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Abstract

The formation control problem for multi-agent systems has been explored in recent years. However, controlling a formation of multiple aerial vehicles in the presence of disturbances has been a challenge for control researchers. To deal with this issue, a robust adaptive formation control algorithm for a group of multiple quadcopters is proposed. A nonlinear model of the dynamics of the formation error is obtained based on a leader–follower scheme. This model considers both the relative position in the x–y plane and the relative heading angle between vehicles in the presence of uncertainties. In addition, by means of a model reference control approach, a robust adaptive formation controller is used to steer the vehicles into a formation pattern and have them maintain the formation shape. Numerical simulations demonstrate the effectiveness of the algorithm.

Keywords

Formation control robust adaptive control leader–follower quadcopter

Introduction

In recent years, the problem of formation control of quadcopters has attracted a great deal of attention in the aerial vehicle control field.^1

–6 Quadcopters have many advantages such as high manoeuvrability and reliability, economic cost, and small size. The copters can be used for several complex tasks that include search and rescue operations, risk and/or hidden zone inspection,^2
–4 aerial mapping, and military applications.^5
–7 Compared with a single quadcopter, a formation of quadcopters increases space for equipping sensors, provides higher payload capacity and larger surveillance coverage, and, as a result, can achieve more difficult tasks in a more efficient manner.^8

–11 However, controlling a formation of multiple quadcopters is posing several difficulties for researchers because of either lacking of explicit mathematic formation dynamics models or the influences of external perturbations. Overcoming these shortcomings is becoming an imperative topic as it promises a broad range of quadcopter application to be developed afterward.

Related works

Several papers relating to the quadcopter formation control problem have been published. Abbas and Wu¹¹ proposed a leader–follower formation control scheme using two controllers. One is a proportional derivative controller used to keep the formation shape, and the other, based on fuzzy logic, is used to achieve the formation pattern. Simulations demonstrated the effectiveness of this algorithm in an environment without uncertainty. In the work of Hua et al.,¹² another leader–follower formation control scheme was proposed for a group of quadcopters. The authors of this approach used a prescribed performance control method to obtain both the formation pattern and formation trajectory. Another approach was proposed Wua et al.,¹³ employing a proportional-integral-derivative (PID) controller and a sliding mode controller (SMC). The authors solved the formation control problem of a team of quadcopters in a simulation environment without including disturbances or communication delays. In the studies by Khaled and Youmin,¹⁴ Mu et al.,¹⁵ and Mercado et al.,¹⁶ the SMC method was also used to solve the leader–follower formation flight problem of multiple autonomous vehicles. This method can provide fast responses of the system. However, a main disadvantage of this method is the chattering phenomenon of control input which can cause the follower to twitch and decrease the lifetime of the entire system.

The leader–follower scheme was also applied in the approaches presented in the study by Abas et al.¹⁷ for solving the circular motion problem of quadcopter formation. An approach presented in the study by Li and Liu¹⁸ described a synchronized position tracking controller based on a proportional-integral (PI) control law for two unmanned aerial vehicles (UAVs). By synchronizing the agents’ positions, the controller preserves the formation shape. In the work of Abdessameud et al.,¹⁹ a distributed control algorithm was obtained to solve the problem of communication delays in a leader–follower formation. In the work of Turpin et al.,²⁰ another nonlinear distributed controller for an aggressive formation of micro-quadcopters was presented. Bayezit and Fidan²¹ proposed a distributed cohesive motion control algorithm for maintaining the three dimensional (3D) formation geometry of autonomous vehicles, whereas in the study by Lee²² and Zhao and Go,²³ the authors utilized a back-stepping technique to obtain a distributed control scheme for the quadcopter formation control problem. In the work of Estevez and Grana,²⁴ a model reference adaptive control algorithm was proposed for a team of three quadcopters. By online tuning the controller gains, this algorithm allows the system to adapt to unexpected perturbations. However, this approach focuses on only wind disturbances while general disturbances were not addressed.

The references indicate several studies have discussed the flight formation of multiple UAVs in theory, simulation, and flight tests. In addition, the leader–follower approach appears in the literature as the most common method used for the formation of flight control of multicopters. Most of the previous studies are only capable of solving the formation control problem without considering the presence of uncertainties, with only very few being robust against perturbations.

Main contributions

In this article, we propose a robust adaptive formation control (RAFC) for quadcopters based on a leader–follower scheme. A nonlinear model of the formation error is obtained based on a leader–follower scheme. This model considers both the relative position in the x–y plane and the relative heading angle between the vehicles in the presence of uncertainties. In addition, using a model reference control approach, a RAFC is applied to drive the vehicles to achieve the formation pattern and maintain the formation shape. Unlike the quadcopter formation control algorithms in some previous studies,^23,25,26 which utilized model predictive control or $H \infty$ robust control, our algorithm requires less computational resources and, as a result, reduces system time delays and the sampling time of the overall controller by applying the model reference control method. The full conventional control scheme of each agent is decoupled into an outer-loop and an inner-loop controller. The proposed algorithm is applied to the outer-loop of position and heading control to generate the referenced translational velocity and heading angular rate. These references are fed into the inner-loop where we use a PID control law to control the velocity and angular rate. Several numerical simulations were carried out to demonstrate the effectiveness of the proposed method.

Organization

The remainder of this article is organized as follows. The second section describes some preliminaries and the problem statement. The third section presents the proposed RAFC algorithm. Numerical simulation results are given in the fourth section, which is followed by conclusions.

Preliminaries and problem statement

Assume that there are n quadcopters in the considered system. Compared with the attitude dynamics, the trajectory dynamics of each quadcopter have much larger time constants, which allows the overall controller can be decoupled into inner-loop and outer-loop corresponding to the quadcopter dynamics and the formation error dynamics, respectively.^23,27,28 In this section, we briefly present the quadcopter dynamics model, the formation error dynamics model, the control objective, and the multi-loop controller’s scheme.

Notation 1: Let $ϕ, θ$ , and ψ denote the three Euler angles roll, pitch, and yaw, respectively. Here, $| ϕ | < π / 2$ , $| θ | < π / 2$ , and $| ψ | \leq π$ . Symbols x, y, and z, respectively, represent the position of the quadcopter along the x, y, and z axes in the earth-fixed reference frame. Let I_x , I_y , and I_z denote the quadcopter’s moments of inertia along the x, y, and z axes, respectively; m represents the mass, l denotes the vehicle’s arm length, and g denotes the gravitational acceleration. Let $ℝ^{m \times n}$ denotes the set of $m \times n$ real matrices. Let $I_{n} \in ℝ^{n \times n}$ denote the n-dimensional identity matrix.

Quadcopter dynamics model

The quadcopter dynamics model was proposed and demonstrated in many previous studies.^29

–33 In this subsection, the dynamics model of the quadcopter is briefly described as follows

{\begin{cases} \ddot{x} = \frac{1}{m} (c o s ϕ s i n θ c o s ψ + s i n ϕ sin ψ) u_{z} \\ \ddot{y} = \frac{1}{m} (c o s ϕ s i n θ sin ψ - s i n ϕ cos ψ) u_{z} \\ \ddot{z} = \frac{1}{m} (c o s ϕ c o s θ) u_{z} - g \\ \ddot{ϕ} = \frac{I_{y} - I_{z}}{I_{x}} \dot{θ} \dot{ψ} + \frac{1}{I_{x}} u_{ϕ} \\ \ddot{θ} = \frac{I_{z} - I_{x}}{I_{y}} \dot{ϕ} \dot{ψ} + \frac{1}{I_{y}} u_{θ} \\ \ddot{ψ} = \frac{I_{x} - I_{y}}{I_{z}} \dot{ϕ} \dot{θ} + \frac{1}{I_{z}} u_{ψ} \end{cases}

where u_z is the total lift applied to the quadcopter in the z direction in the body-fixed frame. $u_{θ}, u_{ϕ}$ , and $u_{ψ}$ represent the torques applied to the quadcopter in the θ, ϕ, and ψ directions, respectively.

The thrust F_i generated by each motor always points upward, parallel to the motor’s rotation axis. When operating, each motor produces a torque $τ_{i}$ (Figure 1). The relations between the thrusts and torques are as follows^29

–32

\begin{array}{l} u_{z} = F_{1} + F_{2} + F_{3} + F_{4} \\ u_{θ} = l (F_{1} - F_{3}) \\ u_{ϕ} = l (F_{2} - F_{4}) \\ u_{ψ} = τ_{1} + τ_{3} - τ_{2} - τ_{4} \end{array}

Figure 1.

Quadcopter configuration.

Formation error dynamics model

As mentioned above, since the trajectory dynamics of each agent is slower than the attitude dynamics, the formation error dynamics model, which is directly obtained from the trajectory information, can be examined separately from the quadcopter dynamics model.^23,27,28 Before moving forward, the following notations are introduced.

Notation 2: Let $Δ_{i}^{d}$ and $Δ_{i}$ denote, respectively, the desired and actual distance between the ith quadcopter and its leader the jth quadcopter $(i, j = 1, ..., n)$ (Figure 2). $Δ_{i, x}^{d}, Δ_{i, y}^{d}$ and $Δ_{i, x}, Δ_{i, y}$ are the projections of $Δ_{i}^{d}$ and $Δ_{i}$ on the leader’s body-fixed x and y axes, respectively. $e_{i, x} = Δ_{i, x}^{d} - Δ_{i, x}$ and $e_{i, y} = Δ_{i, y}^{d} - Δ_{i, y}$ represent the distance error along the leader’s body-fixed x and y axes, respectively. The symbols $ψ_{i}^{d}$ and $ψ_{i}$ denote the desired and actual heading angle, respectively, of the ith quadcopter. $e_{i, ψ} = ψ_{i} - ψ_{i}^{d}$ represents the orientation error. $e_{i} = {[e_{i, x}, e_{i, y}, e_{i, ψ}]}^{T}$ denotes the formation error. $(x_{i}, y_{i})$ is the ith quadcopter’s position in the earth-fixed frame E. $v_{i, x}$ and $v_{i, y}$ are the velocity components of the ith quadcopter along the x and y directions in the body frame B.

Figure 2.

Leader–follower configuration.

The actual distance along the leader’s body-fixed x and y axes is calculated as follows¹⁶

{\begin{cases} Δ_{i, x} = (x_{i} - x_{j}) cos (ψ_{i}) + (y_{i} - y_{j}) sin (ψ_{i}) \\ Δ_{i, y} = - (x_{i} - x_{j}) sin (ψ_{i}) + (y_{i} - y_{j}) cos (ψ_{i}) \end{cases}

Here, it is worth noting that the values of $Δ_{i, x}^{d}, Δ_{i, y}^{d}, Δ_{i, x},$ and $Δ_{i, y}$ can be either positive or negative.

A conventional dynamics model of the formation error was obtained and verified by Wua et al.¹³ and Mercado et al.¹⁶ based on geometric analysis. In this subsection, we briefly present the nonlinear formation error dynamics model in the presence of uncertainties

{\begin{cases} {\dot{e}}_{i, x} = - (Δ_{i, y}^{d} - e_{i, y}) ω^{d} - v_{i, x} cos e_{ψ} + v_{i, y} sin e_{ψ} + v_{j, x} + ξ_{i, x} \\ {\dot{e}}_{i, y} = (Δ_{i, x}^{d} - e_{i, y}) ω^{d} - v_{i, x} sin e_{ψ} - v_{i, y} cos e_{ψ} + v_{j, y} + ξ_{i, y} \\ {\dot{e}}_{ψ} = ω_{i} - ω^{d} + ξ_{i, ψ} \end{cases}

where $ξ_{i} = [ξ_{i, x}, ξ_{i, y}, ξ_{i, ψ}]$ represents the uncertainty acting on the ith follower.

The above formation error dynamics model can be rewritten in matrix form as

{\dot{e}}_{i} = F (e_{i}) + G (e_{i}) u_{i} + ξ_{i}

where

F (e_{i}) = [\begin{matrix} e_{i, y} ω^{d} + α_{1} \\ - e_{i, y} ω^{d} + α_{2} \\ e_{ψ} \end{matrix}]

with

α_{1} = v_{j, x} - ω^{d} Δ_{i, y}^{d}

and

α_{2} = v_{j, y} + ω^{d} Δ_{i, x}^{d}

G (e_{i}) = [\begin{matrix} - cos e_{i, ψ} & sin e_{i, ψ} & 0 \\ - sin e_{i, ψ} & - cos e_{i, ψ} & 0 \\ 0 & 0 & 1 \end{matrix}]

$u_{i} = {[v_{i, x}, v_{i, y}, ω_{i}]}^{T}$ is the control action of the formation controller. These body-frame elements of the control action u_i can be transformed to the earth-fixed frame E as follows

{\begin{cases} {\dot{x}}_{i}^{s p} = v_{i, x} cos (ψ_{i}) - v_{i, y} sin (ψ_{i}) \\ {\dot{y}}_{i}^{s p} = v_{i, x} sin (ψ_{i}) + v_{i, y} cos (ψ_{i}) \\ {\dot{ψ}}_{i}^{s p} = ω_{i} \end{cases}

Control objective

This article focuses on designing a RAFC law for a group of quadcopters against uncertainties to achieve:

The desired formation pattern which is defined through the relative distances between the agents and their orientation, that is, $Δ_{i, x}^{d}$ , $Δ_{i, y}^{d}$ , and $ψ_{i}^{d}$ .

The desired formation trajectory.

In other words, the control law should cause the formation error to asymptotically converge to zero

lim_{t \to \infty} e_{i} = 0

Control scheme

The overall control scheme implemented for each quadcopter consists of an outer-loop in which the RAFC operates and an inner-loop where the velocity and attitude controllers run (Figure 3). Outputs of the outer-loop, including desired horizontal velocity and heading angular rate, thus become inputs of the inner-loop. By employing two PID controllers, the inner-loop controls the velocity and attitude of the vehicle and generates control signals $U_{i} = {[u_{i, z}, u_{i, ϕ}, u_{i, θ}, u_{i, ψ}]}^{T}$ that are sent to the actuators. In Figure 3, we denote the actual position, actual attitude, and the desired roll and pitch angles of the ith quadcopter as $P_{i},$ $Ω_{i}$ , $ϕ_{i d},$ and $θ_{i d}$ , respectively.

Figure 3.

Block diagram of the proposed algorithm.

Before moving on, the following lemma is introduced, which can be found in many text books (see [34]).

Lemma 1: For any vectors $x, y \in ℝ^{n \times 1}$ , the following holds: $t r (y x^{T}) = x^{T} y$ .

Robust adaptive formation controller

The dynamics model in expression (5) can be rewritten as follows

{\dot{e}}_{i} = Λ_{i} F (e_{i}) + G (e_{i}) u_{i}

where $Λ_{i} \in ℝ^{3 \times 3}$ is unknown and depends on the uncertainties $ξ_{i}$ .

The formation error e_i should track the state of the following reference model^1,34

{\dot{e}}_{i, m} = - A_{m} e_{i, m} + B_{m} r_{i}

where $A_{m} \in ℝ^{3 \times 3}$ is a positive definite matrix, and $B_{m} \in ℝ^{3 \times 3}$ , and $r_{i} \in ℝ^{3 \times 1}$ is any bounded reference.

It is easy to prove that $G (e_{i})$ is a non-singular matrix for anye_i . In case of no uncertainty, that is, $Λ_{i} = I_{3}$ , the formation control law becomes

u_{i} = G^{- 1} (e_{i}) [K_{1}^{*} F (e_{i}) + K_{2}^{*} e_{i} + L^{*} r_{i}]

where $K_{1}^{*} = - Λ_{i}$ , $K_{2}^{*} = - A_{m}$ , $L^{*} = B_{m}$ .

In the presence of uncertainties, the control objective in expression (7) can be achieved by applying the following formation control law

u_{i, a d} = G^{- 1} (e_{i}) [K_{1} F (e_{i}) + K_{2} e_{i} + L r_{i}]

where K ₁, K ₂, and L are the estimates of the unknown controller gains to be generated by the following adaptive laws

{\begin{cases} {\dot{K}}_{1} = - sgn (l) B_{m} P ε_{i} F^{T} (e_{i}) \\ {\dot{K}}_{2} = - sgn (l) B_{m} P ε_{i} e_{i}^{T} \\ \dot{L} = - sgn (l) B_{m} P ε_{i} r_{i}^{T} \end{cases}

where $l = 1$ if $L^{*}$ is positive definite and $l = - 1$ if $L^{*}$ is negative definite; $P \in ℝ^{3 \times 3}$ is a symmetric positive definite matrix that satisfies: $P A_{m} + A_{m}^{T} P = - Q$ , for some positive definite $Q \in ℝ^{3 \times 3}$ . The initial values of these controller gains are $K_{1} (0) = K_{1}^{*}$ , $K_{2} (0) = K_{2}^{*}$ , and $L (0) = L^{*}$ . $ε_{i}$ is defined as

ε_{i} \overset{△}{=} e_{i} - e_{i, m}

Theorem 1: If the control law is designed as expression (11) with the adaptive gains as expression (12), then the system (8) is stable and the formation errors are forced to zero.

Proof:

According to the definition of $ε_{i}$ in expression (13), if $ε_{i} \to 0$ then $e_{i} \to e_{i, m}$ . Since the reference model (9) was chosen as an asymptotic stable system, if we set the reference r_i as zero, then the error $e_{i, m}$ converges to zero as time goes. Therefore, instead of proving that $e_{i} \to 0$ , we can prove that $ε_{i} \to 0$ .

Substituting for $Λ_{i} = - K_{1}^{*}$ and adding and subtracting the term $K_{2}^{*} e_{i} + L^{*} r_{i}$ in expression (8) yields

{\dot{e}}_{i} = - K_{1}^{*} F (e_{i}) + G (e_{i}) u_{i} + K_{2}^{*} e_{i} + L^{*} r_{i} - (K_{2}^{*} e_{i} + L^{*} r_{i})

with the consideration of $K_{2}^{*} = - A_{m}$ , $L^{*} = B_{m}$ , (14) becomes

{\dot{e}}_{i} = - A_{m} e_{i} + B_{m} r_{i} + G (e_{i}) u_{i} - [K_{1}^{*} F (e_{i}) + K_{2}^{*} e_{i} + L^{*} r_{i}]

Let ${\tilde{K}}_{1} ≜ K_{1} - K_{1}^{*}$ , ${\tilde{K}}_{2} ≜ K_{2} - K_{2}^{*}$ , $\tilde{L} ≜ L - L^{*}$ , and $\in_{i} ≜ ε_{i} - {\hat{ε}}_{i}$ where ${\hat{ε}}_{i}$ is the estimation value of $ε_{i}$ . From (9), (13), and (15), ${\hat{ε}}_{i}$ can be generated as (with s is the Laplace operator)

{\hat{ε}}_{i} = {(s I_{3} + A_{m})}^{- 1} [G u_{i} - K_{1} F (e_{i}) - K_{2} e_{i} - L r_{i}]

Substituting for $u_{i} = G^{- 1} (e_{i}) [K_{1} F (e_{i}) + K_{2} e_{i} + L r_{i}]$ in (16) yields

{\hat{ε}}_{i} = {(s I_{3} + A_{m})}^{- 1} [K_{1} F (e_{i}) + K_{2} e_{i} + L r - K_{1} F (e_{i}) - K_{2} e_{i} - L r_{i}] = 0

Expression (17) implies that the error $\in_{i}$ can be simply taken to be the tracking error $ε_{i}$ , that is, $\in_{i} = ε_{i}$ . Then, we have

{\dot{\in}}_{i} = {\dot{ε}}_{i} = A_{m} ε_{i} + B_{m} L^{*} [{\tilde{K}}_{1} F (e_{i}) + {\tilde{K}}_{2} e_{i} + \tilde{L} r_{i}]

Considering the candidate Lyapunov function

V_{0} = ε_{i}^{T} P ε_{i} + t r [{\tilde{K}}_{1}^{T} Γ {\tilde{K}}_{1} + {\tilde{K}}_{2}^{T} Γ {\tilde{K}}_{2} + {\tilde{L}}^{T} Γ \tilde{L}] > 0, \forall ε_{i} \neq 0

where P is positive definite and $Γ^{- 1} = L^{*} sgn (l)$ , we have

{\dot{V}}_{0} = - ε_{i}^{T} Q ε_{i} + 2 ε_{i}^{T} P B_{m} L^{* - 1} [{\tilde{K}}_{1} F (e_{i}) + {\tilde{K}}_{2} e_{i} + \tilde{L} r_{i}] + 2 t r [{\tilde{K}}_{1}^{T} Γ {\dot{\tilde{K}}}_{1} + {\tilde{K}}_{2}^{T} Γ {\dot{\tilde{K}}}_{2} + {\tilde{L}}^{T} Γ \dot{\tilde{L}}]

Using the property of trace in Lemma 1 to manipulate expression (20) yields

\begin{array}{l} {\dot{V}}_{0} = - ε_{i}^{T} Q ε_{i} + 2 t r [{\tilde{K}}_{1}^{T} L^{* - 1} B_{m} P ε_{i} F^{T} (e_{i}) \\ + {\tilde{K}}_{2}^{T} L^{* - 1} B_{m} P ε_{i} e_{i}^{T} + {\tilde{L}}^{T} L^{* - 1} B_{m} P ε_{i} r_{i}^{T}] \\ + 2 t r [{\tilde{K}}_{1}^{T} Γ {\dot{\tilde{K}}}_{1} + {\tilde{K}}_{2}^{T} Γ {\dot{\tilde{K}}}_{2} + {\tilde{L}}^{T} Γ \dot{\tilde{L}}] \end{array}

Choosing the law to generate the controller gains as in expression (12), we have

{\dot{V}}_{0} = - ε_{i}^{T} Q ε_{i} < 0, \forall ε_{i} \neq 0

which means that the controller gains are bounded and that $ε_{i} \to 0$ as $t \to \infty$ . This completes the proof.

Remark 1: The control law in expression (11) is robust adaptive as the controller gains are estimated directly from the error with respect to time in the presence of uncertainties.

Remark 2: The proposed formation control algorithm is distributed as the controller is implemented separately on each vehicle and only receives information from its direct neighbours and itself.

Numerical simulation results and discussion

In this section, we demonstrate the effectiveness of the proposed RAFC algorithm. The quadcopter dynamics (expression (1)) and the formation error dynamics (expression (5)) are used to verify the stability and robustness of the proposed controller. The advantages of our algorithm are discussed and highlighted by comparing with some previous studies, such as non-adaptive model reference formation control (normal MRFC) and sliding mode formation control (SMFC).

Simulation assumptions

Consider the RAFC of a group of three agents. The communication topology among the agents (Figure 4) is as follows: the first follower has direct access to the information of the virtual leader whereas the second follower only has access to the information of the first follower.

Figure 4.

Communication topology and desired formation pattern.

The desired formation trajectory is generated by the virtual leader as the following two-dimensional curve: $(x^{d}, y^{d}) = (0.2 t, 5 sin (0.05 t) + 0.2 t)$ . The desired formation pattern is defined through the relative distances between the agents. Between the leader and quadcopter 1, it is $Δ_{1}^{d} = (- 0.7, - 0.5)$ , and it is $Δ_{2}^{d} = (1.4, 0)$ between the quadcopters 1 and 2. The respective initial positions and velocities of the agents are

\begin{array}{l} [x_{1} (0), y_{1} (0), ψ_{1} (0), v_{1, x} (0), v_{1, y} (0), ω_{1} (0)] \\ = [- 0.8, - 0.5, 0, 0, 0, 0] \\ [x_{2} (0), y_{2} (0), ψ_{2} (0), v_{2, x} (0), v_{2, y} (0), ω_{2} (0)] \\ = [0.8, - 0.5, 0, 0, 0, 0] \end{array}

The simulation was conducted based on several assumptions: (1) the attitude of each quadcopter is measured by an inertial navigation system (INS), and the position of the vehicle is determined by the fusion of a global positioning system receiver module and the INS and (2) the uncertainties simultaneously affect all terms of the formation error dynamics of the first quadcopter. Candidate functions of the uncertainties are introduced into the simulations as following expression (24). In practise, these uncertainties may stem from the failure of communication between agents, wind, and/or surrounding magnetic sources that typically act at low frequencies. The remaining system’s parameters used for the numerical simulations are listed in Table 1

ξ_{i} = [\begin{matrix} 0.3 sin (0.1 t) \\ 0.3 sin (0.1 t + π / 3) \\ 0.1 sin (0.1 t - π / 4) \end{matrix}]

Table 1.

System parameters used in the simulations.

Symbols	Value and unit	Description
m	1.80 kg	Total mass of the quadcopter
I_x	0.0121 kg m²	Moment of inertia along x-axis
I_y	0.0119 kg m²	Moment of inertia along y-axis
I_z	0.0223 kg m²	Moment of inertia along z-axis
L	0.23 m	Arm length of quadcopter
g	9.81 m/s²	Gravitational acceleration
A_m	[−1, 0, 0;0, −1, 0;0, 0, −1.5]	Model reference matrix
B_m	[0.3, 0, 0;0, 0.3, 0;0, 0, 0.45]	Model reference matrix

Simulation results

To clearly present the improved performance of our proposed method, the simulations were carried out with two stages: (1) formation control in the absence of uncertainties and (2) formation control in the presence of uncertainties.

Stage 1: Formation control of the quadcopters in the absence of external uncertainties.

For the first 20 s of the experiment we assume that no uncertainty is impacting the formation dynamics and expect our algorithm to perform as well as previously published methods. The quadcopters start performing the formation flight from different initial states as in expression (23).

The results from stage 1 (Figures 5 to 13) illustrate that the proposed controller is capable of driving all following quadcopters to converge to the desired formation pattern after about 5 s and then move along the formation trajectory. Figures 5 and 6 show the position performance of the controllers. It can be seen that all of the controllers are able to control the quadcopters to generally achieve the formation pattern and then maintain the formation shape. To examine these performances, closer views are presented in Figures 7 to 13. Through looking at the position errors as shown in Figures 7 and 8, we know that the position performances are nearly equal among the three types of controller. It is seen that, however, there are some small steady state errors which can be reduced by fine-tuning the inner-loop controller’s PID gains. It is worth noting that this fine-tuning procedure will not be discussed here because it is beyond the scope of this study. The convergences of the velocity and attitude of each quadcopter (Figures 9 to 13) indicate the stability of both the controllers and the entire system.

Figure 5.

Position along the x-axis of all quadcopters.

Figure 6.

Position along the y-axis of all quadcopters.

Figure 7.

Zoom of position errors along the x-axis in stage 1.

Figure 8.

Zoom of position errors along the y-axis in stage 1.

Figure 9.

Velocities along the x-axis in stage 1.

Figure 10.

Velocities along the y-axis in stage 1.

Figure 11.

The roll angles in stage 1.

Figure 12.

The pitch angles in stage 1.

Figure 13.

The yaw angles in stage 1.

The outer-loop controller’s parameters were tuned such that the position, velocity, and attitude performance of the three types of controller are nearly equal during this disturbance-free stage. As a consequence, once uncertainty starts to act, the value of our robust adaptive controller should emerge.

Stage 2: Formation control of the quadcopters in the presence of external uncertainties.

This stage covers the time period t ≥20 s, during which external uncertainties (expression (24)) are applied to the formation error dynamics of the quadcopter 1 (see Figure 4). Under the influence of perturbations, the vehicles’ yaw angles and positions are disrupted and deviations from the tracking formation trajectory or formation shape become apparent. Looking at the simulation results for stage 2 (Figures 14 to 20), it is clear that our algorithm exhibits a significantly superior performance compared to the other approaches. Also, our controller is not only capable of maintaining velocity tracking but also stabilizes the attitude of the vehicles (Figures 17 to 20).

Figure 14.

The yaw angles in stage 2.

Figure 15.

Position error along the x-axis in stage 2.

Figure 16.

Position error along the y-axis in stage 2.

Figure 17.

The velocities along x-axis in stage 2.

Figure 18.

The velocities along x-axis in stage 2.

Figure 19.

The roll angles in stage 2.

Figure 20.

The pitch angles in stage 2.

As per Table 2, the position and heading performance of the MRFC and SMFC exhibit large peak-to-peak amplitudes of oscillation. For the MRFC, it reaches about 0.6 m for x-position, 0.7 m for y-position, and 4° for heading angle. For the SMFC, it reaches about 0.65 m for x-position, 0.8 m for y-position, and 8° for heading angle. In addition, if we look at Figures 17 and 18, it is seen that the velocity performances of these two controllers also oscillate at significantly larger amplitude and higher frequencies than that of the proposed controller.

Table 2.

Comparison of peak-to-peak oscillation amplitude of control performance results in the presence of external uncertainties.

Parameters	x-position (m)	y-position (m)	Heading angle (°)
MRFC	0.6	0.7	4
SMFC	0.65	0.8	8
Proposed controller	0.1	0.15	2

MRFC: non-adaptive model reference formation control; SMFC: sliding mode formation control.

The velocity performances (Figures 17 and 18) show us that the MRFC and SMFC exhibit velocity oscillations which remain at considerable frequencies and amplitudes, and never converge to zero under the impaction of uncertainties. Meanwhile, our proposed controller performs a much improved results in which the velocity converge to a bounded range closed to the leader’s velocity. The attitude performances as shown in Figures 14, 19, and 20 also support the demonstration of the effectiveness of the proposed algorithm.

In summary, the simulation results showed that while all three tested approaches perform equally well in the absence of external uncertainties, that is, they provided both a fast response and good performance for maintaining formation shape, clear performance differences appeared once external disturbances were applied. In this study, the best performer was the proposed RAFC that is relatively simple to design and promises to maintain good control of the vehicles even in the presence of external perturbations, yielding the smallest tracking error, fastest response, and adaptation from among all three tested approaches.

Conclusions

This article presented a RAFC algorithm for a group of quadcopters. The simulation results have shown that all quadcopters are capable of converging to the desired formation pattern and track the desired formation trajectory both with and without the presence of uncertainties. We were able to demonstrate the effectiveness and robustness of the proposed method which presents a significant advancement compared to previous studies. However, we fixed the formation topology and the vehicles were only expected to perform a formation flight in the x–y plane. Our future work will focus on solving the RAFC problem in full 3D space and allow for a switching formation topology.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by The Competency Development Program for Industry Specialists of the Korean Ministry of Trade, Industry and Energy (MOTIE), operated by Korea Institute for Advancement of Technology (KIAT) (Grant No. N0002431), and The Ministry of Science and ICT (MSIT), Korea, under the Information Technology Research Center (ITRC) support program (Grant No. IITP-2018-2019-0-01423) supervised by the Institute for Information & Communications Technology Promotion (IITP).

ORCID iD

Nguyen Xuan-Mung

References

Dydek

Annaswamy

Lavretsky

. Adaptive control of quadrotor UAVs: a design trade study with flight evaluations. IEEE Trans Contr Syst Technol 2013; 21(4): 1400–1406.

Ceccarelli

Cafolla

Russo

. HeritageBot platform for service in cultural heritage frames. Int J Adv Robot Syst 2018; 15(4): 1–13.

Gohl

Burri

Omari

. Towards autonomous mine inspection. Proceedings of the 3rd international conference on applied robotics for the power industry, Foz do Iguassu, 2014, pp. 1–6.

Ceccarelli

Cafolla

Russo

. Design and construction of a demonstrative heritagebot platform. Mech Mach Sci 2018; 49: 355–362.

Tang

Xing

Karimi

. Tracking control of networked multi-agent systems under new characterizations of impulses and its applications in robotic systems. IEEE Trans Ind Electron 2016; 63(2): 1299–1307.

Wang

Xin

. Integrated optimal formation control of multiple unmanned aerial vehicles. IEEE Trans Contr Syst Technol 2013; 21(5): 1731–1744.

Abdessameud

Tayebi

Motion coordination for VTOL unmanned aerial vehicles: attitude synchronisation and formation control. London: Springer-Verlag, 2013.

Kopfstedt

Mukai

Fujita

. Control of formations of UAVs for surveillance and reconnaissance missions. IFAC Proc 2008; 41(2): 5161–5166.

Minaeian

Liu

Son

. Vision-based target detection and localization via a team of cooperative UAV and UGVs. IEEE Trans Syst Man Cybern 2016; 46(7): 1005–1016.

10.

Abbas

. Improved leader follower formation controller for multiple quadrotors based AFSA. Telcomnika 2015; 13(1): 85–92.

11.

Abbas

. Tracking Formation control for multiple quadrotors based on fuzzy logic controller and least square oriented by genetic algorithm. Open Autom Contr Syst J 2015; 7: 842–850.

12.

Hua

Chen

. Leader–follower finite-time formation control of multiple quadrotors with prescribed performance. Int J Syst Sci 2017; 48(12): 2499–2508.

13.

Wua

Chen

Liang

. Leader–follower formation control for quadrotors. IOP Conf Series: Mater Sci Eng 2017; 187: 012016.

14.

Khaled

Youmin

. Formation control of multiple quadrotors based on leader-follower method. International conference on unmanned aircraft systems (ICUAS), Denver, CO, 2015, pp. 1037–1042.

15.

Zhang

Shi

. Integral sliding mode flight controller design for a quadrotor and the application in a heterogeneous multi-agent system. IEEE Trans Ind Electron 2017; 64(2): 9389–9398.

16.

Mercado

D A

Castro

Lozano

. Quadrotors flight formation control using a leader follower approach. In: 2013 European control conference (ECC), Zurich, 2013, pp. 3858–3863.

17.

Abas

MFB

Pebrianti

Azrad

. Circular leader–follower formation control of quadrotor aerial vehicles. J Robot Mech 2012; 25; 60–71.

18.

NHM

Liu

HHT

. Formation UAV flight control using virtual structure and motion synchronization. In: 2008 American control conference, Seattle, WA, 2008, pp. 1782–1787.

19.

Abdessameud

Polushin

Tayebi

. Motion coordination of thrust-propelled underactuated vehicles with intermittent and delayed communications. Syst Control Lett 2015; 79: 15–22.

20.

Turpin

Michael

Kumar

. Trajectory design and control for aggressive formation flight with quadrotors. Autonomous Rob J 2012; 33(1–2): 143–156.

21.

Bayezit

Fidan

. Distributed cohesive motion control of flight vehicle formations. IEEE Trans Ind Electron 2013; 60(12): 5763–5772.

22.

Lee

. Distributed backstepping control of mutiple thrust-propeller vehicles on balanced graph. In: 18th IFAC World congress Milano (Italy), August 28 – September 2, 2011, pp. 8872–8877.

23.

Zhao

. Quadcopter formation flight control combining MPC and robust feedback linearization. J Frankl Inst 2014; 351(3): 1335–1355. ISSN: 0016-0032.

24.

Estevez

Grana

. Improved control of DLO transportation by a team of quadrotors. In: Ferrández

Álvarez-Sánchez

de la Paz

Toledo

Adeli

. (eds) Biomedical Applications Based on Natural and Artificial Computing. IWINAC 2017. Lecture Notes in Computer Science, vol 10338, pp. 117–126.

25.

Tiago

André

GSC

Inkyu

. Nonlinear model predictive formation control for quadcopters. IFAC-Papers OnLine 2015; 48(19): 39–44. ISSN: 2405-8963.

26.

Jasim

. Robust team formation control for quadrotors. IEEE Trans Contr Syst Technol 2018; 26(4): 1516–1523.

27.

Zhou

Kang

. Formation control for unmanned aerial vehicles with directed and switching topologies. Int J Aerospace Eng 2016; 2016: 8.

28.

Dong

Zhou

Ren

. Time-varying formation control for unmanned aerial vehicles with switching interaction topologies. Control Eng Pract 2016; 46: 26–36. ISSN: 0967-0661.

29.

Thanh

HLNN

Hong

. Quadcopter robust adaptive second order sliding mode control based on PID sliding surface. IEEE Access 2018; 6: 66850–66860.

30.

Xuan-Mung

Hong

. A multicopter ground test bed for the evaluation of attitude and position controller. Int J Eng Technol 2018; 7: 65–73.

31.

Xuan-Mung

Hong

. Improved altitude control algorithm for quadcopter unmanned aerial vehicles. Appl Sci 2019; 9: 2122.

32.

Nguyen

Hong

. Sliding mode thau observer for actuator fault diagnosis of quadcopter UAVs. Appl Sci 2018; 8: 1893.

33.

Xiong

Zheng

. Position and attitude tracking control for a quadrotor UAV. ISA Trans. 2014; 53: 725–731.

34.

Petros

Jing

. Robust adaptive control. Upper Saddle River: Prentice-Hall, 1995.

Robust adaptive formation control of quadcopters based on a leader–follower approach

Abstract

Keywords

Introduction

Related works

Main contributions

Organization

Preliminaries and problem statement

Quadcopter dynamics model

Formation error dynamics model

Control objective

Control scheme

Robust adaptive formation controller

Numerical simulation results and discussion

Simulation assumptions

Simulation results

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References