Quaternion Analysis Tools for Engineering and Scientific Applications

Abstract

Q UATERNION algebra has found a number of applications for engineering and scientific problems, including fluidmechanics [1], quantum mechanics [2,3], robotics [4,5], and spacecraft attitude control [5,6]. Unfortunately, the literature available for supporting engineering applications is diffuse for matrix applications, as well as virtually nonexistent for handling scalar linear and quadratic equations. Very little general-purpose software is available for computing quaternion math models. This Note fills an unmet need for providing a convenient reference for quaternion calculations relevant to engineering and scientific applications. Equally important, the Note provides a compendium of examples and techniques for evaluating scalar linear and quadratic equations, as well as linear matrix equations, matrix inversion, and eigensolutions. Basic operations are defined for all intrinsic operations ( , , , =, ) and standard software library function (e.g., sin, cos, ln , exp, cosh,a tan, etc.). A comprehensive solution is presented for handling scalar linear equations, in which many special-case solutions are identified. Particular care is given to identifying the equation coefficient conditions required for generating singular solutions. Numerical algorithms are presented for solving scalar quadratic equations. A previously unknown singularity has been identified that has hampered the solution for quadratic equations, and a new algorithm is presented for analytically eliminating the quadratic matrix-valued singularity for solving scalar quadratic equations. Three strategies are presented for handlingN Nmatrix operations: 1) purely N N quaternion operations, 2) dimension-doubling routines using 2N 2N complex operations, and 3) dimensionquadrupling 4N 4N real operations.Many numerical examples are presented in which Quadpack95, a FORTRAN 95/2003 program developed by the author, is used for evaluating all of the quaternion calculations. II. Quaternion Math Models