Nonlinear tight formation control of multiple UAVs based on model predictive control

Ruiping Zheng ,Yongxi Lyu ,b,*

a School of Automation, Northwestern Polytechnical University, Xi'an 710129, China

b Shaanxi Province Key Laboratory of Flight Control and Simulation Technology, Xi'an 710129, China

Keywords:Unmanned aerial vehicles Tight formation Wake vortex model Model predictive control

ABSTRACT A tight formation of unmanned aerial vehicles (UAVs) has many advantages,such as fuel saving and deceiving enemy radar during battlefield entry.As a result,research on UAVs in close formation has received much attention,and the controller design for formation holding has become a popular research topic in the control field.However,there are many unknown disturbances in tight formation,and the tail aircraft is disturbed by the wake.This paper establishes a mathematical model of wake vortices for tail aircraft that considers uncertainty and strong interference.Two UAVs are simulated by Computational Fluid Dynamics software,followed by the design of a semiphysical simulation model predictive control(MPC) scheme that suppresses uncertainty and interference sufficiently to enable the tail aircraft to accurately track the lead aircraft and maintain a stable,tight formation.The tight formation controller is verified by numerical simulation and semiphysical simulation.The results show that the designed controller has an excellent control effect in the case of disturbance caused by the wake vortex.

1.Introduction

The concept of close formation flight originated from migratory birds[1-3].Birds flying in formation obtain aerodynamic benefits,and when 25 birds fly in a V formation,the range of motion can be increased by about 71%over that of a single bird[4].By measuring the heart rate ofPelecanus onocrotaluswhen they were flying in formation,researchers found that these birds reduce their energy expenditure by flying in formation,thus increasing their range of motion and improving their flight endurance [5].Sekhar et al.analyzed the characteristics of bird formation flight from a thermodynamic point of view,assuming that the V formation is the optimal mode of energy use during flight time [6].

Many experts,inspired by bird formations,have studied mechanisms to reduce energy consumption during formation flight.Ahmad studied the basic parameters of each model of the aircraft wake vortex and pointed out that the vortex intensity varies with the size of the vortex nucleus radius[7].Blake et al.conducted a wind tunnel testing of aircraft in close formation,placing two vertical tailless delta-wing aircraft in close formation in a wind tunnel[8],where the induced drag on the wingman was reduced by 25% when the wings overlapped by 15%-20%.In their paper [9],Kaden et al.investigated the wake vortex model for transport aircraft formations using adaptive generation to fit the variation of vortex intensity.Numerous studies [10,11] and experimental data illustrate that the influence of the wake vortex flow on wingmen varies sharply when they are in different positions relative to the lead aircraft [8,12-14].

Therefore,the issue of control of tight formations is critical to the study of tight formations [15].Keeping the wingman in the optimum position in tight formation reduces the drag on it and minimizes fuel consumption.In paper [16],Daniel B.Wilson et al.propose a quaternion based unscented Kalman filter to improve relative state estimate in the UAV formation flight.In paper[15],Q Zhang et al.proposed a robust control method by using the uncertainty and disturbance estimator to enhance the robustness of aircraft formation control.M.Pachter et al.[17] developed a tight formation flight controller that can enable aircraft to take advantage of the reduction in induced drag brought about by the aerodynamic coupling effects.The paper [18] presented a novel aerodynamic model-based robust adaptive control for close formation at level and straight flight.With the rapid development of technology,more and more sophisticated control methods are being applied to UAVs [19,20].In Ref.[21],an adaptive MPC was used to control a multi-UAV formation,and an extended state observer was incorporated to satisfy the expected control effect,but the case discussed in that paper was a distant formation flight,and the effect of the wake vortex airflow of a tight formation was not considered.With the model predictive control (MPC) scheme,the current state is used as the initial state,an optimal control sequence is generated,and the first set of data in the optimal control sequence is used to act on the controlled object [22].The advantage of the MPC is that a multivariate controller adjusts the output considering all factors simultaneously;another important advantage is that it imposes tight constraints on the states and the control inputs [23,24].

One difficulty faced by tight formation controller design is that it is difficult for a conventional controller to keep the wingman in the proper position stably because of vortex interference.The MPC method can suppress the disturbance well and imposes strict constraints on the states and control inputs [25].Since the controller of a tight formation requires high real-time performance,the MPC cannot be used directly in a tight formation controller[26].However,since the guidance control of the wingman does not require a fast solution frequency,an MPC outer-loop controller was designed in this study,and a conventional control scheme was used for the inner loop of the aircraft.

The primary contributions of this paper are as follows: (1) The non-linear tight formation model built in this paper can incorporate the influence of the wake vortex and describe the formation characteristics more accurately;(2) An outer-loop control law for tight formation is designed based on MPC,which has a certain antiinterference capability while meeting the real-time performance;(3) Numerical simulations and semi-physical experimental simulations are carried out to verify the feasibility and real-time performance of the tight formation system.

This paper is structured as follows: First,the wake vortexinduced velocity model is presented.Then,the paper explains the controller design and discusses the MPC's stability and accuracy.Finally,the results of the tight formation flight simulation are presented and discussed.

2.Modeling the formation flight vortex

The UAV model selected for this study was the XQ7B.Fig.1 shows three views of the UAV,and Table 1 lists its parameters.In close formation flight,the effect of the longitudinal distancelxon the induced force and moment is much smaller than that of the lateral distancelyor the vertical distancelz.Therefore,in this paper,we neglect the effect oflxon the wake vortex [27].

Table 1 Parameters of the UAV.

Fig.1.Three views of the UAV.

Fig.2.Schematic diagram of tight formation induced velocity.

The Proctor vortex model[28]is derived from an analysis of the vortex tangential velocity by LiDAR and is provided by Eq.(1).

whereBis the wingspan length,andrcis the radius of the vortex nucleus (5.82% of the aircraft span).ris the distance from the induced velocity point to the vortex line segment,b0=πB/4,Γ0=4W/πBρV.

Thus,the total induced velocity at each point on the wing of the following aircraft isw（l）=wleft（l）+wright（l）.

The average value of the induced upwash on the wing of the wingman can then be calculated by integratingw（l）,

According to Ref.[29],the values of the variation of forces and moments applied to the wingman can be calculated.The forces are the drag side force and the lift force,and the moments are the roll moment,pitch moment,and yaw moment,created by the upwash and sidewash from the lead aircraft [30].

In a close formation system,the forces on the wingman are changed considerably by a change in the relative positions of the two aircraft.A change of drag will interfere with the speed control of the original flight control system,and a change of lift will affect the altitude control of the wingman;a change of side force will affect the heading control of the wingman [31].

The mathematical model of the wake vortex was established in the previous section.The mathematical model of the aircraft wake vortex presented in this paper has been simulated in Computational Fluid Dynamics software to verify the model’s accuracy.These tasks were verified in previous papers [30] and are not repeated here.The wake vortex is a strong disturbance in formation control,and the following points arise from the formation control law.

(1) The tight formation system is not sufficiently resistant to interference,and the formation holding accuracy is low or unstable.

(2) The dynamic performance of the formation is poor or fails to meet requirements.

Our MPC-based formation outer-loop control law avoids the disadvantages of other control laws.The main contributions of this paper include the following:

(1) The designed control law has a certain suppression effect on disturbances and has strong robustness.

(2) The formation system eliminates steady-state errors and has excellent dynamic performance.

3.Problem formulation

In many scenarios,the wingman must follow a specific position relative to the lead aircraft for the maximum aerodynamic benefit to achieve fuel savings [32].This relative position in the airflow system of the lead aircraft does not change.To facilitate the design of the formation control law,the relative positions of wingman and leader are represented in the inertial coordinate system.As shown in Fig.3,the optimal position changes with the position and attitude of the lead aircraft at a rapid and unknown rate,and the design of the formation controller determines whether the wingman can accurately and consistently track to the optimal position.

Fig.3.Relative position of the lead aircraft and follow aircraft.

Fig.4.3-D Tight formation trajectory.

Fig.5.Longitudinal tracking error.

Fig.6.Lateral tracking error.

Fig.7.Vertical tracking error.

The aircraft kinematics are strongly nonlinear,and the tight formation is a non-complete constrained system.The aircraft generates control inputs that satisfy the nonlinear dynamics constraints and the actuator limit constraints during tracking[33].The designed tight formation control method should overcome the above two difficulties and satisfy the accuracy of tight formation.

In the inertial coordinate system,the position of the leader is assumed to be（xl，yl，zl）,and its optimal position（xd，yd，zd）is constant with respect to the leader.（xf，yf，zf）is the position of the wingman in the inertial coordinate system.When a design engineer develops control laws for a tight formation,the lead aircraft has a set flight path,and the wingman gains aerodynamic formation benefits.The lead aircraft is not affected by the formation airflow,so we need only to design control laws for the wingman.The equations for the nonlinear dynamics of a wingman in the tight formation are shown below.

whereVf，γf，χfare the airspeed,track angle,and heading angle of the following aircraft,respectively.T，L，Dare the thrust force,lift force,and the drag force,respectively.ΔT，ΔL，ΔDare the uncertainties in aerodynamic forces.

The desired tracking position and the longer aircraft position are related as follows:

where LWIis the rotation matrix,

The problem of tight formations translates into wingmen tracking the desired position（xd，yd，zd）.The system can be viewed as a control system with inputs u（v，γ，χ）and states X（x，y，z）that has the general form of

Each point on a given trajectory satisfies this kinematic equation,withdrepresenting the desired value,the general form is

where Xd=［xd，yd，zd］T,ud=［vd，γd，χd］T.

The following equation can be obtained by expanding Eq.(6)at the desired trajectory point using a Taylor series and ignoring higher-order terms.

By subtracting Eq.(6) from Eq.(7),one obtains

To facilitate the design of the model predictive controller,discretize Eq.(8)as follows:

in whichA（k）=I+TA（t），B（k）=TB（t）,andTis the sampling time.

3.1.Design of the objective function

The objective function incorporates the optimization of deviations from the state quantities and control inputs of the formation system to track the desired position quickly and smoothly.The objective function consists of three components.

The first part of the designed objective function considers the wingman's ability to track the desired position.

The second part of the objective function of the design considers constraints on the variation of control inputs

The third part of the objective function considers the collision avoidance problem for wingmen and lead aircraft.As shown in Eq.(12).

wheredfl（t）indicates the distance between the two aircraft.

Dis the safe collision avoidance distance between the two aircraftdfl（t）≥D.σ=105.

The objective function of the design is

3.2.Enhancement of the objective function

The second part of the objective function has certain shortcomings.It cannot limit the control increment in each sampling period or avoid abrupt changes in the control increment of the controlled system.This affects the continuity of the control variables,so adding a relaxation factor to the objective function and replacing the control amount with the control increment can limit the increment directly and avoid implementing the process where no feasible solution exists.This is shown below.

3.3.Stability analysis

The stability of tight formation systems is critical,and this study developed a controller based on the MPC with a nonlinear model.The stability of linear and nonlinear MPCs has been extensively demonstrated in the literature.Here,the stability of the proposed predictive control scheme is investigated by testing the monotonicity of the cost function,where the prediction horizon is finite,and the control horizon is smaller than the prediction horizon[34].The cost function is as follows:

Assume that the control signal is（k）=［u（k|k），…，u（k+Hu-1|k）］Twhen the optimal solutions are taken at momentk.The control signal is（k+1）=［u（k+1|k），…，u（k+Hu-1|k）］Tat momentk+1 when the suboptimal solution is taken,and the cost function at this moment is

Eqs.(17) and (16) are subtracted to obtain

It follows thatJ*（k+1）-J（k）≤0.The cost function of the optimal solution is no greater than the cost function of the suboptimal solution at momentk+1,i.e.,J（k+1）≤J*（k+1）,thereforeJ（k+1）-J（k）≤0.The cost function decreases monotonically with time,so the control system is stable.

4.Experimental results and analysis

This section discusses the experimental simulations of the formation system with the lead and wingman aircraft of type XQ7B.First,the model of the nonlinear dynamics of the two aircraft was constructed.Then,an established aerodynamic model is added to the wingman's dynamics model to describe the aerodynamic effects of close formation flight since only the wingman is affected by the close formation aerodynamic model and not the leader.Finally,experimental validation is discussed.

4.1.Numerical simulation

Numerical simulations were carried out in the MATLAB Simulink module.The lead aircraft flew at a constant speed on one trajectory,while the wingman's initial position was far from the lead aircraft's.The wingman quickly tracked to the desired position,as shown in Fig.4.

The initial state of the lead aircraft is（xl，yl，zl）=（100，0，1000）,vl=27.8 m/s.The initial state of the wingman aircraft is（xf，yf，zf）=（0，100，1000）,vf=27.8 m/s.

With the wake vortex's aerodynamic effects added to the tight formation system,as shown in Figs.5-7,a PID controller cannot suppress the disturbance of the tail vortex,but the designed MPC outer loop meets expectations,is strong and robust,and provides anti-disturbance performance.

Longitudinal,lateral,and vertical tracking errors arexe=xf-xd,ye=yf-yd,ze=zf-zd,respectively.The speed simulation of the UAVs when use the MPC controller is shown in Fig.8.

Fig.8.Speed simulation of UAVs.

4.2.Conducting experiments on the semiphysical simulation platform

After numerical simulation,experimental verification was conducted on a semiphysical simulation platform,shown in Fig.9.The nonlinear dynamics models of the lead and following aircraft were run on xPCs.The flight control system communicated directly with the xPC and transmitted control signals to the lead and following aircraft,while the lead and following aircraft transmitted their states to the flight controller.

Fig.9.Semiphysical simulation of tight formations.

The simulation results are displayed using Tacview,which visualizes the UAV tight formation system shown in Fig.10.Experimental results demonstrated that the designed tight formation controller meets the semiphysical experimental requirements.The following aircraft can track accurately and quickly to the appropriate position relative to the lead aircraft and has excellent antiinterference performance.

Fig.10.Semiphysical simulation flight recordings.

5.Conclusions

In close formation UAV flight,a wingman in the position appropriate for the lead aircraft has favorable benefits,and there is a higher demand on the resistance of the wingman’s control system to interference.This study developed a tight formation control system that is proposed in the paper as an outer-loop nonlinear MPC that meets robustness and real-time performance requirements.The designed control law was subjected to MATLAB mathematical simulation,Tacview simulation,and semiphysical simulation.The results showed that the designed nonlinear control law met the desired requirements.

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.

Acknowledgments

This work was funded by the National Natural Science Foundation of China (Grant Nos.62173277 and 61573286),the Natural Science Foundation of Shaanxi Province (Grant No.2022JM-011),the Aeronautical Science Foundation of China (Grant No.201905053004),and the Shaanxi Province Key Laboratory of Flight Control and Simulation Technology.We thank LetPub(www.letpub.com) for its linguistic assistance during the preparation of this manuscript.