This paper applies the Mixture of Gaussians probabilistic model, combined with Expectation Maximization optimization to the task of summarizing three dimensionals range data for the mobile robot. This provides a flexible way of dealing with uncertainties in sensor information, and allows the introduction of prior knowledge into low-level perception modules. Problems with the basic approach were solved in several ways: the mixture of Gaussians was reparameterized to reflect the types of objects expected in the scene, and priors on model parameters were included in the optimization process. Both approaches force the optimization to find 'interesting' objects, given the sensor and object characteristics. A higher level classifier was used to interpret the results provided by the model, and to reject spurious solutions

Hansen, V.

Murray-Smith, R.

Williamson, M.

English

Murray-Smith, Roderick

Online Research Database In Technology

Robot Docking Using Mixtures of Gaussians

This paper applies the Mixture of Gaussians probabilistic model, combined with Expectation Maximization optimization to the task of summarizing three dimensional range data for a mobile robot. This provides a flexible way of dealing with uncertainties in sensor information, and allows the introduction of prior knowledge into low-level perception modules. Problems with the basic approach were solved in several ways: the mixture of Gaussians was reparameterized to reflect the types of objects expected in the scene, and priors on model parameters were included in the optimization process. Both approaches force the optimization to find `interesting&apos; objects, given the sensor and object characteristics. A higher level classifier was used to interpret the results provided by the model, and to reject spurious solutions. 1 Introduction  This paper concerns an application of the Mixture of Gaussians (MoG) probabilistic model (Titterington et al., 1985) for a robot docking application. We use th..

Matthew Williamson

Roderick Murray-smith

Volker Hansen

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Robot Docking using Mixtures of Gaussians

This paper applies the Mixture of Gaussians probabilistic model, combined with Expectation Maximization optimization to the task of summarizing three dimensional range data for a mobile robot. This provides a flexible way of dealing with uncertainties in sensor information, and allows the introduction of prior knowledge into low-level perception modules. Problems with the basic approach were solved in several ways: the mixture of Gaussians was reparameterized to reflect the types of objects expected in the scene, and priors on model parameters were included in the optimization process. Both approaches force the optimization to find ‘interesting ’ objects, given the sensor and object characteristics. A higher level classifier was used to interpret the results provided by the model, and to reject spurious solutions. 

Robot docking using mixtures of gaussians

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      Williamson, M. and Murray-Smith, R. and Hansen, V. (1999) Robot docking using mixtures of Gaussians. In, Kearns, M.S. and Solla, S.A. and Cohn, D.A., Eds. Conference on Neural Information Processing Systems, 30 November - 5 December 1998 Proceedings of the 1998 conference on Advances in neural information processing systems II, Colorado, USA.         http://eprints.gla.ac.uk/3118/     Glasgow ePrints Service http://eprints.gla.ac.uk Robot Docking using Mixtures of GaussiansMatthew Williamson Roderick Murray-Smithy Volker HansenzAbstractThis paper applies the Mixture of Gaussians probabilistic model, com-bined with Expectation Maximization optimization to the task of sum-marizing three dimensional range data for a mobile robot. This providesa flexible way of dealing with uncertainties in sensor information, and al-lows the introduction of prior knowledge into low-level perception mod-ules. Problems with the basic approach were solved in several ways: themixture of Gaussians was reparameterized to reflect the types of objectsexpected in the scene, and priors on model parameters were includedin the optimization process. Both approaches force the optimization tofind ‘interesting’ objects, given the sensor and object characteristics. Ahigher level classifier was used to interpret the results provided by themodel, and to reject spurious solutions.1 IntroductionThis paper concerns an application of the Mixture of Gaussians (MoG) probabilisticmodel (Titterington et al. 1985) for a robot docking application. We use the Expectation-Maximization (EM) approach (Dempster et al. 1977) to fit Gaussian sub-models to a sparse3d representation of the robot’s environment, finding walls, boxes, etc.. We have modifiedthe MoG formulation in three ways to incorporate prior knowledge about the task, and thesensor characteristics: the parameters of the Gaussians are recast to constrain how they fitthe data, priors on these parameters are calculated and incorporated into the EM algorithm,and a higher level processing stage is included which interprets the fit of the Gaussians onthe data, detects misclassifications, and providing prior information to guide the model-fitting.The robot is equipped with a LIDAR 3d laser range-finder (PIAP 1995) which it uses toidentify possible docking objects. The range-finder calculates the time of flight for a lightpulse reflected off objects in the scene. The particular LIDAR used is not very powerful,making objects with poor reflectance (e.g., dark, shiny, or surfaces not perpendicular to theMIT AI Lab, Cambridge, MA, USA. matt@ai.mit.eduyDept. of Mathematical Modelling, Technical University of Denmark. rod@imm.dtu.dkzDaimler-Chrysler, Alt-Moabit 96a, Berlin, Germany. hansen@dbag.bln.daimlerbenz.comlaser beam) invisible. The scan pattern is also very sparse, especially in the vertical direc-tion, as shown in the scan of a wall in Figure 1. However, if an object is detected, the rangereturned is accurate (1-2cm). When the range data is plotted in Cartesian space it formsa number of sparse clusters, which leads naturally to the use of MoG clustering algorithmsto make sense of the scene. The sparse nature of the data inspired the modifications to theMoG formulation described in this paper.Model-based object recognition from dense range images has been widely reported (see(Arman and Aggarwal 1993) for a review), but is not relevant in this case given the sparse-ness of the data. Denser range images could be collected by combining multiple scans, butthe poor visibility of the sensor hampers the application of these techniques. The advantageof the MoG technique is that the segmentation is “soft”, and perception proceeds iterativelyduring learning. This is especially useful for mobile robots where evidence accumulatesover time, and the allocation of attention is time and state-dependent. The EM algorithm isuseful since it is guaranteed to converge to a local maximum.The following sections of the paper describe the re-parameterization of the Gaussians tomodel plane-like clusters, the formulation of the priors, and the higher level processingwhich interprets the clustered data in order to both move the robot and provide prior infor-mation to the model-fitting algorithm.−101012−1.2−1−0.8−0.6−0.4−0.20xaxis in myaxis in mzaxis in mFigure 1: Plot showing data from a LIDAR scan of a wall, plotted in Cartesian space. Therobot is located at the origin, with the y axis pointing forward, x to the right, and z up. Thesparse scan pattern is visible, as well as the visibility constraint: the wall extends beyondwhere the scan ends, but is invisible to the LIDAR due to the orientation of the wall2 Mixture of Gaussians modelThe range-finder returns a set of data, each of which is a position in Cartesian space xi=(xi; yi; zi). The complete set of data D = fx1: : :xNg is modeled as being generated by amixture densityP (xn) =MXi=1P (xnji; i;i; i)P (i);where we use a Gaussian as the sub-model, with mean i, varianceiand weight i, whichmakes the probability of a particular data point:P (xnj;; ) =MXi=1i(2)3=2jij1=2exp 12(xn  i)T 1i(xn  i)Given a set of data D, the most likely set of parameters is found using the EM algorithm.This algorithm has a number of advantages, such as guaranteed convergence to a localminimum, and efficient computational performance.In 3D Cartesian space, the Gaussian sub-models form ellipsoids, where the size and orien-tation are determined by the covariance matrix i. In the general case, the EM algorithmcan be used to learn all the parameters of i. The sparseness of the LIDAR data makesthis parameterization inappropriate, as various odd collections of points could be clusteredtogether. By changing the parameterization of ito better model plane-like structures, thesystem can be improved. The reparameterization is most readily expressed in terms of theeigenvalues iand eigenvectors Viof the covariance matrix i= ViiV 1i.The covariance matrix of a normal approximation to a plane-like vertical structure willhave a large eigenvalue in the z direction, and in the x–y plane one large and one smalleigenvalue. Since iis symmetrical, the eigenvectors are orthogonal, V  1i= VTi= Vi,and ican be written:i= sin icos i0cos i  sin i00 0 1! ai0 00 ai00 0 bi! sin icos i0cos i  sin i00 0 1!;where iis the angle of orientation of the ith sub-model in the x–y plane, aiscales thecluster in the x and y directions, and biscales in the z direction. The constant  controlsthe aspect ratio of the ellipsoid in the x–y plane.1The optimal values of these parameters (a; b) are found using EM, first calculating theprobability that data point xnis modeled by Gaussian i, (hin) for every data point xnandevery Gaussian i,hin=ijij 1=2exp  12(xn  i)T 1i(xn  i)PMi=1ijij 1=2exp  12(xn  i)T 1i(xn  i):This “responsibility” is then used as a weighting for the updates to the other parameters,^i=PnhinxnPnhin;^i=12tan 12Pnhin(xn1  i1)(xn2  i2)Pnhin[(xn1  i1)2  (xn2  i2)2] = (   1)((xn1  i1) sin  + (xn2  i2) cos )2+ (xn1  i1)2+ (xn2  i2)2a^i=Pnhin2Pnhin;^bi=Pnhin(xn3  n3)2Pnhin;where xn1is the first element of xnetc. and  corresponds to the projection of the data intothe plane of the cluster. It is important to update the means ifirst, and use the new valuesto update the other parameters.2 Figure 2 shows a typical model response on real LIDARdata.2.1 Practicalities of application, and resultsStarting values for the model parameters are important, as EM is only guaranteed to find alocal optimum. The Gaussian mixture components are initialized with a large covariance,allowing them to pick up data and move to the correct positions. We found that initializingthe means ito random data points, rather than randomly in the input space, tended to1By experimentation, a value of  of 0.01 was found to be reasonable for this application.2Intuition for the ^iupdate can be obtained by considering that (xn1 i1) is the x component ofthe distance between xnand i, which is jxn ij cos , and similarly (xn2 i2) is jxn ij sin ,so tan 2 = sin 2cos 2=2 sin  cos cos2 sin2=2(xn1 i1)(xn2 i2)(xn1 i1)2 (xn2 i2)2.−0.5 0 0.500.10.20.30.40.50.60.70.80.9x y plot green;  red;  blue0 0.2 0.4 0.6 0.8 1−0.6−0.5−0.4−0.3−0.2−0.100.10.20.3y z plot green; red; blueFigure 2: Example of a ‘desirable’ clustering of the 3d data points. The left hand graphshows the view from above (the x–y plane), and the right shows the view from the side(the yz plane). The robot is positioned at the origin. The scene shows walls to the frontand the right, and a box in front. All three objects are modeled. The ellipses are drawn at aprobability of 0.5.work better, especially given the sensor characteristics—if the LIDAR returned a rangemeasurement, it was likely to be part of an interesting object.Despite the accuracy of measurement, there are still outlying data points, and it is impos-sible to fully segment the data into separate objects. One simple solution we found wasto define a “junk” Gaussian. This is a sub-model placed in the center of the data, with alarge covariance. This Gaussian then becomes responsible for the outliers in the data (i.e.sparsely distributed data over the whole scene, none of which are associated with a specificobject), allowing the object-modeling Gaussians to work undistracted.The use of EM with the a; b;  parameterization found and represented plane-like data clus-ters better than models where all the elements of the covariance matrix were free to adapt.It also tended to converge faster, probably due to the reduced numbers of parameters in thecovariance matrix (3 as opposed to 6). Although the algorithm is constrained to find planes,the parameterization was flexible enough to model other objects such as thin vertical lines(say from a table leg). The only problem with the algorithm was that it occasionally foundpoor local minimum solutions, such as illustrated in Figure 3. This is a common problemwith least squares based clustering methods (Duda and Hart 1973).−1 −0.5 0 0.5 100.10.20.30.40.50.60.70.80.91x y plot green;  red;  blue−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.800.10.20.30.40.50.60.70.80.91x y plot green;  red;  blueFigure 3: Two examples of ‘undesirable’ local minimum solutions found by EM. Bothgraphs show the top view of a scene of a box in front of a wall. The algorithm has incor-rectly clustered the box with the left had side of the wall.3 Incorporating prior informationAs well as reformulating the Gaussian models to suit our application, we also incorporatedprior knowledge on the parameters of the sub-models. Sensor characteristics are oftenwell-defined, and it makes sense to use these as early as possible in perception, rather thandealing with their side-effects at higher levels of reasoning. Here, e.g., the visibility con-straint, by which only planes which are almost perpendicular to the lidar rays are visible,could be included by writing P (xn) =PMi=1P (xnji; i)P (i)P (visibleji), the updatescould be recalculated, and the feature immediately brought into the modeling process. Inaddition, prior knowledge about the locations and sizes of objects, maybe from other sen-sors, can be used to influence the modeling procedure. This allows the sensor to makebetter use of the sparse data.For a model with parameters  and data D, Bayes rule gives:P (jD) =P ()P (D)YP (xnj):Normally the logarithm of this is taken, to give the log-likelihood, which in the case ofmixtures of Gaussians isL(Dj) = log(p(fi; i; ai; bi; ig))  log(p(D)) +XnlogXip(xnji; i; i; ai; bi; i)To include the parameter priors in the EM algorithm, distributions for the differentparameters are chosen, then the log-likelihood is differentiated as usual to find theupdates to the parameters (McMichael 1995). The calculations are simplified if thepriors on all the parameters are assumed to be independent, p(fi; i; ai; bi; ig) =Qip(i)p(i)p(ai)p(bi)p(i).The exact form of the prior distributions varies for different parameters, both to cap-ture different behavior and for ease of implementation. For the element means (i),a flat distribution over the data is used, specifying that the means should be amongthe data points. For the element weights, a multinomial Dirichlet prior can be used,p(ij) = (+M) (+1)MQMi=1i. When the hyperparameter  > 0, the algorithm favoursweights around 1=M , and when  1 <  < 0, weights close to 0 or 1.3 The expected valueof ai(written as ai) can be encoded as a prior by setting p(aijai) = K exp ( ai=(2ai)),where K is a normalizing factor, independent of ai.4 The prior for bihas the same form.Priors for iwere not used, but could be useful to capture the visibility constraint. Giventhese distributions, the updates to the parameters become^i=PnhinxnPnhin; ^i=Pnhin+ PnPjhjn+ a^i=Pnhin= + ai2Pnhin;^bi=Pnhin(xn3  n3)2+ biPnhin:The update for iis the same as before, the prior having no effect. The update for aiandbiforces them to be near aiand bi, and the update for iis affected by the hyperparameter.The priors on aiand bihad noticeable effects on the models obtained. Figure 4 shows theresults from two fits, starting from identical initial conditions. By adjusting the size of theprior, the algorithm can be guided into finding different sized clusters. Large values of theprior are shown here to demonstrate its effect.3In this paper we make little use of the  priors, but introducing separate i’s for each objectcould be a useful next step for scenes with varying object sizes.4To deal with the case when ai= 0, the prior is truncated, setting p(aijai) = 0 when ai< crit−1 −0.5 0 0.5 100.10.20.30.40.50.60.70.80.91No priorsxy0 0.2 0.4 0.6 0.8 1−0.8−0.6−0.4−0.200.20.4No priors−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.800.10.20.30.40.50.60.70.80.91Priors a0 = 10, b0 = 10, alpha = 0xy0 0.2 0.4 0.6 0.8 1−0.8−0.6−0.4−0.200.20.40.6Priors a0 = 10, b0 = 10, alpha = 0Figure 4: Example of the action of the priors on aiand bi. The photograph shows avisual image of the scene: a box in front of a wall, and the priors were chosen such that adistribution matching that of the wall should be preferred. The two left hand graphs showthe top and side view of the scene clustered without priors, while the two right hand graphsuse priors on aiand bi. The priors give a preference for large values of aiand bi, so biasingthe optimization to find a mixture component matching the whole wall as opposed to justthe top of it.4 Classification and diagnosisMOVE COMMANDFOR ROBOTDATASENSOR MODEL FITTINGEM ALGORITHMHIGHER LEVELPROCESSINGFEATURESPRIOR INFORMATIONFigure 5: Schematic of systemThis section describes how higher-level processing can be used to not only interpret theclusters fitted by the EM algorithm, but also affect the model-fitting using prior information.The processes of model-fitting and analysis are thus coupled, and not sequential.The results of the model fitting are primarily processed to steer the robot. Once the clusterhas been recognized as a box/wall/etc., the location and orientation are used to calculatea move command. To perform the object-recognition, we used a simple classifier on afeature vector extracted from the clustered data. The labels used were specific to docking,and commonly clustered objects – boxes, walls, thin vertical lines, but also included labelsfor clustering errors (like those shown in Figure 3). The features used were the values of theparameters ai, bi, giving the size of the clusters, but also measures of the visibility of theclusters, and the skewness of the within-cluster data. The classification used simple modelsof the probability distributions of the features fi, given the objects Oj(i.e. P (fijOj)),using a set of training data. In addition to moving the robot, the classifier can modify thebehavior of the model fitting algorithm. If a poor clustering solution is found, EM can bere-run with slightly different initial conditions. If the probable locations or sizes of objectsare known from previous scans, or indeed from other sensors, then these can constrain theclustering through priors, or provide initial means.5 SummaryThis paper shows that the Mixture of Gaussians architecture combined with EM optimiza-tion and the use of parameter priors can be used to segment and analyze real data from the3D range-finder of a mobile robot. The approach was successfully used to guide a mobilerobot towards a docking object, using only its range-finder for perception.For the learning community this provides more than an example of the application of aprobabilistic model to a real task. We have shown how the usual Mixture of Gaussiansmodel can be parameterized to include expectations about the environment in a way whichcan be readily extended. We have included prior knowledge at three different levels: 1.The use of problem-specific parameterization of the covariance matrix to find expectedpatterns (e.g. planes at particular angles). 2. The use of problem-specific parameter priorsto automatically rule-out unlikely objects at the lowest level of perception. 3. The results ofthe clustering process were post-processed by higher-level classification algorithms whichinterpreted the parameters of the mixture components, diagnosed typical misclassification,provided new priors for future perception, and gave the robot control system new targets.It is expected that the basic approach can be fruitfully applied to other sensors, to prob-lems which track dynamically changing scenes, or to problems which require relation-ships between objects in the scene to be accounted for and interpreted. A problem com-mon to all modeling approaches is that it is not trivial to determine the number and typesof clusters needed to represent a given scene. Recent work with Markov-Chain Monte-Carlo approaches has been successfully applied to mixtures of Gaussians (Richardson andGreen 1997), allowing a Bayesian solution to this problem, which could provide controlsystems with even richer probabilistic information (a series of models conditioned on num-ber of clusters).AcknowledgementsAll authors were employed by Daimler-Benz AG during stages of the work. R. Murray-Smith gratefully acknowledges the support of Marie Curie TMR grant FMBICT961369.ReferencesArman, Farshid and J. K. Aggarwal (1993). Model-based object recognition in dense-rangeimages—a review. ACM Computing Surveys 25(1), 5–43.Dempster, A. P., N. M. Laird and D. B. Rubin (1977). Maximum likelihood from incom-plete data via the EM algorithm. J. Royal Statistical Society Series B 39, 1–38.Duda, Richard O. and Peter E. Hart (1973). Pattern Classification and Scene Analysis. JohnWiley and Sons.McMichael, D. W. (1995). Bayesian growing and pruning strategies for MAP-optimal es-timation of gaussian mixture models. In: 4th IEE International Conf. on ArtificialNeural Networks.PIAP (1995). Piap impact report on TRC lidar performance. Technical Report 1. IndustrialResearch Institute for Automation and Measurements. 02-486 Warszawa, Al. Jero-zolimskie 202, Poland.Richardson, Sylvia and Peter J. Green (1997). On Bayesian anaysis of mixtures with an un-known number of components. Journal of the Royal Statistical Society B 50(4), 700–792.Titterington, D.M., A.F.M. Smith and U.E. Makov (1985). Statistical Analysis of FiniteMixture Distributions. John Wiley & Sons. Chichester.

Aggarwal (1993).Model-basedobject recognitionin dense-range images—a review.

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Green (1997).On Bayesian anaysisof mixtureswith an unknown number of components.

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