Non-singular practical fixed-time attitude tracking control method based on double mrps

Abstract

The application discloses a non-singular actual fixed-time attitude tracking control method based on double MRPs, and comprises the following steps: step 1, adopting modified Rodrigues parameters (MRPs) and shadow parameters thereof to describe a rigid spacecraft attitude, and establishing error kinematics equations and dynamics equations based on the double MRPs; step 2, designing a non-singular fixed-time sliding mode attitude tracking control law, and realizing fixed-time control by taking a non-singular sliding mode function norm as an exponential coefficient of the attitude tracking control law. The method is aimed at the problems of convergence time dependence on initial conditions and singularity of a sliding mode surface, a non-singular segmented sliding mode function is designed, and then the two-norm of the sliding mode function is taken as a variable to dynamically construct the exponential coefficient of the controller, thereby realizing non-singular fixed-time attitude tracking control of the spacecraft. The attitude controller designed by the application can guarantee convergence to a specified equilibrium point within a fixed time for any initial state.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft attitude tracking and control, and relates to a non-singular attitude tracking and control method for rigid body spacecraft, specifically a non-singular real-time attitude tracking and control method based on dual MRPs. Background Technology

[0002] The convergence time of rigid body spacecraft attitude tracking control algorithms based on traditional terminal sliding mode control design usually depends on the initial state of the system, and the sliding surface has singular problems, which may lead to the challenge of dynamic performance degradation of the control system in practical engineering applications. Summary of the Invention

[0003] To address the convergence time-dependent initial condition problem and the singularity of the sliding surface in attitude tracking control of rigid body spacecraft, this invention provides a non-singular real-fixed-time attitude tracking control method based on dual MRPs. This method designs a non-singular piecewise sliding mode function to address both the convergence time-dependent initial condition problem and the sliding surface singularity problem. Then, the L2 norm of this sliding mode function is used as a variable to dynamically construct the exponential coefficients of the controller, thereby achieving non-singular fixed-time attitude tracking control of the spacecraft.

[0004] The objective of this invention is achieved through the following technical solution:

[0005] A nonsingular real-time attitude tracking and control method based on dual MRPs includes the following steps:

[0006] Step 1: Describe the attitude of the rigid body spacecraft using modified Rodrigues parameters (MRPs) and their shadow parameters. For rigid body spacecraft with external disturbances, establish error kinematic equations and dynamic equations based on dual MRPs:

[0007]

[0008] in, Represented as:

[0009]

[0010] In the formula, Represents the initial time, set Indicates the initial time. and The selection criteria Indicates the attitude of the rigid body spacecraft. With expected posture The attitude tracking error between them express The shadow parameters, Represents the attitude angular velocity of a rigid body spacecraft. With desired attitude angular velocity angular velocity between Here is the rotational inertia matrix of the rigid body spacecraft. This serves as the control input for the attitude tracking and control system of a rigid body spacecraft. External disturbances For vectors, , express The identity matrix, yes The transpose of , for The cross product matrix is ​​as follows:

[0011] .

[0012] here express The One component;

[0013] make and These represent the spacecraft's attitude angle and attitude rotation axis, respectively. Represented as: ;

[0014] Represented as:

[0015] ;

[0016] Furthermore, the following equation holds true:

[0017] ;

[0018] Step 2: For rigid body spacecraft subject to external disturbances, design a non-singular fixed-time sliding mode attitude tracking control law. Fixed-time control is achieved by using the non-singular sliding mode function norm as the exponential coefficient of the attitude tracking control law. The non-singular fixed-time sliding mode attitude tracking control law is as follows:

[0019]

[0020] in, For the system's control input, and These represent the equivalent control term and the disturbance control term, respectively. This is a sliding surface approach control term. For sliding mode function, It is a vector;

[0021] make express The One portion, The expression is:

[0022]

[0023] The expression is:

[0024]

[0025] To ensure and Continuity, parameters The following relationship exists:

[0026]

[0027] , and It is a positive number. Greater than 1, satisfy , satisfy:

[0028]

[0029] And there are Let vector , For constants, the notation is... and Let the following vectors be represented respectively:

[0030]

[0031] in This represents the absolute value operation. To represent a symbolic function, we have:

[0032]

[0033] The final convergence time is:

[0034] .

[0035] Compared with the prior art, the present invention has the following advantages:

[0036] 1. To address the convergence time-dependent initial condition problem in attitude tracking control of rigid body spacecraft, an attitude tracking control law based on dual MRPs is designed with the sliding mode function L2 norm as the exponential coefficient to achieve fixed-time tracking control.

[0037] 2. The sliding mode function designed using this invention is nonsingular when the spacecraft attitude error approaches 0.

[0038] 3. The attitude controller designed using this invention can guarantee convergence to a specified equilibrium point within a fixed time for any initial state. Attached Figure Description

[0039] Figure 1 It is a Simulink model based on a nonsingular real fixed-time attitude tracking controller with dual MRPs;

[0040] Figure 2 yes Time response curve;

[0041] Figure 3 yes The time response curve. Detailed Implementation

[0042] The technical solution of the present invention will be further described below, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0043] This invention provides a nonsingular real-time fixed-time attitude tracking control method based on dual MRPs. First, based on modified Rodrigues parameters (MRPs) and their shadow parameters, i.e., dual MRPs, the dynamic equations representing the attitude tracking error of a rigid body spacecraft are given, and the selection conditions for choosing between MRPs and their shadow parameters at the initial moment are explicitly provided. Then, a nonsingular sliding mode function is designed, and using this sliding mode function, an attitude tracking control law with the sliding mode function norm as the exponential coefficient is designed to achieve fixed-time control. Specifically, the method includes the following steps:

[0044] Step 1: The attitude of the rigid body spacecraft is described using modified Rodrigues parameters (MRPs) and their shadow parameters. For a rigid body spacecraft with external disturbances, the following error kinematic equations and dynamic equations are established based on dual MRPs:

[0045]

[0046] in, Represented as:

[0047]

[0048] In the formula, Indicates the initial time. Indicates the initial time. and The selection criteria Indicates the attitude of the rigid body spacecraft. With expected posture The attitude tracking error between them is:

[0049]

[0050] and The relationship is:

[0051]

[0052] Represents the attitude angular velocity of a rigid body spacecraft. With desired attitude angular velocity The angular velocity error between them, and we have:

[0053]

[0054] in, Let represent the rotation matrix of the rigid body spacecraft from the desired coordinate system to the body coordinate system, and:

[0055]

[0056] For any vector , Let it be an oblique symmetric matrix, defined as follows:

[0057]

[0058] In addition, matrix It is expressed as follows:

[0059]

[0060] Let be the rotational inertia matrix of the rigid body spacecraft, and it is symmetric; This serves as the control input for the attitude tracking and control system of a rigid body spacecraft. Additionally, the vector... for:

[0061]

[0062] In addition, and Let MRPs represent the attitude angle and attitude rotation axis of the spacecraft, respectively. It can be written in the following form:

[0063]

[0064] MRPs shadow parameters It can be written as:

[0065]

[0066] In addition, regarding The following equation holds true:

[0067]

[0068] Step 2: For rigid body spacecraft with external disturbances, design the following non-singular fixed-time attitude tracking control law:

[0069]

[0070] in, , and It is a positive number. Greater than 1, satisfy , satisfy And there are . The expression is:

[0071]

[0072] The expression is:

[0073]

[0074] To ensure and Continuity, parameters Existence Relationship:

[0075]

[0076] because ,Depend on From the expression, we can see that when hour, There is no singularity problem.

[0077] Analyzing the convergence properties, the following Lyapunov function is selected:

[0078]

[0079] Obviously there is. Then we have:

[0080]

[0081] right ,have:

[0082]

[0083] From this, we can obtain ,when Sometimes, .

[0084] right ,have:

[0085]

[0086] Integrating both sides yields That is, when hour, .

[0087] Next, we will construct the Lyapunov function. Analysis on the sliding surface The convergence properties of the surface.

[0088] have:

[0089]

[0090] because From time to time ,but:

[0091]

[0092] It can be known On the sliding surface The value decreases monotonically.

[0093] To further confirm the convergence time, the analysis can be appropriate. hour, Therefore, we take:

[0094]

[0095] but Sometimes, ,and So, for Both sides arrive Points can be earned appropriately. hour, That is, the system achieves convergence in a fixed time, with the upper bound being:

[0096] .

[0097] The following simulation example illustrates the control effect of a nonsingular real fixed-time attitude tracking control law based on dual MRPs designed for a rigid body spacecraft. The main parameters of the rigid body spacecraft are selected as follows:

[0098] Moment of inertia of rigid body spacecraft for:

[0099]

[0100] External disturbances for:

[0101]

[0102] Initial attitude values ​​are:

[0103]

[0104] The desired posture is:

[0105]

[0106]

[0107] The parameters of the nonsingular real fixed-time attitude tracking control law based on dual MRPs are selected as follows:

[0108]

[0109] Other system parameters are selected as follows:

[0110]

[0111] Pick Then, based on the upper bound of the convergence time... Substituting the parameters, we can find that the upper bound of the convergence time is 641.55 seconds.

[0112] Figure 1 It is a Simulink model based on a nonsingular real fixed-time attitude tracking controller with dual MRPs; Figure 2 yes Time response curve; Figure 3 yes The time response curve. (From...) Figure 2 and Figure 3 It can be seen that the actual convergence time of the system is much smaller than the upper bound of the convergence time.

Claims

1. A nonsingular real-time attitude tracking and control method based on dual MRPs, characterized in that... The method includes the following steps: Step 1: Describe the attitude of the rigid body spacecraft using modified Rodriguez parameters (MRPs) and their shadow parameters. For rigid body spacecraft with external disturbances, establish error kinematic equations and dynamic equations based on dual MRPs: in, Represented as: In the formula, Represents the initial time, set Indicates the initial time. and The selection criteria Indicates the attitude of the rigid body spacecraft. With expected posture The attitude tracking error between them express The shadow parameters, Represents the attitude angular velocity of a rigid body spacecraft. With desired attitude angular velocity angular velocity error between Here is the rotational inertia matrix of the rigid body spacecraft. This serves as the control input for the attitude tracking and control system of a rigid body spacecraft. External disturbances For vectors, , express The identity matrix, yes The transpose of , for The cross product matrix; Step 2: For rigid body spacecraft subject to external disturbances, design a non-singular fixed-time sliding mode attitude tracking control law. Fixed-time control is achieved by using the non-singular sliding mode function norm as the exponential coefficient of the attitude tracking control law. The non-singular fixed-time sliding mode attitude tracking control law is as follows: in, and These represent the equivalent control term and the disturbance control term, respectively. This is a sliding surface approach control term. For sliding mode function, It is a vector; make express The One portion, The expression is: The expression is: To ensure and Continuity, parameters The following relationship exists: , and It is a positive number. Greater than 1, satisfy , satisfy: And there are Let vector , For constants, the notation is... and Let the following vectors be represented respectively: in This represents the absolute value operation. To represent a symbolic function, we have: The final convergence time is: 。 2. The non-singular real-time attitude tracking and control method based on dual MRPs according to claim 1, characterized in that... In step 1, Specifically: . here express The Each component.

3. The non-singular real-time attitude tracking and control method based on dual MRPs according to claim 1, characterized in that... In step 1, let and Let represent the attitude angle and attitude rotation axis of the spacecraft, respectively. Then: Represented as: ； Represented as: ； Furthermore, the following equation holds true: 。

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