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Mobile Robot Localization
density function. So the resulting 1000 locations will be concentrated primarily at the high-
est probability locations. This biasing is desirable, but only to a point.
We also wish to ensure that some less likely locations are tracked, as otherwise, if the
robot does indeed receive unlikely sensor measurements, it will fail to localize. This ran-
domization of the sampling process can be performed by adding additional samples from a
flat distribution, for example. Further enhancements of these randomized methods enable
the number of statistical samples to be varied on the fly, based, for instance, on the ongoing
localization confidence of the system. This further reduces the number of samples required
on average while guaranteeing that a large number of samples will be used when necessary
[68].
These sampling techniques have resulted in robots that function indistinguishably as
compared to their full belief state set ancestors, yet use computationally a fraction of the
resources. Of course, such sampling has a penalty: completeness. The probabilistically
complete nature of Markov localization is violated by these sampling approaches because
the robot is failing to update all the nonzero probability locations, and thus there is a danger
that the robot, due to an unlikely but correct sensor reading, could become truly lost. Of
course, recovery from a lost state is feasible just as with all Markov localization techniques.

5.6.3 Kalman filter localization
The Markov localization model can represent any probability density function over robot
position. This approach is very general but, due to its generality, inefficient. Consider
instead the key demands on a robot localization system. One can argue that it is not the
exact replication of a probability density curve but the sensor fusion problem that is key to
robust localization. Robots usually include a large number of heterogeneous sensors, each
providing clues as to robot position and, critically, each suffering from its own failure
modes. Optimal localization should take into account the information provided by all of
these sensors. In this section we describe a powerful technique for achieving this sensor
fusion, called the Kalman filter. This mechanism is in fact more efficient than Markov
localization because of key simplifications when representing the probability density func-
tion of the robot’s belief state and even its individual sensor readings, as described below.
But the benefit of this simplification is a resulting optimal recursive data-processing algo-
rithm. It incorporates all information, regardless of precision, to estimate the current value
of the variable of interest (i.e., the robot’s position). A general introduction to Kalman fil-
ters can be found in [106] and a more detailed treatment is presented in [3].
Figure 5.25 depicts the general scheme of Kalman filter estimation, where a system has
a control signal and system error sources as inputs. A measuring device enables measuring
some system states with errors. The Kalman filter is a mathematical mechanism for produc-
ing an optimal estimate of the system state based on the knowledge of the system and the
measuring device, the description of the system noise and measurement errors and the