverging at 290 milliseconds to double. Yaw control without a fin is now the central configuration problem of combat aircraft design and the X-31 answered it in 1994 as a side experiment made cheap by having already integrated vectoring into the control laws.

And it produced a lesson about instrumentation that cost an aeroplane. A probe fitted to solve a real problem at high angle of attack was not heated, its cockpit switch was not placarded, the hazard analysis was not repeated, and the flight control system had no way to notice that both of its air data computers were being fed the same wrong number. The information needed to catch it was displayed on the head-up display, spoken aloud over the intercom, available on a standby instrument near the pilot’s knee, and computable in one line from the lift equation. None of those was a redundancy the design had asked for, and any one of them would have been enough.

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