Space Partitioning and Classification for Multi-target Search and Tracking by Heterogeneous Unmanned Aerial System Teams Jared Wood∗ and J. Karl Hedrick
†
Center for Collaborative Control of Unmanned Vehicles, University of California, Berkeley, USA
Search and tracking of an unknown number of targets within large search areas is a task performed in settings such as search and rescue, surveillance, area patrol, and reconnaissance. This task can be accomplished by a team of cooperative, heterogeneous, autonomous agents. In order to search for and track targets by a team of agents, target sensing and position estimation are required, as well as algorithms for agent path planning. The estimation of targets is accomplished by fusing detection/no-detection observations. The fusion of these observations forms a distribution of estimated target density over the target search space. The nature of an unknown number of multiple targets and small sensor fields of view results in several maxima in the target density distribution corresponding to potential locations of targets. The goal of the team of agents is to reach system steady-state consisting of the entire target space searched and all targets discovered and being tracked by agents. This corresponds to the target density distribution reaching its lowest possible uncertainty. Assuming mobile targets, once a target has been detected, frequent repeated detections are required to maintain low local uncertainty of the target location. Consequently, as targets are detected, the region of the detections must be tracked. As tracking regions appear, some agents transition from searching to tracking. Extending this to the entire target search area, based on the target density distribution the area is dynamically partitioned to account for regions of potential targets with varying levels of uncertainty. Features are extracted from these partitioned regions to classify the type of search or track that should be performed for each region. The partitioned regions are viewed as tasks and are assigned to the team of agents. The main focus of this paper is to present an approach for dynamically partitioning the target search area, based on the target density distribution, into evolving regions of potential targets of varying local uncertainty. Classification of the partitioned regions into varying levels of search or tracking tasks is also presented. Additionally, estimation of the target density distribution is discussed. Results are provided of a search task consisting of six Unmanned Aerial System (UAS) agents equipped with visual spectrum camera sensors searching for an unknown number of targets.
Nomenclature f (x) Target density distribution Pi Partition number i E Expectation operator Subscript i Variable number ∗ Ph.D. † James
candidate, Vehicle Dynamics Lab, UC Berkeley,
[email protected] Marshall Wells Professor, head of Vehicle Dynamics Lab, Mechanical Engineering Department, UC Berkeley, Member
AIAA
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I.
Introduction
When searching a large area for an unknown number of targets, the field of view of an observation is small relative to the search area. It is necessary to maintain an estimate of target positions and area coverage as the search progresses and use this estimate to optimize the evolving search path.1 As knowledge of the search area changes with new observations, decisions must be made on how to adjust the search strategy. In this paper, the approach toward solving the search planning problem is decomposed as shown in Fig. 1(a). Observations update an estimate of target density over the search area, block (1) in Fig. 1(a). Features computed over the target density are then used to classify regions of the search area into dynamically changing partitions, Fig. 1(a) block (2). Paths are then planned over the search area partitions, Fig. 1(a) block (3). The motivation to decompose search planning in this manner is to separate the planning in two levels as depicted in Fig. 1(b). The first level being a task allocation problem over the partitions (Fig. 1(b) block (1)) and the second level being a path planning problem directly optimizing over the distribution of target density (Fig. 1(b) block (2)). In this paper, an approach toward classifying the search area into dynamic partitions based on a distribution of target density (Fig. 1(a) block (2)) will be presented. Sensor Observations (1) Target Density Estimation
Partitions
Target Density Distribution
(1) Partition Task Allocation
(2) Search Area Dynamic Partitioning
Allocated Partitions
Partitions
(3) Path Planning over Search Area Partitions
(2) Path Optimization Over Partition Target Density Distribution Paths
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(a)
(b)
Figure 1. (a) Structure of search planning decomposition into dynamic partition classification and path planning over the partitions. (b) Structure of path planning separated into task allocation over search partitions and path planning by optimizing directly over the target density distribution.
To motivate the decomposition of search planning (Fig. 1(a)) as presented in this paper, three types of search problems are now considered, along with some current approaches toward solving them. The types of search are 1) it is known that there is only one target, 2) there is a known number of multiple targets, and 3) there is an unknown number of targets.2 For the first type, a single probability distribution can be used to estimate the target position, so implementations of standard recursive Bayesian estimation work well. Consequently, since there’s a single probability distribution, paths can be optimized directly over it.3, 4 And there are several information metrics that can be used for the path optimization, such as probability of detection,3 entropy gain,5 mutual information,6, 7 Kullback-Leibler divergence,8 and other generalizations of information.9, 10 Consequently most path planning strategies have been developed for this case. The effectiveness of the search can be improved by utilizing a team of cooperating agents.6, 7, 11–15 To achieve some level of cooperation there are several approaches. One approach is to have each agent fuse observations from all other agents and choose a path that optimizes over the probability distribution.6, 7, 12 This approach is termed coordinated because the agents do not directly consider the paths of other agents. Another approach is to additionally have each agent receive the predicted paths of all other agents and perform a decentralized optimization for the best path.16, 17 This approach is termed cooperative. Because of the vast amount of communication and compu2 of 11 American Institute of Aeronautics and Astronautics
tation required in fully cooperative approaches, alternative approximately cooperative approaches have been developed.11, 18 The path an agent chooses to follow for each of these methods ultimately is determined by optimization over a probability distribution of the predicted target location. At this point typically either single or multiple-step finite horizon path optimization determines the path for the agent to follow. However, there are performance limitations with finite-horizon optimization. Ways to get around these limitations are proposed in.11, 19 For the second type, variations on Bayesian estimation have been developed, such as multiple hypothesis tracking.20 The most naive approach is to maintain multiple independent probability distributions (one for each target) and assume that the target association of observations is given. However, the target association must be addressed in reality and methods for this case cease to be purely Bayesian estimation because they maintain track tables and require some form of target association.2, 20 Yet in most of these methods a set of distributions are constructed (one for each target) and the path planning optimization strategies developed for the first type can be extended to account for optimization over multiple probability distributions.13, 14 For the third case, the complexity of maintaining a probability distribution for each target (including unknown targets) becomes difficult. This difficulty inspired development of a generalization of the standard recursive Bayesian estimation framework from random variables to random sets that provides a target density distribution over the search area instead of multiple probability distributions.2 With this estimation method, there’s one distribution over the search space that combines all targets into an estimate of target density. Consequently, the path planning strategies developed for the first two cases are no longer applicable. Yet, path planning strategies have been developed that choose paths by maximizing the expected number of targets in the search area.21 However, analysis of the target density distribution can determine regions within the search area that correspond to 1) regions of possible target existence, 2) regions of almost certain target existence, and 3) regions of almost certainly no targets. The search area can then be dynamically partitioned into multiple search and tracking tasks and allocated to the team of agents. Within an allocated partition, agents’ paths can be chosen by applying methods for path planning developed for the first two search types that optimize directly over the target density distribution. In this paper the target density distribution will be defined and an approach toward dynamically classifying the search area into partitions will be presented. This approach involves 1) determining features used to analyze a target density distribution and 2) using the features to dynamically classify regions of the search area into evolving partitions. Results of the dynamic partitioning are presented for a team of six Unmanned Aerial System (UAS) agents equipped with visual spectrum camera sensors searching for an unknown number of targets.
II.
Dynamic Partitions from Target Density Distribution
To motivate dynamic partitioning based on analysis of a target density distribution, first consider the case of estimating a target’s position by a probability distribution. The goal is to have the probability distribution reach the point of having only one mode with low uncertainty. To achieve this, agent paths can be computed by optimizing directly over the probability distribution (e.g., information gain maximization). This method works well because if the target is detected, the probability of the target being outside the observation field of view is reduced. However, for the case of estimating the density of an unknown number of targets, detection of a target does not permit reducing the density of targets outside the observation field of view. Consequently, path planning methods developed for optimizing directly over probability distributions cannot be used on their own. Yet, to take advantage of the development that has gone into these methods, the target density distribution is dynamically partitioned. Parition level target density distributions can then be extracted and path planning within a partition can be achieved by directly optimizing over the partition’s target density distribution. II.A.
Target Density Distribution
The classification of points in the search area into partitions is based on analysis of a target density distribution f (x). Consider the search area S. A target density distribution is defined over the search area as Z f (x) dµ(x) (1) ENA = x∈A⊂S
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where NA is the number of targets in the subarea A. The target density distribution f (x) may be represented by a continuous or discretized function over the search area S. A target density distribution provides an estimated number of targets within regions of the search area. Consequently, given a target density distribution, the expected number of targets within a region can be evaluated simply by integrating f (x) over the region. The expected total number of targets in the search area is then Z f (x) dµ(x) (2) ENS = x∈S
The approach presented in this paper to classify regions in the search area into partitions frequently takes advantage of the ability to quickly evaluate the expected number of targets within an area by integrating the target density distribution over the area. The dimension of the space over which the target density distribution is defined is equal to the dimension of a single target’s position. Note that in this paper, the search is defined over a two dimensional planar area. Because the dimension of the estimation space is small and the same for any number of targets, direct estimation of a target density distribution for an unknown number of targets is an excellent choice.2 How exactly a target density distribution is estimated is beyond the scope of this paper. However, methods for recursively estimating a target density distribution have been developed and applied to target tracking problems.21 In the event that the target positions are estimated by some method other than a target density distribution, a target density distribution can easily be computed. Consider the three types of search discussed in this paper: 1) it is known that there is only one target, 2) there is a known number of multiple targets, and 3) there is an unknown number of multiple targets. For the first type, the target’s location is estimated by a single probability distribution p(x) and we have that f (x) = p(x). For the second type, multiple probability distributions {p1 (x), ..., pN (x)} may be used to estimate each target’s position. Consequently, we have that P f (x) = i pi (x). The same can be done, except the estimation problem is much more difficult, for the third case. II.B.
Dynamic Partitions
Points in the search area are classified into dynamically changing partitions. The partitions are composed of 1. An exploration partition 2. An ordered set of search and tracking partitions The exploration partition consists of 1) areas that have low information content and 2) large repeatedly observed areas with very low target density. Fig. 2 presents an overview of the dynamic partition classification. The process of partitioning the search area according to the target density distribution involves 1) determination of features that describe target density distribution shape and evolution and 2) classification of the search area into partitions descriptive of the features. II.B.1.
Features
Consider seeking for regions within a target density distribution that contain points of similar characterics. Repeatedly observed regions will have high certainty while unsearched regions will have low certainty. Regions in which targets have been repeatedly detected will have a high expected number of targets while repeatedly observed regions in which no targets have been detected will have a low expected number of targets. Some regions will have high certainty and a high expected number of targets. Others will have low certainty and an unaltered expected number of targets. Yet others will have a low expected number of targets. The search area can accordingly be partitioned into regions based on characteristic certainty and expected number of targets. The target distribution is then analyzed by considering features corresponding to local uncertainty, local expected number of targets, and normalized position. Each of these features are based on the local area Sr (x0 ) ⊆ S of some point x0 ∈ S defined as Sr (x0 ) := {x ∈ S : d(x, x0 ) < r} 4 of 11 American Institute of Aeronautics and Astronautics
(3)
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Classify Search Map by Local Entropy & Local Expected Number of Targets Search Map
(2)
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Classify Subpartitions by Normalized Position & Partition Expected Number of Targets
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Figure 2. Structure of the cascade of classifiers for partitioning the search area into an exploration partition and an ordered set of search and tracking partitions.
where d(x, x0 ) is some measure of distance. Local uncertainty is computed by evaluating the local entropy defined as Z 1 dµ(x) (4) fr (x, x0 ) log Hr (x0 ) := fr (x, x0 ) x∈Sr (x0 ) where fr (x, x0 ) is the locally normalized target density function defined by f (x) γ∈Sr (x0 ) f (γ)dµ(γ)
fr (x, x0 ) := R
(5)
A locally normalized density is required so that the maximum value for local uncertainty can be referenced during classification. Local expected number of targets is computed by integrating the target density distribution over the local area as Ir (x0 ) := ENSr (x0 ) Z f (x)dµ(x) =
(6)
x∈Sr (x0 )
The intuition behind each of these features is to imagine an area defined locally around some point in space. Then imagine a distribution defined over the area local to this point. Each feature is then a description of that locally defined distribution. II.B.2.
Classification
Recall the structure of the dynamic partition classification shown in Fig 2. Note that in Fig. 2, the classification is accomplished by a cascade of classifiers. The levels of the cascade classify points in the search area into 1. One partition for exploration and one for search and tracking (Fig. 2 block (1)). 2. Partitions corresponding to expected number of targets (Fig. 2 block (2)). 3. Ordered partitions accounting for level of search and tracking (Fig. 2 block (4)). 5 of 11 American Institute of Aeronautics and Astronautics
Observe Fig. 2 block (1). The search map output from this block labels which regions have at least some information content and potentially contain targets. Regions that have not been searched and initially have no bias of target density will have the least information content available in the search area. These regions will have a characteristic local uniformly distributed density from which can be determined characteristic values of local expected number of targets and uncertainty. Regions that have been searched and in which no targets were found will have a characteristically low expected number of targets. To determine which regions have information, the high entropy map must first be computed. The high entropy map is SHE := {x ∈ S :| Hr (x) − Hmax |< ǫ} (7) where Hr (x) is local entropy as defined in Eq. 4 and Hmax is the maximum local entropy possible defined as Hmax := log | Sr | where | Sr |:=
R
x∈Sr
(8)
dµ(x). When the target density distribution is discretized over the search area and d(x, x0 ) := max ({| x(1) − x0 (1) |, | x(2) − x0 (2) |})
(9)
then Hmax = log N1 N2 , where N1 is the number of columns and N2 is the number of rows representing the grid within a local area. For a particle-based discretization, this value is not constant for each point in the search area. Alternatively, an average value can be determined, resulting in Hmax = log Nave , where Nave is the average number of discrete points within the local area. At the start of the search, the target density distribution is biased by prior knowledge of where targets may be. This prior knowledge consists of • Prior distribution fprior (x) of previously known target positions. • Estimated number of additional targets ENadditional that may exist in the search area. The prior distribution fprior (x) provides a density bias based on where targets have most recently been observed and where they might be now. The estimated number ENadditional of additional targets affects the initial R target density distribution by defining a uniform distribution U (x) such that U (x ∈ S) = ENadditional / x∈S dµ(x). As observations are made, the target density distribution in observed regions with no detected targets will drop below the uniform distribution value. These areas are defined as SN I := {x ∈ S : f (x) < U + ǫ}
(10)
The search map Ssearch is then defined as Ssearch := S \ (SHE ∩ SN I )
(11)
Fig. 3 shows a sample search map along with its corresponding target density distribution. In the figure, black areas represent the search map and white areas represent areas left for exploration. Note that this search map was based on a prior distribution for a search task in which some knowledge was given of eight existing targets and it was estimated that there were two additional targets with no prior knowledge. As time progresses, the target density distribution becomes less smooth and has regions of greater certainty. Consequently the search map breaks into unconnected pieces that specify where the search should be conducted. The search map is then passed on for further classification as shown in Fig. 2. The set of points within the search map are further classified based on local expected number of targets Ir (x) as defined in Eq. 6 and normalized position (Fig. 2 block (2)). At the start of the search task this classifier is initialized with a K-means22 algorithm, whereas through the rest of the search task a Gaussian Mixture Model22 is used. The Gaussian Mixture Model classifier at time t is initialized with the partitions computed at time t − 1. Fig. 4 presents a sample partitioning based on the search map in Fig. 3. In Fig. 4, variations in color are used to represent the partitions of the search area. If some targets are close to each other, there will likely be partitions with local expected number of targets greater than 1. The purpose of the final classifier (Fig. 2 block (4)) is to perform subpartition classification.
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Figure 3. The search map (left) computed from a sample target density distribution (right). In the search map plot, white areas correspond to exploration space and black areas correspond to space that will be partitioned further.
Figure 4. The search and tracking partitions (left) computed from a sample target density distribution (right). Partitions are represented by variations in color over the search area
To accomplish this, the classifier first orders the partitions {P1 , ..., PN } according to partition density ρPi defined as R f (x)dµ(x) R i ρPi := x∈P (12) dµ(x) x∈Pi
where Pi is the ith partition. The classifier then iterates through the partitions in descending order. If a partition Pi has an expected number of targets Z NPi = ceil f (x)dµ(x) (13) x∈Pi
>1
then Pi is subpartitioned into NPi partitions. This classifier continues until a limit on the number of partitions is reached. The limit on the number of partitions can be determined from the current estimate of the total expected number of targets in the search area Z f (x) dµ(x) (14) ENS = x∈S
After the search area is partitioned, agent path planning can then be performed. The type of path planning to pursue is beyond the scope of this paper. However, the partitions generated by this classifier define areas over which subsets of the target density distribution can be extracted. This suggests the application of some kind of task allocation algorithm23 that takes each of the partitioned search areas as tasks with varying level of certainty or priority. As agents are allocated to partitions, their paths within the partitions can then be determined by optimizing over partition level target density distributions. 7 of 11 American Institute of Aeronautics and Astronautics
III.
Results
To test the performance of dynamic partitioning classification, a simulation environment was constructed. In this simulation, the team of agents consisted of six UAS agents equipped with visual spectrum gimballed camera sensors. The capabilities of the UAS agents were designed to closely represent behaviors observed in flight experiments.24 The camera sensors were designed with a field of view corresponding to a view angle of 0.9273 rad. Effect of resolution was included by limiting the distance of allowable observations to 250 m. The UAS were designed to fly at 25 m/s and 100 m altitude with a limited maximum turn rate of 0.2 rad/s. Although the number of targets in the search area was unknown to the search planning of the UAS agents, the total number of targets was six. The targets were allowed to move according to a transition model defined by " # cos(θ) xT,t = xT,t−1 + r (15) sin(θ) where r and θ are distributed as 1 2 r∼ √ exp − 2 (r − µ) 2σ 2πσ 2 θ ∼ U nif orm([0, 2π)) 1
(16)
with µ = 2 m in one second and σ 2 = 2 m2 . Consequently, each of the targets were biased to move from their position at each simulation iteration by an amount that corresponds with a mean of 2 meters in one second. The time interval of each simulation iteration was 4 seconds and the inference time interval was 4 seconds. The type of path planning used for this simulation was simple but effective in showing the performance of dynamic partitioning classification as applied to searching for an unknown number of multiple targets. The structure of the path planning was as presented in Fig. 1(b). Fig. 1(b) block (1) was accomplished by a simple task allocator that allocated partitions in ascending order of partition level entropy and descending order of partition level expected number of targets to the nearest UAS agents. Fig. 1(b) block (2) was accomplished by first computing the position of an allocated partition level target density distribution peak value. The UAS agent turn rate was then computed by minimizing the distance from the predicted UAS agent position and the peak position. The point chosen for sensor observation was chosen to be the position of locally maximum target density distribution value. Many simulations were performed, with various initial target positions, prior target density distributions, and initial UAS agent positions. As the partitioning evolved over time, the general trend was that the sizes of the partitions in which targets existed gradually decreased as is shown in the plot of Fig. 5(a). The cause of the decrease in the size of partitions containing targets was that tracking partitions started to appear. However, the average size of every partition tended to remain fairly constant over time as shown in the plot of Fig. 5(b). This means that, while small tracking partitions were forming around known targets, additional search partitions were being added and growing in size as the exploration partition was decreasing in size. This general behavior is exactly what was desired of the dynamic partitioning classification. As the search area is explored, search partitions grow to account for regions that should be further searched. Meanwhile, regions of known targets appear and gradually decrease in size as the partition certainty increases. A sample sequence of partitions over time will be presented. Before presenting this sequence, see Fig. 6 for labels of the the objects that will be found in the sequence of partitions. As shown in Fig. 6, circles with protruding lines are representations of UAS agents, thin lines extending from a UAS agent are representations of the agent’s sensor field of view, circles with no protruding lines are representations of targets, and the entire search area is partitioned as represented by variations in color. Fig. 7 presents a sequence of a sample simulation in which dynamic partitioning classification was performed. In this sample simulation, the search task consisted of six UAS agents searching an area that contained six targets (although the total number of targets was unknown to the UAS search planning). As observations were made, regions corresponding to no-detections broke up the search area and regions corresponding to detections refined the search area and caused tracking partitions to emerge.
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Figure 5. (a) Sample mean average partition size of partitions containing targets over simulation time. (b) Sample mean average overall partition size over simulation time. Error bars represent sample standard deviation.
IV.
Conclusion
In order to autonomously search for and track an unknown number of multiple targets, dynamic partitioning classification of a target density distribution was presented. Dynamically partitioning the search area allows the search planning to be separated into 1) a layer in which agents are allocated to partitions and 2) a layer in which agent path planning is determined by directly optimizing over partition level target density distributions. Consequently, path planning can be accomplished by combining methods for task allocation and methods for optimizing over distributions. Results of dynamic partitioning classification were presented for a team of six UAS agents equipped with visual spectrum camera sensors in which there were six targets (although the number of targets was unknown to the search planning). Results show that dynamic partitioning performs well to construct search and tracking regions.
References 1 Stone,
L., Theory of Optimal Search, Academic Press, New York, NY, 1975. R. P. S., Statistical Multisource-Multitarget Information Fusion, Artech House, Inc., Norwood, MA, USA, 2007. 3 Bourgault, F., Furukawa, T., and Durrant-Whyte, H., “Optimal Search for a Lost Target in a Bayesian World,” Field and Service Robotics, edited by S. Yuta, H. Asama, E. Prassler, T. Tsubouchi, and S. Thrun, Vol. 24 of Springer Tracts in Advanced Robotics, Springer Berlin / Heidelberg, 2006, pp. 209–222, 10.1007/10991459 21. 4 Kreucher, C., Hero, A., Kastella, K., and Morelande, M., “An Information-Based Approach to Sensor Management in Large Dynamic Networks,” Proceedings of the IEEE , Vol. 95, No. 5, May 2007, pp. 978 –999. 5 Lindley, D. V., “On a Measure of the Information Provided by an Experiment,” The Annals of Mathematical Statistics, Vol. 27, No. 4, 1956, pp. 986–1005. 6 Grocholsky, B., Makarenko, A., and Durrant-Whyte, H., “Information-theoretic coordinated control of multiple sensor platforms,” Robotics and Automation, 2003. Proceedings. ICRA ’03. IEEE International Conference on, Vol. 1, 2003, pp. 1521 – 1526 vol.1. 7 Hoffmann, G. and Tomlin, C., “Mobile Sensor Network Control Using Mutual Information Methods and Particle Filters,” Automatic Control, IEEE Transactions on, Vol. 55, No. 1, 2010, pp. 32 –47. 8 Cover, T. M. and Thomas, J. A., Elements of Information Theory, John Wiley & Sons, Inc., 2001. 9 Csiszar, I., “Information-type measures of difference of probability distributions and indirect observations,” Sudia Sci. Math. Hungar., Vol. 2, 1967, pp. 299–318. 10 Renyi, A., “On measures of entropy and information,” Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability, Vol. 1, 1961, pp. 547–561. 11 Tisdale, J., Kim, Z., and Hedrick, J., “Autonomous UAV path planning and estimation,” Robotics Automation Magazine, IEEE , Vol. 16, No. 2, 2009, pp. 35 –42. 12 Bourgault, F., Furukawa, T., and Durrant-Whyte, H., “Coordinated decentralized search for a lost target in a Bayesian world,” Intelligent Robots and Systems, 2003. (IROS 2003). Proceedings. 2003 IEEE/RSJ International Conference on, Vol. 1, Oct. 2003, pp. 48 – 53 vol.1. 13 Wong, E.-M., Bourgault, F., and Furukawa, T., “Multi-vehicle Bayesian Search for Multiple Lost Targets,” Robotics and Automation, 2005. ICRA 2005. Proceedings of the 2005 IEEE International Conference on, 2005, pp. 3169 – 3174. 2 Mahler,
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Figure 6. Description of the objects that will be presented in a sample simulation’s sequence of dynamic partition classification of simulation time. UAS agents are represented as circles with protruding lines. UAS sensor fields of view are represented as additional thin lines around the UAS. Targets are represented as circles without protruding lines. Partitions are represented by variations in color across the search area.
14 Furukawa, T., Bourgault, F., Lavis, B., and Durrant-Whyte, H., “Recursive Bayesian search-and-tracking using coordinated uavs for lost targets,” Robotics and Automation, 2006. ICRA 2006. Proceedings 2006 IEEE International Conference on, May 2006, pp. 2521 –2526. 15 Ryan, A., Durrant-Whyte, H., and Hedrick, J. K., “Information-theoretic Sensor Motion Control for Distributed Estimation,” Proceedings International Mechanical Engineering Congress and Exposition, 2007. 16 Bourgault, F., Furukawa, T., and Durrant-Whyte, H., “Decentralized Bayesian negotiation for cooperative search,” Intelligent Robots and Systems, 2004. (IROS 2004). Proceedings. 2004 IEEE/RSJ International Conference on, Vol. 3, 2004, pp. 2681 – 2686 vol.3. 17 Fraser, C., Bertuccelli, L. F., and How, J. P., “Reaching Consensus with Imprecise Probabilities Over a Network,” AIAA Guidance, Navigation, and Control Conference, 2009. 18 Mathews, G. M., Durrant-Whyte, H., and Prokopenko, M., “Decentralised decision making in heterogeneous teams using anonymous optimisation,” Robotics and Autonomous Systems, Vol. 57, No. 3, 2009, pp. 310 – 320, Selected papers from 2006 IEEE International Conference on Multisensor Fusion and Integration (MFI 2006), 2006 IEEE International Conference on Multisensor Fusion and Integration. 19 Wood, J., Kehoe, B., and Hedrick, J. K., “Target Estimate PDF-based Optimal Path Planning Algorithm with Application to UAV Systems,” Proceedings Dynamic Systems and Control Conference, ASME, 2010. 20 Stone, L. D., Corwin, T. L., and Barlow, C. A., Bayesian Multiple Target Tracking, Artech House, Inc., Norwood, MA, USA, 1st ed., 1999. 21 Mahler, R., “Unified sensor management using CPHD filters,” Information Fusion, 2007 10th International Conference on, 2007, pp. 1 –7. 22 Duda, R. O., Hart, P. E., and Stork, D. G., Pattern Classification, John Wiley & Sons, Inc., New York, NY, USA, 2001. 23 Godwin, M. and Hedrick, J. K., “Stochastic approximation of an online vehicle routing problem for autonomous aircraft,” Infotech@Aerospace Conference, 2011. 24 Garvey, J., Kehoe, B., Basso, B., Godwin, M., Wood, J., Love, J., Liu, S.-Y., Kim, Z., Jackson, S., Fallah, Y., Fu, T., Sengupta, R., and Hedrick, J. K., “An Autonomous Unmanned Aerial Vehicle System for Sensing and Tracking,” Infotech@Aerospace Conference, 2011.
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Figure 7. Sequence of dynamic partitioning classification over time for a sample simulation involving six UAS agents and six targets (although the number of targets was unknown to the search planning). Follow the sequence from left to right and from top to bottom. In (a) the initial partitions are formed. In (b) tracking partitions start to appear. By (f ) all targets are within tracking partitions and much of the remaining area is assigned to search partitions.
11 of 11 American Institute of Aeronautics and Astronautics
