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Measurement losses adversely affect the performance of target tracking. The sensor network's life span depends on how efficiently the sensor nodes consume energy. In this paper, we focus on minimizing the total energy consumed by the sensor nodes whilst avoiding measurement losses. Since transmitting data over a long distance consumes a significant amount of energy, a mobile sink node collects the measurements and transmits them to the base station. We assume that the default transmission range of the activated sensor node is limited and it can be increased to maximum range only if the mobile sink node is out-side the default transmission range. Moreover, the active sensor node can be changed after a certain time period. The problem is to select an optimal sensor sequence which minimizes the total energy consumed by the sensor nodes. In this paper, we consider two different problems depend on the mobile sink node's path. First, we assume that the mobile sink node's position is known for the entire time horizon and use the dynamic programming technique to solve the problem. Second, the position of the sink node is varied over time according to a known Markov chain, and the problem is solved by stochastic dynamic programming. We also present sub-optimal methods to solve our problem. A numerical example is presented in order to discuss the proposed methods' performance

In a sensor network, a large number of small, low cost sensing devices called sensor nodes are deployed and interconnected to gather information from the field of interest and transmit it to the base station. The gathered information may be used to perform various kinds of tasks, such as environmental monitoring, habit monitoring, intelligent building and military applications. Each sensor node in the sensor field is powered by a battery and consumes energy for various purposes such as sensing, on board signal processing and transmitting. Energy required for transmission is significantly more than other purposes and is a function of the distance between sender and receiver. In general, sensor nodes are stationary and unattended. It is not easy and in most cases impractical to replace or recharge the battery in hazardous environments [

The life span of a sensor network is determined by the time duration until the sensor network fails to function due to inadequate number of sensor nodes [

In target tracking, continuous collection of information about the target improves the tracking quality. The loss of the target and the measurement losses seriously affect the tracking quality [

In this study, we minimize the total energy consumed by the sensor nodes whilst avoiding the additional error in the estimation due to the measurement losses. A mobile sink node is used to collect the measurements and transmit them to the base station (BS). The BS schedules the sensor nodes to send the measurements to the mobile sink node without loss considering the minimal energy consumption. In many situations, multiple sensor nodes cannot be activated at the same time due to the limited bandwidth or avoid interferences between sensor nodes. For example sonar sensor nodes cannot be operated simultaneously in the same frequency band in order to avoid interferences [

The remainder of this paper is organized as follows. Section II constitutes the overview of our problem and analyzes the effect of measurement losses on tracking quality for a linear Gaussian dynamics. In section III, the assumptions and constraints made for this study are presented to formulate the sensor management problem. We describe optimal and sub-optimal methods to solve the sensor management problem for fully and partially known mobile sink node's paths. In Section IV, a numerical example of our problem is presented to analyze the performance and limitations of the proposed methods. Finally, our conclusion and future work are presented in section V.

The base station (BS) schedules the sensor nodes to either _{k}_{2}, …, S_{N}_{k}_{i}_{1}, S_{2}, …, S_{N}} only if its sensing range covers the region Ω_{k}_{k}_{i}_{j}_{i}_{j}_{i}

The power consumption by transmission is denoted by _{t}_{1} represents the power required for sensing and signal processing. The power consumption due to its own timer during the _{2}. In _{1} + _{2}^{2})_{1} denotes the electronic energy required to transmit one bit of data and _{2} is a constant related to the radio energy The transmission range of the sensor node S_{i}_{i}_{i}

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}

The column vector X_{k}_{k} χ̇_{k} ξ_{k} ξ̇_{k}_{k}_{k}_{k}_{k}^{i}_{i}_{i}_{i}^{i}

In this sub-section, we analyze the consequences of measurement losses in target tracking applications. Let X_{k}_{k}

The state of the target evolves after Δ_{0} is known with the error covariance P_{0}, the estimated state of the target _{k∣k}_{k}_{k∣k}_{k}_{k∣k}_{k}_{k∣}k

_{k}_{∣}_{k}_{−1} = F_{k}_{−1∣}_{k}_{−1}

_{k}_{∣}_{k}_{−1} = H_{k}_{−1∣}_{k}_{−1}.

The predicted state and the predicted measurement at time step _{k}_{∣}_{k}_{−1} and _{k}_{∣}_{k}_{−1} respectively. The predicted error of the estimated state P_{k}_{∣}_{k}_{−1} and the Kalman gain K_{k}

It can be inferred from (

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}_{k}_{k}

For observable [F, H] and controllable [F, Q^{½}] the RMSE of the estimated state E_{k}_{k}_{k}

Since P_{k∣k}_{−1} ≥ P_{k∣k}

The variation in RMSE and cumulative RMSE are illustrated in

Let L_{i}_{xi}, l_{yi}, l_{zi}_{k}_{xk}, m_{yk}, m_{zk}_{i}

The position of the target belongs to a known region Ω_{k}_{1}, S_{2}, …, S_{N}_{k}_{k}

Only a single sensor node can be

A sensor node can not be activated during the _{i}_{i}

The transmission range of the activated sensor node S_{i}_{i}_{k}_{i}_{i}

_{i}_{k}

Our objective is to minimize the total energy consumed by the sensor nodes in order to send the measurements to the mobile sink node without loss. We define the cost function _{T} for the entire time horizon {1, 2, .., T} as follows:
_{k}_{k}_{1}, S_{2}, …, S_{N}_{k}

This problem would be easily solved if the mobile sink node were always reachable by a single sensor node with a lower transmission range than the rest of the sensor nodes. However, it may be impossible to find such a sensor node in practical terms. Our aim is to find the optimal sequence of activated sensor nodes
_{T} from time 1 to T:

Since _{T} is a function of the relative position of the mobile sink node, we solve this optimization problem for cases when the mobile sink node's position is known both fully and partially.

Let {M_{1}, M_{2}, …, 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.

We used deterministic backward dynamic programming (DP) to find the optimal sequence of the activated sensor nodes. DP is a recursive technique which divides the problem into a sequence of sub-problems using principle of optimality. In this paper, _{k}_{k}_{k}_{k}

If the _{k}_{k}_{k}_{k}_{i}_{i}

We can rewrite our objective function _{T} as:

The DP proceeds backwards from

K = _{μk}_{(M}_{k}_{)}.

DP solves the sub-problems from _{k}_{k}

The optimal sensor node to be activated at time step 1 and _{j}

The rollout algorithm (RA) is an approximate dynamic programming technique to overcome the curse of dimensionality in DP. The algorithm performs according to the approximate cost-to-go function given by a sub-optimal base heuristic policy. It produces a solution not worse than the solution given by the base heuristic. The computational cost of the RA depends on the problem size and the base heuristic method. In general, RA is computationally more tractable than DP. Since M_{k}

_{1}, S_{2}, …, S_{N1}} and calculate _{1}(S_{i}_{1+}_{ti}_{1+}_{ti}_{i}_{1}, S_{2}, …, S_{N1}}. Here, _{1+}_{ti}_{1+}_{ti}

_{1}(_{1+}_{ti}_{1+}_{ti}

_{1}, S_{2}, …, S_{N}}.

_{j}

_{k}_{i}_{k}_{+}_{ti}_{k}_{+}_{ti}_{i}_{k}_{k}_{+}_{ti}_{k}_{+}_{ti}

_{k}_{i}_{k}_{+}_{ti}_{k}_{+}_{ti}

We use the one-step-look ahead (OSLA) [_{j}

In this section, we assume that the exact position of the mobile sink node M_{k}_{k}

The position of the mobile sink node varies with time according to a known Markov chain. Assume that M_{k}_{1}, …, _{S}_{s}_{1} of the mobile sink node's position are defined as:

A = [_{mn}_{S×S}_{mn}_{k}_{n}∣_{k}_{−1} = _{m}

_{1} = [_{1}(_{S×}_{1} where _{1}(_{1} = _{m}

The Markov chain parameters _{1} are assumed to be known.

Since the exact future position of the mobile sink node is unknown at the current time step, we cannot use the deterministic dynamic programming for this problem. We present the stochastic dynamic programming (SDP) technique to produce the optimal sequence of activated sensor nodes when the mobile sink node' future position is uncertain. Let _{k}_{i}_{k}_{i}_{k}_{i}_{k}_{i}_{k}

The future position of mobile sink node M_{t}_{k}_{i}_{k}

For this problem M_{t}

K = _{μk(Mk)}.

SDP solves the sub-problems from _{k}_{k}_{T} is given by:

The optimal sensor node to be activated at time steps 1 and _{k}_{k}

For a large problem (where the number of _{k}_{k}

Although OSLA algorithm produces a sub-optimal solution for the sensor management problem, it is suitable when the availability of the feasible sensor nodes are unknown a priori or when we need to solve the problem within a short time period.

In this section, we present a numerical example of single target tracking with noisy sensor nodes and the BS is located far away from the sensor field. A mobile sink node flies around the sensor field, collects the measurements from the sensor nodes and transmits them to the BS in order to prolong the life span of the sensor network. We simulate a single target moving in a 2-dimensional Cartesian space according to (3). The system matrix F and the system noise covariance Q are the same as in section 2.2.

In our simulation, we use only 3 sensor nodes to enhance the clarity and simplicity of the problem. Sensor nodes S_{1}, S_{2} and S_{3} 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 S_{1}, S_{2} and S_{3} 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

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 Δ_{1} and _{2} and only consider _{tx}_{tx}_{1} = 50 nJ/b and _{2} = 100 pJ/bm^{2}. The maximum transmission range of
_{0} with the known initial error covariance P_{o}

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
_{i}

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

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 {_{1}, _{2}, _{3}, _{4}} for the position of the mobile sink node. Here _{1} = (250, 250, 100)m, _{2} = (350, 100, 100)m, _{3} = (75, 250, 100)m and _{4} = (150, 400, 100)m.

The initial probability vector of the mobile sink node's position is assumed as _{1} = [0.2 0.4 0.2 0.2]. We choose two different transition probabilities to analyze the performance of our methods as follows:

The optimal expected total energy consumption of the sensor nodes with A_{1} and A_{2} for T = 100s are calculated off-line using stochastic dynamic programming (SDP), and are given by:

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 _{T}} for very many simulated paths of the mobile sink node. For example, we can see that _{100} = 141.27 with A_{1} whereas

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 (_{a}

In this paper, we studied optimal and sub-optimal methods for scheduling the sensor nodes with a mobile sink node to minimize the total energy consumption for target tracking. An activated sensor node increases its transmission range to maximum only if the measurement could not reach the mobile sink node. Moreover, a sensor node cannot be activated while a previously activated sensor node remains active. We studied the sensor management problem when the path of the mobile sink node is known both fully and partially. We solved the problem optimally using deterministic and stochastic dynamic programming techniques and sub-optimally using rollout and one-step-look-ahead algorithms. The deterministic dynamic programming is best suited for the problem where the path of the mobile sink node is fully known. The limitation of dynamic programming technique is that it cannot be used to solve the problem when the available feasible number of sensor nodes are unknown a priori or varying with time. In such cases, rollout and one-step-look-ahead algorithms are useful. Furthermore, for a big problem, stochastic dynamic programming technique does not solve the problem within a feasible time period due to its high computational cost.

(a) Tracking a single target using multiple sensor nodes and a mobile sink node in the sensor network. (b) The Euclidean distances between the sensor nodes and the mobile sink node.

Variation of the RMSE of the estimated state with and without measurement loss.

(a) Variation of the RMSE of the estimated state with measurement losses happening at different times. (b) Variation of cumulative RMSE of the estimated state.

Measurement-loss functions with default transmission ranges of the sensor nodes with time.

Variation of total energy consumption with time obtained by OSLA, RA-OSLA and DP.

Probable positions of the mobile sink and the default transmission ranges and positions of the sensor nodes.

(a) Simulated position of the mobile sink node with time. (b) Total energy consumption obtained by OSLA and SDP for T = 100 s.

Variation of the computational cost required by the algorithms when the number of sensor nodes increases with 4-State Markov chain.

Properties of the sensor nodes.

Sensor Node (S_{i} |
Coordinate (L_{i} |
Transmission Range (_{i} |
Active time period (_{i} |
---|---|---|---|

S_{1} |
(150, 150, 0)m | 200m | 5s |

S_{2} |
(350, 200, 0)m | 300m | 4s |

S_{3} |
(250, 400, 0)m | 200m | 2s |

Comparison of cumulative cost _{T} obtained by OSLA, RA-OSLA and DP for different total time horizons.

T (s) | OSLA (J) | RA-OSLA (J) | DP (J) |
---|---|---|---|

30 | 59.0 | 48.8 | 40.8 |

40 | 72.4 | 62.4 | 49.6 |

50 | 84.0 | 74.4 | 63.2 |

60 | 105.6 | 89.6 | 76.8 |

70 | 123.2 | 101.6 | 89.2 |

80 | 141.6 | 118.4 | 103.2 |

90 | 157.2 | 130.0 | 114.8 |

100 | 175.2 | 148.8 | 130.0 |

Total average energy consumption obtained by OSLA and SDP with different transition matrices.

T (s) | _{T} with A_{1} |
_{T} with A_{2} | ||
---|---|---|---|---|

OSLA (J) | SDP (J) | OSLA (J) | SDP (J) | |

30 | 47.80±5.48 | 42.50±5.01 | 51.33±6.26 | 46.12±5.16 |

40 | 64.12±6.53 | 56.61±5.77 | 67.86±7.43 | 61.50±5.92 |

50 | 80.24±7.36 | 70.70±6.41 | 85.19±8.56 | 77.10±6.75 |

60 | 96.45±7.87 | 85.14±7.05 | 101.79±8.80 | 92.67±7.27 |

70 | 112.55±8.89 | 99.46±7.63 | 118.97±9.27 | 107.80±8.04 |

80 | 128.34±8.93 | 113.26±7.98 | 136.77±9.99 | 123.38±8.48 |

90 | 144.87±9.64 | 127.84±8.48 | 153.43±10.91 | 138.69±9.01 |

100 | 160.90±10.15 | 143.27±9.05 | 170.82±11.35 | 155.74±9.65 |

In page 5, last paragraph:

where _{k}

where _{k}

In page 15,