# Residual-authority policy shield

- ID: 1159
- Canonical URL: https://synthcore.org/idea/1159/residual-authority-policy-shield
- API JSON: https://synthcore.org/api/idea/1159.json
- API Markdown: https://synthcore.org/api/idea/1159.md
- Verification status: unverified
- Source: [arXiv:2607.19708](https://arxiv.org/abs/2607.19708)
- Category: regularization
- Solves: stability, accuracy, generalization
- ML areas: rl, loss, regularization, optimizer
- Math tags: convex-analysis, geometry, optimization, control-theory
- Ratings: usefulness 6/10; difficulty 5/10; novelty 6/10

## Idea description

Augment a neural controller with a differentiable residual-authority margin that measures the distance between the requested task wrench and the boundary of the actuator-feasible wrench polytope. During training, penalize commands with small margin; during deployment, project the policy output onto the largest-margin feasible wrench that remains close to the requested output. This should reduce saturation-induced failures during sustained contact and improve robustness to disturbances that consume available control authority.

## Mathematical statement

Let u in R^n be actuator commands with componentwise limits l_j <= u_j <= h_j, let A in R^(6 x n) be the morphology-dependent wrench map, and define the feasible wrench polytope W = {w in R^6 : w = A u, l <= u <= h}. In half-space form, W = {w : H w <= b}, where H_i is facet normal i and b_i is its offset. For a requested task wrench w_t, the signed residual margin is rho(w_t) = min_i (b_i - H_i w_t)/||H_i||_2. Positive rho means that w_t is interior, zero means boundary operation, and negative rho means infeasibility; when positive, rho is the Euclidean radius of the largest disturbance ball contained in W around w_t. The residual set available for stabilization is R(w_t) = W - w_t = {r : r = A u - w_t, l <= u <= h}. To avoid hidden actuator saturation, compute a slack-maximizing allocation u* using the scalar eta, where eta is the minimum normalized distance of every actuator command from either limit.

## Key formulas

- $$\mathcal{W}=\{\mathbf{w}=A\mathbf{u}\in\mathbb{R}^{6}:\ \boldsymbol{\ell}\leq\mathbf{u}\leq\mathbf{h}\}=\{\mathbf{w}:H\mathbf{w}\leq\mathbf{b}\}.$$
- $$\mathcal{R}(\mathbf{w}_{t})=\mathcal{W}-\mathbf{w}_{t}=\{\mathbf{r}:\mathbf{r}=A\mathbf{u}-\mathbf{w}_{t},\ \boldsymbol{\ell}\leq\mathbf{u}\leq\mathbf{h}\}.$$
- $$\rho(\mathbf{w}_{t})=\min_i\frac{b_i-H_i\mathbf{w}_{t}}{\|H_i\|_2}=\operatorname{dist}_2(\mathbf{w}_{t},\partial\mathcal{W}).$$
- $$\mathbf{u}^{\star}=\arg\max_{\boldsymbol{\ell}\leq\mathbf{u}\leq\mathbf{h},\ A\mathbf{u}=\mathbf{w}_{t}}\ \eta\quad\text{subject to}\quad \eta\leq\frac{u_j-\ell_j}{h_j-\ell_j},\ \eta\leq\frac{h_j-u_j}{h_j-\ell_j}\ \forall j.$$$

## Implementation notes

Integrate this at the output of a neural aerial controller, before the low-level actuator allocator. The policy receives vehicle state, contact-force estimates, and a task command, and outputs a six-dimensional desired wrench w_pi. For a fixed vehicle, enumerate the 2^n actuator-box vertices u in {l_j,h_j}^n, map each vertex through A, and compute a convex-hull half-space representation H w <= b. At each control step, calculate rho = min_i (b_i - H_i w_pi)/||H_i||. Train with L = L_task + lambda_rho softplus((rho_target-rho)/tau)^2, where rho_target is the desired disturbance tolerance and tau smooths the hinge. At inference, solve min_w ||w-w_pi||_2^2 - gamma s subject to H w <= b - epsilon and s <= (b_i-H_i w)/||H_i|| for every facet i; return w and then solve the actuator allocation LP maximizing eta. If contact consumes a known task wrench w_t, apply the margin to the residual command or equivalently use the translated polytope W-w_t. The polytope, facet-distance formula, and allocation constraints are computed exactly; A, actuator limits, contact wrench, and rho_target are supplied by the simulator or vehicle model. First test a small PPO or SAC controller on tilted-hexarotor wall pushing, comparing no shield, margin loss only, and QP shield. Measure contact-loss rate, tracking error, actuator saturation, reward, and minimum rho under disturbances. Success means fewer saturation/contact failures and a positive rho maintained during sustained pushing at comparable task reward.

## Disclaimer

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