# Distance-Based Formation Control for Quadrotors with Collision Avoidance via Lyapunov Barrier Functions

Authors: Jawhar Ghommam; Luis F. Luque-Vega; Maarouf Saad
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

In this paper, group formation control with collision avoidance is investigated for heterogeneous multiquadrotor vehicles. Specifically, the distance-based formation and tracking control problem are addressed in the framework of leader-follower architecture. In this scheme, the leader is assigned the task of intercepting a target whose velocity is unknown, while the follower quadrotors are arranged to set up a predefined rigid formation pattern, ensuring simultaneously interagent collision avoidance and relative localization. The adopted strategy for the control design consists in decoupling the quadrotor dynamics in a cascaded structure to handle its underactuated property. Furthermore, by imposing constraints on the orientation angles, the follower will never be overturned. Rigorous stability analysis is presented to prove the stability of the entire closed-loop system. Numerical simulation results are presented to validate the proposed control strategy.


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## Body

## 1. Introduction

### 1.1. Background and Related Works

Quadrotors are agile mechanical systems which have drawn the attention of researchers for the last decades [1–3], mainly because the use of these types of aerial vehicles is appealing for several real-world applications. Due to their excellent performances, quadrotors are likely to be adopted in various civil potential applications such as inspection of power lines, oil platforms, search and rescue operations, and surveillance [4, 5]. On the other hand, it is worth pointing out that a team of quadrotors has many benefits over using a single quadrotor in several applications where it results in faster and more efficient operations. The coordination of multiple quadrotors to perform cooperative tasks has received considerable attention due to the challenges arising in their control design. Coordinated movement of quadrotors comes with several methodologies and approaches proposed over the last few years [6–10]. From the control structure perspective, the approaches used for coordinated control can be categorized into centralized and decentralized methods. In the centralized method, the control signals for all the agents are generated in the leader agent or in a central station [7, 11]. Whereas, in the decentralized method also known as the distributed method, each member of the team uses its own local controller enhanced with local information from its close neighbors [6, 8, 10].From the control mechanism perspective, there are roughly five approaches to cooperative control of multiple aerial vehicles: the behavioral-based approach [12, 13] where each agent locally reacts to actions of its neighbors. This method is categorized under the decentralized approach and is suitable for a large number of agents in the group. The disadvantage of this approach, however, is that the complexity of the dynamics of the group of robots cannot be amenable to mathematical stability analysis. Another important approach is the virtual structure approach [14–17]. This approach considers all the agents in the group as a single entity, thus limiting the formation scenarios that can be anticipated to merely static formation patterns. Moreover, the priority of agents in the group team is to respect one of the two rules: following their individual trajectories [18] or maintaining the group formation. The artificial potential function-based approach [6, 19, 20] is a powerful tool as it allows the agents to evolve in a cohesive motion while avoiding obstacles. The disadvantage of this approach is that local minima may appear where the vector field vanishes, therefore forcing the agents to get trapped into undesired equilibrium points. The leader-following approach [11, 21, 22], on the other hand, seems to be particularly attractive and preferred over the aforementioned approaches due to its simplicity and scalability. In this approach, some agents are designated as leaders aimed at tracking predefined trajectories, while others are treated as followers whose actions are merely to follow the leader(s) in order to keep the formation. The followers can in turn be designated as leaders for other vehicles in the group resulting in a scalability of the formation. The advantage of this approach is that from the definition of the motion of one entity, the overall group behavior can subsequently be defined. Also, it should be pointed out that the internal formation stability is induced by the control laws of individual vehicles. A major drawback of this approach, however, is that there exists no feedback from the followers to the leaders. Consequently, if the followers are perturbed, the coordination shape of the group team cannot be maintained.A leader-follower formation control algorithm for multiquadrotors was proposed in [22, 23] that involves a linear inner-loop and nonlinear outer-loop control structure, where a combination of sliding mode control and LQR/PD technique was deployed for the trajectory tracking control of a follower to track a leader quadrotor. The formation shape is formed then maintained by keeping a constant distance and a desired angle between each follower and the leader. The authors in [24] proposed a control strategy for a string-like formation of multiple autonomous quadrotor vehicles. In this approach, the objective is to design a maneuvering control for the quadrotor group along a geometric path while respecting an assigned timing law that determines the formation rate of advancement. A formation control strategy has been proposed [11] using the leader-follower approach. The suggested approach differs from common strategies employed for leader-follower schemes and consists in stabilizing, at a predefined distance vector, the position of the leader in the body frame of a virtual follower which in turn generates a trajectory which is used as a reference trajectory to be tracked by the real follower vehicle. The proposed strategy allows for generating curvature-rich formation trajectories. The authors in [25] proposed a 3D leader-follower formation control strategy for a quadrotor with completely unknown dynamics. Their presented strategy relies principally on the use of the spherical coordinate method where the desired position of the follower quadrotor is specified using a desired separation distance along with a desired angle of incidence and bearing. The model unknown nonlinearities are compensated using the neural network techniques. However, a common drawback of many previous leader-follower formation controllers is that they ignore the physical quadrotor flight limitation. In fact, for a safe flight, it is required that the quadrotor avoid singularities in the rotation matrix represented by the Euler angles. Essentially, they often assume that the quadrotor system is always away from the singularity points and that the quadrotor does not rotate too much. From a practical viewpoint, the trajectory generated by the leader vehicle might be aggressive sometimes. Consequently, the follower quadrotor is most likely exposed to a fall in singularity points. This, in turn, may result in poor tracking performance of the proposed controller. An immediate solution to alleviate this problem is to constrain the variation of the quadrotor states within a specified range, then design a controller such that the quadrotor’s state never transgresses these constraints.


### 1.2. Contributions and Novelties

Motivated by the earlier works, the main purpose of this work is to tackle the distance-based formation control problem for a group of homogeneous quadrotor vehicles modeled by an undirected graph. Contrary to the current state of the art, limited explicit communication among neighboring quadrotors is mainly motivated to avoid flooding the communication bandwidth in real-time applications, particularly when the number of cooperating quadrotors increases. To keep communication to a minimum, we propose a new distance-based formation control protocol for a group of multiple quadrotors in a leader-follower architecture, where sensor constraints are explicitly addressed in the design. In this architecture, we assume that the target’s relative position with a leader in the group can be shared among other follower quadrotors, while its velocity is made unavailable. To learn this reference velocity, an estimation mechanism is implemented on each quadrotor so that a predefined formation is recovered so the quadrotors maintain a desired distance among their neighboring followers, while avoiding collision and ensuring relative localization. To deal with underactuation of the quadrotor vehicle in the formation, a decentralized intermediary translational control input is designed for relative position tracking, by which a bounded control thrust and bounded desired orientation are extracted. Then, a control torque for each individual quadrotor is designed to track the desired orientation while avoiding overturn of the quadrotors.In comparison with existing works, the main contributions of this paper are listed as follows:(1)
Compared to [6, 26, 27], our controller is completely designed independently from a global reference coordinate and does not require the local frame of all vehicles to be aligned. Only the distance requirements are used with local sensors to ensure formation maintenance(2)
Our controller is decentralized and ensures simultaneously collision avoidance and self-localization. As opposed to [6, 28], the control input for the translational dynamics is not designed based on a negative gradient of specific local/global potential functions; it is designed using a barrier Lyapunov function ensuring convergence to the appropriate distances with predefined performances(3)
The output constraints on the orientation angles are handled using the integral barrier Lyapunov (iBL) function [29] in the control design, thus relaxing the assumption on small orientation angles in [30–32] and therefore ensuring that the quadrotor will never overturn(4)
Our design is modular in the sense that the stability of the cooperative system is easily shown by the fact that the closed-loop system is regarded as cascaded systems in which the position dynamic subsystem is the driven subsystem while the orientation subsystem is the driving subsystem


### 1.3. Organization

The paper is organized as follows. The upcoming section introduces basic concepts of rigid graph theory and results on the stability of cascade systems. Section3 models the dynamic equations of motion for an individual quadrotor and presents the decentralized distance-based formation control objectives. The controllers for the translational and rotational dynamics of the quadrotors in the group formation are introduced in Section 4. The main result and the convergence analysis for the proposed controller are provided in Section 5. Numerical simulations are presented in Section 6. Finally, the conclusion is presented in Section 7.


## 1.1. Background and Related Works

Quadrotors are agile mechanical systems which have drawn the attention of researchers for the last decades [1–3], mainly because the use of these types of aerial vehicles is appealing for several real-world applications. Due to their excellent performances, quadrotors are likely to be adopted in various civil potential applications such as inspection of power lines, oil platforms, search and rescue operations, and surveillance [4, 5]. On the other hand, it is worth pointing out that a team of quadrotors has many benefits over using a single quadrotor in several applications where it results in faster and more efficient operations. The coordination of multiple quadrotors to perform cooperative tasks has received considerable attention due to the challenges arising in their control design. Coordinated movement of quadrotors comes with several methodologies and approaches proposed over the last few years [6–10]. From the control structure perspective, the approaches used for coordinated control can be categorized into centralized and decentralized methods. In the centralized method, the control signals for all the agents are generated in the leader agent or in a central station [7, 11]. Whereas, in the decentralized method also known as the distributed method, each member of the team uses its own local controller enhanced with local information from its close neighbors [6, 8, 10].From the control mechanism perspective, there are roughly five approaches to cooperative control of multiple aerial vehicles: the behavioral-based approach [12, 13] where each agent locally reacts to actions of its neighbors. This method is categorized under the decentralized approach and is suitable for a large number of agents in the group. The disadvantage of this approach, however, is that the complexity of the dynamics of the group of robots cannot be amenable to mathematical stability analysis. Another important approach is the virtual structure approach [14–17]. This approach considers all the agents in the group as a single entity, thus limiting the formation scenarios that can be anticipated to merely static formation patterns. Moreover, the priority of agents in the group team is to respect one of the two rules: following their individual trajectories [18] or maintaining the group formation. The artificial potential function-based approach [6, 19, 20] is a powerful tool as it allows the agents to evolve in a cohesive motion while avoiding obstacles. The disadvantage of this approach is that local minima may appear where the vector field vanishes, therefore forcing the agents to get trapped into undesired equilibrium points. The leader-following approach [11, 21, 22], on the other hand, seems to be particularly attractive and preferred over the aforementioned approaches due to its simplicity and scalability. In this approach, some agents are designated as leaders aimed at tracking predefined trajectories, while others are treated as followers whose actions are merely to follow the leader(s) in order to keep the formation. The followers can in turn be designated as leaders for other vehicles in the group resulting in a scalability of the formation. The advantage of this approach is that from the definition of the motion of one entity, the overall group behavior can subsequently be defined. Also, it should be pointed out that the internal formation stability is induced by the control laws of individual vehicles. A major drawback of this approach, however, is that there exists no feedback from the followers to the leaders. Consequently, if the followers are perturbed, the coordination shape of the group team cannot be maintained.A leader-follower formation control algorithm for multiquadrotors was proposed in [22, 23] that involves a linear inner-loop and nonlinear outer-loop control structure, where a combination of sliding mode control and LQR/PD technique was deployed for the trajectory tracking control of a follower to track a leader quadrotor. The formation shape is formed then maintained by keeping a constant distance and a desired angle between each follower and the leader. The authors in [24] proposed a control strategy for a string-like formation of multiple autonomous quadrotor vehicles. In this approach, the objective is to design a maneuvering control for the quadrotor group along a geometric path while respecting an assigned timing law that determines the formation rate of advancement. A formation control strategy has been proposed [11] using the leader-follower approach. The suggested approach differs from common strategies employed for leader-follower schemes and consists in stabilizing, at a predefined distance vector, the position of the leader in the body frame of a virtual follower which in turn generates a trajectory which is used as a reference trajectory to be tracked by the real follower vehicle. The proposed strategy allows for generating curvature-rich formation trajectories. The authors in [25] proposed a 3D leader-follower formation control strategy for a quadrotor with completely unknown dynamics. Their presented strategy relies principally on the use of the spherical coordinate method where the desired position of the follower quadrotor is specified using a desired separation distance along with a desired angle of incidence and bearing. The model unknown nonlinearities are compensated using the neural network techniques. However, a common drawback of many previous leader-follower formation controllers is that they ignore the physical quadrotor flight limitation. In fact, for a safe flight, it is required that the quadrotor avoid singularities in the rotation matrix represented by the Euler angles. Essentially, they often assume that the quadrotor system is always away from the singularity points and that the quadrotor does not rotate too much. From a practical viewpoint, the trajectory generated by the leader vehicle might be aggressive sometimes. Consequently, the follower quadrotor is most likely exposed to a fall in singularity points. This, in turn, may result in poor tracking performance of the proposed controller. An immediate solution to alleviate this problem is to constrain the variation of the quadrotor states within a specified range, then design a controller such that the quadrotor’s state never transgresses these constraints.


## 1.2. Contributions and Novelties

Motivated by the earlier works, the main purpose of this work is to tackle the distance-based formation control problem for a group of homogeneous quadrotor vehicles modeled by an undirected graph. Contrary to the current state of the art, limited explicit communication among neighboring quadrotors is mainly motivated to avoid flooding the communication bandwidth in real-time applications, particularly when the number of cooperating quadrotors increases. To keep communication to a minimum, we propose a new distance-based formation control protocol for a group of multiple quadrotors in a leader-follower architecture, where sensor constraints are explicitly addressed in the design. In this architecture, we assume that the target’s relative position with a leader in the group can be shared among other follower quadrotors, while its velocity is made unavailable. To learn this reference velocity, an estimation mechanism is implemented on each quadrotor so that a predefined formation is recovered so the quadrotors maintain a desired distance among their neighboring followers, while avoiding collision and ensuring relative localization. To deal with underactuation of the quadrotor vehicle in the formation, a decentralized intermediary translational control input is designed for relative position tracking, by which a bounded control thrust and bounded desired orientation are extracted. Then, a control torque for each individual quadrotor is designed to track the desired orientation while avoiding overturn of the quadrotors.In comparison with existing works, the main contributions of this paper are listed as follows:(1)
Compared to [6, 26, 27], our controller is completely designed independently from a global reference coordinate and does not require the local frame of all vehicles to be aligned. Only the distance requirements are used with local sensors to ensure formation maintenance(2)
Our controller is decentralized and ensures simultaneously collision avoidance and self-localization. As opposed to [6, 28], the control input for the translational dynamics is not designed based on a negative gradient of specific local/global potential functions; it is designed using a barrier Lyapunov function ensuring convergence to the appropriate distances with predefined performances(3)
The output constraints on the orientation angles are handled using the integral barrier Lyapunov (iBL) function [29] in the control design, thus relaxing the assumption on small orientation angles in [30–32] and therefore ensuring that the quadrotor will never overturn(4)
Our design is modular in the sense that the stability of the cooperative system is easily shown by the fact that the closed-loop system is regarded as cascaded systems in which the position dynamic subsystem is the driven subsystem while the orientation subsystem is the driving subsystem


## 1.3. Organization

The paper is organized as follows. The upcoming section introduces basic concepts of rigid graph theory and results on the stability of cascade systems. Section3 models the dynamic equations of motion for an individual quadrotor and presents the decentralized distance-based formation control objectives. The controllers for the translational and rotational dynamics of the quadrotors in the group formation are introduced in Section 4. The main result and the convergence analysis for the proposed controller are provided in Section 5. Numerical simulations are presented in Section 6. Finally, the conclusion is presented in Section 7.


## 2. Preliminaries

In this section, we present concepts of rigid graph theory for the control formulation taken from [33] and a lemma from [34] for the stability analysis of nonautonomous nonlinear systems, in cascade, which will be useful to establish the stability of the whole closed-loop system.An undirect graphG with pair v,ε is defined as a set of vertices V=1,⋯,n connected with a set of undirected edges E such that if the edge i,j in E,then j,i∈ε. The number of edges that amounts for the connection between the vertices is l=1,⋯,nn−1/2. The set of neighbors of vertex i is defined as Nε such that Nε=j∈vi,j∈ε. A framework of G is the pair F=G,p¯, where p¯∈ℝ3n is the stack vector containing the position of all quadrotors. The rigidity function associated with F is denoted by Λp¯ and is given by Λp¯=⋯,pi−pj22,⋯⊤,i,j∈ε, where ∙22 denotes the Euclidian norm. The rigidity matrix M:ℝ3n⟶ℝl×3nofF=G,p¯ is defined as M=1/2∂Λp¯/∂p¯. It can be shown similarly to [35] that rankM=3n−6. It follows that F is infinitesimally rigid [33] in ℝ3 if the corresponding graph has at least 3n−6 edges. An isometry of ℝ3 is a bijective map I:ℝ3⟶ℝ3 satisfying ν−ϖ=Iv−Iϖ,ν,ϖ∈ℝ3. Two frameworks are said to be isomorphic in ℝ3 if they are related by isometry. The set of all frameworks that is isomorphic to F is denoted by IsoF. Frameworks Gp¯ and Gp¯^ are equivalent if Λp¯=λp^ and are congruent if pi−pj=p^i−p^j,∀i,j∈V. If the infinitesimally rigid frameworks Gp¯ and Gp¯^ are equivalent but not congruent, they are called ambiguous. The collection of frameworks which are ambiguous to the infinitesimally rigid frameworks is denoted by AmbF.Lemma 1 (see [33]).
Given a vectorx∈ℝ3 and let 1n be the n×1 vector of ones, then Mp¯1n⊗x=0.Lemma 2. (see [33]).
If the frameworkF=G,p¯ is minimally and infinitesimally rigid, the the matrix Mp¯Mp¯⊤ is positive definite.Definition 3.
A continuous functionα:0,a⟶0,∞ is said to belong to class K if it is strictly increasing and α0=0. It is said to belong to class K∞ if a=∞ and limr→∞αr=∞. A continuous function β.,.:0,a×0,a⟶0,∞ is said to belong to class KL if, for each fixed s, the mapping βr,s belongs to class K with respect to r and, for each fixed r, the mapping βr,s is decreasing with respect to s and βr,s⟶0 as s⟶∞.Lemma 4. (see [34]).
Consider the following cascade system:(1)ẋ=fx+Φx,z,ż=Gz.
If the equilibriumx=0 is GAS/LES and the equilibrium z=0 is LES and assume also that there exists a constant ℓ1≥0 and a C1‐K function γ., differentiable at z=0 where
(2)Φx,y≤γzx.
If in addition there exists a positive semidefinite radially unlimited functionWx and some positive constants ℓ2 and ℓ3 such that x≥ℓ2(3)∂W∂xfx≤0,∂W∂xx≤ℓ3Wx,then the equilibrium point x,y=0,0 is LES.


## 3. Problem Formulation

The proposed formation scheme is to coordinate the motion ofn quadrotors under the leader-follower architecture. The leader quadrotor in the group is indexed by n, while the rest of the follower vehicles update their state with locally available information. The formation control scheme to be addressed in this paper will be formulated as formation maneuvering and target interception problems as in [33]. Hereafter, the interaction among the vehicles is modeled by undirected graphs G≔V,E where V=1,⋯,i,⋯,n is the set of vertices (quadrotors) and E⊂V×V is the set of edges representing the connectivity among the vehicles. Define the set of neighbors of the i-th quadrotor as Ni=j∈Vi,j∈E.


### 3.1. Quadrotor Model Dynamics

To develop the quadrotor’s equations of motion, we first defined two reference frames. The first one is the earth-fixed inertial frame defined asI=OI,xI,yI,zI, and the second is a body-fixed frame denoted by B=O,xb,yb,zb whose origin O is located at the quadrotor’s center of gravity (CG) as shown in Figure 1.Figure 1
Rigid quadrotor UAV configuration.Letpi∈ℝ3 be the three-dimensional position of the i-th quadrotor and the rotation matrix Ri∈SO3 parameterized with respect to the three Euler angles Θi=ϕi,θi,ψi⊤ with ϕi,θi, and ψi being the roll, pitch, and yaw angles, respectively, and where the special Euclidean group SO3 is the group of 3×3 orthogonal matrices with a determinant of one, i.e., SO3=Ri∈ℝ3×3Ri⊤Ri=I,detRi=1. Following the standard results, the associated orientation dynamics are governed by
(4)Θ̇i=ΨiΘΩi,where
(5)ΨiΘi=1sinϕitanθicosϕitanθi0cosϕi−sinϕi0sinϕicosθicosϕicosθi.The kinematic and dynamic equations of motion of thei-th quadrotor are given by
(6)ṗi=vi,(7)miv̇i=migez−fiRiez+Δxi,(8)Ṙi=RiSΩi,(9)JiΩ̇i=−Ωi×JiΩi+τi+ΔΩi,where vi=vxi,vyi,vzi⊤∈ℝ3 is the linear velocity vector in inertial coordinates, Ωi=ωxi,ωyi,ωzi⊤∈ℝ3, m∈ℝ denotes the total mass of the quadrotor i, Ji∈ℝ3 is the inertia matrix with respect to the frame B, ez=0,0,1⊤ is the unit vector, and g is the gravitational acceleration. The thrust magnitude is denoted as fi∈ℝ, and τi∈ℝ3 is the moment vector; both are considered as the control inputs of the quadrotor. The matrix SΩi is a skew-symmetric matrix containing the elements of the body-fixed angular velocity Ωi. Denote by Δxi and ΔΩi the unstructured uncertainties in the translational dynamics and the rotational dynamics of a quadrotor, respectively. We further assume that the uncertainties Δxi and ΔΩi are small time varying and bounded, that is,
(10)Δxi≤δxi,ΔΩi≤δΩi,where δxi and δωi are known constants. From the dynamic equations (7)–(9), it is worth noting that the quadrotor is an underactuated system with only one degree of freedom for the thrust actuation in the body frame, while the attitude dynamics are fully actuated and can be used to drive the thrust force to any desired orientation.


### 3.2. Control Objective

The leader-follower approach considered in this paper assumes a group ofn quadrotor vehicles with the n-th vehicle set to be the leader which can merely measure the target’s relative position pdt∈ℝ3. Additionally, it is also assumed that the only information that the quadrotors can sense is exclusively acquired from their local sensors. The control objective is to design controllers for the group of vehicles to maintain interdistance specification dij⋆∈ε among the vehicles to attain a rigid desired formation, while guaranteeing formation maintenance and intervehicle collision, that is, if the desired formation is modeled by the framework F∗=G∗,p∗, which is assumed to be infinitesimally and minimally rigid, where F∗=v∗,E∗ and p∗=p1∗⊤,p2∗⊤,⋯,pn∗⊤⊤ are the collection of the targets’ positions. Also, considering that the actual formation of quadrotors is modeled by Ft=G∗,p¯, then the formation control problem boils down to ensure the following:
(11)Ft⟶IsoF∗,ast⟶∞.Mathematically speaking, the control objective reduces to design a desired thrust magnitude for the system dynamics of each quadrotor vehicle (7) such that
(12)limt→∞pi−pj−dij⋆=0,limt→∞vit−vdt=ϵ,∀i,j∈ε,where vdt∈ℝ3 is a bounded, continuous function designating the translational velocity of the formation and is a positive limit upper-bound. An important issue in the formation of multiple quadrotor vehicles is to ensure safe navigation by avoiding collision among interacting vehicles. To this end, the distance that separates the quadrotors should be kept greater than a safety zone. Similarly, the quadrotors should also be retained within the connectivity network to achieve the group tasks. The decentralized controller should also satisfy the following constraint:
(13)−ρ¯ijt<eijt<ρ¯ijt∀i,j∈ε,where eijt=pi−pj−dij⋆ and ρ¯ijt>0,ρ¯ijt>0 are prescribed functions.In addition to the translational motion of the quadrotor, an essential part of the control is the attitude innermost layer of the controller. While the position controller calculates the desired direction of the thrust vector, the attitude controller will ensure the alignment of the real thrust vector with the desired one. A second control objective is to ensure a fast control loop for the attitude innermost layer. Thus, our goal is to satisfy the following limit:
(14)limt→∞Rit=Rdit,∀t>0,where Rdit represents the desired attitude that is calculated based on the desired trust vector fdi which will be derived later in the next section.


## 3.1. Quadrotor Model Dynamics

To develop the quadrotor’s equations of motion, we first defined two reference frames. The first one is the earth-fixed inertial frame defined asI=OI,xI,yI,zI, and the second is a body-fixed frame denoted by B=O,xb,yb,zb whose origin O is located at the quadrotor’s center of gravity (CG) as shown in Figure 1.Figure 1
Rigid quadrotor UAV configuration.Letpi∈ℝ3 be the three-dimensional position of the i-th quadrotor and the rotation matrix Ri∈SO3 parameterized with respect to the three Euler angles Θi=ϕi,θi,ψi⊤ with ϕi,θi, and ψi being the roll, pitch, and yaw angles, respectively, and where the special Euclidean group SO3 is the group of 3×3 orthogonal matrices with a determinant of one, i.e., SO3=Ri∈ℝ3×3Ri⊤Ri=I,detRi=1. Following the standard results, the associated orientation dynamics are governed by
(4)Θ̇i=ΨiΘΩi,where
(5)ΨiΘi=1sinϕitanθicosϕitanθi0cosϕi−sinϕi0sinϕicosθicosϕicosθi.The kinematic and dynamic equations of motion of thei-th quadrotor are given by
(6)ṗi=vi,(7)miv̇i=migez−fiRiez+Δxi,(8)Ṙi=RiSΩi,(9)JiΩ̇i=−Ωi×JiΩi+τi+ΔΩi,where vi=vxi,vyi,vzi⊤∈ℝ3 is the linear velocity vector in inertial coordinates, Ωi=ωxi,ωyi,ωzi⊤∈ℝ3, m∈ℝ denotes the total mass of the quadrotor i, Ji∈ℝ3 is the inertia matrix with respect to the frame B, ez=0,0,1⊤ is the unit vector, and g is the gravitational acceleration. The thrust magnitude is denoted as fi∈ℝ, and τi∈ℝ3 is the moment vector; both are considered as the control inputs of the quadrotor. The matrix SΩi is a skew-symmetric matrix containing the elements of the body-fixed angular velocity Ωi. Denote by Δxi and ΔΩi the unstructured uncertainties in the translational dynamics and the rotational dynamics of a quadrotor, respectively. We further assume that the uncertainties Δxi and ΔΩi are small time varying and bounded, that is,
(10)Δxi≤δxi,ΔΩi≤δΩi,where δxi and δωi are known constants. From the dynamic equations (7)–(9), it is worth noting that the quadrotor is an underactuated system with only one degree of freedom for the thrust actuation in the body frame, while the attitude dynamics are fully actuated and can be used to drive the thrust force to any desired orientation.


## 3.2. Control Objective

The leader-follower approach considered in this paper assumes a group ofn quadrotor vehicles with the n-th vehicle set to be the leader which can merely measure the target’s relative position pdt∈ℝ3. Additionally, it is also assumed that the only information that the quadrotors can sense is exclusively acquired from their local sensors. The control objective is to design controllers for the group of vehicles to maintain interdistance specification dij⋆∈ε among the vehicles to attain a rigid desired formation, while guaranteeing formation maintenance and intervehicle collision, that is, if the desired formation is modeled by the framework F∗=G∗,p∗, which is assumed to be infinitesimally and minimally rigid, where F∗=v∗,E∗ and p∗=p1∗⊤,p2∗⊤,⋯,pn∗⊤⊤ are the collection of the targets’ positions. Also, considering that the actual formation of quadrotors is modeled by Ft=G∗,p¯, then the formation control problem boils down to ensure the following:
(11)Ft⟶IsoF∗,ast⟶∞.Mathematically speaking, the control objective reduces to design a desired thrust magnitude for the system dynamics of each quadrotor vehicle (7) such that
(12)limt→∞pi−pj−dij⋆=0,limt→∞vit−vdt=ϵ,∀i,j∈ε,where vdt∈ℝ3 is a bounded, continuous function designating the translational velocity of the formation and is a positive limit upper-bound. An important issue in the formation of multiple quadrotor vehicles is to ensure safe navigation by avoiding collision among interacting vehicles. To this end, the distance that separates the quadrotors should be kept greater than a safety zone. Similarly, the quadrotors should also be retained within the connectivity network to achieve the group tasks. The decentralized controller should also satisfy the following constraint:
(13)−ρ¯ijt<eijt<ρ¯ijt∀i,j∈ε,where eijt=pi−pj−dij⋆ and ρ¯ijt>0,ρ¯ijt>0 are prescribed functions.In addition to the translational motion of the quadrotor, an essential part of the control is the attitude innermost layer of the controller. While the position controller calculates the desired direction of the thrust vector, the attitude controller will ensure the alignment of the real thrust vector with the desired one. A second control objective is to ensure a fast control loop for the attitude innermost layer. Thus, our goal is to satisfy the following limit:
(14)limt→∞Rit=Rdit,∀t>0,where Rdit represents the desired attitude that is calculated based on the desired trust vector fdi which will be derived later in the next section.


## 4. Control Design

### 4.1. Decentralized Control Design for the Position Outer-Loop

In this section, we will use the backstepping technique to derive a decentralized cooperative controller for the position control loop in the sense that each quadrotor requires the relative position of its neighbors and its own velocity, while ensuring collision avoidance among the vehicles. For that purpose, the vector termfiRiez will be considered as the control input signal for the cooperative position-tracking task. It is common to assign to that input signal a desired value fdiRdiez that we wish to reach in the final stabilization process. Rdi denotes the desired orientation of the follower quadrotor while Rdiez represents the direction of the force amplitude fdiAssumption 5.
The actuator dynamics of the follower quadrotor are negligible with respect to the rigid body dynamics of the quadrotor.Due to the dynamics consideration, Assumption5 reveals that the desired thrust magnitude fdi can be instantaneously reached by the force control input fi; therefore, it is straightforward to conclude that fdiRdiez=fiRdiez. Hence, the first two equations in (7) can be rewritten in terms of Rdi and Ri as follows:
(15)ṗi=vi,(16)miv̇i=migez−fiRdiez+h~ifi,Ri,Rdi+Δxi,where the coupling term h~ifi,Ri,Rdi=−fiRi−Rdiez which can be written as h~ifi,Ri,Rdi=−fiRi~−IRdiez, where R~i=RiRdi⊤ is the rotation matrix error. Thus, system (15) can be viewed as a nominal system, perturbed by a bounded term of the orientation error, which is assumed to converge to zero faster than the closed-loop translational dynamics by ensuring that R~i=I or, in other words, by designing an attitude control law that ensures that Ri converges to Rdi as time goes to infinity. For this reason and to design the position control law, we will make the following assumption.Assumption 6.
Since the closed-loop attitude dynamics is faster than the translational dynamics, then it is logical to consider thatRi=Rdi.Remark 7.
It is important to mention that Assumption6 is only useful to help for the design of the control law position without explicitly accounting for the orientation tracking error R~i in the translational dynamics. The stability analysis of the resulting translational error dynamics with the coupling term will be investigated in detail later on in Section 5.To help for the development of the decentralized cooperative controller, we first define the following state variable:(17)ℓij=p~ij2−dij⋆2,where p~ij=pi−pj from the definition of the relative distance, it can be easily seen that ℓij=eijeij+2dij⋆ which can be shown to be constrained as
(18)ℓ¯ijℓt⋆<ℓij<ℓ¯ijℓ⋆t,where ℓ⋆t=2ℓt is a performance function chosen similar to [36], such that it enforces transient and steady state performance specifications on the state variable ℓij. As such, ℓt=ℓ0−ℓ∞e−at+ℓ∞, where ℓ0,ℓ∞>0 are performance specifications and a>0. Owing to the decreasing property of the performance function, one can see that −ℓ¯ijℓ0 and ℓ¯ijℓ0 serve as the maximum allowed overshoot and the minimum allowed undershot of ℓij, respectively. The control design proceeds as follows:Step 1.
It is important to transform the control problem with the constraint (18) into an unconstrained one. To this end, the following barrier function is defined; it is a subtle treatment for ensuring collision avoidance and connectivity maintenance [37]:
(19)zij=Tℓij2ℓ=lnℓ¯ijℓ¯ij−ℓ¯ijℓij2ℓ−lnℓ¯ijℓ¯ij−ℓ¯ijℓij2ℓ.Clearly, from definition (19), if the variable zij is bounded, then ℓij/ℓ remains within the interval −ℓ¯ij,ℓ¯ij for all t≥0; therefore, the constraints in (18) are satisfied. Furthermore, it is straightforward to show that T−1zij=ℓ¯ijℓ¯ijezij−1/ℓ¯ijezij−ℓ¯ij, then when zij=0, we have ℓij=0.The time derivative of the state variable ofzij considering (17) and (19) is
(20)żij=12ηijℓ̇ij−ϖj=ηijp~ij⊤vi−vj−ϖij,where
(21)ηij=∂T.∂ℓij/ℓ1ℓ,ϖij=ℓ̇ℓℓij.For the dynamic system (20), design the following velocity profile:
(22)ϑdi=−k∑j∈Nεηijp~ij⊤zij+1λminMM⊤∑j∈Nεηijp~ij⊤ϖij+v^d,where k>0 is a design parameter, λmin. denotes the minimum eigenvalue, and v^d is the estimate of the target velocity in which position is only available to the leader vehicle. The estimate of the target velocity is the output of the following filter designed in a similar way as in [38]:
(23)v^d=−ℏ1iφi−ℏ2iφi,rei=φi+ρe,φi0=0,φ̇i=−ℏ1iφiKrei,where ℏ1iφi=c1tanhφ1i/c1,c2tanhφ2i/c2,c3tanhφ3i/c3⊤ and ℏ2iφi=k1tanhφ1i/k1,k2tanhφ2i/c2,k3tanhφ3i/c3⊤, and cm,km>0,m=1,2,3 are design parameters, K=diagk1,k2,k3, and ρe=pnt−p^dt is the interception error between the leader quadrotor and the target, with p^dt=v^dt.Step 2.
In this step, define the velocity tracking error as(24)vei=vi−ϑdi.Taking the time derivative of (24) along the solutions of the second equation of (15) results in
(25)miv̇ei=migez−fiRdiez+Δxi−miϑ̇di.Note that the termh~ifi,Ri,Rdi is temporarily ignored at this stage of the design. It will, however, be investigated in a later stage of the stability analysis. It becomes easy from (25) to select the thrust force fi that stabilizes vei to the origin. It is also important to point out that the total thrust force to be designed must be limited such that actuators’ physical limitations are not violated. By defining the saturated input Ui=fiRdiez and the nominal input as U0i=f0iRdiez, then this unsaturated input is designed as
(26)U0i=k2ivei+migez−miϑ̇di+Δ^xi+k1iξi,where k1i and k2i are design parameters and ξi is an auxiliary state system generated as
(27)ξ̇i=−Kξiξi−giξi+ςiΔUi,ifξmi>vi,m=1,⋯,3,0,ifξmi>vi,m=1,⋯,3,where ξ=ξ1i,ξ2i,ξ3i⊤,υi>0, is a small number; Kξi>1,ςi>0, are design constants; ΔUi=Ui−U0i, and gi=vei⊤ΔUi+0.5ςi2ΔUi⊤ΔUi/ξi2. More discussion about the auxiliary system to overcome the saturation effect of the control input can be found in [39]. Δ^xi is the estimate of the disturbance Δxi, generated using the following disturbance observer [40]:
(28)Δ^xi=ϱxi+Kximivei,(29)ϱxi̇=Kxiϱxi−Kximigez−fiRdiez−miϑ̇di+Kximivei,where Kxi is s a symmetric and positive definite matrix.The following proposition summarizes the distributed cooperative control for the translational dynamics of the set of quadrotors in the formation.Proposition 8.
Consider the formation of a group of quadrotors, given the translational dynamics for each quadrotor in group (15), under Assumptions 5 and 6, if the nominal thrust magnitude f0i and its direction Rdiez are selected according to (30) and (31)
(30)f0i=k2ivei+migez−miϑ̇div^̇d+Δ^xi+k1iξi,(31)Rdiezk2ivei+migez−miϑ̇div^̇d+Δ^xi+k1iξik2ivei+migez−miϑ̇div^̇d+Δ^xi+k1iξi,where v^d=v∧xd,v∧yd,v∧zd⊤∈ℝ3 is the estimate of the target velocity using filter (23), then the equilibrium zij,vei⊤,ξi⊤⊤=0 of the closed-loop system is globally asymptotically stable (GAS). Furthermore, the velocity vector of the leader quadrotor uniformly semiglobally practical asymptotically converges to vdt.Proof.
The proof proceeds in two folds: In a first stage, we show that the desired formation is attained with prescribed transient performance, while avoiding interagent collision and connectivity intermittent by ensuring that the leader vehicle tracks an estimate of the target’s state quite accurately. In a second stage, we show that the estimate of the target’s velocity converges to a vicinity of its true value. To proceed, we first define the following candidate Lyapunov function:(32)V1=12z⊤z,where z=⋯,zij,⋯⊤∈ℝp,i,j∈ε. Differentiating V1 along the solutions of (20) results in
(33)V̇1=z⊤ϒMv¯−z⊤ϒΓ,where ϒ=diagηij∈ℝp×p, Γ=ϖ1j,ϖ2j,⋯,ϖnj,⊤∈ℝl, and v¯=v1⊤,v2⊤,⋯,vn⊤⊤∈ℝ3n. Let ϑ¯d=ϑd1⊤,ϑd2⊤,⋯,ϑdn⊤⊤∈ℝ3n, then adding and subtracting the term z⊤ϒMϑ¯d and substituting the expression ϑdi,i=1,⋯,n, from the velocity protocol (22), using Lemmas 1 and 2, we obtain
(34)V̇1=−kz⊤ϒMM⊤ϒ+zϒMv¯e−MM⊤λminMM⊤z⊤ϒΓ+z⊤ϒΓ+z⊤ϒM1n⊗v^d≤−kz⊤ϒMM⊤ϒ+z⊤ϒMv¯e≤−kb0−12jz⊤z+j2b0v¯e⊤v¯e,where b0=λminϒMM⊤ϒ and j is a parameter chosen such that k>1/2b0j. Note that the second term on the right-hand side of inequality (34) will be dealt with in a further step of the backstepping procedure. In the sequel, let us consider the following positive definite Lyapunov function:
(35)V2=V1+12∑i=1nmivei⊤vei+12∑i=1nΔ~xi⊤Δ~xi+12∑i=1nξi⊤ξi,where Δ~xi=Δxi−Δ^xi represents the estimation error of the disturbances affecting the transitional dynamics of the follower quadrotor. Take the time derivative of (35) along the solutions of (25), (27), (28), and (34) and substituting the control (30) and (31) we obtain
(36)V̇2t=V̇1+∑i=1nmivei⊤v̇ei+∑i=1nΔ~xi⊤Δ~̇xi+∑i=1nξi⊤ξ̇i≤−kb0−12jz⊤z+j1b0∑i=1nvei⊤vei+∑i=1nvei⊤migez−Uio−ΔUi+Δxi−miϑ̇di+∑i=1nΔ~xi⊤Δ~̇xi+∑i=1nξi⊤ξ̇i≤−kb0−12jz⊤z−∑i=1nk2i−j2b0vei⊤vei−∑i=1nKxiΔ~xi⊤Δ~xi−∑i=1nKξξi⊤ξi≤−azV2t,where az=2minkb0−1/2j,k2i−j/2b0,Kxi,Kξ. From (36), it is clear that V̇2≤0; therefore, V2t is decreasing for all t≥0 from which it can be concluded that z,vei,Δxi, and ξi are all bounded. Thus, there exists a positive constant z¯ such that z∈Ψz≜,z∈ℝp:z≤z¯,∀t≥0, which from (19), by noting that ℓij/2ℓ=T−1zij=ℓ¯ijℓ¯ijezij−1/ℓ¯ijezij−ℓ¯ij, we have −ℓ¯ij<ℓ¯ijℓ¯ije−z¯−1/ℓ¯ije−z¯−ℓ¯ij≤ℓij/2ℓ≤ℓ¯ijℓ¯ijez¯−1/ℓ¯ijez¯−ℓ¯ij. That is, ℓij/2ℓ remains within the compact set ℓ¯ijℓ¯ije−z¯−1/ℓ¯ije−z¯−ℓ¯ij≤ℓij/2ℓ,ℓ¯ijℓ¯ijez¯−1/ℓ¯ijez¯−ℓ¯ij of ℓ¯ij,ℓ¯ij, from which it is straightforward to conclude that relative distance performance among the quadrotors in group (18) is guaranteed. Additionally, from (36), the equilibrium point z⊤,vei⊤,Δ~⊤xi,ξi⊤=0⊤ is exponentially stable for all t≥0.
Thus, it can be concluded thatlimt→∞ℓij=0,∀i,j∈E. Also, from (22), it can be deduced that limt→∞ϑdit=v^d.
The second stage of the proof is to show that the estimate of the leader’s velocity converges to a neighborhood of the target’s true velocity. In this direction, following [38], a Lyapunov function candidate is proposed as V3=1/2∑i=1nrei⊤rei+∑i=1nνiφi⊤νiφi, where νφi=kilogcoshφ1i/ki,k2logcoshφ2i/k2,k3logcoshφ3i/k3⊤. The time derivative of V3 along the solutions of filter (23) yields
(37)V̇3=∑i=1nrei⊤ṙei+K−1∑i=1nφ̇i⊤h¯2iφi=∑i=1nrei⊤−Krei+h¯2iφi+v^n+K−1∑i=1nφ̇i⊤h¯2iφi=−K∑i=1nrei⊤rei−K−1∑i=1nh¯1iφi⊤h¯2iφi+∑i=1nrei⊤v^n.
Since we showed thatvei,∀i=1,⋯,n, is bounded and so is ϑdi, also by construction, v^d is bounded. It follows that there exists a constant Vn such thatvn≤Vn; therefore,
(38)V̇3≤−λminK−0.5−3Vnr¯e∑i=1nrei2−K−1∑i=1nh¯1iφi⊤h¯2iφi,where r¯e=re1,⋯,ren⊤. Letting b0 be a given positive constant, we obtain for rei≥b0 the following bound for V̇3(39)V̇3≤−∑i=1nrei2−K−1∑i=1nh¯1iφi⊤h¯2iφi,since h¯1iφi⊤h¯2iφi≥0; therefore, V̇3≤0; this shows that the solution of filter (23) is uniform semipractically asymptotically stable [23]. This completes the proof.


### 4.2. Control Design for the Attitude Inner-Loop

In this stage, the last two equations of (9) are considered. The moment vector τi will be designed to globally asymptotically and locally exponentially stabilize the tracking error ψi−ψdi and the errors between the virtual controls of the roll and pitch angles and their actual values at the origin.Step 1.
Controlling the yaw angle.The desired orientationRdi can be deduced from the expression of the rotation matrix Ri∈SO (2) using its Euler angle parameterization subject to the constraint given by the specification of the desired yaw angle extracted from the translational system as shown in Figure 2. Indeed, in Figure 2, the angle ψ~d represents the desired angle associated with the vector v^dt projected on the x-y plan. Clearly, when vit matches with v^dt, the yaw rate angle ψ~iconverges to zero. Following [41], the desired yaw angle is calculated by extracting yaw rate information from the cross- and inner-product of the desired velocity v^dt and the actual velocity vit as follows:
(40)sinψ~it=v^d×viv^dvi=v∧ydiżi−v∧zdiẏi2+v∧zdiẏi−v∧ydiżi2+v∧xdiẏi−v∧ydiẋi2v^xdi2+v^ydi2+v^zdi2ẋi2+ẏi2+żi2,cosψ~it=v^d⊤viv^dvi=v^xdiẋi+v^ydiẏi+v^zdiżiv^xdi2+v^ydi2+v^zdi2ẋi2+ẏi2+żi2,tanψ~it=v^d×viv^d⊤vi=v∧ydiżi−v∧zdiẏi2+v∧zdiẏi−v∧ydiżi2+v∧xdiẏi−v∧ydiẋi2v^xdiẋi+v^ydiẏi+v^zdiżi,which implies that ψ~it=−arctan2sinψ~it,cosψ~it. Therefore, the desired yaw angle can be obtained by the following expression:
(41)ψdi=ψi−arctan2sinψ~it,cosψ~it⏟ψ~itFigure 2
Planar projection of the quadrotor’s velocity and the desired velocity generated by the guidance system.Note also that the time derivative ofψ~ can be calculated as indicated in (43). Therefore, we can write
(42)ψ̇di=ψ̇i+ψ~̇i=sinϕicosθiωyi+cosϕicosθiωzi+ψ~̇i=sinϕicosθiωyi+cosϕicosθiαψ~i+γωiϕi,θiΩei+ψ~̇i,(43)Si=v∧xdiẏi−vydiẋi2+v∧zdiẏi−v∧ydiżi2+v∧ydiżi−v∧zdiẏi2,Qi=2v^zdiẏi−v^ydiżi−żiv^̇ydi−v^ydiżi+ẏiv^̇zdi+v^zdiy¨i+2v^ydiżi−v^zdiẏiżiv^̇ydi+v^ydiz¨i−ẏiv^̇zdi−v^zdiẏi,ψ~̇it=−v^xdiẋi+v^ydiẏi+v^zdiżi2v^xdiẏi−v^ydiẋiẏiv^̇xdi+v^xdiy¨i−ẋiv^̇ydi−v^ydiẋi+Qi2Siv∧xdiẋi+v∧ydiẏi+v∧zdiżi2+v∧ydiẏi−v∧ydiẋi2+v∧ydiẏi−v∧ydiżi2+v∧ydiżi−v∧zdiẏi2−Siẋiv^̇xdi+v^xdiẋi+ẏiv^̇ydi+v^ydiẏi+żiv^̇zdi+v^zdiżiv∧xdiẋi+v∧ydiẏi+v∧zdiżi2+v∧ydiẏi−v∧ydiẋi2+v∧ydiẏi−v∧ydiżi2+v∧ydiżi−v∧zdiẏi2,where Ωei=Ωi−αi is the angular velocity error, with αi=αϕ~i,αθ~i,αψ~i⊤ yet to be determined later. γωiϕi,θi=00cosϕi/cosθi and αψ~i is the stabilizing function, which will be chosen such that it cancels out unneeded terms in (42). As such, the stabilizing function can be chosen as follows:
(44)αψ~i=cosθicosϕi−sinϕicosθiωyi−ψ~̇i+ψ̇i+κ1ψ̇i,where κ1 is a positive gain. Substituting (44) into (42) results in the following closed-loop dynamics:
(45)ψ~̇i=−κ1ψ~i+ϒωiϕi,θiΩei.Clearly, if the last term of (45) is exponentially vanishing, then the tracking error ψ~i is also exponentially stable; this can be seen by associating the quadratic Lyapunov function Vψi=1/2ψ~i2, whose time derivative along the solutions of (45) gives
(46)V̇ψi=−κ1ψ~i2+γωiϕi,θiψ~iΩei=−κ¯ψ~i2,∀ψ~i>12κ¯γωiϕi,θiΩei,where κ¯=κ1−1/2>0. Hence, the virtual control law (44) renders the closed-loop (45) system input-to-state stability (ISS) from γωiϕi,θiΩei to ψ~i. Therefore, if Ωei is globally exponentially stable GES, then so isψ~i.Remark 9.
It has been shown in [42] that overturn of the quadrotor can only be avoided if the last component of Riez=ρim⊤,m=1⋯3i.e.,ρim=cosθicosϕi is ensured to be strictly positive. This constraint further requires that eζi=ζi−ζdi (with ζi=ρi,m⊤,m=1,2,andζd=ρd,m⊤,m=1,2, is a vector that contains the first two component of Rdez) exponentially decays to zero in order to guarantee that θi⟶θd and ϕi⟶ϕd exponentially. Since ρi,3=1−ρi,12−ρi,22, it is sufficient to show through an adequate control law that ζi≤1.Remark 10.
The virtual control (44) is well defined as long as θi,ϕi≠π/2; this singularity is avoided as long as the term cosθcosϕ is strictly positive or alternatively ζi≤1.Step 2.
Controlling the pitch and roll angles.The pitch and roll angles are contained in the column vectorRiez and are denoted by the vector ζi. The time derivative of ζi along the solutions of the first equation of (9) and (5) gives the following dynamics:
(47)ζ̇i=AiΘiωi,xy=AiΘiαi,xy+AiΘi,02×1Ωei, where ωi,xy=ωi,x,ωi,y⊤,αi,xy=αϕ~i,αθ~i⊤and the matrix AiΘi can be obtained by a long but simple calculation as
(48)AiΘi=sinψicosϕi−cosψisinθisinϕicosψicosθi−cosϕicosψi−sinϕisinθisinψisinψicosθi.The matrixAiΘi is invertible, due to the fact that detAiΘi=1/cosθicosϕi, which implies that as long as the singularity (i.e., θi=ϕi=πi/2) is avoided, the matrix is nonsingular. The control objective is therefore to ensure on top of forcing the pitch and roll angles to track their immediate desired values and to also prevent them from violating the constraint that leads to singularity. In other words, the state variable ζi is required to remain in the set ζi<1 for all t≥0. To address this problem, the integral barrier Lyapunov function iBLF [29] will be employed to guarantee nonviolation of the aforementioned constraint.Motivated by the work in [29], in order to design the virtual control ωi,xy, the following integral barrier Lyapunov candidate functional is proposed:
(49)Vζiζeit,ζdt=∫0ζeiσi⊤1−σi+ζd⊤σi+ζddσi,where ζei=ζi−ζd. It can be seen that Vζiζeit,ζdt is positive definite and continuously differentiable and satisfies the decrescent condition in the set ζi<1 such that for σi=ηiζei, we have
(50)12ζei⊤ζei<Vζiζeit,ζdt<ζei⊤ζei∫01ηi1−ηiζei+ζd⊤ηiζei+ζddηi.Based on the definition in (49), the time derivative of the iBLF Vζiζeit,ζdt along the solutions of (47) yields
(51)V̇ζiζeit,ζdt=ζei⊤1−ζi⊤ζiΑiΘiαi,xy+ΑiΘi,02×1Ωei+∂Vζi∂ζd⊤ζ̇d,where ∂Vζi/∂ζd=ζei⊤/1−ζi⊤ζi−χiζi,ζdζi and χiζi,ζd=∫01ηiζei+ζd⊤1/1−ηiζei+ζd⊤ηiζei+ζddηi with the vector 1 being all components 1. We can obtain the stabilizing function αi,xy as
(52)αi,xy=AiΘi−1−κ2ζei+1−ζi⊤1−ζiχiζ̇d,where κ2 is a positive gain to be designed. Under (52), the time derivative of Vζi is given by
(53)V̇ζi=−κ2ζei⊤ζei1−ζi⊤ζi+ζei⊤1−ζi⊤ζiAiΘi,02×1Ωei,at this step according to (53); the time derivative of Vζi becomes negative definite only when Ωei exponentially converges to zero. The design of the force moment is not finished yet. One more step is required to ensure that Ωei exponentially converges to zero.Step 3.
Force moment control design.The angular velocity error dynamicsΩei based on the second equation of (9) has the following form:
(54)JiΩ̇ei=JΩ̇i−Jiα̇i=−Ωi×JiΩi+τi+ΔΩi−Jα̇i.At this stage, the control input for the force momentτi can be readily designed to compensate for the coupling terms created in the previous steps and the external perturbation to which the quadrotor is subject. To this end, let the control input be designed as follows:
(55)τi=−KiΩei+Jiα̇i+Ωi×JiΩi−ϒωiϕi,θi⊤ψ~i−AiΘi,02×1⊤ζei1−ζi⊤ζi−Δ^Ωi,Δ^Ωi=ϱΩi+KΩiΩi,ϱ̇Ωi=−KΩiϱΩi−KΩi−Ωi×JiΩi+τi+ΔΩi+KΩiϱ,where Ki is a diagonal positive gain matrix and KΩ is a symmetric and positive definite matrix.Remark 11.
The disturbance term is compensated for by an estimateΔ^Ωi which is the output of a nonlinear-observer inspired by [40]. The particularity of this type of observer is to guarantee an exponential convergence of the unknown disturbance to its true value. Furthermore, as we want the inner control loop (i.e., attitude dynamics) to be fast convergent as compared to the dynamic behavior of the outer control loop (i.e., translational dynamics), our choice is hence justified by ensuring complete cancellation of the uncertain terms.
For the convenience of stability analysis, we apply the control input (55) along with the virtual controls (52) and (44) and then the error attitude dynamics becomes
(56)ψ~̇i=−κ1ψ~i+γωiϕi,θiΩei,ζ̇ei=−κ2ζei+1−ζi⊤1−ζiχiζ⋅d+AiΘi,02×1Ωei,JΩ̇ei=−KiΩei−γωiϕi,θi⊤ψ~i−AiΘi,02×1⊤ζei1−ζi⊤ζi−Δ~Ωi,(57)Δ^Ωi=ϱΩi+KΩiJiΩi,(58)ϱ̇Ωi=−KΩiϱΩi−KΩi−Ωi×JiΩi+τi+KΩiΩi.The control design for the attitude inner-loop dynamics has been completed. We summarize the results in the following proposition.Proposition 12.
Consider the attitude dynamics (9), given a desired attitude Rdez extracted from the positioning controller (31), under the proposed attitude control (55), the error vector Ξi=ψ~i,ζei⊤,Ωei⊤,Δ~Ωi⊤⊤ of the attitude dynamics described by equations (56) and (57) is exponentially stable for any initial condition Ξit0∈Qi, where Qi=ℝ×Zi×ℝ3×ℝ3 and Zi=ζi∈ℝ3:ζi≤1. In particular, for any initial condition ζit0∈Zi, the constraint ζit<1 is satisfied ∀t>0.Proof.
The stability analysis of the attitude dynamics is conducted by considering the following Lyapunov function of the associated attitude variables:(59)VATi=Vψi+Vζi+12Ωei⊤JiΩei+δ02Δ~Ωi⊤Δ~Ωi,where δ0 is a positive constant to be determined later. Differentiating both side of (59) along the solutions of the closed-loop system consisting of (56) and (57) gives
(60)V̇ATi=V̇ψi+V̇ζi+Ωei⊤JiΩ̇ei+δ0Δ~Ωi⊤Δ~̇Ωi=−κ1ψ~i2−κ2ζei⊤ζei1−ζi⊤ζi+ζei⊤1−ζi⊤ζiAiΘi,02×1Ωei+γωiϕi,θiψ~iΩei+Ωei⊤−KiΩei−γωiϕi,θi⊤ψ~i−AiΘi,02×1⊤ζei1−ζi⊤ζi−Δ~Ωi−δ0Δ~Ωi⊤Δ^̇Ωi=−κ1ψ~i2−κ2ζei⊤ζei1−ζi⊤ζi−Ωei⊤KiΩei−Ωei⊤Δ~Ωi−δ0Δ~Ωi⊤KΩiΔ~Ωi≤−κ1ψ~i2−κ2ζei⊤ζei1−ζi⊤ζi−λminKi−12⏟k0iΩei2−δ0λminΚΩi−12⏟k1iΔ~Ωi2.
In light of the last inequality in (60) and in order to ensure that V̇ATiis negative definite, one can select the matrix Ki such that λminKi>1/2. Also, the observer gain KΩi is selected such that λminKΩi>21/δ0, whereδ0 can always be picked as a sufficiently small constant. Note, however, that δ0 is not needed in the implementation of the proposed controller.
Now referring to Lemma 2 in [29] to approximate the functional Vζi, it is shown that for all ζit≤1, the following inequality holds:
(61)Vζi≤ζei⊤ζei1−ζi⊤ζi.
Based on (61), the inequality (60) rewrites
(62)V̇ATi≤−2κ1Vψi−κ2Vζi−k0iΩei2−k1iΔ~Ωi2≤−βiVATi,where βi≔min2κ1,2κ2,2k2/Ji,2k2/δ0. It implies that
(63)Ξit≤2VATi0e−βit/2,where Ξit=ψ~i,ζi⊤,Ωei⊤,Δ~Ωi⊤⊤. This implies that the closed-loop system (56) and (57) is locally exponentially stable at the origin. Furthermore, it can be seen that ζi is bounded by an exponentially decreasing function. Then, from (61), it can be concluded that Vζi is also bounded by an exponentially decreasing function such that Vζi≤c0i with c0i≔2VATi0. On the other hand, since by construction Vζi⟶∞ as ζi⟶1, it can be inferred from the fact that Vζi is bounded for all t≥0 and the state ζit always satisfies the constraint ζit≤1, provided that ζi0≤1. This completes the proof.Remark 13.
The controllerτi in (55) involves the derivative of the virtual control αi, which appears to be difficult to calculate. So the robust differentiator [43] can be introduced to handle finite convergence of α̇i to its true value.


## 4.1. Decentralized Control Design for the Position Outer-Loop

In this section, we will use the backstepping technique to derive a decentralized cooperative controller for the position control loop in the sense that each quadrotor requires the relative position of its neighbors and its own velocity, while ensuring collision avoidance among the vehicles. For that purpose, the vector termfiRiez will be considered as the control input signal for the cooperative position-tracking task. It is common to assign to that input signal a desired value fdiRdiez that we wish to reach in the final stabilization process. Rdi denotes the desired orientation of the follower quadrotor while Rdiez represents the direction of the force amplitude fdiAssumption 5.
The actuator dynamics of the follower quadrotor are negligible with respect to the rigid body dynamics of the quadrotor.Due to the dynamics consideration, Assumption5 reveals that the desired thrust magnitude fdi can be instantaneously reached by the force control input fi; therefore, it is straightforward to conclude that fdiRdiez=fiRdiez. Hence, the first two equations in (7) can be rewritten in terms of Rdi and Ri as follows:
(15)ṗi=vi,(16)miv̇i=migez−fiRdiez+h~ifi,Ri,Rdi+Δxi,where the coupling term h~ifi,Ri,Rdi=−fiRi−Rdiez which can be written as h~ifi,Ri,Rdi=−fiRi~−IRdiez, where R~i=RiRdi⊤ is the rotation matrix error. Thus, system (15) can be viewed as a nominal system, perturbed by a bounded term of the orientation error, which is assumed to converge to zero faster than the closed-loop translational dynamics by ensuring that R~i=I or, in other words, by designing an attitude control law that ensures that Ri converges to Rdi as time goes to infinity. For this reason and to design the position control law, we will make the following assumption.Assumption 6.
Since the closed-loop attitude dynamics is faster than the translational dynamics, then it is logical to consider thatRi=Rdi.Remark 7.
It is important to mention that Assumption6 is only useful to help for the design of the control law position without explicitly accounting for the orientation tracking error R~i in the translational dynamics. The stability analysis of the resulting translational error dynamics with the coupling term will be investigated in detail later on in Section 5.To help for the development of the decentralized cooperative controller, we first define the following state variable:(17)ℓij=p~ij2−dij⋆2,where p~ij=pi−pj from the definition of the relative distance, it can be easily seen that ℓij=eijeij+2dij⋆ which can be shown to be constrained as
(18)ℓ¯ijℓt⋆<ℓij<ℓ¯ijℓ⋆t,where ℓ⋆t=2ℓt is a performance function chosen similar to [36], such that it enforces transient and steady state performance specifications on the state variable ℓij. As such, ℓt=ℓ0−ℓ∞e−at+ℓ∞, where ℓ0,ℓ∞>0 are performance specifications and a>0. Owing to the decreasing property of the performance function, one can see that −ℓ¯ijℓ0 and ℓ¯ijℓ0 serve as the maximum allowed overshoot and the minimum allowed undershot of ℓij, respectively. The control design proceeds as follows:Step 1.
It is important to transform the control problem with the constraint (18) into an unconstrained one. To this end, the following barrier function is defined; it is a subtle treatment for ensuring collision avoidance and connectivity maintenance [37]:
(19)zij=Tℓij2ℓ=lnℓ¯ijℓ¯ij−ℓ¯ijℓij2ℓ−lnℓ¯ijℓ¯ij−ℓ¯ijℓij2ℓ.Clearly, from definition (19), if the variable zij is bounded, then ℓij/ℓ remains within the interval −ℓ¯ij,ℓ¯ij for all t≥0; therefore, the constraints in (18) are satisfied. Furthermore, it is straightforward to show that T−1zij=ℓ¯ijℓ¯ijezij−1/ℓ¯ijezij−ℓ¯ij, then when zij=0, we have ℓij=0.The time derivative of the state variable ofzij considering (17) and (19) is
(20)żij=12ηijℓ̇ij−ϖj=ηijp~ij⊤vi−vj−ϖij,where
(21)ηij=∂T.∂ℓij/ℓ1ℓ,ϖij=ℓ̇ℓℓij.For the dynamic system (20), design the following velocity profile:
(22)ϑdi=−k∑j∈Nεηijp~ij⊤zij+1λminMM⊤∑j∈Nεηijp~ij⊤ϖij+v^d,where k>0 is a design parameter, λmin. denotes the minimum eigenvalue, and v^d is the estimate of the target velocity in which position is only available to the leader vehicle. The estimate of the target velocity is the output of the following filter designed in a similar way as in [38]:
(23)v^d=−ℏ1iφi−ℏ2iφi,rei=φi+ρe,φi0=0,φ̇i=−ℏ1iφiKrei,where ℏ1iφi=c1tanhφ1i/c1,c2tanhφ2i/c2,c3tanhφ3i/c3⊤ and ℏ2iφi=k1tanhφ1i/k1,k2tanhφ2i/c2,k3tanhφ3i/c3⊤, and cm,km>0,m=1,2,3 are design parameters, K=diagk1,k2,k3, and ρe=pnt−p^dt is the interception error between the leader quadrotor and the target, with p^dt=v^dt.Step 2.
In this step, define the velocity tracking error as(24)vei=vi−ϑdi.Taking the time derivative of (24) along the solutions of the second equation of (15) results in
(25)miv̇ei=migez−fiRdiez+Δxi−miϑ̇di.Note that the termh~ifi,Ri,Rdi is temporarily ignored at this stage of the design. It will, however, be investigated in a later stage of the stability analysis. It becomes easy from (25) to select the thrust force fi that stabilizes vei to the origin. It is also important to point out that the total thrust force to be designed must be limited such that actuators’ physical limitations are not violated. By defining the saturated input Ui=fiRdiez and the nominal input as U0i=f0iRdiez, then this unsaturated input is designed as
(26)U0i=k2ivei+migez−miϑ̇di+Δ^xi+k1iξi,where k1i and k2i are design parameters and ξi is an auxiliary state system generated as
(27)ξ̇i=−Kξiξi−giξi+ςiΔUi,ifξmi>vi,m=1,⋯,3,0,ifξmi>vi,m=1,⋯,3,where ξ=ξ1i,ξ2i,ξ3i⊤,υi>0, is a small number; Kξi>1,ςi>0, are design constants; ΔUi=Ui−U0i, and gi=vei⊤ΔUi+0.5ςi2ΔUi⊤ΔUi/ξi2. More discussion about the auxiliary system to overcome the saturation effect of the control input can be found in [39]. Δ^xi is the estimate of the disturbance Δxi, generated using the following disturbance observer [40]:
(28)Δ^xi=ϱxi+Kximivei,(29)ϱxi̇=Kxiϱxi−Kximigez−fiRdiez−miϑ̇di+Kximivei,where Kxi is s a symmetric and positive definite matrix.The following proposition summarizes the distributed cooperative control for the translational dynamics of the set of quadrotors in the formation.Proposition 8.
Consider the formation of a group of quadrotors, given the translational dynamics for each quadrotor in group (15), under Assumptions 5 and 6, if the nominal thrust magnitude f0i and its direction Rdiez are selected according to (30) and (31)
(30)f0i=k2ivei+migez−miϑ̇div^̇d+Δ^xi+k1iξi,(31)Rdiezk2ivei+migez−miϑ̇div^̇d+Δ^xi+k1iξik2ivei+migez−miϑ̇div^̇d+Δ^xi+k1iξi,where v^d=v∧xd,v∧yd,v∧zd⊤∈ℝ3 is the estimate of the target velocity using filter (23), then the equilibrium zij,vei⊤,ξi⊤⊤=0 of the closed-loop system is globally asymptotically stable (GAS). Furthermore, the velocity vector of the leader quadrotor uniformly semiglobally practical asymptotically converges to vdt.Proof.
The proof proceeds in two folds: In a first stage, we show that the desired formation is attained with prescribed transient performance, while avoiding interagent collision and connectivity intermittent by ensuring that the leader vehicle tracks an estimate of the target’s state quite accurately. In a second stage, we show that the estimate of the target’s velocity converges to a vicinity of its true value. To proceed, we first define the following candidate Lyapunov function:(32)V1=12z⊤z,where z=⋯,zij,⋯⊤∈ℝp,i,j∈ε. Differentiating V1 along the solutions of (20) results in
(33)V̇1=z⊤ϒMv¯−z⊤ϒΓ,where ϒ=diagηij∈ℝp×p, Γ=ϖ1j,ϖ2j,⋯,ϖnj,⊤∈ℝl, and v¯=v1⊤,v2⊤,⋯,vn⊤⊤∈ℝ3n. Let ϑ¯d=ϑd1⊤,ϑd2⊤,⋯,ϑdn⊤⊤∈ℝ3n, then adding and subtracting the term z⊤ϒMϑ¯d and substituting the expression ϑdi,i=1,⋯,n, from the velocity protocol (22), using Lemmas 1 and 2, we obtain
(34)V̇1=−kz⊤ϒMM⊤ϒ+zϒMv¯e−MM⊤λminMM⊤z⊤ϒΓ+z⊤ϒΓ+z⊤ϒM1n⊗v^d≤−kz⊤ϒMM⊤ϒ+z⊤ϒMv¯e≤−kb0−12jz⊤z+j2b0v¯e⊤v¯e,where b0=λminϒMM⊤ϒ and j is a parameter chosen such that k>1/2b0j. Note that the second term on the right-hand side of inequality (34) will be dealt with in a further step of the backstepping procedure. In the sequel, let us consider the following positive definite Lyapunov function:
(35)V2=V1+12∑i=1nmivei⊤vei+12∑i=1nΔ~xi⊤Δ~xi+12∑i=1nξi⊤ξi,where Δ~xi=Δxi−Δ^xi represents the estimation error of the disturbances affecting the transitional dynamics of the follower quadrotor. Take the time derivative of (35) along the solutions of (25), (27), (28), and (34) and substituting the control (30) and (31) we obtain
(36)V̇2t=V̇1+∑i=1nmivei⊤v̇ei+∑i=1nΔ~xi⊤Δ~̇xi+∑i=1nξi⊤ξ̇i≤−kb0−12jz⊤z+j1b0∑i=1nvei⊤vei+∑i=1nvei⊤migez−Uio−ΔUi+Δxi−miϑ̇di+∑i=1nΔ~xi⊤Δ~̇xi+∑i=1nξi⊤ξ̇i≤−kb0−12jz⊤z−∑i=1nk2i−j2b0vei⊤vei−∑i=1nKxiΔ~xi⊤Δ~xi−∑i=1nKξξi⊤ξi≤−azV2t,where az=2minkb0−1/2j,k2i−j/2b0,Kxi,Kξ. From (36), it is clear that V̇2≤0; therefore, V2t is decreasing for all t≥0 from which it can be concluded that z,vei,Δxi, and ξi are all bounded. Thus, there exists a positive constant z¯ such that z∈Ψz≜,z∈ℝp:z≤z¯,∀t≥0, which from (19), by noting that ℓij/2ℓ=T−1zij=ℓ¯ijℓ¯ijezij−1/ℓ¯ijezij−ℓ¯ij, we have −ℓ¯ij<ℓ¯ijℓ¯ije−z¯−1/ℓ¯ije−z¯−ℓ¯ij≤ℓij/2ℓ≤ℓ¯ijℓ¯ijez¯−1/ℓ¯ijez¯−ℓ¯ij. That is, ℓij/2ℓ remains within the compact set ℓ¯ijℓ¯ije−z¯−1/ℓ¯ije−z¯−ℓ¯ij≤ℓij/2ℓ,ℓ¯ijℓ¯ijez¯−1/ℓ¯ijez¯−ℓ¯ij of ℓ¯ij,ℓ¯ij, from which it is straightforward to conclude that relative distance performance among the quadrotors in group (18) is guaranteed. Additionally, from (36), the equilibrium point z⊤,vei⊤,Δ~⊤xi,ξi⊤=0⊤ is exponentially stable for all t≥0.
Thus, it can be concluded thatlimt→∞ℓij=0,∀i,j∈E. Also, from (22), it can be deduced that limt→∞ϑdit=v^d.
The second stage of the proof is to show that the estimate of the leader’s velocity converges to a neighborhood of the target’s true velocity. In this direction, following [38], a Lyapunov function candidate is proposed as V3=1/2∑i=1nrei⊤rei+∑i=1nνiφi⊤νiφi, where νφi=kilogcoshφ1i/ki,k2logcoshφ2i/k2,k3logcoshφ3i/k3⊤. The time derivative of V3 along the solutions of filter (23) yields
(37)V̇3=∑i=1nrei⊤ṙei+K−1∑i=1nφ̇i⊤h¯2iφi=∑i=1nrei⊤−Krei+h¯2iφi+v^n+K−1∑i=1nφ̇i⊤h¯2iφi=−K∑i=1nrei⊤rei−K−1∑i=1nh¯1iφi⊤h¯2iφi+∑i=1nrei⊤v^n.
Since we showed thatvei,∀i=1,⋯,n, is bounded and so is ϑdi, also by construction, v^d is bounded. It follows that there exists a constant Vn such thatvn≤Vn; therefore,
(38)V̇3≤−λminK−0.5−3Vnr¯e∑i=1nrei2−K−1∑i=1nh¯1iφi⊤h¯2iφi,where r¯e=re1,⋯,ren⊤. Letting b0 be a given positive constant, we obtain for rei≥b0 the following bound for V̇3(39)V̇3≤−∑i=1nrei2−K−1∑i=1nh¯1iφi⊤h¯2iφi,since h¯1iφi⊤h¯2iφi≥0; therefore, V̇3≤0; this shows that the solution of filter (23) is uniform semipractically asymptotically stable [23]. This completes the proof.


## 4.2. Control Design for the Attitude Inner-Loop

In this stage, the last two equations of (9) are considered. The moment vector τi will be designed to globally asymptotically and locally exponentially stabilize the tracking error ψi−ψdi and the errors between the virtual controls of the roll and pitch angles and their actual values at the origin.Step 1.
Controlling the yaw angle.The desired orientationRdi can be deduced from the expression of the rotation matrix Ri∈SO (2) using its Euler angle parameterization subject to the constraint given by the specification of the desired yaw angle extracted from the translational system as shown in Figure 2. Indeed, in Figure 2, the angle ψ~d represents the desired angle associated with the vector v^dt projected on the x-y plan. Clearly, when vit matches with v^dt, the yaw rate angle ψ~iconverges to zero. Following [41], the desired yaw angle is calculated by extracting yaw rate information from the cross- and inner-product of the desired velocity v^dt and the actual velocity vit as follows:
(40)sinψ~it=v^d×viv^dvi=v∧ydiżi−v∧zdiẏi2+v∧zdiẏi−v∧ydiżi2+v∧xdiẏi−v∧ydiẋi2v^xdi2+v^ydi2+v^zdi2ẋi2+ẏi2+żi2,cosψ~it=v^d⊤viv^dvi=v^xdiẋi+v^ydiẏi+v^zdiżiv^xdi2+v^ydi2+v^zdi2ẋi2+ẏi2+żi2,tanψ~it=v^d×viv^d⊤vi=v∧ydiżi−v∧zdiẏi2+v∧zdiẏi−v∧ydiżi2+v∧xdiẏi−v∧ydiẋi2v^xdiẋi+v^ydiẏi+v^zdiżi,which implies that ψ~it=−arctan2sinψ~it,cosψ~it. Therefore, the desired yaw angle can be obtained by the following expression:
(41)ψdi=ψi−arctan2sinψ~it,cosψ~it⏟ψ~itFigure 2
Planar projection of the quadrotor’s velocity and the desired velocity generated by the guidance system.Note also that the time derivative ofψ~ can be calculated as indicated in (43). Therefore, we can write
(42)ψ̇di=ψ̇i+ψ~̇i=sinϕicosθiωyi+cosϕicosθiωzi+ψ~̇i=sinϕicosθiωyi+cosϕicosθiαψ~i+γωiϕi,θiΩei+ψ~̇i,(43)Si=v∧xdiẏi−vydiẋi2+v∧zdiẏi−v∧ydiżi2+v∧ydiżi−v∧zdiẏi2,Qi=2v^zdiẏi−v^ydiżi−żiv^̇ydi−v^ydiżi+ẏiv^̇zdi+v^zdiy¨i+2v^ydiżi−v^zdiẏiżiv^̇ydi+v^ydiz¨i−ẏiv^̇zdi−v^zdiẏi,ψ~̇it=−v^xdiẋi+v^ydiẏi+v^zdiżi2v^xdiẏi−v^ydiẋiẏiv^̇xdi+v^xdiy¨i−ẋiv^̇ydi−v^ydiẋi+Qi2Siv∧xdiẋi+v∧ydiẏi+v∧zdiżi2+v∧ydiẏi−v∧ydiẋi2+v∧ydiẏi−v∧ydiżi2+v∧ydiżi−v∧zdiẏi2−Siẋiv^̇xdi+v^xdiẋi+ẏiv^̇ydi+v^ydiẏi+żiv^̇zdi+v^zdiżiv∧xdiẋi+v∧ydiẏi+v∧zdiżi2+v∧ydiẏi−v∧ydiẋi2+v∧ydiẏi−v∧ydiżi2+v∧ydiżi−v∧zdiẏi2,where Ωei=Ωi−αi is the angular velocity error, with αi=αϕ~i,αθ~i,αψ~i⊤ yet to be determined later. γωiϕi,θi=00cosϕi/cosθi and αψ~i is the stabilizing function, which will be chosen such that it cancels out unneeded terms in (42). As such, the stabilizing function can be chosen as follows:
(44)αψ~i=cosθicosϕi−sinϕicosθiωyi−ψ~̇i+ψ̇i+κ1ψ̇i,where κ1 is a positive gain. Substituting (44) into (42) results in the following closed-loop dynamics:
(45)ψ~̇i=−κ1ψ~i+ϒωiϕi,θiΩei.Clearly, if the last term of (45) is exponentially vanishing, then the tracking error ψ~i is also exponentially stable; this can be seen by associating the quadratic Lyapunov function Vψi=1/2ψ~i2, whose time derivative along the solutions of (45) gives
(46)V̇ψi=−κ1ψ~i2+γωiϕi,θiψ~iΩei=−κ¯ψ~i2,∀ψ~i>12κ¯γωiϕi,θiΩei,where κ¯=κ1−1/2>0. Hence, the virtual control law (44) renders the closed-loop (45) system input-to-state stability (ISS) from γωiϕi,θiΩei to ψ~i. Therefore, if Ωei is globally exponentially stable GES, then so isψ~i.Remark 9.
It has been shown in [42] that overturn of the quadrotor can only be avoided if the last component of Riez=ρim⊤,m=1⋯3i.e.,ρim=cosθicosϕi is ensured to be strictly positive. This constraint further requires that eζi=ζi−ζdi (with ζi=ρi,m⊤,m=1,2,andζd=ρd,m⊤,m=1,2, is a vector that contains the first two component of Rdez) exponentially decays to zero in order to guarantee that θi⟶θd and ϕi⟶ϕd exponentially. Since ρi,3=1−ρi,12−ρi,22, it is sufficient to show through an adequate control law that ζi≤1.Remark 10.
The virtual control (44) is well defined as long as θi,ϕi≠π/2; this singularity is avoided as long as the term cosθcosϕ is strictly positive or alternatively ζi≤1.Step 2.
Controlling the pitch and roll angles.The pitch and roll angles are contained in the column vectorRiez and are denoted by the vector ζi. The time derivative of ζi along the solutions of the first equation of (9) and (5) gives the following dynamics:
(47)ζ̇i=AiΘiωi,xy=AiΘiαi,xy+AiΘi,02×1Ωei, where ωi,xy=ωi,x,ωi,y⊤,αi,xy=αϕ~i,αθ~i⊤and the matrix AiΘi can be obtained by a long but simple calculation as
(48)AiΘi=sinψicosϕi−cosψisinθisinϕicosψicosθi−cosϕicosψi−sinϕisinθisinψisinψicosθi.The matrixAiΘi is invertible, due to the fact that detAiΘi=1/cosθicosϕi, which implies that as long as the singularity (i.e., θi=ϕi=πi/2) is avoided, the matrix is nonsingular. The control objective is therefore to ensure on top of forcing the pitch and roll angles to track their immediate desired values and to also prevent them from violating the constraint that leads to singularity. In other words, the state variable ζi is required to remain in the set ζi<1 for all t≥0. To address this problem, the integral barrier Lyapunov function iBLF [29] will be employed to guarantee nonviolation of the aforementioned constraint.Motivated by the work in [29], in order to design the virtual control ωi,xy, the following integral barrier Lyapunov candidate functional is proposed:
(49)Vζiζeit,ζdt=∫0ζeiσi⊤1−σi+ζd⊤σi+ζddσi,where ζei=ζi−ζd. It can be seen that Vζiζeit,ζdt is positive definite and continuously differentiable and satisfies the decrescent condition in the set ζi<1 such that for σi=ηiζei, we have
(50)12ζei⊤ζei<Vζiζeit,ζdt<ζei⊤ζei∫01ηi1−ηiζei+ζd⊤ηiζei+ζddηi.Based on the definition in (49), the time derivative of the iBLF Vζiζeit,ζdt along the solutions of (47) yields
(51)V̇ζiζeit,ζdt=ζei⊤1−ζi⊤ζiΑiΘiαi,xy+ΑiΘi,02×1Ωei+∂Vζi∂ζd⊤ζ̇d,where ∂Vζi/∂ζd=ζei⊤/1−ζi⊤ζi−χiζi,ζdζi and χiζi,ζd=∫01ηiζei+ζd⊤1/1−ηiζei+ζd⊤ηiζei+ζddηi with the vector 1 being all components 1. We can obtain the stabilizing function αi,xy as
(52)αi,xy=AiΘi−1−κ2ζei+1−ζi⊤1−ζiχiζ̇d,where κ2 is a positive gain to be designed. Under (52), the time derivative of Vζi is given by
(53)V̇ζi=−κ2ζei⊤ζei1−ζi⊤ζi+ζei⊤1−ζi⊤ζiAiΘi,02×1Ωei,at this step according to (53); the time derivative of Vζi becomes negative definite only when Ωei exponentially converges to zero. The design of the force moment is not finished yet. One more step is required to ensure that Ωei exponentially converges to zero.Step 3.
Force moment control design.The angular velocity error dynamicsΩei based on the second equation of (9) has the following form:
(54)JiΩ̇ei=JΩ̇i−Jiα̇i=−Ωi×JiΩi+τi+ΔΩi−Jα̇i.At this stage, the control input for the force momentτi can be readily designed to compensate for the coupling terms created in the previous steps and the external perturbation to which the quadrotor is subject. To this end, let the control input be designed as follows:
(55)τi=−KiΩei+Jiα̇i+Ωi×JiΩi−ϒωiϕi,θi⊤ψ~i−AiΘi,02×1⊤ζei1−ζi⊤ζi−Δ^Ωi,Δ^Ωi=ϱΩi+KΩiΩi,ϱ̇Ωi=−KΩiϱΩi−KΩi−Ωi×JiΩi+τi+ΔΩi+KΩiϱ,where Ki is a diagonal positive gain matrix and KΩ is a symmetric and positive definite matrix.Remark 11.
The disturbance term is compensated for by an estimateΔ^Ωi which is the output of a nonlinear-observer inspired by [40]. The particularity of this type of observer is to guarantee an exponential convergence of the unknown disturbance to its true value. Furthermore, as we want the inner control loop (i.e., attitude dynamics) to be fast convergent as compared to the dynamic behavior of the outer control loop (i.e., translational dynamics), our choice is hence justified by ensuring complete cancellation of the uncertain terms.
For the convenience of stability analysis, we apply the control input (55) along with the virtual controls (52) and (44) and then the error attitude dynamics becomes
(56)ψ~̇i=−κ1ψ~i+γωiϕi,θiΩei,ζ̇ei=−κ2ζei+1−ζi⊤1−ζiχiζ⋅d+AiΘi,02×1Ωei,JΩ̇ei=−KiΩei−γωiϕi,θi⊤ψ~i−AiΘi,02×1⊤ζei1−ζi⊤ζi−Δ~Ωi,(57)Δ^Ωi=ϱΩi+KΩiJiΩi,(58)ϱ̇Ωi=−KΩiϱΩi−KΩi−Ωi×JiΩi+τi+KΩiΩi.The control design for the attitude inner-loop dynamics has been completed. We summarize the results in the following proposition.Proposition 12.
Consider the attitude dynamics (9), given a desired attitude Rdez extracted from the positioning controller (31), under the proposed attitude control (55), the error vector Ξi=ψ~i,ζei⊤,Ωei⊤,Δ~Ωi⊤⊤ of the attitude dynamics described by equations (56) and (57) is exponentially stable for any initial condition Ξit0∈Qi, where Qi=ℝ×Zi×ℝ3×ℝ3 and Zi=ζi∈ℝ3:ζi≤1. In particular, for any initial condition ζit0∈Zi, the constraint ζit<1 is satisfied ∀t>0.Proof.
The stability analysis of the attitude dynamics is conducted by considering the following Lyapunov function of the associated attitude variables:(59)VATi=Vψi+Vζi+12Ωei⊤JiΩei+δ02Δ~Ωi⊤Δ~Ωi,where δ0 is a positive constant to be determined later. Differentiating both side of (59) along the solutions of the closed-loop system consisting of (56) and (57) gives
(60)V̇ATi=V̇ψi+V̇ζi+Ωei⊤JiΩ̇ei+δ0Δ~Ωi⊤Δ~̇Ωi=−κ1ψ~i2−κ2ζei⊤ζei1−ζi⊤ζi+ζei⊤1−ζi⊤ζiAiΘi,02×1Ωei+γωiϕi,θiψ~iΩei+Ωei⊤−KiΩei−γωiϕi,θi⊤ψ~i−AiΘi,02×1⊤ζei1−ζi⊤ζi−Δ~Ωi−δ0Δ~Ωi⊤Δ^̇Ωi=−κ1ψ~i2−κ2ζei⊤ζei1−ζi⊤ζi−Ωei⊤KiΩei−Ωei⊤Δ~Ωi−δ0Δ~Ωi⊤KΩiΔ~Ωi≤−κ1ψ~i2−κ2ζei⊤ζei1−ζi⊤ζi−λminKi−12⏟k0iΩei2−δ0λminΚΩi−12⏟k1iΔ~Ωi2.
In light of the last inequality in (60) and in order to ensure that V̇ATiis negative definite, one can select the matrix Ki such that λminKi>1/2. Also, the observer gain KΩi is selected such that λminKΩi>21/δ0, whereδ0 can always be picked as a sufficiently small constant. Note, however, that δ0 is not needed in the implementation of the proposed controller.
Now referring to Lemma 2 in [29] to approximate the functional Vζi, it is shown that for all ζit≤1, the following inequality holds:
(61)Vζi≤ζei⊤ζei1−ζi⊤ζi.
Based on (61), the inequality (60) rewrites
(62)V̇ATi≤−2κ1Vψi−κ2Vζi−k0iΩei2−k1iΔ~Ωi2≤−βiVATi,where βi≔min2κ1,2κ2,2k2/Ji,2k2/δ0. It implies that
(63)Ξit≤2VATi0e−βit/2,where Ξit=ψ~i,ζi⊤,Ωei⊤,Δ~Ωi⊤⊤. This implies that the closed-loop system (56) and (57) is locally exponentially stable at the origin. Furthermore, it can be seen that ζi is bounded by an exponentially decreasing function. Then, from (61), it can be concluded that Vζi is also bounded by an exponentially decreasing function such that Vζi≤c0i with c0i≔2VATi0. On the other hand, since by construction Vζi⟶∞ as ζi⟶1, it can be inferred from the fact that Vζi is bounded for all t≥0 and the state ζit always satisfies the constraint ζit≤1, provided that ζi0≤1. This completes the proof.Remark 13.
The controllerτi in (55) involves the derivative of the virtual control αi, which appears to be difficult to calculate. So the robust differentiator [43] can be introduced to handle finite convergence of α̇i to its true value.


## 5. Control Law Synthesis: Closed-Loop System Stability Analysis

Before moving on to the stability analysis, we summarize as follows the motion control hierarchy developed in this work. First, a decentralized formation controller is designed for the open-loop translational dynamics of each individual quadrotor, which is based only on exchanging the relative position among neighbor quadrotors to guarantee safe formation maneuver and target interception. Second, the flight controller of the individual quadrotor receives the appropriate signal commands and ensures the relevant aerodynamic thrust and desired attitude necessary to achieve track-following of the leader whose only available information is its position. The resulting guidance and inner- and outer-loop control structures are illustrated in Figure3.Figure 3
Block diagram of the decentralized distance-based formation control.The total motion control system can now be analyzed by considering the complete system composed of the decentralized controller for the translational dynamics (15) and the orientation dynamics (56)–(57). The main result is presented in the following theorem:Theorem 14.
Consider a group ofn quadrotors each of which is described by the dynamics (7)–(9). If the set of quadrotors are organized at an initial minimally and infinitesimally rigid configuration, Assumptions 5 and 6 hold, and if the initial conditions for the roll and pitch angles are away from the singularity angles (i.e., π/2), then the decentralized control law and update laws consisting of (30), (31), (27), and (28) along with the constrained attitude control (55) guarantee achievement of the control objectives (13) and (14). In particular, the combined closed-loop inner- and outer-loop dynamics are forward complete, and the origin of the whole system is GAS.Proof.
The closed-loop of the overall system consists of the translational and the rotational dynamics of then quadrotor in the formation group. In previous sections, each subsystem has been treated separately to investigate its stability property, assuming the interconnection terms among the subsystems are neglected. In this proof, the closed-loop system is rewritten with their corresponding interconnected terms:
(64)ΣX:Ẋ=Fz,v¯e,Δ~x+Het,Xζ¯e,y1=v¯e,(65)Σψ~:Ξ¯̇=GΞ¯,Δ~Ω,ζ¯e,y2=ζ¯e,where X=z⊤,v¯e⊤⊤,Het,z=O3×ndiagf1,f2,⋯,fn,ζ¯e=01×n⊤,ζe1⊤,⋯,ζen⊤⊤. Systems (64) and (65) can be viewed as a cascaded decomposition of two subsystems, where the cooperative translational dynamics are viewed as perturbed nominal systems where the perturbation terms are driven by the attitude dynamics subsystems. In order to proceed with the stability analysis of the cascaded subsystems, it is first necessary to prove that the solutions of the whole systems do not escape to infinity in a finite time. One way to show that is to prove forward completeness [44] of the closed-loop systems (64) and (65).
Consider the following Lyapunov function:(66)Vf=V2+∑i=1nVATi.
Taking the time derivative ofVf along the closed-loop systems (56) and (57) yields
(67)V̇f=V̇2+∑i=1nV̇ATi=azV2+X⊤Het,Xζ¯e−∑i=1nβiVATi≤azV2+12X2Het,X2+12ζ¯e2+β¯∑i=1nVATi≤azV2+β¯+1∑i=1nVATi+12X2Het,X2≤C0Vf+σX,where β¯=minβi,i=1,⋯n; C0 is a positive constant; and σ. is a class-K∞ function. Hence, according to [44], the closed-loop systems (56) and (57) are forward complete. This result implies that we can proceed and analyze the overall system as a cascaded system. Note from (64), it can easily be verified that F0,0,0=0. Furthermore, to show steadiness of systems (64) and (65), it is sufficient to show that both conditions in Lemma 4 are satisfied. Clearly, from previous sections, we showed that the nominal system ΣX (without the coupling term) is locally exponentially stable (LES), also for the subsystem Σψ~. The fact that these two cascaded subsystems are LES is because the term Het,Xζ¯e verifies the linear growth condition (2). Indeed, it is easy to figure out that the coupling term satisfies
(68)Het,Xζ¯e≤Het,XΞ¯≤∑i=1nfi21/2⏟γzΞ¯.
Furthermore, takingV2=W, it can be easily shown that conditions (3) are satisfied. This completes the stability analysis of the cascaded subsystems.


## 6. Numerical Simulations

To show the effectiveness of the proposed constrained distance-based formation controller for a group of quadrotors, numerical simulations are carried out in this section using the MATLAB software platform. In order to make the simulation close to reality, a Gaussian white noise is added to the measured signals including distance measurements among neighboring quadrotors, local velocity, and attitude usingrand. function. In the simulation, we consider three quadrotors, where a reference trajectory is only known by the leader in the group as shown in Figure 4. In this configuration, all quadrotors are equidistantly distributed around the leader, such that dij⋆=1m. For the simulations, the quadrotor follower system parameters are selected as mi=8.2kg, the inertia moment is Ji=diag0.18,0.34,0.28kg·m2, and we assumed g=9.8m/s2. The initial conditions for the quadrotor followers are taken as p1f0=−5,−2,0⊤, p2f0=10,3,4⊤, p3f0=10,3,4⊤, vif0=0,0,0⊤, and Ωif0=0,0,0⊤ where the subscript if stands for follower vehicle i=1,⋯,3. The initial values of Θi0 are chosen such that ϕi0,θi0,ψi0⊤=4,−2,1⊤π/3i,i=1,⋯,3. The initial values chosen for the roll and pitch angles, ϕi0 and θi0,i=1,⋯,3, respectively, are to illustrate the ability of the proposed formation control design to handle large roll and pitch angles without violating the constraints that lead to singularity. The target reference trajectory and the desired heading of the leader quadrotor are chosen as follows:
(69)pTt=2sin0.3t2−2cos0.3t10−10e−0.4t,ψd=0.2e−0.5t.Figure 4
A minimally rigid undirected graph adopted for simulations.To test the robustness of the controller, external disturbances acting on the quadrotor due to wind have been added, which can be modelled by the forcesΔxi=1+0.5cos2t,1+0.5cos2t,0.5cos2t⊤ and ΔΩi=0.5+0.5sin2t,1.5+0.2sin2t,0.1sin2t⊤. The control gain parameters for the translational dynamics are k=10,k1i=5,k2i=15,Kξi=7,Kxi=15. The safety distance before collision and connectivity range specifications are set according to the performance parameters ℓ¯0=0.5, ℓ¯∞=1m, and a=0.1. The control gains for the attitude dynamics are chosen such that κ1=10,κ2=15, Ki=diag10,10,10, and KΩi=diag5,5,5.Figure5 shows the evolution of the three-dimensional position of three quadrotors following the reference trajectory. It can be seen that the proposed controller provides global asymptotic convergence towards the reference trajectory. This can also be verified in Figure 6, where it can be shown how the distance between quadrotors asymptotically converges to the prescribed fixed value. Figure 7 corroborates these results through the convergence of the attitude angles to their desired values. Clearly, the attitude angles of all quadrotors in the group aggregate on the angular value during the formation. It can also be noticed that the pitch and roll angles are within the authorized range of variation and thus do not allow the quadrotor to overturn during the maneuver. Figure 8 depicts the asymptotic convergence of the angular velocity tracking error to zero, which validates the results of Theorem 14. Finally, the thrust forces and the torques applied to the quadrotors can be seen in Figures 9 and 10. From these plots, it can be concluded that the numerical results illustrate that the leader-follower strategy has a successful tracking performance.Figure 5
3D position trajectory of the three quadrotors.Figure 6
Distance errors.Figure 7
Convergence of the backstepping state tracking errors.
(a)
2D formation tracking(b)
Velocity tracking errorsFigure 8
Angular velocity errors.Figure 9
Thrust force.Figure 10
Torque moments.


## 7. Conclusion

In this paper, a strategy for formation maintenance with target interception control of a quadrotor-type aerial vehicle was investigated. The proposed decentralized controller guaranteed formation establishment with prescribed transient and steady state performances while avoiding interagent collision and connectivity losses. A distinctive feature of the proposed approach is that the group of quadrotors does not need explicit communication exchange; it only relies on the relative position of the neighboring vehicles and local sensor measurements. This makes the proposed decentralized control protocol extremely simple and easy to implement. Furthermore, the integral barrier Lyapunov function and the backstepping technique were instrumental to the implementation of the attitude controller which ensured that the quadrotor avoids overturning during the formation maneuver. The future work is to carry out experiments to test the proposed formation control design.



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*Source: 2069631-2020-07-20.xml*