Cartesian Impedance Control Generalized to One-Parameter Splines

Robotic-assisted upper limb rehabilitation has gained significant attention in recent years due to its potential to enhance the recovery process for individuals with motor impairments resulting from neurological conditions and injuries. The main rehabilitation treatments rely on the repetitive execution of a movement of the upper-limb, guided by a therapist to prevent incorrect movements and to provide the necessary support. Many of the exercises performed by therapists can be modeled as a movement in SE(3) space (position and orientation). This movement itself is one-dimensional, as it can be modeled by a one-dimensional curve. To solve a similar problem, some approaches have been proposed in human-robot interaction (HRI) following virtual guides, but are either limited to specific types of curves (e.g. without orientation) or rely on linear control methods with non-intuitive parameters. To address these limitations and enable the use of these methods in physical rehabilitation, this paper extends Cartesian impedance control to splines, which we term path impedance control. It capitalizes on the intrinsic path geometry of end-effector robotic rehabilitation systems. The primary objective of this control algorithm is to emulate the sensation of maneuvering a physical object along a wire, akin to conventional exercise machines; and, in conjunction, provide an intuitive parametrization of rehabilitation exercises. We build on existing virtual guide control strategies using non-linear control and Lie Groups to generalize the control law to any one-parameter SE(3) curve.