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.