Leader-Follower UAV formation flight control based on feature modelling

To solve the problems of backstepping error and poor dynamic tracking approach rate in traditional PID neural network control in UAV formation flight control, a Leader-Follower UAV formation flight control method based on feature modelling is proposed,and the pose relationship model between virtual follower and pilot is established by trajectory tracking and pose dynamic fitting. The pose distribution of thefollower is analyzed in the ground coordinate system, and the parameter information of linear velocity and angular velocity control of UAV is obtained, and the backstepping sliding mode formation controller is formed. The variable structure PID neural network controller is used to design the flight control law of UAV formation, and the fast piecewise power approaching factor is introduced into the PID controller to eliminate the chattering of sliding mode control. The simulation results show that this method can ensure the rapidity of UAV formation flight control also show strong anti-jamming ability. Due to the fast piecewise power approach rate, the UAVs can complete the UAV formation reorganization under disturbance and buffeting in a short time, and the trajectory tracking error approaches zero, and it has good anti-buffeting ability.


Introduction
Unmanned aerial vehicle (UAV) flying formation is used to perform transportation tasks in complex scenes, such as plateau transportation, battlefield transportation, mountain transportation, emergency rescue, and disaster relief transportation, etc., which improves transportation efficiency and effectively meets the accuracy and real-time of material distribution.With the continuous expansion and deepening of drone formation applications, it is necessary to establish an optimized flight formation control model.This model should incorporate the adjustability of drone flight formations to avoid delays and disturbances in multi-drone systems during formation transportation, while ensuring stability and anti-interference capabilities in drone formation flight control (see for example Li et al., 2022;Zou et al., 2019;Zou et al., 2020).
In formation control, the Leader-Follower control has become a popular method due to its simple configuration and high implementation efficiency.In practical applications, the Leader-Follower scheme requires continuous updating of feedback state signals and control commands (Breivik et al., 2008;Duan, 2020).Combined with a centralized control scheme design for multiple CONTACT Yafei Chen 346848735@qq.comThis article has been corrected with minor changes.These changes do not impact the academic content of the article.
unmanned aerial systems, it achieves disturbance rejection control and rigid shape switching for multi-vehicle formations.In addressing the three-dimensional leaderfollower formation control problem for a group of unmanned aerial vehicles (UAVs) with motion constraints and disturbances, the authors Liang et al. (2020) transform the original formation error of follower UAVs into the Frenet-Serret framework.To achieve satisfactory formation error constraint conditions, Yang et al. (2023) proposed a prescribed time performance function based on time scale transformation function, and combined with error conversion to improve the transient performance of formation error.Based on the differential flatness theory, Ai and Yu (2019) proposed an observer-finite-time controller based on the adaptive disturbance suppression method to transform the underactuated quadrotor system into a fully actuated system with four degrees of freedom and four control inputs and this method effectively solves the control problem of multiple quadrotor aircraft.
However, accurate acquisition of model parameters is required for model establishment.When the constraint conditions are complex and the computational burden is excessive, it can lead to delays, oscillations, and poor stability.Yang et al. (2020) proposed a follower vehicle control law based on estimated relative positions to follow the leader.If certain conditions regarding the relative angular velocity between the leader and followers are met, estimation errors will converge to zero, but, it may not be suitable for many scenarios due to model constraints.Considering the potential changes in the external environment and uncertainties within the system, Li et al. (2023) proposed an internal-external loop control method for the formation of drones.However this approach requires stability control for each member of the formation, although optimization was suggested using Active Disturbance Rejection Control (ADRC), it imposes a heavy computational burden and cannot guarantee real-time performance.Hoang et al. (2021) proposed a distributed control law to achieve the desired formation in the presence of continuous uncertainties in the follower model, but it exhibits chattering and lacks strong adaptability when subjected to external disturbances.
At present, the UAV flight formation control mainly adopts sliding mode control, backstepping control, PID control, etc.Most of the current control methods have the problems of backstepping errors and the dynamic tracking approach rate.In the authors of Rebbecca and Yoonsoo (2018) and Wang S. et al. (2022), a PID control is proposed for constant disturbance rejection for the vehicles to achieve a desired formation is proposed to improve the control stability of quadcopters.Hou and Lu (2022) deal with the formation control problem of multiple quadrotor systems based on static/dynamic event-triggered integral sliding mode control (ISMC) and adaptive sliding mode disturbance estimator (ASMDE).Wang F. et al. (2022) and Muslimov and Munasypov (2021) design a distributed formation controller based on an adaptive backstepping control method to control a fixed-wing unmanned aerial vehicle (UAV) swarm to start and keep flying in a parallel formation of a specific geometry.
Inspired by the above research, this paper proposes a Leader-Follower UAV formation flight control method based on feature modelling.Firstly, the pose relationship model of virtual follower and navigator is constructed, and the pose parameters of UAV formation are adjusted and controlled.The fast piecewise power approach factor is introduced into the PID controller to eliminate the chattering phenomenon of sliding mode control, which improves the stability of UAV formation flight control.Finally, the simulation test analysis shows the superior performance of this method in improving the stability and accuracy of UAV formation flight control.

Leader-follower feature modelling
To realize the Leader-Follower UAV formation flight control based on feature modelling, firstly, the Leader-Follower feature model of UAV formation flight is constructed, and the trajectory planning and dynamic parameter tracking identification of the Leader and Follower are carried out respectively.The modified DH parameter method is used to carry out the terminal pose parameter conditions of UAV formation flight, and the spatial coordinate system of UAV formation flight is constructed.Combined with the transformation analysis of the spatial geometric coordinate system, the coordinate point where the centroid and geometric centre of UAV overlap is regarded as the centroid of UAV, and the indirect coordinate system is established with the geometric centre of UAV as the origin.On the base of Wang et al. (2021), under the constraint of dense formation flight, the kinematic model of UAV formation movement is constructed, as shown in Figure 1.
In the kinematic model of UAV formation movement shown in Figure 1, a second-order model with unknown disturbance is used to analyze the parameters of the three-channel operational model of UAV formation fractal, which is expressed as. Where, φ a , ψ a , γ are the time-varying formation flying attitude angles realized by multi-UAV system, φa , ψa , γ are the angular velocity of UAV flying attitude under the constraint of azmuth information, φa , is γ are the angular acceleration of UAV flying attitude under centralized formation control, and b 1 , b 2 , b 3 , d 3 are the known coefficients of formation control of azimuth information.b 1 , b 2 , b 3 , d 3 are uncertain coefficients of flight disturbance of multi-UAV, f d 1 , f d 2 , f d 3 are disturbance parameters of UAV dynamic model parameters, and δ φ , δ ψ , δ γ are control inputs of pitch, yaw and roll channels in UAV body coordinate system.In this paper, the buffeting and fuselage flutter of UAV fuselage are considered.
By using nonlinear system sliding mode parameter identification, combined with sliding mode surface and fast piecewise power trend control, a second-order sliding mode surface is established for UAV pitch angle tracking control, and the control process is described as follows.
Where b > 0, φ a is the indirect coordinate steady-state angle with the geometric centre of the UAV as the origin, φa is the kinematics model parameter of the follower, u is the input of the controlled object, and f d is the external interference in the ground coordinate system XOY, and the formula (2) is rewritten as According to the tracking measurement results of linear velocity and angular velocity of UAV, and based on the analysis of pose parameters of virtual follower and navigator, the flight control object model of Leader-Follower UAV formation is constructed.

Control object analysis
Based on the Leader-Follower control measurement, the PID correlation controller is used to adjust the longitudinal motion parameters of the parameters.The output of the UAV flight formation control system is obtained when the input net j is equal to the output of each branch connected with it, x 1 , x 2 , ... , x n are multiplied by the adaptive learning weights value w 1j , w 2j , ... , w nj respectively.At this time, the roll, pitch, and yaw angles of the UAV are expressed by Euler angles respectively.Under the condition of small disturbance, the output characteristic moments of the variable structure PID neural network are as follows At this time, the Leader-Follower method is used as the control strategy, and the variable structure PID neuron is used to simulate the flight state of the UAV, and the switching model of the flight state of the UAV is obtained as follows Where, θ j is the error surface of neurons, formation control is carried out on the two-dimensional plane at this time.
In this paper, a variable structure PID control weighting vector is introduced into UAV flight control, and the yaw control parameters of UAV are expressed as According to the principle of small disturbance, it is obtained that the output v(k) of PID neural network in UAV flight is related to the reference motion state.At this time, the controller's effective tracking of the expected path is expressed as the second-order differential of the state component, that is Here, the weight ω ij (i = 1, 2; j = 1, 2, 3), when the UAV moves at a small angle, adjust the weight ω j (j = 1, 2, 3) from the hidden layer to the output layer.
Set ω = [ω 1 , ω 2 , ω 3 ], When the UAV yaws, it has yaw moment, ignoring the influence of its pitching and rolling moments, recording the parameter of the disturbed motion as x = [φ a φa ], and transmitting the navigation information of the formation through the information between UAVs, obtaining the spatial distribution estimation value − ρ (x, ω) of the leader and wingman in the ground coordinate system, and then.
Where, σ j (φ a , φa ) is the learning step.Using the centralized formation control structure of pilot-following method, the error state equation between the expected formation spacing is obtained as follows Where, ∫(•)dt is the integrator of multivariable control system controlled by UAV formation, and d(•) is the differential symbol.With variable structure PID weighted learning, the global optimal solution is obtained, and the output characteristic quantity of the attitude controller is obtained as follows The intermediate variables are introduced to construct the deviation control law of position and attitude, and the weights are adjusted online by an adaptive algorithm, taking To ensure that the moving point can quickly reach the convergence state in a limited time, the Lyapunov function is defined as Where, ω = ω * − ω.According to the connection relationship between UAVs in UAV formation, the pose relationship model between virtual Followers and pilots is established by trajectory tracking and pose dynamic fitting.The corresponding upper bound estimation value satisfies the Lyapunov stability principle, and the UAV formation flight controller is constructed by this method.

Design of variable structure PID neural network controller
According to the established UAV formation flight control object model, an m-output multi-dimensional attitude parameter model is used as the multi-variable information fusion centre of UAV formation flight control, and the Leader-Follower formation mode is adopted, and the position information of the Leader is introduced to set the anchor point for the whole formation system, so that the azimuth formation is unique at this time, and there are 2n identical neurons at the input layer of the pilot.
According to the order of Leader to Follower, the dynamic arrangement of UAV formation flight is carried out, and the first follower is set in the Leader-Follower azimuth information formation, which realizes the overall scaling in the movement process of azimuth information formation, and the variable structure controller of UAV formation flight with Leader-Follower structure is designed, in which the Leader UAV flies autonomously according to the expected trajectory.And each Leader UAV meets the expected azimuth constraint.At least two aircraft in the formation are required to act as leaders, and the leader UAVs still account for a small part of the formation system that can be composed of them.With the azimuth and speed of each UAV as input variables, the potential field function between adjacent aircraft is recorded as: By introducing the total potential field function of single node i of UAV formation, taking the deviation of position and attitude as input variables, the adaptive law of attitude system control is obtained as follows According to the control objective of the attitude subsystem, the network input and output vectors are Where, i d_ref , i q_ref , i d , i q are a definite function parameters, which is obtained by measurement, T s and T SW are known parameters, and the variable structure PID inputs the weight value from the layer to the hidden layer Where, η sij is the learning step, and according to the difference between the position of the leader and the wingman, the output weight value of the expected formation of the given formation is obtained Therefore, a Leader-Follower variable structure PID neural network controller for UAV formation flight based on feature modelling is constructed.However, due to the uncertainty of attitude system modelling, UAV formation flight control has chattering, which needs to be suppressed.

Flight buffeting suppression
Based on designing the flight control law of UAV formation with variable structure PID neural network controller, a fast piecewise power approaching factor is introduced into the PID controller to eliminate the chattering phenomenon of sliding mode control, and the balance condition of flight stability is given as f (x 0 , u 0 ) = 0.The variable structure PID neural network is used to self-tune the switching gain flight trajectory, and the estimated value of the state quantity is [φ 0 + φ, φ0 + φ, θ 0 + θ ] T , and the control target estimation equation of the attitude subsystem is.
The measurement equation of linear velocity and angular velocity corresponding to the follower can be expressed as It is assumed that the disturbance control parameters and buffeting characteristics M and h(φ a , φa ) of Leader-Follower UAV formation flight are composed of two parts, namely.
Where, M n and h n (φ a , φa ) are the linear differential uncertainties of the virtual follower under the UAV, and M and h(φ a , φa ) are the control linear differential uncertainties of the expected trajectory.To better control the motion of UAV, the actual linear velocity and angular velocity control quantities are introduced, and the motion constant compensation equation is obtained: Where The dynamic coefficient, transfer function, and frequency domain characteristics of UAV are solved, the fast piecewise power approach factor is introduced, and the upper bound of buffeting control is − ρ (t), that is.
Based on the above-mentioned buffeting suppression algorithm design, an anti-disturbance algorithm is desig ned for each wingman, and the formation stability is maintained by anti-buffeting technology.

Flight control error analysis
Based on constructing the control law of UAV formation flight, the variable structure control method is used to analyze the convergence and stability of the controller design error.Similar to the trajectory tracking controller, the control error is given as follows: The wingman receives the formation controller information and sends it to the error generator error vector together with the expected formation information.The derivation of the error is as follows: The sliding surface designed by exponential reaching law is Then When Then, the sliding mode manifold is designed by using the sliding mode correlation progressive method, and the control law of the UAV can be obtained to solve the longitudinal motion parameters, and the control error convergence of the attitude system can be satisfied It can be seen that the designed control law can ensure that the convergence error tends to zero.

Global stability analysis of control law
For the multi-UAV formation system composed of various UAVs, the expected target formation can meet the uniqueness condition by selecting the appropriate Leader UAV and azimuth constraints.Designing a controller based on feature modelling for each Follower UAV can enable the multi-UAV system to complete the task of forming, maintaining, and tracking the target formation.Lyapunov functional is used to analyze the convergence of the controller.To ensure that the moving point can reach the sliding mode manifold quickly and reduce chattering in a limited time, Lyapunov candidate functions V 1 , V 2 and V 3 are selected, Lyapunov function of UAV flight control is constructed, and the global stability problem is transformed into the problem of deriving V 1 , V 2 and V 3 , which is as follows: Where, V3 (t) is a non-increasing function, V3 (t) > 0, so it is obtained that V3 (t) is bounded, Update the terminal position of UAV flying formation, and use Bayesian estimation method to analyze Lyapunov stability.Because the scalar function V 3 is positive definite and V3 negative semidefinite, and both of them are continuous with time, it can be seen that the virtual speed input designed for the Follower UAV converges to zero with time.According to the Lyapunov stability condition and LMIs convergence condition, the qualitative derivation of Leader-Follower UAV formation flight control is given, and the first Lyapunov function of UAV formation flight control is defined as According to Barbalat's theorem, lim t→∞ e 1 = lim t→∞ e 2 = 0, that is, the orientation between UAVs will gradually converge to the expected value, that is, multiple UAV systems will form an expected time-varying formation determined by the Leader's position and expected orientation.
According to the above analysis, it can be seen that the variable structure PID neural network controller is used to design the flight control law of UAV formation, the reference motion is steady, and the derivative of the Lyapunov function is less than zero, which indicates that the whole formation system is fully asymptotically stable under the action of this control law.

Simulation platform and parameter configuration
To verify the application performance of this method in the flight stability control of UAV formation, an experimental test is carried out.The simulation experiment is realized by MATLAB.Without interference, the expected trajectory starting position (1 m,1.2 m), the terminal heading angle of 0.2 rad and the linear velocity of 12 m/s are given.Under small disturbance, the actual trajectory starting position is set to (5 m,5 m), and the initial rolling and pitching are given.The yaw angles are 0.25°, 0.15°, 0.001°respectively, the expected trajectories of the pilot UAV are x d = 3 cos t, y d = 3 sin t, z d = 2 + 0.5t, the initial measurement state vector is X = [ 0.12 0.25 0.15 0.45 ] T , the inertia parameter of the fusion of PID variable structure neural network is ε 1 = 0.1, the iteration step of error correction is 20, the variable structure PID neural network is set to contain a hidden layer, the number of nodes in the hidden layer is 5, the excitation function is tanh(x), and the flight attitude and mechanical parameter information of the UAV are shown in the table.Attitude and mechanical parameters are set as Leader quality M 1 = 1.2 kg, Follower quality M 2 = 0.933 kg, Flight acceleration g = 9.8 m/s 2 , Triaxial moment of inertia I x , I y , I z are 0.013, 0.032 0.021 N. s/ m, Wheelbase l = 0.25 m, Line velocity v q = 1.3 m/s, Angular velocity v r = 0.5 rad/s.According to the above parameters, the flight stability control simulation test of UAV formation is carried out, including flight attitude stability test, trajectory tracking test and anti-interference test.

Flight attitude stability test of UAV formation
In the simulation process of UAV formation flying attitude stability, the parameters, initial pose and initial error are guaranteed to be the same, and all information in the flight process is represented by unit vector, so that after each UAV flies to the set initial position and reaches the initial expected position point, the UAV as Follower runs the control algorithm designed in this paper to complete the formation and tracking tasks under interference conditions.Under the condition of satisfying the above constraints, the two Leader UAVs move back and forth along the Y direction v d = 0.1 m/s.According to the above parameter settings, the flight attitude angle convergence curves of the Leader UAV and the two Follower UAVs in different control algorithms are given as shown in Figure 2.
By analyzing Figure 2, it is known that the flight attitude angle tends to be stable gradually with the continuous advancement of formation flight.Under the same conditions, the approach effect of the method in this paper is better, and the approach speed and convergence speed are faster.

Trajectory tracking test
On the basis of testing the flight attitude stability of UAV formation, the Leader-Follower formation method is used to test the flight trajectory tracking performance of UAV.For two UAVs as Leader, the position tracking control is carried out through position information, and after reaching the initial expected position, the formation is reorganized according to the expected formation, and the trajectory tracking curves and absolute error comparison results of UAV formation by different methods are shown in Figure 3.
By analyzing the results in Figure 3, it is known that this method is used for UAV Leader-Follower formation flight control.This method has good trajectory tracking performance during long-term formation flight, and the formation formed by three UAVs has good formation maintenance and reconstruction ability after the formation algorithm runs t = 1160s.Comparing the tracking errors of different flight trajectories, see Table 1.According to the analysis of Table 1, the trajectory tracking error of this method is the lowest compared with the traditional method, and the formation system has effectively completed the task of keeping and tracking the expected moving formation.This is because of the fast piecewise power reaching rate, which makes the formation flight control of Leader-Follower UAVs have high control accuracy.
Further, the flight path yaw test is carried out, and the comparison structure is shown in Figure 4.
It can be seen from Figure 4 that the method in this paper has the minimum yaw.By selecting the piecewise power approaching parameter, the curve can converge to the expected formation distance better, and the convergence speed is faster.The formation ability can complete the UAV formation reorganization under disturbance and chattering in a short time, and the trajectory tracking error approaches zero.

Buffeting resistance test
Finally, the anti-buffeting performance of UAV formation flight control is tested, taking the Follower as the test object.In the test, for the formation determined by the Leader-azimuth constraint, through the real-time position and azimuth constraint of any two Leader UAVs, all information in the flight process is represented by the unit vector.The flight attitude angles of UAVs are tested by the improved and improved methods, and the antibuffeting performance is compared and analyzed according to the oscillation analysis of the flight attitude angles.The results are shown in Figure 5.According to the analysis of Figure 5, the Follower UAV gradually converges to the expected position under the action of the designed controller, and always fluctuates in a small range near the expected position, which shows that the method in this paper has good anti-chattering and anti-interference performance.

Conclusions
By constructing an optimized flight formation control model, combined with the reliability control and antidisturbance dynamic adjustment of UAV flight formation, the delay and disturbance during the formation transportation of multi-UAV system are avoided, and the stability and anti-interference of UAV flight control are guaranteed.In this paper, a Leader-Follower UAV formation flight control method based on feature modelling is proposed.A three-layer feedforward neural network is used to construct the control object model of UAV formation, and intermediate variables are introduced to construct the deviation control law of position and attitude.An adaptive algorithm is used to adjust the weights online, and the aircraft bodies are dynamically arranged in the order of Leader to Follower.The variable structure control method is used to analyze the convergence and stability of controller design errors, and a fast piecewise power approach factor is introduced into the PID controller to eliminate the chattering phenomenon of sliding mode control and improve the stability of UAV formation flight control.In the experimental test, the flight attitude stability test, trajectory tracking test, and buffeting resistance test of UAV formation are carried out.The research shows that the trajectory tracking error of this method is the lowest compared with the traditional method, the formation system effectively completes the task of keeping and tracking the expected moving formation, and the Leader-Follower UAV formation flight control has high control accuracy.The method in this paper has the minimum yaw.By selecting the piecewise power approach parameters, the curve can converge to the expected formation distance better, the convergence speed is faster, and the formation ability can complete the formation reorganization of UAV under disturbance and chatter in a short time.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Figure 1 .
Figure 1.Schematic diagram of the kinematic model of UAV formation movement.

Figure 2 .
Figure 2. Flight attitude test results of UAV formation leader and two followers.(a)This method; (b) PID; (c) Sliding mode control.

Figure 3 .
Figure 3. Comparative results of trajectory tracking performance of UAV formation.(a)This method; (b) PID; (c) Sliding mode control.

Table 1 .
Comparison of tracking control errors (unit: m).