  We study the problem of devising a closed-loop strategy to control the
position of a robot that is tracking a possibly moving target. The robot is
capable of obtaining noisy measurements of the target's position. The key idea
in active target tracking is to choose control laws that drive the robot to
measurement locations that will reduce the uncertainty in the target's
position. The challenge is that measurement uncertainty often is a function of
the (unknown) relative positions of the target and the robot. Consequently, a
closed-loop control policy is desired which can map the current estimate of the
target's position to an optimal control law for the robot.
  Our main contribution is to devise a closed-loop control policy for target
tracking that plans for a sequence of control actions, instead of acting
greedily. We consider scenarios where the noise in measurement is a function of
the state of the target. We seek to minimize the maximum uncertainty (trace of
the posterior covariance matrix) over all possible measurements. We exploit the
structural properties of a Kalman Filter to build a policy tree that is orders
of magnitude smaller than naive enumeration while still preserving optimality
guarantees. We show how to obtain even more computational savings by relaxing
the optimality guarantees. The resulting algorithms are evaluated through
simulations.
