Practical Exponential Coordinates Using Implicit Dual Quaternions

Abstract

Modern approaches for robot kinematics employ the product of exponentials formulation, represented using homogeneous transformation matrices. Quaternions over dual numbers are an established alternative representation; however, their use presents certain challenges: the dual quaternion exponential and logarithm contain a zero-angle singularity, and many common operations are less efficient using dual quaternions than with matrices. We present a new derivation of the dual quaternion exponential and logarithm that removes the singularity, and we show an implicit representation of dual quaternions offers analytical and empirical efficiency advantages compared to both matrices and explicit dual quaternions. Analytically, implicit dual quaternions are more compact and require fewer arithmetic instructions for common operations, including chaining and exponentials. Empirically, we demonstrate a 25%–40% speedup to compute the forward kinematics of multiple robots. This work offers a practical connection between dual quaternions and modern exponential coordinates, demonstrating that a quaternion-based approach provides a more efficient alternative to matrices for robot kinematics.