# Regularity-Aware Thrust Head

- ID: 1918
- Canonical URL: https://synthcore.org/idea/1918/regularity-aware-thrust-head
- API JSON: https://synthcore.org/api/idea/1918.json
- API Markdown: https://synthcore.org/api/idea/1918.md
- Verification status: unverified
- Source: [arXiv:2608.06991](https://arxiv.org/abs/2608.06991)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: rl, loss, training-dynamics
- Math tags: dynamical-systems, control-theory, approximation-theory
- Ratings: usefulness 6/10; difficulty 5/10; novelty 8/10

## Idea description

Add an actuator-aware output head to a neural controller that prevents learned thrust references from making generic linear zero crossings. The network predicts a smooth latent reversal coordinate, and thrust is generated with a quadratic signed map, or the training loss penalizes the motor input implied by the predicted thrust trajectory.

## Mathematical statement

The actuator model is $\dot{\omega}=u-k\omega|\omega|$, where $\omega\in\mathbb{R}$ is signed rotor speed, $u=\tau_m/J$ is normalized motor torque, $k=c_\tau/J>0$ is normalized aerodynamic drag, and thrust is $F=c_f\omega|\omega|$ with $c_f>0$. Inverting the thrust map gives $\omega(F)=\operatorname{sign}(F)\sqrt{|F|/c_f}$ and, away from $F=0$, $\dot{\omega}=\dot F/(2\sqrt{c_f|F|})$. Therefore the motor input required to realize a thrust trajectory is $u_{\mathrm{req}}(F,\dot F)=\dot F/(2\sqrt{c_f|F|})+kF/c_f$. For a reversal $F(t)\sim a(t-t_0)^r$, the speed derivative scales as $|t-t_0|^{r/2-1}$: a generic linear crossing with $r=1$ requires unbounded input, whereas an odd cubic crossing with $r=3$ has bounded, vanishing speed derivative. The neural adaptation either predicts a latent speed $s$ and sets $F=c_fs|s|$, constraining $s$ to cross smoothly, or directly regularizes the explicit $u_{\mathrm{req}}$ formula.

## Key formulas

- $$\dot{\omega}=u-k\omega|\omega|,\qquad F=c_f\omega|\omega|,$$
- $$\omega(F)=\operatorname{sign}(F)\sqrt{|F|/c_f},\qquad \dot{\omega}=\frac{\dot F}{2\sqrt{c_f|F|}},$$
- $$u_{\mathrm{req}}(F,\dot F)=\frac{\dot F}{2\sqrt{c_f|F|}}+\frac{kF}{c_f},$$
- $$F(t)=A\,\operatorname{sign}(t-t_0)|t-t_0|^3\quad\Longrightarrow\quad \omega(t)=\operatorname{sign}(t-t_0)\sqrt{A/c_f}\,|t-t_0|^{3/2}$$

## Implementation notes

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.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.