A T-Step Ahead Constrained Optimal Target Detection Algorithm for a Multi Sensor Surveillance System

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1 A T-Step Ahea Constraine Optimal Target Detection Algorithm for a Multi Sensor Surveillance System K Mahava Krishna Henry Hexmoor an Shravan Sogani International Institute of Information Technology CSCE Department, University of Arkansas Hyeraba 5 9, INDIA Fayetteville, AR 727 mkrishna@iiit.ac.in hexmoor@uark.eu Abstract We present a methoology for optimal target etection in a multi sensor surveillance system. The system consists of mobile sensors that guar a rectangular surveillance zone crisscrosse by moving targets. Targets penetrate the surveillance zone with poisson rates at uniform velocities. Uner these conitions we present a motion strategy computation for each sensor such that it maximizes target etection for the next T time-steps. A coorination mechanism among sensors ensures that overlapping an overlooke regions of observation among sensors are minimize. This coorination mechanism is interleave with the motion strategy computation to reuce etections of the same target by more than one sensor for the same time-step. To avoi an exhaustive search in the oint space of all the sensors the coorination mechanism constrains the search by assigning priorities to the sensors an thereby arbitrating among sensory tasks. A comparison of this methoology with other multi target tracking schemes verifies its efficacy in maximizing etections. Sample an time-step are use equivalently an interchangeably in this paper. Inex Terms- multi sensor system, optimal target etection, sensor network, sensor surveillance, multi-agent system I INTRODUCTION We present a framework for maximizing target etections in a surveillance system consisting of multiple mobile sensors (i.e., robots). The sensors collectively guar a rectangular surveillance zone crisscrosse by moving targets. A-priori information about targets is the knowlege of their poisson rates of entry. Current knowlege about all targets within the sensing range of a sensor is available to the sensor in the form of the target velocity information an motion irection. While current information is use in eterministically computing the best places to be at in the subsequent time steps, it oes not take into account the changes in the environment that occur ue to consistent flow of target traffic into an out of the surveillance zone. Knowlege of the statistics of target entry an exit on the other han are use to compute the statistically optimal places to be at in the future from a given location, but oes not consier the current information gleane through sensor observations. Hence a combination of both current target information an target statistics is use to compute a motion strategy of a sensor that maximizes the number of etections for the next T time steps. Computing the best strategy from its iniviual perspective can result in motions for each sensor with overlapping regions of observation. A priority base coorination mechanism is interleave with the motion strategy computation to reuce etections of the same target by more than one sensor. To avoi an exhaustive search in the oint space of all the sensors the coorination mechanism constrains the search by assigning priorities to the sensors. Hence while optimal from a single sensor s perspective the algorithm is not optimal across all sensors a choice that one is reasonable to assume in light of prevailing real time emans. The etails of this metho are presente in section 3 of this paper. Extensive simulation results confirm the efficacy of this strategy in the form of tabular comparisons presente in section 4. Multi-sensor surveillance fins applications such as in borer patrol, guaring of secure areas, search an rescue an warehouse surveillance [, 2]. It involves etection of multiple intrusions an/or tracking through coorination between the sensors. Detection an target tracking has been researche from multiple viewpoints. Some efforts have focuse on the problem of ientifying targets from a given set of ata through particle filters [3], an probabilistic methos [4]. The problem of ata association or assigning sensor measurements to the corresponing targets were tackle by Joint Probabilistic Data Association Filters by the same researchers such as in [3]. Kluge an others [5] use ynamic timestamps for tracking multiple targets. Krishna an Kalra [] presente clustering base approaches for target etection an further extene it to tracking an avoiance. The focus of these approaches has been on builing reliable estimators for preicting target traectories that is ifferent from the obective of this effort to maximize target etections. In the context of istribute task allocation an sensor coorination Parker propose a scheme for elegating an withrawing robots to an from targets through the ALLIANCE architecture [7]. The protocol for allocation was one base on impatience of the robot towars a target while the withrawal was base on acquiescence. Jung an Sukhatme [8] present a strategy for tracking multiple intruers through a istribute mobile sensor network an a strategy for maximizing sensor coverage[8, 9]. Lesser s group have mae significant avances to the area of istribute sensor networks [] an sensor management []. In [2] Parker presents a scheme calle A-CMOMMT where the goal is to maximize the number of targets observe over a time interval of length T base on the same philosophy of behavior-base control as in [7]. The authors of this paper present their scheme for resource allocation an coorination in a istribute sensor

2 system through a set of fuzzy rules in [3] an further compare various resource allocation strategies in terms of their etection performance in [4]. The author of [] has looke at the problem of static placement of sensors in known polygonal environments an [] escribes a istribute sensor approach to target tracking using fixe sensor locations. The current approach is isparate from those of [,] in that in the current scheme the sensors are mobile. Among the approaches that we have encountere the closest to this are [2] an [8]. In [2] a behavior-base approach, A-CMOMMT, is compare with three other heuristic approaches where the sensor s motion strategy is arbitrary or ranom in the first, stationary (the sensor oes not move) in the secon an base on local force control in the thir. In [8] a motion strategy for tracking multiple targets base on ensity estimates is presente. The robot attempts to maximize target etections by maintaining itself at a particular istance from the center of gravity of currently observe targets. In these approaches since no assumption is mae regaring target arrival statistics the motion strategy oes not guarantee sensors move to best possible locations to optimize their etections. In contrast, our approach presents a constraine optimal scheme that moves the sensor to regions in the surveillance region to maximize etections over the next T samples. The optimality of this approach comes with the traeoff of apriori knowlege regaring target statistics. Nonetheless this situation is not new in robotic an multi-robotic literature where optimal path planning an scheuling algorithms require prior knowlege of the workspace in which they operate in terms of their static an ynamic contents vis-à-vis behavior base approaches that o not guarantee optimality or completeness but require no prior knowlege. III. THE METHODOLOGY A. Description of Surveillance Zone, Sensors an Targets: We consier the surveillance system epicte in figure. The sies of the outer rectangle or the biggest rectangle in figure form the bounary of the surveillance zone the area enclose by it is the area of interest where sensors attempt to optimize their rates of etection. The shae circles are effective sensor ranges in their starting positions. The fiel of vision (FOV) of a sensor is 3 egrees. The squares with thick bounaries in figure with the sensors at their center are the inscribe squares of the circular FOV of a sensor. In other wors, the iameter of the FOV of a sensor is the iagonal of one such inscribe square. Purely from the point of view of facilitating easier computations the sensor consiers only those targets that lie within its inscribe square as targets within its FOV. It nees to be emphasize that this simplification oes not have any bearing on the overall philosophy of this approach. In the results section the efficacy of this metho is verifie by uniformly applying this same conition across all other approaches that are compare an the extension to a case of circular FOV is merely one of more involve but computable computations. The entire surveillance region is iscretize into a lattice of cells. The cells are represente as the small squares insie the FOV of the leftmost an topmost sensor. The ashe lines along the length an breath inicate that the cells procee to fill the entire surveillance zone. At each of the cell locations various aspects of target statistics are compute that are escribe later. The crosses outsie the surveillance zone are the source points from where targets emanate as per Poisson statistics. Targets percolate from each of those sources into that horizontal or vertical half-plane that contains the surveillance zone. Therefore, all targets coming from a particular source will be containe with an angular span of π raians. Furthermore, the following assumptions are mae for sensors an targets A sensor can etect all targets within its FOV or occlusion relations are not consiere. The takeoff angle of a target from its source point is uniformly istribute in [,π ] All targets move with the same uniform velocity within the surveillance zone along linear traectories, which can be ascertaine by the sensor. The last assumption allows that the statistical values of various parameters compute at every cell in the lattice to take a unique value rather than a probability istribution. In case of a istribution the expecte values of the parameters nee to be mae use of. B Problem Statement an Approach The problem attacke in this paper is state as follows. Given: N S : The set of all sensors in the system, N S = { s, s,..., sn s }, where si enotes the sensor with label i, orere in a sequence. Hence n s the label of the last enumerate sensor is also the number of sensors in the system, which is a constant. N, : The set of all targets in the system at time t, where, I t { i, i in } N I, t =,..., I, t, an n I, t is the number of targets in the system at time step t that varies at every time step an hence ynamic. gm,t : A binary variable that takes the value if a target im is observe by any one of the ns sensors at t. Obective: To evelop an algorithm such that the T ni, t following cost function J = gm, t is maximize. In t= m= other wors, the number of etections of the targets present in the system at every time step is maximize over T time steps. Approach: ) At each cell, Pi, in the lattice the following are compute. a). λˆ Pi : The expecte rate of target entry into the FOV of a sensor centere at Pi. If there are Q target sources each

3 Q θ emanating targets at rate λ then ˆ λ Pi = λ, where the π = θ ratio enotes the expecte fraction of the total number of π targets that woul enter the FOV of the sensor at Pi from the th of the Q target sources. In figure Q=8. We also enote by θ λpi, = λ the rate at which the sensor at Pi sees targets π from source. In other wors, among the targets taking off at any angle within [,π ], from source only those that takeoff within the angular span [ δ, δ + θ ] woul enter the FOV of the sensor, where δ is the smallest angle forme by the segment connecting a vertex of the FOV square an source an δ + θ is the largest angle. b) Tˆ epi : The expecte escape time for a target through the fiel of vision of the sensor centere at Pi. The escape time is essentially the time for which a target woul be in the FOV before it escapes from a sensor. c) ˆ Pi = ˆ λpitˆ epi : The number of targets expecte to be etecte by a sensor at Pi for any time step. η ) Pi, : The normalization constant at cell Pi ue to target source. This constant reners the summation of the probability istribution function (pf) of the escape time of a target through the FOV of a sensor at Pi ue to source to go to unity. In other wors if te Pi, is the ranom variable that measures the escape time of a target for a sensor at cell Pi ue to source an f ( te Pi, ) be its pf then η Pi, sees to that tb f ( te Pi, ) =, where the lower an uppe bouns of the ta integral are the minimum an maximum possible escape times. It is to be note that in general f ( te Pi, ) is not the same in [ t a, t b ] an nees to be broken into sub functions for respective time intervals for which it has the same close form representation. 2) For every sensor s in the system locate at a particular cell location P the path it takes for the next T time steps is compute such that the number of target etections is maximize base on steps 3 an 4. 3) For s currently at P compute for all cells, Pk, that are at level n, n = {,,2,..., T} from P, the expecte number of etections, n ˆ,, where the first inex, k, in the superscript refers to the cell Pk an the secon inex t enotes the time step for which it was compute. The computation is base on what s currently observes at P an the target arrival statistics, elements of which are compute in step. This computation is carrie out at the current cell P an those cells one time-step away for the next T time steps. For cells at epth two this computation is carrie T- times, corresponing to time steps starting from 2 an ening at T with T inclusive. For a cell at epth T the computation is one once for time step T alone. This results in a tree where each cell at time step t is connecte to its eight nearest cells as well as itself at t +. Thus every noe in the tree except the leaf noe is connecte to nine noes an the epth of the tree is T. Figure : The surveillance zone is represente by the outer rectangle. The sensors by shae circles, the crosses outsie are the target sources an the small squares within the FOV of top an left sensor are the iscretize cells. The ashe lines along the length an breath inicate that the cells procee to fill the entire surveillance zone. 4) On this nine connecte space-time tree perform a epthlimite search to maximize the etections with n ˆ, as the obective value of a linhat connects a current cell noe P at t- to another noe Pk for the next time step t. 5) The paths of the sensors are orere base on the number of etections. The sensor path that has the highest number of etections has the highest priority. Ties are broken ranomly. ) For overlapping FOV between sensors at certain locations in the paths the number of etections is reuce at the corresponing cell for the corresponing time-step for the sensor with lower priority. The reuction is compute as k, t k, t k, t nˆ = nˆ f nˆ. The fraction f is proportional to the area of overlaps given by the area of overlapping FOV ivie by the area of the FOV. 7) The optimal path for the sensor with the lower priority is recompute base on the upate n ˆ, values through epth limite search. For a sensor with priority p, we check for overlaps with all those sensor traectories with priorities, 2,, p-. 8) The steps two to seven are repeate until the en of simulation.

4 Step of the algorithm is performe only once at the start of the simulation across all the cells an is essentially an offline step unless the target statistics changes ynamically. B Computing n ˆ, Computation of n ˆ, varies if it is being carrie out for the same cell location for a future time or for a ifferent cell location. For the case of computing at the same cell the proceure is as follows: Given that the sensor at cell Pi currently or at t= observes n Pi number of targets then the number of targets it is likely to see T time-steps into the future is given by: Q T i, T nˆ = npi k + ˆ λpit ˆ λpi, P( tepi, < T t) = t= () Here ˆ λ Pi, te Pi,, λpi, have the same connotations as iscusse in step an Q is the number of target sources as before. The first term on the right han sie or RHS of equation () is the eterministic part that is compute purely base on what the sensor senses now. It merely states that out of n Pi particle seen currently k of them woul isappear by T time steps. The secon term onwars on the RHS enote the statistical counterpart. It says that λˆ PiT are likely to enter the FOV in T time steps an of which the number in the thir term are likely to leave the FOV by T. The thir term containing the summation is explaine as follows. P( tepi, < T t) enotes the probability that a target from source that entere the FOV at t woul have escape the FOV by T. Hence ˆλ Pi, P( tepi, < T t) is the estimate of the number of particles that entere at t an escape by T, since ˆλ P i, is the estimate of number of particles entering for every t from into the FOV of the sensor at Pi. The summation over t signifies that estimate of the number of particles that woul have left by T nees to be one in corresponence with the time at which they entere between now an T. The summation over Q signifies that the pf of te Pi, is not the same for every source at that cell location. The P( te < T t) Pi, is given below by its pf with notations from step. ; T t < ta T P( tepi, < T t) = f ( tepi, ); ta T t tb. η Pi, ta ; T t > tb The number of caniates likely to be seen for a ifferent cell Pk at T given that n Pi number of targets then the number of targets are currently seen at Pi is given by: nˆ k + κ ˆ λ PiT Q T k, T = n ˆ Pi λpi, = t= ( κ ) ˆ Pk ( < T t) P te Pi, + (2). Equation (2) has similar connotations as () except for the appearance of κ that represents the fraction of area common to FOV erecte at Pi an Pk an ˆ Pk is as iscusse in step. For that fraction of the FOV at Phat is not visible from Pi, ˆ Pk, which is the expecte number of etections at cell Pk at any instant is mae use of. An since ( κ ) is the fraction of FOV at Phat is not visible ˆ Pk is multiplie by that fraction. This makes use of the assumption that the istribution of targets is uniform across an area. In our plots of ˆ Pk one for the surveillance zone variations between cells are steep only towars the eges of the zone. A more rigorous computation involves evaluating the pf for the overlappe an non-overlappe areas separately. This is being avoie since (2) is compute uring the online phase of the algorithm. This assumption is also invoke in step while reucing the number of etections ue to overlapping FOV. C The Coorination Phase: The coorination phase prevents paths of sensors to come close to one another to avoi overlapping FOV. This is one by imposing penalties an reucing values of n ˆ, an recomputing paths for sensors with lower priorities as escribe in steps 5, an 7. The path of the sensor with highest etections is fixe. The path with the secon highest recompute if there are overlapping areas of observation with the first. The recompute path constitutes a motion strategy that is optimal in terms of etections uner the constraint that the path of highest sensor is fixe. Similarly the path of the sensor with the least priority sensor when recompute is optimal uner the constraint that the paths of the sensors with higher priority are fixe. Hence at the coorination phase the optimality of the algorithm is not complete. A fully complete algorithm woul involve a search in the oint space of all sensors that is combinatorially har.. IV SIMULATION RESULTS This section reports results obtaine through simulations on our environment evelope using Borlan s JBuiler IDE. The value of the number of time steps, T, use for these simulations is 3, carrie out on a P4 workstation with clock spee of.8 GHz. Figure 3 shows a snapshot of the environment with sensors. The current position of the sensor is shown through the bigger shae circle, while the targets through smaller ones. The traces of sensor movements are also shown. Target traces are not shown to avoi cluttering.

5 Table compares the performance of the system in the presence an absence of coorination. The experiments one with sensor coorination are labele as a, 2a,, while those without coorination are labele as b, 2b,. Each experiment laste for time steps in total. The first column of the table enotes the inex of the experiment; the secon enotes the number of sensors in the system, the thir the average etections or the number of targets etecte by atleast one sensor per sample. The fourth signifies the average fraction of targets etecte per sample, which is the number of targets etecte per sample by one or more sensors ivie by the number of targets in the surveillance zone per sample. Columns 5, an 7 represent the number of targets etecte exactly by one, two an three sensors per sample. The last column enotes the target velocity. Detections by more than three sensors are not tabulate for they generally assume insignificant values. Of particular interest in comparing the two schemes are the average fraction of target etections as well as the number of targets etecte by exactly one, two an three sensors. It is expecte that the average fraction of etections as well as the number of targets etecte by exactly one sensor to be more for the metho with coorination incorporate. Simultaneously the number of targets etecte by two or more sensors is anticipate to be higher for the metho that oes not use the coorination phase. As the overlapping areas of FOV are higher in the absence of coorination the number of targets etecte by two or more sensors is also higher. This in turn allows more targets to go unetecte an the average fraction of etections to be lesser. The abbreviations use in the column heaings are explaine in the table caption. Figure 3: A simulation system with ten sensors an several targets. The robots are shown through larger circles an targets through smaller ones. The traces of sensors are also shown. EN NS AD ZD AF DS D2S D3S TV a b a 2b a 3b a 4b a 5b Table : Comparison between motion strategies with an without sensor coorination. Experiments with labels ening in a or the first line in every row have coorination incorporate while those ening in b or the secon line in each row are uncoorinate. Abreviations are as follows: EN = experiment number, NS = number of sensors, AD = Average Detections per sample, ZD = Zero Detections per sample, AF = Average Fraction per sample, DS = Detections by exactly sensor, D2S = Detections by exactly 2 sensors, D3S = Detections by exactly 3 sensors, TV = Targets Velocity As seen from table the performance of the coorinate scheme with ecouple optimization is better for all experiments with iffering NS an TV values than the metho sans coorination. For example the number of targets etecte by exactly one sensor per sample, the DS column, is significantly higher in a number of experiments for the coorinate vis-à-vis the uncoorinate scheme. It is higher by, 4 an etections per sample in experiments, 2 an 3. Also the number of targets etecte by two sensors is significantly lesser across all experiments for the coorinate metho. In experiments an 5 the number of targets etecte by three sensors is almost one per sample for the metho lacking coorination. Corresponingly average fraction of targets etecte per sample for the coorinate metho is higher in all the experiments. The results of the table tally with the expectations an the unerlying reasons for these expectations mentione earlier. In table 2 we compare the current strategy with four of our previous methos of target etection an pursuit [4]. The first column signifies the experiments. Inexes a, 2a, etcetera correspon to the current metho. Those with inices b, c, an e correspon to coorinate-istracte, coorinate - eicate, local-istracte an local-eicate methos of resource allocation of sensors to targets for target etection [4]. Table 2 shows that across all experiments the current scheme surpasses all others in terms of average fraction of targets etecte as well as other criteria such as DS, D2S, AD an AF. It is to be note that a ifference of. in average fraction per sample between two methos correspons to a ifference of 4 or 5 target etections per sample (there are 7 to 8 targets in the zone per time-step on an average), which in turn correspons to a ifference of etections in time-steps of simulation. The value of λ was.3 an Q=8 in these simulations. V DISCUSSIONS an COMMENTS Sub-Optimality: The algorithm is mae suboptimal merely to reuce the search space involve in making it

6 globally optimal. A straightforwar implementation woul involve a irect search in the space of all sensors by extening this approach. The metho of assigning priorities to constrain the search has its counterpart in ecouple priority base approaches to multi robot path planning [7,8]. Generality an Assumptions Involve: The current approach retains its utility of optimality for any given arrival profile or arrival statistics an in that sense generalize in principle. The etails vary with respect to how equations () an (2) are compute for ifferent target statistics but the overall approach remains the same. It is not bining the algorithm that the velocities of the targets be constant an uniform. The algorithm shall continue to be optimal in a statistical sense as long as the velocity profile an irection profile is known or can be moele. Varying the velocity of the targets leas to istributions of various parameters at each gri point rather than unique values that in turn makes the optimality of algorithm less eterministic an more statistical. EN NS AD ZD AF DS D2S D3S TV a b c e a 2b 2c 2 2e 3a 3b 3c 3 3e 4a 4b 4c 4 4e 5a 5b 5c 5 5e V CONCLUSIONS Table 2: Comparison of current metho (label a) with four previous methos calle coorinate-istracte (b), coorinate-eicate (c), local-istracte () an local-eicate (e). The abreviations are same as in table. A metho for motion strategy computation of a sensor that maximizes the number of target etections for the next T-time steps is presente. The metho makes use of a-priori known statistics of target arrivals along with etections reporte for the current sample to estimate the number of etections on a lattice of cells. To reuce overlaps a coorination mechanism is specifie that performs a constraine search, where the constraints are in the form of priorities assigne to sensors. The absence of an exhaustive search in the oint space ue to the constraints reners the algorithm sub-optimal from a multi sensor perspective although optimal with respect to a single sensor in presence of those constraints. The tabulations presente in the results section vinicate the performance of the current approach in comparison to previous approaches for target etection an pursuit. The performance enhancement ue to the coorination phase that reuces overlaps in FOV between sensors is also tabulate in section IV. REFERENCES: [] S. A. Stoerer, P. E. Rybski, M. D. Erickson, M. Wyman, M. Gini, D. F. Hougen, N. Papanikolopoulos, A Robot Team for Exploration an Surveillance: Design an Architecture, Proceeings of the International Conference on Intelligent Autonomous Systems, pp , Venice, Italy, July 2. 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