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Using Clifford Algebra to position a test fixture

机译：使用Clifford代数定位测试夹具

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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 Cl3, 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 Cl3, 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.
机译：本文是关于使用Clifford代数来定位万向架测试夹具，该夹具在红外激光（IR）指向系统测试中使用。 Clifford代数Cl3，0（特别称为物理空间代数（APS））可用作物理学中的统一数学框架，可以以一致且连续的方式描述经典物理学，狭义相对论，广义相对论，电动力学和量子力学。简洁的方法。本文力图证明APS在控制简单的2自由度（DOF）万向架系统中的应用，这对于测试2DOF被测装置（UUT）很有用。测试的目的是将UUT旋转到特定的测试位置，并将UUT指向光学测试设备。这要求对多个参考系和坐标系进行准确而可靠的转换。 APS对旋转有用的原因是，APS的偶数子代数与汉密尔顿的四元数代数同构。 Clifford代数Cl3，0可以使用2×2复矩阵的Pauli代数的矩阵表示在计算机上数字实现。本文将介绍APS，四元数和Pauli矩阵之间的关系。本文还将介绍形成超向量，转子/本征转子以及与矩阵形式相互转换的所有基本操作。本文将介绍示例系统中参考帧和矢量的变换和逆变换。

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