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.

On the Configuration Space of Planar Closed Kinematic Chains

“…We develop a new method that explicitly computes configurations of a CKC with n links, which are described by its joint angles. Compared to other methods it does not require linear programming to solve a system of linear inequalities like in [8,10] nor does it rely on probabilistic principles. More precisely, it turns out that a configuration can be computed from new parameters contained in a very simple domain, namely a n − 3 dimensional cube.…”

“…Each C n−1 ∈ SD a ∩ Q a yields 2 n−2 circular configurations which corresponds to the possible choices for the components of ε n−1 . There is a geometric interpretation for the value ε n−k in equation a triangle orientation in [10] when building up a CKC from its diagonal lengths.…”

“…Another geometric approach was recently recognized by Han, Rudolph and Blumenthal. They discovered that it is very beneficial to describe the configuration space of CKC by different parameters than the joint angles, see [8,10,9]. Their idea is to use the length of diagonals from the positions of revolute joints to the origin O as depicted on the right side of Figure 2.…”

“…Given feasible diagonal lengths, several configurations of the CKC can be constructed, since each link of the chain can be flipped over a diagonal. Thus in [8,10] any configuration can be obtained from a set of diagonals and a vector that represents the choices of flipping, which shows that the configuration space is formed by several copies of the polyhedron given by the system of inequalities. This practically convex structure is very useful for motion planning.…”

“…This practically convex structure is very useful for motion planning. In [9,10] paths between CKC with 1000 links are computed very efficiently.…”