Page 230 - Introduction to Autonomous Mobile Robots

Mobile Robot Localization

5.6.2.1 Introduction: applying probability theory to robot localization 215
Given a discrete representation of robot positions, in order to express a belief state we wish
to assign to each possible robot position a probability that the robot is indeed at that posi-
tion. From probability theory we use the term p A() to denote the probability that is true.
A
A
A
This is also called the prior probability of because it measures the probability that is
true independent of any additional knowledge we may have. For example we can use
(
l
p r = l) to denote the prior probability that the robot r is at position at time . t
t
In practice, we wish to compute the probability of each individual robot position given
the encoder and sensor evidence the robot has collected. In probability theory, we use the
term p AB( ) to denote the conditional probability of given that we know . For exam-
B
A
ple, we use p r =( li ) to denote the probability that the robot is at position given that
l
t t
the robot’s sensor inputs . i
The question is, how can a term such as p r =( t li ) be simplified to its constituent parts
t
so that it can be computed? The answer lies in the product rule, which states
p A ∧ B) = p AB)p B() (5.18)
(
(
A
B
Equation (5.18) is intuitively straightforward, as the probability of both and being
B
true is being related to being true and the other being conditionally true. But you should
be able to convince yourself that the alternate equation is equally correct:
(
p A ∧ B) = p BA)p A() (5.19)
(
Using equations (5.18) and (5.19) together, we can derive the Bayes formula for com-
puting p AB( : )
(
p BA)p A()
(
p AB) = ------------------------------ (5.20)
pB()
We use the Bayes rule to compute the robot’s new belief state as a function of its sensory
inputs and its former belief state. But to do this properly, we must recall the basic goal of
L
the Markov localization approach: a discrete set of possible robot positions are repre-
sented. The belief state of the robot must assign a probability p r =( l) for each location l
t
L
in .
The See function described in equation (5.17) expresses a mapping from a belief state
and sensor input to a refined belief state. To do this, we must update the probability asso-
ciated with each position in , and we can do this by directly applying the Bayes formula
L
l
l
to every such . In denoting this, we will stop representing the temporal index for sim-
t
plicity and will further use p l() to mean p r =( l) :