Trajectory Optimization of Stochastic Systems under Chance Constraints via Set Erosion

TL;DR

This work addresses stochastic trajectory optimization for continuous-time systems under trajectory-level chance constraints. It introduces a set-erosion framework that uses a probabilistic tube to bound the deviation between the stochastic trajectory and an associated deterministic trajectory, eroding the safe region accordingly to reduce the problem to a deterministic surrogate. The deterministic problem minimizes a standard cost over the eroded safe set, and feasibility of the stochastic solution follows from a probabilistic tube guarantee; the authors provide bounds on the cost gap between stochastic and deterministic solutions and discuss special cases for linear and L-smooth costs. Case studies on a 3D double integrator and a nonlinear unicycle demonstrate high-probability safety and stochastic performance that closely tracks the deterministic optimum, highlighting the method's scalability and practicality for high-dimensional stochastic systems.

Abstract

We study the trajectory optimization problem under chance constraints for continuous-time stochastic systems. To address chance constraints imposed on the entire stochastic trajectory, we propose a framework based on the set erosion strategy, which converts the chance constraints into safety constraints on an eroded subset of the safe set along the corresponding deterministic trajectory. The depth of erosion is captured by the probabilistic bound on the distance between the stochastic trajectory and its deterministic counterpart, for which we utilize a novel and sharp probabilistic bound developed recently. By adopting this framework, a deterministic control input sequence can be obtained, whose feasibility and performance are demonstrated through theoretical analysis. Our framework is compatible with various deterministic optimal control techniques, offering great flexibility and computational efficiency in a wide range of scenarios. To the best of our knowledge, our method provides the first scalable trajectory optimization scheme for high-dimensional stochastic systems under trajectory level chance constraints. We validate the proposed method through two numerical experiments.

Figures (4)

• Figure 1: Trajectories of $\|X_t-x_t\|$ of the linear system $\mathrm{d} X_t=cX_t\mathrm{d}t+\sigma\mathrm{d}W_t$ with $\sigma=\sqrt{0.1}$ and different $c,~T$. Each figure contains 5000 independent trajectories and the $r_{\delta,t}$ calculated by Theorem \ref{['thm: PT']} with $\delta=10^{-3}$ and $\varepsilon=15/16$. Left: $c=1,~T=2$. Right: $c=-0.5,~T=5$
• Figure 2: Trajectory optimization of the double integrator system \ref{['eq:continuous_double_integrator_3d_compact']} with $1-10^{-4}$ guarantee. Top left: The blue curve is the solution of the deterministic trajectory optimization problem. The solid red objects represent the obstacles, while the transparent regions denote the corresponding set erosion. Top right, Bottom left and Bottom right: Visualization of stochastic trajectories at $t = 1$, $3$, and $5$ seconds, from different viewing angles. Each curve represents an independent trajectory of the stochastic system. The stochastic trajectories are simulated with the optimal control input curve.
• Figure 3: Trajectory optimization of the unicycle system \ref{['eq:unicycle dynamics']} with $1-10^{-3}$ guarantee. Left: The solution of the deterministic trajectory optimization problem. The solid red shapes represent the obstacles. The corresponding set erosion is represented as the transparent red areas. Right: Each curve is an independent trajectory of the stochastic system.
• Figure 4: Cost functions for both cases. The red curve shows the cost of the optimal deterministic trajectory over time, while the blue dashed line indicates the mean cost of the stochastic trajectories. Left: Double integrator. Right: Unicycle.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof
• Corollary 1
• proof