Convex Risk Bounded Continuous-Time Trajectory Planning in Uncertain Nonconvex Environments

Abstract

In this paper, we address the trajectory planning problem in uncertain nonconvex static and dynamic environments that contain obstacles with probabilistic location, size, and geometry. To address this problem, we provide a risk bounded trajectory planning method that looks for continuous-time trajectories with guaranteed bounded risk over the planning time horizon. Risk is defined as the probability of collision with uncertain obstacles. Existing approaches to address risk bounded trajectory planning problems either are limited to Gaussian uncertainties and convex obstacles or rely on sampling-based methods that need uncertainty samples and time discretization. To address the risk bounded trajectory planning problem, we leverage the notion of risk contours to transform the risk bounded planning problem into a deterministic optimization problem. Risk contours are the set of all points in the uncertain environment with guaranteed bounded risk. The obtained deterministic optimization is, in general, nonlinear and nonconvex time-varying optimization. We provide convex methods based on sum-of-squares optimization to efficiently solve the obtained nonconvex time-varying optimization problem and obtain the continuous-time risk bounded trajectories without time discretization. The provided approach deals with arbitrary probabilistic uncertainties, nonconvex and nonlinear, static and dynamic obstacles, and is suitable for online trajectory planning problems.

Non-Gaussian Risk Bounded Trajectory Optimization for Stochastic Nonlinear Systems in Uncertain Environments

“…where ∆ o , ∆ goal ∈ [0, 1] are the given acceptable risk levels, and pr(x 0 ) is the given probability distribution of the initial system states. We assume that either there is a global planner, obtained from the method in [20], for example, which roughly traces out a general path from the initial position to the goal region, or there is a high-level objective function that guides the system to the goal region.…”

Data-Driven Chance Constrained Control using Kernel Distribution Embeddings

“…Guaranteeing constraint satisfaction with high probability typically involves considering a chance constrained problem formulation. The most common formulation enforces pointwise chance constraints that ensure the independent satisfaction of each constraint at each time step with high probability (Castillo-Lopez et al, 2019;Lew et al, 2020;Hewing et al, 2020;Polymenakos et al, 2020;Khojasteh et al, 2020;Jasour et al, 2021). In contrast, joint chance constraints guarantee trajectory-wise constraints satisfaction with high probability (Blackmore et al, 2011;Frey et al, 2020;Schmerling and Pavone, 2017;Koller et al, 2018;Lew et al, 2022) which is of particular interest whenever all constraints should be satisfied at all times jointly.…”

Lie Algebraic Cost Function Design for Control on Lie Groups

“…With the increase of moment relaxation order, research [14] could asymptotically find the collision-free path when a moving obstacle is presented. Research [29] extends such Lasserre's hierarchybased method to develop risk-bounded trajectory planners in the presence of uncertain time-varying obstacles using the notion of risk contours [30]. Moment relaxation methods have been applied to optimal control of hybrid systems [31] in continuous time.…”