Inverse Kinematics by Distance Matrix Completion

INTRODUCTION The Theory of Distance Geometry [1] allows coordinate-free formulations for most position analysis problems. Using such formulations, the inverse kinematics of a serial robot can be performed by using a distance matrix, one whose entries are squared distances between pairs of points selected on the axes of the robot. While some of these distances are known (such as the distances between points on the same axis or between points on consecutive axes), many others are unknown. Then, finding all solutions to an inverse kinematics problem boils down to finding values for these unknown distances that permit completing the matrix into a “proper” Euclidean distance matrix [6]. If, by any means, the unknown distances are obtained, one can then easily assign coordinates to the selected points and trivially derive the possible configurations of the robot. The determination of all values for the unknown distances is usually done via a bound smoothing process: a large range is initially assigned to the unknowns and their bounds are progressively reduced in an iterative manner, by applying triangular inequalities and other necessary conditions [5]. Finding all possible solutions for a given incomplete distance matrix can be extremely complex in general, as this problem is known to be NP-complete. In this paper, we focus on a subclass of distance constraint solving problems where the values of all unknown distances can be derived following a constructive process in which the distance matrix is progressively completed by deducing the value of one unknown at a time. In order to identify all robots whose inverse kinematics can be solved in this way, we first characterize the family of distance matrices that encode all serial robots with six degrees of freedom (DoF). Then, we exhaustively search within this family for those matrices that can be completed in a constructive manner taking as fixed reference either a triangle or a tetrahedron. The result is the identification of a family of serial robots that includes the best-known industrial robots. This paper is organized as follows. First, we describe how to translate an inverse kinematic problem into a distance constraint satisfaction problem. Then, we show how some incomplete distance matrices can be completed by using a sequence of two basic operations. Next, a comprehensive study of all serial robots whose associated distance matrix can be completed using this technique is presented. All these ideas are then applied to the resolution of the inverse kinematics of a Puma 560 manipulator. Finally, we summarize some points deserving further attention.