Planning for Flexible Surgical Robots via Bézier Spline Translation

In a minimally invasive surgery, new flexible instruments enable safer and easier access to difficult-to-reach anatomical regions. However, their introduction into the clinical workflow requires robust replanning because navigation errors during surgery render initially planned trajectories infeasible. Such replanning requires to regularly solve an expensive two-point boundary value problem (BVP) that connects the target pose of the instrument with the currently measured one. We propose a hybrid planning scheme that features both robust and safe replanning. This two-step approach first solves the BVP and then transforms the result to circular arcs that fit the motion of our instruments’ models. We exploit implicitly defined Bézier splines as a robust method for interpolation in the first step. A novel geometric translation of these splines, then, provides a convenient solution for movement along circular arcs. We consider two example applications: 1) planning for a drilling unit in temporal bone surgery; and 2) guidewires in catheter insertion. Evaluation on real patient data of both temporal bone and aorta show that our proposed hybrid two-step approach achieves, on average, ${\text{55}\%}$ higher replanning rates and provides ${\text{31}\%}$ larger clearance to risk structures, thus improving trajectory quality with regard to clinical safety.