Data provided by sensors is always subjected to some level of uncertainty and inconsistency. Multisensor data fusion algorithmsreduce the uncertainty by combining data from several sources. However, if these several sources provide inconsistent data,catastrophic fusion may occur where the performance of multisensor data fusion is significantly lower than the performance ofeach of the individual sensor.This paper presents an approach to multisensor data fusion in order to decrease data uncertainty withability to identify and handle inconsistency.The proposed approach relies on combining amodified Bayesian fusion algorithmwithKalman filtering. Three different approaches, namely, prefiltering, postfiltering and pre-postfiltering are described based on howfiltering is applied to the sensor data, to the fused data or both. A case study to find the position of a mobile robot by estimating itsx and y coordinates using four sensors is presented. The simulations show that combining fusion with filtering helps in handlingthe problem of uncertainty and inconsistency of the data.
1. Introduction
Multisensor data fusion is a multidisciplinary research areaborrowing ideas from many diverse fields such as signalprocessing, information theory, statistical estimation andinference, and artificial intelligence. This is indeed reflectedin the variety of the techniques reported in the literature [1].
Several definitions for data fusion exist in the literature.Klein [2] defines it by stating that data can be providedeither by a single source or by multiple sources. Data fusionis defined by Joint Directors of Laboratories (JDL) [3] asa “multilevel, multifaceted process handling the automaticdetection, association, correlation, estimation, and combi-nation of data and information from several sources.” Bothdefinitions are general and can be applied in different fieldsincluding remote sensing. In [4], the authors present a reviewand discussion of many data fusion definitions. Based onthe identified strengths and weaknesses of previous work, aprincipled definition of data fusion is proposed as the studyof efficient methods for automatically or semiautomaticallytransforming data from different sources and different points
in time into a representation that provides effective supportfor human or automated decision making.
Data fusion is applied in many areas of autonomoussystems. Autonomous systems must be able to perceive thephysical world and physically interact with it through com-puter-controlled mechanical devices. A critical problem ofautonomous systems is the imperfection aspects of the datathat the system is processing for situation awareness. Theseimperfection aspects [1] include uncertainty, imprecision,incompleteness, inconsistency, and ambiguity of the datathat may results in wrong beliefs about system state and/orenvironment state.Thesewrong beliefs can lead consequentlyto wrong decisions. To handle this problem, multisensor datafusion techniques are used for the dynamic integration ofthe multithread flow of data provided by a homogenous orheterogeneous network of sensors into a coherent picture ofthe situation.
This paper discusses how multisensor data fusion canhelp in handling the problem of uncertainty and incon-sistency as common imperfection aspects of the data inautonomous systems. The paper proposes an approach to
2 Advances in Artificial Intelligence
multisensor data fusion that relies on combining a modifiedBayesian fusion algorithm with Kalman filtering [5]. Threedifferent approaches, namely, prefiltering, postfiltering andpre-postfiltering are described based on how filtering isapplied to the sensor data, to the fused data, or both. Theseapproaches have been applied in a simulation to handle theproblem of data uncertainty and inconsistency in a mobilerobot as an example of an autonomous system.
The remainder of this paper is organized as follows:Section 2 reviews the most commonly used multisensor datafusion techniques followed by describing Bayesian and geo-metric approaches in Section 3. The proposed approaches ispresented in Section 4. A case study of position estimation ofamobile robot is discussed in Section 5 to show the efficacy ofthe proposed approaches. Finally conclusion and future workare summarized in Section 6.
2. Multisensor Data Fusion Approaches
Different multisensor data fusion techniques have been pro-posed with different characteristics, capabilities, and limita-tions. A data-centric taxonomy is discussed in [1] to showhow these techniques differ in their ability to handle differentimperfection aspects of the data.This section summarizes themost commonly used approaches to multisensor data fusion.
2.1. Probabilistic Fusion. Probabilistic methods rely on theprobability distribution/density functions to express datauncertainty. At the core of these methods lies the Bayesestimator, which enables fusion of pieces of data, hence, thename “Bayesian fusion” [6]. More details are provided in thenext section.
2.2. Evidential Belief Reasoning. Dempster-Shafer (D-S) the-ory introduces the notion of assigning beliefs and plausi-bilities to possible measurement hypotheses along with therequired combination rule to fuse them. It can be consideredas a generalization to the Bayesian theory that deals withprobability mass functions. Unlike the Bayesian Inference,the D-S theory allows each source to contribute informationin different levels of detail [6].
2.3. Fusion and Fuzzy Reasoning. Fuzzy set theory is anothertheoretical reasoning scheme for dealing with imperfect data.Due to being a powerful theory to represent vague data, fuzzyset theory is particularly useful to represent and fuse vaguedata produced by human experts in a linguistic fashion [6].
2.4. Possibility Theory. Possibility theory is based on fuzzyset theory but was mainly designed to represent incompleterather than vague data. Possibility theory’s treatment ofimperfect data is similar in spirit to probability and D-Sevidence theory with a different quantification approach [7].
2.5. Rough Set-Based Fusion. Rough set is a theory of imper-fect data developed by Pawlak [8] to represent imprecisedata, ignoring uncertainty at different granularity levels. Thistheory enables dealing with data granularity.
2.6. Random Set Theoretic Fusion. The most notable workon promoting random finite set theory (RFS) as a unifiedfusion framework has been done by Mahler in [9]. Com-pared to other alternative approaches of dealing with dataimperfection, RFS theory appears to provide the highestlevel of flexibility in dealing with complex data while stilloperating within the popular and well-studied framework ofBayesian inference. RFS is a very attractive solution to fusionof complex soft/hard data that is supplied in disparate formsand may have several imperfection aspects [10].
2.7. Hybrid Fusion Approaches. The main idea behind devel-opment of hybrid fusion algorithms is that different fusionmethods such as fuzzy reasoning, D-S evidence theory,and probabilistic fusion should not be competing, as theyapproach data fusion from different (possibly complemen-tary) perspectives [1].
3. Handling Uncertainty and Inconsistency
Combining data from several sources using multisensor datafusion algorithms exploits the data redundancy to reducethe uncertainty. However, if these several sources provideinconsistent data, catastrophic fusion may occur where theperformance of multisensor data fusion is significantly lowerthan the performance of each of the individual sensors. Thissection discusses four different approaches with differentlevel of complexity and ability to handle uncertainty andinconsistency.
3.1. Simplified Bayesian Approach (SB). Bayesian inference isa statistical data fusion algorithm based on Bayes’ theoremof conditional or a posteriori probability to estimate an n-dimensional state vector𝑋, after the observation ormeasure-ment 𝑍 has been made. Assuming a state-space represen-taion, the Bayes estimator provides a method for computingthe posterior (conditional) probability distribution/densityof the hypothetical state 𝑥𝑘 at time 𝑘 given the set ofmeasurements 𝑍𝑘 = {𝑧1; . . . ; 𝑧𝑘} (up to time 𝑘) and the priordistribution as follows:
𝑝 (𝑥𝑘 | 𝑍𝑘) =
𝑝 (𝑧𝑘 | 𝑥𝑘) 𝑝 (𝑥𝑘 | 𝑍𝑘−1
)
𝑝 (𝑍𝑘 | 𝑍𝑘−1), (1)
where
(i) 𝑝(𝑧𝑘|𝑥𝑘) is called the likelihood function and is basedon the given sensor measurement model.
(ii) 𝑝(𝑥𝑘|𝑍𝑘−1
) is called the prior distribution and incor-porates the given transition model of the system.
(iii) The denominator is a merely a normalizing term toensure that the probability density function integratesto one.
The probabilistic information contained in 𝑍 about 𝑋
is described by the probability density function 𝑝(𝑍|𝑋),which is a sensor dependent objective function based onobservation. The likelihood function relates the extent towhich a posteriori probability is subject to change and is
Advances in Artificial Intelligence 3
evaluated either via offline experiments or by utilizing theavailable information about the problem. If the informationabout the state 𝑋 is made available independently beforeany observation is made, then likelihood function can beimproved to provide more accurate results. Such a prioriinformation about𝑋 can be encapsulated as the prior proba-bility and is regarded as subjective because it is not based onobserved data.The information supplied by a sensor is usuallymodeled as a mean about a true value, with uncertaintydue to noise represented by a variance that depends onboth the measured quantities themselves and the operationalparameters of the sensor. A probabilistic sensor model isparticularly useful because it facilitates a determinationof the statistical characteristics of the data obtained. Thisprobabilistic model captures the probability distribution ofmeasurement by the sensor 𝑧 when the state of the measuredquantity 𝑥 is known. This distribution is extremely sensorspecific and can be experimentally determined. Gaussiandistribution is one of the most commonly used distributionsto represent the sensor uncertainties and is given by
𝑝 (𝑍 = 𝑧𝑗 | 𝑋 = 𝑥) =1
𝜎𝑗√2𝜋
exp{
{
{
−(𝑥 − 𝑧𝑗)2
2𝜎2
𝑗
}
}
}
, (2)
where 𝑗 represents the sensors. Thus, if there are two sensorsthat are modeled using (2), then from Bayes’ theorem thefused mean of the two sensors is given by the Maximum aposteriori (MAP) estimate as follows:
𝑥𝑓 =𝜎2
2
𝜎2
1+ 𝜎2
2
𝑧1 +𝜎2
1
𝜎2
1+ 𝜎2
2
𝑧2, (3)
where 𝜎1 is the standard deviation of sensor 1 and 𝜎2 is thestandard deviation of sensor 2. The fused variance given by
𝜎2
𝑓=
1
𝜎−2
1+ 𝜎−2
2
. (4)
3.2.Modified BayesianApproach (MB). Sensors often providedata which is spurious due to sensor failure or due tosome ambiguity in the environment. The simplified Bayesianapproach described previously does not handle the spuriousdata efficiently. The approach yields the same weighted meanvalue whether data from one sensor is bad or not, and theposterior distribution always has a smaller variance than thevariance of either of individual distributions beingmultiplied.This can be seen in (3). The simplified Bayesian does nothave a mechanism to identify if data from certain sensor isspurious and thus it might lead to inaccurate estimation. In[11], a modified Bayesian approach has been proposed whichconsiders measurement inconsistency.
Consider
𝑝 (𝑋 = 𝑥 | 𝑍 = 𝑧1, 𝑧2) ∝1
𝜎1√2𝜋
exp{−(𝑥 − 𝑧1)
2
2𝜎2
1𝑓
}
×1
𝜎2√2𝜋
exp{−(𝑥 − 𝑧2)
2
2𝜎2
2𝑓
} .
(5)
As shown in [11], modification observed in (5) causes anincrease in the variance of the individual distribution by afactor given by
𝑓 =𝑀2
𝑀2 − (𝑧1 − 𝑧2)2. (6)
The parameter 𝑀 is the maximum expected differencebetween the sensor readings. Larger difference in the sensormeasurements causes the variance to increase by a biggerfactor. The MAP estimate of state 𝑥 remains unchanged butthe variance of the fused posterior distribution changes.Thus,depending on the squared difference in measurements fromthe two sensors, the variance of the posterior distributionmayincrease or decrease as compared to the individual Gaussiandistributions that represent the sensor models.
The difference between the simplified and the modifiedBayesian can be seen in Figures 1 and 2. In this example,there are two sensors, where sensor 1 has a standard deviationof 2 and sensor 2 has a standard deviation of 4. In the firstcase shown in Figure 1, the two sensors are in agreement. Itcan be seen that fused posterior distribution obtained fromthe proposed strategy has a lower value of variance than thatof each of the distributions being multiplied indicating thatfusion leads to a decrease in posterior uncertainty.
In the second case in Figure 2, the two sensors are indisagreement.The fused posterior distribution obtained fromthe modified Bayesian has a larger variance as compared tothe variance of both sensors. However, the fused variance dueto the simplified Bayesian was the same as the fused variancein Figure 1. This concluded, as in [11], that the modifiedBayesian was highly effective in identifying inconsistencyin sensor data and thus reflecting the true state of themeasurements.
3.3. Geometric Redundant Fusion (GRF). TheGRFmethod isanother method used for fusing uncertain data coming frommultiple sensors. The fusing of 𝑚 uncertain 𝑛-dimensionaldata points is based on a weighted linear sum of the measure-ments as follows:
𝑥𝑓 =
𝑚
∑
𝑖=1
𝑊𝑖𝑧𝑖, (7)
where 𝑥𝑓 is the fused value and 𝑊𝑖 is the weighting matrixfor measurement 𝑖 and 𝑧𝑖. Applying expected values to (7)and assuming no measurement bias yields the followingcondition:
𝑚
∑
𝑖=1
𝑊𝑖 = 𝐼. (8)
For a given set of data, the weighting matrix, 𝑊, and thecovariance matrix, 𝑄, are formed as follows:
𝑊 = [𝑊1 𝑊2 ⋅ ⋅ ⋅ 𝑊𝑚] ,
4 Advances in Artificial Intelligence
0.25
0.2
0.15
0.1
0.05
00 10 20 30 40 50
Prob
abili
ty
Sensor 1Sensor 2
SBMB
x
Figure 1: Two sensors in agreement.
0.25
0.2
0.15
0.1
0.05
00 10 20 30 40 50
Prob
abili
ty
Sensor 1Sensor 2
SBMB
x
Figure 2: Two sensors in disagreement.
𝑄 = (
𝜎2
1⋅ ⋅ ⋅ 0
... d...
0 ⋅ ⋅ ⋅ 𝜎2
𝑚
),
(9)
where 𝜎2𝑖is the uncertainty ellipsoid matrix for measurement
𝑖. Using Lagrange multipliers the following results for theweighting matrices and fused variance are obtained:
𝑊𝑖 = (
𝑚
∑
𝑖=1
𝜎−2
𝑖)
−1
𝜎−2
𝑖,
𝜎2
𝑓= (
𝑚
∑
𝑖=1
𝜎−2
𝑖)
−1
.
(10)
In the one-dimensional case with two measurements thefused result becomes
𝑥𝑓 =𝜎2
1𝑧2 + 𝜎
2
2𝑧1
𝜎2
1+ 𝜎2
2
,
𝜎2
𝑓=
1
𝜎−2
1+ 𝜎−2
2
.
(11)
While the GRF method handles the fusion of 𝑚 mea-surements in 𝑛-dimensional space in an efficient manner,it does not include information about the spacing of themeans of the𝑚measurements in the calculation of the fusedvariance. That is, the magnitude of the spatial separation ofthe means is not used in the calculation of the uncertaintyof the fused result. Figures 3 and 4 shows fusing of two one-dimensional measurements using GRF. It can be observedthat the uncertainty of the GRF remained the same regardlessof the separation or the inconsistency of the measurements.
Therefore, the GRF method provides a fused result withidentical uncertainty independent of whether the measure-ments have identical or highly spatially separated means. Toovercome this problem, a heuristic method was developed toconsider the level and direction of measurement uncertaintyas reflected in the level and direction of disparity between theoriginal measurements. Thus, the output is no longer purelystatistically based but can still provide a reasonable measureof the increase or decrease of uncertainty of the fused data.
3.4. Heuristic Geometric Redundant Fusion (HGRF). Thedesired heuristic wouldmodify the GRF result so that reliableresults are produced for all ranges of measurement dispar-ity or inconsistency. It will contain information about theseparation of the mean of the measurements when findingthe uncertainty of the fused result [12]. The following casesshowhow theHGRF result changes as the separation betweenthe measurements increase. The uncertainty of each mea-surement and of the fused result is shown in the form of anellipsoid.
Figure 5 shows two measurements with no separation orconsistency. For this case, the HGRF uncertainty region isequivalent to the uncertainty region generated by the GRFmethod.
Figure 6 shows two measurements that somehow par-tially agree. The HGRF uncertainty region increased relativeto the GRF which is the same as in Figure 5; however, theHGRF uncertainty decreased relative to the two sensorsuncertainty.
Figure 7 shows two measurements in disagreement orinconsistent. The HGRF uncertainty region is larger than theuncertainty of each sensor and thus the uncertainty increases.It can be observed that the GRF uncertainty remained thesame as in the previous cases.
Figure 8 shows two measurements completely separatedor inconsistent which indicates measurement error. Theresulting HGRF uncertainty ellipsoid covers the entire rangeof the measurement ellipsoids along the dimensions of mea-surement error. In this case, the increase in the uncertaintyindicates the occurrence of measurement error.
Advances in Artificial Intelligence 5
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
00 5 10 15 20 25
Sensor 1Sensor 2
x
GRF
f(x)
Figure 3: One-dimensional fusion of two measurements in agree-ment.
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
00 5 10 15 20 25 30
Sensor 1Sensor 2
x
GRF
f(x)
Figure 4: One-dimensional fusion of two measurements in dis-agreement.
Sensor 1 Sensor 2
HGRF GRF
Figure 5: No separation between measurements.
Sensor 1 Sensor 2
HGRFGRF
Figure 6: Measurements agree.
Sensor 1 Sensor 2
HGRF GRF
Figure 7: Measurements disagree.
Sensor 1 Sensor 2
HGRFGRF
Figure 8: Measurement error.
3.5. Comparative Study. To evaluate the difference betweenthe four approaches mentioned previously, a comparativestudy was carried out. This study considers a mobile robotmoving in a straight line with a constant velocity of 7.8 cm/s.The position of the robot is tracked using two measurementscoming from two sensors: optical encoder and Hall-effectsensor. To detect the position of the robot, the readingscoming from the sensors are being fused using: SB andMB aswell as GRF andHGRFmethods, during 20 seconds of travel.The standard deviations of optical encoder and Hall-effectsensor are 2.378 cm and 2.260 cm, respectively.Therefore, themeasurements coming from the optical encoder have a higheruncertainty.
Figure 9 shows the uncertainty curves of the sensors aswell as the fused results at 𝑡 = 10 sec. It can be observed thatthe variance of the GRF and SB were the same and thus theuncertainty curves were completely overlapping. However,the HGRF result followed the heuristic specification and cov-ered the uncertainty curves of the measurements. Comparedto the SB and the GRF, the MB showed a response to theseparation or inconsistency of the measurements, however,not as much as the HGRF. The fused mean of the fourapproaches was exactly the same value of 78.4 cm, while themean value of the optical encoder and Hall-effect sensor was75.4 cm and 79.1 cm, respectively. The fused mean is morebiased towards the accurate reading of Hall-effect sensor.
To compare the different approaches in terms of thecalculation time, Figure 10 shows how the running time ofeach algorithm changes over 20 seconds. It can be seen thatthe HGRF method is always taking longer time compared tothe other methods.The other approaches have approximatelythe same running time.
6 Advances in Artificial Intelligence
0.7
0.6
0.5
0.4
0.3
0.2
0.1
072 74 76 78 80 82 84 86 88
x
Prob
abili
ty
Optical encoderHall-effectSB
MBGRFHGRF
70
Figure 9: Uncertainty curves.
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
00 5 10 15 20
Time (s)
CPU
runn
ing
time (
ms)
SBMB
GRFHGRF
Figure 10: Time taken to perform each algorithm throughout sim-ulation time.
It can be observed from Figure 11 that the HGRF showsa very big change in the fused variance due to its high sen-sitivity to measurement inconsistency. The MB also showeda response to the separation inconsistency between themeasurements; however, it was not as great as the HGRF.Thefused variance using both the SB and the GRF was exactly thesame as seen by the perfectly overlapping curves, and the vari-ance value did not change throughout the simulation time.
The error curves in Figure 12 show that the fusion resultis generally more accurate with less errors compared to theerror caused by the uncertainty of each measurement. The
Time (s)
102
101
100
10−1
0 5 10 15
Fuse
d va
rianc
e (cm
2)
SBMB
GRFHGRF
20
Figure 11: Trend of the fused variance throughout simulation time.
4
3.5
3
2.5
2
1.5
1
0.5
00 5 10 15 20
Time (s)
Abso
lute
pos
ition
erro
r (cm
)
Optical encoderHall-effectFusion
Figure 12: Measurement and fusion errors.
optical encoder shows a higher range of error due to its higheruncertainty than the Hall-effect sensor.
Table 1 summarizes the main differences between eachof the four multisensor data fusion approaches. In general,the MB outperforms all the other approaches in terms ofaccuracy, time, and variance change.
4. Proposed Approach
The proposed approach to multisensor data fusion in thispaper relies on combining a modified Bayesian fusion algo-rithm with Kalman filtering. The main reason for using themodified Bayesian fusion is its efficient performance that is
Advances in Artificial Intelligence 7
Table 1: Comparison between different fusion techniques.
Fused variance Running time Trend of variance Fused meanSB Not affected by inconsistency Medium No change
Same for all methodsMB Deals with data inconsistency Short Smooth changeGRF Not affected by inconsistency Shortest No changeHGRF Deals with data inconsistency Longest Big change
Unc
erta
in an
d in
cons
isten
t
Kalman
Kalman
Kalman
Sensor-1
Sensor-2 ModifiedEstimateof feature
z1
z2
zn
xf
mea
sure
men
ts
filtering
filtering
filtering Bayesian fusion
Sensor-n
...
Figure 13: Modified Bayesian fusion with prefiltering.
proved in the previous section. Three different techniques,namely, prefiltering, postfiltering and pre-postfiltering aredescribed in the following subsections based on how filteringis applied to the sensor data, to the fused data, or both.
4.1. Modified Bayesian Fusion with Prefiltering (F-MB). Thefirst proposed technique is the prefiltering (F-MB) whichinvolves adding Kalman filters before the modified Bayesianfusion node. As illustrated in Figure 13 and shown inAlgorithm 1, Kalman filter is added to every sensor tofilter out the noise from the sensor measurements. Thefilteredmeasurements are then fused together usingmodifiedBayesian to get a single result that represents the state at aparticular instant of time.
4.2. Modified Bayesian Fusion with Postfiltering (MB-F). Thesecond proposed technique is the postfiltering (MB-F) whichinvolves adding a Kalman filter after the fusion node in orderto filter out the noise from the fused estimate as shownin Algorithm 2 and illustrated in Figure 14. The secondproposed technique is to add Kalman filter after the fusionnode which fuses themeasurements usingmodified Bayesianto produce 𝑥int. Kalman filtering is then applied to thefused state 𝑥int in order to filter out the noise, as shown inAlgorithm 2 and illustrated in Figure 14. The output of theKalman filter represents the state 𝑥𝑓 at a particular instantof time as well as the variance of the estimated fused state 𝑃𝑓.
4.3. Modified Bayesian Fusion with Pre- and Postfiltering (F-MB-F). In this technique, Kalman filter is applied before andafter the fusion node as illustrated in Figure 15.The algorithmof this technique is the integration of Algorithms 1 and 2, asshown in Algorithm 3.
5. A Case Study: Mobile RobotLocal Positioning
In this section, a case study of position estimation of a mobilerobot is presented to proof the efficacy of the proposedapproach. Mobile robot positioning provides answer forthe question: “Where the robot is?” Positioning techniquessolutions can be roughly categorized into relative positionmeasurements (dead reckoning) and absolute position mea-surements. In the former, the robot position is estimated byapplying to a previously determined position the course anddistance traveled since. Later, the absolute position of therobot is computed frommeasuring the direction of incidenceof three or more actively transmitted beacons or usingartificial or natural landmarks or using model matching,where features from a sensor-basedmap and the worldmodelmap are matched to estimate the absolute location of therobot [13].
5.1. Simulation. To test the proposed approaches, a simula-tion was carried out using MATLAB, where it was requiredto locate the position of the robot by finding its x and ycoordinates. It was assumed that two sensors were used tofind how much the robot has moved in the x and anothertwo sensors to find how much the robot has moved in they. The sensors used in the simulation were assumed to haveuncertaintymodelled as a white Gaussian noise.The standarddeviations of the sensors used to find the x-coordinatesof the robot are 4.3 cm and 6.8 cm. However, the standarddeviations of the sensors used to find the y-coordinates are4.5 cm and 6.6 cm.
In addition, it was assumed that the robot is moving witha speed of 7.8 cm/sec in the x and 15.6 cm/sec in the y. Thesampling time is 0.5 sec and the robot was simulated to movefor 20 sec.
Algorithm 3: The pre- and postfiltering algorithm (F-MB-F).
Unc
erta
in an
d in
cons
isten
tm
easu
rem
ents
Kalmanfiltering
ModifiedBayesian fusion
Sensor-1
Sensor-2 Estimateof feature
z1
z2
zn
xf
Sensor-n
...
Figure 14: Modified Bayesian fusion with postfiltering.
Advances in Artificial Intelligence 9
Unc
erta
in an
d in
cons
isten
tm
easu
rem
ents
Kalmanfiltering
Kalmanfiltering
Kalmanfiltering
ModifiedBayesian fusion
Sensor-1
Sensor-2Estimateof feature
z1
z2
zn
xf
Kalmanfiltering
...
Sensor-n
Figure 15: Modified Bayesian fusion with pre- and postfiltering.
5.2. Evaluation Metrics. The performance of the algorithmswas evaluated based on five criteria.
5.2.1. CPURunningTime. This represents the total processingtime of the algorithm to estimate the position of the robotthroughout the travelling time. It is desired to minimize thisrunning time.
5.2.2. Residual Sum of Squares (RSS). This represents thesummation of the squared difference between the theoreticalposition of the robot and the estimated state at each time sam-ple.The smaller the RSS, themore accurate the algorithmwillbe because thismeans that the estimated position of the robotis getting closer to the theoretical position. This is given by
RSS =
𝑛
∑
𝑖=1
(postheoretical,𝑖 − posestimated,𝑖)2. (12)
5.2.3. Variance (P). This represents the variance of the esti-mated position of the robot. The variance will reflect theperformance of the filters in each algorithm.
5.2.4. Coefficient of Correlation. This is a measure of associ-ation that shows how the state estimate of each technique isrelated to the theoretical state. The coefficient of correlationwill always lie between −1 and +1. For example, a correlationclose to +1 indicates the two data are very strongly positivelycorrelated [14].
5.2.5. Criterion Function (CF). A computational decisionmaking method was used to calculate a criterion functionthat is a numerical estimate of the utility associated witheach of the three proposed techniques. A weighting function𝑤 (from 0 to 1) will be defined for each criterion (time,RSS, and variance), depending on its importance. The threeweights should sum up to 1.The cost value 𝑐 (calculated fromthe experiments) of each technique is obtained and finallyCF is calculated as the weighted sum of the utility for eachtechnique as follows:
CF = 𝑤1 ×𝑐1
𝑐1max+ 𝑤2 ×
𝑐2
𝑐2max+ 𝑤3 ×
𝑐3
𝑐3max, (13)
where 𝑤1, 𝑤2, and 𝑤3 are the weights of the time, RSS, andvariance, respectively. These weights are adjusted accordingto the application and desire of the user. In this case study,
Abso
lute
erro
r inx
(mm
)
3
2.5
2
1.5
1
0.5
00 5 10 15 20
Measurement 1Measurement 2MB-F
F-MB-F
Time (s)
F-MB
Figure 16: Measurements and estimated errors to find the x-coordi-nates of the moving robot.
it was assumed that 𝑤1 = 1/3, 𝑤2 = 1/3, and 𝑤3 = 1/3.The values 𝑐1, 𝑐2, and 𝑐3 are the values obtained from theexperiments for the time, RSS, and variance, respectively.The values 𝑐1max, 𝑐2max, and 𝑐3max represent the maximumvalue achieved in each of the criteria: time, RSS, and variance,respectively. The objective is to minimize this function suchthat the algorithm will produce accurate estimates in a shorttime with a minimum variance.
5.3. Results and Discussion. A simulation of 5000 iterationswas carried out to compare the results of the proposed algo-rithms using MATLAB. Figure 16 shows the errors that areproduced due to the noisy uncertain measurements obtainedfrom the sensors. In addition, the figure shows the errors thatare produced due to estimating the x-coordinates of the robotusing the proposed three algorithms.The error represents thedifference between the theoretical state and the output state ofeach algorithm at a particular sample time.Themeasurementerrors are high compared to the estimation errors.
10 Advances in Artificial Intelligence
3
2.5
2
1.5
1
0.5
00 5 10 15 20
Measurement 1Measurement 2MB-F
Time (s)
F-MB-FF-MB
Abso
lute
erro
r iny
(mm
)
Figure 17: Measurements and estimated errors to find the y-coordi-nates of the moving robot.
Figure 17 shows the errors that are produced due to thenoisy uncertain measurements obtained from the sensors aswell as the errors of the proposed three algorithms to getthe y-coordinates of robot. Similar to the results shown inFigure 16, the measurement errors are high compared to theestimation errors.Thus, the first interpretation of these resultsis that the proposed techniques provide better estimates thanrelying on the measurements directly.
Table 2 summarizes the average results of 5000 runsfor the three evaluation metrics described previously. Theminimal value in each criteria has been bolded. AlthoughMBtakes the minimal execution time yet it has the maximumRSS value compared to other techniques and this is clear inthe CF shown in Table 2. Figure 18 shows the CPU runningtime of each algorithm. It is clear that MB takes the shortestcalculation time while F-MB-F takes the longest time.
Figure 19 shows the estimated variance of each technique.It is clear that the variance of the MB-F and F-MB-Fconverged earlier to lower values than MB and F-MB. Thisproves the efficiency of the Kalman filters in MB-F and F-MB-F. Table 2 shows that F-MB-F has a smaller variance thanMB-F. Moreover, the proposed techniques estimates were allpositively and strongly correlated to the theoretical valueswith a correlation coefficient of +0.99.
It can be seen from Figure 20 that MB-F has outper-formed the other techniques, followed by F-MB-F. The leastperformance was observed in MB and thus this provesthat combining fusion with Kalman filtering improves theestimation of the states.
6. Conclusion
Three techniques for multisensor data fusion have beendiscussed in this paper. These techniques combine a modi-fied Bayesian data fusion algorithm with pre- and postdata
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
00 5 10 15 20
MBMB-F F-MB-F
Time (s)
F-MB
CPU
runn
ing
time (
ms)
Figure 18: Time taken to perform each algorithm throughout sim-ulation time.
0 5 10 15 20
20
18
16
14
12
10
8
6
4
2
0
MBMB-F F-MB-F
Time (s)
F-MB
Estim
ated
var
ianc
e (cm
2)
Figure 19: Estimated variance.
filtering in order to handle the uncertainty and inconsistencyproblems of sensory data. A simulation has been carriedout to estimate the position of the robot by applying theproposed techniques to find the x and y coordinates. Thesimulation results proved that combining fusionwith filteringimproves residual sum of squares and variance of the fusedresult. For future research, the case study presented in thispaper will be applied for accurate real-time 2D localizationof a newly built differential drive mobile robot equipped withoptical encoders and Hall-effect sensors for relative positionmeasurements.
Advances in Artificial Intelligence 11
Table 2: Computational decision making chart for the fusion techniques using 2 sensors.
Fusion techniques Criteria 𝑐1, 𝑐2, 𝑐3, respectively Criterion function
As the authors of the paper, we confirm that we do not have adirect financial relation with the commercial identities men-tioned in our paper that might lead to any conflict of interest.
References
[1] B. Khaleghi, A. Khamis, F. O. Karray, and S. N. Razavi, “Multi-sensor data fusion: a review of the state-of-the-art,” InformationFusion, vol. 14, no. 1, pp. 28–44, 2013.
[2] L. A. Klein, “Sensor and data fusion concepts and applications,”in Society of Photo-Optical Instrumentation Engineers (SPIE),1993.
[3] U.S. Department of Defense, “Data Fusion Subpanel of the JointDirectors of Laboratories,” Technical Panel for C3., Data Fusionlexicon, 1991.
[4] H. Bostrm, S. Andler, M. Brohede et al., “On the definition ofinformation fusion as a field of research,” Tech. Rep., Informa-tics Research Centre, University of Skvde, 2007.
[5] G. Welch and G. Bishop, An Introduction to the Kalman Filter,Department of Computer Science, University ofNorthCarolinaat Chapel Hill, 2006.
[6] H.Durrant-Whyte andT.Henderson, Sensor Fusion, chapter 25,Springer, New York, NY, USA, 2008.
[7] J. Llinas, C. Bowman, G. Rogova, A. Steinberg, E. Waltz, andF. White, “Revisiting the JDL data fusion model II,” in Proceed-ings of the 7th International Conference on Information Fusion(FUSION ’04), pp. 1218–1230, July 2004.
[8] Z. Pawlak, Rough Sets: Theoretical Aspects of Reasoning aboutData, Kluwer Academic, Norwell, Mass, USA, 1992.
[9] R. Mahler, Statistical Multisource-Multitarget InformationFusion, Artech-House, Boston, Mass, USA, 2007.
[10] B. Khaleghi, A. Khamis, and F. Karray, “Random finite settheoretic based soft/hard data fusion with application for targettracking,” in Proceedings of the IEEE International Conference onMultisensor Fusion and Integration for Intelligent Systems (MFI’10), pp. 50–55, September 2010.
[11] D. P. Garg, M. Kumar, and R. A. Zachery, “A eneralizedapproach for inconsistency detection in data fusion frommulti-ple sensors,” in Proceedings of the American Control Conference,pp. 2078–2083, June 2006.
[12] J. D. Elliott, Multisensor fusion within an encapsulated logicaldevice architecture [M.S. thesis], University of Waterloo, 1999.
[13] A. Khamis, “Lecture 3—Mobile Robot Positioning. MCTR1002:Autonomous Systems,”Mechatronics EngineeringDepartment,German University in Cairo, 2011-2012.
[14] G. Keller and B. Warrack, Statistics For Management and Eco-nomics, Duxbury Press, 4th edition, 1997.
