Fast Weighted Exponential Product Rules forRobust General Multi-Robot Data Fusion

Nisar Ahmed* Jonathan Schoenberg* Mark CampbellAutonomous Systems Laboratory, Cornell University

Ithaca, New York 14853e-mail:[nra6,jrs55,mc288]@cornell.edu

*both authors contributed equally to this work

Abstract—This paper considers the distributed data fusion(DDF) problem for general multi-agent robotic sensor net-works in applications such as 3D mapping and target search.In particular, this paper focuses on the use of conservativefusion via the weighted exponential product (WEP) rule tocombat inconsistencies that arise from double-counting commoninformation between fusion agents. WEP fusion is ideal forfusing arbitrarily distributed estimates in ad-hoc communicationnetwork topologies, but current WEP rule variants have limitedapplicability to general multi-robot DDF. To address these issues,new information-theoretic WEP metrics are presented along withnovel optimization algorithms for efficiently performing DDFwithin a recursive Bayesian estimation framework. While theproposed WEP fusion methods are generalizable to arbitraryprobability distribution functions (pdfs), emphasis is placed hereon widely-used Bernoulli and Gaussian mixture pdfs. Experi-mental results for multi-robot 3D mapping and target searchapplications show the effectiveness of the proposed methods.

I. INTRODUCTION

The problem of fusing information from an ensemble ofnoisy data streams is critical to many existing and soon-to-be-realized robotic systems operating in uncertain dynamicenvironments. This is particularly true for distributed multi-robot systems requiring distributed perception for applica-tions such as collaborative mapping for exploration, targetsearch/tracking for surveillance, and futuristic unmanned ur-ban transport systems. For sensor agents in a network to shareinformation and perform distributed data fusion (DDF), itis most desirable to establish a scalable, flexible and robustnetwork over which the robots can transmit and receive infor-mation. An ad-hoc and arbitrary connected network providesscalability for fusion agents to join and drop off the network,flexibility to allow agents to join at any point and robustnessto ensure multiple links or agents must fail before the networkbecomes unconnected [5].

Implementation of DDF for general robot sensor networksthus requires conservative data fusion techniques to maintainestimates that avoid inconsistencies due to rumor propaga-tion [3]. A common conservative fusion rule for estimateswith Gaussian probability distribution functions (pdfs) withunknown correlation is Covariance Intersection (CI) [11]. Thisrule is appropriate for certain types of problems, but it isinadequate for handling non-Gaussian distributions that arisein applications such as target search and 3D mapping. A suit-able conservative fusion rule for arbitrary pdfs with unknown

correlation is the weighted exponential product (WEP) [1]. TheWEP is a generalization of CI to non-Gaussian distributions[9] and different cost metrics to determine an optimal WEPfusion weight have been proposed [3, 9]. However, theseexisting WEP fusion approaches have drawbacks that canlimit their usefulness in robotic DDF, including the theoreticalnature of their cost metrics and difficult implementation forarbitrary pdfs.

This paper makes the following contributions: (1) it pro-poses novel information-theoretic metrics for performing WEPfusion that address these issues and are suitable for fusingarbitrary state estimates shared via ad-hoc network communi-cation topologies in a wide range of robotic DDF applications;(2) it presents new formally consistent algorithms for onlineimplementation of our proposed WEP fusion metrics that canbe used to quickly and robustly combine information in arecursive Bayesian estimation framework, with emphasis onthe Bernoulli and Gaussian mixture distribution functions usedwidely in robotics; (3) it demonstrates the effectiveness ofour proposed WEP fusion methods for performing onlinecollaborative 3D mapping and 2D target search with multi-robot networks in loopy communication topologies.

A. DDF Preliminaries

Formally, let xk ∈ Rnx be an nx-dimensional state to beestimated at discrete time steps k = 0, 1, ... by N independentrobotic sensor agents. Assume that each robot i ∈ {1, ..., N}obtains ny local sensor measurements of xk in the vector ykiwith likelihood function pi(yki |xk). Let pi(x

k|Zki ) be the local

posterior pdf for robot i given the set of all information Zki ={

Zk−1i , yki

}available to i before new information Zk

j arrivesfrom robot j 6= i, such that local fusion of yki is given by theposterior pdf from Bayes’ rule,

pi(xk|Zk

i ) ∝ pi(xk−1|Zk−1i )pi(y

ki |xk). (1)

For generality, the robot networks considered here are as-sumed to have arbitrary dynamic node-to-node communicationtopologies, such that: (i) N may vary; (ii) each robot is onlyaware of nodes it connected to; (iii) no robot ever knows thecomplete global network topology; (iv) no robot knows thereceipt status of messages it has sent.

Centralized fusion or raw data sharing can fully recover newinformation in

⋃Ni=1 Z

ki , but such methods scale poorly with

N and are vulnerable to node failures. DDF overcomes suchbottlenecks and can recover the centralized Bayesian fusionresult for a given pair of robots (i, j) [8], since

p(xk|Zki ∪ Zk

j ) ∝pi(x

k|Zki )pj(x

k|Zkj )

p(xk|Zki ∩ Zk

j ), (2)

where the denominator is the common information pdf forthe local pdf pi(x

k|Zki ) ∝ p(xk|Zk

i ∩ Zkj ) · p(xk|Zk

i\j),where Zk

i\j is the exclusive local information set for nodei and similarly for node j. Hence, the common informationpdf must be explicitly known for DDF to optimally extractonly new information in Zk

i ∪ Zkj . Note that if division by

p(xk|Zki ∩ Zk

j ) were ignored in (2), the resulting ‘NaiveBayes’ fusion rule would always double-count p(xk|Zk

i ∩Zkj )

and induce inconsistencies via rumor propagation. However,tracking p(xk|Zk

i ∩ Zkj ) in an arbitrary ad-hoc network re-

quires pedigree-sharing about each distribution to fuse; thisbecomes computationally prohibitive for non-tree communi-cation topologies with loops/cycles [3, 8]. The generality ofassumptions (i)-(iv) above leads to difficulties for implement-ing DDF via graphical model techniques.

An alternative method for consistent DDF considered hereis the conservative weighted exponential product (WEP) fusionrule,

pWEP(xk|Zk

i ∪ Zkj ) =

1

cpωi (xk|Zk

i ) · p1−ωj (xk|Zk

j ), (3)

≡ pWEP(xk)

where ω ∈ [0, 1] and c is a normalizing constant. The WEPrule has two properties that make it well-suited to generalmulti-robot DDF problems: (1) it is applicable to arbitrarynon-Gaussian pdfs, and (2) for any ω ∈ [0, 1], it is guaranteedto avoid double-counting of p(xk|Zk

i ∩ Zkj ) [1]. However,

this also means some exclusive information from i and jis always discarded. Hence, the WEP fusion weight ω mustbe picked carefully to ensure useful information exchangebetween robots i and j.

B. WEP Fusion Metrics

Ref. [9] considers two popular cost metrics on (3) foroptimal selection of ω ∈ [0, 1]. The first cost is the Shannonentropy of pWEP(x

k), which is motivated by the idea that theabsolute uncertainty of (3) should be minimized. Ref. [9]shows that the resulting Shannon fusion rule [3] is a directnon-Gaussian generalization of the Gaussian CI rule. Thesecond cost selects the ω = ω∗ corresponding to the ChernoffInformation between pi(x

k|Zki ) and pj(x

k|Zkj ) [4], and is thus

called Chernoff fusion [6, 10]. It can be shown that this equatesthe Kullback-Leibler divergences (KLDs) between (3) and theindividual robot posteriors, which guarantees that the Chernofffusion result is ‘half-way’ between the original local i and jposteriors in an information gain/loss sense. Both the Shannonand Chernoff fusion rules provide nice information theoreticalreasons for choosing ω, but each rule has potential limitationsin the context of general multi-robot DDF, as explained inSection II.

C. WEP Fusion Algorithms

General WEP fusion can present challenging implementa-tion issues when dealing with arbitrary non-Gaussian pdfs. Forinstance, both Shannon and Chernoff fusion require minimiza-tion of integral costs that are in general analytically intractable,making WEP fusion potentially unsuitable for online roboticDDF problems. Furthermore, (3) is generally not available inclosed-form. Heuristic WEP fusion approximations have beenproposed to address these issues [6, 10], but these have noformal guarantees of either correctly minimizing the desiredWEP cost or accurately approximating (3). Such heuristicapproximations can thus lose information that should bepreserved between robots or introduce spurious information.

II. NEW INFORMATION-BASED WEP METRICS

A. Limitations of Existing Metrics

Shannon fusion is not always justified from a generalBayesian estimation perspective, since acquisition of newinformation can, in many cases, increase the entropy of theexact/centralized Bayesian posterior for non-Gaussian pdfs [1].Chernoff fusion ignores the possibility that an imbalance ininformation gain may exist between robots i and j prior tofusion. To address such scenarios, ref. [9] suggests introducingweighting factors into Chernoff fusion that are equal to thenumber of measurements taken by each robot in order to bias ωtowards the better informed pdf. However, the number of mea-surements is not a complete reflection of information gain foreach robot, as local differences in sensor quality/characteristicsand information obtained from other neighbors in the robotnetwork must also be taken into account.

B. Generalized Information Weighted Chernoff Fusion

The first new fusion rule modifies the measurementweighted Chernoff fusion [9] to apply weighting factorsci, cj ≥ 0 that reflect actual information gain by each robot iand j, instead of the number of measurements. This biases ωtowards the constituent pdf with greater information contentwithout explicitly counting measurements, so that ω satisfies

ci ·DKL

[pWEP(x

k)||pi(xk|Zki )]

= cj ·DKL

[pWEP(x

k)||pj(xk|Zkj )], (4)

where DKL [·||·] is the KLD. For problems yki are alwaysguaranteed to reduce uncertainty (e.g. static linear Gaussianestimation), ci and cj can be conveniently estimated as thereciprocal of the entropy of the constituent distributions,

ci =1

H[pi(xk|Zk

i )], cj =

1

H[pi(xk|Zk

i )], (5)

where H[p(x)] is the Shannon entropy of p(x). This leadsto an Entropy Weighted Chernoff Fusion solution that usesa weight ω∗ ∈ [0, 1] to equate the entropy weighted KLDsbetween the WEP and original distributions. This results ina bias towards the distribution with the lower entropy alongthe chord that connects the two distributions according to theKLDs.

However, as discussed in Section II-A, the Shannon entropyis not always a suitable indicator of information gain. It is alsounsuitable for fusing distributions over continuous states xk,since the differential Shannon entropy of probability densityfunctions can be negative (whereas ci, cj should be non-negative). One possible alternative is to select ci and cj tobe the information gained in the KLD sense with respect to acommon reference distribution pr(xk) for robots i and j. If acommon prior pdf p(x0) is available to all agents at k = 0 forinitializing Bayesian estimation, then this forms a convenientchoice for pr(xk), so that

ci = DKL

[pi(x

k|Zki )||p(x0)

], (6)

and similarly for cj . These weights reflect the total informationgain of each agent over time with respect to p(x0) via thecombined effects of local Bayesian fusion and DDF. Note thata fixed reference distribution must be used with the KLD, sinceit is not a true distance metric and does not obey the triangleinequality. The KLD could be replaced by another divergencethat is a true metric (e.g. the Hellinger divergence [2] or thesymmetric KLD), which removes the need for a common priorand permits ci, cj to be computed incrementally, but that is notdemonstrated here.

C. Minimum Information Loss Weight Fusion

A second cost is now developed to minimize an approxi-mation on the information lost as a result of WEP fusion. Eq.(3) discounts exclusive information at i and j to ensure thatthe unknown common information between the two is countedonly once. Therefore, there is an information loss between theoptimal fusion pdf (2) and the WEP fusion pdf (3), which isdefined

ILOSS∆=DKL

[pOPTIMAL(x|ZK

i

⋃ZKj )||pWEP(x|ZK

i

⋃ZKj )](7)

which cannot be computed without knowing the optimallyfused distribution. However, note that if i and j are knownto have no information in common, then the optimal fusionrule simplifies to the Naive Bayes (NB) fusion,

pNB(x|ZKi

⋃ZKj ) =

1

cp(x|ZK

i )p(x|ZKj ). (8)

In the case of an unknown dependence between the i and jpdfs, the discounting of exclusive information resultant fromWEP fusion leads to information loss if the true dependencebetween the distribution is zero. Therefore, an approximationto the information loss is the KLD between the NB and WEPfusion pdfs,

ILOSS ≈ ILOSS(ω)∆=DKL

[pNB(x|Zi

⋃Zj)||pWEP(x|Zi

⋃Zj)]

(9)

It is now possible to compute the fusion weight ω∗ ∈ [0, 1] tominimize the approximate information loss,

ω∗ = arg minω∈[0,1]

ILOSS(ω). (10)

The advantage of this Minimum Information Loss Fusion(10) scheme is that ω is selected to minimize the possibleinformation loss (7) should the two distributions be trulyuncorrelated. In practice, Minimum Information Loss Fusiondrives the solution towards Naive Bayes fusion in the casewhen the distributions to fuse are significantly different (in aKullback-Leibler sense), which could indicate the two do notshare a significant amount of common information. The Min-imum Information Loss Fusion scheme provides an automaticmethod for trading between the Naive Bayes and conservativefusion rules, without resorting to a heuristic decision.

III. WEP FUSION FOR MULTI-ROBOT OCCUPANCY GRIDSAND GAUSSIAN MIXTURE BELIEFS

The proofs have been omitted due to limited space, but itcan be shown that the newly proposed metrics Section II areall convex in ω for arbitrary pdfs pi(x

k|Zki ) and pj(x

k|Zkj ).

In this section, the problem of implementing WEP fusionusing the (un)weighted Chernoff and Minimum InformationLoss cost metrics is addressed. The focus is on cases whereboth pi(x

k|Zki ) and pj(x

k|Zkj ) are described by either discrete

Bernoulli distributions or continuous finite Gaussian mixtures,which are commonly used in robotics, e.g. for occupancy gridmapping and object tracking/localization, respectively. Withsome minor modifications, the methods described here canbe readily applied in other contexts and to other arbitrarydiscrete/continuous pdfs.

A. Fast WEP Fusion for Bernoulli Distributions

Applying the general WEP fusion rule (3) to two Bernoullidistributions with unknown correlation results in a fuseddistribution that is also Bernoulli,

pi(xk|Zk

i ) = pi, pj(xk|Zk

j ) = pj (11)

pWEP(xk) =

pωi p1−ωj

pωi p1−ωj + (1− pi)ω(1− pj)1−ω

. (12)

In the case of Chernoff fusion, the optimal weight ω∗ isgiven in closed form,

ω∗CF =log

[log(1−pi)−log(1−pj)][log(pi)−log(pj)] − log pj + log(1− pj)

[log pi − log pj − log(1− pi) + log(1− pj)](13)

For Entropy Weighted Chernoff Fusion, ω∗ cannot be foundin closed form, and the following equation must be solvednumerically,

1

H[pi]

[pω log

pωpi

+ (1− pω) log(1− pω)

(1− pi)

]− . . .

1

H[pj ]

[pω log

pωpj

+ (1− pω) log(1− pω)

(1− pj)

]= 0 (14)

where pω is given by (12) and the entropy of a Bernoullidistribution is H[p] = −p log p− (1−p) log(1−p). When theentropy of the two distributions is equal, Entropy WeightedChernoff Fusion collapses to regular (i.e. unweighted) Cher-noff fusion.

In the case of the Minimum Information Loss fusion, ω∗ isgiven in closed form,

ω∗M-L =log pj − log(1− pj)− log

[pipj

(pi−1)(pj−1)

]log pj − log(1− pj)− log pi + log(1− pi)

. (15)

B. Fast WEP Fusion for Finite Gaussian Mixtures

Assume each robot’s pdf is a finite Gaussian mixture (GM),

pi(xk|Zk

i ) = pi(xk) =

Mi∑m=1

wimN (µi

m,Σim) (16)

where (for agent i and similarly for j) M i is the numberof mixands, µi and Σi are the mean and covariance matrixof component m, and wm is the mth mixing weight suchthat wm ∈ [0, 1] and

∑Mm=1 w

m = 1. The WEP cost metricintegrals and the resulting fusion pdf (3) are not closed formfor GMs. A new fast and consistent two-phase Monte Carlo-based optimization and learning procedure is presented toovercome these issues.

1) Particle-based optimization: The Information WeightedChernoff fusion cost can be written as

fkWEP(ω) =

∫hij(x

k) · pWEP(xk)dxk, (17)

where hij(xk) =ci log pi(x

k)− cj log pj(xk)

log pi(xk)− log pj(xk),

(where unweighted Chernoff fusion uses ci = cj = 1); theMinimum Information Loss cost (4) can be written as

fkWEP(ω) = ω · κ+ log

∫hij(x

k) · pWEP(xk)dxk, (18)

hij(xk) = 1, κ =

∫pi(x

k)pj(xk) log

(pj(x

k)

pi(xk)

)dxk

The integral terms with respect to pWEP(xk) in each of these

analytically intractable costs are approximated via importancesampling,∫

hij(xk) · pWEP(x

k)dxk ≈ 1

Ns

Ns∑s=1

θs(ω) · hij(xs), (19)

θs(ω) =pωi (xs)p

1−ωj (xs)

g(xs;ω)(20)

where hij(xs) are sample values of hij(xk) and θs(ω) areunnormalized non-negative weights for Ns i.i.d. samples{xs}Ns

s=1 drawn from a fixed importance pdf g(xk;ω). Theconstants ci and cj in (6) and κ in (18) are generally not avail-able in closed form for GMs, but can be well-approximatedestimated with fast deterministic methods [7].

To ensure that (19) is well-behaved and easy to compute,the importance pdf g(xk;ω) should be easy to sample fromand evaluate, have non-zero support over the entire domainof integration and resemble (3) closely enough to control thevariance of (19). Many choices are possible for g(xk;ω), but

Algorithm 1 GM WEP Fusion via Importance SamplingInputs: robot GM pdfs pi(xk) and pj(xk); number of samples Ns; initialguess ω0; importance pdf exponent ω; update function [ωold, ωcurr] ←R1D[f(ωcurr), ωcurr, ωold] for desired 1D minimization rule; effective samplesize threshold, Nt

Output: ω∗ ∈ [0, 1]; samples {xs}Nss=1 with normalized weights {θs}Ns

s=1

1. Initialize ωcurr ← ω0 and ωold according to R1D

2. construct GM importance sampling pdf g(x; ω) via (21)3. draw Ns samples {xs}Ns

s=1 ∼ g(x;ωcurr)

4. store pdf values pi(xk), pj(xk), g(x;ωcurr) for {xs}Nss=1

while ωcurr not converged do5. compute unnormalized importance weights {θs}Ns

s=1 via (20)6. using (19), compute f(ωcurr) via (17) or (18)7. update [ωold, ωcurr]← R1D[f(ωcurr), ωcurr, ωold]

end while8. if effective sample size < Nt, set ω = ωcurr and go to step 1.9. normalize {θs}Ns

s=1 s.t.∑Ns

s=1 θs = 1 and return ω∗ = ωcurr.

the following ‘first-order’ GM approximation to (3) proposedby [10] is particularly effective,

g(x;ω) =

M F∑m=1

wFmN (µF

m,ΣFm) (21)

ΣFm =

(ω(Σi

q)−1 + (1− ω)(Σjr)−1

)−1(22)

µFm = ΣF

m

(ω(Σi

q)−1µiq + (1− ω)(Σj

r)−1µjr

)(23)

wFm =

(wiq)ω(wj

r)1−ω∑q′,r′ (wi

q′)ω(wj

r′)1−ω

, (24)

which is a component-wise covariance intersection operationon the constituent GMs (with M F resulting mixands).

Eq. (19) is unbiased and converges to the true value of theintegral as Ns → ∞. Hence, a numerical 1D optimizationroutine can obtain an unbiased estimate ω∗ of ω using (19) toevaluate (17) or (18). The 1D search is accelerated by usinga fixed importance pdf g(x;ω) = g(x; ω), where ω is thesame in all optimization steps. As such, the samples {xs}Ns

s=1

are drawn once and the pdf values {pi(xs), pj(xs), g(xs; ω)}are stored as constants for recalculating (20) at new ω values.The resulting optimization procedure is shown in Algorithm1, where R1D[f(ωcurr), ωcurr, ωold] denotes an ω-update rule forany desired 1D optimization algorithm (e.g. bisection searchor golden section search) and ω = 0.5 is typically used inour applications as an initial estimate. Algorithm 1 typicallyconverges quite quickly and therefore may be run again ifit terminates with an effective sample size less than someprescribed threshold Nt. In this case, ω is reset to the currentbest estimate for ω to restart the optimization.

2) WEM Learning: Immediately following optimization,density estimation procedures are applied to the importancesamples to recover an accurate GM to approximate (3) atω∗. The Weighted EM (WEM) algorithm is used for GMsto exploit the normalized importance weights as part of thelearning problem. Details for implementing WEM learning ofGMs from weighted samples are given in [7]. The initial GMparameters for WEM are found via compressing (21) to adesired maximum number of GM terms, MMAX, using [13].

(a) (b)

Fig. 1. (a) Mobile robot set up with Hokuyo lidar and on-board computerfor distributed occupancy grid map fusion experiment. (b) Laboratory envi-ronment for the occupancy grid mapping experiment is 15 × 8 meters andcontains boxes of different sizes.

IV. EXPERIMENTAL RESULTS

A. Multi-Robot Occupancy Grid Map Fusion

The generalized fusion of the Bernoulli distribution (SectionIII-A) is directly applicable to fusion of multi-robot occupancygrid maps. The probability of a voxel of space being occupiedis represented as a Bernoulli probability and each voxel isassumed independent. As a result, the fusion of occupancy gridmaps from multiple robots is accomplished by sequentiallyperforming fusion across each of the grid cells in the map.A laboratory experiment demonstrates the fusion of 3D occu-pancy grid maps based on the different fusion rules derived,including Chernoff Fusion, Entropy Weighted Chernoff Fusionand Minimum Information Loss fusion and compares theresulting approximate information loss as a function of maplocation for the different techniques.

Data was collected using the Pioneer P3-DX differentialdrive mobile robot from Mobile Robots Inc. shown in Fig.1(a). The primary sensor for the occupancy grid mapping isthe Hokuyo URG-04X laser scanner which features a 240o

field-of-view and angular resolution of 0.36o and maximumrange of ≈ 5 m. The laser is pitched downward 45o and scansalong the ground as the robot moves forward.

The test environment is 15× 8 meters and is instrumentedwith a Vicon MX+ precision tracking system for robot lo-calization and 3D object position/attitude determination. Thefeatures in the environment consist of boxes between 10and 25 cm tall that are meant to simulate traffic cones orother similarly sized obstacles for a full-size traffic vehicle.Eight robots are run in different paths around the environmentfor a 120 second data collection (Fig. 2). The robots arerun sequentially to avoid sensing each other during mapconstruction.

The robots each construct a 3D occupancy grid map usingthe Octomap [14] implementation. The occupancy grid reso-lution is 0.05 m. To establish a baseline occupancy grid map,the data from all the mobile robots is processed to make thecentralized solution that is equivalent to all agents sendingeach laser scan and position report to a central server forprocessing. The resulting centralized map is shown in Fig.3(a). The map is rendered displaying only the voxels that areoccupied or have a pOCC > 0.8, the empty and unknown voxelsare not shown and the voxels are falsely colored according to

Fig. 2. The overhead paths of the eight mobile robots exploring theenvironment for distributed occupancy grid mapping experiment. The datacollection interval is 120 seconds and different robots explore different regionsof the environment with some overlap to demonstrate the benefit of thegeneralized fusion rules to distributed mapping. The distributed mappingoptimal network topology is shown as an inset.

(a)

(b)

(c)

Fig. 3. (a) 3D centralized occupancy grid map displaying the occupied voxelswith false coloring based on height. (b) 3D Occupancy grid map from Agent1 using only local updates. The occupied and empty cells (ghost gray) areshown along with the final pose of the robot (red box). (c) 3D Occupancygrid map from Agent 1 after Minimum Information Loss Fusion; this map isqualitatively similar to the centralized solution.

height according to the scale in Fig. 3(b); a 0.25 m grid isshown in gray for reference. The line of small skinny boxesto the left of the environment are clearly visible along with thelarger boxes towards the right of the environment and the outerwalls. There are gaps in the center of the map where no robotexplored. The centralized solution establishes a baseline mapfor use in the distributed occupancy grid mapping approaches.

Distributed Occupancy Grid Fusion on Optimally Con-nected Network: To evaluate the application of the distributedgeneralized fusion rules to occupancy grid mapping, themobile robots are connected in a network. The network isconnected according to the topology with maximum robustness[5]. The topology is symmetric, has equal node and linkconnectivity equal to the minimum degree, and each node is acentral vertex of the graph. The topology is shown as an insetto Fig. 2. This makes it difficult to track common informationduring fusion without data tagging and generalized data fusionfor unknown correlations is used.

The individual sensor nodes collect and process local scansto build a local occupancy grid map that will be updatedwith map data passed along the network from other agents.An example of map constructed via local updates only is inFig. 3(b), which shows the map from Agent 1 (Fig. 2 RedPath). The map is rendered showing the occupied cells falselycolored according to height and the empty cells are shown ina ghost gray; the unknown cells are not shown. The final poseof the robot is shown using a solid red box. The map showsAgent 1 explores only a portion of the map. To enable a fullrepresentation of the environment, the distributed data fusiontechniques are used over the maximally robust sensor network.

The robots share map information across the bi-directionallinks connecting the robots. The agents are required to sharethe following information for each voxel: the center coordi-nate and the log-odds probability. This implies each voxelrequires 32 bytes of data (if all numbers are in doubleprecision) transmitted for each communication. The nodescommunicate aperiodically as they collect information. Thefinal occupancy grid map after Minimum Information LossFusion is shown in Fig. 3(c). The distributed data fusion viaMinimum Information Loss fusion does not have access to thecentralized solution nor the optimal fusion distribution, but issuccessfully utilized to build a map that is qualitatively similarto the centralized solution (Fig. 3(a)). The maps for ChernoffFusion and Entropy Weighted Chernoff Fusion are similar (notshown).

Approximate Information Loss Maps: The approximateinformation loss ILOSS (9) is computed after fusion for each ofthe different fusion rules. The resulting information loss mapscan be used for planning purposes to balance exploration vsverification of cells that may have contained substantial infor-mation loss as a result of fusion. The information loss mapresulting from Chernoff Fusion is shown in Fig. 4(a). The mapis falsely colored according to the approximate informationloss metric ILOSS and the scale is shown in Fig. 4(b). Thiscolor range reflects the maximum range of values accordingto the empirical CDF computed over all possible combinationsof two Bernoulli distributions for the different fusion rules (notshown). The areas where the largest information loss occurswhen information coming from remote nodes clashes withthe content estimated locally at Agent 1 and are in the lefthand portion of the map. The areas of information loss to theright hand portion of the map are resulting from the sequentialapplication of the fusion rule as information is received on the

(a)

(b)

(c)

Fig. 4. (a) Approximate information loss ILOSS on the occupancy grid afterChernoff Fusion for Agent 1. (b) Approximate information loss ILOSS onthe occupancy grid after Entropy Weighted Chernoff Fusion for Agent 1. (c)Approximate information loss ILOSS on the occupancy grid after MinimumInformation Loss Fusion for Agent 1.

three links connected to Agent 1.The resulting information loss map for Entropy Weighted

Chernoff Fusion at Agent 1 is shown in Fig. 4(b). Theresults shows an increase in some areas and a decrease inthe approximate information loss in other areas as a result ofEntropy Weighted Chernoff Fusion. The resulting fused map(not shown) is similar to Fig. 3(b). The Minimum InformationLoss Fusion rule is significantly better than either approachin terms of reducing the need for verification due to potentialinformation loss, especially along the ground.

The resulting information loss map for Minimum Informa-tion Loss Fusion at Agent 1 is shown in Fig. 4(c). The resultsshow a dramatic improvement in the potential informationloss, because the fusion rule has zero loss for 50% of thepossible combinations of Bernoulli distributions. The Mini-mum Information Loss Fusion rule generates consistent andquality occupancy grid maps and has the lowest informationloss. Therefore, this fusion rule is the best for distributed datafusion of occupancy grid maps.

B. GM Fusion for Multi-robot Target Search

In another experiment, three Pioneer 3D-X robots are taskedto search for a static target with location xtarg in an en-vironment similar to the one shown in Fig. 1. Each robotequipped with single forward facing Unibrain Fire-I camera

(a) (b) (c)

Fig. 5. (a) Example GM prior over xTARG (red indicates higher density);(b) model for camera-based binary ‘Detection’/‘No Detection’ likelihood(P (‘No Detection’|xTARG) shown zoomed in; blue indicates probability near0); (c) posterior GM after fusing GM prior in (a) with ‘No Detection’ reportin (b) via eq.(1) using Gaussian sum filter approximation.

with a 42.5 deg field of view, which feeds onboard visualobject detection software that produces binary ‘detection’/‘nodetection’ outputs at 1 Hz with 1 m maximum detection range.The likelihood model for the visual object detectors is shownin Fig. 5(b) and the common GM prior pdf for xTARG atk = 0 is shown in Fig. 6(a). The sensor likelihood model foreach robot is highly non-Gaussian and induces non-Gaussianposterior via (1). As shown in Fig. 5(c), this local Bayesianposterior is well-approximated at each time step k by a finiteGM using a variant of the Gaussian sum particle filter with1000 particles per mixand [12] and a standard GM reductionalgorithm to keep MMAX = 15 for each robot [13].

The robots move along different pre-defined search trajecto-ries for 65 secs, during which they fuse their own camera ob-servations to modify their local copy of the GM pdf for xTARG

(none of the agents finds the target). The robots communicate5 times (once every 13 secs) using the cyclic communicationpattern 1→ 2→ 3→ 1 for time steps k = 26 and k = 39, and1→ 3→ 2→ 1 for the remaining instances. The unweightedChernoff, KLD-gain Information Weighted Chernoff, and Min-imum Information Loss rules were all evaluated for DDF viaAlgorithm 1 with Ns = 1000. Non-DDF benchmark runsusing no sharing and centralized fusion were also evaluated,alongside alternative DDF benchmark runs obtained via NaiveBayes and conservative Bhattacharyya fusion. Bhattacharyyafusion is another WEP fusion rule that always assigns ω = 0.5[3]. This fusion rule requires no optimization but does requireapproximation of the WEP fusion pdf, which is accomplishedby applying WEM to weighted importance samples, as donefor the other WEP rules used here. Fig. 6(b)-(h) show theresulting fusion posterior GM pdfs at k = 65 for robot 1(bottom left of map). Fig. 7 shows the KLD information gains(6) for each robot over time under each fusion scheme.

Fig. 6(b) and Fig. 7(a) show that robot 1 contributes theleast amount of new information, as it remains completelystill at the bottom left of the map throughout the search .When using DDF to update its local copy of the xTARG GM,robot 1 exchanges GMs with robots 2 and 3 without any of therobots being aware of the common information in the network.The Naive Bayes DDF results in Fig. 6(d) and Fig. 7(c) showthat repeated double-counting of common information drivesthe final DDF posterior pdfs for all robots very far from thecentralized fusion posterior in Fig. 6(c) and thus leads toseverely inconsistent results.

In contrast, the final WEP DDF posterior GMs in Fig. 6(e)-(h) and information gains in Fig. 7(d)-(f) all bear much closerresemblance to the centralized fusion result (with KLDs to thecentralized result roughly the same between 0.4-0.55 nats).The posterior GM plots for Information Weighted Chernoffand Minimum Information Loss fusion show that these rulesare better at retaining some of the subtler features of thecentralized robot fusion GM pdf, such as the distinctness of themodes around the outskirts of the map and the ‘plowed lanes’created by robots 2 and 3. On the other hand, the posteriorGMs for unweighted Chernoff and Bhattacharyya fusion aresmeared out, indicating that less novel information is retainedby WEP fusion at the expense of robot 1’s older outdated com-mon prior information (note the frequent information ‘dips’ forunweighted Chernoff). Interestingly, Fig. 7(f) shows that eachrobot is more likely to gain (and less likely to lose) informationvia Information Weighted Chernoff fusion, as the gains foreach robot rarely drop from their previous values, as theydo near the end of the unweighted Chernoff and MinumumInformation Loss results. Further analysis shows that thisbehavior for Information Weighted Chernoff fusion stems froman aggressive ‘winner takes all’ strategy for selecting ω inthe initial stages of the search, so that values of ω closeto 0 or 1 are observed frequently. As the robot informationgains become similar after k = 30, the ω values selectedby Information Weighted Chernoff become less extreme. Thissuggests different approaches are more appropriate at differenttimes during the search process.

V. CONCLUSION

The WEP approach for robust fusing two arbitrary prob-ability distributions with unknown common information wasaddressed for general multi-robot applications. The nominalChernoff fusion rule does not account for differences in con-tent or quality of information sources used to construct con-stituent robot distributions; the Information Weighted ChernoffFusion rule was developed to account for such potential dispar-ities without explicitly counting the number of measurements.To approximately minimize losses incurred by WEP fusionrelative to the exact fusion posterior, a second novel MinimumInformation Loss Fusion rule was also developed using theKLD between the Naive Bayes and WEP-fused distributionsas the cost metric. Novel fast convex implementations ofthe proposed WEP fusion rules were presented for fusion ofBernoulli and arbitrary Gaussian mixture pdfs. These methodswere experimentally demonstrated in loopy multi-robot com-munication topologies for fusion of 3D occupancy grids and2D target location beliefs. Future work will investigate thetheoretical properties and online performance of the proposedWEP methods across a variety of dynamic ad hoc fusionscenarios, spanning scales from dozens to thousands of robots.

REFERENCES

[1] T. Bailey, S. Julier, and G. Agamennoni. On ConservativeFusion of Information with Unknown Non-Gaussian De-

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 6. (a) Prior GM pdf for xTARG at k = 0; (b)-(h) Robot 1’s (bottom left)GM posterior pdf at k = 65 under various fusion rules: (b) no fusion, (c)centralized fusion, (d) Naive Bayes, (e) unweighted Chernoff, (f) InformationWeighted Chernoff, (g) Minimum Information Loss, (h) Bhattacharyya fusion.

pendence. Technical report, Austrailian Center for FieldRobotics, 2011.

[2] F. Bourgault. Decentralized control in a Bayesian World.PhD thesis, University of Sydney, 2005.

[3] K. Chang, Chong Chee-Yee, and S. Mori. Analytical andComputational Evaluation of Scalable Distributed FusionAlgorithms. IEEE Trans. on Aerospace and ElectronicSystems, 46(4):2022–2034, 2010.

[4] T. M. Cover and J. A. Thomas. Elements of informationtheory, volume 6. Wiley Online Library, 1991.

[5] Anthony H. Dekker and Bernard D. Colbert. Networkrobustness and graph topology, 2004.

[6] W. J. Farrell and C. Ganesh. Generalized chernoff fusionapproximation for practical distributed data fusion. In12th Int’l Conf. on Information Fusion., pages 555–562,2009.

[7] J. Goldberger, H. K. Greenspan, and J. Dreyfuss. Simpli-fying Mixture Models Using the Unscented Transform.Pattern Analysis and Machine Intelligence, IEEE Trans.on, 30(8):1496–1502, 2008.

[8] S. Grime and H. F. Durrant-Whyte. Data fusion indecentralized sensor networks. Control EngineeringPractice, 2(5):849–863, 1994.

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Robots 2 and 3 separately and slowly gain new info over search paths

Stationary Robot 1receives no new info

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Optimal info gaincurve for all robots

if all estimates fused

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Information gains ~10x largerthan for centralized fusion

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Large initial info gain for Robot 1, followed by smaller info gains/drops over network

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Info gains drop offand nearly level at later steps

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Info gains more consistentlypositive for each robot

at each fusion step

(e) (f)

Fig. 7. Robot information gains vs. time relative to k = 0 GM priorunder different fusion rules: (a) no fusion, (b) centralized, (c) Naive Bayes(note larger vertical scale) (d) Minimum Information Loss, (e) unweightedChernoff, (f) Information Weighted Chernoff. Results for Bhattacharyya fusion(not shown) are similar to unweighted Chernoff results. Dashed vertical linesdenote interagent DDF time steps.

[9] M. B. Hurley. An Information-Theoretic Justification forCovariance Intersection and Its Generalization. Technicalreport, MIT Lincoln Lab, 2001.

[10] S. Julier. An Empirical Study into the Use of ChernoffInformation for Robust, Distributed Fusion of GaussianMixture Models. In 9th Int’l Conf. on InformationFusion., 2006.

[11] S. J. Julier and J. K. Uhlmann. A non-divergent estima-tion algorithm in the presence of unknown correlations.In Proc. of the American Control Conference., volume 4,pages 2369–2373 vol.4, 1997.

[12] J. H. Kotecha and P. M. Djuric. Gaussian sum particlefiltering. Signal Processing, IEEE Trans. on, 51(10):2602–2612, 2003. 1053-587X.

[13] Andrew R. Runnalls. Kullback-Leibler Approach toGaussian Mixture Reduction. IEEE Trans. on Aerospaceand Electronic Systems., 43(3):989–999, 2007.

[14] K. M. Wurm, A. Hornung, M. Bennewitz, C. Stachniss,and W. Burgard. OctoMap: A probabilistic, flexible, andcompact 3D map representation for robotic systems. InProc. of the ICRA 2010 workshop on best practice in 3Dperception and modeling for mobile manipulation, 2010.