Page 111 - Introduction to Autonomous Mobile Robots

Chapter 4
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Multimodal error distributions. It is common to characterize the behavior of a sensor’s
random error in terms of a probability distribution over various output values. In general,
one knows very little about the causes of random error and therefore several simplifying
assumptions are commonly used. For example, we can assume that the error is zero-mean,
in that it symmetrically generates both positive and negative measurement error. We can
go even further and assume that the probability density curve is Gaussian. Although we dis-
cuss the mathematics of this in detail in section 4.2, it is important for now to recognize the
fact that one frequently assumes symmetry as well as unimodal distribution. This means
that measuring the correct value is most probable, and any measurement that is further
away from the correct value is less likely than any measurement that is closer to the correct
value. These are strong assumptions that enable powerful mathematical principles to be
applied to mobile robot problems, but it is important to realize how wrong these assump-
tions usually are.
Consider, for example, the sonar sensor once again. When ranging an object that reflects
the sound signal well, the sonar will exhibit high accuracy, and will induce random error
based on noise, for example, in the timing circuitry. This portion of its sensor behavior will
exhibit error characteristics that are fairly symmetric and unimodal. However, when the
sonar sensor is moving through an environment and is sometimes faced with materials that
cause coherent reflection rather than returning the sound signal to the sonar sensor, then the
sonar will grossly overestimate the distance to the object. In such cases, the error will be
biased toward positive measurement error and will be far from the correct value. The error
is not strictly systematic, and so we are left modeling it as a probability distribution of
random error. So the sonar sensor has two separate types of operational modes, one in
which the signal does return and some random error is possible, and the second in which
the signal returns after a multipath reflection, and gross overestimation error occurs. The
probability distribution could easily be at least bimodal in this case, and since overestima-
tion is more common than underestimation it will also be asymmetric.
As a second example, consider ranging via stereo vision. Once again, we can identify
two modes of operation. If the stereo vision system correctly correlates two images, then
the resulting random error will be caused by camera noise and will limit the measurement
accuracy. But the stereo vision system can also correlate two images incorrectly, matching
two fence posts, for example, that are not the same post in the real world. In such a case
stereo vision will exhibit gross measurement error, and one can easily imagine such behav-
ior violating both the unimodal and the symmetric assumptions.
The thesis of this section is that sensors in a mobile robot may be subject to multiple
modes of operation and, when the sensor error is characterized, unimodality and symmetry
may be grossly violated. Nonetheless, as we shall see, many successful mobile robot sys-
tems make use of these simplifying assumptions and the resulting mathematical techniques
with great empirical success.