Reconfiguring closed polygonal chains in Euclideand-space

Abstract

Consider the problem of moving a closed chain of n links in two or more dimensions from one given configuration to another. The links have fixed lengths and may rotate about their endpoints, possibly passing through one another. The notion of a "line-tracking motion" is defined, and it is shown that when reconfiguration is possible by any means, it can be achieved by O(n) line-tracking motions. These motions can be computed in O(n) time on real RAM. It is shown that in three or more dimensions, reconfiguration is always possible, but that in dimension two this is not the case. Reconfiguration is shown to be always possible in two dimensions if and only if the sum of the lengths of the second and third longest links add to at most the sum of the lengths of the remaining links. An O(n) algorithm is given for determining whether it is possible to move between two given configurations of a closed chain in the plane and, if it is possible, for computing a sequence of line-tracking motions to carry out the reconfiguration. * The research of the second author was partially supported by an NSERC operating grant.

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

“…Essentially, the greater the singularity depth of a deformation, the more singular the deformation; a non-singular deformation has singularity depth 0. We show that any two closure deformations in the same component can be connected by a piecewise linear path traversing at most n − 2 singular deformations, all of depth 1; they can also be connected via at most 2 singular deformations, one of which has singularity depth at least n − 3 (it is a triangle deformation generalizing one devised by Lenhart and Whitesides [16]). Importantly, the singular deformations are reusable and easily computable.…”

“…In particular, Trinkle, Milgram and Liu [22,20,18,17] have made important discoveries for CSpace of a kinematic chain with fully rotatable joints, either n spherical joints in space (nS) or n revolute joints (nR) in the plane. Building on earlier work by geometers [16,10], they obtained results on the geometry and topology of the set of closure configurations for a closed chain, initially without imposing the collision free constraint but recently allowing point obstacles. Using this information they develop complete path planners, such as an O(n 3 ) accordion planner for a closed chain (ignoring collisions), and a planner for avoiding p point obstacles with conjectural lower and upper bounds Ω(p n−3 ) and O(p 2n−7 ).…”

“…Fig. 4(c) shows a special deformation of the 6R loop used in Example 4, generalized from the concept of "standard triangular form" used by Lenhart and Whitesides [16]. It is defined by finding joint index j satisfying…”

“…Easily, if a loop has m long links, then m ∈ {0, 2, 3}. Prior results [16,10] show that DSpace for a planar nR loop has two connected components or one, according as the loop has 3 long links or fewer. The 5R loops in Fig.…”

“…Such triangles are key to understanding the connectivity of DSpace, and lead to an alternative proof (short, but not short enough to include here!) of the DSpace connectivity results of [22,16,10].…”