Exploiting sensor mobility and covariance sparsity for distributed tracking of multiple sparse targets
 Guohua Ren^{1},
 Vasileios Maroulas^{2} and
 Ioannis D. Schizas^{1}Email author
https://doi.org/10.1186/s136340160354y
© Ren et al. 2016
Received: 2 November 2015
Accepted: 22 April 2016
Published: 4 May 2016
Abstract
The problem of distributed tracking of multiple targets is tackled by exploiting sensor mobility and the presence of sparsity in the sensor data covariance matrix. Sparse matrix decomposition relying on normone/two regularization is integrated with a kinematic framework to identify informative sensors, associate them with the targets, and enable them to follow closely the moving targets. Coordinate descent techniques are employed to determine in a distributed way the targetinformative sensors, while the modified barrier method is employed to minimize proper error covariance matrices acquired by extended Kalman filtering. Different from existing approaches which force all sensors to move, here, local updating recursive rules are obtained only for the targetinformative sensors that can update their location and follow closely the corresponding targets while staying connected. Simulations advocate that the proposed scheme outperforms alternative tracking schemes while accurately tracks multiple targets by imposing mobility only on the targetinformative sensors.
Keywords
1 Introduction
In recent years, potential applications of sensor networks (SN) have expanded due to the low cost of the sensing units, their ability to cover large areas, and the robustness distributed processing offers. One characteristic exploited more and more in sensor networks is sensor mobility and the design of kinematic rules that control sensor movement. Sensor mobility adds extra flexibility to a sensor network making it capable of covering larger areas, as well as being more energy efficient and robust [1]. Mobile sensors have been extensively utilized in target tracking applications to enable sensors to closely follow the moving target(s) and provide accurate target location estimates [2–4]. The aforementioned approaches require all the sensors to keep active [2–4], which may lead to excessive resource consumption despite the targets’ locality and the fact that in practice a small portion of sensors may possess useful information about the present targets. The aim here is to design an adaptive scheme that exploits mobility and covariance sparsity to associate targets with sensors and then properly determine kinematic strategies only for the informative sensors which will closely follow the field targets.
In the absence of sensor mobility, there has been a plethora of approaches for tracking multiple targets while associating targets with sensor measurements. Existing works [5–8] associate measurements acquired at static sensors with targets across time and rely heavily on probability models. A distributed Kalman filtering scheme is proposed in [9] relying on information diffusion strategies. In [9], only neighboring sensors collaborate, though all sensors in the network are utilized to track a single source while sensors have fixed locations. A different approach is followed in [10], where consensusaveraging is employed across the whole sensor network and all the sensors are forced to be active irrespective of the quality of their measurements. In [11], a related singletarget distributed tracking approach is proposed, in which extended Kalman filtering is employed for tracking. A probability model is assumed to determine informative sensors which may lead to instability due to its dependence on the tracking estimates. Different in this paper, distributed tracking of multiple targets will be considered, while sensor mobility will be exploited and combined with a sensortotarget association scheme for selecting targetinformative sensors without the need of relying on model parameters and state estimators that maybe inaccurate and result divergence. It should be pointed out that the distributed characterization here is referring to the fact that (i) only neighboring sensors need to communicate with each other and collaborate for multitarget tracking, while (ii) processing will take place in a few head sensors and will not involve all sensors in the network but only those sensors that bear information about the moving targets.
Singletarget tracking using mobile sensor networks has been studied for a variety of different scenarios [12–14]. Most of these approaches control the movement of all sensors by minimizing the estimation error covariance, [4, 13], while the approach in [12] manages sensor mobility based on a Bayesian estimation model and restricting sensors to move only on a grid of locations. A path planning strategy for a setting involving a fixedlocation target and a single moving sensor is designed in [15] by maximizing the determinant of the Fisher information matrix corresponding to the configuration. In [16], an approach is proposed for controlling the trajectories of multiple UAVs by minimizing the localization uncertainty for a fixedlocation target setting where the target is emitting a radio signal. The work in [17] rigorously presents how sensor mobility can increase spatial resolution when tracking a target with mobile sensors.
When tracking multiple targets with mobile sensors, the approach in [3] proposed an active sensing model, whereas the targetsensor association is based on a nearest neighbor rule which heavily relies on the accuracy of the state estimator while a central processing center is required. The scheme in [18] tackled the problem of moving sensors using a flock control law where all sensors are utilized, while the targets are some of the moving sensors whose position is known. The approach in [19] is utilizing clustering and neural networks to move sensors under the assumption that target locations are available. The scheme in [20] designs a Kalman filtering approach with gradient descentbased kinematic rules under the assumption that it is known which targets every sensor observes bypassing in that way the essential sensortotarget association step. These schemes involve movement of all sensors at every time instant leading to resourcedemanding algorithms that do not exploit spatial locality of the field targets.
Measurements corresponding to sensors which are close to the same target tend to be statistically correlated. Given that targets are spatially localized and affect small portions of the sensor network, an approximately sparse sensor data covariance matrix is emerging. Sparsity (presence of a many zero entries in a vector or matrix) has been exploited in a wide range of applications including sparse regression and statistical inference, e.g., see [21, 22]. The problem of associating targets to sensors, as well as determining the sensors with targetinformative measurements, is formulated here as the task of decomposing a matrix into sparse factors. The sparse matrix factorization techniques in [23, 24] are integrated here with proper sensor kinematic strategies and tracking techniques to exploit sensor mobility. Note that in [23, 24], a stationary (immobile) sensor network is considered where sensors have fixed locations. Tracking in [23, 24] is performed by immobile sensors, whereas here tracking is generalized to a mobile network with the more challenging task of designing and integrating with multitarget tracking, sensor kinematic strategies that improve tracking accuracy while preserving local sensor network connectivity. Normone and normtwo regularization mechanisms are employed to formulate a pertinent minimization framework that recovers sparse covariance factors, while estimates the number of targets on the field. Coordinate descent techniques [25, 26] are employed to derive local updating recursions that allow sensors to associate with targets.
Different from the aforementioned tracking schemes using sensor mobility, here, only the targetinformative sensors will be enabled to move at every time instant and track closely the moving targets. Thus, only targetaffected portions of the sensor network will be used for tracking the moving targets, potentially resulting better resource consumption and prolongation of the network lifetime. Kinematic rules will be designed by minimizing proper error covariance matrices obtained by extended Kalman filtering recursions [27] used to track each of the targets. The minimization will be performed under connectivity constraints that ensure the moving sensors stay connected and are able to communicate. The modified barrier method ([26], pg. 423) is employed to solve a pertinent constrained minimization problem and obtain distributed kinematic rules that the mobile sensors can apply locally without the need of a central controller. In contrast to existing approaches, the novel framework identifies and controls the movement only of targetinformative sensors allowing for accurate tracking.
The novelties of the proposed framework with respect to existing work involves the following: (i) utilization of covariance sparsity and properly designed kinematic rules to perform dynamic spatiotemporal sensortotarget association and tracking using mobile sensors not present in [11, 23, 24]; (ii) different from existing approaches [12–14], it enables only the sensors acquiring informative measurements about the targets to move instead of moving all sensors to track a single target; (iii) different from [11], it does not rely on probabilistic models or state estimates to associate sensors with targets; (iv) it takes local sensor network connectivity into account to ensure that the tracking process carries out continuously; and (v) different from [15, 16] that consider fixedlocation target(s), here, kinematic strategies are developed for tracking multiple moving targets. The proposed approach here achieves this task only by using the sensor data and sparsityimposing mechanisms.
The paper is structured as follows. The problem formulation and setting are given in Section 2. The sensortotarget association scheme which provides the targetinformative sensors is delineated in Section 3.1 and is combined with extended Kalman filtering techniques in Section 3.2. Novel kinematic rules are developed in Section 3.3 after minimizing a pertinent error covariance matrix under connectivity constraints and employing the modified barrier method to derive local kinematic rules that enable the targetinformative sensors to update their location and follow closely to moving targets. The proposed algorithmic framework is detailed in Section 4.1, while the communication and computational costs are discussed in Section 4.2. Extensive numerical tests are carried out in Section 5 that corroborate the advantages of the novel method over existing alternatives.
2 Problem setting
An ad hoc sensor network conformed by p mobile sensors is considered here. The sensors monitor a field where an unknown and possibly timevarying number of moving targets is present. Each sensor communicates only with its neighboring sensors which are within its communication range and are able to exchange information with a singlehop of communication. The singlehop neighborhood of sensor j is denoted as \(\mathcal {N}_{j}(t)\), where t denotes the time index.
where \({\sigma _{u}^{2}}\) is the noise variance and I _{ K } denotes the identity matrix of size K×K, while Δ T denotes the sampling period.

A1: In the measurement model in (3), it is assumed that the targets act as transmitters and each sensor will receive one reflection of the signal emitted from the targets. Signals emitted from the targets propagate via free space, explaining the \(d_{j,\rho }^{2}(t)\) attenuation coefficients, and are superimposed as shown in (3), see e.g., [29].

A2: The signal amplitudes a _{ ρ }(t) are considered to be uncorrelated across the different targets.

A3: Among the summands \(a_{\rho }(t)d_{j,\rho }^{2}(t)\) in (3), only one has a large amplitude when sensor j is close to the ρth target, whereas others are negligible due to the squarelaw attenuation \(d_{j,\rho }^{2}(t)\) caused by the free space propagation.
Note that assumption A3 corresponds to a setting where at most, one target is present within the sensing range of a sensor. Note that this is a more relaxed version of the common assumption that one sensor just contains the measurement of a specific target [6–8]. The signal amplitudes a _{ ρ }(t) will be nonzero for the interval in which the corresponding target is active and moving while is kept at zero when the target is inactive and disappears.
The emitted, from the targets, signals a _{ ρ }(t) could correspond to communication radio signals that possibly the targets are transmitting, e.g., targets could correspond to cell phone users moving in an area or unnamed aerial/ground vehicles or military vehicles that move within the monitored area and need to be tracked, see e.g., [16]. The deployed sensors are listening for these signals to track the moving entities. The targets could correspond to independent entities; thus, it is expected that the information bits they transmit are uncorrelated, giving rise to uncorrelated transmission signals [30]. Thus, the communication radio signals that the targets may be emitting are utilized to perform tracking and move the sensors appropriately. Applications include localization and tracking of mobile users in wireless networks, as well as tracking of vehicles in tactical environments [16].
where Σ _{ a } is the diagonal covariance matrix of a _{ t }, while \(\bar {\mathbf {D}}_{t}:=\mathbf {D}_{t}\mathbf {\Sigma }_{a}^{1/2}\). Among the R entries in a _{ t }, there will be r(t) nonzero entries corresponding to the active targets moving at the sensed field at t. In the setting here, once a target becomes inactive (i.e., a _{ ρ }(t)=0), it remains inactive for the rest of time.
The goal is to enable the mobile sensors to track an unknown number of targets present in the monitored field. Novel target association and sensor mobility strategies will be combined with tracking techniques to enable sensors to accurately track the different target trajectories. Proper kinematic strategies will be developed to allow only a small percentage of targetinformative sensors to move, different from existing approaches [12, 14] where all sensors are moving at every time instant that may be more resource demanding. Judiciously selecting and moving sensors will enable target tracking even when the targets move outside the area originally monitored by the sensors.
3 Distributed association, tracking, and sensor kinematic strategies
3.1 Targetinformative sensor selection
Due to the presence of multiple target in the monitored field, the first goal is to determine sets of sensors, namely \({\mathcal {S}}_{\rho,t}\), that acquire information bearing measurements about the ρth target. From the observation model in (4), note that the strongamplitude entries of the ρ column in D _{ t }, namely \(\{\mathbf {D}_{t,:\rho }\}_{\rho =1}^{R}\), can reveal the sensors within subset \({\mathcal {S}}_{\rho,t}\). Specifically, recall that \(\mathbf {D}_{t}(j,\rho)=d_{j,\rho }^{2}(t)\); thus, when sensor j and target ρ are close in distance then the corresponding entry is expected to have large amplitude, while the further away they get from each other the closer to zero the corresponding entry gets. The matrix D _{ t } can be assumed approximately sparse. Thus, the strongamplitude entries (away from zero) in D _{ t,:ρ } can be used to determine the informative sensor members of \({\mathcal {S}}_{\rho,t}\) at time instant t. Thus, determining \({\mathcal {S}}_{\rho,t}\) boils down to the problem of recovering the support of the columns of D _{ t }.
corresponds to a realtime estimate for the ensemble mean at time instant t. Note that ω in Eqs. (6) and (7) is used in a way that puts more emphasis to the recent data while it gradually forgets the past data, which is exactly what an uptodate estimator needs to do for the timevarying setting considered here. The scaling (1−ω)(1−ω ^{ t+1})^{−1} in (6) and (7) is to ensure that the two estimates \(\hat {\mathbf {\Sigma }}_{{x,t}}\) and \(\bar {\mathbf {x}}_{t}\) for the ensemble quantities Σ _{ x,t } and \(\mathbb {E}[\mathbf {x}_{t}]\) will be unbiased.
where ⊙ denotes the Hadamard operator (entrywise matrix product), while \({\sigma _{j}^{2}}\) is the noise variance estimate at sensor j, and L is an upper bound for the number of active sensed targets r(t) (L≥r(t)) and \(\mathbf {M}_{t}\in \mathbb {R}^{p\times L}\) contains L columns that estimate the sparse columns of \(\bar {\mathbf {D}}_{t}\). M _{ t,:ℓ } denotes the ℓth column of M _{ t }. The formulation was first proposed in [23, 24] to perform targetsensor association in a network of stationary sensors that do not have moving capabilities. Here, this formulation will be utilized to determine the different sets of informative sensors observing different targets before being integrated with kinematic control rules.
The Hadamand operator ⊙ along with the adjacency matrix E in (8) allows only the singlehop covariance entries to be used since they can be calculated by direct communication of the corresponding neighboring sensors. The first term in (8) accounts for the structure in (5). Sparsity is induced in the columns of M _{ t } using the normone term in (8), (see e.g., [22]), while λ _{ ℓ } denotes the nonnegative sparsitycontrolling coefficient used to adjust the number of zeros in \(\hat {\mathbf {M}}_{t,:\ell }\). The coefficient ϕ≥0 in the last term of (8) promotes group sparsity among rows, [32], thus is introduced to adjust the number of nonzero columns of \(\hat {\mathbf {M}}_{t}\) needed to approximate \(\hat {\boldsymbol {\Sigma }}_{{x,t}}\). This is done to zeroout unnecessary columns in \(\hat {\mathbf {M}}\) when the number of active targets in the field is smaller than R. The number of nonzero columns in M _{ t } indirectly estimates the number of targets at time instant t, namely \(\hat {r}(t)\).
The cost in (8) is minimized by an iterative minimization scheme based on coordinate descent [25, 26], where sensor j is responsible for updating the jth row of M _{ t }, namely M _{ t,j:}. Specifically, the cost is minimized wrt one entry of M _{ t } or \(\text {diag}({\sigma _{1}^{2}},\ldots,{\sigma _{p}^{2}})\), while keeping the rest fixed to their most uptodate values. Sensor j updates the entries \(\{\mathbf {M}_{t}(j,\ell)\}_{\ell =1}^{L}\) and variance σ j,t2. During one coordinate cycle, all the entries of M _{ t } and \(\text {diag}(\sigma _{1,t}^{2},\ldots,\sigma _{{p,t}}^{2})\) will be updated.
while δ _{ j,i } denotes the Kronecker delta function, i.e., δ _{ j,i }=1 if j=i, and δ _{ j,i }=0 if j≠i. The roots of the two aforementioned polynomials can be calculated using companion matrices, see e.g., [33].
Sensor j needs to communicate only with its singlehop neighbors in \(\mathcal {N}_{j}\), in order to evaluate the coefficients of the polynomials in (9) and (10) and to update the noise variance estimates in (12). It can be shown that as k→∞, the updates \(\hat {\mathbf {M}}_{t}^{k1}\) converge at least to a stationary point of (8). Further, the sparsitycontrolling coefficients \(\{\lambda _{\ell }\}_{\ell =1}^{L}\) can be set using the strategy proposed in ([23], Sec. V.A). Once the sparse columns \(\{\hat {\mathbf {M}}_{t,:\ell }\}\) are estimated, their support (the indices of relatively strongamplitude entries) is used to determine which sensors sense a specific target at time t.
3.2 Tracking via extended Kalman filtering
Within each informative subset of sensors \(\mathcal {S}_{\rho _{\ell },t}\), the sensor closest in distance to the predicted position of the ρ _{ ℓ }th target, namely \(\hat {\mathbf {s}}_{\rho _{\ell }}(t+1t)\), is set as a the subset head sensor that will gather the measurements of all other sensors in \(\mathcal {S}_{\rho _{\ell },t}\) and perform the EKF tracking recursions.
3.3 Sensor kinematics
The focus in this section is to derive kinematic rules for the targetinformative sensors, which are selected according to the scheme in Section 3.1, such that they follow closely the moving targets and give accurate position estimates. The benefit from having a few sensors moving is that targets can be tracked even when they move away from the original field monitored by the sensors. Having sensors following closely, the moving targets can provide more reliable measurements about the targets than just using static sensors. Note that only informative sensors close to the targets will be responsible for carrying out the tracking procedure leading to resource savings. Toward this end, the informative sensors in each subset \({\mathcal {S}}_{\rho _{\ell }}\) will be placed/move in locations that minimize the trace of the error covariance associated with the estimator \(\hat {\mathbf {s}}_{\rho _{\ell }}(tt)\). This will ensure that the informative sensors associated with each target move to a location that will provide measurements that result good tracking accuracy. The idea of minimizing a scalar function of the predicted error covariance was also applied in moving all sensors in a network for tracking a single target [2, 3, 12]. Here kinematic strategies are derived in the presence of multiple targets, while a judiciously selected small portion of targetinformative sensors will be moving instead of all sensors moving.
which ensures that each subset \({\mathcal {S}}_{\rho _{\ell }}\) of moving sensors will stay connected, as long as the communication range of the sensing units is at least \(\sqrt {2}R\). Thus, R can be set such that the moving sensors stay connected. Connectivity is necessary to elect a head sensor for each moving subset of sensors that will acquire the measurements of all other sensors and perform clustering. Details of the algorithm are given in Section 4.1. Note that existing approaches do not entail mechanisms as the one introduced here to ensure that sensors will be connected.
where {γ ^{ k }} is a positive monotonically increasing scalar sequence [26].
During time instant t+1, each sensor j within the subset \(\mathcal {S}_{\rho _{\ell },t}\) will keep updating their location until the cost function in (19) is not reduced more than a predefined threshold ε within two consecutive updating steps κ,κ+1. The location p _{ j }(t+1) will be set to the last update \(\mathbf {p}_{j}^{K'}(t+1)\) obtained after K ^{′} MBM iterations during time instant t+1. The following steps are carried out during the determination of the sensor’s new location:
S1) The head sensor in each subset \({\mathcal {S}}_{\rho _{\ell },t}\) sends the predicted position estimate of target ρ _{ ℓ }, namely \(\hat {\mathbf {p}}_{\rho _{\ell }}(t+1t)\), to all sensors in \({\mathcal {S}}_{\rho _{\ell },t}\).
The collisionavoidance position modifications in (30) were proposed in [34] to prevent collision of unmanned aerial vehicles with a stationary target. The position updates in (29) and (30) ensure that the updated locations are at distance R _{min,a } and R _{min,b } from another moving sensor or moving target, respectively, satisfying the minimum distance required to prevent collision.
The actual movement can be achieved using for example robotic sensors, see e.g., [35, 36]. Each sensor \(j\in \mathcal {S}_{\rho _{\ell },t}\) updates its location p _{ j }(t+1), in a coordinate fashion while the remaining sensors in \(\mathcal {S}_{\rho _{\ell },t}\) are kept stationary waiting for their turn to update their location.
4 Algorithmic summary
4.1 Implementation
At the startup stage, fast sampling is used to acquire Q measurements fast enough that the initial number of targets r(0) can be assumed stationary. By utilizing the Q acquired data, the subsets of targetinformative sensors \(\{{\mathcal {S}}_{\rho _{\ell },0}\}\) are initialized, where \(\ell =1,\ldots,\hat {r}(0)\) and \(\hat {r}(0)\) is the estimated number of r(0) sensed targets at time t=0 (number of nonzero columns in the sparse matrix \(\hat {\mathcal {M}}_{0}\)). One sensor within each \({\mathcal {S}}_{\rho _{\ell },0}\) will be randomly selected as the head sensor, which will collect the measurements x _{ j }(0) and their positions p _{ j }(0) from all the other sensors \(j\in {\mathcal {S}}_{\rho _{\ell },0}\). Each head sensor \(C_{\rho _{\ell },0}\) averages the positions of the informative sensors in subset \({\mathcal {S}}_{\rho _{\ell },0}\) to be the initial estimate of the corresponding target ρ _{ ℓ }. The latter target location estimate along with the informative measurements x _{ j }(0), for \(j\in {\mathcal {S}}_{\rho _{\ell },0}\) are utilized to initialize the recursions of the extended Kalman filtering carrying out the target tracking in Section 3.2.
At time instant t, every head sensor \(C_{\rho _{\ell },t}\) has available the state estimates for active target ρ _{ ℓ }, namely \(\hat {\mathcal {s}}_{\rho _{\ell }}(tt)\), obtained from EKF in Section 3.2. The target’s estimated position \(\hat {\mathbf {p}}_{\rho _{\ell }}(tt)\) is then used to select a group of “candidate informative” sensors, which are denoted as \({\mathcal {J}}_{\rho _{\ell },t}\) for target ρ _{ ℓ } at time instant t. This set is formed by having the head sensor transmit the estimated state \(\hat {\mathcal {s}}_{\rho _{\ell }}(tt)\) to its singlehop neighboring sensors which then transmit the same information to their own neighbors. Every sensor j who receives \(\hat {\mathcal {s}}_{\rho _{\ell }}(tt)\), from a neighboring sensor, subsequently forwards this estimate only to those sensors \(j'\in {\mathcal {N}}_{j}\) whose present position is within radius R _{ s } from the estimated target location, i.e., \(\\mathbf {p}_{j'}(t)\hat {\mathbf {p}}_{\rho _{\ell }}(tt)\_{2}\leq R_{s}\). The parameter R _{ s } can be set to be sufficiently large in order for all ρ _{ ℓ }targetinformative sensors to be incorporated in subset \({\mathcal {J}}_{\rho _{\ell },t}\). The sensor subset \({\mathcal {J}}_{\rho _{\ell },t}\) by construction is connected.
Since not all sensors within the candidate subsets \({\mathcal {J}}_{\rho _{\ell },t}\) maybe informative, the scheme in Section 3.1 is employed among the sensors in \({\mathcal {J}}_{\rho _{\ell },t}\) to find out the targetinformative sensor subset \({\mathcal {S}}_{{\rho _{\ell }},{t+1}}\subseteq {\mathcal {J}}_{\rho _{\ell },t}\) for all the active targets. Rather than running the targetsensor association scheme in Section 3.1 in the whole sensor network, it is performed independently in the different sensor subsets \({\mathcal {J}}_{\rho _{\ell },t}\) associated with each target.
The head sensor \(C_{\rho _{\ell },{t+1}}\) gathers the sensor measurements x _{ j }(t+1) from the informative sensors \(j\in {\mathcal {S}}_{\rho _{\ell },{t+1}}\) to carry out the extended Kalman filtering recursions at time instant t+1 as outlined in Section 3.2. Then, steps S1 and S2 in Section 3.3 are employed to allow all sensors in \({\mathcal {S}}_{\rho _{\ell },{t+1}}\) to determine and move to their new positions p _{ j }(t+1). Note that connectivity of the sensors in \({\mathcal {S}}_{\rho _{\ell },{t+1}}\) is preserved as explained in Section 3.3. The head sensor \(C_{\rho _{\ell },{t+1}}\) broadcasts the latest state estimate \(\hat {\mathbf {s}}_{\rho _{\ell }}(t+1t+1)\) to its singlehop neighbors and repeats the process described earlier to update the candidate informative sensor subsets \({\mathcal {J}}_{\rho _{\ell },t+1}\).
It is worth mentioning that the kinematic rules implemented in Section 3.3 at each sensor are fully distributed since each sensor requires knowledge only of its location and the estimated target position obtained from the head sensor in \({\mathcal {S}}_{{\rho _{\ell }},{t+1}}\). Connectivity of the candidate informative subsets \({\mathcal {J}}_{\rho _{\ell },t+1}\) is ensured by construction irrespective of the sensor movement. This way the sensortotarget association scheme in Section 3.1 can still be applied in \({\mathcal {J}}_{\rho _{\ell },t+1}\) and determine the informative sensors.
The targetinformative sensor selection scheme in Section 3.1 may also need to be reapplied across the whole sensor network since moving targets may disappear and not being sensed anymore, while new targets may appear at different regions of the sensor network. The following conditions are checked to determine such events: (i) If any of the sensor subsets \({\mathcal {S}}_{\rho _{\ell },{t+1}}\) becomes empty, this implies that some of the targets previously sensed are not present anymore; (ii) If at t+1, the energy of a sensor, not previously selected, exceeds a certain threshold, this indicates that most likely a new target enters the sensed field. The two aforementioned conditions signify that the target configuration has changed and the sensor selection scheme in Section 3.1 needs to be reapplied in the sensor network to update the sensorinformative subsets. The novel tracking scheme is summarized as Algorithm 1.
4.2 Communication and computational expenses
The communication cost of the proposed algorithm is studied next. Note that intersensor communication takes place during (i) the sensortotarget association scheme in Section 3.1; (ii) carrying out the EKF tracking steps in Section 3.2; and (iii) when applying the kinematic strategy in Section 3.3 to move the informative sensors. In detail, at time instant t, sensor j has to receive \({\mathcal {N}}_{j}\) scalar measurements from its neighbors, namely \(\{x_{j'}(t+1)\}_{j'\in {\mathcal {N}}_{j}}\), to update \(\hat {\boldsymbol {\Sigma }}_{{x,t}+1}({j,j}')\). Furthermore, to implement the association scheme in Section 3.1, each sensor j receives the updates \(\{\hat {\mathbf {M}}_{t+1}^{k1}(j',\ell)\}_{\ell =1}^{L}\) from neighborhood \({\mathcal {N}}_{j}\), corresponding to \(L{\mathcal {N}}_{j}\) scalars in total, to form its local updates \(\{\hat {\mathbf {M}}_{t+1}^{k}(j,\ell)\}_{\ell =1}^{L}\). Thus, sensor j receives \((L+1){\mathcal {N}}_{j}\) scalars in total. Similarly, sensor j has to transmit x _{ j }(t+1) and \(\{\hat {\mathbf {M}}_{t+1}^{k1}(j,\ell)\}_{\ell =1}^{L}\), a total of L+1 scalars to its neighbors, per iteration k.
After the targetinformative sensors are determined, each head sensor has to carry out the estimation process about the corresponding target’s states. Thus, the head sensor \(\{C_{\rho _{\ell },t}\}\) will collect the measurements x _{ j }(t) from the sensors within \({\mathcal {S}}_{\rho _{\ell },t+1}\). This involves \({\mathcal {S}}_{\rho _{\ell },t+1}\) scalar exchanges. Further, all sensors in the subset \({\mathcal {S}}_{\rho _{\ell },t+1}\) will receive four scalars corresponding to the current state estimate. Once the state estimation process (Section 3.2) is carried out by the head sensors, sensor communication also occurs among the informative sensors when adjusting their new location to avoid collision with closely located sensors (Section 3.3). Specifically, sensor j receives \(2 \mathcal {N}_{j}\) scalars from its neighbors, corresponding to their two location coordinates, while it sends out its own location. It is worth mentioning that the communication complexity for each sensor is linear with respect to its neighborhood size \(\mathcal {N}_{j}\), and the upper bound number of present targets L. The latter linear cost advocates that the proposed framework is a communicationaffordable distributed approach.
When applying the scheme in Section 3.1 during each coordinate cycle k and time instant t, each sensor j has to form the coefficients in (10), (9) with a computational complexity of the order \({\mathcal {O}}( {\mathcal {N}}_{j})\), while determining the roots of the two thirdorder polynomial in (10), (9) involves determining the eigenvalues of two 3×3 companion matrices whose complexity is fixed and nondependent on any algorithmic parameters. The EKF in Section 3.2 can be carried out at a complexity of the order \({\mathcal {O}}(K^{2}+{\mathcal {S}}_{\rho })\), where K=4 here and \({\mathcal {S}}_{\rho }\) corresponds to the size of the targetinformative subsets. The kinematic rules implemented at the targetinformative sensors in Section 3.3 have a computational complexity \({\mathcal {O}}(K)\).
5 Simulations
6 Conclusions
A novel framework combining sparse matrix factorization with proper kinematic rules enables multiple mobile sensors to track multiple targets. A normone/normtwo regularized matrix decomposition formulation is utilized to perform sensortotarget association and select the targetinformative sensors. Optimal kinematic rules are obtained by minimizing the covariances of parallel extended Kalman filters that track multiple targets using only targetinformative sensors. The modified barrier method is utilized to obtain the sensors’ location updates while ensuring that the moving sensors remain connected. Numerical tests in multisensor networks corroborate that our novel scheme outperforms related approaches and accurately tracks multiple targets utilizing only a small percentage of moving sensors that closely follow the targets.
Declarations
Acknowledgements
Work in this paper is supported by the NSF grant CCF 1218079 and the AirForce grant #FA95501510103, Simons Foundation Award # 279870 and UTA.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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