Human Movement Science 21 (2002) 881–904 www.elsevier.com/locate/humov

A simulation study of the degrees of freedom of movement in reaching and grasping Jianming Jiang, Yueshi Shen, Peter D. Neilson

*

Neuroengineering Laboratory, School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney 2052, Australia

Abstract The question of independently controlled components in the act of reaching and grasping has attracted interest experimentally and theoretically. Data from 35 studies were recently found consistent with simulated kinematic finger and thumb trajectories optimised for minimum jerk. The present study closely reproduces those trajectories using a discrete-time model based on minimum acceleration. That model was further used to generate two-dimensional trajectories for finger and thumb to reach and grasp an elliptical object with varying position and/or orientation. Orthogonalisation of these four trajectories revealed one degree of freedom when direction of reach was constant and two degrees of freedom when direction of reach varied, irrespective of object distance and orientation. These simulations indicate that reach and grasp movements contain redundancy that is removable by formation of task-dependent synergies. As skilled movement can be planned and executed in a low dimension workspace, control of these independent components lessens central workload. Ó 2002 Elsevier Science B.V. All rights reserved. PsycINFO classification: 2330 Keywords: Motor coordination; Degrees of freedom; Synergy; Movement planning; Trajectory simulation

*

Corresponding author. Tel.: +61-2-9385-4024; fax: +61-2-9385-5993. E-mail address: [email protected] (P.D. Neilson).

0167-9457/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-9457(02)00164-1

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1. Introduction 1.1. Background Reaching and grasping is a complex movement involving control of some 58 muscles and some 25 kinematic degrees of freedom of the fingers, thumb, wrist, forearm, elbow, shoulder and scapula. The kinematic description of reaching and grasping suggests that transporting the hand, orienting the wrist and preshaping the fingers into an appropriate grip each involves sets of muscles and joints involving different control modalities. These observations were the basis for the hypothesis of visuomotor channels (Arbib, 1981; Jeannerod, 1981; see also Jeannerod, 1999). It was proposed that each component of reaching and grasping acts as a separate identifiable system characterised by its own input and output and its own intrinsic mechanisms, a point of importance to the present study. Jeannerod advocated two visuomotor channels, one controlling the transport of the hand, the other the size of the grip. Not only was this a convenient way of describing the data but, as Arbib emphasised, the two variables could be regarded as the ones that are actually controlled during reaching and grasping. This ‘‘classical approach’’ led to tremendous development in research, much of which is delineated in Smeets and Brenner (1999) and accompanying commentaries. Smeets and Brenner pointed out that the main reason for the classical description being so attractive lay in ‘‘the two channels correspond[ing] nicely to two distinct anatomical structures and two distinct types of perceptual information’’ (p. 237). They went on to show, however, that this interpretation becomes questionable when the description is formalised according to their alternative approach. They put forward a ‘‘new view’’ in which the trajectories of fingers and thumb are based on the requirements of the grasp. For instance, they state that for a stable grasp, the fingers should be placed at positions on the objectÕs surface in such a way that the line connecting the fingers is perpendicular to the surface on both sides and goes through (or above) the centre of gravity of the object. They note further that the accuracy of digit placement must be highest when the object is heavy or slippery. Thus according to this approach the planning of how to grasp an object begins by determining suitable positions for the fingers and thumb on the objectÕs surface. This is not entirely a new view on grasping. The review by Jeannerod (1999) reveals that a similar idea emerged from his laboratory some years before. In an experiment in which subjects grasped cylindrical objects Paulignan, MacKenzie, Marteniuk, and Jeannerod (1991) noticed that the variability of the spatial paths of index finger and thumb sharply decreased as the digits approached the object, suggesting that the digits were aiming at a predetermined locus on the surface of the object. Similarly, the experiment of Paulignan, Frak, Toni, and Jeannerod (1997) was founded on the hypothesis that, because the digits represent the effector of the reach and grasp movement, their position on the object at the end of movement should be the main parameter to be controlled to achieve an efficient grasp. The important contribution of Smeets and Brenner (1999) was their exploration of the endpoint placement hypothesis by means of kinematic modelling. Their sim-

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ulation started with the assumption that the placements of fingers and thumb on the object were already determined. Thus they focused not on the cognitive processing that precedes this determination, but on the sensorimotor planning of the trajectories that take the fingers and thumb from their initial positions to their specified final positions on the object. Wishing to avoid any consideration of the biomechanics of limbs and joints, they based their simulations on the formulation of trajectories in terms of the kinematics of the end-effector. To do this they chose the minimum jerk model developed by Flash and Hogan (1985) which provides an optimisation with constraints at the beginning and end of the movement. The parameters of a minimum jerk trajectory are the movement time, the initial and final positions, and the velocity and acceleration at the beginning and end of the movement. By setting the endpoint acceleration to be a nonzero deceleration perpendicular to the surface of the object, Smeets and Brenner defined an ‘‘approach parameter’’ that determines the way the trajectory approaches the surface, the larger the deceleration the more perpendicular the approach. Minimum jerk trajectories were computed independently for each digit to move, in a specified time, from its initial position (where it has zero velocity and acceleration) to its predetermined final position on the object surface (where it has zero velocity and a specified finite deceleration in the direction perpendicular to the surface). Smeets and Brenner found that their simulations reproduced most of the kinematic features of reaching and grasping observed experimentally. Specifically, their model generated the following predictions: (1) the transport component is independent of object size, (2) the grip size (maximum aperture of grip) is independent of the object distance, (3) the grip size increases and occurs later for larger objects, and (4) the grip size increases and occurs earlier as the approach parameter increases. They compared features of their trajectories with the experimental results from 35 studies published between 1988 and 1997 and showed that the model reproduced these very well. The authors found predictions (1) and (2) to be particularly interesting since they imply that the model generates grasping behaviour apparently based on two independent visuomotor channels, one for transporting the fingers and one for shaping the grip. In other words, the model predicts exactly the experimental results (Jeannerod, 1981, 1984) that were the basis of the classical hypothesis of two independent visuomotor channels in the brain. Yet the simulations were based not on independent visuomotor channels for transport and grip but on the independent planning of minimum jerk trajectories for each finger and thumb. Aware of subsequent experimental findings that potentially questioned the classical view Smeets and Brenner refrained from claiming that their model demonstrated independent control of any of the variables concerned. Rather, they argued that their model shows how easily one can be deceived, concluding that the independent behaviour of variables in motor control does not necessarily mean that these variables are controlled independently. We have long been interested in the question of the independent control of variables in skilled human performance. Indeed this notion is central to the concept of synergy and the long-discussed ‘‘degrees of freedom’’ problem stemming from the work of Bernstein (1967). The formation of task-dependent synergies is a key part of our general formulation of sensorimotor control, known as adaptive model

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theory (AMT). In a commentary on the Smeets and Brenner study (Neilson, 1999) we pointed out that if the nervous system were to plan optimal trajectories in threedimensional space for each of the five digits, then the act of a single reach and grasp would require the generation of 15 optimal trajectories. This, we suggested, imposes an unnecessarily large workload on the central processes involved in the planning and execution of reaching and grasping movements. We hold that the nervous system can reduce the computational workload by the formation of task-dependent motor synergies that effectively reduce the dimension of the workspace in which the optimal trajectories are generated. The process of synergy formation is a start point for other key aspects of AMT. We argue that during acquisition of motor skills (reaching and grasping being one example), the nervous system forms a task-dependent, adaptive, feedback/feedforward controller (Neilson & Neilson, 2001b; Neilson, Neilson, & OÕDwyer, 1997). Such a controller can render unobservable in the skilled response most, if not all, of the complexities of the neuromuscular and biomechanical systems. In other words, once the appropriate controller is acquired for a task, the planning of required response trajectories no longer demands computations to account for the interactive nonlinear dynamics of the muscle control systems (including tension and length reflex loops), for the multijoint biomechanical loads on those muscles, nor for the complex interactions between muscle tensions and body movements (caused by inertial, viscous, coriolis and gravitational reaction forces within the biomechanical system). All these can be compensated by cross-couplings within the controller once the task-dependent system has been ‘‘learned.’’ As a consequence the task proceeds as if each of the (reduced) degrees of freedom in the response is controlled independently, each degree of freedom of movement having its own ‘‘private channel’’ characterised by its own input and output and its own intrinsic dynamics. Thus AMT provides a conceptualisation of independent sensorimotor channels that has evolved separately from both the classical and the new view on reaching and grasping. By revisiting and extending the modelling of Smeets and Brenner using simulation methods already established in AMT, we seek to clarify the issues that emerge from those two lines of work. 1.2. Aim In this paper we aim to explore the question of independent components in reaching and grasping by means of a degrees of freedom analysis of simulated finger and thumb trajectories in which the position, direction and orientation of the object are varied from reach to reach in various combinations. While this study uses the work of Smeets and Brenner as a departure point, the simulations reported here differ from theirs in two important ways. Firstly, we use trajectories that are optimised by a minimum acceleration criterion rather than a minimum jerk criterion. Secondly, these trajectories are generated by means of discrete-time techniques rather than by continuous-time techniques. These differences reflect modelling choices that we have used successfully many times previously in the development of AMT. We justify these choices briefly below.

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1.3. Minimum acceleration vs. minimum jerk Smeets and Brenner modeled their description using the minimum jerk approach of Flash and Hogan (1985). According to Flash and Hogan, reaching movements are planned as optimally smooth kinematic trajectories. This is achieved by minimising the rate of change of acceleration of the planned trajectories. The rate of change of acceleration (i.e., the third derivative of position) is known as jerk and hence these optimally smooth trajectories are referred to as minimum jerk trajectories. We agree with Flash and Hogan that, in general, motor responses are planned as optimally smooth trajectories (see Neilson, Neilson, & OÕDwyer, 1992). However, in AMT simulations we choose to use minimum acceleration trajectories rather than minimum jerk trajectories for the following reasons. Firstly, in a study carried out in this laboratory (Gibson & Neilson, 1999), visual pursuit tracking responses were simulated using the same trajectory generator described below. Unlike the Flash and Hogan simulations, this implementation allowed intermittent revision of the trajectory during the approach to the target. For comparison, simulations were run for movement to both a fixed and a moving target using both the minimum jerk and the minimum acceleration criterion. For the fixed target, minimum acceleration and minimum jerk trajectories were only marginally different whereas for the moving target, the minimum acceleration criterion produced noticeably smoother trajectories than did the minimum jerk criterion. This was because, when reaching to a moving target using a minimum acceleration trajectory, the position and velocity of the target have to be predicted ahead in time whereas, when planning a minimum jerk trajectory not only do the position and velocity have to be predicted but also the acceleration. Accelerations are known to be more difficult to predict than position and velocity (Neilson, 1993; Poulton, 1974). Consequently, the tracking responses based on the minimum jerk were considerably bumpier than those based on minimum acceleration because the minimum jerk responses included intermittent corrections for acceleration prediction errors. Secondly, for a predominantly inertial system like the arm and hand, minimum acceleration trajectories correspond to minimum energy trajectories. We have written previously about the importance of minimum energy in motor control (Neilson & Neilson, 1999; OÕDwyer & Neilson, 2000). As a general principle within AMT we hold that any organism that can escape predators, capture prey and achieve goals with a minimum demand on metabolic energy has an evolutionary advantage and that consequently, minimum energy solutions to optimisation problems are common throughout the animal world. 1.4. Discrete time vs. continuous time The procedure used in this study for computing minimum acceleration trajectories is based on a zero-order-hold (ZOH) discrete-time model whereas the Flash and Hogan procedure for computing minimum jerk trajectories is based on a continuous-time model. From consideration of both the organisation of cortical columns (Mountcastle, 1978, 1997, 1998), their bursting activity (Braitenberg, 1986; Shaw & Silverman, 1988; von Seelen, Mallot, Krone, & Dinse, 1986; see also

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Sardesai et al., 2001) and the nature of the cortico-cortical and cortico-subcortical connectivity between cortical columns and subcortical structures (Darian-Smith, DarianSmith, & Cheema, 1990; Eccles, 1984; Goldman-Rakic, 1987; Powell, 1981) we argue strongly for the applicability and validity of discrete-time models of neural information processing rather than those based on continuous time. Moreover, it is commonly accepted in neurophysiological practice that the output of an ensemble of neurons, such as a pool of alpha motor neurons driving a muscle, is appropriately measured by the total number of impulses over a brief time window, often set at 50 ms. Consequently, we model the activity of an ensemble of neurons in the same way as it is measured experimentally, that is, by a staircase waveform that represents the average level of activity over each time window and resets its level at periodic (time window) intervals. Such staircase signals are well known in digital signal processing and are referred to as ZOH signals. We note that ZOH signals are necessarily an approximation to the actual variation in the instantaneous level of activity. Their use is readily justified, however, not only by the experimental parallel just mentioned but also by the availability and wide acceptance of powerful discrete-time signal processing algorithms developed for their analysis (e.g., Ogata, 1995).

2. Method 2.1. Trajectory generation The basic algorithm employed in our simulation to generate optimal minimum acceleration trajectories has been described in detail elsewhere (Neilson, Neilson, & OÕDwyer, 1995) and is given briefly here. We write the ZOH discrete-time equivalent state equation: xðk þ 1Þ ¼ GxðkÞ þ HuðkÞ for a double integrator system, where

 1 0:05 0:00125 G¼ ; H¼ ; 0 1 0:05




 hðkÞ xðkÞ ¼ _ ; hðkÞ

uðkÞ ¼ h€ðkÞ:

The aim is to
derive initial  the optimal trajectory xðkÞ to move between
a specified  hð0Þ hðN Þ state xð0Þ ¼ _ at time zero and a specified final state xðN Þ ¼ _ at time N hð0Þ hðN Þ PN 1 2 so that the performance function J ¼ ð1=N Þ k¼0 u ðkÞ (i.e., mean square acceleration) is minimised. The sample frequency is constant at 20 Hz and the sample number is represented by k. The sample time is computed by multiplying the sample number k by the sample interval 0.05 s. The solution to this optimisation problem is given by xðkÞ ¼ Gk xð0Þ þ Cð0; kÞC1 ð0; N ÞfxðN Þ  GN xð0Þg Pk1 where Cð0; N Þ ¼ j¼0 Gj HHT GTðN hjÞ is a discrete-time Grammian.

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Using the software package MATLAB (Version 5.3, The Mathworks, 1999), a subroutine named optimum trajectory generator (OTG) was written to implement the above algorithm. Given a duration N , an initial state xð0Þ and a final state xðN Þ, the OTG subroutine generates the optimal (i.e., minimum acceleration) trajectory of duration N to move from the specified initial position and velocity to the specified final position and velocity. The duration, N , was set to 100 samples (i.e., 5 s at 20 samples/s). 2.2. Trajectories for reaching and grasping The OTG was employed to simulate the following reaching and grasping task in a plane. Imagine an elliptical object (say the handle of a cup) that can be placed anywhere in a plane (as illustrated in Fig. 1). The task for the simulator is to generate the optimal trajectories in time and space for the finger and the thumb to move from their initial state to the final state (i.e., positions and velocities) corresponding to a grasp of the object with the final velocities aligned with the minor axis (as shown in Fig. 1). The size of the final velocity is prespecified and relates to the strength of the grip. We refer to this final velocity as the approach parameter since it serves the same purpose as the final deceleration and approach parameter employed by Smeets and Brenner. The simulator generated the following four trajectories in time: 1. 2. 3. 4.

yf ðkÞ: Trajectory of finger in y-direction as a function of time. yt ðkÞ: Trajectory of thumb in y-direction as a function of time. xf ðkÞ: Trajectory of finger in x-direction as a function of time. xt ðkÞ: Trajectory of thumb in x-direction as a function of time. y Elliptical object Minor axis of ellipse Major axis of ellipse

β Final velocity along minor axis of ellipse

Thumb trajectory

Finger trajectory

a

α

O (Origin equals start position for finger and thumb)

x

Fig. 1. Schematic diagram for the simulation of finger and thumb trajectories for reaching and grasping in a plane. The direction of the reach is defined by the angle a and the distance and orientation of the object are defined by the distance a and the angle b respectively.

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Six parameters were used to define the possible positions of the elliptical object in the plane, the width of the object, the strength of the grasp and the duration of the movement: a a b w v

N

distance to the centre of the elliptical object (varying from 1 to 100 units), angle between the y-axis and a line drawn from the origin to the centre of the ellipse (i.e., direction of reach) varying from p=2 to þp=2 rad, angle between y-axis and major axis of ellipse (i.e., orientation angle of object) varying from p=2 to þp=2 rad, size of the minor axis of the ellipse (i.e., the width of the object) varying from 0 to 8 units, final grasping velocity perpendicular to the surface of the object along the direction of the minor axis (i.e., approach parameter) varying from 1 to 5 units/s, discrete-time number representing the total duration of the movement. N ¼ 100 and sample interval ¼ 0:05 s for all simulations in this study (i.e., movement duration was constant at 5 s).

2.3. Minimum acceleration vs. minimum jerk comparisons To illustrate the similarities and differences between minimum acceleration and minimum jerk trajectories we generated position and velocity waveforms for (i) a minimum acceleration trajectory using the OTG algorithm described above and (ii) a minimum jerk trajectory using an equivalent OTG algorithm for minimum jerk implemented in the previously mentioned Gibson and Neilson (1999) study. The trajectories moved between the same prespecified initial and final positions in the same time. Displacement and velocity waveforms were computed to facilitate comparison of the two optimisation techniques. To compare finger and thumb reaching and grasping trajectories derived from a minimum acceleration model to those of Smeets and Brenner derived from a minimum jerk model we generated two families of finger and thumb trajectories using the method described in the previous section. In the first family of minimum acceleration trajectories the position, direction, orientation and width of the object were held constant (a ¼ 20, a ¼ 0, b ¼ p=4, w ¼ 4) but the approach velocity was varied (v ¼ 1, 2, 3, 4, 5). The positions and tangential speeds of the finger and thumb were computed for each set of parameters. These simulations allow comparison with the similar family of minimum jerk trajectories published by Smeets and Brenner in their Fig. 4 (1999, p. 246). In the second family of minimum acceleration trajectories the position, direction, orientation and approach velocity were held constant (a ¼ 20, a ¼ 0, b ¼ p=4, v ¼ 2), but the width of the object was varied (w ¼ 0, 2, 4, 6, 8). The positions and tangential speeds of the finger and thumb were computed for each set of parameters. These simulations allow comparison with the similar family of minimum jerk finger and thumb trajectories published by Smeets and Brenner in their Fig. 5 (1999, p. 247).

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2.4. Determining the number of degrees of freedom A series of eight simulation experiments was carried out to determine the number of degrees of freedom of movement contained in finger and thumb trajectories generated according to differing constraints. Twenty sets of trajectories were generated in each experiment according to the method of Section 2.2 and these were stored for the analysis described below. In each experiment certain parameters were varied arbitrarily in their defined range while the remaining parameters remained constant. In Experiment 1, only the distance a was varied from one trajectory to the next. In Experiment 2, only the direction angle a was varied. In Experiment 3 only the orientation angle b was varied. In Experiment 4, both the distance a and the orientation angle b were varied. In Experiment 5, both the distance a and the direction angle a were varied. In Experiment 6, both the direction angle a and orientation angle b were varied. In Experiment 7, the distance a, direction angle a, and orientation angle b all were varied. Finally, in Experiment 8, both the direction angle a and the orientation angle b were varied such that a always equaled b. In all these simulation experiments the object width and the approach velocity were kept constant at w ¼ 2 and v ¼ 2 respectively. The constant or varying values of a, a and b for each experiment are given in the results. A MATLAB program was written to analyse the data generated in each simulation experiment. In each experiment data from the 20 repetitions were concatenated to give four signals yf ðkÞ, yt ðkÞ, xf ðkÞ and xt ðkÞ. The analysis program determined the number of degrees of freedom in these four signals within each experiment. This was done by extracting a set of orthogonal feature signals by means of a Gramm– Schmidt orthogonalising procedure (Noble & Daniel, 1988) using a fifth-order polynomial regression analysis, as illustrated in Eq. (1) and Fig. 2. The MATLAB commands Polyfit.m and Polyval.m were used in this procedure. The higher the order of the polynomials used in fitting a relationship the better the fit and the smaller the residual error signals. Extensive trials were carried out using first-, second-, third-, fifth- and 10th-order polynomials. At least a third-order polynomial was required to describe the static nonlinear relationships between trajectories. While higher order polynomials reduced the variance of the residual error signals, the number of degrees of freedom determined on the basis of the 10% criterion described below was always the same for third-, fifth- and 10th-order polynomials. Results from the fifth-order polynomial analysis are presented. The operators rij in Fig. 2 correspond to fifth-order polynomial regressions of the type shown in Eq. (1). ð1Þ

ð2Þ

ð3Þ

ð4Þ

ð5Þ

yt ðkÞ ¼ r21 yf ðkÞ ¼ h21 yf ðkÞ þ h21 yf2 ðkÞ þ h21 yf3 ðkÞ þ h21 yf4 ðkÞ þ h21 yf5 ðkÞ; ð1Þ

ð2Þ

ð3Þ

ð4Þ

ð5Þ

xf ðkÞ ¼ r31 yf ðkÞ ¼ h31 yf ðkÞ þ h31 yf2 ðkÞ þ h31 yf3 ðkÞ þ h31 yf4 ðkÞ þ h31 yf5 ðkÞ; xt ðkÞ ¼ r41 yf ðkÞ ¼

ð1Þ h41 yf ðkÞ

þ

ð2Þ h41 yf2 ðkÞ

þ

ð3Þ h41 yf3 ðkÞ

þ

ð4Þ h41 yf4 ðkÞ

þ

ð1Þ

ð5Þ h41 yf5 ðkÞ:

This equation defines the operators ri1 shown in the first stage of Fig. 2 and similar equations can be written for all the stages of the Gramm–Schmidt orthogonalising

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y f (k )

r21

r31

Y f (k )

Y f (k )

Y f (k )

Yt ( k )

Yt ( k )

r41 Input signal

Yt ( k )

yt ( k )

r32

r42 Input signal

x 'f ( k )

X f (k )

x f (k )

X f (k )

r43

xt' ( k )

xt ( k ) Stage 1

xt'' ( k )

Stage 2

X t (k ) Stage 3

Fig. 2. Schematic diagram of the Gramm–Schmidt orthogonalisation of the two-dimensional finger and thumb trajectories generated from the discrete-time minimum acceleration model. The procedure and symbols are described in Section 2.4 of the text.

procedure. At each stage of the procedure, the signal with the largest variance was chosen as the input signal (see Fig. 2). For example, in stage one in Fig. 2, yf ðkÞ is the input signal. The nonlinear relationships between yf ðkÞ and yt ðkÞ, xf ðkÞ and xt ðkÞ were computed (i.e., fifth-order regressions r21 , r31 and r41 respectively) and the influence of yf ðkÞ subtracted from the signals yt ðkÞ, xf ðkÞ and xt ðkÞ. At each stage, if the variance of an output signal decreased to less than 10% of the variance of the corresponding input signal, it was considered to be negligible, set to zero and effectively dropped from the subsequent stages. The orthogonalised signals at the output of the procedure are represented by upper case symbols Yf ðkÞ, Yt ðkÞ, Xf ðkÞ and Xt ðkÞ in Fig. 2. The number of nonzero orthogonalised output signals indicates the number of degrees of freedom in the four input signals.

3. Results 3.1. Minimum acceleration vs. minimum jerk comparisons Fig. 3 presents position and velocity waveforms for minimum acceleration and minimum jerk trajectories. The upper part of Fig. 3 shows that the minimum acceleration trajectory connecting a specified initial position to a specified final position in a specified time closely resembles the minimum jerk trajectory connecting the same positions in the same time. The small differences in the trajectories can be seen more

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Minimum Acceleration and Minimum Jerk Trajectories Displacement

20 15 10

Min Accel = dashed line Min Jerk = dotted line

5 0

0

0.2

0.4

0.6

0.8

1

0.8

1

Relative Time

Velocity

60 40 Min Accel = dashed line

20 0

0

Min Jerk = dotted line

0.2

0.4

0.6

Relative Time Fig. 3. Comparison of minimum acceleration and minimum jerk trajectories simulated for movement from the same initial position to the same final position in the same time. The upper diagram shows the displacement waveforms and the lower diagram shows the corresponding velocity waveforms.

clearly in the velocity waveforms in the lower part of Fig. 3. At the beginning of the movement the velocity of the minimum jerk trajectory increases more slowly than does the velocity of the minimum acceleration trajectory. In other words, the initial acceleration for a minimum jerk criterion is smaller than that for minimum acceleration criterion. The rate of change of velocity for minimum jerk then increases while that for minimum acceleration decreases. Consequently, the velocity for minimum jerk overtakes that for minimum acceleration. The velocity reaches its maximum at the mid-position for both trajectories but the maximum velocity for minimum jerk is greater than that for minimum acceleration. Both velocity waveforms are symmetrical, so similar comparisons apply for the deceleration phase of the trajectories. The results of the simulations to compare our minimum acceleration trajectories of reaching and grasping with the equivalent minimum jerk trajectories of Smeets and Brenner (1999) are presented in Figs. 4 and 5. Fig. 4 shows the family of finger and thumb trajectories generated with the position, direction, orientation and width of the object held constant (a ¼ 20, a ¼ 0, b ¼ p=4, w ¼ 4) but with the approach velocity varying (v ¼ 1, 2, 3, 4, and 5). Fig. 5 shows the family of finger and thumb trajectories generated with the position, direction, orientation and approach velocity held constant (a ¼ 20, a ¼ 0, b ¼ p=4, v ¼ 2), but with the width of the object varying (w ¼ 0, 2, 4, 6, 8). The finger and thumb trajectories shown in Figs. 4 and 5 are remarkably similar to those presented by Smeets and Brenner in their Figs. 4 and 5 (1999, pp. 246–247).

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Fig. 4. Simulation of minimum acceleration finger and thumb trajectories with the position, direction, orientation and width of the object held constant (a ¼ 20, a ¼ 0, b ¼ p=4, w ¼ 4) but with the approach velocity varying (v ¼ 1, 2, 3, 4, and 5). Part (a) shows the position trajectories for finger and thumb, the inner pair of trajectories corresponding to v ¼ 1 and the outer pair corresponding to v ¼ 5. Part (b) shows the velocity trajectories for finger and thumb, the bottom waveforms for each corresponding to v ¼ 1 and the top waveforms corresponding to v ¼ 5.

Their main predictions about reaching and grasping derived from the minimum jerk model are equally well demonstrated by the minimum acceleration model. As shown in the above figures, (1) the transport component is independent of the object size, (2) the grip size (i.e., maximum aperture of the grip) is independent of distance, (3) the grip size increases and occurs later for larger objects, and (4) the grip size increases and occurs earlier if the approach parameter (i.e., final velocity v) is increased. 3.2. Determining the number of degrees of freedom Data from the eight simulation experiments are shown in Fig. 6. For each simulation experiment, the left-hand side of the figure shows 20 repetitions of the four signals yf ðkÞ, yt ðkÞ, xf ðkÞ and xt ðkÞ obtained by concatenating the 20 repetitions of the simulated minimum acceleration finger and thumb trajectories. The values and

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Fig. 5. Simulation of minimum acceleration finger and thumb trajectories generated with the position, direction, orientation and approach velocity held constant (a ¼ 20, a ¼ 0, b ¼ p=4, v ¼ 2), but with the width of the object varying (w ¼ 0, 2, 4, 6, 8). Part (a) shows the position trajectories for finger and thumb, the inner pair corresponding to w ¼ 0 and the outer pair corresponding to w ¼ 8. Part (b) shows the velocity trajectories for finger and thumb, the bottom waveforms for each corresponding to w ¼ 0 and the top waveforms corresponding to w ¼ 8.

variations of a, a and b for each experiment are shown in the figure. For each simulation experiment, the right-hand side of the figure shows the four orthogonalised signals Yf ðkÞ, Yt ðkÞ, Xf ðkÞ and Xt ðkÞ at the output of the Gramm–Schmidt orthogonalising network. Signals with variance less than 10% of the corresponding input signals that were set to zero in the analysis have not been set to zero in the figure so that the residuals can be examined. When making comparisons in the figure, however, note the scale has been increased for all small signals. Even so, some output signals still appear to be zero in the figure but in fact have a very small variance close to zero. It will be seen that the variance of those signals that were below 10% of the variance of the corresponding input signal were, in all experiments, well below the 10% threshold. From the figure it can be seen that the number of nonzero output signals from the orthogonalising network (i.e., the number of degrees of freedom of movement) was either one or two depending on the experiment. The number of degrees of freedom of movement for all eight simulation experiments is presented in Table 1.

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Fig. 6. Parts (a)–(h) depict the results of the Gramm–Schmidt orthogonalisation of the trajectories generated in the eight simulation experiments. Each experiment involved different values of the parameters a, a and b in Fig. 1. The four left-hand panels for each experiment show the signals obtained by concatenating 20 repetitions of simulated finger and thumb trajectories generated according to the parameter specification. The four left-hand panels show the signals obtained from the orthogonalisation procedure depicted in Fig. 2. The values for a, a and b are given at the top of the results for each experiment, along with the number of degrees of freedom (DOF) indicated in the data.

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Fig. 6 (continued)

The simulated data show that the four minimum acceleration trajectories yf ðkÞ, yt ðkÞ, xf ðkÞ and xt ðkÞ for finger and thumb reaching to an elliptical object

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Fig. 6 (continued)

placed with random distance a, direction angle a, and orientation angle b in a plane, contain only two degrees of freedom of movement, in other words, less than the

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Fig. 6 (continued)

number of trajectories. The four trajectories contained two degrees of freedom of movement whenever the direction angle a was varied from reach to reach, even when

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Table 1 Summary of the degrees of freedom results Experiment Experiment Experiment Experiment Experiment Experiment Experiment Experiment

1 2 3 4 5 6 7 8

Only distance a varied Only direction angle a varied Only orientation angle b varied Distance a and orientation angle b varied Distance a and direction angle a varied Direction angle a and orientation angle b varied Distance a, direction angle a, and orientation angle b varied Direction angle a and orientation angle b varied such that a  b ¼ 0

1 2 1 1 2 2 2 2

DOF DOF DOF DOF DOF DOF DOF DOF

the distance a, and orientation angle b were held constant or when the orientation angle b was varied but kept equal to the direction angle a. Conversely, if the direction angle a was held constant while the distance a and orientation angle b were varied, the four trajectories remained perfectly inter-related and contained only one degree of freedom of movement.

4. Discussion 4.1. Minimum acceleration vs. minimum jerk comparisons Given the similarities between the minimum acceleration and minimum jerk trajectories in Fig. 3, it is understandable that all the important features of reaching and grasping predicted by the Smeets and Brenner (1999) simulations of minimum jerk trajectories are reproduced equally well by our simulations of minimum acceleration trajectories. As indicated previously, these authors compared the relationship between object size, grip size and relative time to maximum grip aperture obtained from their minimum jerk model with experimental data derived from 35 studies reported in the literature between 1990 and 1997. Since these data were found to match the predictions of the minimum jerk model very well, this should be equally so for the minimum acceleration model in light of the comparability demonstrated. Most clear is the fact that two general features of the relationships between object size and grip parameters reported in all 35 experimental studies are reproduced by both the minimum jerk and minimum acceleration models, (i) the maximum grip aperture occurs in the second half of the movement and, (ii) it occurs later for larger objects. In terms of Smeets and BrennerÕs definitions, both models predict that the parameters of the transport component are independent of the size of the object and this matches the average result from the experimental studies. 4.2. Determining the number of degrees of freedom From the simulation experiment described above we find that the four finger and thumb reaching trajectories, yf ðkÞ, yt ðkÞ, xf ðkÞ and xt ðkÞ in a plane contain at most two degrees of freedom of movement despite changes in direction, distance and ori-

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entation. So the results support the prediction that minimum acceleration finger and thumb trajectories are highly interrelated and consequently, the number of degrees of freedom of movement can be considerably less than the number of input signals. As suggested earlier, it follows that this allows the nervous system to plan trajectories in a reduced-dimensional task space and to form task-dependent motor synergies, thereby reducing computational workload. It is interesting to note that it is the change in direction angle a that is responsible for breaking the relationships between the trajectories and thereby increasing the number of degrees of freedom of movement from one to two. Indeed, in all reaching tasks in which the direction angle a was not changed from reach to reach, the number of degrees of freedom was one. So in summary, we find that variations in the distance and the orientation of the object to be grasped can be handled by a single degree of freedom synergy whereas variations in the direction of reach require specification of an additional degree of freedom of movement. With respect to reaching to grasp an object in a plane, rather than finding two independent visuomotor channels controlling (1) transport and (2) grip, we find two independent visuomotor channels controlling (1) the distance and orientation of the reach and grasp and (2) the direction of the reach and grasp.

4.3. The nature of visuomotor channels Thus our simulations confirm certain aspects of the classical view of reaching and grasping. They show that, in JeannerodÕs words (1999, p. 202) ‘‘each of the components of the act of prehension behaves as an identifiable system characterised by its own input and output and its own intrinsic mechanism.’’ However, the anatomical basis for independent visuomotor channels in the classical view is very different from our view in AMT (Neilson et al., 1997). In his article on visuomotor channels Jeannerod (1999) presents a review of neurophysiological evidence supporting the idea that there are separate visuomotor pathways through the brain involved in reaching and grasping. In contrast to JeannerodÕs idea of anatomically independent visuomotor channels, we see independent visuomotor channels as an emergent property of the adaptive processes in the brainÕs movement control system. To explain this view, let us return to the complexity of the control problem. To control a reach and grasp, the nervous system has to generate efferent drive for some 58 skeletal muscles controlling turning forces about 25 joint rotations. The net effect of these joint rotations must transport the finger and thumb along the preplanned trajectories in space and time. The muscles and their reflex systems, the biomechanical loads on the muscles and the kinematic relationships between joint angles and positions of the finger and thumb are all highly interactive. As previewed in Section 1, AMT holds that to perform a reach and grasp (or any other skilled movement for that matter), the nervous system must form, adaptively, a taskdependent feedback/feedforward controller. The controller itself must be strongly interactive in order to compensate or cancel the interactive dynamics of the arm and its muscles. The cascading of these two interactive systems results effectively in a

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noninteractive system with separate visuomotor channels controlling each degree of freedom of movement in the orthogonalised task space. Recent advances in the mathematical analysis of nonlinear dynamical systems (Isidori, 1995; Khalil, 1996; Slotine & Li, 1991) have provided a theoretical basis for such noninteractive control systems. Exactly how a noninteractive controlled system is formed adaptively in the biological context is the key problem addressed by AMT (see Neilson & Neilson, 2001b; Neilson et al., 1997). While our simulations suggest the existence of two independent visuomotor channels, those channels do not correspond to transport and grip as in the classical view. However, even Jeannerod, father of the classical view, has moved away from the idea that transport and grip are independent, acknowledging that almost all recent experiments show some form of dependence of one on the other (Jeannerod, 1999). Why then do the simulations of Smeets and Brenner (and our reproductions in Figs. 4 and 5) show independence of transport and grip? A possible explanation is that these simulations did not include any variation in the direction of the reach. All the simulated trajectories were for reaches in the direction of the y-axis. As we show in Table 1 and Fig. 6, the introduction of the angle of reach breaks up the relationship between finger and thumb trajectories and introduces a second degree of freedom of movement. If the distance a and object orientation angle b vary but the direction a remains constant, the trajectories contain only one degree of freedom. As the simulations of Smeets and Brenner dealt only with reaches in the direction of the y-axis to a circular object, there was no variation of our angles a and b. Rather, they varied either the approach parameter (final deceleration) or the width of the object. These variations were not included in our larger set of simulations for determining degrees of freedom, so whether or not variations in the approach parameter and/or the width of the object introduce an additional degree of freedom into finger and thumb trajectories remains an open question and requires further investigation. The degrees of freedom in any set of movement trajectories depend on the task. For example, as Smeets and Brenner and we in turn have shown, if all the parameters of the optimal trajectories are held constant and only the size of the object (or the approach parameter) is varied, then the transport component is independent of the grip. However more recent experimental studies have stressed the fact that these two components are far from independent of each other. For example, altering the distance to the object affects the formation of the grip (Chieffi & Gentilucci, 1993; Jakobson & Goodale, 1991). Conversely, larger objects yielded faster transport (Bootsma, Marteniuk, MacKenzie, & Zaal, 1994; Jakobson & Goodale, 1991). Smeets and Brenner recognise that the independence of the transport and reach components in their model holds only if the constraints are the same for finger and thumb. For instance, it does not hold if the finger and thumb have different movement time or approach parameters. Indeed, if the distance, direction and orientation of the object are varied from reach to reach it is unlikely that the transport and grip components will remain independent. Thus there is much yet to be done in investigating the question of visuomotor channels in reaching and grasping.

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4.4. Future directions In a commentary (Neilson, 1999) on the paper by Smeets and Brenner (1999) we pointed out that comparisons of minimum acceleration and minimum jerk trajectories with experimentally measured trajectories have failed to take intermittent error correction into account. The concept of intermittency stems from the work of Craik (1947, 1948) and has an important historical place in the theory of movement control. A wide body of research suggests an intermittency interval of about 100–150 ms in the planning and execution of motor responses. Intermittency in response planning is an essential feature of AMT. We hold that during consecutive intermittency intervals there is sufficient time for the preplanned trajectory to be replaced with a completely new trajectory incorporating error correction. Moreover, the amount by which the preplanned trajectory is predicted ahead in time can be varied from one intermittency interval to the next. This provides a control strategy that we have described in detail elsewhere (Neilson et al., 1995) and have called variable horizon predictive control. A similar strategy, referred to as receding horizon predictive control (Clarke, Mohtadi, & Tuffs, 1987; Maciejowski, 2002), is well known technologically. It offers many advantages in the control of complex nonlinear dynamical systems with time delays as found in chemical processing plants and the petroleum industry. When intermittent error corrections are included in simulations of reaching movements (Gibson & Neilson, 1999), the acceleration profiles display a wide variability in contradiction to Kawato (1996). Until the influence of intermittency is fully investigated in simulations of movement, it would seem premature to reject the minimum acceleration criterion in simulated trajectory planning on the basis of experimentally observed acceleration profiles as does Kawato, the more so in view of the present results. In a recent presentation (Neilson & Neilson, 2001a) we showed that movement trajectories generated by tau theory (Lee, 1976) and verified by a variety of experimental observations (e.g., gannet birds diving, frigate birds snatching carrion, snakes striking, babies sucking, people reaching, African drummers drumming, racing car drivers turning into a corner), are almost identical with minimum acceleration trajectories. This provides support for minimum acceleration simulations, but can also be investigated in comparison to minimum jerk. It will also be clear that the present simulations gave no opportunity to examine the influence of duration of movement on endpoint accuracy as described by FittsÕ law. All trajectories simulated in this study are 5 s in duration and can be regarded as the preplanned trajectories to reach from the initial state to the required final state. These preplanned trajectories are assumed to be accurate and so no influence of speed–accuracy tradeoff is included. In the next stage of AMT simulation, execution of the planned trajectories will be simulated including addition of noise in the motor signals and intermittent error corrections at reaction time rates. In a thesis completed recently at this laboratory (Olsen, 2001) it was shown that inclusion of these factors is sufficient to account for both the logarithmic speed–accuracy tradeoff of FittsÕ law and the linear speed–accuracy tradeoff observed in fixed-time movements by Schmidt, Zelaznik, Hawkins, Frank, and Quinn (1979). In the present

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study, however, our focus is on the nature of the nonlinear correlations or couplings between the preplanned optimal trajectories and hence the movement duration was set constant at 5 s. Future simulations that include intermittency will allow the effects of speed–accuracy tradeoff to be investigated in the context of reaching and grasping.

Acknowledgements This work was supported by a doctoral scholarship from the School of Electrical Engineering and Telecommunications, University of New South Wales to the first author and by Australian Research Council Large Research Grant A00106147 to the third author. We thank Megan Neilson for contributions to the revision of this paper.

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