# Polar-Backstepping Policy Residual

- ID: 1445
- Canonical URL: https://synthcore.org/idea/1445/polar-backstepping-policy-residual
- API JSON: https://synthcore.org/api/idea/1445.json
- API Markdown: https://synthcore.org/api/idea/1445.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.26442](https://arxiv.org/abs/2607.26442)
- Category: dynamics
- Solves: stability, accuracy
- ML areas: rl, world-model, training-dynamics, regularization
- Math tags: control-theory, dynamical-systems, geometry, differential-geometry
- Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10

## Idea description

Represent car-like navigation states in the paper's polar coordinates and make a neural policy predict only a residual around an analytic backstepping controller. Add a Lyapunov-decrease penalty so the learned residual can improve trajectory quality without destroying the nominal parking attractor.

## Mathematical statement

For a target at the origin, define \(\rho=\sqrt{x^2+y^2}\), \(\delta=\operatorname{atan2}(y,x)+\pi\), and \(\gamma=\delta-\theta\), where \(\rho\) is range, \(\delta\) is the line-of-sight angle, and \(\gamma\) is heading error. Away from the coordinate cut and target singularity, the kinematic bicycle equations imply \(\dot\rho=-v\cos\gamma\), \(\dot\delta=v\sin\gamma/\rho\), \(\dot\gamma=v\sin\gamma/\rho-(v/L)\tan\phi\), and \(\dot v=a\), with speed \(v\), wheelbase \(L\), acceleration \(a\), and steering angle \(\phi\). Use a positive definite local Lyapunov function \(V=\tfrac12(q_\rho\rho^2+q_\gamma\gamma^2+q_v(v-v_d(\rho))^2)\). Penalize positive drift \([\dot V+\lambda V]_+\), where \(q_\rho,q_\gamma,q_v,\lambda>0\), so the desired certificate is \(\dot V\le-\lambda V\). A bounded nominal choice is \(v_d=k_\rho\rho\), \(a_0=-k_v(v-v_d)-k_\rho v\cos\gamma\), and \(\tan\phi_0=L\sin\gamma/(\rho+\epsilon)+Lk_\gamma\gamma/(v+\epsilon)\), with positive gains \(k_\rho,k_v,k_\gamma\) and safeguard \(\epsilon>0\).

## Key formulas

- $$\rho=\sqrt{x^2+y^2},\qquad \delta=\operatorname{atan2}(y,x)+\pi,\qquad \gamma=\delta-\theta$$
- $$\dot\rho=-v\cos\gamma,\qquad \dot\gamma=\frac{v\sin\gamma}{\rho}-\frac{v}{L}\tan\phi,\qquad \dot v=a$$
- $$V=\frac12\left(q_\rho\rho^2+q_\gamma\gamma^2+q_v(v-k_\rho\rho)^2\right),\qquad \mathcal L_{\mathrm{stab}}=\left[\dot V+\lambda V\right]_+^2$$
- $$a=a_0+\Delta a_\vartheta(s),\qquad \phi=\operatorname{atan}\!\left(\tan\phi_0+\Delta u_{\vartheta}(s)\right)$$

## Implementation notes

1. Integration point: use this controller as a reinforcement-learning or imitation-learning policy for a kinematic bicycle simulator. At every step compute \(s=(\rho,\sin\gamma,\cos\gamma,v,1/\rho_\epsilon)\), where \(\rho_\epsilon=\max(\rho,\epsilon)\). The network outputs bounded residuals \((\Delta a,\Delta u)\) through tanh heads. The analytic controller computes \((a_0,\phi_0)\), and the final actions are clipped to actuator limits. 2. Pseudocode: compute the polar state; compute \(v_d=k_\rho\rho\); compute \(a_0=-k_v(v-v_d)-k_\rho v\cos\gamma\); compute \(u_{\phi0}=L\sin\gamma/\rho_\epsilon+Lk_\gamma\gamma/(|v|+\epsilon)\); evaluate the MLP residual; apply the actions; estimate \(\dot\rho,\dot\gamma,\dot v\) from the analytic equations or finite differences; calculate \(\dot V\); and optimize task loss plus \(\beta[\dot V+\lambda V]_+^2\). 3. Computed from the paper: the polar geometry, transformed dynamics, and strict-feedback structure. Estimated empirically: \(\lambda\), gains, residual bounds, and the fraction of sampled states satisfying the certificate. 4. First cheap experiment: train a two-layer MLP on randomly generated parking goals in a bicycle simulator, comparing residual training against a Cartesian policy and an unconstrained polar policy. Use \(L=2.5\), \(\epsilon=0.05\), and measure \(V(t)\), success rate, and the largest initial-error radius for which \(V(t+1)\le e^{-\lambda\Delta t}V(t)\) holds. The quantitative prediction is an approximately linear semilog decay of \(V\) with slope no greater than \(-\lambda\) over the certified region. Removing the stability loss should sharply increase positive-\(\dot V\) events, particularly when \(|v|<0.2\) or \(\rho<0.5\).

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: Built a fixed-seed polar backstepping residual-policy MVP with bounded MLP residuals, Lyapunov drift penalty, exact polar dynamics, and three parameter sweeps. All three mechanism predictions were confirmed: speed-error decay matched k_v with identity error below 2.3e-16, steering decay matched k_gamma*v/(|v|+epsilon) with relative error below 6.2e-15, and the epsilon transition matched the predicted 1/r-1/epsilon residual below epsilon while vanishing above it. In the toy policy comparison, the Lyapunov penalty reduced positive-drift states from 13.09% to 10.89% and improved mean drift-plus-lambda-V from -0.135 to -0.206, but increased task MSE from 0.00071 to 0.0611, so the stability effect is real but comes with a clear accuracy tradeoff.

### Mechanism check

- Verdict: Built a fixed-seed polar backstepping residual-policy MVP with bounded MLP residuals, Lyapunov drift penalty, exact polar dynamics, and three parameter sweeps. All three mechanism predictions were confirmed: speed-error decay matched k_v with identity error below 2.3e-16, steering decay matched k_gamma*v/(|v|+epsilon) with relative error below 6.2e-15, and the epsilon transition matched the predicted 1/r-1/epsilon residual below epsilon while vanishing above it. In the toy policy comparison, the Lyapunov penalty reduced positive-drift states from 13.09% to 10.89% and improved mean drift-plus-lambda-V from -0.135 to -0.206, but increased task MSE from 0.00071 to 0.0611, so the stability effect is real but comes with a clear accuracy tradeoff.
- Confidence: 8/10
- Limitations: This is a sampled one-step toy study rather than full bicycle trajectory navigation; no success-rate or long-horizon parking evaluation was performed, and no Cartesian-policy baseline was implemented. The certificate is evaluated on sampled states and uses clipped actions, so it is not a formal global stability guarantee.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/565/bench_report.json)
- [polar_backstepping_experiment.py](https://synthcore.org/code/565/polar_backstepping_experiment.py)
- [report.md](https://synthcore.org/code/565/report.md)
- [report_bench_2026-08-31T180039.md](https://synthcore.org/code/565/report_bench_2026-08-31T180039.md)
- [results.json](https://synthcore.org/code/565/results.json)
- [stage2_bench.py](https://synthcore.org/code/565/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/565)

## Disclaimer

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