Article

Constructing informative Bayesian mappriors: A multi-objective optimisationapproach applied to indoor occupancygrid mapping

The International Journal ofRobotics Research1–18© The Author(s) 2017Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0278364916687027journals.sagepub.com/home/ijr

Christina Georgiou1,2, Sean Anderson1 and Tony Dodd1

AbstractThe problem of simultaneous localisation and mapping (SLAM) has been addressed in numerous ways with differentapproaches aiming to produce faster, more robust solutions that yield consistent maps. This focus, however, has resultedin a number of solutions that perform poorly in challenging real life scenarios. In order to achieve improved performanceand map quality this article proposes a novel method to construct informative Bayesian mapping priors through a multi-objective optimisation of prior map design variables defined using a source of prior information. This concept is exploredfor 2D occupancy grid SLAM, constructing such priors by extracting structural information from architectural drawingsand identifying optimised prior values to assign to detected walls and empty space. Using the proposed method a contex-tual optimised prior can be constructed. This prior is found to yield better quantitative and qualitative performance thanthe commonly used non-informative prior, yielding an increase of over 20% in the F2 metric. This is achieved withoutadding to the computational complexity of the SLAM algorithm, making it a good fit for time critical real life applicationssuch as search and rescue missions.

KeywordsBayesian estimation, informative priors, SLAM

1. Introduction

Bayesian methods dominate the estimation algorithmsused in simultaneous localisation and mapping (SLAM)(Durrant-Whyte and Bailey, 2006), yet there has been verylittle research to date into how to effectively construct theBayesian map prior using available information. This is thecase even though the use of priors is a key distinguish-ing feature of Bayesian estimation. The lack of attentiongiven to the map prior can be contrasted to the enormouswealth of raw prior information that is readily availableand untapped for most mapping and localisation scenar-ios, such as architectural drawings for buildings, city mapsfor the urban environment, geographical surveys for thewider outdoor environment and even pipe network mapsfor the underground environment. There exists an impor-tant and unsolved problem, therefore, in how to optimallysynthesize raw prior information into a form that is suit-able for robot navigation. This article addresses this prob-lem for 2D indoors occupancy grid SLAM by propos-ing a novel method to construct contextual priors that canimprove the performance of robot navigation (Figure 1)using architectural drawings and floor plans as a source ofprior information.

Improving navigation performance by constructing andusing meaningful priors can be very beneficial for time andsafety critical applications, such as Urban Search And Res-cue (USAR) missions. Using optimised priors to improveperformance does not increase the computational complex-ity of the SLAM algorithm itself. There is only a small over-head cost of extracting appropriate structural informationand constructing a prior map, but that is a one-off operationthat can be performed offline. Constructing an appropri-ate prior can also help produce better quality maps whena quick exploration is required by providing informationabout any unexplored areas.

The idea of using prior information to improve theperformance of robotics systems has been suggested fora number of systems that operate in real-world, chal-lenging environments. Priors are used in self-driving carsto improve localisation (Maddern et al., 2015) and 3D

1Department of Automatic Control and Systems Engineering,The University of Sheffield, UK2PA Consulting Group, Cambridge Technology Centre, UK

Corresponding author:Christina Georgiou, PA Consulting Group, Cambridge Technology Centre,Melbourn, Herts, SG8 6DP, UK.Email: christina.georgiou@paconsulting.com

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Fig. 1. The structure of the SLAM problem: sensor data, priorinformation and motion model predictions are input to the SLAMalgorithm that recursively maps the environment and localises therobot within it; architectural drawings and floor plans can be usedto extract prior information. The commonly used Bayesian prioris constructed assuming there is no available information aboutthe environment and so is uninformative. The proposed Bayesianprior is constructed using available information that is processedand placed in an optimised prior format.

semantic priors are used to interpret traffic lights (Barneset al., 2015). Prior knowledge can be incorporated infeature-based SLAM by using known landmarks (Burschkaet al., 2003). Williams and Reid (2010) use only a fewknown landmarks as points of reference to correct predic-tions and Parsley and Julier (2011) constrain the possiblelocation of landmarks in GraphSLAM using prior infor-mation. As a result, the observed position error is reducedwhilst maintaining the same or improved consistency com-pared to the no-prior solution (De la Puente and Rodriguez-Losada, 2014). Finally, skeletal SLAM (Milstein, 2005)uses prior information about the shape of a building toconstruct more accurate occupancy grid maps.

In all of the above cases the use of prior informationwas found to be beneficial and yield improved perfor-mance in robot navigation. However, all of the methodsreviewed above define the prior in an ad-hoc way, withoutusing any particular method to choose prior probabilitiesfrom raw prior data. There is a gap, therefore, in con-structing Bayesian map priors from available data usinga formal, methodological approach, that can be optimisedand repeated across different mapping scenarios. The maincontribution of this article is a novel method to constructcontextual Bayesian map priors that improve map quality.

Georgiou et al. (2015) proposed the use of architecturaldrawings and floor plans as a source of prior informa-tion, presenting a novel method to extract wall and emptyspace locations from such drawings. The article presented

a method to extract information that can be used to con-struct the prior map, but did not explore how to constructand use such a prior map using detected walls and emptyspace. Building on Georgiou et al. (2015), we aim to deter-mine optimised values to assign to detected walls and emptyspace. The maps produced using these optimised values aretested in both simulation and experiment. The optimisedBayesian prior map is benchmarked here against the useof a non-informative prior, and the results demonstrate thatthe use of the optimised prior is hugely beneficial in theinitial period of mapping and localisation, and retains animprovement in the final map. These results suggest thatoptimised Bayesian priors have great potential, especiallyin time-critical missions such as search and rescue.

The problem of developing a method for synthesizingoptimal Bayesian map priors from raw data raises a num-ber of key challenges, not least of which is how to measuremap quality in an objective and quantitative way. This prob-lem is addressed here by introducing, for the first time tothis scenario, the approach of measuring map quality by themetrics of precision and recall. The definition and use ofthese metrics enables the optimisation of prior probabilitiesused to construct the map, so that they are not chosen inan ad-hoc way. However, precision and recall tend to con-flict as optimisation objectives, which is a further challengethat is addressed here using multi-objective optimisation.The multi-objective optimisation does not produce a singleoptimal solution, but instead an optimal trade-off surface– a Pareto-optimal front. From the Pareto front, the usercan choose an appropriate trade-off between precision andrecall to define the Bayesian prior map.

In summary, the following four steps are proposed as asolution to constructing informative Bayesian priors: (1)identify a source of prior information and a method toextract useful information from it; (2) determine prior mapdesign variables that affect final map quality; (3) definequantitative measures of map quality – precision and recall;and (4) perform a multi-objective optimisation to identifyprior probabilities that yield high performance metrics.

This article is organised as follows. Section 2 gives anoverview of the role of priors in SLAM and shows howusing an optimised Bayesian prior can yield performanceimprovements. Section 3 presents the proposed strategy toconstruct informative priors and explains each aspect forthe case study explored in this article, indoors occupancygrid mapping. Section 3.1 presents the algorithm used toextract the location of walls from an architectural draw-ing or floor plan. Section 3.2 details a novel quantitativemeasure of map quality and Section 3.3 presents a multi-objective optimisation using a genetic algorithm to identifyPareto-optimal design parameters. Section 4 presents theimage test set used to demonstrate results in this articleand details the simulation used to obtain mapping results.Section 5 compares the maps produced in simulation usingvarious shortlisted priors to determine which are optimaland presents the benefits of the proposed prior construction

Georgiou et al. 3

method. Finally, Section 6 validates the benefits of using aninformative prior experimentally, both in a simulated robotcourse and in a large scale experiment.

2. Bayesian priors in SLAM

SLAM aims to have a robot explore and map an environ-ment whilst simultaneously localising itself within it. Therobot uses sensors to perceive the environment, a motionmodel or odometer to predict robot motion and it can alsouse a prior to incorporate information about the environ-ment (Durrant-Whyte and Bailey, 2006). Each of the ele-ments of the chosen implementation for this article arepresented in this section.

First the probabilistic formulation of SLAM is given,highlighting the role of the prior map. The imple-mentation of choice, occupancy grid FastSLAM (Mon-temerlo et al., 2003), is then presented, showing theseparation of the localisation and mapping tasks usingRao–Blackwellization, and the incorporation of prior infor-mation through the occupancy grid mapping algorithm ishighlighted. This separation allows a study of mappingwithout the need to focus on performing localisation.

2.1. Probabilistic formulation

SLAM performs a Bayesian estimation to determine therobot’s pose and a map of the environment given sensorreadings, control inputs and the robot’s initial pose. For-mally, it aims to compute the joint posterior of the robot’spose xk and the map m (Durrant-Whyte and Bailey, 2006)

p( xk , m|Z0:k , U0:k , x0)∝ A× p( m) (1)

for all times k. xk is the state vector describing the robot’spose, m is the map of the environment (or a selection oflandmarks for feature-based SLAM), p( m) is the prior map,Z0:k are the sensor observations, U0:k the history of controlinputs and x0 is the robot pose at time k = 0. A is given by

A = p( zk , xk|m)

p( xk , m)

p( xk , m|Z0:k−1, U0:k , x0)

p( zk|Z0:k−1, U0:k)(2)

since the joint posterior in (1) can be written

p( xk , m|Z0:k , U0:k , x0)=p( zk|xk , m)

p( xk , m|Z0:k−1, U0:k , x0)

p( zk|Z0:k−1, U0:k)

= p( zk , xk|m)

p( xk , m)p( m)

p( xk , m|Z0:k−1, U0:k , x0)

p( zk|Z0:k−1, U0:k)

∝ A× p( m)

(3)

indicating that the joint posterior is proportional to the priorprobability p( m). Therefore a choice of an optimised priorp( m) can lead to a more accurate estimate of the jointposterior.

The aim is to estimate the robot pose for all times k andthe map of the environment given sensor readings, con-trol inputs and a starting pose. SLAM is solved recursively,producing map and pose estimates at each time step.

In the probabilistic formulation of SLAM the tasks oflocalisation and mapping cannot be viewed separately. Fast-SLAM uses Rao–Blackwellization to separate localisationand mapping as discussed in the following section. There-fore FastSLAM allows the study of the effects of a priormap without the need to address the localisation aspect ofSLAM.

2.2. Occupancy grid FastSLAM formulation

FastSLAM uses Rao–Blackwellization to decompose theSLAM problem into a robot localisation problem and a col-lection of landmark/map estimation problems that are con-ditioned on the robot trajectory estimate (Durrant-Whyteand Bailey, 2006; Montemerlo et al., 2003).

p( X0:k , m|Z0:k , U0:k , x0)

= p( m|X0:k , Z0:k) p( X0:k|Z0:k , U0:k , x0)(4)

This separation of localisation and mapping allows thestudy of mapping priors without the need to explore locali-sation, and is thus the implementation used in this article.

In this case the aim is to compute the joint posterior ofthe map and the complete robot trajectory X0:k rather thanthe single pose xk . That is because landmarks conditionedon the trajectory are independent, allowing the factorisa-tion shown in equation (4). A particle filter can then beused to sample from the motion model and produce the pro-posal distribution but now the estimates for the landmarklocations (conditioned on the robot trajectory estimate) areperformed separately.

Since the pose and map estimates can be performedseparately using Rao–Blackwellization, the occupancy gridmapping algorithm can be used to calculate p( m|X0:k , Z0:k).The starting pose x0 is considered to be deterministic for thepurposes of this article, since for a known prior map we canmeasure pose x0 at t = 0 with respect to the prior map, sim-ilarly to the minimum covariance case in Dissanayake et al.(2001). Therefore, prior information about the environmentcan only be incorporated into p( m|X0:k , Z0:k) through theBayesian prior p( m).

The occupancy grid mapping algorithm (Elfes, 1989) isused in this article to update the probabilities of occupancyof each grid cell in the environment. This representation ischosen since the prior map p( m) is the map used at timek = 0, which is then recursively updated as occupancy mea-surements are taken for map cells as the robot explores theenvironment.

The occupancy grid mapping algorithm (Elfes, 1989)splits the environment to be mapped into a grid of cellsand a prior probability is assigned to each cell. The log oddoccupancy of each grid cell can be updated using

lk,i = InverseSensorModel( mi, xk , zk)+lk−1,i − l0,i (5)

4 The International Journal of Robotics Research

with

lk,i = logp( mi|Z1:k , x1:k)

1− p( mi|Z1:k , x1:k)(6)

A detailed derivation of equation (5) is given in theAppendix.

The Bayesian priors for each cell, p( mi), are incorpo-rated through l0,i, the log odds prior for a given cell. Whenthere is no available prior information a non-informativeBayesian prior is assigned to all grid cells, p( mi)= 0.5,i = 1, . . . , N . Most researchers use such a prior in order toproduce solutions that do not depend on having prior knowl-edge of the environment (Durrant-Whyte and Bailey, 2006).Others claim that access to information such as detailedarchitectural drawings may be difficult (Kumar et al., 2004).Information such as floor plans for buildings like hospitalsand offices is generally available, however, and can be usedto extract useful information and construct SLAM priors.

Incorporating prior information does not add to the com-putational cost of running SLAM and only incurs a one-offcost of extracting prior information and constructing a prior.Since the posterior is proportional to the prior map (equa-tion (3)) and given the recursive nature of SLAM, construct-ing an informative prior map p( m) can help produce a moreaccurate map even if a quick exploration is performed. Inthe case of a quick exploration each grid cell may only bescanned once or twice, making the effect of the prior moresignificant. This is especially useful for time critical appli-cations where the environment needs to be explored quickly,such as USAR missions.

3. Proposed approach

The proposed strategy to convert prior information to aninformative Bayesian prior is presented in this section. Thisconsists of the following:

(a) a method to process a source of prior information suchas an aerial image or architectural drawing to extractrelevant prior information;

(b) a set of prior map design variables that affect final mapquality;

(c) a set of metrics to assess the quality of the final mapwhich can be used to select optimised design variables;

(d) a multi-objective optimisation to identify design vari-ables that yield high map quality metrics.

The rest of this article discusses each of those aspectsfor the case study of indoors occupancy grid mapping usingarchitectural drawings and floor plans as a source of priorinformation.

3.1. Extracting structural information from anarchitectural drawing

In order to construct optimised priors an appropriate sourceof information needs to be identified. This can be an imagesuch as an architectural drawing or floor plan for indoors

SLAM or an aerial photograph or road network map foroutdoors SLAM. These drawings or images then need to beprocessed to extract the location of occupied and unoccu-pied environment sections. For indoors environments, wallsand empty space need to be identified. Similarly, if thesource of prior information is a road map, roads and non-traversable areas can be detected. This article focuses onthe use of architectural drawings as a source of informationand so this section presents a method to extract structuralinformation from them to construct a prior map.

Structural information such as the location of buildingwalls is required in order to construct an indoors prior map.Architectural drawings and floor plans, two-dimensional,top-down drawings of buildings containing structural infor-mation, suggested furniture and annotated text, can be usedto extract such information. These need to be processed toextract structural information in order to construct a mean-ingful prior map of the environment. Once the locationsof structural elements such as walls have been determined,they can be used to assign appropriate prior values p( mi) toall grid cells.

The algorithm used in this article to extract prior infor-mation, Algorithm 1, was proposed by Georgiou et al.(2015) and is designed to extract structural information thatcan be used to construct robot priors. The main problemwith processing drawings to extract this information is thelack of consistent representations between different draw-ings (Figure 2(a)). Depending on the architect and draw-ing method used, doors, walls and other elements can berepresented in a different manner. Therefore, in order toyield reliable results for any drawing, the algorithm usedto extract structural information needs to be independent ofthe drawing style.

The information the algorithm aims to extract is the loca-tion of walls which can be easily put in an occupancy gridformat by assigning detected walls a low prior probabilityof being empty and detected space a high prior probability.Walls are represented as dark lines in the majority of draw-ings but have no other distinct geometric features. Giventhis variation between drawings, using methods such as linethickness to determine wall locations is unreliable. Insteadof searching for walls directly, a feature with distinct geo-metric features and a clear relation to walls is detected.Doors are chosen for their distinctive shape and becausethey connect wall segments. Therefore if the location ofdoors in the image is found walls can also be detected.

There is no universally used symbol for all doors (Baden-Powell et al., 2011), and depending on the type of door (sin-gle, double, sliding, foldable, etc. (see Figure 2(a))) and theway the drawing was produced (different drawing software,hand-drawn) different representations can be used. How-ever, all door symbols share a geometric characteristic: theycan be approximated as isosceles triangles defined by threevertices and (usually) two edges/connected sides as shownin Figure 2(b). The proposed algorithm therefore searchesfor triangles in the image (as defined by their vertices) that

Georgiou et al. 5

Fig. 2. (a) Symbols for different types of doors (doors drawnusing Floorplanner.com). (b) The majority of door types can beapproximated as isosceles triangles (drawn in grey).

Algorithm 1 Drawing processing algorithm.FindHarrisCorners(Drawing)PossDoors← AllUniqueThreeCornerCombinationsDoorsShortlist← [ ]for i=1:size(PossDoors) do

if SideCheck(PossDoors(i)) & AngleCheck(PossDoors(i))then

if ConnectivityCheck(PossDoors(i)) thenDoorsShortlist← [DoorsShortlist, PossDoors(i)]

end ifend if

end forDetectOuterWalls(Drawing)DetectedDoors← RemoveFalsePositiveDoors(DoorsShortlist)FindWalls(Drawing, DetectedDoors)

fulfill the isosceles triangle criteria. This algorithm has acomputational cost of O( n3) where n is the number of Har-ris Corners detected in the image and it is dominated by the

cost of finding the possible combinations nCr = n!r(n−r)! of

r = 3 out of n detected Harris corners.Once wall locations have been extracted from an archi-

tectural drawing, prior occupancy values p( mi) need to beassigned to each grid cell mi accordingly. Three parametershave to be chosen:

• prior probability assigned to detected walls;• prior probability assigned to detected empty space;• occupancy grid cell size (grid resolution).

Each grid cell mi is assigned a Bayesian prior prob-ability p( mi) (Section 2.2). The following notation willbe used throughout the rest of the article. A high prob-ability assigned to a grid cell will mean the cell has ahigh probability of being empty. Conversely, a low prob-ability will signify a low probability that a cell is empty.If wall locations are known, cells that are located wherewalls were detected can be assigned lower prior probabil-ities, p( mwall

i ) < 0.5, and cells located where empty spacewas detected can be assigned higher prior probabilities,p( mspace

i ) > 0.5. For brevity these two probabilities willbe written pwall and pspace, where pwall is the prior prob-ability assigned to grid cells that correspond to locationsof detected walls in the architectural drawing and pspace the

prior probability assigned to cells corresponding to detectedempty space in the drawing. This notation can be usedsince all grid cells mi that correspond to occupied locationswill be assigned a prior probability of pwall and those thatcorrespond to empty space a prior probability pspace.

The alignment of grid cells and detected walls dependson the occupancy grid cell size. In all the results presentedan appropriate grid cell size was chosen for all drawingsused. The effects of using priors when coarser grids are usedare discussed in Section 5.

3.2. Map quality assessment

Using the method described in Section 3.1 walls and emptyspace can be detected in architectural drawings. Prior val-ues of occupancy then need to be assigned to each gridcell. In order to determine which ( pwall, pspace) pair yieldsoptimal performance the problem of assessing map qual-ity is presented as a binary classification problem and aprecision-recall analysis is performed.

The quality of a map produced by an occupancy gridmapping algorithm can be evaluated using a number of met-rics proposed in the literature. Map consistency is a measureof performance but, according to Mazuran et al. (2014),there is no consistent notion of consistency and determin-ing whether or not a SLAM map is consistent is still anopen problem. Global consistency is often used to meanthat the map produced agrees with the ground truth whereaslocal consistency refers to correctly aligning sensor scanslocally. A measure for global consistency is proposed byMazuran et al. (2014) who use the mismatch in the sensordata, but this framework is tailored to SLAM using 2D lasersensors and can struggle in dynamic environments. Collinset al. (2007) compute a correlation between the producedmap and ground truth but this tends to be computationallyexpensive.

In this article we propose a metric that is easy to evaluateand that gives an indication of how successful the systemwas in detecting occupied and empty space in the envi-ronment. The use of the percentage of free and occupiedcells identified correctly is proposed as a performance met-ric in Grewe et al. (2012). The novelty of our proposedapproach is the formulation of the assessment of the qualityof an occupancy grid map as a classification problem wherethe aim is to correctly classify pixels as corresponding tooccupied or unoccupied space. Therefore a precision-recallanalysis can be used to evaluate the quality of the producedmaps. Occupied locations in the environment can then bedefined as true positives and empty space as true negatives.Any other objects detected can then be defined as falsepositives.

The chosen metrics are therefore detailed as follows(Davis and Goadrich, 2006)

Precision ( pre)= TruePositives

DetectedOccupiedSpace

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Recall ( rec)= TruePositives

ActualOccupiedSpace

with precision representing the percentage of correctlydetected walls and recall representing the number of wallscorrectly detected out of all walls in the drawing.

Following these assessment criteria, each point in therobot map is tested against the true environment to deter-mine whether it has been classified correctly. A simple wayto perform this comparison in practice is to find the dif-ference between the true environment image and the pro-duced map to identify false positives and false negativesand thus calculate the above metrics. This is a simple buteffective and computationally inexpensive method to assessquantitative performance.

Constructed priors are tested in a simulation whichassumes perfect knowledge of the robot’s pose, so overor underestimating these values if the map and groundtruth images are misaligned is not a problem. There-fore this method of comparing the map to ground truthto find the number of true/false positives/negatives isused to test the performance of maps produced usingdifferent priors.

If tests are conducted on a map pixel basis, coarse gridswill assign occupied values to all cell pixels that correspondto detected occupied space, leading to a lower precision. Acoarse grid would assign all cell pixels as occupied for acell made up of mostly occupied space. For a large cell sizethis would result in incorrectly assigning many map pix-els to occupied space, thus reducing precision. This wouldresult in an area of misclassified points that are not strictlya classification error, just a limitation of an unsuitable gridresolution. In order to avoid penalising a system with anunsuitably coarse grid resolution, maps that perform wellquantitatively are also tested qualitatively to ensure overallgood performance.

While testing different prior maps for different drawingsit was observed that the objectives of maximising precisionand maximising recall are conflicting. Therefore selectingoptimal prior values to maximise both objectives is nottrivial and a multi-objective optimisation is proposed todetermine optimised prior values.

3.3. Multi-objective optimisation to optimiseprior parameters

In order to test how the values for ( pwall, pspace) that yieldmaximum precision and recall compare in a qualitativesense, maps can be constructed using the prior ( pwall, pspace)values yielding maximum precision and recall. These canthen be tested to assess the qualitative performance of mapsproduced using these prior values. If requirements for eachmetric are conflicting, a multi-objective optimisation can beperformed to determine pairs π = ( pwall, pspace) that yieldboth high precision and high recall. Having both precisionand recall > 40% was empirically determined to be anacceptable threshold.

This multi-objective problem can be formulated as

minπ

F( π ) = [fpre( π ) frec( π ) ]

subject to C =

⎧⎪⎨⎪⎩

0.1 ≤ π < 1

fpre( π ) < 2.5

frec( π ) < 2.5

(7)

where π = ( pwall, pspace), fpre( π )= 1pre and frec( π )= 1

rec(so fpre( π ), frec( π ) < 2.5⇔ pre, rec > 40%).

In order to perform this multi-objective optimisation aPareto-based method was chosen since the relative impor-tance of the objectives is unclear (Giagkiozis et al., 2015)and a controlled elitist genetic algorithm (Deb, 2001)(a variant of NSGA-II (Deb et al., 2002)) was used.

In this minimisation problem a decision vector π̂ withπ̂ ∈ C is Pareto optimal if there is no other π ∈ C forwhich fi( π )≤ fi( π̂ ) , ∀i and at least one fi( π ) < fi( π̂) fori = 1, . . . , K, where K is the number of functions in F( π ).In this case the decision vector π̂ is said to Pareto-dominatevector π . If only the second condition is met the solutionis considered weakly Pareto optimal. The multi-objectiveoptimisation solver used aims to find a subset of Pareto-optimal solutions which is referred to as the Pareto front(Giagkiozis et al., 2015).

The concept of Pareto dominance is applied in order touse a genetic algorithm to solve this multi-objective prob-lem. First the objective function is evaluated for each indi-vidual in the population �. Non-dominated individuals,�nond are then found and removed from the population.This process is repeated until all non-dominated individuals�nond have been identified.

In order to obtain recall and precision variables and per-form the multi-objective optimisation a simulation environ-ment was devised as detailed in the next section.

4. Simulation setup

A simulation was used to produce maps using different( pwall, pspace) pairs. A simulation allows the testing of alarge number of possible prior values to assign to ( pwall,pspace). Thus the multi-objective optimisation can be per-formed without the need to obtain mapping results fordifferent buildings using a robot which would be verytime consuming. The drawing test set, simulation used andassumptions made are discussed in this section.

4.1. Drawing test set selection

A number of different drawings were tested using the sim-ulation environment described in this section. A versionof each drawing containing only structural informationwas created by hand and used as the ground truth. Twofloor plans, drawings (i) and (ii) in Figure 3(a), are presen-ted, with drawing (i) containing labeling and drawing (ii)

Georgiou et al. 7

Fig. 3. Multi-objective optimisation results for all drawings. (a) Drawings used to extract prior information. (b) Interpolated pre-cision and recall colour maps for the five drawings. (c) Pareto fronts for all drawings, with a number of common solutions acrossdrawings meaning there is a set of Pareto-optimal ( pwall, pspace) pairs that yield good performance for all drawings. (d) Pareto-optimal (pspace,pwalls) pairs plotted for all drawings. (e) Clusters detected within Pareto-optimal solutions using the k-means clusteringalgorithm.

containing suggested furniture. Drawings (iii), (iv) and (v)in Figure 3(a), are sections of an architectural drawing ofa University of Sheffield Engineering building. These sec-tions were chosen because they are a representative sam-ple of different outlines and commonly observed elements.Drawing (iv) contains only doors and wall segments making

it an easy drawing to process, and drawings (iii) and (v) arequite challenging to process, containing stairs, text, wallsof varying width and, in the case of drawing (v), a door atan angle. These were chosen as a representative sample ofcommon elements and configurations found in architecturaldrawings and floor plans.

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4.2. Occupancy grid mapping simulation

In order to test the effects of using architectural priors onmap quality a simulation environment was created in MAT-LAB. In this simulation the robot pose is assumed to beknown and accurate. Given the separation of localisationand mapping using Rao–Blackwellization in FastSLAM(Section 2.2), mapping can be studied separately by pro-viding deterministic pose values instead of using MonteCarlo localisation to produce pose estimates. Mathemati-cally, instead of using equation (4) and producing estimatesof both the map and trajectory for each particle, we assumep( X0:k|Z0:k , U0:k , x0) is known and hence only one mapneeds to be updated by determining p( m|X0:k , Z0:k) usingthe occupancy grid mapping algorithm (Elfes, 1989).

The robot is modeled as a point moving through space,with the robot state xk at each time step k given by

xk =( x, y, θ ) (8)

where ( x, y) give the robot position in Cartesian coordinatesand θ the robot orientation. The distance sensor on the robotis modeled as an ultrasound sensor with a conical field ofview defined by an angle φ and range r.

The occupancy grid mapping algorithm (Elfes, 1989) isused to update the log odds of occupancy lk,i for each gridcell at each time step (Section 2.2). The probabilities ofoccupancy can be retrieved from lk,i using

p( mi|Z1:k , x1:k)= 1− 1

1+ exp( lk,i)(9)

The inverse model used is a multiplier, multiplying trueoccupancy values by a factor of 0.9 to introduce someuncertainty. The code used to update the occupancy gridmap was also used to process real sensor and robot posedata collected during experiments.

When prior information is available, p( mi) can beassigned a different prior value based on whether a wall orempty space was detected for grid cell i during the drawingprocessing stage as discussed in Section 3.1. The resultsobtained using this simulator and performing the multi-objective optimisation to determine optimal prior values(pwall, pspace) are presented in the next section.

5. Simulation results

This section presents the results obtained using the simula-tor discussed in Section 4.2, testing different prior values toassign to detected walls pwall and to detected empty spacepspace. The results of the multi-objective optimisation per-formed to determine the (pwall, pspace) pair that yields opti-mised precision and recall are also presented. Finally themap produced using the proposed optimised prior is com-pared to the map produced using a non-informative priorand is found to yield an increase in the F2 metric by at least20%.

5.1. Designing contextual priors that optimiseconflicting performance metrics

In order to evaluate the effect of different priors, possi-ble combinations of prior values (pwall, pspace) between 0.1and 1, multiplied by a factor of 0.9 to avoid values too closeto 1, were examined and linear interpolation was used toproduce continuous values. Figure 4(c) shows colour mapsof the precision and recall metrics for possible combina-tions of pwall and pspace for the drawing in Figure 4(a).Figure 4(b) shows the final maps produced using the priorvalues that yield maximum precision and those that yieldmaximum recall.

A higher precision ensures that it is unlikely free spacewill be incorrectly identified as occupied and a higher recallensures that the building structure is detected. Figure 4(c)shows that the aims of having high precision and also main-taining a high recall are conflicting since pairs that yieldhigh precision tend to yield low recall and vice versa. Thisproblem of determining a suitable pair of ( pwall, pspace) thatensures high precision and high recall (> 40% was empiri-cally determined to be an appropriate threshold) is thereforea multi-objective optimisation problem.

In order to perform this multi-objective optimisation acontrolled elitist genetic algorithm (Deb, 2001) (a variant ofNSGA-II (Deb et al., 2002)) was used. The Pareto front forthe drawing in Figure 4(a) can be seen in Figure 4(c) and thefinal map produced using one of the Pareto-optimal priorvalues is shown in Figure 4(e). This drawing was chosenbecause, due to the the fact that the level of detail, includ-ing labeling, walls at an angle and multiple rooms, high-lights the qualitative difference in performance betweenFigure 4(b) and Figure 4(e).

The map shown in Figure 4(e) avoids detecting thickerwalls or missing doors as can happen using the prior valuesthat optimise recall (Figure 4(b)) and also avoids missingmost of the walls as is done using the prior values that yieldmaximum precision (Figure 4(b)).

5.2. Multi-objective optimisation to determineglobally optimal contextual priors

Figure 3(b) shows the precision and recall colour maps forall representative drawings in Figure 3(a). The shape of thecolour maps for precision and recall does not vary greatlybetween drawings, other than the fact that the region of val-ues that yield good performance is larger/smaller for differ-ent drawings. These results indicate that a globally optimalregion of (pwall, pspace) values that result in high precisioncan be identified.

The locations of maximum precision for three differentgrid resolutions are shown in Table 1. Maximum values areconsistently observed for pwall values between 0.1 and 0.3and pspace = 0.3 regardless of the drawing and grid resolu-tion with the exception of drawing (a) for a 10 × 10 gridcell.

Georgiou et al. 9

Fig. 4. Multi-objective optimisation overview. (a) Architectural drawing used to extract prior information. (b) Maps produced usingprior values that yield maximum precision and maximum recall, neither of which is qualitatively optimal. (c) pre and rec colour mapsfor all combinations of ( pwall, pspace) between 0.1 and 1, with Pareto-optimal values shown as black points. (d) Pareto front producedby the multi-objective optimisation. (e) Map produced using the Pareto-optimal proposed prior values pwall = 0.2, pspace = 0.9.

10 The International Journal of Robotics Research

Table 1. Maximum precision calculated for each of the drawingsin Figure 3; grid cell size given in number of map pixels.

Grid cell 3× 3

Drawing Max pre (%) (pwall, pspace)

a 98.49 (0.3, 0.3)b 99.42 (0.3, 0.3)c 91.77 (0.3, 0.3)d 95.56 (0.1, 0.3)e 93.77 (0.3, 0.3)

Grid cell 5× 5

a 75.00 (0.3, 0.3)b 95.36 (0.3, 0.3)c 79.00 (0.3, 0.3)d 84.94 (0.1, 0.3)e 90.51 (0.1, 0.3)

Grid cell 10× 10

a 48.00 (0.5, 0.6)b 89.14 (0.1, 0.3)c 77.42 (0.1:0.2, 0.3:05)d 92.06 (0.1, 0.3)e 92.68 (0.1, 0.3)

The values that yield optimal recall can be seen inTable 2. These values are in most cases what one wouldintuitively expect to be optimal: a high value for pspace and alower value for pwall. For the architectural drawing sections,optimised values were found to be pspace = 1 with pwall

between 0.1 and 1. Both recall and precision are importantto ensure a high quality map.

As shown in Tables 1 and 2 the values that yieldmaximum precision do not correspond to values that yieldmaximum recall and vice versa. Moreover, as shown inFigure 3(b) regions of high precision correspond to regionsof low recall and vice versa. Therefore multi-objective opti-misation is used to identify prior values that optimise bothprecision and recall.

Multi-objective optimisation using the controlled elitistgenetic algorithm (Deb, 2001) (a variant of NSGA-II (Debet al., 2002)) discussed in Section 5.1 was performed forall five drawings. Figure 3(b) shows the precision and recallinterpolated colour maps for the five drawings used to gen-erate the Pareto fronts in Figure 3(c). Figure 3(c) shows thePareto-optimal fronts for all drawings and floor plans. Thereis a region where the Pareto-optimal fronts for all draw-ings overlap, meaning that there are Pareto-optimal sets of( pwall, pspace) values that yield good performance regardlessof the type of drawing used. Only solutions that fall withinset C as defined in Section 3.3 are of interest.

In the precision plots of Figure 3(b) we observe highervalues for pwall = 0.2. That is because 0.2 is the lowestprobability value for which the log odds value is in the lin-ear section of the log odds plot (Figure 5). Therefore it is

Table 2. Maximum recall calculated for each of the drawings inFigure 3; grid cell size given in number of map pixels.

Grid cell 3× 3

Drawing Max rec (%) (pwall, pspace)

a 80.08 (1,0.2)b 82.77 (0.2,0.1:0.2)c 69.43 (0.1, 1)d 82.70 (0.7:1, 1)e 83.05 (0.5:1, 1)

Grid cell 5× 5

a 66.09 (1,1)b 79.71 (0.2,0.2)c 72.05 (1, 1)d 86.00 (0.4:1, 1)e 88.68 (0.9:1, 1)

Grid cell 10× 10

a 66.53 (1, 1)b 73.42 (0.2, 1)c 86.91 (1, 1)d 91.69 (0.4:0.9, 1)e 95.41 (0.2, 1)

Fig. 5. The log odds plot for probability values 0–1. In the region0.2–0.8 the log odds show linear behaviour, whereas for valuesoutside that range the resulting log odds value increases rapidly.

the lowest value we can assign to detected walls that avoidsthe region close to the asymptote near 0.

In Tables 1 and 2 lower prior values for both pwall andpspace do not yield optimal recall. This seems counter-intuitive given that assigning all cells a low prior probabilitywould be expected to yield a low precision but high recall.However, low pwall and pspace can result in assigning highprobabilities of being empty to grid cells that contain mostlyoccupied space.

Using equation (5), for low pwall and pspace, if a cell con-tains both occupied and empty space the simulator assigns

Georgiou et al. 11

an inverse sensor model value of

InverseSensorModel( mi, xk , zk)=logOdds( 0.9× pelempty

Ncell)

(10)

where pelempty are the number of pixels corresponding todetected empty space in a given cell and Ncell the numberof map pixels in a grid cell. A low prior pspace < 0.2 wouldresult in l0,i < −1.4. For a cell for which 60% of pixels cor-respond to empty space InverseSensorModel( mi, xk , zk)=0.2. Therefore, using equation (5), lk,i > 1.6k − 1.4.This corresponds to a probability of 0.85 for k = 2 andpspace = 0.2, mapping this as an empty cell.

The Pareto-optimal solutions for all drawings were plot-ted as shown in Figure 3(d). The k-means clustering method(Arthur and Vassilvitskii, 2007; Hartigan and Wong, 1979)was then used to assign each point to clusters and the resultsare shown in Figure 3(e).

In order to identify an optimal cluster of solutions, repre-sentatives from each cluster were tested in terms of qualita-tive performance. Solutions in Cluster 3, the green clusterin Figure 3(e), have high values of both pwall and pspace. Thatis the region where maximum recall solutions lie, as shownin Table 2. However, solutions within this region do not pro-vide optimal qualitative results as shown in Figure 4(b).Solutions that fall in Cluster 1, the purple cluster in Fig-ure 3(e) have low values of both pwall and pspace and thuscorrespond to the maximum precision solutions as shownin Table 1. These were also shown to produce non-optimalqualitative results as discussed in Section 5.1 and shown inFigure 4(b). Therefore the optimal solution lies in Cluster2, the light blue cluster.

The values in the light blue cluster are those that intu-itively would be expected to perform well: a low pwall anda high pspace. Converting these back to discrete values, thepossible solutions to investigate are the pairs

Shortlist ={

pwall = 0.2, 0.6 ≤ pspace ≤ 1

0.4 ≤ pwall ≤ 0.5, 0.7 ≤ pspace ≤ 0.8(11)

The maps produced using each of these pairs werecompared in terms of visual quality and the best (pwall,pspace) combination out of the green cluster points in termsof qualitative performance was found to be pwall = 0.2,pspace = 0.9 (Figure 6). The next sections discuss thebenefits of using an informative prior and compare themaps produced using the proposed informative prior tothose produced using an uninformative prior to quantify theimprovement in performance.

5.3. Effects of sensor model on optimised priorvalues

The proposed optimised prior values were obtained for anoptimisation using the simple sensor model described inSection 4.2. The effect of selecting different sensor models

is explored in this section. Different sensor models assigna different probability of occupancy to detected free andoccupied space, setting pfree to detected empty space cellsand poccupied to cells corresponding to occupied space. Morecomplex models might use a normal distribution centredaround the location of a detected obstacle/detected emptyspace cell for poccupied and pfree (Pirker et al., 2011).

The log odds representation is used for the map update,with the initialisation of lk,i = l0,i. Using equation (5), thisleads to the update rule

l1,i = InverseSensorModel( mi, x1, z1)

l2,i = 2× InverseSensorModel( mi, x2, z2)−l0,i

...

lk,i = k × InverseSensorModel( mi, xk , zk)−( k − 1)×l0,i

(12)

for the ith cell at the kth iteration. Therefore, to obtaina larger final map probability for empty cells we needlk,i > 0 corresponding to a probability of occupancypk

mi>0.5 (Figure 5). lk,i > 0 would require

InverseSensorModel( mi, xk , zk) > l0,i (13)

for a large k. Therefore the probability assigned by the sen-sor model to detected free space should be greater thanthe prior value assigned to empty space, pfree > pspace.Conversely, the value assigned to detected occupied spaceby the sensor model should be smaller than the prior,poccupied < pwall.

5.4. Benefits of using an optimised informativeprior

Using an optimised informative prior allows for improvedperformance when a robot quickly explores an area whileoperating in a time-critical mission such as USAR, forexample. In the mapping example presented in Figure 6 therobot has not fully explored the environment. In such casesusing a prior has the added benefit of providing informationabout building sections even if areas are unexplored. More-over, for coarse grid resolutions the commonly used unin-formative prior produces maps of very low quality whereasthe proposed optimised prior of pwall = 0.2, pspace = 0.9produces maps that give an overview of building structure,as shown in Figure 6. Therefore if a quick and computation-ally cheaper exploration is required the proposed optimisedprior yields significantly better maps in terms of visualquality, identifying the majority of building walls.

The maps produced using the proposed prior performbetter in a quantitative sense as shown in Figure 6(c). Giventhe multi-objective nature of the optimisation, merely com-paring the values of precision and recall for maps pro-duced using the informative and non-informative prior doesnot provide enough information. In order to overcome thisproblem the F2 metric (Powers, 2011) is used to allow a

12 The International Journal of Robotics Research

Fig. 6. Comparison of map produced using the proposed optimised prior and the map produced using a non-informative prior. (a) Mapsproduced using a non-informative prior for two different grid resolutions. (b) Processing and converting a drawing to an optimised priorand using it to produce maps for two different grid resolutions. (c) A quantitative comparison of the maps produced using an informativeand a non-informative prior using the F2 metric. (d) Evolution of F2 metric with time when using an informative and a non-informativeprior for the environment shown in (a) and (b).

Georgiou et al. 13

comparison of the precision and recall combination ratherthan individual values for each drawing.

F2 = 5× pre× rec

( 4× pre)+rec(14)

Recall is favoured over precision since it represents thepercentage of correctly detected walls and thus affects themap outline more. As shown in Figure 6(c) using an infor-mative prior yields an increase in F2 measure of at least20% over the uninformative prior. Moreover, it is worthnoting that using an informative prior maintains a highF2 throughout and from the early stages of exploration(Figure 6(d)), greatly outperforming the map producedusing the non-informative prior.

The overall method proposed in this article to process anarchitectural drawing or floor plan, extract structural infor-mation and use it to construct an optimised prior and thusan improved map is shown in Figure 6(b).

6. Experimental results

In order to verify the results obtained using simulated data,distance data were collected using a turtlebot mountedwith a Microsoft Kinect sensor running ROS. Two sets ofexperiments were conducted

• A small scale experiment in a constructed course, aim-ing to simulate a building, Figure 7(a). The robot posewas measured perfectly by hand.

• A large scale experiment in a floor of a University ofSheffield Engineering building. In this case we usedadaptive Monte Carlo localisation and the extractedprior to produce estimates of the robot pose.

In both cases the occupancy grid MATLAB code pre-sented in Section 4.2 was used to update the map.

6.1. Constructed robot course

A known, accurate robot pose was used at each time stepand the occupancy grid map update in the MATLAB simu-lator proposed in Section 4.2 was used to process the datacollected using the Kinect sensor.

An informative prior map produced using perfectlyextracted walls and assigning the proposed prior valuespwall = 0.2, pspace = 0.9 was used (Figure 7(b)). Themaps produced using the proposed informative prior andthe non-informative prior are shown in Figure 7(c) for anexploration that takes three sensor readings per robot pose.The exploration is incomplete with unexplored areas shownin grey. Grey areas in the map that uses the proposed priorcorrespond to a probability of 0.9 and in the map producedusing the non-informative prior to 0.5, providing no infor-mation about the environment. This is an advantage of usingthe proposed prior since, even if areas remain unexplored,there is some information about what we expect to findthere. The map that uses the proposed prior has correctly

detected walls at the bottom of the image as well as the outerright wall. Conversely, using the non-informative prior hasresulted in poorly detected outer walls and multiple greyareas that provide no information about the environment.

Figure 7(d) shows a comparison of the evolution of theF2 metric with time when using the proposed informativeprior and the non-informative prior when three sensor read-ings are used per pose and when only one reading is usedper pose. These results agree with the simulation results,with the proposed method yielding a superior F2 value evenwhen a single sensor reading is used at each location. Notethat the starting F2 value for the maps produced using theinformative prior is very high because the prior used isvery accurate and drops as less accurate sensor readingsare incorporated. Even if the prior information is extractedaccurately the final map should be updated using sensorinformation since floor plans may be outdated (for exam-ple due to a demolished wall) or no longer accurate (due toa collapsed wall in a USAR environment).

The multi-objective optimisation proposed in Section 3.3was repeated for a more realistic sensor model and usingthe real Kinect data collected during the experiment. Thefollowing model was used, based on Pirker et al., 2011.

InverseSensorModel( mi, xk , zk)=⎧⎪⎪⎪⎨⎪⎪⎪⎩

0.9, npixels<zi = npixels

i

0.1, npixels>zi = npixels

i

npixels<zi

npixelsi

, 0 < npixels<zi < npixels

i

(15)

with npixels<zi being the number of map pixels for a cell i

that fall between the robot and the location of the detectedobstacle, z, npixels>z

i the number of map pixels for a cell ithat are located beyond the detected object or at the loca-tion of the detected object and npixels

i the number of mappixels in cell i. For an accurate sensor, cells for which allmap pixels correspond to detected empty space are assigneda value of 0.9, those that correspond to detected objectsa value of 0.1 and those that contain both empty spaceand objects are assigned a value equal to the percentage ofpixels corresponding to detected objects.

A very narrow range of optimal ( pwall, pspace) wasobserved, which mostly agree with the results obtained insimulation, yielding optimised values of pwall = 0.2 andpspace = 1. These values are multiplied by a factor of 0.9before they are used in the prior map as explained in Sec-tion 5.1 to avoid values too close to 1. A less accurate sensormodel assigning 0.8 to detected empty space and 0.2 todetected walls was also tested, yielding optimised valuespwall = 0.4 and pspace = 0.75.

6.2. Larger scale experimental results

A larger scale experiment was conducted, mapping a floorof one of the University of Sheffield Engineering buildings(approximately 43 × 10 metres). The MATLAB version of

14 The International Journal of Robotics Research

Fig. 7. Experimental setup and results. (a) Experimental setup and robot used. (b) Accurate prior map of the environment. (c) Finalmaps produced using the proposed informative prior (left) and the non-informative prior (right). (d) Evolution of the F2 metric with timefor an exploration that takes three measurements at each robot pose (left) and for one that takes only one (right) using the informativeand non-informative priors.

Georgiou et al. 15

Fig. 8. Large scale experimental results. (a) Extracted walls and empty space. (b) Map produced using a non-informative prior.(c) Map produced using the proposed prior, successfully mapping more walls than using the non-informative prior. (d) Map producedusing the proposed informative prior and a slightly less accurate x0. (e) Evolution of the F2 metric with time for an informative andnon-informative prior.

16 The International Journal of Robotics Research

ROS adaptive Monte Carlo localisation was used along-side the floor plan of the building to produce estimates ofthe robot pose. The map was then updated using both theproposed informative prior and the commonly used non-informative prior. The floor plan was used for localisationin both cases to allow a direct comparison of the effectsof different priors on mapping without taking into accountthe effects of the prior on localisation quality. Not using thefloor plan to localise would lead to a less accurate map forthe non-informative prior, which highlights a further advan-tage of using the prior map: more accurate localisation andthe ability to start with a known, accurate x0.

The walls used to construct the prior map in this case(Figure 8(a)) are not perfectly extracted, with some ofthe doors appearing as wall sections on the sixth room in thetop row and the fifth room in the bottom row of offices. Thefinal map is unaffected by these small inaccuracies whichare corrected during mapping.

Some experiments were also run using a less accuratex0, using particles initialised with a non-zero variance. Theresulting maps were less accurate but still better than thoseproposed using a non-informative prior. For a low variancethe results were very similar to those produced using a per-fectly accurate x0. Assuming a known prior pose is realisticsince, given the extracted prior and a floor plan, an opera-tor can align the robot to a known location before startingexploration.

The results of these experiments are shown in Figure 8.Using the proposed prior yields a more accurate map bothin a qualitative and quantitative sense. The map producedusing the proposed prior yields a higher F2 value, even ifparts of the building are unexplored. It also yields morecorrectly mapped walls, demonstrating the advantages ofusing the proposed prior over a non-informative one. Theseresults also demonstrate an added benefit of using a priormap. If exploration of the whole floor were not possible dueto, for example, limited exploration time in a time-criticalmission or certain areas being inaccessible, the proposedprior provides some information about unexplored areas.This may not be completely accurate but it would allow ahuman operator or rescuer to better interpret the mappedsections.

The map shown in Figure 8(b) is less easy to inter-pret: it is unclear what sections of the building the mapareas correspond to and many of the walls have not beenmapped correctly. Conversely, using the proposed informa-tive prior (Figure 8(c)) a human can interpret the map moreeasily. Moreover the majority of corridor walls have beenmapped correctly, unlike Figure 8(b) where many of themare mapped as empty space.

In all cases there appears to be some noise in the map,particularly within the detected rooms. That is due tothe fact that the environment explored was a real world,cluttered office building, containing furniture. Thereforeobjects detected within the rooms mostly correspond todesks, chairs, bookcases and half-open doors. Since our aim

is to improve performance in real world environments wedid not aim to simplify the environment by only explor-ing long corridors (as is often done in the literature) oremptying the rooms.

These results further confirm that using the proposedinformative prior can improve performance especially whena quick exploration is required, yielding consistently betterquantitative and qualitative results.

7. Conclusions

This article presents a method to construct optimisedindoors occupancy grid mapping priors. Priors were con-structed using structural information extracted from archi-tectural drawings and floor plans by assigning appropriateprior probabilities to detected wall and empty space loca-tions. A precision-recall analysis was used to assess quanti-tative performance and a multi-objective optimisation wasused to short-list Pareto-optimal solutions. Both types ofdrawings were found to have a similar region of (pwall,pspace) that yields good performance metrics.

The commonly used uninformative prior was found toperform worse than a prior constructed using an architec-tural drawing and was not one of the Pareto-optimal solu-tions. The values pwall = 0.2, pspace = 0.9 were found toyield optimal results in terms of qualitative performancewhilst also yielding an improvement in the F2 metric of over20%.

The main contribution of this article is thus a method toproduce optimised Bayesian priors that improve map qual-ity in indoors SLAM applications without adding to thecomputational complexity of the SLAM algorithm itself.There is a one-off cost of extracting the prior value but nofurther increase in computational complexity. This methodalso yields improved performance compared to the non-informative prior for coarser grids or explorations that donot cover the entire building. It is therefore very wellsuited to time critical challenging real life applicationssuch as USAR missions where a fast and accurate SLAMimplementation is required.

Acknowledgements

We would like to thank Jon Lipsky, lead developer of ElevenworksLLC, for allowing us to use a floor plan featured in the Eleven-work website (Figure 4(a)) and the University of West Florida forallowing us to use an image of a floor plan of the Heritage Halls ofresidence (Figure 4(b)) to test the mapping algorithms presentedin this article.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of this article:This work was supported by the University of Sheffield and PAConsulting.

Georgiou et al. 17

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Appendix: Occupancy grid mapping

The occupancy grid mapping algorithm (Elfes, 1989) splitsthe environment to be mapped into a grid of cells and a priorprobability is assigned to each cell. Cells are assumed to beindependent in order to update the probability of occupancyof each grid cell. This assumption allows the factorisation

p( m|X0:k , Z0:k)=∏

i

p( mi|X0:k , Z0:k) (16)

where i = 1, . . . , N is the current grid cell and N the totalnumber of grid cells, making this an easier estimation sinceupdating the probability for each cell given the robot poseand sensor readings is merely an update of the probabilityof occupancy of that cell. As the robot moves through the

18 The International Journal of Robotics Research

environment the probabilities of occupancy of cells that arewithin the field of view of the robot sensors are updated.

The probability that a cell mi is occupied given theobservation history is given by

p( mi|Z0:k)= p( zk|mi, Z0:k−1) p( mi|Z0:k−1)

p( zk|Z0:k−1)(17)

which can be written as

p( mi|Z0:k)= p( zk|mi)p( mi|Z0:k−1)

p( zk|Z0:k−1)(18)

using the static world assumption which states that past sen-sor readings are conditionally independent given knowledgeof the map m (Thrun, 2003). Thus equation (18) can bewritten as

p( mi|Z0:k)= p( mi|zk) p( zk)

p( mi)

p( mi|Z0:k−1)

p( zk|Z0:k−1)(19)

using p( zk|mi)= p( mi|zk) p( zk)

p( mi).

Placing equation (19) in the odds form we get

odds( mi|Z0:k)= p( mi|Z0:k)

p(¬mi|Z0:k)

= p( mi|zk) p( mi|Z1:k−1) p(¬mi)

p(¬mi|zk) p(¬mi|Z1:k−1) p( mi)

(20)

Therefore

odds( mi|Z0:k)=odds( mi|zk) odds( mi|Z1:k−1) ( odds( mi) )−1

(21)

where p( mi) is the prior probability of occupancy of theith grid cell, odds( mi|Z1:k−1) is the odds at the previoustime step and odds( mi|zk) represents the inverse sensormodel. Taking the logarithm of equation (21), the log oddsrepresentation of equation (21) is

lk,i = InverseSensorModel( mi, xk , zk)+lk−1, i − l0, i (22)

with

lk,i = logp( mi|Z1:k , x1:k)

1− p( mi|Z1:k , x1:k)(23)

Methods to select a suitable or preferable inverse sen-sor model are proposed in (Thrun, 2003) and (Yaqub andKatupitiya, 2007).