Kalman filtering with optimal sensor motion planning

In this paper the Kalman filter equations are re- derived for systems with mobile sensors. The motion of the sensor impacts the measurement noise statistical properties through the measurement covariance matrix and, consequently, the estimation error covariance matrix. Moreover, since the sensors are mobile, their motion has an associated control cost. Minimizing the control effort expended during the measurement process is of critical importance in many applications, in particular where fuel and energy resources scarce. With this realization, the Kalman filter equations are re-derived in this paper to include sensor dynamics and their impact on the estimation error. While the classical Kalman filter seeks to minimize a measure of the estimation error, in this paper the goal is to minimize both the estimation error as well as control energy expended during the estimation process. Necessary optimality conditions are derived, and are further simplified for the case where the process to be estimated is composed of a set of decoupled and uncorrelated processes. A mathematical continuation approach is proposed to solve the resulting nonlinear two-point value problem. A numerical example is provided to illustrate the work of this paper.