Fixed-wing

A six-degree-of-freedom aircraft with the full non-linear aerodynamic coefficient build-up from

R. W. Beard, T. W. McLain, Small Unmanned Aircraft: Theory and Practice, Princeton University Press, 2012 — Chapter 4 and Appendix E.

Source: uav_sim/vehicles/fixed_wing/

Quick start

from uav_sim.vehicles.fixed_wing import create_fixed_wing, FixedWingPreset

aircraft = create_fixed_wing(FixedWingPreset.AEROSONDE)
controls = aircraft.reset_trimmed(airspeed=35.0, altitude=200.0)

for _ in range(6000):
    aircraft.step(controls, 0.005)

print(aircraft.state[2])   # 200.0
print(aircraft.airspeed)   # 35.0

Thirty seconds of open-loop flight holding altitude to the metre. That is the acceptance test for the whole model: if any force or moment is inconsistent, trim is not an equilibrium and the aircraft wanders.

State and control

state   = [x, y, z, φ, θ, ψ, u, v, w, p, q, r]
control = [elevator, aileron, rudder, throttle]

Position is world ENU; velocity [u, v, w] is body FLU; rates are body-frame. Surfaces are radians, clamped to ±30°; throttle is [0, 1].

Storing velocity in the body frame is deliberate — angle of attack is atan2(-w, u), so the aerodynamics want it there and would otherwise rotate it back on every evaluation.

Derived quantities

aircraft.airspeed            # true airspeed Va [m/s]
aircraft.alpha               # angle of attack [rad]
aircraft.beta                # sideslip [rad]
aircraft.pitch_up            # pitch, aerospace sign convention
aircraft.flight_path_angle   # climb angle [rad]
aircraft.load_factor         # lift / weight — the "g" being pulled
aircraft.is_stalled()        # |alpha| past the stall boundary
aircraft.velocity            # world-frame velocity
aircraft.last_wrench         # lift, drag, thrust from the last step

The aerodynamic model

Forces and moments are built from dimensionless coefficients evaluated at the current flow condition. The model is written in the textbook's Forward-Right-Down frame and converted at the boundary — see Frames and conventions.

Lift, with stall

$$C_L(\alpha )=(1-\sigma (\alpha ))(C_{L_0}+C_{L_{\alpha }}\alpha )+\sigma (\alpha )(2\mathrm{sign}(\alpha ){\sin }^2\alpha \cos \alpha )$$

$\sigma$ is a sigmoid that blends from the linear regime into flat-plate behaviour past the stall angle. Without it, lift grows without bound and the aircraft can "fly" at 60° of incidence.

α	C_L
0°	0.23
10°	1.21
20°	2.18
27°	1.58
45°	0.71

Lift peaks near 20° and falls away — the aircraft can genuinely stall, and recover.

Drag, from the induced-drag polar

$$C_D=C_{D_0}+C_{D_p}+\frac{(C_{L_0}+C_{L_{\alpha }}\alpha {)}^2}{\pi eAR}+C_{D_{\alpha }}{\alpha }^2$$

This is where the Oswald efficiency $e$ and the wing span enter, through the aspect ratio $AR=b^2/S$. A short, stubby wing pays more induced drag than a long one at the same lift, which is why the trim throttle differs between presets.

Moments

$$\begin{array}{rl}m& =\overline{q}Sc(C_{m_0}+C_{m_{\alpha }}\alpha +C_{m_q}\frac{c}{2V_a}q+C_{m_{{\delta }_e}}{\delta }_e)\\ l& =\overline{q}Sb(C_{l_0}+C_{l_{\beta }}\beta +C_{l_p}\frac{b}{2V_a}p+C_{l_r}\frac{b}{2V_a}r+C_{l_{{\delta }_a}}{\delta }_a+C_{l_{{\delta }_r}}{\delta }_r)\\ n& =\overline{q}Sb(C_{n_0}+C_{n_{\beta }}\beta +C_{n_p}\frac{b}{2V_a}p+C_{n_r}\frac{b}{2V_a}r+C_{n_{{\delta }_a}}{\delta }_a+C_{n_{{\delta }_r}}{\delta }_r)\end{array}$$

Three of these terms are what make the aircraft an aircraft rather than a brick with a lift force:

Derivative	Sign	What it does
$C_{m_{\alpha }}$	negative	Static pitch stability. Pitch up, get a nose-down moment back.
$C_{m_q}$	negative	Pitch damping. Without it the short-period mode never settles.
$C_{n_{\beta }}$	positive	Weathercock stability. The nose swings into the relative wind.
$C_{l_p}$	negative	Roll damping.
$C_{l_{\beta }}$	negative	Dihedral effect — sideslip produces roll.

The rate terms are non-dimensionalised by $c/2V_a$ and $b/2V_a$. Those denominators look like they blow up at low airspeed, but $\overline{q}$ carries $V_a^2$, so each product is linear in $V_a$ and vanishes cleanly at rest.

Propulsion

Momentum theory: the propeller accelerates air from $V_a$ to $k_{motor}{\delta }_t$.

$$T=\frac{1}{2}\rho S_{prop}{C}_{prop}((k_{motor}{\delta }_t{)}^2-V_a^2)$$

Thrust therefore falls off with airspeed, which is what makes level flight settle at a finite speed for a given throttle instead of accelerating forever.

Verified behaviour

Each of these has a test in tests/test_fixed_wing_aero.py:

Pitch damping        1 rad/s disturbance: 2.92° swing -> 0.19° after 8 s
Static stability     +0.1 rad alpha -> +19.6 rad/s² restoring (nose-down)
Weathercock          8.1° sideslip -> 7.4 rad/s² yaw into the wind
Side force           8.1° sideslip -> 4.5 m/s² lateral acceleration
Stall                CL peaks at 2.18 near 20°, falls to 0.54 by 35°
Trim                 residual < 1e-11 across the speed envelope
Open-loop trim       200 m held to 0.00 m over 30 s

Every parameter is load-bearing

There is a parametrised test asserting that perturbing Cm0, Cma or e_oswald changes the dynamics. It exists because the previous model declared all three and read none of them — leaving an aircraft with no static pitch stability and no induced drag that still looked complete in the source.

Custom coefficients

from dataclasses import replace
from uav_sim.vehicles.fixed_wing import AeroCoefficients, FixedWing, FixedWingParams

unstable = replace(AeroCoefficients(), Cma=-0.2)   # nearly neutral in pitch
aircraft = FixedWing(FixedWingParams(coeffs=unstable, mass=10.0))

Making an aircraft unstable is a good way to see what the derivatives buy you. Set Cmq = 0.0 and watch the short period ring forever.

Envelope

params = aircraft.fw_params
params.stall_airspeed        # derived from the model's own CL_max
params.aspect_ratio          # b^2 / S
params.wing_loading()        # N/m^2
params.max_lift_coefficient

stall_airspeed is computed as $\sqrt{2mg/(\rho S{C}_{L,max})}$ by sweeping the actual lift curve, rather than being a hand-entered constant that can drift out of sync with the coefficients.

See also

• Trim and equilibrium — solving for steady flight
• Airframe presets — the four supplied aircraft
• Autopilots — altitude, airspeed and course hold
• Fixed-wing flight simulation