An algorithm for the generation of curvilinear wrist motion in an arbitrary plane in three-dimensional space

Human arm redundancy: a new approach for the inverse kinematics problem

The inverse kinematics (IK) problem addresses how both humans and robotic systems coordinate movement to resolve redundancy, as in the case of arm reaching where more degrees of freedom are available at the joint versus hand level. This work focuses on which coordinate frames best represent human movements, enabling the motor system to solve the IK problem in the presence of kinematic redundancies. We used a multi-dimensional sparse source separation method to derive sets of basis (or source) functions for both the task and joint spaces, with joint space represented by either absolute or anatomical joint angles. We assessed the similarities between joint and task sources in each of these joint representations, finding that the time-dependent profiles of the absolute reference frame's sources show greater similarity to corresponding sources in the task space. This result was found to be statistically significant. Our analysis suggests that the nervous system represents multi-joint arm movements using a limited number of basis functions, allowing for simple transformations between task and joint spaces. Additionally, joint space seems to be represented in an absolute reference frame to simplify the IK transformations, given redundancies. Further studies will assess this finding's generalizability and implications for neural control of movement.

Modulation of ellipses drawing by sonification

Most studies on the regulation of speed and trajectory during ellipse drawing have used visual feedback. We used online auditory feedback (sonification) to induce implicit movement changes independently from vision. The sound was produced by filtering a pink noise with a band-pass filter proportional to movement speed. The first experiment was performed in 2D. Healthy participants were asked to repetitively draw ellipses during 45 s trials whilst maintaining a constant sonification pattern (involving pitch variations during the cycle). Perturbations were produced by modifying the slope of the mapping without informing the participants. All participants adapted spontaneously their speed: they went faster if the slope decreased and slower if it increased. Higher velocities were achieved by increasing both the frequency of the movements and the perimeter of the ellipses, but slower velocities were achieved mainly by decreasing the perimeter of the ellipses. The shape and the orientation of the ellipses were not significantly altered. The analysis of the speed-curvature power law parameters showed consistent modulations of the speed gain factor, while the exponent remained stable. The second experiment was performed in 3D and showed similar results, except that the main orientation of the ellipse also varied with the changes in speed. In conclusion, this study demonstrated implicit modulation of movement speed by sonification and robust stability of the ellipse geometry. Participants appeared to limit the decrease in movement frequency during slowing down to maintain a rhythmic and not discrete motor regimen.

The speed-curvature power law of movements: a reappraisal

Several types of curvilinear movements obey approximately the so called 2/3 power law, according to which the angular speed varies proportionally to the 2/3 power of the curvature. The origin of the law is debated but it is generally thought to depend on physiological mechanisms. However, a recent paper (Marken and Shaffer, Exp Brain Res 88:685-690, 2017) claims that this power law is simply a statistical artifact, being a mathematical consequence of the way speed and curvature are calculated. Here we reject this hypothesis by showing that the speed-curvature power law of biological movements is non-trivial. First, we confirm that the power exponent varies with the shape of human drawing movements and with environmental factors. Second, we report experimental data from Drosophila larvae demonstrating that the power law does not depend on how curvature is calculated. Third, we prove that the law can be violated by means of several mathematical and physical examples. Finally, we discuss biological constraints that may underlie speed-curvature power laws discovered in empirical studies.

Drawing ellipses in water: evidence for dynamic constraints in the relation between velocity and path curvature

Several types of continuous human movements comply with the so-called Two-Thirds Power Law (2/3-PL) stating that velocity (V) is a power function of the radius of curvature (R) of the endpoint trajectory. The origin of the 2/3-PL has been the object of much debate. An experiment investigated further this issue by comparing two-dimensional drawing movements performed in air and water. In both conditions, participants traced continuously quasi-elliptic trajectories (period T = 1.5 s). Other experimental factors were the movement plane (horizontal/vertical), and whether the movement was performed free-hand, or by following the edge of a template. In all cases a power function provided a good approximation to the V-R relation. The main result was that the exponent of the power function in water was significantly smaller than in air. This appears incompatible with the idea that the power relationship depends only on kinematic constraints and suggests a significant contribution of dynamic factors. We argue that a satisfactory explanation of the observed behavior must take into account the interplay between the properties of the central motor commands and the visco-elastic nature of the mechanical plant that implements the commands.

Early stages in a sensorimotor transformation

We present a model for several early stages of the sensorimotor transformations involved in targeted arm movement. In psychophysical experiments, human subjects pointed to the remembered locations of randomly placed targets in three-dimensional space. They made consistent errors in distance, and from these errors stages in the sensorimotor transformation were deduced. When subjects attempted to move the right index finger to a virtual target they consistently undershot the distance of the more distal targets. Other experiments indicated that the error was in the sensorimotor transformation rather than in the perception of distance. The error was most consistent when evaluated using a spherical coordinate system based at the right shoulder, indicating that the neural representation of target parameters is transformed from a retinocentric representation to a shoulder-centered representation. According to the model, the error in distance results from the neural implementation of a linear approximation in the algorithm to transform shoulder-centered target parameters into a set of arm orientations appropriate for placing the finger on the target. The transformation to final arm orientations places visually derived information into a frame of reference where it can readily be combined with kinesthetically derived information about initial arm orientations. The combination of these representations of initial and final arm orientations could give rise to the representation of movement direction recorded in the motor cortex by Georgopoulos and his colleagues. Later stages, such as the transformation from kinematic (position) to dynamic (force) parameters, or to levels of muscle activation, are beyond the scope of the present model.

What does body configuration in microgravity tell us about the contribution of intra- and extrapersonal frames of reference for motor control?

The authors report that the reorganization of body configuration during weightlessness is based on an intrapersonal frame of reference such as the configuration of the support surface and the position of the body's center of gravity. These results stress the importance of “knowledge” of the state of internal geometric structures, which cannot be directly signalled by specific receptors responsible for direct dialogue with the physical external world.

Reciprocal and coactivation commands are not sufficient to describe muscle activation patterns

Recent results have shown that the relative activation of muscles is different for isometric contractions and for movements. These results exclude an explanation of muscle activation patterns by a combination ofreciprocal and coactivation commands. These results also indicate that joint stiffness is not uniquely determined and that it may be different for isometric contractions and movements.

Frames of reference interact and are task-dependent

The problem for the CNS in any particular movement task is to coordinate the various frames of reference appropriate to the task. Control variables are determined by this coordination. The coordination problem varies greatly from task to task, and so no single set of control variables is likely to account for a broad range of movement tasks.

The λ model for motor control: More than meets the eye

Understanding of the λ model has greatly increased in recent years as evidenced by most of the commentaries. Some commentators underscored the potential of the model to integrate aspects of different sensorimotor systems in the production of movement. Other commentators focused on not-yet-fully-developed parts of the model. A few persisted in misunderstanding some of its basic concepts, and on these grounds they reject it. In responding to commentaries we continue to elaborate on some fundamental points of the model, especially control variables, the idea of movement production by shifting the positional frame of reference and the hypothesis of biomechanical correspondence in motor control. We also continue to develop our ideas on the intrinsic generation of the frame of reference associated with external space and utilized for the control of arm movement and locomotion. The dynamic principles underlying the model are discussed in light of the dynamical systems approach.

Grip force adjustments during rapid hand movements suggest that detailed movement kinematics are predicted

The λ model suggests that detailed kinematics arise from changes in control variables and need not be explicitly planned. However, we have shown that when moving a grasped object, grip force is precisely modulated in phase with acceleration-dependent inertial load. This suggests that the motor system can predict detailed kinematics. This prediction may be based on a forward model of the dynamics of the loaded limb.

Let us accept a “controlled trade-off” model of motor control

The trade-off between force and length of muscle as adjusted by neural signals is a critical fact in the dynamics of motor control. Whether we call it “length-tension effect,” “feedback-like,” “invariant condition,” or “spring-like” is unimportant. We must not let semantics or details of representation obscure the basic physics of effects introduced by this trade-off in muscle.

Equilibrium-point control? Yes! Deterministic mechanisms of control? No!

The equilibrium-point hypothesis (the λ-model) is superior to all other models of single-joint control and provides deep insights into the mechanisms of control of multi-joint movements. Attempts at associating control variables with neurophysiological variables look confusing rather than promising. Probabilistic mechanisms may play an important role in movement generation in redundant systems.

The unobservability of central commands: Why testing hypotheses is so difficult

The experiments Feldman and Levin suggest do not definitively test their proposed solution to the problem of selecting muscle activations. Their test of the movement directions that elicit EMG activity can be interpreted without regard to the form of the central commands, and their fast elbow flexion test is based on a forward computation that obscures the insensitivity of the predicted trajectory to the details of the putative commands.

Biological variability and control of movements via δλ

Three issues related to Feldman and Levin's treatment of biological variability are discussed. We question the usefulness of the indirect component of δλ. We suggest that trade-offs between speed and accuracy in aimed movements support identification of δλ, rather than λ, as a control variable. We take issue with the authors' proposal for resolving redundancy in multi-joint movements, given recent data.

A few reasons why psychologlsts can adhere to Feldman and Levin's model

We emphasize the relevance to cognitive psychology of Feldman and Levin's theoretical position. Traditional views of motor control have failed to clearly separate “production control” at the level of motor command, based on task-independent CV (control variables), from intentional “product control” based on task-dependent parameters. Because F&L's approach concentrates on the first process (trajectory formation), it can distinguish the product control stage.

Levers to generate movement

The following questions are discussed: (1) Who determines the nature of “control variables”? (2) Is the “positional monopoly” healthy? (3) Does a descending command alter reflex threshold alone without eoncomitantly altering stiffness? (4) How does the CNS deal with history-dependent effects? (5) Should we abandon the idea that the CNS controls classical Newtonian variables such as muscle length?

Kinematic invariances and body schema

Generalizing the notion that muscles are positional frames of reference, a high-dimensional muscle space is defined for multi-muscle systems with an embedded low-dimensional motor manifold of functional articulators. A central representation of such a manifold is proposed as computational body schema. The example of the jaw-tongue system is presented, discussing the relation of functional articulators with kinematic invariances and control problems.

Tendon elasticity and positional control

The spring-like behaviour of a joint following a sudden change of torque is partly a result of the elastic properties of tendons. A large fall in a muscle with a long tendon may be accompanied by tendon recoil causing joint movements as large as 20°.

Command invariants and the frame of reference for human movement

We describe a solution to the redundancy problem related to that proposed in Feldman & Levin's target article. We suggest that the system may use a fixed mapping between commands organized at the level of degrees of freedom and commands to individual muscles. This proposal eliminates the need to maintain an explicit representation of musculoskeletalgeometry in planning movements.

Frameworks on shifting sands

Feldman and Levin present a model for movement control in which the system is said to seek equilibrium points, active movement being produced by shifting frames of reference in space. It is argued that whatever merit this model might have is limited to an understanding of “the how” and not “the why” we move. In this way the authors seem to be forced into a dualistic position leaving the upper level of the proposed control hierarchy “floating.”

Conservative or nonconservative control schemes

The conservative strategy proposed by the authors suggests a solution of the degrees-of-freedom problem of the controller. However, several simple motor control tasks cannot be explained by this strategy. A nonconservative strategy, in which more parameters of the control signal vary, can account for these simple motor tasks. However, the simplicity that distinguishes the proposed model from many others is lost.

The origin and use of positional frames of reference in motor control

A hypothesis about sensorimotor integration (the λ model) is described and applied to movement control and kinesthesia. The central idea is that the nervous system organizes positional frames of reference for the sensorimotor apparatus and produces active movements by shifting the frames in terms of spatial coordinates. Kinematic and electromyographic patterns are not programmed, but emerge from the dynamic interaction among the system s components, including external forces within the designated frame of reference. Motoneuronal threshold properties and proprioceptive inputs to motoneurons may be cardinal components of the physiological mechanism that produces positional frames of reference. The hypothesis that intentional movements are produced by shifting the frame of reference is extended to multi-muscle and multi-degrees-of-freedom systems with a solution of the redundancy problem that allows the control of a joint alone or in combination with other joints to produce any desired limb configuration and movement trajectory. The model also implies that for each motor behavior, the nervous system uses a strategy that minimizes the number of changeable control variables and keeps the parameters of these changes invariant. Examples are provided of simulated kinematic and electromyographic signals from single- and multi-joint arm movements produced by suggested patterns of control variables. Empirical support is provided and additional tests of the model are suggested. The model is contrasted with others based on the ideas of programming of motoneuronal activity, muscle forces, stiffness, or movement kinematics.

Origins of origins of motor control

Examination of infant spontaneous and goal-directed arm movements supports Feldman and Levin's hypothesis of a functional hierarchy. Early infant movements are dominated by biomechanical and dynamic factors without external frames of reference. Development involves not only learning to generate these frames of reference, but also protecting the higher-level goal of the movement from internal and external perturbations.

Spatial frames for motor control would be commensurate with spatial frames for vision and proprioception, but what about control of energy flows?

The model identifies a spatial coordinate frame within which the sensorimotor apparatus produces movement. Its spatial nature simplifies its coupling with spatial reference frames used concurrently by vision and proprioception. While the positional reference frame addresses the performance of spatial tasks, it seems to have little to say about movements involving energy expenditure as the principle component of the task.

Interneurons as backseat drivers and the elusive control variable

It is proposed here that the spinal network of proprioceptive feedback from length and force receptors constitutes the mechanism underlying the coordination of activation thresholds for muscles acting about the same and neighboring joints. For the most part, these circuits come between motoneurons and supraspinal signals, invalidating the idea that the activation thresholds constitute control variables for the motor system.

Two joints are more than twice one joint

An alternative multi-joint extension to the lambda model is proposed. According to this extension, the activity of a muscle depends not only on the difference between lambda and length of that muscle, but also on the difference between lambda and length of other muscles. This 2-D extension can describe more neurophysiological experiments than the extension proposed in the target article.

Control parameters, equilibria, and coordination dynamics

Important similarities exist between the dynamical concepts implicit in Feldman & Levin's extended λ model and those basic to a dynamical systems approach. We argue that careful application of the key concepts of control and order parameters, equilibria, and stability, can relate known facts of neuromuscular processes to the observables of functional, task-specific behavior.

Shifting frames of reference but the same old point of view

Models of central control variables (CVs) that are expressed in positional reference frames and rely on proprioception as the dominant specifier of muscle activation patterns have not yet been shown to be adequate for the description of fast, voluntary movement, even of single joints. An alternative model with illustrative data is proposed.

Inverse kinematic problem: Solutions by pseudoinversion, inversion and no-inversion

Kinematic properties of reaching movements reflect constraints imposed on the joint angles. Contemporary models present solutions to the redundancy problem by a pseudoinverse procedure (Whitney 1969) or without any inversion (Berkenblit et al. 1986). Feldman & Levin suggest a procedure based on a regular inversion. These procedures are considered as an outcome of a more general approach.

Natural unconstrained movements obey rules different from constrained elementary movements

The concept of a conservative control strategy minimizing the number of degrees of freedom used is criticised with reference to 3-D simple reaching and grasping experiments. The vector error in a redundant system would not be the prime controlled variable, but rather the posture for reaching, as exemplified by nearly straight displacements in joint space as opposed to curved ones in task space.

Is λ an appropriate control variable for locomotion?

The lambda model predicts that the command received by each motor nucleus during locomotion is specific for the joint at which its muscle acts and is independent of external conditions. However, investigation of the commands received by motor nuclei during fictive locomotion and of the sensitivity of these commands to feedback from the limb during locomotion indicates that neither condition is satisfied.

How far should we extend the equilibrium point (lambda) hypothesis?

A key feature of the lambda model is the hypothesis of a local spring-like muscle-reflex system defined by a central control variable that has units of position. This is intriguing, especially for a study of postural stability in large-scale systems, but it has limited direct application to skilled everyday movements. If movement is considered as a goal-directed, neuro-optimization problem, however, theavailabilityof lambda-like peripheral models (vs. conventional musculoskeletal models) deserves exploration.

Hypothesis regarding the transformation of the intended direction of movement during the production of graphic trajectories: a study of drawing movements in 8- to 12-year-old children

Children from 8 to 12 years of age drew figure-eights and ellipses at a self-chosen tempo on a digitizing tablet. Global aspects (perimeter and average speed) and local aspects (relation between instantaneous speed and curvature) of performance were analyzed across age groups and types of figures. We tested the predictions of the transformation model, which is based on the hypothesis that changing the intended direction of movement is a time-consuming process that affects the evolution in time of the movement trajectory, and compared how well it fitted the data relative to the power law. We found that the relation between speed and curvature was typically better described by the transformation model than by the power law. However, the power law provided a better description when ellipses were drawn at a fast speed. The analyses of the parameters of the transformation model indicate that processing speed increased linearly with age. In addition, the results suggest that the effects of the spring-like properties of the arm were noticeable when ellipses were drawn at a fast speed. This study indicates that both biomechanical properties and central processes have an effect on the kinematics of continuous movements and particularly on the relation between speed and curvature. However, their relative importance varies with the type of figure and average movement speed. In conclusion, the results support the hypothesis that a time-consuming process of transformation of the intended direction of movement is operating during the production of continuous movements and that this process increases in speed between 8 to 12 years of age.

Kinematic evaluation of gestural and repetitive single joint movements of the arm after posteroventral pallidotomy, subthalamotomy or both procedures combined in two Parkinson's disease patients: two case studies

Two patients with severe Parkinson's disease undergoing partial or complete ablative interruption of basal ganglia (BG) output are presented. One patient who underwent bilateral subthalamotomy, and a second who underwent unilateral posteroventral pallidotomy, followed 7 years later by a bilateral subthalamotomy because of contralateral disease progression, were studied. In addition to the usual clinical evaluation, changes in joint kinematics observed during unconstrained, skilled multi-joint movement and repetitive single joint (RSJ) movement of the wrist were studied. Clinical UPDRS items referred to hand movements contralateral to the procedure, and instrumental measurement of RSJ improved in both patients after either pallidotomy or subthalamotomy. When both BG outflow paths were interrupted as was the case in the second patient (bilateral subthalamotomy after the initial pallidotomy), no added clinical improvement was observed, RSJ even deteriorated slightly. Instrument-based studies for movement alteration detection after simultaneous ablation of the globus pallidus and the subthalamic nucleus of these two patients showed greater sensitivity than clinical evaluation alone. Complex gestural movement performance remained unaffected after partial (subthalamotomy or pallidotomy) or complete interruption of BG outflow (case 2), indicating BG compensatory capacity after total outflow interruption remained intact.

Target interception: hand-eye coordination and strategies

This study was designed to define the characteristics of eye-hand coordination in a task requiring the interception of a moving target. It also assessed the extent to which the motion of the target was predicted and the strategies subjects used to determine when to initiate target interception. Target trajectories were constructed from sums of sines in the horizontal and vertical dimensions. Subjects intercepted these trajectories by moving their index finger along the surface of a display monitor. They were free to initiate the interception at any time, and on successful interception, the target disappeared. Although they were not explicitly instructed to do so, subjects tracked target motion with normal, high-gain smooth-pursuit eye movements right up until the target was intercepted. However, the probability of catch-up saccades was substantially depressed shortly after the onset of manual interception. The initial direction of the finger movement anticipated the motion of the target by approximately 150 ms. For any given trajectory, subjects tended to initiate interception at predictable times that depended on the characteristics of the target trajectories [i.e., when the curvature (or angular velocity) of the target was small and when the target was moving toward the finger]. The relative weighting of various parameters that influenced the decision to initiate interception varied from subject to subject and was not accounted for by a model based on the short-range predictability of target motion.

Manual tracking in three dimensions

Little is known about the manual tracking of targets that move in three dimensions. In the present study, human subjects followed, with the tip of a hand-held pen, a virtual target moving four times (period 5 s) around a novel, unseen path. Two basic types of target paths were used: a peanut-shaped Cassini ellipse and a quasi-spherical shape where four connected semicircles lay in orthogonal planes. The quasi-spherical shape was presented in three different sizes, and the Cassini shape was varied in spatial orientation and by folding it along one of the three bend axes. During the first cycle of Cassini shapes, the hand lagged behind the target by about 150 ms on average, which decreased to 100 ms during the last three cycles. Tracking performance gradually improved during the first 3 s of the first cycle and then stabilized. Tracking was especially good during the smooth, planar sections of the shapes, and time lag was significantly shorter when the tracking of a low-frequency component was compared to performance at a higher frequency (-88 ms at 0.2 Hz vs. -101 ms at 0.6 Hz). Even after the appropriate adjustment of the virtual target path to a virtual shape tracing condition, tracking in depth was poor compared to tracking in the frontal plane, resulting in a flattening of the hand path. In contrast to previous studies where target trajectories were linear or sinusoidal, these complex trajectories may have involved estimation of the overall shape, as well as prediction of target velocity.