I Introduction
Unmanned Aerial Vehicles (UAVs) are increasingly being used in applications such as surveillance [11, 6], package delivery [13, 22], environmental monitoring [5, 29], and precision agriculture [4] due to their ability to monitor large areas in a short period of time. One bottleneck that hinders longterm deployments of UAVs is their limited battery capacity. Recently, there have been efforts in overcoming this bottleneck by using Unmanned Ground Vehicles (UGVs) as mobile recharging stations [30, 27, 31, 21, 17, 16, 28, 14, 8, 10, 12]. However, these works make the simplifying but restrictive assumption that the energy consumption of the UAV is deterministic. In practice, the energy consumption is stochastic and the planning algorithms must be able to deal with the risk of running out of charge. In our recent work [27], we presented a riskaware planning algorithm for planning for a single UAV and UGV. However, that algorithm scales exponentially with the planning horizon and the number of robots. In this paper, we present an efficient riskaware coordination algorithm for scalable, longterm UAVUGV missions.
We consider a scenario where the UAVs and the UGVs are executing a persistent monitoring mission by visiting a sequence of task nodes in the order given (Figure 1). The UAVs can take a detour from the planned mission to rendezvous with some UGV and recharge. The UGV can recharge the UAV while continuing towards its own next task node. We present several algorithms that decide when and where the UAVs should recharge and which UGV they should recharge on.
The rate of battery discharge of a UAV is stochastic in the real world. The planning algorithms should be able to deal with such stochasticity. One approach would be to take recharging detours more frequently, thereby reducing the risk of running out of charge^{1}^{1}1In practice, when a UAV is about to run out of charge, it can land and then be retrieved by a human. By reducing the risk of running out of charge, we aim to reduce the overhead of such costly human interventions.. However, if a UAV takes a detour, then the task nodes visited after the detour will experience a delay, thereby worsening the task performance. Since we are planning over a longer horizon with multiple UAVs and UGVs, there is a complex tradeoff between task performance and risk tolerance.
Given a team of UGVs that move on the road network, and UAVs that can directly fly between any pair of nodes, we are interested in finding a scheduling strategy for the UAVs by treating UGVs as resources such that the sum of detour cost of the UAVs is minimized and the probability that no UAV runs out of charge within the planning horizon is greater than a userdefined tolerance. We formulate this problem as a resource allocation problem with matching and knapsack constraints. The matching constraint captures the resource availability constraints (each UGV can serve a limited number of UAVs at a time). The knapsack constraint captures the risktolerance requirement (probability that no UAV runs out of charge).
The Riskaware Recharging Rendezvous Problem (RRRP) can be formulated as an integer linear program (ILP). While there are sophisticated ILP solvers, we show how to exploit the structural properties of the problem to yield more efficient algorithms with theoretical performance guarantees. Other combinatorial problems with knapsack or budget constraints have been studied in literature. Approximation algorithms for the budgeted version of the minimum spanning tree problem and maximum weight matching problem are presented in [25] and [3] respectively. They both use Lagrangian relaxation which resembles our approach, however, due to different problem structures, we propose a bicriteria approximation algorithm to solve RRRP. We demonstrate the effectiveness of our formulation and algorithm in the context of a persistent monitoring mission.
Our work is built on results from the previous work on the cooperative routing problems, which are usually formulated as variants of the Vehicle Routing Problem (VRP) or the Traveling Salesman Problem (TSP) [22, 30, 13, 17, 31, 15, 18, 19]
. In these works, authors usually assume a deterministic model for the energy consumption of the UAV or use the expected value for simplification. However, when the UAV executes the route, the actual energy consumption may be quite different from that used in route planning. Moreover, in the longduration task, the estimation module may periodically update the energy consumption model. Given the routes of UAVs and UGVs generated by some existing algorithms, we consider the recharging scheduling problem within a planning horizon. Our formulation can be used in a receding horizon fashion to deal with the uncertainties in the longduration tasks. More details will be discussed in the Sec
IV.The main contributions of this paper are:

We introduce the Riskaware Recharging Rendezvous Problem (RRRP) as the first riskaware version of the multirobot recharging problem. We formulate the RRRP in the cooperative aerialground system as a matching problem on a bipartite graph with an additional knapsack constraint.

We present several algorithms to solve the problem, that build on each other. Our main theoretical result is a bicriteria approximation algorithm that yields a solution that violates one of the constraints by a factor of , and has a detour cost within of the optimal cost.

We demonstrate the effectiveness of our formulation and the proposed algorithm in a persistent monitoring application.
Ii Finite Horizon RiskAware Recharging Rendezvous Problem
In this section we first define the Riskaware Recharging Rendezvous Problem on a finite horizon, and then formulate the RRRP as a matching problem on a bipartite graph with an additional knapsack constraint which enforces that the probability of no UAV running out of charge within the horizon is greater than a risktolerance parameter.
Iia Problem Statement
Consider a team of UAVs and UGVs persistently monitoring a set of locations in an environment. The UAVs and UGVs move on the graphs and respectively. The vertex sets and represent the locations to be monitored by the aerial and ground vehicles respectively. The edge set may be complete since the UAVs can move between any of the tasks, whereas the edge set represents the road network on which the ground vehicles can move. We assume that the ordering of the task nodes for the UAVs and the UGVs is given, i.e., we have predefined persistent monitoring tours for the UAVs and UGVs. These tours can be generated by planners that either do not consider recharging [23, 1, 7] or assume deterministic discharge [28, 17]. The tour for UAV is denoted by and the tour for UGV is denoted by . Because of the persistent nature of the problem, the UAVs need to be recharged repeatedly in order for them to keep operating.
A UAV can leave its monitoring tour at any point along the tour to land on one of the UGVs, say UGV , for recharging, while the UGV keeps moving along its tour . The UAV can also wait at a rendezvous location for the UGV if it reaches there before the UGV. The number of UAVs that can simultaneously charge on a given UGV is . Once recharged, the UAV leaves the UGV and goes to the next task node along its monitoring tour. This procedure of a UAV leaving its monitoring tour to rendezvous with a UGV for recharging and then returning to continue its monitoring tour is referred to as a detour.
We assume that the UGVs do not run out of charge.^{2}^{2}2This is a standard assumption in the literature [21, 30, 31, 17, 18, 20] since the runtime of the UGV can be an order of magnitude larger than the UAV. Furthermore, the UGV battery (or fuel, if its fuel powered) can be easily and quickly replenished unlike the UAV. Unlike much of the existing work [17, 21, 31], we do not assume that the discharge rate of by the UAVs is deterministic. Instead, we consider a stochastic discharge model and assume that the probability of a UAV running out of charge within the next time units given the current charge level can be calculated as that in [27]. The stochastic battery discharge is monotonic in the time traveled by the UAV.
Since it is a persistent monitoring scenario where the UAVs need to be recharged indefinitely, we take the approach of solving the problem in a recedinghorizon manner. The problem within a given time window is to find a recharging policy for the UAVs for up to a single recharge for each UAV. Moreover, as the battery discharge rate of UAVs is stochastic, there may be a nonzero probability of some UAVs not being able to reach any of the recharging UGVs. Hence, we also need to have a notion of riskaversion for the UAVs. On the other hand, to avoid frequent recharging, we also consider the detour time spent by the UAVs for recharging.
At a high level, we consider the following problem.
Given a time horizon , a risktolerance probability , task routes for the UAVs and UGVs along with their current locations and state of charge, we seek to solve the problem of finding a recharging schedule for each UAV such that:

each UAV recharges at most once during ,

no UGV can charge more than UAVs at a time,

the probability that no UAV runs out of charge during is at least , and

the total detour time of the UAVs is minimized.
In the following paragraphs, we provide more details regarding the Finite Horizon Recharging Rendezvous Problem and then propose an algorithm to solve the problem in the next section.
IiB Integer Linear Program Formulation
We now formally define the setup of the problem. Since, a UAV can decide at any point along its tour to leave the path in order to rendezvous with a UGV at any point along the UGVs tour, we have a continuous constrained optimization problem. We discretize the tours of the UAVs and UGVs in order to get a combinatorial optimization problem. Given the tour
for UGV , we discretize the tour by introducing vertices every distance starting from the UGV’s current position where is the maximum distance the UGV needs to travel while a UAV completes its charging. These vertices representing possible rendezvous locations are denoted by and are shown as green discs in Figure 2. Similarly the set represents the set of locations from which UAV can leave its monitoring tour for a recharging rendezvous with a UGV. The set contains the current position of UAV and its task nodes that can be visited within next time by UAV . We do not need to further discretize UAVs tours as it can be shown (Lemma 6) that there exists an optimal solution where the UAVs will leave their tours for recharging from either their current position or a task node. In Figure 2, the vertices in are shown as red discs. In the scenario shown in Figure 2 the UAV 1 chooses to leave its task route at the node in order to rendezvous with UGV 1 at vertex . The recharging is complete when the UGV reaches node and the UAV moves to the next task node in its tour . Note that if , no other UAV can recharge on this UGV between and .We now define the problem formally on a bipartite graph , defined as follows.

UAV vertices: The vertex set consists of disjoint node sets, i.e., where . The vertex represents the scenario where the UAV chooses not to rendezvous in the current horizon.

UGV vertices: The vertex set consists of and copies of the set . The vertex for UAV represents the scenario where UAV chooses not to rendezvous for the current horizon. The copies of each vertex in are to represent up to different UAVs recharging at a time on a UGV.^{3}^{3}3We use copies of each vertex in to keep the analysis of the algorithm simple. Note that the problem can be defined without having copies for each vertex in by changing Constraint (4) in Problem 1 to .

Edges: The edge set denotes the set of all feasible recharging detour options. If the UAV , starting from a node , is able to rendezvous with the UGV at , an edge exist between the corresponding two nodes in and . The vertex is connected to for all .

Edge cost: For an edge where and , the edge cost represents the total time needed to finish the task route given that the recharging detour along edge is taken. The time needed for the recharging detour consists of three parts: the time to reach the rendezvous node, the waiting time and the recharging time at the rendezvous node, and the time to go to the next node on the tour. The edge cost along edge is zero.

Edge success probability: For an edge where and , the edge success probability is defined as the overall probability to finish the task route for the current horizon given that the recharging detour along edge is taken. It is the product of the two success probabilities: the probability of a successfully completing the recharging detour, and the probability of finishing the rest of the route after recharging. The probability along edge is the probability of reaching the end of the current horizon without recharging.
One example of the constructed bipartite graph with is shown in Figure 3.
The finite horizon recharge rendezvous problem can now be defined as the following Integer Linear Program (ILP).
Problem 1 (Riskaware Recharging Rendezvous Problem (RRRP)).
Given a bipartite graph where , with edge costs and probabilities for edge , solve:
(1)  
s.t.  (2)  
(3)  
(4)  
(5) 
The decision variable indicates whether edge is in the solution or not. The objective is to minimize the travel time overhead incurred by the recharging detours. Inequality (2) enforces that the probability that no UAV runs out of charge during the current horizon is at least . We can write this as a linear constraint since the stochastic discharging processes of the UAVs are independent. For ease of notation, we will use the following inequality instead of Constraint (2).
(6) 
where and . Constraint (3) enforces that each UAV should be recharged at most once. Constraint (4) enforces that each UGV can recharge at most UAVs at a time (by making sure that at most one UAV recharges at one of the copies of a UGV vertex). Given a solution , let represent the corresponding solution on graph . Let us define , for ease of notation.
Remark 1.
In some applications, it might be desirable to have a constraint on the total detour time while maximizing the probability of robots not running out of charge. Note that although we define the problem with sum of detour times as the objective and with a constraint on probability, the discussion and results in this paper hold with this constraint and the objective swapped as well.
IiC Hardness
We now characterize the hardness of Problem 1 in the following result.
Lemma 1.
Problem 1 is NPhard.
Proof.
We show the hardness by using a reduction from the problem EvenOddPartition which is shown to be NPcomplete in [9].
In an instance of EvenOddPartition, we are given a finite set of positive integers, and the decision problem is to find whether there exists a subset such that and for each .
Given an instance of EvenOddPartition, we construct an instance of Problem 1 as follows. Consider a bipartite graph where consists of one vertex for each and consists of one vertex each for . Connect vertex to vertex and to vertex . Let us denote these edges by and respectively. The cost and weight on edge , i.e., and are and respectively. The cost and weight on edge is and respectively. Exactly one outgoing edge from each vertex in should be selected in the solution. Finally we set the value of maximum allowed weight (equivalent to in Constraint (6)) to .
If the instance of EvenOddPartition is a Yes instance, then there exists a solution in the bipartite graph such that each vertex in has exactly one outgoing edge in the solution and the sum of weights on that solution is . Moreover, by the construction of , the sum of costs of that solution is also .
If the instance of EvenOddPartition is a No instance, then either the weight or the cost on any solution that has exactly one outgoing edge from each , is more than . Since the solutions with a weight of more than are infeasible, the minimum cost of any feasible solution is more than .
Therefore, if the optimal cost in the constructed bipartite graph is more than , then the corresponding EvenOddPartition is a No instance, and it is a Yes instance otherwise. ∎
We present a bicriteria approximation algorithm for Problem 1 in this paper. The main result regarding the algorithm is summarized below and we provide a detailed analysis in the next section.
Theorem 1.
For a given , there is a bicriteria approximation algorithm for Problem 1, i.e., we get a solution such that

, and

, .
Iii Algorithms and Analysis
Although Problem 1 can be solved using ILP solvers, as the number of variables in the problem increases, the runtime of the solver will become intractable. Since we need to solve this problem repeatedly in a receding horizon approach, an efficient solver is desirable. In this section, we present a series of algorithms to solve Problem 1, that build on top of each other. Algorithm 1
is a heuristic algorithm that uses binary search on the Lagrangian multiplier to find a feasible solution. Algorithm
2 uses Algorithm 1 to provide two solutions that are adjacent in the solution polytope, one of them being feasible. A bicriteria approximation algorithm is presented in Algorithm 3, which uses Algortithm 2 as a subroutine. The bicriteria approximation algorithm may violate one of the constraints of the problem (one of the UGVs may need to recharge up to UAVs at a time). In practice, we can use Algorithm 3 to solve the problem and if the solution violates the constraint, we can use the solution returned by Algorithm 2.We start the analysis of our algorithms by considering the following problem :
Let . Also, let denote the polytope formed by the set of constraints (3) and (4). We first show that Probelm 2 is solvable in polynomial time.
Lemma 2.
Problem 2 is solvable in time on graph where .
Proof.
We can solve Problem 2 using a minimum cost network flow problem. Given the bipartite graph , introduce nodes . Connect the vertices in to and connect all vertices to a source vertex . Connect all the vertices in to a sink vertex . The cost of all the edges is set to . The cost of all other edges is zero. The capacity of all edges is one and amount of flow is to be sent from to . This instance has edges and vertices. A solution to this minimum cost network flow problem will satisfy Constraints (3) and (4). Since the total flow and capacities are integers, there exist integer solutions satisfying Constraint (5), that can be found using network flow algorithms such as Orlin’s algorithm [24] which runs in time on a graph with vertices and edges. ∎
Since Problem 2 is solvable in polynomial time, the following observation enables us to use binary search in Algorithm 1 to find two solutions and such that .^{4}^{4}4Note that if no solution exists such that , then solving Problem 2 with solves Problem 1.
Observation 1.
Let and be the optimal solutions to Problem 2 with Lagrangian multipliers and , where . Then , and .
Although the run time of Algorithm 1 depends on the stopping threshold value , an optimal Lagrangian multiplier and two solutions and to Problem 2, with and satisfying and can be found in polynomial time [3], [26, Theorem 24.3]. We can use the procedure from [3] and [26, Theorem 24.3] or Algorithm 1 in Line 2 of Algorithm 2 to find , and . We use Algorithm 1 in experiments to find and as it is simple to implement and works well in practice as seen in Section IV. Since where is the optimal solution to Problem 1 with cost opt, we have,
(8) 
Definition 1.
Given the bipartite graph where , consider the bipartite graph with the vertices for each robot merged together, i.e., where , and ’ has an edge between and if has an edge between some and . Then with abuse of notation, we define a subgraph of as a connected component if the edges of form a connected path or cycle in .
Now we show a property of adjacent extreme points in the solution polytope similar to a property of graph matchings shown in [2].
Lemma 3.
Two vertices and of are adjacent if and only if the symmetric difference of their corresponding solutions and contain exactly one connected component.
Proof.
The symmetric difference of and is a union of connected components (paths and cycles) because each edge in the symmetric difference can have a degree at most two. Suppose that contains exactly one connected component. Then we can construct an objective function (edge costs) such that and are the only two optimal extreme point solutions as follows. Let the number of edges in from and be and respectively. Assign a cost of one to all edges in , a cost of to all edges in , a cost of zero to all edges in , and a negative cost to all other edges in the graph. Then and are the only two extreme point solutions in for this cost function.
Now suppose that the symmetric difference has at least two connected components. Let be one of those connected components and let and . and are also solutions in and
So if there is any objective function for which and are optimal solutions, then the mid point also has the same cost, and hence or also has the same cost. Therefore, and are not adjacent extreme points in . ∎
A procedure to find two adjacent extreme point solutions in the solution polytope for a maximum weight matching problem is given in [3]. We use a similar procedure for our problem to find two adjacent solutions in in Algorithm 2. We have the following result regarding Algorithm 2.
Lemma 4.
Algorithm 2 returns and two solutions , , such that they correspond to adjacent extreme points in and

,

,

.

Proof.
If consists of more than one connected component, we can pick one of those connected components, say , and let . If , replace by . Otherwise replace by . We can repeat this step until consists of only pone connected component. Note that at each step, increases by at least one, so this procedure stops in at most steps. Moreover, always remains feasible, and by the optimality of and with respect to , and do not change during this process. The last inequality follows from the definition of and (8). ∎
The following result shows that given , and from Algorithm 2, we can get a solution that may violate at most one constraint and the cost of is within of the optimal, where is the largest edge cost in . Lines 9 to 23 of Algorithm 3 find such a solution.
Lemma 5.
Let be the optimal solution to Problem 1 with cost opt. There is a polynomial time algorithm that finds a scheduling (corresponding to a solution ) such that

, and

one of the UGVs may have to recharge at most UAVs at a time.
Proof.
From Lemma 4 and Lemma 3 we have two solutions and such that , and contains exactly one connected component . Consider the sequence
where if and if . Since , . Moreover there exists an edge , such that for any cyclic subsequence , [3, Lemma 3]. Therefore,
(9) 
Consider the smallest such subsequence such that . Assume that contains at least two edges, otherwise and we are done. Note that removing one edge from the end of will mean that is the longest sequence such that . Since the connected component is alternating, remove at least one and at most two edges from the end of the sequence to get such that the last edge in belongs to . Also note that by construction, the first edge in also belongs to . Therefore the first and last edges from will appear in and may result in more than one edge connected to some set or some . If one of the robots has more than one connected edges in (because the first and last edges in are from ), we remove one of those edges. Note that can have one more edge connected to a UGV location as compared to , i.e., . Note that this can happen at one of the two ends of as it is alternating. So one of the UGVs may have to recharge at most UAVs at a time. Also note that . Now we lower bound the value of . Since we remove at most one edge from to get ,
where the second inequality is due to the fact that we remove at most two edges from to get .
Now,
∎
Corollary 1.
Now we prove the main result regarding Algorithm 3.
Proof of Theorem 1.
Given , guess edges with highest cost value of in the optimal solution . Let these edges be . Remove and all the edges with costs higher than the lowest cost in from the graph. Also for where , remove the vertices from and from . Also decrease by . Then if the optimal solution to the resulting instance is , is the optimal solution to the original problem. The maximum cost of an edge in the resulting instance will be at most . Then by Lemma 5, we get a solution for the new instance such that
Since one of the UGVs may have to recharge at most UAVs at a time, and because , we get a bicriteria approximation. The algorithm requires guesses for . ∎
Iv Experimental Results
In this section, we first present a qualitative example of the persistent monitoring mission. Next, we study how system parameters (various risk tolerances) influence the recharging behaviors between the UAVs and the UGVs and the task performances of the UAVs. Then, we compare the performance of our scheduling strategy with a baseline (greedy strategy) using the mean time before the first failure and the travel distance overhead as metrics. Moreover, we empirically evaluate the performance of the proposed heuristic algorithm. All experiments in Section IVB are conducted using Python 3.8 on a PC with the i98950HK processor. The baseline solver is Gurobi 9.5.0.
Iva Experimental Setup
We consider a team consisting of two UAVs and two UGVs. The task routes and used in the problem can be either generated jointly by some task planners similar to those in [20, 31] or can be generated separately by different task planners. In our case study, the task of two UGVs is to persistently monitor the road nodes. The setup here is similar to our previous work [27] on Intelligence, Surveillance, and Reconnaissance (ISR) where the focus is on improving highlevel solutions.
The UAV and UGV move at 9.8 m/s and 4.5 m/s respectively based on the field test data [27] in our ongoing project. The recharging process (swapping battery) takes 100 s. The UAV and UGV need to persistently monitor the task nodes on the route. We apply our recharging strategy in a receding horizon fashion: every two minutes, the UAVsUGVs team solves the RRRP problem to decide the UAVs’ recharging schedule for the next seconds. For each UAV, the current position will be the first node when we construct the bipartite graph. If some UAV is on a detour, we do not replan until the UAV has finished its detour.
We consider two sources of stochasticity in the energy consumption model of UAVs: weight and wind velocity contribution to longitudinal steady airspeed. The deterministic energy consumption model of the UAV is a polynomial fit constructed from analytical aircraft modeling data, given as,
(10) 
where to are coefficients, and their experimental values are listed in Table I.
Value  88.77  3.53  0.42  0.043  107.5  2.74 
Weight is randomly selected following a normal distribution with a mean of 2.3 kg and a standard deviation of 0.05 kg,
. Vehicle airspeed, , is the sum of the vehicle ground speed, , and the component of the wind velocity that is parallel to the vehicle ground speed, ignoring sideslip angle and lateral wind components.(11) 
The longitudinal wind speed contribution is derived from two random parameters; wind speed, and wind direction. Wind speed is modeled using the Weibull probability distribution model of wind speed distribution,
, with a characteristic velocity m/s and a shape parameter . This is representative of a fairly mild steady wind near ground level. Wind direction is the heading direction of the wind, and is uniformly randomly selected on a range of degrees.IvB Simulation Results
UAV data  
Avg. travel time before out of charge (s)  39660  27600  24360 
Avg. travel time overhead  19.7 %  18.5 %  17.8 % 
Avg. # of task nodes visited  158  110  105 
Avg. # of rendezvous per s  1.4  1.3  1.3 
Qualitative Example
An illustrative example of the input and the output of the problem considered is shown in Figure 7. The input of the problem is shown in Figure (a)a, which consists of UAV task nodes (red dots), and nodes of the road network (blue nodes). Figure (b)b shows one tour route of one UAV when the system executes the proposed strategy in a receding horizon fashion. The UAV monitors the task route persistently. When the UAV reaches node , it doesn’t move forward to its next task node (connected through a dashed red line). Instead, the new schedule is to rendezvous with UGV at and takes off from the UGV at , and then go to its next task node. Similarly, the UAV will rendezvous with the UGV when it is close to nodes and . Subscriptions and denote the start and the terminal of the recharging process. A sample of the history of the state of charge (SOC) is shown in Figure (c)c. We can observe in Figure (c)c that the UAV’s recharging strategy is more than a simple rule (for example get recharged when the SOC is below 50 ) and may get recharged at various values of SOC.
Effect of Risk Tolerance
Next, we show how different risk tolerances influence the recharging behaviors under our RRRP formulation. In these experiments, we set the risk tolerance to be 0.01, 0.1, and 0.3. Some statistical data are summarized in table II. We use four metrics to quantify the performance of the strategy:

Mean time before the first failure here failure refers to the case when one UAV in the team is out of charge and needs human intervention. This quantity reflects how frequent the system needs human involved and we expect this quantity to be large enough.

Travel time overhead this quantity is computed as
where task time is the time of the route obtained when we project the actual flight route into the planned route. This quantity reflects how well the UAV is performing its task and we expect this quantity to reasonably large.

Avg. # of task nodes visited the average number of task nodes visited by the UAV before its first failure.

Average number of rendezvous per planning horizon if this number is too large, it suggests that the UAV is scheduled too many recharging detours, which should be avoided.
In general, we observe that when the risk tolerance is set to be smaller, the mean time before the first failure will be longer. Similarly, the travel time overhead and the average number of rendezvous per planning horizon will be greater, which implies the UAV spends more portion of flight time in the recharging detours.
Comparison of Algorithm 1 with Baseline
To validate that the scheduling strategy constructed from RRRP, we compare our strategy with a greedy baseline. The greedy policy is set as: choose to rendezvous when stateofcharge drops below a set value. We consider three set values , , and , and the corresponding strategies are denoted as Greedy30, Greedy40, and Greedy50. In this experiment, we set . The first observation is that Algorithm 1 achieves close performances in both metrics compared to that of ILP. Second, as shown in Figure 8, our strategy (obtained by both ILP and Algorithm 1) can achieve longer travel time before first failure on average (left group) and a relatively lower travel distance overhead (right group), which implies that our strategy will not lead too many unnecessary rendezvous. Moreover Algorithm 1 has a better travel time overhead than that of ILP although ILP solves RRRP optimally for a given horizon. This is likely due to solving RRRP repeatedly in a receding horizon manner.
Scalability of Algorithm 2
We also compare the performance of Algorithm 2 with an ILP solver empirically. We used Algorithm 2 for comparison instead of Algorithm 3 as Algorithm 2 and the ILP always return a feasible solution, making the objective value comparison fair. The ILP solver used for this set of experiments is intlinprog function from MATLAB, and Algorithm 2 was also implemented in MATLAB for fair comparison. Since ILP solves Problem 1 optimally, the cost returned by Algorithm 2 is at least that of ILP solver. The percentage difference in the objective function values for different problem sizes is shown in Figure 9. For each problem size, represented by the number of edges or variables, twenty random problem instances were created and the boxplot of resulting objective value difference is shown in the plot. On average, among all the instances, Algorithm 2 was within of the optimal solution. Note that the performance of Algorithm 2 improves as the number of variables increases.
Figure 10 shows the average runtime comparison between Algorithm 2 and the ILP solver. Note that the yscale is logarithmic. For smaller problem instances, both the algorithms solved the problem within a second, with ILP being faster, however, as the number of variables increases, ILP becomes much slower, with the runtime for ILP being up to seven times more than that of Algorithm 2 for variables. Note that there may be other solvers for ILP that have better run time, but since ILP is NPcomplete, the exponential gap between run times is likely to continue as the number of variables increases.
V Conclusion
In this paper, we study a resource allocation problem with matching and knapsack constraints for the UAVsUGVs recharging rendezvous problem. We formulate this problem (Riskaware Recharging Rendezvous Problem (RRRP)) as one Integer Linear Program (ILP) and propose one bicriteria approximation algorithm to solve RRRP. We validate our formulation and the proposed algorithm in one persistent monitoring application. In our current formulation, we assume that the task routes of vehicles are given. In future work, one direction that we will explore is to propose efficient and scalable routing algorithms to generate task routes for vehicles.
References
 [1] (2019) Multirobot routing for persistent monitoring with latency constraints. In 2019 American Control Conference (ACC), pp. 2620–2625. Cited by: §IIA.
 [2] (1974) On the assignment polytope. Siam Review 16 (4), pp. 516–525. Cited by: §III.
 [3] (2011) Budgeted matching and budgeted matroid intersection via the gasoline puzzle. Mathematical Programming 128 (1), pp. 355–372. Cited by: §I, §III, §III, §III, 2.
 [4] (2020) Crop height and plot estimation for phenotyping from unmanned aerial vehicles using 3d lidar. In 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 2643–2649. Cited by: §I.
 [5] (2012) Volcano monitoring with small unmanned aerial systems. In Infotech@ Aerospace 2012, pp. 2522. Cited by: §I.
 [6] (2006) Cooperative air and ground surveillance. IEEE Robotics & Automation Magazine 13 (3), pp. 16–25. Cited by: §I.
 [7] (2019) Efficient computation of optimal uav routes for persistent monitoring of targets. In 2019 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 605–614. Cited by: §IIA.
 [8] (2019) Truckdrone hybrid delivery routing: payloadenergy dependency and nofly zones. International Journal of Production Economics 214, pp. 220–233. Cited by: §I.
 [9] (1979) Computers and intractability: a guide to the theory of npcompleteness. WH Freeman. Cited by: §IIC.
 [10] (2019) The hybrid vehicledrone routing problem for pickup and delivery services. Transportation Research Part C: Emerging Technologies 102, pp. 427–449. Cited by: §I.
 [11] (2008) Decentralized perimeter surveillance using a team of uavs. IEEE Transactions on Robotics 24 (6), pp. 1394–1404. Cited by: §I.
 [12] (2021) Groundvehicle and unmannedaerialvehicle routing problems from twoechelon scheme perspective: a review. European Journal of Operational Research 294 (3), pp. 1078–1095. Cited by: §I.
 [13] (2011) A vehicle routing problem with pickup and delivery time windows, and coordination of transportable resources. Computers & Operations Research 38 (11), pp. 1596–1609. Cited by: §I, §I.
 [14] (2014) Energyaware aerial vehicle deployment via bipartite graph matching. In 2014 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 189–194. Cited by: §I.
 [15] (2019) Cooperative routing problem for ground vehicle and unmanned aerial vehicle: the application on intelligence, surveillance, and reconnaissance missions. IEEE Access 7, pp. 63504–63518. Cited by: §I.
 [16] (2015) On cooperation between a fuel constrained uav and a refueling ugv for large scale mapping applications. In 2015 International Conference on Unmanned Aircraft Systems (ICUAS), pp. 1370–1377. Cited by: §I.
 [17] (2019) Cooperative aerial–ground vehicle route planning with fuel constraints for coverage applications. IEEE Transactions on Aerospace and Electronic Systems 55 (6), pp. 3016–3028. Cited by: §I, §I, §IIA, §IIA, footnote 2.
 [18] (2016) Path planning for cooperative routing of airground vehicles. In 2016 American Control Conference (ACC), pp. 4630–4635. Cited by: §I, footnote 2.
 [19] (2017) Cooperative surveillance in the presence of time sensitive data. In 2017 IEEE Conference on Control Technology and Applications (CCTA), pp. 343–348. Cited by: §I.
 [20] (2019) Cooperative routing for an air–ground vehicle team—exact algorithm, transformation method, and heuristics. IEEE Transactions on Automation Science and Engineering 17 (1), pp. 537–547. Cited by: §IVA, footnote 2.
 [21] (2015) Multirobot rendezvous planning for recharging in persistent tasks. IEEE Transactions on Robotics 31 (1), pp. 128–142. Cited by: §I, §IIA, footnote 2.
 [22] (2015) The flying sidekick traveling salesman problem: optimization of droneassisted parcel delivery. Transportation Research Part C: Emerging Technologies 54, pp. 86–109. Cited by: §I, §I.
 [23] (2008) Persistent surveillance using multiple unmanned air vehicles. In 2008 IEEE Aerospace Conference, pp. 1–14. Cited by: §IIA.

[24]
(1988)
A faster strongly polynomial minimum cost flow algorithm.
In
Proceedings of the Twentieth annual ACM symposium on Theory of Computing
, pp. 377–387. Cited by: §III.  [25] (1996) The constrained minimum spanning tree problem. In Scandinavian Workshop on Algorithm Theory, pp. 66–75. Cited by: §I.
 [26] (1998) Theory of linear and integer programming. John Wiley & Sons. Cited by: §III.

[27]
(2022)
Riskaware uavugv rendezvous with chanceconstrained markov decision process
. 2022 IEEE 61th Annual Conference on Decision and Control (CDC). Cited by: §I, §IIA, §IVA, §IVA.  [28] (2013) Algorithms for routing an unmanned aerial vehicle in the presence of refueling depots. IEEE Transactions on Automation Science and Engineering 11 (1), pp. 287–294. Cited by: §I, §IIA.
 [29] (2021) Environmental hotspot identification in limited time with a uav equipped with a downwardfacing camera. In 2021 IEEE International Conference on Robotics and Automation (ICRA), pp. 13264–13270. Cited by: §I.
 [30] (2016) Sensor planning for a symbiotic uav and ugv system for precision agriculture. IEEE Transactions on Robotics 32 (6), pp. 1498–1511. Cited by: §I, §I, footnote 2.
 [31] (2018) Algorithms for routing of unmanned aerial vehicles with mobile recharging stations. In 2018 IEEE international conference on robotics and automation (ICRA), pp. 5720–5725. Cited by: §I, §I, §IIA, §IVA, footnote 2.