Guiin SUN, Rui ZHOU, Kun XU, Zhi WENG, Yuhng ZHANG,Zhuoning DONG, Yingxun WANG

a School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China

b State Key Laboratory of Software Development Environment, Beihang University, Beijing 100083, China

c College of Electronic Information Engineering, Inner Mongolia University, Hohhot 010021, China

KEYWORDS Complex task environments;Cooperative formation control;Fixed-wing aircraft;Guidance route;Leader-follower structure;Partial integrated formation and control

Abstract In recent years, formation control of multi-agent has been a significant research subject in the field of cooperative control. However, previous works have mainly concentrated on formation control for simple point-mass model and linear model.In contrast,this paper presents a novel cooperative algorithm for multiple air vehicles formation control, which aims to devise a control strategy based on guidance route to achieve precisely coordinated formation control for a group of fixed-wing aircraft in a complex task environment. The proposed method introduces the leader-follower structure for effective organization of the multi-agent coordination. Moreover,the Partial Integrated Formation and Control (PIFC) is adopted to design the control law for Guidance-Route based Formation Control (GRFC). Additionally, the proposed approach designs two guidance-route generation strategies for two special situations to demonstrate the effectiveness of GRFC in complex task environments.Theoretical analysis reveals that the proposed control protocol for guidance command can ensure the overall stability and tracking accuracy of the system.Numerical simulations are performed to illustrate the theoretical results, and verify that the proposed approach can achieve coordinated formation control precisely in a complex task environment.

1. Introduction

Over the past two decades,the topic of coordinated formation control has been vastly studied with application to the coordination of multiple robots,1,2air vehicles3,4, underwater vehicles5,6and spacecraft7,8for practical usage in a broad range of areas, such as surveillance,9,10searching and rescuing11,12target tracking,13,14navigation15,16and so on. The main purpose of formation control is to devise a control strategy that drives all aerial vehicles to the desired formation while guaranteeing both the tracking accuracy and the attitude synchronization. However, there are many theoretical challenges and realistic problems for formation control of a multi-agent system,such as control precision,limited information,time delays and disturbance.In addition,it is more difficult to achieve precisely cooperative formation control for a group of fixed-wing aircraft with complex dynamic characteristics.

In current literatures, various advanced methods, such as leader-follower based,17behavior-based,18consensus-based,19and virtual-structure based approaches,20have been proposed for formation control. Among these techniques, the leaderfollower based approach has been studied vastly in multiagent formation control problems. This architecture emphasizes the role of the leader in the group,who knows the global trajectory information of the group. Other members designated as followers are assumed to have access to the leader’s motion information via internal communication.Then followers are steered to maintain the desired formation with the leader by employing the control protocol. Actually, the leaderfollower structure has been discussed in coordinated formation control of the mobile robots21, surface vehicles22, underwater robots23and aerial vehicles24. However, these works were just limited to the two-dimensional space.

Formation behavior in nature, like flocking and schooling,is beneficial to animals in various ways.25Inspired by the biology, Reynolds developed an efficient behavioral model for multi-agent flocking while each agent only senses their neighbors and local environments.26Improvements to this method have recently been made by Nair et al.27and separately by Sun et al.28Ref.27developed a control strategy for the circular formation of a group of satellites. Ref.28presented an optimized artificial potential field approach for multi-UAV cooperative control. The results of both approaches are more realistic than that of Reynolds’method,because they simulate the mechanics of motion while Reynolds’ approach only uses particle models. However, the behavior-based approaches are only concerned with the formation generation for large numbers of agents with simple kinematic models. Furthermore,they could not achieve formation control precisely, driving multiple agents to a specific geometric formation according to the task demands.

More recently, consensus problems of multi-agent systems have been studied extensively.29,30Consensus means that all agents reach an identity on certain variables of interest.Ref.30presented a time-varying formation method based on consensus protocol to drive m second-order vehicles to a desired formation via harmonizing their positions and velocities. Similar works using this technique can also be found in Refs.31,32However, these literatures solve the formation control problem using point-mass models or linear dynamics models. They ignored the impact of complex dynamic characteristics of specific control objects on formation control,which may lose efficiency when these methods are applied to fixed-wing aircraft with complex dynamic characteristics.

Virtual-structure based approach is another common method for multi-agent formation control problems. In this structure, the entire formation is treated as a single structure.The virtual-structure based method can be derived in three steps.33Firstly,define the desired dynamics of the virtual structure. Secondly, translate the virtual structure’s motion into each agent’s desired motion. Finally, derive the tracking controls for each agent. The virtual-structure approach is applied to multi-agent formation control in Refs.34,35The strength of the virtual-structure based approach is that it is fairly easy to prescribe cooperative behavior for the group.33However, this method requires the formation to act as a virtual structure which limits the potential application of the virtual-structure based approach.

Most of these methods for multi-agent formation control can be classified into three categories.36One class is focused on the Position-based Formation Control (PFC). In this technique, each agent actively controls the deviation of its current position from the desired position to achieve the desired formation effectively without any interaction. However, the PFC approach is more suitable for low-speed and omnidirectional motion model. Improvements to this method have been developed by Gu et al.37and Lee et al.38They separately presented two PFC schemes for multiple fixed-wing aircraft formation control.Ref.37also conducted the flight tests to confirm the performance of the designed controller.Another class is the Displacement-based Formation Control(DFC)strategy.Each agent senses relative positions of its neighbor agents,and actively controls the deviation of its displacement from neighbors’displacements to achieve the desired formation.Relevant works can be found in Refs.39,40. Padhi et al.39developed the Partial Integrated Guidance and Control (PIGC) approach for Line-Of-Sight(LOS)based formation flight using dynamic inversion technique. Similarly, Dehghani and Menhaj40also presented a LOS based formation control algorithm of fixedwing aircraft using sliding mode control. The limitation of the DFC method is that each agent needs to know relative positions of its neighbors via interaction topology in real time.However, the agent usually has limited sensing range and it may take high cost to obtain neighbors’ displacements. The last class is characterized by controlling the inter-agent distances to achieve the desired formation which is specified by the desired distances between any two agents.41,42The agents are assumed to sense relative positions of own neighbor agents with respect to their local coordinate system. Kownacki and Ambroziak proposed an asymmetrical local potential field to accommodate the formation control of multiple fixed-wing aircraft.41The main advantage of the distance-based approach is that each agent requires less global information compared with the PFC and DFC. However, this method could not achieve precise formation control and the majority of the existing researches focus on single-integrator modeled agents.

In recent years, cooperative formation control for complex task environment has been attracting growing interest,such as environmental obstacle avoidance and communication limitation. The common methods for obstacle avoidance are geometry-based,43consensus-based44and potential-based,45methods. Zhang et al. addressed the collision avoidance problem for multiple fixed-wing aircraft formation control with high speed.44,45A neural-network based formation control for multi-agent system in obstacle environment was developed in Ref.46. They combined with the adaptive compensator and adaptive control gain to achieve formation control with obstacle avoidance. At the other end of the spectrum is the formation control with communication limitation, which includes limited data transmission,47communication faults,48,49and communication delays50. Lavaei et al. presented a model predictive control scheme with limited transmission for spacecraft formation.47Izadi et al.utilized decentralized receding horizon technique to address the cooperative formation problem under communication faults.48In Ref.50,researchers developed a formation control strategy for aircraft using virtual structure and adopted motion synchronization strategy to ensure the system stability in the presence of interaction delays.

The purpose of this paper is to present a novel formation control strategy based on guidance route, which aims to achieve precisely coordinated formation control for a group of fixed-wing aerial vehicles in a complex task environment.The novelty of this method is summarized as follows: (A) the Guidance-Route based Formation Control (GRFC) strategy is developed for full nonlinear six-degree-of-freedom (6-DOF)fixed-wing aircraft;(B)the idea of the Partial Integrated Formation and Control(PIFC)is integrated into the GRFC to devise the flight control law for precisely coordinated formation control; (C) two special generation schemes of guidance route are developed to demonstrate the effectiveness of GRFC in the presence of communication failure and environmental obstacle avoidance.

The rest of paper is organized as follows. Section 2 describes the problem formulation of GRFC strategy.Section 3 presents the preliminaries to the nonlinear motion model of 6-DOF fixed-wing aircraft.In Section 4,the GRFC is developed for multiple aerial vehicles coordinated formation control.Some numerical simulations of the proposed strategy are discussed in Section 5. Finally, the conclusions are made and future works are presented in Section 6.

2. Problem formulation

This section makes some basic assumptions and gives a general statement of PFC and GRFC.

2.1. Basic assumptions

In this paper, a new cooperative control strategy is proposed for multiple air vehicles formation flying.The following conditions are assumed to describe the coordinated formation control: (A) The information exchange among aircraft is undirected and connected. (B) The control surfaces of the aircraft for formation flying are only aileron, rudder and tail plane. (C) Each aircraft has the same dynamic characteristics.(D)It is assumed that there is no collision when multiple aerial vehicles perform formation flying.

2.2. Traditional position-based formation control

Consider a multi-agent system with n aerial vehicles performing formation flying. In the formation, there is one leader aircraft and n-1 followers. The leader is steered to track the scheduled trajectory according to the task demands, and each follower aircraft is to track its desired position.

The traditional PFC approach is the most popular one and has been widely used in multi-agent formation control problems.The method can effectively implement formation control in practical applications. However, the traditional PFC approach is more suitable for low-speed and omnidirectional motion aircraft. Moreover, its control accuracy for formation flying of multiple high-speed fixed-wing aircraft is poor due to the existence of inherent dynamic characteristics, namely large inertia and turning radius. To address these issues, several improved methods for the standard PFC have been developed for fixed-wing aerial vehicles.37,38,50Distinguishing from the traditional PFC approach, they introduced the intermediate variables instead of position coordinates,mostly azimuth angle and elevation angle, to overcome the inherent dynamic characteristics of the fixed-wing aircraft,thus achieving the coordinated formation control fleetly and smoothly.

2.3. Guidance-route based formation control strategy

Through the aforementioned analysis and dissection, we present the GRFC strategy to solve multiple fixed-wing aircraft formation control problem. Firstly, the leader aircraft is controlled to track the scheduled trajectory according to the mission demands. Simultaneously, guidance routes are generated for each follower on the basis of desired formation shape.Afterwards, followers track their respective guidance route to maintain the desired formation. Based on the above description, the GRFC strategy can be formulated as follows.

(1) Flight path matching: each aerial vehicle is controlled to track its own guidance route without any deviation,that is to say

where Ridenotes the current flight track of the ith aircraft. It should be noted that the guidance route of the leader is the task-trajectory segment where the leader is currently located.

(2) Forward velocity matching: each air vehicle is steered towards its desired velocity, which can be described as

where Virefers to the current velocity of the ith aircraft.

(3) Collision avoidance: assume that there is no collision happening among aerial vehicles.

Remark 1. Definition 1 reveals that the GRFC can be transformed into the Guidance-Route Generation (GRG)problem and the Guidance-Route Tracking (GRT) problem.GRG problem can be regarded as the route planning at the task decision-making level and GRT problem indicates the guidance-route tracking at the flight control level. The benefit of this construction is that GRFC can make full use of the advantages of route planning to effectively achieve formation control for complex tasks, including formation keeping,obstacle avoidance, formations rendezvous, formation reconfiguration of complex environment and coordinated formation control under time or space constraints. Simultaneously,GRFC introduces the idea of PIFC to design the flight control law for precise formation control. When the guidance route is generated for each aircraft, GRFC utilizes error-free tracking of the route-tracking controller to implement precisely coordinated formation control for a group of fixed-wing aircraft.

Remark 2. It can be found from Definition 1 that the GRFC is actually a collaborative control scheme, that is, the aircraft in the formation cooperate together to achieve their common goal, the desired formation. Each aerial vehicle acquires the nearest aircraft states by communicating with the adjacent aircraft, and calculates its own desired position on the basis of these states and formation shape. Then, each aircraft tracks its guidance route generated by its desired location and current states,to maintain the desired formation.It should distinguish between the guidance route and traditional route. The guidance route is dynamically generated in real time through the interaction with the nearest aircraft, aiming to guide the aircraft to the desired position while satisfying the desired heading and velocity requirements. In this paper, the desired formation shape is actually pre-determined. Certainly, the GRFC is also suitable for the case of non-preset formation shape. The main task of GRFC is to accurately control a group of fixed-wing aircraft to the given formation shape,and the GRFC control process is independent of how the desired formation is given. This implies that regardless of whether the desired formation is pre-determined or dynamically generated, the guidance route can be generated as long as the desired position can be given.

The proposed GRFC is also suitable for distributed cooperative formation control algorithm because the GRG and GRT of each aircraft are calculated and controlled independently.In addition, each follower aircraft can simply communicate with the nearest aircraft to obtain the nearest aircraft states, and generate its own desired position on the basis of these states and formation shape.Then,each aerial vehicle tracks its guidance route generated by its desired location and current states,to maintain the desired formation. Fig. 3 illustrates the schematic diagram of the distributed guidance route generation.Let φibe the heading angle of the ith follower and φLrefer to the leader’s heading angle.Assume that t0,t1and t2are three different moments, and the interval is Δt. At time t0, the follower 1 acquires the leader states through communicating with leader-aircraft, and then calculates its desired position and guidance route.In the same way,the follower 2 communicates with the nearest aircraft, follower 1, to obtain the follower 1 states, and generates its desired position and guidance route.It should be emphasized that the purpose of this paper is to present a novel formation control strategy based on guidance route.For convenience,when doing the simulation,we suppose that all follower aircraft can obtain the leader status,that is,the expected position of each follower is generated by leader state and formation shape. In contrast, the desired location of the follower is calculated according to the nearest aircraft states and formation shape, in the distributed formation control.The distinction is that the aircraft for states reference is different when generating the desired position for each follower.But the guidance-route generation strategy and the formation control process are exactly the same.

Fig. 2 Schematic diagram of guidance-route based formation control.

Fig. 3 Schematic diagram of distributed guidance route generation.

2.4. Control law of guidance-route based formation flying

Another sub-problem of GRFC is the design of formation control law. As depicted in Fig. 4, most of the literatures on formation flying concentrate on three-loop structure, namely Formation Guidance Loop (FGL), Flight Control Loop(FCL) and Attitude Stabilization Loop (ASL).39First, the guidance commands for formation flying are generated in FGL. Then, FCL converts the guidance commands to body rates and throttle setting. Finally, ASL tracks these instructions by generating necessary control surface deflections.

In general,the efficiency of ASL is higher and its response is faster because the ASL is directly controlled by aerodynamical moments and its time constant is smaller. The FCL describes the attitude change of the aerial vehicle. The states of FCL change slowly due to the influences of angular velocity and thrust. In contrast, FGL has a larger time constant, which describes the space motion of the aircraft.In addition to being controlled by aerodynamical moments and thrust, the FGL is also affected by the attitude change. Therefore, the change in states of FGL is the slowest. Actually, the entire control process can be divided into the inner loop (ASL), the outer loop(FCL)and the outermost loop(FGL).Obviously,such a structure introduces time lags among various loops.However,as to time critical applications, such as quick formation and close formation, the three-loop design is usually inappropriate. In addition,these loops are decoupled in the traditional structure and thus it needs excessive design iterations to tune and optimize each loop so that overall system optimality is obtained.39

Fig. 4 Traditional formation control structure.

To address these issues, the Integrated Guidance and Control(IGC)has been proposed for multiple missiles cooperative control,which attempt to integrate these loops in a single loop to reduce the overall loop delay.Therefore,there is no need for IGC to devise and optimize each subsystem separately. However, the IGC ignores the inherent characteristics of the full 6-DOF aircraft, and does not fully utilize the intrinsic timescale separation among three loops. It will result in a small guidance command which may generate excessive moments since the change of attitude component is typically much faster than the one of guidance component. Hence, the control law designed by IGC may only be applicable to guidance control of a specific scenario, which is undesirable.

The difficulty can be overcome by retaining the feature of timescale separation in IGC. In other words, it is a combination of the traditional structure and the IGC framework,namely PIFC. As shown by the dashed box in Fig. 4, the new two-loop PIFC structure fuses the FCL and ASL into a single loop, in which the formation guidance is performed in the slower outer loop and its output behaves as the reference signal for tracking in the inner loop. The PIFC retains the advantage of IGC,that is,minimalizing the overall time delays and reducing the excessive design iterations.Meanwhile,PIFC also reserves the inherent timescale separation of the full 6-DOF aircraft, thus ensuring the inner loop to track the guidance commands generated by outer loop fleetly and stably.Taking into account the essential characteristics of GRFC,the control process of formation flying in this paper is actually the tracking control of the guidance route.Thus,the main role of the outer loop is to track the guidance route and the primary function of the inner loop is to guarantee both stable flying of the aircraft and accurate tracking for reference commands. The detailed presentation of formation control law can refer to Sections 4.2 and 4.3.

3. Preliminaries

In this section, the nonlinear motion model of aircraft is described to support the proposed control algorithm.

3.1. Notations

For the convenience of description, we make the following symbolic comments as shown in Table 1.

3.2. Basic graph theory

It is convenient to use the graph theory to model the internal information exchange of a multi-agent system. Specifically,aerial vehicles can be represented as the nodes of the graph,and the information exchange among aircraft can be denoted by edges of the graph.

In general,the interaction topology among aerial vehicles is naturally modeled by the directed and undirected graph. In this paper, we use undirected graph to describe the internal communication among aircraft. An undirected graphcan be defined as a 2-tuple, that is,, whereis a finite nonempty set of nodes andis an edge set of unor-dered pair of nodes. It is assumed that there is no self-edge,that is,. The set of neighbor nodes of viis

Table 1 Symbolic description in this paper.

Fig. 5 Some related variables defined with reference to earthsurface inertial coordinate frame.

3.3. Nonlinear motion model

In this paper, nonlinear 6-DOF aircraft equations are used to describe the motion of air vehicles for formation flying. The relevant variables are defined with respect to the earthsurface inertial coordinate frame Og-xgygzg,which are shown in Fig. 5. O-xayazais the wind coordinate frame. O-xyz is the aircraft-body coordinate frame.

To facilitate the calculation, we have made the linear approximation of the aerodynamic force and moment coefficients used in Eqs. (3) and (4), which can be defined as

Fig. 7 Schematic diagram of switching process between two route-generation schemes.

The S-GRG scenario is given in the following expression as

The Fig. 8(a) shows the schematic of S-GRG scheme. On the basis of the leader state and desired formation, the guidance route is generated for follower aircraft via Eq. (11). The direction of guidance route Rcis consistent with φL.The starting point psof Rcis generated along φLat a distance of ξ backward fromand the ending point peis produced along φLat a distance of ξ forward fromDistance ξ is proportional to the velocity VFto ensure that the aircraft is always on guidance route Rc.Then follower aircraft are controlled to maintain the desired formation via formation control law. Rccan be changed by the factor k during the control process as depicted in the Fig. 8(b).

Fig.8 Schematic diagram and generation process of short-range generation method.

Fig. 9 Schematic diagram and generation process of long-range generation method.

where φR=φL+σ refers to the heading angle of Rc; ζ is the distance between psand pe, which ensures that the aircraft is always on guidance route Rc. Fig. 9 shows the schematic diagram and generation process of the L-GRG scheme. According to the leader state and desired formation, Rcis produced via Eq. (12) when the follower is far from its desired position.The direction of Rcis determined by p*Fand pF. The starting point psof Rcis generated along φFat a distance of ζ backward from pF. The ending point peis the desired position p*F.The Fig. 9(b) describes the generation process of guidance route during long- range formation flight.

In general, the guidance route generated by Eq. (11) is called Parallel Guidance Route (PGR), and the one generated via Eq. (12) is named as Orientation Guidance Route (OGR).PGR’s heading angle is always in the same direction as the leader heading, and the direction of OGR always points to the expected position of the aircraft.

4.1.2. Guidance-route generation for communication loss

When a certain aircraft in the formation fails and its communication is lost, the faulty aircraft threatens the safety of the entire formation. Therefore, an appropriate separation strategy is needed to move it out of the formation. After the fault vehicle is disengaged from the formation, there will be a position vacancy in the formation, which requires autonomous reconstruction of the formation shape in accordance with a certain reconstruction strategy.If the faulty aircraft communication is restored, it can be re-entered into the formation.The aerial vehicles in the formation are usually at the same level,and hence the fault aircraft can be removed from the current formation using a reduced altitude strategy. The guidance route can be defined as

Fig. 10 Schematic diagram of route generation strategy for communication loss.

where Δh is the safe altitude of the failed aircraft and φRsatisfies φR=φF. Fig. 10 illustrates the schematic of the reduced altitude strategy for communication loss. After the faulty one descends to the specified altitude, the current speed and heading are maintained.It should be noted that the flight height of the entire formation is much larger than the safe altitude Δh.In addition,our research focus is on GRG.For team reconstruction, we simply adopt the principle of proximity to fill vacant positions of the formation in turn.

4.1.3. Guidance-route generation for obstacle avoidance

Environmental barriers are often encountered when groups are formed in complex environments.We suppose that there is such an obstacle which is not on the scheduled route of the leader,but it will affect the flight of other followers. For instance, a group of aerial vehicles pass through a canyon zone.Certainly,if the obstacle is on the scheduled route of the leader,the route planning algorithm can be triggered to re-plan the route for leader. Thus,the guidance route is in the following form:

where Rsrefers to the safe range of obstacle;Δd is the distance between aircraft and barrier center;e indicates the natural constant.c will be limited to between 0 and 1 by limiting measures.λ can change the height of the guidance route. The factor λ is formulated as

Fig. 11 Schematic diagram of route generation strategy for obstacle avoidance and altitude curves of obstacle avoidance process.

where m is the number of followers between leader and current aircraft and μ is a mark.If the aircraft is on the left of the leader, μ=-1, otherwise μ=+1. In order to reduce the unnecessary obstacle avoidance maneuver of the aircraft, the obstacle avoidance maneuver is only performed when the air vehicle approaches the obstacle to a certain extent.The obstacle avoidance process is depicted in Fig. 11. We record four statuses at different moments. It is supposed that a formation encounters the obstacles during formation flying. When the original formation cannot be ensured to pass the obstacle area,the aerial vehicles threatened by the obstacle will generate evasive guidance route via Eq. (14) to make the aircraft avoid obstacles. After crossing the obstacles, the formation will spontaneously return to the original formation.

4.2. Generation of formation control instructions

The control structure diagram of GRFC is illustrated in Fig.12.The picture is mainly composed of three parts,including guidance-route generator, guidance commands generation and nonlinear six-degree-of-freedom aircraft model. It can be seen from the figure that the GRFC consists of two parts,GRG generator and GRT controller. In this paper, the idea of the PIFC is integrated into the GRFC to devise the GRT controller for precisely coordinated formation control. Compared with the conventional three-loop structure for formation flying,the PIFC aims to construct a new two-loop structure by fusing the FCL and ASL into a single loop.First,the guidance commands for formation flying are generated in the slower outer loop according to the guidance route. Then the inner loop tracks these commands to guarantee both stable flying of the aircraft and accurate tracking for reference commands.The advantage of the PIFC is that it retains the inherent timescale separation of the full 6-DOF aircraft while minimalizing the overall time delays and reducing the excessive design iterations.the characteristics of the aircraft are identical.There is one leader aircraft in each formation and each leader tracks a straight task route.Moreover, these two cases use the same GRT control law. The only difference is the generation strategy of formation control instructions, one based on PFC and the other based on GRFC.

It can be found that the formation flight tracks by using PFC scheme are undulatory. It is obvious that each aircraft in the formation cannot converge to their desired positions.It will cause that the entire formation cannot form the expected formation. In contrast, the 3D tracks by distributed GRFC scheme illustrate that the proposed strategy can enable each air vehicle to track the desired position fast, reposefully and without error. Fig. 15 shows the histories of the control instructions. In Case 2, the velocity command of each aircraft can converge to the expected velocity. In addition, attitude control instructions, normal overload and roll angle can decrease to zero. However, the altitude control commands in Case 1 are fluctuating. Furthermore, there is an error between the flight velocity of each aircraft and its expected velocity.This means that the traditional PFC method is not suitable for coordinated formation control of fixed-wing aircraft with complex aerodynamic characteristics.

To further illustrate the effectiveness of the proposed algorithm, each formation is allowed to fly along a spiral route to determine if it can achieve precise formation control. As depicted in Fig. 16, each formation is flying in a wedgeshaped formation.The scheduled trajectory of the leader is spiral route. The routes AB and CD are straight lines, and the route BC is spiral line. Obviously, the formation controlled by GRFC strategy can well maintain the desired formation in flight, while the other formation controlled by PFC scheme does not converge to the expected formation.Fig.17 shows the histories of the control instructions. By comparing with the formation control commands of Case 1, it is found that the control commands of Case 2 all produce a smaller range,and finally can converge. Through the above discussion, the proposed distributed GRFC strategy can effectively implement cooperative formation control for multiple fixed-wing aircraft with complex dynamic characteristics.

Fig. 16 Three-dimensional flight paths of formations in traditional PFC and distributed GRFC strategies.

5.2. Experiments for expansibility description

To justify the expansibility of the proposed formation scheme,this part presents some simulation results with some different kinds of formation control situations.We consider the following cases where the formations have different numbers and shapes. Four aerial vehicles fly in a wedge-shaped formation in Case 1. Six aircraft perform formation flying for a wedge shape in Case 2. Case 3 shows that nine air vehicles are flying in the shape of square formation.For formation flying,the spiral route can well test the performance of the formation control algorithm because its height and heading have been changing. Therefore, we use the spiral route as the tracking trajectory for each formation.

The numerical results by using the distributed GRFC strategy are illustrated in Fig.18.The result of each case shows two flight paths, the Two-Dimensional (2D) and 3D tracks. As shown in Fig. 18, it is found that the proposed method can achieve precise formation control for different formation shapes and different numbers of members. It demonstrates that the proposed GRFC approach is feasible to solve the cooperative formation control of multiple fixed-wing aircraft with complex dynamic characteristics.

Fig. 18 Numerical results by using distributed GRFC strategy.

Through the theoretical analysis and actual experiments of the proposed formation method, it is found that the distributed GRFC scheme can realize the formation control of any given formation shape. Above all, the proposed GRFC is a distributed formation control algorithm. The GRG and GRT of each aircraft are calculated and controlled independently.Therefore,the formation control of any number of aircraft can be achieved as long as each follower aircraft is able to obtain the current position of the leader.It shows that the proposed GRFC strategy has good scalability.

Fig. 19 Three-dimensional flight paths of formation when suffering communication loss.

Fig. 20 Flight states of aerial vehicles in formation when suffering communication loss.

Fig. 21 Three-dimensional flight paths of formation when suffering environmental obstacles. Orange cylinders indicate environmental obstacles.

Fig. 22 Flight states of aerial vehicles in formation when suffering environmental obstacles.

5.3. Experiments for effectiveness demonstration

In order to demonstrate the effectiveness of the proposed GRFC in a complex task environment, we have specially devised two generation schemes of guidance route for communication loss and environmental obstacle avoidance.

Fig. 19 describes a scenario in which a follower aircraft encounters communication loss. The histories of the flight states of aircraft are shown in Fig.20.Initially,a group of aerial vehicles perform formation flying in a wedge-shaped formation. During the flight, the follower 1 suffers communication loss.Then the faulty aircraft is lowered to the specified altitude difference (Δh=300 m). After the faulty one descends to the specified altitude, the current speed and heading are maintained. If the faulty aircraft communication is restored, it can be re-entered into the formation. It can be seen from the simulation diagrams that the detachment and re-entry of the faulty aircraft can be achieved by guidance-route based scheme to avoid its threat to the entire formation.When the faulty aircraft detaches or re-enters the formation, a reconstruction algorithm is needed to reconfigure the formation. Since the focus of this paper is on formation control, the formation reconstruction algorithm will not be described here.

The numerical result of obstacle avoidance is illustrated in Fig.21.The histories of the flight states of aircraft are depicted in Fig.22.It can be seen from the figure that six aircraft fly in a rank formation. During the flight, the formation suffers two environmental obstacles. When the original formation cannot be ensured to pass the obstacle area,the aerial vehicles threatened by the obstacle will generate evasive guidance route to enable the aircraft to avoid obstacles.After crossing the obstacles, the formation will spontaneously return to the original formation.Through the above discussion,it can be found that the guidance-route based strategy can implement cooperative formation control of multiple fixed-wing aircraft in a complex task environment.

6. Conclusions

This paper presents a new solution for cooperative formation control of multiple fixed-wing aircraft in a complex mission environment.

(1) The distributed GRFC strategy can implement coordinated formation control of a group of six-degree-offreedom aircraft with complex dynamic characteristics.The GRFC scheme can enable the fixed-wing aircraft to form the desired formation promptly, accurately and smoothly.

(2) The proposed GRFC can be transformed into the GRG problem and the GRT problem. The benefit of this structure is that GRFC can make full use of the advantages of route planning to effectively achieve formation control in a complex task environment.

(3) The idea of PIFC is introduced to devise the control law of the GRT controller for precisely cooperative formation control. In addition, we have proven that the nonlinear aircraft system is asymptotically stable under the control of the proposed control method.

The future work will focus on the collision avoidance among aircraft and communication topology optimization.The environment disturbance should also be taken into account to perform the cooperative formation flying. Moreover, we will introduce the GRFC into the flocking control.Although there is generally no fixed geometry between agents in flocking flight, the desired location of each agent can be dynamically calculated or predicted. Thus, the guidance route of each agent can be generated based on its desired position and flight states.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (Nos. 61773031 and 61573042) and Graduate Innovation Practice Fund of Beihang University,China(No.YCSJ-01-201915).At the same time,the study was also funded by the State Key Laboratory of Software Development Environment, China.