Optimal Stochastic Vehicle Path Planning Using Covariance Steering

Abstract

This work addresses the problem of vehicle path planning in the presence of obstacles and uncertainties, which is a fundamental problem in robotics. While many path planning algorithms have been proposed for decades, many of them have dealt with only deterministic environments or only openloop uncertainty, i.e., the uncertainty of the system state is not controlled and, typically, increases with time due to exogenous disturbances, which leads to the design of potentially conservative nominal paths. In order to deal with disturbances and reduce uncertainty, generally, a lower-level feedback controller is used. We conjecture that, if a path planner can consider the closed-loop evolution of the system uncertainty, it can compute less conservative but still feasible paths. To this end, in this work we develop a new approach that is based on optimal covariance steering, which explicitly steers the state covariance for stochastic linear systems with additive noise under nonconvex state chance constraints. The proposed framework is verified using simple numerical simulations. arXiv:1809.03380v1 [math.OC]

Input Hard Constrained Optimal Covariance Steering

“…We employ YALMIP [14] along with MOSEK [13] to solve this problem. We first show the results when the controller from [9] is used in Fig. 1.…”

“…We also illustrate randomly picked 100 trajectories with gray lines and observe that the state chance constraints are satisfied. Note, however, that the controller in [9] cannot deal with input hard constraints. Figure 1(b) depicts the acceleration commands of the 100 sample trajectories with gray lines along with the acceleration limits ± 2.9 m/s 2 with red dashed lines.…”

Nonlinear Uncertainty Control with Iterative Covariance Steering

“…Following [7], [8], [9], we will rewrite the discrete system (16) as a single linear equation. Define the state transition matrix from step k 0 to step k 1 as…”

Method for solving chance constrained optimal control problems using biased kernel density estimators

“…In the method of Reference 10, weighted summations are used to approximate the chance constraints as deterministic constraints. Next, in the methods of References 11‐13, the chance constraint is transformed using Gaussian properties to a deterministic form. In addition, Reference 14 transforms the chance constraint to a deterministic form using Gaussian properties.…”