Modulation of Joint Stiffness for Controlling the Cartesian Stiffness of a 2-DOF Planar Robotic Arm for Rehabilitation

This paper presents a method for achieving Cartesian stiffness control at the endpoint of a 2-degree-of-freedom planar robotic arm by modulating joint stiffnesses. Planar robotic arms are widely applied for upper-limb rehabilitation through impedance control, but not generally in Cartesian stiffness control through joint stiffness. A modular robotic actuator with integrated controllers on a robot prototype enables the direct command of desired joint stiffness. A closed-form solution was derived through the Jacobian matrix to map the stiffnesses of a reference equilibrium. In addition, the prediction of the joint displacement corresponding to the endpoint motion is required for computing the needed joint stiffnesses. The proposed method is experimentally validated by recording the Cartesian force against the unidirectional displacement at different robotic arm configurations, showing a linear relationship. The results suggest that the proposed method has the potential for use in rehabilitation tasks when the direction of the endpoint displacement is predetermined. The method allows a precise control of the robotic arm's stiffness, which can help in creating more efficient rehabilitation protocols on an easily accessible and affordable rehabilitation robot. Nonetheless, further work is needed to improve the accuracy and omnidirectional robustness of the control method. The study also highlights the importance of designing a robotic arm to satisfy stiffness requirements in addition to kinematic optimization for sufficient workspaces.