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     

Dynamic optimization of the sit-to-stand movement

The purpose of this study was to clarify criteria that can predict trajectories during the sit-to-stand movement. In particular, the minimum jerk and minimum torque-change models were examined. Three patterns of sit-to-stand movement from a chair, i.e., upright, natural, and leaning forward, were measured in five young participants using a 3-D motion analysis device (200 Hz). The trajectory of the center of mass and its smoothness were examined, and the optimal trajectories predicted by both models were evaluated. Trajectories of the center of mass predicted by the minimum torque-change model, rather than the minimum jerk model, resembled the measured movements in all rising movement patterns. The upright pattern required greater extension torque of the knee and ankle joints at the instant of seat-off. The leaning-forward pattern required greater extension hip torque and higher movement cost than the natural and upright patterns. These results indicate that the natural sit-to-stand movement might be a result of dynamic optimization.     

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     

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     

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.     

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     

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.     

What is “fallback”?: metrics needed to assess telemetry tag effects on anadromous fish behavior

Telemetry has allowed researchers to document the upstream migrations of anadromous fish in freshwater. In many anadromous alosine telemetry studies, researchers use downstream movements (“fallback”) as a behavioral field bioassay for adverse tag effects. However, these downstream movements have not been uniformly reported or interpreted. We quantified movement trajectories of radio-tagged anadromous alewives (Alosa pseudoharengus) in the Ipswich River, Massachusetts (USA) and tested blood chemistry of tagged and untagged fish held 24 h. A diverse repertoire of movements was observed, which could be quantified using (a) direction of initial movements, (b) timing, and (c) characteristics of bouts of coupled upstream and downstream movements (e.g., direction, distance, duration, and speed). Because downstream movements of individual fish were almost always made in combination with upstream movements, these should be examined together. Several of the movement patterns described here could fall under the traditional definition of “fallback” but were not necessarily aberrant. Because superficially similar movements could have quite different interpretations, post-tagging trajectories need more precise definitions. The set of metrics we propose here will help quantify tag effects in the field, and provide the basis for a conceptual framework that helps define the complicated behaviors seen in telemetry studies on alewives and other fish in the field.     

Balance control during an arm raising movement in bipedal stance: which biomechanical factor is controlled?

In order to obtain new insight into the control of balance during arm raising movements in bipedal stance, we performed a biomechanical analysis of kinematics and dynamical aspects of arm raising movements by combining experimental work, large-scale models of the body, and techniques simulating human behavior. A comparison between experimental and simulated joint kinematics showed that the minimum torque change model yielded realistic trajectories. We then performed an analysis based on computer simulations. Since keeping the center of pressure (CoP) and the projection of the center of mass (CoM) inside the support area is essential for equilibrium, we modeled an arm raising movement where displacement of one or the other variable is limited. For this optimization model, the effects of adding equilibrium constraints on movement trajectories were investigated. The results show that: (a) the choice of the regulated variable influences the strategy adopted by the system and (b) the system was not able to regulate the CoM for very fast movements without compromising its balance. Consequently, we suggest that the system is able to maintain balance while raising the arm by only controlling the CoP. This may be done mainly by using hip mechanisms and controlling net ankle torque.     

The Main Sequence of Saccades Optimizes Speed-accuracy Trade-off

In primates, it is well known that there is a consistent relationship between the duration, peak velocity and amplitude of saccadic eye movements, known as the ‘main sequence’. The reason why such a stereotyped relationship evolved is unknown. We propose that a fundamental constraint on the deployment of foveal vision lies in the motor system that is perturbed by signal-dependent noise (proportional noise) on the motor command. This noise imposes a compromise between the speed and accuracy of an eye movement. We propose that saccade trajectories have evolved to optimize a trade-off between the accuracy and duration of the movement. Taking a semi-analytical approach we use Pontryagin’s minimum principle to show that there is an optimal trajectory for a given amplitude and duration; and that there is an optimal duration for a given amplitude. It follows that the peak velocity is also fixed for a given amplitude. These predictions are in good agreement with observed saccade trajectories and the main sequence. Moreover, this model predicts a small saccadic dead-zone in which it is better to stay eccentric of target than make a saccade onto target. We conclude that the main sequence has evolved as a strategy to optimize the trade-off between accuracy and speed.     

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     

Object Grasping using the Minimum Variance Model

Reaching-to-grasp has generally been classified as the coordination of two separate visuomotor processes: transporting the hand to the target object and performing the grip. An alternative view has recently been formed that grasping can be explained as pointing movements performed by the digits of the hand to target positions on the object. We have previously implemented the minimum variance model of human movement as an optimal control scheme suitable for control of a robot arm reaching to a target. Here, we extend that scheme to perform grasping movements with a hand and arm model. Since the minimum variance model requires that signal-dependent noise be present on the motor commands to the actuators of the movement, our approach is to plan the reach and the grasp separately, in line with the classical view, but using the same computational model for pointing, in line with the alternative view. We show that our model successfully captures some of the key characteristics of human grasping movements, including the observations that maximum grip size increases with object size (with a slope of approximately 0.8) and that this maximum grip occurs at 60–80% of the movement time. We then use our model to analyse contributions to the digit end-point variance from the two components of the grasp (the transport and the grip). We also briefly discuss further areas of investigation that are prompted by our model.     

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.     

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     

Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

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.     

Adjustment of the human arm viscoelastic properties to the direction of reaching

The viscoelastic properties of the human arm were measured by means of short force perturbations during fast reaching movements in two orthogonal directions. A linear spring model with time delay described the neuromuscular system of the human arm. The obtained viscoelastic parameters ensured movement stability in spite of the time delay of 50 ms. The stiffness and viscosity ellipses appeared to be predominantly orthogonal to the movement direction, which reduced the effect of force perturbation in the direction orthogonal to the reaching movement. Thus, it can be argued that the viscoelastic properties of the neuromuscular system of the human arm are adjusted to the direction of movement according to a “path preserving” strategy, which minimizes the deviation of the movement path from a straight line, when exposed to an unexpected external force.     

Prediction of a moving target's position in fast goal-directed action

 Subjects made fast goal-directed arm movements towards moving targets. In some cases, the perceived direction of target motion was manipulated by moving the background. By comparing the trajectories towards moving targets with those towards static targets, we determined the position towards which subjects were aiming at movement onset. We showed that this position was an extrapolation in the target’s perceived direction from its position at that moment using its perceived direction of motion. If subjects were to continue to extrapolate in the perceived direction of target motion from the position at which they perceive the target at each instant, the error would decrease during the movements. By analysing the differences between subjects’ arm movements towards targets moving in different (apparent) directions with a linear second-order model, we show that the reduction in the error that this predicts is not enough to explain how subjects compensate for their initial misjudgements. Received: 10 February 1995/Accepted in revised form: 30 May 1995     

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