A group target is moving in an area well covered by a network of passive sensor nods with known positions. Additionally, there are a number of mobile robots with active sensors. In order to obtain a robust estimate of the position of the target and decrease the amount of energy spent on active sensing and communications by the sensor network and the mobile robots a sensor management system optimises the spatial configuration of the mobile robots over time. A tracking algorithm predicts the position of the target over multiple steps. An estimate for the tracking accuracy for each possible sensor action is calculated based on a function of the expected resulting posterior inverse covariance (information) matrix given the position of the nodes of the sensor networks and the feasible position of the mobile robots in future time instants. We propose a novel approach for active sensor management that combines the Rao-Blackwellised particle filter/predictor and multi-objective D-optimal optimisation. The designed decentralised Rao-Blackwellised particle filter (RBPF) is composed of two parts: a decentralised Information or Kalman filter and a particle filter (PF). The sensor management framework that is based on the generalised D-optimal optimisation with slack variables is proposed

Bull, D.

Dolia, A.

Harris, C.

Mihaylova, L.

Page, S.

White, N.

English

Lancaster E-Prints

Multiple Objective D-optimal Sensor Managementfor Group Target TrackingAlexander Dolia∗, Lyudmila Mihaylova∗∗, Scott Page∗, Neil White∗, Chris Harris∗, David Bull∗∗∗University of Southampton ∗∗ University of BristolSouthampton, SO17 1BJ Dept. of Electrical and Electronic Eng.BS8 1UB, England, UK Woodland Road, England, UK Bristol{ad, sfp03r, nmw, cjh} @ecs.soton.ac.uk mila.mihaylova@bristol.ac.ukAbstract—A group target is moving in an area well coveredby a network of passive sensor nods with known positions.Additionally, there are a number of mobile robots with activesensors. In order to obtain a robust estimate of the position of thetarget and decrease the amount of energy spent on active sensingand communications by the sensor network and the mobile robotsa sensor management system optimises the spatial configurationof the mobile robots over time. A tracking algorithm predictsthe position of the target over multiple steps. An estimate for thetracking accuracy for each possible sensor action is calculatedbased on a function of the expected resulting posterior inversecovariance (information) matrix given the position of the nodes ofthe sensor networks and the feasible position of the mobile robotsin future time instants. We propose a novel approach for activesensor management that combines the Rao-Blackwellised particlefilter/predictor and multi-objective D-optimal optimisation. Thedesigned decentralised Rao-Blackwellised particle filter (RBPF)is composed of two parts: a decentralised Information or Kalmanfilter and a particle filter (PF). The sensor management frame-work that is based on the generalised D-optimal optimisationwith slack variables is proposed.I. INTRODUCTIONSensor management has been an active area of research overthe last few years [1], [2]. Efforts are concentrated both ontheoretical and practical issues of sensor networks. Typicaladvantages of using a senor network are the inherent robust-ness to sensor failures and coverage of a larger area. Commonapproaches to sensor management select the most informativeobservations after optimising information measures such as theKullback-Leibler distance or Re´nyi divergence [3], [4], [5], [6].However, most of the requirements for the system as a wholecan be taken into account only by a multi-objective functionwhich can reflect other requirements, e.g. communicationissues, time restrictions, power and other constraints. In thispaper we consider additionally other measures such as energyconsumption and survivability of the sensors in addition toinformation gain. We consider the sensor tasking and the targettracking problems together within a decentralised informationtheoretic sensor management architecture. The chosen config-uration of the mobile sensors in the vicinity of the target isbased on the multi-objective D-optimal optimisation approachin order to satisfy various requirements. The presented ap-proach can be used in many surveillance systems, roboticsapplications, and sensor networks.The paper is organised as follows. Section II presents ourprevious formulation of the sensor management problem. Thesensor management approach based on the RBPF and multi-objective D-optimal optimisation is proposed in Section III.Finally, conclusions are outlined in Section IV.II. PROBLEM FORMULATIONIn this section we formulate the problem of active sensormanagement based on decentralised processing of sensor dataand multi-objective optimisation.Consider scenarios where the sensor data, e.g. range andbearing to the target are related to the states of the targetthrough highly nonlinear relations, which together with non-linear target dynamics makes it difficult or even impossible touse standard techniques such as the extended Kalman filter.A Rao-Blackwellised particle filter (RBPF) is therefore usedwhich estimates the target position, speed and accelerationof the target. The filter also supplies a covariance matrixcharacterising the accuracy of the estimates.The optimisation variables a(t + 1) form the matrix of2D Cartesian coordinates for S sensors at the next time step(t+1) [7]. In our previous work, the optimisation problem wasformulated as follows [7]. Find the action matrix a(t+1) thatis infinite-norm optimal with respect to the weighted objectivesof (minimising) sensor power consumption of moving the sen-sors and (maximising) sensor separation. The selected actionmust satisfy the constraint set Ω which consists of constraintson the maximum sensor displacement, the minimum sensorseparation and the trace of the resulting posterior state estimatecovariance matrix.a∗(t + 1) = arg mina(t+1)(Pow(a(t + 1)), Sep(a(t + 1)),W )(1)s.t. a(t + 1) ∈ Ω,where W is the objective preference weight vector W =[w1, w2] for the two performance objectives. The performanceobjectives are defined as follows. The power consumption ofa particular sensor placement Pow(a(t + 1)) is proportionalto the Euclidean distance travelled by the sensor. The sepa-ration of sensors Sep(a(t + 1)) is defined using the resultingEuclidean distance between sensors for the particular actionProceedings of the 2006 IEEE International Conference on Robotics and AutomationOrlando, Florida - May 20060-7803-9505-0/06/$20.00 ©2006 IEEE 4446a(t + 1) (see [7] for more details). The predicted positions ofthe target along a specified time horizon are calculated by thetracker which coupled with the range-dependant observationinformation contribution provides a mechanism for relatingsensor actions to the resulting posterior estimate covariancematrix. The approach presented can be used for both non-myopic (over several time epochs) [8] and myopic (singleepoch) [7] sensor management.In our previous approach we considered only omnidirec-tional sensors and the single target case [7]. In this paperthe main objective is to introduce a framework for sensormanagement that optimises the sensors field of view. Forexample, two mobile robots can emit signals that can bereflected by the group of targets and is received by sensornetwork (sensors in fixed locations). We also consider the casewhen the field of view covers only a part of the group oftargets. In the next section we briefly outline the proposedmultiple objective D-optimal sensor management frameworkfor group target tracking.III. THE SENSOR MANAGEMENT ALGORITHMA. Tracking AlgorithmThe tracking scheme is based on a Rao-Blackwellsisedparticle filter [9], [10], [11], [12]. Rao-Blackwellisation isa technique for improving particle filtering by analyticallymariginalising out some of the variables (linear, Gaussian)from the joint posterior distribution, and then the linear systemmodel is estimated by a Kalman filter (KF), an optimalestimator, whilst the nonlinear part is estimated by a particlefilter (PF). In this particular problem the positions of the targetcan be estimated by a PF, its speeds and accelerations (in xand y directions) with the KF (see Table 1). As a result ofthe marginalisation, the variance of the estimates is reducedcompared to the standard PF and the computational complexitydecreases as well. Additionally, this RBPF is decentralisedbecause it uses the fused measurements from neighbouringsensors only. The PF can approximate the r-step ahead statepredictive distribution p(xt+ri |z1:t) of positions of the ithtarget. Here, xt+ri denotes the system state vector of the ithtarget at time instant t + r, and z1:t is the vector containingmeasurements up to time instant t. These predictions areused for generating the next best sensor placement. A multi-objective decision making criterion is used composed byseveral terms, accounting for different requirements: trackingaccuracy (through a variance calculated by the particle filter),minimum energy (for the control of the sensors) and spatialrestrictions. Some of the requirements such as minimum powerconsumption and maximum target tracking performance areconflicting. Since we optimise the positions of the sensors withrespect to the positions of the target, only the covariance fromthe PF is required by the optimisation framework, and wedo not use the estimation error covariance from the KF. Otherconstraints such as maintaining minimum distance to obstaclescan be easily accounted for by the optimisation framework.For example, Fig.1 shows the actual, and estimated trajec-tory from the RBPF using measurements from neighbouring0 2000 4000 6000 8000 10000 12000 1400000.511.522.5x 104x coordinate, [m]y coordinate, [m]Sensor Management for TrackingActual trajectoryEstimated trajectoryFig. 1. Sensors’ positions, the actual trajectory of the target, estimatedtrajectoriy by the RBPFsensors.Table 1. A Rao-Blackwellised PF for sensor managementInitialisationGenerate initial samples for the states of the PF and the KF.Set initial weights.For k =1:simulation time1) Particle filter prediction step2) Kalman filtering update step. It uses the states predicted bythe PF as measurements in the KF.3) Kalman filtering prediction step4) Measurement update based on the fused sensor data.Compute the weights and normalise them.5) Output:- estimated target positions by the PF- estimated states (speed and accelerations) by the KF- covariance matrix from the KF (characterises theaccuracy of the target speed and acceleration estimates)- variance matrix from the PF (characterisesthe accuracy of the target position estimates)6) Resampling stependB. Actions GenerationIn order to define the field of view it is necessary tointroduce possible shapes of a covered region. In this paper weassume it is ellipsoidal. In order to define an ellipsoid we needto define its location and volume. The measure of volume ofthe ellipsoid is a determinan of information matrix det(M).The possible locations of the ellipsoid that defines a sensorfield of view at future time t + 1 depends on the previouslocation of the ellipsoid at time t and how fast we can changethe field of view (the center of the ellipsoid). Therefore wearrive at the following optimisation problem for two moving4447sensors[M∗1,M∗2, c∗t+11 , c∗t+12 ] = argmin2∑s=1log detMs +R2s + (ct+1s − cts)′(ct+1s − cts) (2)subject to(xt+1i − ct+11 )′M−11 (xt+1i − ct+11 ) ≤ R21,(xt+1i − ct+12 )′M−12 (xt+1i − ct+12 ) ≤ R22,(ct+11 − ct1)′(ct+11 − ct1) ≤ θ21,(ct+12 − ct2)′(ct+12 − ct2) ≤ θ22,(ct+11 − ct+12 )′(ct+11 − ct+12 ) ≥ d2,i = 1...l,where the solution of this optimisation problem are nonnega-tive definite matrices M∗1 and M∗2 and vectors c∗t+11 , c∗t+12 thatdefine the size and location of two ellipsoids (every ellipsoidcorresponds to space that will be covered by a correspondingsensor at time t + 1), in general, the number of ellipsoidscorresponds to the number of the mobile robots; ct1 and ct2 areknown position of the ellipsoid at time t; θ1, θ2 and d are pre-defined constants given by a user; xi is a predicted position ofthe i−th individual target in the group where l is the numberof individual targets in the group; R1 and R2 are values notequal to zero. In this formulation the size of the ellipsoid canbe changed by varying the choice of the matricesM1 andM2or vectors R1 and R2 but the large values of R1 (R2) shouldbe penalised.If we are interested in tracking the major part of the groupthen we can allow a few individual targets to be outside ofthe coverage space (ellipsoids) by introducing slack variablesξs,i and a parameter ν ∈ [0, 1) (see [13], [14] for more detailsabout slack variables and the parameter ν) which define theupper bound of the proportion of how many individual targetsare permitted be outside of the coverage space.[M∗1,M∗2, c∗t+11 , c∗t+12 ] = argmin2∑s=1log detMs +R2s + (ct+1s − cts)′(ct+1s − cts) +1lνl∑i=1ξs,i (3)subject to(xt+1i − ct+11 )′M−11 (xt+1i − ct+11 ) ≤ R21 + ξ1,i,(xt+1i − ct+12 )′M−12 (xt+1i − ct+12 ) ≤ R22 + ξ2,i,(ct+11 − ct1)′(ct+11 − ct1) ≤ θ21,(ct+12 − ct2)′(ct+12 − ct2) ≤ θ22,(ct+11 − ct+12 )′(ct+11 − ct+12 ) ≥ d2,ξi ≥ 0, i = 1...l,The dual optimisation problem can be defined and solved usinga similar approach described in [13]. In order to incorporatea prediction uncertainty of individual target locations xt+1i inthe optimisation problems (2) and (3) the point xt+1i can bereplaced by an ellipsoid that represents the accuracy of theprediction of the position of the target.IV. CONCLUSIONSThis paper presents a D-optimal procedure for sensormanagement. The approach relies on multiple objective op-timisation and Rao-Blackwellisation. The RBPF brings theadvantages that the highly nonlinear part of the state vectoris estimated by a PF, whilst the linear part is estimated by aKF. This reduces the computational complexity and improvesthe accuracy compared to the case when only a PF-basedprediction is used. In order to find minimum footprint wepropose the sensor management framework that is based on thegeneralised D-optimal optimisation with slack variables. Thiscan reduce the amount of energy required for the active seeingand improve robustness of the sensor management systemagainst outliers.ACKNOWLEDGEMENTSThe authors acknowledge the support of the UK MOD Data andInformation Fusion Defence Technology Centre by projects 8.1: ‘Ac-tive multi-sensor management’ and 2.2 ‘Communication optimisationfor distributed sensor systems’.REFERENCES[1] T. Chung, V. Gupta, J. Burdick, and R. Murray. On a decentralized activesensing strategy using mobile sensor platforms in a network. In Proc. ofthe IEEE Intl. Conf. on Decision and Control, pages 1914–1919, Dec.14-17 2004.[2] R. Gupta and S. R. Das. Tracking moving targets in a smart sensornetwork. In Proc. of the VTC Fall 2003 Symposium, Oct. 2003.[3] C. Kreucher, K. Kastella, and A. Hero III. Multi-target sensor manage-ment using alpha-divergence measures. In Proc. of the First IEEE Conf.on Information Processing in Sensor Networks, Palo Alto, 2003.[4] A. Mutambara. Decentralized Estimation and Control for MultisensorSystems. CRC Press, 1998.[5] H. Durrant-Whyte and B. Grocholsky. Management and control indecentralized networks. In Proc. of the 6th Intl. Conf. on InformationFusion, pages 560–566. ISIF, Queensland, Australia, 2003.[6] H. Durrant-Whyte and M. Stevens. Data fusion in decentralised sensingnetworks. In Proc. of the 4th Intl. Conf. on Information Fusion.Montreal, Canada, 2001.[7] S.F. Page, A.N. Dolia, N.M. White, and C.J. Harris. Multiple objectiveoptimisation for active sensor management. In SPIE European Sympo-sium on Security and Defence, pages 269–280. Orlando, Florida, USA,30-31 March 2005.[8] S.F. Page, A.N. Dolia, N.M. White, and C.J. Harris. Adaptive horizonmodel predictive control based active sensor management for targettracking. In Proceedings of the American Control Conference. Min-nesota, USA, 2006. accepted.[9] G. Casella and C. Robert. Rao-Blackwellisation of sampling schemes.Biometrica, 83(1):84–94, 1996.[10] T. Scho¨n, F. Gustafsson, and P.J. Nordlund. Marginalized particle filtersfor nonlinear state-space models. IEEE Trans. on Signal Processing,Nov. 2005.[11] F. Gustafsson, F. Gunnarsson, N. Bergman, and U. Forssell. Particlefilters for positioning, navigation and tracking. IEEE Trans. on SignalProcessing, 50(2):425–437, 2002.[12] L. Mihaylova, D. Bull, D. Angelova, and N. Canagarajah. Mobilitytracking in cellular networks with sequential Monte Carlo filters. InProc. of the The Eighth Intl. Conf. on Information Fusion, 2005.[13] A.N. Dolia S.F. Page, N.M. White, and C.J. Harris. D-optimality forminimum volume ellipsoid with outliers. In Proceedings of the Sev-enth International Conference on Signal/Image Processing and PatternRecognition, (UkrOBRAZ’2004), pages 73–76. Kiev, Ukraine, 11-15October 2004.[14] Nello Cristianini and John Shawe-Taylor. An introduction to SupportVector Machines and other kernel-based learning methods. CambridgeUniversity Press, 2000.4448

Adaptive horizon model predictive control based active sensor management for target tracking.

An introduction to Support Vector Machines and other kernel-based learning methods.

D-optimality for minimum volume ellipsoid with outliers.

Data fusion in decentralised sensing networks.

Decentralized Estimation and Control for Multisensor Systems.

Management and control in decentralized networks.

Marginalized particle ﬁlters for nonlinear state-space models.

Mobility tracking in cellular networks with sequential Monte Carlo ﬁlters.

Multi-target sensor management using alpha-divergence measures.

Multiple objective optimisation for active sensor management.

On a decentralized active sensing strategy using mobile sensor platforms in a network.

Multiple Objective D-Optimal Sensor Management for Group Target Tracking

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