Steering flexible needles under Markov motion uncertainty

When inserted into soft tissues, flexible needles with bevel tips have been shown experimentally to follow a path of constant curvature in the direction of the bevel. By controlling 2 degrees of freedom at the needle base (bevel direction and insertion distance), these needles can be steered around obstacles to reach targets inaccessible to rigid needles. Motion planning for needle steering is a type of nonholonomic planning for a Dubins car with no reversal. We develop a motion planning algorithm based on dynamic programming where the path of the needle is uncertain due to uncertainty in tissue properties, needle mechanics, and interaction forces. The algorithm computes a discrete control sequence of insertions and direction changes so the needle reaches a target in an imaging plane while minimizing expected cost due to insertion distance, direction changes, and obstacle collisions. We efficiently sample the state space of needle tip positions and orientations and define bounds on the errors due to discretization. We formulate the motion planning problem as a Markov decision process (MDP) and use infinite horizon dynamic programming to compute an optimal control sequence. We first apply the method to the deterministic motion case where the needle precisely follows a path of constant curvature and then to the uncertain motion case where state transitions are defined by a probability distribution. Our implementation generates motion plans for bevel-tip needles that reach targets inaccessible to rigid needles and demonstrates that accounting for uncertainty can lead to significantly different motion plans.