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J Neurophysiol 80: 696-714, 1998;
0022-3077/98 $5.00
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The Journal of Neurophysiology Vol. 80 No. 2 August 1998, pp. 696-714
Copyright ©1998 The American Physiological Society

Smoothness Maximization Along a Predefined Path Accurately Predicts the Speed Profiles of Complex Arm Movements

Emanuel Todorov and Michael I. Jordan

Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Todorov, Emanuel and Michael I. Jordan. Smoothness maximization along a predefined path accurately predicts the speed profiles of complex arm movements. J. Neurophysiol. 80: 696-714, 1998. The speed profiles of arm movements display a number of regularities, including bell-shaped speed profiles in straight reaching movements and an inverse relationship between speed and curvature in extemporaneous drawing movements (described as a 2/3 power law). Here we propose a new model that simultaneously accounts for both regularities by replacing the 2/3 power law with a smoothness constraint. For a given path of the hand in space, our model assumes that the speed profile will be the one that minimizes the third derivative of position (or "jerk"). Analysis of the mathematical relationship between this smoothness constraint and the 2/3 power law revealed that in both two and three dimensions, the power law is equivalent to setting the jerk along the normal to the path to zero; it generates speed predictions that are similar, but clearly distinguishable from the predictions of our model. We have assessed the accuracy of the model on a number of motor tasks in two and three dimensions, involving discrete movements along arbitrary paths, traced with different limb segments. The new model provides a very close fit to the observed speed profiles in all cases. Its performance is uniformly better compared with all existing versions of the 2/3 power law, suggesting that the correlation between speed and curvature may be a consequence of an underlying motor strategy to produce smooth movements. Our results indicate that the relationship between the path and the speed profile of a complex arm movement is stronger than previously thought, especially within a single trial. The accuracy of the model was quite uniform over movements of different shape, size, and average speed. We did not find evidence for segmentation, yet prediction error increased with movement duration, suggesting a continuous fluctuation of the "tempo" of discrete movements. The implications of these findings for motor planning and on-line control are discussed.




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