| Gribble, P.L., Ostry, D.J., Sanguineti, V., Laboissiere, R. Are complex control signals required for human arm movement? Journal of Neurophysiology 1998 (79):1409-1424 [pdf] |
| It has been proposed that the control signals underlying voluntary human arm movement have a "complex" nonmonotonic time-varying form, and a number of empirical findings have been offered in support of this idea. In this paper, we address three such findings using a model of two-joint arm motion based on the lambda version of the equilibrium-point hypothesis. The model includes six one- and two-joint muscles, reflexes, modeled control signals, muscle properties, and limb dynamics. First, we address the claim that "complex" equilibrium trajectories are required to account for nonmonotonic joint impedance patterns observed during multijoint movement. Using constant-rate shifts in the neurally specified equilibrium of the limb and constant cocontraction commands, we obtain patterns of predicted joint stiffness during simulated multijoint movements that match the nonmonotonic patterns reported empirically. We then use the algorithm proposed by Gomi and Kawato to compute a hypothetical equilibrium trajectory from simulated stiffness, viscosity, and limb kinematics. Like that reported by Gomi and Kawato, the resulting trajectory was nonmonotonic, first leading then lagging the position of the limb. Second, we address the claim that high levels of stiffness are required to generate rapid single-joint movements when simple equilibrium shifts are used. We compare empirical measurements of stiffness during rapid single-joint movements with the predicted stiffness of movements generated using constant-rate equilibrium shifts and constant cocontraction commands. Single-joint movements are simulated at a number of speeds, and the procedure used by Bennett to estimate stiffness is followed. We show that when the magnitude of the cocontraction command is scaled in proportion to movement speed, simulated joint stiffness varies with movement speed in a manner comparable with that reported by Bennett. Third, we address the related claim that nonmonotonic equilibrium shifts are required to generate rapid single-joint movements. Using constant-rate equilibrium shifts and constant cocontraction commands, rapid single-joint movements are simulated in the presence of external torques. We use the procedure reported by Latash and Gottlieb to compute hypothetical equilibrium trajectories from simulated torque and angle measurements during movement. As in Latash and Gottlieb, a nonmonotonic function is obtained even though the control signals used in the simulations are constant-rate changes in the equilibrium position of the limb. Differences between the "simple" equilibrium trajectory proposed in the present paper and those that are derived from the procedures used by Gomi and Kawato and Latash and Gottlieb arise from their use of simplified models of force generation. |
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| Lie, W., Todorov, E., Pan, X. Hierarchical Optimal Control of Redundant biomechanical systems 2004 |
| Sensorimotor control occurs simultaneously onmultiple levels. We present a general approach to designing feedback control hierarchies for redundant biomechanical systems, that approximate the (non-hierarchical) optimal control law but havemuch lower computational demands. The approach is applied to the task of reaching, using a detailed model of the human arm. Our hierarchy has two levels of feedback control. The high level is designed as an optimal feedback controller operating on a simplified virtual plant. The low level is responsible for transforming the dynamics of the true plant into the desired virtual dynamics. The new method may be useful not only for modelling the neural control ofmovement, but also for designing Functional Electric Stimulation systems that have to achieve task goals by activating muscles in real time. |
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| Kawato, M. Internal models for motor control and trajectory planning Current Opinion in Neurobiology 1999 (9):718-727 [html] |
| A number of internal model concepts are now widespread in neuroscience and cognitive science. These concepts are supported by behavioral, neurophysiological, and imaging data; furthermore, these models have had their structures and functions revealed by such data. In particular, a specific theory on inverse dynamics model learning is directly supported by unit recordings from cerebellar Purkinje cells. Multiple paired forward inverse models describing how diverse objects and environments can be controlled and learned separately have recently been proposed. The 'minimum variance model' is another major recent advance in the computational theory of motor control. This model integrates two furiously disputed approaches on trajectory planning, strongly suggesting that both kinematic and dynamic internal models are utilized in movement planning and control. |
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| Todorov, E., Jordan, M.I. Optimal feedback control as a theory of motor coordination Nature Neuroscience 2002 (5):11 [pdf] |
| Motor coordination the marshalling of redundant actuators in the service of a desired behavioral outcome is among the most important and least understood facets of motor function. Models that focus on mechanisms for achieving behavioral goals often fail to account for experimental data on movement variability and the exploitation of redundancy. Models that focus on variability and redundancy often fail to explain how goals are achieved in the first place. Here we show that not only are variability and goal achievement compatible, but indeed that allowing variability in redundant dimensions is the optimal strategy in the face of uncertainty. Our approach is based on stochastic optimal control theory, which provides, for a given task, the feedback control law that maximizes expected performance. This control law does not enforce a "desired trajectory" an approach that we show to be suboptimal but instead corrects only those deviations that interfere with the task goals. We find that the resulting behavior exhibits goal-directed adjustments, synergies, controlled parameters , simplifying rules , and discrete coordination modes none of which are built in a priori. Experimentally, we investigate a range of motor tasks and report patterns of variability in close agreement with the model. |
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| Todorov, E. Optimality principles in sensorimotor control Nature Neuroscience 2004 (7)9:907-915 [pdf] |
| The sensorimotor system is a product of evolution, development, learning and adaptation which work on different time scales to improve behavioral performance. Consequently, many theories of motor function are based on optimal performance : they quantify task goals as cost functions, and apply the sophisticated tools of optimal control theory to obtain detailed behavioral predictions. The resulting models, although not without limitations, have explained more empirical phenomena than any other class. Traditional emphasis has been on optimizing desired movement trajectories while ignoring sensory feedback. Recent work has redefined optimality in terms of feedback control laws, and focused on the mechanisms that generate behavior online. This approach has allowed researchers to fit previously unrelated concepts and observations into what may become a unified theoretical framework for interpreting motor function. At the heart of the framework is the relationship between high-level goals, and the real-time sensorimotor control strategies most suitable for accomplishing those goals. |
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| Harris, C.M., Wolpert, D.M. Signal-dependent noise determines motor planning Nature 1998 (394):780-784 [pdf] |
| When we make saccadic eye movements or goal-directed arm movements, there is an infinite number of possible trajectories that the eye or arm could take to reach the target. However, humans show highly stereotyped trajectories in which velocity profiles of both the eye and hand are smooth and symmetric for brief movements. Here we present a unifying theory of eye and arm movements based on the single physiological assumption that the neural control signals are corrupted by noise whose variance increases with the size of the control signal. We propose that in the presence of such signal-dependent noise, the shape of a trajectory is selected to minimize the variance of the final eye or arm position. This minimum-variance theory accurately predicts the trajectories of both saccades and arm movements and the speed accuracy trade-off described by Fitt s law. These profiles are robust to changes in the dynamics of the eye or arm, as found empirically. Moreover, the relation between path curvature and hand velocity during drawing movements reproduces the empirical two-thirds power law. This theory provides a simple and powerful unifying perspective for both eye and arm movement control. |
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