Journal of Experimental l~ycholol~y: Human Perception and Performan~ 1985, Vol. 11, No. 6, 828-845

Copyright 1985 by the American Psychological Ast,ociatiou, Inc. 0096-1523/85/$00.75

Segmentation and Coupling in Complex Movements Paolo Viviani Istituto di Fisiologia dei Centri Nervosi Consiglio Nazionale delle Ricerche, Milan, Italy University of Geneva, Geneva, Switzerland

Marco Cenzato lstituto di Fisiologia dei Centri Nervosi Consiglio Nazionale delle Ricerche, Milan, Italy

In two experiments we explore the structure of complex sequences of drawing movements. We find that in these movements a single parameter--the velocity gain factor--relates the geometrical and kinematic aspects of the movement trajectory via a two-thirds power law. In Experiment I we investigate the relation between the velocity gain factor and the linear extent of the trajectory. In Experiment 2 we demonstrate that the gain factor provides a criterion for segmenting the movement into distinct units of motor action, and we investigate the effects of the speed of execution on this segmentation. A theoretical analysis shows that the results of both Experiments ! and 2 can be given a unitary interpretation by assuming a coupling function of variable strength between segments. The general problem of representing motor programs is discussed within this theoretical framework. Segmentation of Complex Movements Into Units of Motor Action It is generally accepted (cf. Schmidt, 1982) that movements involving fast activation of relatively simple, unidirectional synergies, such as stroking, pointing, ocular saccades, and so forth, are planned as self-contained units of motor action and that their unfolding under constant environmental conditions is largely independent of incoming information. With respect to the more complex motor patterns encountered in handwriting and drawing, the question can then be raised whether these movements are performed by assembling and sequencing simple recipes or motor schemata defined as rules to produce a prototypic elementary action (Greene, 1982). Indeed, the notion of decomposition emerges in many different approaches to the motor control of hand, finger, and speech movements. Several authors (Hulstijn & van Galen, 1983; Keele, 1981; Klapp, 1977; Rosenbaum, Inhoff & Gordon, 1984; Sanders, 1980; Shaffer, 1981; Sternberg, Monsell, Knoll, & Wright, 1978; Wing, 1978) have used chronometric and reThis article includes parts of the doctoral dissertation of Marco Cenzato. The research was supported by Consiglio Nazionaledelle Ricerche. Wegratefullyacknowledgethe help of Lella Orlando in preparing the manuscript. Requests for reprints should be sent to Paolo Viviani, IFCN-CNR, Via Mario Bianco,9, 20131 Milan, Italy.

action time data to argue that the planning of these complex movements is hierarchically organized. From this premise one may conclude that before a movement can be executed it must be represented as the hierarchy of simple discrete components (Crossman & Goodeve, 1983; Klapp, Anderson, & Berrian, 1973; Klapp & Wyatt, 1976; van Galen & Teulings, 1983). Other authors, and more specifically those involved in modeling hand and finger movements (Bizzi & Abend, 1983, Herbst & Liu, 1977; Morasso & Mussa Ivaldi, 1982; Morasso, Mussa Ivaldi, & Ruggiero, 1983; Vredenbregt & Koster, 1971), maintain that the trajectory of the movements can be decomposed into segments (strokes) on the sole basis of kinematic and geometrical criteria. In most cases (cf. Teulings, Thomassen, & van Galen, 1983), a one-to-one correspondence is more or less explicitly assumed between these strokes and an underlying set of units of motor action. Finally, it has been claimed that indirect support in favor of the decomposition hypothesis may be derived from the temporal patterns of electromyographic activity in the agonist and antagonist muscles (Denier van der Gon & Thuring, 1965; Vredenbregt & Rau, 1971). In our view, however, convincing evidence in favor of the decomposition hypothesis has been presented only for such movements as typing (Viviani & Terzuolo, 1983), Morse code tapping (Bryan & Harter, 1897, 1899), and

828

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS

piano playing (Shaffer, 1981), which translate a highly structured symbolic code into a corresponding motor output. In all these cases, in fact, the problem of identifying the criteria of decomposition is greatly simplified by the possibility of concentrating on just one measurable aspect of the movements, the time sequence of the discrete events that compose the gesture. In the general case, however, support for the notion that a discrete structure underlies the apparently continuous modulation of kinematic and dynamic parameters is fairly indirect and not very compelling. Moreover, a case can be made (Kelso, 1981; Kelso, Holt, Rubin, & Kugler, 1981; Kugler, Kelso, & Turvey, 1980) for alternative approaches to the motor control problem that emphasize the morphogenetic potentialities of dynamical systems and take issue with the logical necessity of postulating the existence of identifiable discrete units. To be sure, circumstantial evidence, including that provided by the geometrical analysis of movements trajectories, may and should be considered if one tries to identify a set of hypothetical units of motor action. We maintain, however, that in order for such a notion to advance our understanding of motor programming significantly, the criteria used to isolate these units should ultimately be formulated in terms of quantities directly relevant to the motor plan. The purpose of this report is to propose one such criterion and to test its validity in a relatively simple instance. The rationale for our proposal and for its experimental validation will be introduced next. Velocity Gain Factors and the Planning of Movements In a previous report (Lacquaniti, Terzuolo, & Viviani, 1983) we described an empirical relation, the so-called Two-Thirds Power Law, which applies piecewise in the course of continuous drawing movements. Let V(t) and A(t) denote the instantaneous values of the tangential and angular velocity of the movement, respectively. Moreover, let R(t) and C(t) be the instantaneous values of the radius of curvature and of the curvature of the trajectory. By definition, these four quantities are related in the following ways:

829

R(t) = l/C(t); A(t)= V(t)/R(t)= V(t)C(t). It can then be shown that both (mathematically equivalent) relations, A(t) = K . C(t)2/3

V(t) = K . R(t) 1/3,

(1)

provide a good approximation to the experimental data at all points of the trajectory sufficiently removed from points of inflection (R = ~). Two experimental findings concerning the parameter K are relevant to the issue discussed in this article. 1. Experiments in which complex movements are considered, such as scribbles, elliptic spirals, irregular closed forms, and handwriting (Viviani & Terzuolo, 1982; Lacquaniti, Terzuolo, & Viviani, 1983), suggest that K takes different values over successive segments of the trajectory but remains approximately constant within each segment. 2. In the particular case of simple closed patterns (such as ellipses) the parameter K is constant throughout the entire movement and is related to the perimeter P of the pattern by a power law (Lacquaniti, Terzuolo, & Viviani, 1984): K = KrP ¢ (Kr,/3 > 0). (2) In this expression KT is a parameter that depends only on the general tempo selected by the subject. From the equations in Equation 1 and Point 1 above it follows that at any instant during the movement the velocity of execution V(t) depends on both the instantaneous, everchanging radius of curvature of the trajectory being executed and a term (K), which is invariant and characteristic of each successive segment of the trajectory. For obvious reasons the term K is dubbed the velocity gain factor. From Point 2, the tentative conclusion can be drawn that the specific value taken by K for one segment depends on the linear extent of that segment. With respect to the segmentation issue considered here, one can then propose the following specific hypotheses: 1. In general, the execution of complex movements is decomposed into chunks of motor action. 2. At the level of motor planning, one qualifying attribute of the chunk (i.e., one of the relevant controlled variables) is the linear ex-

830

PAOLO VIVIANI AND MARCO CENZATO

tent o f the corresponding segment o f the intended trajectory. 3. The identification o f the chunks can be carried out by analyzing the covariation o f the velocity gain factor K with the linear extent. Quest for C h u n k s

tematically varied. We use this manipulation first to verify that a subdivision into c h u n k s is indeed borne out by the kinematic analysis o f the movement. Then we study the relation between the timing o f the subunits and their linear extent. In Experiment 2 we vary the general t e m p o o f the execution across trials to demonstrate the presence o f a d y n a m i c interaction between the planning o f the subunits. The nature o f this interaction is described within the framework o f a simple coupling model.

It would appear that the most direct way o f verifying the hypotheses stated above is to calculate the ratio K = V(t)/R(t) 1/3 for the entire time course o f the m o v e m e n t and to use statistical techniques to partition this time course Method into n o n o v e d a p p i n g intervals such that K can be considered constant within each interval Subjects and significantly different across intervals. Three normal adults (2 males, 1 female, including the However, most studies on handwriting and reauthors) volunteered for the experiments. lated movements (for a review, see Viviani & Terzuolo, 1983) point to the presence o f contextual effects in the time structure o f the m o - Apparatus Movements were recorded by a digitizing table Calcomp tor sequences. In particular, Viviani & McCollum (1983) have demonstrated that the av- 9240 (accuracy .025 mm; sampling rate 72 samples/secerage velocity in the two interlaced circles that ond). The pen used is similar in weight and size to an ordinary ballpoint pen. In all cases 1,000 samples (13.85 form the figure eight pattern are not indepen- s) of the X and Y coordinates of the pen's tip were recorded. dent o f each other. Thus, one must in general The task was to draw a geometrical pattern composed of allow for the possibility that the velocity gain two ellipses knotted in a folded-up figure eight (see Figure factor depends on both the linear extent o f the 1, Panel A). The two parts of the figure had the same eccentricity (~ = .90), and the inner ellipse was half the size segment being executed and the linear extent of the outer one. A complete outline of the pattern was o f the other segments. For this reason, a m o r e drawn on the table to provide visual guidance for the heuristic approach to the validation o f the movement. chunking hypothesis seems preferable to the extensive statistical search mentioned above. Experimental Procedure More specifically, it seems sensible to start off The study is based on two experiments. In Experiment with some plausible conjecture concerning the 1 a series of eight patterns of different sizes was provided. segmentation and to verify post factum that In the first element of the series the larger perimeter was indeed the analysis o f the velocity gain factor /'2 = 85.950 cm, and the smaller P~ = 42.957 cm. All is supportive o f the conjecture. Although there successive patterns of the series were scaled down in size according to a geometrical progressionwith ratio ~ . Thus is in general no simple way o f determining a the smallest pattern had the following perimeters: P2 = priori the location o f the c h u n k boundaries, 7.588, PI = 3.794 cm. The smaller patterns required only there are special patterns that, because o f their finger and wrist movements. The larger ones involved structure, suggest the presence o f clearly iden- mostly elbow and shoulder movements. The subject drew each pattern continuously, and the tifiable figural subunits (Viviani & McCollum, actual recording started after a few warming-up cycles. 1983). In these cases, one can tentatively as- The speed of execution and the direction oftbe movement sume that each subunit maps into a corre- were spontaneouslychosen by the subjects. It was required, sponding unit o f m o t o r action. Such patterns however, that the same direction be maintained for the are therefore choice targets for testing the idea entire sequence. A complete experimental session consisted of five identical sections. In each section the sequence of expressed above. patterns was executed twice, first from the larger figure to In this article, two experiments are pre- the smaller, and then in the reverse order. Thus, each of sented that investigate the chunking hypothesis the eight patterns was executed 10 times. The recordings and its implications vis-a-vis the problem o f within a section were closely spaced in time (about 10 s). A couple of minutes of rest were allowedbetween sections. timing control in one o f these special patterns, In Experiment 2 only the second pattern in the sequence namely the folded-up elliptic figure eight. In was used (P2 = 60.766 era, Pi = 30.383 era). The subject Experiment 1, the size o f the patterns is sys- was first provided with a timing signal (2~msacoustic click

831

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS delivered by earphones), which indicated the required period of one complete cycle of the movement. When the subject felt that he or she could reliably match this required period, the timing signal was discontinued and the movement was pursued without interruption for the time necessary for 10 successive recordings. In a complete session we first recorded in a random order all multiples of 0.2 s in the range 1.0 < T < 4.0 s. Because the period actually produced did not always match the required value T, a number of additional recordings were then performed to cover as uniformly as possible the range of interest. Occasionally the observed performances exceeded this range.

Short periods of rest were inserted between successive series of 10 recordings. Moreover, to avoid fatigue, the entire sequence of recordings was distributed over several days. Occasionally in the course of a recording, subjects felt that they were not controlling reliably the period of movement. In these cases the entire sequence of 10 successive recordings was repeated.

Data Processing One movement cycle is defined as the continuous execution of the inner and outer ellipses, starting and ending

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Figure 1. Typical recording for one trial. Top panel: several cycles of execution of the double elliptic pattern (recording period: 13.8 s). (The analysis of the velocity gain factor [see text] is limited to the two half ellipses opposite to the junction point O. The separation of the trajectory [vertical bars] was performed automatically using as a criterion the point with the maximum value of the vertical component of the displacement. During the recording, the major axis of the pattern was actually tilted 45 ° counterclockwise with respect to the horizontal. Cycle-by-cycle geometrical variability in all conditions is of the same order of magnitude as in this typical example.) Bottom panel: Time course (in seconds) of the tangential velocity (upper tracing) and of the cubic root of the radius of curvature (lower tracing) for the complete recording of the movement shown in top panel. (Notice the resemblance of the two traces.)

832

PAOLO VIVIANI AND MARCO CENZATO

at their junction point 0 (Figure 1, Panel A). Thus, in both parts of the experiments,the recordingperiod for one trial ( 13.85 s) comprisesa variablenumber of complete movement cycles.Atterwe smoothedthe raw data with a (doublesided exponential) numerical filter (cut-off: 50 Hz), each complete cycle was divided into two parts, corresponding to the execution of the two ellipses,respectively.The mean values of the perimeter (P), eccentricity (~), and period of execution (T) for each ellipsewerecalculated by averaging separatelythe correspondingvaluesfor each cycleof a trial and all repetitions within an experimentalcondition. The instantaneous values of the tangential velocity(V) and of the radius of curvature (R) were calculated numerically for each entire trial. However,because the junction point 0 represents a transition between two distinct modes of operation (see Results), only the values of V and R in the half ellipses opposite to the junction were retained for analysis. The points used to cut each ellipse in two halves were those with the highest and lowest values of the y coordinates (cf. Figure 1, Panel A). Results As described in the introduction, the basis for the analysis proposed in this report is provided by the empirical relation between the instantaneous values of the tangential velocity V(t) and of the radius of curvature R(t):

V(t) = K" R(t) I/3.

(3)

Panel B in Figure 1 shows the recording of V(t) and R(t) ~/3 for all the movement cycles in the same trial depicted in Panel A. This specific example describes a relatively fast trial in Experiment 2, but the obvious proportionality between V(t) and R(t) ~/3 is typical of all recorded trials. Thus, the empirical relation expressed by Equation 3 is verified also for the double ellipse considered here. No obvious discontinuity is ever apparent in the time course of V and R I/3 at the transition from one ellipse to the other. However, by calculating the instantaneous value of the ratio V(t)/ R(t) ~/3 under various experimental conditions (discussed later), it can be seen that the gain K takes in general two distinct values over the inner and outer ellipse and that the transition between these two values occurs, more or less rapidly, in the vicinity of the junction point 0 (Figure 1, Panel A). These variations of the gain are now examined in detail.

Subdivision o f a Movement Cycle Into Units o f Motor Action First, we demonstrate that, as suggested in the introduction, the subdivision of the double

ellipse into clearly distinct figural elements maps into a structural subdivision of the motor plan. A reliable measure of the average value of K for each ellipse can be obtained by a linear regression analysis of the instantaneous values of V(t) on the corresponding values of R(t) t/3 over the half trajectory opposite to the junction point (cf. Method). Panel A in Figure 2 illustrates the result of this procedure for a typical high-velocity trial from Experiment 2 (average duration of one complete cycle = 1.03 s). The two bundles of data points (the points are too close to be distinguished) describe the covariation of V(t) and R(t) I/3 in the smaller and larger half-ellipse, respectively. The slope of the regression line through the data points supplies the least-square estimate of the velocity gain factor K in the corresponding parts of the complete movement cycle. For all trials in both experiments the linear correlation between V(t) and R(l) 1/3 is very high (.90 _< r < .98; typical values of the 95% confidence interval for K are of the order of 10% of the average value). Moreover, the intercepts CI (large ellipse) and C2 (small ellipse) of the two regression lines are always small relative to the corresponding average values of K (Experiment 1, Subject PV: CI = .551 _+ .358; C2 = -.361 _+ .132; Subject MC: CI = 2.406 + 2.828; C2 = - . 8 4 3 + .231; Subject MF: CI = - 1 . 6 6 2 + .909; C2 = - 1.058 + .567; Experiment 2, Subject PV: CI = 2.537 + 1.057; C2 = - . 1 4 7 + .685; Subject MC: CI = 2.262 + 1.973; C2 = - 1 . 1 4 9 + 1.662; Subject MF: CI = .095 + 1.426; C2 = - 1.096 +__.914). As indicated by both the ratio R K = K2/K~ and the difference S K = K2 - K~, the velocity gain factor in the outer ellipse (1(2) is significantly larger than the gain in the inner ellipse (K0. In the experimental conditions of the trial shown in the Figure 2, the seemingly continuous variation of the kinematic and figural parameters of the movement conceals a discrete modulation of the relational parameter K. Panel B in Figure 2 shows the results for another typical trial in Experiment 2 in which, however, the rhythm of execution was much slower (average duration of one complete cycle = 2.51 s). In this case, the gain factors in the two ellipses are virtually indistinguishable, and their c o m m o n value is much smaller than in Panel A. This difference in the behavior of K between fast and slow trials was observed in all subjects.

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS In conclusion: 1. The rhythm of execution affects the degree of modulation of the velocity gain factor. 2. Whenever a significant modulation is present, this is taken as evidence that the two ellipses that form the complete pattern can be construed as distinct functional units of motor action. Point 1 above will be taken up in more detail when presenting Experiment 2 in a later section.

Relation Between the Velocity Gain Factor and the Perimeter The second point to be considered here concerns the relation between the gains K~ and K2 in the two units of motor action and the

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linear extent of the total trajectory. Figure 3 shows the results of the first experiment, which is directly relevant to this point. For each subject, data points describe the relation between the parameters K~ and/(2 in the two ellipses and the total perimeter o f the pattern P. A quantitative analysis of these data, leading to the fit reported in Table 1, is postponed until a formal model is introduced. Here we simply stress the fact, already discussed previously (Lacquaniti et al., 1984), that both velocity gains KI and K2 are well approximated by power functions of the linear extent P of the trajectory, the exponent/~ of the power function being roughly .6. However, marked differences exist among subjects. In particular, the data from 1 subject (PV) are compatible

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834

PAOLO VIVIANI AND MARCO CENZATO

with the hypothesis that the ratio K2/KI is invariant for all sizes, whereas this is obviously not the case for Subject ME Similar individual differences occur when considering the total execution time T -- T~ + T2 as a function of the pattern size (Figure 4). In 2 subjects (MC and PV) the compensatory covariation of velocity and size is quite effective inasmuch as a more than a tenfold increase in the pattern size (see Methods) results only in a lengthening of the execution time of the order of 50% (we refer to this as isochrony). Moreover, the rhythm of execution spontaneously chosen by these subjects is rather fast (_--1.5 s). By contrast, Subject MF has adopted a considerably slower rhythm, and the compensation is less effective, especially for the larger patterns. In

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Figure4. Total execution time TOn seconds) as a function of the total perimeter of the pattern. (Data in 3 subjects from Experiment 1. Each data point is the average of 10 trials. Bars encompass ___1 standard deviation. Units on the x-axis are log values of the perimeter P expressed in centimeters. A manifold increase in the total linear extent of the pattern produces comparatively small variations in execution time.)

PV

all cases, however, the rhythm chosen by the subjects is fast enough to induce a systematic and significant difference between Kt and/(2. Thus, a segmentation of the movement occurs naturally, even in the absence of time constraints.

Relation Between the Velocity Gain Factors and the Rhythm of Execution

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Figure 3. Velocity gain factor as a function of the total perimeter of the pattern. (Data for 3 subjects [PV, MC, MF] from Experiment 1. Each data point is the average of 10 trials. Bars indicate 1 standard deviation. Units on the x-axis are log values of the perimeter P expressed in centimeters. The dimensions of the gain factor K are cm2/3ls. The very obvious linear trend of the data points in a double logarithmic scale suggests a power-function relation between P and K.)

Taken together, the results displayed in Figures 2, 3, and 4 strongly suggest a relation between the velocity gains KI and Kz and rhythm of execution. This point is specifically dealt with in the second experiment in which a pattern of a given size is executed at different velocities. The rhythm of execution affects the geometrical properties of the patterns very little. In 2 subjects the size of the two ellipses reduces slightly (-10%), at the highest velocity. However, the ratio Pz/P~ is practically constant, and the eccentricities remain very close to the theoretical value ~ = .9. (Subject PV: ~ = .876 +

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS

835

1.41 K?

.014; ~2 = .899 _+ .009. Subject MC: E~ = .890 + .012; ~2 = .903 _+ .014. Subject MF: ~t = .886 ___ .022; ~2 = .900 _ .011.) The relative variability 13 of the total execution time is almost constant at all speeds. The main result o f this experiment is shown in Figure 5, where the average gains K~ and K2 are independently plotted as a function of the total execution time T. As 1.1 expected intuitively from their definition (cf. Equation 3) and also from the data in Figure 2, both gains are a decreasing function of T. 1.0 In agreement with the data of Figure 3, the gain plot for the larger ellipse (K2) is entirely above the plot for the smaller ellipse (K0. However, the difference /(2 - K~ decreases sharply with increasing T, and for T --- 2.8 s the gains are virtually indistinguishable. For 1.0 < T < 2.8 s the ratio K2/KI decreases with the rhythm in a very similar manner for the 3 1.0

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Figure 5. Velocity gain factors as a function of the total execution time (T) in seconds. (Data of 3 subjects [PV, MC, MF] from Experiment 2. Kt and K2 indicate the velocity gain factors in the inner and outer ellipses, respectively. Each data point is the average of 10 trials. Bars indicate 1 standard deviation.)

subjects (Figure 6). No common trends emerge for larger values of T, where the data points are too scattered. In conclusion, Experiment l demonstrated the possibility of decomposing drawing movements performed spontaneously into functionally distinct subunits; Experiment 2 shows that the degree of differentiation between these subunits varies as a function of the overall tempo of the movement.

A Coupling Model In what follows we introduce a formal model to account for the main experimental findings

836

PAOLO VIVIANI AND MARCO CENZATO

presented above. Our starting point is the observation made previously for both isolated circles (Viviani & McCollum, 1983) and ellipses (Lacquaniti et al., 1984) that as long as the movement comprises one single unit of motor action the corresponding (unique) velocity gain factor is a power function of the pattern perimeter: K -- KrP ~, where Kr depends only on the (self)imposed rhythm of the movement and/3 - .6. One can entertain two extreme views for generalizing this observation to the case of the double ellipse. According to one view, a complete cycle of movement would result by merely adjoining two units of action--one for the small, the other for the large ellipse (total uncoupling). According to the opposite view, the entire cycle is planned as a unit (total coupling). If one assumes that in any case the gain K covaries with the total linear extent of the trajectory encompassed by a single unit, simple calculations show that whatever the adopted view, the total duration TI + T2 of one complete cycle is only weakly dependent on the total perimeter of the double ellipse. Thus, the results presented in Figure 4 concerning the relation between size and total duration cannot discriminate clearly between the two alternative hypotheses. However, the two views do yield sharply different predictions concerning the ratio of the execution times. As shown in Appendix A, the Two-Thirds Power Law expressed as in Equation 3 implies that the period of execution of an isolated ellipse is given by the formula T - 27d2(0 p2/3, K

(4)

where P is the perimeter of the ellipse and the term f~(~)depends only on the eccentricity. In the case of total uncoupling, substituting for each ellipse the empirical expression for K (Equation 2) in Equation 4 and remembering that/3 - .6 one gets the following: Tl

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Also the predictions concerning the ratio of the gain factors are quite different. If each unit of action is independently planned according to the same principle that applies for the single ellipse, we would expect that the ratio K2/KI to be equal to (P2/POa ~- 20.6 = 1.516. If, on the contrary, the entire cycle is planned as a unit, we should have instead K1 = K2 = Kr(Pl + P2)~. The first hypothesis considered above is in qualitative agreement with the observed gain values for fast movements, whereas the second one is more in keeping with the results for slow movements (see Figure 6). Neither view, however, can account for the fact that in Experiment 1 the ratio K2/KI is 1.21, 1.21, and 1.12 in the 3 subjects, respectively, and that in Experiment 2 it actually spans the entire interval between 1 and 1.3. Let us suppose that indeed successive units in the motor plan for complex patterns represent contiguous stretches of the trajectory. Then, a variable coupling hypothesis can be considered that stipulates that (a) under certain conditions, contiguous chunks of motor action coalesce into higher order units; (b) the velocity gain K is specified simultaneously by both the fine- and coarse-grained levels of segmentation. If so, the data suggest that the relative weight of each of these two effects can be modulated as a function of the average rhythm of execution. To formalize this intuition, we assume that the velocity gain factors can be represented by the following generalization of Equation 1: Kl(W) = Kr((1 - ~(w))Pq + ~(w)(PI + e 2 : ) K2(w) = KT((I -- ~(6o))P{ + ~(o~)(P, + P2)~).

(5)

In the case of total coupling, the linear extent of the trajectory encompassed by a single unit is Pi + P2, and one gets instead

(7) In these equations, w = 2r/(Tl + T2) represents the overall rhythm of the movement, and 2; is a coupling parameter that ranges between 0 and 1. When 2; = 0 (total uncoupling),

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS KI and K2 depend only on the perimeters PI and P2, respectively. When ~; = 1, both velocity gain factors depend only on the total perimeter PI + P2.

Evaluation of the Model Although the two equations in Equation 7 are supposed to apply under both experimental conditions described above, the interpretation of the term KT differs somewhat according to the parameters actually controlled in the experiment. When the size of the patterns is manipulated (Experiment 1), the parameter KT should in principle reflect the self-imposed rhythm of the movement. When the timing of the movement is imposed (Experiment 2), Kr should instead represent the overall rhythm parameter set by the subject to comply with the experimental requirements. In both cases, however, we can assume without loss of generality that

837

mates of these parameters for the 3 subjects (PV, MC, and MF) are then calculated with a simplex minimization algorithm (Nelder & Mead, 1964): PV MC MF K0 .391 .440 .541 fl .662 .630 .597

Notice that, as predicted by the equations in Equation 5, the estimated value offl is virtually equal to 2/3for subject PV, who displays almost perfect isochrony (cf. Figure 4), whereas fl < 2/3 for Subject MF, who departs considerably from isochrony. The accuracy of the model can be estimated by comparing the experimental values of K2 in Experiment I with those predicted by Equation 9 upon substituting the estimates of Ko and fl (Table 1). The excellent approximation obtained with this procedure provides a first confirmation of the ideas embodied in the model. Additional support for the model results from considering the variations of the parameter Kr as a function KT = 27rKo/(Tl + T2) = Ko~0. (8) of rhythm w. Eliminating again Z(w) between For each experimental condition, the equa- the two equations in Equation 7 and solving tions in Equation 7 relate the observed values for KT we obtain a relation of the type P1, P2, T~, Tz, Kl, and K2 to three unknowns: Kr = f(Kl, K2, P1, P2, fl). (10) the constants K0 and/3, which are supposed to be independent of the controlled variables Since we assume that the value of/3 is inde(rhythm and size of the movement) and the pendent of the experimentally controlled varicoupling factor ~, which depends instead on ables, this relation can be used to predict the the rhythm of execution. Any set of N exper- values of the rhythm parameter Kr in both imental conditions in which the rhythm of ex- Experiments 1 and 2 by inserting the correecution is manipulated (either directly as in sponding values of KI, K2, P~, and P2. The Experiment 2 or indirectly as in Experiment results are shown in Figure 7, where different 1) yields therefore 2N independent equations symbols are used to identify the two condiwith N + 2 unknowns (N values of ~(co) and tions. The lines interpolating the data points the two constants Ko and/3). When N > 4, an were obtained by inserting in Equation 8 the overdetermination results that can be used in parameter Ko estimated from Experiment I. Notice that although the range of variation a variety of ways to verify the assumption of of the execution time in Experiment l is much the model. Our validation strategy is based on the fol- smaller than that in Experiment 2 (cf. Figure lowing steps. First, we get the least-square best 4), one and the same relation applies in both estimates of the parameters Ko and ft. By elim- cases between Kr and ~0. This supports in parinating Z(~o) between the two equations in ticular the contention that the validity of the Equation 7 and substituting KT (Equation 8), simple coupling model expressed by the equations in Equation 7 extends over the two we obtain a relation of the type sharply different experimental conditions. 1£2 = f(Kl, P1, P2, Tl, T2, Ko, /3).1 (9) Figure 8 provides a further confirmation of Inserting in this relation the values KI, K2, the fact that the same basic mechanisms are PI, P2, T~, and T2 measured in the Experiment involved in both cases. Data points represented 1, we obtain a set of 8 nonlinear equations (one for each pattern size) in two unknown 1Explicit expressions for Relations9, 10, and 11 are parameters Ko and /3. The least-square esti- givenin AppendixB.

838

PAOLO VIVIANI AND MARCO CENZATO

Table 1

Experimental and Predicted Values o f the Velocity Gain Factor Pattern size Subject

1

2

3

4

5

6

7

8

8.86 .58 8.88

10.80 .74 10.78

12.78 .65 12.71

15.73 .84 15.68

19.17 .92 18.95

23.11 .76 23.10

27.08 .84 26.98

31.58 2.13 31.66

8.51 .36 8.47

11.78 .89 11.50

! 4.15 1.12 13.82

16.92 1.81 16.54

20.33 2.02 19.73

23.01 2.48 22.87

26.42 1.86 26.20

28.69 1.67 29.14

6.69 .54 6.98

7.86 .95 8.16

10.35 1.20 10.34

1 ! .54 1.37 11.72

13.37 1.37 13.38

15.12 1.32 15.87

16.27 1.91 17.16

18.04 2.08 18.51

PV

K~ M

SD K~' MC

K2 M

SD K~' MF

/(2 M

SD K~'

Note. For each subject are indicated the measured values of the gain/(2 in the larger ellipse (mean and standard deviation) and the corresponding (K~') predicted via Equation 9 in the text.

by open triangles in this figure describe the ratio K2/K~ observed in Experiment 1 as a function of the total execution time. Data points represented by open circles summarize

for comparison the corresponding results from Experiment 2 already shown in Figure 6. Obviously, the trend of K2/KI over the limited range of variations of T induced by changing

-I{T

PV

• o,/2 " * ~

/o&J ~

__./"

I

o

°~°

joJ

• -o~"

I

I

o

MC MF

I~

7I

•

(~ ( r a d / s e c ) Figure 7. Predicted values of the parameter Kr as a function of the overall rhythm Of execution. (Each point and the interpolating lines were calculated, as indicated in the text, from the best firing values of the model parameters. The rhythm of execution on the abscissa is expressed as a pulsation rate (co = 2~r/T) measured in radians per second. (Open circles = predictions for Experiment 2. Shaded triangles = prediction for Experiment 1.)

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS

movement size is quantitatively similar to the trend observed in Experiment 2, in which the execution time is directly and extensively manipulated. As a final step in the exploitation of the model, we focus on the form of the coupling function itself. Simple calculations permit one to derive from the equations in Equation 7 a relation of the general type

1AK2 1.3 -

K1

e

PV

o

12

o o o

•

oe

•

o o

1.1

4) °o o

• o

10 MC

o

o a

1.1

•

o

o o o

1.0

o e

1.2

1.1 o o

o

o

o

o

1D

o

•

o

o

~(oo) = f(Kt, K2, Pi, P2, ~).

839 (11)

In principle, this equation can be used to calculate Z(o~)from any set of data corresponding to a condition in which the overall rhythm varies. However, because the variations in rhythm induced by changing the pattern size are modest, Experiment 1 would allow one to define 2;(o~)only over a limited range of T (see Figure 8). Assuming again that the value of the exponent ~5 is subject-specific, but otherwise independent of the experimental conditions (see above), we can instead calculate the coupling function over a considerable range of T by using the estimates of B calculated from Experiment 1 and inserting in Equation 11 the values of Kl, K2, Pt, and P2 measured in Experiment 2. In Figure 9 different data points describe the values of the coupling function in the 3 subjects, as a function of the logarithm of the execution time T = T] + T2. Despite the considerable quantitative differences in t h e timing parameters of the individual performances, the corresponding coupling functions are almost indistinguishable. This suggests that individual differences are mostly confined to the degree of isochrony (Figure 4) and to the average rhythm spontaneously selected in the first experiment, whereas the notion of coupling function captures a stable feature of the motor control common to all subjects. The degree of coupling progressively increases as a function of the total execution time up to T - 2.8 s. At this rhythm the two parts of the pattern are practically executed as a single unit because the velocity gain parameter is based mostly on the total perimeter P~ + /)2. For larger values of T the results become very confused because of the considerable scatter in the ratio K2/K1 (cf. Figures 5 and 6).

.9

o

i

T

Figure 8. Comparison between Experiment 1 and 2 for 3

subjects(PV,MC,MF).(Datapointsthatare opentriangles represent the ratio K2/K~ between the velocity gain factors in the two ellipses, calculated from the results of Experiment 1. Data points that are open circles reproduce for comparison the results for Experiment 2 [Figure 6]. On the abscissa is indicated the total period of execution T. In Experiment I the variations in total execution time are indirectly produced by changing the total perimeter P and are relatively modest [see Figure 4]. The trend of the ratio K2/KI is nevertheless comparable to that observed in Experiment 2, where large changes in execution time IT, in seconds] were directly induced.)

Discussion The results presented can be summarized in three points: I. One single parameter, the velocity gain factor, summarizes the relation between the kinematic and geometrical aspects of complex drawing movements. This parameter typically takes different values over the two ellipses that form the pattern. 2. The velocity gain factor in each of the two parts depends on (a) the overall rhythm

840

PAOLO

VIVIANI

AND

MARCO

CENZATO 0

0

.oiE

Q

.9

0

.8 L

A ~

I \

o

.7

Io

o

o

•

\

°

•

\

\

\

o

o

•

o

\

49Oo •

o

A

\\

0

.=..

o, °

.6

0

•

\ *\

.4

,3

t

_

.2

.I 0 ! - 0.5

I

I

I

I

I

0

0.5

1.0

1.5

2.0 10

"F

g/

Figure 9. Coupling function (~). (Different symbols identify the 3 subjects: PV = filled circles; MC = open triangles; MF = open circles. The arrow indicates the approximate frequency at which almost perfect coupling occurs [ T ~ 2.8 s]. Beyond this point the scatter of data points makes it difficult to reach any firm conclusion. The dashed line interpolation merely suggests a possible trend.)

of execution, (b) the perimeter of the ellipse being executed, and (c) the total perimeter of the pattern. The combined effects of Points (b) and (c) result in a relative invariance of the execution time vis-a-vis the size of the pattern (isochrony). 3. The ratio of the gain factors in the two ellipses varies as a function of the overall rhythm of execution. A quantitative analysis of the data shows that this relation can be explained if one hypothesizes a coupling of variable strength between the execution of the two ellipses. There can be no doubt that significant and systematic changes in the velocity gain factor provide an empirical and operationally effective criterion for separating the apparently continuous movement pattern into distinct parts. However, as stated in the introduction, the criteria used to support the existence of the units of motor action should be formulated in terms that pertain to the planning of the movements rather than to its implementation. What evidence is there that indeed the velocity gain factor satisfies this condition? How general is the proposed criterion? Although it would be inappropriate at this stage to provide a cat-

egorical answer to these questions, a number of plausible arguments can be put forward to corroborate the claims of structural relevance and generality. As for the first question, we begin by noting that the velocity gain factor is defined as the ratio of two quantities, V(t) and R(t) 1/3, which describe two conceptually distinct aspects of the motion. However, the regression analysis illustrated in Figure 2 Panels A and B, shows that the factor K is considerably less variable than both the kinematic and the geometrical parameters from which it derives. One might explain this correlation between V(t) and R(t) m by assuming that both quantities are jointly specified by some more general parameter of the motor plan. Alternatively, one might propose that the equation V(t) = KR(t) ~/3 should be taken in the procedural sense, which implies that the instantaneous velocity is a dependent variable of the motion while the gain K and the geometry of the trajectory are centrally specified. In the context of drawing movements, where the form of the trajectory has a paramount importance, the second hypothesis is more plausible. The important point, however, is that in both cases the in-

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS

stantaneous relation between form and kinematics is not an emergent feature of the movement at the level of its peripheral instantiation. Thus, the gain K, which quantifies this relation, should in any case be considered a pertinent aspect of the motor plan. A second aspect of the results that bears on the same question is the relation between the extent of the movement and its duration (Figure 4). A covariation of average velocity and movement size, which implies the relative constancy of the duration, is well documented in simple rectilinear hand movements (Binet & Courtier, 1893; Fitts, 1954; Langolf, Chat~n, & Foulke, 1976; Michel, 1971). In agreement with the pioneering work of Freeman (1914) and with a number of previous reports (Lacquaniti et al., 1984; Viviani & McCollum, 1983; Viviani & Terzuolo, 1980), the results of Experiment 1 confirm that such a covariation of velocity and linear extent is present also in complex, curvilinear patterns. When the movement has only one degree of freedom, it is again conceivable that both movement parameters derive concurrently, and without a causal relation, from a common precursor. Following the original idea of Asatryan and Feldman (Asatryan & Feldman, 1965; Feldman, 1966a, 1966b), many authors (e.g., Kelso, 1977; Polit & Bizzi, 1978; Sakitt, 1980) have proposed that movements can be initiated by shifting the equilibrium point of muscular systems arranged in agonist-antagonist pairs, and can be brought to a predetermined final position by a proper setting of the stiffness and resting length in the length-tension muscle characteristics. In this case, velocity and extent of the movement are dependent variables, and their correlation does not imply causality. However, as pointed out elsewhere (cf. Hinton, 1984), this idea cannot explain how curvilinear trajectories involving many degrees of freedom are controlled. Moreover, we cannot say whether the velocity-extent correlation would also be predicted by a mass-spring model suitably generalized to describe these more complex movements. It should be obvious also that the specific representation of the gain factors proposed here cannot be construed as an explicative model. It is still far from clear if and how one could derive the relation between the velocity gain factors and the linear extent of the trajectory (Equation 2) from simpler and more

841

intuitive hypotheses on the organization of motor control. However, the analysis offered here does contribute to a specification of the conceptual framework needed to describe this relation. In particular, two ideas have been introduced that, in the light of the satisfactory approximation of the model to the data, might deserve further scrutiny. The first idea is the postulated factorization of the instantaneous tangential velocity into two multiplicative terms. One term, R(t) I/3, depends on the differential geometrical properties of the trajectory and, in general, varies continuously in the course of the movement. This term may be an emergent feature of mechanisms and constraints that are specific to the execution stage of the movement (Lacquaniti et al., 1983). The other term (the velocity gain factor K) depends on both the metric properties of the trajectory and the self-imposed tempo of the movement. Because, by definition, K is supposed to be approximately constant within each unit of motor action, it is plausible to assume that the value of this parameter is specified in the preparatory stages of the movement (Lacquaniti et al., 1984). The second idea that is supported by the successful application of the model concerns the internal representation of the linear extent. Although the results for simple patterns (e.g., Lacquaniti et al., 1984) are compatible with the hypothesis that the gain factor within a unit is specified only by the corresponding linear extent, the theoretical analysis of our data implies that the total linear extent of the pattern also contributes to this specification. As a matter of speculation, and in line with recent theoretical trends (cf. Keele, 1981; Schmidt, 1982), this may be taken to suggest the presence of two hierarchical levels of representation: a lower level that provides specific information on the unit being executed, and a higher level that provides the global context (plan) for the execution of several units. In the case of nonperiodic, extemporaneous patterns (scribbles), where no global plan exists, this higher level of representation would nevertheless provide look-ahead information on forthcoming units. The general relevance of results obtained for one special pattern can and should be open to question. The criterion based on the velocity gain factor yields a segmentation that coincides with the one naturally suggested by the form

842

PAOLO VIVIANI AND MARCO CENZATO

of the double ellipse. This result was hoped for (cf. introduction), but it is not a trivial consequence of the specific geometry of the pattern. Indeed, for a fixed pattern, the degree of differentiation between the two units that compose the movement is a function of the rhythm of execution, and the segmentation vanishes altogether below a certain rate (Figure 9). The specific hint that proved helpful for identifying the units in the double ellipse (i.e., equating units with closed-loop trajectories) has no intrinsic general relevance and may even be misleading in the case of other more complex patterns with points of inflexion and reversals in the direction of rotation. The analysis of extemporaneous irregular scribbles (Viviani, in press) suggests, however, that even in this rather extreme drawing task a segmentation of the movement can be observed that is congruent with some geometrical regularities of the trajectory. To conclude the discussion on this point, we would like to emphasize again that the degree of differentiation among units is variable, even in one particular pattern, and the segmentation can actually disappear under certain conditions. Thus, if indeed the motor plan has a hierarchical structure, one must assume that the complexity of the intervening levels of descriptions can be modulated according to the operating conditions. Moreover, because a varying degree of differentiation among units is compatible with perfectly smooth path and speed profiles (see Figure 1), a strict correspondence between unit boundaries and path and speed discontinuities (cf. Bizzi & Abend, 1983) seems unwarranted. As a final point, we raise and discuss briefly the following question: What is the functional significance (if any) of the relation between the degree of coupling among units of motor action and the rhythm of execution? We begin by noting that the trend of this relation is counterintuitive and somewhat at odds with at least two current theories of motor response organization. In fact, serial models of stage processing (Kornblum & Requin, 1984; Sternberg, 1969) do not predict any such relation, unless one makes ad hoc stipulations •concerning the program selection and program specification stages. Cascade models (McClelland, 1979) do not fare any better, for, if anything, they would predict a tighter coupling among the components of fast movements.

Clearly, the available data are just not powerful enough to suggest one general principle of organization within which the motion of variable coupling can be satisfactorily framed. We can attempt, however, to provide a post hoc justification for the existence of such variable coupling. The minimum execution time for one unit of motor action is set by biomechanical and nervous factors. Therefore, as long as the homothetic principle applies (Viviani & Terzuolo, 1983), that is, as long as the relative duration of each unit within a complex movement is invariant with respect to the total duration, the fastest possible rhythm for the execution of this movement is dictated by the intrinsically shorter unit. In the particular case of the double ellipse, the shorter unit is always P~, and simple calculations show that for any value of the total duration Ti + T2, the execution time T1 for the shorter unit is inversely proportional to the coupling factor. Because of the inverse relation between TI + T2 and ~, it finally follows that when the rate of execution is increased, the execution time for the shorter unit decreases less than proportionally. In conclusion, the presence of a variable coupling forces a departure from the homothetic principle that makes the execution of very fast movements more comfortable. References Asatryan, D., & Feldman, A. (1965). Functional tuning of the nervous system with control of movement or maintenance of a steady posture: I. Mechanographic analysis of the work of the joint on execution o f a postural task. Biofizika, 10, 925-935. Binet, A., & Courtier, J. (1893). Sur la vitesse des mouvements graphiques. [On the velocity of graphic movements]. Revue Philosophique, 35, 664-671. Bizzi, E., & Abend, W. (1983). Posture control and trajectory formation in single- and multi-joint arm movements. In J. E. Desmedt (Ed.), Motor controlmechanisms in health and disease (pp. 31-45). New York: Raven Press. Bryan, W. L., & Harter, N. (1897). Studies in the physiology and psychology of the telegraphic language. Psychological Review, 4, 27-53. Bryan, W. L., & Harter N. (1899). Studies on the telegraphic language: The acquisition of a hierarchy of habits. Psychological Review, 6, 345-375. Crossman, E. R. F. W., & Goodeve, P. J. (1983). Feedback control of hand-movement and Fitt's law. Quarterly Journal of Experimental Psychology, 47, 381-391. Denier van der Gon, J. J., & Thuring, J. Ph. (1965). The guiding of human writing movements. Kybernetik, 2, 145-148. Feldman, A. G. (1966a). Functional tuning of the nervous

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS system with control of movement or maintenance of a steady posture: 1I. Controllable parameters of the muscles. Biofizika, 11, 565-578. Feldman, A. G. (1966b).,Functional tuning of the nervous system with control of movement or maintenance of a steady posture: II1. Execution by man of simplest motor tasks. Biofizika, 11, 766-775. Fitts, P. M. (1954). The information capacity of the human motor system in controlling the amplitude of the movement. Journal of Experimental Psychology, 47, 381-391. Freeman, E N. ( 1914). Experimental analysis of the writing movement. Psychological Review Monograph Supplement, 17, 1-46. Greene, P. H. (1982). Why it is easy to control your arms? Journal of Motor Behavior, 14, 260-286. Herbst, N. M., & Liu, C. N. (1977, May). Automatic signature verification based on accelerometric data. LB.M. Journal of Research & Development, 245-253. Hinton, G. (1984). Parallel computations for controlling an arm. Journal of Motor Behavior, 16, 171-194. Hulstijn, W., & van Galen, G. P. (1983). Programming in handwriting: Reaction time and movement time as a function of sequence length. Acta Psychologica, 54, 2349. Keele, S. W. ( 1981 ). Behavioral analysis of movement. In V. Brooks (Ed.), Handbook of physiology (Vol. 2, pp. 1391-1414). Baltimore, MD: American Physiological Society. Kelso, J. A. S. (1977). Motor control mechanisms underlying human movement production. Journal of Exper-

imental Psychology: Human Perception and Performance, 3, 529-543. Kelso, J. A. S. (1981). Contrasting perspectives on order and regulation in movement. In J. Long & A. Baddeley (Eds.), Attention and performance IX. Hillsdale, NJ: Erlbaum. Kelso, J. A. S., Holt, K. G., Rubin, P., & Kugler, P. N. (1981). Patterns of human intertimbs coordination emerge from the properties of non-linear, limit cycle oscillatory processes: Theory and data. Journal of Motor Behavior, 13, 226-261. Klapp, S. T. (1977). Reaction time analysis of programmed control. Exercise and Sport Sciences Review, 5, 231253. Klapp, S. T., Anderson, W. G., & Berrian, R. W. (1973). Implicit speech in reading, reconsidered. Journal of Experimental Psychology, 100, 368-374. Klapp, S. T., & Wyatt, E. P. (1976). Motor programming within a sequence of responses. Journal of Motor Behavior, 8, 19-26. Kornblum, S., & Requin, J. (1984). Preparatory states and processes. Hillsdale, N J: Erlbaum. Kugler, P. N., Kelso, J. A. S., & Turvey, M. T. (1980). On the concept of coordinative structure as dissipative structure: 1. Theoretical lines of convergence. In G. E. Stelmach & J. Requin (Eds.), Tutorials in motor behavior (pp. 3--47). Amsterdam: North-Holland. Lacquaniti, F., Terzuolo, C., & Viviani, P. (1983). The law relating the kinematic and figural aspects of drawing movements. Acta Psychologica, 54, 115-130. Lacquaniti, F., Terzuolo, C., & Viviani, P. (1984). Global metric properties and preparatory processes in drawing movements. In S. Kornblum & J. Requin (Eds.), Preparatory states and processes (pp. 357-370). Hillsdale, N J: Erlbaum. Langolf, G. D., Chaffin, D. B., & Foulke, J. A. (1976). An

843

investigation of Fitts' law using a wide range of movement amplitudes. Journal of Motor Behavior, 8, 113-128. McClelland, J. L. (1979). On the time relations of mental processes. An examination of systems of processes in cascade. Psychological Review, 86, 287-330. Michel, E (1971). Etude exp~rimentale de la vitesse du geste graphique [Experimental investigation of the velocity of graphic movements]. Neuropsychologia, 9, 113. Morasso, E, & Mussa Ivaldi, E A. (1982). Trajectory formation and handwriting: A computational model. Biological Cybernetics, 45, 13 l - 142. Morasso, P., Mussa lvaldi, E A., & Ruggiero, C. (1983). How a discontinuous mechanism can produce continuous patterns in trajectory formation and handwriting. Acta Psychologica, 54, 83-98. Nelder, J. A., & Mead, R. (1964). A simplex method for function minimization. The Computer Journal, 7, 308313. Polit, A., & Bizzi, E. (1978). Processes controlling arm movements in monkeys. Science, 201, 1235-1237. Rosenbaum, D. A., Inhoff, A. W., & Gordon, A. M. (1984). Choosing between movement sequences: A Hierarchical Editor Model. Journal of Experimental Psychology: General, 113, 372-393. Sakitt, B. (1980). A spring model and equivalent neural network for arm posture control. Biological Cybernetics, 37, 277-234. Sanders, A. E (1980). Stage analysis of reaction processes. In G. E. Stelmach & J. Requin (Eds.), Tutorials in motor behavior (pp. 331-354). Amsterdam: North-Holland. Schmidt, R. A. (1982). Motor control and learning: Behavioral emphasis. Champaign, 1L: Human Kinetics. Shaffer, L. H. (1981). Performances of Chopin, Bach and Bartok: Studies in motor programming. Cognitive Psychology, 13, 826-376. Sternberg, S. (1969). The discovery of processing stages: Extension of Donder's method. In W. G. Koster (Ed.), Attention and performance H (pp. 276-315). Amsterdam: North-Holland. Sternberg, S., Monsell, S., Knoll, R. L., & Wright, C. E. (1978). The latency and duration of rapid movement sequences: Comparisons of speech and typewriting. In G. E. Stelmach (Ed.), Information processing in motor control and learning. New York: Academic Press. Teulings, H. L., Thomassen, J. W. M., & van Galen, G. P. (1983). Preparation of partly precued handwriting movements: The size of movement units in handwriting. Acta Psychologica, 54, 165-177. van Galen, G. P., & Teulings, H. L. (1983). The independent monitoring of form and scale factors in handwriting. Acta Psychologica, 54, 9-22. Viviani, P. (in press). Do units of motor action really exist?

Experimental Brain Research. Viviani, P., & McCollum, G. (1983). The relation between linear extent and velocity in drawing movements. Neuroscience, 10, 211-218. Viviani, P., & Terzuolo, C. (1980). Space-time invariance in learned motor skills. In G. E. Stelmach & J. Requin (Eds.), Tutorials in motor behavior (pp. 525-533). Amsterdam: North-Holland. Viviani, P., & Terzuolo, C. (1982). Trajectory determines movement dynamics. Neuroscience, 7, 431--437. Viviani, P., & Terzuolo, C. (1983). The organization of movement in handwriting and typing. In B. Butterworth

(Ed.), Language production, Vol.I1: Development, writing

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PAOLO VIVIANI AND MARCO CENZATO

and other language processes (pp. 103-146). New York: Academic Press. Vredenbregt, J., & Koster, W. G. ( 1971). Analysisand synthesis of handwriting.Philips Technical Review, 32, 7378. Vredenbregt,J., & Rau, G. (1971). Coordination in muscle

activityduring simplemovements.Institut voor Perceptie Onderzoek Annual Progress Report, 6, 73-76. Wing, A. M. (1978). Response timing in handwriting. In G. E. Stelmach (Ed.), Information processing in motor control and leaming (pp. 153-172).New York:Academic Press.

Appendix A Relation Between Velocity G a i n Factor a n d Period o f Execution for Ellipses To demonstrate the relation between the velocity gain factor K and the period of execution T of an elliptic pattern (Equation 4 in the text), we begin by rewriting the Two-Thirds Power Law, A(t) = K . C(t) 2/3,

K (AB)I/3 (A2 - x(l)2) ':2,

.f(t) -

(A 1)

in a more explicit form. Let x = x(t) and y = y(t) be the equations of the motion, and let y = f ( x ) be the equation of the trajectory of the movement. From kinematics we know that V(t) = (x(/) 2 + fi(t)2) I/2 (.~(t)2 + 3~(t)2)3/2 R(t) = I~(t)p'(t) - X(t):(t)l "

Upon substitution of this expression into Equation A3, we get the following differential equation:

whose general solution provides the x component of the law of motion, x(t)

KA ~(AB)I/3 cos(~ot) (~0 = 27r/T).

The y component can finally be calculated from the equation of the trajectory

(

(A2)

y(t) = B 1 Remembering that Aft) = V(t)/R(t) and C(t) = 1/ R(t) and substituting Equation A2 into Equation A1, we get

Because, by hypothesis, the composition of the x and y components must result into an ellipse with semiaxes A and B, it necessarily follows that

I:c(t)~(t) - X(t)fi(t)l '/3 = K.

K = oa(AB)l/3;

Because

that is,

df(x)

fi(t) = £ ( t ) -

K2 c°s2(t'°l))1/2"

o~2(AB)2/3

27r I'T = --~ (AB)/'.

dx

d2f(x) df(x) p'(t) = ~(t) z ~ + Y((t) - - - ~ , it follows that the Two-Thirds Power Law can also be expressed in the differential form

(::(xq

:~(t) : r ~ , - ~ - 1

From geometry we know that the semiaxes are related to the perimeter P and the eccentricity E of the ellipse by the relation p2(l AB-

(A3)

(A4)

_ ~2)1/2

16E2(~2)

'

~2 = 1 - B2/A 2 and E denotes the complete elliptic integral of the second kind. Letting [(1 -- ~2)1/2~1/3 where

This general relation, valid for all movements, permits one to predict the law of motion from the knowledge of trajectory. In particular, the equation of an ellipse with semiaxes A and B (B < A) is (BZx 2 + A2y2) 1/2 = AB. Whence, d2f(x) _ dx 2

AB ( A 2 -- X2)3/2 "

and substituting into Equation A4, we finally obtain the desired relation: T = 21rfl(~) p2/3. K

SEGMENTATION AND COUPLING IN COMPLEX MOVEMENTS

845

Appendix B Relations Between Velocity Gain Factors, Perimeters, and Coupling Function Let

K2( PO - pat) - K l ( p a - pa2) P=

P t +P2.

Then, simple algebraic manipulations permit one to derive the following explicit expressions for Equations 9, 10, and 11 in the text, respectively: K r p a ( p a 2 _ p{)

K2 =

(pa _ pat)

( p a _ pa2) ~- K , ( p a _ pat),

K T =

pa( p az ~(~) =

-

pat)

KI POz - K2Pat K 2 ( P a - pat) - K , ( p a - p ~ ) "

Received June 19, 1984 Revision received July 8, 1985 •