Humanoid Reaction Wheel and CMG Balance Sizing

The centroidal-inertia description follows Lee and Goswami (2007), especially the planar RMP in section III.B, attached PDF page 4. Here its leg length and body inertia are fixed, and lean θ is measured from upright. A separate physical rotor exchanges momentum with the body; this is not the paper's full variable-inertia humanoid mapping.

For a horizontal short impulse J at distance ℓp along the body, the support angular impulse is Jℓp cosθ₀. The body rate immediately afterward is ω₀ = Jℓp cosθ₀ / Iₚ, with Iₚ = Ibody + Mℓ². Alternatively ω₀ = vslip/ℓ for the tangential post-slip speed. The body's initial centroidal momentum is Ibody ω₀; support angular impulse and centroidal momentum are different quantities.

A quintic θd(t) matches initial angle and rate, starts with zero angular acceleration, and ends at θ = θ̇ = θ̈ = 0 at the chosen recovery time. The nonlinear inverse torque is u = Mgℓ sinθd − Iₚ θ̈d. Trapezoidal integration gives ΔH(t) = ∫u dt on 801 time points, including gravity throughout recovery.

An annular rotor has Iw = mw Ro²(1 + bore²)/2. A reaction wheel stores H = Iw ωabsolute; its bearing speed is H/Iw − θ̇. Both signed speed limits, motor torque and symmetric motoring/regeneration shaft-power limits are checked. Initial wheel speed is specified before the impulse, while the body is stationary.

For the scissored CMG pair, each rotor has h = Iw Ω and the useful momentum is H = 2h sinδ. Therefore body reaction torque is −2h cosδ δ̇: the finite-angle projection is retained. This is the momentum-vector precession relation treated by Wie (2008), with the paired arrangement verified by Park and Cho (2018), equations (3)–(5). With the same sign convention, reciprocal per-gimbal drive torque is Q = Ig δ̈ − h θ̇ cosδ, consistent with Xu et al. (2022), equation (3). At zero drive, δ̈ = h θ̇ cosδ / Ig: the resulting gyroscopic power gained by the gimbals equals the power lost by the body.

Minimum inertia enforces both signed wheel speed boundaries, or both CMG stroke boundaries and the instantaneous gimbal-rate boundary. The latter is solved analytically as a quadratic inequality in 2IwΩ, retaining the initial gimbal angle. The CMG drive-torque check remains separate because increasing inertia also increases gyroscopic drive load. The diameter maps that inertia onto the entered mass and bore ratio.

The achieved motion solves Iₚ θ̈ = Mgℓ sinθ − u and Ḣ = u using adaptive Dormand–Prince integration (ode45, relative tolerance 10⁻⁶), with feedforward and critically damped tracking at natural frequency 6/T. CMG rate servo dynamics, assembled rotor-plus-gimbal inertia and drive torque saturation are included. The requested gimbal rate is limited using the rated speed and directional braking distance. Actual acceleration always follows δ̈ = (Q + h θ̇ cosδ)/Ig with |Q| ≤ Qmax. If the drive cannot restrain the actual rate, the simulation ends at the rated operating boundary and reports the loss of rate control. Ground reactions follow the exact circular CoM acceleration. The simulation continues to 1.5T or stops at 45° lean, loss of normal contact, CMG stroke exhaustion or an uncontrollable gimbal rate. Initial normal force is checked before a recovery can start. Delivered torque and gimbal acceleration are evaluated directly from the dynamics, rather than differentiating sampled momentum. The plotted duration sweep solves 81 independent recovery plans.

One balance plane, one stationary point support, rigid constant-length body and constant body centroidal inertia. The wheel or symmetric CMG pair is centered at the CoM. No active ankle moment or stepping is credited. The short disturbance acts before the recovery clock starts, with no impulsive motor torque. Slip mode begins after contact has stopped sliding. Positive disturbance and lean share a sign.

CMG rotors are axially symmetric. Spin is regulated and opposite in the two units; gimbals mirror each other. The gimbal-inertia input is the complete rotor-plus-frame inertia about the gimbal axis. It must be at least Iw/2; rotor thickness and frame inertia add to that minimum. Rotor transverse and housing inertia are included in the user-supplied body inertia and treated as constant. The standard high-spin CMG approximation omits nutation, flexibility, spin regulation errors and rotor transverse-inertia variation with gimbal angle. Motor torque and mechanical power limits are symmetric. The feedback controller and recovery trajectory are derived for this tool, not claimed as Lee–Goswami's inertia-shaping controller.

A planned actuator limit pass applies only to this trajectory. The separate recovery verdict requires final lean within 0.5°, body rate within 1°/s, positive normal contact, sufficient entered friction and completion within the gimbal rate and stroke envelope. If the friction check fails, the fixed-contact trajectory is infeasible; sliding is not integrated. Gimbal drive demand is an ideal inverse-trajectory value; the actual servo starts at zero rate and its startup transient is included in the forward simulation.

The minimum rotor inertia covers speed/stroke and gimbal rate only. Check torque, shaft power and gimbal drive demand as well. Enlarging a rotor changes complete robot mass and body inertia: update those inputs and repeat. Equivalent reaction-wheel power and CMG drive outputs permit comparison; the two verdicts apply to the selected actuator. Clipped inverse CMG angles stop at 90° when momentum is impossible; the stroke utilization still identifies the infeasible requirement.

Recovery is one event followed by a short upright hold. Residual momentum must be removed through an external torque, ankle strategy or stepping; a wheel or CMG cannot remove net system momentum by itself. Rotor stress, containment, fatigue, balancing, bearings, electronics, energy conversion and repeated-event heating require separate design.

The adaptive ODE solver uses relative tolerance 10⁻⁶ and reports 1201 samples over 1.5T. An internal physical clock stops at the first contact, lean, CMG stroke or uncontrollable-rate boundary; solver interpolation can overshoot a boundary slightly within tolerance. The rate controller limits its command and its motor torque, while passive precession remains in the acceleration equation. Reaching a gimbal boundary ends the rated recovery trajectory; stop impacts and operation beyond the rated rate are not calculated. The wheel speed controller uses a look-ahead of at most 1.5 ms. The inverse plan has 801 points, and the 81-point duration sweep uses 401 integration points per plan. PCHIP interpolates the cached trajectory for animation only. The view is a normalized schematic; dimension labels carry physical size.