Convexly Stratified Deformation Spaces and Efficient Path Planning for Planar Closed Chains with Revolute Joints

Abstract

Systems involving loops have been especially challenging in the study of robotics, partly because of the requirement to maintain loop closure constraints, conventionally formulated as highly nonlinear equations in joint parameters. In this paper, we present our novel triangle-tree-based approach and parameters for planar closed chains with revolute joints. For such a loop, the loop closure constraints are exactly, not approximately, a set of linear inequalities in our new parameters. Further, our new parameters provide ex plicit parametrization of the system deformation space (configuration space modulo the group of rigid motions of the system's ambient space respecting system specifications) and endow it with a nice geometry. More precisely, the deformation space of a generic planar loop with n revolute joints consists of 2n-2 copies of one and the same convex polytope (which, when all of the link lengths are fixed, is bounded and of dimension n - 3), glued together into either one connected component or two (ignoring collision-free constraints), via proper boundary identification. Such a completely solved, stratified space of convex strata will have profound implications for these sys tems and lead to great simplifications in many kinematics related issues. For example, in essence, our approach makes path planning for planar loops with revolute joints no more difficult than for open chains. We also briefly point out the connection and extension of the work presented here to other systems such as spatial loops with spher ical joints and systems involving multiple loops.