- Email: [email protected]
A model for reaching control
Acta Psychologica 82 (1993) 237-250 North-Holland
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A model for reaching control * David A. Rosenbaum a, Sascha E. Engelbrecht b, Michael M. Bushe a and Loukia D. Loukopoulos a a University of Massachusetts, Amherst, USA b Phillips University, Marburg, Germany
In this paper we propose that reaches are made to target postures which are selected by evaluating stored postures. Target postures are chosen by taking a weighted average of the stored postures, where the weights assigned to the stored postures depend on their effectiveness for the task. Movements from starting postures to target postures are achieved by reducing the distance, in joint space, between the two. The form of the movement depends on drive (assumed to decrease as less distance remains) and inertia. The model predicts the Power Law of learning, compensation for immobility of joints, changes in limb contributions depending on movement speed, asymmetric bell-shaped velocity profiles, velocity-amplitude relations, Fitts’ Law, and position-dependent variations in hand-path curvature. Planned extensions of the model may broaden its application - for example, to handwriting.
How do we decide how to reach for objects when, as is usually the case, infinitely many reaches are possible? Here we address this question within the context of kinematics. The issue we address is how particular postures and transitions between postures (movements) are selected. Classically, the problem of selecting body positions has been referred to as the inverse kinematics problem. This term refers to the fact that there are generally many postures that allow the hand (or any point on the body) to occupy a given location in extrinsic space. This leaves open the question of how particular postures are chosen. The Correspondence to: D.A. Rosenbaum, Dept. of Psychology, Tobin Hall, University of Massachusetts, Amherst, MA 01003, USA. E-mail: [email protected] * The work was supported by grants BNS-8710933 and BNS-9008665 from the National Science Foundation and a Research Scientist Development Award from the National Institute of Mental Health. We thank Horst Krist, Jonathan Vaughan, and two anonymous reviewers for helpful comments. A more complete account of the model and accompanying results appears in Rosenbaum, Engelbrecht, Bushe and Loukopoulos (in press). OOOl-6918/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
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inverse kinematics problem is an instance of the more general degrees-of-freedom problem identified by Bernstein (1967).
The model We have developed a new model to solve the inverse kinematics problem. We call it the Knowledge model. The name refers to the fact that the actor is assumed to possess information that enables him or her to move efficiently. A prime motivation for the development of the Knowledge model is the belief that no previous model adequately accounts for the range of behavioral phenomena that characterize reaching behavior. (Unfortunately, space limitations do not permit a review of the earlier theories. One is given in Rosenbaum et al., in press.) The model we have developed is a computational account that is meant to be biologically and physically plausible. A central assumption is that movement efficiency is a prime element of movement planning. In this respect, our model is similar to others, such as Flash and Hogan’s (1985), which emphasizes minimization of mean squared jerk (a kinematic variable), or Uno et al.‘s (19891, which emphasizes minimization of torque change (a dynamic variable). Our model is implemented as a computer program (on a Macintosh II computer) and is rendered as a stick-figure animation. People interacting with the program see the stick-figure on the computer screen, occupying a seated position. The stick-figure has 3 degrees of freedom: it can bend at the hip, shoulder, and elbow, resulting in movements in the sagittal plane. People interacting with the program can point and click at a location on the screen, designating a target to which the stick-figure should reach. If the target is within the stick-figure’s workspace, it reaches for the target. It can reach up, down, out and back. Reaches can be made no matter what posture the stick-figure occupies at the start of the trial. People using the program generally cannot distinguish animations produced by the model from animations based on digitized movements of university students carrying out comparable reaches. If the latter phenomenon holds up in formal testing, it will be possible to say that the model passes a Turing test for an intelligent action system.
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Overview of the model The model has three parts. We summarize them briefly below and then justify them more fully. One part is concerned with Storage. The stored elements are postures (i.e., static body positions). By hypothesizing stored postures, we wish to see how far we can go without assuming that movements are stored. The second part of the model is concerned with Planning. We assume that when a target is chosen for reaching, the actor has information about the location of the target and his or her starting posture. Each stored posture is evaluated for its potential effectiveness for reaching the target. Weights are assigned to the candidate postures based on the evaluations they receive. A single target posture is then arrived at by taking a weighted average of the postures that were considered. The third part of the model is concerned with Execution. Here the distance between the starting posture and target posture is reduced via movement. Details of the trajectory are not planned explicitly. The kinematics of the movement depend on drive, the tendency to move with greater urgency the greater the distance to be covered, and inertia, the tendency to maintain the velocity achieved in the last time step. Details concerning the three components are given below. Storage As stated above, postures rather than movements are the stored elements in our system. We assume stored postures for the following reasons: (1) If movements are viewed as transitions between postures, fewer elements must be stored with a posture-based storage system than with a movement-based storage system. This is especially true if information must be stored about speed, acceleration, and so forth. Considering the myriad ways in which movements can differ, one sees that the dimensionality of a movement-storage system is unclear, whereas the dimensionality of a posture-based system is not. For a posture-based system, the dimensionality is simply the number of mechanical degrees of freedom of the joints.
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(2) The mass-spring model of movement, which has been widely supported (see Bizzi et al. 1991) demonstrates the primacy of body positions (postures) rather than the means by which those positions are achieved (movements). (3) Similarly, research on ‘memory for movements’ (see Smyth, 1984, for review) shows that memory for positions is more robust than memory for movements. In the current version of the Knowledge model, postures are stored as records indexed by Cartesian hand locations. New postures are added to the storage system either via random adoption of postures (motor ‘babbling’) or through derivation of new target postures via planning (see below). Planning When a target is selected for reaching, each stored posture is evaluated according to its potential for helping to complete the task. The evaluation works by deriving a Total Cost for each posture. The Total Cost equals a weighted sum of two terms: (1) a Spatzd Error Cost and (2) a Travel Cost. The weights assigned to the Spatial Error Cost and Travel Cost correspond, respectively, to the relative importance of minimizing the distance of the hand from the target (the Spatial Error Cost) and minimizing the cost of moving from the starting posture to the candidate posture (the Travel Cost). The Spatial Error Cost for a posture equals the squared Euclidean distance between the Cartesian target location and the Cartesian location occupied by the hand if the posture were adopted. The Travel Cost, VP, for a posture p is:
‘P = C j=l
5j(aj/t)9
(1)
where aj denotes the distance in joint space between the jth component of the candidate posture relative to the start posture (i.e., the jth degree of freedom of the n degrees of freedom characterizing the posture), t denotes movement time (assumed to be the same for all joints), and sj denotes a cost-of-movement factor that depends on the joint’s stiffness, friction, damping, and average moments of inertia of
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its associated limb segments. Thus far, we have not developed an explicit formula for lj. However, in most of the modelling we have done, we have made it highest for the hip, smaller for the shoulder, and smallest for the elbow. We have assumed, for parsimony, that the cost-of-movement factor for a given joint is the same in all postures. However, we have also assumed that the cost-of-movement factor for a given joint can change. As will be seen below, the capacity to modify the cost-of-movement factor for a given joint has dramatic effects on the contributions of other joints. This permits still-healthy joints to compensate for joints that become hurt or stiff. Once a Total Cost is arrived at for each posture, we take a weighted sum of the postures to get a single target posture. Taking a weighted sum is meaningful from a mathematical standpoint because postures can be viewed as vectors in joint space. We take a weighted sum rather than using the posture with the smallest Total Cost so new postures can be adopted. We take a weighted sum rather than a simple unweighted sum to reflect the prominence that should be given to most-adaptive postures (i.e., those given most weight) and to preserve movement amplitudes when the number of stored postures varies. ’ To take a weighted sum of the postures, it is necessary to assign weights to the postures. We have developed an algorithm for obtaining the weights, which we call Gaussian Averaging. The aim of the procedure is to assign large weights to postures that have small Total Costs, and to assign weights to postures of a given cost such that they depend as well on the smallest Total Cost of any posture. The idea is to assign as much weight as possible to any posture that is ideally (or nearly ideally) suited to serve as the target posture, and to assign less weight to other postures the closer the best posture comes to ideal. Computationally, Gaussian Averaging works as follows. The Total Costs of all the postures are passed through a Gaussian function. The
’ Evidence that movements are based on weighted sums rather than simple sums of neural activity has been obtained by Lee et al. (1988). They showed that temporary deactivation of neurons in the monkey superior colliculus does not produce saccades with smaller-than-normal amplitudes. Such hypometric saccades would be expected if neural activity were simply summed. However, directions of saccades were affected by temporary deactivation of the neurons, in a manner consistent with the hypothesis that the ensemble of neurons pool their outputs. The two results together - the presence of a direction effect, and the absence of an amplitude effect are best explained by a weighted-sum model.
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mean of the Gaussian is zero, and its standard deviation is proportional to the smallest Total Cost. 2 After Gaussian values have been obtained for all postures, the weight assigned to each posture is its Gaussian value divided by the sum of all the Gaussian values. Finally, a weighted average is found by multiplying each posture vector by its weight and summing the products. The obtained weighted sum defines the target posture. 3 A final point to be made about the Planning process is that it can be conceptualized as operating in parallel. If posture representations are regarded as cognitive ‘demons’, they can be imagined to work simultaneously to determine how well each can contribute to the task at hand. Execution Movement from the starting posture to the target posture is based on the following equation: @j(t)
=qjoj(t
-
1) + (1_77j)oi*(t),
(2)
where wj(t) denotes the angular velocity for joint component j at time t, qj is an index of inertia (0 I qj I 11, and o:(t) denotes proposed is directly proportional to the angular velocity at time t. 4 w;(t) angular distance still to be covered by the jth joint at time t. We think of the term to the left of the plus sign as representing inertia, and the term to the right of the plus sign as representing drive. Eq. (2) ensures that movement depends both on inertia and error correction (i.e., reducing the discrepancy between the current and desired posture). An important feature of the Execution equation is that all joints start moving together. ’ The mean of the Gaussian corresponds to the ideal Total Cost. All Total Costs input to the Gaussian are greater than zero because negative Total Costs are undefined and a zero Total Cost is impossible for postures requiring movement from the starting posture. 3 Gaussian averaging appears to work well. It produces weighted averages closer to target values than simpler weighted averaging schemes, such as one in which the weight for a posture is based on its Total Cost divided by the sum of the Total Costs of all the candidate postures. One way in which Gaussian averaging does better is that the weighted sums it produces are only slightly biased toward the middle of joint space when postures requiring extreme joint angles are required. 4 Although nj is an index of inertia, it is not meant to denote inertia per se. Inertia has the units N. S, whereas nj is a dimensionless parameter.
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Evaluation
In the following sections, we describe how the Knowledge model performs. Lea ming As mentioned above, postures are learned through motor babbling and storage of newly adopted target postures. Given the nature of Gaussian averaging, performance should improve as the number of stored postures increases. That is, Spatial Error Costs and Travel Costs of target postures should decrease with the number of stored postures. The basis for this expectation is that Gaussian averages should be best (i.e., least costly) when stored postures with very small Total Costs are available. As the number of stored postures increases, there is a higher likelihood that such Gaussian averages will be found. This expectation was borne out in a simulation in which 5,000 randomly chosen postures were added to the storage system, then 50 random targets were tested, then another 5,000 randomly chosen postures were added to the storage system, 50 random targets were tested, and so ‘on. For each simulated reach, we recorded the Spatial Error Cost and Travel Cost. As predicted, both measures improved with ‘practice’. For both measures, the performance data were well fitted by linear functions relating log cost to log number of trials. The fit of the linear function was r2 = 0.97 for the Spatial Error Cost and r2 = 0.91 for the Travel Cost. These results are consistent with the Power Law of learning (Newell and Rosenbloom 19811, which is a hallmark of skill acquisition. Finding that Travel Cost decreases with practice accords with the observation that performance becomes more graceful with experience. 5 Compensation for joint immobility
An important feature of normal motor performance is that actors can compensate immediately for changes in the mobility of individual 5 Adding postures might be expected to slow Planning. This would occur if Planning were not fully parallel and/or if the time to complete the Planning process were unaffected by the availability of low-cost postures. These issues remain to be studied.
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joints. Locking a joint with a cast normally does not leave one paralyzed. Similarly, holding a package against one’s body with one’s upper arm does not leave one unable to use the arm and hand for tasks that normally involve shoulder movement. In such cases, other joints take over for the joint that can no longer move. The Knowledge model provides a simple explanation for such compensation. Suppose it becomes harder than normal to move the hip. In the model, this state of affairs can be represented by increasing the hip’s cost-of-movement factor. When this occurs, the other joints work more than normal because postures that demand large hip displacements are assigned smaller-than-normal weights. As a result, the target posture that is selected requires little or no hip movement. By the same token, if it becomes harder than normal to move the elbow, postures that demand large elbow displacements are assigned smaller-than-normal weights, so the target posture that is selected requires little or no elbow movement. 6 Speed variation Just as actors can compensate for changes in the mobility of individual joints, they can vary the speed of their movements. When they do so, the characteristics of their movements often change. In locomotion, for example, changes in gait typically accompany changes in speed (Alexander’ 1984). In oscillatory movements of the finger, hand, and forearm, the relative contributions of these limb segments depend on the speed with which the fingertip oscillates (Rosenbaum et al. 1991). The Knowledge model allows for speed changes because it explicitly incorporates a movement-time term, t, in its Travel-Cost equation (eq. (1)). More importantly, it predicts changes in the relative contribution of limb segments depending on their suitability for high- versus low-speed performance. It does so for the same reasons that it permits compensation for immobility of selected joints: When t gets small, the Travel Costs of postures that have large costs of movement increase 6 To our knowledge, this account of compensation is simpler than any yet developed. For example, the ‘task dynamics’ framework of Saltzman and Kelso (1987) can achieve compensation, but its assumed computations are so complex as to be biologically questionable (Jordan and Rosenbaum 1989).
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more than the Travel Costs of postures that have small costs of movement. Thus, if the elbow has a higher cost of movement than the wrist or metacarpophalangeal joint, then as required velocity increases (and required amplitude remains small), the postures that are favored are those that do not demand much forearm rotation (i.e., much movement about the elbow). Hence, the relative contributions of the hand and finger increase when velocity is high compared to when velocity is low, as observed by Rosenbaum et al. (1991). Bell-shaped velocity profiles
Another important feature of motor performance is that end-effectors typically follow bell-shaped or right-skewed bell-shaped velocity profiles (see Bullock and Grossberg, 1988, for review). The Knowledge model predicts such profiles, not only for end-effecters (e.g., the hand) but also for more proximal points along the limb-segment chain. Velocity -amplitude relations
Another important feature of motor performance is that the maximum velocity of the end-effector usually increases with the amplitude to be covered. This effect is observed in movements of the limbs, eyes, and speech articulators (see Bullock and Grossberg, 1988, for review). The Knowledge model predicts this positive relation because it contains a term in its Execution equation (07 in eq. (2)) which is positively related to the remaining amplitude of movement. Fitts’ Law
Because the Knowledge model predicts (asymmetric) bell-shaped velocity profiles, it also predicts that the hand approaches the target slowly at first, then more rapidly, and then more slowly again. Depending on the amount of inertia (indexed by 77 in eq. (2)), the hand may overshoot the target position, then undershoot it, then overshoot it by a smaller amount, then undershoot it by a smaller amount, and so on. When the hand actually ‘reaches the target’ depends on the size of the tolerance window around the target; the larger the window, the sooner the hand can be said to arrive at the target location. The latter observation can be coupled with the observation that though
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velocity increases with amplitude, movement time ‘increases with the distance between the hand’s starting and target locations. The fact that movement time increases with movement amplitude and decreases with target tolerance is consistent with Fitts’ Law (see Rosenbaum, 1991, for a review of Fitts’ Law). 7 Hand-path curvature
The Knowledge model also predicts changes in the curvature of hand trajectories depending on the hand’s initial and final locations. Such curvature changes have been reported by Atkeson and Hollerbath (1985). The model predicts such changes because it emphasizes the avoidance of movements demanding large Travel Costs. For tasks requiring movements from particular starting locations to particular target locations, a consequence of avoiding large Travel Costs is that the hand sometimes follows curved paths rather than straight paths. For example, in simulations we have conducted, movements that take the hand from a point in front of the stomach to a point in front of the forehead are curved; Atkeson and Hollerbach observed such paths in human subjects. The Knowledge predicts curved paths for such movements because, of the postures that leave the hand near the forehead, those that have the lowest Travel Costs entail little hip or shoulder rotation but considerable elbow rotation. Hence, reaching the target posture entails considerable elbow movement, which produces curved hand paths. By contrast, movements that take the hand from a point just in front of the stomach to a point horizontally farther from the stomach (at arm’s length) are nearly straight; Atkeson and Hollerbach observed such paths as well in their study. The Knowledge model predicts straight hand paths for such movements because, of the ’ To determine whether the movement-time data from the Knowledge model are quantitatively consistent with Fitts’ Law, we crossed several amplitudes with several tolerance windows, using one set of arbitrary inertia and proposed-angular-velocity parameters. Then we determined, from the obtained position-versus-time functions, how long it took the hand to arrive at a location such that the distance between the hand and the center of the target remained at or within that distance from that time on; we regarded this distance as half the width of the tolerance window. The arrival times were treated as movement times. When we plotted movement times against Fitts’ Index of Difficulty, log, (2A/W), where A denotes the distance between the hand and the center of the target, and W denotes the width of the tolerance window, we found that a straight line accounted for 82% of the variance.
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postures that leave the hand directly in front of the torso, those that have the lowest Travel Cost entail little or no hip rotation but demand considerable shoulder as well as elbow rotation. The two-joint horizontal motion yields a much straighter hand path than the one-joint vertical motion.
Extensions
As implied above, we believe that the Knowledge model is promising. We therefore plan to extend it. Three extensions are described below. Forward kinematics
Recall that in the current version of the Knowledge model, postures are stored as joint vectors indexed by their corresponding hand locations in Cartesian space. The problem with this approach is that it makes it difficult to obtain spatial information about points removed from the hand. For example, it would be difficult to tell where in the space occupied by the body a mosquito has bitten the skin. However, we can swat immediately at mosquitos biting unpredictable skin locations, even if the mosquitos bite on moving limbs. This observation suggests that we can instantly determine where in extracorporeal space a given point on the body is located. It is difficult to imagine how a table look-up system like the one used in the model could support such performance. To remedy this problem, we need only assume that it is possible to compute locations of points of interest based on knowledge of joint angles and limb lengths - that is, to compute forward kinematics and to define Spatial Errors on this basis. The advantage of this approach is that if a task requires bringing some point on the body other than the hand to a spatial target, forward kinematics can be used to determine what postures will be effective. Another advantage is that the system can engage in tool use: if the length of a hand-held tool is known, then postures can be selected that permit the end of the tool (or any point on the tool) to be brought to a desired location.
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Error correction
A second extension of the model concerns error correction. Currently, the model provides no means for the hand to reach a target if it has missed a target after an initial reach. Alternative extensions to the model can correct this problem. The most obvious extension is to try moving from the missed location to the target. The idea is to treat the posture at the end of the first movement as the starting posture for the next movement. Presumably, the hand will get closer to the target as this process is repeated. ’ Another possibility is to allow for mental planning prior to movement. Here a target posture is selected, as in the usual Planning process, but an actual movement to that target posture is initiated only if the Total Cost of the posture is acceptably low. If it is not, a second movement is planned from the first target posture. If the Total Cost of the second target posture is acceptably low, a series of two movements to the first and second target postures is initiated. However, if the Total Cost of the second target posture is not acceptably low, a third target posture is selected, and so forth. This hypothesized process makes two predictions. First, planning times should be long for movements that must have low Total Costs. Second, delays between submovements toward spatial targets should be shorter than delays before the first submovement. The latter prediction follows from the fact that full planning is only necessary before the series of submovements has begun. Whereas the above planning method predicts a series of submovements toward a target, another possible method does not. Here mental planning is used, but once a target posture has been found to be too costly, it is merely added to the stock of stored postures and the problem of reaching the original spatial target from the original starting posture is worked on again, this time with the new stored posture treated as a candidate posture. If the target posture arrived at in the second planning cycle is too costly, it is stored, the problem is worked on again, and so forth. With this procedure, more planning time should occur as more accurate and/or efficient movements are s In using this approach, it may be necessary to store the posture adopted at the end of a given movement so it can contribute to the determination of subsequent weighted averages.
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required, but in contrast to the method described above, a single movement rather than a series of submovements should be made toward the target location. Intentional learning
The final extension to the Knowledge model concerns learning. Recall that in the current version of the model postures are stored both through motor babbling and deliberate aiming. It would be helpful, from the standpoint of parsimony, to dispense with this distinction. Fortunately, it is possible to do so using a fully intentional learning system. Suppose that at birth the motor system can carry out the computations assumed in the Knowledge model and that it has stored postures corresponding only to extreme joint-angle combinations. For the 3 degree-of-freedom system we have modelled, eight postures would be innately specified, corresponding to all possible combinations of extreme hip, shoulder, and elbow angles. 9 These extreme joint-angle combinations would be necessary to constrain the values that can be arrived at through weighted averaging. Spatial targets would be presented one after the other, and for each spatial target a weighted average of the already-stored postures would be found. Because the weighted average arrived at at any given time would be stored, the system would ultimately develop a set of stored postures that is as large as the storage capacity of the system. The density of stored postures in different regions of joint space would reflect the system’s learning history: Those regions that house postures that are well suited for the tasks that have been presented would have many stored postures, whereas those regions that house postures that are less well suited for the tasks that have been presented would have fewer stored postures. A consequence of these differences in storage density would be that when new spatial targets are presented, the accuracy of the movements that can be performed should be high if the relevant regions of joint space have many stored postures, but if the relevant regions of joint space have few stored postures, the accuracy of the movements that can be performed should be low. In effect, the system 9 It is convenient to think of these eight postures whose axes are hip, shoulder, and elbow angles.
as occupying
the corners
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solid
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should show expertise. Depending on the system’s forgetting rules, its expertise should change with the history of tasks with which it has been confronted. lo
References Alexander, R.M., 1984. Walking and running. American Scientist 72, 348-354. Atkeson, C.G. and J.M. Hollerbach, 1985. Kinematic features of unrestrained arm movements. The Journal of Neuroscience 5, 2318-2330. Bernstein, N., 1967. The coordination and regulation of movements. London: Pergamon. Bizzi, E., F.A. Mussa Ivaldi and S. Giszter, 1991. Computations underlying the execution of movement: A biological perspective. Science 253, 287-291. Bullock, D. and S. Grossberg, 1988. Neural dynamics of planned arm movements: Emergent invariants and speed-accuracy properties during trajectory formation. Psychological Review 9549-90. Flash, T. and N. Hogan, 1985. The coordination of arm movements: An experimentally confirmed mathematical model. The Journal of Neuroscience 5, 1688-1703. Jordan, M.I. and D.A. Rosenbaum, 1989. ‘Action’. In: M.I. Posner (ed.), Foundations of cognitive science (pp. 727-767). Cambridge, MA: MIT Press. Lee, C., W.H. Rohrer and D.L. Sparks, 1988. Population coding of saccadic eye movements by neurons in the superior colliculus. Nature 332, 357-360. Newell, A.M. and P.S. Rosenbloom, 1981. ‘Mechanisms of skill acquisition and the law of practice’. In: J.R. Anderson (ed.), Cognitive skills and their acquisition (pp. l-55). Hillsdale, NJ: Erlbaum. Rosenbaum, D.A.,‘1991. Human motor control. San Diego, CA: Academic Press. Rosenbaum, D.A., S.E. Engelbrecht, M.M. Bushe and L.D. Loukopoulos, in press. Knowledge model for selecting and producing reaching movements. To appear in a special theme issue, Control of Arm and Hand Posture and Movement, of Journal of Motor Behavior. Rosenbaum, D.A., J.D. Slotta, J. Vaughan and R.J. Plamondon, 1991. Optimal movement selection. Psychological Science 2, 86-91. Saltzman, E. and J.A.S. Kelso, 1987. Skilled actions: A task-dynamic approach. Psychological Review 94, 84-106. Smyth, M.M., 1984. ‘Memory for movements’. In: M.M. Smyth and A.M. Wing feds.), The psychology of human movement (pp. 83-117). London: Academic Press. Uno, Y., M. Kawato and R. Suzuki, 1989. Formation and control of optimal trajectory in human multijoint arm movement: Minimum torque-change model. Biological Cybernetics 61,89-101. lo A possible forgetting rule is as follows: After a certain number of postures have been learned, forget the oldest stored posture, except the eight postures that were innately specified.
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A model for reaching control * David A. Rosenbaum a, Sascha E. Engelbrecht b, Michael M. Bushe a and Loukia D. Loukopoulos a a University of Massachusetts, Amherst, USA b Phillips University, Marburg, Germany
In this paper we propose that reaches are made to target postures which are selected by evaluating stored postures. Target postures are chosen by taking a weighted average of the stored postures, where the weights assigned to the stored postures depend on their effectiveness for the task. Movements from starting postures to target postures are achieved by reducing the distance, in joint space, between the two. The form of the movement depends on drive (assumed to decrease as less distance remains) and inertia. The model predicts the Power Law of learning, compensation for immobility of joints, changes in limb contributions depending on movement speed, asymmetric bell-shaped velocity profiles, velocity-amplitude relations, Fitts’ Law, and position-dependent variations in hand-path curvature. Planned extensions of the model may broaden its application - for example, to handwriting.
How do we decide how to reach for objects when, as is usually the case, infinitely many reaches are possible? Here we address this question within the context of kinematics. The issue we address is how particular postures and transitions between postures (movements) are selected. Classically, the problem of selecting body positions has been referred to as the inverse kinematics problem. This term refers to the fact that there are generally many postures that allow the hand (or any point on the body) to occupy a given location in extrinsic space. This leaves open the question of how particular postures are chosen. The Correspondence to: D.A. Rosenbaum, Dept. of Psychology, Tobin Hall, University of Massachusetts, Amherst, MA 01003, USA. E-mail: [email protected] * The work was supported by grants BNS-8710933 and BNS-9008665 from the National Science Foundation and a Research Scientist Development Award from the National Institute of Mental Health. We thank Horst Krist, Jonathan Vaughan, and two anonymous reviewers for helpful comments. A more complete account of the model and accompanying results appears in Rosenbaum, Engelbrecht, Bushe and Loukopoulos (in press). OOOl-6918/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
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inverse kinematics problem is an instance of the more general degrees-of-freedom problem identified by Bernstein (1967).
The model We have developed a new model to solve the inverse kinematics problem. We call it the Knowledge model. The name refers to the fact that the actor is assumed to possess information that enables him or her to move efficiently. A prime motivation for the development of the Knowledge model is the belief that no previous model adequately accounts for the range of behavioral phenomena that characterize reaching behavior. (Unfortunately, space limitations do not permit a review of the earlier theories. One is given in Rosenbaum et al., in press.) The model we have developed is a computational account that is meant to be biologically and physically plausible. A central assumption is that movement efficiency is a prime element of movement planning. In this respect, our model is similar to others, such as Flash and Hogan’s (1985), which emphasizes minimization of mean squared jerk (a kinematic variable), or Uno et al.‘s (19891, which emphasizes minimization of torque change (a dynamic variable). Our model is implemented as a computer program (on a Macintosh II computer) and is rendered as a stick-figure animation. People interacting with the program see the stick-figure on the computer screen, occupying a seated position. The stick-figure has 3 degrees of freedom: it can bend at the hip, shoulder, and elbow, resulting in movements in the sagittal plane. People interacting with the program can point and click at a location on the screen, designating a target to which the stick-figure should reach. If the target is within the stick-figure’s workspace, it reaches for the target. It can reach up, down, out and back. Reaches can be made no matter what posture the stick-figure occupies at the start of the trial. People using the program generally cannot distinguish animations produced by the model from animations based on digitized movements of university students carrying out comparable reaches. If the latter phenomenon holds up in formal testing, it will be possible to say that the model passes a Turing test for an intelligent action system.
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Overview of the model The model has three parts. We summarize them briefly below and then justify them more fully. One part is concerned with Storage. The stored elements are postures (i.e., static body positions). By hypothesizing stored postures, we wish to see how far we can go without assuming that movements are stored. The second part of the model is concerned with Planning. We assume that when a target is chosen for reaching, the actor has information about the location of the target and his or her starting posture. Each stored posture is evaluated for its potential effectiveness for reaching the target. Weights are assigned to the candidate postures based on the evaluations they receive. A single target posture is then arrived at by taking a weighted average of the postures that were considered. The third part of the model is concerned with Execution. Here the distance between the starting posture and target posture is reduced via movement. Details of the trajectory are not planned explicitly. The kinematics of the movement depend on drive, the tendency to move with greater urgency the greater the distance to be covered, and inertia, the tendency to maintain the velocity achieved in the last time step. Details concerning the three components are given below. Storage As stated above, postures rather than movements are the stored elements in our system. We assume stored postures for the following reasons: (1) If movements are viewed as transitions between postures, fewer elements must be stored with a posture-based storage system than with a movement-based storage system. This is especially true if information must be stored about speed, acceleration, and so forth. Considering the myriad ways in which movements can differ, one sees that the dimensionality of a movement-storage system is unclear, whereas the dimensionality of a posture-based system is not. For a posture-based system, the dimensionality is simply the number of mechanical degrees of freedom of the joints.
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(2) The mass-spring model of movement, which has been widely supported (see Bizzi et al. 1991) demonstrates the primacy of body positions (postures) rather than the means by which those positions are achieved (movements). (3) Similarly, research on ‘memory for movements’ (see Smyth, 1984, for review) shows that memory for positions is more robust than memory for movements. In the current version of the Knowledge model, postures are stored as records indexed by Cartesian hand locations. New postures are added to the storage system either via random adoption of postures (motor ‘babbling’) or through derivation of new target postures via planning (see below). Planning When a target is selected for reaching, each stored posture is evaluated according to its potential for helping to complete the task. The evaluation works by deriving a Total Cost for each posture. The Total Cost equals a weighted sum of two terms: (1) a Spatzd Error Cost and (2) a Travel Cost. The weights assigned to the Spatial Error Cost and Travel Cost correspond, respectively, to the relative importance of minimizing the distance of the hand from the target (the Spatial Error Cost) and minimizing the cost of moving from the starting posture to the candidate posture (the Travel Cost). The Spatial Error Cost for a posture equals the squared Euclidean distance between the Cartesian target location and the Cartesian location occupied by the hand if the posture were adopted. The Travel Cost, VP, for a posture p is:
‘P = C j=l
5j(aj/t)9
(1)
where aj denotes the distance in joint space between the jth component of the candidate posture relative to the start posture (i.e., the jth degree of freedom of the n degrees of freedom characterizing the posture), t denotes movement time (assumed to be the same for all joints), and sj denotes a cost-of-movement factor that depends on the joint’s stiffness, friction, damping, and average moments of inertia of
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its associated limb segments. Thus far, we have not developed an explicit formula for lj. However, in most of the modelling we have done, we have made it highest for the hip, smaller for the shoulder, and smallest for the elbow. We have assumed, for parsimony, that the cost-of-movement factor for a given joint is the same in all postures. However, we have also assumed that the cost-of-movement factor for a given joint can change. As will be seen below, the capacity to modify the cost-of-movement factor for a given joint has dramatic effects on the contributions of other joints. This permits still-healthy joints to compensate for joints that become hurt or stiff. Once a Total Cost is arrived at for each posture, we take a weighted sum of the postures to get a single target posture. Taking a weighted sum is meaningful from a mathematical standpoint because postures can be viewed as vectors in joint space. We take a weighted sum rather than using the posture with the smallest Total Cost so new postures can be adopted. We take a weighted sum rather than a simple unweighted sum to reflect the prominence that should be given to most-adaptive postures (i.e., those given most weight) and to preserve movement amplitudes when the number of stored postures varies. ’ To take a weighted sum of the postures, it is necessary to assign weights to the postures. We have developed an algorithm for obtaining the weights, which we call Gaussian Averaging. The aim of the procedure is to assign large weights to postures that have small Total Costs, and to assign weights to postures of a given cost such that they depend as well on the smallest Total Cost of any posture. The idea is to assign as much weight as possible to any posture that is ideally (or nearly ideally) suited to serve as the target posture, and to assign less weight to other postures the closer the best posture comes to ideal. Computationally, Gaussian Averaging works as follows. The Total Costs of all the postures are passed through a Gaussian function. The
’ Evidence that movements are based on weighted sums rather than simple sums of neural activity has been obtained by Lee et al. (1988). They showed that temporary deactivation of neurons in the monkey superior colliculus does not produce saccades with smaller-than-normal amplitudes. Such hypometric saccades would be expected if neural activity were simply summed. However, directions of saccades were affected by temporary deactivation of the neurons, in a manner consistent with the hypothesis that the ensemble of neurons pool their outputs. The two results together - the presence of a direction effect, and the absence of an amplitude effect are best explained by a weighted-sum model.
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mean of the Gaussian is zero, and its standard deviation is proportional to the smallest Total Cost. 2 After Gaussian values have been obtained for all postures, the weight assigned to each posture is its Gaussian value divided by the sum of all the Gaussian values. Finally, a weighted average is found by multiplying each posture vector by its weight and summing the products. The obtained weighted sum defines the target posture. 3 A final point to be made about the Planning process is that it can be conceptualized as operating in parallel. If posture representations are regarded as cognitive ‘demons’, they can be imagined to work simultaneously to determine how well each can contribute to the task at hand. Execution Movement from the starting posture to the target posture is based on the following equation: @j(t)
=qjoj(t
-
1) + (1_77j)oi*(t),
(2)
where wj(t) denotes the angular velocity for joint component j at time t, qj is an index of inertia (0 I qj I 11, and o:(t) denotes proposed is directly proportional to the angular velocity at time t. 4 w;(t) angular distance still to be covered by the jth joint at time t. We think of the term to the left of the plus sign as representing inertia, and the term to the right of the plus sign as representing drive. Eq. (2) ensures that movement depends both on inertia and error correction (i.e., reducing the discrepancy between the current and desired posture). An important feature of the Execution equation is that all joints start moving together. ’ The mean of the Gaussian corresponds to the ideal Total Cost. All Total Costs input to the Gaussian are greater than zero because negative Total Costs are undefined and a zero Total Cost is impossible for postures requiring movement from the starting posture. 3 Gaussian averaging appears to work well. It produces weighted averages closer to target values than simpler weighted averaging schemes, such as one in which the weight for a posture is based on its Total Cost divided by the sum of the Total Costs of all the candidate postures. One way in which Gaussian averaging does better is that the weighted sums it produces are only slightly biased toward the middle of joint space when postures requiring extreme joint angles are required. 4 Although nj is an index of inertia, it is not meant to denote inertia per se. Inertia has the units N. S, whereas nj is a dimensionless parameter.
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Evaluation
In the following sections, we describe how the Knowledge model performs. Lea ming As mentioned above, postures are learned through motor babbling and storage of newly adopted target postures. Given the nature of Gaussian averaging, performance should improve as the number of stored postures increases. That is, Spatial Error Costs and Travel Costs of target postures should decrease with the number of stored postures. The basis for this expectation is that Gaussian averages should be best (i.e., least costly) when stored postures with very small Total Costs are available. As the number of stored postures increases, there is a higher likelihood that such Gaussian averages will be found. This expectation was borne out in a simulation in which 5,000 randomly chosen postures were added to the storage system, then 50 random targets were tested, then another 5,000 randomly chosen postures were added to the storage system, 50 random targets were tested, and so ‘on. For each simulated reach, we recorded the Spatial Error Cost and Travel Cost. As predicted, both measures improved with ‘practice’. For both measures, the performance data were well fitted by linear functions relating log cost to log number of trials. The fit of the linear function was r2 = 0.97 for the Spatial Error Cost and r2 = 0.91 for the Travel Cost. These results are consistent with the Power Law of learning (Newell and Rosenbloom 19811, which is a hallmark of skill acquisition. Finding that Travel Cost decreases with practice accords with the observation that performance becomes more graceful with experience. 5 Compensation for joint immobility
An important feature of normal motor performance is that actors can compensate immediately for changes in the mobility of individual 5 Adding postures might be expected to slow Planning. This would occur if Planning were not fully parallel and/or if the time to complete the Planning process were unaffected by the availability of low-cost postures. These issues remain to be studied.
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joints. Locking a joint with a cast normally does not leave one paralyzed. Similarly, holding a package against one’s body with one’s upper arm does not leave one unable to use the arm and hand for tasks that normally involve shoulder movement. In such cases, other joints take over for the joint that can no longer move. The Knowledge model provides a simple explanation for such compensation. Suppose it becomes harder than normal to move the hip. In the model, this state of affairs can be represented by increasing the hip’s cost-of-movement factor. When this occurs, the other joints work more than normal because postures that demand large hip displacements are assigned smaller-than-normal weights. As a result, the target posture that is selected requires little or no hip movement. By the same token, if it becomes harder than normal to move the elbow, postures that demand large elbow displacements are assigned smaller-than-normal weights, so the target posture that is selected requires little or no elbow movement. 6 Speed variation Just as actors can compensate for changes in the mobility of individual joints, they can vary the speed of their movements. When they do so, the characteristics of their movements often change. In locomotion, for example, changes in gait typically accompany changes in speed (Alexander’ 1984). In oscillatory movements of the finger, hand, and forearm, the relative contributions of these limb segments depend on the speed with which the fingertip oscillates (Rosenbaum et al. 1991). The Knowledge model allows for speed changes because it explicitly incorporates a movement-time term, t, in its Travel-Cost equation (eq. (1)). More importantly, it predicts changes in the relative contribution of limb segments depending on their suitability for high- versus low-speed performance. It does so for the same reasons that it permits compensation for immobility of selected joints: When t gets small, the Travel Costs of postures that have large costs of movement increase 6 To our knowledge, this account of compensation is simpler than any yet developed. For example, the ‘task dynamics’ framework of Saltzman and Kelso (1987) can achieve compensation, but its assumed computations are so complex as to be biologically questionable (Jordan and Rosenbaum 1989).
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more than the Travel Costs of postures that have small costs of movement. Thus, if the elbow has a higher cost of movement than the wrist or metacarpophalangeal joint, then as required velocity increases (and required amplitude remains small), the postures that are favored are those that do not demand much forearm rotation (i.e., much movement about the elbow). Hence, the relative contributions of the hand and finger increase when velocity is high compared to when velocity is low, as observed by Rosenbaum et al. (1991). Bell-shaped velocity profiles
Another important feature of motor performance is that end-effectors typically follow bell-shaped or right-skewed bell-shaped velocity profiles (see Bullock and Grossberg, 1988, for review). The Knowledge model predicts such profiles, not only for end-effecters (e.g., the hand) but also for more proximal points along the limb-segment chain. Velocity -amplitude relations
Another important feature of motor performance is that the maximum velocity of the end-effector usually increases with the amplitude to be covered. This effect is observed in movements of the limbs, eyes, and speech articulators (see Bullock and Grossberg, 1988, for review). The Knowledge model predicts this positive relation because it contains a term in its Execution equation (07 in eq. (2)) which is positively related to the remaining amplitude of movement. Fitts’ Law
Because the Knowledge model predicts (asymmetric) bell-shaped velocity profiles, it also predicts that the hand approaches the target slowly at first, then more rapidly, and then more slowly again. Depending on the amount of inertia (indexed by 77 in eq. (2)), the hand may overshoot the target position, then undershoot it, then overshoot it by a smaller amount, then undershoot it by a smaller amount, and so on. When the hand actually ‘reaches the target’ depends on the size of the tolerance window around the target; the larger the window, the sooner the hand can be said to arrive at the target location. The latter observation can be coupled with the observation that though
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velocity increases with amplitude, movement time ‘increases with the distance between the hand’s starting and target locations. The fact that movement time increases with movement amplitude and decreases with target tolerance is consistent with Fitts’ Law (see Rosenbaum, 1991, for a review of Fitts’ Law). 7 Hand-path curvature
The Knowledge model also predicts changes in the curvature of hand trajectories depending on the hand’s initial and final locations. Such curvature changes have been reported by Atkeson and Hollerbath (1985). The model predicts such changes because it emphasizes the avoidance of movements demanding large Travel Costs. For tasks requiring movements from particular starting locations to particular target locations, a consequence of avoiding large Travel Costs is that the hand sometimes follows curved paths rather than straight paths. For example, in simulations we have conducted, movements that take the hand from a point in front of the stomach to a point in front of the forehead are curved; Atkeson and Hollerbach observed such paths in human subjects. The Knowledge predicts curved paths for such movements because, of the postures that leave the hand near the forehead, those that have the lowest Travel Costs entail little hip or shoulder rotation but considerable elbow rotation. Hence, reaching the target posture entails considerable elbow movement, which produces curved hand paths. By contrast, movements that take the hand from a point just in front of the stomach to a point horizontally farther from the stomach (at arm’s length) are nearly straight; Atkeson and Hollerbach observed such paths as well in their study. The Knowledge model predicts straight hand paths for such movements because, of the ’ To determine whether the movement-time data from the Knowledge model are quantitatively consistent with Fitts’ Law, we crossed several amplitudes with several tolerance windows, using one set of arbitrary inertia and proposed-angular-velocity parameters. Then we determined, from the obtained position-versus-time functions, how long it took the hand to arrive at a location such that the distance between the hand and the center of the target remained at or within that distance from that time on; we regarded this distance as half the width of the tolerance window. The arrival times were treated as movement times. When we plotted movement times against Fitts’ Index of Difficulty, log, (2A/W), where A denotes the distance between the hand and the center of the target, and W denotes the width of the tolerance window, we found that a straight line accounted for 82% of the variance.
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postures that leave the hand directly in front of the torso, those that have the lowest Travel Cost entail little or no hip rotation but demand considerable shoulder as well as elbow rotation. The two-joint horizontal motion yields a much straighter hand path than the one-joint vertical motion.
Extensions
As implied above, we believe that the Knowledge model is promising. We therefore plan to extend it. Three extensions are described below. Forward kinematics
Recall that in the current version of the Knowledge model, postures are stored as joint vectors indexed by their corresponding hand locations in Cartesian space. The problem with this approach is that it makes it difficult to obtain spatial information about points removed from the hand. For example, it would be difficult to tell where in the space occupied by the body a mosquito has bitten the skin. However, we can swat immediately at mosquitos biting unpredictable skin locations, even if the mosquitos bite on moving limbs. This observation suggests that we can instantly determine where in extracorporeal space a given point on the body is located. It is difficult to imagine how a table look-up system like the one used in the model could support such performance. To remedy this problem, we need only assume that it is possible to compute locations of points of interest based on knowledge of joint angles and limb lengths - that is, to compute forward kinematics and to define Spatial Errors on this basis. The advantage of this approach is that if a task requires bringing some point on the body other than the hand to a spatial target, forward kinematics can be used to determine what postures will be effective. Another advantage is that the system can engage in tool use: if the length of a hand-held tool is known, then postures can be selected that permit the end of the tool (or any point on the tool) to be brought to a desired location.
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Error correction
A second extension of the model concerns error correction. Currently, the model provides no means for the hand to reach a target if it has missed a target after an initial reach. Alternative extensions to the model can correct this problem. The most obvious extension is to try moving from the missed location to the target. The idea is to treat the posture at the end of the first movement as the starting posture for the next movement. Presumably, the hand will get closer to the target as this process is repeated. ’ Another possibility is to allow for mental planning prior to movement. Here a target posture is selected, as in the usual Planning process, but an actual movement to that target posture is initiated only if the Total Cost of the posture is acceptably low. If it is not, a second movement is planned from the first target posture. If the Total Cost of the second target posture is acceptably low, a series of two movements to the first and second target postures is initiated. However, if the Total Cost of the second target posture is not acceptably low, a third target posture is selected, and so forth. This hypothesized process makes two predictions. First, planning times should be long for movements that must have low Total Costs. Second, delays between submovements toward spatial targets should be shorter than delays before the first submovement. The latter prediction follows from the fact that full planning is only necessary before the series of submovements has begun. Whereas the above planning method predicts a series of submovements toward a target, another possible method does not. Here mental planning is used, but once a target posture has been found to be too costly, it is merely added to the stock of stored postures and the problem of reaching the original spatial target from the original starting posture is worked on again, this time with the new stored posture treated as a candidate posture. If the target posture arrived at in the second planning cycle is too costly, it is stored, the problem is worked on again, and so forth. With this procedure, more planning time should occur as more accurate and/or efficient movements are s In using this approach, it may be necessary to store the posture adopted at the end of a given movement so it can contribute to the determination of subsequent weighted averages.
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required, but in contrast to the method described above, a single movement rather than a series of submovements should be made toward the target location. Intentional learning
The final extension to the Knowledge model concerns learning. Recall that in the current version of the model postures are stored both through motor babbling and deliberate aiming. It would be helpful, from the standpoint of parsimony, to dispense with this distinction. Fortunately, it is possible to do so using a fully intentional learning system. Suppose that at birth the motor system can carry out the computations assumed in the Knowledge model and that it has stored postures corresponding only to extreme joint-angle combinations. For the 3 degree-of-freedom system we have modelled, eight postures would be innately specified, corresponding to all possible combinations of extreme hip, shoulder, and elbow angles. 9 These extreme joint-angle combinations would be necessary to constrain the values that can be arrived at through weighted averaging. Spatial targets would be presented one after the other, and for each spatial target a weighted average of the already-stored postures would be found. Because the weighted average arrived at at any given time would be stored, the system would ultimately develop a set of stored postures that is as large as the storage capacity of the system. The density of stored postures in different regions of joint space would reflect the system’s learning history: Those regions that house postures that are well suited for the tasks that have been presented would have many stored postures, whereas those regions that house postures that are less well suited for the tasks that have been presented would have fewer stored postures. A consequence of these differences in storage density would be that when new spatial targets are presented, the accuracy of the movements that can be performed should be high if the relevant regions of joint space have many stored postures, but if the relevant regions of joint space have few stored postures, the accuracy of the movements that can be performed should be low. In effect, the system 9 It is convenient to think of these eight postures whose axes are hip, shoulder, and elbow angles.
as occupying
the corners
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should show expertise. Depending on the system’s forgetting rules, its expertise should change with the history of tasks with which it has been confronted. lo
References Alexander, R.M., 1984. Walking and running. American Scientist 72, 348-354. Atkeson, C.G. and J.M. Hollerbach, 1985. Kinematic features of unrestrained arm movements. The Journal of Neuroscience 5, 2318-2330. Bernstein, N., 1967. The coordination and regulation of movements. London: Pergamon. Bizzi, E., F.A. Mussa Ivaldi and S. Giszter, 1991. Computations underlying the execution of movement: A biological perspective. Science 253, 287-291. Bullock, D. and S. Grossberg, 1988. Neural dynamics of planned arm movements: Emergent invariants and speed-accuracy properties during trajectory formation. Psychological Review 9549-90. Flash, T. and N. Hogan, 1985. The coordination of arm movements: An experimentally confirmed mathematical model. The Journal of Neuroscience 5, 1688-1703. Jordan, M.I. and D.A. Rosenbaum, 1989. ‘Action’. In: M.I. Posner (ed.), Foundations of cognitive science (pp. 727-767). Cambridge, MA: MIT Press. Lee, C., W.H. Rohrer and D.L. Sparks, 1988. Population coding of saccadic eye movements by neurons in the superior colliculus. Nature 332, 357-360. Newell, A.M. and P.S. Rosenbloom, 1981. ‘Mechanisms of skill acquisition and the law of practice’. In: J.R. Anderson (ed.), Cognitive skills and their acquisition (pp. l-55). Hillsdale, NJ: Erlbaum. Rosenbaum, D.A.,‘1991. Human motor control. San Diego, CA: Academic Press. Rosenbaum, D.A., S.E. Engelbrecht, M.M. Bushe and L.D. Loukopoulos, in press. Knowledge model for selecting and producing reaching movements. To appear in a special theme issue, Control of Arm and Hand Posture and Movement, of Journal of Motor Behavior. Rosenbaum, D.A., J.D. Slotta, J. Vaughan and R.J. Plamondon, 1991. Optimal movement selection. Psychological Science 2, 86-91. Saltzman, E. and J.A.S. Kelso, 1987. Skilled actions: A task-dynamic approach. Psychological Review 94, 84-106. Smyth, M.M., 1984. ‘Memory for movements’. In: M.M. Smyth and A.M. Wing feds.), The psychology of human movement (pp. 83-117). London: Academic Press. Uno, Y., M. Kawato and R. Suzuki, 1989. Formation and control of optimal trajectory in human multijoint arm movement: Minimum torque-change model. Biological Cybernetics 61,89-101. lo A possible forgetting rule is as follows: After a certain number of postures have been learned, forget the oldest stored posture, except the eight postures that were innately specified.