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     

A theory for cursive handwriting based on the minimization principle

 We propose a trajectory planning and control theory which provides explanations at the computation, algorithm, representation, and hardware levels for continuous movement such as connected cursive handwriting. The hardware is based on our previously proposed forward-inverse-relaxation neural network. Computationally, the optimization principle is the minimum torque-change criterion. At the representation level, hard constraints satisfied by a trajectory are represented as a set of via-points extracted from handwritten characters. Accordingly, we propose a via-point estimation algorithm that estimates via-points by repeating trajectory formation of a character and via-point extraction from the character. It is shown experimentally that for movements with a single via-point target, the via-point estimation algorithm can assign a point near the actual via-point target. Good quantitative agreement is found between human movement data and the trajectories generated by the proposed model. Received: 23 June 1994 / Accepted in revised form: 3 February 1995     

Optimal trajectory formation of constrained human arm reaching movements

Opening a door, turning a steering wheel, and rotating a coffee mill are typical examples of human movements that are constrained by the physical environment. The constraints decrease the mobility of the human arm and lead to redundancy in the distribution of actuator forces (either joint torques or muscle forces). Due to this actuator redundancy, there is an infinite number of ways to form a specific arm trajectory. However, humans form trajectories in a unique way. How do humans resolve the redundancy of the constrained motions and specify the hand trajectory? To investigate this problem, we examine human arm movements in a crank-rotation task. To explain the trajectory formation in constrained point-to-point motions, we propose a combined criterion minimizing the hand contact force change and the actuating force change over the course of movement. Our experiments show a close matching between predicted and experimental data.     

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     

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.     

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.     

A minimum energy cost hypothesis for human arm trajectories

 Many tasks require the arm to move from its initial position to a specified target position, but leave us free to choose the trajectory between them. This paper presents and tests the hypothesis that trajectories are chosen to minimize metabolic energy costs. Costs are calculated for the range of possible trajectories, for movements between the end points used in previously published experiments. Calculated energy minimizing trajectories for a model with biarticular elbow muscles agree well with observed trajectories for fast movements. Good agreement is also obtained for slow movements if they are assumed to be performed by slower muscles. A model in which all muscles are uniarticular is less successful in predicting observed trajectories. The effects of loads and of reversing the direction of movement are investigated. Received: 11 July 1995 / Accepted in revised form: 7 October 1996     

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.     

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     

Movement curvature planning through force field internal models

Human motion studies have focused primarily on modeling straight point-to-point reaching movements. However, many goal-directed reaching movements, such as movements directed towards oneself, are not straight but rather follow highly curved trajectories. These movements are particularly interesting to study since they are essential in our everyday life, appear early in development and are routinely used to assess movement deficits following brain lesions. We argue that curved and straight-line reaching movements are generated by a unique neural controller and that the observed curvature of the movement is the result of an active control strategy that follows the geometry of one’s body, for instance to avoid trajectories that would hit the body or yield postures close to the joint limits. We present a mathematical model that accounts for such an active control strategy and show that the model reproduces with high accuracy the kinematic features of human data during unconstrained reaching movements directed toward the head. The model consists of a nonlinear dynamical system with a single stable attractor at the target. Embodiment-related task constraints are expressed as a force field that acts on the dynamical system. Finally, we discuss the biological plausibility and neural correlates of the model’s parameters and suggest that embodiment should be considered as a main cause for movement trajectory curvature.     

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.     

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.     

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     

Credit Assignment during Movement Reinforcement Learning

We often need to learn how to move based on a single performance measure that reflects the overall success of our movements. However, movements have many properties, such as their trajectories, speeds and timing of end-points, thus the brain needs to decide which properties of movements should be improved; it needs to solve the credit assignment problem. Currently, little is known about how humans solve credit assignment problems in the context of reinforcement learning. Here we tested how human participants solve such problems during a trajectory-learning task. Without an explicitly-defined target movement, participants made hand reaches and received monetary rewards as feedback on a trial-by-trial basis. The curvature and direction of the attempted reach trajectories determined the monetary rewards received in a manner that can be manipulated experimentally. Based on the history of action-reward pairs, participants quickly solved the credit assignment problem and learned the implicit payoff function. A Bayesian credit-assignment model with built-in forgetting accurately predicts their trial-by-trial learning.     

End-point constraints in aiming movements: effects of approach angle and speed

 The present study focuses on two trajectory-formation models of point-to-point aiming movements, viz., the minimum-jerk and the minimum torque-change model. To date, few studies on minimum-jerk and minimum torque-change trajectories have incorporated self- or externally imposed end-point constraints, such as the direction and velocity with which a target area is approached. To investigate which model accounts best for the effects on movement trajectories of such – in many circumstances – realistic end-point constraints, we adjusted both the minimum-jerk and the minimum torque-change model so that they could generate trajectories of which the final part has a specific direction and speed. The adjusted models yield realistic trajectories with a high curvature near movement completion. Comparison of simulated and measured movement trajectories show that pointing movements that are constrained with respect to final movement direction and speed can be described in terms of minimization of joint-torque changes. Received: 7 July 1999 / Accepted in revised form: 8 January 2001     

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     

Hedging Your Bets: Intermediate Movements as Optimal Behavior in the Context of an Incomplete Decision

Existing theories of movement planning suggest that it takes time to select and prepare the actions required to achieve a given goal. These theories often appeal to circumstances where planning apparently goes awry. For instance, if reaction times are forced to be very low, movement trajectories are often directed between two potential targets. These intermediate movements are generally interpreted as errors of movement planning, arising either from planning being incomplete or from parallel movement plans interfering with one another. Here we present an alternative view: that intermediate movements reflect uncertainty about movement goals. We show how intermediate movements are predicted by an optimal feedback control model that incorporates an ongoing decision about movement goals. According to this view, intermediate movements reflect an exploitation of compatibility between goals. Consequently, reducing the compatibility between goals should reduce the incidence of intermediate movements. In human subjects, we varied the compatibility between potential movement goals in two distinct ways: by varying the spatial separation between targets and by introducing a virtual barrier constraining trajectories to the target and penalizing intermediate movements. In both cases we found that decreasing goal compatibility led to a decreasing incidence of intermediate movements. Our results and theory suggest a more integrated view of decision-making and movement planning in which the primary bottleneck to generating a movement is deciding upon task goals. Determining how to move to achieve a given goal is rapid and automatic.     

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