Multirotor With Flap Effects — Robotics/Rotorcraft

Multirotor With Flap Effects

Robotics / Rotorcraft

A four-rotor machine as one block, with each rotor's disc allowed to tilt: every rotor sees its own hub velocity, flaps its own way, and contributes its tilted thrust and its torque to one force and one moment about the vehicle's centre.

One rotor's thrust and torque, with k_T = C_T·R²·π·R² and k_Q = −C_Q·R³·π·R²:

• T = (k_T·Ω²)·ρ, so thrust does not care which way the rotor turns
• M_z = ρ·(k_Q·Ω·|Ω|), which does – a rotor spun backwards reverses its torque, and that is how a quadcopter's counter-rotating pairs cancel

With flap effects the rotor disc tilts, and the thrust tilts with it:

• F = T·[sin(a_1s)·(−cos(b_1s)); sin(b_1s); cos(b_1s)·(−cos(a_1s))]
• M = [0; 0; M_z] + arm × F

a_1s and b_1s are the steady-state longitudinal and lateral flapping angles, solved from the hub's own velocity V_r = V_b + ω_b × arm: the advance ratio μ = |V_r,xy|/(|Ω|·R), the inflow ratio λ_c = |V_r,z|/(|Ω|·R), the sideslip angle ψ = atan2(V_r,y, V_r,x) that rotates the pair back into body axes, the blade pitch, the Lock number and the hinge offset.

The sums are F = ΣF_i and M = Σ([0; 0; M_z,i] + arm_i × F_i), and the arm is what makes each rotor's hub velocity different: V_r,i = V_b + ω_b × arm_i.

Ports

• Omega – the four rotational speeds in rad/s, a [4,1] column in the arm order below; each sign is that rotor's sense of rotation.
• rho – ρ, the air density, [1,1], in kg/m³ or slug/ft³.
• Vb – the body velocity of the vehicle's centre, a [3,1] column in m/s, ft/s or knots, as the Units say.
• wb – the body angular velocity ω_b, a [3,1] column in rad/s. It moves every hub at a different velocity, so a rolling or pitching machine flaps its four discs differently.
• Fxyz – the summed force in body axes, a [3,1] column in N or lbf.
• Mxyz – the summed moment about the centre, a [3,1] column in N·m or lbf·ft.

Parameters

• Units – Metric (MKS), English (velocity in ft/s) or English (velocity in kts). The arithmetic is unit-consistent in both systems, so the first two run the same numbers; kts is the only one that converts anything – the body velocity, by 1852/3600/0.3048 ≈ 1.68781 ft/s per knot.
• Configuration – where the four arms point, with l the arm length and h the rotor height:
  • Quadcopter X – the diagonals: [l/√2, −l/√2, h], [l/√2, l/√2, h], [−l/√2, l/√2, h], [−l/√2, −l/√2, h]
  • Quadcopter + – the axes: [l, 0, h], [0, l, h], [−l, 0, h], [0, −l, h]
• Arm Length – l, the distance from the centre to a rotor hub, in the length unit.
• Rotor Height – h, the rotors' common z in body axes (negative is above the centre), in the length unit.
• Coefficient Source – where the two coefficients come from:
  • Dialog – the two values below, used as given
  • Compute using BEMT – blade element momentum theory over the blade geometry below, solved once when the configuration is loaded, so the export carries the resulting pair as constants
• Thrust Coefficient – C_T, used when the source is Dialog: one number for all four rotors, or a four-element vector, one per rotor. It also reaches each rotor's flapping, through the coning term.
• Torque Coefficient – C_Q, the same two shapes.
• Number Of Blades – N_b, used by the momentum solver (through the solidity σ = N_b·c/(π·R) and Prandtl's tip loss).
• Blade Radius – R, in the length unit; it sets the whole scale of the block, thrust going as R⁴ and torque as R⁵.
• Blade Chord – c, in the length unit.
• Hinge Offset – e, the flapping hinge's distance from the shaft, in the length unit; at least zero and less than the radius.
• Lift Curve Slope – the blade section's lift slope per radian.
• Mean Drag Coefficient – c_d0, the profile drag the momentum solver adds to the torque coefficient as σ·c_d0/8. It reaches nothing else.
• Lock Number – γ, the ratio of the blade's aerodynamic to inertial flapping moments. It reaches the flapping only, and it is in every one of its hinge-offset terms.
• Twist Distribution – Linear or Ideal, which decides the one blade pitch the block uses: (2/3)·θ_root + (1/2)·θ_twist on a Linear blade, θ_tip on an Ideal one.
• Blade Root Angle – θ_root, in radians (Linear twist).
• Blade Twist Angle – θ_twist, in radians (Linear twist).
• Blade Tip Angle – θ_tip, in radians (Ideal twist).
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. Every parameter is baked into the arithmetic at export, the two coefficients included – so a core exported with Compute using BEMT carries the pair the solver found and does not re-solve anything at run time.

The three HDL targets are simulation-only: an arctangent, a square root, two divisions and the trigonometry of the flapping angles does not belong in a Q16.16 datapath, so they compute in real and quantize only at the ports – which also bounds what can cross one, ±32768. PLC Structured Text has no ATAN2 in IEC 61131-3, so where one is needed it is rebuilt from ATAN(y/x) with the quadrant reconstruction.

The four rotors are written out one after another rather than looped, so every arm and every coefficient is a constant in the emitted core.

Simulink bridge

Export only, onto Aerospace Blockset's aerolibrotordyn/Multirotor with modelMode fixed at With flap effects. The Multirotor block owns the import of that library block, because two entries claiming one path can only differ by a parameter the importer cannot see. Units → units, Configuration → config, Arm Length → armLength, Rotor Height → rotorHeight, Coefficient Source → CTCQMode, Thrust Coefficient → CT, Torque Coefficient → CQ, Number Of Blades → Nb, Blade Radius → radius, Blade Chord → chord, Hinge Offset → hingeOffset, Lift Curve Slope → clalpha, Mean Drag Coefficient → cd0, Lock Number → gamma, Twist Distribution → twistType, Blade Root Angle → thetaRoot, Blade Twist Angle → thetaTwist and Blade Tip Angle → thetaTip. controlInput = off and CTout = off are written fixed, because each adds a port; Simulink's Ports coefficient source is not offered for the same reason. The Simulink block has no SampleTime (measured), so the rate stays on the ICore side.

Notes

• Algebraic and stateless; nonlinear, so no state space.
• Measured, not derived. Every rule above was read off Aerospace Blockset R2026a's own block, which is a built-in block type whose source does not ship, and the transcription agrees with it to 1e-16 on about 1600 random states.
• The inflow ratio uses the MAGNITUDE of the hub's vertical velocity, so a climb and a descent of the same speed flap identically. Measured on the Simulink block.
• A rotor at Ω = 0 contributes zero, not a NaN: its flapping is undefined (every term divides by |Ω|) and its thrust is zero anyway. The Simulink block answers zero too.
• Four rotors, always. Both Simulink configurations are quadcopters; the rotor count is not a parameter on either side.
• A negative density is not refused here. The Simulink block stops the run on one.