Stratified Deformation Space and Path Planning for a Planar Closed Chain with Revolute Joints

Abstract

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.

A Unified Geometric Approach for Inverse Kinematics of a Spatial Chain with Spherical Joints

“…Our recently developed anchored triangle parameters [10], [11], consisting of anchored diagonal lengths, anchored triangle orientations (such as dihedral angles for the linkages studied here), and variable link lengths if any, are particularly well suited for many broad classes of planar and spatial linkages, including planar chains and loops with revolute joints, spatial chains and loops with spherical joints, chains with variable link lengths (which can model some prismatic joints), and some kinematic structures more complicated than a chain or single loop. What is striking is that the loop closure constraints, arising from the physical loops as well as virtual loops for IK problems, are linear inequalities in this set of geometric parameters.…”

“…Yet past research has led to a large body of impressive results; a small set of representative work appears in We recently introduced a new set of linkage parameters tailored to the study of inverse kinematics. These anchored triangle parameters consist of certain inter-joint distances (called diagonal lengths) and certain triangle orientation parameters (signs or dihedral angles); they were described in some detail for serial chains with spherical joints in space or revolute joints in the plane in [10], and for closed planar chains with revolute joints in [11]. It was shown that for the linkages under study the addressed IK problem can be formulated exactly, not approximately, as a set of linear inequalities in the anchored triangle parameters.…”

“…More specifically, in space the two-endpoint-position constraint is 5-dimensional: it constrains the last joint's position in space (3-dimensional constraint) and the penultimate joint's position on the sphere centered at the last joint with radius the last link length (2-dimensional constraint). On the other hand, the common end-effector constraint in space imposes a 6-dimensional constraint, and the encoding of the sixth constraint in the new parameters is not addressed in [10]; nor do [10], [11] give many details on how to deal with singular configurations, especially super-singular configurations (to be explained later). Yet singular configurations play an important role in reconstructing the global topological structure of the solution set, especially for planar chains with revolute joints [11].…”

“…On the other hand, the common end-effector constraint in space imposes a 6-dimensional constraint, and the encoding of the sixth constraint in the new parameters is not addressed in [10]; nor do [10], [11] give many details on how to deal with singular configurations, especially super-singular configurations (to be explained later). Yet singular configurations play an important role in reconstructing the global topological structure of the solution set, especially for planar chains with revolute joints [11].In this paper, we address these issues and study three types of IK problems for a serial spatial chain consisting of n rigid links connected by spherical joints. IK Problem 1 is commonly called the "reaching" problem and poses a constraint on the position of the tip of the chain; IK Problem 2 is that studied in [10]; and IK Problem 3 imposes a 6 dimensional constraint on the chain similar to the constraints imposed by a required configuration of the last link frame.…”

Motion Planning for Highly Constrained Spaces

“…Analytical approaches construct explicit geometrical and topological representation of the closure set [4], [9], [13], but are usually inefficient in practice. Practical sampling-based methods [3], [16] usually project the closure set on the subset of parameters, on which the planning is performed [5], [8].…”