Bell-shaped Velocity Profiles Research Articles

Understanding voluntary human movement variability through data-driven segmentation and clustering

Recently, we proposed a novel approach where movements are decomposed into sub-segments, termed movement elements. This approach, to date, provides a robust construct of how the brain may generate simple as well as complex movements. Here, we address the issue of motor variability during voluntary movements by applying an unsupervised clustering algorithm to group movement elements according to their morphological characteristics. We observed that most movement elements closely match the theoretical bell-shaped velocity profile expected from goal-directed movements. However, for those movement elements that deviate from this theoretical shape, a small number of defined patterns in their shape can be identified. Furthermore, we observed that the axis of the body from which the movement elements are extracted (i.e., medio-lateral, antero-posterior, and vertical) affect the proportion of the movement elements matching the theoretical model. These results provide novel insight into how the nervous system controls voluntary movements and may use variability in movement element properties to explore the environment.

Bioinspired smooth neuromorphic control for robotic arms

Beyond providing accurate movements, achieving smooth motion trajectories is a long-standing goal of robotics control theory for arms aiming to replicate natural human movements. Drawing inspiration from biological agents, whose reaching control networks effortlessly give rise to smooth and precise movements, can simplify these control objectives for robot arms. Neuromorphic processors, which mimic the brain’s computational principles, are an ideal platform to approximate the accuracy and smoothness of biological controllers while maximizing their energy efficiency and robustness. However, the incompatibility of conventional control methods with neuromorphic hardware limits the computational efficiency and explainability of their existing adaptations. In contrast, the neuronal subnetworks underlying smooth and accurate reaching movements are effective, minimal, and inherently compatible with neuromorphic hardware. In this work, we emulate these networks with a biologically realistic spiking neural network for motor control on neuromorphic hardware. The proposed controller incorporates experimentally-identified short-term synaptic plasticity and specialized neurons that regulate sensory feedback gain to provide smooth and accurate joint control across a wide motion range. Concurrently, it preserves the minimal complexity of its biological counterpart and is directly deployable on Intel’s neuromorphic processor. Using the joint controller as a building block and inspired by joint coordination in human arms, we scaled up this approach to control real-world robot arms. The trajectories and smooth, bell-shaped velocity profiles of the resulting motions resembled those of humans, verifying the biological relevance of the controller. Notably, the method achieved state-of-the-art control performance while decreasing the motion jerk by 19% to improve motion smoothness. Overall, this work suggests that control solutions inspired by experimentally identified neuronal architectures can provide effective, neuromorphic-controlled robots.

Extending the kinematic theory of rapid movements with new primitives

The Kinematic Theory of rapid movements, and its associated Sigma-Lognormal, model 2D spatiotemporal trajectories. It is constructed mainly as a temporal overlap of curves between virtual target points. Specifically, it uses an arc and a lognormal as primitives for the representation of the trajectory and velocity, respectively. This paper proposes developing this model, in what we call the Kinematic Theory Transform, which establishes a mathematical framework that allows further primitives to be used. Mainly, we evaluate Euler curves to link virtual target points and Gaussian, Beta, Gamma, Double-bounded lognormal, and Generalized Extreme Value functions to model the bell-shaped velocity profile. Using these primitives, we report reconstruction results with spatiotemporal trajectories executed by human beings, animals, and anthropomorphic robots.

Pen-point Trajectory Analysis During Trail Making Test Based on a Time Base Generator Model.

The Trail Making test (TMT) is a widely used neuropsychological test to assess the cognitive function of patients. This paper presents the analysis method of pen-point trajectory during the TMT based on a time base generator (TBG). In the proposed method, the movement segments between targets are first extracted from pen-point trajectories, which are measured during performance of the TMT on an iPad. By fitting the extracted trajectories with a TBG-based trajectory generation model, the proposed method can then calculate quantitative indices representing the shape and collapse of the velocity profile. In the experiment, we analyzed TMT data from 25 stroke patients who were classified into three groups according to their scores on the Mini-Mental State Examination (MMSE). The results revealed that most of the measured inter-target trajectories had unimodal bell-shaped velocity profiles, as seen in reaching movements. Furthermore, we found that the degree of collapse in the velocity profile shape increased significantly when the cognitive function decreased.

Hierarchical Control of Visually-Guided Movements in a 3D-Printed Robot Arm.

The control architecture guiding simple movements such as reaching toward a visual target remains an open problem. The nervous system needs to integrate different sensory modalities and coordinate multiple degrees of freedom in the human arm to achieve that goal. The challenge increases due to noise and transport delays in neural signals, non-linear and fatigable muscles as actuators, and unpredictable environmental disturbances. Here we examined the capabilities of hierarchical feedback control models proposed by W. T. Powers, so far only tested in silico. We built a robot arm system with four degrees of freedom, including a visual system for locating the planar position of the hand, joint angle proprioception, and pressure sensing in one point of contact. We subjected the robot to various human-inspired reaching and tracking tasks and found features of biological movement, such as isochrony and bell-shaped velocity profiles in straight-line movements, and the speed-curvature power law in curved movements. These behavioral properties emerge without trajectory planning or explicit optimization algorithms. We then applied static structural perturbations to the robot: we blocked the wrist joint, tilted the writing surface, extended the hand with a tool, and rotated the visual system. For all of them, we found that the arm in machina adapts its behavior without being reprogrammed. In sum, while limited in speed and precision (by the nature of the do-it-yourself inexpensive components we used to build the robot from scratch), when faced with the noise, delays, non-linearities, and unpredictable disturbances of the real world, the embodied control architecture shown here balances biological realism with design simplicity.

A dot that went for a walk: People prefer lines drawn with human-like kinematics.

A dominant theory of embodied aesthetic experience (Freedberg & Gallese, 2007, Trends in Cognitive Sciences, 11, 197) posits that the appreciation of visual art is linked to the artist’s movements when creating the artwork, yet a direct link between the kinematics of drawing actions and the aesthetics of drawing outcomes has not been experimentally demonstrated. Across four experiments, we measured aesthetic responses of students from arts and non‐arts backgrounds to drawing movements generated from computational models of human writing. Experiment 1 demonstrated that human‐like drawing movements with bell‐shaped velocity profiles (Sigma Lognormal [SL] and Minimum Jerk [MJ]) are perceived as more natural and pleasant than movements with a uniform profile, and in both Experiments 1 and 2 movements that were perceived as more natural were also preferred. Experiment 3 showed that this effect persists if lower‐level dynamic stimulus features are fully matched across experimental and control conditions. Furthermore, aesthetic preference for human‐like movements were associated with greater perceptual fluency in Experiment 3, evidenced by unbiased estimations of the duration of natural movements. In Experiment 4, line drawings with visual features consistent with the dynamics of natural, human‐like movements were preferred, but only by art students. Our findings directly link the aesthetics of human action to the visual aesthetics of drawings, but highlight the importance of incorporating artistic expertise into embodied accounts of aesthetic experience.

Motor skill acquisition during a balance task as a process of optimization of motor primitives.

It has been argued that the central nervous system relies on combining simple movement elements (i.e. motor primitives) to generate complex motor outputs. However, how movement elements are generated and combined during the acquisition of new motor skills is still a source of debate. Herein, we present results providing new insights into the role of movement elements in the acquisition of motor skills that we obtained by analysing kinematic data collected while healthy subjects learned a new motor task. The task consisted of playing an interactive game using a platform with embedded sensors whose aggregate output was used to control a virtual object in the game. Subjects learned the task over multiple blocks. The analysis of the kinematic data was carried out using a recently developed technique referred to as "movement element decomposition." The technique entails the decomposition of complex multi-dimensional movements in one-dimensional elements marked by a bell-shaped velocity profile. We computed the number of movement elements during each block and measured how closely they matched a theoretical velocity profile derived by minimizing a cost function accounting for the smoothness of movement and the cost of time. The results showed that, in the early stage of motor skill acquisition, two mechanisms underlie the improvement in motor performance: 1)a decrease in the number of movement elements composing the motor output and 2)a gradual change in the movement elements that resulted in a shape matching the velocity profile derived by using the above-mentioned theoretical model.

Spatiotemporal Parameterization of Human Reaching Movements Based on Time Base Generator

Human reaching movement is characterized by the invariant features of linearity and the symmetrical bell-shaped velocity profile of the hand trajectory. Reaching movement has been widely evaluated in fields such as rehabilitation and interface design; however, no standardized protocols exist for data collection, processing, or analysis. This paper proposes a spatiotemporal parameterization method that evaluates reaching movements with a trajectory generation model based on a time base generator. The proposed method extracts motion information as a set of model parameters, such as motion duration, distance, and asymmetry of the bell-shaped velocity profile from the reaching data, that are estimated by fitting the model to reaching data with a robust parameter estimation method. The reaching trajectory generation model, for which a straight trajectory in a three-dimensional space is assumed, is also derived from the proposed model. Experiments were conducted using simulated data and real-world three-dimensional and one-dimensional human reaching movement datasets. The results show that the model assuming a straight trajectory in the three-dimensional space is reasonable for evaluating reaching movements. In addition, the model parameters were estimated with relatively greater robustness than were those of the conventional model and were highly correlated with physical features extracted from measured datasets. The results also showed that the parameters representing the asymmetry of the bell-shaped velocity profile differed between individuals. Therefore, the characteristics of reaching movements can be parameterized and evaluated with the proposed method.

Complex Upper-Limb Movements Are Generated by Combining Motor Primitives that Scale with the Movement Size

The hand trajectory of motion during the performance of one-dimensional point-to-point movements has been shown to be marked by motor primitives with a bell-shaped velocity profile. Researchers have investigated if motor primitives with the same shape mark also complex upper-limb movements. They have done so by analyzing the magnitude of the hand trajectory velocity vector. This approach has failed to identify motor primitives with a bell-shaped velocity profile as the basic elements underlying the generation of complex upper-limb movements. In this study, we examined upper-limb movements by analyzing instead the movement components defined according to a Cartesian coordinate system with axes oriented in the medio-lateral, antero-posterior, and vertical directions. To our surprise, we found out that a broad set of complex upper-limb movements can be modeled as a combination of motor primitives with a bell-shaped velocity profile defined according to the axes of the above-defined coordinate system. Most notably, we discovered that these motor primitives scale with the size of movement according to a power law. These results provide a novel key to the interpretation of brain and muscle synergy studies suggesting that human subjects use a scale-invariant encoding of movement patterns when performing upper-limb movements.

A recurrent neural network model for generation of humanlike reaching movements

ABSTRACTWe constructed a recurrent neural network model that can generate human reaching motion. Given a target position, the neural network model is capable of producing muscular activation signals that move the hand to the target endpoint, while solving the problem of musculoskeletal redundancy by dynamic relaxation of the energy functions embedded in the network. The proposed neural network model was integrated with a two-dimensional three-link eight muscle musculoskeletal model of the human arm to simulate arm reaching movements in the horizontal and sagittal planes. Our results demonstrate that the model is capable of generating natural arm movements that have spatiotemporal features, such as a slightly curved hand path and the characteristic bell-shaped velocity profile, that are similar to those of actual human movements. Some aspects of the proposed computational framework might be utilized in the central nervous system for generation of reaching movements.

Quantitative evaluation of age-related decline in control of preprogramed movement.

In this paper, we examined the age-related changes in control of preprogramed movement, with emphasis on its accuracy. Forty-nine healthy subjects participated in this study, and were divided into three groups depending on their ages: the young group (20–39 years) (n = 16), the middle-age group (40–59 years) (n = 16), and the elderly group (60–79 years) (n = 17). We asked the subjects to perform step-tracking movements of the wrist joint with a manipulandum, and recorded the movements. We evaluated the accuracy of control of preprogramed movement in the three groups in terms of the primary submovement, which was identified as the first segment of the step-tracking movement based on the bell-shaped velocity profile, and calculated the distance between the end position of the primary submovement and the target (i.e. error). The error in the young group was found to be significantly smaller than that in the middle-age and elderly groups, i.e., the error was larger for the higher age groups. These results suggest that young subjects have better control of preprogramed movement than middle-age or elderly subjects. Finally, we examined the temporal property of the primary submovement and its age-related changes. The duration of the primary submovement tended to be longer for the aged groups, although significance was reached only for the elderly group. In particular, the ratio of the duration of the primary submovement to total movement time tended to be lower for the aged groups, suggesting that the proportion of additional movements that are required to compensate for the incomplete control in the preprogramed movement, which are under feedback control, was higher for the aged groups. Consequently, our results indicate that the distance between the end point of the primary submovement and the target center (i.e. error) in the step-tracking movement is a useful parameter to evaluate the age-related changes in control of preprogramed movement.

Evidence for deficient motor planning in ADHD

We compare motor planning mechanisms of ADHD and control subjects based on their effect on later observed kinematic characteristics. We monitor hand movement following planning conditions that differ in preparation time, and evaluate the differences across conditions and participants with/without ADHD. Our findings show that when there is sufficient planning time, people without ADHD seem to have a motor plan ready, and immediately initiate a planned movement after a ‘GO’ cue, with a bell shaped velocity profile. When planning time is not sufficient, they start the movement in a delayed time, possibly indicating that they needed to complete a movement plan. However, people with ADHD, did not start movement immediately after the cue, even when provided with a long preparation time, possibly indicating that even for this planning interval they did not have a motion plan ready. The movement was not only delayed, its velocity profile was not bell shaped and had several peaks. We further found differences between control and ADHD participants in the velocity profile, variability and jitter of movements. Our results suggest that ADHD motion characteristics, are associated with an immature motor plan. Based on the results we propose a paradigm to evaluate deficiencies in motor planning.

From the motor cortex to the movement and back again.

The motor cortex controls motor behaviors by generating movement-specific signals and transmitting them through spinal cord circuits and motoneurons to the muscles. Precise and well-coordinated muscle activation patterns are necessary for accurate movement execution. Therefore, the activity of cortical neurons should correlate with movement parameters. To investigate the specifics of such correlations among activities of the motor cortex, spinal cord network and muscles, we developed a model for neural control of goal-directed reaching movements that simulates the entire pathway from the motor cortex through spinal cord circuits to the muscles controlling arm movements. In this model, the arm consists of two joints (shoulder and elbow), whose movements are actuated by six muscles (4 single-joint and 2 double-joint flexors and extensors). The muscles provide afferent feedback to the spinal cord circuits. Cortical neurons are defined as cortical "controllers" that solve an inverse problem based on a proposed straight-line trajectory to a target position and a predefined bell-shaped velocity profile. Thus, the controller generates a motor program that produces a task-specific activation of low-level spinal circuits that in turn induce the muscle activation realizing the intended reaching movement. Using the model, we describe the mechanisms of correlation between cortical and motoneuronal activities and movement direction and other movement parameters. We show that the directional modulation of neuronal activity in the motor cortex and the spinal cord may result from direction-specific dynamics of muscle lengths. Our model suggests that directional modulation first emerges at the level of muscle forces, augments at the motoneuron level, and further increases at the level of the motor cortex due to the dependence of frictional forces in the joints, contractility of the muscles and afferent feedback on muscle lengths and/or velocities.

Manuscript shapes generated by novel bi-axis control algorithm based on a mathematical handwriting model

<span lang="EN-US">Handwriting movement is one of the most complex activities of human motions. It’s a blend of kinesthetic, cognitive, perceptual and motor components. The study of this biological process shows that bell-shaped velocity profiles are generally observed in the handwriting motion. In this paper, an identification technique, based on Recursive Least Square algorithm (RLS), is proposed to identify the pen-tip movement in human handwriting process, by using input and output data which present EMG signals and velocities according to x and y coordinates. Using the estimated coordinates that have resulted from the velocity model; we propose a novel algorithm to generate handwritten graphic traces, which is inspired from the idea of tracing circles by Bresenham bi-axis control algorithm. The effectiveness of this approach should be observed on predicting cursive Arabic letters and Arabic word written on (</span><em><span lang="EN-US">x,y</span></em><span lang="EN-US">) plane, these shapes constituting a recorded experimental basis.</span>

Whole-Body Reaching Movements Formulated by Minimum Muscle-Tension Change Criterion.

It is well known that planar reaching movements of the human shoulder and elbow joints have invariant features: roughly straight hand paths and bell-shaped velocity profiles. The optimal control models with the criteria of smoothness or precision, which determine a unique movement pattern, predict such features of hand trajectories. In this letter on expanding the research on simple arm reaching movements, we examine whether the smoothness criteria can be applied to whole-body reaching movements with many degrees of freedom. Determining a suitable joint trajectory in the whole-body reaching movement corresponds to the optimization problem with constraints, since body balance must be maintained during a motion task. First, we measured human joint trajectories and ground reaction forces during whole-body reaching movements, and confirmed that subjects formed similar movements with common characteristics in the trajectories of the hand position and body center of mass. Second, we calculated the optimal trajectories according to the criteria of torque and muscle-tension smoothness. While the minimum torque change trajectories were not consistent with the experimental data, the minimum muscle-tension change model was able to predict the stereotyped features of the measured trajectories. To explore the dominant effects of the extension from the torque change to the muscle-tension change, we introduced a weighted torque change cost function. Considering the maximum voluntary contraction (MVC) force of the muscle as the weighting factor of each joint torque, we formulated the weighted torque change cost as a simplified version of the minimum muscle-tension change cost. The trajectories owing to the minimum weighted torque change criterion also showed qualitative agreement with the common features of the measured data. Proper estimation of the MVC forces in the body joints is essential to reproduce human whole-body movements according to the minimum muscle-tension change criterion.

Modeling human target reaching with an adaptive observer implemented with dynamic neural fields

Humans can point fairly accurately to memorized states when closing their eyes despite slow or even missing sensory feedback. It is also common that the arm dynamics changes during development or from injuries. We propose a biologically motivated implementation of an arm controller that includes an adaptive observer. Our implementation is based on the neural field framework, and we show how a path integration mechanism can be trained from few examples. Our results illustrate successful generalization of path integration with a dynamic neural field by which the robotic arm can move in arbitrary directions and velocities. Also, by adapting the strength of the motor effect the observer implicitly learns to compensate an image acquisition delay in the sensory system. Our dynamic implementation of an observer successfully guides the arm toward the target in the dark, and the model produces movements with a bell-shaped velocity profile, consistent with human behavior data.

Do muscle synergies reduce the dimensionality of behavior?

The muscle synergy hypothesis is an archetype of the notion of Dimensionality Reduction (DR) occurring in the central nervous system due to modular organization. Toward validating this hypothesis, it is important to understand if muscle synergies can reduce the state-space dimensionality while maintaining task control. In this paper we present a scheme for investigating this reduction utilizing the temporal muscle synergy formulation. Our approach is based on the observation that constraining the control input to a weighted combination of temporal muscle synergies also constrains the dynamic behavior of a system in a trajectory-specific manner. We compute this constrained reformulation of system dynamics and then use the method of system balancing for quantifying the DR; we term this approach as Trajectory Specific Dimensionality Analysis (TSDA). We then investigate the consequence of minimization of the dimensionality for a given task. These methods are tested in simulations on a linear (tethered mass) and a non-linear (compliant kinematic chain) system. Dimensionality of various reaching trajectories is compared when using idealized temporal synergies. We show that as a consequence of this Minimum Dimensional Control (MDC) model, smooth straight-line Cartesian trajectories with bell-shaped velocity profiles emerged as the optima for the reaching task. We also investigated the effect on dimensionality due to adding via-points to a trajectory. The results indicate that a trajectory and synergy basis specific DR of behavior results from muscle synergy control. The implications of these results for the synergy hypothesis, optimal motor control, motor development, and robotics are discussed.

Bio-inspired Trajectory Planning for Robot Catching Movements Based on Tau Theory

A bio-inspired trajectory planning method for robot reaching movement to catch an object is presented. The method is based on the Tau theory which has been established by cognitive science researchers from studying the natural motion patterns when animals(including humans) approaching and catching a target object. Based on the Tau theory, two fundamental motion strategies, namely, the Tau coupling strategy and the intrinsic Tau-g guidance strategy have been analyzed. In order to solve the problem that the initial acceleration value is nonzero in the intrinsic Tau-g guidance strategy, a new strategy, namely, the intrinsic constant jerk Tau guidance strategy is presented. The trajectory planning formulae based on the Tau theory are derived. Trajectory planning, kinetics analysis, dynamic analysis, computer simulation and experiments are performed on a 3-DOF planar manipulator. The results show that, by using the proposed method, the end-effector can achieve reaching the target position and attitude at the same definite time with good continuity of joint velocity and acceleration, and the trajectory in Cartesian space is a straight path with a smooth bell-shaped velocity profile.

Kinematic feedback control laws for generating natural arm movements

We propose a stochastic optimal feedback control law for generating natural robot arm motions. Our approach, inspired by the minimum variance principle of Harris and Wolpert (1998 Nature 394 780–4) and the optimal feedback control principles put forth by Todorov and Jordan (2002 Nature Neurosci. 5 1226–35) for explaining human movements, differs in two crucial respects: (i) the endpoint variance is minimized in joint space rather than Cartesian hand space, and (ii) we ignore the dynamics and instead consider only the second-order differential kinematics. The feedback control law generating the motions can be straightforwardly obtained by backward integration of a set of ordinary differential equations; these equations are obtained exactly, without any linear–quadratic approximations. The only parameters to be determined a priori are the variance scale factors, and for both the two-DOF planar arm and the seven-DOF spatial arm, a table of values is constructed based on the given initial and final arm configurations; these values are determined via an optimal fitting procedure, and consistent with existing findings about neuromuscular motor noise levels of human arm muscles. Experiments conducted with a two-link planar arm and a seven-DOF spatial arm verify that the trajectories generated by our feedback control law closely resemble human arm motions, in the sense of producing nearly straight-line hand trajectories, having bell-shaped velocity profiles, and satisfying Fitts Law.

Submovement Composition of Head Movement

Limb movement is smooth and corrections of movement trajectory and amplitude are barely noticeable midflight. This suggests that skeletomuscular motor commands are smooth in transition, such that the rate of change of acceleration (or jerk) is minimized. Here we applied the methodology of minimum-jerk submovement decomposition to a member of the skeletomuscular family, the head movement. We examined the submovement composition of three types of horizontal head movements generated by nonhuman primates: head-alone tracking, head-gaze pursuit, and eye-head combined gaze shifts. The first two types of head movements tracked a moving target, whereas the last type oriented the head with rapid gaze shifts toward a target fixed in space. During head tracking, the head movement was composed of a series of episodes, each consisting of a distinct, bell-shaped velocity profile (submovement) that rarely overlapped with each other. There was no specific magnitude order in the peak velocities of these submovements. In contrast, during eye-head combined gaze shifts, the head movement was often comprised of overlapping submovements, in which the peak velocity of the primary submovement was always higher than that of the subsequent submovement, consistent with the two-component strategy observed in goal-directed limb movements. These results extend the previous submovement composition studies from limb to head movements, suggesting that submovement composition provides a biologically plausible approach to characterizing the head motor recruitment that can vary depending on task demand.