M5-T2: Quadrotor dynamic modelling

Learning objectives

After this lesson, you should be able to:

• List the parameters of a quadrotor model and how each is measured.
• Estimate inertia from a pendulum test and the motor time constant from a step response.
• Choose an integration scheme and step size for the stiff motor dynamics.
• Validate the model against flight data by comparing step responses.

Conceptual explanation

The model needs: mass $m$, centre of mass and inertia $J$, arm geometry, the propeller thrust and drag-torque as functions of rotor speed ($T$ = $k$_T Ω², $Q$ = $k$_Q Ω² as the standard approximation), the motor and ESC time constant $\tau _{m}$ and limits, aerodynamic drag, and sensor models. Parameters from CAD are a starting point; measured values replace them.

Engineering intuition

Not all parameters matter equally for the controller you are tuning: the attitude loop is dominated by inertia and the motor lag; altitude hold by thrust-to-weight and the thrust curve; position hold by drag. Measure those first. The validation step is the honest one: command a small attitude step in flight, compare it with the same step in the model; mismatch in rise time points to inertia or motor lag, mismatch in overshoot to damping or delay.

Mathematical formulation

Thrust and drag torque from rotor speed, first-order motor dynamics, and the discretisation:

Inertia about a horizontal axis from a pendulum test with period $P$ (small oscillations), mass $m$, suspension height $d$ from the centre of mass and the suspension’s own inertia negligible:

Motor time constant from a step response: the time to reach 63.2% of the final value. Integration: the fastest dynamics set the step; with $\tau _{m}=30\,\mathrm{ms}$ an explicit Euler step above about $2\tau _{m}$ is unstable, while the exact exponential update is stable for any $\Delta t$; choose $\Delta t\le \tau _{m}/10$ for accuracy of the rest.

Thrust, weight and the sign of gravity

The rotor model feeds the translational equation of lesson M1-T5. With $z$ up, the total thrust is $T = \sum_i k_T\,\Omega_i^2$ along the body $z$ axis and the equation is

The weight is not negative: its size $mg$ is positive (14.7 $N$ for a 1.5 kg vehicle). The minus sign is its direction. $\hat{\mathbf{z}} = [0,0,1]^{\top}$ points up and gravity pulls down, so the weight vector is $[0,0,-mg]^{\top}$. On the vertical axis, level thrust gives +T and the weight −mg, so $m\dot v_z = T - mg$. In a simulator this is one line, a = R @ [0, 0, T] / m - [0, 0, g]. In a z-down frame (NED) $\hat{z}$ points down, the weight is +mg $\hat{z}$ and thrust is −$T$ $\hat{z}$; the physics is the same, so pick one convention for the whole simulator and convert at the boundary.

Hover is the case $m\dot v_z = 0$: the four rotors must together produce $T = mg$. With equal rotor speeds this fixes the hover speed, and it is the number to check first when a simulated or real vehicle climbs or sinks at zero command:

$k_{T}$, $k_{Q}$

thrust and torque coefficients (N/(rad/s)$^{2}$, $N$·m/(rad/s)$^{2}$).

$\tau _{m}$

motor-ESC time constant.

$P$, $d$

pendulum period and distance from pivot to the centre of mass.

Architecture and data flow

Parameter file (YAML) shared by simulator, controller design scripts and planner; versioned with the airframe (lesson M7-T19). Gazebo vehicle models used with PX4 SITL have their own parameter sets: keep them aligned with your measured values if you want SITL to predict your flight.

Real-world UAV applications

• Tuning in simulation before the first flight.
• Checking that a payload change leaves the controller stable.
• Model-based estimators and MPC.
• Detecting a damaged motor from a changed time constant.

Implementation example

This runs as shown: inertia from a simulated pendulum test, motor time constant fitted from a noisy step response, and the stability of explicit Euler versus the exact update.

import numpy as np
rng = np.random.default_rng(0)
g = 9.81

# 1) inertia from a pendulum test: the airframe hangs on a pivot 0.12 m above its centre of mass
m, d, J_true = 1.5, 0.12, 0.020
P_true = 2 * np.pi * np.sqrt((J_true + m * d * d) / (m * g * d))                   # the period the experiment would give
P_meas = P_true * (1 + 0.005 * rng.standard_normal())                              # 0.5 % timing error
J_est = m * g * d * P_meas ** 2 / (4 * np.pi ** 2) - m * d * d
print(f"inertia: true {J_true:.4f}, estimated {J_est:.4f} kg m^2")
assert abs(J_est - J_true) / J_true < 0.10

# 2) motor time constant from a noisy step response (command 0 -> 1000 rad/s at t = 0)
tau_true, dt = 0.030, 0.002; t = np.arange(0, 0.3, dt); omega = 1000 * (1 - np.exp(-t / tau_true)) + 8 * rng.standard_normal(len(t))
A = np.stack([t, np.ones_like(t)], axis=1); y = np.log(np.clip(1 - omega / 1000, 1e-3, None)); m_ = t < 0.09                 # fit the early part of the decay
slope = np.linalg.lstsq(A[m_], y[m_], rcond=None)[0][0]; tau_est = -1 / slope
print(f"motor time constant: true {tau_true * 1000:.0f} ms, fitted {tau_est * 1000:.1f} ms")
assert abs(tau_est - tau_true) / tau_true < 0.2

# 3) explicit Euler versus the exact update for the first-order motor
def euler(dt, tau, n=200):
    w = 0.0
    for _ in range(n): w += (1000 - w) / tau * dt
    return w
def exact(dt, tau, n=200):
    w = 0.0
    for _ in range(n): w = 1000 + (w - 1000) * np.exp(-dt / tau)
    return w
assert abs(euler(0.001, 0.03, 300) - 1000) < 1.0 and abs(euler(0.07, 0.03) - 1000) > 1e6          # explicit Euler diverges when dt > 2 tau
assert abs(exact(0.07, 0.03) - 1000) < 1e-6                                                           # the exact update is stable for any step

Common engineering mistakes

Failure modes

Debugging approach

1. Compare step responses of roll rate between model and log.
2. Check hover throttle in the log against the model.
3. Run the simulation at half the step and see whether results change.
4. Plot ESC command and measured rpm to estimate $\tau _{m}$.

Practical exercises

1. Compute the inertia error if the centre-of-mass offset $d$ is mis-measured by 5 mm.
2. Design a step test for the yaw axis and the fit to extract $k_{Q}/k_{T}$.
3. Show that the exact update equals the analytic step response.
4. Measure your own vehicle’s hover throttle and compare with the model.

Interview questions

Write an answer in your own words, then reveal the model answer. Nothing is sent anywhere.

References and further reading

Primary papers, official documentation and standards. Each claim above that is specific to a tool or version points here.

• R. Mahony, V. Kumar, P. Corke, IEEE RAM 2012
• PX4: simulation (vehicle model parameters)
• $R$. $W$. Beard, $T$. $W$. McLain, Small Unmanned Aircraft, Princeton University Press, 2012, appendix on parameter identification.

What’s next