Sensing and Communication in UAV Cellular Networks: Design and Optimization

Recently, the use of uncrewed aerial vehicles (UAVs) in joint sensing and communication applications has received a lot of attention. However, integrating UAVs in current cellular systems presents major challenges related to trajectory optimization and interference management among others. This paper considers a multi-cell network including a UAV, which senses and forwards the sensory data from different events to the central base station. Particularly, the current manuscript covers how to design the UAV’s (<inline-formula> <tex-math notation="LaTeX">${i}$ </tex-math></inline-formula>) 3D trajectory, (ii) power allocation, and (iii) sensing scheduling such that (a) a set of events are sensed, (b) interference to neighboring cells is kept at bay, and (c) the amount of energy required by the UAV is minimized. The resulting nonconvex optimization problem is tackled through a combination of (<inline-formula> <tex-math notation="LaTeX">${i}$ </tex-math></inline-formula>) low-complexity binary optimization, (ii) successive convex approximation, and (iii) the Lagrangian method. Simulation results over a range of various key parameters have shown the merits of our approach, which consumes 33%-200% less energy compared to different benchmarks.


Sensing and Communication in UAV Cellular
Networks: Design and Optimization Carles Diaz-Vilor , Member, IEEE, Mojtaba Ahmadi Almasi , Member, IEEE, Amr M. Abdelhady , Member, IEEE, Abdulkadir Celik , Senior Member, IEEE, Ahmed M. Eltawil , Senior Member, IEEE, and Hamid Jafarkhani , Fellow, IEEE Abstract-Recently, the use of uncrewed aerial vehicles (UAVs) in joint sensing and communication applications has received a lot of attention.However, integrating UAVs in current cellular systems presents major challenges related to trajectory optimization and interference management among others.This paper considers a multi-cell network including a UAV, which senses and forwards the sensory data from different events to the central base station.Particularly, the current manuscript covers how to design the UAV's (i) 3D trajectory, (ii) power allocation, and (iii) sensing scheduling such that (a) a set of events are sensed, (b) interference to neighboring cells is kept at bay, and (c) the amount of energy required by the UAV is minimized.The resulting nonconvex optimization problem is tackled through a combination of (i) low-complexity binary optimization, (ii) successive convex approximation, and (iii) the Lagrangian method.Simulation results over a range of various key parameters have shown the merits of our approach, which consumes 33%-200% less energy compared to different benchmarks.

I. INTRODUCTION
T HE transition towards 6G networks has started with a potential use case of combining sensing and communications under the same umbrella [1], [2], [3].While static sensing and/or communication networks have been greatly studied, efficient sensing and reliable communication in dynamic networks result in new challenges not perceived by the static counterparts.To circumvent these challenges, the uncrewed Carles Diaz-Vilor, Mojtaba Ahmadi Almasi, and Hamid Jafarkhani are with the Center for Pervasive Communications and Computing, University of California at Irvine, Irvine, CA 92697 USA (e-mail: cdiazvil@uci.edu;maalmasi@uci.edu;hamidj@uci.edu).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TWC.2023.3326457.
Compared to earlier results, this manuscript covers unsolved fundamental challenges that naturally arise in joint sensing and communication networks such as the 3D UAV trajectory optimization, interference management, and sensing scheduling.One important defining aspect of these networks is data collection versus sensing.In data collection UAV networks, sensing is done on the ground and a communication link between the ground device and the UAV is needed.On the other hand, in sensing UAV networks, UAVs sense the events cooperatively whilst guaranteeing a successful sensing probability.For example, UAVs can be equipped with a variety of cameras and sensors which, after sensing an event, generate a certain amount of data that is transmitted to the base station (BS) [30], [31], [32].
In fact, the vast majority of the literature separates sensing from communications, especially within UAV networks given the complexity of achieving optimal deployments.Therefore, this work aims at filling this gap by focusing on the UAVenergy minimization problem with respect to (i) 3D UAV trajectory, (ii) power allocation, and (iii) sensing in multi-cell UAV-aided networks constrained to a set of sensing, communications, and mechanical technical specifications.Concretely, each cell provides service to a set of ground users (GUs).
In addition, the central cell also contains a set of events that must be sensed by the UAV, that can be regarded as an aerial user which always reports to the central BS. 1 After sensing an event, usually represented through a probabilistic sensing model [33], [34], [35], the UAV generates the sensory data which is transmitted to the central BS over a certain resource block.
Unlike the existing works in the literature, we envision a time-dependent probability of sensing, i.e., longer periods flying close to an event result in higher sensing probabilities.In addition, the sensory data transmission is mainly constrained by the information causality, UAV mobility, and interference.In fact, compared to [22] and [24], and similar works, the constraints that preserve the causality of the information need to be redefined.Two major differences arise: (i) the UAV uses the same resource block at all times without any time or frequency scheduling, and (ii) there is no link between the events and UAV as previously mentioned.As a consequence, the spectral efficiency of this system is higher although a reformulation of the causality constraint is needed.Moreover, a significant issue that is often ignored when designing UAV networks, as in [14], [15], [16], [19], [20], [21], and [22] among others, relates to interference.Particularly, it is assumed that the devices within each cell are assigned an orthogonal resource block; therefore, intra-cell interference is negligible, which can be easily achieved for example by orthogonal frequency division multiple access (OFDMA).However, frequency reuse among cells gives rise to intercell interference.Specifically, interference in aerial radio links is predominant compared to the ground counterparts given the Rician nature of air-to-ground channels [36], [37].Consequently, interference arising from the sensory data transmission from the UAV to the BS must be accounted for.Otherwise, correct decoding at certain GUs or BSs may fail.There are two main solutions to mitigate interference: (a) using coding or beamforming at the transmitter [38], [39], [40], [41] and/or (b) perform power control at the UAV [42], [43].In this work, to handle inter-cell interference, power control is performed at the UAV, to make it possible to carry a single antenna.
Finally, we broaden the model by considering a variable flying time.This is in contrast with most of the UAV literature, in which the flying time is fixed and predetermined.By exploiting the path discretization technique [21], we allow non-uniform time slots, which are shown to boost the UAV's energy efficiency.
The optimization problem we aim to solve is highly nonconvex; hence, it is decomposed into four subproblems and is solved utilizing the block coordinate descend (BCD) approach [44].The first subproblem deals with the logic-based sensing variables while the second jointly optimizes the UAV trajectory and time slot length by leveraging the well-known successive convex approximation method (SCA) [45], which is also applied to optimize the UAV altitudes.Finally, it is shown that the analytical expression for the UAV transmit power can be obtained through the Lagrangian method.Therefore, the main contributions of this paper can be summarized as: • A novel scenario that integrates sensing and communications is presented in UAV-aided multi-cell cellular networks based on realistic channels, sensing models, and subject to sensing, communications, and mechanical constraints.
Fig. 1.A multi-cell network with UAV-aided communications and sensing.
• The UAV energy minimization problem is studied as a function of the 3D UAV path, sensing, and transmit power which results in a mixed-integer nonlinear programming problem.
• Capitalizing on four low-complexity subproblems, suboptimal solutions minimizing the on-board UAV energy are obtained.The sensing is handled through a lowcomplexity binary optimization algorithm while the SCA technique is used to optimize the UAV's path.Finally, the optimal UAV transmit power is analytically obtained.• The impact and tradeoffs between a variety of parameters is well established.Our results show that the proposed scheme outperforms other benchmark methods and reduces the energy consumption between 33%-200%.The remainder of the paper is organized as follows.Section II presents the complete system model for the UAVenabled sensing and communications network.The problem formulation and solution are studied in Sections III and IV, respectively.Numerical results are presented and discussed in Section V while concluding remarks are set forth in Section VI.
Notation: lowercase letters, lowercase bold letters, and capital bold letters denote scalars, vectors, and matrices, respectively.A circularly symmetric complex Gaussian random variable (r.

II. SYSTEM MODEL
Consider a cellular network composed of multiple cells as shown in Fig. 1.Each cell contains one BS while the central cell also features one UAV and M events that need to be sensed with M = {1, . . ., M } denoting the set of events of interest.Particularly, the mth event of interest is located inside the central cell2 at ℓ m = (l m , 0) with l m ∈ R 2×1 whose coordinates are known a priori via different techniques such as Global Positioning System (GPS). 3 The flying/mission time is represented by T and the corresponding UAV coordinates at time t are given by ℓ(t) = (q(t), H(t)) where q(t) ∈ R 2×1 represents the ground coordinates and H(t) denotes the flying altitude.The BSs are located at the center of each cell.However, provided that the UAV reports to the BS in the central cell, only the location of such BS is relevant, whose coordinates are ℓ B = (q B , H B ). 4 For the kth GU outside the central cell, its coordinates are ℓ k = (w k , 0).Particularly, {q B , w k } ∈ R 2×1 and H B is the BS altitude, common for all of them.The optimization with respect to continuous time variables would yield to an intractable problem since the number of optimization variables is infinite.Therefore, the mission time T is discretized into N non-uniform time slots, denoted by δ(n), whose lengths are included in the optimization problem.Consequently, T = N n=1 δ(n) and thus the UAV's path is discretized, where the 3D coordinates at slot n are represented by ℓ(n) = (q(n), H(n)).Usually, discretizing the trajectory only specifies the locations whereas discretizing the path involves both the locations and the time dimension [21].To maintain the sampling accuracy, an appropriate value for N must be chosen by imposing the following constraint where d max is a proper value such that the UAV is assumed to fly at a constant velocity within each segment and the parameter variation between two successive time slots is small.In fact, when d max → 0, then N → ∞ and a nearly continuous trajectory is obtained.To obtain a scalable N that satisfies (1), one can first obtain an upper bound on the flying distance, given by D max , and require (N + 1)d max ≫ D max .Consequently, under the worst case scenario of the UAV flying D max , the sampling accuracy is still preserved.For ease of exposition, we define d B (n) and d k (n) as the Euclidean distance from the UAV to the central BS and User k, respectively.

A. Channel Model
The channel coefficient between the UAV and any of the network elements, denoted by g i (n) for i ∈ {B, k}, follows a Rician distribution, which encompasses two elements: (a) the LoS component and (b) a Rayleigh-distributed small-scale fading component [46,Sec. 3.4.1] where β 0 and κ are the path loss at a reference distance of 1m and the path loss exponent, respectively.The Rician factor is K i (n), which depends on the geometry between transmitter and receiver and environmental parameters [37].Finally, ψ i (n) ∼ U[0, 2π] and a i (n) ∼ N C (0, 1) account for the phase rotation and the small-scale fading, respectively.

B. Rate Calculation
The UAV can be regarded as an aerial user within the central cell.Therefore, it is assigned an orthogonal resource block, avoiding interference to the GUs within the central cell.Upon sensing an event, the UAV generates and transmits the sensory data to the BS with an instantaneous rate of where σ 2 denotes the noise power and p(n) is the UAV's transmit power, which must not exceed a certain value: where p max is the maximum transmit power.A common approach to manage the randomness of the rate is to consider the ergodic capacity, i.e., E{R(n)}.Furthermore, application of Jensen's inequality to E{R(n)} removes the effect of the small-scale fading given that As shown in [16], to keep the contribution of the LoS and NLoS channel components, the channel coefficient can be well approximated by a logistic regression.Therefore, as suggested in [16], the rate is approximated as where C 1 , C 2 , B 1 , and B 2 are obtained from the logistic model and u .

C. Interference Management
A key enabler in wireless cellular networks is interference management.While intra-cell interference can be avoided through well-known techniques such as OFDMA, neighboring cells will reuse some of the resources. 5Consequently, given the dominance of the LoS component in aerial channels, intercell interference arises at both the BSs and the GUs originated by the UAV transmissions. 6Therefore, to keep interference at a bay, a joint optimization of the UAV trajectory and power is considered, in recognition that an isolated study of one aspect may be misleading because of potential bottlenecks in the other.By using the same logistic regression approximation for the channels between the UAV and GUs (or neighboring BSs), the amount of interference that GU k receives from the UAV is: Therefore, the following interference-related constraint must be met for each of the GUs: where I th is the maximum level of interference each GU can tolerate without compromising the decoding.

D. Energy Consumption Model
We consider a realistic energy consumption model for the UAV comprised of (i) flying power p f n, ℓ(n), δ(n) , (ii) sensing power p s , and (iii) communication power p(n) [47], [48], [49].Following [21], for rotatory-wing UAVs, the total UAV energy, spent over the N time slots, is where α m (n) is a logic binary variable defined as Based on [21] and [50], p f n, ℓ(n), δ(n) is a function of the UAV locations and the time slots as follows where v(n) is the 3D velocity at time slot n defined as: τ c (n) denotes the climb angle, which is a function of ℓ(n), and the rest of parameters are defined in the aforementioned references.Finally, other constraints related to the UAV locations include a minimum and maximum flying altitude, the initial and final positions and a maximum velocity constraint:

E. Sensing
The goal of sensing is to collect data from different events.Indeed, optimizing the UAV's trajectory will improve the efficiency and accuracy of the sensing.Particularly, we consider M events and utilize the probabilistic sensing model [33], [34], [35] where the dependency between sensing probability and UAV trajectory is through a distance-based exponential function.In other words, the probability of sensing event m at time n is where µ determines the sensing capability of the UAV.Other relevant sensing models can be found in [13].However, note that (15) avoids the dependency with respect to the time-slot length, i.e., longer δ(n) might provide better sensing accuracy.
In this work, we also study the impact of a time-dependent sensing probability.Inspired by [29], assume that the nth time slot is divided into X sub-slots of equal length t n , i.e., δ(n) = Xt n , and that the UAV carries out X trials to sense an event.A failure in the sensing occurs when all X trials fail to correctly sense the event.Hereby, the probability of successful sensing can be obtained through its complementary: where (i) a longer δ(n) results in a higher probability and (ii) δ(n) ≥ t n to ensure at least one trial.In addition, only one event can be sensed at a time, modeled by Moreover, the sensing of the M events follows a certain order, discussed in Sec.IV-F, determined by S = {s 1 , . . ., s M } with s m ∈ M. Therefore, the following constraints must be met Furthermore, it is required to sense each event once, which can be mathematically modeled by where a certain amount of data, C m , is assumed to be generated by the UAV if α m (n) = 1.Altogether, based on the logic variables α m (n), the constraint that must be met in terms of sensing probability is where P s,th is the minimum sensing probability the UAV requires to correctly sense an event and A is chosen to be a constant larger than P s,th .Note that a sensing variable can be active, i.e., α m

F. Causality of the Information
As stated in the introduction, compared to the existing works in the literature, the constraint that relates to the causality of the information needs to be redefined given that (i) the UAV uses the same resource block at all times without any time or frequency scheduling, and (ii) there is no link between the events and UAV.In fact, upon sensing an event, the UAV generates the sensory data, e.g. a picture or a measurement among others.Let us assume the UAV senses the events in a certain order, as defined by S in Section II-E, and the amount of data generated by the s m th event is C sm .Successful reception of the sensory data at the BS is ensured by the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
following sensing constraints: where B is chosen to be a constant larger than M m=1 C sm .For example, before sensing any event or between two events, the binary variables are zero but the constraints are met due to C sm .When the last event is sensed, α s M (n) = 1 and therefore after D samples, the amount of data that the BS receives has to be larger than C s M .Whenever the penultimate event is sensed, Hence, the BS must be able to receive the data from the last two events in the last N − n M −1 − D samples.A similar procedure is followed to generate the causality constraints for the remaining events.

III. PROBLEM FORMULATION
With the aim of producing energy-efficient trajectories for the sensing and communication problem, the objective function in our work is presented in Eq. (8).Constraints ( 4) and (7) determine the maximum transmit power and maximum GU interference, respectively.The set of constraints presented in ( 11)-( 14) relate to the UAV mechanical capabilities and starting/ending point of the mission.In addition, introducing the set of logic variables α m (n) allows us to combine the communication and sensing constraints under the same umbrella.More specifically, constraints ( 17)-( 21) ensure all M events are sensed and the corresponding sensory data is successfully received at the central BS.To this end, the optimization variables are: (i) sensing variables α m (n), (ii) UAV trajectory q(n), H(n) , (iii) length of the time slots δ(n), and (iv) transmit UAV power p(n).Therefore, the problem presented in (22), shown at the bottom of the page, can be formulated, as in (22) which falls within the class of nonconvex mixed-integer non-linear programming problems whose solution requires prohibitive time and computational complexity.Accordingly, we split the problem into four subproblems: (i) optimizing sensing with path, length of the time slots, and power fixed; (ii) optimizing 2D trajectory and time slot length with fixed sensing, altitude and power; (iii) optimizing altitude with sensing, 2D trajectory, length of the time slots, and power fixed; and (iii) optimizing power with sensing, path, and time slots fixed.Once the solution to each of the four subproblems is obtained, a BCD procedure [44] is followed until convergence is achieved.

IV. JOINT OPTIMIZATION
In this section, we conduct a thorough analysis of the subproblems that arise from (22).More specifically, the optimization of the binary sensing variables is covered first.Second, an SCA-based approach for the joint trajectory and time slot optimization is presented.Finally, the optimal UAV transmit power can be derived through the Lagrangian method.

A. Sensing Optimization
For fixed q(n), δ(n), H(n), and p(n), we first aim at solving the sensing optimization subproblem.Provided that the only contribution of the sensing variables, α m (n), in the cost function is through the term that depends on the sensing power, p s , the first optimization problem reduces to: The binary nature of α m (n) results in an NP hard problem.
Relaxing the binary assumption would result in constraint violations, and therefore the problem should be kept in the binary domain.To cope with the increased complexity, note that α m (n) = 0 when P m (n) ≤ P s,th .Hence, there is no need to search over the entire solution space, but only in the points satisfying P m (n) ≥ P s,th .Consequently, the branch and cut is an appealing technique to solve this problem [51].The complexity of this approach is discussed in Sec.IV-E

B. Trajectory and Slot Length Optimization
Given any feasible α m (n), H(n) and p(n), optimizing the 2D trajectory and the length of the time slots reduces to the following problem: ), ( 7), ( 11) − ( 14), ( 20), ( 21), which is nonconvex.Therefore, we leverage the SCA technique to create an approximated version of (24).To full-fill this goal, we first obtain an equivalent problem by adding different slack variables such as , ( 4), ( 7), ( 11) − ( 14), ( 17) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
be re-written as: where y(n), being a second slack variable, is defined below: After some algebraic manipulations over the previous equation, we have Finally, a third slack variable is needed to deal with the raterelated constraints: Consequently, the set of constraints in ( 21) can be expressed as a function of β(n) in a compact manner for a given event sensing order S = {s 1 , . . ., s M }: As a result, the problem in ( 24) can be re-written as presented in (30), shown at the bottom of the page.Proposition 1: The optimization problems presented in ( 24) and (30) are equivalent.
Proof: The proof can be found in App. A. Still, some of the constraints in (30) are nonconvex which makes SCA relevant in this work.More precisely, SCA (a) locally convexifies the initial problem and (b) solves a convex approximated version by alternating between two steps: (i) upper (lower) bound a concave (convex) function by its firstorder Taylor expansion and (ii) find the optimal solution of the approximated convex problem.In the subsequent, we derive the necessary approximations for the nonconvex constraints in (30).For ease of exposition, we denote variables by x and the value of the variable in the approximation point by x and first cope with the expressions that relate to R(n).
Proposition 2: R(n) is jointly convex with respect to e −(B1+B2u B (n)) and d 2 B (n). Proof: The proof can be found in App.B. Using Prop.2, the following lower bound for R(n) can be derived as a function of λ(n) = B 1 + B 2 u B (n) and q(n).
Lemma 1: At any local point for the UAV trajectory q(n) and λ(n), R(n) accepts the following lower bound: Proof: The proof and the values of ϕ(n) and ζ(n) can be found in App. C.
Hence, the constraint related to R(n) in ( 30) is locally convex given R lb (n).However, provided that λ , a lowerbound for u B (n) is required.
Lemma 2: At any local point for the UAV trajectory q(n), u B (n) accepts the following lower bound: Proof: The proof and the value of ψ(n) can be found in App.D.
Therefore, the constraint that λ(n) must satisfy is The set of constraints that takes into account the maximum interference tolerated by out-of-central cell GUs is also nonconvex with respect to q(n).However, an upper bound on the interference can be derived by considering the worst-case scenario of the UAV flying on top of User k: where , ( 7), ( 11), ( 13), ( 14), ( 20), ( 29).
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Following the previous equation and rearranging terms, we obtain which is still non-convex.Further application of the SCA results in: A similar procedure applies to (29), which results in the following convex set of constraints: Armed with the respective upper or lower bounds and after applying the same technique to the expressions that involve y(n) 2 , ∆(n) 2 , and, if needed ( 16), the local approximation of the problem in ( 30) is introduced in ( 41), shown at the bottom of the page, where, compared to (30), we add c z ≤ 1 given that sin τ c (n) ≤ 1 and the sin-function is neither convex nor concave.It can be verified that both the cost function and the constraints in (41) are convex.Therefore, (41) can be solved using standard optimization solvers such as CVX [52].

C. Altitude Optimization
A similar methodology is followed to optimize the altitudes, H(n).Given the space limitations, we skip some derivations since the process is very similar to the 2D optimization.Consequently, the approximated optimization problem for the altitude is presented in (42), shown at the bottom of the page, where R lb (n), (35), and (39) are slightly modified to account for the gradients w.r.t. to the altitude.Finally, ( 42) is a convex optimization problem and therefore can be efficiently solved [52].
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Proposition 3: The optimal power allocation for the UAV is given by where a(n) depends on the Lagrangian multipliers and is defined in Eq. ( 62), , and the operator [x] + = max(x, 0).
Proof: The proof can be found in App.E.

E. Algorithm Analysis
Based on the solutions to the previous subproblems, we propose an iterative method for the initial nonconvex problem in which we optimize four sets of variables: sensing variables, 3D UAV trajectories, time slot length, and power allocation.The convergence of the proposed BCD approach in Alg. 1 is guaranteed by the following proposition.
Proposition 4: The sequence of objective values generated by Alg. 1 is monotonically non-increasing with a lower bound, and therefore converges.
Proof: The proof can be found in App.F.

Algorithm 1 BCD Updates for the Optimization Variables
Require: Initial sensing, trajectory, time slots and power variables at the first iteration, j = 0, given by {α 0) (n)} and define by η (0) the respective cost function.
Compute the cost function η (j+1) .end while The complexity of the previous algorithm is given by the combination of individual complexities for solving each of the subproblems.Particularly, it can be shown via induction that, given a sensing order, the maximum number of combinations for α m (n) in ( 23) is upper bounded by corresponding to the case in which the M events can be sensed during the N time slots, i.e., a very conservative bound.Additionally, given that (41) involves logarithmic forms, the interior point method presents a complexity of O (7N ) 3.5 log(1/ϵ 1 ) where ϵ 1 relates to the convergence accuracy [53].Similarly, solving (42) has a complexity of O (5N ) 3.5 log(1/ϵ 2 ) .Finally, solving (43) has a complexity of O((N + N 2 ) log(1/ϵ 3 ) where the term in N 2 arises after applying the ellipsoid to the dual problem [52].Consequently, the overall complexity is dominated by the SCA-based subproblems.

F. Algorithm Initialization
Given initial and final trajectory points, the first step is to determine the sensing order S. Different criteria may apply, e.g. based on priority, distance or random.In our case, we set the ordering based on distance.Consequently, Next, the value of D max is obtained, serving as an upper bound on the maximum distance the UAV has to cover, i.e., As a consequence, for a fixed d max , the number of slots can be set as (N +1)d max ≫ D max .Next, we turn to generating the UAV trajectory.Initially, the UAV path coordinates will lie on the lines joining the departure point, events locations, and the destination point.Therefore, we ensure the initial trajectory is capable of sensing all events.Particularly, until the first event is sensed, i.e., P s1 (n) ≥ P s,th and therefore α s1 (n) = 1, the UAV coordinates lie on the line between the departure point and the first event as: where θ and ϕ are the inclination and azimuth angles between ℓ i and ℓ s1 , respectively.The trajectory between two events lies on the line connecting the location where the former is sensed and the subsequent, with the corresponding recalculation of θ and ϕ while satisfying the altitude constraints.Finally, after sensing s M , the same formulation applies with the angles corresponding to the points associated to ℓ s M and ℓ f .To maintain the accuracy in the sampling while satisfying the maximum velocity constraint, we have α ≤ min{d max , V max δ(n)}.Once the trajectory and sensing variables are initialized, the powers are set to the minimum value between p max and the value that satisfies the interference constraint (7).To verify that the causality constraints are met, a search is performed.Otherwise, either more slots can be added or the problem can be declared infeasible.

V. NUMERICAL RESULTS
For the purpose of performance evaluation, we consider a cellular network composed by one central and six neighboring cells, each following a hexagonal shape of radius R = 200m, although our work is independent of the cell shape.Unless otherwise specified, Table I lists the parameters used in the simulations, selected from the UAV and sensing literature [12], [15], [26].Observe that the parameters related to Eq. ( 8) are not included since they are the same as the ones in [21].To maintain the accuracy in the path discretization process, we consider d max = 2 and an upper bound on the flying distance D max = 400m.Therefore, (N + 1)d max > D max is

TABLE I SIMULATION PARAMETERS
satisfied with N > 300.In this case, we use N = 400 where the initial length of the slots is δ(n) = 0.25s with a resulting initial flying time of T = 100s.The typical scenario that we use in our simulations and the corresponding events and GU locations are presented in Fig. 2a, with M = 8 events and K = 6 GUs, i.e., one per each neighboring cell.The starting and final points are ℓ i = [130, −30, 50] and ℓ f = [65, 100, 50], respectively.The UAV trajectory for two cases of µ = 10 −3 and µ = 2 • 10 −3 using the above typical scenario is included in Figs.2a and 2b, with the former showing the 2D projection and the latter plotting the 3D trajectories to gain intuition on the altitude variations.Though more insight will be provided about the influence of µ, it is shown that higher values, i.e., worse sensing capability, require the UAV to fly closer to the events.In fact, from Fig. 2b it can be shown that the UAV tends to fly at lower altitudes for µ = 2 • 10 −3 to satisfy the sensing requirements compared to µ = 10 −3 .
We first verify Prop. 4 in Fig. 3 for different initializations and BCD orderings.Particularly, in O1, the optimization order in Alg. 1 is:  In addition, 2D initializes the altitude at a constant value H(n) = 50m while 3D uses the method described in Sec.IV-F.Finally, different values of c z are tested as well Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.to see its effect in the final solution.First, note that no matter what ordering and initialization setup we utilize, convergence in Alg. 1 is achieved only after a few iterations.Additionally, the difference in terms of cost function is minimal.However, the solutions differ.This can be seen in Fig. 4, where for a variety of initializations, the UAV altitudes may be different though requiring a similar energy budget.Such a tendency is mainly because all methods converge to a solution where the UAV flies at the velocity that minimizes the required energy, as described in [21, Sec.II-B].More details on this phenomena are provided in subsequent paragraphs where the UAV velocity is also a matter of study.
Next, in Figs.5a and 5b we compare the energy and flying time, respectively, for different benchmarks parametrized by M .Particularly, "Proposed" stands for the algorithm presented in this work while "Min-Time" solves the flying time minimization problem.Additionally, "Equal-δ(n)" utilizes a similar algorithm as ours, but with fixed δ(n) = 0.25s for all n.Finally, the "Max-Vel" benchmark is an heuristic algorithm based on the one described in Sec.IV-F, where the UAV flies at maximum velocity between each pair of points ensuring the rest of constraints are met.As expected, the more events, i.e., increasing M , the higher the energy and flying times are.Clearly, in terms of required on-board energy, our method outperforms the rest of benchmarks by saving at least 25% of the energy, whereas the minimum flying time is attained by the "min-Time" benchmark, with our method being close.Note that given the correlation between the "Proposed" and "min-Time" algorithms, their performance is similar, where shorter flying time results in less energy.In addition, the gap between the "Proposed" and "Equal-δ(n)" curves arises by adding δ(n) into the optimization problem.Therefore, it is clear that non-uniform slots make a big difference in terms of energy-efficiency and flying time given the added degrees of freedom and the enlarged feasibility region compared to a fixed time slot.Fig. 6 studies the impact of N in the simulation environment for the proposed algorithm and a variety of benchmarks.
While Figs. 6a and 6b plot the energy and flying time, respectively, parametrized by N , Fig. 6c presents the UAV velocity for N = 400 under three algorithms.It can be concluded from Figs. 6a and 6b that the methods that include δ(n) in the optimization, i.e., "Proposed" and "Min-Time", converge to the same cost function independent of the value of N .This is in conjunction with the flying velocities presented in Fig. 6c, where the "Proposed" method adjusts the UAV path to fly at the velocity that minimizes the energy, as explained in [21, Sec.II-B], achieved if v(n) = 17.7 m/s.In addition, to minimize the flying time, the UAV flies at its maximum velocity, in this case v(n) = 30 m/s.To the contrary, if δ(n) is not included in the optimization: (i) the energy and flying time tend to be higher than the other methods and (ii) after a certain value of N , both energy and time stabilize given that adding more slots is no longer helpful.In this case, it can be concluded from the three pictures that N > 260 does not provide any change given that the UAV can finalize its mission in less slots and therefore will hover at the destination point spending unnecessary energy.Fig. 7 examines the dependency of the amount of needed energy and time to complete the sensing mission with respect to the sensing probability, P s,th .The blue curves correspond to the required energy while the red curves relate to the flying time.Figs.7a and 7b utilize the model in (15), with Fig. 7b considering a fixed flying altitude of H(n) = 50m.Finally, Fig. 7c considers the model in (16).In addition, different values for the sensing capability, µ, are also presented.Clearly, a smaller µ allows the UAV to sense the events at larger distances compared to a higher µ, as shown in Figs.2a and 2b.Therefore, the use of smaller µ results in trajectories whose energy and time serve as a lower bound for higher values of µ.Additionally, increasing the value of P s,th requires the UAV to fly closer to the events.As a consequence, both energy and flying time tend to increase with the threshold probability.By comparing Figs.7a and 7b, it is shown that for higher values of P s,th , optimizing the altitude, as in Fig. 7a, reduces the required energy and time given that the distance between Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the UAV and the event can be smaller if the UAV has freedom to adapt its altitude.Finally, the difference between the values in Figs.7a and 7c is minimal.Although the latter considers a sensing model that depends on the length of the time-slot, i.e., higher δ(n) results in a higher sensing probability, the energy and flying time mainly depend on the velocity.In fact, as shown in Fig. 7d, which plots the velocity for the models in (15) (solid-blue) and (16) (dashed-red), in both cases the UAV velocity converges to the value that minimizes the required energy, i.e., v(n) = 17.7 m/s.As a consequence, the energy and time obtained using ( 15) and ( 16) are similar.
Next, we study the effects of the maximum interference tolerated by the GUs, given by I th .In Figs.8a and 8b, we present the variation of energy and time parametrized by I th , respectively.We present a variety of algorithms, as in previous figures.While the "Proposed" and "min-Time" refer to the same systems as in other figures, the "max-Vel" also adjusts the power as in (43) provided that the initialization algorithm might use more power than what is needed for the communication part.As in previous figures, the proposed method outperforms the rest in terms of required energy by at least 25% though needs more time to complete the mission.Additionally, given that the cost function is dominated by the flying energy, variations in the transmit power are not distinguishable in Fig. 8a.Second, in general, the amount of needed energy and time tend to increase as I th decreases given that the feasible regions shrink for decreasing I th .Note that for I th → 0, the problem might be infeasible.
To gain more insight on the effects of interference, we include Fig. 9. Particularly, Fig. 9a plots the gap in the interference constraint, defined as: for the GU at [302.8,−50.7].The definition of ∆ arises from calculating the difference between the square root of the left and right hand sides of the constraint presented in (38).Note that in fact, (48) provides a notion on how (38) is met.Higher values of ∆ mean the UAV easily meets the constraint while smaller values of ∆ mean the UAV finds it harder to meet the interference constraint.More particularly, by looking at Fig. 9a, it can be verified that smaller I th yields a smaller ∆ since the regions where the UAV can fly, meeting the interference constraints, become smaller.In addition, note that in fact, the "min-Time" algorithm provides smaller ∆ given that higher transmit powers can still meet the constraints since its goal is to minimize Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the flying time, not the flying energy/power.Additionally, Figs.9b and 9c plot the transmit power, p(n), for different values of I th using the "Proposed" and "min-Time" algorithms, respectively.Note that these curves compare an easy-to-meet interference constraint, i.e., I th = 10 −8 , versus limiting values of I th .More importantly, the orders of magnitude for the transmit power are very different in Figs.9b and 9c.While the former adjusts its power to satisfy the communication requirements with minimum power, the latter does not perform power minimization, which yields to a

VI. CONCLUSION
This paper has considered energy-efficient communications and sensing where an aerial vehicle senses multiple events of interest.After generating the sensory data, the UAV ensures its reception by the BS while managing the interference effect to GUs located at neighboring cells.We have considered a generic cellular network with Rician channel models and presented mechanical-related, communication-related, and sensing-related constraints that must be satisfied to complete the mission.We presented a novel logic-based approach to formulate (a) the 3D path planning, (b) sensing, and (c) transmit power subproblems.This formulation has allowed us to use classic optimization techniques.Most remarkably, we have studied the dependency of the UAV trajectory with respect to different parameters and benchmarks, namely: (i) maximum velocity, (ii) minimum flying time, and (iii) fixed-slot duration.Comparative studies across various number of events have demonstrated that our proposed approach results in a reduction of energy consumption between 33%-50%.Moreover, the proposed scheme outperformed the minimum flying time and maximum velocity benchmarks by consuming 33%-41% less energy, depending on the maximum level of interference the GUs can tolerate.Finally, the number of time slots and time slot duration have a significant impact on overall performance.
Our solution consumes 33% and 200% less energy compared to the minimum flying time and fixed-slot duration benchmarks, respectively.APPENDIX A Given that we have added three slack variables, we analyze their behavior individually.First, ||ℓ(n + 1) − ℓ(n)|| ≤ ∆(n) will be met with equality at the optimal, otherwise the value of ∆(n) can be reduced to achieve a better cost function.A similar argument is used to show that y(n , otherwise the value of y(n) can be decreased to reduce the cost function.Finally, at the optimal, Eq. ( 28) is met, or else the value of β(n) can be increased.

APPENDIX B
We first define x = 1 + e −(B1+B2u B (n)) and y = d .The Hessian matrix of f (x, y) is included in Eq. ( 49), shown at the bottom of the page.For any vector s = (s 1 , s 2 ) T , it can be shown that s T ∇ 2 f (x, y)s ≥ 0 as included in Eq. (50).Therefore, f (x, y) is convex with respect to x, y, which ultimately implies the convexity of R(n) as well.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

APPENDIX C
By exploiting the convexity of R(n) as shown in Prop.2, such a term accepts a lower bound of the type: where coefficients ϕ(n) and ζ(n) are provided in Eqs. ( 52) and ( 53), shown at the bottom of the page, respectively.

APPENDIX D
A similar procedure as the one followed in App.B is considered to show the convexity of u B (n) = (H(n)−H B ) d B (n) with respect to ||q(n) − q B || 2 .As a consequence, u B (n) accepts the following lower bound: where the value of ψ(n) is Solving the previous equation for p(n), we obtain: for all n = D + 1, . . ., N .The value of a(n) in Eq. ( 44) is therefore: s T ∇ 2 f (x, y)s = s Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Manuscript received 23
November 2022; revised 17 March 2023 and 16 August 2023; accepted 18 October 2023.Date of publication 30 October 2023; date of current version 12 June 2024.The work of Carles Diaz-Vilor, Mojtaba Ahmadi Almasi, and Hamid Jafarkhani was supported in part by the NSF under Award CNS-2209695.The work of Amr M. Abdelhady, Abdulkadir Celik, and Ahmed M. Eltawil was supported in part by the NEOM under Grant 4849 and in part by the King Abdullah University of Science and Technology.The associate editor coordinating the review of this article and approving it for publication was M. Kaneko.(Corresponding author: Carles Diaz-Vilor.) v.) is denoted by N C (a, b), with mean a and variance b, while U[a, b] denotes a uniform r.v. on the range [a, b].

Fig. 4 .
Fig. 4. Final UAV flying altitude for a variety of initializations and cz.

Fig. 5 .
Fig. 5. (a) Energy and (b) flying time for different M and a variety of algorithms.

Fig. 6 .
Fig. 6.(a) Energy and (b) flying time for different N and a variety of algorithms.In (c), the UAV velocity for N = 400.

Fig. 9 .
Fig. 9.For different values of I th and algorithms, (a) ∆; (b) p(n) for the "Proposed" algorithm; and (c) p(n) for the "Min-Time" algorithm.

= H(n) − H B 2 (
||q(n) − q B || 2 + (H(n) − H B ) 2 ) power allocation problem, we first formulate the Lagrangian as in (58), shown at the bottom of the page.Defining w m (n) = n−D i=1λ i,m α sm (i), the Lagrangian can be rewritten as presented in (59), shown at the bottom of the page, where K ct is a constant term that does not depend on p(n).Taking the derivative with respect to the optimization variables, p(n), for fixed values of the multipliers, we obtaindL(p(n), λ) dp(n) = δ(n) + M m=1 w m (n)δ(n) ln(2)