A stochastic optimized-submovement model is proposed for Pitts' law, the classic logarithmic tradeoff between the duration and spatial precision of rapid aimed movements. According to the model, an aimed movement toward a specified target region involves a primary submovement and an optional secondary corrective submovement. The submovements are assumed to be programmed such that they minimize average total movement time while maintaining a high frequency of target hits. The programming process achieves this minimization by optimally adjusting the average magnitudes and durations of noisy neuromotor force pulses used to generate the submovements. Numerous results from the literature on human motor performance may be explained in these terms. Two new experiments on rapid wrist rotations yield additional support for the stochastic optimizedsubmovement model. Experiment 1 revealed that the mean durations of primary submovements and of secondary submovements, not just average total movement times, conform to a square-root approximation of Pitts' law derived from the model. Also, the spatial endpoints of primary submovements have standard deviations that increase linearly with average primary-submovement velocity, and the average primary-submovement velocity influences the relative frequencies of secondary submovements, as predicted by the model. During Experiment 2, these results were replicated and extended under conditions in which subjects made movements without concurrent visual feedback. This replication suggests that submovement optimization may be a pervasive property of movement production. The present conceptual framework provides insights into principles of motor performance, and it links the study of physical action to research on sensation, perception, and cognition, where psychologists have been concerned for some time about the degree to which mental processes incorporate rational and normative rules. An enduring issue in the study of the human mind concerns of mathematical probability theory and statistical decision thethe rationality and optimality of the mental processes that guide ory (e.g., see Edwards, 1961; Edwards, Lindman, & Savage, thought and action. Philosophers and psychologists have long 1963;Hollandetal., 1986;Kahneman&Tversky, 1973;Nisbett debated about the extent to which people's reasoning and prob-& Ross, 1 980; Tversky & Kahneman, 1974, 1 98 1 ). It is generlem solving adhere to the rules of formal logic (e.g., see Braine, ally agreed now that without explicit training, people do not 1978; Holland, Holyoak, Nisbett, & Thagard, 1986 ; Osherson, always behave exactly as formal logic, probability theory, or 1975). Similarly, there has been much interest in how well decidecision theory would prescribe. Nevertheless, their behavior sion making and choice behavior adhere to the normative rules may approximate the ideal, subject to limitations in memory capacity and information-processing resources (Simon, -1957 ). Portions of this research were first presented at a meeting of the PsyThe present article extends this concern with rationality and chonomic Society (Abrams, Kornblum, Meyer, & Wright, 1983) . Fundoptimality to the domain of human motor performance. We foing was provided by National Institute of Mental Health Grant R01 cus especially on the production of rapid aimed movements in MH37 1 45 to the University of Michigan, Sylvan Kornblum and David which a part of the body (e.g., finger, arm, leg, head, eye, tongue) E. Meyer, principal investigators. Richard A. Abrams participated unmust be shifted quickly and accurately from some initial posider a traineeship from the National Institutes of Health (Grant T32 tion to another desired terminal p OS i tion . Such movements EY07022) and a fellowship from the Horace Rackham School of Gradmerit careful scruti ^^ th constitute the bask buj , di ments. Carol Huff and Peggy Kwun provided superb technical assiscontribute to many complex physical skills. As tor higher menCorrespondence concerning this article should be addressed to David ^ Processes, our goal is to demonstrate how rapid aimed move-E. Meyer, Human Performance Center, Department of Psychology, ments may be understood on the basis of ideal operating characUniversity of Michigan, 330 Packard Road, Ann Arbor, Michigan teristics that are conditioned by inherent limitations in system 48104. capacities and resources. Moreover through this demonstra-  where A and B are positive constants, and log 2 (2D/ W) is a measure of spatial precision scaled in terms of information bits. Equation 1 has become known as Fitts'law. Numerous studies have shown it to hold for a variety of movements produced under different environmental conditions. The logarithmic speed-accuracy trade-off applies not only to stylus tapping but also to wrist flexion and rotation (Crossman & Goodeve, 1963 / 1983 Jagacinski, Repperger, Moran, Ward, & Glass, 1980) , arm extention (B. A. Kerr & Langolf, 1977) , and finger manipulation (Langolf, Chaffin, & Foulke, 1976  Yet studies regarding Fitts' law have not proceeded entirely without controversy. Some investigators have proposed different logarithmic speed-accuracy trade-off functions to characterize rapid aimed movements (e.g., Crossman & Goodeve, 1963 /1983 Sheridan, 1979; Welford, 1968) , and others have suggested that either power or linear functions are more appropriate under certain circumstances (e.g., Howarth, Beggs, & Bowden, 1971; Kvalseth, 1980; Meyer, Smith, & Wright, 1982; Schmidt, Zelaznik, & Frank, 1978; Schmidt, Zelaznik, Hawkins, Frank, & Quinn, 1979) . The exact form of the obtained trade-off may depend on specific task demands. For example, Wright and Meyer (1983) have drawn a distinction between tasks involving spatially constrained movements and tasks involving temporally constrained movements (cf. Fitts, 1954). Spatially constrained movement tasks are ones in which subjects must terminate their movements within a specified target region while attempting to minimize their average movement times, as required by Fitts' stylus-tapping task. Temporally constrained movement tasks are ones in which subjects must produce movements having precisely specified durations while terminating the movements as close as possible to a discrete target point, not a region (e.g., trying to hit a target point with a movement that lasts exactly 200 ms). A linear trade-off function has been found to fit better than a logarithmic function for temporally constrained movements (Schmidt et al., 1978 (Schmidt et al., , 1979 Wright, 1983a Wright, , 1983b Zelaznik, Shapiro, & McColsky, 1981) , whereas a logarithmic function ' Following conventional terminology (Keele, 1968) , we use the term movement time to designate the duration of the time interval from when a movement begins overtly to when it ends. This term should not be confused with the terms reaction time and movement latency, which designate the duration of the time interval between the onset of a response signal and the beginning of an overt movement. & Goodeve, 1963 & Goodeve, /1983 Keele, 1968; Schmidt et al., 1979) , whereas others have attributed it to the nature of the initialimpulse phase (e.g., Jagacinski et al., 1980; Langolfetal., 1976; Meyer et al., 1982) . Some data support the latter viewpoint over the former, but details of the initial-impulse and current-control phases still need to be explicated definitively. The remainder of this article further considers the empirical and theoretical bases of Pitts' law, relating them more closely to the linear speed-accuracy trade-off and other aspects of human motor performance. First, we summarize one major account, the deterministic iterative-corrections model (Grossman & Goodeve, 1963 /1983 Keele, 1968) , that has been formulated to rationalize Pitts' law for rapid spatially constrained movements. Several significant deficiencies of this model will be noted. Next, we introduce an alternative account, the stochastic optimized-submovement model, that shares some features with the deterministic iterative-corrections model but goes beyond it to handle a considerably broader range of data. Explicit connections will be drawn between the stochastic optimized-submovement model and previous work (Meyer et al., 1982; Schmidt et al., 1978 Schmidt et al., , 1979 Wright, 1983a Wright, , 1983b on the random variability of noisy neuromotor force pulses used to generate aimed movements. We hypothesize that the production of rapid spatially constrained movements may be characterized in terms of an ideal compromise between the durations of component submovements made under the influence of such noise. It will be shown that this hypothesis explains several previously puzzling results in the motorperformance literature.  For about 25 years, the best known explanation of Pitts' law has been the deterministic iterative-corrections model, which was formulated originally by Grossman and Goodeve (1963/ 1983) and elaborated subsequently by Keele (1968) Either visual or kinesthetic feedback could, in principle, be used to guide the submovements of the deterministic iterativecorrections model (Grossman & Goodeve, 1963 /1983 . However, there is considerable evidence that subjects do not produce corrective submovements very effectively on the basis of kinesthetic feedback (Abrams, Kornblum, Meyer, & Wright, 1983; Carlton, 1981a Carlton, , 1981b Prablanc, Echallier, Komilis, & Jeannerod, 1979; Vince, 1948; Wallace & Newell, 1983; Woodworth, 1899) . So it seems likely that visual feedback may pro-3 Operationally, a submovement is denned as a change in the position of a body part during a time interval such that (a) at the beginning of the interval, the velocity of the body part crosses a near-zero threshold or certain marked reversals occur in its acceleration, (b) one of these types of events occurs again at the end of the interval, (c) the absolute value of the velocity throughout the interval exceeds a specified criterion, and (d) the total distance traveled from the beginning to the end of the interval exceeds a specified criterion. The criteria for this definition may be selected to discriminate voluntary submovements from passive involuntary activity (e.g., tremor). For a more complete treatment of such matters, see Experiment 1. 4 If more than three submovements are needed to reach the target, then the submovement series would continue onward in the same fashion; for example, the nth submovement would travel a distance of . In this expression, the term />(! -paY~l represents the amount of distance left between the end of the n -1st submovement and the target center. The term p a is the proportion of this distance that the nth submovement would travel. Figure 2 . Outline of assumptions for the deterministic iterative-corrections model (Grossman & Goodeve, 1963 /1983 Keele, 1968) . (  OPTIMAL MOTOR PERFORMANCE DISTANCE Using the deterministic iterative-corrections model, it is possible to account quantitatively for Pitts' law. The account stems directly from the fact that for a specified target region, the num- The deterministic iterative-corrections model is also consistent with some other observations regarding rapid spatially constrained movements. Crossman and Goodeve (1963/1983) found periodic fluctuations in the velocity profiles of wrist rotations to specified target regions, suggesting the occurrence of multiple submovements. Multiple submovements likewise appeared at least occasionally during studies by Carlton (1979 Carlton ( , 1980 , Langolf et al. (1976, Experiment 1), and Jagacinski et al. (1980) . This is what one would expect if feedback-guided iterative corrections mediate the logarithmic speed-accuracy trade-off (Pitts' law). Jagacinski et al., 1980; Langolf et al., 1976) . Such variability, which subjects cannot completely eliminate, has been studied mainly in experiments on temporally constrained movements (e.g., Schmidt et al., 1978 Schmidt et al., , 1979 Wright, 1983a Wright, , 1983b Zelaznik et al., 1981) , while remaining relatively ignored in experiments on spatially constrained movements (cf. Welford, 1968). Yet it could have an important bearing on the source of the logarithmic speed-accuracy trade-off (Schmidt etal., 1978). 5 More specifically, suppose that n is the expected number of submovements. Then n must be the smallest integer such that 2 PiD(l - where the floor symbols (L J) denote the greatest integer less than or equal to the enclosed ratio of the two logarithms (Keele, 1968) . 'On the basis of reports by Vince (1948) and by Keele and Posner (1968) , Keele (1968) estimated that each submovement travels about 93% of the remaining distance to the center of the target (i.e., p t = .93) and that the duration of each submovement equals about 200 ms. These estimates allowed him to predict the slope of the logarithmic trade-off function (i.e., the constant B in Equation 1) with good accuracy. His predictions, like some available data (e.g., Fitts, 1954) , placed this slope at about 70 ms per unit of ID (i.e., Iog 2 (2£>/ W)) for aimed stylus-tapping movements. Carlton, 1979 Carlton, , 1980 Jagacinskietal., 1980). Instead, the error rate tends to increase as the target distance increases or width decreases (Fitts, 1954; Fitts & Peterson, 1964; Wallace & Newell, 1983) . On the other hand, the deter- Prablanc et al., 1979; Wallace & Newell, 1983) . Fins' law may also hold for movements whose durations are sufficiently short to preclude visual feedback from being used to correct them. As Schmidt et al. (1979) have pointed out, such results raise additional doubts about the rationale for the deterministic iterative-corrections model, which depends heavily on the availability and use of visual feedback (Keele, 1968) . Given the preceding empirical and theoretical considerations, we have developed an alternative account for the production of rapid spatially constrained movements. This account involves a stochastic optimized-submovement model that stems from work on impulse variability in temporally constrained movements (Meyer et al., 1982; Schmidt et al., 1978 Schmidt et al., , 1979 Wright, 1983a Wright, , 1983b . Our approach modifies some assumptions of the classical deterministic iterative-corrections model (Crossman & Goodeve, 1963 /1983 Keele, 1968) and replaces other assumptions with more plausible ones. A key feature of the present conceptual framework is that it represents the movement-production process as an ideal compromise between the durations of primary and secondary submovements. The form of the compromise depends on assumptions about neuromotor noise and the spatial variability of submovement endpoints. Using these assumptions and the hypothesis that submovement durations are appropriately optimized to cope with the noise, we can reconcile a number of phenomena not handled by earlier accounts. Moreover, the present framework builds a solid bridge between Fitts' law and the linear speed-accuracy trade-off observed in studies of temporally constrained movements. An outline of the stochastic optimized-submovement model appears in Figure 3 Neuromotor noise. Under the stochastic optimized-submove-7 According to Vince (1948) and Carlton (1979) , a primary submovement typically travels 90% or more of the distance from its starting location to the center of the target. Thus, if each subsequent corrective submovement travels an equally large proportion (i.e., p a & .9) of whatever distance remains at the end of the immediately preceding submovement, then the target region should always be reached within two submovements except when D/Wis extremely large (e.g., D/W> 64, and ID = log 2 (2Z)/H / ) > 7). Under the deterministic iterative-corrections model, this would allow no freedom for DfWio affect movement times logarithmically within the range of target distances and widths (i. 1978 Wright, 1983a Wright, , 1983b Zelaznik et al., 1981, in press). Given a constant target distance and goal duration for each of several movements, subjects do not (and apparently cannot) produce exactly the same movement repeatedly across a series of trials. Instead, the movement distances and movement times deviate randomly about their intended objectives. Randomness likewise appears in other aspects of the movement trajectories (Meyer et al., 1982; Wright, 1983a Wright, , 1983b and in bursts of electromyographic (EMG) activity associated with them (Wallace, 1981). The source of the neuromotor noise postulated here is not yet known, and controversies have arisen over its detailed microscopic properties (Carlton & Newell, 1988; Meyer et al., 1982; Schmidt, Sherwood, Zelaznik, & Leikind, 1985; Schmidt et al., 1979 (Meyer et al., 1982; Schmidt et al., 1978 Schmidt et al., , 1979 . Although some results leave this proposal open to question (Schmidt et al., 1985; Zelaznik & Schmidt, 1983; Zelaznik, Schmidt, & Gielen, 1986) , other data support it substantially (Schmidt etal., 1978(Schmidt etal., , 1979 Wright, 1983a Wright, , 1983b . For current purposes, the stochastic optimized-submovement model does not require a complete description of the underlying mechanisms that cause neuromotor noise. We simply claim that the noise exists and that it leads to systematic variability in the endpoints of component submovements executed during the production of spatially constrained movements. Following this claim, some additional assumptions are outlined below. Number of submovements. We assume that a rapid spatially constrained movement toward a target region includes either one or two component submovements, regardless of the target distance and width. The primary (initial) submovement is assumed to be programmed to end at the center of the target, as illustrated by one of the solid curves that start from the home position in Figure 3 . If the primary submovement ends within the target region, then no secondary submovement follows it. However, the endpoint of a primary submovement may miss the target because of perturbations caused by neuromotor noise, as two of the other solid curves in Figure 3 illustrate. If a miss occurs, then a secondary submovement, prepared through visual or other (e.g., kinesthetic) feedback, is assumed to correct the error, as the dashed curves in Figure 3 illustrate. Our motivation for proposing a model with only two submovements is partly didactic. We want to demonstrate that many aspects of the speed-accuracy trade-off associated with spatially constrained movements can be explained by hypothesizing a relatively small number of submovements. This demonstration may provide useful insights into the source of the logarithmic trade-off function (Pitts' law) and create an instructive contrast with alternative views, such as those embodied in the deterministic iterative-corrections model. Empirical justification also exists for assuming only a primary and secondary submovement. The two-submovement assumption is supported by some results of Langolf et al. (1976) and Carlton (1979) . They found a large preponderance of trials with just one or two submovements in subjects' overall movements toward a target region. The absence of additional higher 346 MEYER, ABRAMS, KORNBLUM, WRIGHT, AND SMITH order (e.g., tertiary) submovements seems reasonable given that a primary submovement may typically travel over 90% of the distance from the home position to the center of the target (Carlton, 1979; Vince, 1948) . With this amount of travel and with the usual distance-width combinations selected for spatially constrained movement tasks, a combination of two submovements would almost always suffice to reach the target region successfully (Footnote 7). Also, when three or more submovements do occur (e.g., Grossman & Goodeve, 1963 /1983 Jagacinski et al., 1980) , the present theoretical approach may still be appropriate in that our model can assimilate within its secondary submovements the contributions from all higher order submovements.  where D t is the mean distance traveled by the primary submovements, T, is their mean duration, and AT is a positive constant. This assumption stems directly from the conceptual framework developed previously to characterize temporally constrained movements in terms of impulse-variability mechanisms (Meyer etal., 1982; Schmidt etal., 1978 Schmidt etal., ,1979. Equation in press) and extended to wrist rotations by Wright and Meyer (1983; also see Wright, 1983a also see Wright, , 1983b.  8 The occurrence of higher order corrective submovements may depend on the extent to which a spatially constrained movement task involves extremely difficult targets (Carlton, 1980) and requires virtually errorless performance (Jagacinski, Repperger, Moran, Ward, & Glass, 1980) . In studies of disc and pin transfer, Fitts (1954) and Annett, Golby, and Kay (1958) had subjects continue moving until they were always properly aligned with a target region. Similarly, Jagacinski et al. (1980) had subjects perform wrist movements that required a cursor to be superimposed continuously on a displayed target region for at least 360 ms. Such stringent accuracy requirements could induce additional finer and finer corrections until sufficient spatial accuracy is achieved. 9 However, it should be stressed that many of the predictions made by the stochastic optimized-submovement model do not depend heavily on the present normality assumption. For example, average total movement time (T) should exhibit the same form of speed-accuracy tradeoff (viz. a quasi square-root function of DIW) regardless of whether the Midpoint distributions are normal or uniform. Combined with the model's other assumptions, any well-behaved symmetric unimodal endpoint distribution would lead to essentially the same predictions as a normal distribution does. 10 A few studies of temporally constrained movements have suggested relations other than a linear speed-accuracy trade-off between the standard deviation of movement endpoints and the average movement velocity (e.g., Hancock & Newell, 1985; Schmidt, Sherwood, Zelaznik, & Leikind, 1985) . Under some circumstances, spatial endpoint variability may appear to increase with the mean movement distance but be independent of movement time. Also, the constant K in Equations 2 and 3 may differ for primary and secondary submovements, given that they are intended to achieve markedly different degrees of spatial precision. Such subtleties would require modifications of these equations and could lead to a somewhat different account of spatially constrained movements than outlined here. OPTIMAL MOTOR PERFORMANCE 347 of the secondary corrective submovements (7* 2 ), as specified more fully later (see Quantitative Predictions). Issues regarding the optimality of motor performance have also arisen in other related contexts. The production of aimed movements requires selecting parameters for movement trajectories along a number of related physical dimensions, including ones associated with space, time, force, and so on. An ideal actor may seek to minimize various quantities involving these parameters, such as the total distance traveled to reach some spatial position, the energy expended during movement, and the stress exerted on a moving body part. Theorists who study natural (e.g., human) and artificial (e.g., robot) movement systems have analyzed how this kind of minimization can be accomplished when movements are produced through noise-free deterministic mechanisms (e.g., Hogan, 1984; Hollerbach, 1982; Nelson, 1983) . There has been some work in mathematical control theory on optimal behavior for manual tracking by nondeterministic (i.e., stochastic) mechanisms where noise plays a role (e.g., Kleinman, Baron, & Levison, 1970 and acceleration) of the primary submovement. The process that prepares secondary submovements may also use "feedforward" (efference copy) from the primary submovements (von Hoist, 1954) ." We assume that feedback and feedforward are processed "on the fly" during movement production, so that essentially no delay ensues between the end of a primary submovement and the beginning of a subsequent secondary submovement. As indicated already, the stochastic optimized-submovement model simply sums the durations of the primary and secondary submovements in order to represent the total movement time on a trial. The model does not allow for "dead time" (i.e., intermediate pauses during which movement velocity is zero over an extended time interval) after a primary submovement has begun. This constraint simplifies computation of the mean submovement durations that minimize the average total movement time (see Appendix), and it is consistent with most of the movement trajectories found in the experiments reported here. Using the stochastic optimized-submovement model, we have derived a number of quantitative predictions about rapid spatially constrained movements. The predictions concern the average total movement time, mean durations of primary and secondary submovements, spatial variability (standard deviation) of primary-submovement endpoints, relative frequency of secondary submovements, and relative frequency of errors (i.e., target misses). For each of these dependent variables, a prediction can be made as a function of target distance and width. To facilitate our presentation, we will outline the model's predictions in two complementary forms. First, in the main text, several of the predictions are introduced as simplified approximations involving relatively nontechnical notation. Second, in the accompanying footnotes and Appendix, the predictions are stated more precisely with technically exact notation. Table 1 summarizes both the approximate and exact forms of the pre-" This would involve comparing a set of intended motor commands against copies of the actual motor commands sent to the effectors for producing a desired primary submovement. If the intended and actual commands differ from each other, then their difference could provide a way of deciding what correction(s) to make during subsequent secondary submovements. The efference-copy correction mechanism might be entirely central in nature, not requiring any peripheral feedback (von Hoist, 1954 The square-root trade-off function predicted by the stochastic optimized-submovement model is also consistent with previous results in the literature. Several investigators have reported data for which a power function with an exponent of about onehalf describes the speed-accuracy trade-off in spatially constrained movements better than a logarithmic function does (Kvalseth, 1980) . Observed deviations of empirical data from Pitts' law have tended to reveal a slight upward curvature when average total movement times (7") are plotted versus log 2 (2Z)/ W), as expected from Equation 4. 14 There are some circumstances in which Equation 4 might not apply, and T might instead be a linear function of D/W for spatially constrained movements. This would happen if subjects restricted their movement attempts to lone primary submovements, omitting secondary and higher order corrective submovements. Using a one-submovement strategy, they could perform a spatially constrained movement task by programming the primary submovements to have a mean duration 7\ such that T, = T= 2Kz a (D/ W), where c 2 is the desired frequency of correct movement trials (i.e., target hits), and z c2 is a unit-normal deviate such that a proportion c 2 of the area under a standardized normal probability-density function falls inside the interval (-z c2 , z c2 ). Then performance would mimic the results typically observed for temporally constrained movements (e.g., Schmidt et al., 1979; . Under a broad range of conditions, however, the one-submovement strategy would not be as optimal as a two-submovement strategy with respect to movement-time minimization. When D/W > 4/ z c2 V5ir, two (i.e., a primary and optional secondary) submovements of the sort incorporated in the stochastic optimized-submovement model can yield an average total movement time (7") that is shorter than would be possible with one primary submovement whose accuracy achieves the same desired level as the two-submovement strategy does (see Appendix). Mean primary-submovement duration. For primary submovements, the stochastic optimized-submovement model predicts that the mean duration is approximated by
The most popular explanation of Fitts' Law in the HCI community is Meyer et al.'s sub-movements model REF : first, a fast sub-movement towards the target (often called the ballistic phase of the movement) is performed, then if this movement does not hit the target, another (probably smaller) sub-movement towards the target is performed, and this process continues until the target is reached.
Optimality in human motor performance: ideal control of rapid aimed movements
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