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Passive Stability and Active Control in a Rhythmic Task

¹ Department of Kinesiology and Integrative Biosciences, Pennsylvania State University, University Park, Pennsylvania; and ² University Medical Center, Leiden, The Netherlands

Submitted 4 July 2007; accepted in final form 14 September 2007

	ABSTRACT

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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.

	INTRODUCTION

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Human actions necessarily express themselves as physical interactions with the environment where the biological system creates and is subject to forces that constrain the execution and the control of action. The stability of these forces is key to their control. Stability in postural and movement control is achieved by many different means: physiologically, it can arise from the stiffness of the muscular and tendonous tissues mediated by stretch reflexes; from a control perspective, stability is attained through feedback-based responses that correct for errors; a third source of stability is of mechanical and dynamical origin: the stability of the actor-environment system as it is set up by the task. This system may provide stability "for free" that is not brought about by the active compensations of errors, i.e., feedback control, but rather it is brought about by the inherent stability of the task. A prominent example that has demonstrated the importance of this source of stability is the passive dynamic walker. McGeer (1990) showed, by using both an analytic model and a multilink robot, that a purely mechanical system without actuators and control could maintain a stable walking pattern when walking down a gentle slope.

Models of this type achieve dynamic stability without active control, relying solely on the passive dynamics of the physical constructs. Over the past two decades, a lot of attention and effort has been directed to extending the idea of passive dynamics to human-like biped robots walking on level ground and in 3D (Coleman and Ruina 1998; Collins et al. 2001). Several models demonstrated that walking in two dimensions has inherent stability, but to achieve walking in three dimensions, additional actuation was needed. However, the fundamental reliance on passive dynamics has offered these 3D walkers efficiency in energy expenditure, reduced demands in control, and elegant mimicry of human motion (Collins et al. 2005; Tedrake et al. 2004). The striking similarity of these minimally controlled walking machines with human walking suggests that passive dynamics may play an important role in shaping coordinated human behavior. Without downplaying the importance of muscular forces and their tuning by perceptual information in determining behavior, the passive dynamic walkers show that our understanding about control can be deepened by studying the motion that may emerge without control.

To tease apart the contributions of passive dynamics and active control, a motor task is wanted that first affords such a passively stable solution. Sternad and colleagues have shown this to be the case for the task of rhythmically bouncing a ball on a racket (de Rugy et al. 2003; Dijkstra et al. 2004; Schaal et al. 1996; Sternad et al. 2000, 2001). This task requires an actor to bounce the ball with a racket up in the air to a consistent height over repeated bounces. By viewing the racket as an oscillating planar surface and the ball as a point mass colliding with the racket with an inelastic impact, a simple mechanical model was derived for the ball-bouncing task. Stability analyses of this model yielded predictions about criteria for which this passive stability is achieved. Specifically, when the acceleration of the racket's upward movement is negative when contacting the ball, dynamic stability is indicated for the model.

Empirical studies of the ball-bouncing task confirmed that humans exploit the passive stability properties as predicted by the model. Experienced actors hit the ball with negative racket accelerations in a variety of experimental conditions: when different ball amplitudes were required (Schaal et al. 1996), when the movement of the ball was confined to the vertical dimension only (Sternad et al. 2000, 2001), or when the ball and the racket moved freely in three dimensions; when the ball was bounced by using a paddle moving downward instead of a hand-held racket moving upward to hit the ball (Schaal et al. 1996), or when the experiment was conducted in a virtual reality setup (de Rugy et al. 2003). In contrast, novice actors performed with positive impact acceleration and only gradually, within 30 min of practice, tuned their movements to use negative impact acceleration (Dijkstra et al. 2004). This result was accompanied by decreasing variability supporting the interpretation that performance had improved with this change of strategy. This latter result highlighted that hitting the ball with negative impact acceleration was not an intuitive or trivial solution for actors. The task offers the advantage of stability, but it has to be learned. This was also shown by Siegler and colleagues in a study on learning new phase relations between ball and racket where subjects frequently performed the task with positive accelerations (Morice et al. 2007). In sum, the empirical results on the ball-bouncing task support the interpretation that actors exploit the stability properties of the task.

This ball bouncing model shares many features with the passive dynamic walking model. Both models are formulated over the actor-environment system with collisions as the primary form of interaction. Given that control of the continuous system is confined to intermittent moments of contact, it is a hybrid control system. Importantly, both tasks have multiple stable solutions with period-1 to period-n and chaotic solutions. The presence of multistability immediately raises the question about the boundary between these multiple solutions. How large can a perturbation be before the system ends up in another solution? How large is the basin of attraction? Results of passive dynamic walkers show that the basin of attraction is not very large, even with meticulous tuning of the parameters (Garcia et al. 1998; Schwab and Wisse 2001; Wisse and van Frankenhuyzen 2003). As a result, the bipedal robots are sensitive to initial conditions and demand a careful launch. These observations are in contrast to human walking, which is much more robust to perturbations and adaptive to different conditions pointing to the presence and importance of active control in human walking.

Although multistability similarly exists in the ball-bouncing map, only period-1 performance has been investigated thus far. Further understanding about the relationship between passive stability and active control can be obtained by examining the basin of attraction. Hence, the present study investigates the ball-bouncing task with periodic solutions under systematic perturbations that are designed in view of the basin of attraction for the period-1 solution. Will the actor discard the strategy of using passive stability and turn to active control with error feedback on a cycle-to-cycle basis? Alternatively, will the actor rely on passive stability without active error corrections when the ball is only slightly perturbed with perturbations inside the basin of attraction? Or will the actor adopt a mixture strategy with both the exploitation of passive stability and of perception-guided error corrections (Warren 2006)? In sum, are actors sensitive to the boundaries of the basin of attraction of the period-1 solution?

These questions can be answered by applying perturbations of different magnitudes. In a previous study, de Rugy et al. (2003) applied perturbations by randomly changing the coefficient of restitution of the ball-racket system on impacts leading to unexpected under- or overshooting of the ball with respect to the target height. Results showed that actors quickly reestablished negative acceleration at impact, indicating the use of passive stability. Yet for all the applied perturbations, modulations of racket movements also indicated signs of active control. Specifically, the study found that directly following perturbation the periods of the racket cycles shortened or lengthened for smaller or larger ball amplitudes, respectively. The amplitudes of the racket movements remained largely unchanged. Although this study gave a first indication that actors "actively tracked passive stability," the sensitivity to the basin of attraction could not be tested because no theoretical analyses of the basin of attraction were available. Post hoc analyses revealed that the perturbations mostly took the system outside the basin of attraction. In fact, very small perturbations were excluded by design. To extend these first investigations, the current study provides a derivation of the basin of attraction for the period-1 attractor of the ball bouncing map. The experimental perturbations were designed such that they covered a wide range both inside and outside the basin. Furthermore, the coefficient of restitution of the racket, a critical variable influencing the shape of the basin of attraction, was also systematically varied. Based on the locations of perturbations in the basin, predictions about the response of a purely passive strategy can be made. These predictions are then compared with human performance to elucidate the relationship between passive dynamics and active control.

	MODEL

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The predictions are based on the same model that was derived in earlier work (Dijkstra et al. 2004; Guckenheimer and Holmes 1983; Holmes 1982; Sternad et al. 2001; Tuffilaro et al. 1992). As mentioned in the preceding text, the task of periodically bouncing a ball with a racket to a target height affords a passively stable period-1 solution. This solution is entirely open-loop and, once initiated, requires no control or error correction. The period-1 solution co-exists with other solutions, in particular sticking solutions and period-n solutions. The latter solutions give rise to the period-doubling route to chaos and were an important motivation for the initial studies of the map (see Tuffilaro et al. 1992). However, they are of no concern in the current context.

The ball bouncing map is based on the following three assumptions: 1) Ballistic flight: between the kth and the k +1th bounce the vertical ball position x_b(t) follows the ballistic flight equation

(1)

with x_b(t_k) the vertical ball position at the time of the last (kth) impact, v_b⁺, the ball velocity immediately after impact, and g the acceleration due to gravity (9.81 m/s²). 2) Instantaneous impact: the impact is instantaneous such that the ball velocity immediately after impact v_b⁺ is determined by

(2)

where v_b^– denotes ball velocity just before impact. v_r denotes the velocity of the racket at impact, and denotes the coefficient of restitution, which captures the energy loss at the impact. The velocity of the racket does not change during impact because the mass of the racket is much larger than the mass of the ball. 3) Sinusoidal racket movement: the racket movement is a pure sinusoid

(3)

with amplitude a_r and angular frequency _r (Bapat et al. 1986).

The validity of these assumptions for bouncing a physical ball is discussed in Dijkstra et al. (2004) and Brody et al. (2002). Because the task is performed in a virtual set-up (see METHODS in the following text), the ballistic flight and the assumption of instantaneous impact are satisfied by design. The assumption of a pure sinusoid is not obeyed, not even in the virtual set-up: actors pick a periodic waveform that is slightly steeper than a sine wave in the ascending phase before the racket meets the ball. Unfortunately, there is no simple mathematical description of this waveform. However, we note that only position and velocity with which the racket hits the ball determine the ball trajectory. Thus for mathematical simplicity, we use an equivalent sinusoid that is close to the actual waveform at the impact: its equivalent frequency _r is calculated from the period between bounces and its equivalent amplitude a_r is calculated from the stationary phase of impact as (see Dijkstra et al. 2004, Eq. 8)

(4)

From these assumptions, the ball-bouncing map can be derived as

(5)

(6)

This is an implicit map with the two state variables v_k, the ball velocity just after impact, and _k, the racket phase of impact. The ball bouncing map has a period-1 attractor which is locally linearly stable when the acceleration at impact, denoted by AC, is bounded by

(7)

The period-1 attractor co-exists with other attractors: for AC more negative than the lower boundary, there exist attractors where the ball sticks to the racket for part of the cycle; for positive AC several attractors exist, among them the period-doubling route to chaos (Tuffilaro et al. 1992). It is important to stress that the constraint on AC in Eq. 7 is based on the assumption of stationarity. Moreover, stability also depends on other parameters such as amplitude or period; however, only when other parameters are kept constant. The only single parameter that concisely captures the possible bifurcations is the acceleration at impact AC. When comparing these predictions with experimental data, one needs to keep in mind that we only report trial means and not trial-to-trial data. Given the ubiquitous variability, it occasionally happened that positive racket accelerations are interspersed.

The domain of attraction of the period-1 attractor depends on the parameters of the map: g, the acceleration of gravity, , the coefficient of restitution, _r, the racket frequency, and a_r, the racket amplitude. These parameters can be tightly controlled or determined in the experiment to obtain a good quantitative match with the model. The first two parameters, g and , are independent of the actor and can be experimentally manipulated. Because the linear stability in the model and the domain of attraction depend strongly on (see Fig. 1), this parameter was varied as an independent measure in the current study. _r and a_r are more difficult to control experimentally because they do depend on the actors' performance. However, _r can be fixed by having actors bounce to a visual target. Because the ball amplitude determines the racket period (through the flight equation) and actors hit the ball at an approximately constant height relative to the floor, having a target at a fixed height relative to the floor fixes the racket period. The racket amplitude was not prescribed but the actual movement amplitude was estimated from the impact phase, using Eq 4. The model parameter a_r was set accordingly for the calculations.

View larger version (14K): [in this window] [in a new window]   FIG. 1. Basin of attraction of the ball bouncing map. Dashed line, stationary release velocity. Gray shading, how many bounces or cycles are needed for the perturbations to die out. For example, the light gray means that the map took >2 and <7 bounces to return within a band of the stationary state (as defined in the text); the darker gray denotes that the map took any number of cycles between 7 and 100 to return to steady state. Black area, all those perturbations where the system assumes states other than period-1. White dots, perturbations used in the experiment: for each of the 4 coefficients of restitution, 14 different magnitudes of perturbations were applied. The 3 larger dots highlight 3 selected perturbations for = 0.6 for which the time course is illustrated in Fig. 2, following the ordering from the top to the bottom.

The results of these computations are presented in Fig. 1, where the white and gray shaded areas denote the domain of attraction with increasing relaxation times (counted in number of cycles). The black area indicates initial conditions that did not converge to the stationary state within 100 iterations or that led to a sticking solution where the time between impacts was <1 ms.

The four vertical columns of dots in Fig. 1 indicate the different perturbation magnitudes that were used for the four different values in the experiment. Figure 2 shows some simulation results for the three marked perturbation magnitudes in Fig. 1. A and B illustrate the time course of the two state variables at impact following a large perturbation that takes seven bounces to return to steady state. Equivalently, C and D show the state variables for a smaller perturbation that relaxes back within four bounces. The continuous time series on E and F illustrates how a sticking solution occurs. Note the perturbations were first designed based on preliminary calculations of the basin of attraction using parameter estimates from previous experiments. After having analyzed the data of the present experiment, the basin of attraction was recalculated as described in the preceding text to provide a better evaluation of the perturbations for the present data.

View larger version (19K): [in this window] [in a new window]   FIG. 2. Simulations for 3 selected perturbations of the ball bouncing map. A and B: time evolution of the state variables impact phase and ball release velocity, respectively. The shaded area denotes the bandwidth when the system is considered to be back at equilibrium. These 2 panels correspond to the upper perturbation in Fig. 1. C and D: equivalent plots for the middle perturbation in Fig. 1. E and F: time series of continuous position and velocity of both the racket and the ball, showing a sticking solution. This was obtained for the perturbation in the black area of Fig. 1.

PREDICTION 1. The basin of attraction has a boundary separating stable period-1 solutions from sticking and period-n solutions. Actors are sensitive to this boundary and rely on passive stability when perturbations are inside the basin of attraction. For perturbations outside the basin, two outcomes are conceivable: actors lose period-1 stability and switch to period-n (with n > 1) or sticking solutions or they adopt an active strategy that aims to correct for errors. In either case, a qualitative change in the behavior is expected. Note that sticking or period-n solutions have never been observed in the current experimental set-up (de Rugy et al. 2003). Hence active corrections are the more likely case that will be analyzed from modulations of the racket kinematics for destabilizing perturbations.

However, even if discontinuous changes in strategy across the boundary will not be observed, the basin of attraction still predicts qualitative changes in behavior as a function of perturbation magnitude and coefficient of restitution. Three qualitative predictions can be formulated.

PREDICTION 2A. With increasing perturbation magnitude, the time for returning to steady-state performance increases for all coefficients of restitution. Larger perturbations that take the system further out of the basin of attraction lead to longer relaxation times.

PREDICTION 2B. The relaxation time is longer for negative perturbations than for positive perturbations for all coefficients of restitution (i.e., for release velocities smaller than the average release velocity). This follows from the observation that the lower boundary of the basin of attraction is closer to the stationary state than the upper boundary.

PREDICTION 2C. For positive perturbations, the smaller coefficients of restitution have a wider basin of attraction. Hence for positive perturbations, the smaller coefficients of restitution should show faster returns than the higher coefficients of restitution. For negative perturbations, there should be no difference in relaxation time for different coefficients of restitution.

ALTERNATIVE PREDICTION 3. Actors apply active control for all errors regardless of their magnitude to ensure stable performance. Hence modulations of the racket trajectory should scale continuously with the magnitude of the perturbation. While in opposition to the previous results supporting the exploitation of passive stability, this statement also recognizes the fact that active control is needed to tune the system into this "passive dynamic" strategy.

	METHODS

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Participants

Seven volunteers participated, with ages ranging from 23 to 47 yr. With the exception of one participant, all others had some prior experience in performing this task. All participants reported to be right-handed and used their preferred right hand to bounce the ball with the racket. Before the experiment, all participants were informed about the procedure and signed the consent form approved by the Regulatory Committee of the Pennsylvania State University.

Experimental apparatus

In the virtual reality set-up, participants manipulated a real table tennis racket to bounce a virtual ball that was projected on a screen in front of them (Fig. 3A). Participants stood 0.5 m behind a back-projection screen with width 2.5 m and height of 1.8 m. A PC (2.4 GHz Pentium CPU, Windows XP) controlled the experiment and generated the visual stimuli with a graphics card (Radeon 9700, ATI). The same PC also acquired the data using a 16 bit A/D card (DT322, DataTranslation). The images were projected by a Toshiba TLP 680 TFT-LCD projector and consisted of 1,024 x 768 pixels with a 60-Hz refresh rate. Accelerations of the racket were measured using a solid-state piezoresistive accelerometer mounted on top of the racket (T45-10, Coulbourne). The mechanical brake acted on the rod that was attached to the racket and was controlled by a solenoid (Magnet-Schultz type R 16 x 16 DC pull, subtype S-07447). A light rigid rod with three hinge joints was attached to the racket surface and ran through a wheel whose rotation was registered by an optical encoder (Fig. 3B). Its accuracy was one pulse for 0.27 mm of racket movement. The pulses from the optical encoder were counted by an onboard counter (DT322). The racket could move and tilt with minimal friction in three dimensions, but only the vertical displacement was measured. Images of racket and ball position were shown on-line on the back projection screen using custom-made software. The maximal velocity of the ball was estimated to be 30 pixels/s on average. However, around the apex of the ball trajectory, where participants tend to focus their attention, the ball velocity is an order of magnitude smaller. The delay between real and virtual racket movement was measured in a separate experiment and found to be 22 ± 0.5 (SD) ms on average.

View larger version (24K): [in this window] [in a new window]   FIG. 3. A: sketch of the virtual reality setup for the ball bouncing task, a side view and a front view of the screen display. B: photographic view of the hardware used for the acquisition of the racket position and acceleration and the brake. Arrow 1, accelerometer mounted on the racket; arrow 2, brake and the optical encoder mounted on the hardware.

The computer program controlling the experiment would read the latest racket position from the optical encoder and racket acceleration from the accelerometer. When the racket was away from the ball, the program would update the ball positions based on the ballistic flight equation. On the 2.4-GHz computer under Windows XP this led to an update rate of 800 Hz. When the ball and racket were close, the computer program would keep a running estimate of time to contact and control the brake accordingly. The increased computational load led to a slow-down of the update rate to 250 Hz in the 30 ms surrounding an impact. The update rate was not fixed because Windows XP is not a real-time operating system and thus timing is not deterministic. Hence all data were time-stamped using the high-resolution timer on the Pentium CPU with an accuracy better than 1 ms.

Procedure and experimental conditions

Prior to each experiment, the participant was placed on a support base to adjust for height differences. The support height was adjusted such that the height of the hand-held racket, when held with the forearm horizontally, was 10 cm above its lowest position. Each trial began with a ball appearing at the left side of the screen and rolling on a horizontal line extending to the center of the screen (see Fig. 3A, inset). On reaching the center, the ball dropped from the horizontal line (0.7 m high). The task instruction was to rhythmically bounce the ball for the duration of a trial (55 s) as accurately as possible to the target line (the same line that the ball started on). The experiment consisted of a total of 80 trials, which were collected in two sessions. Each session lasted 1 h.

The entire experiment was divided into four blocks of 20 trials, one block for each value of the coefficient of restitution : 0.5, 0.6, 0.7, and 0.8. The blocks were presented in either ascending or descending order, counterbalanced among participants. The first two and the last two trials of each block were control trials without any perturbation. In the remaining 16 experimental trials, perturbations were applied at random impact times. A perturbation was created by an abrupt change of the ball release velocity immediately after the ball-racket impact. This led to an unexpected ball amplitude without any other noticeable change in the ball trajectory. This perturbation in velocity was chosen randomly from 14 magnitudes which were added or subtracted from the current release velocity –1.0, –0.86, –0.71, –0.57, –0.43, –0.29, –0.14, 0.14, 0.29, 0.43, 0.57, 0.71, 0.86, and 1.0 m/s (see Fig. 1). Negative values led to smaller ball amplitudes, positive values to larger ball amplitudes. With an average ball amplitude of 0.55 m and a corresponding average bounce period of 650 ms, the effect of these perturbations can be converted to deviations from the target height. The largest positive perturbation caused an overshoot of 0.37 m above the target, and the largest negative perturbation an undershoot of 0.27 m. The smallest perturbation of 0.14 m/s caused an overshoot of 0.047 m and –0.14 m/s an undershoot of 0.045 m. The 14 different magnitudes of perturbations ranging from –1 to +1 m/s were labeled as P–7, P–6, P–5, ... , to P+6, P+7.

The complete set of 14 perturbations for each was delivered on two successive trials with 7 perturbations within one trial in randomized order. As each trial had 80–90 bounces, the perturbations occurred randomly on the 8th, 9th, or 10th bounce relative to the previous perturbation. Across the 16 experimental trials, the set of 14 perturbations could be administered eight times. With randomization of both time and magnitude of perturbation, participants were unable to anticipate the ball amplitude or the time of the perturbations.

Data reduction and analysis

The raw data of the racket displacement and acceleration were resampled at a fixed frequency of 500 Hz and filtered with a fourth-order Savitzky-Golay filter with a window size of 0.01 s on both sides (Gander and Hrebicek 2004). The filter order and window size were chosen empirically to remove measurement noise while not excessively smoothing the signals. The Savitzky-Golay filter is superior for smoothing data that have abrupt changes as compared with conventional filters like Butterworth filters. These abrupt changes occurred in the data when the racket exhibited a sudden drop in acceleration caused by the brake. The ball displacement was generated by the computer, so it contained no measurement noise. Therefore no filtering was necessary. As a verification of our filtering procedure, the racket displacement was double-differentiated using a Savitzky-Golay filter and compared with the acceleration data collected by the accelerometer. Figure 4A illustrates that the two signals showed a good match, supporting the validity of the data acquisition. The time series shows two cycles with two ball-racket impacts and indicates the moment of impact as estimated from the double-differentiated position signal and the raw accelerometer signal. On average, the accelerometer signal rendered estimates that were 0.52 m/s² more positive than the position signal. To be conservative and given that the variability of the impact accelerations from the raw acceleration signal was smaller, we opted to report the estimates from the raw acceleration data.

View larger version (21K): [in this window] [in a new window]   FIG. 4. A: segments of time series of the double-differentiated position signal and the raw accelerometer signal together with the racket trajectory to illustrate their match. B: segment of an exemplary trial with a large perturbation (P–7). The cycles following the perturbed impact are marked as C0, C1, C2, etc. For the definitions of the dependent measures, see text for details.

Figure 4B illustrates the primary dependent measures. Performance was evaluated by the ball height error, HE, which was defined as the signed difference between the maximum ball height and the target height. Height error was equivalent to the state variable ball velocity at the moment of release from the racket, as this velocity determined the subsequent ball amplitude in the gravitational field if the impact position was relatively constant. The racket amplitude, A, was calculated as half the distance between the minimum to the maximum of the racket trajectory during one cycle. The racket period, T, was calculated from the intervals between the times of peak velocities of successive bounces. The acceleration of the racket at impact, AC, was determined from the accelerometer signal one sample before the time of impact.

Relaxation time after perturbation

To quantify the return to stationary bouncing after a perturbation, we fitted the return of the height error over cycles with an exponential function. The cycles were labeled in sequential order starting from the perturbed cycle directly following the perturbed impact, labeled C0; the following cycles were labeled C1, C2, and so on (Fig. 4B). To estimate the relaxation time after the perturbation at C0, a Levenberg-Marquardt least-squares fitting of an exponential function was performed using the following functional form (Matlab 6.5, Mathworks)

(8)

where y_k was the dependent measure and k was the cycle number with k = 0 denoting the perturbed cycle, C0. Fitted parameters were the amplitude of the perturbation y₀, the final level y and the relaxation time . We fitted the grand mean over eight repetitions and seven participants of each dependent measure as a function of cycle for each of the four conditions. Although the exponential fits were satisfactory in general, the smallest perturbations P–1 and P+1 could not be reliably fitted as the return took effect within one cycle. For P–2, P+2, and P–3, a portion of the fits was acceptable and was included in the figures and further analyses. The reported means were calculated over the values that were reliably obtained.

Exit times

To calculate whether and when the racket trajectory was actively modified due to the perturbations, the racket cycle directly following the perturbation was compared with the unperturbed racket trajectories during the control trials. The time series of racket position of the two control trials at the start of each block (total 170 cycles) was parsed into cycles at the moment of impact, then overlaid and averaged. The median and interquartile range were calculated to render a representative control cycle with an error band for comparison. The exit times were defined as the time from the application of the perturbation at impact to the time when the median racket trajectory exited the error band (1.5 times the interquartile range).

	RESULTS

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Performance during control trials

RACKET ACCELERATION. Given that the objective of the study examined perturbations away from stable performance the data first had to be examined whether actors indeed performed the task in accord with the criteria of passive stability. Hence, performance was evaluated in the control trials (4 trials for each of the 4 condition). The primary measure indicating performance at passive stability is the mean impact accelerations AC across all bounces of one trial (typically 70–80 bounces during 55-s-long trials). The mean values across the four control trials, determined separately for each of the four conditions, are listed for all seven participants in Table 1. Overall, participants showed negative AC values as predicted by the model and seen in previous studies, with only two exceptions: participants 1 and 2 had small positive values for = 0.8 and 0.5, respectively. Excluding these two cases, these results verified that all participants indeed performed the task consistent with criteria for passive stability. A 4 () x 7 (participant) ANOVA was performed on these data with participant treated as a random factor. The results showed no significant differences between different values, F(3,84) = 0.63, P = 0.607. The main effect of participant and the interaction were significant, F(6,84) = 9.98, P < 0.0001, and F(18,84) = 3.02, P < 0.0001, respectively. The grand mean of these control data (–1.75m/s²) was used as the baseline for the design of the perturbation magnitudes and the calculations of the basin of attraction.

RACKET PERIOD AND AMPLITUDE. For a better characterization of the task performance, the continuous racket trajectories during steady state were assessed by their mean period and amplitudes per trial. The same 4 () x 7 (participant) ANOVA performed on period yielded significant differences between different values, F(3,84) = 8.91, P < 0.005, and between different participants, F(6,84) = 20.56, P < 0.0001. The interaction between participants and was also significant, F(18,84) = 4.44, P < 0.0001; the racket periods tended to increase for larger conditions (634 ± 38, 642 ± 44, 654 ± 45, 679 ± 44 ms). One cause for this trend was that in the higher conditions participants impacted the ball at slightly lower positions but this observation was not statistically significant.

The equivalent ANOVA on amplitudes revealed a decreasing trend for higher , F(6,84) = 173.22, P < 0.0001, indicating that subjects moved the racket less when the ball-racket contact was bouncier: 0.067 ± 0.007, 0.049 ± 0.005, 0.036 ± 0.003, and 0.025 ± 0.005 m for conditions 0.5, 0.6, 0.7, and 0.8, respectively. The main effect for participant and the interaction were significant, F(6,84) = 3.87, P < 0.05, F(18,64) = 6.76, P < 0.0001.

Performance after perturbations

HEIGHT ERROR. We first report the perturbations of the height error as the task demanded a minimization of height error. Figure 5 displays the grand averages of HE over all repetitions and all seven participants as a function of cycle number directly before and after the perturbation. The error bars were calculated as the average over all individuals' SDs across trials. Due to space limitations, only eight perturbation magnitudes are displayed. As to be expected, large effects of the perturbations were observed at the perturbed cycle C0 and HE deviated from the baseline level with a magnitude that scaled with the perturbation; larger perturbations lead to larger HE values, indicating that the experimental perturbation had a significant effect on the performance. During subsequent cycles, HE showed an approximately exponential return back to preperturbation values. This return was not symmetrical for positive and negative perturbations of the same magnitudes. Comparing the largest negative perturbation, P–7, with the largest positive perturbation, P+7, it was apparent that P+7 showed a faster return, even though for P+7, the ball amplitude deviated more from target (approximately +0.37 vs. –0.27 m for P+7 vs. P–7, respectively).

View larger version (16K): [in this window] [in a new window]   FIG. 5. Grand averages over all repetitions and participants of height error (HE) plotted as a function of cycle number. The 8 panels show the data for 8 perturbation magnitudes. Different conditions are shown by different gray shades. The error bars denote the average of the individuals' SDs across trials. The data points for the 4 conditions are slightly shifted for better visibility.

RELAXATION TIMES OF HEIGHT ERROR. To quantify the rate of return , an exponential function was fitted to the HE data (Eq. 8). Although HE is not a state variable, it is proportional to the square of the ball velocity after impact, which is a state variable. The exponential curve fits are illustrated for 10 of the 14 perturbation magnitudes for = 0.6 in Fig. 6. The two smallest perturbations P+1/P–1 and P+2/P–2 could not be fitted. The different lines represent the fitted curves for the grand average of HE over all eight repetitions and seven actors and shown over eight cycles after the perturbation (from C0 to C7). We chose to fit the grand average rather than per-trial or per-subject fits because the resulting relaxations were quite variable. The curve fits rank-order with perturbation magnitude, with larger perturbations producing larger perturbation amplitudes (y₀) and longer relaxation times, . The R² values for the 10 fits for all participants ranged between 0.84 and 0.99, with an average of 0.96.

View larger version (14K): [in this window] [in a new window]   FIG. 6. Exponential curve fits of the height error HE for 10 of the 14 different perturbation magnitudes for a coefficient of restitution of 0.6. The dots are grand averages over repetitions and participants. The height errors for P+1/P–1 and P+2/P–2 are not fitted with exponential curves.

View larger version (18K): [in this window] [in a new window]   FIG. 7. Relaxation times obtained from the exponential fits of (HE). Different conditions are shown by different lines and symbols. The gray shaded box indicates that the values for the smallest perturbations have to be interpreted with caution (see text for discussion).

View larger version (19K): [in this window] [in a new window]   FIG. 8. Impact acceleration AC (averages over participant and repetition) plotted as a function of cycle number. The error bars denote the average of the individuals' SDs across trials. Each of the 8 panels shows the data for 1 perturbation magnitude. Different conditions are shown by different lines. The data points for the 4 conditions are slightly shifted for better visibility.

ACTIVE MODULATION OF RACKET TRAJECTORIES. A first assessment of the strategy that actors applied to deal with perturbations can be obtained from Fig. 9, which shows illustrative racket trajectories of one participant. For the selected eight perturbations, the median of the racket cycles directly after the perturbation is shown for 1 s (going slightly beyond the impact). For comparison, the median trajectory and its interquartile range from two control trials were calculated. As can be seen, the racket trajectory was modified depending on the magnitude of the perturbation. For P–7 the racket cycle shortened such that the racket trajectory exited the error band after approximately half a cycle as indicated by the white dot in Fig. 8. Analogously, when the positive perturbations became larger, the racket cycle immediately "stretched" and exited the error band after the valley. With smaller perturbations, the exit times became longer and in this participant no deviations were observed for P–1 and P+1. (For this participant, the conditions P–2 and P+2 induced modulations with exit times). Over all participants, the smallest perturbations P–1 and P+1 lead to deviations in only 24% of the trials. For P–2 and P+2, 80% of the trials showed significant deviations measured by exit times. For all larger perturbations, every trial showed exit times.

View larger version (17K): [in this window] [in a new window]   FIG. 9. Exemplary trajectories from 1 participant showing the adaptations of the racket trajectories to the perturbed ball trajectory in the cycle directly after the perturbation. For selected perturbations, the median perturbed racket trajectory (calculated over 8 trials for the same perturbation condition) is shown by the solid black line. The dashed gray lines are the median and 1.5 of the interquartile range calculated from the control trials of the same subject. The white circles denote the moment where the median trajectory exits the error band of the control trials. The beginning of the trajectory segment is at the perturbed impact, the interval is 1 s.

View larger version (9K): [in this window] [in a new window]   FIG. 10. Grand average of exit times for the perturbation magnitudes averaged over the conditions. The error bars denote the SDs.

View larger version (17K): [in this window] [in a new window]   FIG. 11. Period averages (over participant and repetition) between impacts T as a function of cycle number. The error bars denote the average of the individuals' SDs across trials. Each of the 8 panels shows the data for 1 perturbation magnitude. Different conditions are shown by different lines.

View larger version (22K): [in this window] [in a new window]   FIG. 12. Racket amplitude averages A (over participant and repetition) plotted as a function of cycle number. The error bars denote the average of the individuals' SDs across trials. Each of the 8 panels shows the data for 1 perturbation magnitude. Different conditions are shown by different lines.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL METHODS RESULTS DISCUSSION GRANTS ACKNOWLEDGMENTS REFERENCES

 
The present experiment examined the role of passive stability and active control in a rhythmic perceptual-motor task where participants aimed to steadily bounce a ball to a given target height. A series of previous studies on steady-state performance of the same experimental task established that actors are sensitive to and exploit the stability properties defined by the dynamics of the racket and ball movements (Dijkstra et al. 2004; Schaal et al. 1996; Sternad et al. 2000, 2001). This strategy implies that small errors, such as unintended deviations of the ball from the target height, require no corrections as the ball trajectory passively relaxes back to its steady state. Large perturbations, however, may take the system outside its basin of attraction and stable period-1 solutions are lost if the errors are not actively corrected for. In extension of these studies, Siegler and colleagues have shown in the same virtual task as in this study that novice actors can also perform the task with positive accelerations, a strategy that appears less efficient but also leads to continued successful performance of the task (Morice et al. 2007). These results may be taken as an indication that active control was applied. The present study presented perturbations to examine whether actors changed their strategy to correct for such induced errors and whether their strategy is dependent on the magnitude of perturbations.

A set of 14 perturbations of different magnitudes was designed for each of the four coefficients of restitution based on the basin of attraction that was determined from the ball bouncing model. As the stability boundaries were different for different coefficients of restitution , the experiment was conducted with four values. The predictions about relaxation behaviors after perturbations were based solely on the passive dynamics of the ball-racket system as the ball bouncing model does not include any active control of the racket movements. The first prediction was that if actors exploit this passive stability and do not change their racket movements according to perceived error information as long as errors are small, a sharp change in behavior should be seen for perturbation inside and outside the basin of attraction; for larger perturbations, actors either lose period-1 stability or apply active corrections to their racket movements to regain stability. Such sensitivity to the boundary of stability is present in purely passive dynamical walking models where disturbances outside the narrow basin of attraction make the bipeds fall (Garcia et al. 1998; Schwab and Wisse 2001). A second set of predictions formulated more qualitative expectations. a) With increasing perturbation magnitude, the time for returning to baseline performance increases for all . b) The relaxation time is longer for negative perturbations than for positive perturbations for all . And c) for positive perturbations, the smaller should show faster returns than the higher . The alternative hypothesis was that actors perceived all applied errors, even small errors, and actively adjusted their racket movements to regain steady state performance. This third prediction adopts the extreme position that actors always correct for errors in their performance and compensatory actions should continuously scale with the magnitude of the perturbation.

The previous study by de Rugy et al. (2003) already examined perturbations in the ball-bouncing task and identified active modulations of the racket trajectory. However, several essential aspects were different in this previous study. First, the perturbations were applied in terms of changes in the coefficient of restitution , a parameter of the model, and not in terms of ball release velocity, a state variable of the model. Therefore the actual perturbation effects on the height error depended also on the ball and racket velocity. As perturbation magnitudes were not as accurately controlled as in the present study, the resulting effects were not analyzed as a function of perturbation magnitude but rather pooled over all perturbations. Further, small perturbations were excluded to ensure that effects were observable. As no analyses of the basin of attraction were available, the effect of the stability boundary on behavior could not be addressed. Also the virtual set-up only provided a visual interface and was therefore not as realistic as the current development with the haptic contact. Despite these differences, the findings clearly indicated that actors adjusted their racket movement periods not amplitudes to re-establish the stable pattern, i.e., they actively tracked passive stability (see also Dijkstra et al. 2004) (Fig. 11). In contrast, the present analyses revealed significant modulations of both racket period and amplitude. This difference in results can be explained by noticing that the perturbations of de Rugy et al. were on average 0.45 m/s in absolute value with a range from 0.3 to 0.6 m/s (the changes were designed as random changes in values which can be converted into velocities assuming an average ball velocity). This roughly corresponds to the perturbations P–3 to P+3. In Figs. 11 and 12, it can be observed that the modulations in racket period were indeed relatively larger than the modulations in amplitude for that range of perturbations with = 0.5. Thus we believe the difference in results can be explained by the relatively small range of perturbations in the de Rugy et al. study.

The present work built on this experiment but significantly developed the theoretical framework and fine-tuned the experimental approach to afford the testing of model-based quantitative and qualitative predictions about the effects of perturbations on dynamically stable behavior.

Overall the relaxation behavior was considerably faster than the model predicted. Based on the assumption that the racket trajectory remains unchanged in the face of perturbations, the model predicted that the return to steady state should take two to several tens of bounces for perturbations, even when the perturbation was still inside the basin of attraction. Solutions other than period-1 or the sticky solutions as predicted for perturbations outside the basin of attraction were never observed. This shows that in all cases, participants actively accelerated their returns to the preperturbation steady state to as fast as one to three bounces. This fast return behavior was accomplished by active modulation of the racket trajectory in both amplitude and period. Analysis of the continuous trajectory showed deviations in the first cycle after the perturbation. Interestingly, the results of the exit times provided no signs of sensitivity to the boundary of the basin of attraction (prediction 1). Both the performance measure height error and also the racket trajectories showed gradually more pronounced adaptations to increasing perturbation magnitudes as stated in prediction 3. Support for this gradual change in racket kinematics was seen in the analysis of exit times, i.e., when the racket trajectory deviated from the typical trajectory in unperturbed conditions (Figs. 8 and 9). This indicated that for all applied errors some compensatory behavior is seen despite the dynamic stability afforded by the task. The only caveat to this conclusion is presented by the smallest perturbations as the return back to steady state was very fast and did not lead to significant deviations of the racket trajectory.

Despite this clear indication of active error compensation, the focal-dependent measure, acceleration of the racket at contact, exhibited very little changes for all but the three largest negative perturbations. Consistent with previous studies this demonstrated that conditions for passive stability, negative acceleration at impact, were maintained or immediately reestablished. This permits the conclusion that the racket trajectory is modulated to optimize the next ball contact, which also implies that actors may set up conditions for passive stability to assist return to steady state.

Further support for the sensitivity to passive dynamic stability properties are provided by the results that showed qualitative agreement with the second set of predictions of the model. First, the return to steady state after a perturbation took longer for larger perturbation magnitudes in all conditions. This was evidenced by the increasingly longer relaxation times of the ball height errors over successive cycles after the perturbation with larger perturbation magnitudes. Second, relaxation times were shorter for positive compared with negative perturbations of corresponding magnitudes, mirroring the asymmetry in the basin of attraction. Third, lower conditions exhibited faster returns for positive perturbations than higher conditions. This is consistent with the topology of the basin of attraction, which is wider for higher conditions but only on the positive side.

For the three most negative perturbations (P–7, P–6, P–5) high positive impact accelerations were observed in the two cycles after the perturbation, coincident with relatively large changes in the racket amplitudes. A first conj