The velocity profiles for temporally constrained aimed movements have a symmetrical bell shape. No matter the speed, timed movements produce symmetrical velocity profiles.

Results#

Behavioural evidence

Velocity profiles

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Robin T. Bye* and Peter D. Neilson

Variable horizon control predicts speed-accuracy tradeoffs and velocity profiles in aimed movements

* Corresponding author. E-mail: robin.bye@student.unsw.edu.auNEUROENGINEERING @ UNSW

UNSW

5. SimulatorA Simulink model has been developed that by simulating intermittent adaptive optimal control in the central nervous system reproduces the logarithmic and linear speed-accuracy tradeoff with corresponding asymmetrical and symmetrical velocity profiles by employing receding and fixed horizon control, respectively.

6. Speed-accuracy tradeoffWhen humans perform aimed movements, a speed-accuracy tradeoff occurs: Increasing movement speed normally decreases accuracy whereas increased accuracy requires the movement to slow down. A variety of speed-accuracy tradeoffs has been found. Apart from the two most important tradeoffs, namely the logarithmic and linear speed-accuracy tradeoffs, variants of these two as well as power laws, quadratic laws, and the delta-lognormal law have been proposed.

2. Adaptive Model TheoryAdaptive model theory (AMT) is a computational theory about information processing within the perceptual-motor loop involved in the control of purposive movement. It proposes the adaptive formation of task-dependent feedback/feedforward controllers able to generate stable, noninteractive control and render nonlinear interactions unobservable in sensory-motor relationships. These controllers exist in the central nervous system as neural adaptive filters based on cerebellar circuitry.

6a. Logarithmic tradeoffSpatially constrained movements (movements that must end within a target region, typically as fast as possible) yield a logarithmic tradeoff formalised as Fitts’ law:

T = a + b log2(2D/W) = a + b Id

The movement time T increases logarithmically with an increase in target distance D and target width W. The log-term is often substituted with the index of difficulty Id = log2(2D/W).

6b. Linear tradeoffTemporally constrained movements (movements that must reach target as well as have a pre-specified duration) result in a linear tradeoff:

We = a + b (D/T) T ≈ b D/We (a small)

The standard deviation of movement endpoint We, termed effective target width analogous to target width W in Fitts’ law, increases linearly with increased target distance D and decreased movement time T.

7a. Receding horizon controlThe prediction horizon is held constant throughout the movement leading to a logarithmic speed-accuracy tradeoff and an asymmetrical (left-skewed) velocity profile that match that of spatially constrained aimed movements.

7b. Fixed horizon controlAt every planning interval the prediction horizon is reduced with the distance moved during the interval until the target is reached, leading to a linear tradeoff and a symmetrical velocity profile that match that of temporally constrained movements.

Behavioural evidence for the linear tradeoff that occurs in temporally constrained movements exists not only in single tapping tasks, but also

Fitts’ law holds for tapping, throwing, and rotating movements both on land, under water, and in aircraft flights. Subjects include children, young and

The velocity profiles for spatially constrained aimed movements are asymmetrically (left-skewed) bell-shaped. The profile becomes increasingly asymmetrical as movement is slowed.

3. Intermittent response planningThe response planning (RP) system in AMT operates intermittently. During each planning interval (100 ms) an optimal minimum acceleration trajectory is generated while at the same time the previously planned trajectory is being executed as well as afferent signals being stored. This is achieved by using parallel array processing readily available in the parallel structures of the central nervous system.

7. Variable horizon controlThe RP system can vary the prediction horizon for the response to be generated at each planning interval. At the extremes lie the receding and fixed horizon control strategies, which yield a logarithmic and linear speed-accuracy tradeoff, respectively. Combinations of the two strategies can lead to other desired goals such as minimising the infinity norm of the error or the integral of the error squared used in LQR optimal control.

Logarithmic tradeoff Asymmetric profile Symmetric profile Linear tradeoff# Simulation results for a 500 ms prediction horizon

Neuroengineering Laboratory EE223, Systems & Control Research Group, School of Electrical Engineering & Telecommunications, University of New South Wales, UNSW Sydney NSW 2052 AUSTRALIA

elderly adults, mentally challenged people, Parkinsonians, and drugged people.

in saccadic eye movements, wrist rotations, and most other time-matching tasks.

4. Noise in the nervous systemAMT suggests that broadband noise is introduced at the level of motor commands and that its standard deviation increases with the size of the motor command. It is this stochastic noise that causes every movement to be a little different from the previous in a repetitive movement task.

1. IntroductionThe principles underlying the speed-accuracy tradeoffs in rapid aimed movements have attracted major interest and controversy both experimentally and theoretically for over a century. While the logarithmic tradeoff, Fitts' law, holds for movements emphasizing spatial accuracy, the linear tradeoff occurs for movements emphasizing both spatial and temporal accuracy. Typically, the first kind of movements produces asymmetrical velocity profiles, whereas the latter kind produces symmetrical velocity profiles. We propose that the combination of intermittent response planning using variable horizon predictive control and the existence of signal-dependent noise in the nervous system predicts both kinds of speed-accuracy tradeoffs as well as both kinds of velocity profiles.