Tether tension and end effector based coordinated control method for tethered aerial vehicles

Abstract

The application discloses a tethered aerial vehicle coordination control method based on tether tension and end effector, and aims at the accurate tracking control problem of a tethered aerial vehicle system under the conditions of coupling nonlinearity, unknown disturbance and position constraint, establishes a tethered aerial vehicle dynamics model, estimates unknown tether tension and external unknown air flow disturbance by using a fixed time extended state observer, and finally designs a tethered aerial vehicle coordination control method by using an obstacle Lyapunov method and a dynamic surface method. The application can make the tethered aerial vehicle accurately track an expected trajectory without violating the position constraint.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aircraft, and particularly relates to a tethered aircraft coordinated control method. BACKGROUND

[0002] Soft autonomous aerial refueling system, towed decoy system and tethered towed target system are widely used in the field of aviation, and they can be abstracted as a large main aircraft dragging a small aircraft without forward power through a tether, and the systems are collectively referred to as tethered aircraft systems. During the execution of a task, the tethered aircraft system often needs to accurately track the desired trajectory under complex and changeable airflow disturbance and on the premise of not violating the position constraint.

[0003] The flexibility and elasticity of the tether bring great coupling nonlinearity to the tethered aircraft system, and the airflow environment in which the system is located is complex and changeable, so the characteristics of the system itself and the position constraint problem required by the task all cause great difficulty in the design of the controller. SUMMARY

[0004] In order to overcome the deficiencies of the prior art, the application provides a tethered aircraft coordinated control method based on tether tension and end effector, which is aimed at the accurate tracking control problem of the tethered aircraft system under the conditions of coupling nonlinearity, unknown disturbance and position constraint, establishes a tethered aircraft dynamics model, then estimates the unknown tether tension and unknown external airflow disturbance by using a fixed-time extended state observer, and finally designs a tethered aircraft coordinated control method by using the obstacle Lyapunov method and the dynamic surface method. The application can enable the tethered aircraft to accurately track the desired trajectory on the premise of not violating the position constraint.

[0005] The technical solution adopted by the application to solve the technical problems comprises the following steps:

[0006] Step 1: tethered aircraft dynamics model establishment;

[0007] The tethered aircraft dynamics equation is:

[0008]

[0009] In the formula, x1=[l,β r ,α r ] T ; x3=[T n ,ψ,θ] T ; (·) T represents the transpose of a vector or matrix (·); denotes the derivative with respect to time; l is the distance from the connection point of the tether and the main aircraft to the center of mass of the tethered aircraft; β r , a r is the azimuth angle of l; ψ, θ are the yaw angle and the pitch angle of the tethered aircraft; ω y , ω z are the rotational angular velocities of the tethered aircraft; T n is the tether tension; u1 and u2 are the actuators of the tethered aircraft; A2 is a function related to x1 and x3; B2, A3, A4, B4 are system known items; C2, C3, C4 are system unknown items;

[0010] Step 2: Fixed-time extended observer design;

[0011] From equation (1), C2, C3, C4 are system unknown items, in which C3 is processed by the robustness of the controller; C2, C4 are estimated by the designed observer, which is estimated by the designed fixed-time extended state observer as shown in equations (2) and (3):

[0012]

[0013]

[0014] In the formula, is the estimated vector of x i , C i ; a1, a2, a3, a4 are positive real numbers; κ 1,1 , κ 1,2 , κ 2,1 , κ 2,2 , κ 2,3 , κ 3,1 , κ 3,2 , κ 4,1 , κ 4,2 , κ 4,3 are diagonal positive definite matrices; κ 23 = diag(κ 23,1 , κ 23,2 , κ 23,3 ); κ 43 = diag(κ 43,1 , κ 43,2 , κ 43,3 ) are positive real numbers, and diag(·) represents a diagonal matrix, in which the diagonal elements of the matrix are the elements of (·); sig a (·) means sig a (x) = [sig a (x1) siga (x2)sig a (x3)] T ; sign(·) denotes the sign function; |·| denotes the absolute value of ·;

[0015] Step 3: Controller Design;

[0016] The following coordinate transformation is used:

[0017]

[0018] In the formula, x d τ1, τ2, and τ3 are the expected tracking trajectories of x1, x2, A2, and x4, respectively;

[0019] Introduce a first-order low-pass filter:

[0020]

[0021] In the formula, λ i α is a diagonal positive definite matrix, and is the time constant. i τ is a virtual control variable; i For filter output;

[0022] Define the filtering error as:

[0023] e i =τ i -α i ,i=1,2,3 (6)

[0024] According to equations (5) and (6):

[0025]

[0026] In the formula, for The 2-norm, M i It is a positive real number; For λ i The inverse matrix;

[0027] Step 3-1: According to equations (1), (4) and (6), we know that:

[0028]

[0029] Design the Lyapunov function as follows:

[0030]

[0031] In the formula, z1=[z 1,1 ,z 1,2 ,z1,3 ] T ; k b,i is a positive real number satisfying z 1,i < k b,i ;

[0032] Differentiating equation (9) and substituting equations (7) and (8) into it, we have

[0033]

[0034] where

[0035] According to equation (10), the virtual control law can be chosen as

[0036]

[0037] where k1is a diagonal positive definite matrix;

[0038] Step 3-2: According to equations (1), (2), (4) and (6), we have

[0039]

[0040] The Lyapunov function is designed as

[0041]

[0042] Differentiating equation (13) and substituting equations (7) and (12) into it, we have

[0043]

[0044] According to equation (14), the virtual control law can be chosen as

[0045]

[0046] where k2is a diagonal positive definite matrix;

[0047] Step 3-3: According to equations (1), (4) and (6), we have

[0048]

[0049] where is the partial derivative of (*) with respect to (·);

[0050] The Lyapunov function is designed as

[0051]

[0052] Differentiating equation (17) and substituting equations (7) and (16) into it, we have

[0053]

[0054] The virtual control law is selected according to formula (18):

[0055]

[0056] In the formula, k3 is a diagonal positive definite matrix;

[0057] Step 3-4: According to formula (1), (3), (4) and (6), it can be known that:

[0058]

[0059] The Lyapunov function is designed as:

[0060]

[0061] Derivation is carried out on formula (21), and formula (20) is substituted into formula (21):

[0062]

[0063] The actual control law is selected as:

[0064]

[0065] In the formula, k4 is a diagonal positive definite matrix.

[0066] The beneficial effects of the present application are as follows:

[0067] 1. The present application and the standard backstepping method are different in coordinate transformation mode, the present application takes the system coupling nonlinear whole as a virtual control item, and fully utilizes model information;

[0068] 2. The present application and the traditional extended observer are different, the present application estimates the lumped disturbance by using the fixed time extended state observer, and solves the problem of slow convergence speed of the traditional extended observer;

[0069] 3. In view of the position constraint problem of the tethered aircraft system, the present application combines the barrier Lyapunov method and the dynamic surface method to design a coordinated controller, and ensures accurate tracking of the tethered aircraft under the premise of meeting the position constraint. BRIEF DESCRIPTION OF DRAWINGS

[0070] Figure 1 The present application is a schematic diagram of a tethered aircraft system.

[0071] Figure 2 The present application is a schematic diagram of airflow disturbance.

[0072] Figure 3 The present application is a tethered aircraft tracking trajectory diagram.

[0073] BRIEF DESCRIPTION OF DRAWINGS: 1 - main aircraft, 2 - tether, 3 - tethered aircraft. DETAILED DESCRIPTION

[0074] The application is further illustrated below in combination with the drawings and examples.

[0075] The application takes into account the coupling nonlinearity, unknown disturbance and position constraint problems of the system, and proposes a coordinated control method based on a fixed-time extended state observer, which ensures that the tethered aircraft accurately tracks the desired trajectory without violating the position constraint.

[0076] As shown in Figure 1 , a tethered aircraft coordinated control method based on tether tension and end effector includes the following steps:

[0077] 1. Mechanical model establishment

[0078] The tethered aircraft dynamics equation is:

[0079]

[0080] In the formula, x1 = [l, β r , α r ] T ; x3 = [T n , ψ, θ] T ; (·) T denotes the transpose of the vector or matrix (·); denotes the differential with respect to time; l is the distance from the connection point of the tether and the main aircraft to the center of mass of the tethered aircraft; β r , α r are the azimuth angles of l; ψ, θ are the yaw angle and pitch angle of the tethered aircraft; ω y , ω z are the rotation angular velocities of the tethered aircraft; T n is the tether tension; u1 and u2 are the actuators of the tethered aircraft; A2 is the relevant function of x1 and x3; B2, A3, A4, B4 are known items of the system; C2, C3, C4 are unknown items of the system, which are related to unknown air flow disturbance and unmeasurable tension and tension distance.

[0081] 2. Design of time-invariant extended observer

[0082] As can be seen from formula (1), C2, C3, C4 are unknown items of the system, among which the value of C3 is small and can be handled by the robustness of the controller, while the values of C2 and C4 are large and need to be estimated by an observer, and the fixed-time extended state observer shown below is designed to estimate them:

[0083]

[0084]

[0085] where, is x i ,C i the estimated vector of x a1,a2,a3,a4 are positive real numbers sufficiently small; κ 1,1 ,κ 1,2 ,κ 2,1 ,κ 2,2 ,κ 2,3 ,κ 3,1 ,κ 3,2 ,κ 4,1 ,κ 4,2 ,κ 4,3 is a diagonal positive definite matrix; κ 23 =diag(κ 23,1 ,κ 23,2 ,κ 23,3 ); κ 43 =diag(κ 43,1 ,κ 43,2 ,κ 43,3 ) are positive real numbers, diag(·) denotes a diagonal matrix whose diagonal elements are the elements of (·); sig a (·) has the meaning sig a (x) = [sig a (x1) sig a (x2) sig a (x3)] T ; sign(·) denotes the sign function; |·| denotes the absolute value of ·.

[0086] 3. Controller design

[0087] Unlike the standard backstepping method, the present application uses the following coordinate transformation:

[0088]

[0089] where x d , τ1, τ2, τ3 are the desired tracking trajectories of x1, x2, A2, x4, respectively;

[0090] In order to solve the "derivative explosion" problem existing in the backstepping method, a first-order low-pass filter is introduced:

[0091]

[0092] In the formula, λ i α is a diagonal positive definite matrix, and is the time constant. i τ is a virtual control variable; i This is the filter output.

[0093] Define the filtering error as:

[0094] e i =τ i -α i ,i=1,2,3 (29)

[0095] According to equations (5) and (6):

[0096]

[0097] In the formula, for The 2-norm, M i It is a positive real number; For λ i The inverse matrix.

[0098] Step 1: According to equations (1), (4) and (6), we know that:

[0099]

[0100] Design the Lyapunov function as follows:

[0101]

[0102] In the formula, z1=[z 1,1 ,z 1,2 ,z 1,3 ] T ;k b,i Let z be a positive real number that satisfies 1,i <k b,i .

[0103] Differentiating equation (9) and substituting it into equations (7) and (8), we get:

[0104]

[0105] In the formula,

[0106] According to equation (10), the virtual control law can be selected as:

[0107]

[0108] In the formula, k1 is a diagonal positive definite matrix.

[0109] Step 2: According to equations (1), (2), (4) and (6), we have

[0110]

[0111] Design the Lyapunov function as

[0112]

[0113] Take the derivative of equation (13) and substitute equations (7) and (12) into it, we have

[0114]

[0115] According to equation (14), we can choose the virtual control law as

[0116]

[0117] where k2 is a diagonal positive definite matrix.

[0118] Step 3: According to equations (1), (4) and (6), we have

[0119]

[0120] where is the partial derivative of (*) with respect to (·).

[0121] Design the Lyapunov function as

[0122]

[0123] Take the derivative of equation (17) and substitute equations (7) and (16) into it, we have

[0124]

[0125] According to equation (18), we can choose the virtual control law as

[0126]

[0127] where k3 is a diagonal positive definite matrix.

[0128] Step 4: According to equations (1), (3), (4) and (6), we have

[0129]

[0130] Design the Lyapunov function as

[0131]

[0132] Differentiating equation (21) and substituting it into equation (20), we get:

[0133]

[0134] The actual control law is selected as follows:

[0135]

[0136] In the formula, k4 is a diagonal positive definite matrix.

[0137] like Figure 2 The figure shown is a diagram of airflow disturbance, with the vertical axis being v. x v y v z These represent the components of the external airflow velocity along the x, y, and z axes, respectively.

[0138] like Figure 3 The image shows the tracking trajectory of the tethered aircraft, with the vertical axis representing l and β. r α r The desired trajectory and the tracking trajectory are respectively located on the l-axis and β-axis. r Axis, α r The components of the axis are shown, where the dotted line represents the desired trajectory, the solid line represents the tracking trajectory, and the dashed line represents the position constraint boundary. The results show that the tethered aircraft can still track the desired trajectory well under the influence of external airflow disturbances without violating the constraint boundaries.

Claims

1. A tether tension and end effector based coordinated control method for tethered aerial vehicles, characterized by, Comprising the following steps: Step 1: Establishing the dynamics model of the tethered aerial vehicle; The dynamics equation of the tethered aerial vehicle is: （1） wherein ; ; ; ; ; denotes the transpose of a vector or matrix ; denotes the derivative with respect to time ; is the distance from the tether and main aircraft connection point to the tethered aircraft center of mass , is the azimuth angle of ; , is the yaw and pitch angle of the tethered aircraft , is the rotational angular velocity of the tethered aircraft is the tether tension and is the actuator of the tethered aircraft is the correlation function of and ; , , , are the system knowns , , are the system unknowns Step 2: Designing the fixed-time extended state observer; From equation (1), , , are unknown terms of the system, where robustness of the controller; , Estimation is performed by designing an observer, and a fixed-time extended state observer as shown in equations (2) and (3) is designed to estimate it: （2） （3） wherein ; is an estimate vector of ; ; ; ; ; ; ; ; ; ; ; ; ; is a positive real number; , , , , , , , , , is a diagonal positive definite matrix; ; ; ; is a positive real number, denotes a diagonal matrix, wherein the diagonal elements of the matrix are the elements of ; ; has the meaning of ; ; denotes the sign function; denotes the absolute value of ; Step 3: Designing the controller; The following coordinate transformation is adopted: （4） wherein , , , are respectively , , , desired tracking trajectories; A first-order low-pass filter is introduced: （5） wherein is a time constant and is a diagonal positive definite matrix; is a virtual control variable; is a filter output; The filtering error is defined as: （6） According to equations (5) and (6), we have: （7） wherein , is the 2-norm of is a positive real number; is the inverse matrix of Step 3-1: According to equations (1), (4) and (6), we have: （8） The Lyapunov function is designed as: （9） In the formula, ; is a positive real number satisfying ; The derivative of equation (9) is taken and substituted into equations (7) and (8), we have: （10） In the formulae, ; According to equation (10), the virtual control law is selected as: （11） In the formula, is a diagonal positive definite matrix; Step 3-2: According to equations (1), (2), (4) and (6), we have: （12） The Lyapunov function is designed as: （13） The derivative of equation (13) is taken and substituted into equations (7) and (12), we have: （14） According to equation (14), the virtual control law is selected as: （15） In the formula, is a diagonal positive definite matrix; Step 3-3: According to equations (1), (4) and (6), we have: （16） wherein is to partial derivative; The Lyapunov function is designed as: （17） The derivative of equation (17) is taken and substituted into equations (7) and (16), we have: （18） According to equation (18), the virtual control law is selected as: （19） In the formula, is a diagonal positive definite matrix; Step 3-4: According to equations (1), (3), (4) and (6), we have: （20） The Lyapunov function is designed as: （21） The derivative of equation (21) is taken and substituted into equation (20), we have: （22） The actual control law is selected as: （23） In the formula, is a diagonal positive definite matrix.

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