TY - JOUR

T1 - Instabilities of robot motion

AU - Farber, Michael

N1 - Funding Information:
✩ Partially supported by a grant from the Israel Science Foundation and by the H. Minkowski Center for Geometry; part of this work was done while the author visited ETH in Zürich. E-mail address: mfarber@tau.ac.il (M. Farber).

PY - 2004/5/28

Y1 - 2004/5/28

N2 - Instabilities of robot motion are caused by topological reasons. In this paper we find a relation between the topological properties of a configuration space (the structure of its cohomology algebra) and the character of instabilities, which are unavoidable in any motion planning algorithm. More specifically, let X denote the space of all admissible configurations of a mechanical system. A motion planner is given by a splitting X×X=F1∪F2∪⋯∪ Fk (where F1,...,Fk are pairwise disjoint ENRs, see below) and by continuous maps sj:Fj→PX, such that E○sj=1Fj. Here PX denotes the space of all continuous paths in X (admissible motions of the system) and E:PX→X×X denotes the map which assigns to a path the pair of its initial-end points. Any motion planner determines an algorithm of motion planning for the system. In this paper we apply methods of algebraic topology to study the minimal number of sets Fj in any motion planner in X. We also introduce a new notion of order of instability of a motion planner; it describes the number of essentially distinct motions which may occur as a result of small perturbations of the input data. We find the minimal order of instability, which may have motion planners on a given configuration space X. We study a number of specific problems: motion of a rigid body in R3, a robot arm, motion in R3 in the presence of obstacles, and others.

AB - Instabilities of robot motion are caused by topological reasons. In this paper we find a relation between the topological properties of a configuration space (the structure of its cohomology algebra) and the character of instabilities, which are unavoidable in any motion planning algorithm. More specifically, let X denote the space of all admissible configurations of a mechanical system. A motion planner is given by a splitting X×X=F1∪F2∪⋯∪ Fk (where F1,...,Fk are pairwise disjoint ENRs, see below) and by continuous maps sj:Fj→PX, such that E○sj=1Fj. Here PX denotes the space of all continuous paths in X (admissible motions of the system) and E:PX→X×X denotes the map which assigns to a path the pair of its initial-end points. Any motion planner determines an algorithm of motion planning for the system. In this paper we apply methods of algebraic topology to study the minimal number of sets Fj in any motion planner in X. We also introduce a new notion of order of instability of a motion planner; it describes the number of essentially distinct motions which may occur as a result of small perturbations of the input data. We find the minimal order of instability, which may have motion planners on a given configuration space X. We study a number of specific problems: motion of a rigid body in R3, a robot arm, motion in R3 in the presence of obstacles, and others.

UR - http://www.scopus.com/inward/record.url?scp=2342634325&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2003.07.011

DO - 10.1016/j.topol.2003.07.011

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AN - SCOPUS:2342634325

SN - 0166-8641

VL - 140

SP - 245

EP - 266

JO - Topology and its Applications

JF - Topology and its Applications

IS - 2-3

ER -