Dynamics and Control

Optimal mechanical design, model-based control, and robot dynamic calibration mainly rely on the analytical formulation of robot dynamics. In general, dynamics analysis of parallel robots is more complicated than serial robots due to the existence of closed kinematic chains. In the explicit dynamic model, robot dynamics are required to be derived by three principal components, mainly the mass matrix, the centrifugal and Coriolis matrix, and the robot gravity vector. With the advantages of explicit dynamics extraction, many advanced model-based controllers can be designed and implemented on parallel robots. Nonetheless, in the absence of sufficient knowledge of the explicit dynamic model of the robot, only simple and effective model-free controllers such as PD and PID can be utilized. These controllers are configuration-dependent, and their tuning for the whole workspace is very difficult. 

Generally, derivation of the explicit form of parallel robots dynamics using the Newton-Euler method is not recommended since it requires derivation and elimination of the constraint forces. Likewise, the Lagrange method necessitates taking partial derivatives of the mass matrix analytically to extract Coriolis terms, which is a prohibitive task. In the virtual work approach, first, each link’s linear and angular accelerations are extracted, and the inertial forces and moments are derived. Next, by using D’Alembert’s principle, the entire robot may be assumed to be at static equilibrium, and by utilizing the principle of virtual work, the actuator forces may be derived. This approach has lower computational cost in comparison to that of other methods.

The main contribution of this work is to derive an explicit dynamics model of the 3-UPU translational parallel robot applicable in the robot’s controller design. However, it should be noted that due to the presence of universal joints in the robot, there are some angular velocity terms for each limb, which are wrongly ignored in the literature, to simplify the kinematic and dynamic equations of the robot. Although this assumption simplifies the robot’s kinematic analysis, the robot’s dynamics model would lose its precision.