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Evolutionary Particle Filter: Re-sampling from the

Genetic Algorithm Perspective∗

N. M. Kwok1, Gu Fang2and Weizhen Zhou1

1ARC Centre of Excellence for Autonomous Systems

University of Technology, Sydney

Broadway, NSW, 2007, Australia

{ngai.kwok,wzhou}@eng.uts.edu.au

2School of Engineering and Industrial Design

University of Western Sydney

Penrith, NSW, 2747, Australia

g.fang@uws.edu.au

Abstract—The sample impoverishment problem in particle

filters is investigated from the perspective of genetic algorithms.

The contribution of this paper is in the proposal of a hy-

brid technique to mitigate sample impoverishment such that

the number of particles required and hence the computation

complexity are reduced. Studies are conducted through the use

of Chebyshev inequality for the number of particles required.

The relationship between the number of particles and the

time for impoverishment is examined by considering the take-

over phenomena as found in genetic algorithms. It is revealed

that the sample impoverishment problem is caused by the re-

sampling scheme in implementing the particle filter with a

finite number of particles. The use of uniform or roulette-wheel

sampling also contributes to the problem. Crossover operators

from genetic algorithms are adopted to tackle the finite particle

problem by re-defining or re-supplying impoverished particles

during filter iterations. Effectiveness of the proposed approach

is demonstrated by simulations for a monobot simultaneous

localization and mapping application.

Index Terms—particle filter, re-sampling, genetic algorithms,

selection.

I. INTRODUCTION

Particle filters [1] had been widely applied in estimation

problems containing non-linear system and non-Gaussian

noise models. The particle filter (PF) is, in principle, a

sample based implementation of Bayesian estimation [2].

Applications of PFs include those in mobile robot localization

and mapping, [3], fault diagnosis in nonlinear stochastic sys-

tems [4], user detection in wireless telecommunications [5],

speaker tracking in auditory applications [6] and many other

areas. Although the application of PFs to non-linear/non-

Gaussian systems has demonstrated satisfactory results, the

implementation complexity is prohibitively high for systems

with limited computing resources especially for dynamical

and high-dimensional systems. This difficulty arises because

∗This work is supported by the ARC Centre of Excellence programme,

funded by the Australian Research Council (ARC) and the New South Wales

State Government.

of the so-called sample impoverishment, i.e., the loss of

diversity for the particles to adequately represent the solution

space. An obvious solution to this problem is to use a large

number of particles at the beginning of the filtering process.

However, this increases the computational complexity.

In order to reduce the impoverishment effect, or the num-

ber of particles required, several approach had been proposed

in the literature. In [7], it is suggested to include, as imple-

mentation measures, sample boosting, smoothing and prior

editing. In sample boosting, the number of particles is in-

creased in an order of magnitude during an intermediate cal-

culation stage, then re-sampled to restore to the original size.

This method rather increases the computational complexity.

For the smoothing technique, particles are perturbed as the

virtual result of sampling from a continuous approximation

of the discrete states represented by the particles. However,

there may be difficulties in designing the continuous approx-

imation when the system is highly non-linear or cannot be

accurately modelled. In prior editing, the number of particles

is increased in regions of high likelihoods. This approach is

an advance from sample boosting by concentrating on the

promising solution regions, but the lack of knowledge on

locations and numbers of high likelihood regions may hinder

the success of this method. Since sample impoverishment is

mostly contributed from re-sampling, a test of the effective

particle number is checked before re-sampling is carried out,

see [8]. This method partially avoids the lost of particle

diversity, but there is still no recovery of particles once they

have impoverished.

It has been noted in many occasions in the literature, that

particle filters bear certain similar implementation charac-

teristics to that of genetic algorithms (GA), [9] and [10].

In [11], the application of sampling algorithms is treated

as the survival of the fittest inspired by the Theory of

Evolution from which the GA is developed. Furthermore, in

[12], the connection between PF and GA was established

from the Monte Carlo simulation view point. On the other

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hand, incorporation of Bayesian framework into evolutionary

computation was proposed in [13] with performance im-

provements in function optimizations. More recent work in

the hybridization of PF and GA can be found in [14] and [15]

where function optimization problems are addressed. In these

research works, the application of hybridizedGA/PF has indi-

cated an attractive research direction in combining estimation

and optimization. Within the mobile robotics research area,

application of GA was found in [16]. It adopted the GA

to enhance the estimations from an extended Kalman filter

(EKF) but the implementation of the GA was not specifically

addressed. In [17], a GA with a simple fitness function design

was applied in mobile robot localization and mapping but

insights into the algorithm were not reported either.

In the PF implementation,uniform re-sampling or selection

is frequently employed. This scheme, unavoidably, intro-

duces the sample impoverishment problem and an analysis

is available in [18]. This scheme also contains unlimited

error spread as proved in [19] where the stochastic universal

sampling (SUS) was proposed which bounds the sampling

error. Apart from uniform re-sampling, there are alternative

selection methods available in the GA literature including

the tournament and truncation selection schemes [20]. It is

also noted that the complexity of a PF depends critically

on the number of particles required. This observation was

considered in [21], where the choice of the number of parti-

cles was guided by the Chebyshev inequality. Another major

problem in PF implementation is that particles are not being

supplied in the high probability regions as needed. Although

there may not be known a priori on where the region is in

the solution space, the GA approach re-supplies or re-defines

particles via the crossover and mutation operators [22]. These

techniques may suggest an attractive hybridization approach

by combining the advantages of PF and GA. In this paper,

the operation of a particle filter will be re-studied from the

GA perspective. The major contribution of this paper are

in characterizing the sample impoverishment problem and

proposing an alternative re-sampling scheme.

The rest of the paper is organized as follows. In Section

II, the implementations of the particle filter and genetic

algorithm are briefly reviewed. The combination of the two

techniques in mobile robot localization is developed in Sec-

tion III. Simulation results are presented and discussed in

Section IV. A conclusion is drawn in Section V.

II. PARTICLE FILTER AND GENETIC ALGORITHM

A. System Description

Assume a mobile robot being deployed in its operation

area. The robot moves along a straight line with two land-

marks being observed1. In state space description, the robot

1This is a simple one-dimensional problem and the robot is termed a

monobot.

transition is given by the process model,

xv,k+1= xv,k+ vk∆T + ηv,k,

(1)

where xv is the robot state, k is the time index, v is the

velocity control, ∆T is the discrete time interval and ηvis the

process noise assumed as ηv∼ N(0,Q) and is a stationary

sequence. The complete system state is

xk= [xv,k,xm1,k,xm2,k]T,

(2)

Note that the landmarks are assumed stationary, so they

are not included into the process model for presentation

simplicity. While the robot moves, it observes or measures

the distance (range) from the landmarks. The measurement

model is,

zi,k= xmi,k− xv,k+ ηz,k,

where ziis the range measurement to the i−th landmark xmi,

ηzis the measurement noise assumed to be ηz∼ N(0,R).

B. Particle Filter

The particle filter is developed on the basis of Bayes’ Rule

which states that the posterior is proportional to the product

of the likelihood and prior, given as

(3)

p(xk|z1:k) ∝p(zk|xk)p(xk|xk−1)p(xk−1|z1:k−1)

p(zk|z1:k−1)

(4)

where p(·) is a probability density function (pdf), xk is the

current state to be estimated and z1:k is the measurement

up to time index k. The operation of the particle filter is as

follows, see [1] and [2].

1) Initialize:

• Generate xi

initial location of the robot (assumed at the origin of the

coordinate frame), where N is the number of particles.

• Generate random numbers xi

distributed, (a 2 × N matrix) representing the initial

unknown location of landmarks. xmaxis the maximum

operating space.

2) Measure:

• Make range measurements to landmarks, giving z1,2,

which are corrupted by noise.

• Calculate the importance weights,

wi

v= 0, for i = 1···N, representing the

m1,2in [0,xmax] uniformly

j= exp(−0.5νiT

j= zj− (xi

superscript (T) stands for transpose.

• Calculate the normalized overall importance weight2,

jR−1νi

j),j = 1,2;

(5)

where νi

mj− xi

v) is the innovation and

˜ wi= Π2

j=1wi

j,

¯ wi=

˜ wi

i=1˜ wi,

ΣN

(6)

2Note that the product and summation is performed component-wise for

the weights and there are 2 landmarks assumed.

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3) Update:

• Perform re-sampling to form new particles ˜ xi, such that

the probability of selection is proportional to the weights

¯ wi.

• Calculate the estimate and the uncertainty covariance

ˆ x = ΣN

P = ΣN

i=1¯ wi˜ xi,

i=1¯ wi(˜ xi−ˆ x)(˜ xi−ˆ x)T. (7)

C. Genetic Algorithm

The genetic algorithm, as a stochastic searching algorithm,

is widely treated as a function optimizer and is developed

by the inspiration from Darwin’s Theory of Evolution. GA

simulates the evolution of individuals in competing for sur-

vival. Fitter3individuals cross-breed and produce better off-

springs hence promoting the fitness of the whole population.

Moreover, mutation also occurs during the production of

off-springs. In computer implementations, GA is governed

by the Schema Theorem4originally derived from binary

string representation of the genes of a chromosome within

an individual5. The Schema Theorem can be expressed as

(see [10] for details),

m(?,t + 1) ≥ m(?,t)f(?)

¯f

(1 − pc

δ(?)

L − 1)(1 − pm)o(?), (8)

where m(?,t) is the number of schema ? at generation t,

f(?) is the average fitness of chromosomes having the same

schema,¯f is the average fitness of the whole population, pc

is the crossover probability, δ(?) is the length of a schema,

L is the chromosome length, pmis the mutation probability

and o(?) is the order of a schema.

The Schema Theorem says that the fitness of individuals

having contributing characteristics will increase over genera-

tions (iterations) and finally converge to the optimal solution.

The implementation procedures of the GA are described in

the following.

1) Initialize:

• Generate random numbers (chromosomes) describing

the solution, the number corresponds to the size of the

population and bounded within the solution space.

2) Iteration:

• Calculate the fitness of chromosomes based on measure-

ments made (equivalent to equ. 6 in PF implementation).

• Select chromosomes into an intermediate population

according to their fitness.

• Perform crossover and mutation to mix/perturb the in-

termediate population.

3Fitness can be viewed as the closeness of a candidate solution from the

optimum.

4A schema is a defining characteristic of the encoding that contributes to

the optimal solution.

5A chromosome is an encoding of the solution to an optimization problem

which is equivalent to a particle in the particle filter. Hence, the two terms

will be used interchangeably in the rest of the paper.

3) Termination:

• If some termination condition is met

. Calculate the estimation (e.g., equ. 7)

. Otherwise, repeat iterations.

III. EVOLUTIONARY PARTICLE FILTER

Based on the similarities and differences between PF and

GA, an evolutionary particle filter is proposed (EPF). This

algorithm complements PF and GA. In particular, solves the

sample impoverishment problem found in the PF. The number

of particles required, the cause of sample impoverishmentand

the time for impoverishment will be investigated in the sequel

from the perspective of the GA selection process.

A. Number of Particles Required

Consider the one dimensional case, a particle x which

is the one left after impoverishment. It was initialized by

drawing a sample from a distribution with mean µ and

variance σ2. The final probability of estimation error ? is

bounded by the Chebyshev inequality given by

P(|x − µ| ≥ ?) ≤σ2

?2.

(9)

When the particles were initialized to cover a certain range

in the solution space, the chance of the particle that falls in

the vicinity of the true solution is increased by generating

more samples. Now consider that there were N particles

initialized identically independently distributed, and arrange

the particles in a sequence x. Then the Chebyshev inequality

gives

P(|x

Hence, for a specified error ?, the error probability is inversely

proportional to the number of particles required. However,

there is always a limitation in the computational resources

and the use of a small number of particles is very desirable.

N− µ| ≥ ?) ≤

σ2

N?2.

(10)

B. Impoverishment from Re-sampling

The sample impoverishment phenomenon may be studied

via the gambler’s ruin problem. Consider two particles as

gamblers A and B. When they were initialized, capitals,

cA and cB respectively, were assigned according to their

closeness to the true solution. In most particle filter re-

sampling process, a pointer is generated from a uniform

distribution emulating a spin from a roulette wheel. However,

if a small number of particles are used, a true uniform

distribution cannot be guaranteed in practice. In the gambler’s

view, this becomes an unfair game as p ?= q, where p is

the winning probability for gambler A and q is the winning

probability for B.

Consider when a particle is duplicated for gambler A

and a particle is removed from B, which corresponds to

the winning and losing outcomes. The probability that, say,

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gambler A eventually wins (sample impoverished) can be

derived from the theory of random walks as

PA=

(q/p)cA− 1

(q/p)cA+cB− 1,

(11)

where cA is the initial capital of gambler A and a similar

expression applies to gambler B.

It is evident that as far as the game is unfair, say, gambler

A is favorable, then A will ultimately win all the wealth of

gambler B. Since exact uniform distribution for the selection

pointer is not available in practice, this stochastic effect will

accelerate the impoverishment process.

From the genetic algorithm literature [19], a linear spaced

pointer is generated in the selection process called the sto-

chastic universal sampling (SUS). These pointers satisfy a

uniform distribution and guarantee the same interval between

pointers. Hence, reduces the selection bias and the adverse

effect of impoverishment is reduced. A set of pointers are

generated by a single roulette wheel spin as follows,

Pt= N−1((1···N) − r),

(12)

where r ∈ [0,1] is a random number.

C. Impoverishment Time

In the PF re-sampling procedure, particles are copied or

removed according to their weights ¯ wi, the change in copies

is proportional to multiples of 1/N. If the weights are sorted

and cumulatively summed, they can be approximated by the

power law such as uc, where u is the normalized index, u ∈

[0,1], resulted from sorting and c is the power constant (see

details in [18]). For example, Fig. 1(a) shows the normalized

weights of particles in the space range ±2.5m. The weights

are generated from a Gaussian distribution with µ = 0 and

σ2= 0.2. The corresponding cumulative sum in normalized

index is depicted in Fig. 1(b) which is the approximation by

the power law (here, c = 6.5 is determined experimentally).

−2.5−2 −1.5−1 −0.500.511.522.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Range(m)

Weight

(a) Distribution of particles

0 0.10.2 0.30.40.50.60.7 0.80.91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ranking index

Cumulative weight

(b) Cumulative sum of weights

Fig. 1.Power law approximation

By considering a general range in u and u − 1/N, the

proportion of particles at iteration k is

Pu,k= uck+1− (u − 1/N)ck+1.

(13)

This expression indicates the growth of particles within the

range around u. Sample impoverishment occurs when the

particle with the highest weight dominates, i.e., u = 1. The

trace of its growth becomes

?N − 1

Setting this proportion to N − 1)/N, which represents the

highest ranked weight, and after some manipulations, the

impoverishment time k∗is approximately

P1,k= 1 −

N

?ck+1

.

(14)

k∗≈ c−1(N lnN − 1).

(15)

For re-sampling or selection to be effective, the power law

constant must be c > 1 and a larger c imposes larger

selectivity. The above equation shows that in this case,

the impoverishment time k∗is finite and is extended by

the number of particles in proportional to N lnN. Hence,

impoverishment is inevitable when the re-sampling process

is adopted in implementing a particle filter.

D. Proposed Approach

In order to mitigate the sample impoverishment problem

when implementing a particle filter with re-sampling, an

evolutionary approach is proposed where re-sampling is

conducted implicitly thus avoiding the impoverishment. The

algorithm hybridizes the particle filter and genetic algorithm

procedures while complements the advantages of each other.

In the monobot scenario, the system state initially contains

the robot location and the landmark states are augmented

when they are firstly observed to form an overall system

state. The system state is partitioned such that floating point

numbers are used to represent individual state and there are N

copies, or chromosomes, to form a population. The states are

modifies by the genetic crossover operator depending on the

range measurements made between the robot and landmarks

during each time step. The algorithm can be described as

follows.

1) System initialization at time k = 0:

• Generate N chromosomes6to represent the robot state,

all are set to zero representing the origin of the coordi-

nate frame.

2) Measurements:

• Make range measurements from the robot to landmarks,

giving zifor each landmark.

• If the landmarks are firstly seen, generate sets of chro-

mosomes for each landmark7.

• Otherwise, calculate and normalize the fitness of each

chromosome such that they sum to unity.

6Floating point numbers are used in this work to gain a better resolution

of the estimation.

7The chromosomes are locations around the first range measurement with

some arbitrary distribution, e.g., uniform distribution.

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3) Update at k > 0:

• Select chromosomes using stochastic universal sam-

pling.

• Compute the estimates of the robot and landmark states.

• Re-normalize fitness to f ∈ [0,1].

• Loop through N times.

- Randomly pick two chromosomes c1 and c2 with

fitness f1and f2.

- If f1< γ, then set

c1← c1+ r∆c,

- If f2< γ, then set

c2← c2+ r∆c,

where γ ∈ [0.1] is some fitness threshold (e.g., γ =

0.05), r ∈ [0,1] is a random number, ∆c is the

distance between the chromosomes.

• Repeat from the measurement step until user specified

termination of the filtering process.

∆c = c1− c2

(16)

∆c = c2− c1

(17)

E. Justifications

In the proposed approach, the initialization and measure-

ment stages follow that of standard PF or GA implementa-

tions. In the update stage, chromosomes are selected accord-

ing to their fitness. The resultant chromosomes need to be

separated to prevent impoverishment. A pair of chromosomes

are manipulated when the fitness of one chromosome in

the pair that is lower than some threshold γ. The distance

between the two chromosomes is calculated. The adjusted

chromosome is then repelled from the one of higher fitness.

Moreover, the adjustment is also moderated by the distance

and the weighting given by r. This technique may be viewed

as re-supplying chromosomes or particles to locations in

the solution space not yet being explored while preventing

sample impoverishment.

IV. SIMULATION

Simulations were conducted for a monobot initially located

at the coordinate origin then moves repeatedly from left to

right and vice versa in a 5m range. Two landmarks were

placed at 3m and 4m respectively. The robot moves at 0.2m/s

and the odometer measures the velocity with an error of

standard deviation at 0.03m/s, the range measurement to

landmarks carries an error of 0.1m standard deviation, the

noises are assumed Gaussian.

Two cases were simulated: 1) standard particle filter imple-

mentation with re-sampling and 2) the proposed evolutionary

approach. In both cases, the use of a small number of

particles, 500, and a relatively larger number of 5000 are

tested. Fig. 2(a) plots the particles corresponding to the robot

and landmarks and their associated weights in case 1. Due

to sample impoverishment, particles concentrated on discrete

locations. The improvement from adopting the evolutionary

approach is illustrated in Fig. 2(b) with the use of 500

0.4 0.50.60.70.80.9

1

1.5

2

2.5

3x 10

−3

Robot

Weight

2.82.933.1

1

1.5

2

2.5

3x 10

−3

Mark1

3.83.944.1

1

1.5

2

2.5

3x 10

−3

Mark2

(a) PF re-sampling approach

0.20.30.40.50.6 0.7

0

1

2

3

4x 10

−3

Robot

Weight

2.93 3.13.2

0

1

2

3

4x 10

−3

Mark1

3.944.14.2

0

1

2

3

4x 10

−3

Mark2

(b) Proposed approach

Fig. 2.Particles distribution vs. weights: top - robot, bottom - landmarks

particles. It is clear that particles are able to represent the pdf

which can be noted from a trace of the envelope. A more

concentrated region of particles is also observable which

indicates the convergence to the solution.

Time traces of the spread of the particles in case 1 are

plotted in Fig. 3(a) and 3(b) respectively. The top trace is

for the robot location error while the lower two are for the

landmarks, the corresponding 3σ error bound is also shown.

It is clearly seen that for 500 particles used, the particles

collapsed to a single one at about 250 time steps. The location

estimations becomes un-reliable. The sample impoverishment

is also noticeable even when 5000 particles are used, see Fig.

3(b).

Results from case 2, which adopts the proposed evolution-

ary particle filter approach, are depicted in Fig. 4(a) and 4(b).

The results from the use of 500 particles show acceptable

results while the use of 5000 particles clearly removes the

sample impoverishment problem.

V. CONCLUSION

In this paper, the sample impoverishment problem in a par-

ticle filter is resolved by hybridizing techniques using genetic

algorithms. It has been shown by analysis and simulations

that the proposed method produces better estimation results

than the conventional particle filter. This is because that the

proposed method maintains the diversity of particles in the

re-sampling process. Further work will be done on a real

robot.

Page 6

0 50100 150200250300350 400450500

−0.1

−0.05

0

0.05

0.1

Robot err(m)

Estimation Error

0 50100 150200250300350 400450500

−0.2

−0.1

0

0.1

0.2

M1 err(m)

0 50100150200 250300350400450 500

−0.2

−0.1

0

0.1

0.2

M2 err(m)

(a) 500 particles

0 50100150 200250 300 350400450500

−0.1

−0.05

0

0.05

0.1

Robot err(m)

Estimation Error

050 100 150200 250 300350400 450500

−0.2

−0.1

0

0.1

0.2

M1 err(m)

0 50100150200250300350400450500

−0.2

−0.1

0

0.1

0.2

M2 err(m)

(b) 5000 particles

Fig. 3.Simulation results from standard PF implementation

0 50100150200 250300350400450 500

−0.1

−0.05

0

0.05

0.1

Robot err(m)

Estimation Error

050100150 200250300350400450500

−0.2

−0.1

0

0.1

0.2

M1 err(m)

050100150200250300350400450500

−0.2

−0.1

0

0.1

0.2

M2 err(m)

(a) 500 particles

050100150200 250300350400450500

−0.1

−0.05

0

0.05

0.1

Robot err(m)

Estimation Error

050 100150200250300 350400450500

−0.2

−0.1

0

0.1

0.2

M1 err(m)

050100150200 250300350400450 500

−0.2

−0.1

0

0.1

0.2

M2 err(m)

(b) 5000 particles

Fig. 4.Simulation results from evolutionary particle filter approach

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