A redundantly actuated parallel kinematic machine (PKM) can be used to avoid singularities, normalize manipulability, and increase the stiffness of anon-redundant mechanism. In this study, a redundantly actuated symmetrical PKM with five revolute (5R) joints is optimized for isotropic stiffness in the workspace. The stiffness of the 5R symmetrical PKM is calculated by the superposition of the actuator stiffness and the structural stiffness. We compared the stiffness of anon-redundant PKM and a redundant PKM. Compliance ellipses of the actuator stiffness and the structural stiffness of the non-redundant PKM resulted in the same configurations in the workspace, while those of the redundant PKM resulted in very different configurations. Optimization was performed by determining the optimal actuator torques that are needed to maximize the conditioning index. Optimal results considering structural stiffness can provide a more uniform directional stiffness than optimal results considering the index. When the strength of the linkages in a PKM is weak, the structural stiffness affects the actual stiffness considerably. We believe that the results of this study can be used to help design and control redundant PKMs.

Redundant actuation for the parallel kinematic machine (PKM) is a well-known technique for overcoming general drawbacks of the PKM by helping it to avoid singularity and enhance stiffness characteristics, among others. Torque distribution plays a critical role in redundant actuation because this actuation causes the PKM to consume too much energy or put a substantial amount of stress on joints and links. This paper proposes a new torque distribution method for reducing the maximum torque of the actuator of a planar PKM. Here the main idea behind the proposed method is the use of superposition of a particular solution for a non-redundant case and an optimized null-space solution for a redundant case with a constant coefficient. The optimal value of a null-space solution can be easily determined by checking only the intersection points of the profile of the actuator's torque as the coefficient varies. We consider three cases of planar PKMs—2-, 3-, and 4-RRR PKMs—and present a detailed procedure for deriving a kinematic solution for the 2-RRR PKM based on Screw theory. We compare the proposed method with the minimum-norm pseudo-inverse method and assess a limitation of the proposed method. The torque distribution algorithm can be used to determine the number of actuators in an efficient manner and to reduce energy consumption.

An adaptive computed torque (ACT) controller in the task space is proposed for the trajectory tracking of a parallel manipulator with redundant actuation. The dynamic model, including the active joint friction, is established in the task space for the parallel manipulator, and the linear parameterization expression with respect to the dynamic and friction parameters is formulated. On the basis of the dynamic model, a new control law, which contains adaptive dynamics compensation, friction compensation, and tracking error elimination terms, is designed. After defining the state-space model of the error system, the parameter adaptation law is derived by using the Lyapunov method, and the convergence of the tracking error and the error rate is proved by using the Barbalat's lemma. The ACT controller is implemented in the trajectory tracking experiments of an actual 2-DOF parallel manipulator with redundant actuation, and the experiment results are compared with the computed torque controller.