Energy-E ﬃ cient Mobile Sensing in Distributed Multi-Agent Sensor Networks

In this paper, we exploit an integration between Compressive sensing (CS) and the random mobility of sensors in distributed mobile sensor networks (MSNs). A small number of distributed mobile sensors are deployed randomly in a sensing area to observe a large number of positions. The distributed mobile sensors sparsely sample the sensing area for data collection. At each sampling time, the sensors collect data at their random positions and exchange their readings to the others through their neighbors within the sensor transmission range to form one CS measurement at each sensor. After a certain number of rounds for moving, sensing and sharing data, each mobile sensor creates a su ﬃ cient CS measurements to be able to reconstruct all readings from all positions that need to be observed. Network performance is analyzed considering the number of sensors deployed in the networks, the convergence time and the sensor transmission range. Expressions for transmission power consumption are formulated and optimal low power cases are identified.


Motivation
Energy efficiency is the most important issue for mobile sensor networks (MSN) that can be useful for measuring data in many applications including environmental monitoring, event detection, intrusion detection, etc. These networks are constructed from sensors, control algorithms and other dynamic factors which depend on specific purposes or application scenarios [1,2]. We define the vector X = [x 1 x 2 . . . x N ] T to represent all sensor readings from the locations to be observed. These readings are typically highly correlated and compressible and could be an object for energy saving.
Compressive sensing (CS) [3,4,5] offers to sample and to reconstruct sparse or compressible signals using fewer samples than the Nyquist-Shannon theorem would suggest. CS can be applied effectively with wireless sensor networks (WSN) and MSNs to reduce the amount of data and power required by the network. The technique can recover all data from X based on a small number of CS measurements (Y = [y 1 y 2 . . . y M ] T ) compared to the number of nodes or positions (M N ) asX = arg min X 1 , subject to Y = ΦX, where Φ is the measurement matrix, also called routing matrix in wireless networks. Each CS measurement can be collected from all sensing nodes or from some random nodes.
In this paper we propose a novel frame work that exploits CS sampling by mobile sensors deployed randomly in a sensing area. At each sampling time, each sensor samples its own data as x i which is shared with others L connected sensor nodes in the MSN. We assume the mobile sensors move into random positions in the area. After each round of moving, data sampling, and sharing, a linear combination of readings is computed as y i = L j=1 x j . To achieve a desired errortarget, each mobile sensor needs to move M times and share data to obtain M CS measurements (Y ∈ R M ). A CS recovery algorithm can be applied to reconstruct all readings from positions in the sensing area (X ∈ R N ).
With a set transmission range, R, mobile sensors may move out of range and become disconnected from the group. With the proposed method the unequal CS measurements do not negatively impact CS performance due to a sparse binary measurement matrix.

Related work
The integration between CS and data collection methods in WSNs is being exploited effectively [6,7]. It offers to reconstruct all readings from the networks based on a small number of CS measurements. Each CS measurement is collected from some random sensors. Some data collection methods utilizing CS are proposed as energy efficient algorithms to reduce power consumption for sensors. In [8,9,10] random walks with CS provide distributed routing methods for WSNs. Cluster based [11,12] and tree based [13,14] data collection methods significantly show the power reduced based on the combination with CS.
Recent research studies have exploited the mobility of sensors and CS. Wang [15] applied CS to monitoring vehicle networks. Mostofi built maps in mobile networks [16] and robot networks [17] while the mobile sensors and robots were deployed outside the sensing areas. Huang [18] reconstructed a scalar field using MSNs and information fusion. Nguyen [19,20] used flocking control to lead a group of distributed robots to sample data and applied CS to recover data from the field based on a certain number of CS measurements.
The previous work has not focused on the integration between the random mobility of sensors and CS. In this paper, we complete our previous work [21]. The mobile sensors move only a certain number of times and share data among them to generate a desired number of CS measurements. Each sensor can reconstruct all raw data based on the measurements. The important results in this paper include: 1. A proposed new distributed compressive mobile sensing data collection method. 2. Formulations and analysis for estimating network transmission power consumption. 3. Analysis and simulation of factors such as the number of mobile sensors, the convergence time and the sensor transmission range leading to choices of minimizing power consumption.

Network Model
We assume an area to be observed with N unknown positions corresponding to N readings for collection. There are L mobile sensors randomly distributed in the sensing area to sense data. They are allowed to move with random direction and random velocities. The connections between sensors are created by radio links having a maximum sensor transmission range denoted as R. An appropriate value of R is chosen for all sensors. As shown in Figure 1, all mobile sensors are connected as an undirected graph G(L, E), where L is the set of vertices representing the mobile sensors and E is the set of edges representing the connections between sensors.
We further assume that the distributed sensors move randomly in the area. Each mobile sensor shares data to the others through its neighbors within the transmission range R.

Compressive Sensing (CS)
Compressed Sensing techniques [3,4,5] bring us amazing work to recover a compressible signal from undersampled random projections. They are also called measurements. A compressible vector signal is k-sparse (X has k non zero elements) or dense but sparse in Ψ domain X = Ψ Θ (where Θ is k-sparse vector) will be sampled and then recovered precisely with CS algorithm. The huge gain when we apply CS is the number of measurements are much less than the number of original vector values. Vector Y, called the measurement vector, contains data sampled from N sensor readings; Signal Sampling: The random measurements are generated by Y = ΦX where Φ ∈ R M×N is often full-Gaussian matrices or binary matrices, called projection matrices. Y ∈ R M is the measurement vector with y i = n j=1 ϕ i,j x j , where ϕ i,j are all entries on the i th row of our projection matrix Φ.
Signal Recovery: The number of CS measurements required to reconstruct the original signal perfectly with high probability is M = O (k log N /k) following the l 1 optimization problem given by [3].
or in case we need sparsifying matrix to have X sparse in Ψ domain (Ψ can be Wavelet or DCT depending on our signal properties) as follows. where The l 1 optimization problem can be solved with linear programming techniques such as Basis Pursuit (BP) [3].
In reality, we have to consider the noise while sampling and sending the measurements (in our case we collect measurements and send to the base-station) : Y = ΦX + e , with || e || 2 < and recover

Data Collection in Distributed MSNs utilizing CS
The idea of applying CS into MSNs is that the sensing field can be sampled randomly based on the mobility of mobile sensors randomly deployed in the field. In general, each CS measurement is a collection from some random positions as shown in Figure 2. L mobile sensors are deployed randomly in the sensing area that they can collect sensing data randomly. If these sensors exchange their own data to each other, they can all achieve the data sampled randomly as shown  in Figure 2. At this point, one CS measurement is collected at each distributed mobile sensor. After M moving steps, each sensor has M measurements. This number of measurements defined in CS background in Section 2.2 helps to reconstruct all sensory readings from N positions in the area.
After M periods of time, L distributed mobile sensors visit approximately N unknown positions. The readings are contained in vector X which is compressible or sparse in proper domains as where Θ = [θ 1 θ 2 . . . θ N ] T has k non-zero elements and Ψ referred to as the sparsifying matrix, is an orthogonal basis of R N . If X is a k-sparse vector, Ψ is an identity matrix. Otherwise, Ψ can be chosen from another domain such as wavelet, DCT (discrete cosine transform), etc. In order to verify this problem, Figures 3 and 4 are provided. All raw sensor readings from 1000 points of interest are shown in Figure 3. This real data is not sparse but it is highly correlated due to the positions next to each other. DCT Ψ matrix has been chosen to sparsify the real data. Figure 4 shows all the coefficients in DCT domain. We can see that signal energy would be reduced following Equation 5. At time instant t, mobile sensor l at position i collects sensory data x l i (i = 1, . . . N ). All L mobile sensors share their readings (x l i ), identified by the corresponding position indices, with the other mobile sensors through their neighbors as shown in the graph G(L, E). After a convergence time 1 , denoted as I, each mobile sensor has one CS measurement as a sum of L sensor readings where e t is additive noise dependent on the system. φ ti is a binary vector (dimensioned 1 × N ) that represents which positions are sampled at time instant t. It is also the t th row in the measurement matrix Φ M×N after M periods of time. The total measurements collected at each sensor are According to [22], the number of measurements required to reconstruct perfectly all the raw reading from the network is where k < M N . So instead of collecting N readings, one for each sampling position in the sensing area, each mobile sensors needs only M measurements and reconstructs all data from the area. If the network is fully connected all the time, each sensor has the same binary measurement matrix Φ with a constant row weight of L.
The restricted isometry property (RIP) of the sparse binary matrix has been studied in [23] where it is shown that the matrix can satisfy RIP and therefore can be used as an efficient measurement matrix. This matrix (Φ M×N ) can perform as well as a full Gaussian matrix for the CS recovery process. is set as < e 2 . The recovery algorithm is addressed aŝ X = arg min X 1 , subject to Y − ΦX 2 < .    As mentioned, each distributed mobile sensor must collect M measurements to be able to reconstruct all readings from N positions in the sensing area. With L distributed mobile sensors fully connected as given by the graph G(L, E) and based on the sensor transmission range R, the proposed algorithm is summarized as Algorithm 1.
Due to the limitation of R and sensor motion, some mobile sensors maybe disconnected from others and separated at time instant t + k. This means that the measurements y t+k collected at each sensor are not from all L sensors. This does not affect CS performance when the measurement matrix has different row weights [22]. By applying CS, each mobile sensor only has to visit M positions to collect all data from the sensing area (M N ). This significantly reduces consumed power not only for communications but also for sensor movements.

Analysis of Power Consumption for Communications
The total consumed power for communications in the network consists of three main elements: the intraneighborhood consumed power denoted as P nei , the convergence time I and the number of measurements required (M). The total power consumption can be calculated as follows P nei represents the communication cost associated with L mobile sensors transmitting data to their neighbors and can be calculated as where ω is the average number of neighbors of each sensor corresponding to the available communication links, and α is the path loss exponent. It is shown in [24] that α = 2 and α = 4 for free space and multipath fading channels, respectively. We choose α = 2 throughout this paper. In the following sections both circular and square sensing areas are examined and analyzed.

Circular Sensing Area
If sensors are randomly distributed in a circular sensing area with radius R 0 , the average number of sensors deployed in the area covered by each sensor transmission range R can be found as where L πR 2 0 is the sensor density. We can calculate approximately the mean value of ω as Hence, the total consumed power for sharing data between L sensors is calculated as

Square Sensing Area
We assume a square sensing area with dimensions H × H. From Equation (13), the average number of sensors deployed in the area covered by each sensor transmission range R is β = L H 2 × πR 2 . Hence In Figure 6, both the true and the reconstructed data are shown together to illustrate the accuracy of the CS recovery algorithm with the number of CS measurements given by M = 300 stored at each sensor. The larger the number of measurements, the better the accuracy of the reconstruction.
The total consumed power for sharing data for L sensors to achieve M converged CS measurements in the square sensing area can be written as The convergence time I depends on both the sensor density ( L πR 2 0 or L H 2 ) and the connections between the sensors. The smaller the number of mobile sensors, the smaller the convergence time. However, since the transmission range R determines the connections, increasing the transmission range R to maintain the connections increases the transmission power consumption. This means that increasing or reducing R may www.astesj.com   increase or reduce the number of connections between the sensors in each neighborhood. Increasing the connections between sensors reduces the convergence time I. Thus a trade-off exists between these various parameters.

Simulation Results
We assume a fixed number of 50 or 100 mobile sensors deployed in a square sensing area of 100 square units (H = 100). We also assume that there are approximately 1000 positions where temperature needs to be observed. We consider the transmission range of 15 ≤ R ≤ 20 units in which the network is always connected. Without loss of generality, it is assumed that the power for transmitting 1 unit of data is 1 unit of power. The simulations were performed using real temperature sensor data from Sensorscope [25]. The reconstruction error related to CS signal recovery is the normalized reconstruction error given as Figure 5 compares the ability to reconstruct data in the CS recovery process between three measurement matrices. Both sparse binary matrices with row weight L = 50 and 100 are shown to perform as well as the full Gaussian matrix which corresponds to full sampling. Figure 7 illustrates that as R is increased, the convergence time I is reduced as discussed in Section 4. Figure 8 depicts the corresponding total power consumption in the network, and we see the power consumption is reduced as R is decreased. R cannot be reduced without limit as at some point the L sensors will become disconnected and fragmented resulting in an excessively sparse measurement matrix.
We chose the smallest transmission range R * = 15 and the corresponding convergence time I * = 94 for the optimal total power consumption for sensor communications in the network in both analysis and simulation which are shown in Figure 9.
Finally, in Figure 6, both the true and the reconstructed data are shown together to the show the accuracy of the CS recovery algorithm at the number of CS measurements as M = 300 stored at each sensor. The more number of measurements, the better accuracy of the reconstruction.

Conclusions and Future Work
This paper proposed a framework for distributed MSNs utilizing CS for efficient data sensing and recovery and investigated the dependence of power consumption on various MSN parameters. The algorithm exploits the random mobility of sensors for sparse sampling the sensing area. CS based sampling and reconstruction of the sensory data with a sparse binary measurement matrix were compared with dense Gaussian and shown to be equal under the simulation conditions, illustrating that CS may be used to advantage in MSNs. Expressions for transmission power consumption in the MSN were developed, simulated, and analyzed. The results show the trade-off between the number of sensors, the transmission range, and the convergence time that can further reduce the power consumption for data transmission in the networks.
In future work, we consider the correlation of sensing data in the field. This could help to improve the performance of CS in signal recovery.