Bouncing a ball: tuning into dynamic stability

Human arm endpoint-impedance in rhythmic human-robot interaction exhibits cyclic variations

Estimating the human endpoint-impedance interacting with a physical environment provides insights into goal-directed human movements during physical interactions. This work examined the endpoint-impedance of the upper limb during a hybrid ball-bouncing task with simulated haptic feedback while participants manipulated an admittance-controlled robot. Two experiments implemented a force-perturbation method to estimate the endpoint parameters of 31 participants. Experimental conditions of the ball-bouncing task were simulated in a digital environment. One experiment studied the influence of the target height, while the other explored the impedance at three cyclic phases of the rhythmic movement induced by the task. The participants' performances were analyzed and clustered to establish a potential influence of endpoint impedance on performance in the ball-bouncing task. Results showed that endpoint-impedance parameters ranged from 45 to 445 N/m, 2.2 to 17.5 Ns/m, and 227 to 893 g for the stiffness, damping, and mass, respectively. Results did not support such a critical role of endpoint impedance in performance. Nevertheless, the three endpoint-impedance parameters described significant variations throughout the arm cycle. The stiffness is linked to a quasi-linear increase, with a maximum value reached before the ball impacts. The observed damping and mass cyclic variations seemed to be caused by geometric and kinematic variations. Although this study reveals rapid and within-cycles variations of endpoint-impedance parameters, no direct relationship between endpoint-impedance values and performance levels in ball-bouncing could be found.

Towards functional robotic training: motor learning of dynamic tasks is enhanced by haptic rendering but hampered by arm weight support

Background

Current robot-aided training allows for high-intensity training but might hamper the transfer of learned skills to real daily tasks. Many of these tasks, e.g., carrying a cup of coffee, require manipulating objects with complex dynamics. Thus, the absence of somatosensory information regarding the interaction with virtual objects during robot-aided training might be limiting the potential benefits of robotic training on motor (re)learning. We hypothesize that providing somatosensory information through the haptic rendering of virtual environments might enhance motor learning and skill transfer. Furthermore, the inclusion of haptic rendering might increase the task realism, enhancing participants’ agency and motivation. Providing arm weight support during training might also enhance learning by limiting participants’ fatigue.

Methods

We conducted a study with 40 healthy participants to evaluate how haptic rendering and arm weight support affect motor learning and skill transfer of a dynamic task. The task consisted of inverting a virtual pendulum whose dynamics were haptically rendered on an exoskeleton robot designed for upper limb neurorehabilitation. Participants trained with or without haptic rendering and with or without weight support. Participants’ task performance, movement strategy, effort, motivation, and agency were evaluated during baseline, short- and long-term retention. We also evaluated if the skills acquired during training transferred to a similar task with a shorter pendulum.

Results

We found that haptic rendering significantly increases participants’ movement variability during training and the ability to synchronize their movements with the pendulum, which is correlated with better performance. Weight support also enhances participants’ movement variability during training and reduces participants’ physical effort. Importantly, we found that training with haptic rendering enhances motor learning and skill transfer, while training with weight support hampers learning compared to training without weight support. We did not observe any significant differences between training modalities regarding agency and motivation during training and retention tests.

Conclusion

Haptic rendering is a promising tool to boost robot-aided motor learning and skill transfer to tasks with similar dynamics. However, further work is needed to find how to simultaneously provide robotic assistance and haptic rendering without hampering motor learning, especially in brain-injured patients.

Trial registrationhttps://clinicaltrials.gov/show/NCT04759976

Supplementary Information

The online version contains supplementary material available at 10.1186/s12984-022-00993-w.

Preparing to move: Setting initial conditions to simplify interactions with complex objects

Humans dexterously interact with a variety of objects, including those with complex internal dynamics. Even in the simple action of carrying a cup of coffee, the hand not only applies a force to the cup, but also indirectly to the liquid, which elicits complex reaction forces back on the hand. Due to underactuation and nonlinearity, the object's dynamic response to an action sensitively depends on its initial state and can display unpredictable, even chaotic behavior. With the overarching hypothesis that subjects strive for predictable object-hand interactions, this study examined how subjects explored and prepared the dynamics of an object for subsequent execution of the target task. We specifically hypothesized that subjects find initial conditions that shorten the transients prior to reaching a stable and predictable steady state. Reaching a predictable steady state is desirable as it may reduce the need for online error corrections and facilitate feed forward control. Alternative hypotheses were that subjects seek to reduce effort, increase smoothness, and reduce risk of failure. Motivated by the task of 'carrying a cup of coffee', a simplified cup-and-ball model was implemented in a virtual environment. Human subjects interacted with this virtual object via a robotic manipulandum that provided force feedback. Subjects were encouraged to first explore and prepare the cup-and-ball before initiating a rhythmic movement at a specified frequency between two targets without losing the ball. Consistent with the hypotheses, subjects increased the predictability of interaction forces between hand and object and converged to a set of initial conditions followed by significantly decreased transients. The three alternative hypotheses were not supported. Surprisingly, the subjects' strategy was more effortful and less smooth, unlike the observed behavior in simple reaching movements. Inverse dynamics of the cup-and-ball system and forward simulations with an impedance controller successfully described subjects' behavior. The initial conditions chosen by the subjects in the experiment matched those that produced the most predictable interactions in simulation. These results present first support for the hypothesis that humans prepare the object to minimize transients and increase stability and, overall, the predictability of hand-object interactions.

Collective Variables and Task Constraints in Movement Coordination, Control and Skill

In this paper we review studies that have identified collective variables (order parameters) in movement coordination, control and skill with emphasis on whole-body multiple joint degree of freedom (DF) tasks. Collective variables of a dynamical system have been proposed formally and informally from a diverse set of perceptual-motor tasks, from which we emphasize: bimanual coordination, locomotion (pedalo, walking, running, bicycle riding), roller ball task, static (quiet standing) and dynamic (moving on a ski-simulator) balance, grasping, and juggling. Several types of candidate collective variables have been identified, including: relative phase, frequency ratio, number of hands active in grasping, synchrony, learning rate and relative timing. There is a strong influence of the task goal in determining the collective variable that can be body or environment relative. The emergence of the task relevant collective variable is typically in the early stage of skill learning where subjects through practice adapt movement organization to realize a never previously produced movement coordination pattern. Throughout, the paper elaborates on open theoretical, experimental and analysis issues for collective variables in the context of task constraints and Bernstein's (1967) view of skill acquisition as learning to master redundant DF.

Learning and transfer of complex motor skills in virtual reality: a perspective review

The development of more effective rehabilitative interventions requires a better understanding of how humans learn and transfer motor skills in real-world contexts. Presently, clinicians design interventions to promote skill learning by relying on evidence from experimental paradigms involving simple tasks, such as reaching for a target. While these tasks facilitate stringent hypothesis testing in laboratory settings, the results may not shed light on performance of more complex real-world skills. In this perspective, we argue that virtual environments (VEs) are flexible, novel platforms to evaluate learning and transfer of complex skills without sacrificing experimental control. Specifically, VEs use models of real-life tasks that afford controlled experimental manipulations to measure and guide behavior with a precision that exceeds the capabilities of physical environments. This paper reviews recent insights from VE paradigms on motor learning into two pressing challenges in rehabilitation research: 1) Which training strategies in VEs promote complex skill learning? and 2) How can transfer of learning from virtual to real environments be enhanced? Defining complex skills by having nested redundancies, we outline findings on the role of movement variability in complex skill acquisition and discuss how VEs can provide novel forms of guidance to enhance learning. We review the evidence for skill transfer from virtual to real environments in typically developing and neurologically-impaired populations with a view to understanding how differences in sensory-motor information may influence learning strategies. We provide actionable suggestions for practicing clinicians and outline broad areas where more research is required. Finally, we conclude that VEs present distinctive experimental platforms to understand complex skill learning that should enable transfer from therapeutic practice to the real world.

Complementary spatial and timing control in rhythmic arm movements

Volitional rhythmic motor behaviors such as limb cycling and locomotion exhibit spatial and timing regularity. Such rhythmic movements are executed in the presence of exogenous visual and nonvisual cues, and previous studies have shown the pivotal role that vision plays in guiding spatial and temporal regulation. However, the influence of nonvisual information conveyed through auditory or touch sensory pathways, and its effect on control, remains poorly understood. To characterize the function of nonvisual feedback in rhythmic arm control, we designed a paddle juggling task in which volunteers bounced a ball off a rigid elastic surface to a target height in virtual reality by moving a physical handle with the right hand. Feedback was delivered at two key phases of movement: visual feedback at ball peaks only and simultaneous audio and haptic feedback at ball-paddle collisions. In contrast to previous work, we limited visual feedback to the minimum required for jugglers to assess spatial accuracy, and we independently perturbed the spatial dimensions and the timing of feedback. By separately perturbing this information, we evoked dissociable effects on spatial accuracy and timing, confirming that juggling, and potentially other rhythmic tasks, involves two complementary processes with distinct dynamics: spatial error correction and feedback timing synchronization. Moreover, we show evidence that audio and haptic feedback provide sufficient information for the brain to control the timing synchronization process by acting as a metronome-like cue that triggers hand movement. NEW & NOTEWORTHY Vision contains rich information for control of rhythmic arm movements; less is known, however, about the role of nonvisual feedback (touch and sound). Using a virtual ball bouncing task allowing independent real-time manipulation of spatial location and timing of cues, we show their dissociable roles in regulating motor behavior. We confirm that visual feedback is used to correct spatial error and provide new evidence that nonvisual event cues act to reset the timing of arm movements.

The primacy of rhythm: how discrete actions merge into a stable rhythmic pattern

This study examined how humans spontaneously merge a sequence of discrete actions into a rhythmic pattern, even when periodicity is not required. Two experiments used a virtual throwing task, in which subjects performed a long sequence of discrete throwing movements, aiming to hit a virtual target. In experiment 1, subjects performed the task for 11 sessions. Although there was no instruction to perform rhythmically, the variability of the interthrow intervals decreased to a level comparable to that of synchronizing with a metronome; furthermore, dwell times shortened or even disappeared with practice. Floquet multipliers and decreasing variability of the arm trajectories estimated in state space indicated an increasing degree of dynamic stability. Subjects who achieved a higher level of periodicity and stability also displayed higher accuracy in the throwing task. To directly test whether rhythmicity affected performance, experiment 2 disrupted the evolving continuity and periodicity by enforcing a pause between successive throws. This discrete group performed significantly worse and with higher variability in their arm trajectories than the self-paced group. These findings are discussed in the context of previous neuroimaging results showing that rhythmic movements involve significantly fewer cortical and subcortical activations than discrete movements and therefore may pose a computationally more parsimonious solution. Such emerging stable rhythms in neuromotor subsystems may serve as building blocks or dynamic primitives for complex actions. The tendency for humans to spontaneously fall into a rhythm in voluntary movements is consistent with the ubiquity of rhythms at all levels of the physiological system. NEW & NOTEWORTHY When performing a series of throws to hit a target, humans spontaneously merged successive actions into a continuous approximately periodic pattern. The degree of rhythmicity and stability correlated with hitting accuracy. Enforcing irregular pauses between throws to disrupt the rhythm deteriorated performance. Stable rhythmic patterns may simplify control of movement and serve as dynamic primitives for more complex actions. This observation reveals that biological systems tend to exhibit rhythmic behavior consistent with a plethora of physiological processes.

The self-organization of ball bouncing

The hybrid rhythmic ball-bouncing task considered in this study requires a participant to hit a ball in a virtual environment by moving a paddle in the real environment. It allows for investigation of the online visual control of action in humans. Changes in gravity acceleration in the virtual environment affect the ball dynamics and modify the ball-paddle system limit cycle. These changes are shown to be accurately reproduced through simulation by a model integrating continuous information-movement couplings between the ball trajectory and the paddle trajectory, giving rise to a resonance-tuning phenomenon. On the contrary, the tested models integrating only intermittent sensorimotor couplings were unable to replicate the observed human behavior. Results suggest that the visual control of action is achieved online, in a prospective way. Human rhythmic motor control would benefit from the timing and phase control emerging from the low-level continuous coupling between the central pattern generator and the visual perception of the ball trajectory. This control strategy, which precludes the need for internal clock and explicit environmental representation, is also able to explain the empirical result that the bounces tend to converge toward a passive stability regime during human ball bouncing.

Stability and predictability in human control of complex objects

Previous research on movement control suggested that humans exploit stability to reduce vulnerability to internal noise and external perturbations. For interactions with complex objects, predictive control based on an internal model of body and environment is needed to preempt perturbations and instabilities due to delays. We hypothesize that stability can serve as means to render the complex dynamics of the body and the task more predictable and thereby simplify control. However, the assessment of stability in complex interactions with nonlinear and underactuated objects is challenging, as for existent stability analyses the system needs to be close to a (known) attractor. After reviewing existing methods for stability analysis of human movement, we argue that contraction theory provides a suitable approach to quantify stability or convergence in complex transient behaviors. To test its usefulness, we examined the task of carrying a cup of coffee, an object with internal degrees of freedom. A simplified model of the task, a cart with a suspended pendulum, was implemented in a virtual environment to study human control strategies. The experimental task was to transport this cart-and-pendulum on a horizontal line from rest to a target position as fast as possible. Each block of trials presented a visible perturbation, which either could be in the direction of motion or opposite to it. To test the hypothesis that humans exploit stability to overcome perturbations, the dynamic model of the free, unforced system was analyzed using contraction theory. A contraction metric was obtained by numerically solving a partial differential equation, and the contraction regions with respect to that metric were computed. Experimental results showed that subjects indeed moved through the contraction regions of the free, unforced system. This strategy attenuated the perturbations, obviated error corrections, and made the dynamics more predictable. The advantages and shortcomings of contraction analysis are discussed in the context of other stability analyses.

Model of rhythmic ball bouncing using a visually controlled neural oscillator

The present paper investigates the sensory-driven modulations of central pattern generator dynamics that can be expected to reproduce human behavior during rhythmic hybrid tasks. We propose a theoretical model of human sensorimotor behavior able to account for the observed data from the ball-bouncing task. The novel control architecture is composed of a Matsuoka neural oscillator coupled with the environment through visual sensory feedback. The architecture's ability to reproduce human-like performance during the ball-bouncing task in the presence of perturbations is quantified by comparison of simulated and recorded trials. The results suggest that human visual control of the task is achieved online. The adaptive behavior is made possible by a parametric and state control of the limit cycle emerging from the interaction of the rhythmic pattern generator, the musculoskeletal system, and the environment.NEW & NOTEWORTHY The study demonstrates that a behavioral model based on a neural oscillator controlled by visual information is able to accurately reproduce human modulations in a motor action with respect to sensory information during the rhythmic ball-bouncing task. The model attractor dynamics emerging from the interaction between the neuromusculoskeletal system and the environment met task requirements, environmental constraints, and human behavioral choices without relying on movement planning and explicit internal models of the environment.

The critical phase for visual control of human walking over complex terrain

To walk efficiently over complex terrain, humans must use vision to tailor their gait to the upcoming ground surface without interfering with the exploitation of passive mechanical forces. We propose that walkers use visual information to initialize the mechanical state of the body before the beginning of each step so the resulting ballistic trajectory of the walker's center-of-mass will facilitate stepping on target footholds. Using a precision stepping task and synchronizing target visibility to the gait cycle, we empirically validated two predictions derived from this strategy: (1) Walkers must have information about upcoming footholds during the second half of the preceding step, and (2) foot placement is guided by information about the position of the target foothold relative to the preceding base of support. We conclude that active and passive modes of control work synergistically to allow walkers to negotiate complex terrain with efficiency, stability, and precision.

Mapping Time-to-Contact and Time-to-Peak Effector Velocity in Interceptive Striking Tasks

PURPOSE

This study investigated perception and action coupling as reflected in a mapping between time-to-contact (TTC) and time-to-peak-velocity (TPV) for children and youths aged 7 to 8, 11 to 12, and 15 to 16 years old and adults aged 19 to 20 years old performing a task that allowed the participants to self-select the interception position in a baseball-like batting game on a graphics tablet.

METHOD

The moving object to be hit for distance had curvilinear trajectories toward the hitter that were determined by simulated weightings of gravity and velocity.

RESULTS

Successful interceptions in all age groups were characterized by a mapping of mean TTC and TPV and a higher correlation of these variables with increasing age. The object was hit at approximately 90% TPV so that the implement was still accelerating on contact-an adaptive task feature for a striking distance criterion.

CONCLUSION

The results are consistent with theories of perception-action coupling that are congruent with the ecological and dynamical approaches to action where TTC, a source of environmental information, is mapped relative to TPV, largely independent of the particular kinematic task demands of the approaching object. The organization of the coupling of TTC prior to TPV was evident across all age groups and is evidence for a common strategy for this type of interceptive action. The age-related changes in the strength of the relationship between TTC and TPV are evidence of increased sensitivity to the coupling between TTC and TPV.

Development of information-movement couplings in a rhythmical ball-bouncing task: from space- to time-related information

We studied the development of information-movement couplings in a ball-bouncing task with a special interest in how space- and time-related information is used by people of different ages. Participants from four age groups (children aged 7-8, 9-10 and 11-12 years, and adults) performed a virtual ball-bouncing task in which space- and time-related information were independently manipulated. Task performance and information-movement couplings were analyzed. Our results confirm a clear use of time-related information in adults, while children demonstrated a predominant relationship between space-related information and the period of movement. In the course of development, however, the children become progressively more capable of using time-related information in order to control the rhythmic ball-bouncing task. A second and weaker coupling, between ball height information and racket velocity at impact, also appears in the course of development. The data seem to show that the development of children follows the freezing-freeing-exploiting sequence proposed by Savelsbergh and Van der Kamp (Int J Sport Psychol 31:467-484, 2000), with a significant change in how information is used to control movement related to age.

Implicit guidance to stable performance in a rhythmic perceptual-motor skill

Feedback about error or reward is regarded essential for aiding learners to acquire a perceptual-motor skill. Yet, when a task has redundancy and the mapping between execution and performance outcome is unknown, simple error feedback does not suffice in guiding the learner toward the optimal solutions. The present study developed and tested a new means of implicitly guiding learners to acquire a perceptual-motor skill, rhythmically bouncing a ball on a racket. Due to its rhythmic nature, this task affords dynamically stable solutions that are robust to small errors and noise, a strategy that is independent from actively correcting error. Based on the task model implemented in a virtual environment, a time-shift manipulation was designed to shift the range of ball-racket contacts that achieved dynamically stable solutions. In two experiments, subjects practiced with this manipulation that guided them to impact the ball with more negative racket accelerations, the indicator for the strategy with dynamic stability. Subjects who practiced under normal conditions took longer time to acquire this strategy, although error measures were identical between the control and experimental groups. Unlike in many other haptic guidance or adaptation studies, the experimental groups not only learned, but also maintained the stable solution after the manipulation was removed. These results are a first demonstration that more subtle ways to guide the learner to better performance are needed especially in tasks with redundancy, where error feedback may not be sufficient.

The Effect of Haptic Guidance on Learning a Hybrid Rhythmic-Discrete Motor Task

Bouncing a ball with a racket is a hybrid rhythmic-discrete motor task, combining continuous rhythmic racket movements with discrete impact events. Rhythmicity is exceptionally important in motor learning, because it underlies fundamental movements such as walking. Studies suggested that rhythmic and discrete movements are governed by different control mechanisms at different levels of the Central Nervous System. The aim of this study is to evaluate the effect of fixed/fading haptic guidance on learning to bounce a ball to a desired apex in virtual reality with varying gravity. Changing gravity changes dominance of rhythmic versus discrete control: The higher the value of gravity, the more rhythmic the task; lower values reduce the bouncing frequency and increase dwell times, eventually leading to a repetitive discrete task that requires initiation and termination, resembling target-oriented reaching. Although motor learning in the ball-bouncing task with varying gravity has been studied, the effect of haptic guidance on learning such a hybrid rhythmic-discrete motor task has not been addressed. We performed an experiment with thirty healthy subjects and found that the most effective training condition depended on the degree of rhythmicity: Haptic guidance seems to hamper learning of continuous rhythmic tasks, but it seems to promote learning for repetitive tasks that resemble discrete movements.

Rhythmic manipulation of objects with complex dynamics: predictability over chaos

The study of object manipulation has been largely confined to discrete tasks, where accuracy, mechanical effort, or smoothness were examined to explain subjects' preferred movements. This study investigated a rhythmic manipulation task, which involved continuous interaction with a nonlinear object that led to unpredictable object behavior. Using a simplified virtual version of the task of carrying a cup of coffee, we studied how this unpredictable object behavior affected the selected strategies. The experiment was conducted in a virtual set-up, where subjects moved a cup with a ball inside, modeled by cart-and-pendulum dynamics. Inverse dynamics calculations of the system showed that performing the task with different amplitudes and relative phases required different force profiles and rendered the object's dynamics with different degrees of predictability (quantified by Mutual Information between the applied force and the cup kinematics and its sensitivity). Subjects (n = 8) oscillated the virtual cup between two targets via a robotic manipulandum, paced by a metronome at 1 Hz for 50 trials, each lasting 45 s. They were free to choose their movement amplitude and relative phase between the ball and cup. Experimental results showed that subjects increased their movement amplitudes, which rendered the interactions with the object more predictable and with lower sensitivity to the execution variables. These solutions were associated with higher average exerted force and lower object smoothness, contradicting common expectations from studies on discrete object manipulation and unrestrained movements. Instead, the findings showed that humans selected strategies with higher predictability of interaction dynamics. This finding expressed that humans seek movement strategies where force and kinematics synchronize to repeatable patterns that may require less sensorimotor information processing.

Major changes in a rhythmic ball-bouncing task occur at age 7 years

The aim of the study was to investigate the development of a rhythmical skill of children aged from 5 to 12 years old. Five age groups (5–6, 7–8, 9–10, 11–12, and young adults) performed a virtual ball bouncing task (16 forty-second long test trials). Task performances, racket oscillation, ball-racket impacts as well as the ball-racket coupling were analysed. The results showed a change in both performance and behaviour at the age of 7 years old. Before this age, children exhibited restricted perceptual-motor coordination with a high frequency of racket oscillation and a poor level of performance. After the age of 7, cycle-to-cycle adaptive coordination based on visual information was progressively acquired leading to increasing performance levels with age. Overall these results revealed a rapid change in capability to perform the ball bouncing task across age with a late emergence of the required coordination and significant change in the coordination at the age of 7.

Dynamic primitives of motor behavior

We present in outline a theory of sensorimotor control based on dynamic primitives, which we define as attractors. To account for the broad class of human interactive behaviors-especially tool use-we propose three distinct primitives: submovements, oscillations, and mechanical impedances, the latter necessary for interaction with objects. Owing to the fundamental features of the neuromuscular system-most notably, its slow response-we argue that encoding in terms of parameterized primitives may be an essential simplification required for learning, performance, and retention of complex skills. Primitives may simultaneously and sequentially be combined to produce observable forces and motions. This may be achieved by defining a virtual trajectory composed of submovements and/or oscillations interacting with impedances. Identifying primitives requires care: in principle, overlapping submovements would be sufficient to compose all observed movements but biological evidence shows that oscillations are a distinct primitive. Conversely, we suggest that kinematic synergies, frequently discussed as primitives of complex actions, may be an emergent consequence of neuromuscular impedance. To illustrate how these dynamic primitives may account for complex actions, we briefly review three types of interactive behaviors: constrained motion, impact tasks, and manipulation of dynamic objects.

The role of haptic feedback when manipulating nonrigid objects

Humans can learn to manipulate objects with complex dynamics, including nonrigid objects with internal degrees of freedom. The first aim of this study was to assess the contribution of haptic feedback when learning to manipulate a nonrigid object. The second aim was to evaluate how learning without haptic feedback influences subsequent learning with haptic feedback and vice versa. The task involved moving a simulated mass-attached to a grasped handle via a simulated, damped spring-to a target as quickly as possible. In the haptic plus vision (HV) condition, appropriate forces were applied to the handle, which was attached to a robot. In the vision only (V) condition, these forces were turned off. Participants completed 80 trials in each condition, with one-half starting with the HV condition. Both groups exhibited significant learning, as measured by movement time, in both conditions. For the condition performed first, initial performance, learning rate, and final performance were better with haptic feedback. Prior experience in the HV condition led to faster learning and better final performance in the V condition. However, prior experience in the V condition led to slower learning and worse final performance in the HV condition. In the V condition, all participants tended to keep the mass close to the hand. In the HV condition, participants who started with the HV condition allowed the mass to move away from the hand, whereas participants who started with the V condition continued to keep the mass close to the hand. We conclude that haptic feedback as well as prior experience with haptic feedback enhance the ability to control nonrigid objects and that training without haptic feedback can lead to persisting detrimental effects when subsequently dealing with haptic feedback.

Bouncing between model and data: stability, passivity, and optimality in hybrid dynamics

Rhythmically bouncing a ball with a racket is a seemingly simple task, but it poses all the challenges critical for coordinative behavior: perceiving the ball's trajectory to adapt position and velocity of the racket for the next ball contact. To gain insight into the underlying control strategies, the authors conducted a series of studies that tested models with experimental data, with an emphasis on deriving model-based hypotheses and trying to falsify them. Starting with a simple dynamical model of the racket and ball interactions, stability analyses showed that open-loop dynamics affords dynamical stability, such that small perturbations do not require corrections. To obtain this passive stability, the ball has to be impacted with negative acceleration--a strategy that subjects adopted in a variety of conditions at steady state. However, experimental tests that applied perturbations revealed that after perturbations, subjects applied active perceptually guided corrections to reestablish steady state faster than by relying on the passive model's relaxation alone. Hence, the authors derived a model with active control based on optimality principles that considered each impact as a separate reaching-like movement. This model captured some additional features of the racket trajectory but failed to predict more fine-grained aspects of performance. The authors proceed to present a new model that accounts not only for fine-grained behavior but also reconciles passive and active control approaches with new predictions that will be put to test in the next set of experiments.

Optimal control of a hybrid rhythmic-discrete task: the bouncing ball revisited

Rhythmically bouncing a ball with a racket is a hybrid task that combines continuous rhythmic actuation of the racket with the control of discrete impact events between racket and ball. This study presents experimental data and a two-layered modeling framework that explicitly addresses the hybrid nature of control: a first discrete layer calculates the state to reach at impact and the second continuous layer smoothly drives the racket to this desired state, based on optimality principles. The testbed for this hybrid model is task performance at a range of increasingly slower tempos. When slowing the rhythm of the bouncing actions, the continuous cycles become separated into a sequence of discrete movements interspersed by dwell times and directed to achieve the desired impact. Analyses of human performance show increasing variability of performance measures with slower tempi, associated with a change in racket trajectories from approximately sinusoidal to less symmetrical velocity profiles. Matching results of model simulations give support to a hybrid control model based on optimality, and therefore suggest that optimality principles are applicable to the sensorimotor control of complex movements such as ball bouncing.

Open-loop, closed-loop and compensatory control: performance improvement under pressure in a rhythmic task

According to explicit monitoring theories, the phenomenon of choking under pressure is due to actors focusing their attention on the execution of the skill. This step-by-step perceptually guided control may then interfere with automatic execution. In order to examine the changes in control at the sensorimotor level, we examined the rhythmic task of ball bouncing which affords detailed quantification of indicators of control based on previous research. The hypothesis was that under psychological pressure perceptually guided control should lead to decreased performance due to over-emphasis on closed-loop control and decreased compensatory control. In two experiments of different difficulty psychological stress was induced via setting up a fake competition. Results showed that, contrary to the hypothesis, performance accuracy and consistency improved together with an increase in compensatory control. Indicators for open- and closed-loop processes did not change. Only under more challenging conditions in Experiment 2, enhanced performance under pressure was accompanied by more active, closed-loop and less passive control. The results are discussed in light of task demands and the continuous rhythmic nature of the task: in more challenging tasks, control appears to be more prone to disturbance due to psychological stress. The different control demands in continuous rhythmic tasks may be less prone to interference due to psychological stress than in discrete tasks.

Passive and Active Discrimination of Natural Frequency of Virtual Dynamic Systems

It has been shown that humans use combined feedforward and feedback control strategies when manipulating external dynamic systems, and when exciting virtual dynamic systems at resonance, that they can tune their control parameters in response to changing natural frequencies. We present a study to determine the discrimination thresholds for the natural frequency of such resonant dynamic systems. Weber fractions (WFs in %) are reported for the discrimination of 1, 2, 4, and 8 Hz natural frequencies. Participants were instructed either to passively perceive or actively excite the virtual system via a one degree-of-freedom haptic interface with visual and/or haptic feedback. The average WF for natural frequency ranged from 4% to 8.5% for 1,2, and 4 Hz reference natural frequencies, while the WF was approximately 20% for systems with a reference natural frequency of 8 Hz. Results indicate that sensory feedback modality has a significant effect on WF during passive perception, but no significant effect in the active perception case. The data also suggest that discrimination sensitivity is not significantly affected by excitation mode. Finally, results for systems with equivalent natural frequencies but different spring stiffness indicate that participants do not discriminate natural frequency based on the maximum force magnitude perceived.

Stability and Variability: Indicators for Passive Stability and Active Control in a Rhythmic Task

Using a rhythmic task where human subjects bounced a ball with a handheld racket, fine-grained analyses of stability and variability extricated contributions from open-loop control, noise strength, and active error compensation. Based on stability analyses of a stochastic-deterministic model of the task—a surface contacting the ball by periodic movements—open-loop or dynamic stability was assessed by the acceleration of the racket at contact. Autocovariance analyses of model and data were further used to gauge the contributions of open-loop stability and noise strength. Variability and regression analyses estimated active error compensation. Empirical results demonstrated that experienced actors exploited open-loop stability more than novices, had lower noise strength, and applied more active error compensations. By manipulating the model parameter coefficient of restitution, task stability was varied and showed that actors graded these three components as a function of task stability. It is concluded that actors tune into task stability when stability is high but use more active compensation when stability is reduced. Implications for the neural underpinnings for passive stability and active control are discussed. Further, results showed that stability and variability are not simply the inverse of each other but contain more quantitative information when combined with model analyses.

Robotics and neuroscience: a rhythmic interaction

At the crossing between motor control neuroscience and robotics system theory, the paper presents a rhythmic experiment that is amenable both to handy laboratory implementation and simple mathematical modeling. The experiment is based on an impact juggling task, requiring the coordination of two upper-limb effectors and some phase-locking with the trajectories of one or several juggled objects. We describe the experiment, its implementation and the mathematical model used for the analysis. Our underlying research focuses on the role of sensory feedback in rhythmic tasks. In a robotic implementation of our experiment, we study the minimum feedback that is required to achieve robust control. A limited source of feedback, measuring only the impact times, is shown to give promising results. A second field of investigation concerns the human behavior in the same impact juggling task. We study how a variation of the tempo induces a transition between two distinct control strategies with different sensory feedback requirements. Analogies and differences between the robotic and human behaviors are obviously of high relevance in such a flexible setup.

Control of bimanual rhythmic movements: trading efficiency for robustness depending on the context

This paper investigates how the efficiency and robustness of a skilled rhythmic task compete against each other in the control of a bimanual movement. Human subjects juggled a puck in 2D through impacts with two metallic arms, requiring rhythmic bimanual actuation. The arms kinematics were only constrained by the position, velocity and time of impacts while the rest of the trajectory did not influence the movement of the puck. In order to expose the task robustness, we manipulated the task context in two distinct manners: the task tempo was assigned at four different values (hence manipulating the time available to plan and execute each impact movement individually); and vision was withdrawn during half of the trials (hence reducing the sensory inflows). We show that when the tempo was fast, the actuation was rhythmic (no pause in the trajectory) while at slow tempo, the actuation was discrete (with pause intervals between individual movements). Moreover, the withdrawal of visual information encouraged the rhythmic behavior at the four tested tempi. The discrete versus rhythmic behavior give different answers to the efficiency/robustness trade-off: discrete movements result in energy efficient movements, while rhythmic movements impact the puck with negative acceleration, a property preserving robustness. Moreover, we report that in all conditions the impact velocity of the arms was negatively correlated with the energy of the puck. This correlation tended to stabilize the task and was influenced by vision, revealing again different control strategies. In conclusion, this task involves different modes of control that balance efficiency and robustness, depending on the context.

Passive Stability and Active Control in a Rhythmic Task

Rhythmically bouncing a ball with a racket is a task that affords passively stable solutions as demonstrated by stability analyses of a mathematical model of the task. Passive stability implies that no active control is needed as errors die out without requiring corrective actions. Empirical results from human performance demonstrated that actors indeed exploit this passive dynamics in steady-state performance, thereby reducing computational demands of the task. The present study investigated the response to perturbations of different magnitudes designed on the basis of the model's basin of attraction. Humans performed the task in a virtual reality set-up with a haptic interface. Relaxation times of the performance errors showed significantly faster returns than predicted from the purely passive model, indicative of active error corrections. Systematic adaptations in the racket trajectories were a monotonic function of the perturbation magnitudes, indicating that active control was applied in proportion to the perturbation. These results did not indicate any sensitivity to the boundary of stability. Yet the influence of passive dynamics was also seen: the pattern of relaxation times in the major performance variable ball height was consistent with qualitative predictions derived from the basin of attraction and racket accelerations at contact were generally negative signaling use of passive stability. These findings suggest that the fast return back to steady state was assisted by passive properties of the task. It was concluded that actors used a blend of active and passive control for all sizes of perturbations.

On rhythmic and discrete movements: reflections, definitions and implications for motor control

At present, rhythmic and discrete movements are investigated by largely distinct research communities using different experimental paradigms and theoretical constructs. As these two classes of movements are tightly interlinked in everyday behavior, a common theoretical foundation spanning across these two types of movements would be valuable. Furthermore, it has been argued that these two movement types may constitute primitives for more complex behavior. The goal of this paper is to develop a rigorous taxonomic foundation that not only permits better communication between different research communities, but also helps in defining movement types in experimental design and thereby clarifies fundamental questions about primitives in motor control. We propose formal definitions for discrete and rhythmic movements, analyze some of their variants, and discuss the application of a smoothness measure to both types that enables quantification of discreteness and rhythmicity. Central to the definition of discrete movement is their separation by postures. Based on this intuitive definition, certain variants of rhythmic movement are indistinguishable from a sequence of discrete movements, reflecting an ongoing debate in the motor neuroscience literature. Conversely, there exist rhythmic movements that cannot be composed of a sequence of discrete movements. As such, this taxonomy may provide a language for studying more complex behaviors in a principled fashion.

Learning new perception-action solutions in virtual ball bouncing

How do humans discover stable solutions to perceptual-motor tasks as they interact with the physical environment? We investigate this question using the task of rhythmically bouncing a ball on a racket, for which a passively stable solution is defined. Previously, it was shown that participants exploit this passive stability but can also actively stabilize bouncing under perceptual control. Using a virtual ball-bouncing display, we created new behavioral solutions for rhythmic bouncing by introducing a temporal delay (45 degrees -180 degrees ) between the motion of the physical racket and that of the virtual racket. We then studied how participants searched for and realized a new solution. In all delay conditions, participants learned to maintain bouncing just outside the passively stable region, indicating a role for active stabilization. They recovered the approximate initial phase of ball impact in the virtual racket cycle (half-way through the upswing) by adjusting the impact phase with the physical racket. With short delays (45 degrees , 90 degrees ), the impact phase quickly shifted later in the physical racket upswing. With long delays (135 degrees , 180 degrees ), bouncing was destabilized and phase was widely visited before a new preferred phase gradually emerged, during the physical downswing. Destabilization was likely due to the loss of spatial symmetry between the ball and physical racket motion at impact. The results suggest that new behavioral solutions may be discovered and stabilized through broad irregular sampling of variable space rather than through a systematic search.

The dynamics of perception and action

How might one account for the organization in behavior without attributing it to an internal control structure? The present article develops a theoretical framework called behavioral dynamics that integrates an information-based approach to perception with a dynamical systems approach to action. For a given task, the agent and its environment are treated as a pair of dynamical systems that are coupled mechanically and informationally. Their interactions give rise to the behavioral dynamics, a vector field with attractors that correspond to stable task solutions, repellers that correspond to avoided states, and bifurcations that correspond to behavioral transitions. The framework is used to develop theories of several tasks in which a human agent interacts with the physical environment, including bouncing a ball on a racquet, balancing an object, braking a vehicle, and guiding locomotion. Stable, adaptive behavior emerges from the dynamics of the interaction between a structured environment and an agent with simple control laws, under physical and informational constraints.

The Cinderella of Psychology: The Neglect of Motor Control in the Science of Mental Life and Behavior

One would expect psychology--the science of mental life and behavior--to place great emphasis on the means by which mental life is behaviorally expressed. Surprisingly, however, the study of how decisions are enacted--the focus of motor control research--has received little attention in psychology. This article documents the neglect and considers possible reasons for it. The hypotheses considered include three that are raised and then rejected: (a) no famous psychologists have studied motor control, (b) cognitive psychologists are mainly interested in uniquely human functions, and (c) motor control is simply too hard to study. Three other hypotheses are more viable: (d) cognitive psychologists have been more interested in epistemology than in action, (e) psychologists have disfavored motor control because overt responses were the only admissible measure in behaviorism, and (f) psychologists have felt that neuroscientists have the market cornered when it comes to motor control research. There are signs that motor control's Cinderella status is changing.

On the Road to Automatic: Dynamic Aspects in the Development of Expertise

One of the important steps on the road to becoming expert in a motor skill occurs when the individual can perform the movements in a seemingly effortless and automatic fashion. The authors review two lines of investigations, namely, fMRI and mathematically guided studies of the dynamics of skill acquisition, that suggest that this road to automatic involves two steps: (1) an increasing reliance on the self-regulatory aspects of the motor task, and (2) a minimization of the role of mechanisms based on intentionally directed corrective movements. The interplay between these two mechanisms implies that, at a given skill level, performance decreases whenever intention intervenes. The observation that psychological factors may be as important as mechanical repetition for the development of expertise has important implications for the design of neurorehabilitative strategies.

Dynamical Stability in the Acquisition and Performance of Rhythmic Ball Manipulation: Theoretical Insights With a Clinical Slant

Three experiments illustrate how a task-based approach and dynamical modeling of a perceptual-motor task can provide a useful framework for understanding functional and dysfunctional behavior. In the chosen task, subjects held a racket and bounced a ball rhythmically in the air with invariant ball amplitude. As such, the task could be cast into a mechanical model that encompassed the movements of the actor (racket) and the manipulandum (ball). In this form, the movement task is a dynamical system that displays dynamical stability, i.e., performance where perturbations die out by themselves. The hypothesis is that skilled actors seek to perform with this "passive" stability as it alleviates the control demands because perturbations do not require explicit corrections. In the experimental data, this strategy could be characterized by a single parameter, the acceleration of the racket at impact, which provided quantitative predictions. Experiment 1 established that subjects with normal sensorimotor functions indeed performed with racket acceleration values that were predicted to provide passive stability. Experiment 2 showed that subjects improved their skill over a course of 40 practice trials, as evidenced in decreased variability and accompanied by a change in racket acceleration toward values that provided optimal stability. In experiment 3, perturbations were applied and the subjects' adaptability was tested. When perturbations were large enough, subjects altered their racket timing to resume contact that provided stability. The results are discussed with their relevance to clinical contexts: How can such a task-based approach provide insights into the control of functional and dysfunctional movements? Can such behavioral results serve as a diagnostic tool? How can this approach to sensorimotor behavior stimulate physicians and therapists to progress therapeutic measures?

Stability and phase locking in human soccer juggling

We explored a variant of juggling in which human adults were asked to rhythmically bounce a soccer ball with their dominant foot while standing on the other foot. Eight subjects performed the task at three prescribed heights and one free height condition. Kinematic analyses of foot movement at ball-foot impact showed that, for the smallest height, foot acceleration was positive or zero at impact, which indicates an active stabilization regime. Increasing juggling height resulted in foot acceleration becoming increasingly negative at impact, which is required for a passive dynamical stability regime. These results show that skilled soccer jugglers exploit the passive stability regime afforded by the task, but that similar stability can be achieved with an active stabilization strategy.

Actively tracking 'passive' stability in a ball bouncing task

This study investigates the control involved in a task where subjects rhythmically bounce a ball with a hand-held racket as regularly as possible to a prescribed amplitude. Stability analyses of a kinematic model of the ball-racket system revealed that dynamically stable solutions exist if the racket hits the ball in its decelerating upward movement phase. Such solutions are resistant to small perturbations obviating explicit error corrections. Previous studies reported that subjects' performance was consistent with this 'passive' stability. However, some 'active' control is needed to attune to this passive stability. The present study investigates this control by confronting subjects with perturbations where stable behavior cannot be maintained solely from passive stability. Six subjects performed rhythmic ball bouncing in a virtual reality set-up with and without perturbations. In the perturbation trials the coefficient of restitution of the ball-racket contact was changed at every fifth contact leading to unexpected ball amplitudes. The perturbations were compensated for within 2-3 bouncing cycles such that ball amplitudes decreased to initial values. Passive stability was reestablished as indicated by negative racket acceleration. Results revealed that an adjustment of the racket period ensured that the impacts occurred at a phase associated with passive stability. These findings were implemented in a model consisting of a neural oscillator that drives a mechanical actuator (forearm holding the racket) to bounce the ball. Following the perturbation, the oscillator's period is adjusted based on the perceived ball velocity after impact. Simulation results reproduced the major aspects of the experimental results.

Control of ball-racket interactions in rhythmic propulsion of elastic and non-elastic balls

Ball-racket interactions were investigated in a task where participants propelled a ball rhythmically into the air. The study contrasted two ball-racket conditions: (1) an elastic impact where the ball was able to rebound due to the elasticity of the colliding objects and participants bounced the ball, and (2) a non-elastic impact where the coefficient of restitution was zero and the ball did not rebound such that the participants had to throw the ball. The goal of the study was to contrast the situations where haptic information about the ball-racket interactions is either secondary (elastic bouncing) or becomes a primary factor for control (non-elastic propulsion). In the elastic condition, the performers controlled the parameters for ball-racket contact prior to contact: In agreement with the criteria for dynamical stability defined by a model, racket accelerations immediately before the contacts were negative, racket positions and velocities at the instant of the initial contact correlated negatively, contact durations were short (30+/-9 ms), and during the collision interval racket velocity and acceleration decreased monotonically. In the non-elastic condition, the parameters of ball release were primarily controlled during the collision phase: Racket accelerations before contact were positive, racket positions and velocities at initial contact showed weak correlations, and the contact intervals were significantly longer (116+/-15 ms) with a clear segmentation into two segments. Negative correlations were observed between the integrals of the velocity and acceleration computed over the two consecutive segments, giving evidence that in the non-elastic condition the CNS is able to introduce corrections during the very short collision interval. The results are discussed with respect to physiological mechanisms of movement corrections available during such short time intervals.