Human Movement Science 22 (2003) 13–36 www.elsevier.com/locate/humov

On-line trajectory modifications of planar, goal-directed arm movements q Evert-Jan Nijhof

*

Perception-Motor Integration Group, Helmholtz Institute, Utrecht University, BBL-370, P.O. Box 80000, 3508 TA Utrecht, The Netherlands

Abstract Sometimes a goal-directed arm movement has to be modified en route due to an unforeseen perturbation such as a target displacement or a hand displacement by an external force. In this paper several aspects of that modification process are addressed. Subjects had to perform a point-to-point movement task on a computer screen using a mouse-coupled pointer as the representation of the hand position. Trajectory modifications were imposed by unexpectedly changing the position of the target or by changing the relation between mouse and screen pointer. In the first series of experiments, we examined how often a trajectory is updated. Here, trajectory modifications were imposed by unexpectedly changing the normal relation between mouse and pointer to a shear-like relation, where a percentage of the forward/backward position of the hand was added to the pointer position in the left/right direction. Withdrawal of visual feedback during the movement revealed that trajectories were updated at interval times shorter than 200 ms. From the similarity with experiments where the original relation between mouse and pointer was restored during the movements, we conclude that motor plans are updated on-line to move the hand from its current perceived position to the target. In a second series of experiments, we studied whether a continuous change in target position yields similar trajectory modifications as a continuous hand displacement. To mimic the latter perturbation, we used the above-mentioned distortion of the mouse–pointer relation. We found that the resulting hand paths did not differ for the two visual perturbations and conclude that the perturbed, goal-directed movements are modified in a consistent way,

q

Part of this research was presented at the International Conference ‘‘Plasticity and Adaptation in Motor Control’’, September 1998, Aussois, France, and at the International Conference ‘‘Progress in Motor Control––II’’, August 1999, University Park, PA, USA. * Corresponding author. Tel.: +31-30-2532274; fax: +31-30-2522664. E-mail address: [email protected] (E.J. Nijhof). 0167-9457/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-9457(02)00140-9

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irrespective of whether the position of the target or hand was perturbed. Simulations of the experimental data with a kinematic reaching model support this conclusion. Ó 2002 Elsevier Science B.V. All rights reserved. PsycINFO classification: 2330 Keywords: Reaching; Perturbation; Kinematic model; Motor control; Human

1. Introduction Goal-directed arm movements are so common that we easily forget their inherent redundancies (Bernstein, 1967). For example, the spatial path of the hand between starting and target positions is usually not prescribed, thereby introducing a kinematic redundancy. A dynamic redundancy also exists because we have more muscles acting about a particular joint than are required to cause movement of that joint. These redundancies gave rise to the long standing question: on what basis does the central nervous system (CNS) select (i) a particular trajectory and (ii) the muscles necessary to generate the appropriate joint torques? Fortunately, from a kinematic point of view, it is irrelevant which muscles generate the torques about a joint, because only the time sequence of their total vector sum gives rise to a particular hand path. In the current paper, we address the former problem of trajectory formation and, in particular, the modification of an on-going movement on the basis of new visual information. A solution to BernsteinÕs problem is found by using invariant movement characteristics. In the early eighties, it was shown that the hand paths of horizontal pointto-point movements are generally straight with smooth, bell-shaped velocity profiles (e.g. Flash & Hogan, 1985; Morasso, 1981). Later on, it was noticed, however, that the amount of straightness depends on whether the hand is free to move or in contact with an external device, a so-called compliant movement (Desmurget, Jordan, Prablanc, & Jeannerod, 1997). These authors found that free movements are significantly more curved than compliant ones and hypothesized that the two involve different control strategies. Moreover, free pointing movements in the vertical plane were found to be more curved too, although this observation may also originate from the noncompliant experimental conditions (Atkeson & Hollerbach, 1985; Soechting & Lacquaniti, 1981). It has been proposed that the invariances are the outcome of an optimization or a minimization process, although it is still an open question which quantity is optimized or minimized. Inspired by the observed smoothness of hand movements, minimizing the mean squared value of the jerk of the movement (the time derivative of the acceleration) seemed very promising to describe experimentally recorded hand movements (Flash & Hogan, 1985). That theory predicts a unimodal, symmetric velocity profile with its maximum velocity at half the movement time. However, slow and very fast movements showed an asymmetric velocity profile (Nagasaki, 1989; Zalaznik, Schmidt, & Gielen, 1986; Zhang & Chaffin, 1999), thereby demanding for refined models. Nagasaki (1989) extended the mini-

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mum jerk model to a constrained jerk model that predicts asymmetric velocity profiles as well. Nevertheless, his model was still unable to cover the entire velocity range with one, single cost function for the optimization process. Uno, Kawato, and Suzuki (1989) used a model where the change in joint torques was minimized. Kawato (1996) proposed other quantities to be minimized such as the change in muscle tensions or the change in motor commands, and Alexander (1997) tested the minimum energy cost hypothesis. Others support the idea that the trajectory results from the internal dynamics of the system. For example, the equilibrium-point hypothesis assumes that the target position of the effector––the equilibrium state–– is defined in terms of muscles commands, and that the spring-like properties of the muscles bring the arm to that equilibrium position (e.g. Feldman, 1986 and references therein; Latash, 1993). In the vector-integration-to-endpoint model, it is assumed that the invariants emerge through the implicit interactions in a realtime neural network rather than through an explicitly precomputed trajectory (Bullock & Grossberg, 1988). One of the advantages of that model is that it correctly predicts the above-mentioned asymmetry in velocity profiles at low and high speeds. If goal-directed arm movements are indeed the outcome of some kind of optimization process, it is obvious that certain aspects of the arm trajectory have to be specified in advance. For example, minimum jerk or minimum torque change trajectories are determined on the basis of movement duration and movement amplitude. Most of the time, the optimization process yields a path that ends up at the target with the appropriate boundary conditions, i.e. target accuracy, end velocity and end acceleration. From this reasoning a tempting question arises: how are trajectories modified if the CNS concludes that the current trajectory will not accomplish the given task? For example, the current path is not going to end up at the target at all due to an unforeseen external perturbation. The objective of this paper is to obtain more insight in the way how trajectories are on-line modified in order to compensate for (external) perturbations. In the last two decades, the double-step target displacement paradigm has been extensively used to study on-line arm trajectory modifications. In this paradigm, subjects were instructed to move their hand from a starting position to a visual target, and either before movement initiation or during the movement, the target was unexpectedly displaced to another position. In one class of experiments, the time between the presentation of the first and the second target was varied and the subjects were asked to move either at maximum speed (Flanagan, Ostry, & Feldman, 1993; Soechting & Lacquaniti, 1983; Van Sonderen & Denier van der Gon, 1990; Van Sonderen, Denier van der Gon, & Gielen, 1988; Van Sonderen, Gielen, & Denier van der Gon, 1989) or at a preferred rate (Flanagan et al., 1993; Flash & Henis, 1991; Flash, Henis, Inzelberg, & Korczyn, 1992; Henis & Flash, 1995). It was found that the observed trajectories strongly depended on that interstimulus interval, where the starting direction was more towards the second target if the interval was shorter. Van Sonderen et al. (1988, 1989, 1990) concluded that muscle activations are continuously updated in order to move the hand towards an internal representation of the target, and that this internal target gradually moves from the first to the second position. Flanagan

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et al. (1993) came to similar observations but attributed that gradual shift to the natural dynamics of the system. At lower speeds and longer interval times, the hand moves first towards the original target and is during the movement corrected to move towards the displaced one (Flanagan et al., 1993; Flash & Henis, 1991; Flash et al., 1992; Henis & Flash, 1995). In another series of experiments, the target switch itself occurred during an eye saccade and was thereby unnoticed by the subjects. Nevertheless, the movements were modified towards the displaced target, irrespective of whether during the movement the hand was visible or not (Goodale, P
elisson, & Prablanc, 1986; P
elisson, Prablanc, Goodale, & Jeannerod, 1986; Prablanc & Martin, 1992). Recently, Turrell, Bard, Fleury, Teasdale, and Martin (1998) showed that vision during the movement is of importance. In a dark environment, they found the gradual shift from the first towards the second target, inline with the results of Van Sonderen et al. (1988, 1989, 1990). In a lightened room, on-line error reductions with respect to the new target were observed which they attributed to proprioceptive feedback. They confirmed that latter hypothesis by repeating the same experiments with a deafferented subject (Bard et al., 1999). Feedback processes, on the other hand, implies time delays and can therefore only be effective in the second stage of the movement. This was confirmed by several experiments that showed that the acceleration part of the movement is highly automatic and unaffected by a target change, whereas the deceleration phase can be modified (Day & Lyon, 2000; Heath, Hodges, Chua, & Elliott, 1998; Komilis, P
elisson, & Prablanc, 1993). Note that there is ample evidence that aspects of both feedforward and feedback control can be found in reaching experiments (for reviews see Desmurget & Grafton, 2000; Rabes, 2000). The conventional view was that the long time delays of visual and proprioceptive feedback impose feedforward control during very fast movements while slower movements can use sensory feedback information to compensate for perturbations. More recent ideas are that internal feedback loops can control very fast movements as well, as they may use a forward model to predict the motor output on the basis of efferent and afferent signals. Other external influences during target displacements have been addressed too, such as attention and motor preparation (Boulinguez & Nougier, 1999), movement with the dominant or the non-dominant hand and perturbation of the left or right side of the workspace (Elliott, Lyons, Chua, Goodman, & Carson, 1995), and the possible use of velocity information of the moving target (Brenner & Smeets, 1997). Finally, it should be noted that the ability to modify trajectories is not exclusive to man as was already shown by Georgopoulos, Kalaska, and Massey (1981) with their study on rhesus monkeys. During a double-step target displacement experiment, a single modification of the hand trajectory may be sufficient. In the present paper, we shall extend the double-step to a continuously moving target experiment, and study how often the trajectory is updated. In a second series of experiments, we shall address the question how a trajectory is updated if not the target but the hand is displaced from its track towards the target. Finally, we want to capture the observed trajectory modifications of the two experiments in one, simple, kinematic reaching model.

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2. Methods 2.1. Subjects Three subjects (2 males, 1 female, aged 26–43) participated in the first series of experiments and 10 subjects (9 males, 1 female, aged 21–43) in the second series. They did not have any known neuromuscular disorders and all had normal or correctedto-normal vision. All subjects, except EN (Exps. 1 and 2) and SH (Exp. 2), were naive with regard to the purpose of the experiments. For most of the subjects, it was the first time they participated in these kinds of experiments. 2.2. Experimental set-up The hand position was determined by the position of an ordinary computer mouse connected to a Macintosh PowerPC. The mouse could move on dedicated mousepad of 40
40 cm2 . Starting and target positions were presented as 5
5 mm2 squares on a 17-inch monitor and a 3-mm dot (hereafter referred to as the pointer) represented the position of the mouse (and thus of the hand). The sensitivity of the mouse was adjusted such that mouse and pointer had a linear, one-to-one relationship. 1 The movements were recorded at 75 Hz (13.3 ms interval time) and with an 82 dots-per-inch (0.31 mm) resolution. A custom-made mouse driver was written to obtain a constant sampling rate. Before each recording session the computer mouse was carefully cleaned and a slip-free rolling condition was ensured. 2.3. Experimental procedure The subjects were seated at a table about 60 cm in front of the computer screen. The height of the chair was adjusted rather high such that the subjectÕs forearm was just free from the table. The subjects held the mouse between their fingertips and their hand did not touch the mousepad. Before a recording session, they practiced the movements 10–20 times to become familiar with the (unusually) slow mouse– pointer relation. All subjects were daily computer users and had no trouble learning to control the screen pointer by moving the mouse. The instruction to the subjects was to move the pointer from the starting square in the bottom half of the screen to the target square located 14 cm vertically above the starting position in one, fluent, self-paced movement. They had to press the mouse button when they were ready for take off, make their movement and press the mouse button once again to end the trial. The subjects were instructed not to make explicit endpoint corrections when they missed the target. Data recording took place between the two button presses. After the trial, the pointer should be returned to its starting position.

1 Subjects still had to make the usual 90° rotation about the horizontal axis so that a horizontal mouse movement from/to the body is converted to an up/down movement of the pointer on the screen.

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2.4. Data analysis Our coordinate system to indicate positions uses the starting position as its origin. Positive X -values run to the right and positive Y -values are away from the subjectÕs body for the mouse and hand, and upwards for the pointer. To obtain the velocity, the recorded position data were numerically differentiated off-line by taking the analytical derivative of the second-order polynomial fitted through a symmetrical moving time window (67 ms) of five points (Press, Vetterling, Teukolsky, & Flannery, 1992). The differentiated signal was digitally low-pass filtered to 10 Hz. This procedure was repeated to calculate the second-order derivative of the position signal. Averaged trajectories were calculated in three steps. First, the position data of every trial in a block was normalized in time to obtain a movement of standard duration. These resampled trajectories were then vectorially averaged. Finally, the average trajectory was scaled to the mean movement time of the trials in the block. During the averaging process, spatial standard deviations in the direction perpendicular to the major movement direction were also calculated. 2.5. Perturbations Two visual perturbations were introduced that were designed to induce more than one trajectory modification. One perturbation mimics a displacement of the hand position and the other of the target. They have in common that the same hand movement can bring the screen pointer to the target. The first type of perturbation, referred to as target move (TM), is an extension of the double-step target displacement paradigm (left panel of Fig. 1). During TM, the position of the target ðxt ; yt Þ is no longer fixed but coupled to the position of the mouse ðxm ; ym Þ and thus to the hand according to ( ( xp ¼ xm xt ¼ kym ; ð1Þ yp ¼ ym yt ¼ yt;0 where k is a constant and ðxp ; yp Þ denotes the position of the screen pointer. The unperturbed situation is characterized by k  0, a rightward perturbation by k > 0 and a leftward perturbation by k < 0. The unperturbed position of the target is denoted by ð0; yt;0 Þ, where yt;0 ¼ 14 cm. Note that TM is not a classical interception task, because the subject moves the target her/himself. The other perturbation, referred to as pointer move (PM), is similar in nature to TM, but now the position of the pointer ðxp ; yp Þ instead of the target is coupled to the position of the mouse according to ( ( xp ¼ xm  kym xt ¼ 0 ; ð2Þ yp ¼ ym yt ¼ yt;0 with k the same constant (right panel of Fig. 1). The two perturbations were designed such that the same oblique hand movement, ym ¼ k1 xm , could bring the pointer to

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Fig. 1. Schematic diagram of the two perturbations. Left panel: Target move, where the horizontal position of the target is coupled to the forward position of the mouse; Right panel: Pointer move, where the extra horizontal displacement of the pointer is coupled to the forward position of the mouse. The path denoted ‘‘Pointer’’ is the path as shown on the screen and the path denoted ‘‘Mouse’’ is the path as made by the subject.

the target, either at ðkyt;0 ; yt;0 Þ during TM or at ð0; yt;0 Þ during PM. By comparing trajectory modifications induced by these special kinds of perturbations, we can study whether it makes a difference if the target (TM) or the hand (PM) is perturbed. In order to validate whether the above-mentioned perturbations do indeed induce more than one trajectory modification, an adapted version of PM was used, referred to as pointer off (PO). Here, after a certain amount of time, the Toff time, the screen pointer was hidden so subjects were deprived of visual feedback. They were instructed to always continue their movement even when the pointer disappeared until they assumed their hand was at the target. The Toff time was varied and target errors were recorded. To study how the trajectory was completed after the pointer was switched off, we compared the paths during PO with a fourth perturbation referred to as disturbance off (DO). Here, the PM perturbation was removed after the Toff time and the normal one-to-one relation between pointer and mouse restored. The rationale behind these perturbations is that during PO, the visual feedback is removed and so subjects cannot modify their trajectories any longer on the basis of new visual input. They are left with the current motor plan and may continue that. During DO, the normal relation between pointer and mouse is restored and subjects do not have to modify their trajectories any longer. In this case, they can also continue with the current motor plan, provided that plan will bring the hand at the target. If for the two perturbations the hand paths are the same and the DO path arrives at the target, we have evidence that hand path corrections are made on the basis of its current position relative to the target and are updated on the basis of new visual information about target and hand positions.

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2.6. Experiments In the first series of experiments, we tested whether the trajectory is modified more than once as was designed in the perturbations. This was examined using PO and DO, the two modified versions of the PM perturbation. The two perturbations were applied on the same day during two separate sessions of 200 movements. After each session, the subjects were allowed to relax for a couple of minutes. Within a given session, 40% of the trials were perturbed in a random order. The magnitude of the perturbation was constant and randomly switched between þ0.5 (rightwards) or )0.5 (leftwards). The Toff times of the perturbed trials were randomly selected from the set 1.0, 0.8 and 0.6 s, yielding 13 perturbed trials per condition on average. In the second series of experiments, we compared the TM and PM perturbations in two separate sessions within the same day with a rest period between the sessions. Here, each session consisted of 100 movements with 30% of the trials perturbed in a random order. The magnitude of the perturbation was again fixed to 0.5, yielding 15 trials per condition on average.

3. Results During data analysis, each trial was visually inspected and excluded from further analysis if any of the following criteria was met: (i) the mouse button was not pressed immediately before movement initiation or immediately after movement completion, (ii) the mouse ran off the mousepad, (iii) the pointer was more than 2 cm from the target at movement completion, or (iv) the velocity profile showed more than one substantial peak to exclude trials with an explicit second correction stage. In order to meet the first criterion, the velocity in the Y -direction should be smaller than 0.5 cm/s either 100 ms after the first button press or 100 ms before the second button press. The criteria required the rejection of less than five trials per experimental condition. 3.1. Typical examples In this subsection, some typical examples of the perturbed movements will be presented. We shall concentrate on the new PM paradigm, although most features are similar for TM (see next section). During these examples, the magnitude of the perturbation was slightly smaller than during the actual experiments and constant with its direction either to the right (k ¼ þ0:4) or left (k ¼ 0:4). Typical examples of mean hand paths during the PM perturbation are shown in Fig. 2. The dots denote the sampled hand paths at 13.3 ms time interval. To make the figure easier to read, the left and right paths were horizontally displaced by )2 and þ2 cm, respectively. The solid lines mark the 1 standard deviation in X -direction of the mean paths. Note that the unperturbed path was almost straight with only a very slight curvature to the left, and that this small curvature also appeared in the two perturbed paths. Because the perturbations occurred unpredictably, hand paths of

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Fig. 2. An example of the hand paths during the PM perturbation k ¼ 0:4 for subject EN. The left traces have been shifted 2 cm leftwards and the right traces 2 cm rightwards. The dots represent the average hand trajectories sampled at 75 Hz. The solid lines mark the 1 standard deviation in the X -direction.

the perturbed trials also started in the direction of the target (as indicated by the thin, dotted lines). After about 250 ms the paths bend away from the initial direction and smoothly curve towards the virtual target (gray square) to bring the pointer on the screen at the real target. From Fig. 2, it appears that, apart from having an obviously different end point, perturbed trials have features in common with unperturbed trials. This is further expressed in the next two figures. In Fig. 3, the profiles of the two velocity components are shown for three subjects. It can be seen that they only modified the component in the perturbed X -direction (upper panel), whereas the velocity component in the unperturbed Y -direction (lower panel) was virtually the same. The mean durations and path lengths of the trials during the PM perturbations are presented in Fig. 4 for the same three subjects together with the number of averaged trials. To compare perturbed and unperturbed trials, we normalized the former ones by the latter, where the duration of the mean unperturbed trial equaled 0:95  0:05 s and its path length 13:9  0:2 cm for subject EN (left three columns), 1:32  0:09 s and 14:0  0:2 cm for JB (middle three columns) and 1:15  0:11 s and 13:9  0:3 cm for EB (right three columns). The perturbed trials lasted about 3% longer for subject EN, 6% longer for JB, and 10% longer for EB, than the unperturbed trials. For all subjects, the path lengths were about 13% longer. It can be seen that EN compensated for the longer path by increasing his mean velocity. On the other hand, EB kept her mean velocity constant which resulted in an increased movement time. JB increased both movement time and mean velocity.

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Fig. 3. Velocity profiles of the mean trajectories for three subjects. The upper panel shows the velocity component in the perturbed X -direction and the lower panel in the unperturbed Y -direction. The corresponding hand paths for subject EN are shown in Fig. 2.

Fig. 4. Durations and path lengths of the mean trajectories as shown in Fig. 3 normalized to their unperturbed values. Error bars denote the standard deviations and numbers the amount of trials per condition.

3.2. Pointer off versus disturbance off The perturbations PO and DO were introduced to study the nature of the modification process during PM. They were designed to answer the question how often

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Fig. 5. Mouse paths (outer four curves) and the pointer paths (inner four curves) during the DO (solid curves) and the PO perturbations (dotted curves) for three different Toff times. The arrows indicate the position of the pointer at time Toff . The mouse paths are displaced 7 cm to the left and right.

the trajectory is modified and also to yield information on the modification process itself. In Fig. 5 we have plotted the mean paths of the mouse and pointer during PO and DO for k ¼ 0:5 and for one subject. The solid curves and solid arrows refer to DO and the dotted curves and open arrows to PO. Each panel corresponds to one Toff time that either the pointer was visible (during PO) or that the PM disturbance was effective (during DO). The four inner curves of each panel show the paths of the screen pointer, and the four outer curves indicate the underlying mouse (or hand) paths. To separate the traces during the first part of the movement, the mouse paths in the left part of the panels (related to the nearest pointer paths with k ¼ 0:5) have been displaced 7 cm leftwards and the mouse paths in the right part of the panels (related to the nearest pointer paths with k ¼ þ0:5) 7 cm rightwards. The arrows indicate the position of the pointer at the instant of PO (open arrows) or DO (solid arrows). The mean movement time of the unperturbed trials was 1:1  0:1 s. Let us first concentrate on PO (dashed curves and open arrows). Note that to compare the results with the other perturbation, we have drawn the paths of the pointer even beyond the point where they became actually invisible. By comparing the three panels from left to right, it can be seen that any additional visual input––due to a later Toff ––is used by the subject to modify the trajectory and to arrive closer to the target. Hence, the trajectory is indeed modified more than once and at a rate faster than 200 ms. A comparison with the DO experiments yields additional information about how the trajectory is completed after the visual information is blocked. We compared the X -positions of the two perturbations at four Y -positions (3, 6, 9 and 12 cm from the starting position) using a 2
2
3
4 ANOVA (analysis of variance) with perturbation (2), direction (2), Toff time (3) and position (4) as fixed factors. An a priori power analysis showed that in order to detect medium size effects according to CohenÕs effect size convention (f ¼ 0:25) with a power 1  b ¼ 0:9,

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a total sample size of less than 250 is sufficient. Therefore we analyzed 240 trials, i.e. the first five trials per experimental condition. This overall analysis revealed no significant differences between PO and DO (F1;192 ¼ 3:4, p > 0:05) nor between the Toff times (F2;192 ¼ 1:8, p > 0:1) and nor between position (F3;192 ¼ 2:6, p > 0:05). Unsurprisingly, direction (F1;192 ¼ 1452, p 6 0:0001) was a significant factor. So we conclude that there is resemblance between the hand paths if the pointer or the perturbation is switched off. Recall that after PO, visually based hand path modifications are not possible anymore due to the lack of information. After DO, hand path modifications are not required anymore because the normal relation between mouse and pointer is restored, provided that the current motor plan brings the hand to the target. Since the two perturbations yield similar hand paths, it is reasonable to assume that during the unperturbed last part of a DO trajectory no visually guided additional modifications have been made and thus that the running motor plan before Toff brings the hand to the target. Based on these experiments, we hypothesize that motor plans are frequently, perhaps (quasi) continuously, updated on the basis of new visual information in such a way that the hand is going to move from its current position to the target. In the second set of experiments, we tested this pointto-point control hypothesis explicitly. 3.3. Target move versus pointer move In this subsection, we present the results of the PM and TM perturbations for 10 subjects. The mean mouse paths of five representative subjects are shown in Fig. 6, where the solid curves refer to PM and the dashed curves to TM. To increase the visibility, the paths of subjects DT and SH have been displaced 8 cm to the left and right, and those of subjects LD and RS 4 cm to the left and right. We found that only subject LD failed to reach the targets in one, smooth movement from start to finish (hatched targets). In particular, for the perturbations that required a leftward correction, she overshot the target line and had to reverse her movement direction. The other subjects reached or nearly reached the targets.

Fig. 6. Mouse paths during the PM (solid curves) and the TM (dashed curves) perturbations for five subjects. The paths of subjects DT and SH were displaced for 8 cm and those of subjects LD and RS for 4 cm.

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To compare the two perturbations quantitatively, ANOVAs with repeated measures were used, having subjects (10) as a random factor. First, we studied the movement duration using a 10
ð2
2Þ ANOVA with perturbation (PM/TM) and direction (2) as fixed factors. It is found that movement duration did not depend on the perturbation (F1;9 ¼ 0:087, p > 0:5) nor on the direction (F1;9 ¼ 4:5, p > 0:05). Also the target error––defined as the shortest distance between the end of the path and the center of the target––did not depend on the perturbation (F1;9 ¼ 0:45, p > 0:5) nor on the direction (F1;9 ¼ 0:10, p > 0:5). The two hand paths were compared using a 10
ð2
2
4Þ ANOVA between the handÕs X -positions at four Y -positions (3, 6, 9 and 12 cm from the starting position). We found a significant difference between the two directions (F1;9 ¼ 122, p 6 0:0001), although again no significant differences between the two perturbations (F1;9 ¼ 0:41, p > 0:5) nor between the four positions (F3;27 ¼ 2:3, p > 0:1). A post-hoc power analysis revealed that the power, 1  b ¼ 0:86, was high enough to detect at least a medium effect. From this second set of experiments we conclude that modifications of a perturbed, goal-directed arm movement are the same if the target position or the hand position is perturbed.

4. Model 4.1. Introduction We want to describe the perturbations as found in the previous section using a simple phenomenological model and capture the observed hand paths in the smallest number of parameters. As was already stated in the Introduction part, as yet no single model has been proposed that describes even unperturbed arm movements in full detail. Nevertheless, some simple models have been used to describe double-step target displacement paradigms to some extent: the abort-replan (AR) and the superposition (SP) model. The former model assumes that the initial motion plan to the first target is aborted and replaced by a second plan that brings the hand smoothly to the displaced target location (Hoff & Arbib, 1991, 1993). The SP model, on the other hand, assumes that the initial plan continues to its intended completion and that an additional plan is superimposed on the first plan in order to correct for the target displacement (Flash & Henis, 1991; Henis & Flash, 1995). At first sight, the SP model seems favorable to describe the double-step experiments, since the AR model assumes that accurate knowledge of the dynamics of the hand (e.g. position, velocity, acceleration or even higher time derivatives) at the instant of the trajectory modification is available in order to connect the two motion plans smoothly together. In the SP model, the hand kinematics are not required because the superimposed plan represents the path between the two stationary target positions. At a closer look, however, the SP has some severe drawbacks. If more than one trajectory modification is required, the sum of all the previous motion plans should still be available and continued to be executed. Moreover, the superposition should take place at hand-space level where the trajectories of the end effector should

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be added vectorially. How this is translated to other levels in the motor hierarchy remains obscured. For example, if the additional motion plan requires activation of an antagonistic muscle, superposition might result in undesired co-contraction. The AR model has the advantage that the target will be reached regardless of the nature of the perturbation and the number of required trajectory modifications. Of importance is that the SP model cannot describe our PM perturbations, because SP is a feedforward model. Its motor plans are determined in advance on the basis of (static) starting and ending positions only and it is assumed that these plans are indeed realized by the motor system. If the hand position is externally perturbed, however, the target will be missed. The AR model is fundamentally feedback-driven as it generates hand paths on the basis of the difference between the actual hand position as provided by proprioception and the target. In the next subsection, we shall use an extended version of the AR model to describe our PM perturbation experiments. 4.2. Description The original AR model of Hoff and Arbib (1991) describes the kinematics of hand movements under a variety of experimental paradigms, such as simple point-to-point movements, target jumps and direction reversals. We have used that model as the building blocks of our modified version as is shown in Fig. 7. It is important to note that, in the absence of perturbations and sensory delays, the AR model predicts symmetrical velocity profiles. At first sight, this property seems to make the AR model less appropriate to describe our data. However, inevitable sensory delays and noisy feedback signals will destroy the symmetry and more elaborate trajectory generators can always be included in the model to describe second-order effects in a subsequent stage. We extended the original model in several aspects to make it applicable to our experimental data. First, the original model was used to describe one dimensional (1D) movements only. Following Flash and Hogan (1985) and Hoff and Arbib (1993),

Fig. 7. Schematic diagram of the feedback model to describe goal-directed arm movements. The hand position X is compared with the target position T using two delayed feedback loops, a visually based loop (indicated by subscript v) and proprioceptively based loop (indicated by subscript p).

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we assume that a planar, 2-D movement can be decoupled into two independent 1-D movements. This assumption is further justified by our own observation that the velocity component in the unperturbed direction is invariant to the perturbation, as was shown in Fig. 3. Hence, we use here two independent trajectory generators, one for the unperturbed Y -direction and another for the perturbed X -direction. Second, the original model uses estimates of the hand position and hand velocity based on (delayed) kinesthetic data only. Consequently, it describes movements performed without vision of the arm itself. Although we are quite able to move without seeing our arm, it is known from daily-life experience that goal-directed movements with visual feedback are performed more accurately. Moreover, if during a visual task, proprioceptive and visual cues are conflicting, like in our PM experiments, the visual cue supersedes the proprioceptive one. Therefore, we extended the model from having only (delayed) visual information of the target position to having (delayed) visual information of the hand position as well. As in the original model, our trajectory unit uses two delayed feedback signals to generate a hand trajectory X ðtÞ: a visually based estimate of the target position T ðt  Dv Þ, and a proprioceptively based estimate of the momentary kinematic hand state (i.e. position, velocity and acceleration). Here Dv denotes the visual time delay. The difference between the target position T and the estimated hand position xFB , the estimated hand velocity v~ðtÞ and acceleration a~ðtÞ, and the remaining movement time D ¼ Tmove  t are used to calculate the new acceleration, according to da=dt ¼ 9~ a=D  36~ v=D2 þ 60ðT  ~xÞ=D3 :

ð3Þ

The output signals of the trajectory unit, i.e. the current acceleration, velocity and position, are found by straightforward integration of Eq. (3). Note that in the absence of noise and time delays, a~  a, v~  v, ~x  x, and Eq. (3) can be solved to yield a minimum jerk trajectory   T 2 2 xðtÞ ¼ T 6s5  15s4 þ 10s3 ; vðtÞ ¼ 30 s ð1  sÞ ; ð4Þ Tmove where s ¼ t=Tmove . In order to obtain the estimates for the current hand position ~xp ðtÞ and current velocity v~p ðtÞ on the basis of delayed proprioceptive information, a lookahead unit is incorporated similar as in the original model. This look-ahead unit compensates for the sensorimotor delays and takes the form of two leaky integrators Z t   ð5Þ aðtÞ dt þ v p t  Dp v~p ðtÞ ¼ tDp

and ~xp ðtÞ ¼

Z

t

  v~ðtÞ dt þ xp t  Dp ;

ð6Þ

tDp

where v p ðt  Dp Þ is the delayed and noisy velocity signal, xp ðt  Dp Þ the delayed position signal and Dp represents the proprioceptive time delay. In our adapted model, an additional feedback loop of the hand position is present. This new loop with time delay Dv is based on vision of the hand as is indicated

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by subscript v. In the comperator, one of the feedback signals of the hand is compared with the (delayed) target position T ðt  Dv Þ to generate the input signal T  xFB to the trajectory unit. The switch ensures that if visual feedback is available, it is used and supersedes the less accurate proprioceptive signal. If the visual feedback is removed, the trajectory generator relies on the proprioceptive signal. Furthermore, subjects had to move a certain distance in the Y -direction in order to detect the presence of the visual perturbation. Therefore, we introduced a dead time Tdead that marked the switch between proprioceptively and visually based feedback, so  ~xp ðtÞ t < Tdead or t > Toff xFB ðtÞ ¼ : ð7Þ xv ðtÞ Tdead 6 t 6 Toff Note that in the absence of delays and noise, xFB equals x for all tÕs, irrespective of the presence of visual feedback. Finally, during simulations of the PM perturbations, the visual feedback is manipulated according to Eq. (2) for Tdead 6 t 6 Toff . 4.3. Simulations In this subsection the experiments PO and DO as presented in Fig. 5 will be simulated using the proposed model. Apart from the overall simulation time step (fixed at 13.3 ms), each of the two components of the model contains three fixed parameters, the two time delays Dv and Dp , and the amplitude of the velocity noise N . The sensorimotor time delay Dp was fixed at 250 ms (Hoff & Arbib, 1991), whereas a value of 40 ms for the visual time delay Dv yielded the best results. The velocity noise Nv is assumed to be proportional to the velocity itself, i.e. Nv ¼ N Rnd½1; þ1 , where Rnd is a uniformly distributed random number between 1. In our simulations we have tolerated a noise amplitude N of 5%, although even higher amplitudes yielded acceptable results. In the original model, the remaining movement time D, i.e. the difference between the planned duration of the entire movement and the elapsed time, is used as one of the input signals (see Eq. (3)). In a preliminary version of our modified model, we used this single quantity as the input for both the X - and the Y -trajectory generator, thereby assuming that the perturbation is completely compensated at the end of the initially planned movement. This resulted in simulated paths of the perturbed trials that were unrealistic: the hand moved away from its initial direction only in the very last part of the movement, in contrast with the experimental results that showed much earlier reactions. Therefore, we used two independent times: one for the unperturbed direction equal to its original definition, and another denoted by Tcomp that represented the compensation time for the perturbation. Furthermore, we have used for each simulation the experimentally found movement time in the Y -direction as well as the final Y -position as the target input to the model. Finally, the simulations of the set of three trials with the same perturbation were simultaneously optimized by changing the dead time Tdead and the compensation time in the perturbed X -direction Tcomp as summarized in Table 1. Two representatives of the model predictions are shown in Fig. 8, where we see experiment PO in the left and experiment DO in the right panel both with

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Table 1 Optimized timing parameters for the model and the mean and maximum absolute differences between the experimental and simulated X -positions associated with a given Y -position Perturbation

Pointer off

k

)0.5

Disturbance off

Toff (s)

0.6

0.8

1.0

0.6

0.8

1.0

0.6

0.8

1.0

0.6

0.8

1.0

Tdead (s) Tcomp (s) Mean (cm) Max (cm)

0.30 0.35 0.17 0.79

0.30 0.35 0.06 0.15

0.30 0.35 0.25 0.53

0.50 0.45 0.14 0.56

0.50 0.45 0.22 0.57

0.50 0.45 0.22 0.69

0.35 0.30 0.12 0.35

0.35 0.30 0.12 0.34

0.35 0.30 0.14 0.51

0.40 0.35 0.12 0.56

0.40 0.35 0.12 0.64

0.40 0.35 0.24 0.69

)0.5

+0.5

+0.5

Fig. 8. Experimental hand paths (solid lines) and model predictions (dots) of two perturbations. Left panel: Pointer off, right panel: disturbance off, both with Toff ¼ 0:6 s. The layout conforms to Fig. 5.

Toff ¼ 0:6 s, i.e. the condition as shown in the left panel of Fig. 5. The solid lines represent the experimental hand paths and the dots the model predictions at a time interval of 13.3 ms. The fits between the predictions and experimental data were quite good. The small discrepancies near the target positions during PO were due to the restriction of having equal Tdead and Tcomp for the three Toff times and could be reduced if that restriction was removed. As was already noticed before, there exists a small asymmetry between leftward and rightward perturbations. From the data in Table 1 we see that hand corrections to the left on perturbations to the right (k ¼ þ0:5) started slightly later than corrections to leftward perturbations (k ¼ 0:5). Moreover, the compensation times for the DO perturbations are slightly shorter than for the PO perturbations. To quantify the fidelity of the model predictions, we have calculated the mean and the maximum absolute differences between the X -positions of the model and the experiment at 28 equidistant Y -positions. Averaged over the twelve experimental conditions, the mean absolute difference

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Fig. 9. Velocity profiles of two of the four hand paths as shown in Fig. 8 (solid curves), together with the model predictions (dashed curves). Thick curves represent the PO experiments, thin curves the DO experiments.

between experiment and simulation (ÔMeanÕ) was only 0.18 cm and the maximum difference (ÔMaxÕ) was 0.55 cm. The velocity profiles of two of the four hand paths as shown in Fig. 8 are depicted in Fig. 9 (solid curves), together with the model predictions (dashed curves), where the thick curves represent the PO experiments and the thin curves the DO experiments. It can be seen that the maximum velocities are reasonably well predicted by the model. The instants of maximum velocity are predicted to occur at a somewhat later stage in the movement. This is a consequence of the trajectory generator that tends to produce rather symmetrical profiles. The introduction of a more sophisticated trajectory generator would be worthwhile although beyond the scope of this article. Hoff and Arbib (1991) added another parameter in their model to account for an extension of the movement duration after a perturbation. Because the duration extensions in our experiments are small and we wanted to keep the number of free parameters as small as possible, we have chosen not to include such a parameter although some improvement of the timing can be expected.

5. Discussion In this paper we have studied how subjects adapt an ongoing, goal-directed movement to an external perturbation. Several aspects of this process can be addressed,

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such as the number of times a motor plan can be updated in a given amount of time, the information that is or can be used to update the ongoing motor plan, and the very nature of the update process itself. In this paper we addressed the former two aspects experimentally and we shall discuss the latter aspect. Hoff and Arbib (1991) raised the interesting question ‘‘. . . whether reaching is controlled by intermittent or continuous feedback, and if it is intermittent, then at what frequency’’. We tried to answer that question by using a paradigm that requires continuous visual feedback throughout the entire movement in order to be successful. At random instants, however, the visual feedback was removed and subjects had to finish their movement under proprioceptive control only. Our PM paradigm mimics a situation where the hand gets off its track towards the target. Evidently, during the perturbation the visual feedback of the hand is in conflict with the proprioceptive feedback. If this discrepancy sustains, a new visuo-motor transformation will be established. With a sustained perturbation strength of k ¼ 0:4, it took the hand path between 10 and 20 trials to become as straight as it was in the unperturbed situation. We therefore used a low number of perturbed trials (6 40%) to prevent this adaptation from occurring. We found that all of our subjects coped with the PM perturbation quite well with target errors only slightly larger than during the unperturbed trials. Another interesting finding was that subjects were unable to conceive the perturbation and were therefore unable to compensate for future pointer movements. In the first set of experiments, we tested how often the trajectory was modified and whether the control was purely sequential or had some parallel components. During a PM perturbation, either the perturbation (DO) or the visual feedback (PO) was turned off after a random amount of time. Subjects were instructed to continue their movement until they had the idea they were on the target. From these experiments it is clear that more than one modification was performed, because additional visual feedback in a subsequent stage resulted in smaller target errors. Moreover, the hand paths during trials where the visual feedback was suddenly removed, were not significantly different from those when the perturbation was switched off (Fig. 5). Based on these two results and on the way we designed the two perturbations, we conclude that the motor plan is updated in such a way that the new plan moves the hand from its perceived current position to the target. If the trajectory is modified more than once, the question remains how often. Our results showed that the trajectories are updated at intervals shorter than 200 ms. This is consistent with other experiments that showed that an ongoing movement can be modified within 125–160 ms after a change in the visual stimulus (Day & Lyon, 2000), i.e. faster than the time required for the stimulus change to become aware to the subject. This short interval time makes a sequential process of visual evaluation of the motor action followed by a modified new one unlikely. A simple model can illustrate this. In Fig. 10, a hypothetical intermittent control scheme based on perception followed by action is shown. The line in the right panel shows a cascade of path segments of the mouse and the line in the left panel represents the corresponding pointer path on the screen. The dashed arrows indicate the initial directions of the submovements. Because the perturbations occur unpredictably, the hand starts to move at instant 0 from the starting position S in the direction of

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Fig. 10. Hypothetical cascade of path segments of the pointer and the mouse (solid lines). During PM, pointer path in left panel and mouse path in the right panel; during TM, both pointer and mouse paths in the right panel. The dashed arrows indicate the movement directions at the various instants of time. S is the starting position and Ti the (intermediate) target position.

the initial target T0 . Due to the PM perturbation, the pointer does not move in the direction of the screen target T , but obliquely. At instant 1, the perturbation is noticed and a trajectory modification is performed that would bring the pointer from position 1 to target T in the remaining movement time according to our earlier conclusion. However, the modification does not incorporate any future pointer displacements and therefore the pointer moves towards position 2. This update process can be continued until the target position is eventually reached. During the TM perturbation, where the target movement is controlled by the hand, mouse and pointer follow the same path as is shown in the right panel of Fig. 10. At instant 0, the hand starts to move towards the initial target T0 . At instant 1, the target has moved to position T1 and a trajectory modification towards T1 is performed. This process is repeated until the final target position is reached. Of course, the individual segments will not show up in the ultimate hand path due to the low-pass filter characteristics of the limb. It can be easily shown, however, that even the quasi-continuous version of this sequential process––where the update interval becomes negligibly short––cannot describe the general features of our measured data. The curvature of the simulated hand path in the first part of the movement is too small and in the last part too large. Interestingly, only the very first perturbed trials of a subject look like the path shown in Fig. 10, and already after a few more perturbed trials, the curvature in the first part of the movement increases at the expense of the last part. A control process where perception and action have components in parallel (Van Sonderen et al., 1989) is more consistent with the present experiments. Also recent results by Heath et al. (1998) point to a parallel control process. These authors studied trajectory modifications of fast pointing movements imposed by either a change in

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target width or a change in target distance. They observed that the first half of the movement is primarily determined by the properties of the original target and rather insensitive to the perturbation. The modification is manifested in the second half of the movement. They hypothesized that the feedback signals during a movement are continuously monitored, but only used to effect modifications if the need arises. In this way, movements can still be fast and unaffected by the long sensorimotor delays, but also easily modified on the basis of feedback signals. Our results with the multitrajectory modifications showed that new feedback signals can be used at least every 200 ms. Thus, it may be argued that if a distinction between monitoring and using feedback signals is possible, the two processes cannot be entirely sequential but have to occur at least partially simultaneous. In our second set of experiments, we wanted to test whether a displacement of the hand is similarly corrected as a displacement of the target. The design of the perturbations were such that they could be corrected by one and the same hand movement. ANOVA tests for 10 subjects revealed no significant differences between the total movement times of the two perturbations nor between the target errors. The same holds for four representative hand positions along the path towards the target. This similarity of PM and TM suggests that, from a control point of view, it makes no difference whether the target or the pointer has been perturbed. Therefore, we can generalize our earlier conclusion to one that states that for the update of the hand paths during a visual reaching task, the control system uses the current hand position in relation to the current target position. In order to capture the results in a simple phenomenological model with a small number of parameters, we have used the kinematic reaching model of Hoff and Arbib (1991) as the basis. We have extended that model with a direct visual feedback loop having its own time delay to obtain the hand position relative to the target. Moreover, it was assumed that as long as visual information of the hand is available, it supersedes the proprioceptive signal even for the case when the two sources are conflicting. This assumption is justified because the percentage of perturbed trials was relatively small and the perturbations themselves were relatively small. It is comparable to the well-known reaching experiments performed by subjects wearing goggles with prism glasses (e.g. Welch, 1986). The very first movements with the glasses on are also primarily controlled by visual feedback, although vision and proprioception are in conflict. A similar approach with respect to visual feedback was chosen by Goodbody and Wolpert (1999). These authors also modified Hoff and ArbibÕs model to include the effect of a mismatch between visually and proprioceptively perceived hand position. Instead of superseded by vision, they assumed that the internal estimate of the hand position is biased towards the visually perceived position, where the bias is proportional to the distance between the hand and shoulder. With only three fixed, global parameters and two optimized parameters to obtain the best fit between simulation and particular experimental condition, our adapted model was able to describe the experimental hand paths. We found that the results of the simulations were rather insensitive to two of the three fixed parameters––the sensorimotor delay and the noise on the velocity signal––because the look-ahead module effectively compensated for them. The third parameter, the visual time delay, had to be short, because

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quickly after the perturbation was switched off the shape of the path changed. A fixed value of 40 ms was the best for all the simulations. This parameter is different from the more commonly encountered visuomotor delay. It represents the delay between the occurrence of the visual stimulus on the retina and the instant it becomes unconsciously available to the central nervous system. The time required to detect the perturbation and the time to compensate the perturbation were optimized for each of the four experimental conditions, i.e. pointer/disturbance off and left/right perturbation. The optimum detection times were different for each condition and somewhat shorter for perturbations to the left. The reason for this asymmetry may originate from the intrinsic shape of the subjectÕs paths that were slightly skewed to the right, thereby effectively extending its initial straight part. Interestingly, the time to compensate for the perturbation should be shorter than the difference between movement time and detection time. This suggests that subjects want to quickly compensate for the perturbation and want to return to the original track before reaching the target. In a future PM experiment, we shall address this question in more detail and study whether and under which conditions a target modification is made towards the original track or to the target itself. Although the model describes only kinematics, we think it can be useful in studies on motor control of the arm and in particular to quantify differences between hand trajectories using only a small number of parameters. 6. Conclusions In summary, we found that subjects modified goal-directed arm movements on the basis of the momentary perceived hand position relative to the momentary perceived target position. These two positions are (quasi) continuously monitored and used to update the hand trajectories if necessary at interval times shorter than 200 ms. Moreover, we argued that this process of perception and action is not completely sequential but should have components in parallel. Acknowledgements The author thanks Mr. A.J. Hardeman for collecting some of the data, and Prof. Dr. C.J. Erkelens for his stimulating discussions. References Alexander, R. McN. (1997). A minimum energy cost hypothesis for arm trajectories. Biological Cybernetics, 76, 97–105. Atkeson, C. G., & Hollerbach, J. M. (1985). Kinematic features of unrestrained vertical arm movements. Journal of Neuroscience, 5, 2318–2330. Bard, C., Turrell, Y., Fleury, M., Teasdale, N., Lamarre, Y., & Martin, O. (1999). Deafferentation and pointing with visual double-step perturbations. Experimental Brain Research, 125, 410–416.

E.J. Nijhof / Human Movement Science 22 (2003) 13–36

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Bernstein, N. (1967). The coordination and regulation of movements. London: Pergamon. Boulinguez, P., & Nougier, V. (1999). Control of goal-directed movements: The contribution of orienting of visual attention and motor preparation. Acta Psychologica, 103, 21–45. Brenner, E., & Smeets, B. J. (1997). Fast responses of the human hand to changes in target position. Journal of Motor Behavior, 29, 297–310. Bullock, D., & Grossberg, S. (1988). Neural dynamics of planned arm movements: Emergent invariants and speed-accuracy properties during trajectory formation. Psychological Review, 95, 49–90. Day, B. L., & Lyon, I. N. (2000). Voluntary modification of automatic arm movements evoked by motion of a visual target. Experimental Brain Research, 130, 159–168. Desmurget, M., & Grafton, S. (2000). Forward modeling allows feedback control for fast reaching movements. Trends in Cognitive Sciences, 4, 423–431. Desmurget, M., Jordan, M., Prablanc, C., & Jeannerod, M. (1997). Constrained and unconstrained movements involve different control strategies. Journal of Neurophysiology, 77, 1644–1650. Elliott, D., Lyons, J., Chua, R., Goodman, D., & Carson, R. G. (1995). The influence of target perturbation on manual aiming asymmetries. Cortex, 31, 685–697. Feldman, A. G. (1986). Once more on the equilibrium-point hypothesis (k model) for motor control. Journal of Motor Behavior, 18, 17–54. Flanagan, J. R., Ostry, D. J., & Feldman, A. G. (1993). Control of trajectory modifications in targetdirected reaching. Journal of Motor Behavior, 25, 140–152. Flash, T., & Henis, E. (1991). Arm trajectory modifications during reaching towards visual targets. Journal of Cognitive Neuroscience, 3, 220–230. Flash, T., Henis, E., Inzelberg, R., & Korczyn, A. D. (1992). Timing and sequencing of human arm trajectories: Normal and abnormal motor behaviour. Human Movement Science, 11, 83–100. Flash, T., & Hogan, N. (1985). The coordination of arm movements: An experimentally confirmed mathematical model. Journal of Neuroscience, 5, 1688–1703. Georgopoulos, A. P., Kalaska, J. F., & Massey, J. T. (1981). Spatial trajectories and reaction times of aimed movements: Effects of practice, uncertainty, and change in target location. Journal of Neurophysiology, 46, 725–743. Goodale, M. A., P
elisson, D., & Prablanc, C. (1986). Large adjustments in visually guided reaching do not depend on vision of the hand or perception of target displacement. Nature, 320, 748–750. Goodbody, S. J., & Wolpert, D. M. (1999). The effect of visuomotor displacements on arm movements paths. Experimental Brain Research, 127, 213–223. Heath, M., Hodges, N. J., Chua, R., & Elliott, D. (1998). On-line control of rapid aiming movements: Unexpected target perturbations and movement kinematics. Canadian Journal of Experimental Psychology, 52, 163–173. Henis, E. A., & Flash, T. (1995). Mechanisms underlying the generation of averaged modified trajectories. Biological Cybernetics, 72, 407–419. Hoff, B., & Arbib, M. A. (1991). A model of the effects of speed, accuracy and perturbation on visually guided reaching. In R. Caminiti (Ed.), Control of arm movement in space: Neurophysiological and computational approaches (Experimental brain research series) (pp. 285–306). New York: SpringerVerlag. Hoff, B., & Arbib, M. A. (1993). Models of trajectory formation and temporal interaction of reach and grasp. Journal of Motor Behavior, 3, 175–192. Kawato, M. (1996). Trajectory formation in arm movements: Minimization principles and procedures. In H. N. Zelaznik (Ed.), Advances in motor learning and control (pp. 225–259). Leeds: Human Kinetics. Komilis, E., P
elisson, D., & Prablanc, C. (1993). Error processing in pointing at randomly feedbackinduced double-step stimuli. Journal of Motor Behavior, 4, 299–308. Latash, M. L. (1993). Control of human movement. Champaign, IL: Human Kinetics Publishers. Morasso, P. (1981). Spatial control of arm movements. Experimental Brain Research, 42, 223–227. Nagasaki, H. (1989). Asymmetric velocity and acceleration profiles of human arm movements. Experimental Brain Research, 74, 319–326. P
elisson, D., Prablanc, C., Goodale, M. A., & Jeannerod, M. (1986). Visual control of reaching movements without vision of the limb. II. Evidence of fast unconscious processes correcting the

36

E.J. Nijhof / Human Movement Science 22 (2003) 13–36

trajectory of the hand to the final position of a double-step stimulus. Experimental Brain Research, 62, 303–311. Prablanc, C., & Martin, O. (1992). Automatic control during hand reaching at undetected twodimensional target displacements. Journal of Neurophysiology, 67, 455–469. Press, W. H., Vetterling, W. T., Teukolsky, S. A., & Flannery, B. P. (Eds.). (1992). Numerical recipes in C: The art of scientific computing (2nd ed). Cambridge: Cambridge University Press. Rabes, P. N. (2000). The planning and control of reaching movements. Current Opinion in Neurobiology, 10, 740–746. Soechting, J. F., & Lacquaniti, F. (1981). Invariant characteristics of a pointing movement in man. Journal of Neuroscience, 1, 710–720. Soechting, J. F., & Lacquaniti, F. (1983). Modification of trajectory of a pointing movement in response to a change in target location. Journal of Neurophysiology, 49, 548–564. Turrell, Y., Bard, C., Fleury, M., Teasdale, N., & Martin, O. (1998). Corrective loops involved in fast aiming movements: Effect of task and environment. Experimental Brain Research, 120, 41–51. Uno, Y., Kawato, M., & Suzuki, R. (1989). Formation and control of optimal trajectory in human arm movement––Minimum torque-change model. Biological Cybernetics, 61, 89–101. Van Sonderen, J. F., & Denier van der Gon, J. J. (1990). A simulation study of a programme generator for centrally programmed fast two-joint arm movements: Responses to single- and double-step target displacements. Biological Cybernetics, 63, 35–44. Van Sonderen, J. F., Denier van der Gon, J. J., & Gielen, C. C. A. M. (1988). Conditions determining early modification of motor programmes in response to changes in target location. Experimental Brain Research, 71, 320–328. Van Sonderen, J. F., Gielen, C. C. A. M., & Denier van der Gon, J. J. (1989). Motor programmes for goaldirected movements are continuously adjusted according to changes in target location. Experimental Brain Research, 78, 139–146. Welch, R. B. (1986). Adaptation of space perception. In K. Boff, L. Kaufman, & J. Thomas (Eds.), Handbook of perception and performance (Vol. 1). New York: Wiley Interscience. Zalaznik, H. N., Schmidt, R. A., & Gielen, S. C. A. M. (1986). Kinematic properties of rapid aimed hand movement. Journal of Motor Behavior, 18, 353–372. Zhang, X., & Chaffin, D. B. (1999). The effects of speed variation on joint kinematics during multisegment reaching movements. Human Movement Science, 18, 741–757.