Chance Constrained Motion Planning for High-Dimensional Robots

Abstract

This paper introduces Probabilistic Chekov (p-Chekov), a chance-constrained motion planning system that can be applied to high degree-of-freedom (DOF) robots under motion uncertainty and imperfect state information. Given process and observation noise models, it can find feasible trajectories which satisfy a user-specified bound over the probability of collision. Leveraging our previous work in deterministic motion planning which integrated trajectory optimization into a sparse roadmap framework, p-Chekov shows superiority in its planning speed for high-dimensional tasks. P-Chekov incorporates a linear-quadratic Gaussian motion planning approach into the estimation of the robot state probability distribution, applies quadrature theories to waypoint collision risk estimation, and adapts risk allocation approaches to assign allowable probabilities of failure among waypoints. Unlike other existing risk-aware planners, p-Chekov can be applied to high-DOF robotic planning tasks without the convexification of the environment. The experiment results in this paper show that this p-Chekov system can effectively reduce collision risk and satisfy user-specified chance constraints in typical real-world planning scenarios for high-DOF robots.

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

“…1) Time-Varying SOS Optimization: we use time-varying SOS optimization, introduced in Section II, to solve the time-varying deterministic polynomial optimization in (14). In implementation, we use the heuristic algorithm introduced by [7].…”

“…3) RRT-SOS: In this method, we use sampling-based motion planning algorithms like RRT to construct the risk bounded trajectory of the deterministic polynomial optimization in (14). To ensure safety along the edges of the RRT, we use an SOS-based continuous-time technique to verify the constraints of the optimization in (14b) as follows:…”

“…We want to find a risk bounded trajectory between the points x(0) = [−1, −1] and x(1) = [1, 1] by solving the probabilistic optimization in (13) with ∆ = 0.1. For this purpose, we solve the deterministic optimization problem in (14) with respect to the 0.1-risk contour of the uncertain obstacle using the discussed 3 methods as shown in Figure 4. More precisely, we use standard SOS optimization to obtain a polynomial trajectory of order 2 and also a piecewise linear trajectory with 2 pieces.…”

“…Trajectory planning problems under uncertainty look for trajectories with a bounded probability of collision with uncertain obstacles. Existing methods to address trajectory planning problems under uncertainty either are limited to Gaussian uncertainties and convex obstacles [8]- [14] or rely on sampling-based methods [15]- [17]. For example, chance constrained RRT * algorithm in [11] assumes Gaussian uncertainties and linear obstacles and performs a probabilistic collision check for the nodes of the search tree.…”

Fast-Reactive Probabilistic Motion Planning for High-Dimensional Robots

“…To address these difficulties, we propose probabilistic Chekov (p-Chekov), a combined sampling-based and optimization-based approach that takes advantage of the fact that most obstacles in a lot of practical motion planning tasks are static and only a small number of objects are dynamic during deployment. In these cases, we can construct sparse roadmaps based on our prior knowledge about the static environment to cache feasible trajectories offline, so that during plan execution, we only need to optimize solution trajectories according to new observations [19,49] and adjust plans to satisfy safety requirements [20]. Combining ideas from risk allocation [46,47] and supervised learning, p-Chekov can effectively reason over uncertainties and provide motion plans that satisfy constraints over the probability of plan failure, i.e., chance constraints [47].…”