Nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function

Abstract

The invention discloses a nonlinear proportional-differential attitude tracking control method based on a hyperbolic sine function. The nonlinear proportional-differential attitude tracking control method comprises the following steps: firstly, representing a kinetic equation of an attitude tracking error of a rigid spacecraft by adopting a modified Rodrigues parameter (MRPs); secondly, designing a nonlinear proportional-differential attitude tracking controller based on a hyperbolic sine function by utilizing a proportional-differential control technology, and explicitly giving a subset of an attraction domain of two stable equilibrium points; and finally, verifying the effectiveness of the designed control method by using a simulink module in MATLAB shown in the figure 1. By adopting the attitude controller designed by the invention, a closed-loop rigid spacecraft attitude tracking control system has an anti-unwinding performance, and under the condition that an initial attitude angular velocity error is zero, the spacecraft can be ensured to reach any expected attitude by rotating an angle less than 180 degrees.

Description

Technical Field

[0001] The present invention belongs to the field of spacecraft attitude tracking control, and relates to a non-rewinding attitude tracking control method for a rigid spacecraft, and specifically to a nonlinear proportional-differential attitude tracking control method based on a hyperbolic sine function. Background Art

[0002] Traditional rigid spacecraft attitude tracking control algorithms can only ensure that the closed-loop rigid spacecraft attitude tracking control system contains a stable equilibrium point, which may cause the spacecraft's attitude unwinding problem, resulting in the spacecraft consuming too much fuel to reach the desired attitude when completing certain attitude control tasks. Summary of the invention

[0003] In order to solve the unwinding problem in the attitude tracking control of rigid spacecraft, the present invention provides a nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function. This method designs a nonlinear proportional-differential attitude tracking control law for the unwinding problem in the attitude tracking control of rigid spacecraft. By introducing the hyperbolic sine function, it ensures that the closed-loop system has two stable equilibrium points, and gives the attraction domain estimation of the two stable equilibrium points, so as to realize the attitude tracking control without unwinding.

[0004] The objective of the present invention is achieved through the following technical solutions:

[0005] A nonlinear proportional-differential attitude tracking control method based on a hyperbolic sine function comprises the following steps:

[0006] Step 1: Use modified Rodriguez parameters (MRPs) to describe the attitude of rigid spacecraft. For rigid spacecraft with external interference, the error kinematic equations and dynamic equations are established based on MRPs as follows:

[0007]

[0008] Among them, σ e Denotes the rigid spacecraft body attitude σ and the desired attitude σ d The posture tracking error between e Represents the rigid spacecraft body attitude angular velocity ω and the desired attitude angular velocity ω d The angular velocity error between e ) represents the following matrix:

[0009]

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

[0011] Let θ(t) and e represent the attitude angle and attitude rotation axis of the spacecraft, respectively, and σ e Written in the following form:

[0012] σ e =etan(θ(t) / 4);

[0013] In addition, the following holds:

[0014]

[0015] 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 E1 and E2, where:

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

[0017] u=-τ-k1Jω e -k2Jρσ e ;

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

[0019]

[0020] p=2arctan||σ e ||;

[0021] The two stable equilibrium points E1 and E2 are of the following form:

[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 invention has the following advantages:

[0025] 1. Aiming at the unwinding problem existing in the attitude tracking control of rigid spacecraft, 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 present invention enables the closed-loop rigid spacecraft attitude tracking control system to have anti-unwinding performance. When the initial attitude angular velocity error is zero, it can ensure that the spacecraft reaches any desired attitude by rotating at an angle less than 180 degrees. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 It is the Simulink model of the nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function;

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

[0029] Figure 3 is the time response curve of ω;

[0030] Figure 4 is the time response curve of u;

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

[0032] The technical solution of the present invention is further described below in conjunction with the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be included in the protection scope of the present invention.

[0033] The present invention provides a nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function. Firstly, modified Rodrigues parameters (MRPs) are used to represent the dynamic equation of the attitude tracking error of a rigid spacecraft. Then, a nonlinear proportional-differential attitude tracking controller based on the hyperbolic sine function is designed by using proportional-differential control technology, and a subset of the attraction domain of two stable equilibrium points is explicitly given. Finally, the method is applied to Figure 1 The Simulink module in MATLAB shown in the figure verifies the effectiveness of the designed control method. The specific steps include the following:

[0034] Step 1: Use modified Rodriguez parameters (MRPs) to describe the attitude of a rigid spacecraft. For a rigid spacecraft with external interference, the error kinematic equations and dynamic equations established based on MRPs are as follows:

[0035]

[0036] Where, represents the attitude of the rigid spacecraft With expectation attitude The posture tracking error between , and:

[0037]

[0038] Represents the angular velocity of the rigid spacecraft and the expected attitude angular velocity The angular velocity error between and has:

[0039] ω e =ω-Rω d

[0040] Among them, 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 a skew-symmetric matrix, defined as follows:

[0043]

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

[0045]

[0046] is the moment of 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 represent the attitude angle and attitude rotation axis of the spacecraft respectively, then the MRPs parameter σ e It can be written as follows:

[0049] σ e =etan(θ(t) / 4) (3)

[0050] In addition, the following holds:

[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-unwinding attitude tracking control law for a rigid spacecraft with external interference:

[0055] The designed anti-unwinding attitude tracking control law is as follows:

[0056] u=-τ-k1Jω e -k2Jρσ e (7)

[0057] Where k1 and k2 are positive numbers and ρ is:

[0058]

[0059] in:

[0060] p=2arctan||σ e || (9)

[0061] Choose the following Lyapunov function:

[0062]

[0063] in,

[0064] η=max(cosh(cosp)) (11)

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

[0066]

[0067] According to formula (3) and (9), we can get:

[0068]

[0069] Combining this formula with formula (11), we can get:

[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] By taking the derivative of formula (10), we can get:

[0074]

[0075] Substituting the control law (7) into the above equation, we can obtain:

[0076]

[0077] Obviously, when Sometimes, there is Combining this equation with equation (14), we can see that both equilibrium points E1 and E2 are stable equilibrium points.

[0078] Estimate the attraction domain of two stable equilibrium points to achieve attitude tracking control without unwinding:

[0079] The design parameter k2 satisfies the following constraints:

[0080]

[0081] in,

[0082]

[0083] g(t)=ρ||σ e || (18)

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

[0085]

[0086] definition:

[0087]

[0088] Obviously, the proof

[0089]

[0090] Establishment is equivalent to proof

[0091]

[0092] According to formula (4), we can get:

[0093]

[0094] According to formula (16), we can further obtain:

[0095]

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

[0097]

[0098] in,

[0099]

[0100] because

[0101] e T ωe ≤||e||||ω e ||=||ω e ||

[0102] Established, so the formula Then, for l in formula (26), the following property holds:

[0103]

[0104] The area and They are divided into the following two large subsets:

[0105]

[0106] Additionally, define:

[0107]

[0108] According to the above formula, we can get:

[0109]

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

[0111]

[0112] According to this relationship, we can get that the subsets of the attraction domains of the stable equilibrium points E1 and E2 are A and B respectively.

[0113] The following example simulation illustrates the control effect of the nonlinear proportional-differential control law based on the hyperbolic sine function designed for rigid spacecraft. The main parameters of the rigid spacecraft are selected as follows:

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

[0115]

[0116] The initial value of the posture is:

[0117] ω(0)=

[000] T

[0118] σ(0)=

[000] T

[0119] The expected posture is:

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

[0121] σ d (0) = [0.10.2-0.3] T

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

[0123] k1=2,k2=2,ε=0.2

[0124] In addition, the control law (13) in reference [1] (MR Binette, CJ 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 its 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 its adjustable parameters are selected as:

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

[0128] The closed-loop system simulation diagram obtained is Figure 2-Figure 5In these four figures, NNPD represents the control law designed by the present invention, and ControllerA and ControllerB represent the control laws in references [1] and [2] respectively. Figure 2-5 It can be seen that: when the tracking error convergence speed is roughly the same, the control law proposed in the present invention requires less energy consumption to complete the attitude control task.

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 Rodriguez parameters (MRPs) to describe the attitude of a rigid spacecraft. For a rigid spacecraft with external interference, the error kinematic equations and dynamic equations established based on MRPs are as follows: Among them, σ e Denotes the rigid spacecraft body attitude σ and the desired attitude σ d The posture tracking error between e Represents the rigid spacecraft body attitude angular velocity ω and the desired attitude angular velocity ω d The angular velocity error between e ) represents a matrix; J is the moment of inertia matrix of the rigid spacecraft, u is the control input of the attitude tracking control system of the rigid spacecraft, and τ is a vector; Let θ(t) and e represent the attitude angle and attitude rotation axis of the spacecraft, respectively, and σ e Written in the following form: s e =etan(θ(t) / 4); 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 E1 and E2, where: The designed anti-unwinding attitude tracking control law is as follows: u=-τ-k1Jω e -k2Jрс e ; Where k1 and k2 are positive numbers and ρ is: p=2arctan||σ e ||; The two stable equilibrium points E1 and E2 are of the following form: E1={(σ e ,oh e ):4arctan||σ e ||=0,ω e =0}; E2={(σ e ,oh e ):4arctan||σ e ||=2π,ω e =0}; 2. The nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function according to claim 1 is characterized in that In step 1, ω e =ω-Rω d , where R represents the rotation matrix of the rigid 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 Said 4. The nonlinear proportional-differential attitude tracking control method based on hyperbolic sine function according to claim 1 is characterized in that In step 1, for any vector x=[x1 x2 x3] T , x × is a skew-symmetric matrix, defined as follows:

5. 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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