Autopilot - Wikiparency

# Autopilot > [[PORTAL_Aviation|Aviation]] · [[PORTAL_Avionics|Avionics]] spine.  ## Microsims — three.js ### Autopilot (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Autopilot.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Autopilot — three.js microsim"></iframe> </div> **Open it full-screen:** [Autopilot.html](https://wikitube-3d-microsims.netlify.app/Autopilot.html) · library `threejs` · route `microsim/threejs/` ### Related microsims Live sims on neighbouring articles: - [[Fly-by-wire]] - [[Aircraft_flight_dynamics]] - [[Inertial_navigation_system]] - [[Radio_navigation]] - [[Head-up_display]] - [[Avionics]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4).*  ## Overview An autopilot is not one controller. It is a stack of controllers, each wrapped around the last, and the reason it is built that way is the whole subject. Ask what an autopilot does and the answer is *it flies the aeroplane along a route* — a statement about cross-track error, which changes over tens of seconds. Ask what a control surface does and the answer is about roll rate, which changes in fractions of a second. Between those timescales lies a factor of a hundred, and no single proportional-integral loop spans it gracefully. So the function is split into tiers: - a **navigation** loop, the slowest, which looks at how far the aeroplane is from the commanded path and asks for a heading; - an **outer** (guidance) loop, which looks at the difference between that commanded heading and the actual one and asks for a bank angle — and, in the vertical axis, looks at altitude error and asks for a vertical speed and thence a pitch attitude; - an **inner** (attitude) loop, the fastest, which looks at the difference between the commanded bank and the actual bank and moves the aileron. Only the inner loop ever touches a surface. Everything above it produces setpoints for the tier beneath. The justification for this arrangement is **timescale separation**. The outer loop is allowed to be simple — a proportional gain and a limiter — because by the time it notices anything has changed, the inner loop underneath it has already finished responding. From the outer loop's point of view the inner closed loop is a wire with a gain of one. That approximation is what makes the design tractable, and it holds only while the inner loop is substantially faster than the loop above it. Collapse the separation and the stack does not degrade politely: it oscillates, and then it diverges. The microsim above renders this as three concentric rings of circulating packets around the aeroplane, one per tier, each turning at that loop's own gain-crossover frequency. In trim the inner ring is a blur at about 0.62 rad/s, the outer ring turns at 0.12 rad/s, and the navigation ring barely creeps at 0.03 rad/s: a visible five-to-one and four-to-one separation. Raise the guidance gain and you watch the middle ring accelerate toward the inner one — and the aeroplane starts to hunt at the same moment, weaving a couple of tenths of a mile either side of the track with the wings rocking to the 25-degree bank limit. The idea is not new. On 18 June 1914, at the *Concours de la Sécurité en Aéroplane* organised by the Aéro-Club de France and the French war ministry, Lawrence Sperry flew a Curtiss C-2 past the grandstand with his hands off the controls while his mechanic Emil Cachin walked out along the wing. The gyroscopic stabiliser that held the aeroplane level used four gyroscopes spinning at 7,000 rpm, driving pneumatic servos on all three axes, in a box eighteen inches on a side weighing forty pounds. It won the 50,000-franc first prize. That machine was an inner loop and nothing else: it held attitude and had no idea where it was. Everything since — heading and altitude hold, VOR and localiser tracking, the digital flight management systems of the 1980s that put a whole route into the outer tiers, and today's required navigation performance approaches — has been the progressive addition of slower loops on top of Sperry's fast one. The move from analogue to digital flight guidance through the late 1970s and 1980s did not change that loop structure — a digital attitude loop closes on the same error a pneumatic one did. What it changed was what could sit on top. Once the guidance computer was a programmable machine reading a digital air data and inertial reference, the outer tiers could hold an entire flight plan, schedule their gains against airspeed and altitude, sequence waypoints with computed turn anticipation, and — crucially — report their own state to the crew. Both the capability and the mode-confusion problem discussed below arrive with that same change. ## The physics ### The airframe, reduced The model in the sim is a 220 kt transport, 113.2 m/s true airspeed, held constant: there is no autothrottle and no drag model. The roll axis is reduced to **roll subsidence** alone, p_dot = L_da·δa + L_p·p , φ_dot = p with a roll subsidence time constant of 1.4 s, so *L*p = −0.714 s⁻¹, and *L*δa chosen so that full aileron (25°) produces a steady 15°/s roll — representative transport figures, not manufacturer data. The pitch axis is reduced the same way: a first-order pitch-rate response plus attitude integration, which drops the short period's own natural frequency. That is legitimate here only because the pitch *attitude* loop under study is far slower than a transport's short period; stated plainly, the short period is approximated by its rate-damping term alone. The flight path lags the pitch attitude. In flight dynamics the pitch-attitude-to-flight-path transfer has a numerator time constant conventionally written *T*θ2, a couple of seconds for a transport in the cruise. The sim uses γ_dot = (θ − α_trim − γ) / T_θ2 , T_θ2 = 2.0 s , α_trim = 2.5° so vertical speed is *V*·sin γ and an aeroplane does not begin to climb the instant it pitches up. ### The inner loop, designed by placing poles The attitude loop is proportional on attitude error with rate feedback for damping: δa = K_φ(φ_c − φ) − K_p·p Substituting into the plant gives a second-order closed loop directly: φ̈ = L_δa·K_φ(φ_c − φ) − (L_δa·K_p − L_p)·φ̇ so that ωn² = *L*δa*K*φ and 2ζωn = *L*δa*K*p − *L*p. The sim exposes ωn and ζ and solves back for the gains, which is how attitude loops are actually designed: choose where the closed-loop poles go, then read off what gains put them there. The nominal ωn is 1.0 rad/s, deliberately at the gentle end, and the choice is worth defending. At ωn = 2.2 rad/s the same algebra returns *K*φ = 11.3 rad of aileron per rad of bank error, putting the aileron on its 25-degree stop at a bank error of two degrees. An inner loop saturated for every real manoeuvre is a relay, not a linear loop, and its phase margin stops describing anything. At 1.0 rad/s, *K*φ = 2.33 and *K*p = 1.83, and the aileron stays linear out to about eleven degrees of bank error. ### The outer loop, and the kinematics that make it easy Above the attitude loop sit the heading and vertical channels: φ_c = clamp( K_ψ(ψ_c − ψ), ±25° ) v_s,c = clamp( K_h(h_c − h), ±v_s,sel ) θ_c = clamp( α_trim + asin(v_s,c/V) + K_vs(v_s,c − v_s), ±12° ) The heading kinematics are the coordinated-turn relation, ψ̇ = (*g*/*V*)·tan φ, exact for a level ball-centred turn. Linearised, that is a pure integrator of gain *g*/*V* = 0.0866 s⁻¹, so the outer crossover is approximately *K*ψ·*g*/*V*. With *K*ψ = 1.4 that lands at 0.121 rad/s. It also keeps the bank command inside its 25-degree limit for heading errors up to about eighteen degrees, so an ordinary heading change is a linear event and the phase margin printed in the HUD actually applies to it. The vertical gain is set from the same target crossover, *K*vs = ωc,out·*T*θ2/*V*, so one slider scales both channels and means the same thing in each. The altitude gain *K*h = 0.05 s⁻¹ asks for roughly 300 ft/min per 100 ft of error. ### The navigation loop The slowest tier converts cross-track error *e*y into a heading: ψ_c = track − clamp( atan(k_x·e_y), ±30° ) The arctangent is the limiter: far from the track it saturates at the intercept angle, close in it is linear with slope *k*x. Since ėy = *V*·sin(ψ − track) ≈ *V*(ψ − track), the linearised loop is first-order with crossover *k*x·*V*, set here to 0.030 rad/s — four times slower again than the outer tier. In flight-deck units *k*x is 26.4 degrees of intercept per nautical mile of error, saturating at the 30-degree limit about 1.2 NM out. Leg sequencing uses the standard turn-anticipation geometry. A coordinated level turn at bank φ has radius *r* = *V*²/(*g*·tan φ), which for this aeroplane at the 25-degree limit is 2802 m; the tangent distance for a track change Δ is *r*·tan(Δ/2), so the 70-degree turn in the route is begun 1962 m — 1.06 NM — before the waypoint. Both expressions are exact, which is why every flight management system uses them. ### Phase margin, computed rather than asserted The stability readout in the HUD is not a lookup table. The same coefficients that integrate the motion are evaluated on the imaginary axis: L_in(s) = K_φ · [A(s) / (1 + K_p·A(s))] / s , A(s) = L_δa / [(τ_a s + 1)(s − L_p)] L_out(s) = K_ψ · T_in(s) · (g/V) / s L_nav(s) = k_x · V · T_out(s) / s with T(s) = L(s)/(1 + L(s)) Crossover is bracketed on a logarithmic sweep and bisected; the phase is unwrapped along the sweep from the known low-frequency asymptote; the margin is 180° plus the loop phase at crossover. One subtlety is easy to get wrong and is worth naming: the asymptote is −90° for **all three** loops, not −90°, −180°, −270°. Each tier's *closed* loop *T*(*s*) has unity gain and zero phase at DC because of its own integrator, so an outer loop does not inherit the inner loop's integrator — it only ever sees its own. Anchor the unwrap at −180° for the outer tier, as an early version of this sim did, and every margin comes out ninety degrees too low and a perfectly healthy cascade reports as unstable. At the nominal settings the three margins are 66°, 80° and 77°, with crossovers of 0.62, 0.121 and 0.030 rad/s — a comfortable cascade. ### Three ways to break it, all reachable **Guidance gain.** Turning up the outer-loop gain scales the heading, vertical-speed and cross-track gains together, as a real system schedules them as a set. What does *not* scale is the inner attitude loop — and that is precisely the point. At ten times nominal, *K*ψ = 14 and the outer crossover has climbed to 0.912 rad/s, within touching distance of the inner loop's 1.0 rad/s natural frequency. The outer phase margin falls to 6°. In flight the aeroplane weaves about ±0.15 NM across the track with a period near twenty seconds while the wings rock to the bank limit, and the oscillation builds from nothing over several cycles before the limiter bounds it. It is not divergent forever — a real aeroplane's authority limits always turn a divergence into a limit cycle — but it is divergent until those limits arrive, and the linear margin is what predicts it. **Actuator lag.** A first-order lag of time constant τ contributes −atan(ωτ) of phase, so it only threatens a loop whose crossover is near 1/τ. This attitude loop crosses over at 0.62 rad/s, which is why the slider must reach whole seconds before anything visible happens: at 2.2 s the inner margin falls to 26° and the loop rings on every gust, with no gain changed anywhere. That is the lesson rather than a weakness of the demonstration. The same 0.06 s actuator that is invisible here would be fatal to a loop ten times faster; the rule that fast loops need fast actuators is exactly this inequality. **Inner damping.** The rate-feedback term *K*p·*p* is the damping. Turn ζ down to 0.10 and the closed-loop pair moves toward the imaginary axis: the inner margin falls to 9°, the attitude loop rings at about 0.97 rad/s, and the outer and navigation loops — whose own margins are untouched at 89° and 77° — faithfully inherit every wobble underneath them. This is the most instructive of the three, because the loop that is *reported* as healthy is the one visibly failing to hold the aeroplane steady. ### Modes, armed and captured A mode that is **armed** is watching for a condition. A mode that is **captured** is flying the aeroplane. In between, the two look almost identical from the window and completely different on the panel, which is the entire reason the panel exists. The sim annunciates lateral and vertical modes in fixed columns — active in green on the top line, armed in white below — and boxes and flashes a mode for ten seconds after it becomes active. The convention is not decorative: FAA Advisory Circular 25.1329-1C, which is the certification guidance for flight guidance systems under 14 CFR §25.1329, requires that active and armed modes be clearly distinguished with consistent discriminators and colouring, and that operationally relevant mode changes carry attention-getting features. Altitude capture is the clearest case. Selecting an altitude while in vertical-speed mode arms ALT; the aeroplane keeps climbing at whatever the wheel is set to and the annunciator says V/S with ALT armed. Capture begins when the altitude error falls inside a band computed from the closure rate. A capture is a constant-vertical-acceleration arc, so the distance needed to arrest a vertical speed *v* at deceleration *a* is *v*²/2*a*; at *a* = 0.04 *g*, gentle enough that passengers do not notice, 1500 ft/min needs about 240 ft. Inside that band the annunciation changes to ALT\* — arriving — and only when the aeroplane has settled does it become plain ALT. The sim draws the band in the world as two outlines above and below the commanded-altitude sheet, so the transition happens somewhere you can see rather than somewhere you have to trust. The approach shows the same structure one level up. Arming APP does not change what the aeroplane is doing: it keeps flying HDG or NAV while LOC sits armed underneath, waiting for the beam. Only when the localiser deviation falls inside about 0.7 NM with a sensible intercept angle does the aeroplane change hands. The glidepath — 3.0°, the near-universal ICAO value — arms behind the localiser and captures only from below and only close in, which is why crews level at a platform altitude and let the beam come down to them. ## Controls -> what each maps to | Control | Symbol | Range and units | What it actually does | |---|---|---|---| | Lateral mode | — | HDG / NAV / APP | Selects which lateral tier owns the aeroplane. NAV and APP *arm*; they do not engage until their capture condition is met, and until then the aeroplane flies the selected heading. | | Vertical mode | — | V/S alone / V/S with ALT armed / APP | Chooses whether an altitude capture is armed at all. "V/S alone" is a deliberate trap: it will fly straight through the selected altitude, and the HUD says so. | | Commanded heading | ψc | 0–359°, magnetic-style 3-digit | Setpoint for the outer heading loop when HDG is active, and the intercept heading when NAV or LOC is armed but not captured. | | Commanded altitude | *h*c | 1000–14000 ft, 100 ft steps | Setpoint for the altitude loop. Moving it far from the current altitude drops a captured ALT back to V/S and re-arms, which is what a real mode control panel does. | | Commanded vertical speed | *v*s,sel | −3000 to +3000 ft/min, 100 ft/min steps | The V/S wheel. Signed: set it away from the selected altitude and ALT will stay armed forever without capturing. | | Outer-loop gain | ×(*K*ψ, *K*vs, *k*x) | 0.2–12 ×, dimensionless | Scales the whole guidance tier. *K*ψ is shown live in deg of bank per deg of heading error (1.40 at nominal). The inner loop is untouched, which is what collapses the separation. | | Inner-loop damping | ζ | 0.05–1.20, dimensionless | Damping ratio of the attitude loop's closed-loop pole pair; the code solves back to the rate-feedback gain *K*p = (2ζωn + *L*p)/*L*δa. | | Actuator lag | τa | 0.02–3.0 s, first-order | Surface actuator time constant, plus a rate limit of 60°/s in roll and 40°/s in pitch. 0.02–0.10 s is a healthy transport actuator; beyond that is a fault injection. | | Turbulence | — | 0–1, arbitrary | Scales white noise through a first-order filter at 1.6 rad/s driving roll and pitch acceleration. Not a Dryden or von Kármán spectrum. | | Failure presets (F) | — | Trim / Hunt / Lag / Ragged | One click to each of the three failure routes and back. Each breaks a different tier; the phase-margin readout names which. | | Pause (Space) / Reset (R) | — | — | Freeze the integrator without touching state; or restore every gain, mode and the initial condition. | ## Learning objective Fly a commanded three-dimensional route with a cascade of three nested loops; see the timescale separation between them rendered directly as three ring speeds; and then destroy the cascade three different ways — guidance gain, actuator lag, inner-loop damping — while a numerically computed phase margin counts down to zero and names the tier that is failing. The payoff is the reason autopilots are cascades at all: each loop is simple only because the loop inside it is much faster, and when that stops being true the failure is a property of the structure and the timing, not of any broken component. A secondary objective, arguably the one with more lives attached to it, is the armed/captured distinction. Watching ALT sit armed for a full minute while the aeroplane climbs, then change to ALT\* and finally to plain ALT, is the cheapest available demonstration of why the flight mode annunciator is the most important piece of text on a modern flight deck. ## Limits and connections This is a linearised cascade model, not a certified autopilot and not a certified flight model. The honest list: - **The gains are fixed.** This is the largest single difference from a real system. A flight guidance computer schedules its gains continuously against airspeed, altitude and configuration, precisely because a gain that is right at 220 kt is wrong at 140 kt on approach: *g*/*V* alone changes the outer-loop crossover by 60% across that range. Holding the gains fixed is what lets one slider expose the instability; it is not what a real system does. - **The airframe is reduced to two decoupled single-degree-of-freedom axes.** Roll subsidence only — no spiral mode, no dutch roll, no aileron-induced yaw. Pitch attitude with a first-order rate response — no short period as such, no phugoid. Sideslip is zero throughout, and the turn is assumed perfectly coordinated. - **Airspeed is constant.** No thrust, no drag, no autothrottle. Pitching for vertical speed costs nothing in speed here, which is emphatically not true of an aeroplane. - **All three loops run in continuous time.** A real flight guidance computer samples them at genuinely different rates — attitude at tens of hertz, guidance at a few hertz, navigation at one hertz or slower — and that sampling contributes its own half-sample of phase lag. Keeping them continuous is what makes the printed phase margin exactly right for the model as written rather than approximately right for something else. - **Phase margin describes the linear model only.** Once a surface or the bank limiter is on its stop, the loop is no longer linear and the printed margin no longer describes what you are watching. The HUD flags saturation when it happens, which is itself worth seeing: the reason the divergence turns into a bounded limit cycle rather than running away is the limiter, not the margin. - **Turbulence is coloured noise**, not a Dryden or von Kármán gust spectrum, and makes no claim to that spectrum's shape. - **Model time runs at three times wall-clock time** so the route turn arrives within a minute. Every rate quoted anywhere — crossover frequencies, margins, vertical speeds, ring speeds — is in model seconds and internally consistent; only the mapping to your wall clock is stretched. **Mode confusion and automation surprise** are recognised categories in aviation human factors, not informal complaints. Sarter and Woods framed mode error and mode awareness as a supervisory-control problem in 1995, arguing that the flexibility bought by many modes creates a monitoring load which can exceed what a crew can sustain; the literature since treats "automation surprise" — the aeroplane doing something correct but unexpected — as a distinct failure class. The 1996 FAA Human Factors Team report identified insufficient mode awareness as a systemic issue across the fleet rather than a training deficiency at any one operator. The canonical illustration is Air Inter Flight 148, an A320 that struck high ground near Mont Sainte-Odile on 20 January 1992. The investigation concluded that a descent intended as a 3.3-degree flight path angle was entered while the flight control unit was in vertical-speed mode, where the same two digits meant 3300 ft/min; the aircraft descended at roughly four times the intended rate. The display showed only two digits, so "33" was ambiguous between the two modes. Airbus subsequently changed the unit to show four digits for vertical speed, which removes the ambiguity at a glance. Care is warranted here: the account of the mode selection is an inference from the recorded flight path rather than a direct readout of the crew's intent, and other contributing factors have continued to be debated. The design lesson — that two modes sharing one display field is a latent trap — is not in dispute. **Connections.** The inner attitude loop is the same loop a [[Fly-by-wire]] system runs continuously whether or not the autopilot is engaged; the difference is that fly-by-wire closes it around the pilot's stick while the autopilot closes it around a computed setpoint, and on most modern types they are literally the same computers. The plant those loops act on is the subject of [[Aircraft_flight_dynamics]]: the roll subsidence, short period and *T*θ2 used here are its standard reduced forms. The navigation tier needs a position, which comes from an [[Inertial_navigation_system]] blended with satellite and ground-based fixes, and its approach modes track beams described in [[Radio_navigation]]. The mode annunciation discussed above appears again on the primary flight display and, on aircraft so equipped, in the [[Head-up_display]]. And the whole stack — computers, buses, redundancy, certification — is the domain of [[Avionics]]. ## References - Blakelock, J. H. *Automatic Control of Aircraft and Missiles*, 2nd edition. John Wiley & Sons, 1991. The standard treatment of autopilot loop structure, including attitude, heading and altitude hold synthesised as a cascade. - McRuer, D., Ashkenas, I. and Graham, D. *Aircraft Dynamics and Automatic Control*. Princeton University Press, 1973. The reference for the reduced-order longitudinal and lateral approximations used here, including the *T*θ2 flight-path lag. - Stevens, B. L., Lewis, F. L. and Johnson, E. N. *Aircraft Control and Simulation: Dynamics, Controls Design, and Autonomous Systems*, 3rd edition. John Wiley & Sons, 2015. - Nelson, R. C. *Flight Stability and Automatic Control*, 2nd edition. McGraw-Hill, 1998. - Franklin, G. F., Powell, J. D. and Emami-Naeini, A. *Feedback Control of Dynamic Systems*, 8th edition. Pearson, 2019. For gain and phase margin, cascade (inner/outer loop) design, and the bandwidth-separation rule of thumb. - Åström, K. J. and Murray, R. M. *Feedback Systems: An Introduction for Scientists and Engineers*, 2nd edition. Princeton University Press, 2021. Freely available from the authors; chapters on loop analysis and robustness cover the margin computation used in the sim. - Sarter, N. B. and Woods, D. D. "How in the World Did We Ever Get into That Mode? Mode Error and Awareness in Supervisory Control." *Human Factors*, volume 37, issue 1, pages 5–19, 1995. DOI [10.1518/001872095779049516](https://doi.org/10.1518/001872095779049516). - Sarter, N. B. "Investigating Mode Errors on Automated Flight Decks: Illustrating the Problem-Driven, Cumulative, and Interdisciplinary Nature of Human Factors Research." *Human Factors*, 2008. DOI [10.1518/001872008X312233](https://doi.org/10.1518/001872008X312233). - Billings, C. E. *Aviation Automation: The Search for a Human-Centered Approach*. Lawrence Erlbaum Associates, 1997. - Federal Aviation Administration Human Factors Team. *Report on the Interfaces Between Flightcrews and Modern Flight Deck Systems*. FAA, Washington DC, 1996. - Federal Aviation Administration. Advisory Circular AC 25.1329-1C, *Approval of Flight Guidance Systems*, issued 27 October 2014. [faa.gov](https://www.faa.gov/documentLibrary/media/Advisory_Circular/AC_25_1329-1C.pdf) — see paragraph 44 on mode annunciation, the distinction between active and armed modes, and attention-getting features for operationally relevant mode changes. - Title 14, Code of Federal Regulations, §25.1329, *Flight guidance system*. The airworthiness requirement that AC 25.1329-1C provides means of compliance for. - International Civil Aviation Organization. *Annex 10 to the Convention on International Civil Aviation, Volume I — Radio Navigation Aids*. ICAO, Montréal. Specifies the ILS glide path, whose nominal 3.0-degree angle the sim uses. - Bureau d'Enquêtes et d'Analyses. Report of the commission of inquiry into the accident of 20 January 1992 near Mont Sainte-Odile to the Airbus A320 registered F-GGED operated by Air Inter. Ministère de l'Équipement, des Transports et du Tourisme, France, 1993. - Davenport, W. W. *Gyro! The Life and Times of Lawrence Sperry*. Charles Scribner's Sons, New York, 1978. The standard biography, and the source for the June 1914 demonstration. - Hughes, T. P. *Elmer Sperry: Inventor and Engineer*. Johns Hopkins University Press, Baltimore, 1971. Context for the Sperry Gyroscope Company's aircraft stabiliser work. **On the spine:** [[Avionics]] · [[Fly-by-wire]] · [[Autopilot]] · [[Head-up_display]] · [[Traffic_alert_and_collision_avoidance_system]] · [[Terrain_awareness_warning_system]] · [[Radio_navigation]] · [[Radiation_hardening]] · [[Radar]] · [[Air_traffic_control]] · [[Aviation]].  **Part of the [[Avionics]] hub** — main article for section X15, *Autopilot*. Related sections: [[Fly-by-wire]] · Instrument landing system · Flight management system.  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Autopilot) : [Wikitube](https://en.wikitube.io/wiki/Autopilot) --- *PORTAL_Avionics three.js batch · 2026-08-05 · sim staged in `_3d_deploy_stage/`.*