SPEED/ACCURACY TRADEOFFS IN TARGET DIRECTED MOVEMENTS

Keywords

Abstract

I.INTRODUCTION

Speed/accuracy tradeoffs in target-directed movement have been studied for more than a century. Fullerton and Cattell (1892) referenced some experimental work by both German and French investigators on the psychophysics of movement. Woodworth (1899) is often credited with being the founder of research on movement speed/accuracy tradeoffs. The main contribution of Woodworth's doctoral dissertation (1899) is the proposition of a cohesive account of the accuracy of voluntary movements, which had never been done before. Woodworth's observations implied an intricate relationship between movement duration, amplitude and velocity in the determination of movement accuracy. He suggested the separation of a rapid movement into two phases, called the initial adjustment phase and the current control phase. Furthermore, he attributed the effect of movement speed on accuracy to the characteristics of current control and eliminated any effects on the accuracy of the initial adjustment.

After Woodworth's work, many investigators pursued descriptions of the speed/accuracy functions (Brown & Slater-Hammel 1949; Craik & Vince 1943/1963a, 1944/1963b; Garrett 1922; Philip 1936; Searle & Taylor 1948; see Meyer et al. 1990 for a historical review). The next major contribution to research on speed/accuracy tradeoffs was done by Fitts (1954) who was the first to propose a formal relationship linking movement time (MT) and the inverse of the relative spatial error: where A represents the amplitude of the movement, and W the target width; a and b being empirically determined constants. In this formulation, the logarithmic term is called the index of difficulty of the movement:and is linked to the maximum rate of information transmission of the human motor system.

In subsequent years, Fitts' tapping tasks or Fitts' paradigms were used in a systematic way for numerous studies, and equation (1) was found to be so general that it became known as Fitts' law (Crossman & Goodeve 1963/1983; Keele 1968). In spite of its generality, Fitts' relationship was found to be of limited use for spatially constrained tasks of low precision, and even inappropriate in timely constrained tasks (see Schmidt 1988 for a survey). Moreover, from a theoretical point of view, a few models have been developed to explain the origin of these phenomena (see Meyer et al. 1988 for a review), but so far no satisfying interpretation has been proposed that covers the various aspects of the speed/accuracy tradeoffs.

The present paper constitutes an attempt toward such a goal, which is to explain the origin of the different speed/accuracy tradeoffs by linking them to some fundamental properties of the kinematic behavior of a large number of coupled neuromuscular networks (Plamondon 1995a). The text is divided into four parts. First, a list of domains and contexts where Fitts' law has been studied is reported. In this short review, the previous theoretical models used to explain speed/accuracy tradeoffs are presented. This part also summarizes the various experimental limitations that have been reported concerning Fitts' law as well as the different mathematical formulations that have been proposed to take these discrepancies into account. In the second part, the kinematic theory of rapid human movements (Plamondon 1993b,c; 1995a,b) is summarized to highlight the origin of speed/accuracy tradeoffs in complex neuromuscular systems. This new theory is used in the context of spatially-constrained and time-constrained tasks to predict the different relationships between movement time and the inverse of the relative spatial error (Plamondon 1993c,d; 1995b). In the third part, the resulting laws are tested using the data available from the numerous studies published in the field. The implications of the kinematic theory for movement control and understanding are discussed in the fourth section.

II.SPEED/ACCURACY TRADEOFFS: A SHORT SURVEY

The majority of experimental procedures used in the study of speed-accuracy tradeoffs involve two classes of protocols: spatially constrained movements and temporally constrained movements. In movements with spatial constraints, subjects are asked to move as quickly as possible to a target placed at a distance (A). The target width (W) may or may not be given. In movements with temporal constraints, subjects are asked to move to a fixed target at a specified time (MT). The required timing precision (~MT) may or may not be given.

In this section, we report on these two classes of protocols. For spatially constrained movements, we present Fitts' law (1954) for the case where both W and A are given and MT is measured, and Howarth et al.'s (1971) approach for the case where only A is specified and results on W and MT are reported. For temporally constrained movements, A and MT are specified. We present the work of Schmidt et al. (1979) who report data on spatial variability and the work of Newell et al. (1979) who report data on temporal variability.

1.Spatially constrained movements where both A and W are given

The popularity of Fitts' law is mainly due to the large number of experiments and studies that have been reported to support it. It has been roughly validated for a variety of movements, and some investigators have adopted the model as a tool for investigating other issues. The experimental validation of Fitts' law has been confirmed, totally or partially, for a variety of movements, a variety of limbs and muscle groups, a variety of experimental conditions and manipulating devices, a variety of subjects, and a wide range of performance indices (see Table 1).

Insert Table 1 about here

It should be noted that not all these studies support Fitts' law. Many of them report violations, new formulations or new explanations of this law. Indeed, although Fitts' law has been generally accepted by many research groups as a good and practical working tool, many investigators have worked on its experimental verification, and several modifications to the original equation (1) have been advanced to take specific experimental conditions into account and to provide for better data fitting.

There are several factors indicating limitations to the potential of Fitts' law as a general description of the movement speed/accuracy relationship. An upward curvature of MT data, away from the regression line, has been observed for low ID values, so the lawful relationship fails at very low IDs (Buck 1986; Crossman 1957; Crossman & Goodeve 1963/1983; Drury 1975; Klapp 1975; Langolf et al. 1976; Meyer et al. 1988, 1990; Wallace et al. 1978; Welford 1960). In Fitts' reciprocal tapping experiment, the constant a becomes negative when movement time is plotted against his index of difficulty, as it does for zero information per response as well; and further, the best regression line through the data is not straight, but curved slightly upwards (Knight & Dagnall 1967).

Insert Table 2 about here

Several variations between Fitts' experimental data and his theoretical formulation have been deduced by many authors, who changed the definition of the index of difficulty (see Table 2). Welford (1968) suggested that subjects utilize only the near half of the target area, and he modified the equation accordingly. Subsequently, Welford et al. (1969) advanced an equation relating movement time to amplitude and target width separately. When studying moving targets (see note 1), Jagacinski et al. (1980b) proposed an alternate index of difficulty which explicitly incorporates a velocity factor. Recently, Hoffmann (1991a) has proposed two models for capturing the time of a moving target which give an excellent fit on the experimental data of Jagacinski et al. (1980b). The main improvement of the Hoffmann models over that of Jagacinski et al. is the theoretical explanation given by Hoffmann. For situations where there is a transmission delay (see note 3) between control movements and the feedback of the system response to the operator, Hoffmann (1992) proposed a model incorporating the delay in its formulation. Recently, MacKenzie (1989; 1992) showed that Fitts' choice of an equation that deviates slightly from the fundamental Theorem 17 of Shannon is unfounded and proposed a corrected equation. Gan and Hoffmann (1988) found that when the index of difficulty is small, MT can be predicted by an equation depending only on A. Johnsgard (1994) proposed a modification to the equation of MacKenzie (1989; 1992) which includes the effect of the device gain (G).

Another problem stems from the relative contributions of W and A in the equation for MT. In contrast to the implicit assumption of Fitts' law, it has been suggested, following the example of Welford et al. (1969), that amplitude and target width do not possess equal weighting in the determination of movement time (Sheridan 1979). The disproportionate increase in movement time caused by reductions in target width when compared to similar increases in target amplitude has also been noted directly from data by some investigators (Buck 1986; Jagacinski et al. 1980a; Jagacinski & Monk 1985; Keele 1973; Meyer et al. 1988; Welford et al. 1969) or indirectly from the analysis of the error-rate, which was found to increase as target width decreases independently of target amplitude (Card et al. 1978; Wade et al. 1978).

A few models have been proposed to give a theoretical explanation for speed/accuracy tradeoffs, and particularly for Fitts' law. We briefly review these in the sequel.

a)The Information-Theory Hypothesis of Fitts (1954)

Using the information theory (Miller 1953; Pierce 1961; Shannon 1948; Shannon & Weaver 1949), Fitts (1954) hypothesized that the task difficulty could be measured in bits using an information metric and that, in carrying out a movement task, information is transmitted through a noisy stochastic communication channel which models the behavior of the human motor system. Theorem 17 of Shannon (1948) expresses the effective information capacity C (in bits/s) of a communications channel of bandwidth B (in Hz) as: where N is the noise power and S is the signal power. Fitts claimed that, at the maximum rate of information transmission, the human motor system behaves in accordance with the logarithmic relation by identifying 1/MT with B, A with S+N and W/2 with N to obtain the speed/accuracy tradeoff relation proposed in equation (1).

One of the major hypotheses in Fitts' information-theory interpretation is that the human motor system behaves like a stochastic communication channel. This theoretical framework has been criticized. Crossman and Goodeve (1963/1983) have pointed out "the empirical difficulty of establishing the existence of the postulated "noise" or initial uncertainty". Kvalseth (1979) has claimed that "the Fitts' ID measure does indeed yield false estimates of the information capacity of the human motor system". Recently, MacKenzie (1989) showed that Fitts' law is in fact derived from Goldman's equation 39 (Goldman 1939), which is an "approximation" of Shannon's theorem, instead of the original Theorem 17 of Shannon (1948) given by equation (3):

The resulting variation of Fitts' law, (see Table 2) has been shown recently (Brogmus 1991; Welford 1990) to be one of the best, although the origin of the constant a is still problematic and not predicted by the theory.

b)The Deterministic Iterative-Corrections Model of Crossman and Goodeve (1963/1983)

An alternative to Fitts' information-theory approach which originated in the model of Crossman and Goodeve (1963/1983), is known as the deterministic iterative-corrections model and was subsequently developed by Keele (1968) and Langolf et al. (1976). Under this model, movements intended to reach a target region quickly and accurately are executed through iterations of feedback-guided corrective submovements. A submovement is assumed to take a constant time t to cover a distance equal to (1-p) times the remaining distance. By applying these assumptions, the submovement sequence continues under either visual or kinesthetic feedback until the target region has been reached. So, if Xi denotes the distance remaining up to the target center after the ith submovement, then:

Since the movement time is MT = nt, it can be expressed as:

Since the first move should take less than t (by a constant a) because the time it takes to decide how far to move initially occurs before a move begins (Keele 1968), then:

To verify these assumptions, the time it takes to process visual feedback was determined to be in the 135 ms to 290 ms range (Beggs & Howarth 1970; Carlton 1981; Crossman & Goodeve 1963/1983; Keele & Posner 1968; Zelaznik et al. 1983) and the proportional error constant p was estimated to be between 0.04 and 0.07 (Langolf et al. 1976; Meyer et al. 1988; Pew 1974; Schmidt 1988; Vince 1948), giving b in the 29.1 ms/bits to 75.6 ms/bits range. This estimate of the slope of the logarithmic tradeoff relation was sufficiently impressive that the deterministic iterative-corrections models soon became accepted as the best available account of Fitts' law (Meyer et al. 1990). Furthermore, with this model, Crossman (1956) suggested that the ID could be interpreted as the difference between two fundamental quantities, log2(W) measuring the entropy of the endpoint distribution, and log2(2A), measuring the entropy of a hypothetical initial distribution of motion amplitudes.

Despite its simplicity, the assumptions of the deterministic iterative-corrections model have been found to be suspect, and since the seventies it has become increasingly clear that this model is seriously flawed. Langolf et al. (1976) and Jagacinski et al. (1980a) have found considerable variation in the duration of the initial submovement, which is contrary to the model hypothesis of constant-duration submovements. Langolf et al. (1976) found that some movements exhibiting the logarithmic tradeoff relationship have only one correction despite the model prediction of several corrective submovements for large-movement- difficulty indices. Wallace and Newell (1983) observed that aimed hand movements produced in the absence of visual feedback obey Fitts' law, which gives additional evidence against the dichotomy of feedback-controlled movements obeying Fitts' law and feedforward-controlled movements obeying a linear speed/accuracy relation. More fundamentally, one handicap of this model is the fact that it is completely deterministic. This implies that for a fixed target distance and width the sequence of submovements would always be the same. So, the model cannot explain why subjects sometimes miss a target and commit an error (Fitts & Peterson 1964; Meyer et al. 1988; Wallace & Newell 1983).

c)The Control Model of Connelly (1984)

Connelly (1984) proposed an alternative interpretation to Fitts' law based on a control model linking the error rate of a movement to the error. To illustrate the mathematical development for a simple control law, he assumed that the error rate is a linear function of the error X.

The solution to this equation is:

It can be written to obtain: where Finally, by identifying X(0) with A and X(MT) with W/2, the expression for MT is obtained:

which is the same equation as Fitts' law except for the constant a which again is not predicted here. Connelly (1984) concluded his study by noting that a specific model has not yet been identified and that numerous models and control laws, both linear and nonlinear, can be formulated.

If we analyze the equations given above, it can be seen that the control law proposed in (10) is no more than a generalization of Crossman and Goodeve's deterministic iterative-corrections model. In fact, Crossman and Goodeve's model is discrete, while Connelly's model is continuous, and Crossman and Goodeve's model has a specific control law (Xi = pXi-1) which is the discrete representation of , while Connelly's model does not yet have a specific control law.

d)The Unifying Noise/Velocity Relationship of Chan and Childress (1990)

Chan and Childress (1990) have proposed a relationship that relates the variance of the human- machine noise to the mean square velocity of the human-machine output. Furthermore, they showed that this noise/velocity relationship is verified for some human-machine models (McReur's crossover model, Elkind's human-machine model and Fitts' law) and that it can be considered as a more fundamental human-machine behavior property.

Noting R as the radius of the region from the target center where human-machine output is most likely to be at any time t, the derivation of their relationship from Fitts' law is obtained by writing: where R = W/2; c = b/ln(2.0) for t = MT.

In this context, the authors represented the random variable, corresponding to the position from the target center at any time t, by a zero-mean Gaussian random variable with a standard deviation en related to R by:

where k is a proportionality constant.

From this,

since the position is a Gaussian random variable, the velocity is also a random variable with mean square v2 and:

to finally obtain a unifying noise/velocity relationship:

Chan and Childress (1990) did not provide any theoretical support for this new relationship.

e)The Stochastic Optimized-Submovement Model of Meyer et al. (1988, 1990)

This model (Meyer et al. 1988, 1990) represents the movement production process as an ideal compromise between the duration of primary and secondary submovements. The model assumes the existence of noise in the neuromotor system which may affect the primary submovement, causing it either to overshoot or undershoot the target. The stochastic optimized-submovement model assumes that the effect of the motor noise increases with the velocity of the submovements and that the relationship between primary submovement endpoint standard deviation S1 and the average velocity V1 of the primary submovement is: where K is a positive constant, D1 is the mean distance travelled by the primary submovements and T1 is their mean duration. Another assumption of the model is that for the secondary submovement,

where ~ is the distance travelled in a mean time T2~ by the secondary submovement. K is the same constant in equations (21) as in (22). Finally, another key assumption is that the average velocities of the primary and secondary submovements are programmed to minimize the average total movement duration (MT).

Under these assumptions, the stochastic optimized-submovement model

predicts that:

where A and B are non negative constants. Many other interesting predictions are made by this model with respect to the mean primary submovement durations, the proportion of secondary submovements and the values of error rates (see Meyer et al. 1988, 1990 for more details). Using computer simulations of this model, these authors have suggested that for multiple submovements, a quasi-power function might be a better predictor of MT:

where n is the number of submovements.

As stated previously, a few studies of temporally constrained movements have suggested relationships that are other than linear between the standard deviation of movement endpoints and the average velocity (Hancock & Newell 1985; Schmidt et al. 1985). Also, the constant K in equations (21) and (22) may differ for primary and secondary submovements. In these cases, the polynomial speed/accuracy tradeoff represented by equation (23) will not be verified, and equation (24) neither.

f)The VITE Model of Bullock and Grossberg (1988)

The VITE model consists of a set of channels controlling the length of a particular muscle (Bullock and Grossberg 1988). Motor planning occurs in the form of a target position command (TPC) that specifies the length to which all trajectory-controlling muscles are intended to move, and a command signal GO which ties together all these channels and which specifies the movement's overall speed. Mutual interactions exist between channels for antagonistic muscles. The present position command (PPC) subtracted from the TPC command specifies the difference vector (DV) which is integrated through time by the VITE circuit after being multiplied by the GO signal. Thus the PPC is gradually updated and generates an outflow movement command to reach the target. To generate a movement, a TPC different from the PPC generates a non-zero DV which is multiplied by the GO signal to generate an input to the PPC. By integrating this signal through time, the PPC will be updated until it equals TPC (Bullock & Grossberg 1988).

In its simplest form, the VITE circuit obeys these equations (Bullock

& Grossberg 1988):

where DV+ = max(DV,0); ` is a given parameter of the model.

Bullock and Grossberg (1988) showed that:

where E is the amount of overshoot error in the VITE command. In this relation TPC(0) - PPC(0) represents the distance to be moved. It should be noted that this relationship was mathematically proved in only one specific case (Bullock & Grossberg 1988, Appendix I) and, using computer simulations, the authors suggest that it occurs with greater generality (Bullock & Grossberg 1988).

2.Movements with spatial constraints where only A is given

Howarth et al. (1971) have proposed a theory of intermittent visual control which is not based on linear control assumptions. Their theory specifies that the hand should be as close as possible to the target at the time of initiation of the last corrective movement. In an attempt to determine the precise relationship between distance and time as the hand approaches the target, they found experimentally that:

where d is the distance to the hand from the target in mm, t is the time remaining before the target is hit in seconds, MT is the time for the total movement in seconds.

They found, from experimental analysis, that

where E2 is the mean square error from the target center, E02 is a square error due to tremor, eo2 is the angular error of the final corrective movement, and tu is the uncontrolled movement time evaluated at tu = 290 ms by Beggs and Howarth (1970).

As presented above, equation (29) is empirically determined and so does not specify the nature of the control processes contained therein (see Langolf et al. 1976 for an extensive criticism of this theory). It should be noted that in their experimental procedure Howarth et al. (1971) considered only one target distance, A = 50 cm. If equation (29) is written differently, we obtain:

which is similar to a power law (see Kvalseth, Table 2) where corresponds to ; and, because A is constant (50 cm) here, corresponds to: a(A)b. So, the empirical relationship of Howarth et al. can be seen as a manifestation of a power law in their experiment.

3.Movements with temporal constraints where MT is given: Focus on spatial variability

Another fundamental problem is the dependency of the speed/accuracy tradeoff formulation on the subjects' temporal and spatial goals (Schmidt et al. 1979). It was claimed that a linear tradeoff relationship is more appropriate for time-matching tasks or temporally constrained tasks, while a logarithmic tradeoff relationship better explains time-minimization movement or spatially constrained tasks. The linear speed/accuracy tradeoff can be characterized by: where We is the standard deviation of the endpoint coordinates.

Two main hypotheses have been considered for characterizing the conditions under which a linear rather than a logarithmic tradeoff will occur for aimed movements (see Wright & Meyer 1983 for a review).

The first hypothesis, named the movement-brievity hypothesis (Wright & Meyer 1983), was proposed by Schmidt et al. (1979) as a motor-output variability theory for ballistic movements. The proportional relationship between the within-subject variability in movement amplitude, called the effective target width (We), and the average velocity of the single aiming movement (A/MT) is derived in this model from two proportionality relationships: the first between the impulse and the average velocity: impulse ~ velocity (32)

and the second between the within-subject variability of the impulse

and the size of the impulse: eimpulse ~

impulse (33)

Since We is proportional to the variability of the impulse (Schmidt et

al. 1978; 1979): We ~ impulse (34)

We is also proportional to the average velocity of the movement (see Schmidt et al. 1979 for more details): We ~ velocity ~ A/MT (35)

The second hypothesis, named the temporal-precision hypothesis (Wright & Meyer 1983), was proposed by Meyer et al. (1982) as the symmetric impulse-variability model. This second approach provides a way to unify the linear and logarithmic tradeoffs by attributing precisely timed movements to a single pair of opposing force pulses that minimizes temporal variability, and spatially precise movements to a pre-programmed series of overlapping force pulses which increases temporal variability. This approach includes a number of assumptions about the shapes of force pulses used to produce movements and about the stochastic variation of pulses across different movements (see Meyer et al. 1982 for more details). The linear speed/accuracy tradeoffs have been observed by other investigators: stylus- tapping movements (Zelaznik et al. 1981, 1988); wrist rotations (Wright 1983; Wright & Meyer 1983); saccadic eye movements (Abrams et al. 1989). The scaling property of the force assumed by these models has been reported by Abrams et al. (1989), Armstrong (1970), Freund and Budingen (1978), Ghez (1979), Ghez and Vicario (1978), Gordon and Ghez (1987), etc. However, some problems have been raised. Most of the detailed studies have reported nonproportional relationships between force and force variability in both isometric (Carlton & Newell 1985; Fullerton & Cattell 1892; Jenkins 1947; Newell & Carlton 1985; Noble & Bahrick 1956; Provins 1957) and anisometric tasks (Newell et al. 1982). Furthermore, this relationship was found to have an inverted-U shape (Newell et al. 1984; Sherwood & Schmidt 1980; Sherwood et al. 1988).

A few investigators have questioned the model's assumptions about the scaling of force pulses (Schmidt et al. 1985; Zelaznik et al. 1986). Zelaznik et al. (1986) have reported that "the acceleration- time functions were not symmetrical and that symmetricality decreased as MT decreased", which caused difficulties for the Meyer et al. (1982) symmetric model (see Plamondon et al. 1993 for a review on the asymmetry of velocity profiles).

4.Temporally constrained movements where MT is given: Focus on temporal variability

Newell et al. (1979) conducted three experiments in which subjects were asked to move, as soon as they were ready, through a target distance, and not to stop directly opposite the target, in a time as close as possible to a target time MT. Newell et al. (1979) examined MTs of 100, 200, 500, 600 and 1,000 ms over distances of 0.75, 2.5, 5 and 15 cm. A controversial finding of their experiment, still without any convincing explanation, was that "the faster one moves, the more accurate the timing of the response" (Newell et al. 1979, see Hancock & Newell 1985 for a synthesis on the space-time approach to the speed/accuracy tradeoff). They also found that the velocity effect was independent of the values of movement time, and that for very slow movements there was a reduction in timing accuracy and movement control broke down (Newell et al. 1979).

III. THE KINEMATIC THEORY OF PLAMONDON (1993b,c; 1995a,b)

As we can see, the speed/accuracy tradeoffs in rapid human movements are far from being completely understood. None of the theoretical explanations proposed to date is able to take into account the major experimental observations in the field under a single scheme. Moreover, the majority of the resulting mathematical equations presented in Table 2 produce good data fitting over a limited range. If a mathematical expression has to be considered as a law in a specific domain, it should be general enough to describe all the phenomena occurring in that domain and, ideally, should be supported both theoretically and experimentally.

Over the last five years, a research group led by the first author has been studying the theoretical and practical interest of using lognormal functions to describe speed/accuracy tradeoffs (Alimi & Plamondon 1993a,b; 1994; 1995; Plamondon 1990a,b; 1991a,b; 1992a,b; 1993a; Plamondon et al. 1993). Recently, Plamondon (1993b,c; 1995a,b) has come up with a kinematic theory that seems to fulfill most of the previous requirements.

The major claim of this theory is that speed/accuracy tradeoffs are inherent constraints that emerge directly from the delta-lognormal impulse response of the global neuromuscular system involved in a synergy. The intrinsic properties of this impulse response, combined with a simple perceptivo-motor condition that has to be met at some sensory-motor level, is sufficient to ensure the effective production of any rapid movements. In this context, the kinematic relationships that have been reported in the field can be taken into account under a single paradigm.

1. The delta-lognormal law

The production of a rapid movement involves the activation of a complex neuromuscular system made up of several components organized both hierarchically and in parallel (see Ghez (1991) for an extensive survey). The primary motor cortex, the premotor cortex and the supplementary motor area constitute the highest levels of control. These components contain somatotopic maps and receive information from the periphery via sensory relay nuclei. The next level of the hierarchy is the brain stem. It is made up of three neuronal systems (medial, lateral and aminergic) that modulate motor neurons and interneurons in the spinal cord. This latter component constitutes the next level of the hierarchy. Its motor neurons interact, directly or indirectly, with proximal and distal muscles. Moreover, two other components regulate motor functions: the cerebellum and the basal ganglia. At lower levels, various musculoskeletal networks are involved. Indeed, even the simplest movements require the coordination of several skeletal muscles acting in groups rather than individually. Such a group is generally referred to as a synergy (Bernstein 1967). The muscles that cause the desired action are called the agonist muscles, and those causing the opposite effect, the antagonist muscles. Any of these muscles acts on bones via tendons by contracting parallel bundles of muscle fibers. For simple rapid movements, it is generally accepted that sensory feedback is not continuously used to control the trajectory, but that advance information from sensory events is used as a feedforward control to adjust the properties of the neuromuscular system with respect to the task objective.

According to the kinematic theory of Plamondon (1993b,c; 1995a,b) a schematic view of the complete neuromuscular synergy involved in the production of a rapid-aimed movement can be represented as in Figure 1.

Insert Figure 1 about here

This synergy is composed of two parallel systems, each made up of numerous components that represent the sets of neural and muscular networks involved in the generation of the agonist and antagonist neuromuscular activities resulting in a specific movement. What is of interest here is that it is possible to come up with an analytical description of a synergy output if a few basic hypotheses are made.

First of all, a representation space must be selected for this output. One of the most well-accepted invariants in rapid-aimed movements is the shape of the absolute velocity profile. Several authors (Abend et al. 1982; Atkeson & Hollerbach 1985; Beggs & Howarth 1972; Georgopoulos et al. 1981; Morasso 1981; Nagasaki 1989; Soechting & Laquantini 1981; Uno et al. 1989, etc.) have shown that the velocity profiles of rapid-aimed movement have a global "asymmetric bell shape" that is invariant over a wide range of movement sizes and speeds. This invariance suggests that velocity might play a key role in movement control and that it is reasonable to assume that a synergy output can be described in the velocity domain.

Second, since we are concerned with the description of well-learned and- practiced movements, it is assumed that for a specific task each subsystem works in a linear mode around some steady state conditions. In this context, the agonist and antagonist systems described in Figure 1 can be considered globally as linear time-invariant systems producing a velocity output (v1(t) or v2(t)) from an impulse command (U0(t-t0)) of amplitude D1 or D2, occurring at t0. Although proprioceptive feedback as well as various forms of interaction and coupling exist in several places between these two systems, we assume that the global effect of all these mechanisms can be taken into account at the very end of the process by subtracting the two outputs. The resulting velocity of the end-effector of the synergy is thus represented by:

where subscripts 1 and 2 stand for the agonist and the antagonist systems respectively and H(t-t0) represents the impulse response of the each system.

Equation (37) describes the output of the synergy as the difference between the impulse responses of the agonist and antagonist neuromuscular systems, weighted by the respective amplitude of their input activation commands. For each of these systems, their internal architecture, as depicted in Figure 1, is quite complex. Each component interacts serially to its nearest neighbour, but also in a hierarchically parallel fashion with a large number of more distant components. If this model could be simplified, an analytical expression could be obtained for the global impulse response of both the agonist and antagonist systems using asymptotic predictions. On the one hand, if emphasis could be put only on the parallel interactions between the components of a system, its impulse response could be derived using the product of the integrals of the impulse response of each component, and, under some specific conditions, it could converge asymptotically toward different types of exponential functions: double exponential, power of exponential, Weibull function (Galambos 1978; Leadbetter et al. 1983). For example, using a purely parallel model, Ulrich and Wing (1991) have proposed a specific sum of weighted exponential functions as an impulse response for a force-generating system. On the other hand, if emphasis could be put only on the sequential interactions between components, the impulse response of the system would be the convolution of the impulse responses of each component, and its mathematical description could be specified from the prediction of the central-limit theorem, as applied to the convolution of a large number of positive functions (Papoulis 1987). So, if all the neural and muscular networks composing a system were to form an independent sequence of subsystems, the impulse response (H(t-t0)) of the resulting linear system could be described by a gaussian function (Plamondon 1991a; 1993b; 1995a).

The kinematic theory of Plamondon stays in between these two extreme cases using an argument based on the time delay introduced in a system by the different components that have to react to a specific command. In the purely parallel model, the total time delay characterizing a system, agonist or antagonist, would be limited by the distribution of the maximum time delays of the different components. In the purely sequential system, the total time delay would be defined by the sum of the individual time delays of each component.

With the mixed architecture depicted in Figure 1, the time delay associated with each component taken individually will affect the total time delay of the global system in a more complex way to reflect both the parallel and the sequential coupling between the components. One simple way to link the time delay of each component is to assume that the cumulative time delay (Tj) after j subprocessing steps is related to the cumulative delay time (Tj-1) of the previous j-1 components by a law of proportional effect (Gibrat 1931), or, in other words, by a Weber law:

where ~j is a proportionality factor reflecting the coupling between step j and all the previous steps, each ~j being independent of the others and independent of Tj.

This hypothesis can be interpreted as assuming that both the agonist and antagonist systems depicted in Figure 1 can be represented by an equivalent sequential system made up of n components where the jth is linked to the j-1 previous ones. Then, making an analogy with the predictions of the central-limit theorem (Papoulis 1987; 1991), it is predicted that under hypothesis (38) the impulse response of a neuromuscular system will converge toward a lognormal curve (Plamondon 1991; 1993b,c; 1995a) provided that the individual impulse response hij(t) of each component meets some very general conditions (real, normalized, non negative with a finite third moment and scaled dispersion). So, under these conditions, the complete velocity profile of the total synergy will be described by the weighted difference of two lognormals (Plamondon 1993b,c; 1995a):

As will be seen in the rest of the paper, equation (39) is very general, and we will refer to it as a delta-lognormal law or a ~~ law from this point on (Plamondon 1993b; 1995a). Similarly, we will refer to f and e as the total logtime delay and logresponse time respectively, since these parameters reflect the total time delay and response time of a lognormal impulse response on a logarithmic time scale (see note 4).

In other words, the synergetic execution of a rapid movement can be seen as resulting from the simultaneous activation (at t = t0) of an agonist and an antagonist neuromuscular system, with commands of amplitude D1 and D2 respectively. Both systems react to their specific commands with an impulse response described asymptotically by a lognormal function, whose parameters f1, e1 and f2, e2 characterize the logtime delay and the logresponse time of the agonist and antagonist neuromuscular system involved in the synergy. From this point of view, t0, D1 and D2 can be considered as command parameters and f1, f2, e1, e2 as synergetic or system parameters.

We have shown in previous studies that the ~~ law is actually the most powerful equation for reproducing complete velocity profiles of simple movements (Alimi & Plamondon 1993a; 1994; Plamondon et al. 1993). Figure 2 shows a few examples of the optimum reconstruction of velocity profiles as obtained in these studies. The solid lines represent the curvilinear velocity of a pentip movement as computed from digitizer data, while the crosses represent the best fitting ~~ law in each case. The specific parameter values are given in the figure caption. Similar results were also obtained for wrist flexions and extensions on a monkey subject as well as visual saccades, head rotations and hand movements on a human subject (Plamondon 1995a; Plamondon et al. 1995a,b). Insert Figure 2 about here

Moreover, Plamondon (1993 b,c; 1995a) has demonstrated that the ~~ law predicts the majority of phenomena consistently reported by many research groups studying these types of velocity profiles. First of all, the theory predicts that from a single pair of synchronous input commands D1 and D2 occurring at t0, single, double or triple peak velocity profiles can be generated. The main peak has an asymmetric bell shape; its asymmetry depends upon the velocity and can be inverted at very high speeds. All these phenomena have been regularly and consistently reported (Beggs & Howarth 1972; Zelaznik et al. 1986; Georgopoulos et al. 1981; Morasso 1981; Soechting & Laquaniti 1981; Abend et al. 1982; Atkeson & Hollerbach 1985; Nagasaki 1989, Uno et al. 1989, etc.).

For movements where subjects are asked to produce displacements of different amplitudes with the same duration, the ~~ law predicts that the maximum velocity of the dominant peak will increase almost proportionally with the distance covered, that the time to peak velocity will be constant and that the different velocity profiles will be perfectly superimposable after displacement rescaling. All these facts have been reported in detail by Gielen et al. (1985).

For movements where subjects are asked to cover a constant distance at different speeds, the ~~ law predicts that the maximum velocity of the dominant peak will increase for faster movements and that the time to peak velocity will decrease as the maximum velocity increases. These predictions are consistent with the results of Gielen et al. (1985), Corcos et al. (1989), Nagasaki (1989), Lestienne (1979). Moreover, the velocity profiles of a specific family of curves will be approximately superimposable after appropriate amplitude and time rescaling, as reported by Gielen et al. (1985), Mustard and Lee (1987) and Corcos et al. (1989).

For movements where subjects are asked to move as quickly as possible to a target zone, computer simulation using the ~~ law predicts that the maximum velocity of the dominant peak will increase with movement time and that the time to peak velocity will increase with an increase of maximum velocity. These predictions can be verified from the data reported by Brown and Cook (1974), Gielen et al. (1985), Mustard and Lee (1987) and Goggin (1990). It is also predicted that the maximum velocity increases with displacement, as observed by many researchers (Binet & Courtier 1893; Freeman 1914; Jeannerod 1984; Hoffman & Stick 1986; Milner 1986). For a specific distance interval, the latter relationship can be approximated by a straight line with a positive intercept; this is consistent with Wadman et al. (1979) and Milner (1986). Moreover, if the relationship is plotted on a log-log scale, it can also be approximated with a straight line for some specific distance intervals. Such an observation has been reported by Brown and Slater-Hammel (1949). Similar predictions are also made for the relationship between the mean velocity of the dominant peak as a function of the distance covered, in accordance with the linear approximation reported by Brook (1974), Freund and Beudingen (1978), Brown and Slater- Hammel (1949).

2. The quadratic and power laws

Moreover, one of the most striking predictions of Plamondon's kinematic theory is related to its predictions concerning the practical duration of a movement. Although, a single lognormal curve reaches a null value after an infinite time (Plamondon 1991a), the subtraction of two lognormals may result in one or two zero crossings in the velocity profile (Plamondon 1993c; 1995b). The time occurrence (t1) of these zero crossings can be calculated by canceling out the ~~ law. This automatically leads to a quadratic law that links the logarithm of the movement time as defined here by MT = t1 - t0 (see note 5) and the logarithm of the ratio of the agonist-to-antagonist input commands:

So, depending on the general aspect of the velocity curve, zero, one or two values will be observed for MT (see note 6), and these will be linked to the ratio D1/D2. In other words, the velocity profile will encompass one, two or three peaks. These different cases can be analytically described under various parameter conditions (Plamondon 1993c; 1995b). For example, if e = e = e2, equation (41) reduces to a linear equation with respect to lnMT, so there is a single zero crossing in this case, with a time occurrence defined by a power law:

As previously stated, the system parameters (fi, ei) reflect the global timing properties of the different neural and muscular networks recruited for the production of a specific movement of an end- effector. Since the limb inertial and viscoelastic properties are fairly constant in adults, it is concevable that for a specific type of movement, the fi and ei will remain relatively constant from one movement to another. So, for a set of synergetic movements of an end-effector characterized either by fixed values of f1, f2, e1, e2, or by the fact that f1, f2, e1, e2 covary in such a way that ai in (41) or K and ` in (46) are constants, the logarithm of the movement time will be linked to the logarithm of the ratio of the amplitude of the input commands by a quadratic law in general, or by a linear equation (a power law in MT) in some specific cases (Plamondon 1993c; 1995b).

These latter conclusions highlight an efficient strategy that can be used by humans to control movement amplitude and movement time at the command level (see note 7). Indeed, the movement amplitude (MA) (associated with the dominant pulse in the case of one velocity zero crossing or with the first two pulses when two zero crossings are observed in the velocity profile) is obtained by integrating the delta-lognormal law over movement time:

Moreover, as can be seen from equations (41) and (45), the movement time can be controlled either at the input level, at the system level or by both mechanisms together. The input control of the movement time is reflected by the ratio D1/D2 of the amplitudes of the agonist and antagonist input commands. In other words, for one set of experimental conditions, if it can be assumed that the system parameters f1, f2, e1 and e2 are fixed or covary in such a way that ai in (41) or K and ` in (46) are constant, then the quadratic law directly predicts the duration of the dominant peak of the velocity profile (as well as the duration of the second pulse when the profile has two zero crossings).

From this input level perspective, the theory explains how easy it can be to generate either one set of movements with different amplitudes and the same duration or movements with the same amplitude but of different durations. In other words, if a subject is instructed to generate movements of different amplitudes within the same movement time, he should use a pair of input commands having a constant ratio (D1/D2 = constant) while changing D1 - D2 according to the required movement amplitude. If the goal is to produce movements with the same amplitude but of different duration, the subject must use a pair of input commands having a constant difference (D1 - D2 = constant), while modifying the ratio D1/D2 according to the required movement time.

3. Speed/accuracy tradeoffs

In a Fitts task, a subject has to move a pointer to a target positioned at a distance D from an origin, and reach the target within an absolute spatial precision of q~D as quickly as possible. The first condition to be met by the subject can be expressed as putting a limit ~1 on the maximum relative spatial error:

where ~1 is a constant for a specific task. The second condition is described by equation (41):

where it is assumed that the ai are constants for an experiment where the same neuromuscular synergy is involved. The second constraint can thus be interpreted as putting a limit ~2 on the maximum ratio of the antagonist to agonist input commands:

To meet both conditions simultaneously, a subject has to map the spatial constraints with the command constraint. In this case, equations (48) and (50) are combined to obtain the general expression of the constraints of a Fitts task on the ratio of antagonist to agonist input commands:

that is, to succeed in a Fitts task the ratio D2/D1 must be proportional to the relative spatial error required by the experimental protocol. In this context, substituting (51) into (49), the kinematic theory predicts that in general a quadratic law will be observed between the logarithm of the movement time and the logarithm of the inverse of the relative spatial error:

This quadratic law also reduces to a power law when a1 = 0.

In Figure 3, we have reproduced typical velocity profiles that are predicted by the kinematic theory (Plamondon 1993c; 1995b) as a function of different values of D/~D. Figure 3a shows some profiles corresponding to the quadratic law (equation 52), while Figure 3b depicts some examples of the power law (equation 54). These curves are similar to those reported by Corcos et al. (1988), Goggin (1989) and Soechting (1984). The maximum velocity decreases as D/~D increases. Moreover, it is predicted that the asymmetry of the profile increases as the accuracy demand becomes greater, that is, as D/~D increases. This is consistent with the results of Corcos et al. (1988), Jeannerod (1984), MacKenzie et al. (1987), Marteniuk et al. (1987), Milner & Ijaz (1990) and Soechting (1984). Figures 3c and 3d also show that the profiles cannot be perfectly rescaled, but their rising phase can be made almost similar after appropriate amplitude rescaling, as many others have reported as well. Insert Figure 3 about here

So, the kinematic theory predicts that under experimental conditions where the system parameters are held constant or covary in such a way that ai in (52) or K and ` in (54) are constants, an efficient strategy for producing for example, a specific reaching movement would be to evaluate visually the distance D to be covered by the end-effector as well as the relative spatial precision ~D/D needed to accomplish the task. By coupling some sensory motor maps in such a way that D = D1 - D2 and D2/D1 ~ ~D/D, the target will be reached in a movement time predicted by (52) and (54) for each specific case.

The fundamental claim made by Plamondon (1991a; 1992a; 1993a,b,c; 1995a,b) is that these relationships hold even when the D or ~D information is not directly specified, in which case the subject performs some visual estimate and uses some kind of virtual or default values for the missing information. In this perspective, the same equation, (52) or (54), can be used to analyze any movements made under different spatiotemporal conditions.

In the Schmidt et al. (1979) protocol, for example, subjects have to reach a target at distance D within a certain movement time. The information about ~D is not specified. If we assume that subjects are roughly estimating ~D to succeed in this experiment, a strategy similar to the one described above can be used and the predictions of the kinematic theory can thus be found for these conditions if equation (52) is rewritten as:

Since MT is fixed here, this latter equation predicts a proportional relationship between ~D and D for experiments requiring subjects to perform aiming tasks with the same neuromuscular synergy (fi, e constant) within a certain specified movement time (MT). Moreover, in this context, equation (56) predicts that MT will play the role of a scaling factor, decreasing the slope of the proportionality relationship for a longer movement time.

Similarly, for an experimental protocol like the one used by Howarth et al. (1971), where subjects are asked to move from a home position to a constant distance (D) while measuring the error (~D) for different movement times (MT), the kinematic theory predicts a quadratic relationship between ln~D and lnMT in the general case:

and a linear relationship between them for the specific case where a1 = 0 (power law):

Since D, ~D and MT are fixed, it can be seen that to succeed in such a protocol the subjects will have to adapt their fi and ei to meet the constraint described by (57) or (58).

Finally, for experiments dealing with time accuracy, such as those reported by Newell et al. (1979), where subjects are required to reach a target of fixed width (which is equivalent to assuming ~D = constant) under different distance and movement time conditions, if the subject uses a strategy governed by (52), the predictions of the kinematic theory concerning timing precision will be found under these conditions by estimating the movement time errors ~MT. Differentiating equation (52) and assimilating the absolute errors to the differentials (Topping 1972), we obtain:

So, for experiments requiring the same absolute spatial accuracy ~D at different movement times, the absolute timing error will be related to the movement time by a nonlinear relationship. Moreover, the distance (D) will act as a scaling factor under these conditions. For the specific case where e = e (a1 = 0; power law), equation (59) reduces to a proportionality relationship:

This summarizes the major list of predictions made by Plamondon (1993b,c; 1995a,b) concerning the kinematic theory. In the next section, we will report some validation results based on the data directly available from the numerous papers published on this topic.

IV. EXPERIMENTAL RESULTS

Among the numerous studies dealing with speed/accuracy tradeoffs are many which provide tables of numerical data that can be used directly or indirectly to test the predictions of the kinematic theory concerning speed/accuracy tradeoffs. We have grouped the tests under the four general headings previously used in section II.

1. Movements with spatial constraints where both D and ~D are given

a) The data and the protocol

To test the validity of the above relationships between movement time and the inverse of the relative spatial accuracy, we have made an extensive survey of the studies dealing with Fitts' task. For any study where sufficient numerical data were available (see note 8), we have run a regression analysis using the following three equations:

The first two equations are the quadratic and power laws previously proposed by Plamondon (1993c; 1995b), while the latter is the well-known equation generally referred to as Fitts' law (Fitts 1954) rewritten in our own notation for the relative spatial error (2A/W = D/~D) and using a natural logarithm instead of the base 2 logarithm. Eleven studies were thus found, providing 50 datasets. Their protocols are summarized in Appendix I and some of their basic features can be found in Table 3.

b) Results

Table 3 reports the results of our regression analysis over the eleven studies that were found with sufficient numerical data. For each study, the value of the fitting correlation coefficients for the quadratic law (R ) , the power law (R ) , and the Fitts' law (R ) are listed, under their specific headings. Insert Table 3 about here

As can be seen from these values, the quadratic law is the most powerful in predicting all the data. It is always better than (or sometimes equivalent to) the Fitts' law. Moreover, the power law outperforms Fitts' law in about 86% of the cases. These results are in accordance with the fact that a power law can be considered as a sufficient approximation of the quadratic law in many cases (Plamondon 1993a,b) and that a logarithmic law can be seen as the first-order approximation of a power law:

To better illustrate the differences in the prediction capacity of equations (61) and (62) over (63), we have plotted in Figure 4 a few examples of the data reported by some of the authors of these studies on a log-log scale, describing the predictions of the quadratic, the power and the Fitts laws. Even with a small, 3% differences in R , the predictions of Fitts' law can lead to large errors, particularly in curve extremities.

Insert Figure 4 about here

One might argue that some part of the improvement gained with the quadratic law could be due to the fact that it encompasses three parameters instead of two. First, this argument is not valid when comparing the performances of the power and the Fitts laws where the number of parameters is the same in both cases. Figure 5 summarizes the results of this comparison. It shows that on a mean basis the quadratic law gives a 2 % increase in R as compared to the R of the power law, which is itself 2 % higher than the R of Fitts' law. A similar analysis using the equation proposed by MacKenzie (1989; 1992) gives results that are slightly better than the Fitts' predictions (1 % better), but generally not as good as those of the power law or the quadratic law (1 % and 3 % worse respectively).

Second, it should be remembered that a larger number of parameters is not an automatic guarantee of a better fit (Alimi & Plamondon 1993a; Plamondon et al. 1993). For example, a regression analysis on the previous 50 datasets was executed using eight equations: quadratic law (3 parameters), power law (2 parameters), Welford's equation (1 parameter), Fitts' law (2 parameters), Welford et al.'s equation (3 parameters), Jagacinski et al.'s equation (3 parameters), MacKenzie's equation (2 parameters), and Meyer et al.'s equation (2 parameters). The results of this analysis (Alimi & Plamondon 1995) show that the quadratic law is always the best among the three parameter equations and that the 3- parameters equations of Welford et al.'s and of Jagacinski et al.'s perform less well than Meyer et al.'s equation, or the power law equation, both of which have only two parameters. Thus, a larger number of parameters with a less descriptive equation does not systematically improve the performances of a model in explaining real data.

Insert Figure 5 about here

When sufficient numerical data were available, we ran a regression analysis using the same equations as above, but with the data grouped by amplitude values. The results of this regression analysis are reported in Table 4. A similar pattern is observed here again that reflects the superiority of the quadratic law and the power law over Fitts' law.

Insert Table 4 about here

In 4 out of 11 cases in Table 3, and in 2 out of 5 cases in Table 4, sufficient data were available, and a statistical analysis was conducted which shows that these differences are significant.

~ In the Andres and Hartung study (1989, see Table 1), R > R (T(17)

= 2.6025, Prob > 3T3 = 0.0186) and R > R (T(17) = 2.3108, Prob >

3T3 = 0.0337).

~ In the Fitts study (1954, see Table 1), R > R (T(3) = 11.0000,

Prob > 3T3 = 0.0016) but R and R were not significantly

different.

~ In the Gan and Hoffmann study (1988, see Table 1), R > R (T(3) =

12.2574, Prob > 3T3 = 0.0012) but R and R were not significantly

different.

~ In the Kerr and Langolf study (1977, see Table 1), R > R (T(8) =

2.8000, Prob > 3T3 = 0.0232) and R > R (T(8) = 2.3570, Prob > 3T3

= 0.0462).

~ In the Fitts study (1954, see Table 2), R > R (T(16) = 4.2885,

Prob > 3T3 = 0.0006) and R > R (T(16) = 3.8454, Prob > 3T3 =

0.0014).

~ In the Kerr and Langolf study (1977, see Table 2) R > R (T(35) =

2.5966, Prob > 3T3 = 0.0137) and R > R (T(35) = 4.2642, Prob > 3T3

= 0.0001).

Table 4 also highlights another phenomenon. As can be seen from the R values, a better data fit is obtained when data are grouped under the same movement amplitudes than when they are grouped for all experimental conditions. This suggests that depending on the distance to be covered a subject might use a slightly different neuromuscular synergy, and therefore slightly different f1, f2, e1, e2 parameters from one distance to another. The data collected for each amplitude condition will obey a specific quadratic (or power) law and the grouped data will be a mixture of the slightly different laws due to slightly different parameters. From an external point of view, this could be misinterpreted as different relative contributions of ~D and D in these conditions, as has been suggested by Sheridan (1979) and Welford et al. (1969).

2. Movements with spatial constraints where only D is given

The predictions of the kinematic theory (Plamondon 1993c; 1995b) are summarized in equations (57) and (58) for these conditions. Howarth et al. (1971) have reported such an experiment where subjects were required to move from a home position to a constant distance target of varying size (~D) at different movement times (MT) (see note 10). These authors could not predict their results from Fitts' law. Figure 6 presents their results, as plotted in a graph of ln~D vs lnMT. The solid line shows the quadratic law predictions (equation 57), and the dotted line the power law approximation (equation 58). As can be seen, an almost perfect fit is obtained in the former case (R = 99 %) and a very good approximation is still reached in the second (R = 97 %). In other words, the Howarth et al. (1971) experiment just reflects another aspect of the quadratic law. Insert Figure 6 about here

3. Movements with temporal constraints where MT is given: Focus on spatial variability

The predictions of the kinematic theory in these conditions are summarized in equation (56) (Plamondon 1993c; 1995b). However, most of the papers dealing with this type of movement study the relationship between the movement distance and the standard deviation of the spatial error along or perpendicular to the overall direction of movement, not the absolute error (Schmidt et al. 1979; Wright 1983; Zelaznik et al. 1981; 1988). Two studies (Abrams et al. 1989; Wright & Meyer 1983) report both the standard deviation (SD) and the constant error (CE), from which the absolute error can be estimated under the assumption of a gaussian distribution of errors (Schutz & Roy 1973):

where Ay represents the area between CE and the desired target in a normal distribution.

Substituting (68) in (56), a strong correlation is predicted by the kinematic theory, between E{~D} and D, if the gaussian distribution hypothesis (Schutz & Roy 1973) holds in these experiments. This prediction is confirmed by both studies. Moreover, the Wright and Meyer (1983) data are consistent with the scaling effect of MT, that is E{~D} decreases as MT increases. Figure 7 highlights these predictions by plotting the best linear regression between E{~D} and D over the different sets of data reported by Wright & Meyer (1983).

Insert Figure 7 about here

4. Movements with temporal constraints where MT is given: Focus on temporal variability

The predictions of the kinematic theory can be found under these conditions by equations (59) and (60). The study performed by Newell et al. (1979) provides a data set which includes the absolute error ~MT as a function of MT and for two distance values. Both a nonlinear regression based on equation (59) and a linear regression based on (60) can be used to describe these data, as can be seen from Figure 8. The nonlinear relationship provides a better fit, although equation (60) gives consistent predictions. The scaling effect of the movement amplitude is also clearly apparent in this graph. Similar predictions can also be derived for ~tVmax as a function of tVmax in the same context. So, if the target is reached at zero speed or at maximum velocity, a proportionality relationship is predicted between the time to contact and its absolute error.

Insert Figure 8 about here

5. Other experimental protocols

Finally, the kinematic theory can also explain why consistent and simple patterns do not always emerge under some specific experimental conditions. For example, in the second experiment by Zelaznik et al. (1988), subjects are required to land in a spatial target zone and within a temporal bandwidth, in other words, D, ~D and ~MT are fixed in this experiment. This protocol forces the subjects to change their synergetic parameters f1, f2, e1, e2 from one condition to another, and the data collected reflect a behavior still obeying the quadratic law, but where non constant ai are used from one task to another. Moreover, since too many constraints are put on the system and since f1, f2, e1 and e2 certainly have some upper and lower bounds, it is predicted that some task requirements will not be met by some subjects. This is clearly apparent in the data reported by these authors, particularly for large and precise movements to be executed very fast. Similar reasoning can be applied to explain the nonlinear speed/accuracy functions recently reported by Newell et al. (1993) for both spatial and temporal errors. The different tasks cannot be executed under a single set of f1, f2, e1 and e2 and a mixture of data from different quadratic laws are collected depending on the experimental groupings. According to the kinematic theory, a complete analysis of these data would first require analysis of each velocity profile separately with a specific ~~ law, extracting the different parameters that allow an optimal reconstruction of that profile, and then analysis of the variations of these parameters under different experimental groupings. As long as the ~~ law parameters for each specific movement are not extracted and analyzed individually and statistically, these nonlinear observations will remain difficult to explain, although they probably reflect the effects of these specific experimental conditions on the synergetic parameters.

V. DISCUSSION

As one can see from these results and predictions, the kinematic theory of Plamondon (1993b,c; 1995a,b) can take into account the various forms of the velocity profiles of an end-effector as well as its different properties as a function of various experimental conditions. Using a basic ~~ law (equation 41), all the resulting observations can be described and analytical predictions about movement time, time to maximum velocity, maximum velocity etc. can be performed. Using a quadratic law (or a power law approximation) that derives from the delta-lognormal law, all the observations dealing with speed/accuracy tradeoffs can be described under a single framework. Using a single model that schematizes the global asymptotic behavior of a neuromuscular synergy, a simple analytical description of very complex phenomena can be reached. This is consistent with the fact that for a specific set of tasks it is always the same neuromuscular system that is being used, under different spatial or temporal constraints. One basic system, one basic description; but analyzed under the numerous points of view provided by the different experimental protocols that have been so cleverly designed over the last century.

Taking each of these various curve fittings individually, an improvement of a few percent would probably not justify the adoption of a new specific theory. What is of interest here is that only one basic equation is sufficient to explain all the data and to provide a new paradigm to analyze them from a new perspective. Here are some examples of the new points of view provided by the theory.

1. Origin of speed/accuracy tradeoffs

The kinematic theory of Plamondon (1993b,c; 1995a,b) provides a new insight into what might ultimately be responsible for the speed/accuracy tradeoffs: the asymptotic impulse response of a neuromuscular synergy, as described by the ~~ law (equation 41). This law predicts that even if there was absolutely no noise, no variability in the neuromuscular systems, a tradeoff would have to be taken into account by a subject planning a rapid movement at the command level because the distance to be covered is predicted by the difference between the agonist and antagonist input commands (D = D1 - D2), and the movement time is linked to the ratio of these two commands (see eq. 41 or 45). Assuming that a coupling exists between some sensorimotor maps in such a way that the ratio of antagonist to agonist activities is made proportional to the desired relative spatial error ~D/D of the planned movement, the different speed/accuracy tradeoffs follow. These tradeoffs result basically from the fact that rapid-aimed movements are executed by an ensemble of networks that produce very specific velocity profiles.

For example, using proprioceptive feedback, a subject can estimate the initial position of the end- effector to be moved. Using visual feedback, the distance to be covered, as well as the required accuracy can be estimated. By coupling this information with D1 - D2 and D1/D2 respectively, an appropriate pair of input commands can be fed into a specific neuromuscular system. Depending on the pathways and units involved, the overall synergy will react with logtime delays f1 and f2 and logresponse times e1 and e2. Once the specific commands are fed into the agonist and antagonist systems, the subject "already knows" from previous learning that the target will be reached with a certain precision ~D/D within a time MT, depending on the acuity and integrity of the sensory information available (visual, kinesthetic or virtual). No visual feedback is needed during the movement, except as will be discussed below for the preparation of the next movement, if a complex sequence is being executed.

In this context, the variability of the different parameters describing a ~~ synergy is not the intrinsic cause of the speed/accuracy tradeoffs. This variability constitutes a noise that is superimposed on the basic processes to produce motor output fluctuations around some mean values, each of which is a specific movement being described by the kinematic theory. In this context, the relationships that have been reported to depict the variability of space or time targets just emerge from the variability of the command and system parameters according to basic experimental constraints that are put on a subject who has to cope with its intrinsic neuromuscular limitations, as described by the ~~ law (Plamondon 1993b; 1995a).

2. Control variables

The kinematic theory also suggests an answer to the problem of control variables in rapid-aimed movement. Several variables have been proposed for controlling limb movements: force, velocity, length, stiffness, viscosity, etc. (see Stein 1982 for an extensive review). The kinematic theory shows that for target-directed movements, the input control variables D1 and D2 have to be coupled with distance information (D and ~D) because the goal of the subject is to exploit the asymptotic shape and properties of the impulse response of its neuromuscular system. This approach seems in accordance with the results of Houk & Gibson (1987) and Gibson et al. (1985) who have shown that short high-frequency bursts transmitted from the red nucleus to the spinal cord in the rubrospinal track, as recorded from monkey subjects that were trained to perform a visual tracking task, code movement velocity, in terms of the burst frequency of the firing cells. Moreover, in these experiments, burst duration correlates closely with movement duration, and the number of spikes in a burst correlate with movement amplitude. Based on these observations, it was suggested that red nucleus signals may serve to command velocity (Houk 1989).

Moreover, Georgopoulos et al. (1984) have reported a strong similarity between the asymmetry of the velocity profile and the asymmetry of the vector cell population profile. This has been interpreted by Bullock and Grossberg (1988) as a manifestation of the properties of the neural networks that control movement.

A rapid-aimed movement may be seen as a motor task producing a certain spatial output within relatively stringent time limits. From a physics point of view, controlling velocity seems to be the simplest way to perform such a task. In this way, more complex movements like handwriting can be segmented into strokes with relatively less activity at the beginning and at the end (Plamondon & Maarse 1989). These segmentation points can be interpreted in terms of spatial targets and a control strategy based on these virtual targets can be learned from the visual inspection of a trajectory (Plamondon & Privitera 1995).

Moreover, the velocity vector is the sole dynamic information which is uniquely related to the end- effector trajectory: it is always tangent to the trajectory and can thus be recovered at least partially from visual information, for example in handwriting. Other representations, like acceleration or force vectors, cannot. This is a major point to bear in mind when considering the importance of the visual system in the learning and control of more complex movements.

Some indirect evidence also support the importance of velocity in movement generation. For example, in a comparative simulation of 14 simplified handwriting models, it has been shown (Plamondon & Maarse 1989) that models controlling the velocity output yield the best reconstruction in an analysis-by-synthesis experiment. Similarly, in a study of comparative performance of position, velocity and acceleration signals for automatic signature verification, it has been suggested that the velocity domain is one of the best representation spaces for a 2D signature verification system (Plamondon & Parizeau 1988).

3. Ballistic movements

The generation of a multiple peak velocity profile from a single pair of synchronous impulse commands is one of the most powerful characteristics of the ~~ law (Plamondon 1993b; 1995a). Most of the models published to date, if not all, ignore or neglect secondary peaks in velocity profiles or require specific commands for each peak (see Plamondon et al. 1993 for a comparative review). The ~~ law shows that secondary peaks emerge naturally from a single pair of synchronous impulse commands when the agonist and antagonist systems composing a synergy have different logresponse times, different logtime delays or both.

In this context, the kinematic theory provides a practical definition of a simple ballistic movement: a rapid movement produced by a synergy made up of an agonist and an antagonist system synchronously activated by a pair of impulse commands D1U0(t-t0) and D2U0(t-t0), each system being asymptotically characterized by a lognormal impulse response. As can be seen in the examples shown in Figure 2, a simple movement does not necessarily result in a single peak velocity profile. Depending on the state of the neuromuscular system, as depicted globally by the values of f1 and f2, e1 and e2, a small velocity reversal or a double reversal may occur. These secondary peaks do not have to be associated with any closed-loop corrective submovements. They are not the result of new input commands for making trajectory corrections, rather they are part of a single ballistic movement and result from the differences in the timing properties of the impulse responses of the agonist and antagonist systems to a single pair of synchronous inputs. In fact, the occurrence of these vanishing oscillations in the system response to a single pair of synchronous commands seriously questions the concept of corrective movements or submovements that have been put forward in the past by several researchers to explain phenomena related to the speed/accuracy tradeoffs of simple movements. Moreover, the kinematic theory provides a way to check if there are real closed-loop homing-in phenomena in more complex trajectories. Indeed, using equation (41), a complex experimental velocity profile can be reconstructed by concatenating and superimposing a few basic units (that might incorporate secondary peaks). The optimal solution to this signal reconstruction process would provide the number and the relative importance of the submovements needed to synthesize the complex velocity profile. Moreover, if multiple solutions emerge in analyzing these data (Guerfali & Plamondon 1994), the ~~ law could serve as a basis for studying strategies and optimizing principles that might be involved in the production and control of complex movements (Guerfali & Plamondon 1995) (see note 11).

4. Complex movements

In the light of the kinematic theory, many previous results can be reanalyzed and new experiments planned to study the effect of the command and system parameters on movement kinematics. This will require the measurement of the complete velocity profiles and the extraction of movement parameters (Alimi & Plamondon 1994; Plamondon et al. 1995a, b). With such an approach, a set of simple ballistic movements can be studied under different experimental conditions in terms of changes in the input commands and of changes in the global state of the system, instead of focussing on data-driven interpretations like single/multiple movements, accelerating/decelerating phases, closed and open-loop portions, etc. Since the theory does not require continuous feedback, it can also be useful in providing criteria to discriminate between open-loop and closed-loop conditions. If the velocity profile of a movement cannot be fitted by a ~~ law (as could happen, for example, for complex visual tracking tasks), this might suggest that the corresponding movement, or part of it, has probably involved some kind of continuous feedback and that the ~~ law does not hold in these conditions. What is of interest here is that the theory provides some cues on the necessity or the possibility of using feedback information. Unless default or virtual values are used for D and ~D, feedback is necessary prior to movement initiation to evaluate D and ~D and to recruit the proper neuromuscular networks (as characterized by fi, ei). Once a movement is initiated, a specific ~~ velocity profile is generated and the end-effector will reach its target within MT with a relative spatial accuracy ~D/D and timing accuracy ~MT/MT. If some new sensory information is provided during that first movement, which requires a change in the planned trajectory to a new target, another movement with more or less similar ~~ characteristics (depending on the new neuromuscular networks recruited) will be initiated and a new velocity profile will start to become superimposed vectorially on the previous one. In this perspective, more complex movements can be analyzed by vector superimposition of different ballistic units (Plamondon 1992b,c; Guerfali & Plamondon 1994, 1995).

5. Sequence generation and control

Assuming that the built-in properties of the ~~ law are known and exploited by a subject, we have recently developed a neural network model that generates and learns rapid movement sequences, each of these being described by a ~~ velocity profile (Plamondon & Privitera 1995). In other words, we have used the kinematic theory to describe a neuromuscular synergy that is coupled to a neural map representing the planning space and composed of leaky integrator elements. In light of the basic arguments previously introduced, the generation of a structured motor plan is then seen as the outcome of a predefined mental image of the movement where only the principal targets and their corresponding time sequence is already represented before execution, with an indication related to the evaluation of the required spatial accuracy of the movement.

A movement sequence is instantiated by a recall of the movement sequencing plan from the long-term memory and the corresponding positioning of this plan on the surface of the planning space. In other words, all the virtual targets composing the sequence are settled on the surface of the leaky integrator grid and the map finally represents a sort of virtual imagery of the movement. By means of a simple competition mechanism, it is possible to define a threshold process able to detect the descending part of the velocity of each motor stroke, and consequently to instantiate the next ballistic stroke. This point of synchronization depends on the time requested for the movement: the higher the velocity of the movement the faster the next motor impulse has to be anticipated, finally risking losing the intended form of the movement. On the basis of this virtual imagery, which is kept for the entire course of the external movement sequencing, the command generator is able to activate the corresponding sequence of impulse commands D1 and D2 for the neuromuscular synergy. During movement learning, the same process is exploited: in this case the movement is executed by an external subject and the occurrence of a synchronization point is interpreted as the location of one of the virtual targets composing the new movement. The goal of the learning phase is to build a central representation of the observed movement by means of storing the sequence of corresponding virtual targets.

6. Motor-perception interaction

One key interpretation which is highlighted by the kinematic theory of Plamondon (1993b, c; 1995a,b) is the inherent relationship that must exist between the perceptual information and motor commands. Indeed, to execute a target-directed movement, at least two basic cues must be perceived from the environment or from mental imagery previously acquired from learning: the distance (D) to be covered and the absolute error (~D) that is required in executing that movement. The knowledge of the absolute position of the target and the end-effector is not necessary at this stage, although it might be needed at another level to estimate the two previous pieces of information. Once this information is extracted from the optic flow, it is matched via different sensorimotor maps into two commands, D1 and D2, whose difference, D1 - D2, predicts the movement amplitude (D) and whose ratio, D1/D2 ~ D/~D, predicts the movement time (MT) (Plamondon 1993b,c; 1995a,b). Depending on the status of the neuromuscular synergy recruited in the process, as described by the parameters f1, f2, e1 and e2, the resulting ~~ velocity profile will be more or less complex, in terms of its number of pulses (up to three for a single pair of commands). Proper coupling between the different stages involved in the continuous activation process will result in a faster or a slower movement, depending on the relative precision that is required by the task.

In other words, once a subject has learned how to control a specific neuromuscular synergy to execute a spatially constrained task with a specific end-effector, a specific ~~ velocity profile does emerge due to the coupling that is built between the different components of the agonist and antagonist systems involved (equation 38). This allows the subject to "forget" about the synergy itself and concentrate on the goal of the action at a higher level via a direct coupling between the global activation commands and the relevant perceptual information. In this context, the movement time does not have to be planned or programmed in advance as a specific goal since it will automatically emerge from the selection of the ratio of two basic activation commands. Even when the maximum absolute error (~D) is not specified by the experimenter, the subject is probably able to extract from mental imagery a significant or a default value for this cue. In any case, the spatial target will generally be reached within a movement time MT, as predicted by the quadratic law.

How is this mapping done? What types of maps are necessary, from a retinotopic to a motor representation? How are intermodality sensorimotor commands encoded as the target position and as spatial resolution? How does the learning affect the time delays of the different components? These are some of the numerous questions that are raised by the new point of view that is provided by the kinematic theory. It is hoped that some constructive suggestions will be discussed by the commentators in order to shed some light on these potential mechanisms.

VI.CONCLUSION

This paper had two specific goals. First, we wanted to raise the point that speed-accuracy phenomena, as observed in simple rapid-aimed movements, are still without a fully comprehensive interpretation. Second, we wanted to show that the kinematic theory recently proposed by Plamondon (1993b,c; 1995a,b) provides a general framework in which such phenomena can be described and further studied.

To reach our first goal, a systematic survey of the scientific literature dealing with speed/accuracy tradeoffs was presented to highlight the numerous mathematical and theoretical interpretations that had emerged over recent decades from the various studies that had been conducted on this topic. Although reflecting the richness of the studies in the field and the high degree of interest that such basic phenomena represent for the understanding of human movement, such a variety of points of view questions the validity of many of the models with respect to their capacity to explain all the basic observations consistently reported in the field.

In the second part of this paper, we have summarized Plamondon's kinematic theory of rapid human movement in which the basic properties of the proposed model emerge as a fundamental consequence of its architecture. Indeed, by considering the asymptotic behavior of a large number of dependent linear systems, a mathematical expression describing the velocity profile of an end-effector driven by the action of a synergy made up of an agonist and an antagonist system has been derived. The resulting equation which describes the velocity profile, referred to as a ~~ law (Plamondon 1993b; 1995a), can be used to describe the kinematic properties of simple ballistic movements and, more specifically, the various aspects of the speed/accuracy tradeoffs, from a quadratic or a power law (Plamondon 1993c; 1995b) that immediately emerges from it. So, within a single framework, it is possible to take into account the basic observations consistently reported in the classical studies in the field.

The kinematic theory (Plamondon 1993b,c; 1995a,b) encompasses both similarities and dissimilarities with respect to the other models previously published. For example, unlike the minimum jerk (Hogan 1984) or the minimum torque (Uno et al. 1989) models, the theory does not require any minimization criterion to generate "bell-shaped" velocity profiles. The form invariance of the profile emerges through the asymptotic behavior of a large number of coupled neuromuscular networks. However, for the study of a more complex trajectory where many ~~ equations will have to be superimposed to reproduce the complete velocity profile, it is expected that optimization principles will have to be incorporated into the theory to reduce the number of potential solutions. Like the impulse timing models (see Plamondon & Maarse 1989 for a review), the kinematic theory succeeds in separating out command and system parameters, except that the end of the activation does not have to be directly specified at the command level in terms of a specific parameter (Plamondon 1993c; 1995b). Unlike the equilibrium point models (Feldman 1966a, 1986; Polit & Bizzi 1979), where the focus is mainly on the modeling of the mechanical properties of the muscles, or the neural network models (Bullock & Grossberg 1988), where the emphasis is on the modeling of the architectural properties of the neural systems, the kinematic theory provides a global view that is based on the timing properties of both the neural and the muscular networks. The price to pay for such generality is that it is difficult, without further experiments to provide a direct biological interpretation for the system parameters f1, f2, e1, e2 and further study will be needed in this context. Unlike the neural network models, the kinematic theory is based on the linear system theory. It provides an analytical solution to describe the trajectory of a well-learned rapid movement, but it does not provide too many cues on the learning process itself. However, using the basic knowledge that emerges from the theory, it is possible to construct higher-level neural networks that can learn to generate complex movements (Plamondon & Privitera 1995). Finally, the model can be considered to stand in between purely parallel (e.g. Ulrich & Wing 1993) and purely sequential models (e.g. Plamondon 1991). The proportionality relationship (equation (38)) that is assumed to exist between the cumulative time delays of the different stages of the equivalent sequential representation of the synergy is a way to take into account the different hierarchical couplings (both sequential and parallel) that exist in a synergy.

As will probably be seen in the forthcoming commentaries, although the kinematic theory is very powerful at a descriptive level, numerous questions remain unanswered in the study of well-learned tasks. To answer most of them, new experiments will have to be designed or old data will have to be reanalyzed. To progress in our understanding of rapid human movements and to take advantage of the new window provided by the kinematic theory summarized here, a large set of velocity profiles will have to be studied with an analysis-by-synthesis methodology based on an optimal parameter fitting of the ~~ law to these profiles (Alimi & Plamondon 1993a,b; 1994; Plamondon et al. 1993; Plamondon et al. 1995a,b). It is hoped that the analysis of the parameters that will be extracted from these studies will provide some answers to these unsolved problems.

ACKNOWLEDGEMENTS

This work was supported partially by grants NSERC-000915 and ICR046114 from NSERC Canada, grant ER1220 from FCAR Qubec and a grant from the Office Franco-Qubcois to Rjean Plamondon. Adel M. Alimi has benefited from a scholarship from ACDI. The authors wish to thank Mrs. Hl ne Dallaire for her invaluable assistance in typing and editing the various versions of this manuscript. Part of this work was written while the first author was a visiting scientist at the Laboratoire de Neurosciences Cognitives, Centre National de la Recherche Scientifique, Marseille, France.

APPENDIX I

Eleven studies were selected to study the performance of the quadratic law vs the power law and Fitts' law. These studies provide 50 datasets. Their protocols are summarized in the following paragraphs.

Andres and Hartung (1989):

The study consisted of two experimental sessions. Nine male subjects were asked to move a chin stylus reciprocally between targets of various widths and separations. The subjects' head movement capability in a Fitts reciprocal tapping task was predicted by recording the time intervals between movements of the chin stylus repetitively tapped between two target plates located along the horizontal axis of a reciprocal tapping apparatus. The experimental protocol consisted of nine A/W conditions (A = 7.6, 15.2 and 30.5 cm; W = 1.3, 2.5 and 3.8 cm) giving a Fitts index of difficulty ID = log2(2A/W) in the 2.00 to 5.58 bits range. The mean movement times were reported for the nine subjects and the two sessions for each A/W condition, providing eighteen datasets.

Drury (1975):

Ten male subjects participated in experiment 2 of this study. Each subject started reciprocal tapping using his preferred foot. Six different amplitudes (A = 150, 225, 300, 375, 525 and 675 mm) were crossed with two pedal sizes (W = 25 and 50 mm), giving a Fitts index of difficulty ID = log2(2A/W) in the 0.918 to 3.335 bits range. The mean width of subjects' shoes (108.8 mm) was added to the target width as a reasonable adjustment because any portion of a shoe touching the target was recorded as a hit. The mean movement times over subjects for experiment 2 were reported for each A/W condition, providing one dataset. Drury and Hoffmann (1992):

In this study, experiment I dealt with a standard Fitts paradigm. Ten male subjects were asked to make discrete movements to hit simulated keyboard keys. The amplitude of movement A was kept constant at 160 mm (light key spacings) and five target boards were used onto which were attached target sets of width (B = 2, 6, 10, 14 and 18 mm). Four metal probes were held by the subjects in the hand, flat across the palm so that they produced a natural extension of the index finger. These probes had tip widths of (P = 0, 5, 10 and 15 mm). A further condition, in which the subjects used their index fingers as the probe, was also included (the mean pad width in this case was 11 mm).

To evaluate W in Fitts' index of difficulty ID = log2(2A/W), the authors used W = B + 0.6 P in the probe condition (B is the target width and P the probe width) and W = B + 10 in the finger condition. The mean movement times over subjects for single targets were reported for each A/W condition for the probe and the finger condition, providing two datasets.

Fitts (1954):

Four experiments were reported in this original study consisting of a reciprocal tapping task with a stylus of 1-oz and 1-lb, a disk-transfer task and a pin-transfer task. Sixteen male subjects participated in the first three experiments and twenty (ten men and ten women) in the last one. In the tapping task, four distances (A = 2, 4, 8 and 16 in) and four target widths (W = 0.25, 0.5, 1.0 and 2.0 in) were used, giving a Fitts index of difficulty ID = log2(2A/W) in the 1 to 7 bits range. In the disk-transfer task, four distances (A = 4, 8, 16 and 32 in) and four target widths (W = 0.0625, 0.125, 0.25 and 0.5 in) were used, giving a Fitts ID in the 4 to 10 bits range. Finally, in the last pin-transfer task, five distances (A = 1, 2, 4 8 and 16 in) and four target widths (W = 0.03125, 0.0625, 0.125, and 0.25 in) were used, giving a Fitts ID in the 3 to 10 bits range. The mean movement times over subjects were reported for each A/W condition and for all the tasks, providing four datasets.

Gan and Hoffmann (1988):

Six male and six female subjects were asked to make discrete tapping movements about the elbow in a left-to-right direction, and to make these as rapidly as possible. The apparatus consisted of boards having a starting plate and a target plate at a distance of (A = 4, 9, 16 or 25 cm) from the starting plate and each board had a constant Fitts ID = log2(2A/W) of 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 and 6.0 bits. The mean movement times over the subjects were reported for each ID and each A condition, providing four datasets.

Hoffmann and Sheikh (1991):

Five males and five females took part in the experiment. There were 15 experimental conditions (A = 100, 200 and 400 mm; W = 2, 6, 10, 14 and 18 mm). Movements were made by the subjects' either holding a sharp-pointed probe or using the finger as a probe. The probe was pointed at its tip. The mean finger pad size of the 10 subjects was P = 10.2 mm. Times for the discrete movements were measured starting when the probe left the starting plate and ending when it contacted the target. The mean movement times over subjects were reported for each A/W condition for the sharp-probe condition and the finger condition, providing two datasets. For the finger condition, the value of the finger width P was added to the target width W to obtain the effective target width used in the calculus of the indices of difficulty.

Johnsgard (1994):

Eighteen subjects were asked to move a cursor from a starting position to a rectangular bar target drawn on the computer screen using a mouse or a virtual reality glove. Three target amplitudes were chosen (A = 2, 4, and 8 in), fully crossed with three target widths (W = 0.5, 1 and 2 in). For each device three values of gain were tested (G = 1, 2 and 3). The mean movement times over the subjects were reported for each ID and each G condition, providing three datasets for each device.

Kerr and Langolf (1977):

Eight male subjects participated in the experiment. The task consisted of moving a stylus forward from a back contact plate to hit a target placed at specified distances in front of the subject. Four levels of movement distance were chosen (A = 8, 12, 16 and 20 in) and, for each of these, four target widths were chosen (W = 0.25, 0.5, 1.0 and 2.0 in), giving a Fitts ID in the 3.0 to 7.32 bits range. The mean movement times were reported for the eight subjects for each A/W condition. So nine datasets (see note 9) were available (the eight subjects and the mean of all the subjects).

MacKenzie et al. (1987):

Six subjects participated in the experiment. The task was a discrete tapping task. There were twelve experimental conditions with the same IDs as in experiment 1 of Fitts and Peterson (1964). Four target diameters (W = 2.54, 1.27, 0.64 and 0.32 cm) were crossed with three amplitudes (A = 7,62, 15.24 and 30.48 cm), giving a Fitts ID in the 2.58 to 7.58 bits range. The mean movement times over all subjects were reported for each A/W condition, providing one dataset.

Nineteen subjects performed two batteries of tests where unusual body dynamics were imposed on subjects in order to assess altered environment performance. The results of the first battery of tests are not used here because only two points were given in the ID vs MT dataset. Eleven subjects (six females and five males) participated in the second battery of tests where an altered environment was electronically created by introducing a first-order lag characteristic between the graphic input device and the computer. The dynamics of the arm motion/input device with a lag bear little resemblance to human arm motion/input device dynamics in microgravity. However, an astronaut performing tasks in the weightless state is in an analogous situation in being exposed to unusual body dynamics and responses. The average movement times over the subjects were reported for a Fitts ID in the 1.5 to 3.7 bits range, for two conditions (with no lag and with lag), providing two datasets.

Repperger and Remis (1990):

Five subjects ran 5 trials each of the 5 different stylus diameters selected with the same taskhole. Subjects were placed in an exaskeleton device to investigate the feasibility of using such devices in the performance of a Fitts' task. The mean movement times over all subjects were reported for a Fitts ID in the 6.35 to 11.73 bits range, providing one dataset.

Note 1:It should be noted that if V = 0, we obtain an expression of MT for the case of static targets, which is still different from Fitts' formulation.

Note 2:It should be noted that if V = 0, we obtain an expression of MT for two cases of static targets, which is similar to Fitts' formulation.

Note 3:It should be noted that if D = 0, we obtain an expression of MT for the no-delay case, which is similar to Fitts' formulation.

Note 4:The reader should not confuse what we call hereafter the logtime delay (f) and the logresponse time (e) which refer to (logt), a logarithmic time scale, with the time delay and the response time of a lognormal on a conventional time scale (Plamondon 1995a).

Note 5:To be more precise, we should use the term "activation time" here, since the definition of (t1-t0) is more general than the operational definition of the movement time that is normally used in most studies. According to the kinematic theory, the movement time refers to the whole activation time, starting at the initiation of the input command t0 and finishing at t1, when the velocity reaches zero. So, the time reference for computing movement time is not the reaction time (tr), that is, the time where the end-effector starts to move, but t0, the time of occurrence of the commands. Since, for a specific set of identical movements produced by the same synergy, t1 - tr should be proportional to t1 - t0, we will not distinguish between activation time and movement time from now on, to avoid confusion.

Note 6:A supplementary zero velocity value is also predicted for t1 = l. See Plamondon (1993b; 1995a) for a discussion about this asymptotic prediction.

Note 7:Depending on the relative importance of the surface under the velocity curve from the longest time occurrence of the zero crossing to tf = l, the approximation will be more or less realistic. For example, for the results reported in Figure 2, approximation (47) would certainly be acceptable. In general, since the remainder will be a function of the system parameters (f1, f2, e1, e2), control from both input command and system processes might have to be taken into account in some extreme cases.

Note 8:A few studies were eliminated for the following reasons:

~ Beggs and Howarth (1972): they reported only three values of D/~D;

~ Carlton (1980): he reported only two values of log2 D/~D;

~ Kerr (1973): he included the trials which missed the target in the calculation of MT (p.177);

~ Kvalseth (1977): for the same reason as Kerr (1973) (see Kvalseth 1977, p. 43);

~ Spitz (1990): he included in the measure of MT the time to stabilize the cursor on the target (p. 407).

Note 9:For subject 7, the value 000.3 (probably a typing error) in Table I was replaced by 600.3 which is consistent with the mean value reported (Kerr & Langolf 1977).

Note 10:The ~D values were computed from the effective target values (We) reported by the authors using their method: ~D = 4.11 ~ We (Howarth et al. 1971).

Note 11:We always assume here that the recording devices or the manipulanda used by the subjects have a negligible effect on the neuromuscular impulse response. If these mechanical devices strongly interferes with the end effector under study, the ~~ velocity profile might be partially masked by the impulse response of the recording devices itself, making data analyzing really more complex.

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LIST OF TABLES

Table 1-Domains of application of Fitts' law.

Table 2-Mathematical formulation of Fitts' law.

Table 3-Results of a regression analysis comparing the predictive power of a quadratic law and a power law vs Fitts' law.

Table 4-Results of a regression analysis comparing the predictive power of a quadratic law and a power law vs Fitts' law. The data are grouped under movements of similar amplitude.

FIGURE CAPTIONS

Figure 1- Schematic view of the neuromuscular synergy involved in the production of a rapid-aimed movement. Figure 2-

Typical results of an analysis-by-synthesis experiment for three typical velocity profiles.

Figure 2a D1 = 3.45 ;D2 = 1.44 ;f1 = -1.59 ;f2 = -1.45 e12 = 2.77~10-2 ;e22 = 5.20~10-2; t0 = 2.04~10-1 ; MSE = 4.53~10-2

Figure 2b D1 = 2.33; D2 = 3.36~10-1; f1 = -1.28; f2= -1.10 e12 = 2.13~10-2 ;e22 = 9.10~10-3;t0= 2.88~10-1; MSE= 2.50~10-2

Figure 2c D1 = 2.45; D2 = 4.39~10-1; f1 = -1.59~10-1; f2= -1.35 e12 = 1.13~10-2 ; e22 = 3.80~10-3; t0= 2.84~10-1; MSE= 6.60~10-2

Figure 3-Properties of velocity profiles for movements with different accuracy demands.

Figure 3a Typical velocity profiles corresponding to the quadratic law.

Figure 3b Typical velocity profiles corresponding to the power law.

Figure 3c and 3d Approximate superimposition of the profiles of 3a and 3b respectively, after amplitude rescaling.

Figure 4-Fitting of the data from a few typical experiments with a quadratic law (solid line), a power law (truncated line) and Fitts' law (dotted line).

Figure 4a Data from the Fitts (1954) experiment.

Figure 4b Data from the Drury (1975) experiment.

Figure 4c Data from the Mackenzie et al. (1987) experiment.

Figure 5-Summary of the comparison analysis between the quadratic power and Fitts' laws for the eleven studies reported in the text. DDDDD Quadratic, DD DD DD power, - - - - Fitts.

Figure 6-Fitting of the data from Howarth et al. (1971) with a quadratic law (solid line, R = 0.99) and a power law (truncated line, R = 0.97).

Figure 7-Best linear regression between E{~D} (equation 68) and D resulting from the proportionality between ~D and D as predicted by equation (56) for movements of the same duration. Data from Wright & Meyer (1983, Tables 2 & 4). DDD MT = 239 ms, R2 = 0.97 --- MT = 317 ms, R2 = 0.93

Figure 8-Best nonlinear regression (equation 77, solid line) and proportional regression (equation 78, truncated line) between ~MT and MT. Data from Newell et al. (1979, Experiment 1). ~ D = 5 cm, ~ D = 15 cm, MSE = 9.79 , MSE = 1.02~10-1 ,R2 = 0.99 R2 = 0.99