Monte Carlo Motion Planning for Robot Trajectory Optimization Under Uncertainty

Abstract

This article presents a novel approach, named MCMP (Monte Carlo Motion Planning), to the problem of motion planning under uncertainty, i.e., to the problem of computing a low-cost path that fulfills probabilistic collision avoidance constraints. MCMP estimates the collision probability (CP) of a given path by sampling via Monte Carlo the execution of a reference tracking controller (in this paper we consider LQG). The key algorithmic contribution of this paper is the design of statistical variance-reduction techniques, namely control variates and importance sampling, to make such a sampling procedure amenable to real-time implementation. MCMP applies this CP estimation procedure to motion planning by iteratively (i) computing an (approximately) optimal path for the deterministic version of the problem (here, using the FMT * algorithm), (ii) computing the CP of this path, and (iii) inflating or deflating the obstacles by a common factor depending on whether the CP is higher or lower than a target value. The advantages of MCMP are threefold: (i) asymptotic correctness of CP estimation, as opposed to most current approximations, which, as shown in this paper, can be off by large multiples and hinder the computation of feasible plans; (ii) speed and parallelizability, and (iii) generality, i.e., the approach is applicable to virtually any planning problem provided that a path tracking controller and a notion of distance to obstacles in the configuration space are available. Numerical results illustrate the correctness (in terms of feasibility), efficiency (in terms of path cost), and computational speed of MCMP.

Provably Safe Robot Navigation with Obstacle Uncertainty

“…This means that they have trouble scaling to situations where approaches zero and failed samples are very rare. For example, Janson et al's method takes seconds to evaluate a simple trajectory with = 0.01, even with variance reduction techniques [6].…”

“…Monte-Carlo methods can evaluate the probability of collision by sampling, but can be computationally expensive when the likelihood of failure is very small [6]. When the uncertainty is restricted to Gaussian uncertainty on the robot's pose, probabilistic collision checking can yield notable performance improvements [17][13] [12].…”

Sequential Chance Optimization For Flow-Tube Based Control Of Probabilistic Nonlinear Systems

“…Note that in chance optimization (8), instead of maximizing the probability in terms of trajectory of the system, we maximize the probability in term of the states at single time step k. Hence, to design a time-varying polynomial controller over k = 0, ..., T − 1, we need to find the probability distribution of states px k and solve a chance optimization of the form (8) at each time step k. In the next sections, we will provide the convex relaxation of the chance optimization (8) and address the uncertainty propagation to obtain the probability distribution of states px k .…”

“…Sampling-based approaches such as Monte Carlo based techniques ( [7], [8]) look for controllers that satisfy control objectives for the all sampled uncertainties. Being a randomized approach, no analytical guarantees can be provided.…”

Deep Sequential Models for Sampling-Based Planning

“…Unlike our approach, the GAN does not model the dynamics of other objects or capture uncertainty when sensing. Janson et al [24] show a Monte-Carlo planning approach that, like the model described here, incorporates uncertainty in the observation of obstacles. It does not, however, consider the dynamics of obstacles or learn to extract relevant local features automatically.…”