Abstract
The movements of the human arm have been extensively studied for a variety of goaldirected experimental tasks. Analyses of the trajectory and velocity of the arm have led to many hypotheses for the planning strategies that the CNS might use. One family of control hypotheses, including minimum jerk, snap and their generalizations to higher orders, comprises those that favor smooth movements through the optimization of an integral cost function. The predictions of each order of this family are examined for two standard experimental tasks: pointtopoint movements and the periodic tracing of figural forms, and compared both with experiment and the twothirds power law. The aim of the analyses is to generalize previous numerical observations as well as to examine movement segmentation. It is first shown that contrary to recent statements in the literature, the only members of this family of control theories that match reaching movement experiments well are minimum jerk and snap. Then, for the case of periodic drawing, both the ellipse and cloverleaf are examined and the experimentally observed power law is derived from a firstprinciples approach. The results for the ellipse are particularly general, representing a unification of the twothirds power law and smoothness hypotheses for ellipses of all reasonable eccentricities. For complex shapes it is shown that velocity profiles derived from the costfunction approach exhibit the same experimental features that were interpreted as segmented control by the CNS. Because the cost function contains no explicit segmented control, this result casts doubt on such an interpretation of the experimental data.
Movements of the human body are restricted both by mechanics and computational resources. However, the repertoire of movements that is actually observed comprises a small part of those that are possible, because the CNS uses planning strategies that impose further restrictions. The determination of these strategies is nontrivial because they are not measurable directly. The approach has been therefore to observe the behavior under various conditions and to construct mathematical theories that describe the output of the motor system. The past few decades of experimentation have lead to a number of (apparently) different hypotheses for planning strategies; for example, it has been proposed that the CNS plans in the coordinate system of the joint angles (Uno et al., 1989;Nakano et al., 1999) or in the coordinates of the hand's position (Morasso, 1981; Flash and Hogan, 1985) or that the CNS plans movements that are robust against the inherent noise in the motor system (Harris, 1998;Harris and Wolpert, 1998).
A successful description of an experiment by a theory is usually taken as evidence that the CNS does indeed operate in such a manner. But what if two different theories have the same predictions? Obviously in this case no conclusions can be firmly drawn until further experiments are performed, outside the region of agreement. However, hypotheses can also be compared mathematically to reach an understanding of their interrelation. In this paper such a comparative study is undertaken with respect to two simple, but widely applicable theories: the optimization of smoothnessbased cost functions and the twothirds power law between curvature and velocity. Predictions of these control hypotheses will be examined for two standard experimental paradigms: pointtopoint reaching movements and the periodic tracing of simple figures. For the case of pointtopoint movements it is shown that only a few members of the family of smoothnessbased cost functions can match experiments well. For the case of the periodic drawing of figural forms, a number of novel results will be derived. A mathematical form for the exponent relating hand velocity to curvature of the twothirds power law will be obtained from the cost functions for an ellipse and shown to be indistinguishable from the experimental value of ∼0.33 (the name “twothirds” power law comes from the original formulation in terms of curvature and angular velocity). This result holds for all practical eccentricities, generalizing and formalizing previous numerical evidence obtained for specific examples (Wann et al. 1988; Harris and Wolpert, 1998). As has been pointed out previously, the ellipse is a special case with respect to the twothirds power law because it implies coupled harmonic motion. For this reason, the same analysis was performed for the cloverleaf, and again, an exponent was found that was in good agreement with experiment. Finally, the case of movement segmentation in complex shapes is examined. The velocitycurvature relationship is derived from the smoothnessbased cost function for a figureofeight. It is shown that the same features of segmentation seen in experiment are seen in the mathematically derived velocity profile. Because the cost function contains no segmented control, this suggests that this apparent segmentation is an epiphenomenum of smooth movements, in agreement with the experimental work of Sternad and Schaal (1999).
MATERIALS AND METHODS
In this section, the two motorcontrol hypotheses to be studied will be defined mathematically, and the notion of segmentation with respect to the twothirds power law will be reviewed.
The mean squared derivative cost functions. Each member of this family of motionplanning hypotheses is a generalized version of the minimumjerk approach used initially to model velocity profiles generated by elbow movements (Hogan, 1984) and later extended to trajectory prediction for reaching movements between visual targets in the horizontal plane and to curved and obstacleavoidance movements (Flash and Hogan, 1985). Since then, the minimumjerk hypothesis has been applied to a great many other motor tasks, including the drawing of complicated figural patterns (Viviani and Flash, 1995; Todorov and Jordan, 1998) for which a numerical approach was used to obtain velocity profiles in excellent agreement with experiment. The predictions of these mean squared derivative (MSD) hypotheses are derived through the minimization of the time integral of the squared n thderivative of the coordinates of the hand (x(t), y(t)):
For completeness it should be stated that the trajectories derived miss some of the fine detail seen in pointtopoint movements. The paths measured are actually slightly, but systematically, curved depending on their position with respect to the body. This observation has led to further hypotheses (Flash, 1987; Uno et al., 1989; Harris and Wolpert, 1998) that deal with the problem of motor execution and include explicit details of the mechanical properties of the arm. Nevertheless, the simpler cost function of minimum jerk gives a satisfactory prediction of the observed movements and is more than sufficiently accurate to provide the basis for the analyses below.
Given that minimum jerk is just one member of the MSD cost function family (with n = 3), it could be asked if the theoretical velocity profiles derived for other orders give an equally good prediction of experiment. Recently it was stated in the literature (Harris, 1998) that this was indeed the case. If this were true it would of course imply a fundamental problem with MSDgenerated velocity profiles: if they all fit the data equally well there are no criteria for choosing one hypothesis over another. In the first part of Results it will be shown that in fact this is not the case. By calculating the predicted velocity profiles for each order, it will be shown that only minimum jerk (and to a lesser extent minimum snap) is compatible with the experimental data.
The twothirds power law. In the extensive analysis of handwriting and drawing motion, where the trajectories of the hand are curved, it was seen that the velocity is not constant but varies strongly with the instantaneous curvature. It was found (Viviani and Terzuolo, 1982; Lacquaniti et al., 1983) that this phenomena could be expressed in terms of a power law between velocity υ and curvature κ as follows:
An intriguing aspect of the power law is seen in the tracing of extended shapes and also in free scribbling. In such cases a single value of the gain factor g is not sufficient to fit the data. However, a power law between velocity and curvature can still be shown to exist if a piecewise constant gain factor is used. A second and separate issue is that the regions of zero curvature in such shapes are incompatible with the power law in its usual form, and so an altered form involving an effective radius of curvature R* (the standard radius of curvature is R = 1/κ) was also introduced to avoid divergences (Viviani and Stucchi, 1992). Thus the velocity υ, in this case, takes the form:
RESULTS
In this section, explicit mathematical forms for the order n predicted velocity profiles will be given for the two contrasting experimental tasks of pointtopoint and periodic drawing movements.
Reaching movements in the plane
The aim of this section is to analyze the forms of the predicted velocity profiles for each order n of the derivative of the coordinates of the hand. The mathematical form for the n thorder profile will be obtained, and the limit n → ∞ will be examined. It will be shown that the profiles diverge in this limit, and therefore not all MSD profiles can fit the data equally well.
The position vector of the hand is written as r = (x, y). Using the Euler–Lagrange formalism for the functional minimization of the cost function given in Equation 1 yields the following differential equation for the trajectory r as a function of time t:
However, the conclusions reached above need to be reconciled with the results previously reported that stated that the MSD velocity profiles converge to an experimentally more realistic Gaussian function in the large n limit (Harris, 1998). There the velocity profiles found at each order n were rescaled by different ( n dependent) amounts so that their peaks all coincided at the same point and their widths passed through a value of 25% of the normalized peak velocity at the same times. Such a rescaling of the profiles in Figure 1 has been performed, with the results plotted in Figure 2. In this way, it appears that the curves become similar as n takes higher values, with the n = ∞ limit converging to a Gaussian. Nevertheless, because the velocity tails become increasingly large, a cutoff must also be introduced to fix the time of the movement to be T (which is given by experiment). Essentially what that approach amounted to was to treat each order n MSD velocity profile as a curve that can be fitted to experiment by stretching the time axis and compressing the velocity axis. Curvefitting is a valid and useful approach for examining the shape of the experimental curves (Plamondon et al., 1993). However, it is in spirit a very different approach to the derivation of a curve from a hypothesis as was done here and elsewhere (Flash and Hogan, 1985), with a minimal number of free parameters.
Periodic tracing of figural forms
In this section the case of continuous drawing of closed shapes is analyzed. The approach will be to substitute into the MSD cost function, written in Equation 1, a specific path (the template of the shape) and then use a minimization procedure to derive the predicted velocity profile. It is important to note that this approach differs from the one taken in an earlier study (Viviani and Flash, 1995) where the experimental values of the velocity of the hand and acceleration at several viapoints were used in the minimumjerk prediction. By contrast, in the present study the velocity profiles are derived from a firstprinciples approach for general classes of shapes, necessitating minimal experimental input (only the path itself and the tempo). In fact the mathematical approach used here is more closely related to the detailed numerical study used in Todorov and Jordan (1998) in which the jerk was minimized along a prescribed path.
In contrast to the previous experimental task, the motion is periodic, and because of this it is natural to describe the motion in the language of Fourier series. The time dependence of the x and y coordinates of the hand can each be expanded in Fourier series, where for convenience the normalized time τ = t/T is again used (where T is now the period of the motion):
The ellipse
The general parametric equations for an ellipse are:
This equivalence between the two laws only holds for the somewhat abstract limit of n → ∞. It would be more interesting to know what the predictions for the exponent β are from MSD hypotheses with orders that are more closely related to quantities with physical meaning (as was shown for pointtopoint movements) like jerk or snap. In the remainder of this section, the results of such a calculation are presented with the mathematical details given in Appendix . The method used is a perturbative approach whereby the calculation is performed for ellipses that are very close to circles, with a small parameter ε measuring deviation of the ellipse from a circle. After performing this calculation, the powerlaw exponent predicted from the order n MSD cost function can be found to have the following form:
The cloverleaf
In the previous section, the case of an ellipse was analyzed. Although the choice of this shape is ubiquitous in the experimental literature it is, from the theoretical point of view, a special case. This is because choosing the exponent to be exactly
Unfortunately, the form of the cloverleaf used in experiment is far removed from a circle. Because of this, a similar perturbative approach to that used above yields bad results. For example, a general form for a cloverleaf that extrapolates between a circle (ε = 0) and the figure used in experiment (ε = 1) is:
General onesegment shapes
Two specific shapes were tested above. However, it is possible to generalize to an arbitrary onesegment shape. The approach is to write the curvature as a Taylor expansion in a small parameter ε and solve the optimization problem in terms of the derivatives of the curvature with respect to this variable. Details of the method of obtaining this generalized form for the exponent β are given in Appendix .
The drawing of complex figures
In the previous section the cases of the ellipse and the cloverleaf were analyzed. Both of these shapes have a common feature: their curvaturevelocity profiles are well matched by the twothirds power law with a single gain factor. This means that in a log–log plot of the velocity versus curvature only a single segment is seen. In this section the more complex case of a multiplesegment shape is analyzed, specifically the asymmetric lemniscate, or figureofeight with two different sized lobes. Again comparison is made with experimental data, but it should be stressed that here quantitative accuracy is not required, just a qualitative reproduction of the features of segmentation. Because of the complexity of the analysis of multiplesegment shapes, the approach in this case is numerical. An algorithm similar to that described in Todorov and Jordan (1998) was used, in which the time the movement passes through 10 viapoints is varied until the trajectory with the lowest minimumjerk cost is found. The optimal times were obtained using a stochastic descent algorithm, to an accuracy of 0.25% of the total movement duration.
Before examining the results, it is worth underlining again the importance of using a firstprinciples approach in the derivation. In a related paper (Viviani and Flash, 1995), similar shapes were analyzed, but experimental information in the form of velocity and acceleration was also supplied at the via points. The argument put forward here is that the minimization of an integral cost function (e.g. minimum jerk) gives a movement that mimics the segmentation seen in experiment. This argument would be undermined if experimentally derived values of the velocity and curvature for the viapoints were used as they could contain information about possible segmented planning of the movements by the CNS.
The figureofeight
The shape to be analyzed has the following mathematical form:
The numerically derived minimumjerk trajectory is plotted in Figure6, in velocity υ versus (R*)^{1/3} form. In this format, different segments lie on radial lines passing through the origin, with gradients corresponding to the different gain factors g_{L}and g_{S}. The duration of the movement has been scaled to agree with the average experimental period of T = 1.43s. As can be seen, the theoretically derived curve shows qualitatively the same features of segmentation as the experimental data. The β exponents of the power law are not too well defined for the theoretical curve because of the curvature of the two segments (unlike the experimental data, the theoretical curve has not been smoothed, a process which tends to increase the likelihood of measuring an exponent close to 0.33), and, as stressed above the aim here is to show that features of segmentation are seen in theoretical predictions, not to show close quantitative agreement. However, to provide a level of comparison, the tangential dotted lines in Figure 6 (meant as guides for the eye) represent gain factors corresponding to g_{L} = 20.5 and g_{S} = 18.4, which implies a ratio g_{L}/g_{S} = 1.1 (the choice here of using tangents, rather than a fit, tends to underestimate the value of this ratio). Though this value is about 10% smaller than the average found experimentally (∼1.2, the corresponding values of g_{L} = 21 and g_{S} = 17 are represented as dashed lines in Fig. 6), it is not inconsistent with the range of experimental values seen across the three subjects.
The theoretically generated trajectory captures all the qualitative features of segmentation, despite the absence of any piecewise algorithm in the minimumjerk hypothesis. In fact even the hooklike feature (corresponding to the region of the trajectory between the curvature maxima M2 and M3 via the curvature minimum m1) can be seen in a detailed analysis of the experimental data. In examining this feature, it becomes clear what criteria determine the apparent segmentation of the velocity versus curvature plots: it is the extrema of the curvature and the symmetries of the shape. Thus, the segmentation features seen in such loglog plots can be directly accounted for by the geometric aspects (curvature profile) of the drawn path: segmented planning in the CNS need not be assumed.
DISCUSSION
A family of movementplanning strategies for the human arm was analyzed in mathematical detail and compared with experiment, with a number of results derived. The planning strategies are defined through the minimization of cost functions that favor smoothness, to differing degrees, and are known collectively as the MSD cost functions. Two standard and contrasting motor tasks were analyzed: pointtopoint reaching movements and the periodic tracing of closed shapes. For the first task of pointtopoint movements the predicted velocity profile from each order n of the MSD family was derived explicitly and compared with experiment. By comparing the ratio of the peak to average velocities it was concluded that the best fit (from the MSD family) was by the minimumjerk cost function, corresponding to order n = 3. It was further shown that as the order n was increased the corresponding velocity profile diverged, reaching the highly unphysical limit of a Dirac delta function when n → ∞. This result corrects conclusions previously reached in the literature (Harris, 1998). However, it is acknowledged here that this was not the main point of that paper, and therefore the current work should not be mistaken as a criticism of the minimumvariance principle, for which the paper of Harris (1998) laid the foundation. Rather, the aim here was to clarify the mathematical understanding of the smoothnessbased cost functions.
The theory of minimum variance was applied to both arm and eye movements and may seem, at first view, to be at odds with the smoothnessbased cost function approach. However, there are many overlaps between the two theories. An obvious example is the independence of the velocity profiles for fast movements from physical parameters of the arm. This suggests an interpretation of the minimumjerk theoretical framework as a simplified limit of a more complete theory. As has already been noted, minimumjerk predictions for pointtopoint movements miss some of the fine structure of the trajectories. Experimentally measured trajectories show a small but systematic curvature, a feature captured by other models that take into account both the problems of trajectory planning and motor execution (Flash, 1987; Uno et al., 1989;Harris and Wolpert, 1998). In this context, it would be very interesting to examine theoretically the similarities between the MSD cost functions and other more detailed hypothesis such as the minimumvariance hypothesis (Harris and Wolpert, 1998) or the optimal feedback control hypothesis of Todorov (2001). In the latter work, it is demonstrated that greater control of a noisy goaldependent variable can be obtained at the expense of allowing variance to increase in redundant degrees of freedom. In their formulation, features such as simplifying rules, control parameters, and synergies emerge as epiphenomena from the optimal feedback control.
Several neural implementations have been proposed for the minimumjerk cost function (Hoff and Arbib, 1992; Jordan et al., 1994) and neurophysiological studies have shown that the fine kinematic details of the movements are represented within cortical neuronal populations (Schwartz and Moran, 1999). Nevertheless, given the present level of knowledge of the neural correlates of movement and also the success that different modeling approaches have in describing the same psychophysical experiments (as was shown here in the context of minimumjerk and the twothirds power law), it is clear that the data are not yet sufficiently fine to restrict the space of the possible neurally based algorithms that might generate the observed behavior. As an example, the optimal feedback hypothesis produces movements that are smooth, and in some limits may be reproduced by the minimumjerk predictions, but the underlying structure of the controlling system is very different.
The second experimental task analyzed was the periodic drawing of closed shapes. In contrast to pointtopoint movements the order n velocity profile was shown to converge to a smooth function as n was increased. For the limit n → ∞ it was shown that the predicted velocity profile is the one for which the Fourier series for the trajectory of the hand has the lowest values in the highfrequency range. For the specific examples of an ellipse and cloverleaf it was shown that the powerlaw exponent derived from a MSD cost function, with any order greater than minimum acceleration, gives values that are indistinguishable from the accepted value of β ≃ 0.33. Further comparison was then made between the two theories for the case of a complex figural form (a figureofeight with different sized lobes). The velocity profile derived from the MSD smoothnessbased cost functions was seen to have features that implied segmented control. However, the cost function contains no explicit segmented control. Therefore, the conclusion must be that these types of experimental features can no longer be taken as evidence for segmented control in the CNS, in the absence of further supporting evidence.
Despite the success of the smoothnessbased cost function in fitting different aspects of the twothirds power law, there are still many questions unanswered. In earlier studies several different reasons and explanations for the observed twothirds power law were offered. These reasons included either the mechanical properties of muscles (Gribble and Ostry, 1996), acceleration, or noise constraints (Harris and Wolpert, 1998) or the coupling between oscillatory joint rotations (Sternad and Schaal, 1999). The analysis performed by Schwartz (1994)and more recently by Schwartz and Moran (1999), has shown that the twothirds power law fits neural trajectories derived by using a population vector analysis of the neural activity data recorded from motor cortical cells in monkeys during drawing movements. The fact that the twothirds power law has been found in motor cortical representations of drawing movements suggests that this relation is manifested in the planning stages of hand trajectories (Schwartz and Moran, 2000), consistent with the idea that central neural representations have evolved or are acquired with age (Viviani and Schneider, 1991) or through learning to achieve the smoothest and/or the least variable movement. Nevertheless, explanations for the twothirds power law based on central mechanisms or on peripheral factors, such as the mechanical properties of muscles or neuromotor noise, may not be mutually exclusive, and both peripheral and central factors might have evolved to work in tandem to guarantee the smoothest possible or most accurate movement. Other questions of interest that relate to the work presented here concern the nature of the relation between the twothirds power law and visual perception of motion. There is compelling evidence from a number of studies that the twothirds power law is related to both the motor production and visual perception of movement (Pollick and Sapiro, 1996;de'Sperati and Viviani, 1997). Such connections were not examined here and represent an interesting case for further study.
Another significant topic for future study that was addressed here is that of movement segmentation. Many earlier hypotheses concerning the nature of the underlying movement segments have been made (Krebs et al., 1999; Doeringer and Hogan, 1998). Among them, one of the most compelling hypotheses, which attempted to suggest a criterion for motion segmentation, is that based on the existence of a piecewise constant gain factor in the relation between velocity and curvature (Lacqaniti et al., 1983). Using a similar approach, the boundaries of movement segments during drawing movements were defined kinematically as places in the trajectory where speed is maximal (Schwartz and Moran, 1999). These are points at which the velocity gain factor in the power law changes instantaneously, where curvature is minimal, and where curvature inflections also occur. The conclusion, based on the mathematical analysis presented here is that the presence of sharp changes in the velocity gain factor does not necessarily suggest a noncontinuous mode of motion planning and therefore cannot provide sufficient evidence for segmented control [supporting the findings reached in the experimental study of Sternad and Schall (1999)]. This conclusion does not undermine the idea that complex movements are constructed from simpler units of action or movement segments that are then concatenated together or are temporally overlapped to generate longer movement sequences. It does, however, raise the need for new ideas concerning the nature of the underlying movement segments or strokes.
Appendix
This appendix contains the details of the calculation leading to the predicted exponent for the power law of the ellipse, Equation 10, as derived from the general order n MSD cost function. One route to obtaining the exponent β_{n} of the power law is to substitute the velocity form given in Equation 2 into the integral (Eq. 1) and change the integration variable from time to arclength. This can be done directly for very small values of n by using the Frenet–Serret formulation for curves (see Appendix ). However, to get general results for arbitrary order, a less direct approach is necessary. The route taken to find β_{n} is to obtain the Fourier coefficients for the cost function (Eq. 7) in terms of the unknown exponent β. Once this has been done it is straightforward to minimize C_{n} by differentiation with respect to the exponent.
The first step is to write the coordinates of the hand (quite generally) as parametric functions of a time dependent angle θ(τ):
What remains now is to obtain the Fourier coefficients in Equation 6 of the x and y coordinates. This is achieved by substituting Equation 16 with the results (Eq. 17) into Equation 14, for example:
Appendix
The first step is to rewrite the cost function (Eq. 1) with n = 3 for minimum jerk in terms of velocity and curvature. This step can be done conveniently by using Frenet's relations, see for example Todorov and Jordan (1998). The result is:
Again, perturbations from a circle to a shape of interest are considered. By symmetry the extremizing velocity is a constant for a circle, and hence this particular example is easy to solve. It is assumed that there exists some parameter ε that measures how close the shape in question is to a circle, for which ε = 0. The curvature and differential ξ can be expanded in terms of this parameter:
Because no specific values of the curvature or differential ξ have been specified, the method described here is quite general. After some algebra, the minimizing β can be found and has the following value:
Footnotes

This work was supported in part by the Minerva Foundation (Germany), by a grant from the Israeli Ministry of Science, Culture, and Sport, and by the Moross Laboratory. M.J.E.R. acknowledges the support from a Feinberg research fellowship.

Correspondence should be addressed to Magnus J. E. Richardson, Laboratoire de Physique Statistique, Ecole Normale Superieure, 24 Rue Lhomond 75231, Paris Cedex 05, France. Email:Magnus.Richardson{at}lps.ens.fr.