Huiming LI， Hao CHEN， Xiangke WANG

College of Intelligence Science and Technology， National University of Defense Technology， Changsha 410073， China E-mail: huiminglhm@163.com; chenhao09@nudt.edu.cn; xkwang@nudt.edu.cn

Received Feb. 28， 2021; Revision accepted Aug. 24， 2021; Crosschecked Jan. 25， 2022; Published online Mar. 29， 2022

Abstract: The affine formation tracking problem for fixed-wing unmanned aerial vehicles (UAVs) is considered in this paper， where fixed-wing UAVs are modeled as unicycle-type agents with asymmetrical speed constraints. A group of UAVs are required to generate and track a time-varying target formation obtained by affinely transforming a nominal formation. To handle this problem， a distributed control law based on stress matrix is proposed under the leader-follower control scheme. It is proved， theoretically， that followers can converge to the desired positions and achieve affine transformations while tracking diverse trajectories. Furthermore， a saturated control strategy is proposed to meet the speed constraints of fixed-wing UAVs， and numerical simulations are executed to verify the effectiveness of our proposed affine formation tracking control strategy in improving maneuverability.

Key words: Affine formation; Fixed-wing unmanned aerial vehicles; Multi-agent system

1 Introduction

With the rapid development of computation and communication capabilities，increasing attention has been paid to multi-agent systems for their huge potential value in industry， agriculture， national defense， and many other fields (Paranjape et al.，2018;Wang XK et al.， 2019). For a widely applicable system with multiple unmanned aerial vehicles(UAVs)，the ability to achieve various formations is critical.

In recent years， diverse formation control approaches have been proposed， including the leaderfollower method (Miao et al.， 2018; Chen H et al.，2021) and consensus theory (Ren et al.， 2007; Zhao SY， 2018). Oh et al. (2015) divided existing formation control approaches based on consensus theory into three categories: displacement-， distance-， and bearing-based methods. The displacement-based formation control method uses the displacement constraints between agents to design control laws，so it is difficult to track target formations based on scale or orientation changes. Similarly， the distance-based control method can track target formations with time-varying translations and orientations instead of time-varying scales. The bearing-based control method performs well in tracking formations with time-varying translations and scales，rather than orientations. Consequently，these control methods have different problems related to solving the time-varying formation tracking problem (Lin ZY et al.， 2016;Zhao SY， 2018). It is difficult to control the relative distance and position among agents without increasing the complexity of the protocol for these methods. As an alternative，an affine formation control scheme can drive the multi-agent system to track time-varying affine formation shape transformations，which enables the system to have better maneuverability in dynamic environments(Lin ZY et al.，2016;Zhao SY， 2018;Lin YJ et al.， 2021).

The affine formation control scheme is inspired by affine transformation， which can deal with geometric distortions such as translation， rotation，scale， and their combinations. Therefore， the application of an affine formation control scheme provides a new way to track and reshape formations while preserving straightness and parallelism. Affine formation control strategies were proposed with different dynamic models， including the linear time-invariant model (Xu et al.，2018，2019a;Onuoha et al.，2019a，2019b;Chen LM et al.，2020)，Euler-Lagrange model(Xu et al.， 2019b， 2019c)， and unicycle model with symmetrical constraints (Zhao SY， 2018; Xu et al.，2020). In Zhao SY (2018)， some distributed controllers were designed based on the consensus protocol for agents described by the unicycle model. The convergence of the multi-agent system was achieved and analyzed， but the tracking control problem was not considered. In Xu et al. (2020)， the affine formation tracking problem was taken into account.Agents can move along a straight line and achieve affine transformations in the process of trajectory tracking，but the inflexible trajectory tracking ability is not strong enough to meet the high-performance maneuverability requirements of fixed-wing UAVs.

In general， fixed-wing UAVs are usually described by the unicycle model， which is limited by asymmetrical input constraints. The classic model of small fixed-wing UAVs can be described by 12-dimensional， state-coupled， first-order ordinary differential nonlinear equations (Wu， 2013)， and it is too complex to be directly used in guidance control problems. In the flight process， a minimum positive linear velocity， i.e.， the stall speed， is necessary to generate sufficient lift for fixed-wing UAVs. In addition， the angular speeds have great influence on the UAVs’ turning radius. Therefore， a simplified nonholonomic agent model， i.e.， the unicycle model，is usually applied to describe fixed-wing UAV kinematics(Wang YZ et al.，2020;Zhao SL et al.，2020);its applicability has been proved in both simulations and flight experiments(Beard et al.，2014;Liu et al.，2020;Wang YZ et al.，2020;Zhao SL et al.，2020).

In this paper，we adopt the leader-follower strategy to solve the affine formation tracking control problem for the system with multiple fixed-wing UAVs. Theoretical analysis and simulation results are provided to illustrate the effectiveness of the proposed control strategy. The main contributions are as follows:

1. We propose an original distributed control law based on stress matrix to achieve the desired time-varying formation pattern and to track affine transformations along diverse trajectories，which improves the maneuverability of the fixed-wing UAV formation in dynamic environments.

2. The convergence of the distributed control law is analyzed for unicycle-type agents， which lays a foundation for the application of the affine formation control scheme in the system with multiple fixed-wing UAVs. In addition， a saturated control strategy is proposed to meet the asymmetrical speed constraints of fixed-wing UAVs，and the effectiveness is verified by numerical simulations.

2 Problem description

2.1 Notations of formations and graphs

Suppose that there arenfixed-wing UAVs in Rd，whered ≥2 andn ≥d+1. Denotepi=[xi，yi]Tthe position of theithUAV andRdnthe configuration of the whole UAV formation. Naturally， an undirected graph is used to model the communication topology among UAVs，i.e.，G=(V，E)，whereV={1，2，···，n}denotes the vertex set andE ?V ×Vdenotes the edge set. The edge (i，j)∈Emeans that theithUAV can receive information from thejthUAV. In the undirected graph， (i，j)∈E ?(j，i)∈E. The neighbors of theithUAV are defined asNi={j ∈V:(i，j)∈E}.

A formation is expressed as(G，p)by associating the system with multiple fixed-wing UAVs and the undirected graphG. Inspired by the leader-follower control strategy， suppose that there arenlUAVs playing the role of leaders and that the remainingnfUAVs are followers，wherenf=n-nl. Thus，the leader set isVl={1，2，···，nl}and the follower set is

2.2 Stress matrix

A stress is a set of scalars that are assigned to all the edges， that is，?ijfor (i， j)∈ E.?ij=?jiis reasonable for an undirected graph.For a formation (G，p)， an equilibrium stress (Connelly， 2005) is established only if the stress satisfies which can be reworded inanother form as

where “?” represents the Kronecker product andIdis the identity matrix.Ω ∈Rn×nis called the stress matrix satisfying

Obviously， the stress matrix form is similar to the Laplacian matrix， while the weight in a stress matrix，?ij，may be positive，negative，or even zero.

2.3 Affine formation control

Geometrically，affine transformation means that a vector space is transformed into another vector space through a linear transformation followed by a translation. As shown in Fig. 1， affine transformations can realize rotation，translation，scaling，shear，and their combinations.

Fig. 1 An illustration of affine transformations of a nominal configuration: (a) nominal; (b) rotation;(c) scaling; (d-f) shear

Similarly， a nominal configurationr=is given asthe affine transformation baseline of the fixed-wing UAV formation in Rd. Thus， the nominal formation is expressed as (G，r)， and the time-varying target formation for(G，p) is defined as follows:

Definition 1(Target formation) The time-varying target formation has the form of

where 1n= [1，1，···，1]Tand the variable with the superscript“*”represents the target value. The nominal configurationris constant， whileA(t)∈Rd×dandb(t)∈Rdare continuous ont. In the target formation，the desired position of theithUAV is expressed as

In fact， all the affine transformations ofrare included in the affine imageA(r) (Zhao SY， 2018)，including the time-varying target formationp*(t).In addition， fornfixed-wing UAVs in Rd， define the

which means that three non-collinear points can affinely span the two-dimensional(2D)plane. Without loss of generality， suppose that the nominal formation(G，r)can meet the following condition:

Assumption 1For nominal formation (G， r)，assume thataffinely span Rd.

Based on the leader-follower control strategy，a notion termed affine localizability is defined to make clear that sufficient and appropriate leaders can guide the entire formation to the target affine formation.

Definition 2(Affine localizability)(Zhao SY，2018)The nominal formation (G，r) is affinely localizable by the leaders if for anycan be uniquely determined bypl.

Lemma 1(Zhao SY，2018) Under Assumption 1，the nominal formation(G，r)is affinely localizable if and only if{ri}i∈Vlaffinely span Rd.

Lemma 1 means that UAVs in (G，r) can be chosen as leaders to guarantee affine localizability as long as they can affinely span Rd. Accordingly， at leastd+1 UAVs inrmust be selected as leaders to affinely span Rd.

2.4 Problem formulation

Suppose that allnfixed-wing UAVs are flying at the same altitude andd= 2. The kinematic equations of theithfixed-wing UAV are described as

wherevminandvmaxrepresent the minimum and maximum of the linear speed respectively，andωmaxis the maximum angular speed.

Remark 1In fact，Eq.(5)is widely used to study the formation control problem of multiple fixed-wing UAVs. The control of flight altitude is usually ignored in the cooperative control problem. In addition， Eq. (5) simplifies the actuator’s dynamic characteristics. Consequently， the navigation control inputs，viandωi， need to be transformed to guide the bottom flight control of fixed-wing UAVs.

whereδpf(t) represents the position error andδhf(t)represents the heading error.

Problem 1Given a team ofnfixed-wing UAVs，each of which is modeled by Eq. (5)， our objective is to design a control lawμ= [vi，ωi] for each follower to track the time-varying target positions and to achieve affine transformations，i.e.，

3 Main results

In this section， we adopt the affine formation control scheme to solve Problem 1. Define

Theorem 1Under Assumptions 1-3， the control law designed for theithfollower is established

as

where ki∈R2×2nlis a matrix consisting the 2i-1 and 2i rows of matrixAs a result，the following results hold: the followers’ tracking errors δpfand δhfcan converge to zero when vi＞0 for i ∈Vf.

Proof Control law (9)can be rewritten as

Accordingly，the closed-loop dynamics is shown as

We now prove the convergence of the proposed control law (9). Consider a Lyapunov function denoted as

Combined with Eq.(12)，the derivative of V at time t is

In conclusion，δpfandδhfwould converge to zero provided thatvi ＞0，?i ∈Vf.

Remark 2Control law (9) consists of two parts:affine formation generation and velocity matching.The first term in the brackets[]in Eq.(9)is the consensus protocol of positions， and the last two terms aim at achieving velocity matching among the leaders and followers. Therefore，the system with multiple fixed-wing UAVs can track different trajectories and achieve any time-varying affine transformations.As a result，the maneuverability in dynamic environments is improved.

In fact， the speeds of fixed-wing UAVs are bounded， as described in inequality (6). The linear speeds during flight must be positive so thatvi ＞0 fori ∈Vfis an achievable requirement. To meet the speed constraints in inequality (6)， a saturated control strategy is proposed as follows:

Define a saturation function sat(x，a，b) to limit the input values. Supposinga ＜b， the saturation function is defined as

Therefore，the proposed affine formation control law (9)can be modified as

Simulation results presented in Section 4 verify that the saturated control strategy is effective in constraining the control inputs， and that the tracking errors would like to converge to zero under control law (16). In other words， fixed-wing UAVs can achieve target affine formation while tracking different trajectories without breaking the speed constraints.

Remark 3The modified control law(16)is closely related to the physical characteristics of fixed-wing UAVs. It is meaningful but challenging to analyze the convergence of the nonlinear system with asymmetrical input constraints. The proposed saturated control strategy is a simple and practical method to limit the value range of speeds， which is inspired by Fathian et al.(2018)and Zhao SY(2018). Next，further stability analysis of the nonlinear system with speed constraints is carried out.

4 Simulations

In this section， we present simulation examples of six fixed-wing UAVs to verify the effectiveness of the proposed affine formation tracking control law (16). The nominal formation (G，r) is given in Fig. 2. In a 2D space， the first three UAVs are chosen as the leaders and the other three UAVs are the followers. The nominal configuration is set asandAccordingly，the stress matrix is calculated using the algorithm proposed in Zhao SY(2018)，and the weights are labeled at the corresponding edges in Fig.2. The speed constraints are set asvmin=10 m/s，vmax=25 m/s，andωmax=0.5 rad/s.

Fig. 2 Nominal formation of six fixed-wing UAVs

Fig. 3 A simulation example to illustrate the affine formation tracking control law (16). The leaders move along different trajectories: (a) straight line; (b) circle; (c) sine curve (L: leader; F: follower)

4.1 Trajectory tracking

In Fig. 3b， leaders move in a circle， andωc= 0.1 rad/s is set to declare the rotation of the nominal formation. The initial states of the leaders are [p1;θ1] = [0，0，0]T，Because the initial positions of the leaders are not in the nominal configuration， the scale of the target formation is different from the nominal formation. To keep the target configuration， the leaders move along different circular trajectories so that their linear speeds are not the same. Accordingly， the linear speeds of the followers cannot converge to the same constant， but the angular speeds are homogeneous， as shown in Fig. 5b. Trajectories of the six UAVs are described in Fig. 3b. It can be clearly seen that the six UAVs can form target formation and realize rotation. The distance errors in Fig. 4b converge to zero， which illustrates the generation of target formation and affine transformation. In Fig. 3c， the speeds of leaders are set asvi=15-cos(0.12t)m/s andωi= 0.1 cos(0.12t) rad/s for alli ∈Vl， which are time-varying and reasonable in reality. It is clear that the formation tracking errorδpfconverges to zero (Fig. 4c)， and that the linear speedviand angular speedωiare bounded(Fig. 5c).

Fig. 4 Tracking errors of the three following fixed-wing UAVs in the affine formation. The leaders move along different trajectories: (a) straight line; (b) circle; (c) sine curve (F: follower)

In the simulation， the speeds and trajectories of the three leaders are generated in advance. The results demonstrate that the saturated affine formation tracking control law(16)can drive UAVs to converge to the target states and to adapt to diverse trajectories， which is helpful in tracking the realtime trajectories generated from the tasks in practical applications.

To make the results more comparable， two different control laws proposed in Zhao SY (2018) and Xu et al.(2020)are applied to the formation tracking problem， where the leaders have the same speeds as shown in Fig. 3c. Fig. 6 exhibits the simulation results of the control law proposed in Zhao SY(2018)，and Fig.7 shows the simulation results of the method proposed in Xu et al.(2020). Obviously，tracking errors of the three followers cannot converge to zero and the control objectives are unachievable，proving that the proposed control law in this study has a higher adaptability.

Fig. 5 Constrained linear and angular speeds of the fixed-wing UAVs in the affine formation. The leaders move along different trajectories: (a) straight line; (b) circle; (c) sine curve (L: leader; F: follower)

Fig.6 Formation tracking simulation results under the control law proposed in Zhao SY(2018): (a)trajectories of the six fixed-wing UAVs; (b) tracking errors of the three followers (L: leader; F: follower)

Fig.7 Formation tracking simulation results under the control law proposed in Xu et al.(2020): (a)trajectories of the six fixed-wing UAVs; (b) tracking errors of the three followers (L: leader; F: follower)

4.2 Affine transformations

The second simulation example demonstrates that the control law (16) can achieve affine transformations when the multi-UAV system maneuvers，as shown in Fig. 8. As can be seen in Fig. 9， the formation keeps maneuvering while changing its configuration，including the centroid，orientation，scale，and geometric pattern. In Fig.10，the tracking error remains zero when the formation transforms its configuration affinely， except for two clockwise turnings fromt=220 s and 290 s， because the configuration of the three leaders is not affinely transformed from the nominal configuration in those two moments. It is observed that the tracking error appears but converges to zero quickly. Accordingly，the affine formation tracking control law (16) shows favorable maneuverability，which can realize affine transformation of the fixed-wing UAV formation without breaking speed constraints.

Fig. 8 A simulation example to illustrate the affine transformations of the six fixed-wing UAVs， including translation， zooming in， zooming out， and shearing

Fig. 9 Six fxied-wing UAVs with affinely transformed nominal configuration at t=0 s (a)， 110 s (b)， 150 s (c)，260 s (d)， 305 s (e)， 350 s (f)， 400 s (g)， 450 s (h)， and 530 s (i) (L: leader; F: follower)

Fig. 10 Tracking errors of the three following fixedwing UAVs in the formation during affine transformation (F: follower)

5 Conclusions

To improve the maneuverability of fixed-wing UAV formations，the unicycle model with asymmetrical input constraints was applied to study the affine formation tracking control problem in this paper.Under the affine formation control scheme， the target formations were obtained from the affine transformations of the defined nominal formation，and the distributed formation tracking control strategy was designed based on the stress matrix. Based on the leader-follower control strategy，the leaders used the predesigned formation information to fly at a desired state and to determine the target configuration; the followers were driven by the affine formation control laws to track the leaders along different trajectories and to converge to the target positions. The Lyapunov theory was applied to analyze the stability of the control laws， and a saturated control strategy was proposed to ensure that the speed constraints of fixed-wing UAVs are satisfied. Simulation results showed that the proposed control strategy can adapt well to diverse trajectories and time-varying affine formations. In the future， it is necessary to analyze the convergence of the nonlinear system with input constraints.

Contributors

Huiming LI designed the research and drafted the paper.Hao CHEN and Xiangke WANG helped organize the paper.All the authors revised and finalized the paper.

Compliance with ethics guidelines

Huiming LI， Hao CHEN， and Xiangke WANG declare that they have no conflict of interest.