Motor algebra approach for visually guided robotics

Abstract

This paper presents the formulation of the 3D kinematics in the geometric algebra framework. We show that this approach is extremely useful for solving problems in the field of visual guided robotics. In this algebraic system the 3D Euclidean motion of points, lines and planes can be advantageously represented using the algebra of motors. The computational complexity of the direct and indirect kinematics and other problems concerning robot manipulators depends on its degrees of freedom as well on its geometric characteristics. Our approach makes possible a direct algebraic formulation of the concrete problem in such a way that it reflects the underlying geometric structure. This is achieved by switching to a description of parts of the problem based on motor representations of points, lines and planes where necessary. The first robotics task this paper deals with is the formulation and computation of closed-form solutions of the direct and indirect kinematics of standard robot manipulators and a simple example of a grasping task. The flexible method presented here is new and it widens the current standard point or line representation based approaches for the treatment of problems related to robot manipulators. The second challenging task presented in this paper is the solution of the hand–eye calibration problem when cameras are attached to robot arms. The solution of this problem in the motor algebra framework turns to be linear. Both tasks are crucial for visual guided robotics, that is why we are strongly motivated to present them to show how useful it is to apply motor algebra for solving geometric problems in visual guided robotics.