A hyperbolic sine function-based nonlinear proportional-derivative attitude tracking control method

Abstract

The application discloses a nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function, and first adopts modified Rodrigues parameters (MRPs) to express a dynamic equation of a rigid spacecraft attitude tracking error; then utilizes proportional-differential control technology to design a nonlinear proportional-differential attitude tracking controller based on hyperbolic sine function, and explicitly gives a subset of an attraction domain of two stable equilibrium points; finally, the effectiveness of the designed control method is verified by using a simulink module in MATLAB shown in Fig. 1. The attitude controller designed by the application makes the closed-loop rigid spacecraft attitude tracking control system have anti-unwinding performance, and can ensure that the spacecraft reaches any desired attitude by rotating an angle less than 180 degrees under the condition that the initial attitude angular velocity error is zero.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of spacecraft attitude tracking control, and relates to a rigid spacecraft unwinding-free attitude tracking control method, in particular to a hyperbolic sine function-based nonlinear proportional-derivative attitude tracking control method. BACKGROUND

[0002] Traditional rigid spacecraft attitude tracking control algorithms can only guarantee that the closed-loop rigid spacecraft attitude tracking control system contains one stable equilibrium point, which may cause the attitude unwinding problem of the spacecraft, resulting in the need for excessive fuel consumption for the spacecraft to reach the desired attitude when completing certain attitude control tasks. SUMMARY

[0003] In order to solve the unwinding problem in rigid spacecraft attitude tracking control, the present application provides a hyperbolic sine function-based nonlinear proportional-derivative attitude tracking control method. The method designs a nonlinear proportional-derivative attitude tracking control law for the unwinding problem existing in rigid spacecraft attitude tracking control, ensures that the closed-loop system has two stable equilibrium points by introducing a hyperbolic sine function, and gives the attraction domain estimation of the two stable equilibrium points, realizing unwinding-free attitude tracking control.

[0004] The purpose of the present application is achieved by the following technical solutions:

[0005] A hyperbolic sine function-based nonlinear proportional-derivative attitude tracking control method, comprising the following steps:

[0006] Step 1, using modified Rodrigues parameters (MRPs) to describe the attitude of a rigid spacecraft, for a rigid spacecraft with external disturbance, establishing the error kinematics equation and the dynamics equation based on MRPs as follows:

[0007]

[0008] wherein σ e represents the attitude tracking error between the body attitude σ of the rigid spacecraft and the desired attitude σ d , ω e represents the angular velocity error between the body angular velocity ω of the rigid spacecraft and the desired angular velocity ω d , M(σ e ) represents the following matrix:

[0009]

[0010] J is the rotational inertia matrix of the rigid spacecraft, u is the control input of the rigid spacecraft attitude tracking control system, and τ is a vector.

[0011] Let θ(t) and e represent the attitude rotation angle and the attitude rotation axis of the spacecraft respectively, σ e be written as follows:

[0012] σ e = etan(θ(t) / 4);

[0013] In addition, the following formula is true:

[0014]

[0015] Step 2, for a rigid spacecraft with external disturbance, the following anti-unwinding attitude tracking control law is designed, and by introducing the hyperbolic sine function, it is guaranteed that the closed-loop system of the rigid spacecraft attitude tracking control system has two stable equilibrium points E1 and E2, wherein:

[0016] The designed anti-unwinding attitude tracking control law is as follows:

[0017] u = -τ - k1Jω e -k2Jρσ e ;

[0018] Wherein, k1 and k2 are positive numbers, and ρ is:

[0019]

[0020] p = 2arctan||σ e ||;

[0021] The forms of the two stable equilibrium points E1 and E2 are as follows:

[0022] E1 = {(σ e ,ω e ):4arctan||σ e || = 0, ω e = 0};

[0023] E2 = {(σ e ,ω e ):4arctan||σ e || = 2π, ω e = 0}.

[0024] Compared with the prior art, the present application has the following advantages:

[0025] 1. For the unwinding problem existing in the rigid spacecraft attitude tracking control, a nonlinear proportional-differential attitude tracking control law is designed to ensure that the closed-loop system has two stable equilibrium points.

[0026] 2. The attitude controller designed by the application makes the closed-loop rigid spacecraft attitude tracking control system have anti-unwinding performance, and can ensure that the spacecraft reaches any desired attitude by rotating an angle less than 180 degrees under the condition that the initial attitude angular velocity error is zero. BRIEF DESCRIPTION OF DRAWINGS

[0027] Figure 1 A simulink model of the nonlinear proportional-derivative attitude tracking control method based on hyperbolic sine function;

[0028] Figure 2 A time response curve of the spacecraft attitude rotation angle θ(t);

[0029] Figure 3 A time response curve of ω;

[0030] Figure 4 A time response curve of u;

[0031] Figure 5 A time response curve of energy E. DETAILED DESCRIPTION

[0032] The technical solutions of the application are further described below in combination with the drawings, but are not limited thereto, and any modification or equivalent replacement of the technical solutions of the application without departing from the spirit and scope of the technical solutions of the application shall be included in the protection scope of the application.

[0033] The application provides a nonlinear proportional-derivative attitude tracking control method based on hyperbolic sine function, which comprises the following steps: Figure 1 The effectiveness of the designed control method is verified by using the simulink module in MATLAB. Specifically, the steps include the following:

[0034] Step 1, the modified Rodrigues parameters (MRPs) are used to describe the attitude of the rigid spacecraft, and the error kinematics equation and the dynamic equation based on the MRPs are established for the rigid spacecraft with external disturbance as follows:

[0035]

[0036] wherein represents the attitude tracking error between the attitude of the rigid spacecraft body and the desired attitude, and has:

[0037] ​​

[0038] represents the angular velocity error between the rigid spacecraft body attitude angular velocity and the desired attitude angular velocity , and has

[0039] ω e = ω - Rω d

[0040] where R represents the rotation matrix of the rigid spacecraft from the desired coordinate system to the body coordinate system, and

[0041]

[0042] For any vector x × is skew-symmetric, defined as follows:

[0043]

[0044] In addition, the matrix is represented as follows:

[0045]

[0046] is the rotational inertia matrix of the rigid spacecraft, and is symmetric; is the control input of the rigid spacecraft attitude tracking control system. In addition, the vector τ is:

[0047]

[0048] In addition, let θ(t) ∈ [0, 2π] and respectively represent the attitude rotation angle and the attitude rotation axis of the spacecraft, then the MRP parameter σ e can be written in the following form:

[0049] σ e = e tan(θ(t) / 4) (3)

[0050] In addition, the following formula is true:

[0051]

[0052] In addition, the closed-loop system of the rigid spacecraft attitude tracking control system has the following two equilibrium points:

[0053]

[0054] Step 2, design an anti-reel attitude tracking control law for the rigid spacecraft with external disturbance:

[0055] The designed anti-backlash posture tracking control law is in the form of

[0056] u = -τ - k1Jω e - k2Jpσ e (7)

[0057] where k1 and k2 are positive numbers, and p is

[0058]

[0059] where

[0060] p = 2arctan||σ e || (9)

[0061] The Lyapunov function is chosen as

[0062]

[0063] where

[0064] η = max(cosh(cosp)) (11)

[0065] Obviously, V(t) ≥ 0. In addition, the following equation holds:

[0066]

[0067] According to equations (3) and (9), we have

[0068]

[0069] Combining this equation with equation (11), we have

[0070] η = cosh(cosp)| p＝0 = cosh(cosp)| p＝ π (13)

[0071] According to equations (5), (6), (9) and (10), the following equation holds:

[0072]

[0073] Taking the derivative of equation (10), we have

[0074]

[0075] Substituting the control law (7) into the above equation, we have

[0076]

[0077] Obviously, when , we have E1 and E2 are both stable equilibrium points.

[0078] Estimate the attraction domain of the two stable equilibrium points, and realize the no-backlash posture tracking control:

[0079] The design parameter k2 satisfies the following restriction condition:

[0080]

[0081] where,

[0082]

[0083] g(t) = p||σ e || (18)

[0084] p is given in equation (8). Under the premise that the parameter k2 of the control law (7) satisfies the restriction condition (15), the subsets of the attraction domain of the stable equilibrium points E1 and E2 are respectively:

[0085]

[0086] Definition:

[0087]

[0088] Obviously, proving equation

[0089]

[0090] is equivalent to proving

[0091]

[0092] According to equation (4), we have:

[0093]

[0094] According to equation (16), we further have:

[0095]

[0096] The above equation can be further written as:

[0097]

[0098] where,

[0099]

[0100] Since

[0101] e T ωe ||e|| < ||ω|| < ||e|| + ||ω|| e || = ||ω e || = ||ω

[0102] holds, so that the equation holds. Then, for l in equation (26), the following property holds:

[0103]

[0104] The region and are divided into two large subsets as follows, respectively:

[0105]

[0106] In addition, define:

[0107]

[0108] According to the above equation, we have:

[0109]

[0110] This shows that for and the following holds: Therefore, when and , decreases. In addition, when θ(t) ∈ (0, π), as θ(t) increases, decreases; when θ(t) ∈ (π, 2π), as θ(t) decreases, increases. Further, the following equation holds:

[0111]

[0112] According to this relationship, it can be obtained that the subsets of the attraction domain of stable equilibrium points E1 and E2 are A and B, respectively.

[0113] The control effect of the hyperbolic sine function-based nonlinear proportional-derivative control law designed for a rigid spacecraft is illustrated by example simulation. The main parameters of the rigid spacecraft are selected as follows:

[0114] The moment of inertia J of the rigid spacecraft is:

[0115]

[0116] The initial value of the attitude is:

[0117] ω(0) =

[000] T

[0118] σ(0) = [0 0 0] T

[0119] The desired attitude is:

[0120] ω d = 0.05[sin(0.01t) sin(0.02t) sin(0.03t)] T

[0121] σ d (0) = [0.1 0.2 -0.3] T

[0122] The parameters of the nonlinear proportional-derivative control law based on hyperbolic sine function are selected as:

[0123] k1 = 2, k2 = 2, ε = 0.2

[0124] In addition, the control law (13) in reference [1] (M. R. Binette, C. J. Damaren, and L. Pavel, “Nonlinear H∞ attitude control using modified rodrigues parameters,” Journal of Guidance, Control, and Dynamics, vol. 37, no. 6, pp. 2017-2021, 2014.) is used for comparison, and the adjustable parameters are selected as:

[0125] γ = 2, q1 = 523.4076, q2 = 7500, a = 100, b = 100

[0126] The control law (15) in reference [2] (K. Subbarao, “Nonlinear pid-like controllers for rigid-body attitude stabilization,” The Journal of the Astronautical Sciences, vol. 52, no. 1, pp. 61-74, 2004.) is used for comparison, and the adjustable parameters are selected as:

[0127] k v = 25, k p = 200, k i = 0.1

[0128] The simulation diagram of the closed-loop system is obtained as Figures 2-5In the four figures, NNPD represents the control law designed in this paper, Controller A and Controller B represent the control laws in [1] and [2] respectively. By Figures 2-5 It can be seen that the control law designed in this paper needs less energy to complete the attitude control task while the convergence speed of tracking error is roughly the same.

Claims

1. A nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function, characterized in that The method comprises the following steps: Step 1: Use modified Rodrigues parameters (MRPs) to describe the attitude of a rigid spacecraft. For a rigid spacecraft with external interference, the error kinematic equations and dynamic equations based on MRPs are as follows: in, Represents the attitude of the rigid spacecraft and expected posture The posture tracking error between Indicates the angular velocity of the rigid spacecraft body and the desired attitude angular velocity The angular velocity error between represents a matrix; is the moment of inertia matrix of the rigid spacecraft, is the control input of the rigid spacecraft attitude tracking control system, is a vector; make and represent the attitude angle and attitude rotation axis of the spacecraft respectively, Written as follows: ； In addition, the following holds: ； Step 2: For rigid spacecraft with external interference, the following anti-unwinding attitude tracking control law is designed. By introducing the hyperbolic sine function, the closed-loop system of the rigid spacecraft attitude tracking control system has two stable equilibrium points. and ,in: The designed anti-unwinding attitude tracking control law is as follows: ； in, and is a positive number, for: ； ； Two stable equilibrium points and The form is as follows: ； 。 2. The nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function according to claim 1 is characterized in that In the step 1, ,in, Represents the rotation matrix of the rigid body spacecraft from the desired coordinate system to the body coordinate system.

3. The nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function according to claim 2 is characterized in that described .

4. The nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function according to claim 1 is characterized in that In the step 1, .

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