An Evolutionary MAP Filter for Mobile Robot Global Localization

Abstract

The proposed algorithm has a range of interesting characteristics: Evolutive filters can deal with non-linear state space dynamics and noise distributions. Due to the set of solutions does not try to approximate posterior density distributions, it does not requires any assumptions on the shape of the posterior density as parametric approaches do. Evolutive filter focus computational resources in the most relevant areas, by addressing the set of solutions to the most interesting areas according to the fitness function obtained. The number of tentative solutions required in the evolving set is much lower than those required in particle filters, and similarly to those filters the evolving set can be reduced when the algorithm has converged to a reduced area around the best estimate. The size of the minimum solution's set required to guaranty the convergence of the evolutive filter to the true solution is low. Due to the stochastic nature of the algorithm search of the best robot's pose estimate the algorithm is able to cope a high level of sensor noise with low degradation of the estimation results. The algorithm is easy to implement, and the computational cost makes it able to operate on line even in relatively big areas. The proposed method has been tested under real conditions in a B-21 mobile robot. Acknowledgment This research is sponsored by Spanish Government (MICYT) under agreement number DPI2003-01170.

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An Evolutionary MAP Filter for Mobile Robot
Global Localization
L. Moreno1, S. Garrido1 , M. L. Muñoz2, and D. Blanco1
1 Robotic's Laboratory, Universidad Carlos III, Madrid
2 Facultad de Informática, Universidad Politécnica, Madrid
Spain
1. Introduction
This chapter presents a new evolutionary algorithm for Mobile Robot Global Localization.
The Evolutive Localization Filter (ELF) presented in this paper is a non linear filter
algorithm which is able to solve the global localization problem in a robust and efficient
way. The proposed algorithm searches along the configurations space for the best robot
pose estimate. The elements of each generation are the set of pose solutions and represent
the areas with more probability according to the perception and motion information up to
date. The population evolves according to the observation and the motion error derived
from the comparison between observed and predicted data obtained from the probabilistic
perception and motion model. The algorithm has been tested using a mobile robot with a
laser range finder to demonstrate the effectiveness, robustness and computational efficiency
of the proposed approach.
Mobile robot localization finds out a robot’s coordinates relatives to its environment,
assuming that one is provided with a map of the environment. Localization is a key
component in navigation and required to execute successfully a trajectory. We can
distinguish two different cases: the re-localization case and the global localization case. Re-
localization or tracking problem tries to keep track of mobile robot´s pose, where the robot
knows its initial position (at least approximately) and therefore has to maintain localized the
robot along the given mission. The global localization problem does not assume any
knowledge about the robot’s initial position and therefore has to globally localize itself.
The two most important aspects that have to be dealt with when designing a localization
system is how to represent uncertain information of the environment and the robot’s pose.
Among the many ways to represent the knowledge about an environment, this article deals
with geometrical localization methods and assumes that the environment is modelled
geometrically as an occupancy grid map.
In the robot’s pose uncertainty representation and estimation techniques, the vast majority
of existing algorithms address only the position tracking problem. In this case the small
incremental errors produced along the robot motion and the initial knowledge of the robot’s
pose makes classical approaches such as Kalman filters or Scan Matching techniques
applicable. If we consider the robot’s pose estimation as a Bayesian recursive problem,
Kalman filters estimate posterior distribution of robot’s poses conditioned on sensor data.
Source: Frontiers in Evolutionary Robotics, Book edited by: Hitoshi Iba, ISBN 978-3-902613-19-6, pp. 596, April 2008, I-Tech Education
and Publishing, Vienna, Austria
Open Access Database www.intehweb.com

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Based on the Gaussian noise assumption and the Gaussian-distributed initial uncertainty
this method represents posteriors distributions by Gaussians. Kalman filter constitutes an
efficient solution for re-localization problems. However, the assumption’s nature of the
uncertainty representation makes Kalman filters not robust in global localization problems.
Scan Matching techniques are also an iterative local minimization technique and can not be
used for global localization.
Different families of algorithms can solve the global localization problem, some frequently
used are: multi-hypothesis Kalman filters, grid-based probabilistic filters and Monte Carlo
localization methods. Those methods can be included in a wider scope group of Bayesian
estimation methods. Multi-hypothesis Kalman filters (Arras et al., 2002, Austin & Jensfelt,
2000, Jensfelt & Kristensen, 1999, Cox & Leonard, 1994, Roumeliotis et. al, 2000) represent
distributions using mixtures of Gaussians, enabling them to keep track of multiple
hypothesis, each of which is represented by a separate Gaussian. This solution presents
some initialization problems: one of them is the determination of the initial hypotheses (the
number can be very high and it is not bounded), this leads the algorithm to a high
computational cost at initial stages. Besides, the Kalman filter is essentially a gradient based
method and consequently poorly robust if the initial hypothesis is bad or noise assumptions
fails. Grid-based localization algorithms (Fox et al., 1999, Burgard et al., 1996, Reuter, 2000)
represent distributions by a discrete set of point probabilities distributed over the space of
all possible poses. This group of algorithms are capable of representing multi-modal
probabilities distributions. A third group is the Monte Carlo localization algorithms (Jensfelt
et al., 2000, Thrun et al., 2001, Dellaert et al., 1999). These algorithms represent the
probability distribution by means of a set of samples drawn according to the posterior
distribution over robot’s poses. These algorithms can manage arbitrary noise distributions
and non-linearities in the system and observation models. These methods present a high
computational cost due to its probabilistic nature requires a high number of samples to
draw properly the posterior probability density function. The main advantage is its
statistical robustness.
This article presents a localization algorithm based on a non-linear filter called Evolutionary
Localization Filter (ELF). ELF solves the global localization robot problem in a robust and
efficient way. The algorithm can deal with arbitrary noise distributions and non-linear space
state systems. The key idea of ELF is to represent the uncertainty about the robot’s pose by a
set of possible pose estimates weighted by a fitness function. The state is recursively
estimated using set of solutions selected according on the weight associated to each possible
solution included in the set. The set of solutions evolve in time to integrate the sensor
information and the robot motion information. The adaptation engine of the ELF method is
based on an evolutive adaptation mechanism which combines a stochastical gradient search
together with probabilistic search to find the most promising pose’s candidates.
2. Differential evolutionary filter
The evolutive optimization techniques constitute a series of probabilistic search methods
that avoid derivatives or probability density estimations to estimate the best solution to a
localization problem. In the method proposed here each individual in the evolutive
algorithm will represent a possible solution to the localization problem and the value of the
loss function represent the error to explain the perceptual and motion data. The search of
this solution is done stochastically employing an evolutive search technique based on the

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differential evolution method proposed by Storn and Price (Storn & Price, 2001) for global
optimization problems over continuous spaces. The Evolutive Filter uses a a parallel direct
search method which utilizesn dimensional parameter vectors
each candidate solutioni to the optimization problem at iteration stepk . This method
utilizes N parameter vectors{ 0 1}i… N
; = , , ,
optimization process. Each element of the population set represents a possible solution, but
it hasn’t associated a probability value to each one (in the particle filter case, each element of
the particle set has associated a probability value).
The initial population is chosen randomly to cover the entire parameter space uniformly. In
absence of a priori information the entire parameter space has the same probability of
containing the optimum parameter vector, and a uniform probability distribution is
assumed. The differential evolution filter generates new parameter vectors by adding the
weighted difference vector between two population members to a third member. If the
resulting vector yields a lower objective function value than a predetermined population
member, the newly generated vector replaces the vector with which it was compared;
otherwise, the old vector is retained. This basic idea is extended by perturbing an existing
vector through the addition of one or more weighted difference vectors to it.
The perturbation scheme generate a variation v according to the following expression,
1
()
kkkT
iii n
,
xx … x
,
,
=,
to point
k
ix
as a population for each generation t of the

23
()()
k
i
k
b
k
i
k
r
k
r
vx L xx F xx
=+−+−
(1)
where
k
ix is the parameter vector to be perturbed at iteration k ,
k
bx is the best parameter
vector of the population at iteration k ,
2
k
rx and
3
k
rx are parameter vectors chosen randomly
from the population and are different from running index i . L andF are real and constant
factors which controls the amplification of the differential variations()
k
b
k
i
xx
−

and
23
()
kk
rr
xx
−
. This expression has two different terms: the()
k
b
k
i
xx
−
term is a kind of
stochastical gradient while
23
()
k
r
k
r
xx
−
is a kind of random search.
In order to increase the diversity of the new generation of parameter vectors, crossover is
()
iii i D
uuu … u
,,,
=, , ,
the new parameter vector with
introduced. Denote by
12
kkkkT

if
otherwise
k
k
i j
,
i j
,
k
i j
,
k
i j
,
vp
u
x
δ<
=⎧
⎨
⎩
(2)
where
population member i at step k and δ is the crossover probability and constitutes the
crossover control variable. The random values
k
i j
p, is a randomly chosen value from the interval [0 1]
,
for each parameter j of the
k
i j
p, are made anew for each trial vectori .

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To decide whether or not vector
k
iu should become a member of generation
1
i + , the new
vector is compared to
k
ix . If vector
k
iu yields a better value for the objective fitness function
than
generation. The general idea of the previous mechanism: mutation, crossover and selection
are well known and can be found in literature (Goldberg 1989).
k
ix , then is replaced by
1
k
iu
+; otherwise, the old value
k
ix is retained for the new
xt
xt
r3
xt
b
i
new parameter vector
v
L( - )
xt
bxt
i
K( - )
r2
xt
r3
xt
xt
r2
minimum

Figure 1. New population member generation
A. Fitness function
Due to we are trying to localize a mobile robot, the natural choice for fitness function is the
sum of squared errors function. If the observation vector at time t is
1
()
T
tt p t
zz … z
, ,
,,
=

and the predicted observations according the estimated robots pose is
then the penalty function can be stated as
1
()
ˆ
z
ˆ
z
ˆ
z
T
tt p t
…
, ,
,,
=


()( ) ()
ˆ
x
ˆ
z
ˆ
z
T
tt
t
ttt
L xzz
−=−−
(3)
In the global localization problem there exist some aspects that make this fitness function
difficult to manage:
•
The range and accuracy of the sensor limits the possibility of discriminate between
different poses, leading the fitness function to a high number of global maxima.
•
The number of sensors limits the possibility of discriminate between robot’s poses
leading to multiple global maxima in the fitness function.

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•
The geometrical similarities in the environment due to the repetition of the space
distribution originate the presence of a high number of possible robot’s pose solutions
to the mean square loss function.
The loss function defined in this way provides us with an optimization mechanism which
obtain an unstable parameter solution and by extension an unstable filter. The instability of
an evolutive filter based on this penalty function derives from the environment ambiguities.
Due to that the evolutive filter can move from one local minimum to other, originating
abrupt changes in the pose estimate. This problem comes from the fact we are not using all
information we know about the system at the loss function. In fact, we only use the
observation model to predict the sensor measurements at each position, and these predicted
measurements together with the actual measurements are introduced in the loss function to
evaluate the robot’s pose estimate. The instability originated by the existence of multiple
solutions to the loss function, can not be solved by considering only the available sensor
observations. One possibility consists of introducing in the loss function all the information
available about the system.
The solution proposed introduced all available information in two ways:
•
State space model information.
State transition model and observation model are used to estimate the robot’s pose and
robot’s observations next cycle. If we define
obtained in the last iteration of the algorithm at stept . According to the robot motion model
the estimated position at step
1
t + according to the odometric information available will be
i t
x, as a possible robot’s pose solution i

1
()
i t
, +
i t
,
t
x g xu
=,
(4)
This information is used for moving the poses of all population members {
the last iteration m of the evolutive filter at stept . A faster approach, from a computational
point of view, consist of estimating only the movement of the best estimate
ˆ
ˆˆ
tt
t
xx
d
+
=−
and then to apply to all population member. The
initial population at step
1
t + will be
}
m
it
x
obtained in
1
()
ˆ
x
ˆ
x
tt
t
gu
+=,

to estimate a displacement
1

0
i1
ˆ
{ }
x
{}
m
ittt
x
d
+=+
(5)
In a similar form the observation model together with the state estimate let us predict the
sensor observation values
( )
ˆ
ˆt
t
hx
z =
.
•
Probabilistic information.
To eliminate the ambiguity, the statistical information available have to be considered. This
information is included into the loss function. One possible choice is to include the distance
ˆt x + and each candidate solution
step ; in the loss function together with the observation error between sensor prediction
and sensor measurement.
between the estimated robot’s pose
1
k
ix at iteration k at