FLIGHT CONTROL & AUTOPILOT / CHAPTER 03

Heading & Track Hold

The nose points one way. The ground path can point another.

Coordinated-turn guidance · steady uniform wind · original synthetic teaching example · ideal airspeed and altitude hold

02 / LIVE B · 3D ATTITUDE

0 s

North reference · body direction (amber) · ground track (turquoise)

A · no saved case

CONTROL → RESPONSE → FEEDBACK

Preparing the model

03 / THE CALCULATED TRAJECTORY

One aircraft, two directions

A · dashed amber B · solid turquoise

Equal-scale north/east map. Drag the turquoise compass handle to set the selected heading or track. True bearings; no magnetic variation.

Air + wind = motion over the ground

Vectors use one common velocity scale. The ground-velocity arrow is the vector sum. The 3D direction arrows above indicate direction only; their lengths are decorative.

Follow the arrows

Air velocity plus wind velocity gives ground velocity. The aircraft symbol follows heading; the ground trail follows track.

Crab, then hold

Track mode calculates a wind-corrected heading. Heading mode keeps the nose direction even when wind carries the aircraft sideways.

Direction is not a route

Holding a track does not remove an existing cross-track offset. Chapter 04 adds a reference line and capture guidance.

04 / ONE PLAYBACK CLOCK

Follow the whole response

A · dashed amber / B · turquoise
Dotted: command. Click or use arrow keys to scrub.

Compare A and B

Quantity	A · saved	B · live

A stays fixed for this page session. Export includes settings and sampled histories.

Equations, assumptions and sources

Model contract

Level-turn motion with uniform wind

\dot{E} = V \sin (ψ) + w_{E}

\dot{N} = V \cos (ψ) + w_{N}

\dot{ψ} = \frac{g \tan (φ)}{V}

L = \frac{m g}{\cos (φ)}

Ideal airspeed and altitude hold; zero sideslip. Bank lag represents a pre-stabilised inner loop, not an aerodynamic actuator.

Symbols & units

$E, N$: East and north coordinates, m; these are not energy or force symbols.
$ψ, φ$: True heading clockwise from north and bank to the right, rad.
$w_{E}, w_{N}$: Wind towards east and north, m/s. The input is a towards bearing, not the meteorological from bearing.
$V, L$: Ideal constant true airspeed, m/s; required lift, N. Available thrust, stall and roll inertia are not solved.

Model context and notation

The aircraft is a reduced-order level coordinated-turn model. True airspeed and altitude are held ideally; sideslip is zero. The bank controller has first-order lag and a 15°/s rate limit. With constant wind: Ė = V sin ψ + wE, Ṅ = V cos ψ + wN, ψ̇ = g tan φ / V. Lift is mg/cos φ. Available thrust, stall, roll inertia and Dutch roll are outside this model. Bank lag represents a pre-stabilised inner loop; it is not an aerodynamic actuator model.

Track or heading command to bank response

w_{cross} = w_{E} \cos (χ_{c}) - w_{N} \sin (χ_{c})

ψ_{c} = χ_{c} - asin (\frac{w_{cross}}{V})

φ_{c} = clip (atan (\frac{V 0.08 R wrap (ψ_{c} - ψ)}{g}); - φ_{limit}, φ_{limit})

\dot{φ} = clip (\frac{φ_{c} - φ}{τ_{roll}}; - 15 ° /s, 15 ° /s)

Holding track does not remove an existing route offset. The selected bank limit applies to the command, not a certified aircraft limit.

Symbols & units

$R$: Dimensionless heading-response multiplier; 0.08R has units s⁻¹.
$φ_{limit}, τ_{roll}$: Selected command limit, rad; first-order bank time constant, s. Roll rate is limited to ±15°/s.
$wrap / clip$: wrap selects the principal heading error in [−π, π]. clip bounds a command to the stated lower and upper limits.
$χ_{c}, ψ_{c}$: Ground-track and nose-direction targets, rad. Heading mode selects ψc directly; the wind triangle applies to track guidance.

Model context and notation

Limits and sources

Bank command limited to the selected 10–35°; uniform wind 0–25 m/s and TAS 45–85 m/s. No vertical wind, turbulence or geodetic Earth model. The route is a local synthetic plane, not an operational airport procedure.

Teaching connections: reference frames, turning performance, flight control and flight management systems. Original extension with synthetic parameters.

MIT OCW · lateral autopilots supports the control concepts; its aircraft coefficients are not used.

Numerical verification does not establish flight-test validity.