Page 162 - Introduction to Autonomous Mobile Robots

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Perception
µ = EX[] = ∫ ∞ xf x() x (4.53)
d
– ∞
Note in the above equation that calculation of EX[] is identical to the weighted average
x
of all possible values of . In contrast, the mean square value is simply the weighted aver-
age of the squares of all values of : x

∞ 2
2
[
EX ] = ∫ x fx() x (4.54)
d
– ∞
Characterization of the “width” of the possible values of is a key statistical measure,
X
and this requires first defining the variance σ 2 :
∞
2 2
Var X() = σ = ∫ ( x – µ) fx() x (4.55)
d
– ∞
Finally, the standard deviation is simply the square root of variance , and σ 2 will
σ
σ
play important roles in our characterization of the error of a single sensor as well as the error
of a model generated by combining multiple sensor readings.

4.2.1.1 Independence of random variables.
With the tools presented above, we often evaluate systems with multiple random variables.
For instance, a mobile robot’s laser rangefinder may be used to measure the position of a
feature on the robot’s right and, later, another feature on the robot’s left. The position of
each feature in the real world may be treated as random variables, X and X .
1 2
Two random variables X 1 and X 2 are independent if the particular value of one has no
bearing on the particular value of the other. In this case we can draw several important con-
clusions about the statistical behavior of X and X . First, the expected value (or mean
1 2
value) of the product of random variables is equal to the product of their mean values:
[
[
[
EX X ] = EX ]EX ] (4.56)
1 2 1 2
Second, the variance of their sums is equal to the sum of their variances:
Var X +( X ) = Var X ) + Var X ) (4.57)
(
(
1 2 1 2
In mobile robotics, we often assume the independence of random variables even when
this assumption is not strictly true. The simplification that results makes a number of the
existing mobile robot-mapping and navigation algorithms tenable, as described in