NON-CENTRALIZED TARGET TRACKING WITH MOBILE AGENTS Petar M. Djuri´c, Jonathan Beaudeau, and M´onica F. Bugallo∗ Department of Electrical & Computer Engineering Stony Brook University, Stony Brook, NY 11794 (USA) email:
[email protected],
[email protected],
[email protected]
ABSTRACT In this paper we consider the problem of target tracking in a network of mobile agents. We propose a scheme with agents that are endowed with processing and decision-making capabilities and without a central unit that controls them and/or fuses information. The agents measure received signal strengths from the targets and communicate it to the remaining agents engaged in the tracking. Each agent applies particle filtering for tracking and uses an algorithm for optimal agent deployment for the next time instant. We describe the details of the tracking from its initialization to its completion. We demonstrate the proposed method by computer simulations. Index Terms— Particle filtering, target tracking, mobile agents. 1. INTRODUCTION Advances in robotics are making a strong impact on mobile sensor networks [1]. This in turn raises the interest in addressing problems of target tracking with mobile agents (sensors). Target tracking with mobile agents poses a different set of problems than tracking with stationary sensor networks. Some of them include issues related to where and how the processing of the measurements of the sensors is carried out, and who and how controls the movement of the mobile agents. This problem gets the attention of the research community with increased interest. For example, in [2] the authors have described a distributed mobility management scheme for mobile sensor networks, where the mobile agents make decisions for movement by following a carefully designed optimization scheme. The dynamic models are linear and the tracking is based on Kalman filtering. In [3] targets are tracked with mobile sensor networks by using a distributed Kalman filtering. More recently, the tracking of a single target in a hybrid sensor network has been addressed in [4]. There, the emphasis is on a target tracking algorithm based on distributed particle filtering in a network consisting of both static and mobile nodes. In a previous work, we have presented a problem of tracking with mobile agents where the agents are controlled by a central unit. The central unit is also in charge of processing the measurements made by each of the agents [5, 6]. The tracking at the central unit is carried out by applying particle filtering [7] or cost-reference particle filtering [8]. In this paper, we propose for tracking a completely different strategy. First, the tracking is carried out without a central unit. In other words, each mobile agent processes all the available measurements. Second, the agents make decisions without a central unit too. This work was supported by the NSF under Awards CCF-0953316 and CCF-1018323 and by the ONR under Award N00014-09-1-1154.
978-1-4577-0539-7/11/$26.00 ©2011 IEEE
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These decisions include determining if the mobile agents should start tracking a new target, and if they start tracking, where they move so that they do not collide and the target is successfully tracked. Thus, we propose an approach for target tracking with mobile agents where we relinquish the need for a central unit. In our scheme, the agents that track the target collaborate by broadcasting their measurements to the other agents that track. Thus, each of the tracking agents have a complete set of measurements for estimating the state parameters of the target. In other words, we allow for each of the mobile agents to act like a central unit for processing of the data. Note that here we adopt a completely different approach of cooperation among the agents than in other recent efforts where the underlying principle is achieving a consensus [9]. The paper is organized as follows. In the next section we formulate the problem. In Section 3 we describe the proposed solution. Simulation results are shown in Section 4. In Section 5, we have some final conclusions. 2. PROBLEM FORMULATION A certain area is under surveillance by N mobile agents. When they are idle, they are static and on alert to detect a target that enters the area and track it. The agents are equipped with sensors that acquire signals from the target that are used for tracking. Once the targets leave the area of surveillance, the agents stop tracking the target and possibly hand it off to a group of agents that are tasked to monitor a neighboring area. We assume that a target moves in a 2-D plane according to the Markovian model xt
=
Axt−1 + But
(1)
where xt is a state vector defined by xt = [x1,t x2,t x˙ 1,t x˙ 2,t ] with x1,t and x2,t being the coordinates of the target in the 2-D Cartesian coordinate system, and x˙ 1,t and x˙ 2,t , the components of the target’s velocity. The symbol A denotes a known 4 × 4 matrix, defined by I 2 Ts I 2 A= 02 I 2 where Ts is the sampling period, and I 2 and 02 are the identity and zero 2 × 2 matrices, respectively. The symbol B is a 4 × 2 known matrix given by 2 Ts I2 . 2 B= Ts I 2
ICASSP 2011
The state noise is represented by the 2 × 1 vector ut whose distribution is known and is not necessarily Gaussian. The agents that track the target are denoted by n = 1, 2, · · · , N . They receive signals from the target modeled by y n,t
=
gn (xt ) + v n,t
(2)
where, in general gn (·) is a function that describes how the state of the target maps into a measured signal by the n−th agent (in absence of observation noise) and v n,t denotes the observation noise of the n−th agent. The distribution of the noise is known, and again, it does not have to be Gaussian. In this formulation, we allow that the agents may acquire different types of signals, for example, direction of arrival, or signal strength and so on. For simplicity, here we assume that the function gn (·) produces a scalar value and is given by gn (xt )
=
Ψdα 0 r n,t − lt α
(3)
where lt = [x1,t x2,t ] is the location of the target at time instant t; r n,t is the location of the n−th agent at time instant t, Ψ is the emitted signal power by the target measured at distance d0 , and α is a path-loss coefficient that depends on the transmission medium and is assumed known. Once a target enters the monitored area of the agents, the objective is to quickly initiate and continue the tracking of the target by available agents without a central unit. The initialization has to be performed based on the signals that were used for initial detection of the target and according to a predefined protocol. Each of the agents must have a complete estimate of the state of the target and should self-deploy without the assistance of a central unit. We also want to study the problem where the mobile agents during tracking may experience a degradation of their measurements due to a stationary interfering source. In that case, the agents must detect the interfering source and temporarily include its location to the state of unknown parameters. The observation model is then y n,t
=
gn (xt ) + gn (λ) + v n,t
(4)
where λ is the location of the interfering source. 3. PROPOSED SOLUTION According to the problem statement, each agent measures the strength of a signal emitted by the target with some error. First we describe the tracking of the target once it is initialized, and then we explain the process of initialization. Due to lack of space, we do not explain management of agent mobility. For information on this subject, see [5]. 3.1. Tracking There are two important issues that we need to resolve. One is the tracking of the target by all the agents, and the other is the deployment of the agents for the next time instant. The agents perform tracking by employing particle filtering. We assume that the mobile agents exchange their measurements upon making them, and therefore each agent has all the measurements available for updating the latest estimate of the target’s state. Thus, unlike in distributed processing schemes, where the agents exchange their estimates and the uncertainties about them followed up by a fusion of the received information, here we simply allow that the agents share their measurements.
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In an ideal scenario, the agents operate synchronously, and the exchanged messages are received without errors. In that case, the agents broadcast to the agents of their group (which track the target) their identification (ID) number and the made measurement only. They do not need to send their current locations because (under the ideal scenario) the agents are programmed in a way so that they can compute these locations. Besides the ID and the measurement, the agents may transmit additional information such as the ID of the target, which is assigned during initialization. Thus, mobile agents which are in the proximity of the tracking agents can get information about the group that is tracking a particular target. Let the random measure of the target’s state at time instant t be (m) (m) (m) χt = {xt , wt }M are the particles of the ranm=1 , where xt (m) dom measure, wt the weights associated to the particles and M the total number of particles. Since an agent has its own measurement of the target and those of the remaining N − 1 agents from the group, it proceeds with updating the random measure χt to χt+1 in a way that is identical to that of the other agents. If all the agents start processing the measurements by using the same random seed for generation of random numbers, it is clear that all the agents will have exactly the same random measures that approximate the posterior of the target’s state. Thus, we assume that all the agents use the same proposal function for generating particles and the same resampling scheme. In the end, the agents have identical replicas of the random measures. The computation of the weights of the particles require knowledge of the locations of the remaining agents from the group. Since the new locations are computed by using a deployment algorithm shared by all the agents, we deduce that all the agents know these locations. 3.2. Initialization Suppose that a target enters the field under surveillance by the mobile agents. The agents that are closest to the target get the strongest signal from the target. As soon as they measure the signal from the target, the agents broadcast it to their neighbors. This information is accompanied with the current locations of the agents and their IDs. Once this information is received by the agents, they make decision if they should be part of the group that will track the target. They rank all the measurements and if their measurement is in the top N , they are part of the group. Based on the same prior knowledge about the state of the target, the agents produce their first random measures χ0 , and from then on proceed operating as described in the previous subsection. In the case when we have a stationary interferer, we consider two cases. One is when the interferer has been detected by the mobile agents and its location estimated before the appearance of the target, and the other, when the agents have been tracking the target for some time, and they approach and pass by an interferer. Here we explain the initialization for the second case, which is more challenging. Suppose that at time instant t − 1, the agents have (m) (m) the random measure χt−1 = {xt−1 , wt−1 }. They propagate the (m) particles for the next time instant, xt . From the particles and the (m) weights wt−1 , the sensors predict the next measurements by yn,t =
M
(m)
(m)
wt−1 gn (lt
),
n = 1, 2, · · · , N.
(5)
m=1
From the differences between the predicted and the actual measurements, statistics are formed which are then used for deciding if there
is an additional source in the sensors’ proximities. If based on the statistics, the sensors detect such source, they first remove the target signal from yn,t to obtain yn,t = yn,t − yn,t ,
n = 1, 2, · · · , N
(6)
and then from yn,t , n = 1, 2, · · · , N , they quickly estimate the location of the interfering source. 3.3. Tracking in presence of stationary interference Here we assume that during tracking of the target, the mobile agents may receive interfering signals from a stationary source. Therefore, the observation equation is modified to gn (xt )
=
Ψdα 0
r n,t − lt α
+
Ψi dα 0
r n,t − λα
(m)
Step 1: Propagate the particles xt−1 by using p(xt |xt−1 ), i.e, gen(m) erate xt by (m)
xt
∼
(m)
p(xt |xt−1 ),
m = 1, 2, · · · , M.
Step 2: Generate particles of λt by (m) t−1 , Σ t−1 , ∼ N λ λt
m = 1, 2, · · · , M
(m)
∝
(m)
wt−1
N
(m) (m) . p yn,t |xt , λt
((m−1)M1 +m1 )
wt
∝
N
(m) ((m−1)M1 +m1 ) p yn,t |xt , λt n=1
×
((m−1)M1 +m1 ) p λt ((m−1)M1 +m1 ) q λt
(13)
((m−1)M1 +m1 ) t−1 , Σ t−1 comis the Gaussian N λ where p λt ((m−1)M1 +m1 ) t−1 and , and where parameters λ puted at λt the t−1 are obtained at t − 1. Finally, q λ(m) is another Gaussian, Σ t (m) 1 +m1 ) t , Σt , also computed at λ((m−1)M . N λ t t, t, λ Step 4: Normalize the weights and compute the estimates x and Σt , from all the M1 M particles, i.e.,
t x t λ t Σ
= = = ×
M M1 ((m−1)M1 +m1 ) (m) wt xt 1 =1 m m=1 ((m−1)M1 +m1 ) ((m−1)M1 +m1 ) M M1 w λt−1 t−1 1 =1 m m=1 ((m−1)M1 +m1 ) M M1 w m=1 m1 =1 t ((m−1)M1 +m1 ) 1 +m1 ) t λ((m−1)M t λt −λ −λ t (m)
(8)
Step 5: Resample M times by using the weights wt , m = 1, 2, · · · , M1 M . Thus, in this step we reduce the number of particles from M1 M back to M .
(9)
We note that in the above two schemes we approximate the pos t, Σ t [10]. terior of λ at time instant t by a Gaussian, N λ
t−1 , Σ t−1 ) stands for normal distribution with mean where N (λ t−1 . λt−1 and covariance Σ Step 3: Compute the particle weights according to wt
Thus, here we allow each particle to have M1 children so that at the end of this step we have a total of M1 M particles. The covariance matrix, Σt , is prespecified. Step 3: Compute the particle weights
(7)
where λ is the location of the interfering source and Ψi is its emitted power measured at d0 . The method for tracking in the presence of interfering source can be implemented as follows: At time instant t − 1, we have the random measure χt−1 =
M (m) (m) (m) (m) , where λt−1 is the m−th particle of the xt−1 , λt−1 , wt−1 m=1 location of the stationary source (the subscript t − 1 should not be interpreted as if the source is moving, but rather, as the m−th parti t−1 cle of the (t − 1) random measure). We also have the estimates λ t−1 is the minimum mean square error (MMSE) t−1 , where λ and Σ t−1 is the covariance of the estimate of λ obtained from χt−1 and Σ estimate of λ computed also from χt−1 (see Step 4 below). Then (m)
Then for each m, m = 1, 2, · · · , M , generate M1 particles of λt according to (m) ((m−1)M1 +m1 ) t , Σt , m1 = 1, 2, · · · , M1 . (12) λt ∼N λ
(10)
n=1
We note that the measurements are scalars. t, (m) ,λ Step 4: Normalize the weights and compute the estimates x t t by and Σ M (m) (m) t = x xt m=1 wt (m) (m) M t = w λt λ (11) m=1 t M (m) (m) t λ(m) t Σt = λ w − λ − λ . t t t m=1 Step 5: Resample if necessary. An alternative approach would be the following. Step 1: Same as above. (m) (m) Step 2: Estimate λ from yn,t = yn,t − gn (lt ), n = 1, 2, · · · , N using a nonlinear estimation scheme (Newton-Raphson, for exam(m) (m) . ple, with initial value λ ). Denote the obtained estimates as λ t t−1
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4. SIMULATIONS A simulation scenario, implementing the first of the two methods in subsection 3.3, was developed consisting of nine sensors distributed across a 3 × 3 grid, with the actual position of each sensor randomly jittered to avoid problems due to sensor collinearity and with nominal sensor spacing of 10m. The target approached the sensor grid from a number of possible locations with a fixed velocity and was subject to random perturbations. The remaining parameters were set to Ψ = 10, d0 = 1m, Ts = 1, α = 2. The sensors maintained a distance of 2m from the target. The number of particles were M = 80 and M1 = 60. The state and observation noises were zero-mean Gaussian with variances σu2 = 0.00035 and σv2 = 0.05, respectively. Four out of the nine sensors were initialized to track the target once it came within the detection range. This initial detection was based on comparing each sensor measurement to an empirically determined threshold. During the initial transient period of acquiring the target and maneuvering the sensors to their designated locations to maintain tracking, the interference source was not present. Some time after tracking was stabilized (t = 140 s), the interference source (with Ψi = 10) was switched on and activated within
25
root square error (m)
Interference Compensated Filter Non Compensated Filter
12 .01 .04 .25
10 Root mean square error (m)
the sensor grid. This could occur in reality in a situation where a jamming source or decoy could be deployed in order to derail the sensors from continued tracking. Upon detection of an interfering source, the filter dynamics were modified and the system was placed into interference compensation mode. Figure 1 shows the tracking error plotted as a function of time for both the interference-compensated filter and the conventional uncompensated filter. The large initial error is present as the sensors have not yet detected the target and the filter has not been initialized. As expected, once the target has come sufficiently close, there is a large drop in error as the sensors acquire the target. At the point at which interference is turned on, one can see a substantial rise in error as the system reacts to the detected interference and acts to localize the source. Eventually, the error descends back to its previous levels as the interference source is successfully localized and proper tracking is maintained. An examination of the mean square error (MSE) for three different sensor noise variance values was conducted over 125 trials, and it is shown in Figure 2. It can be seen that higher noise variance generally causes initialization of the sensors at an earlier point in time indicating that the detection threshold may need to be adjusted in subsequent revisions to maintain the same estimate quality. It also clear that the system sensitivity to interference is higher with increasing noise levels; this is expected as higher levels of noise will degrade the target estimate, resulting in lower interference detection performance and longer interference localization times.
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4
2
0
0
100
200
300 400 Time (s)
500
600
700
Fig. 2. Root mean square error. The numbers in the legend refer to the different variances of sensor noise.
6. REFERENCES [1] K. Dantu, M. Rahimi, H. Shah, S. Babeland, A. Dhariwal, and G. S. Sukhatme, “Robomote: enabling mobility in sensor networks,” in IPSN ’05: Proceedings of the 4th International Symposium on Information Processing in Sensor Networks, 2005.
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[2] Y. Zou and K. Chakrabarty, “Distributed mobility management for target tracking in mobile sensor networks,” IEEE Transactions on Mobile Computing, vol. 6, no. 8, pp. 872–887, 2007.
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[3] R. Olfati-Saber, “Distributed tracking for mobile sensor networks with information-driven mobility,” in Proceedings of American Control Conference, New York, NY, 2007, pp. 4606– 4612.
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[4] T. Wimalajeewa and S. K. Jayaweera, “Mobility assisted distributed tracking in hybrid sensor networks,” in the Proceedings of IEEE International Conference on Communications (ICC), 2010.
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[5] Y. Li and P. M. Djuri´c, “Particle filtering for target tracking with mobile sensors,” in the Proceedings of ICASSP, 2007. [6] Y. Li and P. M. Djuri´c, “Target tracking with mobile sensors using cost-reference particle filtering,” in the Proceedings of ICASSP, 2008.
Fig. 1. Single run error comparison.
[7] A. Doucet, N. de Freitas, and N. Gordon, Eds., Sequential Monte Carlo Methods in Practice, Springer, New York, 2001. 5. CONCLUSIONS In this paper we proposed a non-centralized scheme for target tracking with mobile agents. The agents that track the target broadcast their measurements to the other tracking agents, and they each perform particle filtering for tracking. The agents are also capable of making decisions for their movement which is in agreement with the movement of the other agents. We simulated a scenario with one target and an interfering stationary source. The simulation results show good performance of the proposed scheme.
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[8] J. M´ıguez, M. F. Bugallo, and P. M. Djuri´c, “A new class of particle filters for random dynamical systems with unknown statistics,” EURASIP Journal on Applied Signal Processing, vol. 2004(15), pp. 2278–2294, 2004. [9] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proceedings of the IEEE, vol. 95, no. 10, pp. 319–339, 2007. [10] J. Kotecha and P. M. Djuri´c, “Gaussian particle filtering,” IEEE Transactions on Signal Processing, vol. 51, no. 10, pp. 2592– 2601, 2003.
