This paper investigates the tracking control performance regulation and actuation allocation of mechatronic systems subject to coupling motions. In particular, the kinematic and dynamic model is described by dual quaternion, which captures the coupling effect between translation and rotation movements. Considering external disturbances and system uncertainties, a non-singular fast terminal sliding controller is then developed to ensure finite-time tracking performance. In addition, the unwinding problem caused by the redundancy of dual quaternion is addressed with the help of a novel attitude error function. Furthermore, an improved simplex method is designed for distributing the developed control commands to proper actuators. Numerical simulations demonstrate the effectiveness with respect to disturbance suppression, fast tracking, high accuracy, and finite-time stability of the proposed method.