Representing the Motion of Object in Contact using Dual Quaternions and its Applications

Abstract

This report presents a general method for representing the motion of an object while maintaining contact with other fixed objects. The motivation for our work is the assembly plan from observation (APO) system. The APO system observes a human perform an assembly task. It then analyzes the observations to reconstruct the assembly plan used in the task. Finally, the APO converts the assembly plan into a program for a robot which can repeat the demonstrated task. The position and orientation of an object, known as its configuration can be represented using dual vectors. We use dual quaternions to represent the configuration of objects in 3D space. When an object maintains a set of contacts with other fixed objects, its configuration will be constrained to lie on a surface in configuration space called c-surface. The c-surface is determined by the geometry of the object features in contact. We propose a general method to represent c-surfaces in dual vector space. Given a set of contacts, we choose a reference contact and represent the c-surface as a parametric equation based on it. The contacts other than the reference contact will impose constraints on these parameters. The reference contact and the constraints constitute the representation of the c-surface. We show that the use of dual quaternions simplifies our representation considerably. Once we define our c-surface representation, we propose methods to compute the projection of a point in configuration space onto the c-surface and to interpolate between points on the c-surface. We used our theory to implement the APO system. We used our c-surface representation to correct approximate configurations of the objects at each observed instant of the demonstrated task. We interpolated between corrected points on the c-surface to obtain segments of the assembly path. The complete assembly path used in the observed task is then a concatenation of these path segments. Finally, we used the reconstructed assembly path to program a six axis robot arm to repeat the observed assembly task.