The authors describe a computational framework in which a probabilistic method for noisy sensor-based robotic navigation in dynamic environments can be devised. The aim of the method is to generate an optimal trajectory by considering an optimality criteria the probability of not colliding with the obstacles and the probability of accessing an operational position with respect to a moving target object. A formal framework in which the probability of collision associated with an elementary robot displacement can be calculated is discussed. Estimates on the obstacle kinematic parameters and measures of confidence on these estimates are used to produce the probability of collision associated with any robot displacement. The probability of collision is derived in two steps: a stochastic model is defined in the kinematic state space of the obstacles and collision events are given a simple geometric characterization in this state space.