A Theory of Impedance Control based on Internal Model Uncertainty

Efficient human motor control is characterised by an extensive use of joint impedance modulation, which to a large extent is achieved by co-contracting antagonistic muscle pairs in a way that is beneficial to the specific task. Studies in single and multi joint limb reaching movements revealed that joint impedance is increased with faster movements [1] as well as with higher positional accuracy demands [2]. A large body of experimental work has investigated the motor learning processes in tasks with changing dynamics conditions (e.g., [3]) and it has been shown that subjects generously make use of impedance control to counteract destabilising external force fields (FF). In the early stage of dynamics learning humans tend to increase co-contraction. As learning progresses in consecutive reaching trials, a reduction in co-contraction with a parallel reduction of the reaching errors made can be observed. While there is much experimental evidence available for the use of impedance control in the CNS, no generally-valid computational model of impedance control derived from first principles have been proposed so far. Many of the proposed computational models have either focused on the biomechanical aspects of impedance control [4] or have proposed simple low level mechanisms to try to account for observed human co-activation patterns [3]. However these models are of a rather descriptive nature and do not provide us with a general and principled theory of impedance control in the nervous system. Here we develop a powerful computational model for impedance control, which describes muscle coactivation in human arm reaching tasks as an emerging mechanism from first principles of optimality. We hypothesise that, in conjunction with an appropriate antagonistic arm and motor variability model, impedance control emerges from an optimisation process that minimises prediction uncertainties of the internal model. Our model is formalized within the theory of stochastic Optimal Feedback Control (OFC) [5], in which an actor’s goal is expressed as a solution to an optimisation process that typically minimises energy consumption and reaching error. Unlike previous OFC models, which required analytic dynamics model formulations, here we postulate that the dynamics model is acquired as a motor learning process based on continuous sensorimotor feedback. Because this stochastic OFC with learned dynamics (OFC-LD) updates the internal dynamics model from plant data during control, it can be used to model adaptation processes due to changing dynamics conditions. This allows us to investigate human FF adaptation paradigms within the powerful framework of optimality. The human sensorimotor system is known to exhibit a highly stochastic behaviour and a complete motor control theory must be able to deal with the detrimental effects of signal dependent noise (SDN) 1 . Previously proposed stochastic optimal control models [6] assumed naive SDN (Fig. 1B), which ignored the impedance to noise characteristics of the musculoskeletal system and therefore failed to model co-contraction. Here however we model an extended type of SDN (Fig. 1C), the magnitude of which realistically decreases with higher coactivation levels [7]. The learner interprets this motor variability as prediction uncertainty, which is given algorithmically in form of heteroscedastic (i.e., locally valid) prediction variances. With these ingredients we formulate a minimum-uncertainty optimal control model that introduces this stochastic information into OFCLD. Due to the realistic nature of the motor variability our system exhibits, the OFC-LD solutions will, while still trying to reduce energetic costs and endpoint error, favour co-contraction in order to reduce the negative effects of the SDN. In summary, this normative model of impedance control for antagonistic limb systems is based on the quality of the learned internal model and therefore leads to the intuitive requirement that impedance will be increased in cases where the actor is uncertain about his model predictions. This is of special importance during adaptation tasks, where prediction uncertainty also increases, leading to similar increases in cocontraction. We evaluated our model in several simulation experiments with stationary dynamics (Fig. 2) as well as in adaptation tasks (Fig. 3). The results show that our computational model is able to predict many well-known impedance control phenomena from the first principles of optimality, and that the minimum-uncertainty approach can conceptually explain the origins of co-activation in volitional human reaching tasks. We believe that our model will generalise well to more complex (multi-joint) plant models.