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We consider nonlinear stochastic systems that arise in path planning and control of mobile robots. As is typical of almost all nonlinear stochastic systems, the optimally solving problem is intractable. We provide a design approach which yields a tractable design that is quantifiably nearoptimal. We exhibit a "separation" principle under a small noise assumption consisting of the optimal open-loop design of nominal trajectory followed by an optimal feedback law to track this trajectory, which is different from the usual effort of separating estimation from control. As a corollary, we obtain a trajectory-optimized linear quadratic regulator design for stochastic nonlinear systems with Gaussian noise.

Abstract-Planning under process and measurement uncertainties is a challenging problem. In its most general form it can be modeled as a Partially Observed Markov Decision Process (POMDP) problem. However POMDPs are generally difficult to solve when the underlying spaces are continuous, particularly when beliefs are non-Gaussian, and the difficulty is further exacerbated when there are also non-convex constraints on states. Existing algorithms to address such challenging POMDPs are expensive in terms of computation and memory. In this paper, we provide a feedback policy in non-Gaussian belief space via solving a convex program for common non-linear observation models. The solution involves a Receding Horizon Control strategy using particle filters for the non-Gaussian belief representation. We develop a way of capturing non-convex constraints in the state space and adapt the optimization to incorporate such constraints, as well. A key advantage of this method is that it does not introduce additional variables in the optimization problem and is therefore more scalable than existing constrained problems in belief space. We demonstrate the performance of the method on different scenarios.

Abstract-Optimizing measures of the observability Gramian as a surrogate for the estimation performance may provide irrelevant or misleading trajectories for planning under observation uncertainty. I. INTRODUCTIONThe Observability Gramian (OG) is used to determine the observability of a deterministic linear time-varying system [1]- [3]. For such systems, the properties of the OG have been well-studied [1], [4], [5]. When sensors provide noisy stochastic measurements, the state is only partially observed. The general problem of planning under process and observation uncertainties has been formulated as such a stochastic control problem with noisy observations. The solution of this problem provides an optimal policy via the Hamilton-Jacobi-Bellman equation [6], [7]. However, the computational hurdle for finding a solution to these equations has necessitated the study of a variety of methods to approximate the solution [8]- [11]. One approach has been to maximize the estimation performance by planning for trajectories that can exploit the properties of observation, process and a priori models. We examine the appropriateness or lack thereof of methods based on the OG, and show that they can provide misleading trajectories.Borrowed from deterministic control theory, the OG has been exploited in order to provide more observable trajectories, particularly in trajectory planing problems [12]-[18]. In the special case of a diagonal observation covariance with the same uncertainty level in each direction [1], the Standard Fisher Information Matrix (SFIM) does reduce to the OG. Indeed the usage of the OG in filtering problems has been justified through its connections to the SFIM and its relations to the parameter estimation problem [13], [19]. In fact, tailored to the parameter estimation problem, the SFIM only addresses the amount of information in the measurements alone [1], and neglects both the prior information and process uncertainty. Closely-related approaches are the methods that base their planning on the observation model or the likelihood function [8], [20], and the analysis of this paper can be helpful in providing a better understanding of those problems.

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