Non-singular actual fixed time attitude tracking control method based on double MRPs

Abstract

The invention discloses a non-singular actual fixed time attitude tracking control method based on double MRPs, and the method comprises the following steps: 1, describing the attitude of a rigid spacecraft through employing a modified Rodrigues parameter MRPs and a shadow parameter thereof, and building an error kinematics equation and a kinetic equation based on the double MRPs; and 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 index coefficient of the attitude tracking control law. According to the method, aiming at the problem that convergence time depends on initial conditions and the problem that a sliding mode surface is singular, a non-singular segmented sliding mode function is designed, then an exponential coefficient of a controller is dynamically constructed by taking a two-norm of the sliding mode function as a variable, and non-singular fixed-time attitude tracking control of a spacecraft is realized. By adopting the attitude controller designed by the invention, for any initial state, the attitude controller can be converged to a specified balance point within a fixed time.

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. angular velocity between the desired attitude angular velocity , is the inertia matrix of the rigid spacecraft, is the control input of the attitude tracking control system of the rigid spacecraft, is the external disturbance, is a vector, , denotes the identity matrix, is the transpose vector of , is the skew-symmetric matrix of , specifically:

[0011] .

[0012] Here denotes the th component of ;

[0013] Let and denote the attitude rotation angle and the attitude rotation axis of the spacecraft, respectively, denotes: ;

[0014] denotes:

[0015] ;

[0016] In addition, the following formula holds:

[0017] ;

[0018] Step 2, for a rigid spacecraft with external disturbance, a non-singular fixed-time sliding mode attitude tracking control law is designed, and the norm of the non-singular sliding mode function is used as the exponential coefficient of the attitude tracking control law to realize fixed-time control, the non-singular fixed-time sliding mode attitude tracking control law is:

[0019]

[0020] wherein, is the control input of the system, and denote the equivalent control term and the disturbance control term, respectively, is the sliding mode surface approaching control term, is the sliding mode function, is a vector;

[0021] Let denote the th component of 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 non-singular 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 non-singular actual fixed-time attitude tracking control method based on double MRPs, characterized by The method comprises the following steps: Step 1, rigid spacecraft attitude is described by using modified Rodrigues parameters (MRPs) and their shadow parameters, for rigid spacecraft with external disturbance, error kinematics equation and dynamics equation are established based on double MRPs: wherein is represented by: wherein denotes the initial time instant, the set denotes the initial time instant and the selection condition, denotes the attitude tracking error between the rigid spacecraft body attitude and the desired attitude , denotes the shadow parameter of , denotes the angular velocity between the rigid spacecraft body attitude angular velocity and the desired attitude angular velocity , is the rotational inertia matrix of the rigid spacecraft, is the control input of the rigid spacecraft attitude tracking control system, is the external disturbance, is the vector, , denotes the identity matrix of , is the transpose vector of , is the skew matrix of ; Step 2, for rigid spacecraft with external disturbance, a nonsingular fixed-time sliding mode attitude tracking control law is designed, and the fixed-time control is realized by taking the norm of nonsingular sliding mode function as the exponential coefficient of the attitude tracking control law.

2. The dual-MRP based non-singular actual fixed-time attitude tracking control method according to claim 1, characterized in that In step 1, Specifically: . Here denotes the first component of 3. The dual-MRP based non-singular actual fixed-time attitude tracking control method according to claim 1, characterized in that In step 1, let and denote the attitude rotation angle and the attitude rotation axis of the spacecraft, respectively, then: is represented by: ； is represented as: ； In addition, the following formula is established: 。 4. The dual-MRP based non-singular actual fixed-time attitude tracking control method according to claim 1, characterized in that In the step 2, the nonsingular fixed-time sliding mode attitude tracking control law is: wherein, and respectively represent equivalent control terms and disturbance control terms, is a sliding mode surface approaching control term, is a sliding mode function, is a vector; Let denote the th component of The expression is: To ensure and continuity, the parameters exist the following relationship: , and is positive, greater than 1, satisfies , satisfies: and also , let the vector , be constant, the notation and denotes the following vectors, respectively: wherein denotes an absolute value operation, denotes a sign function, with: The final convergence time is: 。

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