A Modeling Study of Potential Sources of Curvature in Human Reaching Movements

The authors of this article suggest that the slight but consistent posture-dependent curvature of the spatial paths in the kinematic transformation between intrinsic and extrinsic coordinates may result in a systematic curvature of movements initially planned as straight-line trajectories toward the target. A kinematic planning model is presented that takes into account the anisotropy of the intrinsic and extrinsic transformation and tends to avoid movements that require excessive joint rotations by introducing slight deviations from a straight-line trajectory. Preliminary simulations showed reasonably good agreement with experimental data, especially considering that the current model is strictly based on kinematics. A quantitative analysis showed that the strategy used in the model achieves a favorable compromise between straight-line movements and angular joint changes: By slightly increasing the spatial length of the movement (i.e., by introducing curvature), an individual can greatly reduce the total amount of joint rotation required to produce the movement.     

Multidimensional scaling in riemannian space

A metric which is a function of position is proposed for the analysis of the intrinsic geometry involved in preference or similarity judgments. Variation in the distance function or metric is characteristic of the Riemannian spaces and may be interpreted as curvature, stress or distortion in distance estimates and thus in the subjective perceptual space. It is possible to find the coefficients in the distance function at selected points by fitting a least-squares Riemannian surface to the Euclidean plane. The functional form of the distance can then be obtained by an application of the Laplace equation. Several examples are worked out for the two-dimensional solution but extension to higher spaces appears to be quite feasible.     

Eye-hand coupling during closed-loop drawing: evidence of shared motor planning?

Previous paradigms have used reaching movements to study coupling of eye-hand kinematics. In the present study, we investigated eye-hand kinematics as curved trajectories were drawn at normal speeds. Eye and hand movements were tracked as a monkey traced ellipses and circles with the hand in free space while viewing the hand's position on a computer monitor. The results demonstrate that the movement of the hand was smooth and obeyed the 2/3 power law. Eye position, however, was restricted to 2-3 clusters along the hand's trajectory and fixed approximately 80% of the time in one of these clusters. The eye remained stationary as the hand moved away from the fixation for up to 200 ms and saccaded ahead of the hand position to the next fixation along the trajectory. The movement from one fixation cluster to another consistently occurred just after the tangential hand velocity had reached a local minimum, but before the next segment of the hand's trajectory began. The next fixation point was close to an area of high curvature along the hand's trajectory even though the hand had not reached that point along the path. A visuo-motor illusion of hand movement demonstrated that the eye movement was influenced by hand movement and not simply by visual input. During the task, neural activity of pre-motor cortex (area F4) was recorded using extracellular electrodes and used to construct a population vector of the hand's trajectory. The results suggest that the saccade onset is correlated in time with maximum curvature in the population vector trajectory for the hand movement. We hypothesize that eye and arm movements may have common, or shared, information in forming their motor plans.     

The Three-Dimensional Curvature of Straight-Ahead Movements

Three-dimensional curvature of point-to-point hand movements in the forward direction was examined. Subjects (N = 4) moved their hand from a position above the start point to a forward position above targets of different size and distance. Paths were curved as a result of an initial lateral and downward movement that was compensated for in the second half of the movement. The downward component of motion had a bell-shaped velocity profile and was temporally coupled to the forward motion. Curvature was greater for movements to near targets. Examination of the relation between kinematics and geometry revealed that velocity was related to radius of curvature by a power law with an exponent of 0.59. Simulations of the component of motion in the vertical plane reproduced the qualitative behavior of curvature and fit a power law relationship between velocity and radius of curvature     

Is perceptual space inherently non-Euclidean?

It is often assumed that the space we perceive is Euclidean, although this idea has been challenged by many authors. Here we show that if spatial cues are combined as described by Maximum Likelihood Estimation, Bayesian, or equivalent models, as appears to be the case, then Euclidean geometry cannot describe our perceptual experience. Rather, our perceptual spatial structure would be better described as belonging to an arbitrarily curved Riemannian space.     

Exocentric pointing to opposite targets

We use an exocentric pointing task to study exocentric visual directions to targets that are opposite to a pointer relative to the observer. (The apparent distance between the target and the pointer always exceeded 90 degrees of visual angle.) All pointing takes place in the horizontal plane at eye height. Observers could not see both target and pointer at a single glance. They had to look back and forth between them, using combinations of eye movements, head turns, twists at the waist and turning on the feet. In the limit of diametrically opposite targets we find that the observers pick either one of two distinct orientations of the pointer as equally "visually correct". Which one results depends on the stance assumed by the observer. The difference between the two equally acceptable pointings is between 5 degrees and 10 degrees. Such a result is predicted from earlier measurements in the context of a model that describes the geometry of the horizon as a Riemannian space with varying intrinsic curvature. The present results thus fit--perhaps surprisingly--very well in such a picture.     

A new view on visuomotor channels: the case of the disappearing dynamics

A considerable body of kinematic data supports the proposal that independent visuomotor channels are involved in the control of the transport and grip components of reach and grasp. These channels are seen as having separate perceptual inputs, outputs and internal processing and are thought by some to correspond to independent neuroanatomical pathways. The idea that different groups of muscles and biomechanical structures can be controlled independently is attractive, but this kinematically-inspired hypothesis fails to take into account the complexity of the dynamic relationships and their interactions within the neuromusculoskeletal system. Inertial, viscous, centrifugal, coriolis, gravitational and reflex cross couplings exist between efferent drives to muscles and resulting body movements. Rotation at even a single joint generates a complex set of dynamic reaction forces and requires coordinated activation of many muscles throughout the body to maintain posture and balance. In this theoretical paper we present a new view of independent visuomotor channels in the form of an adaptive neural controller that can compensate for the above interactions and decouple the relationships between efferent drives to muscles and resulting body movements. At the same time, the neural controller renders all the dynamics (linear and nonlinear), other than time delays, of the neuromusculoskeletal system, unobservable in the visuomotor relationships. Using the geometry of nonlinear dynamical systems we show that, providing certain constraints on the structure of time delays within the system are satisfied, there exists a neural controller that can render all the dynamics of the neuromusculoskeletal system (except for time delays) unobservable in the responses. The controller simultaneously decouples all the interactive dynamics so that each of the m independent inputs controls one and only one degree of freedom of the response. This means that each degree of freedom in a multi-joint response can be controlled by an independent component of the visual input, a behaviour that has long been observed in visual tracking experiments. The controller effectively establishes m independent visuomotor channels. However, rather than reflecting separate neuroanatomical pathways, the independent channels result from a neural controller with convergent and divergent connections to compensate for the interactive nonlinear dynamics within the neuromusculoskeletal system. This new view of visuomotor channels has implications for neural control processes involved in the acquisition and adaptability of skilled perceptual-motor behaviour in general, as well as for the design of robotic controllers.     

Target selection during bimanual reaching to direct cues is unaffected by the perceptual similarity of the targets

Investigations of bimanual movements have shed considerable insight on the constraints underlying our ability to perform coordinated actions. One prominent limitation is evident when people are required to produce reaching movements in which the two trajectories are of different amplitudes and/or directions. This effect, however, is only obtained when the movements are cued symbolically (e.g., letters indicate target locations); these planning costs are absent when the target locations are directly cued (J. Diedrichsen, E. Hazeltine, S. Kennerley, & R. B. Ivry, 2001). The present experiments test whether the absence of planning costs under the latter condition is due to the perceptual similarity of the direct cues. The results demonstrate that measures of response planning and execution do not depend on the perceptual similarity of the direct cues. Limitations in our ability to perform distinct actions with the two hands appear to reflect interactions related to response selection involving the translation of symbolic cues into their associated movements rather than arise from interactions associated with perception, motor programming, and motor execution.     

Visual control of hand action

Recent investigations in normal and brain-damaged individuals have begun to identify the types of visual information used to plan and guide reaches. Binocular visual cues have been shown to be important for both movement planning and on-line guidance of hand movements, while emerging evidence suggests that dynamic visual analysis of the moving limb may provide a rich source of information for precise control of the hand in flight. Reaching movements appear to be planned to follow what is perceived to be a straight trajectory in peripersonal space. Furthermore, the process of selecting visual targets appears to influence hand trajectories, with hand movements curving away from non-target objects. This behaviour may be explained most effectively by a dynamic representation of space which is sculpted by attentional mechanisms into selected (target) and inhibited (non-target) regions. The role of attention in movement planning in individuals with attentional disorders is controversial. Patients with visual neglect have impairments of visuomotor control including reaches that, under certain conditions, are significantly more curved than those of normal individuals. The representations of space that neglect patients use to plan reaches may be distorted by impairments in the mechanisms that normally act to select target regions and inhibit non-target zones.     

Performance in haptic geometrical matching tasks depends on movement and position of the arms

Previous research on the properties of haptic space has shown systematic deviations from Euclidean parallelity in haptic parallelity tasks. The mainstream explanation for these deviations is that, in order to perform the task, participants generate a spatial representation with a frame of reference that integrates egocentric and allocentric components. Several studies have shown that the amount and type of deviations are affected by the configurations with regard to the arms and the rods to be matched. The present study reports 4 experiments that further address the effects of task configurations and body movements. Experiments 1 and 2 replicate and extend previous results concerning haptic matching task and acoustic pointing tasks. The third experiment includes acoustic cues aligned differentially to the reference and test bars. The fourth experiment concerns a geometrical matching task performed in the rear peripersonal space. Results show that haptic deviations from the Euclidean space are modulated by the available cues and by the body configurations. This indicates the need for further analysis on the role of body, arm and shoulder positions, and movement effects in haptic space perception.     

The effects of instructions to subjects on the programming of visually directed reaching movements

Numerous studies of human motor control have examined the effects of constraints on the programming and execution of visually directed limb movements. Only a few studies, however, have explored how the subject's objective in making the movement affects the coordinated sequence of eye and limb movements that unfolds as the subject points to or grasps an object in space. In the present study, the characteristics of the targets and the environment remained constant while the demands for speed and accuracy were varied across blocks of trials by changing the instructions to the subject. In other words, the constraints operating in the situation were kept constant, but the objective of the movement was systematically varied by changing the relative demands for speed and accuracy. All subjects were required to point to visual targets presented on a screen in front of them. Eye position was monitored by infrared reflection. The position of each subject's hand in three-dimensional space was reconstructed by a computer-assisted analysis of the images provided by two rotary-shutter video cameras. The speed and accuracy demands of the task were varied in blocks of trials by requiring the subjects to point to the target "as quickly as you can" (speed condition); "as accurately as you can" (accuracy condition); or both "quickly and accurately" (speed/accuracy condition). The time to initiate an eye movement to the target was found to be reduced by increasing either the speed or accuracy demands of the task although the time to initiate the hand movement was reduced only in the speed condition. While the duration of the acceleration phase of the reach remained constant in real time, the duration of the deceleration phase was increased with increased demands for accuracy. As expected, both variable and absolute errors were largest in the speed condition. The findings indicated that the programming of the limb movement and its coordination with the associated eye movements were affected by varying the objective of the task.     

Relative damping improves linear mass-spring models of goal-directed movements

A limitation of a simple linear mass-spring model in describing goal directed movements is that it generates rather slow movements when the parameters are kept within a realistic range. Does this imply that the control of fast movements cannot be approximated by a linear system? In servo-control theory, it has been proposed that an optimal controller should control movement velocity in addition to position. Instead of explicitly controlling the velocity, we propose to modify a simple linear mass-spring model. We replaced the damping relative to the environment (absolute damping) with damping with respect to the velocity of the equilibrium point (relative damping). This gives the limb a tendency to move as fast as the equilibrium point. We show that such extremely simple models can generate rapid single-joint movements. The resulting maximal movement velocities were almost equal to those of the equilibrium point, which provides a simple mechanism for the control of movement speed. We further show that peculiar experimental results, such as an 'N-shaped' equilibrium trajectory and the difficulties to measure damping in dynamic conditions, may result from fitting a model with absolute damping where one with relative damping would be more appropriate. Finally, we show that the model with relative damping can be used to model subtle differences between multi-joint interceptions. The model with relative damping fits the data much better than a version of the model with absolute damping.     

Seeing movement: Dancing bodies and the sensuality of place

This paper investigates how the study of dance and the visibility of the moving body can be used to rethink the conceptualisation of place. Examining the dancing body reveals an unspoken corporeal knowledge that subsists in our engagement with the everyday world, and exceeds our ways of talking or writing about dance as an art form. The body inscribes the space it moves across, leaving trajectories and choreographic lines and, due to its tangibility, also comes to corporeally fill or occupy a place. It is about understanding the movement of bodies in terms of the physical expression of the performer but also from the perspective of an audience, which serves as a third person perspective on corporeal expression. When watching a body move, the observer becomes attentive to the gestures, rhythm, intentionality, and the crowd of virtual movements that accompany it. Through the work of dance theorists such as Rudolf Laban, we examine how the dancing body oscillates between the intentional and the observed in the constitution of the sensual dimensions of space. Most importantly we explore how dance shifts the point of focus away from place as a locale to place as a site of corporeal witnessing.     

A quantitative evaluation of the AVITEWRITE model of handwriting learning

Much sensory-motor behavior develops through imitation, as during the learning of handwriting by children. Such complex sequential acts are broken down into distinct motor control synergies, or muscle groups, whose activities overlap in time to generate continuous, curved movements that obey an inverse relation between curvature and speed. The adaptive vector integration to endpoint handwriting (AVITEWRITE) model of Grossberg and Paine (2000) [A neural model of corticocerebellar interactions during attentive imitation and predictive learning of sequential handwriting movements. Neural Networks, 13, 999-1046] addressed how such complex movements may be learned through attentive imitation. The model suggested how parietal and motor cortical mechanisms, such as difference vector encoding, interact with adaptively-timed, predictive cerebellar learning during movement imitation and predictive performance. Key psychophysical and neural data about learning to make curved movements were simulated, including a decrease in writing time as learning progresses; generation of unimodal, bell-shaped velocity profiles for each movement synergy; size scaling with isochrony, and speed scaling with preservation of the letter shape and the shapes of the velocity profiles; an inverse relation between curvature and tangential velocity; and a two-thirds power law relation between angular velocity and curvature. However, the model learned from letter trajectories of only one subject, and only qualitative kinematic comparisons were made with previously published human data. The present work describes a quantitative test of AVITEWRITE through direct comparison of a corpus of human handwriting data with the model's performance when it learns by tracing the human trajectories. The results show that model performance was variable across the subjects, with an average correlation between the model and human data of 0.89+/-0.10. The present data from simulations using the AVITEWRITE model highlight some of its strengths while focusing attention on areas, such as novel shape learning in children, where all models of handwriting and the learning of other complex sensory-motor skills would benefit from further research.     

Movement trajectory smoothness is not associated with the endpoint accuracy of rapid multi-joint arm movements in young and older adults

The minimum variance theory proposes that motor commands are corrupted by signal-dependent noise and smooth trajectories with low noise levels are selected to minimize endpoint error and endpoint variability. The purpose of the study was to determine the contribution of trajectory smoothness to the endpoint accuracy and endpoint variability of rapid multi-joint arm movements. Young and older adults performed arm movements (4 blocks of 25 trials) as fast and as accurately as possible to a target with the right (dominant) arm. Endpoint accuracy and endpoint variability along with trajectory smoothness and error were quantified for each block of trials. Endpoint error and endpoint variance were greater in older adults compared with young adults, but decreased at a similar rate with practice for the two age groups. The greater endpoint error and endpoint variance exhibited by older adults were primarily due to impairments in movement extent control and not movement direction control. The normalized jerk was similar for the two age groups, but was not strongly associated with endpoint error or endpoint variance for either group. However, endpoint variance was strongly associated with endpoint error for both the young and older adults. Finally, trajectory error was similar for both groups and was weakly associated with endpoint error for the older adults. The findings are not consistent with the predictions of the minimum variance theory, but support and extend previous observations that movement trajectories and endpoints are planned independently.     

The influence of advance information about target location and visual feedback on movement planning and execution.

This study was designed to determine if movement planning strategies incorporating the use of visual feedback during manual aiming are specific to individual movements. Advance information about target location and visual context was manipulated using precues. Participants exhibited a shorter reaction time and a longer movement time when they were certain of the target location and that vision would be available. The longer movement time was associated with greater time after peak velocity. Under conditions of uncertainty, participants prepared for the worst-case scenario. That is, they spent more time organizing their movements and produced trajectories that would be expected from greater open-loop control. Our results are consistent with hierarchical movement planning in which knowledge of the movement goal is an essential ingredient of visual feedback utilization.     

The effect of abnormal displacement of the retinal image during eye movements

The displacement of the images on the retina that results from a turning of the eye does not lead to an apparent motion of what is seen, It has been generally assumed that this is due to a compensating process which takes eye movement into account and serves to discount those image displacements that result from eye movements, It follows from this view that an abnormal image displacement, that is, an image displacement that is larger or smaller than the causing eye movement would warrant, should lead to an experienced displacement of the target. Abnormal image displacement was produced by placing the eye in the converging or diverging bundle of rays from a point source that form behind a strong positive lens; this arrangement yielded a disc-shaped image, the projection of the pupil onto the retina, which displaced abnormally during eye movements. By changing the position of the eye along the axis of the lens in relation to the crossing point of the bundle, the degree to which the displacement was abnormal could be varied, For various displacement rates ranging from 25% to 120 and 400% of normal, abnormal displacements produced by incidental eye movements remained unnoticed, Only where eye movements were intentional did some of our Ss report shifts of the perceived image. It is suggested that the organism copes with the image displacement resulting from the ever-present incidental eye movements not by compensation but by ignoring them.     

The impact of perceptual, cognitive and motor factors on bimanual coordination

Bimanual coordination is governed by constraints that permit congruent movements to be performed more easily than incongruent movements. Theories concerning the origin of these constraints range from low level motor-muscle explanations to high level perceptual–cognitive ones. To elucidate the processes underlying coordinative constraints, we asked subjects to use a pair of left–right joysticks to acquire corresponding pairs of congruent and incongruent targets presented on a video monitor under task conditions designed to systematically modulate the impact of several perceptual–cognitive processes commonly required for bimanual task performance. These processes included decoding symbolic cues, detecting goal targets, conceptualizing movements in terms of goal target configuration, planning movement trajectories, producing saccades and perceiving visual feedback. Results demonstrate that constraints arise from target detection and trajectory planning processes that can occur prior to movement initiation as well as from inherent muscle properties that emerge during movement execution, and that the manifestation of these constraints can be significantly altered by the ability to visually monitor movement progress.     

Locomotor body scheme

The concept of body schema has been introduced and widely discussed in the literature to explain various clinical observations and distortions in the body and space representation. Here we address the role of body schema related information in multi-joint limb motion. The processing of proprioceptive information may differ significantly in static and dynamic conditions since in the latter case the control system may employ specific dynamic rules and constraints. Accordingly, the perception of movement, e.g., estimation of step length and walking distance, may rely on a priori knowledge about intrinsic dynamics of limb segment motion and inherent relationships between gait parameters and body proportions. The findings are discussed in the general framework of space and body movement representation and suggest the existence of a dynamic locomotor body schema used for controlling step length and path estimation.     

Fitts' Law in two dimensions with hand and head movements

Subjects performed two-dimensional discrete movements either with a helmet-mounted sight or with a joystick. Fitts' Law was found to be a good predictor of the speed-accuracy tradeoff for both systems. The joystick produced faster movement times than the helmet-mounted sight. For both systems, horizontal and vertical movements were slightly faster than diagonal movements. Two dimensional generalizations of Fitts' Law were discussed in terms of multidimensional scaling. The obtained pattern of movement times was found to be intermediate to the predictions of Euclidean and City-block models of the movement space. Muscle coordination strategies were considered, and a strictly serial coordination model was rejected. A strictly parallel model was also rejected for the helmet-mounted sight, but not for the joystick.