20
07
Variable horizon control predicts speed-accuracy
tradeoffs and velocity profiles in aimed movements
Robin T.
*
Bye
and Peter D. Neilson
1. Introduction
3. Intermittent response planning
The 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.
The 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.
2. Adaptive Model Theory
Adaptive 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.
4. Noise in the nervous system
AMT 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.
5. Simulator
A 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 tradeoff
6a. Logarithmic tradeoff
Spatially
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 logterm is often substituted with the index
of difficulty Id = log2(2D/W).
When
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 deltalognormal law have been proposed.
The velocity profiles for spatially constrained
aimed movements are asymmetrically (leftskewed) bell-shaped. The profile becomes
increasingly asymmetrical as movement is slowed.
Velocity
profiles
6b. Linear tradeoff
Temporally 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.
The velocity profiles for temporally constrained
aimed movements have a symmetrical bell
shape. No matter the speed, timed movements
produce symmetrical velocity profiles.
Fitts’ law holds for tapping, throwing, and rotating
Behavioural evidence for the linear tradeoff that
Behavioural
movements both on land, under water, and in
occurs in temporally constrained movements
evidence
aircraft flights. Subjects include children, young and
exists not only in single tapping tasks, but also
elderly adults, mentally challenged
in saccadic eye movements, wrist
people,
Parkinsonians,
and
rotations, and most other time7.
Variable
horizon
control
drugged people.
matching tasks.
The RP system can vary the prediction horizon for
the response to be generated at each planning
7b. Fixed horizon control
7a. Receding horizon control
interval.
At
the
extremes
lie
the
receding
and
At every planning interval the
The prediction horizon is held
fixed horizon control strategies, which yield a
prediction horizon is reduced with
constant throughout the movement
logarithmic and linear speed-accuracy tradeoff,
the distance moved during the
leading to a logarithmic speedrespectively.
Combinations
of
the
two
strategies
interval until the target is reached,
accuracy
tradeoff
and
an
can
lead
to
other
desired
goals
such
as
minimising
leading to a linear tradeoff and a
asymmetrical (left-skewed) velocity
the
infinity
norm
of
the
error
or
the
integral
of
the
symmetrical velocity profile that
profile that match that of spatially
error squared used in LQR optimal control.
match
that
of
temporally
constrained aimed movements.
constrained movements.
Results#
UNSW
Logarithmic tradeoff
# Simulation
Asymmetric profile
results for a 500 ms prediction horizon
* Corresponding author. E-mail: robin.bye@student.unsw.edu.au
Symmetric profile
Linear tradeoff
NEUROENGINEERING @ UNSW
Neuroengineering Laboratory EE223, Systems & Control Research Group, School of Electrical Engineering & Telecommunications, University of New South Wales, UNSW Sydney NSW 2052 AUSTRALIA