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Feed-forward control of a redundant motor system.

We describe a model of feed-forward control of a redundant motor system and validate it using, as examples, tasks of multi-finger force production. The model assumes the existence of two input signals at an upper level of the control hierarchy, related and unrelated to a task variable. Knowledge of the Jacobian of the system is assumed at the level of generation of elemental variables (variables at the level of effectors). Variance at the level of elemental variables is considered as the sum of two components, related and unrelated to variability in the task variable. An index of stabilization of the task variable is similarly introduced as to how it was done in several studies using the framework of the uncontrolled manifold hypothesis. Several phenomena have been simulated including data point distributions corresponding to presence and absence of force-stabilizing synergies in two-finger tasks, changes in synergies with practice, and changes in synergy indices in preparation to a fast action. The model is discussed in comparison to other models of control of multi-element systems based on feedback processes. It shows that patterns of structured variability in the space of elemental variables can result from feed-forward processes. Relations of the model to the equilibrium-point hypothesis are also discussed.

Feedback↗

On the role of recurrent inhibitory feedback in motor control.

This article reviews presumed roles of recurrent inhibition in motor control, that have been proposed over the past five decades. The discussion is structured in an order of increasing complexity. It starts out with the simplest and earliest circuit, that is recurrent self-inhibition of skeleto-motoneurons, and related functions. It soon becomes clear that in order to understand recurrent inhibition, we must look beyond the simple self-inhibitory CNS circuit. First, recurrent inhibition must be seen in the context of other neural circuits. Second, some quantitative features appear to be correlated with features of the neuromusculo-skeletal periphery. Third, the aspect of lateral inhibition between different members of a motoneuron pool as well as between different motoneuron pools points to the essential multiple input-multiple output structure of recurrent inhibition that again can be understood only by correlating it with features of the neuromusculo-skeletal periphery. Another extension results from the discovery that recurrent inhibition affects not only skeleto-motoneurons, but also gamma-motoneurons, Ia inhibitory interneurons mediating reciprocal inhibition between antagonist motoneurons, other Renshaw cells and cells of origin of the ventral spinocerebellar tract (VSCT). Then the view broadens again, investigating the potential role that recurrent inhibition plays in two far-ranging theories of motor control, the inverse-dynamics approach and the equilibrium-point hypothesis. Finally, the present author tries to formulate, in broad strokes, a personal functional interpretation of recurrent inhibition. All the functional considerations, right or wrong, should yield ideas for new experiments, and this then is the last objective of this review.

Animals↗

Reconstruction of shifting elbow joint compliant characteristics during fast and slow movements.

The purpose of this study was to experimentally investigate the applicability of the equilibrium-point hypothesis to the dynamics of single-joint movements. Subjects were trained to perform relatively slow (movement time 600-1000 ms) or fast (movement time 200-300 ms) single-joint elbow flexion movements against a constant extending torque bias. They were instructed to reproduce the same time pattern of central motor command for a series of movements when the external torque could slowly and unpredictably increase, decrease, or remain constant. For fast movements, the total muscle torque was calculated as a sum of external and inertial components. Analysis of the data allowed reconstruction of the elbow joint compliant characteristics at different times during execution of the learned motor command. "Virtual" trajectories of the movements, representing time-varying changes in a central control parameter, were reconstructed and compared with the "actual" trajectories. For slow movements, the actual trajectories lagged behind the virtual ones. There were no consistent changes in the joint stiffness during slow movements. Similar analysis of experiments without voluntary movements demonstrated a lack of changes in the central parameters, supporting the assumption that the subjects were able to keep the same central motor command in spite of externally imposed unexpected torque perturbations. For the fast movements, the virtual trajectories were N-shaped, and the joint stiffness demonstrated a considerable increase near the middle of the movement. These findings contradict an hypothesis of monotonic joint compliant characteristic translation at a nearly constant rate during such movements.

Adult↗

Virtual trajectories, joint stiffness, and changes in the limb natural frequency during single-joint oscillatory movements.

In the framework of the equilibrium-point hypothesis, virtual trajectories and patterns of joint stiffness were reconstructed during voluntary single-joint oscillatory movements in the elbow joint at a variety of frequencies and against two inertial loads. At low frequencies, virtual trajectories were in-phase with the actual joint trajectories. Joint stiffness changed at a doubled frequency. An increase in movement frequency and/or inertial load led to an increase in the difference between the peaks of the actual and virtual trajectories and in both peak and averaged values of joint stiffness. At a certain, critical frequency, virtual trajectory was nearly flat. Further increase in movement frequency led to a 180 degree phase shift between the actual and virtual trajectories. The assessed values of the natural frequency of the system "limb + manipulandum" were close to the critical frequencies for both low and high inertial loads. Peak levels and integrals of the electromyograms of two flexor and two extensor muscles changed monotonically with movement frequency without any special behavior at the critical frequencies. Nearly flat virtual trajectories at the natural frequency make physical sense as hypothetical control signals, unlike the electromyographic recordings, since a system at its natural frequency requires minimal central interference. Modulation of joint stiffness is assumed to be an important adaptive mechanism attenuating difference between the system's natural frequency and desired movement frequency. Virtual trajectory is considered a behavioral observable. Phase transitions between the virtual and actual trajectories are illustrations of behavioral discontinuities introduced by slow changes in a higher level control parameter, movement frequency. Relative phase shift between these two trajectories may be considered an order parameter.

Adult↗

Control processes underlying elbow flexion movements may be independent of kinematic and electromyographic patterns: experimental study and modelling.

Using a non-linear dynamic model based on the lambda version of the equilibrium-point hypothesis, we investigated the shape and duration of the control patterns underlying discrete elbow movements. The model incorporates neural control variables, time-, position- and velocity-dependent intrinsic muscle and reflex properties. Two control variables (R and C) specify a positional frame of reference for activation of flexor and extensor motoneurons. The variable R (reciprocal command) specifies the referent joint angle (R) at which the transition of net flexor to extensor active torque or vice versa can be observed during changes in the actual joint angle elicited by an external force. The variable C (coactivation command) surrounds the transition angle by an angular range in which flexor and extensor muscles may be simultaneously active (if C > 0) or silent (if C < or = 0). An additional, time-dimensional control variable (mu command) influences the dependency of the threshold of the stretch reflex on movement velocity. This control variable is responsible for the reflex damping. Changes in the R command result in shifts in the equilibrium state of the system, a dynamical process leading to electromyographic modifications and movement production. Commands C and mu provide movement stability and effective energy dissipation preventing oscillations at the end of movement. A comparison of empirical and model data was carried out. A monotonic ramp-shaped pattern of the R command can account for the empirical kinematic and electromyographic patterns of the fastest elbow flexion movements made with or without additional inertia, as well as of self-paced movements. The rate of the shifts used in simulation was different for the three types of movements but independent of movement distance (20-80 degrees). This implies that, for a given type of movement, the distance is encoded by the duration of shift in the equilibrium state. The model also reproduces the kinematic and electromyographic patterns of the fastest uncorrected movements opposed in random trials by a high load (80-90% of the maximal) generated by position feedback to a torque motor. The following perturbation effects were simulated: a substantial decrease in the arm displacement (from 60-70 degrees to 5-15 degrees) and movement duration (to about 100 ms) so that these movements ended near the peak velocity of those which were not perturbed; a prolongation of the first agonist electromyographic burst as long as the load was applied; the suppression of the antagonist burst during the dynamic and static phases of movements: the reappearance of the antagonist burst in response to unloading accompanied by a short-latency suppression of agonist activity. These kinematic and electromyographic features of both perturbed and non-perturbed movements were reproduced by using the same control patterns which elicited a monotonic shift in the equilibrium state of the system ending before the peak velocity of non-perturbed movements. Our results suggest that the neural control processes underlying the fastest unopposed changes in the arm position are completed long before the end of the movement and phasic electromyographic activity. Neither the timing nor the amplitude of electromyographic bursts are planned but rather they represent the long-lasting dynamic response of central, reflex and mechanical components of the system to a monotonic, short-duration shift in the system's equilibrium state.

Adult↗

Effects of contraction history on control and stability in explosive actions.

In this model study, the effect of contraction history in skeletal muscle on joint movement and stability was investigated. A joint system was constructed with two identical (antagonistic) muscles. The muscles were modelled either according to the Hill tradition or as a modified Hill system including history-dependent properties. The joint system underwent movements with full activity of both muscles, mimicking explosive actions with high stability demands. Movements starting away from a balanced mid-position, as well as perturbation experiments, were simulated. The comparison between the Hill and modified Hill systems showed that contraction history improved stability (stiffness under perturbation) and, under certain conditions, caused a shift in the final joint position, which depended on the task characteristics (starting position and perturbations characteristics). This result indicates that modulations of muscle activity, required to move a joint to a particular end-position, do not only depend on the end-position but also on the preceding movements. This finding does not agree with the equilibrium-point hypothesis and is discussed accordingly.

Biomechanical Phenomena↗

An equilibrium-point model for fast, single-joint movement: I. Emergence of strategy-dependent EMG patterns.

We describe a model for the regulation of fast, single-joint movements, based on the equilibrium-point hypothesis. Limb movement follows constant rate shifts of independently regulated neuromuscular variables. The independently regulated variables are tentatively identified as thresholds of a length sensitive reflex for each of the participating muscles. We use the model to predict EMG patterns associated with changes in the conditions of movement execution, specifically, changes in movement times, velocities, amplitudes, and moments of limb inertia. The approach provides a theoretical neural framework for the dual-strategy hypothesis, which considers certain movements to be results of one of two basic, speed-sensitive or speed-insensitive strategies. This model is advanced as an alternative to pattern-imposing models based on explicit regulation of timing and amplitudes of signals that are explicitly manifest in the EMG patterns.

Journal Article↗

On the problem of adequate language in motor control.

An adequate language is a prerequisite for progress in any area of science, including movement science. Notions of structural units and synergies and the principle of minimal interaction are revisited, discussed, and illustrated with a few examples from recent studies. Equilibrium-point hypothesis is considered an example of identifying significant variables in the control of a voluntary movement.

Humans↗

Are complex control signals required for human arm movement?

It has been proposed that the control signals underlying voluntary human arm movement have a "complex" nonmonotonic time-varying form, and a number of empirical findings have been offered in support of this idea. In this paper, we address three such findings using a model of two-joint arm motion based on the lambda version of the equilibrium-point hypothesis. The model includes six one- and two-joint muscles, reflexes, modeled control signals, muscle properties, and limb dynamics. First, we address the claim that "complex" equilibrium trajectories are required to account for nonmonotonic joint impedance patterns observed during multijoint movement. Using constant-rate shifts in the neurally specified equilibrium of the limb and constant cocontraction commands, we obtain patterns of predicted joint stiffness during simulated multijoint movements that match the nonmonotonic patterns reported empirically. We then use the algorithm proposed by Gomi and Kawato to compute a hypothetical equilibrium trajectory from simulated stiffness, viscosity, and limb kinematics. Like that reported by Gomi and Kawato, the resulting trajectory was nonmonotonic, first leading then lagging the position of the limb. Second, we address the claim that high levels of stiffness are required to generate rapid single-joint movements when simple equilibrium shifts are used. We compare empirical measurements of stiffness during rapid single-joint movements with the predicted stiffness of movements generated using constant-rate equilibrium shifts and constant cocontraction commands. Single-joint movements are simulated at a number of speeds, and the procedure used by Bennett to estimate stiffness is followed. We show that when the magnitude of the cocontraction command is scaled in proportion to movement speed, simulated joint stiffness varies with movement speed in a manner comparable with that reported by Bennett. Third, we address the related claim that nonmonotonic equilibrium shifts are required to generate rapid single-joint movements. Using constant-rate equilibrium shifts and constant cocontraction commands, rapid single-joint movements are simulated in the presence of external torques. We use the procedure reported by Latash and Gottlieb to compute hypothetical equilibrium trajectories from simulated torque and angle measurements during movement. As in Latash and Gottlieb, a nonmonotonic function is obtained even though the control signals used in the simulations are constant-rate changes in the equilibrium position of the limb. Differences between the "simple" equilibrium trajectory proposed in the present paper and those that are derived from the procedures used by Gomi and Kawato and Latash and Gottlieb arise from their use of simplified models of force generation.

Arm↗

[Hypothesis on the equilibrium point and variability of amplitude, speed and time of single-joint movement].

Problems of single-joint movement variability are analysed in the framework of the equilibrium-point hypothesis (the lambda-model). Control of the movements is described with three parameters related to movement amplitude speed, and time. Three strategies emerge from this description. Only one of them is likely to lead to a Fitts' type speed-accuracy trade-off. Experiments were performed to test one of the predictions of the model. Subjects performed identical sets of single-joint fast movements with open or closed eyes and some-what different instructions. Movements performed with closed eyes were characterized with higher peak speeds and unchanged variability in seeming violation of the Fitt's law and in a good correspondence to the model.

Humans↗

Paradoxical effects of practice of fast single-joint movements.

We studied the effects of extensive practice of fast, unidirectional, single-joint elbow flexions against a small extending torque bias upon the kinematic and electromyographic (EMG) characteristics of the movements as well as upon the reconstructed hypothetical control patterns (equilibrium trajectories). The subjects were tested at different distances, both with and without the bias torque prior to and after the practice sessions. The basic finding was paradoxical: The subjects did not improve their performance at the practiced task (against the bias) and at other distances in the same condition; however, they showed an increase in movement speed and a decrease in movement time at all distances in unpracticed conditions (without the bias). Changes in the EMG patterns were similar in both conditions. We hypothesize that the principle of learning the dynamics of interaction with the experimental setup in combination with a very steep learning curve form the basis for the observed paradoxical effects of practice. The equilibrium-point hypothesis of movement control provides the least controversial description of these effects as compared to the force-control and EMG-control approaches.

Adult↗

The timing of control signals underlying fast point-to-point arm movements.

It is known that proprioceptive feedback induces muscle activation when the facilitation of appropriate motoneurons exceeds their threshold. In the suprathreshold range, the muscle-reflex system produces torques depending on the position and velocity of the joint segment(s) that the muscle spans. The static component of the torque-position relationship is referred to as the invariant characteristic (IC). According to the equilibrium-point (EP) hypothesis, control systems produce movements by changing the activation thresholds and thus shifting the IC of the appropriate muscles in joint space. This control process upsets the balance between muscle and external torques at the initial limb configuration and, to regain the balance, the limb is forced to establish a new configuration or, if the movement is prevented, a new level of static torques. Taken together, the joint angles and the muscle torques generated at an equilibrium configuration define a single variable called the EP. Thus by shifting the IC, control systems reset the EP. Muscle activation and movement emerge following the EP resetting because of the natural physical tendency of the system to reach equilibrium. Empirical and simulation studies support the notion that the control IC shifts and the resulting EP shifts underlying fast point-to-point arm movements are gradual rather than step-like. However, controversies exist about the duration of these shifts. Some studies suggest that the IC shifts cease with the movement offset. Other studies propose that the IC shifts end early in comparison to the movement duration (approximately, at peak velocity). The purpose of this study was to evaluate the duration of the IC shifts underlying fast point-to-point arm movements. Subjects made fast (hand peak velocity about 1.3 m/s) planar arm movements toward different targets while grasping a handle. Hand forces applied to the handle and shoulder/elbow torques were, respectively, measured from a force sensor placed on the handle, or computed with equations of motion. In some trials, an electromagnetic brake prevented movements. In such movements, the hand force and joint torques reached a steady state after a time that was much smaller than the movement duration in unobstructed movements and was approximately equal to the time to peak velocity (mean difference < 80 ms). In an additional experiment, subjects were instructed to rapidly initiate corrections of the pushing force in response to movement arrest. They were able to initiate such corrections only when the joint torques and the pushing force had practically reached a steady state. The latency of correction onset was, however, smaller than the duration of unobstructed movements. We concluded that during the time at which the steady state torques were reached, the control pattern of IC shifts remained the same despite the movement block. Thereby the duration of these shifts did not exceed the time of reaching the steady state torques. Our findings are consistent with the hypothesis that, in unobstructed movements, the IC shifts and resulting shifts in the EP end approximately at peak velocity. In other words, during the latter part of the movement, the control signals responsible for the equilibrium shift remained constant, and the movement was driven by the arm inertial, viscous and elastic forces produced by the muscle-reflex system. Fast movements may thus be completed without continuous control guidance. As a consequence, central corrections and sequential commands may be issued rapidly, without waiting for the end of kinematic responses to each command, which may be important for many motor behaviours including typing, piano playing and speech. Our study also illustrates that the timing of the control signals may be substantially different from that of the resulting motor output and that the same control pattern may produce different motor outputs depending on external conditions.

Adult↗

Control of Trajectory Modifications in Target-Directed Reaching.

Human reaching movements to fixed and displaced visual targets were recorded and compared with simulated movements generated by using a two-joint arm model based on the equilibrium-point (EP) hypothesis (lambda model) of motor control (Feldman, 1986). The aim was to investigate the form of central control signals underlying these movements. According to this hypothesis, movements result from changes in control variables that shift the equilibrium position (EP) of the arm. At any time, muscle activations and forces will depend on the difference between the arm's EP and its actual position and on the limb's velocity. In this article, we suggest that the direction of EP shift in reaching is specified at the hand level, whereas the rate of EP shift may be specified at the hand or joint level. A common mechanism underlying reaching to fixed and displaced targets is proposed whereby the EP of the hand shifts in a straight line toward the present target. After the target is displaced, the direction of the hand EP shift is modified toward the second target. The results suggest that the rate of shift of the hand EP may be modified for movements in different parts of the work space. The model, with control signals that vary in a simple fashion over time, is able to generate the kinematic patterns observed empirically.

Journal Article↗

Model Predictive Impedance Control: A Model for Joint Movement.

Impedance control has been suggested as the strategy employed by the central nervous system to control human postures and movements. A realization of this strategy is presented that uses a model predictive control algorithm as a higher motor controller. External disturbances are explicitly included in the model. The combination of 3 key factors-joint impedance control, model predictive controller, and external disturbance input-forms the basis for the generality of this model. The model was applied to 3 different types of joint movements: a tracking movement with an unpredicted disturbance, a rhythmic movement, and an unstable biped model of human walking. Computer simulation results showed excellent performance of the model in all 3 cases for optimal values of active joint impedances and an exact match between the musculoskeletal system and the model internal to the model predictive controller. The controller was also able to maintain acceptable performance in the presence of a 25% mismatch between the musculoskeletal system and its internal model.

equilibrium-point hypothesis↗

Analysis of the role of proprioceptive information during arm movements using a model of the human arm.

Using a model for the neuromuscular control of human arm movements, the possible roles of different proprioceptive signals are analyzed. The control model is represented by a neural network and includes both feedback and feedforward control modes. After a learning process, the controller regulates a wide range of arm movements. Evaluation of the roles of different afferent signals shows that sensed muscle forces are important to achieve accurate control of fast movements. For a moderately high look delay (50 ms), velocity feedback is not essential, but for small look delays (0 to 25 ms) an increased performance is attained by feedback of velocity. Position sense is essential to prevent steady-state errors. The arm impedance is affected considerably by the delay in the control loop and by the configuration of the motor control system. The achieved relation between muscle length and force is similar to the invariant characteristics laying at the basis of the equilibrium-point (EP) hypothesis. However, control of fast movements on the basis of EP along is not feasible, but required feedforward control. During training in a velocity-dependent force field, the impedance of the arm increases at first, due to enhanced cocontraction. Subsequently, both impedance and movement errors decrease, indicating a successful representation of the changed inverse dynamics.

Arm↗

Human arm stiffness and equilibrium-point trajectory during multi-joint movement.

By using a newly designed high-performance manipulandum and a new estimation algorithm, we measured human multi-joint arm stiffness parameters during multi-joint point-to-point movements on a horizontal plane. This manipulandum allows us to apply a sufficient perturbation to subject's arm within a brief period during movement. Arm stiffness parameters were reliably estimated using a new algorithm, in which all unknown structural parameters could be estimated independent of arm posture (i.e., constant values under any arm posture). Arm stiffness during transverse movement was considerably greater than that during corresponding posture, but not during a longitudinal movement. Although the ratios of elbow, shoulder, and double-joint stiffness were varied in time, the orientation of stiffness ellipses during the movement did not change much. Equilibrium-point trajectories that were predicted from measured stiffness parameters and actual trajectories were slightly sinusoidally curved in Cartesian space and their velocity profiles were quite different from the velocity profiles of actual hand trajectories. This result contradicts the hypothesis that the brain does not take the dynamics into account in movement control depending on the neuromuscular servo mechanism; rather, it implies that the brain needs to acquire some internal models of controlled objects.

Adult↗

An equilibrium-point model for fast, single-joint movement: II. Similarity of single-joint isometric and isotonic descending commands.

The model for isotonic movements introduced in the preceding article in this issue is used to account for isometric contractions. Isotonic movements and isometric contractions are analyzed as consequences of one motor program acting under different peripheral conditions. Differences in isotonic and isometric EMG patterns are analyzed theoretically. Computer simulation of the EMG patterns was performed both with and without the inclusion of possible effects of reciprocal inhibition. A series of experiments was performed to test the model. The subjects made fast isotonic movements that were unexpectedly blocked at the very beginning in some of the trials. The observed differences in the EMG patterns between blocked and unblocked trials corresponded to the model's predictions. The results suggest that these differences are due to the action of a tonic stretch reflex rather than to preprogrammed reactions. The experimental and simulation findings, and also the data from the literature, are discussed in the framework of the model and the dual-strategy hypothesis. They support the hypothesis that the motor control system uses one of a few standardized subprograms, specifying a small number of parameters to match a specific task.

Journal Article↗