Closed-loop system identification of ankle dynamics using a subspace method with reference input as instrumental variable

Joint stiffness, the dynamic relationship between the angular position of a joint and the torque acting about it, can be used to describe the dynamic behavior of the human ankle during posture and movement. Joint stiffness can be separated into intrinsic stiffness and reflex stiffness, which are modeled as linear and LNL systems, respectively. For most functional tasks, the ankle interacts with a compliant load. The joint stiffness can be viewed as being operated in closed-loop because the torque is fed back to change the position of the ankle. Consequently, standard open loop identification methods will give biased results. In this paper, we present a new method to estimate intrinsic and reflex stiffness from the total torque measured in closed-loop. A MOESP (multivariable output-error state-space) subspace system identification method is used to estimate the dynamics of each pathway directly from measured data. The past reference input is used as an instrumental variable to eliminate noise fed back via the controller loop. Simulation and experimental studies demonstrate that the method produces accurate results.