Techniques for concatenating, interpolating and
upsampling pose transforms represented as dual quaternions are described, including: (1) derivation of a complex-valued
matrix form of dual quaternions and
dual quaternion operations; (2) derivation of a
transformation operator on position vectors which obviates an explicit conversion to a classical 4×4
spatial transformation matrix and keeps results in complex-valued
matrix space; (3) design for a generic
lookup table system for functions to supply logarithm and exponentiations of the
dual quaternion in its native format with trigonometry lookup tables to avoid precision issues when denominators tend to zero; and (4) a mechanism for wrapping the complex-exponentiation together with a simple complex arithmetic unit for computing
dual quaternion macro-operations in both native dual
quaternion space and through simplifications of the equivalent complex-valued matrix to compute dual
quaternion operations such as inverses, multiplications, logarithms and exponentials in order to chain the
pose transformations encoded within.