Motion control of musculoskeletal systems with redundancy

Motion control of musculoskeletal systems for functional electrical stimulation (FES) is a challenging problem due to the inherent complexity of the systems. These include being highly nonlinear, strongly coupled, time-varying, time-delayed, and redundant. The redundancy in particular makes it difficult to find an inverse model of the system for control purposes. We have developed a control system for multiple input multiple output (MIMO) redundant musculoskeletal systems with little prior information. The proposed method separates the steady-state properties from the dynamic properties. The dynamic control uses a steady-state inverse model and is implemented with both a PID controller for disturbance rejection and an artificial neural network (ANN) feedforward controller for fast trajectory tracking. A mechanism to control the sum of the muscle excitation levels is also included. To test the performance of the proposed control system, a two degree of freedom ankle–subtalar joint model with eight muscles was used. The simulation results show that separation of steady-state and dynamic control allow small output tracking errors for different reference trajectories such as pseudo-step, sinusoidal and filtered random signals. The proposed control method also demonstrated robustness against system parameter and controller parameter variations. A possible application of this control algorithm is FES control using multiple contact cuff electrodes where mathematical modeling is not feasible and the redundancy makes the control of dynamic movement difficult.     

Static optimization underestimates antagonist muscle activity at the glenohumeral joint: A musculoskeletal modeling study

Static optimization is commonly employed in musculoskeletal modeling to estimate muscle and joint loading; however, the ability of this approach to predict antagonist muscle activity at the shoulder is poorly understood. Antagonist muscles, which contribute negatively to a net joint moment, are known to be important for maintaining glenohumeral joint stability. This study aimed to compare muscle and joint force predictions from a subject-specific neuromusculoskeletal model of the shoulder driven entirely by measured muscle electromyography (EMG) data with those from a musculoskeletal model employing static optimization. Four healthy adults performed six sub-maximal upper-limb contractions including shoulder abduction, adduction, flexion, extension, internal rotation and external rotation. EMG data were simultaneously measured from 16 shoulder muscles using surface and intramuscular electrodes, and joint motion evaluated using video motion analysis. Muscle and joint forces were calculated using both a calibrated EMG-driven neuromusculoskeletal modeling framework, and musculoskeletal model simulations that employed static optimization. The EMG-driven model predicted antagonistic muscle function for pectoralis major, latissimus dorsi and teres major during abduction and flexion; supraspinatus during adduction; middle deltoid during extension; and subscapularis, pectoralis major and latissimus dorsi during external rotation. In contrast, static optimization neural solutions showed little or no recruitment of these muscles, and preferentially activated agonistic prime movers with large moment arms. As a consequence, glenohumeral joint force calculations varied substantially between models. The findings suggest that static optimization may under-estimate the activity of muscle antagonists, and therefore, their contribution to glenohumeral joint stability.     

A Method for Numerical Simulation of Single Limb Ground Contact Events: Application to Heel-Toe Running

The objective of this work was to develop a method to simulate single-limb ground contact events, which may be applied to study musculoskeletal injuries associated with such movements. To achieve this objective, a three-dimensional musculoskeletal model was developed consisting of the equations of motion for the musculoskeletal system, and models for the muscle force generation and ground contact elements. An optimization framework and a weighted least-squares objective function were presented that generated muscle stimulation patterns that optimally reproduced subject-specific movement data. Experimental data were collected from a single subject to provide initial conditions for the simulation and tracking data for the optimization. As an example application, a simulation of the stance phase of running was generated. The results showed that the average difference between the simulation and subject's ground reaction force and joint angle data was less than two inter-trial standard deviations. Further, there was good agreement between the model's muscle excitation patterns and experimentally collected electromyography data. These results give confidence in the model to examine musculoskeletal loading during a variety of landing movements and to study the effects of various factors associated with injury. Limitations were examined and areas of improvement for the model were presented.     

Combined finite-element and rigid-body analysis of human jaw joint dynamics

The jaw joint plays a crucial role in human mastication. It acts as a guidance for jaw movements and as a fulcrum for force generation. The joint is subjected to loading which causes tensions and deformations in its cartilaginous structures. These are assumed to be a major determinant for development, maintenance and also degeneration of the joint. To analyze the distribution of tensions and deformations in the cartilaginous structures of the jaw joint during jaw movement, a dynamical model of the human masticatory system has been constructed. Its movements are controlled by muscle activation. The articular cartilage layers and articular disc were included as finite-element (FE) models. As this combination of rigid-body and FE modeling had not been applied to musculoskeletal systems yet, its benefits and limitations were assessed by simulating both unloaded and loaded jaw movements. It was demonstrated that joint loads increase with muscle activation, irrespective of the external loads. With increasing joint load, the size of the stressed area of the articular surfaces was enlarged, whereas the peak stresses were much less affected. The results suggest that the articular disc enables distribution of local contact stresses over a much wider area of the very incongruent articular surfaces by transforming compressive principal stress into shear stress.     

Modulation of shoulder muscle and joint function using a powered upper-limb exoskeleton

Robotic-assistive exoskeletons can enable frequent repetitive movements without the presence of a full-time therapist; however, human-machine interaction and the capacity of powered exoskeletons to attenuate shoulder muscle and joint loading is poorly understood. This study aimed to quantify shoulder muscle and joint force during assisted activities of daily living using a powered robotic upper limb exoskeleton (ArmeoPower, Hocoma). Six healthy male subjects performed abduction, flexion, horizontal flexion, reaching and nose touching activities. These tasks were repeated under two conditions: (i) the exoskeleton compensating only for its own weight, and (ii) the exoskeleton providing full upper limb gravity compensation (i.e., weightlessness). Muscle EMG, joint kinematics and joint torques were simultaneously recorded, and shoulder muscle and joint forces calculated using personalized musculoskeletal models of each subject’s upper limb. The exoskeleton reduced peak joint torques, muscle forces and joint loading by up to 74.8% (0.113 Nm/kg), 88.8% (5.8%BW) and 68.4% (75.6%BW), respectively, with the degree of load attenuation strongly task dependent. The peak compressive, anterior and superior glenohumeral joint force during assisted nose touching was 36.4% (24.6%BW), 72.4% (13.1%BW) and 85.0% (17.2%BW) lower than that during unassisted nose touching, respectively. The present study showed that upper limb weight compensation using an assistive exoskeleton may increase glenohumeral joint stability, since deltoid muscle force, which is the primary contributor to superior glenohumeral joint shear, is attenuated; however, prominent exoskeleton interaction moments are required to position and control the upper limb in space, even under full gravity compensation conditions. The modeling framework and results may be useful in planning targeted upper limb robotic rehabilitation tasks.     

OpenSim Moco: Musculoskeletal optimal control

Musculoskeletal simulations are used in many different applications, ranging from the design of wearable robots that interact with humans to the analysis of patients with impaired movement. Here, we introduce OpenSim Moco, a software toolkit for optimizing the motion and control of musculoskeletal models built in the OpenSim modeling and simulation package. OpenSim Moco uses the direct collocation method, which is often faster and can handle more diverse problems than other methods for musculoskeletal simulation. Moco frees researchers from implementing direct collocation themselves—which typically requires extensive technical expertise—and allows them to focus on their scientific questions. The software can handle a wide range of problems that interest biomechanists, including motion tracking, motion prediction, parameter optimization, model fitting, electromyography-driven simulation, and device design. Moco is the first musculoskeletal direct collocation tool to handle kinematic constraints, which enable modeling of kinematic loops (e.g., cycling models) and complex anatomy (e.g., patellar motion). To show the abilities of Moco, we first solved for muscle activity that produced an observed walking motion while minimizing squared muscle excitations and knee joint loading. Next, we predicted how muscle weakness may cause deviations from a normal walking motion. Lastly, we predicted a squat-to-stand motion and optimized the stiffness of an assistive device placed at the knee. We designed Moco to be easy to use, customizable, and extensible, thereby accelerating the use of simulations to understand the movement of humans and other animals.     

Alterations of musculoskeletal models for a more accurate estimation of lower limb joint contact forces during normal gait: A systematic review

Musculoskeletal modelling is a methodology used to investigate joint contact forces during a movement. High accuracy in the estimation of the hip or knee joint contact forces can be obtained with subject-specific models. However, construction of subject-specific models remains time consuming and expensive. The purpose of this systematic review of the literature was to identify what alterations can be made on generic (i.e. literature-based, without any subject-specific measurement other than body size and weight) musculoskeletal models to obtain a better estimation of the joint contact forces. The impact of these alterations on the accuracy of the estimated joint contact forces were appraised.The systematic search yielded to 141 articles and 24 papers were included in the review. Different strategies of alterations were found: skeletal and joint model (e.g. number of degrees of freedom, knee alignment), muscle model (e.g. Hill-type muscle parameters, level of muscular redundancy), and optimisation problem (e.g. objective function, design variables, constraints). All these alterations had an impact on joint contact force accuracy, so demonstrating the potential for improving the model predictions without necessarily involving costly and time consuming medical images. However, due to discrepancies in the reported evidence about this impact and despite a high quality of the reviewed studies, it was not possible to highlight any trend defining which alteration had the largest impact.     

Contribution of muscle short-range stiffness to initial changes in joint kinetics and kinematics during perturbations to standing balance: A simulation study

Simulating realistic musculoskeletal dynamics is critical to understanding neural control of muscle activity evoked in sensorimotor feedback responses that have inherent neural transmission delays. Thus, the initial mechanical response of muscles to perturbations in the absence of any change in muscle activity determines which corrective neural responses are required to stabilize body posture. Muscle short-range stiffness, a history-dependent property of muscle that causes a rapid and transient rise in muscle force upon stretch, likely affects musculoskeletal dynamics in the initial mechanical response to perturbations. Here we identified the contributions of short-range stiffness to joint torques and angles in the initial mechanical response to support surface translations using dynamic simulation. We developed a dynamic model of muscle short-range stiffness to augment a Hill-type muscle model. Our simulations show that short-range stiffness can provide stability against external perturbations during the neuromechanical response delay. Assuming constant muscle activation during the initial mechanical response, including muscle short-range stiffness was necessary to account for the rapid rise in experimental sagittal plane knee and hip joint torques that occurs simultaneously with very small changes in joint angles and reduced root mean square errors between simulated and experimental torques by 56% and 47%, respectively. Moreover, forward simulations lacking short-range stiffness produced unreasonably large joint angle changes during the initial response. Using muscle models accounting for short-range stiffness along with other aspects of history-dependent muscle dynamics may be important to advance our ability to simulate inherently unstable human movements based on principles of neural control and biomechanics.     

Are Subject-Specific Musculoskeletal Models Robust to the Uncertainties in Parameter Identification?

Subject-specific musculoskeletal modeling can be applied to study musculoskeletal disorders, allowing inclusion of personalized anatomy and properties. Independent of the tools used for model creation, there are unavoidable uncertainties associated with parameter identification, whose effect on model predictions is still not fully understood. The aim of the present study was to analyze the sensitivity of subject-specific model predictions (i.e., joint angles, joint moments, muscle and joint contact forces) during walking to the uncertainties in the identification of body landmark positions, maximum muscle tension and musculotendon geometry. To this aim, we created an MRI-based musculoskeletal model of the lower limbs, defined as a 7-segment, 10-degree-of-freedom articulated linkage, actuated by 84 musculotendon units. We then performed a Monte-Carlo probabilistic analysis perturbing model parameters according to their uncertainty, and solving a typical inverse dynamics and static optimization problem using 500 models that included the different sets of perturbed variable values. Model creation and gait simulations were performed by using freely available software that we developed to standardize the process of model creation, integrate with OpenSim and create probabilistic simulations of movement. The uncertainties in input variables had a moderate effect on model predictions, as muscle and joint contact forces showed maximum standard deviation of 0.3 times body-weight and maximum range of 2.1 times body-weight. In addition, the output variables significantly correlated with few input variables (up to 7 out of 312) across the gait cycle, including the geometry definition of larger muscles and the maximum muscle tension in limited gait portions. Although we found subject-specific models not markedly sensitive to parameter identification, researchers should be aware of the model precision in relation to the intended application. In fact, force predictions could be affected by an uncertainty in the same order of magnitude of its value, although this condition has low probability to occur.     

Modeling and dynamic simulation of astronaut’s upper limb motions considering counter torques generated by the space suit

Extravehicular activity (EVA) is an inevitable task for astronauts to maintain proper functions of both the spacecraft and the space station. Both experimental research in a microgravity simulator (e.g. neutral buoyancy tank, zero-g aircraft or a drop tower/tube) and mathematical modeling were used to study EVA to provide guidance for the training on Earth and task design in space. Modeling has become more and more promising because of its efficiency. Based on the task analysis, almost 90% of EVA activity is accomplished through upper limb motions. Therefore, focusing on upper limb models of the body and space suit is valuable to this effort. In previous modeling studies, some multi-rigid-body systems were developed to simplify the human musculoskeletal system, and the space suit was mostly considered as a part of the astronaut body. With the aim to improve the reality of the models, we developed an astronauts’ upper limb model, including a torque model and a muscle-force model, with the counter torques from the space suit being considered as a boundary condition. Inverse kinematics and the Maggi-Kane’s method was applied to calculate the joint angles, joint torques and muscle force given that the terminal trajectory of upper limb motion was known. Also, we validated the muscle-force model using electromyogram (EMG) data collected in a validation experiment. Muscle force calculated from our model presented a similar trend with the EMG data, supporting the effectiveness and feasibility of the muscle-force model we established, and also, partially validating the joint model in kinematics aspect.     

Are fixed limb inertial models valid for dynamic simulations of human movement?

During human movement, muscle activation and limb movement result in subtle changes in muscle mass distribution. Muscle mass redistribution can affect limb inertial properties and limb dynamics, but it is not currently known to what extent. The objectives of this study were to investigate: (1) how physiological alterations of muscle and tendon length affect limb inertial characteristics, and (2) how such changes affect dynamic simulations of human movement. To achieve these objectives, a digital model of a human leg, custom software, and Software for interactive musculoskeletal modeling were used to simulate mass redistribution of muscle–tendon structures within a limb segment during muscle activation and joint movement. Thigh and shank center of mass and moments of inertia for different muscle activation and joint configurations were determined and compared. Limb inertial parameters representing relaxed muscles and fully active muscles were input into a simulated straight-leg movement to evaluate the effect inertial parameter variations could have on movement simulation results. Muscle activation and limb movement altered limb segment center of mass and moments of inertia by less than 0.04 cm and 1.2%, respectively. These variations in limb inertial properties resulted in less than 0.01% change in maximum angular velocity for a simulated straight-leg hip flexion task. These data demonstrate that, for the digital human leg model considered, assuming static quantities for segment center of masses and moments of inertia in movement simulations appear reasonable and induce minimal errors in simulated movement dynamics.     

Robotics-based synthesis of human motion

The synthesis of human motion is a complex procedure that involves accurate reconstruction of movement sequences, modeling of musculoskeletal kinematics, dynamics and actuation, and characterization of reliable performance criteria. Many of these processes have much in common with the problems found in robotics research. Task-based methods used in robotics may be leveraged to provide novel musculoskeletal modeling methods and physiologically accurate performance predictions. In this paper, we present (i) a new method for the real-time reconstruction of human motion trajectories using direct marker tracking, (ii) a task-driven muscular effort minimization criterion and (iii) new human performance metrics for dynamic characterization of athletic skills. Dynamic motion reconstruction is achieved through the control of a simulated human model to follow the captured marker trajectories in real-time. The operational space control and real-time simulation provide human dynamics at any configuration of the performance. A new criteria of muscular effort minimization has been introduced to analyze human static postures. Extensive motion capture experiments were conducted to validate the new minimization criterion. Finally, new human performance metrics were introduced to study in details an athletic skill. These metrics include the effort expenditure and the feasible set of operational space accelerations during the performance of the skill. The dynamic characterization takes into account skeletal kinematics as well as muscle routing kinematics and force generating capacities. The developments draw upon an advanced musculoskeletal modeling platform and a task-oriented framework for the effective integration of biomechanics and robotics methods.     

Predictions of co-contraction depend critically on degrees-of-freedom in the musculoskeletal model

In biomechanics, the calculation of individual muscle forces during movements is based on a model of the musculoskeletal system and a method for extracting a unique set of muscle forces. To obtain a unique set of muscle forces, non-linear, static optimisation is commonly used. However, the optimal solution is dependent on the musculoskeletal geometry, and single joints may be represented using one, two or three degrees-of-freedom. Frequently, a system with multiple degrees-of-freedom is replaced with a system that contains a subset of all the possible degrees-of-freedom. For example, the cat ankle joint is typically modelled as a planar joint with its primary degree-of-freedom (plantar-dorsiflexion), whereas, the actual joint has three rotational degrees-of-freedom. Typically, such simplifications are justified by the idea that the reduced case is contained as a specific solution of the more general case. However, here we demonstrate that the force-sharing solution space of a general, three degrees-of-freedom musculoskeletal system does not necessarily contain the solutions from the corresponding one or two degrees-of-freedom systems. Therefore, solutions of a reduced system, in general, are not sub-set solutions of the actual three degrees-of-freedom system, but are independent solutions that are often incompatible with solutions of the actual system. This result shows that representing a three degrees-of-freedom system as a one or two degrees-of-freedom system gives force-sharing solutions that cannot be extrapolated to the actual system, and vice-versa. These results imply that general solutions cannot be extracted from models with fewer degrees-of-freedom than the actual system. They further emphasise the need for precise geometric representation of the musculoskeletal system, if general force-sharing rules are to be derived.     

Automated estimation of ankle muscle EMG envelopes and resulting plantar-dorsi flexion torque from 64 garment-embedded electrodes uniformly distributed around the human leg

The design of personalized movement training and rehabilitation pipelines relies on the ability of assessing the activation of individual muscles concurrently with the resulting joint torques exerted during functional movements. Despite advances in motion capturing, force sensing and bio-electrical recording technologies, the estimation of muscle activation and resulting force still relies on lengthy experimental and computational procedures that are not clinically viable. This work proposes a wearable technology for the rapid, yet quantitative, assessment of musculoskeletal function. It comprises of (1) a soft leg garment sensorized with 64 uniformly distributed electromyography (EMG) electrodes, (2) an algorithm that automatically groups electrodes into seven muscle-specific clusters, and (3) a EMG-driven musculoskeletal model that estimates the resulting force and torque produced about the ankle joint sagittal plane. Our results show the ability of the proposed technology to automatically select a sub-set of muscle-specific electrodes that enabled accurate estimation of muscle excitations and resulting joint torques across a large range of biomechanically diverse movements, underlying different excitation patterns, in a group of eight healthy individuals. This may substantially decrease time needed for localization of muscle sites and electrode placement procedures, thereby facilitating applicability of EMG-driven modelling pipelines in standard clinical protocols.     

Fibre operating lengths of human lower limb muscles during walking

Muscles actuate movement by generating forces. The forces generated by muscles are highly dependent on their fibre lengths, yet it is difficult to measure the lengths over which muscle fibres operate during movement. We combined experimental measurements of joint angles and muscle activation patterns during walking with a musculoskeletal model that captures the relationships between muscle fibre lengths, joint angles and muscle activations for muscles of the lower limb. We used this musculoskeletal model to produce a simulation of muscle-tendon dynamics during walking and calculated fibre operating lengths (i.e. the length of muscle fibres relative to their optimal fibre length) for 17 lower limb muscles. Our results indicate that when musculotendon compliance is low, the muscle fibre operating length is determined predominantly by the joint angles and muscle moment arms. If musculotendon compliance is high, muscle fibre operating length is more dependent on activation level and force-length-velocity effects. We found that muscles operate on multiple limbs of the force-length curve (i.e. ascending, plateau and descending limbs) during the gait cycle, but are active within a smaller portion of their total operating range.     

Effects of lumbo-pelvic rhythm on trunk muscle forces and disc loads during forward flexion: A combined musculoskeletal and finite element simulation study

Previous in-vivo studies suggest that the ratio of total lumbar rotation over pelvic rotation (lumbo-pelvic rhythm) during trunk sagittal movement is essential to evaluate spinal loads and discriminate between low back pain and asymptomatic population. Similarly, there is also evidence that the lumbo-pelvic rhythm is key for evaluation of realistic muscle and joint reaction forces and moments predicted by various computational musculoskeletal models. This study investigated the effects of three lumbo-pelvic rhythms defined based on in-vivo measurements on the spinal response during moderate forward flexion (60°) using a combined approach of musculoskeletal modeling of the upper body and finite element model of the lumbosacral spine. The muscle forces and joint loads predicted by the musculoskeletal model, together with the gravitational forces, were applied to the finite element model to compute the disc force and moment, intradiscal pressure, annular fibers strain, and load-sharing. The results revealed that a rhythm with high pelvic rotation and low lumbar flexion involves more global muscles and increases the role of the disc in resisting spinal loads, while its counterpart, with low pelvic rotation, recruits more local muscles and engages the ligaments to lower the disc loads. On the other hand, a normal rhythm that has balanced pelvic and lumbar rotations yields almost equal disc and ligament load-sharing and results in more balanced synergy between global and local muscles. The lumbo-pelvic rhythm has less effect on the intradiscal pressure and annular fibers strain. This work demonstrated that the spinal response during forward flexion is highly dependent on the lumbo-pelvic rhythm. It is therefore, essential to adapt this parameter instead of using the default values in musculoskeletal models for accurate prediction of muscle forces and joint reaction forces and moments. The findings provided by this work are expected to improve knowledge of spinal response during forward flexion, and are clinically relevant towards low back pain treatment and disc injury prevention.     

A reduced muscle model and planar musculoskeletal model fit for the simulation of whole-body movements

Musculoskeletal models are made to reflect the capacities of the human body in general, and often a specific subject in particular. It remains challenging to both model the musculoskeletal system and then fit the modelled muscles to a specific human subject. We present a reduced muscle model, a planar musculoskeletal model, and a fitting method that can be used to find a feasible set of active and passive muscle parameters for a specific subject. At a minimum, the fitting method requires inverse dynamics data of the subject, a scalar estimate of the peak activation reached during the movement, and a plausible initial estimate for the strength and flexibility of that subject. While additional data can be used to result in a more accurate fit, this data is not required for the method solve for a feasible fit. The minimal input requirements of the proposed fitting method make it well suited for subjects who cannot undergo a maximum voluntary contraction trial, or for whom recording electromyographic data is not possible. To evaluate the model and fitting method we adjust the musculoskeletal model so that it can perform an experimentally recorded stoop-lift of a 15 kg box.     

Experimental quadriceps muscle pain impairs knee joint control during walking.

Pain is a cardinal symptom in musculoskeletal diseases involving the knee joint, and aberrant movement patterns and motor control strategies are often present in these patients. However, the underlying neuromuscular mechanisms linking pain to movement and motor control are unclear. To investigate the functional significance of muscle pain on knee joint control during walking, three-dimensional gait analyses were performed before, during, and after experimentally induced muscle pain by means of intramuscular injections of hypertonic saline (5.8%) into vastus medialis (VM) muscle of 20 healthy subjects. Isotonic saline (0.9%) was used as control. Surface electromyography (EMG) recordings of VM, vastus lateralis (VL), biceps femoris, and semitendinosus muscles were synchronized with the gait analyses. During experimental muscle pain, the loading response phase peak knee extensor moments were attenuated, and EMG activity in the VM and VL muscles was reduced. Compressive forces, adduction moments, knee joint kinematics, and hamstring EMG activity were unaffected by pain. Interestingly, the observed changes persisted when the pain had vanished. The results demonstrate that muscle pain modulated the function of the quadriceps muscle, resulting in impaired knee joint control and joint instability during walking. The changes are similar to those observed in patients with knee pain. The loss of joint control during and after pain may leave the knee joint prone to injury and potentially participate in the chronicity of musculoskeletal problems, and it may have clinically important implications for rehabilitation and training of patients with knee pain of musculoskeletal origin.     

Reduction of neuromuscular redundancy for postural force generation using an intrinsic stability criterion

Postural control requires the coordination of multiple muscles to achieve both endpoint force production and postural stability. Multiple muscle activation patterns can produce the required force for standing, but the mechanical stability associated with any given pattern may vary, and has implications for the degree of delayed neural feedback necessary for postural stability. We hypothesized that muscular redundancy is reduced when muscle activation patterns are chosen with respect to intrinsic musculoskeletal stability as well as endpoint force production. We used a three-dimensional musculoskeletal model of the cat hindlimb with 31 muscles to determine the possible contributions of intrinsic muscle properties to limb stability during isometric force generation. Using dynamic stability analysis we demonstrate that within the large set of activation patterns that satisfy the force requirement for posture, only a reduced subset produce a mechanically stable limb configuration. Greater stability in the frontal-plane suggests that neural control mechanisms are more highly active for sagittal-plane and for ankle joint control. Even when the limb was unstable, the time-constants of instability were sufficiently great to allow long-latency neural feedback mechanisms to intervene, which may be preferential for movements requiring maneuverability versus stability. Local joint stiffness of muscles was determined by the stabilizing or destabilizing effects of moment-arm versus joint angle relationships. By preferentially activating muscles with high local stiffness, muscle activation patterns with feedforward stabilizing properties could be selected. Such a strategy may increase intrinsic postural stability without co-contraction, and may be useful criteria in the force-sharing problem.     

Analysis of kinematic invariances of multijoint reaching movement

 There is a no unique relationship between the trajectory of the hand, represented in cartesian or extrinsic space, and its trajectory in joint angle or intrinsic space in the general condition of joint redundancy. The goal of this work is to analyze the relation between planning the trajectory of a multijoint movement in these two coordinate systems. We show that the cartesian trajectory can be planned based on the task parameters (target coordinates, etc.) prior to and independently of angular trajectories. Angular time profiles are calculated from the cartesian trajectory to serve as a basis for muscle control commands. A unified differential equation that allows planning trajectories in cartesian and angular spaces simultaneously is proposed. Due to joint redundancy, each cartesian trajectory corresponds to a family of angular trajectories which can account for the substantial variability of the latter. A set of strategies for multijoint motor control following from this model is considered; one of them coincides with the frog wiping reflex model and resolves the kinematic inverse problem without inversion. The model trajectories exhibit certain properties observed in human multijoint reaching movements such as movement equifinality, straight end-point paths, bell-shaped tangential velocity profiles, speed-sensitive and speed-insensitive movement strategies, peculiarities of the response to double-step targets, and variations of angular trajectory without variations of the limb end-point trajectory in cartesian space. In humans, those properties are almost independent of limb configuration, target location, movement duration, and load. In the model, these properties are invariant to an affine transform of cartesian space. This implies that these properties are not a special goal of the motor control system but emerge from movement kinematics that reflect limb geometry, dynamics, and elementary principles of motor control used in planning. All the results are given analytically and, in order to compare the model with experimental results, by computer simulations. Received: 6 April 1994/Accepted in revised form: 25 April 1995