Measurement 46 (2013) 3737–3744

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Measurement

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Bayes filter for dynamic coordinate measurements – Accuracyimprovment, data fusion and measurement uncertaintyevaluation

0263-2241/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.measurement.2013.04.001

⇑ Corresponding author. Tel.: +49 91318526548.E-mail address: garcia@fmt.fau.de (E. Garcia). 1 Neither the GUM nor the GS1 deal with dynamic measureme

E. Garcia a,⇑, T. Hausotte a, A. Amthor b

a Institute of Manufacturing Metrology, University Erlangen-Nuremberg, Germanyb Siemens AG, Erlangen, Germany

a r t i c l e i n f o

Article history:Available online 15 April 2013

Keywords:Dynamic coordinate measurementsBayesian filteringParticle filterSequential Monte Carlo methodsOnline measurement uncertainty evaluationand data fusion

a b s t r a c t

This paper presents a novel methodology to improve the measurement accuracy ofdynamic measurements. This is achieved by deducing an online Bayes optimal estimateof the true measurand given uncertain, noisy or incomplete measurements within theframework of sequential Monte Carlo methods. The estimation problem is formulated asa general Bayesian inference problem for nonlinear dynamic systems. The optimal estimateis represented by probability density functions, which enable an online, probabilistic datafusion as well as a Bayesian measurement uncertainty evaluation corresponding to the‘‘Guide to the expression of uncertainty in measurement‘‘. The efficiency and performanceof the proposed methodology is verified and shown by dynamic coordinate measurements.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Traditional coordinate measuring machines (CMMs)were increasingly completed or superseded by mobile,dynamic coordinate measuring machines (DMMs), suchas laser tracker or photogrammetry systems (i.e. camera-based systems). They allow non-contact, in situ and onlinecoordinate measurements of complex moving objects overlarge measurement volumes along with excellent precisionand high measurement rate directly in the shop-floor [1,2].

Typical applications are dynamic motion measurements(traveled paths, positions, displacements, velocities, accel-erations, jerks, orientations and vibrations), guidance andcalibration of machines, robot trajectory measurements,deformation analysis as well as assembly and inspectionof parts and structures.

Often, DMM are portable optical measurement systems,which perform online shop-floor measurements, mostly inunstable environments. However, it is in general not possi-ble to model, quantify or compensate significant error

sources, such as optical aberrations (e.g. reflection errors,changing refractive index), imaging errors or dynamic er-rors (e.g. motion artifacts), during dynamic measurements.As a result the obtained measurements exhibit a lower sig-nal-to-noise ratio and reduced measurement accuracy.Furthermore it is not possible to evaluate the measure-ment uncertainty according to the ‘‘Guide to the expres-sion of uncertainty in measurement’’ (GUM) due to themodel imperfection and dynamic characteristic of themeasurement process.1

To overcome these shortcomings and to extend theapplication spectrum of DMM, this paper presents a meth-odology to increase the measurement accuracy by deduc-ing the Bayes optimal estimate of the true measurandgiven uncertain, noisy or incomplete measurements inreal-time. In every time step the measurand is representedby a probability density function (PDF). Thus, the Bayesianframework enables an online, probabilistic data fusion aswell as measurement uncertainty evaluations correspond-ing to the GUM.

nts.

3738 E. Garcia et al. / Measurement 46 (2013) 3737–3744

The rest of the paper is structured as follows. In Section2 the basic principles of measurement uncertainty evalua-tion as described in the GUM and in the ‘‘Supplement 1 tothe GUM’’ (GS1) [3,4] are briefly restated. Section 3 intro-duces the Bayesian inference and the probabilistic modelof the measurement process using the state-space repre-sentation. The Bayesian estimation problem is formulatedand a general solution to this problem within the frame-work of sequential Monte Carlo methods is devised. Sec-tion 4 presents experimental results of Bayes filterapplied to dynamic coordinate measurements. In the enda summary and conclusion are given.

2 The state transition function is also named system, process, dynamic,evolution or likelihood function.

2. Basic principles of uncertainty evaluation

A measurement result is generally expressed as a singlemeasured quantity value and a measurement uncertainty[5, definition 2.9]. The de facto international standards toevaluate, calculate and express uncertainty in all kinds ofmeasurements are the GUM and its extension the GS1[3,4]. These documents provide guidance on the uncer-tainty evaluation as a two-stage process: formulation andcalculation. The formulation stage involves developing ameasurement model relating output (measurand) y to in-put quantities x1, . . .,xN, incorporating corrections andother effects as necessary and finally assigning probabilitydistributions to the input quantities on the basis of avail-able knowledge. The calculation stage consists of propagat-ing the probability distributions for the input quantitiesthrough the measurement model y = f(x1, . . .,xN) to obtainthe probability distribution for the output quantity. Fromthis PDF the expected value and the standard deviationof the output quantity as well as the coverage intervalare calculated. Several approaches are considered for thepropagation of distributions:

(a) the GUM uncertainty framework, constituting theapplication of the law of propagation of uncertainty,

(b) analytic methods, in which mathematical analysis isused to derive an algebraic form for the probabilitydistribution for the output quantity and

(c) a Monte Carlo method (MCM), in which an approxi-mation to the distribution function for the outputquantity is established numerically by making ran-dom draws from the probability distributions forthe input quantities, and evaluating the model atthe resulting values.

Any of these procedures can be applied in manysituations and leads to valid statements of uncertainty.But they also exhibit their respective merits and draw-backs as well as application limitations. The GUM uncer-tainty framework is a general approximation that mightnot be satisfactory for nonlinear measurement functions,asymmetric or non-Gaussian probability distributions forthe input or output quantities or significantly differentsensitivity of the uncertainty of input quantities. The ana-lytic methods theoretically yield to exact solutions, but inpractice it is often not possible to analytically derive analgebraic form for the probability distribution of the

output quantity. Finally, the Monte-Carlo methods are onlyexact within the limits of the numerical accuracy and comeup with the burden of computational complexity. In orderto not go beyond the scope of this paper, it is referred toGUM and GS1 [3,4] for further details of the propagationof distribution.

In summary, the GUM and the GS1 specify a method tocalculate the measurement uncertainty by using the prop-agation of probability distributions through a mathemati-cal model of the measurement to derive the resultingprobability distribution for the output quantity. The mea-surement uncertainty is evaluated based on this PDF andstated as uncertainty budget, which should include themeasurement model, estimates, and measurement uncer-tainties associated with the quantities in the measurementmodel, covariances, type of applied probability densityfunctions, degrees of freedom, type of evaluation of mea-surement uncertainty and the coverage interval. Hereby,the method of choice would be an analytical solution ofthe output density function. But as this is not possible ingeneral, especially for nonlinear dynamic processes,sequential Monte Carlo methods are utilized to achieve aBayes optimal estimation of the output density, as de-scribed in the next section.

3. Bayesian inference

A measurement process generates disperse (uncertain)observations of a true (but unknown) measurand. This isan erroneous transformation due to ubiquitous errors.Thus, a complete, exact or unique reconstruction of thetrue measurand is always impossible. But using Bayes’ the-orem it is possible to estimate the true measurand fromobservations, since the functional relation between bothis preserved.

This estimation problem can be solved as a sequentialprobabilistic inference problem for nonlinear dynamic sys-tems. The unknown measurand (e.g. coordinates) is mod-elled as a state of a dynamic system (e.g. moving object)that evolves over time and is observed with a particularmeasurement process. This allows the estimation of mea-surable states (e.g. position) as well as derived or not di-rectly measurable states (e.g. velocities, acceleration andorientations) of arbitrary dynamic systems in a statisticallysound way given uncertain, noisy or incompleteobservations.

The nonlinear dynamic system is described by the gen-eral discrete-time stochastic state space model

xk ¼ f kðxk�1;vk�1Þ;yk ¼ hkðxk;wkÞ;

ð1Þ

where xk 2 Rnx is the hidden system state vector withdimension nx evolving over time k 2 N (discrete time in-dex) according to the possibly nonlinear state transitionfunction2 f(�), the state xk�1 and the process noise vk�1.The process noise is used to describe uncertainties or mis-modeling effects in the process model. The observations

1k−y ky 1k+y

1k−x kx 1k+x

1( | )k kp −x x

( | )k kp y x

1k − k 1k +

Measurements / Observations(observable)

Hidden states(unobservable and must be estimated)

Time

Fig. 1. First order hidden Markov model.

1( , )

( , )k k k

k k k

−==

x f x v

y h x w

1:1:

1:

Bayesian inference( | ) ( )

( | )( )

k k kk k

k

p pp

p=

y x xx y

y

kxk

ySystem or Process

Fig. 2. Bayesian inference problem for dynamic systems.

E. Garcia et al. / Measurement 46 (2013) 3737–3744 3739

yk 2 Rny with dimension ny are related to the state vector viathe nonlinear observation function h(�) and the measure-ment or observation noise wk, which is corrupting the obser-vation of the state. The dynamic equation f(�) describes theprocess dynamic or objects motion and the measurementequation h(�) defines the mapping between object statesand measurements. This state space model corresponds toa first order hidden Markov model [6,7] with an initial prob-ability density p(x0), the state transition probability p(xk|xk-�1) and the observation probability density p(yk|xk). Both thestate transition density and the observation likelihood arefully specified by f(�) and the process noise probabilityp(vk) and by h(�) and the observation noise distributionp(wk), respectively. The stochastic state-space model, to-gether with the known statistics of the noise random vari-ables as well as the prior distributions of the systemstates, defines a probabilistic model of how the systemsevolves over time and how we inaccurately observe the evo-lution of this hidden state (see Fig. 1).

The objective is to estimate the hidden system states xk

in a recursive fashion (i.e. sequential update of previousestimate) as measurement yk becomes available.3 This isthe central issue of the sequential probabilistic inferencetheory. From a Bayesian perspective, the complete solutionto this problem is given by the conditional posterior densityp(xk|y1:k) of the state xk, taking all observations y1:k ={y1,y2, . . .,yk} into account (see Fig. 2). The knowledge ofthe posterior allows one to calculate an optimal stateestimate with respect to any criterion. For example theminimum mean-square error estimate is the conditionalmean x̂k ¼ E½xkjy1:k� ¼

Rxk pðxkjy1:kÞdxk. Other estimates,

3 This is also referred to as online, sequential or iterative filtering.

such as median, modes and confidence intervals, can alsobe computed from the posterior. That enables the Bayesianevaluation of the measurement uncertainty correspondingto the GUM and its supplement.

The computation of the aforementioned Bayes posteriordensity requires the computation with all measurementsy1:k in one step (batch processing). A method to recursivelyupdate the posterior density as new observations arrive isgiven by the recursive Bayesian estimation algorithm,which is faster and allows an online processing of datawith lower storage costs and a quick adaption to changingdata characteristics.

Using the Bayes theorem and the discrete-time stochas-tic state space model (1), the posterior density can be de-rived and factored into the following recursive update form

pðxkjy1:kÞ ¼pðy1:kjxkÞpðxkÞ

pðy1:kÞ¼ pðykjxkÞpðxkjy1:k�1Þ

pðykjy1:k�1Þ: ð2Þ

The computation of the Bayesian recursion essentiallyconsists of two steps: the prediction step p(xk�1|y1:k�1) ?p(xk|y1:k�1) and the update step p(xk|y1:k�1, yk) ? p(xk|y1:k).The prediction step involves using the process function andthe previous posterior density4 p(xk�1|y1:k�1) to calculatethe prior density of the state xk at time k via the Chap-man-Kolmogorov equation

pðxkjy1:k�1Þ ¼Z

pðxkjxk�1Þpðxk�1jy1:k�1Þdxk�1; ð3Þ

where the state transition PDF is given by5

pðxkjxk�1Þ ¼Z

dðxk � f ðxk�1;vkÞÞpðvkÞdvk: ð4Þ

The prior of the state xk at time k does not incorporatethe latest measurement yk, i.e. the probability for xk is onlybased on previous observations. When the new uncertainmeasurement yk becomes available the update step is car-ried out and the posterior PDF of the current state xk iscomputed from the predicted prior (3) and the new mea-surement via the Bayes theorem (2), where the observationlikelihood PDF and the normalizing constant in the denom-inator are given by

pðykjxkÞ ¼Z

dðyk � hðxk;wkÞÞpðwkÞdwk ð5Þ

and pðykjy1:k�1Þ ¼Z

pðykjxkÞpðxkjy1:k�1Þdxk: ð6Þ

Eqs. (2)–(6) formulate the Bayesian solution to recur-sively estimate the unknown system states (measurands)of a nonlinear dynamic system from uncertain, noisy orincomplete measurements. But this is only a conceptualsolution in the sense that in general the integrals cannotbe determined analytically. Furthermore, the algorithmicimplementation of this solution requires the storage ofthe entire (non-Gaussian) posterior PDF as an infinitedimensional matrix, because generally it cannot becompletely described by a sufficient statistic of finite

4 At time k = 1 the previous posterior is the initial density of the statevector, i.e. pðx0j£Þ ¼ pðx0Þ.

5 d(�) is the Dirac-delta function.

Fig. 3. Experimental setup consisting of close range photogrammetricCMM and high accuracy parallel kinematic (hexapod).

Table 1Error in position measurements averaged over three independent runs forv = 1, 3, 5 mm/s. The position error is defined as Euclidean distance in

3740 E. Garcia et al. / Measurement 46 (2013) 3737–3744

dimension. Only in some restricted cases a closed-formrecursive solution is possible. For example with restrictionto linear, Gaussian systems a closed-form recursive solu-tion is given by the famous Kalman filter [8]. In most situ-ations, either both the dynamic and the measurementprocess or only one of them are nonlinear. Thus the mul-ti-dimensional integrals are not tractable and approxima-tive solutions such as sequential Monte Carlo methodsmust be used [9]. They make no explicit assumption aboutthe form of the posterior density and approximate theBayesian integrals with finite sums. These methods havethe advantage of not being subject to the curse of dimen-sionality as well as not being constrained to linear orGaussian models. The resulting filter implementation iscalled particle filter [10]. It will be used in the nextexperiments to estimate the unknown trajectories andcorresponding uncertainties of dynamic coordinatemeasurements with an optical DMM.

reference to the least square circle fitted in the measured coordinates of theCMM.

Mean (error) in lm Max (error) inlm

Std (error) inlm

CMM 10.1, 16.8, 24.1 19.8, 25.8, 40.1 0.9, 2.1, 3.1EKF 8.96, 15.4, 20.7 15.9, 19.8, 30.3 0.7, 1.8, 2.1SIR PF 8.90, 13.8, 17.1 15.6, 19.8, 23.7 0.7, 1.5, 1.9

6 More particles were computed but did not change the results signif-icantly, due to the well-known process dynamic and process noise PDF(Gaussian).

4. Experimental results for known dynamics

Bayes filter is a general term for all algorithmic imple-mentations of the Bayesian recursion Eqs. (2)–(6). In thissection the extended Kalman filter (EKF) and the sequen-tial importance resampling particle filter (SIR PF) – as thetwo mainly used Bayes filters – were analyzed and verifiedwith respect to efficiency and performance applied to dy-namic coordinate measurements of a marker-based closerange photogrammetric (i.e. stereo vision based) CMM.

The experiments were conducted in a high precisionmeasuring room (accuracy class 1 according to VDI/VDE2627 part 1 [11]) by constant temperature # = 20 ± 0.1 �C.The dynamic CMM consists of two Gigabit Ethernet inter-line progressive scan charge-coupled device (CCD) camerasusing infrared illumination. To block out visible light, opti-cal infrared band-pass filters, cutting off at 810 nm and aspectral bandwidth of 35 nm, were used. Both camerasare mounted on a mechanical frame fulfilling the normalstereo case. The system calibration, i.e. estimating the in-ner and outer orientation of each camera, was performedby bundle adjustment using a 3D field of calibrated refer-ence points. Both cameras were synchronized by hardwaretrigger and acquire images with 776 � 582 pixels at maxi-mal 64 frames per seconds. All tracking targets were cali-brated with a traditional CMM that is specified with abidirectional Maximum Permissible Error E3 = (0.75 + L/500) lm, where the measuring length L is given in mmcomparable to ISO 10360 [12]. The targets were movedby a high accuracy parallel kinematic (hexapod) that isspecified with a smallest linear and angular increment of1 nm and 1 lrad as well as a linear and angular repeatabil-ity of 200 nm and 10 lrad. The target was moved along acircular trajectory r = 5 mm in an inclined plane of 45�according to ISO 9283 [13]. The experimental setup is de-picted in Fig. 3.

The measurement rate of the camera system was set tof = 30 Hz. Three independent measurement runs were per-formed with v = 1, 3, 5 mm/s. The SIR PF and EKF were usedfor recursive forward–backward smoothing with a slidingwindow of 500 ms length of time. The measurement

process was modelled as nonlinear estimation problem of3-dimensional Cartesian coordinates. The measured quan-tity yk (i.e. coordinates of the circular trajectory) is gener-ated by

xk ¼xk

yk

zk

264

375 ¼

r cos uk sin hk

r sinuk sin hk

r cos hk

264

375þ vk;

yk ¼ xk þwk:

The PF with stratified resampling (based on the effec-tive sample size), 200 particles6 Gaussian process noisewith standard deviation 5 lm, and uniform measurementnoise with standard deviation 50 lm was used. The EKF al-ways uses Gaussian noise with the same noise parameters asthe PF. The applied process noise based on an uncertaintybudget that incorporates the positioning error of the parallelkinematic, the calibration uncertainty and environmentalinfluences. The measurement noise corresponds to the Max-imum Permissible Error of the dynamic CMM, which wasdetermined by calibration measurements as specified inVDI 2634 (derived from ISO 10360) [11,12]. The realizedmeasurement error is given in Table 1. It was computed asEuclidean distance between the least square circles fittedin the measured coordinates as well as the unfiltered and fil-tered coordinates of the moved tracking target. The mea-sured coordinates, the fitted circular trajectory and thehistogram of deviations for a single measurement run areshown in Fig. 4. An example of an estimated posterior Bayesposterior PDF is given in Fig. 5.

Fig. 4. Analysis of a measurement run. A least square circle was fitted in the measured coordinates of the CMM.

Fig. 5. Example of an estimated Bayes posterior PDF in a time step. Due tothe sake of visualization the z-coordinate is neglected and only the 2DPDF approximation for the quantity xk = [sx, sy]T is shown.

E. Garcia et al. / Measurement 46 (2013) 3737–3744 3741

The results show that the PF outperforms the EKF, espe-cially in the case of high velocities. This is due to theincreasing nonlinear dependence of consecutive measuredcoordinates, which yields to poor prediction accuracy ofthe Taylor approximation of the process dynamic of theEKF.

5. Experimental results for unknown dynamics

The Bayesian estimation is an extremely powerful andeasy to implement method for inferencing in dynamic sys-tems, i.e. data processing with the purpose of prediction,filtering or smoothing. However, it is very difficult toappropriately determine the process equation and evenmore to specify the noise sources, systematically in a gen-eral setting. Especially the dynamic function and the statis-tical characterization of the process noise were key designparameters with crucial impact on the achievableaccuracy.

In case of coordinate measurements of standard geo-metrical features, as points, lines and circles7, it is easilypossible to derive an appropriate state space models, likeautoregressive (AR) models [14,15] xk = xk�1 + vk, linear or

7 That means continuously measuring a point in space or measuring alinear or circular trajectory.

circular models (e.g. as the one given in the last section). Itis also quite feasible to develop process models in caseswhere the motion path is completely known a priori (e.g.drill trajectory) and will be realized by a mechatronic posi-tioning system (e.g. robot). In such cases the dynamic equa-tion xk = f(�) can either be implemented as look-up tablecontaining kinematic quantities for every time step or asanalytical expression, e.g. by an nth-degree polynomial inCartesian coordinates.

But for the general case of measuring the motion (coor-dinates) of an object that is freely moving in arbitrary coor-dinates, it is almost impossible to determine a preciseprocess model. Here it is only feasible to apply universallykinematic white noise models [16], which are derived fromthe simple equations of motion.These models assume thata derivative of the position s(t) is a white noise processw(t). The simplest model is the white-noise accelerationmodel, where the acceleration €sðtÞ is specified as whitenoise. The second simplest model is the Wiener-processacceleration model (WPAM). This model is commonly usedas so called white-noise jerk model that assumes that theacceleration derivative, i.e. jerk j(t), is a white noise pro-cess: j(t) = _aðtÞ ¼ s

vðtÞ ¼ wðtÞ. The corresponding state space

representation for coordinate measurements is given by

_xðtÞ ¼0 1 00 0 10 0 0

264

375xðtÞ þ

001

264

375vðtÞ;

yðtÞ ¼ ½1 0 0 �T xðtÞ þwðtÞ;

with x = [s, v, a]T along a generic Cartesian axis.The above models are simple but crude. Actual dynam-

ics seldom have constant accelerations that are uncoupledacross coordinate directions. Furthermore it is difficult tospecify adequate noise sources. Usually, the process andmeasurement noise should be chosen such that they repre-sent the dynamic uncertainty and the uncertainty of mea-surement system. But this information is mostly notavailable for all kinematic quantities and can also only beobtained to some extent for the quantity position by cali-bration measurements.

Fig. 6. Collapse of PF with WPAM after few time steps in the estimation ofa highly nonlinear helix trajectory.

3742 E. Garcia et al. / Measurement 46 (2013) 3737–3744

Therefore an adaptive particle filter (APF) for arbitrarytrajectories and kinematics was developed. It based onthe assumption that the measurement target (i.e. evolutionof coordinates in time) behaves locally like a polynomial ofsome degree d in time, where d depends on the order of thestate space model. For example, if a 3rd order model isused for estimating s, v, a, then d = 4 should be chosen.For an arbitrary time window T = (t1, . . ., tn) and a fixed

Fig. 7. Superior accuracy of proposed APF for the est

degree d of a polynomial with n P dþ 1, a least squaresinterpolation polynomial can be fitted in the measuredcoordinates, continuously. Then, the extrapolation of thepolynomial and the Markov-chain information in the parti-cles is used to generate a process function. This approachsolved several problems at once: (a) the complete statevector can be reconstructed (i.e. analytically calculatehigher derivatives of the measured position), (b) the stan-dard deviations for the likelihood function can be com-puted sequentially, (c) the process noise can be adaptedvia the extrapolation error, continuously and (d) there isno need to develop or laboriously parameterize a statespace model.

In order to analyze the proposed methodology a highlynonlinear helix trajectory was generated by cylinder coor-dinate transformation with sampling period T = 1/30 s, 3rdorder polynomial, NP = 1000 particles and initial true statex0 ¼ ½u; r; z; _u; _r; _z; . . . �T ¼ ½0;1;0;p=ð100TÞ;0;1=ð10TÞ;0;1;0�T :

Gaussian process and measurement noise withr ¼ ½10 10 50 �lm for x, y and z were used, which weredetermined by calibration measurements of the photo-grammetric CMM (see last section). In the conductedexperiments SIR PF with WPAM collapsed after a few sec-

imation of a highly nonlinear helix trajectory.

Fig. 8. Root-mean-square error (RMSE) with increasing number ofparticles for PF with WPAM and APF.

Fig. 9. Workflow of Bayesian filtering and data fusion for dynamiccoordinate measurements. For the sake of visualization, the case ofmeasuring a two-dimensional motion of two objects with a photogram-metric CMM is illustrated, i.e. the state vector xk = [sx, sy]T.

E. Garcia et al. / Measurement 46 (2013) 3737–3744 3743

onds (see Fig. 6), whereas the APF shows superior estima-tion accuracy (see Fig. 7) for all dynamics and kinematics.In order to study the WPAM the experiments were re-peated with a less nonlinear trajectory generated withthe initial true state x0 = [0,1,0,2p/(100T),0,1/(100T),0,0,0.4]T and the same noise sources. The resulting root-mean-square error (RMSE) with increasing number ofparticles NP = 300, . . .,10,000 is depicted in Fig. 8. The APFrealized always significant better estimation accuracycompared to SIR-PF with WPAM. Especially for low num-ber of particles the WPAM model failed to compute an esti-mate. That results in NaN estimations. These results showthe robustness, outstanding estimation accuracy and easi-ness of the proposed methodology for arbitrary nonlineardynamics and kinematics (see Fig. 8).

6. Fusing dynamic coordinate measurements

The usage of the Bayesian estimation for dynamicalcoordinate measurements requires the software integra-tion in existing coordinate measurement systems as wellas straightforward and easy to use models for various mea-surement tasks without laborious determination of modelparameters. Since Bayes filter need as inputs only sequen-tial coordinate measurements, their addition and integra-tion in the data processing chain of existing coordinatemeasurement systems should be possible withoutproblems.

The above given APF allows the measurement of arbi-trary trajectories and kinematics of multiple objects and/or sensors by simply extending the state vector. This re-sults automatically in Bayesian estimation for the com-bined state vector, i.e. a Bayesian data fusion is achieved.Hereby, the number of the state dimensions is only limitedby the available computing power. The resulting workflowand data processing chain is given in Fig. 9.

Another kind of data fusion can be achieved by combin-ing the estimated posterior densities of separate filterbased on probabilistic, information theoretical or othermetrics. The most simple data fusion approach would bethe multiplication or convolution of multiple posterior

density functions and subsequent evaluation of the result-ing density function, see e.g. [17]. This could be further im-proved by using maximum a posteriori, maximumlikelihood, minimax, mean-shift, kernel density estimationor entropy based criterions. In general the data fusionproblem can be considered as an optimization problemfor a given metric. Several approaches exist and can befound in the literature, e.g. [18,19].

Unfortunately, there is no general solution to this kindof data fusion since every fusion approach highly dependson the specific measurement task.

7. Summary and conclusions

In this paper the problem of improving and fusing coor-dinate measurements of an arbitrary dynamic process wasconsidered. In order to deal with uncertainty and estimatethe true measurand, it was solved as a Bayesian sequentialprobabilistic inference problem for nonlinear dynamic sys-tems. Benefit of the developed methodology is a statisti-cally sound, real-time and data-driven estimation ofdynamic system variables of interest (measurand, parame-ters, etc.) in form of the Bayes posterior PDF. Based on theposterior density the measurement accuracy can be im-proved by filtering of ubiquitous noise and deducing anoptimal quantity value with respect to any optimality cri-teria. Furthermore it is possible to realize a measurementuncertainty evaluation and probabilistic data fusion inevery time step. This approach allows a Bayesian uncer-tainty analysis. Depending on the process under investiga-

3744 E. Garcia et al. / Measurement 46 (2013) 3737–3744

tion this measurement uncertainty evaluation yields toidentical posterior PDFs like the GUM, which is also basedon the model-based Bayesian evaluation of probabilities.

In future work it is planned to apply this methodologyto simultaneous utilization of multiple sensors with subse-quent data fusion and measurement uncertainty evalua-tion based on the estimated probability density functionsfor each sensor. The presented APF will be enhanced to acompletely data-driven and self-adapting tracking filterwithout the need of laborious modelling and parameteri-zation of the measurement process under investigation. Fi-nally, all developed software will be released as open-source software for reducing the burden of implementingBayesian filters.

Acknowledgement

The research was supported by the German ResearchFoundation (Deutsche Forschungsgemeinschaft e.V.) underthe DFG-Project Number WE 918/34-1.

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