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Author’s version (published in IEEE Transactions on Haptics Issue No.2, vol.8, pp: 222-234, 2015)DOI: 10.1109/TOH.2014.2375173.

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

Laura Marchal-Crespo, Mathias Bannwart, Robert Riener, Heike Vallery

Abstract —Bouncing a ball with a racket is a hybrid rhythmic-discrete motor task, combining continuous rhythmic racket movementswith discrete impact events. Rhythmicity is exceptionally important in motor learning, because it underlies fundamental movementssuch as walking. Studies suggested that rhythmic and discrete movements are governed by different control mechanisms at differentlevels of the Central Nervous System. The aim of this study is to evaluate the effect of fixed/fading haptic guidance on learning to bouncea 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. Althoughmotor learning in the ball-bouncing task with varying gravity has been studied, the effect of haptic guidance on learning such a hybridrhythmic-discrete motor task has not been addressed. We performed an experiment with thirty healthy subjects and found that the mosteffective training condition depended on the degree of rhythmicity: Haptic guidance seems to hamper learning of continuous rhythmictasks, but it seems to promote learning for repetitive tasks that resemble discrete movements.

Index Terms —Haptic guidance, fading guidance, motor learning, hybrid rhythmic-discrete task

1 INTRODUCTION

T THE long-standing Guidance Hypothesis states thatphysically guiding a movement will impair motor

learning1, because it changes the input-output relation-ship of the task to be learned [2], [3]. A number ofstudies have found that haptic guidance that reducedperformance errors during training did not aid motorlearning but in fact hampered it [4], [5]. Nevertheless,sophisticated robotic mechanisms have been developedto support motor training of complex movements, suchas walking or multi-joint arm movements [6]. Thesedevices can haptically guide subjects to move their limbsin a correct kinematic pattern. The rationale is that hapticguidance provides the Central Neural System (CNS)with additional proprioceptive and somatosensory cues,which may facilitate movement planning and enable theCNS to attempt more advanced movement strategies.

Recent studies with therapy robots contradict theguidance hypothesis, suggesting instead that guidanceshould fade as learning progresses, providing justenough guidance to allow participants to practice the

• L. Marchal-Crespo, H. Vallery and R. Riener are with the Sensory-Motor Systems (SMS) Lab, Institute of Robotics and Intelligent Systems(IRIS), ETH Zurich, Switzerland and Medical Faculty, Balgrist UniversityHospital, University of Zurich, Switzerland.E-mail: laura.marchal@hest.ethz.ch

• H. Vallery is also with the Department of BioMechanical Engineering,Faculty of Mechanical, Maritime and Materials Engineering, Delft Uni-versity of Technology, The Netherlands.

• M. Bannwart is with the Institute of Neuroinformatics (INI), ETH Zurich,Switzerland.

1. Motor learning is defined as a set of processes associated withpractice or experience that lead to relatively permanent changes in theability to perform a skill [1].

task, while decreasing the guidance to encourage learn-ing [4], [7], [8], [9]. For example, an experiment con-ducted with unimpaired children who practiced learningto drive a power wheelchair with fading guidance [7]showed that children who practiced with fading hapticguidance from a robotic joystick learned better thanchildren who practiced without physical guidance. Sim-ilarly, subjects trained to perform a tennis stroke with arobotic tennis trainer that faded the guidance as learningprogressed learned to time their movements better thansubjects who trained with visual feedback [10].

This apparent contradiction with the guidance hy-pothesis can be explained by emerging evidence thathaptic guidance may be specifically useful for learningtiming [11], [12], [13]. Several studies showed a benefit ofhaptic guidance in learning to reproduce the temporal,but not spatial, characteristics of complex spatiotemporalcurves [14], [15]. The effect of haptic guidance on learn-ing a visuo-manual tracking task has been evaluated infurther studies, which found a positive effect of guidanceon the time-related components of a dynamic task, suchas an increase in movement speed and smoothness [16],and better learning of temporal force patterns [17].Training with haptic guidance also benefited learning toplay a pinball-like game [18]. In these studies, hapticdemonstration of optimal timing, rather than movementmagnitude, may have facilitated skill transfer. Theseresults suggest that training with appropriately designedhaptic guidance can enhance learning of some motorskills, such as time-critical tasks. However, few studiestested for long-term retention [12], [11], [8], and there-fore conclusions on the effect of haptic guidance onmotor learning should be taken cautiously [19]. In thefollowing, we refer to short-term training as any training

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protocol to achieve motor-learning that is limited to threesessions or less.

Timing of an action plays a crucial role in theproper accomplishment of many meaningful tasks, suchas hitting a moving object (discrete timing task), orrhythmically moving an object (continuous timing task).Research in motor control has focused on two maintask classes: (i) discrete movements (e.g. target-orientedreaching movements) that include well-defined posturesat the beginning and the end of the movement, which aremaintained for a non-negligible duration (dwell time),and (ii) rhythmic movements (e.g. walking) that are pe-riodic or at least repetitive movements [20]. Several stud-ies have demonstrated that discrete and rhythmic move-ments are driven by different control primitives [21],[22], [23]. Indeed, brain imaging studies have shownthat the production of rhythmic and discrete movementsinvolve different brain areas [24]. Specifically, rhythmicmovements require significantly less cortical and sub-cortical activation, suggesting that discrete tasks mayrequire more complex control mechanisms located atdifferent levels of the CNS.

Regarding rhythmic tasks, different haptic and visualguidance paradigms have been tested for their effect onlearning [25]. An extensively studied rhythmic task con-sists in hitting fixed targets by exciting a virtual second-order (mass-spring-damper) system at its resonant fre-quency [26], [27]. The effect of different haptic trainingparadigms on learning this rhythmic task was recentlycompared to learning a path-following task [28]; resultsshowed that the benefits of the different paradigmsdepended strongly on the task type.

However, not all motor tasks fall into one of the twoclasses “discrete” or “rhythmic”, but may lie somewherebetween the two. Therefore, one may better designatea movement by its “degree of rhythmicity”. Severalmeasures have been proposed to characterize rhyth-micity, for example the “index of harmonicity” [29],which is based on local extrema in the acceleration timeseries, or the mean-squared jerk [20]. As a robust andintuitive measure, here we characterize the “degree ofrhythmicity” of an “almost periodic” [20] movement bythe “activity period” as defined in [30], which is 100%minus the percentage of dwell time [20] during onecycle. This means that periodic tasks with negligibledwell time between consecutive movements approach adegree of rhythmicity of 100%, while repetitive discretetasks with negligible movement time compared to dwelltime approach 0% rhythmicity.

A classical example of an ambiguous task is bouncinga ball to a target apex height. The classification of thistask depends on the apex height: bouncing a ball to ahigh apex results in an almost discrete task with longdwell times, while a lower apex can be classified asa continuous task with a high degree of rhythmicity.Modifying gravity in a virtual environment has the sameeffect on the racket trajectory as changing the apexheight: the higher the value of gravity, the more rhythmic

the task becomes. This discrete versus rhythmic behaviorin the bouncing ball task was observed in previousstudies, which demonstrated that the increase in cycleperiod did not lead to a simple scaling of the rackettrajectory, but to a change in the racket trajectory shapefrom an approximately sinusoidal profile to a sequenceof discrete movements separated by dwell times [30],[31].

Bouncing a ball requires sophisticated motor control.Mathematical analysis of the bouncing-ball model re-vealed that feedforward control yields satisfactory per-formance once the racket-ball interaction has settled tothe correct steady-state movement [32]. Passive stabilityrelated to the bouncing movement implies that smalldeviations from the desired apex attenuate over succes-sive bounces without involving active corrective motoractions from subjects. This property depends on nega-tive racket acceleration at ball impact. In line with thisobservation, the racket acceleration at impact has beensuggested to be a good performance variable to quantifyfeedforward control [33] – better performance in feedfor-ward control seems to be related to a lower accelerationat impact. Recent studies have demonstrated that, inaddition to feedforward control, subjects may also applyfeedback control when an error is observed after aperturbation, and actively correct the racket movementto accelerate the return to the steady state [33].

Motor learning in the ball-bouncing task with varyinggravity has been extensively studied [30]. However, theeffect of haptic guidance on learning such a hybridrhythmic-discrete motor task has not been addressed.Here, we use a robotic device to apply different typesof haptic guidance while subjects bounce a ball in avirtual-reality paradigm in different virtual gravity en-vironments, in order to investigate the effect of hapticguidance on learning, both in time-critical continuousrhythmic movements (high gravity) and in discrete tasks(low gravity). In particular, we investigate (i) a fixedamount of haptic guidance, (ii) no guidance, and (iii) anamount of guidance that fades as learning progresses.The stated task objective was to reach a target apexheight. As reference for the guidance, a model-predictivecontroller continuously calculates the racket’s optimalmovement based on the two-layered hybrid-discrete taskmodel introduced in [30]. The selection of performancemeasures is crucial to validate training [25]. Here, learn-ing is assessed by two measures: First, a significantreduction of the overall error (difference between theactual ball apex and the target apex). Second, the racketacceleration at ball impact (a time-critical variable), toquantify feedforward control performance [33].

Based on the previous work on time-critical motorlearning experiments, we hypothesized that fading guid-ance may be especially suited to help people learn thetask under all gravity levels. Furthermore, we expectedbetter generalization (i.e. transfer of learning gains) tountrained gravities in subjects trained without guidanceand, to a lower extent, with fading guidance [34].

adjustment slide

handle

load cell

elbow rest

potentiometer

Fig. 1. Virtual display for the 3 gravities trained (g1, g3 and g4) and assistive robotic device with one degree of freedom.

2 METHODS

2.1 Bouncing-ball simulator

We designed a bouncing-ball game and an assistive robotto perform the desired experiments. The task to be per-formed was to bounce a virtual ball with a virtual racket.The experiment was implemented using a one-degree-of-freedom haptic device (Fig 1, right) developed in ourlab [35]. The robotic device consists of an actuated leverwith a handle and an elbow rest. The lever is actuated bya Maxon RE 35 DC Motor (maxon motor, Switzerland),with U = 48V nominal voltage, combined with a Har-monic Drive gearbox (HFUC-2UH, size 14, HarmonicDrive AG, Germany) with a transmission ratio of i = 100.The combined moment of inertia of motor and gearbox is9.86·10−2 kgm2 at the output shaft. The robot is equippedwith an incremental encoder on the motor shaft and apotentiometer on the joint (GL60 Contelec, Switzerland)that redundantly measure the elevation angle, and aforce sensor (KD 140 200N, ME-Meßsysteme GmbH,Germany) that measures the torque τuser produced bythe subject on the robot. Precision and accuracy of theangle measurement are 0.007◦ and 0.1◦, respectively, andprecision and accuracy of the torque measurement are0.2 Nm and 0.1 Nm, respectively. Coulomb friction in thedrive train is approximately 0.1 Nm. The lever can beadjusted to the length of the subject’s forearm. In closed-loop zero-impedance control, the residual reflected iner-tia of the device was identified to be 2.5 ·10−2 kgm2, andthe reflected damping to be 1.4 Nms/rad. Force controlbandwidth is 8.8 Hz.

The bouncing-ball simulator (Fig 1, left) was devel-oped in MATLAB/Simulink, using the xpc target real-time operating system and the Virtual Reality Toolbox.The simulator consists of a 3-D graphical presentationof a ball that falls under the influence of gravity onto aracket. The rotation of the lever with respect to the elbowjoint in supination forearm position is mapped to thevertical motion of the virtual racket. The angular positionof the lever θ was linked to the vertical position r of

the virtual racket by a linear mapping where θ = rm−1.This is a compromise given that we emulated a multi-joint, combined rotational and linear task (hitting a ballwith a racket) using a single-joint device. The kinematiclimitations of the haptic device only allow a rotationalmovement of the forearm about the elbow joint, whichdoes not realistically represent racket movements in realtennis playing, where vertical translations would play arole. By mapping linearly, we are consistent with [30],and we make sure that subjects do not reach a near-singular configuration for larger angles, which wouldreduce their ability to influence the impact location. Thismapping does not influence performance or evaluationthereof, as long as subjects are capable of understandinghow the haptic lever movement relates to the virtualracket’s movement.

The aim of the game was to rebound the virtual ballto a certain apex height shown on the screen, whichremained constant through the experiment. Impact withthe racket was modeled identical to [30]: The ball veloc-ity b+ after impact is given as a function of ball velocityb− before impact, the coefficient of restitution α and theracket velocity r− before impact:

b+ = −αb− + (1 + α)r−. (1)

The dynamics of the ball during free flight are ballisticand only a function of the initial conditions and thechosen gravity. This implies that once the ball impactedwith the racket, the trajectory of the ball is fully deter-mined and the subject has no more influence on it. It alsoimplies that the task is not a trajectory tracking task; taskperformance is exclusively determined by the final stateof the racket at the instant of impact with the ball.

The game consisted of performing the task under theinfluence of 5 different gravity values. All environmentshad the same layout but different background color, tohelp participants identify the relative gravity by the colorof the environment.

2.2 The two-layered hybrid-discrete task model

Rhythmically bouncing a ball is a hybrid task thatcombines continuous movement of the racket with thecontrol of discrete racket-ball impacts.

To obtain references for haptic guidance, the optimaltime till the next impact event, the final racket positionand velocity (immediately before impact), as well asthe racket’s continuous reference position, velocity andacceleration trajectories until impact were calculated ateach sampling instant using predictive optimization.

Changing the value of gravity in the virtual environ-ment affects the racket’s optimal movement: The higherthe value of gravity, the more rhythmic the task becomes,while a lower gravity reduces the bouncing frequency,changing the motor command to a discrete task (Fig 2).

0 1 2 3 4 5 6 7 8 9

time(s)

Gravity g4=9.81m/s2

BallRacketDesired apex

0 1 2 3 4 5 6 7 8 9

time(s)

Gravity g1=0.61m/s2

BallRacketDesired apex

Fig. 2. Examples of racket and ball trajectories under alow gravity level (on top, g1 = 0.61m/s2) and under “Earth”gravity (bottom, g4 = 9.81m/s2), while the robotic devicestiffly guided the movement. The dashed line representsthe desired ball apex height (hdes = 0.8m).

The hybrid control architecture was based on a two-layered framework [30], which minimizes a cost func-tional J over the movement between discrete impactevents [30] (To simplify notation, variables are consid-ered dimensionless):

J = (rf − rref,f)2 + (rf − rref,f)

∫ tf

[wr(r(t) − rref,0)2 + weu(t)

2]dt.(2)

with reference values denoted by the index ref , initialvalues by 0, and final values (immediately before impact)

by f. The control input u is the input to a simplifiedmuscle model [30]. The cost functional accounts for theneed to reach a final racket position rref,f and velocityrref,f at impact with little control effort (penalized bythe weight we). Furthermore, the weight wr enforces theracket (i.e. the elbow) to remain at the resting positionrref,0 when possible, so that the trivial solution wherethe racket is raised until it contacts the ball at thetarget apex (and remains there) is avoided. As weightingparameters, the same values as in [30] were employed:wr = 0.012

√g, we = 0.00002. The task is to find the

optimal duration T = tf − t0 of the movement (i.e. theremaining time to impact), the final reference values, aswell as the corresponding continuous racket trajectory.

The optimization problem was split into two layers,as in [30]: In an outer loop, the time duration T wasiteratively adjusted and final reference values were cal-culated. In an inner loop, standard LQ control was usedto calculate the reference continuous movement betweenimpact events for a given T and final reference values.

In [30], the optimization for the racket movement wasrun only once between the discrete events. One conclu-sion of that paper was that it is unlikely that humansre-optimize their internal controller at each impact ortime step. However, replicating this for a haptic guidanceparadigm is only meaningful when the racket is notallowed to deviate from the a priori calculated trajectory,so in stiff guidance. For guidance with low stiffness,where subjects may deviate from this trajectory, theoptimal time to impact and the corresponding optimalmovement may change drastically. Compliant guidancealong an originally planned trajectory without takinginto account de-synchronization between subject androbot can lead to undesired effects, such as the “beat phe-nomenon” [36] observed for gait rehabilitation robots.Only consistent re-optimization of the trajectory ensuresmeaningful assistance also at low guidance forces. Thus,a model-predictive controller that performs the mini-mization at each sampling instant was needed.

The calculations needed in the outer loop are simple:As the ball’s movement is ballistic, each T uniquelydetermines the ball and racket height at impact and theimpact equations (1) determine the necessary velocity ofthe racket to bounce the ball to the desired apex height(so rref,f = r−). For this outer loop, we used a goldensection algorithm, which guarantees convergence withinone sampling step.

For the inner loop, however, the need for on-lineoptimization led to a challenge: The desired samplingtime of 1 ms required a very efficient implementation ofthe predictive optimization (detailed in the appendix):In contrast to standard algorithms to solve the optimalfollower problem, it is not necessary to perform time-consuming online integration (to solve the Ricatti equa-tion). The simplification is made possible by recogniz-ing that the inner loop does not need to compute theentire trajectories for each iteration of T , only the costassociated with a given T is needed until the optimal T

is found. With this idea, the cost functional can be re-formulated such that it only depends on initial and finalconditions, which are found by transforming the contin-uous differential equations into a discrete mapping frominitial to final conditions and vice versa.

2.3 Training conditions

Three control modes were needed for the different train-ing paradigms: First, the robot should allow the subjectto try the task without guidance. Second, the robot hadto stiffly move the virtual racket to follow the optimalreference trajectory. Finally, the robot should reduceguidance as learning progressed. In the following sub-sections, the three control strategies designed to achievethese behaviors are described in detail.

2.3.1 No guidance

In the no-guidance condition, the robot followed the sub-ject’s movements in closed-loop force control. Thus, therobot was compliant and the subject was able to achieveany elbow flexion with minimal interaction forces causedby the dynamics of the robot. The virtual racket wasrendered as a 1 kg object (always experiencing Earthgravity). This force control loop was realized by a pro-portional force feedback on the measured moment τmeas,combined with feed-forward compensation of the leverweight. The reference moment τrac was calculated byprojecting the virtual racket weight onto the lever. Theball-racket impact was rendered as a 2 Nm torque thatwas applied at impact and lasted 50 ms. Although theimpact duration in real elastic ball-racket interactionexperiments was observed to be around 30±9ms [37], weincreased this time to ensure that all subjects perceivedthe impact forces.

2.3.2 Fixed guidance

A stiff PD position controller with proportional gainKs = 600Nm/rad and derivative gain Kd = 60Nms/radwas implemented to track the optimal reference racketposition θref and velocity θref . A PD controller is apractical solution to prevent errors during a tracking taskdespite subject interactions. The reference position andvelocity at each sampling instant were calculated usingthe two-layered hybrid-discrete optimization of Sec. 2.2.The ball-racket impact was again rendered as a 2 Nmtorque that was applied at impact and lasted 50 ms.

2.3.3 Fading guidance

Fading assistance is a common strategy to include ac-curate trajectory demonstration of the fixed guidancemode, while still allowing subjects to try the task bythemselves [38]. Here, fading guidance was realized byan impedance controller on the measured angle θmeas

that guided the movement by a moment τguid usingclosed-loop force control (Fig. 3), with

τguid = Ks(θref − θmeas) +Kd(θref − θmeas). (3)

The stiffness Ks and the damping Kd are equivalent tothe gains of the position controller. The only difference isthat in position control, no inner force feedback loop wasused, to allow maximal values for Ks and Kd (withoutrisk of instability). Guidance of the impedance controllerwas systematically faded on a hit-by-hit basis, until animperceptible force had settled [11]:

Ki,k+1 = fRKi,k, i ∈ s, d, (4)

where Ki,k and Ki,k+1 represent the impedance gainsKs and Kd in the k-th and k + 1-st hit, respectively,and fR is the robot forgetting factor (fR = 0.997). Theinitial values of the impedance stiffness and dampinggains were set to Ks,0 = 300Nm/rad, Kd,0 = 35Nms/rad.As the gains were in the same order of magnitude, theguidance during these first hits was similar to the stiffposition controller.

Fading-guidance controllers typically settle the finalassistance force to a value that allows subjects to movefreely, and the same was done here. Simultaneously, alsothe virtual racket rendering τrac, calculated as in the no-guidance condition, was faded hit by hit at the same rate.By the end of the training phase, subjects felt the taskdynamics without any assistance. The ball-racket impactwas again rendered as a 2 Nm torque lasting 50 ms.

2.4 Experimental protocol

The experiments were approved by the Research EthicsCommittee of ETH Zurich, and all subjects providedinformed consent. Thirty right-handed subjects (11 fe-males) with no known motor deficits took part in theexperiment. The subjects were between 18–31 yearsold (mean age 25.7 ± 2.5). They were randomly as-signed to one of three groups, of 10 members each: the“no-guidance”, the “fixed-guidance” and the “fading-guidance” groups.

The robot was rigidly attached onto a table, and theseat height was adjusted so subjects could rest theirelbow in a comfortable position. Subjects sat in frontof a computer screen and were instructed to keep theelbow on the elbow rest and grasp the robot handlein supinated posture. Then, they were asked to bouncethe virtual ball shown on the screen to match the ballmidpoint with a depicted apex in the virtual environ-ment, by moving the virtual racket through the rotationof the robot handle (Fig 1). The virtual apex remainedconstant throughout the experiment. No other indicationof movement accuracy (e.g. number of times that the ballreached the desired apex) was provided, in order notto divert the subjects’ attention from the bouncing taskitself.

Each subject participated in two experimental sessionson different days (Fig. 4). During the second experi-mental session (3-4 days after training), only long-termretention was evaluated, to allow for time-dependentmemory consolidation [39], [40]. Practicing during asingle-day session can already have quasi-permanent

control user

sensor

racket

rendering

actuated

racket

optimi-

zation

dynamics

variable

guidance

θmeas

τmeas

θref τguid

τuser

τactτrefθ

Fig. 3. Impedance controller for fading guidance: In each time step, predictive optimization provides a reference angleθref in function of the current ball position b, the current lever angle θ, and the desired apex height hdes. Correspondingguiding moments τguid to the user are rendered by means of an inner force control loop.

effects on the ability to perform a skill [3]. We decidedto run a single-day training session because the ball-bouncing task was a relatively easy task that we believedcould be mastered in a short time. Therefore, performingmultiple-day training sessions could increase the unde-sirable boredom effect, i.e. performance degradation dueto lack of attention.

The first session started with a warm-up period inno-guidance mode, to allow subjects to familiarize withthe robot and the virtual reality. During baseline, sub-jects performed under 5 different gravity levels (g1 =0.61m/s2, g2 = 1.09m/s2, g3 = 2.45m/s2, g4 = 9.81m/s2,g5 = 15m/s2) with no guidance, hitting the ball 26times under each gravity condition. The gravity valueswere selected from the gravity levels presented in [30]so that the difference between gravities could be easilyrecognized. The different gravities have associated dif-ferent “degrees of rhythmicity”: When performing thetask with the lower gravity level g1, the dwell time (de-fined as time between impacts where the racket velocityremains below 5% of its peak value) is 50% of thetotal time between impacts, whereas the percentage ofdwell time is reduced to 10% when performing with the“Earth” gravity level g4, and to 25% when performingwith the “medium” gravity g3 [30]. The cycle periodsassociated with the different gravities are: T1 ≈ 2800ms,T2 ≈ 2100ms, T3 ≈ 1400ms, T4 ≈ 700ms, T5 ≈ 550ms.The gravity levels were presented in random order, witha pause of 3 s between them. The time to completethe baseline condition was 7 minutes. Subjects wereinformed about the current gravity level through differ-ent colors in the virtual environment and through thegravity name (g1 to g5) on the screen (Fig 1). Betweenthe baseline and the following training conditions, weallowed for 1–5 minutes rest.

During the training phase, subjects performed thetask with 3 different gravities (g1, g3, g4). Training wasdivided into 9 blocks, each consisting of 50 hits underthe same gravity level (3 gravities × 3 blocks). The 9

Warm up

Baseline26 hits per gravity

Short-Term Retention26 hits per gravity

Training sub-phase 1

(no, fixed, or fading guidance)

Long-Term Retention26 hits per gravity

Wash-out20 hits per gravity

x 5 gravities

x 3 gravities

(g1, g2, g3, g4, g5)

(g1, g3, g4)

Fig. 4. Experimental protocol

blocks were presented in randomized order, to promotelearning of the different gravity levels [41]. To avoidfatigue, subjects were allowed to rest for 1-2 minutesafter completing training blocks 3 and 6. We denoteas training sub-phases 1, 2 and 3 the resulting trainingblocks separated by the resting times. During training,subjects in the no-guidance group did not receive as-sistance from the robot. Subjects in the fixed-guidancegroup, who were trained with stiff position control,were instructed to actively follow the robot’s movement.The fading-guidance group, which trained while thestiffness of the impedance controller faded throughoutthe training blocks, were also instructed to cooperatewith the robot. Following the training phase, subjectswere asked to perform the task again with no guidanceunder the 3 gravity levels, in order to wash out possibleaftereffects (20 hits per trained gravity).

After a short break of 5–10 minutes, subjects pro-ceeded with a short-term retention test. All subjects alsoreturned after 3–4 days for a long-term retention test.Generalization is a crucial aspect of motor learning.It is a type of transfer of learning that occurs fromone task to another very similar task [1]. In order toevaluate motor learning generalization, subjects playedagain under the influence of the 5 gravity levels duringthe retention tests. We selected two different gravitylevels to test for generalization: a gravity level that wasbetween two trained gravities (g2 = 1.09m/s2), anda second gravity that was higher than the maximumtrained gravity (g5 = 15m/s2). The aim was to testfor differences in generalization between a gravity levelclose to two consecutive trained gravity levels and agravity level outside the range of the trained gravitylevels. Both retention tests followed the same structureas baseline, with 26 hits per gravity.

During baseline, training and retention tests, pertur-bations were introduced in a pseudo-random manner. Aperturbation was defined as a change in the coefficientof restitution–with default value α = 0.6–similar to [42].A change in α resulted in a ball that rebounded more(α > 0.6) or less (α < 0.6) than expected. The goal of theperturbations was to compel subjects to practice theirfeedback control. During baseline and retention tests,two perturbations (α = {0.75, 0.45}) were randomlyintroduced after the 5th and 13th trials. During eachtraining block, the perturbing hits were presented asfollows: No perturbations occurred during the first 5hits, neither during the last 12 hits, then each of these5 perturbations (α = {0.45, 0.5, 0.7, 0.75, 0.8}) was ran-domly introduced every 7 hits, in intervals long enoughto stabilize the movement before a new perturbation waspresented.

The time required to finish the first experimentalsession was about 30 minutes.

2.5 Data processing and statistical analysis

The overall error during each protocol (sub-)phase wascalculated as the mean absolute difference between theactual ball apex height h and the target apex (hdes =0.8m), without considering any first hit after a pertur-bation. We also calculated the mean absolute interactiontorque (τmeas) during each protocol block and the meanracket acceleration at impact of the last 5 hits in eachprotocol block. Note that there were 12 disturbance-freehits after the last perturbation, so subjects had enoughtime to settle to the steady-state movement. Data fromone subject in the fading-guidance group and one subjectin the fixed-guidance group were discarded because theydid not follow the instructions despite coaching, andinstead systematically rebounded the ball to a differentapex.

A two-sided t-test was used to evaluate whether sub-jects in different groups performed differently duringbaseline. An analysis of variance (ANOVA) with the

overall error during baseline was employed to evaluate ifperforming with any one gravity level was more difficultthan with the others. A two-sided paired t-test wasused to evaluate if learning occurred after training: Theoverall error and racket acceleration at impact over thelast 5 baseline trials were compared with the values atretention tests.

The percentage of overall error reduction from base-line to retention test was employed to normalize thedata. A repeated-measures ANOVA with the four mainprotocol phases (baseline, training, short-term and long-term retention) as within-subjects variables, and trainingconditions as between-subjects condition was performedto evaluate the racket acceleration at impact in thetrained gravity levels. We used a linear-mixed modelto test the effect that the training conditions and non-trained gravity levels (and their interaction) had onthe performance variables during baseline and retentiontests. We used a repeated measure ANOVA with baselineand the short- and long-retention test as within-subjectsvariables and gravity level as between-subject factor todetermine whether the gravity level had an effect onthe measured torque, and if subjects reduced the torqueafter training. We performed a univariate analysis withthe measured torque during training with the trainingstrategy and the gravity level as fixed factors to evaluatethe influence of the training strategy on the measuredtorque for different gravity levels. Correlations betweenmean overall error and racket acceleration at impactwere tested using Pearson correlation tests. Pairwisefollow-up comparisons were performed using Tukeycorrection. The significance value was set to p = 0.05. Allstatistical analyses were performed using the softwarepackage SPSS.

3 RESULTS

3.1 Overall performance

During baseline, we did not find significant differencesbetween training groups in the mean overall error(Fig. 5A) and racket acceleration (Fig. 5B). We founda significant difference between gravity levels in theoverall error during baseline (p < 0.001); in particular,subjects performed better with g2 and g3, as suggestedby a significantly smaller overall error compared to theother gravity levels (p < 0.05).

All training groups significantly reduced the overallerror from baseline to short-term retention (Fig. 5D, no-guidance: p < 0.001, fading-guidance: p = 0.002, fixed-guidance: p < 0.007) and long-term retention (Fig. 5D, allconditions: p < 0.001). We found a significant main effectof training condition on error reduction at long term(p = 0.023). Contrast revealed that subjects who trainedwithout guidance reduced the error to a greater percent-age than subjects who practiced with fading guidance(p = 0.006). We also found a significant interactionbetween the training condition and gravity (p = 0.028).

Fixed GuidanceFading

Guidance

No Guidance

Fixed GuidanceNo Guidance

Long retentionShort retention

Fading

Guidance

D: Percentage of mean baseline error reduced E: Reduction of racket acceleration at impact

Fixed Guidance

Fading Guidance

No Guidance

A. Mean overall error B. Mean racket acceleration at impact

at long and short retention from baseline to long and short retention

Long retention

Short retention

Training 3

Training 2

Training 1

Baseline

-10.00

-20.00

Long retention

Short retention

Training 3

Training 2

Training 1

Baseline

Long retention

Short retention

Training 3

Training 2

Training 1

Baseline

C. Mean Torque

Fig. 5. A: Mean overall error during baseline, training, and short- and long-term retention for the three differentconditions. B: Mean racket acceleration at impact. C: Mean interaction torque between subject and robotic device.D: Percentage of baseline mean overall error reduced at short- and long-term retention. E: Reduction of racketacceleration at ball impact from baseline to short- and long-term retention. Error bars show ±1 CI. Bars with asterisksshow a significant reduction. Significantly different training condition pairs are identified with an asterisk (p < 0.05).

Subjects did not significantly reduce their racket ac-celeration at impact. Only subjects who trained withoutguidance showed a tendency to reduce the acceleration(Fig. 5E, p = 0.110).

We found that subjects significantly reduced themeasured torque from baseline to short-term retention(Fig. 5C, p < 0.001), and long-term retention (p < 0.001).We also found a significant main effect of gravity levelon the measured torque (p < 0.001). In particular, wefound that subjects needed more torque to perform withgravity g4 than at any of the lower gravity levels (p <0.001), and even more torque to perform g5 (p < 0.001).We found a significant interaction between gravity andprotocol phase. In particular, subjects seem to reduce thetorque between baseline and retention tests to a higherextent for the higher gravities g4 and g5. We also foundthat the torque measured during training depended onthe training condition (Fig. 5C, p < 0.001). In particular,training with fixed guidance increased the torque com-pared to no guidance (p = 0.020) and fading guidance(p < 0.001). There was a tendency of more torque whentraining with no guidance compared to fading guidance(p = 0.147). The torque measured during training also

depended on the gravity level (p < 0.001), as observedduring baseline and retention tests. In particular, trainingwith Earth gravity g4 resulted in a higher torque thantraining with g1 (p < 0.001) or g3 (p = 0.005). We founda significant interaction between the training conditionand the gravity level: training with fixed guidance re-sulted in consistently more torque the higher the gravitylevel was, while the torque measured during trainingwith no guidance and fading guidance did not increaseas acutely.

3.2 Learning performance at trained gravity levels3.2.1 Mean overall errorAt the lowest trained gravity (g1 = 0.61m/s2), subjectsin the no-guidance and fixed-guidance groups signif-icantly reduced the error from baseline to short-term(no-guidance p = 0.007, fixed-guidance: p = 0.001) andlong-term retention (Fig. 6A, no-guidance: p = 0.005,fixed-guidance: p = 0.006). The percentage of errorreduction at long term was different between traininggroups (p = 0.049). Subjects trained without guidancereduced the error significantly more than subjects in thefading-guidance group (p = 0.038).

retention

Fixed Guidance

Fading Guidance

No Guidance6.00

-10.00

-20.00

Racket acceleration at ball impact in trained gravity levels

D: Low gravity g1=0.61 m/s2 E: Medium gravity g

3=2.45 m/s2 F: High gravity g

4=9.81 m/s2

retention

Baseline Training Long

retention Short

retention

Baseline Training Long

retention Short

retention

Baseline Training

Guidance

No Guidance

) 80.00

Fixed GuidanceNo Guidance

Percentage of baseline mean error reduced at short and long retention in trained gravity levels

Fading

Guidance

Fading

Guidance

A: Low gravity g1=0.61 m/s2 B: Medium gravity g

3=2.45 m/s2 C: High gravity g

4=9.81 m/s2

Fig. 6. Percentage of baseline mean overall error reduced at short- and long-term retention for the different trainingconditions in trained low gravity g1 = 0.61m/s2 (A), medium gravity g3 = 2.45m/s2 (B) and high gravity g4 = 9.81m/s2

(C). Racket acceleration at impact for the different conditions in low trained gravity g1 (D), medium gravity g3 (E) andhigh gravity g4 (F). Error bars show ±1 CI.

At the medium gravity (g3 = 2.45m/s2), subjectstrained without guidance (p = 0.013) and with fixedguidance (p = 0.005) significantly reduced the error atshort term. Subjects of all groups reduced the overallerror at long term (Fig. 6B, no-guidance: p = 0.014,fading-guidance: p = 0.046, fixed-guidance: p = 0.007),but with a significant difference in error reduction be-tween training groups (p = 0.043). Subjects in the fixed-guidance group reduced the error significantly morethan subjects in the fading-guidance group (p = 0.034).

At the high gravity (g4 = 9.81m/s2), all subjectssignificantly reduced the error at short term (Fig. 6C,no-guidance: p = 0.006, fading-guidance: p = 0.006 ,fixed-guidance: p = 0.006). Subjects in the no-guidance(p = 0.001) and fading-guidance (p = 0.001) groupsalso significantly reduced the errors at long term. Thepercentage of error reduction at long term was differ-ent between groups (p = 0.032). In particular, the no-guidance group reduced the error significantly morethan the fixed-guidance group (p = 0.035).

3.2.2 Racket acceleration at impact

Figure 7 shows the ball and racket trajectories, as wellas racket acceleration at impact over the last five hitsfor a representative subject while training with fixedguidance and during short-term retention. A repeated-measures ANOVA with the four main protocol phases(baseline, training, short-term and long-term retention)was performed to further investigate the limited learningfor racket acceleration (Fig. 5E).

At the lowest trained gravity, we found a significantdifference in the racket acceleration between protocolphases (Fig. 6D, p < 0.001): Subjects reduced the accel-eration between baseline and training (p < 0.001), wheretraining with fixed guidance reduced the accelerationsignificantly more than no guidance or fading guidance(p < 0.001). We neither found a significant accelerationreduction between baseline and retention, nor betweengroups.

When practicing with the medium gravity, we alsofound significant differences between protocol phases(Fig. 6E, p < 0.001): Subjects reduced the acceleration

14.5 15 15.5 16 16.5 17 17.5 18

B. Racket position and acceleration at short retention

time (s)

Racket accelerationAcceleration at impactBall positionRacket positionDesired apex

34.5 35 35.5 36 36.5 37 37.5 38 38.5-20

time (s)

Racket accelerationAcceleration at impactBall positionRacket positionDesired apex

A. Racket position and acceleration during !xed guidance training

Fig. 7. Ball and racket trajectories and racket accelerationover the last five hits for a representative person underthe Earth gravity g4 = 9.81m/s2 while training with fixedguidance (A), and at short-term retention (B).

significantly from baseline to training (p < 0.001), andalmost significantly from baseline to long-term retention(p = 0.057). The acceleration reduction during trainingwas different between groups: Subjects trained withfixed guidance reduced the acceleration significantlymore than subjects in the no-guidance (p = 0.008) andfading-guidance groups (p = 0.002). However, we didnot find significant differences between groups in theacceleration reduction at retention.

When practicing with the high gravity, we also founda significant difference between protocol phases (Fig. 6F,p < 0.001). We found a significant acceleration reductionfrom baseline to training (p < 0.001), which was differentbetween groups (p = 0.012): Training with fixed guid-ance reduced the acceleration significantly more thanfading guidance (p = 0.014). The acceleration reductionfrom baseline to long-term retention was significantlydifferent between groups (p = 0.035). The no-guidancegroup tended to reduce acceleration more than boththe fading-guidance group (p = 0.061) and the fixed-guidance group (p = 0.066).

3.3 Learning performance at non-trained gravities

3.3.1 Mean overall error

Subjects in the no-guidance (p = 0.002), and fixed-guidance groups (p = 0.001) significantly reduced theoverall error at short term in the non-trained grav-ity levels. Subjects in all training groups improvedtheir hitting performance at long term (Fig. 8A, no-guidance:p < 0.001, fading-guidance: p = 0.004, fixed-guidance: p = 0.001). We found neither a main effect ofthe training condition, nor of the gravity in the linearmixed model analysis. The interaction between trainingconditions and gravities was also non-significant.

3.3.2 Racket acceleration at impact

None of the training groups significantly reduced theiracceleration at impact in the non-trained gravity levels(Fig. 8B). We found neither a main effect of the trainingcondition, nor of the gravity in the mixed linear model.The interaction between conditions and gravities wasalso non-significant.

3.4 Correlation between performance measures

The overall error reduction from baseline to short- andlong-term retention tests was not accompanied by a re-duction of the racket acceleration at impact, as suggestedby a lack of correlation between the reduction of the twoperformance measures; we neither found correlations forany gravity level, nor for all gravities together.

4 DISCUSSION

This study investigated which form of robotic guidance— no, fixed, and fading haptic guidance— would bemore beneficial in learning a hybrid rhythmic-discretemotor task. We found that the most effective trainingcondition in terms of error reduction depended on thedegree of rhythmicity of the movement. Further, trainingwith haptic guidance seems not to have any effect onracket acceleration at impact. These findings contradictresults from previous studies, which found that fadingguidance was more beneficial to learn time-related con-tinuous tasks [15], [16]. In the following subsections, wediscuss the results in detail.

4.1 The most effective training condition dependson the degree of rhythmicity

The Challenge Point theory states that optimal learningis achieved when the difficulty of a task is adjusted to theindividual participant’s level of expertise (i.e. when thechallenge point is reached). Thus, low-skilled subjectswould learn less when training a too difficult task thanwhen training at an appropriate level of difficulty. Like-wise, proficient subjects would not learn optimally whenpracticing a task that is easy for them. Thus, reducingtask difficulty via haptic guidance could have a positiveimpact on learning when subjects are training an overly

Guidance

No Guidance Fixed GuidanceNo Guidance Fading

Guidance

A: Percentage of mean baseline error reduced B: Reduction of racket acceleration at impact

Fig. 8. Performance changes in non-trained gravities. A: Percentage of baseline’s mean overall error reduced atshort- and long-term retention for no-guidance, fading-guidance and fixed-guidance groups. B: Reduction of racketacceleration at ball impact from baseline to short- and long-term retention. Error bars show ±1 CI.

difficult task. In fact, haptic guidance has been found tobe more beneficial for less skilled participants in somerecent studies [10], [18], [11], supporting the ChallengePoint theory. Likewise, haptic guidance could have anegative impact when applied during training of tasksthat are already easy for subjects.

In our study, the higher gravities seemed more “chal-lenging”, because subjects performed significantly worseduring baseline with g4 and g5 than with the lowergravities g2 and g3 (A possible rationale may be thatfaster movements allow less opportunity for online errorcorrection, and thus the performance under higher gravi-ties degraded, especially given that subjects encounteredthe task for the first time during baseline). According tothe Challenge Point theory, a positive impact of hapticguidance on learning could be expected when practicingg4, and a negative impact for g3.

Our results contradict the Challenge Point theory:Training with fixed guidance improved learning in termsof error reduction in the gravity level that subjects mas-tered better during baseline (g3), while training withoutguidance resulted in better learning in gravity g4 thantraining with fixed guidance.

Thus, the differences in the most effective training con-dition do not seem to depend on the “level of difficulty”,but rather on the “degree of rhythmicity”: For the peri-odic discrete task (low gravity), no guidance and fixedguidance resulted in better performance after training.Fixed guidance also improved learning in more mixedtasks that fall in between discrete and rhythmic classes(medium gravity). However, for the continuous rhythmicmovement (higher gravity), no guidance seemed to beespecially suitable to learn the task. The positive effect offixed guidance is in line with recent studies that showedthat guidance is especially beneficial to learn discretetiming tasks, such as the best moment to turn in sharpcurves [11] or the moment to start a tennis stroke [10].

Fixed guidance seems inappropriate for tasks with ahigh degree of rhythmicity, i.e. continuous, repetitivemovements with negligible dwell times.

4.2 Haptic guidance did not benefit learning of theracket acceleration at impact

Subjects systematically hit the ball with a racket accel-eration higher than the negative acceleration observedin experienced subjects [33]. Subjects hit the ball using anegative racket acceleration only during training withfixed guidance (Fig. 6). However, when the guidancewas removed, subjects systematically hit the ball at avery low position, shortly before the racket reachedmaximum velocity, so with small positive acceleration.This strategy contradicts previous work [30], [33]; in-stead it is more consistent with what the hypothesis ofenergy minimization would predict: The energeticallyoptimal contact should be at maximum racket velocity(i.e. acceleration zero) [31]. A possible rationale for thisdiscrepancy could be that parameters of the cost functionmight have to be further adjusted to match humanbehavior well enough to be suitable for haptic guidance.These parameters were taken from [30], where the aimwas to show a similarity between model predictionsand human behavior concerning the varying degree ofrhythmicity with varying gravity. However, [30] did notgive a quantitative comparison between predicted andobserved activity periods. Furthermore, discrepanciesmay arise from the differences between the devicesemployed: We used a transparent actuated lever withrotational movement about the elbow as a pivot, whilesubjects in [30], [33] moved a sensorized racket withoutconstraints on the arm, and apparent dynamics of theapparatus were not reported. Difference in setups mighthave led to a biased trade-off between energy optimalityand passive stability [31].

The finding that subjects did not reduce the racket ac-celeration at impact contradicts recent evidence that hap-tic guidance may be specifically useful to reproduce thetemporal characteristics of spatiotemporal patterns [14],[15], [16]. One might argue that we tested for long-termeffects of haptic guidance, while previous work mainlytested performance gains at short term. However, we didnot find any benefit at short-term retention either, so thiscannot explain the differences.

The lack of any significant correlation between racketacceleration at impact and mean overall error, duringbaseline and during the retention tests, suggests thatracket acceleration at impact may not be a time-criticalfeature of the ball-bouncing task. In fact, we did notfind a correlation between the reduction from baseline toretention of the mean overall error and the accelerationat impact, thereby suggesting that subjects exploredstrategies to improve their hitting success different fromreducing the racket acceleration at impact as observedin previous research [30], [33]. This is in line with recentwork on the ball-bouncing paradigm, which suggeststhat subjects actively regulate their movements on everycycle based on perceived variables, i.e. they employfeedback control more actively, in detriment of feedfor-ward control [43], [44]. Thereby, the lack of benefit fromhaptic guidance may be explained by the inappropriateselection of racket acceleration at impact as performancemeasure.

4.3 Subjects generalized to non-trained gravities

We hypothesized that training without guidance wouldgeneralize better to untrained gravities. Indeed, sub-jects in the no-guidance group significantly reduced theoverall error at both untrained gravities. However, alsosubjects trained with fixed and fading guidance signifi-cantly reduced the overall error at untrained gravities.This result contradicts our recent study, which foundthat haptic guidance limited generalization in a timing-based task [34], while training without guidance didbenefit generalization. In this previous experiment, wespeculated that the limited generalization arose becausesubjects who received guidance experienced a narrowerrange of training examples. A possible rationale for thediscrepancy is that bouncing a ball at the non-trainedgravity levels was indeed similar to the experiencegained when practicing with the trained gravities, andthus training with guidance also allowed to create a “richand varied experience” [34] that improved learning ofthe untrained gravity levels. In fact, a cycle with the lowuntrained gravity g2 was only 25% shorter than a cyclewith the low gravity g1, and a cycle with the untrainedhigh gravity g5 was only 21% shorter than with the hightrained gravity g4.

4.4 Why did fading guidance not improve learning?

We hypothesized that fading guidance would improvelearning at all gravities. However, we did not find any

positive effects of this special form of guidance onlearning the bouncing task.

Fading guidance might be seen as compliant guidancethat leaves subjects freedom to choose a strategy, but stillto benefit from the error-correcting functionality of thedevice. While guidance faded, instead of adopting theinitially demonstrated robust, self-stabilizing strategywith negative acceleration at impact, subjects consis-tently converged towards a more energy-efficient, butless robust strategy with racket acceleration close tozero, like subjects trained without guidance (as seen inFig. 5B). The fading guidance, however, allowed themto employ this strategy during training and still reachhigh performance, as can be seen in the low overallerror during training in Fig. 5A. This could be explainedby the constant re-calculation of the optimal trajec-tory, which kept performance high even without self-stabilizing properties of the strategy. This gave subjectsan erroneously good impression of their performance,reducing error as driving factor for learning to occur.Further, the dynamics of the task were not displayedin full, due to the compliant assistance, which is alsoreflected in the lower interaction moments as comparedto no guidance conditions (Fig. 5C). Only the last hitsof the training phase, when assistive forces approachedzero, gave subjects a realistic impression of the taskdynamics, so they had only limited training possibilities.When the correction from the device was not availableanymore during retention trials, subjects accordinglydisplayed bad performance.

The selection of a fixed fading gain in (4), independentof the subjects’ individual performance, was done toassure that all subjects were exposed to the same levelof guidance during the experiment. The reduction of therobotic guidance may have been too fast or too slow fora subject’s specific learning rate. A guidance-as-neededalgorithm that aims to systematically reduce guidancebut can also increase guidance based on a subject’scurrent performance level may have resulted in betterlearning [11].

4.5 Technical limitations

The present study is influenced by several technical lim-itations, in particular concerning the haptic performanceof the robotic device, the realism of the virtual reality, aswell as the cost function used to calculate the optimalracket trajectory. The haptic device does not perfectlyreflect a freely moving racket. Friction and sensor noiseplay a role and reduce achievable precision and accuracyin all conditions. For the cost function, we used thesame cost function and weighting parameters as [30], asthis approximated the fraction of dwell times observedin human movement in preliminary experiments. Moreevidence would be needed in how far this particular costfunction and its parameters indeed reflect or reproducethe human’s internal strategies. However, this is outsidethe scope of the present study. Finally, the displayed vir-tual reality is inspired by the task of playing tennis, but it

does not reflect this task in a realistic manner. However,we believe that specific features of the task, in particularthe rhythmic-discrete nature, are still maintained andworth investigating even in this more artificial task.

5 CONCLUSION

In this study, we investigated the influence of hapticguidance on learning of hybrid rhythmic-discrete mo-tor tasks: Subjects played a ball-bouncing game withvarying amounts of haptic guidance at different gravitylevels, provoking different degrees of rhythmicity.

We found that the most effective training conditionin fact depended on the degree of rhythmicity of themovement: Fixed haptic guidance seems inappropriatefor tasks with a high degree of rhythmicity, thus tasksthat are close to continuous, repetitive movements withnegligible dwell times. However, training with hapticguidance seems to be beneficial in improving perfor-mance for tasks that resemble discrete movements, sim-ilarly to training without guidance.

Additional studies are needed to further validate theseobservations. It is still an open question whether thebenefit of haptic guidance in fact only depends onthe movement class analyzed, independent of the taskdifficulty level.

Furthermore, we could not find benefits of fadingguidance compared to fixed or no guidance, and wecould not reproduce previous findings on the dominanceof negative accelerations at impact, which indicate sub-jects’ strategies for passive self-stabilization. Instead, oursubjects seemed to systematically prefer more energy-efficient movement patterns. We could not find a rela-tionship between task performance and accelerations atimpact either.

APPENDIX AA highly efficient algorithm for model-predictive controlof the racket position in the ball-bouncing task wasderived, in order to allow fast on-line computation ineach sampling step. First, the cost functional (2) wasrewritten in the form of a standard linear-quadraticregulator (LQR) problem:

J = ||xf − xref,f ||2S

∫ tf

(||x(t)− xref (t)||2Q + ||u(t)||2R)dt (5)

with the weighting matrices R = we, Q = diag(wr , 0, 0),and S = diag(1, 1, 0), and the state vector

r r am)T

where am is muscle activation, r is racket position, andr is racket velocity. The system is governed by thedynamics [30]:

x = Ax+Bu, with (7)

0 −γ/I 1

0 0 −1/τm

0 0 1/τm)

The parameters are: Damping coefficientγ = 0.25N/(m/s), moment of inertia of arm andracket I = 0.05N/(m/s2), and muscle time constantτm = 50ms. By suitable coordinate definitions, rref,0 = 0and thus xref (t) = 0 ∀t ∈ {t ∈ R : t0 ≤ t < tf}, and

xref,f =(

rref,f rref,f 0)T

.The solution to this optimal follower-controller prob-

lem is given by the control law [45]:

u(t) = −R−1BTPx(t) +R−1BTp(t), (8)

where the matrix function P(t) is the solution of theRicatti equation, and the vector function p(t) reflects theinfluence of the reference trajectory:

P = −(PA+ATP−PBR−1BTP+Q), (9)

P(tf ) = Pf = S, (10)

p = −((A−BR−1BTP)Tp+Qxref ), (11)

p(tf ) = pf = Sxref,f . (12)

Because of the two-step optimization, it is not desir-able to compute the complete solution in each iterationof T . Instead, only the optimal cost for a specific durationT is needed, until the optimal T is found.

This can be achieved in a very efficient manner by re-writing the cost functional (5) in the form (The derivationfollows [46]):

J = xTref,fSxref,f − x

Tf Sxref,f + x

T0 P0x0 − x

T0 p0. (13)

The fundamental advantage of this expression is that itdoes not involve the entire trajectories of x nor u, butinstead only needs some initial and final values: Thematrix P0 only depends on the duration T and can befound by backward integration of (10). The value of pf isa function of the reference final value xref,f , as given in(11). The value of p0 depends both on the duration T andon the final value pf . However, as this is a linear system,transition matrices Φp can be found that immediatelymap pf to p0 for a particular T :

p0 = Φp(T )pf (14)

These transition matrices are found off-line: First, (11)is integrated backward for a maximally expected dura-tion Tmax, with three arbitrary, but linearly independentvalues for pf , in order to find the corresponding valuesfor p0. Then, for each T , the transition matrix is foundby solving (14) for the entries of Φp. Similarly, the finalracket states xf , immediately before the next impact, arefound via convolution:

xf = Φx(T )x0 + Γ(T )pf (15)

with the transition matrices

Φx(T ) = ΦTp (T ), Γ(T ) = −

Φx(τ)BR−1BTΦp(τ)dτ

The transition matrices only depend on the durationT , they are independent of initial and final condi-tions. Therefore, the values for P0,p0,Γ,Φp can be pre-computed and stored as matrix or vector functions of T ,respectively, with a sufficiently large domain of T valuesfor the specific gravity condition. As a resolution for T ,we choose the sampling frequency of the haptic system,1 ms.

Also when the optimal T is found, the predictivecontroller only needs the first values of u and x, whichare also found immediately from (8) and (7), respectively.

This procedure reduces computational requirements toa minimum; no on-line integration is necessary.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the contributionof Peter Wolf and Alexander Duschau-Wicke for theirenriching comments.

The contents of this publication were partially de-veloped under two grants from the US Departmentof Education, NIDRR grant numbers H133E070013 andH133E120010. However, those contents do not neces-sarily represent the policy of the US Department ofEducation, and you should not assume endorsementby the Federal Government of the United States ofAmerica. Laura Marchal-Crespo holds a Marie-Curie in-ternational income fellowship PIIF-GA-2010-272289, andHeike Vallery holds a Marie-Curie career integrationgrant PCIG13-GA-2013-618899.

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Laura Marchal-Crespo received her Dipl. Ing.degree in Industrial Engineering with a spe-cialization in Automatics from the UniversitatPolitecnica de Catalunya (UPC), Spain in 2003.She obtained her M.S and PhD in Mechanicaland Aerospace Engineering at the Universityof California at Irvine in 2006 and 2009, re-spectively. During her PhD, she did research inthe general areas of human-machine interfacesand biological learning, and, specifically, in theuse of robotic/haptic simulation to aid people in

learning motor tasks. In January 2010, she joined the Sensory-MotorSystems Lab, where she continues her research on motor learning andfMRI compatible robotic devices.

Mathias Bannwart obtained his B.Sc. in Biologyin 2010 and his M.Sc. in Neural Systems andComputation in 2013 both from the Swiss Fed-eral Institute of Technology, ETH Zurich. Duringhis M.Sc., he worked on projects involving virtualreality environments for neurorehabilitation andthe use of haptic feedback for learning motortasks. He completed his degree with a thesisabout the usability of low-cost vibration motorsfor sensory and motor rehabilitation.

Robert Riener is the head of the Sensory-MotorSystems Lab, ETH Zurich. He is Full Profes-sor at the Department of Health Sciences andTechnology, ETH Zurich and holds a Double-Professorship with the Balgrist University Hos-pital. He received the Dipl.-Ing. degree and theDr. degree from the TU Munchen in 1993 and1997, respectively. In 1993 he joined the Instituteof Automatic Control Engineering, where he haspursued research into neuroprosthetics. Afterpostdoctoral work at the Centro di Bioingegne-

ria, Politecnico di Milano from 1998 to 1999, he returned to the TUMunchen, where he finished his Habilitation in the field of Biomecha-tronics about multi-modal VR applied to medicine. Since his activityin Zurich, Riener develops robots and interaction methods for motorlearning in rehabilitation and sports.

Heike Vallery received her Dipl.-Ing. in Mechan-ical Engineering from RWTH Aachen Univer-sity in 2004. In 2009, she received her PhDfrom Technische Universitat Munchen, whereshe had investigated compliant actuation andcontrol concepts for gait rehabilitation robotics.As a postdoc at ETH Zurich, she continued thisresearch and also worked on leg exoprosthetics.Currently, she holds an assistant professorshipat Delft University of Technology, and she alsoremains affiliated with ETH Zurich. Her research

interests are in the areas of bipedal locomotion, neurorehabilitation, andhaptic human-robot interaction. She has published more than 40 peer-reviewed publications and filed 4 patents.

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