Nonlinear summation of force in cat soleus muscle results primarily from stretch of the common-elastic elements.

The complex connective tissue structure of muscle and tendon suggests that forces from two parts of a muscle may not summate linearly. This study measured the nonlinear summation of force (F(nl)) in whole cat soleus during isometric and ramp movements. In six anesthetized cats, the soleus was attached to a servomechanism to control muscle length and record force. The ventral roots were divided into two bundles, each innervating about half the soleus; thus the two parts could be stimulated alone or together. In all experiments, F(nl) was small (<6% of maximum tetanic tension). Peak F(nl) occurred during changes in muscle force, either as a result of imposed muscle movement or the onset or offset of a stimulus train. The data were fit to a model in which both parts of the muscle were assumed to stretch to a common elasticity. The servomechanism was programmed to compensate for reduced stretch of the common elasticity during partial compared with whole muscle activation. These compensatory movements showed how the model could account for most, but not all, of F(nl).     

A control theoretic model of the forearm

In this paper, a control theoretic model of the forearm is developed and analyzed, and a computational method for predicting muscle activations necessary to generate specified motions is described. A detailed geometric model of the forearm kinematics, including the carrying angle and models of how the biceps and the supinator tendons wrap around the bones, is used. Also, including a dynamics model, the final model is a system of differential equations where the muscle activations play the role of control signals. Due to the large number of muscles, the problem of finding muscle activations is redundant, and this problem is solved by an optimization procedure. The computed muscle activations for ballistic movements clearly recaptures the triphasic ABC (Activation-Braking-Clamping) pattern. It is also transparent, from the muscle activation patterns, how the muscles cooperate and counteract in order to accomplish desired motions. A comparison with previously reported experimental data is included and the model predictions can be seen to be partially in agreement with the experimental data.     

Stability and motor adaptation in human arm movements

In control, stability captures the reproducibility of motions and the robustness to environmental and internal perturbations. This paper examines how stability can be evaluated in human movements, and possible mechanisms by which humans ensure stability. First, a measure of stability is introduced, which is simple to apply to human movements and corresponds to Lyapunov exponents. Its application to real data shows that it is able to distinguish effectively between stable and unstable dynamics. A computational model is then used to investigate stability in human arm movements, which takes into account motor output variability and computes the force to perform a task according to an inverse dynamics model. Simulation results suggest that even a large time delay does not affect movement stability as long as the reflex feedback is small relative to muscle elasticity. Simulations are also used to demonstrate that existing learning schemes, using a monotonic antisymmetric update law, cannot compensate for unstable dynamics. An impedance compensation algorithm is introduced to learn unstable dynamics, which produces similar adaptation responses to those found in experiments.     

Neuromusculoskeletal modeling: estimation of muscle forces and joint moments and movements from measurements of neural command

This paper provides an overview of forward dynamic neuromusculoskeletal modeling. The aim of such models is to estimate or predict muscle forces, joint moments, and/or joint kinematics from neural signals. This is a four-step process. In the first step, muscle activation dynamics govern the transformation from the neural signal to a measure of muscle activation-a time varying parameter between 0 and 1. In the second step, muscle contraction dynamics characterize how muscle activations are transformed into muscle forces. The third step requires a model of the musculoskeletal geometry to transform muscle forces to joint moments. Finally, the equations of motion allow joint moments to be transformed into joint movements. Each step involves complex nonlinear relationships. The focus of this paper is on the details involved in the first two steps, since these are the most challenging to the biomechanician. The global process is then explained through applications to the study of predicting isometric elbow moments and dynamic knee kinetics.     

A general-purpose framework to simulate musculoskeletal system of human body: using a motion tracking approach*

Computation of muscle force patterns that produce specified movements of muscle-actuated dynamic models is an important and challenging problem. This problem is an undetermined one, and then a proper optimization is required to calculate muscle forces. The purpose of this paper is to develop a general model for calculating all muscle activation and force patterns in an arbitrary human body movement. For this aim, the equations of a multibody system forward dynamics, which is considered for skeletal system of the human body model, is derived using Lagrange–Euler formulation. Next, muscle contraction dynamics is added to this model and forward dynamics of an arbitrary musculoskeletal system is obtained. For optimization purpose, the obtained model is used in computed muscle control algorithm, and a closed-loop system for tracking desired motions is derived. Finally, a popular sport exercise, biceps curl, is simulated by using this algorithm and the validity of the obtained results is evaluated via EMG signals.     

How Variability and Effort Determine Coordination at Large Forces

Motor control is a challenging task for the central nervous system, since it involves redundant degrees of freedom, nonlinear dynamics of actuators and limbs, as well as noise. When an action is carried out, which factors does your nervous system consider to determine the appropriate set of muscle forces between redundant degrees-of-freedom? Important factors determining motor output likely encompass effort and the resulting motor noise. However, the tasks used in many previous motor control studies could not identify these two factors uniquely, as signal-dependent noise monotonically increases as a function of the effort. To address this, a recent paper introduced a force control paradigm involving one finger in each hand that can disambiguate these two factors. It showed that the central nervous system considers both force noise and amplitude, with a larger weight on the absolute force and lower weights on both noise and normalized force. While these results are valid for the relatively low force range considered in that paper, the magnitude of the force shared between the fingers for large forces is not known. This paper investigates this question experimentally, and develops an appropriate Markov chain Monte Carlo method in order to estimate the weightings given to these factors. Our results demonstrate that the force sharing strongly depends on the force level required, so that for higher force levels the normalized force is considered as much as the absolute force, whereas the role of noise minimization becomes negligible.     

Learning the dynamics of reaching movements results in the modification of arm impedance and long-latency perturbation responses

 Some characteristics of arm movements that humans exhibit during learning the dynamics of reaching are consistent with a theoretical framework where training results in motor commands that are gradually modified to predict and compensate for novel forces that may act on the hand. As a first approximation, the motor control system behaves as an adapting controller that learns an internal model of the dynamics of the task. It approximates inverse dynamics and predicts motor commands that are appropriate for a desired limb trajectory. However, we had previously noted that subtle motion characteristics observed during changes in task dynamics challenged this simple model and raised the possibility that adaptation also involved sensory–motor feedback pathways. These pathways reacted to sensory feedback during the course of the movement. Here we hypothesize that adaptation to dynamics might also involve a modification of how the CNS responds to sensory feedback. We tested this through experiments that quantified how the motor system's response to errors during voluntary movements changed as it adapted to dynamics of a force field. We describe a nonlinear approach that approximates the impedance of the arm, i.e., force response as a function of arm displacement trajectory. We observe that after adaptation, the impedance function changes in a way that closely matches and counters the effect of the force field. This is particularly prominent in the long-latency (>100 ms) component of response to perturbations. Therefore, it appears that practice not only modifies the internal model with which the brain generates motor commands that initiate a movement, but also the internal model with which sensory feedback is integrated with the ongoing descending commands in order to respond to error during the movement. Received: 10 January 2001 / Accepted in revised form: 30 May 2001     

Foot and ankle forces during an automobile collision: the influence of muscles

Muscles have a potentially important effect on lower extremity injuries during an automobile collision. Computational modeling can be a powerful tool to predict these effects and develop protective interventions. Our purpose was to determine how muscles influence peak foot and ankle forces during an automobile collision. A 2-D bilateral musculoskeletal model was constructed with seven segments. Six muscle groups were included in the right lower extremity, each represented by a Hill muscle model. Vehicle deceleration data were applied as input and the resulting movements were simulated. Three models were evaluated: no muscles (NM), minimal muscle activation at a brake pedal force of 400 N (MN), and maximal muscle activation to simulate panic braking (MX). Muscle activation always resulted in large increases in peak joint force. Peak ankle joint force was greatest for MX (10120 N), yet this model also had the lowest peak rearfoot force (629 N). Peak force on the Achilles tendon was 4.5 times greater, during MX (6446 N) compared to MN (1430 N). We conclude that (1). external and internal forces are dependent on muscles, (2). muscle activation level could exacerbate axial loading injuries, (3). external and internal forces can be inversely related once muscle properties are included.     

Which data should be tracked in forward-dynamic optimisation to best predict muscle forces in a pathological co-contraction case?

The choice of the cost-function for predicting muscle forces during a movement remains a challenge, especially in patients with neuromuscular disorders. Forward dynamics-based optimisations mainly track joint kinematics or torques, combined with a least-excitation criterion. Tracking marker trajectories and/or electromyography (EMG) has rarely been proposed. Our objective was to determine the best tracking objective-function to accurately predict the upper-limb muscle forces. A musculoskeletal model was created and EMG was simulated to obtain a reference movement – a shoulder abduction. A Gaussian noise (mean = 0; standard deviation = 15%) was added to the simulated EMG. Another noise – corresponding to the actual soft tissue artefacts (STA) of experimental shoulder abduction movements – was added to the trajectories of the markers placed on the model. Muscle forces were estimated from these noisy data, using forward dynamics assisted by six non-linear least-squared objective-functions. These functions involved the tracking of marker trajectories, joint angles or torques, with and without EMG-tracking. All six approaches used the same musculoskeletal model and were solved using a direct multiple shooting algorithm. Finally, the predicted joint angles, muscle forces and activations were compared to the reference values, using root-mean-square errors (RMSe) and biases. The force RMSe of the approach tracking both marker trajectories and EMG (18.45 ± 12.60 N) was almost five times lower than the one of the approach tracking only joint angles (82.37 ± 66.26 N) or torques (85.10 ± 116.40 N). Therefore, using EMG as a complementary tracking-data in forward dynamics seems to be promising for the estimation of muscle forces.     

Similar movements are associated with drastically different muscle contraction velocities

We investigated how kinematic redundancy interacts with the neurophysiological control mechanisms required for smooth and accurate, rapid limb movements. Biomechanically speaking, tendon excursions are over-determined because the rotation of few joints determines the lengths and velocities of many muscles. But how different are the muscle velocity profiles induced by various, equally valid hand trajectories? We used an 18-muscle sagittal-plane arm model to calculate 100,000 feasible shoulder, elbow, and wrist joint rotations that produced valid basketball free throws with different hand trajectories, but identical initial and final hand positions and velocities. We found large differences in the eccentric and concentric muscle velocity profiles across many trajectories; even among similar trajectories. These differences have important consequences to their neural control because each trajectory will require unique, time-sensitive reflex modulation strategies. As Sherrington mentioned a century ago, failure to appropriately silence the stretch reflex of any one eccentrically contracting muscle will disrupt movement. Thus, trajectories that produce faster or more variable eccentric contractions will require more precise timing of reflex modulation across motoneuron pools; resulting in higher sensitivity to time delays, muscle mechanics, excitation/contraction dynamics, noise, errors and perturbations. By combining fundamental concepts of biomechanics and neuroscience, we propose that kinematic and muscle redundancy are, in fact, severely limited by the need to regulate reflex mechanisms in a task-specific and time-critical way. This in turn has important consequences to the learning and execution of accurate, smooth and repeatable movements—and to the rehabilitation of everyday limb movements in developmental and neurological conditions, and stroke.     

Generalization in Adaptation to Stable and Unstable Dynamics

Humans skillfully manipulate objects and tools despite the inherent instability. In order to succeed at these tasks, the sensorimotor control system must build an internal representation of both the force and mechanical impedance. As it is not practical to either learn or store motor commands for every possible future action, the sensorimotor control system generalizes a control strategy for a range of movements based on learning performed over a set of movements. Here, we introduce a computational model for this learning and generalization, which specifies how to learn feedforward muscle activity in a function of the state space. Specifically, by incorporating co-activation as a function of error into the feedback command, we are able to derive an algorithm from a gradient descent minimization of motion error and effort, subject to maintaining a stability margin. This algorithm can be used to learn to coordinate any of a variety of motor primitives such as force fields, muscle synergies, physical models or artificial neural networks. This model for human learning and generalization is able to adapt to both stable and unstable dynamics, and provides a controller for generating efficient adaptive motor behavior in robots. Simulation results exhibit predictions consistent with all experiments on learning of novel dynamics requiring adaptation of force and impedance, and enable us to re-examine some of the previous interpretations of experiments on generalization.     

Agreement of measured and calculated muscle activity during highly dynamic movements modelled with a spherical knee joint

The inclusion of muscle forces into the analysis of joint contact forces has improved their accuracy. But it has not been validated if such force and activity calculations are valid during highly dynamic multidirectional movements. The purpose of this study was to validate calculated muscle activation of a lower extremity model with a spherical knee joint for running, sprinting and 90°-cutting. Kinematics, kinetics and lower limb muscle activation of ten participants were investigated in a 3D motion capture setup including EMG. A lower extremity rigid body model was used to calculate the activation of these muscles with an inverse dynamics approach and a cubic cost function. Correlation coefficients were calculated to compare measured and calculated activation. The results showed good correlation of the modelled and calculated data with a few exceptions. The highest average correlations were found during walking (r = 0.81) and the lowest during cutting (r = 0.57). Tibialis anterior had the lowest average correlation (r = 0.33) over all movements while gastrocnemius medius had the highest correlation (r = 0.9). The implementation of a spherical knee joint increased the agreement between measured and modelled activation compared to studies using a hinge joint knee. Although some stabilizing muscles showed low correlations during dynamic movements, the investigated model calculates muscle activity sufficiently.     

Inverse and forward dynamics: models of multi-body systems

Connected multi-body systems exhibit notoriously complex behaviour when driven by external and internal forces and torques. The problem of reconstructing the internal forces and/or torques from the movements and known external forces is called the 'inverse dynamics problem', whereas calculating motion from known internal forces and/or torques and resulting reaction forces is called the 'forward dynamics problem'. When stepping forward to cross the street, people use muscle forces that generate angular accelerations of their body segments and, by virtue of reaction forces from the street, a forward acceleration of the centre of mass of their body. Inverse dynamics calculations applied to a set of motion data from such an event can teach us how temporal patterns of joint torques were responsible for the observed motion. In forward dynamics calculations we may attempt to create motion from such temporal patterns, which is extremely difficult, because of the complex mechanical linkage along the chains forming the multi-body system. To understand, predict and sometimes control multi-body systems, we may want to have mathematical expressions for them. The Newton-Euler, Lagrangian and Featherstone approaches have their advantages and disadvantages. The simulation of collisions and the inclusion of muscle forces or other internal forces are discussed. Also, the possibility to perform a mixed inverse and forward dynamics calculation are dealt with. The use and limitations of these approaches form the conclusion.     

An improved inverse dynamics formulation for estimation of external and internal loads during human sagittal plane movements

Planar musculoskeletal models are common in the inverse dynamics analysis of human movements such as walking, running and jumping. The continued interest in such models is justified by their simplicity and computational efficiency. Related to a human planar model, a unified formulation for both the flying and support phases of the sagittal plane movements is developed. The actuation involves muscle forces in the lower limbs and the resultant muscle torques in the other body joints. The dynamic equations, introduced in absolute coordinates of the segments, are converted into useful compact forms using the projective technique. The solution to a determinate inverse dynamics problem allows for the explicit determination of the external reactions (presumed to vanish during the flying phases) and the resultant muscle torques in all the model joints. The indeterminate inverse dynamics problem is then focused on the assessment of muscle forces and joint reaction forces selectively in the supporting lower limb. Numerical results of the inverse dynamics simulation of sample sagittal plane movements are reported to illustrate the validity and effectiveness of the improved formulation.     

Biological arm motion through reinforcement learning

The present paper discusses an optimal learning control method using reinforcement learning for biological systems with a redundant actuator. It is difficult to apply reinforcement learning to biological control systems because of the redundancy in muscle activation space. We solve this problem with the following method. First, we divide the control input space into two subspaces according to a priority order of learning and restrict the search noise for reinforcement learning to the first priority subspace. Then the constraint is reduced as the learning progresses, with the search space extending to the second priority subspace. The higher priority subspace is designed so that the impedance of the arm can be high. A smooth reaching motion is obtained through reinforcement learning without any previous knowledge of the arms dynamics.     

Normalized Index of Synergy for Evaluating the Coordination of Motor Commands

Humans perform various motor tasks by coordinating the redundant motor elements in their bodies. The coordination of motor outputs is produced by motor commands, as well properties of the musculoskeletal system. The aim of this study was to dissociate the coordination of motor commands from motor outputs. First, we conducted simulation experiments where the total elbow torque was generated by a model of a simple human right and left elbow with redundant muscles. The results demonstrated that muscle tension with signal-dependent noise formed a coordinated structure of trial-to-trial variability of muscle tension. Therefore, the removal of signal-dependent noise effects was required to evaluate the coordination of motor commands. We proposed a method to evaluate the coordination of motor commands, which removed signal-dependent noise from the measured variability of muscle tension. We used uncontrolled manifold analysis to calculate a normalized index of synergy. Simulation experiments confirmed that the proposed method could appropriately represent the coordinated structure of the variability of motor commands. We also conducted experiments in which subjects performed the same task as in the simulation experiments. The normalized index of synergy revealed that the subjects coordinated their motor commands to achieve the task. Finally, the normalized index of synergy was applied to a motor learning task to determine the utility of the proposed method. We hypothesized that a large part of the change in the coordination of motor outputs through learning was because of changes in motor commands. In a motor learning task, subjects tracked a target trajectory of the total torque. The change in the coordination of muscle tension through learning was dominated by that of motor commands, which supported the hypothesis. We conclude that the normalized index of synergy can be used to evaluate the coordination of motor commands independently from the properties of the musculoskeletal system.     

Dissociating Variability and Effort as Determinants of Coordination

When coordinating movements, the nervous system often has to decide how to distribute work across a number of redundant effectors. Here, we show that humans solve this problem by trying to minimize both the variability of motor output and the effort involved. In previous studies that investigated the temporal shape of movements, these two selective pressures, despite having very different theoretical implications, could not be distinguished; because noise in the motor system increases with the motor commands, minimization of effort or variability leads to very similar predictions. When multiple effectors with different noise and effort characteristics have to be combined, however, these two cost terms can be dissociated. Here, we measure the importance of variability and effort in coordination by studying how humans share force production between two fingers. To capture variability, we identified the coefficient of variation of the index and little fingers. For effort, we used the sum of squared forces and the sum of squared forces normalized by the maximum strength of each effector. These terms were then used to predict the optimal force distribution for a task in which participants had to produce a target total force of 4–16 N, by pressing onto two isometric transducers using different combinations of fingers. By comparing the predicted distribution across fingers to the actual distribution chosen by participants, we were able to estimate the relative importance of variability and effort of 17, with the unnormalized effort being most important. Our results indicate that the nervous system uses multi-effector redundancy to minimize both the variability of the produced output and effort, although effort costs clearly outweighed variability costs.     

Impedance characteristics of a neuromusculoskeletal model of the human arm I. Posture control

The mechanical impedance of neuromusculoskeletal models of the human arm is studied in this paper. The model analysis provides a better understanding of the contributions of possible intrinsic and reflexive components of arm impedance, makes clear the limitations of second-order mass-viscosity-stiffness models and reveals possible task effects on the impedance. The musculoskeletal model describes planar movements of the upper arm and forearm, which are moved by six lumped muscles with nonlinear dynamics. The motor control system is represented by a neural network which combines feedforward and feedback control. It is optimized for the control of movements or for posture control in the presence of external forces. The achieved impedance characteristics depend on the conditions during the learning process. In particular, the impedance is adapted in a suitable way to the frequency content and direction of external forces acting on the hand during an isometric task. The impedance characteristics of a model, which is optimized for movement control, are similar to experimental data in the literature. The achieved stiffness is, to a large extent, reflexively determined whereas the approximated viscosity is primarily due to intrinsic attributes. It is argued that usually applied Hill-type muscle models do not properly represent intrinsic muscle stiffness. Received: 14 October 1997 / Accepted in revised form: 18 May 1999     

Static and dynamic optimization solutions for gait are practically equivalent

The proposition that dynamic optimization provides better estimates of muscle forces during gait than static optimization is examined by comparing a dynamic solution with two static solutions. A 23-degree-of-freedom musculoskeletal model actuated by 54 Hill-type musculotendon units was used to simulate one cycle of normal gait. The dynamic problem was to find the muscle excitations which minimized metabolic energy per unit distance traveled, and which produced a repeatable gait cycle. In the dynamic problem, activation dynamics was described by a first-order differential equation. The joint moments predicted by the dynamic solution were used as input to the static problems. In each static problem, the problem was to find the muscle activations which minimized the sum of muscle activations squared, and which generated the joint moments input from the dynamic solution. In the first static problem, muscles were treated as ideal force generators; in the second, they were constrained by their force-length-velocity properties; and in both, activation dynamics was neglected. In terms of predicted muscle forces and joint contact forces, the dynamic and static solutions were remarkably similar. Also, activation dynamics and the force-length-velocity properties of muscle had little influence on the static solutions. Thus, for normal gait, if one can accurately solve the inverse dynamics problem and if one seeks only to estimate muscle forces, the use of dynamic optimization rather than static optimization is currently not justified. Scenarios in which the use of dynamic optimization is justified are suggested.