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Statistical Contraction for Chance-Constrained Trajectory Optimization of Non-Gaussian Stochastic Systems

Rihan Aaron D'Silva, Hiroyasu Tsukamoto

TL;DR

A novel method for distribution-free robust trajectory optimization and control of discrete-time, nonlinear, and non-Gaussian stochastic systems, with closed-loop guarantees on chance constraint satisfaction, with statistical guarantees that are non-diverging and can be computed with finite samples of the underlying uncertainty.

Abstract

This paper presents novel method for distribution-free robust trajectory optimization and control of discrete-time, nonlinear, and non-Gaussian stochastic systems, with closed-loop guarantees on chance constraint satisfaction. Our framework employs conformal inference to generate coverage-based confidence sets for the closed-loop dynamics around arbitrary reference trajectories, by constructing a joint nonconformity score to quantify both the validity of contraction (i.e., incremental stability) conditions and the impact of external stochastic disturbance on the closed-loop dynamics, without any distributional assumptions. Via appropriate constraint tightening, chance constraints can be reformulated into tractable, statistically valid deterministic constraints on the reference trajectories. This enables a formal pathway to leverage and validate learning-based motion planners and controllers, such as those with neural contraction metrics, in safety-critical real-world applications. Notably, our statistical guarantees are non-diverging and can be computed with finite samples of the underlying uncertainty, without overly conservative structural priors. We demonstrate our approach in motion planning problems for designing safe, dynamically feasible trajectories in both numerical simulation and hardware experiments.

Statistical Contraction for Chance-Constrained Trajectory Optimization of Non-Gaussian Stochastic Systems

TL;DR

A novel method for distribution-free robust trajectory optimization and control of discrete-time, nonlinear, and non-Gaussian stochastic systems, with closed-loop guarantees on chance constraint satisfaction, with statistical guarantees that are non-diverging and can be computed with finite samples of the underlying uncertainty.

Abstract

This paper presents novel method for distribution-free robust trajectory optimization and control of discrete-time, nonlinear, and non-Gaussian stochastic systems, with closed-loop guarantees on chance constraint satisfaction. Our framework employs conformal inference to generate coverage-based confidence sets for the closed-loop dynamics around arbitrary reference trajectories, by constructing a joint nonconformity score to quantify both the validity of contraction (i.e., incremental stability) conditions and the impact of external stochastic disturbance on the closed-loop dynamics, without any distributional assumptions. Via appropriate constraint tightening, chance constraints can be reformulated into tractable, statistically valid deterministic constraints on the reference trajectories. This enables a formal pathway to leverage and validate learning-based motion planners and controllers, such as those with neural contraction metrics, in safety-critical real-world applications. Notably, our statistical guarantees are non-diverging and can be computed with finite samples of the underlying uncertainty, without overly conservative structural priors. We demonstrate our approach in motion planning problems for designing safe, dynamically feasible trajectories in both numerical simulation and hardware experiments.
Paper Structure (8 sections, 5 theorems, 18 equations, 3 figures)

This paper contains 8 sections, 5 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Setup & Preliminaries
  3. Uncertainty Quantification & Contraction
  4. Deterministic Reformulation
  5. Case Studies
  6. Numerical Simulations
  7. Hardware Experiments
  8. Conclusion

Key Result

Lemma 1

Consider a uniformly bounded matrix function $M_k \coloneqq M(x,k)$, i.e., $\underline{m}I \preceq M_k \preceq \overline{m}I$ for some $\overline{m} \geq \underline{m} >0$. If $F_k^\top M_{k+1} F_k \preceq \lambda M_k$ with $\lambda\in [0,1)$ holds for all $x\in\mathbb{R}^{n_x}$ and $k\in \mathbb{I}

Figures (3)

  • Figure 1: Illustration of our proposed approach. Given a finite dataset $\mathcal{D}_w$ of non-Gaussian disturbance samples, from predictions of the control contraction metrics $\mathbf{\hat{M}}$, tracking policy $\hat{\pi}(\cdot)$ for contraction rate $\lambda$, we infer high probability chance constraint satisfaction guarantees for safe motion planning. Code : https://github.com/Rihan24/SCC-TrajOpt.
  • Figure 2: Closed-loop realizations for the Dubin's car system with our approach (shown on the left) and linearization approach with Gaussian approximation of noise and LQR feedback (shown on the right). Case (i) Uniform noise and Case (ii) Gaussian mixture noise are shown in green and blue, respectively. The corresponding target trajectory $\mathbf{x}^\ast$ is shown in dashed white lines, and the respective confidence sets are shown in orange.
  • Figure 3: Left: Closed-loop realizations of the Crazyflie drone using our approach. The 3D confidence set $\mathcal{B}^{1-\delta}(\hat{x}_k, \eta(\delta)^2 M^{-1})$ for the positions is shown in green. Right: One iteration of Crazyflie trajectory in the cluttered environment.

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Definition 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Corollary 1
  • proof
  • ...and 2 more