Using Clifford Algebra to position a test fixture

This paper is about using Clifford Algebra to position a gimbal test fixture that is used during infrared laser (IR) pointing system testing. The Clifford Algebra Cl_{3, 0} specifically known as the Algebra of Physical Space (APS) is useful as a unifying mathematical frame work in physics that can describe classical physics, special relativity, general relativity, electro-dynamics, and quantum mechanics in a consistent and concise way. This paper seeks to demonstrate the use of APS in controlling a simple 2 degree of freedom (DOF) gimbal system that would be useful for testing a 2DOF unit under test (UUT). The goal of the test is to rotate the UUT to a specific test position and point the UUT towards optical test equipment. This requires the accurate and reliable transformation of multiple reference frames and coordinate systems. The reason why APS is useful for rotations is because the even sub-algebra of APS is homomorphic to Hamilton´s quaternion algebra. Clifford Algebra Cl_{3, 0} can be implemented numerically on a computer using the matrix representation of the Pauli algebra of 2×2 complex matrices. This paper will show the relationship between APS, quaternions, and Pauli matrices. The paper will also cover all of the basic operations of forming paravectors, rotors/eigenspinors, and converting to and from the matrix form. The paper will cover transformation and inverse transformations of reference frames and vectors in our example system.