Distributed formation control of multiple aerial vehicles based on guidance route

Jinyong CHEN, Rui ZHOU, Guiin SUN,*, Qingwei LI, Ning ZHANG

a School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

b Science and Technology on Information Systems Engineering Laboratory, Nanjing 210007, China

c AVIC Xi’an Flight Automatic Control Research Institute, Xi’an 710076, China

KEYWORDS Formation shape control;Fixed-wing aircraft;Guidance route;Hop-count estimation;Model predictive control

Abstract Formation control of fixed-wing aerial vehicles is an important yet rarely addressed problem because of their complex dynamics and various motion constraints,such as nonholonomic and velocity constraints.The guidance-route-based strategy has been demonstrated to be applicable to fixed-wing aircraft.However,it requires a global coordinator and there exists control lag,due to its own natures.For this reason,this paper presents a fully distributed guidance-route-based formation approach to address the aforementioned issues.First, a hop-count scheme is introduced to achieve distributed implementation, in which each aircraft chooses a neighbor with the minimum hop-count as a reference to generate its guidance route using only local information.Next, the model predictive control algorithm is employed to eliminate the control lag and achieve precise formation shape control.In addition, the stall protection and collision avoidance are also considered.Finally, three numerical simulations demonstrate that our proposed approach can implement precise formation shape control of fixed-wing aircraft in a fully distributed manner.

1.Introduction

Formation shape control has received tremendous research attention in the past few decades due to their potential applications, such as target tracking1,2, surveillance3,4, and rescue missions5,6.The single-integrator and second-integrator models have been widely considered in formation control due to their simplicity.However, these models usually cannot well approximate real dynamics because the control inputs of these models can be arbitrarily assigned regardless of the motion states.In contrast, the control input of a real aircraft may be subject to various constraints,such as nonholonomic dynamics and velocity constraints.If not handled properly, most of the existing works are difficult to extend to real objects, such as fixed-wing aerial vehicles.

Nomenclature Symbol Meaning φ roll angle nz normal overload δp throttle opening S wing area CD drag coefficient CL lift coefficient CD0 zero-lift CD Fmax maximum thrust g gravity acceleration D aerodynamic drag m mass v ground speed χ azimuth angle γ path angle ρ air density Tc control horizon Tp prediction horizon Ts sampling time Nc control step Np prediction step F engine thrust

The existing formation control approaches can be categorized according to the type of controlled variables, i.e.,position-based7,8, distance-based9,10, and bearing-based11,12approaches.In position-based approaches,each agent controls the derivation between its current position and desired position to achieve the predefined formation pattern13.This kind of method is suitable for low-speed and omnidirectional vehicles14.However,it may not be applied to fixed-wing aerial vehicles because arbitrary changes in control quantities could result in unstable flight.Different from position-based techniques,distance-based and bearing-based approaches form the desired formation by controlling the relative distances and bearings between agents, respectively.Most of the existing works using these two methods focus on the single-integrator models,whose velocity can be arbitrarily assigned15,16.Motivated by this,many researchers have studied cooperative formation control subject to the nonholonomic dynamics17–19and velocity saturation20–22.Ref.23proposed a general framework for multi-agent coordination control, and considered the nonholonomic and velocity saturation constraints.These works, however, ignore the positive-minimum velocity constraint that is unique to fixed-wing aircraft and instead limit the maximum motion velocity.Recently, several works do consider the positiveminimum velocity constraint in formation shape control.Ref.24proposed a distributed formation control approach to nonholonomic vehicles whose velocity lies between two positive constants.Ref.25also considered the positive-minimum velocity constraint when designing the formation algorithm for fixed-wing aerial vehicles.However,these works only use a single integrator to approximate the real object.

Actually,the dynamics of fixed-wing aerial vehicles is more complex than nonholonomic single integrators.First, fixedwing aircraft rely on forward speed to generate enough lift to balance their gravity.Consequently, they must maintain a sufficient velocity at all times, or they will stall and crash.Thus, immediate reverse and stop-and-wait strategies are forbidden in formation control of fixed-wing aircraft.Second,as nonholonomic vehicles that can only move forward, fixedwing aircraft cannot achieve instantaneous lateral movements.As a result, lateral and forward position errors may result in disproportional restoration efforts26.Finally, the deceleration of most fixed-wing aircraft is slow and uncontrollable because they are not equipped with deceleration devices and only lean upon air resistance to decelerate.Therefore, it is still a challenge to control a group of fixed-wing aircraft to achieve precise formation shape and maintain their velocities within the required safety range.

Our previous work14presented a guidance-route-based formation control strategy.The main idea of this strategy is to govern each aircraft to track its own guidance route instead of desired position,which is more compatible with the dynamics of fixed-wing aerial vehicles.However, there are still some problems to be addressed.First, it requires a global coordinator, that is, a global leader that can communicate with all followers.This connection topology is strict and thus possesses less robustness.Second, the guidance route is generated based on the leader position in the last period.As a result, there is hysteresis between the real and desired positions, which may result in steady state errors in formation control.Third, speed safety and collision avoidance are not taken into consideration in Ref.14.

In this paper, we propose a distributed coordination approach combined with model predictive techniques to address the aforementioned problems.The main contributions can be summarized as follows.First, we propose a distributed guidance route generation strategy to achieve the distributed control of multiple aerial vehicles formations.Each aircraft first estimates its hop-count and then generates a guidance route using the neighbor with minimum hop-count as reference.In this way, the formation control problem is decomposed into a route tracking problem for each aircraft, which is naturally distributed and well suited to the dynamics of fixed-wing aircraft.Second, a model predictive route tracking controller is developed to eliminate the control lag of the original guidance-route-based approach.We derive the prediction model from the nonlinear aircraft model and finally transform the finite-time domain optimization problem in route tracking into a quadratic programming problem.Third,we achieve precise formation shape control of multiple fixed-wing aircraft in a distributed manner.Our proposed method takes full account of characteristics unique to fixed-wing aircraft, such as complex dynamics, nonholonomic motion, and positive-minimum velocity constraint, while also providing stall protection and collision avoidance guarantees.

The rest of this paper is organized as follows.Section 2 presents the preliminaries and problem statement of our proposed method.Then in Section 3, we introduce the distributed guidance route generation strategy.Section 4 develops a model predictive controller for precise and safe formation control.Numerical simulation results and conclusions are given in Section 5 and Section 6, respectively.

2.Preliminaries and problem statement

This section presents the nonlinear motion model of fixed-wing aerial vehicles and the definition of desired formation.Moreover, a formal description of our proposed approach is given.

2.1.Nonlinear motion model

For simplicity,we refer to fixed-wing aerial vehicles as aircraft in the following paragraphs.The dynamics is described by a four-degree-of-freedom nonlinear model as follows:

where f(ξ,u ) is a nonlinear function.

2.2.Desired formation

A graphical illustration is given in Fig.1.The arrow symbol at the origin indicates that the z-axis points to the inside of the paper.The red, blue, and gray aircraft represent the informed aircraft, uninformed aircraft, and their desired positions,respectively.The red star represents the target to be tracked.

In practical applications, it is unreasonable that all aircraft know the position of the target.For this reason, we introduce two types of roles:one is the informed aircraft who knows the time-varying target to be tracked; the other is the uninformed aircraft who only perceives local information.As a result,uninformed aircraft cannot employ Eq.(5)to directly calculate their desired positions as informed aircraft do.Later in Section 3, we will specify how uninformed aircraft approximate their desired positions based on the hop-count estimation strategy, which is an important part of the distributed implementation scheme.

2.3.Guidance-route-based formation control problem

In position-based formation control, each aircraft typically controls the position error to achieve the desired formation.However, the direction of position error may change drastically as aircraft approaches its desired position.As a result,an aircraft needs to change its head orientation frequently to reduce the position error because it cannot move sideways directly, which could cause unstable formation flying.

To overcome this problem, we employ the guidance-routebased formation control strategy,in which each aircraft tracks its own guidance route instead of desired position.Compared to traditional methods, guidance-route-based formation control allows aircraft to maintain a sufficient speed for all time and smoothly eliminate the lateral distance error through minor adjustments in direction.Therefore, it is ideal for fixed-wing aircraft with nonholonomic and positiveminimum velocity constraints.See Fig.2 for an illustration.

Fig.1 Description of formation pattern.

Fig.2 Examples to explain difference between position-based and guidance-route-based formation strategies.

To make the formation control scheme practical and completely distributed, each aircraft is required to generate and track its guidance route independently and safely using available information.Therefore, the key issues of the distributed guidance-route-based formation control can be summarized in three aspects: (A) formulate a distributed guidance route generation strategy; (B) design a high-performance route tracking algorithm;(C)deal with formation flight safety issues,such as inter-vehicle collision avoidance and stall protection.

3.Distributed guidance route generation strategy

This section develops a distributed guidance-route generation strategy to get rid of the need for a global coordinator in Ref.14.First,a hop-count strategy is proposed for uninformed aircraft to choose their own reference aircraft among neighbors in a distributed manner.Then, each uninformed aircraft approximates its desired position according to the relative position with respect to the reference aircraft in the formation configuration.Next, two guidance route generation strategies based on desired position are designed for each aircraft to achieve formation gathering and formation maintenance,respectively.

3.1.Distributed hop-count estimation

Fig.3 Description of deviation variables.

where λi=1 if i is an informed aircraft and λi=0 otherwise.

In reality,an uninformed aircraft is unaware of its real hopcount, which depends on the communication topology of the formation.Therefore, we present an estimation strategy for each uninformed aircraft to approximate its hop-count by communicating with neighbors.The update formula is designed as

Theorem 1.Consider a formation of m informed aircraft and n-m uninformed aircraft.For any aircraft i ?V, its real and current estimated hop-count distribution are hiand ^hi, respectively.Using the update formula in Eq.(9), the hop-count distribution will converge to real distribution with no more than n-m+1 steps, that is ^hki=hi,?i ?V,k ≥n-m+1.

Proof.Aircraft in the formation can be divided into different sets according to their hop-counts, that i,

This completes the proof of Theorem 1.

According to Theorem 1, all aircraft can obtain their real hop-counts after exchanging information with neighbors a few times.For a formation of 10 aircraft,if the communication period is 0.01 s,the convergence time of the hop-count estimation does not exceed 0.1 s.For conciseness,in the following,we assume that each aircraft already knows its own hop-count,that is, ^hi=hi,?i ?V.

3.2.Distributed approximation of desired position

Informed aircraft can directly employ Eq.(5)to calculate their desired positions, whereas uninformed aircraft cannot due to insufficient perceived information.To this end, we propose a hop-count based strategy for uninformed aircraft to approximate their desired positions.Each uninformed aircraft chooses a neighbor with minimum hop-count as its reference aircraft,denoted as βi, that is

Proof.We continue to divide the aircraft into different sets according to their hop-counts as above.Suppose there are m informed aircraft and n-m uninformed aircraft.Since all m informed aircraft have the same zero hop-count,the formation can be divided into at most n-m+1 distinct sets, that is,

According to mathematical induction,Eq.(24)holds.Combining Eq.(23), Theorem 2 is proved.

As a result, the formation control problem is transformed into the position and azimuth control problem of each aircraft,which is completely distributed and ideal for controllers based on guidance route.

3.3.Guidance route generation

As shown in Fig.4,Aircraft 1 generates a parallel guidance based on the moving target;Aircraft 2 and Aircraft 3 generate a parallel guidance route and a gathering guidance-routebased on Aircraft 1, respectively; Aircraft 4 chooses Aircraft 2 as its reference aircraft and keeps its previous guidance route generation strategy.

The gathering guidance route is designed for aircraft to quickly form the predefined geometric pattern when they are far away from desired positions.Each aircraft flies directly to its estimated desired position at the time of arrival.As shown in Fig.5, the gathering guidance route is given by

The parallel guidance route is designed for aircraft to smoothly reach their desired positions and precisely maintain the formation pattern.Each aircraft tries to maintain the estimated desired azimuth while approaching the estimated desired position.As shown in Fig.6, the parallel guidance route is a horizontal straight segment passing through the estimated desired position and parallel to the velocity of its reference aircraft.The parallel guidance route of aircraft i is given by

Fig.4 Guidance route generation strategy.

Fig.5 Gathering guidance route generation.

where tpis a given time constant.

4.Guidance-route tracking via model predictive control

There are steady errors in Ref.14due to the control lag that followers can only obtain the leader position at last control period.To this end, we develop a guidance-route tracking controller based on Model Predictive Control (MPC).In this way,each uninformed aircraft can predict the current position of its reference aircraft to eliminate control lag, and improve formation maintenance accuracy during maneuvers through trajectory prediction.This section first introduces in detail the implementation of the MPC algorithm and then deal with input and safety constraints,including stall protection and collision avoidance.

4.1.Implementation of MPC algorithm

4.1.1.Derivation of prediction model in finite horizon

The dynamics of the aircraft in this paper is described by a four-degree-of-freedom nonlinear model.To obtain control constructions through model predictive control scheme, it requires a nonlinear program to be solved on-line at each time step if we directly use the nonlinear aircraft model as the prediction model.It will take formidable effort to find the optimal solution and aborting optimization can have unpredictable consequences for the performance27.For this reason, we use at each time step a different linear model derived from a local linearization and then employ standard linear dynamic matrix control, which can significantly reduce the computational effort.

Fig.6 Parallel guidance route generation.

where tcrepresents the control period.Informed aircraft calculate their reference state sequences using the same way based the target’s future state sequence,which is either received directly from global information or linearly extrapolated from current state.

4.1.2.Optimization problem of route tracking

Fig.7 Conversion of state variables.

The computational effort required to solve the optimization problem Eq.(49)is crucial to the implementation of the model predictive controller.As long as Hi,kis proved to be a positive semi-definite matrix, Eq.(49) is a convex quadratic programming problem, whose optimal solution is unique and easy to obtain.Since diagonal elements of diagonal matrices K and M are non-negative, they are positive semi-definite, that is

Therefore, Hi,kis positive semi-definite and Eq.(49) is a convex quadratic programming problem.

4.2.Design of input constraints and safety strategies

In this subsection, we focus on handling input constraints in practical applications and stall protection issue unique to fixed-wing aircraft, and then develop a collision avoidance strategy.

4.2.1.Input constraints

Since the input constraints are linear constraints,it is still a convex quadratic programming problem.

4.2.2.Stall protection

In current literature, many formation control approaches focus on handling velocity saturation constraints,while allowing agents to move slowly or even stand still.Unlike quadcopters or surface vehicles, fixed-wing aircraft flying too slowly will stall and crash due to lack of lift.Therefore, it is a basic requirement for formation method of multiple fixedwing aircraft to ensure that the flying speed of each aircraft is always higher than its stall speed.In addition,the path angle γ should also be not greater than the maximum allowable path angle γmaxduring the formation flight.Thus, at time k, the velocity of aircraft i should satisfy

After adding the positive-minimum velocity constraint,Eq.(58) is still a convex quadratic programming problem.It should be noted that our proposed method is not merely applicable to controlled objects with this type of velocity constraint.More importantly, it can prevent aircraft from stalling due to the inability to maintain sufficient speed under limited input conditions.

4.2.3.Collision avoidance

Inter-vehicle collision avoidance is a key research topic within the field of multi-agent cooperative control.For fixed-wing aircraft, the difficulty of collision avoidance increases significantly due to their positive-minimum velocity constraint,nonholonomic constraint, and extremely slow deceleration.

In this study,each aircraft predicts its own future trajectory through the model predictive method and receives the predicted trajectories of its neighbors.These predicted trajectories can be fully utilized to effectively avoid collision.Each aircraft is marked with a unique number to indicate the priority of collision avoidance.Without loss of generality,we directly use the index i to indicate the priority.The smaller the i,the higher the priority.Then the set of senior neighbors of aircraft i is denoted as

Algorithm 1.Decision process of aircraft i.

Decision process of aircraft i Input: ξi,k: current state; ξβi,k,ξβi,k+1,...,ξβi,k+Np■■: predicted state trajectory of reference aircraft;pj,k,pj,k+1,...,pj,k+Np■■,j ?N+i : predicted position trajectory of senior neighbors.Output: Optimal control input ui,k.1.Generate guidance route Ri,k,Ri,k+1,...,Ri,k+Np■based on ξi,k and ξβi,k,ξβi,k+1,...,ξβi,k+Np■■■;2.collision ←False;3.while True do 4.Calculate reference state ξr ■i,k,ξr i,k+1,...,ξr ■i,k+Np ;5.～ξi,k ←ξi,k-ξr i,k;6.U*i,k ←solution to optimization problem Eq.(58);■■7.Predict position pi,k,pi,k+1,...,pi,k+Np ;8.for j ?N+i do 9.for k′ ←k to k+Np do 10.if |pj,k′-pi,k′ |

5.Numerical simulations

In this section, three numerical simulations are conducted to assess our proposed method.In the first simulation, we compare our proposed method with the original guidance-routebased formation strategy to demonstrate the advantages of our method in formation gathering efficiency and formation maintenance accuracy.Then, the second simulation proves that our proposed method can coordinate a multiple aircraft formation with changing communication topology and kinematic constraints to complete given tasks without collisions and stalls.Finally,simulations of formation control with more aircraft are executed to verify the expansibility of our proposed method.

5.1.Comparison simulation

In the first scenario, five fixed-wing aerial vehicles, including one informed aircraft and four uninformed aircraft, gradually form the predefined formation geometric pattern while performing the scheduled task,and then make a 90°turn to verify the control accuracy of our proposed method under maneuvering conditions.The control period and desired speed are set to 0.03 s and 30 m/s respectively.In order to simply compare the performance of the formation control method itself, the stall protection and collision avoidance strategies of our proposed method are not enabled in this simulation.

As shown in Fig.8, compared with the original guidanceroute-based strategy in Ref.14, our proposed method is faster in forming the desired pattern and has less distortion in turn maneuvering.In our proposed method, the time used to form the formation and the position error at turning are about 6 s and 14 m, respectively, while in the original guidance-route-based strategy, they are above 10 s and 24 m (see Figs.8(a) and 8(b)).This is mainly attributed to the predictability of our model predictive route tracking controller.The controller considers the comprehensive performance for a period of time in the future (i.e.prediction horizon Tp), thus effectively improving the route tracking accuracy of the fixed-wing aircraft that cannot arbitrarily change velocity due to their complex dynamics.In addition,the control command is an approximate optimal solution obtained from the quadratic programming solution at each control period, which simultaneously improves the performance of our controller compared to the traditional PID controllers with fixed parameters.

Figs.8(c)and 8(d)demonstrate the curves of position errors in formation shape maintenance.Our proposed method can maintain the formation without steady-error, while the original guidance-route-based strategy has a steady state error of about 0.9 m.Thanks to the predicted state trajectories shared among neighbors, aircraft in our control scheme can obtain the reference aircraft information at the correct time by linear interpolation (see Eq.(43)), thereby eliminating the steadystate error caused by the control lag.

Figs.8(e) and 8(f) show that, with our proposed method,the speed of aircraft achieves consistency and converges to the desired speed faster,suggesting that aircraft fly more stably in the forward direction.In addition to the predictability,another important reason for the faster convergence of speed is that the cost function of the model predictive control contains the velocity error.

Fig.8 Comparison of our proposed method and the original guidance-route-based formation control strategy.

Considering that the minimum distance between any two vehicles is an important indicator for evaluating the safety and stability of formation flight, the minimum distance comparison between the two methods is also given in Figs.8(g)and 8(h).During formation gathering and maneuvering turn,the minimum distance of our proposed approach is more stable and does not decrease significantly.This is due to the better route tracking accuracy of our model predictive controller.

It should be noted that our algorithm is completely distributed, and the comparison method requires a centralized coordinator.Even so, it still has better performance than the latter.

To summarize,the superiority of our proposed algorithm is mainly attributed to its two important differences from the traditional Guidance-Route-based Formation Control strategy(GRFC), i.e., distributed guidance generation strategy and model predictive route tracking controller.The traditional GRFC strategy is centralized because it requires a global leader that can communicate with all followers, while our proposed algorithm is fully distributed.Moreover, aircraft in traditional GRFC strategy generate guidance routes based on leader information at last control period and track the routes using traditional PID controllers,leading to control lags and formation maintenance errors.These problems are eliminated in our approach using model predictive techniques.Meanwhile, the model predictive controller brings better control performance and additional capabilities,such as stall protection and collision avoidance.

5.2.Effectiveness simulation

This scenario is designed to demonstrate that our proposed formation strategy is capable of coordinating a multiple aircraft formation with changing communication topology while providing collision avoidance and stall protection guarantees.Five aircraft, including one informed aircraft and four uninformed aircraft, first gather together to form a wedge formation.Then, they make a left turn maneuver, and reconfigure the formation geometric pattern as well as the communication topology twice.The stall speed and safety distance are set to 24 m/s and 5 m, respectively.The other parameters are the same as the previous simulation.

As shown in Fig.9, a group of aircraft take off from a chaotic initial position, and form a wedge-shaped formation after about 15 s.Each aircraft smoothly converges to its desired position, and there is no steady-state error in the formation.Then, they make a left turn maneuver.During the maneuver, the position disturbance and speed change are less than 8 m and 6 m/s, respectively.These derivations are quickly eliminated without vibration phenomenon.At time t=60 s, the formation mirrors the geometric pattern, that is,each aircraft exchanges desired position with its symmetrical individual.Without the collision avoidance strategy, aircraft will move symmetrically and collide during this maneuver.On the contrary,in our proposed method,each aircraft is capable of anticipating the danger of collision in advance and planning a collision-free trajectory on-line.Consequently, the minimum distance between any two aircraft in the maneuvering process is not less than the safety distance,which is verified in simulation results(Fig.10(a)).Finally,at time t=100 s,the formation pattern changes from a wedge to a straight line without any inter-vehicle collision.

Our proposed method is capable of preventing aircraft from stalling and crashing due to low flying speeds.The simulation result in Fig.10(b)shows that the minimum speed of the aircraft in the formation is always higher than the stall speed during the entire formation flight.However, without the stall protection strategy, the minimum speed will drop to less 20 m/s,which means that the aircraft has stalled and crashed.

5.3.Expansibility simulation

Fig.9 Simulation of our proposed method for maneuvering and formation reconfiguration.

Fig.10 Minimum distance and speed during formation.

Fig.11 Formation simulations with nine and sixteen aircraft.

Fig.12 Minimum distance and speed of nine aircraft.

Fig.13 Minimum distance and speed of sixteen aircraft.

To assess the expansibility of our proposed method, two formation simulations with nine and sixteen aircraft are performed in this part.Except for the increase in the number of aircraft, the rest of the simulation parameter settings are the same as the previous simulation.As shown in Fig.11, these aircraft form a wedge-shaped formation from a disordered initial position.Then, the formation changes its geometric pattern as well as communication topology twice during the flight.It first changes from a wedge to a straight line,and then changes from a straight line to a diamond.Each aircraft works cooperatively to form these given desired formation quickly and successfully completes the scheduled flight mission, suggesting that our proposed formation method is capable of controlling a fixed-wing aircraft formation of 16 or even more individuals.It is worth mentioning that each aircraft only solves its own optimization problem, and thus the time they spend on calculating control instructions will not increase significantly as the number of aircraft increases.The minimum velocity and minimum distance between aircraft during formation flight are given in Fig.12 and Fig.13, showing that the minimum speed and minimum distance are always larger than the allowable minimum values, which indicates that the safety of the flight is guaranteed.

6.Conclusions

This paper proposed a distributed formation control method for multiple aerial vehicles.Our proposed method is capable of handling nonholonomic constraint and positive-minimum velocity constraint.It has been shown in simulation results that our proposed method has better performance than the original guidance-route-based strategy and allows the communication topology to change over time.In addition, the stall protection and collision avoidance strategies have also been presented to guarantee the safety of formation flying.It also has been verified that our proposed method can be applied to formations with a larger number of aircraft.In this paper,the communication topology of the formation is described by an undirected graph, which is assumed to be always connected.In future work, it is meaningful to study the maintenance of formation connectivity.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was partially supported by the STI 2030-Major Projects (No.2022ZD0208804) and the Postdoctoral Fellows of Beihang‘‘Zhuoyue”Program, China.