## Section: New Results

### Path planning for navigation in a geographic information system

Participants : Jean–Pierre Le Cadre, Francis Céleste.

The problem we considered is the optimization of the navigation of an intelligent mobile in a real world environment, described by a map. The map is composed of features representing natural landmarks in the environment. The vehicle is equipped with sensors which allows it to obtain landmark parameter estimates. These measurements are correlated with the map so as to estimate the mobile position. The optimal trajectory must be designed in order to control a measure of the performance for the filtering algorithm used for mobile navigation. As the mobile state and the measurements are random, a well–suited measure can be a functional of the posterior Cramer–Rao bound (PCRB). In many applications, it is crucial to be able to estimate accurately the state of the mobile during the execution of the plan. So it is necessary to couple the planning and the execution stages.

A classical tool is the constrained Markov decision process (MDP) framework. However, our optimality criterion is based on the posterior Cramer–Rao bound, and the nature of the objective function for path planning makes it impossible to perform complete optimization within the MDP framework. Indeed, the reward in one stage of our MDP depends on all the history of the trajectory. To overcome this problem, the cross–entropy method [69] , [34] , originally used for rare–event simulation, is a valuable tool. Its principle is to translate a classical optimization method into an associated stochastic problem and then to solve it adaptively as the simulation of rare events. This approach has been tested on various (simple) geographic environments and performs satisfactorily. Our contribution, conducted in the context of Francis Céleste PhD work, has been devoted to the derivation of closed–form approximations of the information we can gain from an elementary motion. Using them, it is possible to immerse the problem within an optimal control framework and to use efficiently the maximum principle.