Dynamic analysis of simultaneous adaptation of force, impedance and trajectory

When carrying out tasks in contact with the environment, humans are found to concurrently adapt force, impedance and trajectory. Here we develop a robotic model of this mechanism in humans and analyse the underlying dynamics. We derive a general adaptive controller for the interaction of a robot with an environment solely characterised by its stiffness and damping, using Lyapunov theory. The dynamics of a n-degree-of-freedom (n-DOF) robot in the operational space are given by M (q) ¨ x + C(q, ˙ q) ˙ x + G(q) = u + f (1) where x is the position of the robot and q the vector of joints angle. M (q) denotes the inertia matrix, C(q, ˙ q) ˙ x the Coriolis and centrifugal forces, and G(q) the gravitational force, which can be identified using e.g. nonlinear adaptive control [1]. u is the control input and f the interaction force. In [2], we have described the control input u in two parts: u = v + w , (2) with v to track the reference trajectory x r by compensating for the robot's dynamics, i.e.