The base station (BS) schedules the sensor nodes to either sleep or active mode to perform the tasks efficiently. In order to satisfy the limited bandwidth, one sensor node (or generally a set of sensor nodes) is in active mode at any given instant. We assume that the position of the target belongs to a region denoted by Ω_k at time step k and it is known a priori. Let {S1, S₂, …, S_Nk} denote the set of feasible sensor nodes for scheduling at time step k and sensor node S_i is an element in the set {S₁, S₂, …, S_N} only if its sensing range covers the region Ω_k. Here, N_k is the total number of feasible sensor nodes available for the scheduling at time step k. If a sensor node S_i is activated at time step k, the BS cannot activate another sensor node S_j for a period of t_i time steps (ie, BS can only activate the S_j sensor node at k + t_i time step or later). Each sensor node consumes energy for sensing, signal processing and transmitting. The energy consumption of the sensor node during its active and sleep modes is assumed as follows: (1) ψ = { ψ i + ψ t if active ψ 2 if sleep

The power consumption by transmission is denoted by ψ_t and ψ₁ represents the power required for sensing and signal processing. The power consumption due to its own timer during the sleep mode is denoted by ψ₂. In active mode the sensor node consumes more energy than in sleep mode, due to particularly the transmission. In order to transmit r bits of data to a distance of d, the sensor node requires (α₁ + α₂d²)r energy at each time step [25], where α₁ denotes the electronic energy required to transmit one bit of data and α₂ is a constant related to the radio energy The transmission range of the sensor node S_i is set at default range r_i. However, it can be set at r i max ( > r i ) only when the mobile sink node is not reachable by the sensor node with r_i. The intuition behind this method is to prolong the life span of the sensor network by avoiding unnecessary long distance transmissions.

Moreover, the sensor nodes' measurements are linearly related to the state of the target and corrupted by white Gaussian noise. The measurement from the sensor node S_i at time step k is given by: (2) y k i = H i X k + υ k i

The column vector X_k = [χ_k χ̇_k ξ_k ξ̇_k]′ denotes the state of the target at time step k where χ_k and ξ_k represent its positions in the x and y directions, χ̇_k and ξ̇_k represent its velocities in the x and y directions. The observation matrix is denoted as Hⁱ. The measurement noise υ k i for each sensor node is assumed to be independent of each other and υ k i ∼ N ( 0 , R i ). Here R_i denotes the error covariance of the sensor node S_i. In this study, we use homogeneous sensor nodes and therefore, the error covariance R_i and the observation matrix Hⁱ are the same for all the sensor nodes and are known a priori.

In this sub-section, we analyze the consequences of measurement losses in target tracking applications. Let X_k represent the state of the target at time step k. Then the motion of the target can be modeled by a stochastic equation. In many applications, the stochastic equation is assumed linear [16, 26], non linear [27] and Markov process [28, 29]. In this study, we assume that the target evolves according to the linear stochastic equation impaired by white Gaussian noise and given by: (3) X k + 1 = FX k + w kwhere w_k is the white Gaussian process noise with a known covariance Q and the intensity of the process noise is determined by the scalar quantity q. F denotes the system matrix which models the state kinematics of the target. For a nearly constant velocity model of the target, F and Q are given by [30]: F = [ 1 Δ t 0 0 0 1 0 0 0 0 1 Δ t 0 0 0 1 ] , Q = q [ Δ t 3 3 Δ t 2 2 0 0 Δ t 2 2 Δ t 0 0 0 0 Δ t 3 3 Δ t 2 2 0 0 Δ t 2 2 Δ t ] .

The state of the target evolves after Δt time steps. Since the state of the target and the measurements are assumed to have linear Gaussian dynamics, the Kalman filter is used to optimally estimate the state and calculate its error covariance [31]. If the initial state of the target X₀ is known with the error covariance P₀, the estimated state of the target ξ_k∣k = {X_k} and its error covariance P_k∣k = {(X_k − ξ_k∣k) (X_k − ξ_k∣k)′} at time step k are given by: (4) ξ k ∣ k = ξ k ∣ k − 1 + K k ( y k − z k ∣ k − 1 ) (5) P k ∣ k = ( I − K k H ) P k ∣ k − 1where

ξ_k∣k−1 = Fξ_{k−1∣k−1}

z_k∣k−1 = Hξ_{k−1∣k−1}.

The predicted state and the predicted measurement at time step k − 1 are denoted as ξ_k∣k−1 and z_k∣k−1 respectively. The predicted error of the estimated state P_k∣k−1 and the Kalman gain K_k at time steps k − 1 and k are given as: (6) P k ∣ k − 1 = FP k − 1 ∣ k − 1 F ′ + Q , (7) K k = P k ∣ k − 1 H ′ ( R + HP k ∣ k − 1 H ′ ) − 1 .

It can be inferred from (5) and (7) that the error covariance of the estimated state is influenced by the error covariance R of the sensor node. In [16], we studied the problem of sensor management to enhance the tracking quality, and showed that the tracking quality depends on the sequence of the activated sensor nodes where the error covariances of the sensor nodes are different. However, the tracking quality does not depend on the sequence of activated sensor nodes if the error covariances of the sensor nodes are the same and the measurements are available at each time step.

In state estimation, measurements may be absent for many reasons, such as that the target is not covered by any sensor nodes or the sensed measurement is lost. Considering the limited transmission range of each sensor node, it is possible that measurements may not be collected by the mobile sink node when it is outside the transmission range of the activated sensor node. Let E_k and E k lost denote the RMSE (root mean square error) of the estimated state at time step k without and with measurement loss, respectively. Thus E_k is the square root of the estimated error for availability of continuous measurements and E k lost is the square root of estimation error for availability of noncontinuous measurements. We can calculate E_k and E k lost as follows: (8) E k = P k ∣ k (9) E k lost = { P k ∣ k if measurement is received P k ∣ k − 1 if measurement is lost .

For observable [F, H] and controllable [F, Q^½] the RMSE of the estimated state E_k asymptotically reaches a steady state if there is no measurement loss. The variation of E_k and E k lost with time are illustrated in Figure 2, and it can be seen that E_k reaches to a steady state error, but not E k lost. Here, we used Δt = 1s, q = 10, observation matrix as 4 × 4 identity matrix and the error covariance of the sensor nodes as: R = diag [ 2.25 × 10 3 100 1.5 × 10 5 100 ]

Since P_k∣k−1 ≥ P_k∣k, we can say that E k lost ≥ E k at any time step k as it can be seen in Figure 2. The additional E k lost − E k error in estimation is due to the measurement loss. Moreover, we consider three different cases where the measurements are lost at different time steps and the number of measurement losses are not equal. The measurement losses occurred at k = 5, 6, 7, 8, k = 14, 16, 18, 20, 22 and k = 84, 86, 88, 90, 92 for case 1, case 2 and case 3 respectively. We denote the RMSE of the estimated error as E k lost 1, E k lost 2 and E k lost 3 for case 1, case 2 and case 3 respectively.

The variation in RMSE and cumulative RMSE are illustrated in Figure 3. Even though the number of measurement losses are equal for case 2 and case 3, the cumulative RMSE are not the same. Furthermore, the number of measurement losses for case 1 is less than that of case 2 and case 3, but produces a higher cumulative RMSE than case 2 and case 3. This indicates that the time steps of measurement losses are more important than the number of measurement losses for target tracking. Moreover, it can be seen in Figures 2 and 3 that measurement losses creates unnecessary error in estimation resulting in degraded tracking quality. Therefore, we make sure that the measurements are not lost at any time step to avoid the additional error in estimation. We increase the transmission range of the sensor nodes to avoid measurement loss only if necessary as this has to be done with minimal energy consumption of the sensor nodes. The following section describes the sensor management strategies according to the mobile sink node positions.

Let {M₁, M₂, …, M_T} denote the sequence of the position of the mobile sink node. We assume that it is known a priori. The base station (BS) collects information about the sensor nodes, such as location and the energy availability. We solve this problem using a dynamic programming technique and rollout algorithm as follows.

In this section, we assume that the exact position of the mobile sink node M_k is known only at time step k otherwise it is only partially known. In other words, M_k is known ∀k ≤ c and partially known ∀k > c at current time step c. We assume that the BS immediately gets to know the current position of the mobile sink node at time step k.

The position of the mobile sink node varies with time according to a known Markov chain. Assume that M_k is an S-state Markov chain with the state space {e₁, …, e_S}. Here, e_s denotes a position in 3-dimensional Cartesian space. The transition probability matrix A and the initial probability vector π₁ of the mobile sink node's position are defined as:

A = [a_mn]_S×S where a_mn = P(M_k = e_n∣M_k−1 = e_m), m, n ∈ {1, …,S}.

π₁ = [π₁(m)]_S×1 where π₁(m) = P(M₁ = e_m), m ∈ {1, …,S}.

The Markov chain parameters A and π₁ are assumed to be known.

In our simulation, we use only 3 sensor nodes to enhance the clarity and simplicity of the problem. Sensor nodes S₁, S₂ and S₃ are deployed in a sensor field of 500 m × 500 m and for the experimental purpose, we assume that each sensor node covers the target such that the feasible sensor set has S₁, S₂ and S₃ sensor nodes at each time step. A mobile sink node collects the measurements and immediately transmits to the BS. The properties of the sensor nodes are given in Table 1.

It is assumed that the measurements are sent to the mobile sink node in a single-hop communication without any time delay. The observation matrix H is a unit matrix, and therefore the measurements are linearly related to the state of the target. The time interval between the measurements is considered as Δt = 1s. We assumed the measurement data size as 1 MB and the data rate as 8 Mbps. Without loss of generality, we neglect the energy consumption ψ₁ and ψ₂ and only consider ψ_tx. The parameters in ψ_tx are set α₁ = 50 nJ/b and α₂ = 100 pJ/bm². The maximum transmission range of r i max = 500 m was chosen for all sensor nodes. The target starts at X₀ with the known initial error covariance P_o and moves for T s.

In this section, we assume that the position of the mobile sink node is known at each time step and the mobile sink node flies horizontally at a height of 100 m above the ground. We plot the measurement-loss function δ i M k with the r_i for the total time horizon of 100s, shown in Figure 5.

We compare the sub-optimal solutions obtained by the rollout algorithm (RA) and one-step-look-ahead (OSLA) method with the optimal solution obtained by dynamic programming (DP). We use OSLA method as the base heuristic method for RA. We increase the total time horizon from T = 30s to 100s. The results for all three methods are shown in Table 2. The variation of total energy consumption with time is shown in Figure 6 for T = 100s. We can see from the results in Table 2 that RA always produces better results than the base heuristic method (OSLA). Since the positions of the mobile sink node are known a priori, the computational cost of DP is not large and therefore DP is best suited for this particular problem.

In this section, we assume that the position of the mobile sink node varies according to a known Markov chain and that the exact position of the mobile sink node is known only at the current time step but not before. We have chosen a 4-State Markov chain with the state space {e₁, e₂, e₃, e₄} for the position of the mobile sink node. Here e₁ = (250, 250, 100)m, e₂ = (350, 100, 100)m, e₃ = (75, 250, 100)m and e₄ = (150, 400, 100)m. Figure 7 shows the probable positions of the mobile sink node and of the sensor nodes, and the default transmission ranges of the sensor nodes

The initial probability vector of the mobile sink node's position is assumed as π₁ = [0.2 0.4 0.2 0.2]. We choose two different transition probabilities to analyze the performance of our methods as follows: A 1 = [ 0.2 0.3 0.2 0.3 0.3 0.3 0.2 0.2 0.2 0.3 0.2 0.3 0.3 0.1 0.2 0.4 ] , A 2 = [ 0.1 0.5 0.1 0.3 0.1 0.3 0.1 0.5 0.3 0.1 0.5 0.1 0.1 0.3 0.1 0.5 ]

The optimal expected total energy consumption of the sensor nodes with A₁ and A₂ for T = 100s are calculated off-line using stochastic dynamic programming (SDP), and are given by: E { J 100 A 1 } = ∑ a = 1 4 π 1 ( a ) J 1 ( e a ) = 142.54 J , E { J 100 A 2 } = ∑ a = 1 4 π 1 ( a ) J 1 ( e a ) = 154.45 J

Since the future positions of the mobile sink node are uncertain, the OSLA schedules the sensor nodes on-line. Therefore, we simulate the mobile sink node's path according to the known Markov parameters to obtain the results for OSLA.

The results in Table 3 were averaged over 100 independently simulated paths for the mobile sink node using the known Markov parameters. The mean and standard deviation of the total energy consumption are shown in Table 3. The results in Table 3 obtained from SDP become equal to E{J_T} for very many simulated paths of the mobile sink node. For example, we can see that J₁₀₀ = 141.27 with A₁ whereas E { J T A 1 } = 142.54. Figure 8 shows a simulated path for the mobile sink node using the known Markov parameters and the total energy consumption obtained by OSLA and SDP.

Since OSLA is a sub-optimal method, on average it consumes more energy than SDP. Even though the results are not promising, the computational cost of the OSLA is very low compared to that of SDP. The number of probable positions (e_a) of the mobile sink node does not affect the computational cost of the OSLA whereas the computational cost of the SDP increases. In order to analyze the effect of the number of sensor nodes on the computational cost of the algorithms, we increase the sensor nodes from 3 to 350 whilst keeping a constant number of probable positions of the mobile sink node. It can be seen from Figure 9 that the computational cost of SDP increases as the number of sensor nodes increases, whereas no significant variation for OSLA was observed.