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     

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.     

Two-arm trajectory planning in a manipulation task

This paper addresses the trajectory planning problem for a task which requires positioning and orienting an object firmly grasped by two hands at a visually specified goal configuration in the horizontal plane. The motor task involves three degrees of freedom (two translational and one rotational), and the motions of the arms are constrained by the physical coupling through the held object. Experimentally measured trajectories of two arms in the coordinated positioning/orienting task are presented. The hypothesis that the rotational and translational components of motions are decoupled and independently planned is tested. Two explicit mathematical models to account for the kinematic features of the two-arm motions are formulated, and the predictions of the models are compared with the experimental data. Both models extend the minimum-jerk model to the two-arm coordinated motions case. The trajectories predicted by the models were found to be in qualitative agreement with the experimental data. However, neither model could account for the observed configuration dependence of the motions, nor for some of the properties of the measured velocity components of the motions. Our findings support the idea that the rotational and translational components of two-arm motions in the positioning/orienting task are independently planned in extra-personal space, and are further combined in a hierarchical fashion to produce the observed motions. The tested models may serve as a basis for further investigations of issues pertinent to the generation of two-arm trajectories. Received: 27 March 1995 / Accepted in revised form: 17 September 1996     

Invariant characteristics of horizontal-plane minimum-torque-change movements with one mechanical degree of freedom

 Uno et al. (1989) suggested that movements are organized such that the squared change of torque is minimized over time. Although influential, this theory has attracted much less attention from experimental researchers than the competing minimum-jerk model (Flash and Hogan 1985). One reason for this relative neglect has been the lack of general quantitative predictions, which results from the belief that minimum-torque-change trajectories have to be computed numerically for individual movements and arm-dynamical parameters. In the present paper, we show that for an important special case, that of planar horizontal movements with one mechanical degree of freedom (DOF), it is actually possible to find an analytic expression for the predicted minimum-torque-change trajectories. Based on this mathematical result, we derive a set of properties which are characteristic of these trajectories and compare them to experimental data which have not previously been related to the minimum-torque-change model. Certain discrepancies between these experimental data and minimum-torque-change model predictions are revealed. Received: 3 September 1996 / Accepted in revised form: 7 January 1997     

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.     

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.     

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     

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.     

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.     

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

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

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.     

An equilibrium-point model of electromyographic patterns during single-joint movements based on experimentally reconstructed control signals

The purpose of this work has been to develop a model of electromyographic (EMG) patterns during single-joint movements based on a version of the equilibrium-point hypothesis, a method for experimental reconstruction of the joint compliant characteristics, the dual-strategy hypothesis, and a kinematic model of movement trajectory. EMG patterns are considered emergent properties of hypothetical control patterns that are equally affected by the control signals and peripheral feedback reflecting actual movement trajectory. A computer model generated the EMG patterns based on simulated movement kinematics and hypothetical control signals derived from the reconstructed joint compliant characteristics. The model predictions have been compared to published recordings of movement kinematics and EMG patterns in a variety of movement conditions, including movements over different distances, at different speeds, against different-known inertial loads, and in conditions of possible unexpected decrease in the inertial load. Changes in task parameters within the model led to simulated EMG patterns qualitatively similar to the experimentally recorded EMG patterns. The model's predictive power compares it favourably to the existing models of the EMG patterns.     

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     

Knee medial and lateral contact forces in a musculoskeletal model with subject-specific contact point trajectories

Contact point (CP) trajectory is a crucial parameter in estimating medial/lateral tibio-femoral contact forces from the musculoskeletal (MSK) models. The objective of the present study was to develop a method to incorporate the subject-specific CP trajectories into the MSK model. Ten healthy subjects performed 45 s treadmill gait trials. The subject-specific CP trajectories were constructed on the tibia and femur as a function of extension-flexion using low-dose bi-plane X-ray images during a quasi-static squat. At each extension-flexion position, the tibia and femur CPs were superimposed in the three directions on the medial side, and in the anterior-posterior and proximal-distal directions on the lateral side to form the five kinematic constraints of the knee joint. The Lagrange multipliers associated to these constraints directly yielded the medial/lateral contact forces. The results from the personalized CP trajectory model were compared against the linear CP trajectory and sphere-on-plane CP trajectory models which were adapted from the commonly used MSK models. Changing the CP trajectory had a remarkable impact on the knee kinematics and changed the medial and lateral contact forces by 1.03 BW and 0.65 BW respectively, in certain subjects. The direction and magnitude of the medial/lateral contact force were highly variable among the subjects and the medial-lateral shift of the CPs alone could not determine the increase/decrease pattern of the contact forces. The suggested kinematic constraints are adaptable to the CP trajectories derived from a variety of joint models and those experimentally measured from the 3D imaging techniques.     

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.     

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