A model of open-loop control of equilibrium position and stiffness of the human elbow joint

Increased velocity feedback gains in the presence of sensory noise can explain paradoxical changes in trunk motor control related to back pain

Literature reports paradoxical findings regarding effects of low-back pain (LBP) on trunk motor control. Compared to healthy individuals, patients with LBP, especially those with high pain-related anxiety, showed stronger trunk extensor reflexes and more resistance against perturbations. On the other hand, LBP patients and especially those with high pain-related anxiety showed decreased precision in unperturbed trunk movement and posture. These paradoxical effects might be explained by arousal potentially increasing average and variance of muscle spindle firing rates. Increased average firing rates could increase resistance against perturbations, but increased variance could decrease precision. We performed a simulation study to test this hypothesis. We modeled the trunk as a 2D inverted pendulum, stabilized by two antagonistic Hill-type muscles, based on their open-loop muscle activation dependent intrinsic stiffness and damping and through 25 ms-delayed, noisy contractile element length and velocity feedback. Reference feedback gains and sensory noise levels were tuned based on previously reported experimental data. We assessed the effect of increasing feedback gains on precision of trunk orientation at different perturbation magnitudes and assessed sensitivity of the effects to open-loop muscle stimulation and noise levels. At low perturbation magnitudes, increasing reflex gains consistently caused an increase in the variance of trunk orientation. At larger perturbation magnitudes, increasing reflex gains consistently caused a decrease in the variance of trunk orientation. Our results support the notion that LBP and related anxiety may increase reflex gains, resulting in an increase in the average and variance of spindle afference, which in turn increase resistance against perturbations and decrease movement precision.

A geometry- and muscle-based control architecture for synthesising biological movement

A key problem for biological motor control is to establish a link between an idea of a movement and the generation of a set of muscle-stimulating signals that lead to the movement execution. The number of signals to generate is thereby larger than the body's mechanical degrees of freedom in which the idea of the movement may be easily expressed, as the movement is actually executed in this space. A mathematical formulation that provides a solving link is presented in this paper in the form of a layered, hierarchical control architecture. It is meant to synthesise a wide range of complex three-dimensional muscle-driven movements. The control architecture consists of a 'conceptional layer', where the movement is planned, a 'structural layer', where the muscles are stimulated, and between both an additional 'transformational layer', where the muscle-joint redundancy is resolved. We demonstrate the operativeness by simulating human stance and squatting in a three-dimensional digital human model (DHM). The DHM considers 20 angular DoFs and 36 Hill-type muscle-tendon units (MTUs) and is exposed to gravity, while its feet contact the ground via reversible stick-slip interactions. The control architecture continuously stimulates all MTUs ('structural layer') based on a high-level, torque-based task formulation within its 'conceptional layer'. Desired states of joint angles (postural plan) are fed to two mid-level joint controllers in the 'transformational layer'. The 'transformational layer' communicates with the biophysical structures in the 'structural layer' by providing direct MTU stimulation contributions and further input signals for low-level MTU controllers. Thereby, the redundancy of the MTU stimulations with respect to the joint angles is resolved, i.e. a link between plan and execution is established, by exploiting some properties of the biophysical structures modelled. The resulting joint torques generated by the MTUs via their moment arms are fed back to the conceptional layer, closing the high-level control loop. Within our mathematical formulations of the Jacobian matrix-based layer transformations, we identify the crucial information for the redundancy solution to be the muscle moment arms, the stiffness relations of muscle and tendon tissue within the muscle model, and the length-stimulation relation of the muscle activation dynamics. The present control architecture allows the straightforward feeding of conceptional movement task formulations to MTUs. With this approach, the problem of movement planning is eased, as solely the mechanical system has to be considered in the conceptional plan.

Morphological Computation Increases From Lower- to Higher-Level of Biological Motor Control Hierarchy

Voluntary movements, like point-to-point or oscillatory human arm movements, are generated by the interaction of several structures. High-level neuronal circuits in the brain are responsible for planning and initiating a movement. Spinal circuits incorporate proprioceptive feedback to compensate for deviations from the desired movement. Muscle biochemistry and contraction dynamics generate movement driving forces and provide an immediate physical response to external forces, like a low-level decentralized controller. A simple central neuronal command like "initiate a movement" then recruits all these biological structures and processes leading to complex behavior, e.g., generate a stable oscillatory movement in resonance with an external spring-mass system. It has been discussed that the spinal feedback circuits, the biochemical processes, and the biomechanical muscle dynamics contribute to the movement generation, and, thus, take over some parts of the movement generation and stabilization which would otherwise have to be performed by the high-level controller. This contribution is termed morphological computation and can be quantified with information entropy-based approaches. However, it is unknown whether morphological computation actually differs between these different hierarchical levels of the control system. To investigate this, we simulated point-to-point and oscillatory human arm movements with a neuro-musculoskeletal model. We then quantify morphological computation on the different hierarchy levels. The results show that morphological computation is highest for the most central (highest) level of the modeled control hierarchy, where the movement initiation and timing are encoded. Furthermore, they show that the lowest neuronal control layer, the muscle stimulation input, exploits the morphological computation of the biochemical and biophysical muscle characteristics to generate smooth dynamic movements. This study provides evidence that the system's design in the mechanical as well as in the neurological structure can take over important contributions to control, which would otherwise need to be performed by the higher control levels.

Optimality Principles in Human Point-to-Manifold Reaching Accounting for Muscle Dynamics

Human arm movements are highly stereotypical under a large variety of experimental conditions. This is striking due to the high redundancy of the human musculoskeletal system, which in principle allows many possible trajectories toward a goal. Many researchers hypothesize that through evolution, learning, and adaption, the human system has developed optimal control strategies to select between these possibilities. Various optimality principles were proposed in the literature that reproduce human-like trajectories in certain conditions. However, these studies often focus on a single cost function and use simple torque-driven models of motion generation, which are not consistent with human muscle-actuated motion. The underlying structure of our human system, with the use of muscle dynamics in interaction with the control principles, might have a significant influence on what optimality principles best model human motion. To investigate this hypothesis, we consider a point-to-manifold reaching task that leaves the target underdetermined. Given hypothesized motion objectives, the control input is generated using Bayesian optimization, which is a machine learning based method that trades-off exploitation and exploration. Using numerical simulations with Hill-type muscles, we show that a combination of optimality principles best predicts human point-to-manifold reaching when accounting for the muscle dynamics.

Predicting Perturbed Human Arm Movements in a Neuro-Musculoskeletal Model to Investigate the Muscular Force Response

Human movement is generated by a dynamic interplay between the nervous system, the biomechanical structures, and the environment. To investigate this interaction, we propose a neuro-musculoskeletal model of human goal-directed arm movements. Using this model, we simulated static perturbations of the inertia and damping properties of the arm, as well as dynamic torque perturbations for one-degree-of freedom movements around the elbow joint. The controller consists of a feed-forward motor command and feedback based on muscle fiber length and contraction velocity representing short-latency (25 ms) or long-latency (50 ms) stretch reflexes as the first neuronal responses elicited by an external perturbation. To determine the open-loop control signal, we parameterized the control signal resulting in a piecewise constant stimulation over time for each muscle. Interestingly, such an intermittent open-loop signal results in a smooth movement that is close to experimental observations. So, our model can generate the unperturbed point-to-point movement solely by the feed-forward command. The feedback only contributed to the stimulation in perturbed movements. We found that the relative contribution of this feedback is small compared to the feed-forward control and that the characteristics of the musculoskeletal system create an immediate and beneficial reaction to the investigated perturbations. The novelty of these findings is (1) the reproduction of static as well as dynamic perturbation experiments in one neuro-musculoskeletal model with only one set of basic parameters. This allows to investigate the model's neuro-muscular response to the perturbations that-at least to some degree-represent stereotypical interactions with the environment; (2) the demonstration that in feed-forward driven movements the muscle characteristics generate a mechanical response with zero-time delay which helps to compensate for the perturbations; (3) that this model provides enough biomechanical detail to allow for the prediction of internal forces, including joint loads and muscle-bone contact forces which are relevant in ergonomics and for the development of assistive devices but cannot be observed in experiments.

CONTINUOUS MOTION AND TIME-VARYING STIFFNESS ESTIMATION OF THE HUMAN ELBOW JOINT BASED ON SEMG

The flexibility of body joints plays an important role in daily life, particularly when performing high-precision rapid pose switching. Importantly, understanding the characteristics of human joint movement is necessary for constructing robotic joints with the softness of humanoid joints. A novel method for estimating continuous motion and time-varying stiffness of the human elbow joint was proposed in the current study, which was based on surface electromyography (sEMG). We used the Hill-based muscle model (HMM) to establish a continuous motion estimation model (CMEM) of the elbow joint, and the genetic algorithm (GA) was used to optimize unknown parameters. Muscle short-range stiffness (SRS) was then used to characterize muscle stiffness, and a joint kinetic equation was used to express the relationship between skeletal muscle stiffness and elbow joint stiffness. Finally, we established a time-varying stiffness estimation model (TVSEM) of the elbow joint based on the CMEM. In addition, five subjects were tested to verify the performance of the CMEM and TVSEM. The total average root-mean-square errors (RMSEs) of the CMEM with the optimal trials were 0.19[Formula: see text]rad and 0.21[Formula: see text]rad and the repeated trials were 0.24[Formula: see text]rad and 0.25[Formula: see text]rad, with 1.25-kg and 2.5[Formula: see text]kg-loads, respectively. The values of elbow joint stiffness ranged from 0–40[Formula: see text]Nm/rad for different muscle activities, which were estimated by the TVSEM.

Comparative Sensitivity Analysis of Muscle Activation Dynamics

We mathematically compared two models of mammalian striated muscle activation dynamics proposed by Hatze and Zajac. Both models are representative for a broad variety of biomechanical models formulated as ordinary differential equations (ODEs). These models incorporate parameters that directly represent known physiological properties. Other parameters have been introduced to reproduce empirical observations. We used sensitivity analysis to investigate the influence of model parameters on the ODE solutions. In addition, we expanded an existing approach to treating initial conditions as parameters and to calculating second-order sensitivities. Furthermore, we used a global sensitivity analysis approach to include finite ranges of parameter values. Hence, a theoretician striving for model reduction could use the method for identifying particularly low sensitivities to detect superfluous parameters. An experimenter could use it for identifying particularly high sensitivities to improve parameter estimation. Hatze's nonlinear model incorporates some parameters to which activation dynamics is clearly more sensitive than to any parameter in Zajac's linear model. Other than Zajac's model, Hatze's model can, however, reproduce measured shifts in optimal muscle length with varied muscle activity. Accordingly we extracted a specific parameter set for Hatze's model that combines best with a particular muscle force-length relation.

Spinal circuits can accommodate interaction torques during multijoint limb movements

The dynamic interaction of limb segments during movements that involve multiple joints creates torques in one joint due to motion about another. Evidence shows that such interaction torques are taken into account during the planning or control of movement in humans. Two alternative hypotheses could explain the compensation of these dynamic torques. One involves the use of internal models to centrally compute predicted interaction torques and their explicit compensation through anticipatory adjustment of descending motor commands. The alternative, based on the equilibrium-point hypothesis, claims that descending signals can be simple and related to the desired movement kinematics only, while spinal feedback mechanisms are responsible for the appropriate creation and coordination of dynamic muscle forces. Partial supporting evidence exists in each case. However, until now no model has explicitly shown, in the case of the second hypothesis, whether peripheral feedback is really sufficient on its own for coordinating the motion of several joints while at the same time accommodating intersegmental interaction torques. Here we propose a minimal computational model to examine this question. Using a biomechanics simulation of a two-joint arm controlled by spinal neural circuitry, we show for the first time that it is indeed possible for the neuromusculoskeletal system to transform simple descending control signals into muscle activation patterns that accommodate interaction forces depending on their direction and magnitude. This is achieved without the aid of any central predictive signal. Even though the model makes various simplifications and abstractions compared to the complexities involved in the control of human arm movements, the finding lends plausibility to the hypothesis that some multijoint movements can in principle be controlled even in the absence of internal models of intersegmental dynamics or learned compensatory motor signals.

The cost of moving optimally: kinematic path selection

It is currently unclear whether the brain plans movement kinematics explicitly or whether movement paths arise implicitly through optimization of a cost function that takes into account control and/or dynamic variables. Several cost functions are proposed in the literature that are very different in nature (e.g., control effort, torque change, and jerk), yet each can predict common movement characteristics. We set out to disentangle predictions of the different variables using a combination of modeling and empirical studies. Subjects performed goal-directed arm movements in a force field (FF) in combination with visual perturbations of seen hand position. This FF was designed to have distinct optimal movements for muscle-input and dynamic costs while leaving kinematic cost unchanged. Visual perturbations in turn changed the kinematic cost but left the dynamic and muscle-input costs unchanged. An optimally controlled, physiologically realistic arm model was used to predict movements under the various cost variables. Experimental results were not consistent with a cost function containing any of the control and dynamic costs investigated. Movement patterns of all experimental conditions were adequately predicted by a kinematic cost function comprising both visually and somatosensory perceived jerk. The present study provides clear behavioral evidence that the brain solves kinematic and mechanical redundancy in separate steps: in a first step, movement kinematics are planned; and in a second, separate step, muscle activation patterns are generated.

Theoretical Hill-type muscle and stability: numerical model and application

The construction of artificial muscles is one of the most challenging developments in today's biomedical science. The application of artificial muscles is focused both on the construction of orthotics and prosthetics for rehabilitation and prevention purposes and on building humanoid walking machines for robotics research. Research in biomechanics tries to explain the functioning and design of real biological muscles and therefore lays the fundament for the development of functional artificial muscles. Recently, the hyperbolic Hill-type force-velocity relation was derived from simple mechanical components. In this contribution, this theoretical yet biomechanical model is transferred to a numerical model and applied for presenting a proof-of-concept of a functional artificial muscle. Additionally, this validated theoretical model is used to determine force-velocity relations of different animal species that are based on the literature data from biological experiments. Moreover, it is shown that an antagonistic muscle actuator can help in stabilising a single inverted pendulum model in favour of a control approach using a linear torque generator.

Control of position and movement is simplified by combined muscle spindle and Golgi tendon organ feedback

Whereas muscle spindles play a prominent role in current theories of human motor control, Golgi tendon organs (GTO) and their associated tendons are often neglected. This is surprising since there is ample evidence that both tendons and GTOs contribute importantly to neuromusculoskeletal dynamics. Using detailed musculoskeletal models, we provide evidence that simple feedback using muscle spindles alone results in very poor control of joint position and movement since muscle spindles cannot sense changes in tendon length that occur with changes in muscle force. We propose that a combination of spindle and GTO afferents can provide an estimate of muscle-tendon complex length, which can be effectively used for low-level feedback during both postural and movement tasks. The feasibility of the proposed scheme was tested using detailed musculoskeletal models of the human arm. Responses to transient and static perturbations were simulated using a 1-degree-of-freedom (DOF) model of the arm and showed that the combined feedback enabled the system to respond faster, reach steady state faster, and achieve smaller static position errors. Finally, we incorporated the proposed scheme in an optimally controlled 2-DOF model of the arm for fast point-to-point shoulder and elbow movements. Simulations showed that the proposed feedback could be easily incorporated in the optimal control framework without complicating the computation of the optimal control solution, yet greatly enhancing the system's response to perturbations. The theoretical analyses in this study might furthermore provide insight about the strong physiological couplings found between muscle spindle and GTO afferents in the human nervous system.

Conclusions on motor control depend on the type of model used to represent the periphery

Within the field of motor control, there is no consensus on which kinematic and kinetic aspects of movements are planned or controlled. Perturbing goal-directed movements is a frequently used tool to answer this question. To be able to draw conclusions about motor control from kinematic responses to perturbations, a model of the periphery (i.e., the skeleton, muscle-tendon complexes, and spinal reflex circuitry) is required. The purpose of the present study was to determine to what extent such conclusions depend on the level of simplification with which the dynamical properties of the periphery are modeled. For this purpose, we simulated fast goal-directed single-joint movement with four existing types of models. We tested how three types of perturbations affected movement trajectory if motor commands remained unchanged. We found that the four types of models of the periphery showed different robustness to the perturbations, leading to different predictions on how accurate motor commands need to be, i.e., how accurate the knowledge of external conditions needs to be. This means that when interpreting kinematic responses obtained in perturbation experiments the level of error correction attributed to adaptation of motor commands depends on the type of model used to describe the periphery.

In vivo dynamics of the musculoskeletal system cannot be adequately described using a stiffness-damping-inertia model

BACKGROUND

Visco-elastic properties of the (neuro-)musculoskeletal system play a fundamental role in the control of posture and movement. Often, these properties are described and identified using stiffness-damping-inertia (KBI) models. In such an approach, perturbations are applied to the (neuro-)musculoskeletal system and subsequently KBI-model parameters are optimized to obtain a best fit between simulated and experimentally observed responses. Problems with this approach may arise because a KBI-model neglects critical aspects of the real musculoskeletal system.

METHODOLOGY/PRINCIPAL FINDINGS

The purpose of this study was to analyze the relation between the musculoskeletal properties and the stiffness and damping estimated using a KBI-model, to analyze how this relation is affected by the nature of the perturbation and to assess the sensitivity of the estimated stiffness and damping to measurement errors. Our analyses show that the estimated stiffness and damping using KBI-models do not resemble any of the dynamical parameters of the underlying system, not even when the responses are very accurately fitted by the KBI-model. Furthermore, the stiffness and damping depend non-linearly on all the dynamical parameters of the underlying system, influenced by the nature of the perturbation and the time interval over which the KBI-model is optimized. Moreover, our analyses predict a very high sensitivity of estimated parameters to measurement errors.

CONCLUSIONS/SIGNIFICANCE

The results of this study suggest that the usage of stiffness-damping-inertia models to investigate the dynamical properties of the musculoskeletal system under control by the CNS should be reconsidered.

The central nervous system does not minimize energy cost in arm movements

It has been widely suggested that the many degrees of freedom of the musculoskeletal system may be exploited by the CNS to minimize energy cost. We tested this idea by having subjects making point-to-point movements while grasping a robotic manipulandum. The robot created a force field chosen such that the minimal energy hand path for reaching movements differed substantially from those observed in a null field. The results show that after extended exposure to the force field, subjects continued to move exactly as they did in the null field and thus used substantially more energy than needed. Even after practicing to move along the minimal energy path, subjects did not adapt their freely chosen hand paths to reduce energy expenditure. The results of this study indicate that for point-to-point arm movements minimization of energy cost is not a dominant factor that influences how the CNS arrives at kinematics and associated muscle activation patterns.

Inverse biomimetics: how robots can help to verify concepts concerning sensorimotor control of human arm and leg movements

Simulation test, hardware test and behavioral comparison test are proposed to experimentally verify whether a technical control concept for limb movements is logically precise, physically sound, and biologically relevant. Thereby, robot test-beds may play an integral part by mimicking functional limb movements. The procedure is exemplarily demonstrated for human aiming movements with the forearm: when comparing competitive control concepts, these movements are described best by a spring-like operating muscular-skeletal device which is assisted by feedforward control through an inverse internal model of the limb--without regress to a forward model of the limb. In a perspective on hopping, the concept of exploitive control is addressed, and its comparison to concepts derived from classical control theory advised.

The influence of visual perturbations on the neural control of limb stiffness

To adapt to novel unstable environments, the motor system modulates limb stiffness to produce selective increases in arm stability. The motor system receives information about the environment via somatosensory and proprioceptive signals related to the perturbing forces and visual signals indicating deviations from an expected hand trajectory. Here we investigated whether subjects modulate limb stiffness during adaptation to a purely visual perturbation. In a first experiment, measurements of limb stiffness were taken during adaptation to an elastic force field (EF). Observed changes in stiffness were consistent with previous reports: subjects increased limb stiffness and did so only in the direction of the environmental instability. In a second experiment, stiffness changes were measured during adaptation to a visual perturbing environment that magnified hand-path deviations in the lateral direction. In contrast to the first experiment, subjects trained in this visual task showed no accompanying change in stiffness, despite reliable improvements in movement accuracy. These findings suggest that this sort of visual information alone may not be sufficient to engage neural systems for stiffness control, which may depend on sensory signals more directly related to perturbing forces, such as those arising from proprioception and somatosensation.

Equilibrium point control cannot be refuted by experimental reconstruction of equilibrium point trajectories

In the literature, it has been hotly debated whether the brain uses internal models or equilibrium point (EP) control to generate arm movements. EP control involves specification of EP trajectories, time series of arm configurations in which internal forces and external forces are in equilibrium; if the arm is not in a specified EP, it is driven toward this EP by muscle forces arising due to central drive, reflexes, and muscle mechanics. EP control has been refuted by researchers claiming that EP trajectories underlying movements of subjects were complex. These researchers used an approach that involves applying force perturbations during movements of subjects and fitting a mass-spring-damper model to the kinematic responses, and then reconstructing the EP trajectory using the estimated stiffness, damping, and measured kinematics. In this study, we examined the validity of this approach using an EP-controlled musculoskeletal model of the arm. We used the latter model to simulate unperturbed and perturbed maximally fast movements and optimized the parameter values of a mass-spring-damper model to make it reproduce as best as possible the kinematic responses. It was shown that estimated stiffness not only depended on the "true" stiffness of the musculoskeletal model but on all of its dynamical parameters. Furthermore it was shown that reconstructed EP trajectories were in agreement with those presented in the literature but did not resemble the simple EP trajectories that had been used to generate the movement of the model. It was concluded that the refutation of EP control on the basis of results obtained with mass-spring-damper models was unjust.