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climbyaw

fwdroll

Attitude (deg): actual / commanded

Rotor speeds (rad/s) and limits

Battery voltage and state of charge

Electrical power vs airspeed (W)

Quadcopter Flight Dynamics Lab

6-DOF rigid body · rotor thrust & mixer · cascaded PID · wind · ground effect · battery endurance. A generic civilian survey / rescue multirotor, educational flight physics only.

Fly

Hover altitude

Set-point step:

Manual: W/S pitch, A/D roll, ↑/↓ climb, ←/→ yaw (or the two on-screen sticks). Angle mode: sticks command tilt angle, climb rate and yaw rate; releasing the sticks levels the aircraft and holds altitude.

Parcel mass carried (never released)

Survey rows Grid size (m) Survey altitude (m) Survey speed (m/s)

point the nose along the track

Rotor failure:

force vectors flight trail gimbal camera inset

Tune the controller

Cascade: position PD → desired thrust vector → attitude (geometric) P → body-rate PID → mixer. The outer loop is a second-order system with natural frequency ω_n and damping ζ; the step test compares the real 6-DOF response with that ideal double-integrator answer.

Position ωn (rad/s) Position ζ Altitude ωn Attitude P (1/s) Rate Kp Rate Ki Rate Kd Max tilt (deg) Position Ki

drag feed-forward (cancels the quadratic body drag)

Wind, air and ground

Steady wind (m/s) Wind from heading (deg) Turbulence σ (m/s) Gust amplitude (m/s)

Density altitude (m) ground effect (Cheeseman–Bennett)

Wind direction is the heading the wind blows from (0 = north, 90 = east).

Experiments: measured vs theory

1. Hover thrust and power vs formula

Closed-loop hover for 8 s: mean thrust per rotor vs mg/n and electrical power vs n·T^1.5/√(2ρA)/FM/η+P_av.

2. Ground effect vs Cheeseman–Bennett

Thrust at fixed rotor speed measured from the vehicle's acceleration at several heights, divided by the out-of-ground-effect value.

3. Position step vs linearised double integrator

Full nonlinear 6-DOF step (with your gains) vs the analytic second-order response; see the Tune section for the plot.

4. Energy bookkeeping

75 s of manoeuvring: ∫P dt + ∫I²R dt vs the energy implied by the state-of-charge drop (independent quadrature of the open-circuit-voltage curve).

5. Best-endurance and best-range speed

Steady flights at 0–16 m/s in the full simulation; parabola through the lowest points vs the analytic minimum of P(V) and P(V)/V. Points appear in the power graph.

6. Torque-free rigid body (integrator check)

Motors off in vacuum: kinetic energy and |L| must stay constant and the body-frame precession rate must equal (I_zz−I_xx)/I_xx·ω_z.

7. Endurance to the 20 % reserve

Hover until the pack reaches 20 % SoC (computed at 100 Hz in a second or two) vs E_usable/P_hover with and without the I²R loss.

The science

Rigid body

State: position p, velocity v (world), attitude quaternion q, body rates ω. Integrated with RK4 at 400 Hz.

m v̇ = R(q)[ F_drag,b + (0,0,ΣT_i) ] − m g ẑ I ω̇ = τ − ω × Iω − ω × h_rotors q̇ = ½ q ⊗ (0,ω) I = diag(I_xx, I_xx, I_zz)

Rotors and mixer

T_i = k_T ω_i² Q_i = k_Q ω_i² = c_Q T_i k_T ∝ ρ ω̇_i = (ω_cmd − ω_i)/τ_m (τ_m = 30 ms motor lag) τ_x = Σ y_iT_i τ_y = −Σ x_iT_i τ_z = −Σ s_iQ_i − J_rΣ s_iω̇_i

The four commands (thrust, roll, pitch, yaw torque) are mapped to rotor thrusts with the pseudo-inverse of the 4×n mixing matrix; if a rotor would leave [T_min, T_max] the yaw demand is given up first, then roll/pitch authority is reduced (priority: thrust > roll/pitch > yaw), and integrators are frozen (anti-windup). Max rotor speed falls with pack voltage (ω_max ∝ V).

Cascaded control

a = K_p(p_ref−p) + K_d(v_ref−v) + K_i∫e, K_p = ω_n², K_d = 2ζω_n f = m(a + g ẑ) − F_drag(v) T = f·z_b, z_b,des = f/|f| e_R = ½ (R_dᵀR − RᵀR_d)^∨ ω_des = −K_att e_R α = K_p(ω_des−ω) + K_i∫(ω_des−ω) − K_dω̇ τ = Iα + ω×Iω

Geometric attitude error after Lee, Leok & McClamroch (2010); the position/attitude structure follows Mellinger & Kumar (2011). For small angles and a fast inner loop, the horizontal channel reduces to ẍ = K_p(x_ref−x) − K_dẋ, the double integrator used in check 3.

Ground effect

T_IGE/T_OGE = 1 / (1 − (R/4z)²) (Cheeseman & Bennett 1955, constant rotor power)

Applied per rotor using its own height, so a tilted aircraft feels a rolling moment near the floor. The classic form is an empirical single-rotor law; multirotor overlap and wall effects are ignored, and it is clamped for z < R/2.

Power, battery and endurance

hover: P_ideal = T^3/2/√(2ρA) P = P_ideal/FM (FM = 0.65 figure of merit) forward flight: v_i = v_h²/√(V_∥² + (V_⊥+v_i)²) (Glauert) P_rotor = κT(v_i+V_⊥) + c_pT^3/2/√(2ρA)(1+4.65μ²), c_p = 1/FM − κ P_elec = ΣP_rotor/η_m + P_avionics V_pack = n_s·OCV(SoC) − I R_int, P = V I ⇒ I = (OCV − √(OCV² − 4RP))/2R t_end = 0.8 E_pack / P_hover

The P(V) curve is the trimmed level-flight solution (thrust tilts to balance quadratic drag). The best-endurance speed minimises P(V); the best-range speed minimises P(V)/V (tangent from the origin). At altitude ρ falls, so hover power rises as ρ^−1/2 and the rotors must spin faster.

Wind

Steady wind + first-order Gauss–Markov turbulence (time constant 2 s, vertical component 0.6×), a stand-in for Dryden turbulence (Beard & McLain), and a 1−cos discrete gust. The airframe drag uses the air-relative velocity.

One motor out

A quad has four unknowns for four demands, so losing a rotor removes a degree of freedom: this controller gives up yaw (the nose spins, limited only by aerodynamic yaw damping) and tries to hold position. A hexacopter with the pseudo-inverse keeps all four controls: with rotor k gone the opposite rotor must drop to zero thrust, which is exactly the boundary of the feasible set, so there is no margin for further disturbance. Real quad survivors use special reduced-attitude controllers (Mueller & D'Andrea 2014); this lab uses the ordinary one on purpose.

What is simplified

No blade flapping, no rotor-hub forces, no vortex-ring-state model in descent (induced velocity is clamped), constant air density with height for a given density-altitude slider, no propeller inflow lag, identical motors, perfect state estimation (no sensor noise or delay). Frame inertia and drag areas are generic. The aircraft is a generic civilian survey / aid-logistics platform and the parcel mass is only ballast; this is a flight-physics teaching model.

Self-checks in this page

Hover thrust and power vs formula, ground effect vs Cheeseman–Bennett, the 6-DOF step response vs the linearised double integrator, energy bookkeeping, best-endurance speed vs the analytic minimum, torque-free precession, and endurance vs E/P. All run the same engine headless.

References: Lee, Leok & McClamroch, CDC 2010; Mellinger & Kumar, ICRA 2011; Cheeseman & Bennett, ARC R&M 3021 (1955); Leishman, Principles of Helicopter Aerodynamics; Beard & McLain, Small Unmanned Aircraft; Mueller & D'Andrea, ICRA 2014. 3D models and terrain built procedurally in Blender 5.2 with baked textures.