Given a linkage belonging to any of several broad classes (both planar and spatial), we have defined parameters adapted to a stratification of its deformation space (the quotient space of its configuration space by the group of rigid motions) making that space "practically piecewise convex". This leads to great simplifications in motion planning for the linkage, because in our new parameters the loop closure constraints are exactly, not approximately, a set of linear inequalities. We illustrate the general construction in the case of planar nR loops (closed chains with revolute joints), where the deformation space (link collisions allowed) has one connected component or two, stratified by copies of a single convex polyhedron via proper boundary identification. In essence, our approach makes path planning for a planar nR loop essentially no more difficult than for an open chain. OverviewMotion planning is important to the study of robotics [13,3,15] and is also relevant to other fields as diverse as computer-aided design, computational biology, and computer animation. A unifying concept for motion planning is the set of all configurations of a system under study, called the configuration space of the system and here denoted CSpace. In terms of CSpace, motion planning amounts to finding a valid curve connecting two given points, where a system configuration is valid if it satisfies the underlying constraints of the system-e.g., the collision free constraint for rigid objects, joint limit constraints for linkage systems, and loop closure constraints for closed chains. Thus all the complexity of motion planning is encoded in CSpace and its partition into subsets CFree and CObstacle of valid and invalid configurations.In many practical systems, CSpace has high dimension and a complicated structure in its own right. For some constraints, the partition introduces much greater complication; for instance, the fastest complete planner taking into account the ubiquitous collision free constraint [2] has exponential runningThe authors acknowledge computing support from NSF award DBI-0320875.
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.
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