Given some geometric elements such as points and lines in R3, subject to a set of pairwise distance constraints, the problem tackled in this paper is that of finding all possible configurations of these elements that satisfy the constraints. Many problems in robotics (such as the position analysis of serial and parallel manipulators) and CAD/CAM (such as the interactive placement of objects) can be formulated in this way. The strategy herein proposed consists of looking for some of the a priori unknown distances, whose derivation permits solving the problem rather trivially. Finding these distances relies on a branch-and-prune technique, which iteratively eliminates from the space of distances entire regions which cannot contain any solution. This elimination is accomplished by applying redundant necessary conditions derived from the theory of distance geometry. The experimental results qualify this approach as a promising one.

This work was supported by projects: 'Design and implementation of efficient parallelizable algorithms with applications to robotics and proteomics' (J-00869), 'Grup de recerca consolidat - ROBÒTICA' (8007), 'Resolución de sistemas de ecuaciones cinemáticas para la simulación de mecanismos, posicionado interactivo de objetos y conformación de moléculas' (070-722). This work was supported in part by the Spanish CICYT under Contracts TIC2000-0696 and TIC2003-03396, and in part by the Catalan Research Commission through the “Robotics and Control” group. The work of J. M. Porta and L. Ros was supported by a Ramón y Cajal contract from the Spanish Ministry for Science and Technology.

This paper was presented in part at the IEEE International Conference on Robotics and Automation, Taipei, Taiwan, R.O.C., September 2003.

Peer Reviewed