Inverse Dynamics of a Redundantly Actuated Four-Bar Mechanism Using an Optimal Control Formulation

This paper presents an approach to estimating joint torques in a four-bar closed-chain mechanism with prescribed kinematics and redundant actuation, i.e., with more actuators than degrees of freedom. This problem has several applications in industrial robots, machine tools, and biomechanics. The inverse dynamics problem is formulated as an optimal control problem (OCP). The dynamical equations are derived for an open-chain mechanism, what keeps the formulation simple and straightforward. Sets of constraints are explored to force the three-link open-chain to behave as a four-bar mechanism with a crank rotating at a constant velocity. The controls calculated from the OCP are assumed to be the input joint torques. The standard case with one torque actuator is solved and compared to cases with two and three actuators. The case of two actuators presented the smallest peak and mean torques, using one specific set of constraints. Such torques were smaller than the solution obtained using an alternative method existing in literature that solves the redundancy problem by means of the pseudo-inverse matrix. Comparison with inverse dynamics solutions using well-established methods for the one-actuator closed-loop four-bar were equal. Reconstructed kinematical trajectories from forward integration of the closed-loop mechanism with the OCP obtained torques were essentially similar. The results suggest that the adopted procedure is promising, giving solutions with lower torque requirements than the regularly actuated case and redundantly actuated computed with other approaches. The applicability of the method has been shown for the four-bar mechanism. Other classes of redundantly actuated, closed-loop mechanisms could be tested using a similar formulation. However, the numerical parameters of the OCP must be chosen carefully to achieve convergence.