Clifford Algebras in Engineering

Abstract

In this chapter, we look at some applications of Clifford algebra in engineering. These applications are geometrical in nature concerning robotics and vision mainly. Most engineering applications have to be implemented on a computer these days so we begin by arguing that Clifford algebra are well suited to modern microprocessor architectures.

Our first application is to satellite navigation and uses quaternions. The rotation of the satellite is to be found from observations of the fixed stars. The same problem occurs in many other guises throughout the natural sciences and engineering.

Next we use biquaternions to write down the kinematic equations of the Stewart platform. This parallel robot is used in aircraft simulators and in novel machine tools. The problem is to determine the position and orientation of the platform from the lengths of the hydraulic actuators.

In the next section, we introduce a less familiar Clifford algebra Cℓ(0, 3, 1). The homogeneous elements of this algebra can be used to represent points, lines and planes in three dimensions. Moreover, meets joins and orthogonal relationships between these linear subspaces can be modelled by simple formulas in the algebra. This algebra is used in the following sections to discuss a couple of problems in computer vision and the kinematics of serial robots.