A computational model of limb impedance control based on principles of internal model uncertainty

Abstract

Efficient human motor control is characterized by an extensive use of joint impedance modulation, which is achieved by co-contracting antagonistic muscles in a way that is beneficial to the specific task. While there is much experimental evidence available that the nervous system employs such strategies, no generally-valid computational model of impedance control derived from first principles has been proposed so far. Here we develop a new impedance control model for antagonistic limb systems which is based on a minimization of uncertainties in the internal model predictions. In contrast to previously proposed models, our framework predicts a wide range of impedance control patterns, during stationary and adaptive tasks. This indicates that many well-known impedance control phenomena naturally emerge from the first principles of a stochastic optimization process that minimizes for internal model prediction uncertainties, along with energy and accuracy demands. The insights from this computational model could be used to interpret existing experimental impedance control data from the viewpoint of optimality or could even govern the design of future experiments based on principles of internal model uncertainty.

Figure 1. Schematic representation of our OFC-LD approach.

The optimal controller requires a cost function, which here encodes for reaching time, endpoint accuracy, endpoint velocity (i.e., stability), and energy efficiency. Further a forward dynamics function is required, which in OFC-LD is learned from plant feedback directly. This learned internal dynamics function not only allows us to model changes in the plant dynamics (i.e., adaptation) but also encodes for the uncertainty in the dynamics data. The uncertainty itself, visible as kinematic variability in the plant, can originate from different sources, which we here classify into external sources and internal sources of uncertainty. Most notably OFC-LD identifies the uncertainty directly from the dynamics data not making prior assumptions about its source or shape.

Figure 2. Illustration of the effects of standard and extended SDN on kinematic variability in the end-effector.

Standard SDN scales proportionally to the muscle activation, whereas the extended SDN takes into account the stabilizing effects of higher joint impedance when co-contracting (see Methods), producing a “valley of reduced SDN” along the co-contraction line . The colors represent the noise variance as a function of muscle activations, whereas the dark lines represent muscle activations that exert the same joint torque computed for joint angle position . (a) Only muscle is activated, producing Nm joint torque with a Gaussian kinematic variability of . (b) The same torque with higher co-contraction produces significantly higher kinematic variability of under standard SDN. (c) Same conditions as in (a) in the case where only muscle is activated. In contrast to (b) the extended SDN in (d) favors co-contraction leading to smaller kinematic variability of and to more stable reaching.

Figure 3. Comparison of the results from stochastic OFC using standard SDN (a) and extended SDN (b).

We performed 50 OFC reaching movements (only 20 trajectories plotted) under both stochastic conditions. The shaded green area indicates the region and amount of co-contraction in the extended SDN solution. The plots in (c) quantify the results (mean +/− standard deviation). Left: average joint angle error (absolute values) at final time T = 500 msec. Middle: Joint angle velocity (absolute values) at time T. Right: integrated muscle commands (of both muscles) over trials. The extended SDN outperforms the reaching performance of the standard SDN case at the expense of higher energy consumption.

Figure 4. Experimental results from stochastic OFC-LD for different accuracy demands.

The first row of plots shows the averaged joint angles (left), the averaged joint velocities (middle) and the averaged muscle signals (right) over 20 trials for the five conditions A, B, C, D, and E. The darkness of the lines indicates the level of accuracy; the brightest line indicates condition A, the darkest condition E. The bar plots in the second row average the reaching performance over 20 trials for each condition. Left: The absolute end-point error and the end-point variability in the trajectories decreases as accuracy demands are increased; Middle: End-point stability also increases (demonstrated by decreasing error in final velocities); Right: The averaged co-contraction integrated during 500 msec increases with higher accuracy demands, leading to the reciprocal relationship between accuracy and impedance control as observed in humans.

Figure 5. Experimental results from stochastic OFC-LD for different peak joint velocities.

The first row of plots shows the averaged joint angles (left), the averaged joint velocities (middle) and the averaged muscle signals (right) over 20 trials for reaches towards the three target conditions “near”, “medium” and “far”. The darkest line indicates “far”, the brightest indicates the “near” condition. The bar plots in the second row quantify the reaching performance averaged over 20 trials for each condition. The end-point errors (left) and end-velocity errors (middle) show good performance but no significant differences between the conditions, while co-contraction during the motion as expected increases with higher velocities, due to the higher levels of muscle signals.

Figure 6. Optimal reaching movement, before, during and after adaptation.

Clearly the solution is being re-optimized with the learned dynamics (including the FF).

Figure 7. Adaptation results.

(a) Accumulated statistics during 25 adaptation trials using stochastic OFC-LD. Trials 1 to 5 are performed in the NF condition. Top: Muscle activations and co-contraction integrated during 500ms reaches. Middle: Absolute joint errors and velocity errors at final time T = 500ms. Bottom: Integrated (internal model) prediction uncertainties along the current optimal trajectory, after this has been updated. (b) The same statistics for the adaptation using deterministic OFC-LD, meaning no uncertainty information is used for the optimization. This leads to no co-contraction and therefore worse reaching performance during adaptation.