a solution to the simultaneous ion and

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A Solution to the Simultaneous Localisation and

Map Building (SLAM) ProblemM.W.M.G. Dissanayake, P.Newman, S. Clark,

H.F. Durrant-Whyte and M. Csorba

Australian Centre for Field Robotics

Department of Mechanical and Mechatronic Engineering

The University of Sydney

NSW 2006, Australia

Abstract The simultaneous localisation and map building

(SLAM) problem asks if it is possible for an autonomous ve-hicle to start in an unknown location in an unknown environ-ment and then to incrementally build a map of this environ-ment while simultaneously using this map to compute abso-lute vehicle location. Starting from the estimation-theoreticfoundations of this problem developed in [1], [2], [3], this pa-per proves that a solution to the SLAM problem is indeedpossible. The underlying structure of the SLAM problem isfirst elucidated. A proof that the estimated map convergesmonotonically to a relative map with zero uncertainty isthen developed. It is then shown that the absolute accuracyof the map and the vehicle location reach a lower bound de-fined only by the initial vehicle uncertainty. Together, theseresults show that it is possible for an autonomous vehicle tostart in an unknown location in an unknown environmentand, using relative observations only, incrementally build a

perfect map of the world and to compute simultaneously abounded estimate of vehicle location.

This paper also describes a substantial implementationof the SLAM algorithm on a vehicle operating in an out-door environment using millimeter-wave (MMW) radar toprovide relative map observations. This implementation isused to demonstrate how some key issues such as map man-agement and data association can be handled in a practicalenvironment. The results obtained are cross-compared withabsolute locations of the map landmarks obtained by sur-veying. In conclusion, this paper discusses a number of keyissues raised by the solution to the SLAM problem includingsub-optimal map-building algorithms and map management.

I. Introduction

The solution to the simultaneous localisation and mapbuilding (SLAM) problem is, in many respects, a HolyGrail of the autonomous vehicle research community. Theability to place an autonomous vehicle at an unknown lo-cation in an unknown environment and then have it builda map, using only relative observations of the environment,and then to use this map simultaneously to navigate wouldindeed make such a robot autonomous. Thus the mainadvantage of SLAM is that it eliminates the need for artifi-cial infrastructures or a priori topological knowledge of theenvironment. A solution to the SLAM problem would beof inestimable value in a range of applications where abso-lute position or precise map information is unobtainable,including, amongst others, autonomous planetary explo-ration, subsea autonomous vehicles, autonomous air-bornevehicles, and autonomous all-terrain vehicles in tasks such

as mining and construction.

The general SLAM problem has been the subject ofsubstantial research since the inception of a robotics re-search community and indeed before this in areas such asmanned vehicle navigation systems and geophysical sur-veying. A number of approaches have been proposed toaddress both the SLAM problem and also more simpli-fied navigation problems where additional map or vehiclelocation information is made available. Broadly, these ap-proaches adopt one of three main philosophies. The mostpopular of these is the estimation-theoretic or Kalman-filter based approach. The popularity of this approach isdue to two main factors. Firstly, it directly provides both arecursive solution to the navigation problem and a meansof computing consistent estimates for the uncertainty invehicle and map landmark locations on the basis of statis-tical models for vehicle motion and relative landmark ob-servations. Secondly, a substantial corpus of method andexperience has been developed in aerospace, maritime andother navigation applications, from which the autonomousvehicle community can draw. A second philosophy is to es-chew the need for absolute position estimates and for pre-cise measures of uncertainty and instead to employ morequalitative knowledge of the relative location of landmarksand vehicle to build maps and guide motion. This general

philosophy has been developed by a number of differentgroups in a number of different ways; see [4][5] and [6].The qualitative approach to navigation and the generalSLAM problem has many potential advantages over theestimation-theoretic methodology in terms of limiting theneed for accurate models and the resulting computationalrequirements, and in its significant anthropomorphic ap-peal. The third, very broad philosophy, does away withthe rigorous Kalman filter or statistical formalism while re-taining an essentially numerical or computational approachto the navigation and SLAM problem. Such approaches in-clude the use of iconic landmark matching [7], global mapregistration [8], bounded regions [9] and other measuresto describe uncertainty. Notable are the work by Thrunet. al. [10] and Yamauchi et. al. [11]. Thrun et. al.use a bayesian approach to map building that does notassume Gaussian probability distributions as required bythe Kalman filter. This technique while very effective for


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localisation with respect to maps, does not lend itself toprovide an incremental solution to SLAM where a map isgradually built as information is received from sensors. Ya-

mauchi et. al. use a evidence grid approach that requiresthat the environment is decomposed to a number of cells.

An estimation-theoretic or Kalman filter based approachto the SLAM problem is adopted in this paper. A majoradvantage of this approach is that it is possible to developa complete proof of the various properties of the SLAMproblem and to study systematically the evolution of themap and the uncertainty in the map and vehicle location.A proof of existence and convergence for a solution of theSLAM problem within a formal estimation-theoretic frame-work also encompasses the widest possible range of navi-gation problems and implies that solutions to the problem

using other approaches are possible.The study of estimation-theoretic solutions to the SLAM

problem within the robotics community has an interestinghistory. Initial work by Smith et al. [12] and Durrant-Whyte [13] established a statistical basis for describing re-lationships between landmarks and manipulating geomet-ric uncertainty. A key element of this work was to showthat there must be a high degree of correlation betweenestimates of the location of different landmarks in a mapand that indeed these correlations would grow to unity fol-lowing successive observations. At the same time Ayacheand Faugeras [14] and Chatila and Laumond [15] were un-dertaking early work in visual navigation of mobile robotsusing Kalman filter-type algorithms. These two strandsof research had much in common and resulted soon af-ter in the key paper by Smith, Self and Cheeseman [1].This paper showed that as a mobile robot moves throughan unknown environment taking relative observations oflandmarks, the estimates of these landmarks are all neces-sarily correlated with each other because of the commonerror in estimated vehicle location. This paper was fol-lowed by a series of related work developing a number ofaspects of the essential SLAM problem ( [2] and [3] for ex-ample). The main conclusion of this work was two fold.Firstly accounting for correlations between landmarks in a

map is important if filter consistency is to be maintained.Secondly that a full SLAM solution requires that a statevector consisting of all states in the vehicle model and allstates of every landmark in the map needs to be maintainedand updated following each observation if a complete solu-tion to the SLAM problem is required. The consequenceof this in any real application is that the Kalman filterneeds to employ a huge state vector (of order the num-ber of landmarks maintained in the map) and is in gen-eral, computationally intractable. Crucially, this work didnot look at the convergence properties of the map or itssteady-state behaviour. Indeed, it was widely assumed atthe time that the estimated map errors would not convergeand would instead execute a random walk behaviour withunbounded error growth. Given the computational com-plexity of the SLAM problem and without knowledge ofthe convergence behaviour of the map, a series of approx-imations to the full SLAM solution were proposed which

assumed that the correlations between landmarks could beminimised or eliminated thus reducing the full filter to aseries of decoupled landmark to vehicle filters (see Renken

[16], Leonard and Durrant-Whyte [3] for example). Alsofor these reasons, theoretical work on the full estimation-theoretic SLAM problem largely ceased, with effort insteadbeing expended in map-based navigation and alternativetheoretical approaches to the SLAM problem.

This paper starts from the original estimation-theoreticwork of Smith, Self and Cheeseman. It assumes an au-tonomous vehicle (mobile robot) equipped with a sensorcapable of making measurements of the location of land-marks relative to the vehicle. The landmarks may be ar-tificial or natural and it is assumed that the signal pro-cessing algorithms are available to detect these landmarks.

The vehicle starts at an unknown location with no knowl-edge of the location of landmarks in the environment. Asthe vehicle moves through the environment (in a stochasticmanner) it makes relative observations of the location of in-dividual landmarks. This paper then proves the followingthree results:

1. The determinant of any submatrix of the map covari-ance matrix decreases monotonically as observations aresuccessively made.2. In the limit as the number of observations increases, thelandmark estimates become fully correlated.3. In the limit, the covariance associated with any singlelandmark location estimate is determined only by the ini-

tial covariance in the vehicle location estimate.These three results describe, in full, the convergence

properties of the map and its steady state behaviour. Inparticular they show that

The entire structure of the SLAM problem critically de-pends on maintaining complete knowledge of the cross cor-relation between landmark estimates. Minimizing or ignor-ing cross correlations is precisely contrary to the structureof the problem. As the vehicle progresses through the environment theerrors in the estimates of any pair of landmarks becomemore and more correlated, and indeed never become less

correlated. In the limit, the errors in the estimates of any pair oflandmarks becomes fully correlated. This means that giventhe exact location of any one landmark, the location of anyother landmark in the map can also be determined withabsolute certainty. As the vehicle moves through the environment takingobservations of individual landmarks, the error in the esti-mates of the relative location between different landmarksreduces monotonically to the point where the map of rela-tive locations is known with absolute precision. As the map converges in the above manner, the errorin the absolute location of every landmark (and thus thewhole map) reaches a lower bound determined only by theerror that existed when the first observation was made.

Thus a solution to the general SLAM problem existsand it is indeed possible to construct a perfectly accu-rate map and simultaneously compute vehicle position esti-


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mates without any prior knowledge of vehicle or landmarklocations.

This paper makes three principal contributions to the so-

lution of the SLAM. Firstly, it proves three key convergenceproperties of the full SLAM filter. Secondly, it elucidatesthe true structure of the SLAM problem and shows howthis can be used in developing consistent SLAM algorithms.Finally, it demonstrates and evaluates the implementationof the full SLAM algorithm in an outdoor environment us-ing a millimeter-wave radar sensor.

Section 2 of this paper introduces the mathematicalstructure of the estimation-theoretic SLAM problem. Sec-tion 3 then proves and explains the three convergence re-sults. Section 4 provides a practical demonstration of animplementation of the full SLAM algorithm in an outdoor

environment using MMW radar to provide relative obser-vations of landmarks. An algorithm addressing pertinentissues of map initialisation and management is also pre-sented. The algorithm outputs are shown to exhibit theconvergent properties derived in Section 3. Section 5 dis-cusses the many remaining problems with obtaining a prac-tical, large scale, solution to the SLAM problem includingthe development of sub-optimal solutions, map manage-ment and data association.

II. The Estimation-Theoretic SLAM Problem

This section establishes the mathematical framework em-ployed in the study of the SLAM problem. This framework

is identical in all respects to that used in Smith et. al. [1]and uses the same notation as that adopted in Leonard andDurrant-Whyte [3].

A. Vehicle and Land-Mark Models

The setting for the SLAM problem is that of a vehi-cle with a known kinematic model, starting at an un-known location, moving through an environment contain-ing a population of features or landmarks. The terms fea-ture and landmark will be used synonymously. The vehicleis equipped with a sensor that can take measurements ofthe relative location between any individual landmark and

the vehicle itself as shown in Figure 1. The absolute lo-cations of the landmarks are not available. Without prej-udice, a linear (synchronous) discrete-time model of theevolution of the vehicle and the observations of landmarksis adopted. Although vehicle motion and the observation oflandmarks is almost always non-linear and asynchronous inany real navigation problem, the use of linear synchronousmodels does not affect the validity of the proofs in Sec-tion 3 other than to require the same linearisation assump-tions as those normally employed in the development ofan extended Kalman filter. Indeed, the implementationof the SLAM algorithm described in Section 4 uses non-linear vehicle models and non-linear asynchronous obser-vation models. The state of the system of interest consistsof the position and orientation of the vehicle together withthe position of all landmarks. The state of the vehicle ata time step k is denoted xv(k). The motion of the vehiclethrough the environment is modeled by a conventional lin-

xv

pi

Features and Landmarks

Global Reference Frame

Mobile Vehicle

Vehicle-Feature Relative

Observation

Fig. 1. A vehicle taking relative measurements to environmental

landmarks

ear discrete-time state transition equation or process modelof the form

xv (k + 1) = Fv (k)xv (k) + uv (k + 1) + vv (k + 1), (1)

where Fv(k) is the state transition matrix, uv(k) a vectorof control inputs, and vv(k) a vector of temporally uncor-related process noise errors with zero mean and covarianceQv(k) (see [17] and [18] for further details). The locationof the ith landmark is denoted pi. The state transitionequation for the ith landmark is

pi(k + 1) = pi(k) = pi, (2)

since landmarks are assumed stationary. Without loss ofgenerality the number of landmarks in the environment isarbitrarily set at N. The vector of all N landmarks isdenoted

p =

pT1 . . . pTN

T(3)

where T denotes the transpose and is used both inside andoutside the brackets to conserve space. The augmentedstate vector containing both the state of the vehicle and

the state of all landmark locations is denotedx(k) =

xTv (k) p

T1 . . . p

TN

T. (4)

The augmented state transition model for the complete sys-tem may now be written as

xv (k + 1)p1

...pN

=

Fv (k) 0 . . . 00 Ip1 . . . 0...

.... . . 0

0 0 0 IpN

xv (k)p1

...pN

+

uv (k + 1)0p1

...0pN

+

vv (k + 1)0p1

...0pN

(5)

x(k + 1) = F(k)x(k) + u(k + 1) + v(k + 1) (6)


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where Ipi is the dim(pi) dim(pi) identity matrix and 0piis the dim(pi) null vector.

The case in which landmarks pi are in stochastic motion

may easily be accommodated in this framework. However,doing so offers little insight into the problem and further-more the convergence properties presented by this paperdo not hold.

B. The Observation Model

The vehicle is equipped with a sensor that can obtain ob-servations of the relative location of landmarks with respectto the vehicle. Again, without prejudice, observations areassumed to be linear and synchronous. The observationmodel for the ith landmark is written in the form

zi(k) = Hix(k) + wi(k) (7)= Hpip Hvxv(k) + wi(k) (8)

where wi(k) is a vector of temporally uncorrelated observa-tion errors with zero mean and variance Ri(k). The termHi is the observation matrix and relates the output of thesensor zi(k) to the state vector x(k) when observing thei(th) landmark. It is important to note that the observa-tion model for the ith landmark is written in the form

Hi = [Hv , 0 0, Hpi, 0 0] (9)

This structure reflects the fact that the observations arerelative between the vehicle and the landmark, often inthe form of relative location, or relative range and bearing(see Section 4).

C. The Estimation Process

In the estimation-theoretic formulation of the SLAMproblem, the Kalman filter is used to provide estimatesof vehicle and landmark location. We briefly summarisethe notation and main stages of this process as a neces-sary prelude to the developments in Sections 3 and 4 ofthis paper. Detailed descriptions may be found in [17],[18]and [3]. The Kalman filter recursively computes estimatesfor a state x(k) which is evolving according to the processmodel in Equation 5 and which is being observed accord-ing to the observation model in Equation 7. The Kalmanfilter computes an estimate which is equivalent to the con-ditional mean x(p|q) = E [x(p)|Zq] (p q), where Zq isthe sequence of observations taken up until time q. The er-ror in the estimate is denoted x(p|q) = x(p|q)x(p). TheKalman filter also provides a recursive estimate of the co-

variance P(p|q) = E

x(p|q)x(p|q)T

|Zq

in the estimate

x(p|q). The Kalman filter algorithm now proceeds recur-sively in three stages:

Prediction: Given that the models described in equa-tions 5 and 7 hold, and that an estimate x(k|k) of thestate x(k) at time k together with an estimate of the co-variance P(k|k) exist, the algorithm first generates a pre-diction for the state estimate, the observation (relative tothe ith landmark) and the state estimate covariance at time

k + 1 according to

x(k + 1|k) = F(k)x(k|k) + u(k) (10)

zi(k + 1|k) = Hi(k)x(k + 1|k) (11)

P(k + 1|k) = F(k)P(k|k)FT(k) + Q(k), (12)

respectively. Observation: Following the prediction, an observationzi(k + 1) of the i

th landmark of the true state x(k + 1) ismade according to Equation 7. Assuming correct landmarkassociation, an innovation is calculated as follows

i(k + 1) = zi(k + 1) zi(k + 1|k) (13)

together with an associated innovation covariance matrix

given by

Si(k + 1) = Hi(k)P(k + 1|k)HTi (k) + Ri(k + 1). (14)

Update: The state estimate and corresponding state es-timate covariance are then updated according to:

x(k + 1|k + 1) = x(k + 1|k) + Wi(k + 1)i(k + 1) (15)

P(k + 1|k + 1) = P(k + 1|k) Wi(k + 1)

S(k + 1)WTi (k + 1) (16)

Where the gain matrix Wi(k + 1) is given by

Wi(k + 1) = P(k + 1|k)HTi (k)S

1i (k + 1) (17)

The update of the state estimate covariance matrix is ofparamount importance to the SLAM problem. Under-standing the structure and evolution of the state covariancematrix is the key component to this solution of the SLAMproblem.

III. Structure of the SLAM problem

In this section proofs for the three key results underlyingstructure of the SLAM problem are provided. The impli-cations of these results will also be examined in detail. Theappendix provides a summary of the key properties of pos-itive semi-definite matrices that are invoked implicitly inthe following proofs.

A. Convergence of the map covariance matrix

The state covariance matrix may be written in blockform as

P(i|j) =

Pvv (i|j) Pvm(i|j)PTvm(i|j) Pmm(i|j),

where Pvv (i|j) is the error covariance matrix associatedwith the vehicle state estimate, Pmm(i|j) is the map covari-ance matrix associated with the landmark state estimates,and Pvm(i|j) is the cross-covariance matrix between vehi-cle and landmark states.

Theorem 1: The determinant of any submatrix of themap covariance matrix decreases monotonically as succes-sive observations are made.


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The algorithm is initialised using a positive semi-definite(psd) state covariance matrix P(0|0). The matrices Q andRi are both psd, and consequently the matrices P(k+1|k),

Si(k+1), Wi(k+1)Si(k+1)WTi (k+1) and P(k+1|k+1)are all psd. From Equation 16, and for any landmark i,

det P(k + 1|k + 1) = det[P(k + 1|k)

Wi(k + 1)Si(k + 1)WT(k + 1)]

det P(k + 1|k) (18)

The determinant of the state covariance matrix is a mea-sure of the volume of the uncertainty ellipsoid associatedwith the state estimate. Equation 18 states that the totaluncertainty of the state estimate does not increase duringan update.

Any principal submatrix of a psd matrix is also psd (seeAppendix 1). Thus, from Equation 18 the map covariancematrix also has the property

det Pmm(k + 1|k + 1) det Pmm(k + 1|k) (19)

From Equation 12, the full state covariance predictionequation may be written in the form

Pvv (k + 1|k) Pvm(k + 1|k)PTvm(k + 1|k) Pmm(k + 1|k)

= Y1

where

Y1 =

FvPvv (k|k)Fv

T + Qvv FvPvm(k|k)

PTvm(k|k)FvT Pmm(k|k)

.

Thus, as landmarks are assumed stationary, no processnoise is injected in to the predicted map states. Conse-quently, the map covariance matrix and any principal sub-matrix of the map covariance matrix has the property that

Pmm(k + 1|k) = Pmm(k|k). (20)

Note that this is clearly not true for the full covariancematrix as process noise is injected in to the vehicle location

predictions and so the prediction covariance grows duringthe prediction step.

It follows from Equations 19 and 20 that the map covari-ance matrix has the property that

det Pmm(k + 1|k + 1) det Pmm(k|k). (21)

Furthermore, the general properties of psd matrices ensurethat this inequality holds for any submatrix of the mapcovariance matrix. In particular, for any diagonal element2ii of the map covariance matrix (state variance),

2

ii

(k + 1|k + 1) 2ii

(k|k).

Thus the error in the estimate of the absolute location ofevery landmark also diminishes.

Theorem 2: In the limit the landmark estimates becomefully correlated

As the number of observations taken tends to infinitya lower limit on the map covariance limit will be reachedsuch that

limk

[Pmm(k + 1|k + 1)] = Pmm(k|k) (22)

Writing P(k + 1|k) as P and P(k + 1|k + 1) as P fornotational clarity, the SLAM algorithm update stage canbe written as

P = P Wi(k + 1)SiWiT(k + 1)

= P PHiTS1i HiP

= P

M1M2

S1i

M1

T M2T

= P

M1S1

i

M1T M1S

1

i

M2T

M2S1i M1

T M2S1i M2

T

(23)

where

M1 = P

vvHvT + PvmHpi

T

M2 = P

vmHvT + PmmHpi

T (24)

The update of the map covariance matrix Pmm can bewritten as

Pmm(k + 1|k + 1) = Pmm(k + 1|k) M2S1i M2

T

= Pmm(k|k) M2S1i M2

T (25)

Equations 22 and 25 imply that the matrix M2S1i M2

T =0. The inverse of the innovation covariance matrix S1i isalways p.s.d because the observation noise covariance Ri ispsd, therefore Equation 22 requires that M2 = 0

Pmm(k|k)HpiT = Pvm(k|k)Hv

T i (26)

Equation 26 holds for all i and therefore the block columnsof Pmm are linearly dependent. A consequence of this factis that in the limit the determinant of the covariance matrixof a map containing more than one landmark tends to zero.

limk [det Pmm(k|k)] = 0 (27)

This implies that the landmarks become progressively morecorrelated as successive observations are made. In the limitthen, given the exact location of one landmark the locationof all other landmarks can be deduced with absolute cer-tainty and the map is fully correlated.

Consider the implications of Equation 26 upon the esti-mate d of the relative position between any two landmarkspi and pj of the same type.

d = pi pj

= Gijx

The covariance Pd of d is given by

Pd = GijPGijT

= Pii + Pjj Pij PijT


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With similar landmarks types, Hpi = Hpj and so Equation26 implies that the block columns of Pmm are identical.Furthermore, because Pmm is symmetric it follows that

Pii = Pjj = Pij = PijT i, j (28)

Therefore, in the limit, Pd = 0 and the relationship be-tween the landmarks is known with complete certainty. Itis important to note that this result does not mean that thedeterminants of the landmark covariance matrices tend tozero. In the limit the absolute location of landmarks maystill be uncertain.

Theorem 3: In the limit, the lower bound on the covari-ance matrix associated with any single landmark estimateis determined only by the initial covariance in the vehicleestimate P0v at the time of the first sighting of the first

landmark.As described previously, the covariance of the landmarklocation estimates decrease as successive observations aremade. The best estimates are obviously obtained whenthe covariance matrices of the vehicle process noise Q andthe observation noise Ri are small. The limiting value andhence lower bound of the state covariance matrix can be ob-tained when the vehicle is stationary giving Q = 0. Underthese circumstances it is convenient to use the informationform of the Kalman filter to examine the behaviour of thestate covariance matrix. Continuing with the single land-mark environment,the state covariance matrix after thissolitary landmark has been observed for k instances can bewritten as

P(k|k)1 = P(k|k 1)1 +

Hv

T

Hp1T

TR1

1

Hv Hp1

(29)

where because Q = 0,

P(k|k 1)1 = P(k 1|k 1)1 (30)

Using Equations 29 and 30 P(k|k)1 can now be writtenas

P(

k|k

)

1

=P0v

1 0

0 0

+Y2 (31)

where

Y2 =

kHv

TR11Hv kHv

TR11Hp1

kHp1TR1

1Hv kHp1TR1

1Hpi

Invoking the matrix inversion lemma for partitioned ma-trices,

P(k|k) =

P0v P0vHvT

Hp1

T1

H1p1 HvP0v H1p1 HvP0vH

1p1 Hv

T+ Y3

(32)

where

Y3 =H1p1 R1

H1p1

Tk

.

Examination of the lower right block matrix of equation 32shows how the landmark uncertainty estimate stems onlyfrom the initial vehicle uncertainty P0v . To find the lower

bound on the state covariance matrix and in turn the land-mark uncertainty the number of observations of the land-mark is allowed to tend to infinity to yield in the limit

limk

P(k|k) =

P0v P0vHvT

Hp1

T1

H1p1 HvP0v H1p1 HvP0vH

1p1 Hv

T

(33)

Equation 33 gives the lower bound of the solitary landmark

state estimate variance as H1p1 HvP0vH1p1 Hv

T. Examine

now the case of an environment containing N landmarks,

The smallest achievable uncertainty in the estimate of theith landmark when the landmark has been observed at theexclusion of all other landmarks is H1pi HvP0vH

1pi Hv

T.

If more than one landmark is observed as k , as willbe the case in any non-trivial navigation problem, thenit is possible for the landmark uncertainty to be furtherreduced by theorem 1. In the limit the lower bound on theuncertainty in the ith landmark state is written as

Ppi,pi = min

i[1,N]

H1pi HvP0vH

1pi Hv

T

(34)

and is determined only by the initial covariance in the ve-hicle location estimate P0v . Note that because Q was setto zero in search of the lower bound the vehicle uncertaintyremains unchanged at P0v as k . In the simple casewhere Hpi and Hv are identity matrices in the limit thecertainty of each landmark estimate achieves a lower boundgiven by the initial uncertainty of the vehicle.

When the process noise is not zero the two competingeffects of loss of information content due to process noiseand the increase in information content through observa-tions, determine the limiting covariance. The problem isnow analytically intractable, although the limiting covari-ance of the map can never be below the limit given by theabove equation and will be a function of P0v, Q and R.

It is important to note that the limit to the covarianceapplies because all the landmarks are observed and ini-tialised solely from the observations made from the vehi-cle. The covariances of landmark estimates can not befurther reduced by making additional observations to pre-viously unknown landmarks. However, incorporation of ex-ternal information, for example using an observation madeto a landmark whose location is available through externalmeans such as GPS, will reduce the limiting covariance.

In summary, the three theorems derived above describe,in full, the convergence properties of the map and its steadystate behaviour. As the vehicle progresses through the en-vironment the total uncertainty of the estimates of land-mark locations reduces monotonically to the point wherethe map of relative locations is known with absolute pre-cision. In the limit, errors in the estimates of any pairof landmarks become fully correlated. This means thatgiven the exact location of any one landmark, the location


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of any other landmark in the map can also be determinedwith absolute certainty. As the map converges in the abovemanner, the error in the absolute location estimate of every

landmark (and thus the whole map) reaches a lower bounddetermined only by the error that existed when the firstobservation was made.

Thus a solution to the general SLAM problem exists andit is indeed possible to construct a perfectly accurate mapdescribing the relative location of landmarks and simul-taneously compute vehicle position estimates without anyprior knowledge of landmark or vehicle locations.

IV. Implementation of the simultaneous

localisation and map building algorithm

This section describes a practical implementation of the

simultaneous localisation and map building (SLAM) algo-rithm on a standard road vehicle. The vehicle is equippedwith a millimeter-wave radar (MMWR) which provides ob-servations of the location of landmarks with respect to thevehicle. The implementation is aimed at demonstratingkey properties of the SLAM algorithm; convergence, con-sistency and boundedness of the map error.

The implementation also serves to highlight a numberof key properties of the SLAM algorithm and its practicaldevelopment. In particular, the implementation shows howgenerally non-linear vehicle and observation models may beincorporated in the algorithm, how the issue of data associ-ation can be dealt with, and how landmarks are initialised

and tracked as the algorithm proceeds.The implementation described here is, however, only a

first step in the realisation and deployment of a fully au-tonomous SLAM navigation system. A number of substan-tive further issues in landmark extraction, data association,reduced computation and map management are discussedfurther in following sections.

A. Experimental setup

Figure 2 shows the test vehicle; a conventional utilityvehicle fitted with a MMWR wave radar system as the pri-mary sensor used in the experiments. Encoders are fitted

to the drive shaft to provide a measure of the vehicle speedand a Linear Variable Differential Transformer (LVDT) isfitted to the steering rack to provide a measure of vehicleheading. A differential GPS system and an inertial mea-surement unit are also fitted to the vehicle but are not usedin the experiments described below. In the environmentused for the experiments, DGPS was found to be prone tolarge errors due to the reflections caused by nearby largemetal structures (usually known as multi-path errors).The radar employed in the experiments is a 77 GHz FMCWunit. The radar beam is scanned 360 degrees in azimuth ata rate of 1-3Hz. After signal processing the radar providesan amplitude signal (power spectral density), correspond-ing to returns at different ranges, at angular increments ofapproximately 1.5o. This signal is thresholded to provide ameasurement of range and bearing to a target. The radaremploys a dual polarisation receiver so that even and oddbounce specularities can be distinguished. The radar is

Fig. 2. The test vehicle, showing mounting of the MMWR and GPSsystems

capable of providing range measurements to 250m with aresolution of 10cm in range and 1.5o in bearing. A detaileddescription of the radar and its performance can be foundin [19]. Figure 3 shows the test vehicle moving in an envi-

Fig. 3. Test vehicle at the test site. A radar point landmark can beseen in the left foreground

ronment that contains a number of radar reflectors. Thesereflectors appear as omni-directional point landmarks inthe radar images. These, together with a number of nat-ural landmark point targets, serve as the landmarks to be

estimated by the SLAM algorithm.The vehicle is driven manually. Radar range and bearing

measurements are logged together with encoder and steerinformation by an on-board computer system. In the evalu-ation of the SLAM algorithm, this information is employedwithout any a priori knowledge of landmark location todeduce estimates for both vehicle position and landmarklocations.

To evaluate the SLAM algorithm, it is necessary to havesome idea of the true vehicle track and true landmark lo-cations that can be compared with those estimated by theSLAM algorithm. For this reason the true landmark lo-cations were accurately surveyed for comparison with theoutput of the map building algorithm. A second navigationalgorithm, that employs knowledge of beacon locations isthen run on the same data set as used to generate the mapestimates. (This algorithm is very similar to that describedin [20].) This algorithm provides an accurate estimate of


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true vehicle location which can be used for comparison withthe estimate generated from the SLAM algorithm.

In the following, the vehicle state is defined by xv =

[x, y, ]T where x and y are the coordinates of the centreof the rear axle of the vehicle with respect to some globalcoordinate frame and is the orientation of the vehicleaxis. The landmarks are modeled as point landmarks andrepresented by a cartesian pair such that pi = [xi, yi], i =1 . . . N. Both vehicle and landmark states are registered inthe same frame of reference.

A.1 The process model

Figure 4 shows a schematic diagram of the vehicle in theprocess of observing a landmark. The following kinematicequations can be used to predict the vehicle state from the

steering and velocity inputs V;

x = V cos()

y = V sin()

=V tan()

L,

where L is the wheel-base length of the vehicle. These equa-tions can be used to obtain a discrete-time vehicle processmodel in the form

x(k + 1)y(k + 1)

(k + 1)

=

x(k) + T V(k)cos((k))y(k) + T V(k)sin((k))

(k) +TV(k) tan((k))

L

(35)

for use in the prediction stage of the vehicle state estima-tor. The landmarks in the environment are assumed to bestationary point targets. The landmark process model isthus

xi(k + 1)yi(k + 1)

=

xi(k)yi(k)

(36)

for all landmarks i = 1 . . . N . Together, equations 35 and36 define the state transition matrix F() for the system.

A.2 The observation model

The millimeter wave radar used in the experiments re-turns the range ri(k) and bearing i(k) to a landmark i.Referring to Figure 4, the observation model can be writtenas

ri(k) =

(xi xr(k))2 + (yi yr(k))2 + wr(k)

i(k) = arctan

yi yr(k)

xi xr(k)

(k) + w(k) (37)

where wrand w are the noise sequences associated withthe range and bearing measurements, and [xr(k), yr(k)] isthe location of the radar given, in global coordinates, by

xr(k) = x(k) + a cos((k)) b sin((k))

yr(k) = y(k) + a sin((k)) + b cos((k))

Equation 37 defines the observation model Hi() for a spe-cific landmark.

a

b

ir

i

(x,y)

pi(x

i,y

i)

Global Reference Frame

L

(xr,y

r)

Fig. 4. Vehicle and observation kinematics

A.3 Estimation equations

The theoretical developments in this paper employedonly linear models of vehicle and landmark kinematics.This was necessary to develop the necessary proofs of con-vergence. However, the implementation described here re-quires the use of non-linear models of vehicle and landmarkkinematics f(.) and non linear models of landmark obser-vation h(.).

Practically an Extended Kalman Filter (EKF) ratherthan a simple linear Kalman filter is employed to gener-ate estimates. The EKF uses linearised kinematic and ob-

servation equations for generating state predictions. Theuse of the EKF in vehicle navigation and the necessary as-sumptions needed for successful operation is well known(see for example the development in [20]), and is thus notdeveloped further here.

A.4 Map initialisation and management

In any SLAM algorithm the number and location of land-marks is not known a priori. Landmark locations mustbe initialised and inferred from observations alone. Theradar receives reflections from many objects present in theenvironment but only the observations resulting from re-

flections from stationary point landmarks should be usedin the estimation process. Figure 5 shows a typical testrun and the locations that correspond to all the reflec-tions received by the radar. Only about 30% of the radarobservations corresponded to identifiable point landmarksin the environment. In addition to these, a large freightvehicle and a number of nearby buildings also reflect theradar beam and produced range and bearing observations.This data set illustrates the importance of correct land-mark identification, initialisation and subsequent data as-sociation. In this implementation a simple measure of land-mark quality is employed to initialise and track potentiallandmarks. Landmark quality implicitly tests whether thelandmark behaves as a stationary point landmark. Rangeand bearing measurements which exhibit this behaviour areassigned a high quality measure and are incorporated as alandmark. Those that do not are rejected.

The landmark quality algorithm is described in detail


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100 50 0 50

40

20

0

20

40

60

80

100

X(m)

Y(m)

Radar Observations

Vehicle Path

Radar Refelectors

Fig. 5. A typical test run. Only 30% of radar observations correspond

to identifiable landmarks

in Appendix 2. The algorithm uses two landmark lists torecord tentative and confirmed targets. A tentativelandmark is initialised on receipt of a range and bearingmeasurement. A tentative target is promoted to a con-firmed landmark when a sufficiently high quality measureis obtained. Once confirmed, the landmark is inserted intothe augmented state vector to be estimated as part of theSLAM algorithm. The landmark state location and covari-ance is initialised from observation data obtained when thelandmark is promoted to confirmed status. Figure 6 shows

the computed landmark quality obtained at the end of thetest run described in this paper.

40 30 20 10 0 10 20 30

10

0

10

20

30

40

0.69

0

0

0

0.8

0.81 0.8

0.77

0.42

0

0.74 0.63

0.75

0.85

0.79

0

0

0

X (m)

Y(m)

Feature Quality

Fig. 6. Calculated landmark qualities at the end of the test run.Landmark quality ranges from 1 (for highest quality) to 0 (forlowest quality). Landmarks are marked with a star. Landmarkswhich are also circled are the artificial landmarks whose locationswere surveyed so that the accuracy of the map building algorithmcan be evaluated.

B. Experimental Results

The vehicle starts at the origin, remaining stationary forapproximately 30 seconds and then executing a series ofloops at speeds up to 10m/s. Figure 7 shows the overallresults of the SLAM algorithm. The estimated landmark

locations are designated by stars. The actual (true) sur-veyed landmark locations are designated by circles. Thevehicle path is shown with a solid line. In order to com-

pute the true vehicle path, surveyed locations of the ar-tificial point landmarks were used for running a map basedestimation algorithm [20]. The absolute accuracy of thevehicle path obtained using this algorithm was computedto be approximately 5cm.

B.1 Vehicle localisation results

40 30 20 10 0 10 20 30

10

0

10

20

30

40

X(m)

Y(m)

Measured Feature Locations

Estimated Feature Locations

Vehicle Path

Fig. 7. The true vehicle path together with surveyed (circles) andestimated (starred) landmark locations

60 70 80 90 100 110 1200.8

0.6

0.4

0.2

0

0.2

0.4

0.6

Time (sec)

ErrorinX(m)

Error in vehicle path (fixed features slam)

60 70 80 90 100 110 1201

0.5

0

0.5

1

Time (sec)

ErrorinY(m)

Fig. 8. Actual error in vehicle location estimate in x and y (solidline), together with 95% confidence bounds (dotted lines) derivedfrom estimated location errors. The reduction in the estimatedlocation errors around 110 seconds is due to the vehicle slowingdown and stopping.

The differences between the true vehicle path shown inFigure 7 and the path estimated from the SLAM algorithmare too small to be seen on the scale used in Figure 7. Fig-ure 8 shows the error between true and SLAM estimatedposition in more detail. The figure shows the actual errorin estimated vehicle location in both x and y as a func-tion of time (the central solid line). The figure also shows


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95% (2) confidence limits in the estimate error. Theseconfidence bounds are derived from the state estimate co-variance matrix and represent the estimated vehicle error.

The actual vehicle error is clearly bounded by the confi-dence limits of estimated vehicle error. The estimate pro-duced by the SLAM algorithm is thus consistent (and in-deed is conservative). The estimated vehicle error as de-fined by the confidence bounds does not diverge so the es-timates produced are stable. The jump in x error near thestart of the run is caused by the vehicle accelerating whenthe model implicitly assumes a constant velocity model.The slight oscillation in errors and estimated errors aredue to the vehicle cornering and thus coupling long-travelwith lateral errors. Selecting a suitable model for the ve-hicle motion requires giving consideration to the tradeoff

between the filter accuracy and the model complexity. It ispossible to improve the results obtained by assuming a con-stant acceleration model, and relaxing the non-holonomicconstraint used in the derivation of the vehicle kinematicequation 35 by incorporating vehicle slip. This clearly in-creases the computational complexity of the algorithm.

The SLAM algorithm thus generates vehicle location es-timates which are consistent, stable and have bounded er-rors.

B.2 Map building results

Figure 9 shows the innovations in range and bearing ob-servations together with the estimated 95% confidence lim-its. The innovations are the only available measure foranalysing on-line filter behaviour when true vehicle statesare unavailable. The innovations here indicate that thefilter and models employed are consistent.

In addition to the ten radar reflectors placed in the en-vironment, the map building algorithm recognises a fur-ther seven natural landmarks as suitable point landmarks.These can be seen in Figure 7 as stars (confirmed land-marks) without circles (without a survey location). Someof these natural landmarks correspond to the legs of a largecargo moving vehicle parked in the test site. The othersdo not correspond to any obvious identifiable point land-

marks, but were recognised as landmarks simply becausethey returned consistent point-like radar reflections. Thelandmark qualities Q calculated at the end of the test runare shown in Figure 6.

Figures 10 and 11 show the error between the actual andestimated landmark locations for two of the radar reflec-tors. One of these landmarks (landmark 1) was observedfrom the initial vehicle location at the start of the testrun. The second landmark (landmark 3) was first observedabout 30 seconds later into the run. Figures 10 and 11also show the associated 95% confidence limits in the lo-cation estimates. As before, these are calculated using theestimated landmark location covariances. The landmarklocation estimates are thus consistent (and conservative)with actual landmark location errors being smaller thanthe estimated error.

It can be seen that the initial variance of landmark 3 ismuch greater than that of landmark 1. This is due to the

30 40 50 60 70 80 90 100 110 1201.5

1

0.5

0

0.5

1

1.5

Time (sec)

Innovation

inrange(m)

30 40 50 60 70 80 90 100 110 1200.2

0.1

0

0.1

0.2

Time (sec)

Innovationinbearing(rad)

Fig. 9. Range and bearing innovations together with associated 95%confidence bounds

40 50 60 70 80 90 100 1100.5

0

0.5Feature Number 1

Time (sec)

ErrorinX(m)

40 50 60 70 80 90 100 1101.5

1

0.5

0

0.5

1

1.5

Time (sec)

ErrorinY(m)

Fig. 10. Difference between the actual and estimated location forlandmark 1. The 95% confidence limit of the difference is shownby dotted lines

40 50 60 70 80 90 100 1103

2

1

0

1

2

3Feature Number 3

Time (sec)

ErrorinX

(m)

40 50 60 70 80 90 100 1101

0.5

0

0.5

1

Time (sec)

ErrorinY(m)

Fig. 11. Difference between the actual and estimated location forlandmark 3. The 95% confidence limit of the difference is shownby dotted lines


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fact that the uncertainty in the vehicle location is smallwhen landmark 1 is initialised, whereas landmark 3 is ini-tialised while the vehicle is in motion and uncertainty in

vehicle location is relatively high. These figures also showthat there is some bias in the landmark location estimates.However, this bias is well within the accuracy of the truemeasurement (estimated to be 0.1 m).

Figures 12 and 13 show the estimated standard devia-tions in x and y of all landmark location estimates pro-duced by the filter (for graphical purposes, the variancesare set to zero until the landmark is confirmed). As pre-dicted by theory, the estimated errors in landmark locationdecrease monotonically; and thus the overall error in themap reduces at each observation. Visually, the errors inlandmark location estimates reach a common lower bound.

As predicted by theory, this lower bound corresponds tothe initial uncertainty in vehicle location.

40 50 60 70 80 90 100 1100

0.5

1

1.5

2

2.5

Time (sec)

StandardDeviationinX(m)

Fig. 12. Decreasing uncertainty in landmark location estimates

40 50 60 70 80 90 100 1100

0.5

1

1.5

2

2.5

3

Time (sec)

StandardDeviationinY(m)

Fig. 13. Standard deviation of the landmark location estimate for alldetected landmarks. As predicted, the uncertainty in landmarklocation estimates decrease monotonically

V. Discussion and Conclusions

The main contribution of this paper is to demonstratethe existence of a non-divergent estimation theoretic so-lution to the SLAM problem and to elucidate upon thegeneral structure of SLAM navigation algorithms. Thesecontributions are founded on the three theoretical resultsproved in section 2; that uncertainty in the relative mapestimates reduces monotonically, that these uncertaintiesconverge to zero, and that the uncertainty in vehicle andabsolute map locations achieves a lower bound.

Propagation of the full map covariance matrix is essen-tial to the solution of the SLAM problem. It is the cross-correlations in this map covariance matrix which maintainknowledge of the relative relationships between landmarklocation estimates and which in turn underpin the exhib-

ited convergence properties. Omission of these cross corre-lations destroys the whole structure of the SLAM problemand results in inconsistent and divergent solutions to themap building problem.

However, the use of the full map covariance matrix ateach step in the map building problem causes substantialcomputational problems. As the number of landmarks Nincreases, the computation required at each step increasesas N2, and required map storage increases as N2. As therange over which it is desired to operate a SLAM algo-rithm increases (and thus the number of landmarks in-creases), it will become essential to develop a computa-tionally tractable version of the SLAM map building algo-rithm which maintains the properties of being consistentand non-divergent. There are currently two approachesto this problem; the first uses bounded approximations tothe estimation of correlations between landmarks, the sec-ond method exploits the structure of the SLAM problem totransform the map building process into a computationallysimpler estimation problem.

Bounded approximation methods use algorithms whichmake worst-case assumptions about correlatedness betweentwo estimates. These include the covariance intersectmethod [21], and the bounded region method [22]. Thesealgorithms result in SLAM methods which have constant

time update rules (independent of the number of landmarksin the map), and which are statistically consistent. How-ever, the conservative nature of these algorithms meansthat observation information is not fully exploited and con-sequently convergence rates for the SLAM method are of-ten impracticably slow (and in some cases divergent).

Transformation methods attempt to re-frame the mapbuilding problem in terms of alternate map or landmarkrepresentations which particularly have relative indepen-dence properties. For example, it makes sense that land-marks which are distant from each other should have esti-mates that are relatively independent and so do not need tobe considered in the same estimation problem. One exam-ple of a transformation method is the relative filter [23][24]which directly estimates the relative, rather than absolute,location of landmarks. The relative landmark location er-rors may be considered independent thus resulting in a mapbuilding algorithm which has constant time complexity re-


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gardless of the number of landmarks. More generally, anumber of approaches are being developed for constructinglocal maps and embedding these in a map management

process. Here, consistent (full filter) local maps are linkedby conservative transformations between local maps to gen-erate and maintain larger scale maps. This embodies theidea that local landmarks are more important to immediatenavigation needs than distant landmarks, and that land-marks can naturally be grouped into localised sets. Suchtransformation methods can exploit relatively low degreesof correlation between landmark elements to generate rela-tively decoupled sub-maps. The advantage of these trans-formation methods is that they highlight the real issue oflarge-scale map management.

The theoretical results described in this paper are es-

sential in developing and understanding these various ap-proaches to map building. As discussed in the introductionthere are a number of existing approaches to the localisa-tion and map building problem. The important contribu-tion of this paper is the proof that a solution exists. Fur-thermore it presents an algorithm that is efficient in thesense that it makes optimal use of the observations of rel-ative location of landmarks for estimating landmark andvehicle locations, a property inherent in the Kalman filter.The value of all other alternative real-time SLAM algo-rithms that use similar information can be evaluated withrespect to this full solution. This is particularly truein the case where simplifications are made to SLAM algo-

rithms in order to increase the computational efficiency.The most effective algorithm for SLAM depends much

on the operating environment. For example, the relativesparseness of occupied regions in the environment usedfor the example shown in section 4, or the grid based ap-proaches proposed in [11] and [10]. In an indoor environ-ment the use of only point landmarks as in section 4 willbe inefficient as the information such as the ranges to wallswill not be utilised. The framework proposed in this pa-per, however, can incorporate geometric features such aslines (for example see [25]). In environments where geo-metric features are difficult to detect, for example in an

underground mine, the proposed strategy will not be fea-sible. Many navigation systems used in outdoor environ-ments rely on exogenous systems such as GPS. Clearly ifexternal information is available these can be incorporatedto the framework proposed in this paper. This is partic-ularly useful in the situations where the exogenous sensoris unreliable, for example GPS not being able to observesufficient numbers of satellites or multi-path errors causedwhen operating in cluttered environments.

The implementation described in this paper is relativelysmall scale. It does, however, serve to illustrate a range ofpractical issues in landmark extraction, landmark initiali-sation, data association, maintenance and validation of theSLAM algorithm. The implementation and deployment ofa large-scale SLAM system, capable of vehicle localisationand map building over large areas, will require further de-velopment of these practical issues as well as a solution tothe map management problem. However, such a substan-

tial deployment would represent a major step forward inthe development of autonomous vehicle systems.

Appendix 1: Properties of positive semi-definitematrices

1. Diagonal entries of a psd matrix are non-negative2. If A Rnn is psd then for any matrix B Rnm,BABT is psd.3. IfA Rnn and C Rnn t are both psd then det(A+C) det(A).4. All principal submatrices of psd matrices are psd.5. For A Rmm, B Rnm and C Rnn , if

D =

A B

BT C

is psd, then

det(D) det(A)det(C)

Appendix 2: Landmark Initialisation Algorithm

The following describes the procedure used to initialisethe landmark locations and associate observations to par-ticular landmarks. This procedure also evaluates the qual-ity of the landmarks. This algorithm is an essential pre-cursor to the estimation process. It is not specific to theradar sensor but can be generalised to any sensor capableof observing landmarks in the environment.

Two landmark lists are maintained. One list stores land-

marks that are confirmed to be valid pi, i = 1 . . . N andthe other stores potential landmarks yet to be validatedqi, i = 1 . . . M. Initially both lists are empty. The mapmanagement algorithm proceeds as follows.1. Given an observation [rf, f] at time instant k from theradar, the location of the landmark possibly responsible forthis observation pf = [xf, yf]

T = g(x, y, , rf, f) and itscovariance Pf is calculated using the following relationship.

xfyf

=

x(k)y(k)

+

xvf(k)yvf(k)

wherexvf(k)yvf(k)

=

a cos((k)) b sin((k)) + rf cos((k) + f)a sin((k)) + b cos((k)) + rf sin((k) + f)

and

Pf = gxyPvgTxy + grffRg

Trff

where Pv is the covariance matrix of the vehicle locationestimate extracted from the state covariance matrix P(k|k)and R is the measurement noise covariance.2. An observation is associated with a landmark pi in the

confirmed landmark set if

dfi = (pf pi)T(Pf + Pi)

1(pf pi) < dmin

where Pi is the covariance of the landmark location es-timate pi, extracted from the state covariance matrix


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P(k|k). Note that dfi is the Mahalanobis distance betweenpf and pi, and its probability distribution in this case isthat of a 2 variable with two degrees of freedom. There-

fore, a suitable value for dmin can be selected such thatthe null hypothesis that pf and pi are the same is not re-jected at some desired confidence level. Also if the abovecondition results in an observation being associated withmore than one landmark p, the observation is rejected. Ifaccepted, the observation is then used to generate a newstate estimate.3. If an observation cannot be associated with any con-firmed landmark, then it is checked against the set of po-tential landmarks for possible association. Mahalanobisdistance is again used as the criterion for association. Ifan association with the potential landmark qj is justified

the new observation is used to update the location of thepotential landmark qj and its covariance Pj. In addition,a counter cj indicating the number of associations withlandmark qj is also incremented.4. An observation that is not associated with either a con-firmed or a potential landmark can be considered as a newlandmark. In this case pf is added to the list of potentiallandmarks as qM+1 , a counter cM+1 is initialised and thenumber of the time step k is assigned to a timer tM+1.5. The potential landmark list qi, i = 1 . . . M is then ex-amined against the following criteria.

(a) Ifcj is greater than a predetermined number of asso-ciations cmin, the landmark j is considered to be sufficiently

stable and therefore is transferred to the confirmed land-mark list.

(b) If (k tj) is greater than a predetermined tmax thenthe landmark j has not achieved the desired minimum num-ber of associations over a sufficient length of time. Land-mark j is therefore removed from the potential landmarklist.6. The probability density function (PDF) of the observa-tions associated to a given landmark can be used to esti-mate its quality. As suggested in [26] the quality Qi oflandmark i is calculated using the following equation.

Qi =

lj=1

12 |Sj|

12 exp(jS1j

Tj

2 )lj=1

12

|Sj|

1

2

(38)

where l is the number of observations so far associatedwith the landmark i, where vj is the innovation of the ob-servation j associated with landmark i observed at timek. Sj is the innovation covariance as defined by equation14.The landmark quality Qi is the ratio between the sum ofthe probability densities of the observations and the maxi-mum value of the probability density that is achieved whenall the observations coincide with their predicted values.

Therefore landmark qualities Q lie between 0 and 1. Atreasonable intervals, landmark qualities can be calculatedand landmarks that do not achieve a predetermined Qmincan be deleted from the map.7. Return to step 2 when the next observation is received.

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a solution to the simultaneous ion and

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