Traffic monitoring using an adaptive sensor power scheduling algorithm
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Abstract
Using sensors to monitor surface or subsurface traffic requires sensor placement, detection of traffic changes, and sensor power scheduling for improved efficiency. Of these capabilities, sensor power scheduling is one of the most important as the appropriate sensors must be selected for activation to respond to changes in the traffic. We present an adaptive power scheduling algorithm that uses the homogeneous equilibrium of a potentialfieldbased dynamical system to determine which sensors should be active. Our algorithm assumes a nearest neighbor topology, which makes additional assumptions about the placement of sensors. We formalize these conditions and construct a sensor placement algorithm to support our scheduling algorithm. To demonstrate the efficacy of our scheduling approach, we provide two distinctive traffic detection algorithms that we combine with our placement and scheduling algorithm to test via simulation. We provide the simulation results that show in both cases, the adaptive scheduling algorithm behaves efficiently as compared to an area coverage approach , as well as an allactive path coverage approach.
Keywords
Autonomous power scheduling Traffic monitor Nearest neighbor topology Mixedinteger linear programming (MILP) Potential field1 Introduction
The idea of using statically placed sensors to monitor geographical regions is well understood and has been applied to monitoring marine species [2], underwater vehicles [1], or more specifically, port security [16, 18]. Three facets to the problem of traffic monitoring are where to place the sensors, how to detect traffic distribution changes, and how to schedule sensors in response to traffic distribution changes. With respect to sensor placement, this problem is either solved as an area coverage problem [6, 7, 11, 19] or path coverage problem [14, 20]. As for the remaining facets, McIntyre and Hintz use information gain to determine sensor schedules for searching for unknown entities [17]. Others have considered the tradeoff between accuracy and power with regard to scheduling sensors for tracking individual entities [12, 23]. Our scheduling approach, which we discuss in the next paragraph, considers detections from multiple entities.
Our main contribution to the traffic monitoring problem is to focus on the scheduling facet of traffic monitoring. Simply stated, when a traffic distribution change is detected, a sensor scheduling algorithm must determine which sensors should be active and which should be inactive. Our approach, which uses statically placed sensors, borrows ideas from robotic path planning for mobile sensors [3, 15]. For mobile path planning, the velocity of the vehicle is proportional to the gradient of some potential function. Analogously, our approach assumes that the velocity of activating sensors is proportional to the gradient of an attractive potential field. Solving the resulting ordinary differential equation (ODE) for mobile path planning yields a path, while the solution of our ODE yields the indices of the sensors to be active. Thus, our approach combines elements of static and mobile sensors to create a new, unique sensor scheduling method.
However, if sensors are not present where the traffic has shifted, then no scheduling of sensors can satisfactorily monitor the traffic. Thus, sensor placement clearly impacts the ability of any sensor schedule algorithm. In this paper, we more formally state placement assumptions and use them to construct a sensor placement algorithm to support our adaptive scheduling algorithm. In particular, our sensor placement algorithm uses a nearest neighbor topology, which is a modification of [20]; our placement approach considers the placement of only sensors.
Sensor scheduling begins with detections of changes to traffic distributions. Without such indications, the sensor scheduling algorithm cannot proceed. Therefore, we provide two unique traffic change detection algorithms that we now describe.
Our first traffic change detection approach relies on the following assumption: If traffic continuously shifts away from active sensors, then the total number of detections, which are the number of detections made by all active sensors, should decrease. To identify this event, we first use cubic bsplines as in [21] to smooth the noisy data of total detections. Using a uniform knot sequence, which guarantees \(C^2\) smoothness of the bspline everywhere over its domain [9], we are able to compute all of the inflection points and choose the inflection point with the highest number of detections. If the number of detections drops below the number of detections associated with the inflection point, we label that event as a traffic change event.
Our second detection approach identifies a traffic change event as one in which the Hellinger distance between the previous and current detection distributions meets a specified threshold. The idea of using the Hellinger distance to compute the distance between distributions is an accepted technique as found in [13], in which the authors apply the Hellinger distance to distinguish distributions for classification problems.
Our paper is organized as follows. Section 2 describes the symbols that we use in this paper. Section 3 describes assumptions of our approach, while Sect. 4 describes the adaptive scheduling algorithm itself. In Sect. 5, we use the assumptions of 3 to construct a sensor placement algorithm. We describe two traffic detection algorithms in 6. In Sect. 7, we include the simulation results and conclude with a summary in Sect. 8.
2 Nomenclature
 B

Region in \(\mathbb {R}^3\) to be monitored
 \({\mathscr {E}}\)

Entry boundary of traffic into the B
 \(\mathscr {X}\)

Exit boundary of traffic from the B
 \(\mathscr {L}\)

Left boundary of the B
 \(\mathscr {R}\)

Right boundary of the B
 \(\varvec{\tau }\)

Element of B
 \(\varvec{p_i}\)

Position of ith sensor
 \(E_{env}\)

\(m\)vector representing the environment
 \({\mathbf {e}}\)

Environmental vector
 \(\rho _i\)

ith sensor detection function
 \(R_i\)

Maximum sensing range of the ith sensor
 \(\mathscr {S}_i\)

Sensing volume of the ith sensor
 M

Number of sensors
 \(\mathcal {A}\)

Arrival traffic distribution
 \(T(\varvec{\tau })\)

Traffic trajectory of a single entrant into B
 \(s_{\xi }\)

Sensor with largest number of detections
 \(s_{\xi +w}\)

Rightmost active sensor
 \(x_{\xi +w}\)

Index of rightmost active sensor
 \(s_{\xi w}\)

Leftmost active sensor
 \(x_{\xi w}\)

Index of leftmost active sensor
 U

Arbitrary subset of \(\mathbb {R}^n\)
 f

Potential field
 \(C^2(U)\)

Space of twice continuously differentiable functions over U
 g

Function belonging to \(C^2(U)\)
 \(\varvec{x^*}\)

Desired goal of potential field
 \(\kappa\)

Potential field proportionality constant
 \(G(\varvec{x})\)

Jacobian of g
 \(d_i\)

Number of detections for a sensor \(s_i\)
 \(\nu _i\)

Number of iterations with no detections for sensor \(s_i\) in active mode
 \(\nu\)

Maximum of \(\nu _i\)
 \(\zeta _i\)

Number of iterations with no detections for a sensor \(s_i\) in standby mode
 \(\zeta\)

Maximum of \(\zeta _i\)
 d

Depth of traffic
 \(\mathscr {P}\)

Plane of traffic at depth d
 \(\mathscr {S}'_i\)

Intersection of \(\mathscr {S}_i\) and \(\mathscr {P}\).
 \(\beta\)

Intersection of \({\mathscr {E}}\) and \(\mathscr {P}\)
 \(\mathcal {P}_{\beta }\)

Orthogonal projector of planar sensing regions onto \(\beta\)
 \(\psi _i\)

Projection of \(\mathscr {S}'\) onto \(\beta\)
 \(I_k\)

Domain of \(\psi _k\)
 \(\left I_k\right\)

Length of interval \(I_k\)
 \(\alpha _k\)

Binary variable that indicates if elements of \(I_k\) belong to composite interval
 \(I_{\alpha _1, \ldots , \alpha _M}\)

Composite interval
 \(L_j\)

Set containing intervals formed from the overlap of j intervals
 F

Objective function that estimates the probability of detection of all placed sensors
 a

Constant for uniform traffic distribution
 \(F_j\)

F evaluated over \(L_j\)
 T

Triangular mesh representing environmental effects on performance
 \(T_{jk}\)

Triangle belonging to T
 \(N_T\)

Number of triangles in T
 \((x_i,y_i)\)

Planar position of sensor \(s_i\)
 \(b_{ijk}\)

Binary variable that indicates if \((x_i,y_i)\) belongs to \(T_{jk}\)
 \((x_{ijk},y_{ijk})\)

Contribution from triangle \(T_{jk}\) to \((x_i,y_i)\)
 \(X^{min}_{jk}\)

Minimum value of all \(x_{ijk}\)
 \(X^{max}_{jk}\)

Maximum value of all \(x_{ijk}\)
 \(Y^{min}_{jk}\)

Minimum value of all \(y_{ijk}\)
 \(Y^{max}_{jk}\)

Maximum value of all \(y_{ijk}\)
 \(\alpha _{jk}\)

xCoefficient describing the hypotenuse of \(T_{jk}\)
 \(\beta _{jk}\)

yCoefficient describing the hypotenuse of \(T_{jk}\)
 \(\gamma _{jk}\)

Constant describing the hypotenuse of \(T_{jk}\)
 \(A_{ijk}\)

Constant describing sensor performance
 \(B_{ijk}\)

xCoefficient describing sensor performance
 \(C_{ijk}\)

yCoefficient describing sensor performance
 \(d_1(\dot{)}\)

\(L_1\)metric
 R

Sensor radius
 \(M_{max}\)

Maximum difference of all \(y\in B\)
 \(\delta ^{+}_{i,i+1}\)

\(y_{i+1}  y_i\), if \(y_{i+1} \ge y_i\)
 \(\delta ^{}_{i,i+1}\)

\(y_i  y_{i+1}\), if \(y_{i+1} < y_i\)
 \(g_j(x_i,y_i)\)

General linear constraint function
 \(N_d\)

Sequence of detections from discrete time events \(d_1\) to \(d_N\)
 \(\mathcal {D}\)

Smooth approximation of \(N_d\)
 S

Set of inflection points of \(\mathcal {D}\)
 \(p^*\)

Time event for which the number of detections is maximal
 h

Distribution of detections over all sensors
 \(\delta ^{(k)}\)

Set of new detections over all sensors at \(k\)th time step
 \(h^{(k)}\)

Aggregated distribution of detections
 \(d^2(h,h^{(k)})\)

Hellinger distance
 \(\alpha\)

Order of Hellinger distance
 \(U[x,x+2]\)

Uniform random distribution defined over \([x,x+2]\)
 \(\upsilon\)

Homotopy parameter that varies influence between a static and dynamic uniform random distribution
 e

Detection efficiency
 \(N_a\)

Total amount of active time
 \(N_t\)

Total number of ships
 \(N_a\)

Total number of active time units
3 Assumptions
The overall goal of our adaptive scheduling algorithm is to use statically placed sensors in some volume \(B \subset \mathbb {R}^3\) to adjust the sensor modes to monitor the movement of the traffic distribution. We now describe our sensor, field, and traffic distribution assumptions.
3.1 Sensors and field
We identify the sensor modes as active, standby, and off. Our algorithm explicitly decides which sensors should be placed in the active state, thus allowing a sensor to make detections. Once active, the sensor remains active as long as it receives detections within \(\nu\) time units or is explicitly selected again by the scheduling algorithm. However, if no detections are received within \(\nu\) time units, the sensor shifts to the standby state. Once in the standby state, if the sensor receives detections, it can again be placed in the active state. If no detections are received for \(\zeta\) time units, the sensor is placed in the off state, in which it cannot receive any detections. However, once in the off state, a sensor cannot be placed in the active state again without being explicitly chosen by the scheduling algorithm.
For \(B = [0,d_x]\times [0,d_y]\times [0,d_z]\), we are assuming that traffic enters a single face of B denoted as \({\mathscr {E}}\) and exits a different face denoted as \(\mathscr {X}.\) Let us assume for \(\varvec{\tau } = (\tau _x,\tau _y,\tau _z)\), the entrance face is given as \({\mathscr {E}} = \{\varvec{\tau } \in B: \tau _y = d_y\}\) and the exit face is given as \(\mathscr {X}= \{\varvec{\tau } \in B: \tau _y = 0\}.\) Furthermore, we label the left face as \(\mathscr {L}= \{\varvec{\tau } \in B: \tau _x = 0 \}\) and the right face as \(\mathscr {R}= \{\varvec{\tau } \in B : \tau _x = d_x\}\).
Using the field B, we define our sensors in the following way
Definition 1
3.2 Traffic
The set of configurations that satisfy (3) would include both area coverage and path coverage algorithms. We consider path coverage algorithms, and in particular, we focus on path coverage as provided by the nearest neighbor topology as it seems to provide adequate coverage of the traffic, but with a smaller number of sensors. We confirm this idea with our simulation results that we present later.
We assume that the sensors will be able to communicate while not necessarily powered on with regard to sensing. Note that the nearest neighbor sensor topology also has the benefit of supporting energyefficient communication peertopeer queries [10].
4 Adaptive power scheduling algorithm
Proposition 1
For \(\mathbf{x}^* \in U \subset \mathbb {R}^n\), suppose that for \(g \in C^2(U),\) \(\frac{\partial g}{\partial x_i}\) is a strictly decreasing function for all \(1\le i \le n\). For \({\dot{\mathbf{x}}} = \nabla g(\kappa \left\ \mathbf{x}\mathbf{x}^*\right\ ^2_2)\) the rest point \(\mathbf{x}^*\) is asymptotically stable.
Proof
Proposition (1) shows that under a small number of assumptions, the model governed by Eq. (8) will produce solutions that in the limit, will converge to \(x^* = x_{max}\) in a sufficiently small neighborhood, where \(x_{max}\) is the index corresponding to the sensor with the most detections. Therefore, when a traffic change is detected, we can assign \(x_{\xi } = x_{max},\) as \(x_{max}\) is the asymptotic solution for Eq. (8) when f is of the form \(g( \kappa \left\ x  x_{max} \right\ ^2_2 ).\) Algorithm (1) uses the asymptotically stable rest point for scheduling and shows the major steps of the scheduling algorithm.
As earlier stated, we now describe a placement algorithm that adheres to the conditions described by (4)–(3). This algorithm is later used to demonstrate the utility of our adaptive scheduling algorithm.
5 Sensor placement
The purpose of this section is to demonstrate how to create a placement algorithm that adheres to the constraints of our adaptive algorithm. Thus, we use the coverage Condition (3) to create an objective function that we constrain using Conditions (4)–(7) to determine the placement of sensors. We consider approaches for both homogeneous and heterogeneous environments.
5.1 Objective function
5.2 Homogeneous environment
In a homogeneous environment, \(\psi _i(x)\) remains constant with respect to traffic at a particular depth. Furthermore, the length of the interval \(I_i\) is given as \(\left I_i\right = I\) for all \(\psi _i.\) Consequently, all of the \(\psi _i\) are simple linear translations of each other. The next proposition shows that if we only place two sensors, and \(\left I_1\right + \left I_2\right \le \left \beta \right ,\) then the two sensors must be placed such that \(I_1 \cap I_2 = \emptyset\) to achieve an optimal placement.
Proposition 2
Proof
This proposition guarantees optimal placement as long as the sensors do not overlap. Satisfaction of the nearest neighbor topology Conditions (4)–(7) imply necessary optimality if the sensors overlap only the boundaries. In other words, no complex planning technique is required for placement for this case. However, for the situation in which Conditions (4)–(7) can only be satisfied with overlapping sensors, or the environment is heterogeneous, we resort to more tractable computational methods.
5.3 Heterogeneous environment
6 Traffic detection algorithms
We now describe two very distinctive traffic change detection algorithms. The first algorithm identifies traffic changes by using inflection points of a cubic bspline representation of the data. The second algorithm uses the Hellinger distance to determine when the traffic distribution changes. These two different approaches, when combined with our adaptive scheduling and placement algorithms, demonstrates the consistency of our adaptive approach as discussed in the next section.
6.1 Inflection point detection algorithm
6.2 Hellinger detection algorithm
7 Simulation results
Figure 8 shows no significant difference for the PF and PFH method. However, Fig. 9 shows that if we decrease \(\alpha\) to .1, we see a change in performance; the PFH method performs more efficiently than the PF method. As \(\alpha\) controls the sensitivity of the Hellingerbased detector algorithm, it is tempting to always set the sensitivity to something small. If the sensor placement algorithm is developed to favor sensor performance, as is the case with our sensor placement algorithm, then small \(\alpha\) values are desired. However, in the case where another sensor placement algorithm chooses sensor positions that yield a large variance in the quality of sensor performance, smaller \(\alpha\) values may be problematic for the following reason. If \(\alpha\) is chosen where any change in the distribution results in a traffic change detection, then the scheduling algorithm could unnecessarily choose to activate a sensor of a poorer quality than the currently active sensor. This is a conjecture whose validation is the subject of future work. Nonetheless, the higher efficiency results of both PF and PFH methods as compared to both MMC and AA methods demonstrate the consistent efficiency of our adaptive scheduling algorithm.
8 Summary
We have provided a potential field based algorithm that works very well with regard to efficient monitoring of traffic. This algorithm is general and requires a sensor placement algorithm and a traffic detection algorithm. To demonstrate the utility of this algorithm, we created a sensor placement algorithm, two traffic change detection algorithms, and combined these algorithms to create two distinctive traffic monitoring solutions that use the same placement algorithm. We compared our traffic monitoring solutions to an area coverage method and allactive path coverage method using a homotopy method to vary the influence of a static and a mobile distribution. Our results demonstrated the efficiency of our approach, which we believe has wide application to both sea and land domains.
Notes
Author contributions
RDT, JCH, and MJB originally developed and matured the original concept of adaptive scheduling. RDT and BH developed and discussed proofs of the main results. RDT implemented the software to test the algorithms and drafted the manuscript. All authors contributed to numerous edits and versions of this draft.
Funding
This work is funded by the Office of Naval Research (ONR), Code 32, and the Independent Applied Research (IAR) program at NSWC PCD.
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
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