Smoothness Maximization Along a Predefined Path Accurately Predicts the Speed Profiles of Complex Arm Movements
Abstract
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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This mixture of weighted error variance is related formally to phenomenological cost functions proposed to account for movement trajectories.
“This mixture of weighted error variance is related formally to phenomenological cost functions proposed to account for movement trajectories (cf.Todorov and Jordan 1998;Nakano et al. 1999;Berret et al. 2008).”
From Action and behavior: a free-energy formulation (2010) · cites this paper as Todorov and Jordan 1998
This mixture is formally similar to cost-functions in the motor-control literature that try to account for smooth motor trajectories observed empirically.
“This mixture is formally similar to cost-functions in the motor-control literature that try to account for smooth motor trajectories that are observed empirically (Todorov and Jordan 1998;Nakano et al. 1999;Berret et al. 2008).”
From Action and behavior: a free-energy formulation (2010) · cites this paper as Todorov and Jordan 1998
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- Authors
- Todorov, E., & Jordan, M. I.
- Affiliations
- Massachusetts Institute of Technology (US)
- ORCID
- Jordan, Michael I.
- Venue
- Journal of Neurophysiology
- Publisher
- American Physiological Society
- Published
- 1998-08-01
- Volume
- 80
- Issue
- 2
- Pages
- 696-714
- ISSN
- 0022-3077 (ISSN-L)
- 1522-1598
- OpenAlex
- W1851172369
- Cited by
- 241
- References
- 29
- Citations by year
- 2026: 8 · 2025: 12 · 2024: 14 · 2023: 6 · 2022: 8 · 2021: 9 · 2020: 7 · 2019: 5 · 2018: 7 · 2017: 5 · 2016: 12 · 2015: 13 · 2014: 11 · 2013: 10 · 2012: 11
- FWCI
- 1.06
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Neurophysiology (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Motor Control and Adaptation
- Physical Sciences › Engineering › Control and Systems Engineering › Human Motion and Animation
- Physical Sciences › Engineering › Biomedical Engineering › Muscle activation and electromyography studies
- Keywords
- 2/3 power law, smoothness constraint, jerk minimization, path curvature, reaching movements, drawing movements, motor planning, movement segmentation, movement tempo
- MeSH terms
- Arm, Humans, Models, Neurological, Movement, Psychomotor Performance, Reaction Time
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