Optimal feedback control of a flexible needle under anatomical motion uncertainty

Bevel-tip flexible needles allow for reaching remote/ inaccessible organs while avoiding the obstacles (sensitive organs, bones, etc.). Motion planning and control of such systems is a challenging problem due to the uncertainty induced by needle-tissue interactions, anatomical motions (respiratory and cardiac induced motions), imperfect actuation, etc. In this paper, we use an analogy where steering the needle in a soft tissue subject to the uncertain anatomical motions is compared to the Dubins vehicle traveling in the stochastic wind field. Achieving the optimal feedback control policy requires solution of a dynamic programming problem that is often computationally demanding. Efficiency is not central to many optimal control algorithms that often need to be computed only once for a given system/noise statistics. However, intraoperative policy updates may be required for adaptive or patient-specific models. We use the method of approximating Markov chain to approximate the continuous (and controlled) process with its discrete and locally consistent counterpart. We examine the linear programming method of solving the imposed dynamic programming problem that significantly improves the computational efficiency in comparison to the state-of-the-art approaches. In addition, the probability of success and failure are simply the variables of the linear optimization problem and can be directly used for different objective definitions. A numerical example of the 2D needle steering problem is considered to investigate the effectiveness of the proposed method.