Advanced Robotics-24 s6

832019 Advanced Robotics-24 s6

httpslidepdfcomreaderfulladvanced-robotics-24-s6 120

Advanced Robotics 24 (2010) 585ndash604brillnlar

Full paper

Analysis of Rank-Based Resampling Based on Particle

Diversity in the RaondashBlackwellized Particle Filter for

Simultaneous Localization and Mapping

Nosan Kwak alowast Kazuhito Yokoi a and Beom-Hee Lee b

a

Humanoid Research GroupJoint Robotics Laboratory UMI3218CRT National Institute of Advanced Industrial Science and Technology Central 2 AIST Umezono 1-1-1 Tsukuba

Ibaraki 305-8568 Japanb

School of Electrical Engineering and Computer Science Seoul National University Gwanak 599

Gwanak-gu Seoul 151-742 South Korea

Received 6 March 2009 accepted 10 June 2009

Abstract

In order to solve the simultaneous localization and mapping (SLAM) problem of mobile robots the Raondash

Blackwellized particle filter (RBPF) has been intensively employed However it suffers from particle

depletion problem ie the number of distinct particles becomes smaller during the SLAM process As

a result the particles optimistically estimate the SLAM posterior meaning that particles tend to underesti-

mate their own uncertainty and the filter quickly becomes inconsistent The main reason of loss of particle

diversity is the resampling process of RBPF-SLAM Standard resampling algorithms for RBPF-SLAM can-

not preserve particle diversity due to the behavior of their removing and replicating particles Thus we

propose rank-based resampling (RBR) which assigns selection probabilities to resample particles based on

the rankings of particles In addition we provide an extensive analysis on the performance of RBR includ-

ing scheduling of resampling Through the simulation results we show that the estimation capability of

RBPF-SLAM by RBR outperforms that by standard resampling algorithms More importantly RBR pre-

serves particle diversity much longer so it can prevent a certain particle from dominating the particle set

and reduce the estimation errors In addition through consistency tests it is shown that RBPF-SLAM by the

standard resampling algorithms is optimistically inconsistent but RBPF-SLAM by RBR is so pessimisti-

cally inconsistent that it gives a chance to reduce the estimation errors

copy Koninklijke Brill NV Leiden and The Robotics Society of Japan 2010

Keywords

SLAM particle diversity resampling ranking consistency

To whom correspondence should be addressed E-mail nosan-kwakaistgojp

copy Koninklijke Brill NV Leiden and The Robotics Society of Japan 2010 DOI101163016918610X487126



586 N Kwak et al Advanced Robotics 24 (2010) 585ndash604

1 Introduction

The RaondashBlackwellized particle filter (RBPF) was introduced about a decade ago

as an effective means to solve the simultaneous localization and mapping (SLAM)

problem by Murphy [1] and Doucet et al [2] This solution to SLAM is RBPF-

SLAM which is also known as FastSLAM [3] The SLAM posterior is normally

the joint estimation of a robotrsquos path and a map However RBPF-SLAM factors the

joint estimation using RaondashBlackwellization which factors a state into a sampled

part (path) and an analytical part (map) In RBPF-SLAM the robotrsquos path is esti-

mated by a particle filter and the map by low-dimensional extended Kalman filters

(EKFs)

RBPF-SLAM has two major advantages compared to other approaches (i) By

factoring the SLAM posteriors RBPF-SLAM has linear time complexity (ii) Un-like EKF-SLAM RBPF-SLAM allows each particle to perform its own data asso-

ciation which implements multi-hypothesis data association [3] The ability to si-

multaneously pursues multi-hypothesis data association makes RBPF-SLAM more

robust to data association problems than algorithms based on incremental maxi-

mum likelihood data association such as EKF-SLAM This special characteristic

is dependent on particle diversity The bigger the number of distinct particles the

more chance to close a loop because new observations can affect the locations of

the landmark [4] Thus it is crucial for RBPF-SLAM to maintain particle diversityas long as possible

However the innate disadvantage of RBPF-SLAM is that past pose estimation

errors of a robot are not forgotten which means that they are recorded in the feature

estimates Whenever the resampling process is conducted the entire robot paths

and feature estimates of rejected particles are lost forever As a result the number

of particles representing past paths and feature estimates severely decreases This

is called the particle depletion problem [5] In other words as the particle set loses

its diversity it becomes over-confident which means it tends to underestimate itsown uncertainty According to the work by Bailey et al [4] the particle diversity is

drastically attenuated when a robot closes a large traverse loop

Another critical issue in RBPF-SLAM is the consistency which is the ability of

the filter to accurately estimate uncertainty A filter can be inconsistent in either

an optimistic or a pessimistic way According to Ref [6] a filter is optimistic or

over-confident if there is significant bias in the estimates the errors are too large

compared to the filter-calculated covariance or the covariance is too small On the

other hand a filter is pessimistic or conservative if the covariance is too large It hasbeen shown in simulation works [4 7] that the current RBPF-SLAM algorithm is in-

consistent in an optimistic way and Stachniss et al [8] showed in their experiments

that RBPF-SLAM is consistent only in a local area (or in the short term) According

to Ref [4] RBPF-SLAM in its current form cannot produce consistent estimates

in the long term although it stays reasonably consistent for a few tens of seconds

after starting Beevers et al [7] improved the consistency by applying block pro-

posal distribution in the sampling process which exploits future information Their



N Kwak et al Advanced Robotics 24 (2010) 585ndash604 587

modified strategies however cannot guarantee a consistent filter in the long term

The consistency of a filter is closely related to particle depletion which is directly

related to the resampling process in RBPF-SLAM Thus we extensively conducted

analysis on standard resampling algorithms to investigate the relationship between

particle diversity and resampling algorithms in the previous work [9] According toour results all resampling algorithms cannot preserve particle diversity

The existing results show that the loss of particle diversity causes critical prob-

lems such as poor data association and inconsistent estimates Even though RBPF-

SLAM thanks to accurate sensors is applicable to practical problems it is desirable

to guarantee its performance over the long term One-particle RBPF-SLAM some-

times shows results as good as one with 100 particles This can be interpreted that

the single particle estimates the SLAM posterior with the help of accurate sensors

However if the only particle has large estimation errors then there is no way to

compensate for the errors Besides according to our previous work [10] perfor-

mance by taking the mean of particles is better than that by the most weighted

particle Thus the only way for consistent estimates over the long term is to keep

particle diversity as long as possible

To preserve particle diversity in this work we propose rank-based resampling

(RBR) which is an indirect resampling algorithm since it uses the ranking of a par-

ticle for resampling Actually we briefly introduced the RBR in our previous work [9] but we underestimated its effectiveness for keeping particle diversity In this

work we thoroughly analyze RBR in terms of particle diversity and consistency

of RBPF-SLAM and emphasize its capabilities For the organization of this work

a brief introduction to RBPF-SLAM including its particle diversity and consistency

will be presented in Section 2 RBR will be described in Section 3 and its per-

formance will be investigated in Section 4 with estimation errors particle diversity

and consistency of RBPF-SLAM Finally concluding remarks on the capability of

RBR will be given in Section 5

2 RBPF-SLAM

The structure of SLAM enables particle filters to be applicable since the SLAM

problem is characterized by a conditional independence between any two disjoint

sets of landmarks in the map given the robotrsquos pose [11] It means if the robotrsquos true

path was given locations of all landmarks would be estimated independently Thisspecial particle filter is known as RBPF In this section the algorithm of SLAM

using RBPF (RBPF-SLAM) is briefly introduced in terms of particle diversity and

consistency of RBPF-SLAM

21 RBPF-SLAM Algorithm

RBPF-SLAM enables us to factor the SLAM posterior into a product of simpler

terms The key mathematical insight of RBPF-SLAM pertains to the fact that the




full SLAM posterior can be factored as [11]

p(x1t M |z1t u1t c1t )= p(x1t |z1t u1t c1t )

N f n=1

p(mn|x1t z1t c1t ) (1)

where x1t is the robot path up to time t M (mn is nth landmark and there are N f

landmarks) is the map and z1t u1t and c1t are the measurements controls and

correspondences up to time t respectively RBPF-SLAM uses a particle filter to

estimate the robotrsquos pose and EKFs to estimate the robotrsquos map More specifically

the mapping problem can be factored into separate low-dimensional EKFs using

the conditional independence among landmarks [3]

A particle at time t Y [k]t is denoted by

Y [k]t =

x

[k]1t μ

[k]1t

[k]1t μ

[k]N f t

[k]N f t

(2)

where the [k] indicates the index of the particle and x[k]1t is the robot path estimate

of the kth particle at time t μ[k]nt and

[k]nt are mean and covariance of the Gaussian

distribution representing the nth feature location relative to the kth particle respec-

tively Altogether these elements form the kth particle Y [k]t and there are a total of

N p particles and N f features in a particle set The RBPF-SLAM algorithm consists

of four steps as follows [11]

(i) Sampling x[k]t sim p(xt |x

[k]t minus1 z1t u1t c1t )

(ii) Measurement update For each observed feature zit identify the correspon-

dence j for the measurement zit and incorporate the measurement zit into the

corresponding EKF by updating the mean μ[k]jt and covariance

[k]jt

(iii) Importance weight Calculate the importance weight w[k] for the new particle

(iv) Resampling Sample N p particles with replacement where each particle is

sampled with a probability proportional to w[k]

22 Resampling Algorithms

In common particle filtering resampling is used to reduce the particle degener-

acy which occurs because particles or samples have negligible weights over time

Through resampling (removing particles with low weights and replicating more

particles in more probable regions) a particle set can better reflect the true poste-rior of SLAM However resampling makes particles with high weights be selected

more and more often As a result after a few iterations of the algorithm the par-

ticles with high weights dominate the particle set Thus a particle cannot perform

its own function because it is a copy of the dominant particle at a certain point

The most commonly used resampling algorithms in SLAM are different variants

of stratified sampling such as residual resampling (RR) and systematic resampling

(SR) SR is the most commonly used since it is the fastest resampling algorithm




Table 1

RSR algorithm

Algorithm RSR(wN in N out)

Input A set of normalized weights w

and number of inputs N in and outputs N out

Output A set of numbers to replicate each particle N R

Generate a random number U 0 sim U([0 1N out

])

for i = 1 to N in

N [i]R = [(w[i] minusU iminus1) middotN out] + 1

U i =U iminus1 +N [i]R N out minusw[i]

end

for computer simulations Bolic et al [12] proposed the residual systematic re-

sampling (RSR) which produces an identical resampling result as SR with fewer

operations and less memory access In the previous work [9] we confirmed that

RSR for RBPF-SLAM shows the best performance among the variants of stratified

resampling approaches Thus we will compare our resampling algorithm with RSR

The algorithm of RSR is presented in Table 1 where RSR draws the first uniform

random number U 0 =U 0 and updates it by U i =U iminus1 +N [i]R N out minusw

[i]n

The output N R is an array of indices which means how many times each particle is

replicated for the next particle set In the RSR algorithm the updated uniform ran-

dom number is formed in a different fashion compared to the standard SR That is

it requires only one iteration loop In addition in RSR resampling is performed in

fixed time whereas in SR it is not performed in fixed time because the number of

replicated particles is random which makes an unspecified number of operations

23 Particle Diversity

In the resampling step particles are resampled based on their importance weights

which are computed by the ratio of the target or posterior distribution and the pro-

posal distribution for sampling as [11]

w[k]t =

target distribtuion

proposal distribution

= p(x[k]

1t |u1t z1t c1t )p(x

[k]1t minus1|u1t minus1 z1t minus1 c1t minus1)p(x

[k]t |x

[k]1t minus1 u1t z1t c1t )

(3)

where it is assumed that paths in x[k]1t minus1 have been generated according to the tar-

get distribution one step earlier p(x[k]1t minus1|u1t minus1 z1t minus1 c1t minus1) Note that the most

recent measurement zt is used to construct the proposal distribution from which

particles are sampled If the sensor is very accurate relative to the motion model

the target distribution will be sharply peaked relative to a flat proposal distribution




Figure 1 Number of distinct particles over time

After resampling a small percentage of particles are assigned non-negligible im-

portance weights causing significant duplication of a few dominant particles Once

the particles are removed in the set particle diversity cannot be recovered because

particles share the robot path and feature estimates at some point This is the parti-cle depletion problem Over time particle depletion could result in particles drifting

away from the true state [13]

A measure for the rate of loss of particle diversity is obtained by recording the

number of distinct particles having different estimates for a landmark in the set

Once a landmark goes out of the robotrsquos sight resampling causes some particles to

be rejected and others to be replicated At first all of the particles are distinct which

means they have different feature estimates about a landmark As time passes only

particles with high weights survive and particles with low weights disappear to-gether with their feature estimates Thus the number of distinct estimates of the

landmark becomes smaller The number of distinct particles is counted after every

resampling process and its transition (the result is obtained from simulations in the

environment of Fig 6 with the condition in Section 41) in the case of using RSR

is shown in Fig 1 Soon after closing a loop the ratio of distinct particles becomes

smaller than 3 of the initial distinct particles As is seen by this example parti-

cle depletion often occurs and due to this particle depletion RBPF-SLAM might

produce very inaccurate estimates

24 Consistency of RBPF-SLAM

The χ2 distribution is often used to check state estimators for consistency ie

whether their actual errors are consistent with the variances calculated by the esti-

mator [6] For the RBPF-SLAM algorithm to measure if a filter is consistent one

would compare its estimate with the probability density function (PDF) obtained

from an ideal Bayesian filter The PDF is however not available for the RBPF-




SLAM algorithm Instead the true pose of the robot can be known in simulations

but not in real experiments With this information the normalized estimation error

squared (NEES) defined in (4) can be used to investigate the consistency of a filter

[4 6 7] NEES is defined as

εt = (xt minus xt )TP minus1t (xt minus xt ) (4)

where xt P t are the estimated mean and covariance of particles at time t A mea-

sure of filter consistency is obtained by examination of the average NEES over N

Monte-Carlo runs of the filter Under the assumptions that the filter is consistent

and is approximately linear Gaussian εt is χ2 distributed with dim(xt ) degrees

of freedom The consistency of RBPF-SLAM is evaluated by conducting several

Monte-Carlo runs and computing the average NEES Given N runs the average

NEES is obtained as

εt =1

N

N i=1

εit (5)

Given the hypothesis of a consistent linear Gaussian filter N εt has a χ2 density

with N dim(xt ) degrees of freedom Thus in case of three-dimensional robot pose

with N = 50 the 95 probability concentration region for εt is bounded by the in-

terval [236372] [6] If εt rises significantly higher than the upper bound the fileris optimistic or over-confident If it tends below the lower bound the filter is pes-

simistic or conservative The average NEES of the current RBPF-SLAM framework

presented in Fig 2 shows that the filter is not consistent More precisely at first the

filter is pessimistic but after about 3000 time steps it suddenly becomes optimistic

Figure 2 Average NEES of the standard RBPF-SLAM algorithm over 50 Monte-Carlo runs Two

horizontal red lines indicate the upper and the lower bounds of χ2 which are obtained with the

assumption that the filter is consistent




3 RBR

31 Ranking for Selection

Direct resampling with importance weights causes loss of particle diversity and

as a result RBPF-SLAM becomes inconsistent in an optimistic way If the opti-mistic estimation becomes erroneous there would be no way to correct the error

in RBPF-SLAM Thus for any particle filters including RBPF preserving particle

diversity is of paramount importance since each distinct particle represents a dif-

ferent hypothesis of the SLAM posterior In other words as the number of distinct

particles becomes smaller several probable regions of the posterior cannot be esti-

mated because the particles that are assigned in the regions have been removed in

the resampling process

The diversity issue also occurs in a genetic algorithm [14] which is a search

technique used in computing to find exact or approximate solutions to optimization

and search problems In a genetic algorithm each gene that represents a solution to

a problem is reproduced using its own fitness The mechanism of the reproduction

process is very similar to the resampling process in particle filtering Most of the

schemes in genetic algorithms cannot overcome premature convergence or diversity

better than the rank-based reproduction [15] A ranking is used as a transformation

function that assigns a new value to a gene based on its fitness By using not a fitness

but a ranking it is possible to slow down the premature convergence Furthermore

it is possible to control the speed of the convergence with varying the ranking func-

tion To take advantage of the rank-based reproduction scheme it is modified in

this work and used as RBR In addition we thoroughly analyze RBR in terms of

particle diversity and consistency of RBPF-SLAM

32 RBR for RBPF-SLAM

In order to employ the rank-based reproduction scheme in the RBPF-SLAM frame-work RBR is described in this section RBR consists of two parts The first part is

assigning a selection probability of a particle using a ranking function The ranking

can be easily obtained by sorting the particles by the magnitude of their importance

weights The second part is standard resampling with the selection probability of

each particle This ranking approach seems to discard information of importance

weights but it actually discards the information about the magnitude of importance

weights and assigns relative magnitude instead Therefore RBR can be called an

indirect resampling algorithm In the current RBPF-SLAM framework the impor-tance weight is the only measure to evaluate the performance of a particle However

when an accurate sensor such as a laser range finder is used the differences of im-

portance weights between the most weighted particle and the others having slight

pose differences are so large that only the most weighted particle dominates the

particle set As a result several particles are suddenly rejected in the particle set

With RBR these ill-balanced performance measures are linearly re-assigned using

the ranking function




With the selection probabilities of all the particles the RBR performs the stan-

dard resampling RSR In this work a linear ranking function is used to assign the

selection probability of a particle When the ranking function is linear the mean

of the selection probabilities will correspond to the median rank in the particle set

[15] One can think of a non-linear ranking function but it will shift the mean to-ward the top of the set This is not desirable to preserve particle diversity as long

as possible because of larger differences of selection probabilities than in the linear

ranking function Thus the non-linear ranking function is not considered in this

work The slopes of linear functions are adjusted to control the selection pressure

which is the ratio of the best particlersquos selection probability over the average selec-

tion probability of all particles in the set The following linear equation is used as

the ranking function for the selection probability of the kth particle p[k]s

p[k]s =

1

N p

ηmax minus (ηmax minus ηmin)

(rank (k)minus 1)

N p minus 1

(6)

where N p is the number of particles ηmaxN p is the maximum selection probabil-

ity of the highest weight and ηminN p is the minimum selection probability of the

lowest weight The particle at the first ranking gets the highest selection probability

whereas the particle at the last ranking gets the lowest selection probability When

the number of particles is fixed ηmin = 2 minus ηmax 0 should be satisfied and ηmax

usually has a value between 1 and 2 [16] Depending on ηmax the selection proba-bility varies The larger ηmax which means the selection pressure is getting larger

the larger the differences between selection probabilities The relation between se-

lection probabilities and rankings of particles with varying ηmax is plotted in Fig 3

in the case of six particles in the set This ranking approach is inserted into the RBR

Figure 3 Selection probabilities over rankings of particles with varying ηmax




Table 2

RBR algorithm for resampling in RBPF-SLAM

Algorithm RBR (w N in N out)

Input A set of normalized weights w and N in and N out


1 Set a value between [12] to ηmax

2 ηmin larr 2 minus ηmax

3 [wsorted I sorted] larr Sort w in a descending order

4 for i = 1 to N in

5 k larr I sorted(i) i is the ranking

5 p[k]s larr (ηmax minus (ηmax minus ηmin) lowast (i minus 1)(N in))N in

6 end for7 N R larr Call RSR( ps N in N out)

algorithm shown in Table 2 where I sorted stores indices of particles in a descending

order ie the first element of I sorted has the highest ranking

33 A Biased Resampler

It is worth noting that RBR has a big difference from the standard resampling algo-rithms such as RSR since RBR uses the indirect information of importance weights

Particles drawn from RBR construct a different distribution from the true posterior

due to the indirect usage of the importance weights In this sense adding new ran-

dom particles also distorts the particle distribution This kind of resampler is called

a biased resampler which violates the lsquounbiasednessrsquo or lsquoproper weightingrsquo condi-

tion defined as follows A random variable X drawn from a proposal distribution q

is said to be properly weighted [17] by a weighting function w(X) with respect to

the target distribution π if for any integrable function hEqh(X)w(X) = Eπ h(X) (7)

A set of random samples and weights (x[k]w[k]) is said to be properly weighted

with respect to π if

limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)

The effect of RBR seems like that of regularization which attempts to create a morediverse posterior density approximation by relocating the particles to a more con-

tinuous distribution [13] However RBR does not draw a new particle Instead it

selects particles taking into account the indirect information of the posterior the

ranking Figure 4 shows the normalized importance weights of all the particles in

case of the RSR In Fig 4 few particles have very high weights whereas most of the

particles have negligible weights even though the weights are normalized There-

fore after RSR only the particles with high weights survive and are replicated




Figure 4 Importance weights of all the particles just before the loop-closure when applying RSR

Table 3

Number of replicas of the dominant particles

Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8

as shown in Table 3 As shown in the above example peaked weight distribution

severely damages particle diversity and particle depletion often occurs in RBPF-

SLAM since the measurement of an accurate sensor is used to build the samplingdistribution (proposal distribution) It is known that the standard resampling satis-

fying the proper weighting condition cannot resolve the particle depletion problem

[4 9] According to our previous works [10] in the current RBPF-SLAM frame-

work keeping particle diversity is very important because all the particles drawn

from the proposal distribution are valuable When particle diversity is preserved we

showed that mean particle data gives the better estimation results It is worth testing

how RBPF is biased using RBR instead of the unbiased RSR in the perspective

of unbiasedness of an estimator Generally means of particle poses and variancesare unbiased and asymptotically unbiased respectively [6] The unbiasedness of an

estimator is defined as

E[x] = 0 (9)

where x is the estimation error A simulation is conducted for the bias test and the

result is provided in Fig 5 where means of particle paths and features are pre-

sented with the true path and landmarks According to the simulation result the




Figure 5 Path mean and feature mean of all the particles for checking the bias of RBR

particle mean estimates the path and the landmarks correctly The estimation per-

formance of RBR is usually better than that of RSR thanks to the particle diversity

In this sense RBR can be a solution to keep particle diversity even though it does

not satisfy the proper weighting condition In addition results in Refs [18 19] in-

dicate that the proper weighting condition is unnecessary to obtain convergence

results [5]

In this paper strategies that reallocate particles such as artificial evolution [20]

are not considered since the particle filter is used for SLAM which has to deal

with robot pose and the map at the same time Perturbation to the particles cannot

influence the map data that each particle stores

4 Simulation Results

In this work we only conducted simulations of RBPF-SLAM with a mobile robot

since comparing the estimation error of the filters requires the true values of poseand features In the simulation it was assumed that a mobile robot detects point fea-

tures using a laser range scanner that produces the range and the bearing to a feature

Also it was assumed that data association between measurements and features is

known in order to effectively investigate the performance of the filter The simula-

tion works were focused on the consistency and particle diversity of RBPF-SLAM

For this purpose NEES particle diversity and rms estimation errors including

scheduling of RBR and ranking functions were analyzed in this work




Figure 6 Sparse environment 54 landmarks The point C is the only loop closing point

41 Simulation Set-up

RBPF-SLAM simulations were performed on a sparse environment shown in Fig 6

(and Fig 9) Simulations were run with 100 particles In Fig 6 there is only one

outer loop-closure the point C In every simulation the mobile robot closed the

large loop twice Note that the resampling process is not conducted at every iter-ation of RBPF-SLAM It is conducted only when the effective sample size [21]

falls below a threshold to keep particle diversity as long as possible We conducted

simulations to compare the performance of several thresholds In addition we also

conducted simulations with varying ηmax in (6) The weights of all the particles are

initialized with the same weight after every resampling The motion noise and the

observation noise of the robot were set to (03 ms 3 s) and (01 m 1) respec-

tively Control and observation times were set to 25 and 200 ms respectively Every

result in the following simulations was obtained by averaging over 50 Monte-Carloruns since results of RBPF-SLAM are different at each run

42 Estimation Errors

In order to compare the localization and mapping performance of RBR we mea-

sured estimation results with varying ηmax of (6) Estimation errors with different

ηmax in the environment of Fig 6 are summarized in Table 4 where rms position

and orientation of the robot pose and feature errors are denoted RMSE P RMSE O




Table 4

Summary of estimation errors with different ηmax

ηmax RMSE P (m) RMSE O (rad) RMSE F (m)

11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5

Comparison of estimation errors between RSR and RBR in the environment of Fig 6

Resampling RMSE P (m) RMSE O (rad) RMSE F (m)

RSR 03099 00468 03590

RBR 01267 00543 01201

Remarks minus59 16 minus66

and RMSE F respectively The estimation errors over 50 Monte-Carlo runs were

collected after the robot closed the large loop twice These results were obtained bytaking the mean of all the particles since we confirmed that the mean of particles

produces the best results of RBPF-SLAM [10] According to the results of Table 4

the case of ηmax = 13 showed the least errors overall even though its position er-

ror was slightly larger than the case of ηmax = 11 From now on every result for

RBR was from the simulations with ηmax = 13 In order to compare the estima-

tion performance of RBR with that of RSR simulation results are summarized in

Table 5 Note that again these results are obtained by taking the mean of all the

particles since for instance the error variance of feature errors are much smaller(00033 m) when taking the mean of particles than when taking the most weighted

particle (00381 m) In Table 5 estimation errors of RBPF-SLAM by RBR were

much reduced compared to those of RBPF-SLAM by RSR The bigger error in

the orientation by 16 (=04286) is small compared to the improvements in the

position and the feature errors These estimation improvements come from parti-

cle diversity The estimation improvements in RMSE P and RMSE F were about 59

and 66 respectively In addition the standard deviation of the estimation errors

over 50 runs were much reduced The standard deviations by RBR were 00537( RMSE P) and 00574 m ( RMSE F) while those by RSR were 01049 ( RMSE P) and

01774 m ( RMSE F)


Comparison of the loss of particle diversity between RBPF-SLAM by RSR and

RBPF-SLAM by RBR is shown in Fig 7 where the time step is presented when

the robot closed the large loop The rate of the loss of particle diversity by RBR is




Figure 7 Particle diversity of RBPF-SLAM by RBR decreases almost linearly which is significantly

different from the result of RBPF-SLAM by RSR In both resampling approaches no resampling

occurred after the first loop-closure which is presented as the time step

Table 6

Comparison of estimation errors varying a threshold for resampling

Threshold () RMSE P (m) RMSE O (rad) RMSE F (m)

25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563

almost linear whereas that by RSR is exponential After the loop-closure the num-

ber of distinct particles by RBR was more than 50 of the particle size The reason

why the graphs keep the constant value after the loop-closure is because no resam-

pling was conducted after the first loop-closure We confirmed that resampling after

the large loop-closure is not effective for RBPF-SLAM performance Even though

the loss of particle diversity by RBR cannot be prevented RBR makes more than

half of the particle size survive after the robot closes the large loop Related to par-

ticle diversity we also conducted simulations for the time to instigate the RBR TheRBR was conducted when the effective sample size falls below the thresholds as

shown in Table 6 where estimation errors are presented A threshold of 25 for

instance means that the RBR was conducted whenever the ratio of the effective

sample size falls below 25 of the particle size According to the results in Table 6

the case of the 25 threshold showed the most accurate results overall Also note

that the lower the threshold the less computational cost since the lower threshold

means that the resampling occurs less often




Figure 8 Average NEES over 50 Monte-Carlo runs of RBPF-SLAM by RSR and RBR Two horizon-

tal red lines indicate the upper and the lower bounds of χ2 which are obtained with the assumption

that the filter is consistent Both approaches show that they are not consistent but that by RBR is

inconsistent in a pessimistic way and that by RSR is inconsistent in an optimistic way

44 ConsistencyWe measured the average NEES over 50 Monte-Carlo runs to test whether RBPF-

SLAM by RBR is consistent over the long term and compared the results with the

average NEES of RBPF-SLAM by RSR which are shown in Fig 8 According to

the average NEES of RBPF-SLAM by RBR although it is not consistent (the NEES

is not always inside the two bounds red lines) RBR produces a very different graph

from that of RSR More specifically RBPF-SLAM by RSR is inconsistent in an

optimistic way meaning that the estimated uncertainty is smaller than the true un-

certainty On the other hand RBPF-SLAM by RBR is inconsistent in a pessimisticway meaning that the uncertainty of particles is larger than the true uncertainty

Several SLAM literature [4 7] only report inconsistency of RBPF-SLAM as op-

timistic estimates No instance of pessimistic estimates of RBPF-SLAM has been

reported In our previous work [10] we confirmed that after several loop-closures

one can obtain an accurate map and path by taking the mean of particles when the

particle diversity is preserved even though RBPF-SLAM is pessimistically incon-

sistent When RBPF-SLAM is optimistically inconsistent however there is no way

to induce the better map and path than those of the most weighted particle since the

uncertainty of particles is too small to keep the particle diversity

45 Analysis in a Large Environment

We also analyzed the performance of RBR in a large environment 240 m times 240 m

as shown in Fig 9 The resulting data were also obtained by averaging over 50

Monte-Carlo runs In Table 7 the resulting data such as estimation errors and num-

ber of distinct particles are compared with those of RSR RBR produced about 50




Figure 9 Large sparse environment 35 landmarks 240 m times 240 m

Table 7

Summary of simulation results in the large environment

Resampling RMSE P RMSE O RMSE F No distinct

(m) (rad) (m) particles ()

RSR 44857 00696 44575 29

RBR 19060 00873 19306 243

Remarks minus57 25 minus57 84

less rms errors and about 8 times higher particle diversity than RSR and RMSE Oslightly increased The degradation in RMSE O by 25 (=10115) is small com-

pared to the improvement in RMSE P and RMSE F In addition the average NEES

graphs of both RBR and RSR are presented in Fig 10 to examine the consistency

of RBPF-SLAM The average NEES of RBPF-SLAM by RBR is similar to that in

the smaller environment of Fig 8 The RBPF-SLAM by RBR is pessimistically in-

consistent during the SLAM process but its SLAM estimation is much better than

that of when RSR is used for the resampling process for RBPF-SLAM

5 Conclusions

RBPF has been employed in several robotic problems such as SLAM thanks to the

robust data association and the lower computational complexity However it suffers

from the particle depletion problem ie the number of distinct particles becomes

smaller over the RBPF-SLAM process As a result the particle set to estimate the




Figure 10 Average NEES over 50 Monte-Carlo runs of RBPF-SLAM by RSR and RBR Also in this

environment that by RBR is inconsistent in a pessimistic way and that by RSR is inconsistent in an

optimistic way

SLAM posterior becomes over-confident which means it tends to underestimate its

own uncertaintyEven though RBPF-SLAM thanks to accurate sensors is applicable to practical

problems it is desirable to guarantee its performance as long as possible However

the standard resampling algorithms of RBPF-SLAM cannot preserve particle diver-

sity after a large loop-closure Thus we analyzed on RBR which assigns selection

probabilities to resample particles based on the rankings of importance weights

The estimation capability of RBPF-SLAM by RBR outperformed that by RSR

which is the commonly used resampling algorithm More specifically RBR pre-

serves particle diversity much longer than RSR so it can prevent some particlesfrom dominating the particle set Through our results we confirmed that when

particle diversity is preserved the estimation performance is almost always better

than the particle depletion case In addition through the consistency test although

RBPF-SLAM by RBR is not consistent the average NEES of RBPF-SLAM by

RBR is bounded during the RBPF-SLAM process whereas that of RBPF-SLAM

by RSR quickly diverges just after the loop-closure

Consequently RBPF-SLAM by RBR can preserve particle diversity much longer

than usual while reducing the estimation errors Although RBR cannot guaranteethe consistency of RBPF-SLAM it can make the filter pessimistically inconsistent

which prevents the filter from diverging

Acknowledgements

This work has been supported by a JSPS Postdoctoral Fellowship for Foreign Re-

searchers (20-08610) and the Brain Korea 21 Project Korea Science and Engineer-




ing Foundation (KOSEF) NRL Program grant funded by the Korean government

(MEST) (R0A-2008-000-20004-0) and the Growth Engine Technology Develop-

ment Program funded by the Ministry of Knowledge Economy

References

1 K Murphy Bayesian Map Learning in Dynamic Environments (Advances in Neural Information

Processing Systems) MIT Press Cambridge MA (1999)

2 A Doucet N de Freitas K Murphy and S Russell RaondashBlackwellized particle filtering for

dynamic Bayesian networks in Proc Conf on Uncertainty in Artificial Intelligence Stanford

CA pp 176ndash183 (2000)

3 M Montemerlo FastSLAM a factored solution to the simultaneous localization and mapping

problem with unknown data association PhD Dissertation Carnegie Mellon University (2003)

4 T Bailey J Nieto and E Nebot Consistency of the FastSLAM algorithm in Proc IEEE Int

Conf on Robotics and Automation Orlando FL pp 424ndash429 (2006)

5 R Merwe A Doucet N de Freitas and E Wan The unscented particle filter Technical Report

CUEDF INFENGTR380 Cambridge University Engineering Department (2000)

6 Y Bar-Shalom X R Li and T Kirubarajan Estimation with Applications to Tracking and Navi-

gation Wiley New York NY (2001)

7 K R Beevers and W H Huang Fixed-lag sampling strategies for particle filtering SLAM in

Proc IEEE Int Conf on Robotics and Automation Rome pp 2433ndash2438 (2007)

8 C Stachniss D Hahnel W Burgard and G Grisetti On actively closing loops in grid-basedFastSLAM Adv Robotics 19 1059ndash1080 (2005)

9 N Kwak G W Kim and B H Lee A new compensation technique based on analysis of resam-

pling process in FastSLAM Robotica 26 205ndash217 (2008)

10 N Kwak B H Lee and K Yokoi Result representation of RaondashBlackwellized particle filter for

Mobile Robot SLAM J Korea Robotics Soc 3 308ndash314 (2008) (in Korean)

11 S Thrun W Burgard and D Fox Probabilistic Robotics MIT Press Cambridge MA (2005)

12 M Bolic P M Djuric and S J Hong Resampling algorithms for particle filters a computational

complexity perspective Eurasip J Appl Signal Process 15 2267ndash2277 (2004)

13 A D Anderson Recovering sample diversity in RaondashBlackwellized particle filters for simultane-

ous localization and mapping Master Thesis MIT Cambridge MA (2006)

14 D E Goldberg Genetic Algorithms in Search Optimization and Machine Learning Kluwer

Boston MA (1989)

15 D Whitley The GENITOR algorithm and selection pressure why rank-based allocation of repro-

ductive trials is best in Proc 3rd Int Conf on Genetic Algorithms Fairfax pp 116ndash121 (1989)

16 K-K Jin Genetic Algorithms and Their Applications Kyo Woo Sa Seoul (2002) (in Korean)

17 J S Liu and R Chen Sequential Monte-Carlo methods for dynamic systems J Am Stat Assoc

93 1032ndash1044 (1998)18 G Kitagawa Monte-Carlo filter and smoother for non-Gaussian nonlinear state space models

J Computat Graph Stat 5 1ndash25 (1996)

19 D Crisan P Moral and T Lyons Discrete filtering using branching and interacting particle sys-

tems Markov Process Related Fields 5 293ndash318 (1999)

20 A Doucet N de Freitas and N Gordon Sequetial Monte-Carlo Methods in Practice Springer

Berlin (2000)

21 J S Liu and R Chen Blind deconvolution via sequential imputations J Am Stat Ass 90

567ndash576 (1995)




About the Authors

Nosan Kwak received the PhD degree (MS and PhD integrated course) in Elec-

trical Engineering and Computer Science from Seoul National University South

Korea in 2008 and the BE in Mechanical Engineering from Ajou University

South Korea in 2001 He is currently a Japan Society for the Promotion of Sci-ence (JSPS) Postdoctoral Fellow at Humanoid Research Group National Institute

of Advanced Industrial Science and Technology (AIST) at Tsukuba Japan He

was a postdoctoral researcher at AIST until September 2008 He also belongs to

the Joint Robotics Laboratory where he researches Visual SLAM for a humanoid

robots using particle filters His major research fields are SLAM particle filtering game theory and

altruistic behaviors Overall his goal in robotics is to develop pragmatic robots in order to use them

in daily life

Kazuhito Yokoi received his BE degree in Mechanical Engineering from NagoyaInstitute of Technology in 1984 and the ME and PhD degrees in Mechanical

Engineering Science from Tokyo Institute of Technology in 1986 and 1994 re-

spectively In 1986 he joined the Mechanical Engineering Laboratory Ministry

of International Trade and Industry He was the Co-director of the ISAIST-CNRS

Joint Robotics Laboratory He is currently the Deputy Director of the Intelli-

gent Systems Research Institute National Institute of Advanced Industrial Science

and Technology at Tsukuba and is the Group Leader of the Humanoid Research

Group He is also an Adjunctive Professor of the Cooperative Graduate School at University Tsukuba

From November 1994 to October 1995 he was a Visiting Scholar at the Robotics Laboratory Com-

puter Science Department Stanford University His research interests include humanoids human-

centered robotics and intelligent robot systems He is a Member of the IEEE Robotics and Automation

Society

Beom-Hee Lee received the BS and MS degrees in Electronics Engineering from

Seoul National University in 1978 and 1980 respectively and the PhD degree in

Computer Information and Control Engineering from the University of Michigan

Ann Arbor MI USA in 1985 Since then he was associated with the School of

Electrical Engineering at Purdue University as an Assistant Professor until 1987

He is now with the School of Electrical Engineering and Computer Science as a

Professor and with the Office of Information Systems as Dean at Seoul National

University His major research interests include motion planning and control of

robot manipulators multi-robot operation sensor fusion applications and factory automation He has

been the President of the Korea Robotics Society since December 2008




1 Introduction

The RaondashBlackwellized particle filter (RBPF) was introduced about a decade ago

as an effective means to solve the simultaneous localization and mapping (SLAM)

problem by Murphy [1] and Doucet et al [2] This solution to SLAM is RBPF-

SLAM which is also known as FastSLAM [3] The SLAM posterior is normally

the joint estimation of a robotrsquos path and a map However RBPF-SLAM factors the

joint estimation using RaondashBlackwellization which factors a state into a sampled

part (path) and an analytical part (map) In RBPF-SLAM the robotrsquos path is esti-

mated by a particle filter and the map by low-dimensional extended Kalman filters

(EKFs)

RBPF-SLAM has two major advantages compared to other approaches (i) By

factoring the SLAM posteriors RBPF-SLAM has linear time complexity (ii) Un-like EKF-SLAM RBPF-SLAM allows each particle to perform its own data asso-

ciation which implements multi-hypothesis data association [3] The ability to si-

multaneously pursues multi-hypothesis data association makes RBPF-SLAM more

robust to data association problems than algorithms based on incremental maxi-

mum likelihood data association such as EKF-SLAM This special characteristic

is dependent on particle diversity The bigger the number of distinct particles the

more chance to close a loop because new observations can affect the locations of

the landmark [4] Thus it is crucial for RBPF-SLAM to maintain particle diversityas long as possible

However the innate disadvantage of RBPF-SLAM is that past pose estimation

errors of a robot are not forgotten which means that they are recorded in the feature

estimates Whenever the resampling process is conducted the entire robot paths

and feature estimates of rejected particles are lost forever As a result the number

of particles representing past paths and feature estimates severely decreases This

is called the particle depletion problem [5] In other words as the particle set loses

its diversity it becomes over-confident which means it tends to underestimate itsown uncertainty According to the work by Bailey et al [4] the particle diversity is

drastically attenuated when a robot closes a large traverse loop

Another critical issue in RBPF-SLAM is the consistency which is the ability of

the filter to accurately estimate uncertainty A filter can be inconsistent in either

an optimistic or a pessimistic way According to Ref [6] a filter is optimistic or

over-confident if there is significant bias in the estimates the errors are too large

compared to the filter-calculated covariance or the covariance is too small On the

other hand a filter is pessimistic or conservative if the covariance is too large It hasbeen shown in simulation works [4 7] that the current RBPF-SLAM algorithm is in-

consistent in an optimistic way and Stachniss et al [8] showed in their experiments

that RBPF-SLAM is consistent only in a local area (or in the short term) According

to Ref [4] RBPF-SLAM in its current form cannot produce consistent estimates

in the long term although it stays reasonably consistent for a few tens of seconds

after starting Beevers et al [7] improved the consistency by applying block pro-

posal distribution in the sampling process which exploits future information Their






























2 RBPF-SLAM















N f n=1









Y [k]t =

x

[k]1t μ

[k]1t

[k]1t μ

[k]N f t

[k]N f t

(2)













[k]jt


















Table 1

RSR algorithm






])

for i = 1 to N in



end








[i]n











w[k]t =

target distribtuion


= p(x[k]



[k]t |x


(3)
















































NEES is obtained as

εt =1

N

N i=1

εit (5)














3 RBR




















































p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society








































2 RBPF-SLAM















N f n=1









Y [k]t =

x

[k]1t μ

[k]1t

[k]1t μ

[k]N f t

[k]N f t

(2)













[k]jt


















Table 1

RSR algorithm






])

for i = 1 to N in



end








[i]n











w[k]t =

target distribtuion


= p(x[k]



[k]t |x


(3)
















































NEES is obtained as

εt =1

N

N i=1

εit (5)














3 RBR




















































p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society
















N f n=1









Y [k]t =

x

[k]1t μ

[k]1t

[k]1t μ

[k]N f t

[k]N f t

(2)













[k]jt


















Table 1

RSR algorithm






])

for i = 1 to N in



end








[i]n











w[k]t =

target distribtuion


= p(x[k]



[k]t |x


(3)
















































NEES is obtained as

εt =1

N

N i=1

εit (5)














3 RBR




















































p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society














Table 1

RSR algorithm






])

for i = 1 to N in



end








[i]n











w[k]t =

target distribtuion


= p(x[k]



[k]t |x


(3)
















































NEES is obtained as

εt =1

N

N i=1

εit (5)














3 RBR




















































p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society





















































NEES is obtained as

εt =1

N

N i=1

εit (5)














3 RBR




















































p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society

























NEES is obtained as

εt =1

N

N i=1

εit (5)














3 RBR




















































p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society














3 RBR




















































p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society

























p[k]s =

1

N p


(rank (k)minus 1)

N p minus 1

(6)














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society














Table 2








4 for i = 1 to N in

















limN prarrinfin

N pk=1h(x

[k])w[k]

N pk=1w

[k]=Eπ h(X) (8)












Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society















Table 3


Particle index 1 2 3 21 46 49 83 87 88 96 100

No replicas 29 1 5 1 2 9 1 1 1 42 8












E[x] = 0 (9)













results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society




















results [5]






































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society



































Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society














Table 4



11 01239 00554 01239

13 01267 00543 01201

20 01637 00491 01529

Table 5



RSR 03099 00468 03590

RBR 01267 00543 01201


















01774 m ( RMSE F)











Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society

















Table 6



25 01267 00543 01201

50 01312 00537 01324

75 01513 00534 01563















































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society











































Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society















Table 7




RSR 44857 00696 44575 29

RBR 19060 00873 19306 243









5 Conclusions










optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society
















optimistic way


















Acknowledgements









References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society

















References





CA pp 176ndash183 (2000)






















Boston MA (1989)










Berlin (2000)


567ndash576 (1995)




About the Authors










in daily life












Society














About the Authors










in daily life












Society











Documents

Advanced Robotics-24 s6