Energy-Aware Stochastic UAV-Assisted Surveillance

2021, IEEE Transactions on Wireless Communications

Abstract

With the ease of deployment, capabilities of evading the jammers and obscuring their existence, unmanned aerial vehicles (UAVs) are one of the most suitable candidates to perform surveillance. There exists a body of literature in which the inspectors follow a deterministic trajectory to conduct surveillance, which results in a predictable environment for malicious entities. Thus, introducing randomness to the surveillance is of particular interest. In this work, we propose a novel framework for stochastic UAV-assisted surveillance that i) inherently considers the battery constraints of the UAVs, ii) proposes random moving patterns modeled via random walks, and iii) adds another degree of randomness to the system via considering probabilistic inspections. We formulate the problem of interest, i.e., obtaining the energy-efficient random walk and inspection policies of the UAVs subject to probabilistic constraints on inspection criteria of the sites and battery consumption of the UAVs, which turns out to be signomial programming that is highly non-convex. To solve it, we propose a centralized and a distributed algorithm along with their performance guarantee. This work contributes to both UAV-assisted surveillance and classic random walk literature by designing random walks with random inspection policies on weighted graphs with energy limited random walkers.

Figures (12)

should be satisfied; ii) the UAVs should have enough energy stored in their batteries to perform the surveillance and come back to their bases. Due to the inherent randomness in the movement of the UAVs and the inspections patterns of the sites, these constraints are of probabilistic nature. We first state these constraints and then convert them into tractable mathematical expressions to be used in our optimization framework later on. Table I: Major notations.

It can be verified that although the objective function and all the constraints in problem (10) can be expressed as monomials and posynomials w.r.t. the optimization variables, problem (10) does not obey the standard form of GP in (11). In fact, none of the constraints (C1), (C2), (C3), and (C4) can be directly expressed as inequalities on posynomials or equalities on monomials, which is required in GP.? Thus, the problem fits SRe 4 denotes the strictly positive quadrant of n-dimensional Euclidean space.

Algorithm 1: Centralized random walk and inspection policy design of stochastic surveillance

into the category of signomial programming, for which direct derivation of a solution is intractable [36]. In the following, we apply two methods, namely penalty functions and monomial approximations, on problem (10), to approximate the problem as a series of GP problems. Afterward, we propose an effective algorithm to solve the problem along with its performance guarantee. We first use the method of penalty functions and auxiliary variables [38]. To this end, we consider each equality on a posynomial in the format of g(a) = c as two inequality constraints: i) g(a) < c, and ii) Age) < c, where A is an auxiliary variable, which will later be fureed to be 1. Aiming to make problem (10) as close to a GP as possible, we perform some algebraic manipulation and rewrite it as:

q,l considering the new variables. Writing problem (20) w.r.t. these new variables and taking the log from all the inequality constraints will result in a convex programming problem. We omit the resulting problem in the interest of space; however, we derive the Lagrangian function of the problem in (21), which is of particular interest, where A, ¢, 0, p, €,0,v, x, GB, and é€ are the vector/matrix of Lagrangian multipliers. Note that we assumed 6; — 1, Vi, for convenience (see Footnote 5). Also, h;(a) is expanded using (15) since except h(a) all the monomial approximations are functions of the UAV index 7 and can be locally computed without the knowledge of other UAV parameters. This expansion results in the last term in the second line and the term in the third line of (21). The corresponding dual function of the problem can be written as: D(A, ¢, , p.€,5, v, x, B, €) . r/f(~ wr Bw F ne © nT \ (23) the solution. [his raises two concerns: 1) the processor should have a global knowledge about all the parameters of all the UAVs, i.e., at each iteration, it should have the knowledge of the entire sets gq = [qi,q2,--~ , Qe] and = [t1, 42,--> , dyer]; and the inspection and transition costs of all the UAVs. This is due to the fact that solving the convex programming in line 7 of Algorithm 1 is carried out using the gradient descent method [43], in which the iterative update of the Lagrangian multipliers requires global knowledge of the current values of all the Lagrangian multipliers and all the UAVs’ parameters. Obtaining this knowledge might be cumbersome/infeasible in some scenarios. ii) The size of the problem is |2/| x (|V|? + |V]), which, for a given map, escalates quickly as the number of UAVs increases.!° Based on the above two considerations, we aim to develop a distributed algorithm that firstly eliminates the requirement on global knowledge and secondly is scalable w.r.t. the number of UAVs. To achieve this, we aim to break down the problem into |//| individual sub-problems, each of which can be solved using a single processor. A processor can refer to the UAV’s computing devices, the computing facilities of the UAV’s base node, or any third party computing facility.

Fig. 3: The average consumed energy per surveillance cycle (SC) for different num- bers of UAVs w.r.t. the iteration count of our algorithm (top plot). The corresponding comparison with the baselines (bottom plot). Fig. 2: The long-term average consumed energy during the surveillance for different numbers of UAVs w.r.t. the iteration count of our algorithm (top plot). The corresponding comparison with the baselines (bottom plot).

Fig. 4: Top: The comparison of the average energy used for movement per surveillance cycle (SC) between our method and the baselines. Bottom: The comparison of the average energy used for inspection per SC between our method and the baselines.

Considering (1), to obtain a tractable expression, define the Bernoulli random variable Ty fOs =i ( . Considering the left hand side (1.h.s) of inequality (1), we welt

E|L;] refer to the expected value of the movement energy and the expected value of the inspection energy per surveillance cycle, respectively. In the following, we derive a closed-form expression for each of them in order. For UAV u,;, the long term average expected energy of movement is given by: We first prove that algorithm 1 generates a sequence of improved feasible solutions that converge to a point 2x* satisfying the KKT conditions of (13). Note that (20) is in fact an inner approximation of (13) [60]. Hence, it is sufficient to prove the following three characteristics for (20) [60]: