Standard Dual Quaternion Functions and Standard Dual Quaternion Optimization

Dual quaternions now have wide applications in robotics, 3D motion modelling and control, and computer graphics. The magnitudes of dual quaternions and the 2-norms of dual quaternion vectors are dual numbers. A total order was defined for dual numbers. Thus, dual quaternion optimization problems, where objective and constraint functions have dual quaternion variables and dual number function values naturally arise. In this paper, we show that several common dual quaternion functions, such as the power function, the magnitude function, the 2-norm function and the kth largest eigenvalue function of dual quaternion Hermitian matrices, are standard dual quaternion functions, i.e., the standard parts of their function values depend upon only the standard parts of the dual quaternion variables. Furthermore, the sum, product, minimum, maximum and composite functions of two standard dual functions, the logarithm and the exponential of a standard unit dual quaternion functions, are still standard dual quaternion functions. To solve a standard dual quaternion optimization problem, we only need to solve two quaternion optimization problems. Thus, if the dual quaternion functions are standard, the related dual quaternion optimization problem is solvable.