FITTS' LAW
Fitts' Law is one of the most successful formulas in human factors research.
This law describes the time taken to acquire a visual target using some
kind of manual input device. Although there are many variants on Fitts'
law the most commonly used is
Mean Time = C1 + C2log2(D/W + 0.5) (1)
where D is the distance to the center of the target, W is the target width,
C1 and C2 are experimentally determined constants. Fitts' Law was originally
derived from information theory and recently MacKenzie has argued from this
perspective that a slight variation on this formula is more satisfying.
He replaced the 0.5 constant with a 1.0 constant so that the formula becomes:
Mean Time = C1 + C2log2(D/W + 1.0). (2)
Whichever variant on Fitts' Law is chosen, the value of the logarithmic
part of the expression, log2(D/W + 0.5) or log2(D/W + 1.0) is called the
index of difficulty (ID). Thus Fitts' Law can be expressed as
Mean Time = C1 + C2ID (3)
The quantity 1/C2 is called the index of performance, the units are bits
per second.
There is some evidence that the process modeled by Fitts' Law is a series
of movements each of which gets the hand guided probe closer to the target,
until the probe actually falls within the target area . In reality, the
hand will not come to be a complete stop, instead a series of corrections
will be applied in a dynamic feedback loop. This loop is illustrated in
Figure 1, where it can be seen that both human and machine components are
performed iteratively in series. According to this model the ID portion
of Fitts' Law can be interpreted as a measure of the average number of movements
(or movement corrections) required to acquire the target, or in other words
the number of times the main human-machine processing loop is executed.
Most Fitts' Law studies have assumed the machine processing lag to be zero.
However, this is clearly not the case for computer graphics or telerobotics
applications. We therefore modify Equation 3 so that it becomes:
Mean Time = C1 + C2(C3 + MachineLag)ID (4)
where C3 represents the human processing time required to make a corrective
movement, MachineLag represents the machine processing time, C2 ID represents
the average number of iterations of the control loop and C1 represents the
sum of the initial response time and the time required to confirm the acquisition
of the target. If an additional sensory or motor processing load is introduced
because the human operator is highly stressed (or tired) then any of the
human processing components C1, C2 or C3 may be increased. MacKenzie
and Ware found a three parameter model of this kind to be an excellent description
of the data from a one dimensional Fitts' Law experiment with lag, although
they did not interpret it in terms of a control loop. In a much earlier
study Sheridan and Ferrell proposed a similar open loop control model to
account for data derived from a task with machine lags of between zero and
three seconds.
2D and 3D Fitts' Law
The classical Fitts Law is model of one dimensional movement. MacKenzie
and Buxton tested a number of two dimensional variations on Fitts Law on
rectangular targets. They found two of these to be successful. In the
first the index of difficulty was modified by taking target width in the
two dimensions into account.
ID = log2(D/min(W1,W2) + 1.0) (5)
where W1 are W2 are the target sizes in the X direction and the Y direction
respectively, and D is the distance to the center of the target. Essentially
this rule states that performance is determined by the smaller of the two
target dimensions. This variation on Fitts' Law can be trivially extended
to three dimensional data.
ID = log2(D/min(W1,W2,W3) + 1.0) (6)
MacKenzie and Buxton's second model also modified the index of difficulty.
ID = log2(D/W'+ 1.0) (7)
where W' represents the thickness of the target in the direction of hand
motion.
Effective target width
With large targets the subject may always group the position of the target
hits well inside the target boundaries, whereas with a small target the
distribution may overlap the target boundaries. There is a variant on Fitts'
Law which is based on the idea of an "effective target width".
In calculating the index of difficulty the actual target width is replaced
by 4.13 times the standard deviation of the distribution of hits (representing
a 5% error rate).
IDs = log2(D/4.13s+ 1.0) (8)
where s represents the standard deviation of hits in the direction of movement.
This metric may provide a more accurate measure of the rate of information
processing achieved in the performance of controlled movement tasks; however,
if the goal is to predict performance in some particular situation, models
of performance which include the actual target dimensions may be preferable.
Lag and the Display Cycle
The basic display cycle used in interactive 3D graphics is as follows.
An input device is sampled immediately following the buffer swap. This
value is then used to construct the graphical image for the next frame of
the display and after this frame is constructed the buffers are switched
at the next available vertical blanking interval. If the image construction
time is 100 msec then a minimum of a 100 msec lag occurs before the effects
of that input are made visible. That image remains on the screen for another
100 msec. If we assume that perception occurs in the middle of the frame
interval then the total lag becomes:
MachineLag = DeviceLag + FrameInterval*1.5 (9)
At the current state of technology a display with a 10 Hz update rate and
a device lag of 60 msec (including communication delays) is fairly typical;
this will yield a total lag of 210 msec.
While the assumptions in the above estimate are probably reasonable for
rapid frame rates they become questionable when the frame-rate is low.
In this case it is probable that perception of the effect of a movement
occurs at some time before the middle of the frame interval, and in addition
the low rate of sampling the hand position may have adverse effects. For
example, at a 1 Hz frame rate an entire corrective movement may be missed.
Evidence suggests that the maximum rate of controlled forearm movement is
approximately 3 Hz and the Nyquist theorem requires that to sample this
we need at least a 6 Hz sampling rate, preferably more.
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