Regularity-Aware Thrust Head

Integrate this method at the final thrust-output head of a neural policy or learned aerial-interaction controller. Let the policy receive the current state and output either latent rotor-speed commands $s_t$ or a short horizon of $H$ future thrust commands. The preferred implementation is: (1) produce unconstrained network outputs $z_{t:t+H}$; (2) fit a cubic Hermite or cubic B-spline interpolant $s(\tau)$ over the horizon, imposing bounded derivatives and zero slope at detected sign changes; (3) generate thrust by $F(\tau)=c_fs(\tau)|s(\tau)|$; and (4) execute the first command in receding-horizon fashion. The simpler MVP retains direct thrust outputs $F_t$, estimates $\dot F_t=(F_t-F_{t-1})/\Delta t$, and computes $u_{\mathrm{req},t}=\dot F_t/[2\sqrt{c_f(|F_t|+\epsilon)}]+kF_t/c_f$. Add $\lambda_u\operatorname{Huber}(u_{\mathrm{req},t},u_{\max})$ to the policy or supervised-control loss, plus a reversal penalty activated when $F_{t-1}F_t<0$. Estimate $c_f$ and $k$ from motor-propeller calibration; estimate $\dot F$ by finite differences or a differentiable Savitzky-Golay filter. The paper-derived quantities are the inverse map and implied input; $\epsilon$ is only numerical stabilization. First test a small MLP or PPO controller on a simulated single bidirectional rotor tracking sinusoidal force references with random reversal times. Compare direct thrust, tanh-bounded thrust, and the proposed head at equal parameter count and training budget. Success is lower peak and 99th-percentile $|u|$, fewer motor saturations, lower tracking error in a window around reversals, and equal or higher overall reward without degrading non-reversal tracking.