Distributed Coverage Hole Prevention for
Visual Environmental Monitoring with Quadcopters
via Nonsmooth Control Barrier Functions

Cooperative environmental monitoring has been studied extensively in the context of coverage control, which yields an optimal deployment of mobile sensors with respect to a spatial field in a distributed fashion [12, 13, 14]. The coverage control problem has been considered in a variety of scenarios, including networks of Pan-Tilt-Zoom (PTZ) cameras [15, 16, 17, 18] as well as teams of ground [19, 20, 21, 22] and aerial robots [23, 24, 25, 26].

In the context of aerial robots, the work in [23] presents a coverage strategy for a heterogeneous robot team where the cameras present various degrees of freedom depending on the robot type, e.g., PTZ cameras, quadcopters, etc. The proposed control law captures a 2D environment with a maximal resolution, where a better coverage quality is allocated to an area monitored by multiple cameras as opposed to the same area covered by a single camera. The work [24] proposes a solution that reduces FOV overlaps while guaranteeing the altitude of quadcopters within the predefined range. A more challenging scenario is considered in [25], where each quadcopter is mounted with a PTZ camera, making their FOVs become ellipses on the 2D environment. The work [26] presents the unified framework for both exploration and coverage problems in an unknown 3D environment with quadcopters. Both problems are formulated to maximize the portion of observed environments while minimizing the overlap of FOVs. Despite the variety of advances presented in these works, the proposed controllers do not address the appearance of unmonitored areas in between FOVs, which may result in overlooking critical events occurring in the environment.

The work in [27, 28] reduces the unmonitored area in the whole environment, rather than eliminating unobserved regions in-between FOVs, by maximizing the area of the covered region. Both works consider the agents with heterogeneous and fixed sensing regions. The uncovered areas are detected by utilizing a multiplicatively weighted Voronoi diagram having curved boundaries, where the weight of each sensor is equal to its sensing radius. The algorithms guarantee increased area coverage, but the elimination of unmonitored areas is not assured. The rejection of unmonitored regions is also considered in [29], where the total moving distance of mobile sensors is minimized. Still, this work assumes the sensing radius of all agents is fixed and identical. These methods, which assume a fixed or same measurement range, might not be readily extendable to cooperative monitoring with a team of quadcopters, where the size of FOVs dynamically evolves according to the changes of the zoom level or altitude. The work [30] proposes a solution based on a nonsmooth gradient flow for a disk-covering problem, where each agent can control its sensing radius. Because a disk-covering problem corresponds to completely covering the entire environment with disks of minimum radius, the proposed control law eliminates unobserved areas. The resulting final deployment renders the radius of sensing fields of all agents identical. However, it is difficult to cover the entire environment with high-resolution images in case the whole mission space is large compared with FOVs of quadcopters, or the number of quadcopters is limited. For instance, the monitoring area of [31] reaches $2\,\textrm{km}^{2}$. In such cases, rather than overextending FOVs for covering a whole area, preventing unmonitored areas around targeted regions under a team’s surveillance could yield more detailed information for contributing to the monitoring purposes. Furthermore, if the environment has non-uniform importance, allowing each quadcopter to decide on its sensing radius could lead to better monitoring performance because a team can adjust the resolution of the image according to the importance of the area. To achieve such a visual monitoring strategy, the team should maximize the objective function for visual monitoring quality, as in [23, 24, 25, 26]. However, in case the reduction or elimination of unmonitored areas is already encoded as the objective of the team as in [27, 28, 29, 30], the problem becomes a multi-objective optimization, with the associated challenges of balancing two different requirements.

In order to address the aforementioned issue, this paper prevents the appearance of a hole by means of a constraint-based approach, where the visual monitoring quality is maximized as long as unmonitored areas in-between a team are prevented. More specifically, the behavior of the quadcopters is constrained to prevent unsurveilled areas in-between quadcopters by CBFs. CBFs provide the forward invariance property to the admissible set of the robot state space [32, 33, 34]. CBFs have been employed in various applications ranging from bipedal robots [35, 36] to swarm robots [37, 38, 39, 40] and seeing success also in the context of environmental monitoring [41, 42, 43]. The work [41] proposes a control framework that guarantees the robot’s battery is never depleted during a monitoring task. Several charging models, such as solar lights and charging stations, are interpreted as a time-varying function defined in the mission space. Then, a time-varying CBF is developed to ensure energy constraints of the robots. The work [42] also utilizes the time-varying CBF to consider time-varying coverage control, where a density function specifying the importance of field changes dynamically. The proposed methods do not impose any requirements on the rate of change of the density functions and can be executed in a distributed fashion. In [43], a notion of information reliability is encoded through a time-varying density function as in [44], which decays if a region is not surveilled for a long period. Then, a patrolling motion is achieved by a time-varying CBF together with other safety constraints. In [45, 46], an extended version of CBFs are utilized to a quadcopter having nonlinear dynamics.

In this paper, we propose a control method that prevents unmonitored areas to arise in-between FOVs of quadcopters equipped with downward facing cameras while monitoring a mission area. First, CBFs preventing the appearance of unmonitored areas in between a team is designed by utilizing a power diagram. The power diagram [47] is a weighted Voronoi diagram defined by the different radii of FOVs, where the boundaries of the diagram are composed of lines known as the radical axis. We leverage this favorable property of the power diagram, compared with the multiplicatively weighted Voronoi partitions in [27, 28] that generate more complicated Voronoi cells composed of Appollonian circles, to analyze its feasibility and develop distributed algorithm. Furthermore, we provide the coverage control law that intends to monitor important regions while preventing too much overlap between the fields of view of quadcopters. The presented coverage controller is synthesized with the aforementioned CBFs to achieve the maximum visual monitoring quality while preventing the unmonitored areas in-between a team.

The contributions of this paper are as follows.

• (a)

  We develop a distributed algorithm preventing an unmonitored area in between a team of quadcopters based on CBFs. For this goal, the appearance of unsurveilled areas is encoded by the vertices of a power diagram. Because the derived constraints contain a Boolean logic, we leverage a nonsmooth control barrier function (NCBF) [48] to synthesize the controller that prevents an unmonitored area. We then formulate an algorithm that allows the jumps in the value of NCBF caused by the changes in the Delaunay graph induced by the power diagram.

• (b)

  We verify the feasibility of the proposed CBF-based methods. The existence of a control input that satisfies the constraints is closely related to how the CBF evolves together with the system dynamics. For example, in [49, 40], feasibility was established by showing that the gradient of the CBF does not vanish. In our case, this computation is not feasible and instead, we rely on the special structure of the power diagram to derive a simplified form of the designed CBFs’ derivatives. Then, we show that the set yielding the derivatives of CBF zero vectors can be neglected in practice. This result guarantees that the proposed CBF-based controller can generically generate the control input to prevent an unmonitored area in between a team during monitoring mission.

• (c)

  The coverage control law in our conference paper [50] is combined with the above CBF-based methods to showcase its utility. This visual coverage control serves two purposes: (i) improving the performance of the monitoring, considering the spatial sensing quality of cameras as in [17], and (ii) reducing the overlap between FOVs of the quadcopters to cover a large-scale area by a team.

• (d)

  The practical usefulness of the proposed monitoring strategy is demonstrated by simulations and experiments.

In our previous work [51], we proposed the preliminary results of (a), which does not allow a distributed computation without a strong assumption. More specifically, when the algorithm calculates the derivatives of CBFs for Quadcopter $i$, it presumes the other quadcopters are fixed in the environment. In this paper, the algorithm is improved to achieve a distributed computation without the assumption by utilizing the symmetric properties of the proposed CBFs. We newly analyze the validity of the proposed NCBFs in (b), which was not discussed in our previous conference papers [51, 50]. The coverage control law discussed in (c) is proposed in our previous work [50], but we only show the utility of the control law with the preliminary algorithm that overly restricts a teams’s movement for preventing the appearance of unmonitored area in between quadcopters. Finally, this paper provides comprehensive simulation and experimental results demonstrating the effectiveness of the proposed methods.

In this paper, we consider the scenario illustrated in Fig. 1, where $n$ quadcopters $\mathcal{N}=\{1,\cdots n\}$ equipped with a downward facing camera are distributed in the 3-D space to monitor a 2-D region $\mathcal{Q}$. We denote the region $\mathcal{Q}$ to be surveilled as the mission space hereafter, and assumed to be a closed and bounded convex subspace of the environment $\mathcal{E}\subset{\mathbb{R}}^{3}$. The relative importance of points on $\mathcal{E}$ is specified by a density function $\phi:\mathcal{E}\to{\mathbb{R}}_{\geq 0}$, where the higher the importance of a point $q\in\mathcal{E}$ is, the higher the value of $\phi(q)$, with $\phi(q)=0,~{}q\in\mathcal{E}\backslash\mathcal{Q}$. The goal of this paper is to monitor the important region while preventing the appearance of unmonitored areas in-between a team of quadcopters, shown as hatched areas in Fig. 1.

The world coordinate frame $\Sigma_{w}$ is placed so that its $x_{w}y_{w}$-plane is coplanar with the environment $\mathcal{E}$, where $\{{\bf e}_{x},{\bf e}_{y},{\bf e}_{z}\}$ describes the standard basis of $\Sigma_{w}$. Then, the set of coordinates of all points in the environment is defined as $\mathcal{E}:=\{q\in{\mathbb{R}}^{3}\mid{\bf e}_{z}^{\top}q=0\}$. We suppose that all quadcopters stay in the common half space $\{q\in{\mathbb{R}}^{3}\mid{\bf e}_{z}^{\top}q>0\}$ during the operation.

Let $\Sigma_{i}$ be the coordinate frame of Quadcopter $i\in\mathcal{N}$, where its position vector with respect to $\Sigma_{w}$ is $p_{i}=[x_{i}~{}y_{i}~{}z_{i}]^{\top}$ with $z_{i}\in{\mathbb{R}}_{>0}$, and its attitude is fixed to be consistent with that of $\Sigma_{w}$ regardless of the attitude of Quadcopter $i$. We assume each quadcopter equips with a gimbal that can keep an optical axis of the camera corresponds with the axis of the $z_{i}$-axis of $\Sigma_{i}$. Namely, Quadcopter $i$ points its camera to the downward direction. In addition, each quadcopter is assumed to be equipped with localization devices, such as the GNSS, gyro sensors, and IMU, to estimate its own pose accurately, as in [54]. Let us express the image projection of the camera with a perspective projection model [55]. Then, the coordinate frame $\Sigma_{i}$ is located above the image plane with a distance the same as the focal length $\lambda_{i}\in{\mathbb{R}}_{>0}$. We assume that all quadcopters have a circular image plane with radius $r\in{\mathbb{R}}_{>0}$. Then, Quadcopter $i$ can monitor all points inside of its field of view (FOV) $\mathcal{F}_{i}$ described as

where its side view is illustrated in Fig. 2(a) with $I=[x_{i}~{}y_{i}~{}0]^{\top}$ denoting Quadcopter $i$’s horizontal position projected on the environment $\mathcal{E}$.

The state of Quadcopter $i$ is denoted as ${\bf p}_{i}=[p_{i}^{\top}~{}\lambda_{i}]^{\top}$, where the focal length $\lambda_{i}$ specifies the zoom level of the camera. We assume that the dynamics of each quadcopter can be modeled as

which can be converted to reflect the dynamics of a quadcopter [56]. Note that higher-order dynamics could represent more aggressive motions, as in [45]. However, good quality images are typically paramount in environmental monitoring, which means quadcopters rarely fly with large accelerations, e.g., to prevent image blurs. Hence, we opt for first-order dynamics. Furthermore, this paper does not consider the disturbance, such as wind, assuming a controller implemented in each quadcopter can successfully cancel the effect as in [57].

Every quadcopter is capable of sending its state to the neighboring quadcopters, where the inter-robot communication graph $\mathcal{G}(t)$ is modeled by the two undirected graphs $\mathcal{G}_{\mathcal{F}}(t)$ and $\mathcal{G}_{\mathcal{P}}(t)$. First, $\mathcal{G}_{\mathcal{F}}(t)$ is described as the graph which has an edge between the agents $i$ and $j$ if their FOVs are overlapped, namely $\mathcal{F}_{i}\cap\mathcal{F}_{j}\neq\emptyset$. Second, $\mathcal{G}_{\mathcal{P}}(t)$ is the graph generated by the power diagram, which is a form of the weighted Voronoi diagram, specified by the weighted distance

where $rz_{i}/\lambda_{i}$ in the second term signifies the radius of $\mathcal{F}_{i}$ as shown in Fig. 2LABEL:sub@fig:test1. Hereafter, we call the distance (14) the power distance following [47]. Then, the communication graph is expressed as $\mathcal{G}(t)=\mathcal{G}_{\mathcal{F}}(t)\cap\mathcal{G}_{\mathcal{P}}(t)$, which preserves the edges of power diagram only between those neighbors whose FOVs overlap, depicted by blue lines in Fig. 2LABEL:sub@fig:test2. The neighbors of Quadcopter $i$ in graph $\mathcal{G}(t)$ is denoted as $\mathcal{N}_{i}(t)$.

An example of a power diagram is shown in Fig. 3 with the graph $\mathcal{G}(t)$. The bisector of two agents $i$ and $j$ is composed of a straight line known as the radical axis, along which both circles $\mathrm{bd}(\mathcal{F}_{i})$ and $\mathrm{bd}(\mathcal{F}_{j})$, with the symbol $\mathrm{bd}(\cdot)$ to mean boundary of a set, have the same power distance. Namely, if we denote the radical axis between agent $i$ and $j$ as $L_{ij}$, $d_{\mathcal{P}}(\mathbf{p}_{i},q)=d_{\mathcal{P}}(\mathbf{p}_{j},q),~{}q\in L_{ij}$ holds. In addition, if $\mathrm{bd}(\mathcal{F}_{i})\cap\mathrm{bd}(\mathcal{F}_{j})\neq\emptyset$ with $\mathbf{p}_{i}\neq\mathbf{p}_{j}$, then $\mathrm{bd}(\mathcal{F}_{i})\cap\mathrm{bd}(\mathcal{F}_{j})\in L_{ij}$ is satisfied. It is known that the radical axes defined by three circles whose centers are not collinear intersect in a common point termed the radical center, depicted with red dots in Fig. 3. The radical center is generated by three agents and forms a vertex of the weighted Voronoi cell with the weighted distance (14). Let us denote the radical center generated by $\mathrm{bd}(\mathcal{F}_{i})$, $\mathrm{bd}(\mathcal{F}_{j})$, and $\mathrm{bd}(\mathcal{F}_{k})$ as $v_{ijk}=[v_{ijk}^{x}~{}v_{ijk}^{y}~{}0]^{\top}\in{\mathbb{R}}^{3}$. Then, from the property of the radical axis, the following condition is satisfied.

Since the graph $\mathcal{G}_{\mathcal{P}}(t)$ is generated in a similar fashion to how the ordinary Voronoi diagram yields the Delaunay triangulation, the graph $\mathcal{G}(t)$ considered throughout this paper also provides triangular subgraphs as observed in Fig. 3. Let us denote the set of triangular subgraphs containing Quadcopter $i$ as its vertex as $\mathcal{T}_{i}(t)=\{\Delta_{1},\cdots,\Delta_{m_{i}}\}$, where $\Delta_{\iota},\iota\in\mathcal{I}_{i}(t)=\{1,\cdots,m_{i}(t)\}$ stands for $m_{i}(t)$ triangular subgraphs¹¹1Although we should define a triangular subgraph of Quadcopter $i$ with $\Delta_{i,\iota}$, we use $\Delta_{\iota}$ for notational simplicity.. Note that $\mathcal{T}_{i}(t)$ can be regarded as a subset of the Delaunay triangulation generated by the power diagram. Without loss of generality, we introduce the following assumption for a trio constituting a triangular subgraph $\Delta_{\iota}\in\mathcal{T}_{i}(t)$.

Under Assumption 1, the existence of the radical axes $\{L_{ij},L_{jk},L_{ki}\}$ is assured, and all three axes intersect at a single point. Moreover, $\triangle{IJK}$ does not become a degenerate triangle, where $\triangle{IJK}$ denotes the triangle with vertices $I$, $J$, and $K$. Note that $\triangle{IJK}$ becomes a degenerate triangle only when the pathological situation, and even if it happens, the proposed method can easily manage such an ill condition with a heuristic method, as shown in the simulation studies in Section VI-A.

Notice that the graph $\mathcal{G}(t)$ dynamically changes according to the quadcopters’ motion. We suppose that the set of triangular subgraphs $\mathcal{T}_{i}(t)$ does not change over intervals specified by a switching sequence $\{\tau_{i}^{\mathrm{k}}\}_{\mathrm{k}=1}^{\infty}$ as introduced in Definition 1.

The proposed scenario requires each quadcopter to exchange its state with the neighboring quadcopters. While one could consider employing a state estimation technique to obtain the neighbors’ state [58], the zoom level of the mounted camera, included in the state vector, is deemed challenging to estimate externally. Therefore, this paper presumes that each quadcopter is equipped with a communication device to send and receive its state. The communication devices and the challenges involved with outdoor communication, such as guaranteeing reliability by dedicating redundant networks and fail-safety, are detailed in [59].

Noticing that the power diagram generates convex cells, this paper prevents the formation of unmonitored areas in-between FOVs by including the Voronoi vertices $v_{ijk}$, which exist in-between FOVs of teams, in each FOV, as shown in Fig. 3. Because each Voronoi vertex is generated by three agents except for degenerate Voronoi diagrams [60], we can separate the problem into triangular subgraphs $\Delta_{\iota}\in\mathcal{T}_{i}(t)$. Note that this degeneracy does not cause serious problems because degenerate Voronoi diagrams can be divided into triangular subgraphs.

For the above reasons, we restrict our attention to the unmonitored regions that may appear among a trio determined by triangular subgraphs $\mathcal{T}_{i}(t),\forall i\in\mathcal{N}$, illustrated in Fig. 4(a). This unsurveilled area is denoted as a hole hereafter and formally defined as follows.

Since a hole in Definition 6 depends on a triangular subgraph $\Delta_{\iota}\in\mathcal{T}_{i}(t)$ that dynamically changes according to the switching sequence $\{\tau_{i}^{\mathrm{k}}\}_{\mathrm{k}=1}^{\infty}$, the condition for preventing a hole also switches. Let us suppose that the zero superlevel set of a function $h_{i,\Delta_{\iota}}^{\mathrm{k}}(\mathbf{p}_{l\in\Delta_{\iota}})$ with $\mathbf{p}_{l\in\Delta_{\iota}}:=[\mathbf{p}_{i}^{\top}\mathbf{p}_{j}^{\top}\mathbf{p}_{k}^{\top}]^{\top}$, $\Delta_{\iota}=\{i,j,k\}$, which we will derive in Section IV, describes the condition for Quadcopter $i$ to eliminate a hole in $\Delta_{\iota}\in\mathcal{T}_{i}(\tau_{i}^{\mathrm{k}})$. By incorporating $h_{i,\Delta_{\iota}}^{\mathrm{k}}(\mathbf{p}_{l\in\Delta_{\iota}}),\forall\Delta_{\iota}\in\mathcal{T}_{i}(\tau_{i}^{\mathrm{k}})$ together, we obtain

the zero-superlevel set of which signifies the constraint Quadcopter $i$ must satisfy to prevent the formation in-between all triangular subgraphs $\mathcal{T}_{i}(\tau_{i}^{\mathrm{k}})$. Then, the set

with HNCBF

signifies the prevention of a hole is satisfied for Quadcopter $i$ during the entire mission.

The goal of this paper is to present a visual coverage framework that prevents the appearance of a hole by guaranteeing the hybrid forward invariance of the set $\mathcal{C}_{i},\forall i\in\mathcal{N}$. More specifically, during a monitoring task, we ensure no hole appears among FOVs of quadcopters if the changes of the graph do not make holes instantaneously in-between $\mathcal{T}_{i}(t),\forall i\in\mathcal{N}$. Note that, in the proposed scenario, there is a possibility that changes of a graph $\mathcal{G}(t)$ yield a sudden appearance of a hole. For instance, as illustrated in the right part of Fig. 3, $\mathrm{bd}(\bigcup_{i=1}^{n}\mathcal{F}_{i})$ can have a dent. If the right-upper and right-lower quadcopters’ FOVs make overlap, a new subtrianglar graph is created. Although this graph change could alter the dent into a hole, the robustness of the forward invariance of the set $\mathcal{C}_{i}$ can eliminate this suddenly appeared hole in most cases as will be shown in the simulation.

In this subsection, we formalize the condition to be satisfied for preventing a hole during a time interval $[\tau_{i}^{\mathrm{k}},\tau_{i}^{\mathrm{k}+1})$. As discussed in Section III-C, an appearance of a hole in-between $\Delta_{\iota}=\{i,j,k\}\in\mathcal{T}_{i}(\tau_{i}^{\mathrm{k}})$ can be prevented by satisfying $v_{ijk}\in\mathcal{F}_{i}$. However, as shown in Fig. 4(c), this condition is too conservative because certain movements of quadcopters are restricted, even if such movements do not form a hole. To alleviate this condition, let us consider the differences between Figs. 4(a) and (c), where a hole only appears in Fig. 4(a) although $v_{ijk}$ is outside of $\mathcal{F}_{i}$ in either case. Then, we realize that, in Fig. 4(c), $v_{ijk}$ is outside of $\triangle{IJK}$ while the deployment in Fig. 4(a) is not. This observation leads us to introduce the following condition to prevent a formation of a hole

which requires Quadcopter $i$ to satisfy $v_{ijk}\in\mathcal{F}_{i}$ only when $v_{ijk}\in\triangle{IJK}$ holds. The following theorem in our previous work [51] formalizes the relationship between (20) and the existence of a hole.

The condition $v_{ijk}\in\triangle{IJK}$ appeared in (20) can be described as

where

is a candidate CBF satisfying $h_{i,\Delta_{\iota},IJK}^{\mathrm{k}}>0$ if and only if $v_{ijk}$ and $K$ exist in the same half-plane separated by the line $IJ$. Note that the denominator of (22) does not become $0$ under Assumption 1. By definition in (22),

holds for $\Delta_{\iota}=\{i,j,k\}\in\mathcal{T}_{i}(\tau_{i}^{\mathrm{k}})$, where the same relationship is satisfied for $h_{i,\Delta_{\iota},JKI}^{\mathrm{k}}$ and $h_{i,\Delta_{\iota},KIJ}^{\mathrm{k}}$. The other condition $v_{ijk}\in\mathcal{F}_{i}$ in (20) is expressed with another CBF candidate as

The condition (20) preventing a hole in a triangular subgraph $\Delta_{\iota}=\{i,j,k\}\in\mathcal{T}_{i}(\tau_{i}^{\mathrm{k}})$ can be written as

From (7) and (25), the NCBFs of Quadcopter $i$ to be considered for preventing the formation of holes in-between all triangular subgraphs $\Delta_{\iota}\in\mathcal{T}_{i}(t)$ on the $\mathrm{k}$-th interval are described as

Here, we define $h_{i,\Delta_{\iota},\ell}^{\mathrm{k}}$, $\ell\in\{1,\cdots,4\}$ for each triangular subgraph $\Delta_{\iota}=\{i,j,k\}\in\mathcal{T}_{i}(\tau_{i}^{\mathrm{k}})$, where they correspond with $-h_{i,\Delta_{\iota},IJK}^{\mathrm{k}}$, $-h_{i,\Delta_{\iota},JKI}^{\mathrm{k}}$, $-h_{i,\Delta_{\iota},KIJ}^{\mathrm{k}}$, and $h_{i,\Delta_{\iota},\mathcal{F}}^{\mathrm{k}}$ in order for notational simplicity.

This subsection presents the distributed algorithm that prevents the appearance of a hole in-between a team even under the graph $\mathcal{G}(t)$ switches. Namely, the set $\mathcal{C}_{i}$ in (18) defined with the HNCBF (19) and the NCBF (26) is rendered hybrid forward invariance by the algorithm. To calculate the control input in a distributed manner, we will utilize the symmetric property of the designed CBFs.

The hybrid forward invariance of the set $\mathcal{C}_{i}$ can be guaranteed by satisfying the following constraint during each switching interval $t^{\prime}\in[\tau_{i}^{\mathrm{k}},\tau_{i}^{\mathrm{k}+1})$

for every Carathéodry solution, for some locally Lipschtz extended class-$\mathcal{K}$ function $\alpha:{\mathbb{R}}\to{\mathbb{R}}$ [53]. However, the nonsmoothness exists in (26) hinders us from obtaining the time derivative $\dot{h}_{i,\Delta_{\iota}}^{\mathrm{k}}$ via the usual chain rule.

As a remedy for this problem, we reformulate the constraint (27) by incorporating the gradients of all the functions comprising the almost-active set of functions as discussed in Remark 1. Then, the appearance of holes can be prevented by satisfying the following constraint on each interval

with the almost-active set of functions

However, the input $u_{i}$ satisfying (28) cannot be calculated in a distributed manner because it requires input information of other agents $l\in\Delta_{\iota}\backslash\{i\}$. Furthermore, in order to evaluate the second term of the left side of (28), it requires the information of both $h_{i,\Delta_{\iota},\ell}^{\mathrm{k}}$ and $u_{j}$; namely, neither Quadcopter $i$ nor $j$ can calculate it alone. The same issue occurs in the third term. The following theorem proves that condition (28) can be divided into three conditions, where each of them can be evaluated in a distributed manner by Quadcopters $i$, $j$, and $k$, respectively.