A Study of Singularities in Homotopy Continuation for Kinematic Applications

Task requirements and positions of robotic kinematic systems can be represented by a system of polynomial equations. The use of homotopy continuation, a numerical method from algebraic geometry, solves these systems and analyzes their solution sets. However, these systems are plagued by branch points, and homotopy paths passing within proximity to these points experience ill-conditioning leading to computational burdens and even numerical failures. Although branch points negatively impact homotopy continuation methods, one can leverage knowledge regarding these points for solving and analyzing kinematic systems. First, we present a holistic consideration of the presence of branch points in homotopy systems of generic cases and an applied kinematic problem. We study the distribution of the image of branch points, ramification points, compactified to the Riemann sphere for select combinations of start and end systems of a homotopy. We compute metrics to assess the uniformity of the ramification point distribution and observe methods that impact that distribution, including the use of specially structured start systems and scaling coefficients. Next, we consider the complete computation of solution sets for the four-bar optimal path synthesis problem in kinematics. As task requirements become more complex, so do their resulting systems, ultimately leading one to consider an optimization formulation such as the least-squares approximation. Homotopy continuation methods, namely the use of random monodromy loops, can be employed to these polynomial formulations to yield a nearly, if not totally, complete solution set starting from a single seed solution. We consider three scenarios of the optimal path synthesis with no, one, and two pivots pre-specified and applied examples for each scenario. Lastly, we propose a method for kinematic path planning that incorporates topological ideas into computing a radius graph representation of the workspace for a five-bar mechanism. The workspace of a five-bar mechanism is a manifold possessing extra folds that, when projected into a 2D view, appears to admit additional internal boundaries. The mechanism satisfies additional constraints at these boundaries, but only a portion of those boundaries should be avoided due to unfavorable transmission qualities displayed by the five-bar. Our method identifies and removes these boundaries in a workspace discretization while preserving characteristics of the workspace manifold. We sample a sufficient number of points from the configuration space such that we preserve topological features of the manifold. Then, through homotopy applications, we identify and remove problematic boundaries from our viable workspace discretization for use in path-planning algorithms. We apply our method to two path-planning examples and show the resulting path and joint actuations taken by the five-bar mechanism.