Finding Largest Empty Circles with Location Constraints

Let S be a set of n points in the plane and let CH(S) represent the convex hull of S. The Largest Empty Circle (LEC) problem is the problem of finding the largest circle centered with CH(S) such that no point of S lies within the circle. Shamos and Hoey (SH75) outlined an algorithm for solving this problem in time O(n log n) by first computing the Voronoi diagram, V(S), in time O(n log n), then using V(S) and CH(S) to compute the largest empty circle in time O(n). In a recent paper [Tou83], Toussaint pointed out some problems with the algorithm as outlined by Shamos and presented an algorithm which, given V(S) and CH(S), solves the LEC problem in time O(n log n). In this note we show that Shamos\u27 original claim was correct: given V(S) and CH(S), the LEC problem can be solved in time O(n). More generally, given V(S) and a convex k-gon P, the LEC centered within P can be found in time O(k+n). We also improve on an algorithm given by Toussaint for computing the LEC when the center is constrained to lie within an arbitrary simple polygon. Given a set S of n points and an arbitrary simple k-gon P, the largest empty circle centered within P can be found in time O(kn + n log n). This becomes O(kn) if the Voronoi diagram of S is already given

Multi-contact Walking Pattern Generation based on Model Preview Control of 3D COM Accelerations

We present a multi-contact walking pattern generator based on preview-control of the 3D acceleration of the center of mass (COM). A key point in the design of our algorithm is the calculation of contact-stability constraints. Thanks to a mathematical observation on the algebraic nature of the frictional wrench cone, we show that the 3D volume of feasible COM accelerations is a always a downward-pointing cone. We reduce its computation to a convex hull of (dual) 2D points, for which optimal O(n log n) algorithms are readily available. This reformulation brings a significant speedup compared to previous methods, which allows us to compute time-varying contact-stability criteria fast enough for the control loop. Next, we propose a conservative trajectory-wide contact-stability criterion, which can be derived from COM-acceleration volumes at marginal cost and directly applied in a model-predictive controller. We finally implement this pipeline and exemplify it with the HRP-4 humanoid model in multi-contact dynamically walking scenarios