M10-T4 — Approach and Automatic Landing

Learning objectives

After this lesson, you should be able to:

• Derive the loop gain of an angular glide-slope loop in metres and its dependence on range.
• Schedule the gain on range and verify the damping stays constant.
• Write the exponential flare law, choose its start height and time constant from the sink-rate requirement, and find the touchdown point.
• Estimate the effect of a downdraft or a lag on the touchdown sink rate.

The intuition

UAV M6-T6: Flight modes lists an automatic landing as one of the modes a stack offers, and M10-T2: The Autopilot by Successive Loop Closure built the altitude loop that this lesson commands. Hopper's own touchdown has the same shape: a path that converges to a point, a final phase that reduces the sink rate and a requirement on the speed at contact, and the terminal law of M8-T3: Explicit Guidance to a Terminal State has the same ingredients in a different form.

A beam in angle

An instrument landing system gives the aircraft the angle between the line to the antenna and the glide slope (3° here), as an error $\varepsilon$. The autopilot wants to reduce the distance $d = r\varepsilon$ from the beam. If the commanded flight path angle is proportional to $\varepsilon$, the flight path responds to a distance error with a gain $K V / r$ per second: it grows as the range $r$ shrinks. With two lags of 1 s and 0.5 s between command and flight path (the autopilot and the aircraft), the third-order loop is stable only while $K V/r < (T_1 + T_2)/(T_1T_2) = 3$ per second. With a gain of 2 rad/rad at 25 m/s that fails at 16.7 m of range: the damping ratio of the dominant pair is 0.85 at 200 m, 0.51 at 100 m, 0.27 at 50 m, zero at 17 m and negative at 12 m.

Schedule the gain on range

The cure follows from the same arithmetic: make the angular gain proportional to the range, $K(r) = K_{\mathrm{ref}}\,r/R_{\mathrm{ref}}$. The loop gain in metres is then constant, $K_{\mathrm{ref}}V/R_{\mathrm{ref}}$, and the poles do not move with range: the damping stays at 0.51 everywhere. In practice the schedule is capped at the far end (the gain cannot grow without bound) and the gain is reduced further in the last few seconds.

The flare

The glide slope would bring the aircraft to the ground at its sink rate, 1.31 m/s for 3° at 25 m/s, which is more than a landing gear or a small aircraft's structure wants. The flare reduces it: command a sink rate proportional to height, $\dot{h}_c = -(h + h_{\mathrm{off}})/\tau_f$. The height decays exponentially, and since the law aims at a point $h_{\mathrm{off}}$ below the ground, the aircraft reaches the ground with a sink rate $h_{\mathrm{off}}/\tau_f$ instead of zero. The flare starts when the commanded sink rate equals the glide sink rate, at $h_s = \tau_f \dot{h}_0 - h_{\mathrm{off}}$.

For this aircraft with the two lags, a flare time constant of 1 s touches down at 1.26 m/s (the lags prevent it flaring in time), 2 s at 0.97 m/s, 3 s at 0.63 m/s and 4 s at 0.37 m/s, with flare starts at 0.71, 2.0, 3.3 and 4.6 m and ground runs of 14, 42, 78 and 128 m. A longer time constant gives a softer touchdown and a longer float.

Schedule angular gains on range so that the loop gain in metres is constant. Design the flare as an exponential in height with a time constant long enough for the response lags, and check the sink rate with a downdraft.

Definitions

Localizer and glide slope

Radio beams that give the lateral and vertical angular deviation from the approach path to the runway.

Beam error $\varepsilon$ and distance $d$

$\varepsilon$ is the angle between the aircraft's bearing from the antenna and the beam; $d \approx r\varepsilon$ the corresponding distance at range $r$.

Range-scheduled gain

A gain $K(r) = K_{\mathrm{ref}}r/R_{\mathrm{ref}}$ proportional to the range from the antenna.

Flare

The pitch-up manoeuvre that reduces the sink rate to a touchdown value; an exponential flare commands $\dot{h} = -(h + h_{\mathrm{off}})/\tau_f$.

Sink rate

The vertical speed at the ground contact.

Derivation: the stability limit of the angular loop, and the flare

From the geometry of the beam to the stability boundary of the loop.

1. $$d = r\,\varepsilon,\qquad \varepsilon = d/r$$

   The deviation in metres is the range times the beam error in radians (small angles).

2. $$\dot{d} = V\,\delta\gamma$$

   The distance from the beam changes with the vertical speed relative to the beam: the speed times the flight path error $\delta\gamma = \gamma - \gamma_{GS}$.

3. $$\delta\gamma_c = -K\varepsilon = -Kd/r$$

   The proportional guidance command on the beam error, in angle.

4. $$\dfrac{\delta\gamma}{\delta\gamma_c} = \dfrac{1}{(T_1s + 1)(T_2s + 1)}$$

   The path response lags the command through the autopilot and the aircraft.

5. $$T_1T_2s^3 + (T_1 + T_2)s^2 + s + g = 0,\quad g = KV/r$$

   Close the loop: $s\,d = V\delta\gamma$ and the lags give a third-order characteristic polynomial with the gain g.

6. $$\text{stable} \iff 0 < g < \dfrac{T_1 + T_2}{T_1T_2}$$

   The Routh condition for a cubic $a_3s^3 + a_2s^2 + a_1s + a_0$: $a_2a_1 > a_3a_0$, i.e. $(T_1+T_2) > T_1T_2\,g$. For $T_1 = 1$, $T_2 = 0.5$ the limit is 3 per second, reached at $r = KV/3$ = 16.7 m for K = 2 and V = 25.

7. $$K(r) = K_{\mathrm{ref}}\,r/R_{\mathrm{ref}} \Rightarrow g = K_{\mathrm{ref}}V/R_{\mathrm{ref}} = \text{const}$$

   Scheduling the gain on range makes the metre-gain constant: the poles do not move.

The flare

1. $$\dot{h} = -(h + h_{\mathrm{off}})/\tau_f \Rightarrow h(t) = (h_s + h_{\mathrm{off}})\,e^{-t/\tau_f} - h_{\mathrm{off}}$$

   The solution of the first-order equation: height decays exponentially toward the aim point below the ground.

2. $$h(t_{\mathrm{td}}) = 0 \Rightarrow t_{\mathrm{td}} = \tau_f\ln\dfrac{h_s + h_{\mathrm{off}}}{h_{\mathrm{off}}},\quad \dot{h}_{\mathrm{td}} = -\dfrac{h_{\mathrm{off}}}{\tau_f}$$

   Touchdown when the height reaches zero; the sink rate there is the aim offset over the time constant, independent of the start height.

3. $$h_s + h_{\mathrm{off}} = \tau_f\,V\sin\gamma_0$$

   Continuity of sink rate at the flare start: the commanded sink rate equals the glide sink rate there.

The angular glide-slope loop has a gain in metres of K V / r, which rises without bound toward the runway and eventually breaks the Routh condition. Scheduling the angular gain on range holds the loop gain, and the damping, constant. The flare sets the touchdown sink rate by the aim offset over the time constant.

A worked example

An aircraft at 25 m/s on a 3° slope (glide sink rate 1.31 m/s) with two lags of 1.0 s and 0.5 s between command and flight path.

1. Stability limit. The loop is stable while $KV/r < 3$ per second. With K = 2: $KV/r$ = 0.08, 0.25, 0.50, 1.00, 2.50 and 5.00 at 600, 200, 100, 50, 20 and 10 m. The limit is passed at 16.7 m.
2. Poles, fixed gain. The dominant pair has damping 0.85 at 200 m, 0.51 at 100 m, 0.27 at 50 m, 0.00 at 17 m and −0.06 at 12 m (real part +0.10 1/s). At 600 m the roots are real (−2.07, −0.83, −0.10): slow convergence.
3. Poles, scheduled gain. With $K = 2\,r/100$ the poles are −0.34 ± 0.56j (damping 0.51) at every range.
4. Simulation. From a 30 m offset at 900 m with a 1.5 m/s downdraft crossing 80 m to 60 m: the fixed-gain loop has a deviation of 5.7 m at 400 m (slow far out) against 0.5 m for the scheduled loop; after the gust both settle within 1.2 m at 60 m.
5. Flare. For $\tau_f$ = 1, 2, 3, 4 s the flare starts at 0.71, 2.02, 3.33, 4.63 m, touches down at 1.26, 0.97, 0.63, 0.37 m/s after a ground run of 13.7, 42.0, 78.3, 127.9 m (0.6, 1.7, 3.1, 5.1 s).
6. Downdraft. With a 0.5 m/s downdraft from the flare start the touchdown sink rates are 1.78, 1.60, 1.36, 1.15 m/s for the four time constants: the downdraft adds 0.5 to 0.8 m/s.

import math
import gnckit as g

V, GS = 25.0, math.radians(3.0)
T1, T2 = 1.0, 0.5                    # two lags between the commanded and the achieved flight path angle: autopilot and aircraft response

def stability_limit():
    return (T1 + T2) / (T1 * T2)
print(f'glide slope tracking: with an angular deviation ε = d / r and a gain K (rad of flight path angle per rad of beam error), the loop gain in metres is K V / r')
print(f'  two lags {T1} s and {T2} s: the third-order loop is stable while K V / r < (T1 + T2) / (T1 T2) = {stability_limit():.2f} per second')
K0 = 2.0
for r in (600.0, 200.0, 100.0, 50.0, 20.0, 10.0):
    gain = K0 * V / r
    print(f'  range {r:5.0f} m: fixed angular gain {K0} gives K V / r = {gain:5.2f}' + ('  UNSTABLE' if gain > stability_limit() else '') + f'; scheduled gain (K proportional to r) holds {K0 * V / 100.0:.2f}')

print('poles of T1 T2 s^3 + (T1 + T2) s^2 + s + K V / r = 0, dominant pair, for the fixed gain and the scheduled gain:')
for r in (600.0, 200.0, 100.0, 50.0, 17.0, 12.0):
    row = []
    for gain in (K0 * V / r, K0 * V / 100.0):
        roots = [complex(z) for z in g.poly_roots([T1 * T2, T1 + T2, 1.0, gain])]
        pair = [z for z in roots if z.imag > 1e-9]
        slow = min(roots, key=lambda z: abs(z))
        if pair:
            z0 = pair[0]; row.append(f'pair {z0.real:+.2f} +- {z0.imag:.2f}j (damping {-z0.real / abs(z0):.2f}), slow root {slow.real:+.2f}')
        else:
            row.append('real roots ' + ', '.join(f'{z.real:+.2f}' for z in roots))
    print(f'  range {r:5.0f} m: fixed: {row[0]:62s} | scheduled: {row[1]}')
def approach(scheduled, d0=30.0, r0=900.0, dt=0.02, Kref=2.0, Rref=100.0):
    r = r0
    h = r * math.tan(GS) + d0
    g1 = g2 = -GS                       # achieved flight path angle through two lags
    log = []
    while r > 8.0:
        eps = math.atan2(h, r) - GS
        K = Kref * (r / Rref) if scheduled else Kref
        K = min(K, 4.0) if scheduled else K
        gc = -GS - K * eps
        g1 += (gc - g1) / T1 * dt
        g2 += (g1 - g2) / T2 * dt
        gust = -1.5 if 60.0 < r < 80.0 else 0.0                    # a downdraft of 1.5 m/s while crossing 80 m to 60 m of range
        h += (V * g2 + gust) * dt
        r -= V * math.cos(g2) * dt
        log.append((r, h - r * math.tan(GS)))
    return log
for scheduled in (False, True):
    log = approach(scheduled)
    def at(rq): return min(log, key=lambda p: abs(p[0] - rq))[1]
    peak_late = max(abs(d) for r, d in log if r < 60)
    print(f'  {"range-scheduled gain" if scheduled else "fixed gain          "}: deviation from the beam {at(400):+6.2f} m at 400 m, {at(60):+6.2f} m at 60 m (after the gust), {at(40):+6.2f} m at 40 m, {at(25):+6.2f} m at 25 m, {at(12):+6.2f} m at 12 m; largest inside 40 m: {max(abs(d) for r, d in log if r < 40):.2f} m')

print('flare: exponential law h_dot_c = -(h + h_off) / tau_f, started when it equals the glide sink rate; same two lags; touchdown at h = 0')
sink0 = V * math.sin(GS)
print(f'  glide sink rate {sink0:.2f} m/s at {V:.0f} m/s on a {math.degrees(GS):.0f} degree slope')
def flare(tau_f, h_off=0.6, wind_down=0.0, dt=0.005):
    hs = tau_f * sink0 - h_off
    h = hs; x = 0.0
    g1 = g2 = -GS
    t = 0.0
    while h > 0.0 and t < 60:
        hdot_c = -(h + h_off) / tau_f
        gc = math.asin(max(-1.0, min(1.0, hdot_c / V)))
        g1 += (gc - g1) / T1 * dt
        g2 += (g1 - g2) / T2 * dt
        hd = V * g2 - wind_down
        h += hd * dt
        x += V * math.cos(g2) * dt
        t += dt
    return hs, hd, x, t
print('  tau_f   flare start height   touchdown sink rate   ground run from the flare start   time')
for tau_f in (1.0, 2.0, 3.0, 4.0):
    hs, hd, x, t = flare(tau_f)
    print(f'  {tau_f:4.1f} s   {hs:8.2f} m          {-hd:8.2f} m/s          {x:8.1f} m                    {t:5.1f} s')
print('  the same flares in a 0.5 m/s downdraft that begins at the flare start:')
for tau_f in (1.0, 2.0, 3.0, 4.0):
    hs, hd, x, t = flare(tau_f, wind_down=0.5)
    print(f'  tau_f = {tau_f:3.1f} s: touchdown sink rate {-hd:.2f} m/s, ground run {x:6.1f} m')

glide slope tracking: with an angular deviation ε = d / r and a gain K (rad of flight path angle per rad of beam error), the loop gain in metres is K V / r
  two lags 1.0 s and 0.5 s: the third-order loop is stable while K V / r < (T1 + T2) / (T1 T2) = 3.00 per second
  range   600 m: fixed angular gain 2.0 gives K V / r =  0.08; scheduled gain (K proportional to r) holds 0.50
  range   200 m: fixed angular gain 2.0 gives K V / r =  0.25; scheduled gain (K proportional to r) holds 0.50
  range   100 m: fixed angular gain 2.0 gives K V / r =  0.50; scheduled gain (K proportional to r) holds 0.50
  range    50 m: fixed angular gain 2.0 gives K V / r =  1.00; scheduled gain (K proportional to r) holds 0.50
  range    20 m: fixed angular gain 2.0 gives K V / r =  2.50; scheduled gain (K proportional to r) holds 0.50
  range    10 m: fixed angular gain 2.0 gives K V / r =  5.00  UNSTABLE; scheduled gain (K proportional to r) holds 0.50
poles of T1 T2 s^3 + (T1 + T2) s^2 + s + K V / r = 0, dominant pair, for the fixed gain and the scheduled gain:
  range   600 m: fixed: real roots -2.07, -0.83, -0.10                                 | scheduled: pair -0.34 +- 0.56j (damping 0.51), slow root -0.34
  range   200 m: fixed: pair -0.40 +- 0.25j (damping 0.85), slow root -0.40            | scheduled: pair -0.34 +- 0.56j (damping 0.51), slow root -0.34
  range   100 m: fixed: pair -0.34 +- 0.56j (damping 0.51), slow root -0.34            | scheduled: pair -0.34 +- 0.56j (damping 0.51), slow root -0.34
  range    50 m: fixed: pair -0.24 +- 0.86j (damping 0.27), slow root -0.24            | scheduled: pair -0.34 +- 0.56j (damping 0.51), slow root -0.34
  range    17 m: fixed: pair -0.01 +- 1.40j (damping 0.00), slow root -0.01            | scheduled: pair -0.34 +- 0.56j (damping 0.51), slow root -0.34
  range    12 m: fixed: pair +0.10 +- 1.61j (damping -0.06), slow root +0.10           | scheduled: pair -0.34 +- 0.56j (damping 0.51), slow root -0.34
  fixed gain          : deviation from the beam  +5.74 m at 400 m,  -1.21 m at 60 m (after the gust),  -1.00 m at 40 m,  -0.61 m at 25 m,  -0.15 m at 12 m; largest inside 40 m: 1.00 m
  range-scheduled gain: deviation from the beam  +0.50 m at 400 m,  -1.18 m at 60 m (after the gust),  -1.04 m at 40 m,  -0.83 m at 25 m,  -0.63 m at 12 m; largest inside 40 m: 1.04 m
flare: exponential law h_dot_c = -(h + h_off) / tau_f, started when it equals the glide sink rate; same two lags; touchdown at h = 0
  glide sink rate 1.31 m/s at 25 m/s on a 3 degree slope
  tau_f   flare start height   touchdown sink rate   ground run from the flare start   time
   1.0 s       0.71 m              1.26 m/s              13.7 m                      0.6 s
   2.0 s       2.02 m              0.97 m/s              42.0 m                      1.7 s
   3.0 s       3.33 m              0.63 m/s              78.3 m                      3.1 s
   4.0 s       4.63 m              0.37 m/s             127.9 m                      5.1 s
  the same flares in a 0.5 m/s downdraft that begins at the flare start:
  tau_f = 1.0 s: touchdown sink rate 1.78 m/s, ground run    9.9 m
  tau_f = 2.0 s: touchdown sink rate 1.60 m/s, ground run   28.8 m
  tau_f = 3.0 s: touchdown sink rate 1.36 m/s, ground run   49.9 m
  tau_f = 4.0 s: touchdown sink rate 1.15 m/s, ground run   73.4 m

In robotics

Automatic landing systems are among the most scrutinized pieces of flight control.

• Gain scheduling on range or height. Every glide-slope coupler schedules its gain on radio altitude or range, and lowers the gain toward the threshold; the lateral localizer loop is scheduled in the same way as the lateral deviation is also an angle.
• Flare design. The flare time constant and offset are set by the sink-rate limit (landing gear strength), the dispersion of touchdown point (a longer flare floats farther) and the response lags. A sink-rate requirement of 0.4 m/s at touchdown with this aircraft needs a 4 s time constant and a 128 m float: a short runway rules it out.
• Disturbances. A downdraft or shear in the last seconds (a microburst or a ground-effect change) adds to the sink rate as shown. Autoland certification includes dispersions of wind, shear, sensor error and lag, with probabilities of landing outside the box, as in the Monte Carlo of module 12.
• Radar altimeter. The flare uses height from a radar altimeter, not the barometer, because the barometric error (tens of metres, from the aiding-sensors lesson) is far larger than the flare height of 5 m.
• Hopper. The same structure carries over to the rocket: a converging descent whose guidance depends on range, a final phase with a touchdown speed limit (the 0.3 m/s hold in the explicit-guidance lesson), and a measure of height from a radar altimeter.

The lab M10-L2: Close the Loops in Order builds the loops and lands the survey aircraft in turbulence. The requirements an autoland is judged against are the subject of M10-T5: Handling Qualities, Margins and Envelope Protection.

Common misconceptions

Key takeaways

1. An angular glide-slope loop has loop gain K V / r in metres, which rises as the range falls and breaks the Routh limit (K V / r < (T₁ + T₂)/(T₁T₂)) near the runway.
2. With K = 2, V = 25 m/s and lags of 1 and 0.5 s the loop is unstable inside 16.7 m; damping 0.85, 0.51, 0.27, 0.00 at 200, 100, 50, 17 m.
3. Scheduling K in proportion to range holds the damping at 0.51 at every range.
4. The exponential flare $\dot{h} = -(h + h_{\mathrm{off}})/\tau_f$ gives a touchdown sink rate $h_{\mathrm{off}}/\tau_f$ ideally; with lags, 1.26, 0.97, 0.63 and 0.37 m/s for time constants 1 to 4 s.
5. Softer touchdown costs ground run: 14 m at 1 s and 128 m at 4 s.
6. A downdraft of 0.5 m/s adds 0.5 to 0.8 m/s to the touchdown sink rate.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• B. L. Stevens, F. L. Lewis and E. N. Johnson, Aircraft Control and Simulation, 3rd edition, Wiley, 2015, the sections on approach and landing and the glide-slope coupler. Paid.
• D. McRuer, I. Ashkenas and D. Graham, Aircraft Dynamics and Automatic Control, Princeton University Press, 1973, the classic text on autoland. Paid.
• FAA Advisory Circular 120-28D, Criteria for Approval of Category III Weather Minima for Takeoff, Landing, and Rollout, the performance requirements for automatic landing. Free to read from the FAA; US government work.
• MIT 16.333 Aircraft Stability and Control (OpenCourseWare) , the lectures on autopilots and landing. Free; licence CC BY-NC-SA 4.0.

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What’s next

Related content

Nothing below is required to finish this lesson.

• Before
  M10-T3: Air Data, Angle of Attack and Wind

  Compute airspeed, altitude, angle of attack and sideslip from air-data measurements, and estimate the wind from air data and inertial velocity.

• Lab
  M10-L2: Close the Loops in Order

  Build the longitudinal autopilot of the survey aircraft from pitch rate to glide slope, one loop at a time, and land within the sink-rate limit in turbulence.

Other courses on this site treat neighbouring topics from their own angle:

• UAV M1-T2: Multirotor, fixed-wing and VTOL architectures
• UAV M5-T1: UAV equations of motion (wind triangle)
• UAV M6-T6: Flight modes
• UAV M6-T3: PX4 architecture
• Navigation M9-T5: Other Platforms: Legged, Aerial, Marine and Off-Road

This course leaves the following to the pages that already teach them:

• Multirotor flight control
• The wind triangle
• PX4 and ArduPilot software, flight modes and tooling