Topological Complexity of Motion Planning

Abstract

In this paper we study a notion of topological complexity TC(X) for the motion planning problem. TC(X) is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, TC(X) is the minimal number k such that there are k different "motion planning rules", each defined on an open subset of X × X, so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik -Schnirelman theory) to study the topological complexity TC(X) . We give an upper bound for TC(X) (in terms of the dimension of the configuration space X) and also a lower bound (in the terms of the structure of the cohomology algebra of X). We explicitly compute the topological complexity of motion planning for a number of configuration spaces: for spheres, two-dimensional surfaces, for products of spheres. In particular, we completely calculate the topological complexity of the problem of motion planning for a robot arm in the absence of obstacles. 1Keywords: topological complexity, motion planning, configuration spaces, Lusternik -Schnirelman theory 1 Definition of topological complexity Let X be the space of all possible configurations of a mechanical system. In most applications the configuration space X comes equipped with a structure of topological space. The motion planning problem consists in constructing a program or a devise, which takes pairs of configurations (A, B) ∈ X × X as an input and produces as an output a continuous path in X, which starts at A and ends at B, see [3], [4], [6]. Here A is the initial configuration, and and B is the final (desired) configuration of the system.We will assume below that the configuration space X is path-connected, which means that for any pair of points of X there exists a continuous path in X connecting them. Otherwise, the motion planner has first to decide whether the given points A and B belong to the same path-connected component of X. * Partially supported by a grant from the Israel Science Foundation 1 I am thankful to D. Halperin, M. Sharir and S. Tabachnikov for a number of very useful conversations.

Complexity of the forward kinematic map

“…Thus, by Theorem 3.1, together with the computations of complexity for T and T 2 (see [7,Theorem 12]) we have…”

“…A more geometrical measure for the complexity of motion planning was introduced by M. Farber [7] who observed that algorithmic solutions normally yield robust mapping plans. He then defined the concept of the topological complexity of motion planning in the working space of a mechanical device as the minimal number of continuous partial solutions to the motion planning problem.…”

“…The configuration space of a n-joint robot arm is the cartesian product of n-circles, C = T n , whose topological complexity is exactly n + 1 (see [7,Theorem 13]), so if there exists a global robust inverse kinematic map for F , then cx(F ) ≤ n + 1. On the other side, the complexity of the standard working space W = R 3 × SO(3) is 4 (see [9,Theorem 4.61]), so in the presence of a global inverse kinematic map the motion planning requires at least 4 robust partial roadmaps.…”

“…By [7,Theorem 1] spaces with complexity one are contractible (i.e continuously deformable to a point). This is possible only if the robot working space is a spatial domain without obstacles, and its movement does not involve planar or spatial rotations (cf.…”

“…Then the complexity of F equals the complexity of the interval, which is 1 by [7,Theorem 1]. However, if the end-effector is connected to a transmission mechanism so that it takes several full rotations of the joint to move the end-effector between two position in the working space, then there does not exist a continuous inverse kinematic map and one still needs two robust roadmaps to navigate this simple mechanism.…”