Virtual trajectory and stiffness ellipse during multijoint arm movement predicted by neural inverse models

We predict the virtual trajectories and stiffness ellipses during multijoint arm movements by computer simulations. A two-link manipulator with four single-joint muscles and two double-joint muscles is used as a model of the human arm. Physical parameters of the model are derived from several experimental data. Among them, special emphasis is put on low values of the dynamic hand stiffness recently measured during single joint and multijoint movements. The feedback-error-learning scheme to acquire the inverse dynamics model and the inverse statics model is utilized for this prediction. The virtual trajectories are much more complex than the actual trajectories. This indicates that planning the virtual trajectory is as difficult as solving the inverse dynamics problem for medium and fast movements, and simply falsifies the advocated computational advantage of the virtual trajectory control hypothesis. Thus, we conclude that learning inverse models is essential even in the virtual trajectory control framework. Finally, we propose a new computational model to learn the complicated shape of the virtual trajectories by integrating the virtual trajectory control and the feedback-error-learning scheme.     

Analysis of an optimal control model of multi-joint arm movements

 In this paper, we propose a model of biological motor control for generation of goal-directed multi-joint arm movements, and study the formation of muscle control inputs and invariant kinematic features of movements. The model has a hierarchical structure that can determine the control inputs for a set of redundant muscles without any inverse computation. Calculation of motor commands is divided into two stages, each of which performs a transformation of motor commands from one coordinate system to another. At the first level, a central controller in the brain accepts instructions from higher centers, which represent the motor goal in the Cartesian space. The controller computes joint equilibrium trajectories and excitation signals according to a minimum effort criterion. At the second level, a neural network in the spinal cord translates the excitation signals and equilibrium trajectories into control commands to three pairs of antagonist muscles which are redundant for a two-joint arm. No inverse computation is required in the determination of individual muscle commands. The minimum effort controller can produce arm movements whose dynamic and kinematic features are similar to those of voluntary arm movements. For fast movements, the hand approaches a target position along a near-straight path with a smooth bell-shaped velocity. The equilibrium trajectories in X and Y show an ‘N’ shape, but the end-point equilibrium path zigzags around the hand path. Joint movements are not always smooth. Joint reversal is found in movements in some directions. The excitation signals have a triphasic (or biphasic) pulse pattern, which leads to stereotyped triphasic (or biphasic) bursts in muscle control inputs, and a dynamically modulated joint stiffness. There is a fixed sequence of muscle activation from proximal muscles to distal muscles. The order is preserved in all movements. For slow movements, it is shown that a constant joint stiffness is necessary to produce a smooth movement with a bell-shaped velocity. Scaled movements can be reproduced by varying the constraints on the maximal level of excitation signals according to the speed of movement. When the inertial parameters of the arm are altered, movement trajectories can be kept invariant by adjusting the pulse height values, showing the ability to adapt to load changes. These results agree with a wide range of experimental observations on human voluntary movements. Received: 4 December 1995 / Accepted in revised form: 17 September 1996     

On the possibility of linear modelling the human arm neuromuscular apparatus

It has been widely claimed that linear models of the neuromuscular apparatus give very inaccurate approximations of human arm reaching movements. The present paper examines this claim by quantifying the contributions of the various non-linear effects of muscle force generation on the accuracy of linear approximation. We performed computer simulations of a model of a two-joint arm with six monarticular and biarticular muscles. The global actions of individual muscles resulted in a linear dependence of the joint torques on the joint angles and angular velocities, despite the great non-linearity of the muscle properties. The effect of time delay in force generation is much more important for model accuracy than all the non-linear effects, while ignoring this time delay in linear approximation results in large errors. Thus, the viscosity coefficients are rather underestimated and some of them can even be paradoxically estimated to be negative. Similarly, our computation showed that ignoring the time delay resulted in large errors in the estimation of the hand equilibrium trajectory. This could explain why experimentally estimated hand equilibrium trajectories may be complex, even during a simple reaching movement. The hand equilibrium trajectory estimated by a linear model becomes simple when the time delay is taken into account, and it is close to that actually used in the non-linear model. The results therefore provide a theoretical basis for estimating the hand equilibrium trajectory during arm reaching movements and hence for estimating the time course of the motor control signals associated with this trajectory, as set out in the equilibrium point hypothesis. Received: 17 February 1999 / Accepted in revised form: 22 October 1999     

Computational model of motor learning and perceptual change

Motor learning in the context of arm reaching movements has been frequently investigated using the paradigm of force-field learning. It has been recently shown that changes to somatosensory perception are likewise associated with motor learning. Changes in perceptual function may be the reason that when the perturbation is removed following motor learning, the hand trajectory does not return to a straight line path even after several dozen trials. To explain the computational mechanisms that produce these characteristics, we propose a motor control and learning scheme using a simplified two-link system in the horizontal plane: We represent learning as the adjustment of desired joint-angular trajectories so as to achieve the reference trajectory of the hand. The convergence of the actual hand movement to the reference trajectory is proved by using a Lyapunov-like lemma, and the result is confirmed using computer simulations. The model assumes that changes in the desired hand trajectory influence the perception of hand position and this in turn affects movement control. Our computer simulations support the idea that perceptual change may come as a result of adjustments to movement planning with motor learning.     

Kinematic construction of the trajectory of sequential arm movements

A kinematic construction rule determining the trajectory of human sequential movements is formulated using minimum-jerk and minimum-angular-jerk trajectories. The kinematic construction rule states that the observed trajectory of sequential movements coincides with a weighted average of the minimum-jerk trajectory and the segmented minimum-angular-jerk trajectory. This rule covers not only point-to-point movements but also simple sequential movements. Five kinds of experiments that measure the trajectories in planar, multijoint sequential arm movements were conducted. The measured trajectories coincide with the predictions made on the basis of the kinematic construction rule presented here. Moreover, predictions of previous models such as the minimum-jerk, the equilibrium-trajectory, and the minimum-torque-change models are shown to be incompatible with our observations of sequential movements. Received: 31 October 1997 /Accepted in revised form: 18 November 1998     

Kinematic invariants during cyclical arm movements

It has been observed that the motion of the arm end-point (the hand, fingertip or the tip of a pen) is characterized by a number of regularities (kinematic invariants). Trajectory is usually straight, and the velocity profile has a bell shape during point-to-point movements. During drawing movements, a two-thirds power law predicts the dependence of the end-point velocity on the trajectory curvature. Although various principles of movement organization have been discussed as possible origins of these kinematic invariants, the nature of these movement trajectory characteristics remains an open question. A kinematic model of cyclical arm movements derived in the present study analytically demonstrates that all three kinematic invariants can be predicted from a two-joint approximation of the kinematic structure of the arm and from sinusoidal joint motions. With this approach, explicit expressions for two kinematic invariants, the two-thirds power law during drawing movements and the velocity profile during point-to-point movements are obtained as functions of arm segment lengths and joint motion parameters. Additionally, less recognized kinematic invariants are also derived from the model. The obtained analytical expressions are further validated with experimental data. The high accuracy of the predictions confirms practical utility of the model, showing that the model is relevant to human performance over a wide range of movements. The results create a basis for the consolidation of various existing interpretations of kinematic invariants. In particular, optimal control is discussed as a plausible source of invariant characteristics of joint motions and movement trajectories.     

Equilibrium point control of a monkey arm simulator by a fast learning tree structured artificial neural network

A planar 17 muscle model of the monkey's arm based on realistic biomechanical measurements was simulated on a Symbolics Lisp Machine. The simulator implements the equilibrium point hypothesis for the control of arm movements. Given initial and final desired positions, it generates a minimum-jerk desired trajectory of the hand and uses the backdriving algorithm to determine an appropriate sequence of motor commands to the muscles (Flash 1987; Mussa-Ivaldi et al. 1991; Dornay 1991b). These motor commands specify a temporal sequence of stable (attractive) equilibrium positions which lead to the desired hand movement. A strong disadvantage of the simulator is that it has no memory of previous computations. Determining the desired trajectory using the minimum-jerk model is instantaneous, but the laborious backdriving algorithm is slow, and can take up to one hour for some trajectories. The complexity of the required computations makes it a poor model for biological motor control. We propose a computationally simpler and more biologically plausible method for control which achieves the benefits of the backdriving algorithm. A fast learning, tree-structured network (Sanger 1991c) was trained to remember the knowledge obtained by the backdriving algorithm. The neural network learned the nonlinear mapping from a 2-dimensional cartesian planar hand position {x, y} to a 17-dimensional motor command space {u ₁, ..., u ₁₇}. Learning 20 training trajectories, each composed of 26 sample points {{x y{,{u ₁, ..., u ₁₇} took only 20 min on a Sun-4 Spare workstation. After the learning stage, new, untrained test trajectories as well as the original trajectories of the hand were given to the neural network as input. The network calculated the required motor commands for these movements. The resulting movements were close to the desired ones for both the training and test cases.     

Formation and control of optimal trajectory in human multijoint arm movement

In this paper, we study trajectory planning and control in voluntary, human arm movements. When a hand is moved to a target, the central nervous system must select one specific trajectory among an infinite number of possible trajectories that lead to the target position. First, we discuss what criterion is adopted for trajectory determination. Several researchers measured the hand trajectories of skilled movements and found common invariant features. For example, when moving the hand between a pair of targets, subjects tended to generate roughly straight hand paths with bell-shaped speed profiles. On the basis of these observations and dynamic optimization theory, we propose a mathematical model which accounts for formation of hand trajectories. This model is formulated by defining an objective function, a measure of performance for any possible movement: square of the rate of change of torque integrated over the entire movement. That is, the objective function CT is defined as follows: (formula; see text) We overcome this difficult by developing an iterative scheme, with which the optimal trajectory and the associated motor command are simultaneously computed. To evaluate our model, human hand trajectories were experimentally measured under various behavioral situations. These results supported the idea that the human hand trajectory is planned and controlled in accordance with the minimum torque-change criterion.     

Optimal control of antagonistic muscle stiffness during voluntary movements

This paper presents a study on the control of antagonist muscle stiffness during single-joint arm movements by optimal control theory with a minimal effort criterion. A hierarchical model is developed based on the physiology of the neuromuscular control system and the equilibrium point hypothesis. For point-to-point movements, the model provides predictions on (1) movement trajectory, (2) equilibrium trajectory, (3) muscle control inputs, and (4) antagonist muscle stiffness, as well as other variables. We compared these model predictions to the behavior observed in normal human subjects. The optimal movements capture the major invariant characteristics of voluntary movements, such as a sigmoidal movement trajectory with a bell-shaped velocity profile, an N-shaped equilibrium trajectory, a triphasic burst pattern of muscle control inputs, and a dynamically modulated joint stiffness. The joint stiffness is found to increase in the middle of the movement as a consequence of the triphasic muscle activities. We have also investigated the effects of changes in model parameters on movement control. We found that the movement kinematics and muscle control inputs are strongly influenced by the upper bound of the descending excitation signal that activates motoneuron pools in the spinal cord. Furthermore, a class of movements with scaled velocity profiles can be achieved by tuning the amplitude and duration of this excitation signal. These model predictions agree with a wide body of experimental data obtained from normal human subjects. The results suggest that the control of fast arm movements involves explicit planning for both the equilibrium trajectory and joint stiffness, and that the minimal effort criterion best characterizes the objective of movement planning and control.     

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.     

A method for measuring endpoint stiffness during multi-joint arm movements

Current methods for measuring stiffness during human arm movements are either limited to one-joint motions, or lead to systematic errors. The technique presented here enables a simple, accurate and unbiased measurement of endpoint stiffness during multi-joint movements. Using a computer-controlled mechanical interface, the hand is displaced relative to a prediction of the undisturbed trajectory. Stiffness is then computed as the ratio of restoring force to displacement amplitude. Because of the accuracy of the prediction (< 1 cm error after 200 ms) and the quality of the implementation, the movement is not disrupted by the perturbation. This technique requires only 13 as many trials to identify stiffness as the method of Gomi and Kawato (1997, Biological Cybernetics 76, 163-171) and may, therefore, be used to investigate the evolution of stiffness during motor adaptation.     

Characterizing and Predicting Submovements during Human Three-Dimensional Arm Reaches

We have demonstrated that 3D target-oriented human arm reaches can be represented as linear combinations of discrete submovements, where the submovements are a set of minimum-jerk basis functions for the reaches. We have also demonstrated the ability of deterministic feed-forward Artificial Neural Networks (ANNs) to predict the parameters of the submovements. ANNs were trained using kinematic data obtained experimentally from five human participants making target-directed movements that were decomposed offline into minimum-jerk submovements using an optimization algorithm. Under cross-validation, the ANNs were able to accurately predict the parameters (initiation-time, amplitude, and duration) of the individual submovements. We also demonstrated that the ANNs can together form a closed-loop model of human reaching capable of predicting 3D trajectories with VAF >95.9% and RMSE ≤4.32 cm relative to the actual recorded trajectories. This closed-loop model is a step towards a practical arm trajectory generator based on submovements, and should be useful for the development of future arm prosthetic devices that are controlled by brain computer interfaces or other user interfaces.     

Three-dimensional model to predict muscle forces and their relation to motor variances in reaching arm movements

A three-dimensional (3-D) arm movement model is presented to simulate kinematic properties and muscle forces in reaching arm movements. Healthy subjects performed reaching movements repetitively either with or without a load in the hand. Joint coordinates were measured. Muscle moment arms, 3-D angular acceleration, and moment of inertias of arm segments were calculated to determine 3-D joint torques. Variances of hand position, arm configuration, and muscle activities were calculated. Ratios of movement variances observed in the two conditions (load versus without load) showed no differences for hand position and arm configuration variances. Virtual muscle force variances for all muscles except deltoid posterior and EMG variances for four muscles increased significantly by moving with the load. The greatly increased variances in muscle activity did not imply equally high increments in kinematic variances. We conclude that enhanced muscle cooperation through synergies helps to stabilize movement at the kinematic level when a load is added.     

Trajectory formation based on physiological characteristics of skeletal muscles

Human arm trajectories in natural unrestricted reaching movements were studied. They have particular properties such that a hand path is a rather simple straight or curved line, and a tangential velocity profile of hand is bell-shaped. Also these properties are invariant, independent of movement duration and hand-held load. In this study, trajectory formation is investigated on the basis of physiological characteristics of skeletal muscles, and a criterion prescribed by a derivative of isometric muscle torque is proposed. Subsequently, optimal trajectories are formulated under various conditions of movement to account for a planning strategy of human arm trajectories. In addition to such a theoretical approach, human arm trajectories are experimentally observed by a measuring system which provides a visual sensor and a target tracking device, enabling totally unrestricted movements. Then, optimal trajectories are quantitatively evaluated in comparison with experimental data in which essential properties of human arm trajectories are demonstrated. These results support the idea that human arm trajectories are planned in order to minimize the proposed criterion which is determined from physiological aspects. Finally, the physiological advantages of human arm trajectories are discussed with regard to the analysis of observed and optimal trajectories. Received: 2 December 1997 / Accepted in revised form: 20 March 1998     

The Relation between Geometry and Time in Mental Actions

Mental imagery is a cognitive tool that helps humans take decisions by simulating past and future events. The hypothesis has been advanced that there is a functional equivalence between actual and mental movements. Yet, we do not know whether there are any limitations to its validity even in terms of some fundamental features of actual movements, such as the relationship between space and time. Although it is impossible to directly measure the spatiotemporal features of mental actions, an indirect investigation can be conducted by taking advantage of the constraints existing in planar drawing movements and described by the two-thirds power law (2/3PL). This kinematic law describes one of the most impressive regularities observed in biological movements: movement speed decreases when curvature increases. Here, we compared the duration of identical actual and mental arm movements by changing the constraints imposed by the 2/3PL. In the first two experiments, the length of the trajectory remained constant, while its curvature (Experiment 1) or its number of inflexions (Experiment 2) was manipulated. The results showed that curvature, but not the number of inflexions, proportionally and similarly affected actual and mental movement duration, as expected from the 2/3PL. Two other control experiments confirmed that the results of Experiment 1 were not attributable to eye movements (Experiment 3) or to the perceived length of the displayed trajectory (Experiment 4). Altogether, our findings suggest that mental movement simulation is tuned to the kinematic laws characterizing actions and that kinematics of actual and mental movements is completely specified by the representation of their geometry.     

Velocity-Based Planning of Rapid Elbow Movements Expands the Control Scheme of the Equilibrium Point Hypothesis

According to the equilibrium point hypothesis of voluntary motor control, control action of muscles is not explicitly computed, but rather arises as a consequence of interaction between moving equilibrium position, current kinematics and stiffness of the joint. This approach is attractive as it obviates the need to explicitly specify the forces controlling limb movements. However, many debatable aspects of this hypothesis remain in the manner of specification of the equilibrium point trajectory and muscle activation (or its stiffness), which elicits a restoring force toward the planned equilibrium trajectory. In this study, we expanded the framework of this hypothesis by assuming that the control system uses the velocity measure as the origin of subordinate variables scaling descending commands. The velocity command is translated into muscle control inputs by second order pattern generators, which yield reciprocal command and coactivation commands, and create alternating activation of the antagonistic muscles during movement and coactivation in the post-movement phase, respectively. The velocity command is also integrated to give a position command specifying a moving equilibrium point. This model is purely kinematics-dependent, since the descending commands needed to modulate the visco-elasticity of muscles are implicitly given by simple parametric specifications of the velocity command alone. The simulated movements of fast elbow single-joint movements corresponded well with measured data performed over a wide range of movement distances, in terms of both muscle excitations and kinematics. Our proposal on a synthesis for the equilibrium point approach and velocity command, may offer some insights into the control scheme of the single-joint arm movements.     

Examination of the λ equilibrium point hypothesis when applied to single degree of freedom movements performed with different inertial loads

 One of the theories of human motor control is the λ Equilibrium Point Hypothesis. It is an attractive theory since it offers an easy control scheme where the planned trajectory shifts monotionically from an initial to a final equilibrium state. The feasibility of this model was tested by reconstructing the virtual trajectory and the stiffness profiles for movements performed with different inertial loads and examining them. Three types of movements were tested: passive movements, targeted movements, and repetitive movements. Each of the movements was performed with five different inertial loads. Plausible virtual trajectories and stiffness profiles were reconstructed based on the λ Equilibrium Point Hypothesis for the three different types of movements performed with different inertial loads. However, the simple control strategy supported by the model, where the planned trajectory shifts monotonically from an initial to a final equilibrium state, could not be supported for targeted movements performed with added inertial load. To test the feasibility of the model further we must examine the probability that the human motor control system would choose a trajectory more complicated than the actual trajectory to control. Received: 20 June 1995 / Accepted in revised form: 6 August 1996     

A model of handwriting

The research reported here is concerned with hand trajectory planning for the class of movements involved in handwriting. Previous studies show that the kinematics of human two-joint arm movements in the horizontal plane can be described by a model which is based on dynamic minimization of the square of the third derivative of hand position (jerk), integrated over the entire movement. We extend this approach to both the analysis and the synthesis of the trajectories occurring in the generation of handwritten characters. Several basic strokes are identified and possible stroke concatenation rules are suggested. Given a concise symbolic representation of a stroke shape, a simple algorithm computes the complete kinematic specification of the corresponding trajectory. A handwriting generation model based on a kinematics from shape principle and on dynamic optimization is formulated and tested. Good qualitative and quantitative agreement was found between subject recordings and trajectories generated by the model. The simple symbolic representation of hand motion suggested here may permit the central nervous system to learn, store and modify motor action plans for writing in an efficient manner.     

Motor control of voluntary arm movements

The motor control of pointing and reaching-to-grasp movements was investigated using two different approaches (kinematic and modelling) in order to establish whether the type of control varies according to modifications of arm kinematics. Kinematic analysis of arm movements was performed on subjects' hand trajectories directed to large and small stimuli located at two different distances. The subjects were required either to grasp and to point to each stimulus. The kinematics of the subsequent movement, during which subject's hand came back to the starting position, were also studied. For both movements, kinematic analysis was performed on hand linear trajectories as well as on joint angular trajectories of shoulder and elbow. The second approach consisted in the parametric identification of the black box (ARMAX) model of the controller driving the arm movement. Such controller is hypothesized to work for the correct execution of the motor act. The order of the controller ARMAX model was analyzed with respect to the different experimental conditions (distal task, stimulus size and distance). Results from kinematic analysis showed that target distance and size influenced kinematic parameters both of angular and linear displacements. Nevertheless, the structure of the motor program was found to remain constant with distane and distal task, while it varied with precision requirements due to stimulus size. The estimated model order of the controller confirmed the invariance of the control law with regard to movement amplitude, whereas it was sensitive to target size.     

Mechanisms underlying the generation of averaged modified trajectories

In this work we have studied what mechanisms might possibly underlie arm trajectory modification when reaching toward visual targets. The double-step target displacement paradigm was used with inter-stimulus intervals (ISIs) in the range of 10-300 ms. For short ISIs, a high percentage of the movements were found to be initially directed in between the first and second target locations (averaged trajectories). The initial direction of motion was found to depend on the target configuration, and on : the time difference between the presentation of the second stimulus and movement onset. To account for the kinematic features of the averaged trajectories two modification schemes were compared: the superposition scheme and the abort-replan scheme. According to the superposition scheme, the modified trajectories result from the vectorial addition of two elemental motions: one for moving between the initial hand position and an intermediate location, and a second one for moving between that intermediate location and the final target. According to the abort-replan scheme, the initial plan for moving toward the intermediate location is aborted and smoothly replaced by a new plan for moving from the hand position at the time the trajectory is modified to the final target location. In both tested schemes we hypothesized that due to the quick displacement of the stimulus, the internally specified intermediate goal might be influenced by both stimuli and may be different from the location of the first stimulus. It was found that the statistically most successful model in accounting for the measured data is based on the superposition scheme. It is suggested that superposition of simple independent elemental motions might be a general principle for the generation of modified motions, which allows for efficient, parallel planning. For increasing values of the inferred locations of the intermediate targets were found to gradually shift from the first toward the second target locations along a path that curved toward the initial hand position. These inferred locations show a strong resemblance to the intermediate locations of saccades generated in a similar double-step paradigm. These similarities in the specification of target locations used in the generation of eye and hand movements may serve to simplify visuomotor integration. Received: 22 June 1994 / Accepted in revised form: 15 September 1994