Constant-Curvature Motion Planning Under Uncertainty with Applications in Image-Guided Medical Needle Steering

We consider a variant of nonholonomic motion planning for a Dubins car with no reversals, binary left/right steering, and uncertainty in motion direction. We develop a new motion planner and apply it to steerable needles, a new class of flexible bevel-tip medical needles that clinicians can steer through soft tissue to reach targets inaccessible to traditional stiff needles. Our method explicitly considers uncertainty in needle motion due to patient differences and the difficulty in predicting needle/tissue interaction: the planner computes optimal turning points to maximize the probability that the needle will reach the desired target. Given a medical image with segmented obstacles and target, our method formulates the planning problem as a Markov Decision Process (MDP) based on an efficient discretization of the state space, models motion uncertainty using probability distributions, and computes turning points to maximize the probability of successfully reaching the target using infinite horizon Dynamic Programming (DP). This approach has three features particularly beneficial for medical planning problems. First, the planning formulation only requires parameters that can be directly extracted from images. Second, we can compute the optimal needle insertion point by examining the DP look-up table of optimal controls for every needle state. Third, intra-operative medical imaging can be combined with the pre-computed DP look-up table to permit optimal control of the needle in the operating room without requiring time-consuming intra-operative re-planning. We apply the method to generate motion plans for steerable needles to reach targets inaccessible to stiff needles and illustrate the importance of considering uncertainty during motion plan optimization.