Unit Quaternions: A Mathematical Tool for Modeling, Path Planning and Control of Robot Manipulators

Abstract

Among the different parameterizations of the orientation manifold, Euler parameters are of interest because they are unit quaternions, and form a Lie group. This chapter was intended to recall some of the properties of the quaternion algebra, and to show the usage of Euler parameters for modeling, path planning and control of robot manipulators. For many years, the Denavit?Hartenberg parameters have been employed for obtaining the direct kinematic model of robotic mechanisms. Instead of using homogeneous matrices for handling the D-H parameters, we proposed an algorithm that uses the quaternion multiplication as basis, thus being computationally more efficient than matrix multiplication. In addition, we sketched a general methodology for obtaining the dynamic model of serial manipulators when Euler parameters are employed. The so?called spherical linear interpolation (or slerp) is a method for interpolating curves in the orientation manifold. We reviewed such an algorithm, and proposed its application for motion planning in robotics. Finally, we studied the stability of two task space controllers that employ unit quaternions as state variables; such analysis allows us to conclude that both controllers are suitable for motion control tasks in pose space. As a conclusion, even though Euler parameters are still not very known, it is our believe that they constitute a useful tool, not only in the robotics field, but wherever the description of objects in 3-D space is required. More research should be done in these topics.