A precision improvement method for rigid-flexible coupling space robot system

Abstract

The application belongs to the technical field of robot control, and particularly discloses a precision improvement method for a rigid-flexible coupling space robot system. The method constructs a rigid-flexible coupling dynamics model of the space robot, designs a sliding mode controller based on a non-singular sliding surface, estimates vibration of a flexible structure and space environment disturbance by using a radial basis function (RBF) neural network, and overcomes total disturbance and estimation error of the neural network by using an adaptive gain, so as to improve robustness and response speed of the system. The method can effectively ensure high precision and stability of the space robot when performing tasks such as capture operation, on-orbit maintenance and space garbage cleaning, and is suitable for various space task scenarios with flexible structures.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, specifically relating to a method for improving the accuracy of a rigid-flexible coupled space robot system. Background Technology

[0002] Space robots play a crucial role in tasks such as satellite maintenance, space debris cleanup, and on-orbit construction. During their on-orbit operation, they typically need to perform high-precision manipulations of target objects in complex external environments (such as microgravity and space radiation). With the increasing frequency of space exploration activities and the continuous growth in the demand for on-orbit operations, traditional space robot systems relying solely on rigid robotic arms and fixed bases are proving inadequate in the face of complex operating environments. This challenge has led to more complex structural designs for space robots, resulting in the introduction of flexible structural components such as solar panels. This not only significantly enhances the environmental adaptability and flexibility of space robots but also effectively improves the autonomous operation capability and mission execution efficiency of the entire system.

[0003] In space robotic systems, the coupling between flexible solar panels and rigid robotic arms poses challenges to the system's tracking control accuracy and stability. While the introduction of flexible structures brings many advantages, it also easily induces structural vibrations. These vibrations introduce additional dynamic uncertainties, potentially leading to reduced accuracy during mission execution and increasing the complexity and difficulty of the control system. Traditional control methods struggle to maintain high-precision operation while effectively addressing the vibration coupling effects of flexible structures and the interference from the space environment. Therefore, in space capture missions, there is an urgent need for an advanced control method that can adapt to and compensate for the dynamic characteristics of flexible structures to ensure the stability and accuracy of the robotic arm during capture and manipulation tasks. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention proposes a method for improving the accuracy of a rigid-flexible coupled space robot system, in order to address the impact of flexible structure vibration and external environmental interference on the operational accuracy of the space robot, thereby enhancing its stability and precision.

[0005] The method for improving the accuracy of a rigid-flexible coupled space robot system according to the present invention includes the following steps:

[0006] S1. Establish a dynamic model of a rigid-flexible coupled space robot system based on recursive set theory and velocity variational method. The rigid-flexible coupled space robot system includes a central rigid body base, rigid links of the robotic arm, and a flexible sail. The establishment of the dynamic model specifically includes the following steps:

[0007] S1.1 Define the generalized coordinates of the system. For the central rigid body base, the generalized coordinates are defined as follows: r1 and ψ1 are the position vector of the center of mass of the rigid body base and the Euler angle vector expressed in xyz form, respectively; for the rigid link of the robotic arm, the generalized coordinates are defined as y i =q i (i = 2, 3, ..., n + 1); For a flexible solar panel, the generalized coordinates are defined as follows: Where a i Let q be the modal coordinates of the flexible solar panel. i The hinge's rotation angle;

[0008] S1.2, Define moving body B i The generalized velocity is as follows:

[0009]

[0010] Among them, v i and w i These are the linear velocity and angular velocity of the moving body in its body coordinate system relative to its inertial coordinate system, respectively.

[0011] S1.3, Obtain the rigid body B i and B j The relationship of motion between them, B i and B j H j connect, and They are respectively moving body B i hinge point and moving body B j hinge point The coordinate system at that location and This represents the transformation matrix of each hinge point relative to the floating coordinate system of the object. Here, the coordinate system of the hinge points is set to be the same as the floating coordinate system of the object. Will and The transformation relationship between them is defined as R j Movement B j The posture can be represented as:

[0012]

[0013] Among them, C j =C j0 R j C j0 The initial mounting matrix represents the two hinge points;

[0014] Fixed coordinate system O of space robot system j x j y j z j Position relative to an inertial frame of reference is expressed as:

[0015]

[0016] Differentiating equation (3) yields:

[0017]

[0018] Movement B j The angular velocity of the moving body B i The angular velocity is expressed as:

[0019] w j =w i +w rj (5)

[0020] Differentiating equation (5) yields:

[0021]

[0022] Among them, h j ′ represents a rotating hinge H j The rotation vector in the inertial coordinate system, h j =A 0i h j ′;h j ′ represents moving body B j Relative to moving body B i The rotation vector;

[0023] Substituting equation (5) into equation (4), we get:

[0024]

[0025] Differentiating equation (4):

[0026]

[0027] Substituting equations (5) and (6) into equation (8) yields:

[0028]

[0029] Combining equations (6) and (7), we can obtain:

[0030]

[0031] in,

[0032] Combining equations (6) and (9), we can obtain:

[0033]

[0034] Equations (10) and (11) are kinematic relationships between rigid bodies;

[0035] S1.4 Analyze the kinematic relationship between the central rigid body base and the flexible sail, which is the recursive relationship between the velocity and acceleration of the rigid body and the flexible structure:

[0036]

[0037] in,

[0038] S1.5. Obtain the kinematic recursive relationship between the moving bodies of the spatial rigid-flexible coupled robot; for a multibody system, the relationship can be expressed as:

[0039]

[0040] Among them, i=i + (j) represents the moving body B j the internal body of

[0041]

[0042] According to equation (13), the velocity change of the system can be calculated as follows:

[0043]

[0044] The system dynamic equations are established based on Jourdain's variational velocity principle:

[0045]

[0046] Among them, M i f i ω f i o f i u These are respectively called the generalized mass matrix, the generalized inertial force matrix, the generalized external force matrix, and the generalized deformation force matrix; ΔP is the virtual power of the ideal constraint force and the non-ideal constraint force.

[0047] Substituting equations (13) and (14) into equation (15), we get:

[0048]

[0049] Using the robot's joint coordinates and the flexible body's modal coordinates as generalized variables, the dynamic model of the rigid-flexible coupled space robot can be expressed as follows:

[0050]

[0051] Equation 1 of the spatial rigid-flexible coupled robot dynamics model (17) can be transformed into:

[0052]

[0053] Where, q r Let q be the generalized coordinate system of the central rigid body base and the rigid robotic arm. f Let u be the generalized coordinates of the flexible solar panel, and u be the control vector.

[0054] From the first row of equation (18):

[0055]

[0056] Considering various uncertainties, equation (19) can be written as:

[0057]

[0058] Where, d e For external disturbances, d is the set of various disturbance forces that will affect the control accuracy of the robotic arm; Equation (20) includes all disturbances to the central rigid body base and the rigid robotic arm. The effect of the control force on the system is also affected by the generalized inertia tensor, and the system has strong nonlinearity.

[0059] S2. Controller design: A control strategy combining sliding mode control and radial basis function neural network is adopted. The specific steps are as follows:

[0060] S2.1, Define the non-singular terminal sliding surface:

[0061]

[0062] Where e = q r -q d Let q be the tracking error vector of the system state. d Given the desired trajectory, Let qp be the rate of change of error in system state tracking. 1 < qp < 2, where p and q are positive odd numbers;

[0063] Differentiating the sliding mode function (21) and substituting the first line of equation (20) into it, we obtain the expression:

[0064]

[0065] Where, diag(*) means constructing a diagonal matrix with the elements of the vector as the main diagonal;

[0066] S2.2. The following control law is designed for the space robot system:

[0067] u = u1 + u2(23)

[0068] make The equivalent control law u1 can be obtained:

[0069]

[0070] in, For the estimation of uncertainty by the RBF neural network;

[0071] Set the switching control law as follows:

[0072] u2 = -Z 11 (k0s+diag(k1)sat(s))(25)

[0073] Where k0 is a positive definite diagonal matrix; sat(*) is the saturation function with a saturation boundary thickness of υ; k1 is the adaptive gain, and the adaptive law is designed as follows:

[0074]

[0075] Where η>0 is the adaptive learning rate;

[0076] S2.3 Setting the adaptive law for the neural network:

[0077]

[0078] in, Let be the estimated weights of the RBF neural network; ζ is a positive definite diagonal matrix representing the learning rate of the neural network; h is the radial basis vector of the RBF neural network, which is a column vector, and the i-th expression can be represented as:

[0079]

[0080] Where x is the input signal of the neural network; b i c is the width of the basis functions; i It serves as the data center for neural nodes.

[0081] The beneficial effects of this invention are:

[0082] (1) High nonlinear approximation capability: Through rigid-flexible coupling dynamic modeling, this invention comprehensively considers the coupling effects of the vibration of the flexible solar panel and the disturbances of the external space environment on the rigid base and rigid robotic arm system at the space center. Utilizing the online adaptive estimation capability of the radial basis function (RBF) neural network, the system can dynamically learn and update the estimates of uncertainties caused by the flexible structure and external disturbances in real time. The nonlinear approximation capability of the RBF neural network ensures the high adaptability and robustness of the system in complex dynamic environments, thereby improving control accuracy.

[0083] (2) High-efficiency adaptive and robust control: The sliding mode controller of this invention combines an adaptive control strategy based on neural networks. By effectively estimating the impact of external environmental disturbances caused by the vibration of the flexible structure on robot control, and combining the robust characteristics of sliding mode control, these disturbances are suppressed. This combination not only ensures the stability of the system under uncertain conditions, but also endows the system with the ability to quickly respond to changes in system state, achieving high-efficiency control in dynamic task environments. In addition, the system can adjust control parameters in real time according to the actual operating state, ensuring high-precision operation of the robotic arm under various task conditions, demonstrating excellent adaptive capabilities.

[0084] (3) High applicability and wide range of applications: This invention is not only applicable to space missions with flexible structures such as space capture, on-orbit maintenance, and space debris removal, but can also be extended to spacecraft operation, flexible robotic arms, industrial automation, and other fields involving flexible-rigid coupling systems. This broad application prospect makes this invention of great technical promotion value. Attached Figure Description

[0085] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0086] Figure 1 This is a schematic diagram of a rigid-flexible coupled space robot system.

[0087] Figure 2 This is a schematic diagram of the kinematic relationship between adjacent objects.

[0088] Figure 3 It is the fixed coordinate system of the space robot system.

[0089] Figure 4 These are control effect diagrams for a space robot, including (a) the position and orientation of the base, (b) the trajectory of the robot arm's joint angles, (c) the position tracking error of the base, (d) the position tracking error of the base, (e) the tracking error of the robot arm's joint angles, (f) the control force of the base, (g) the control torque of the base, and (h) the control torque of the robot arm.

[0090] Figure 5 This is a diagram of the nodal displacements at the edge of the windsurfing board. Detailed Implementation

[0091] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0092] Example 1

[0093] like Figure 1 As shown, the space robot system includes a central rigid body base B1, an n-degree-of-freedom robotic arm with rigid links, and two flexible solar panels B1. n+2 and B n+3 Virtual body B0 and virtual hinge H1 are used to describe the motion of the central rigid body B1. Virtual hinge H1 is fixed to virtual body B0 and has three translational degrees of freedom and three rotational degrees of freedom. The inertial coordinate system O0x0y0z0 is fixed to the virtual body. The floating coordinate system O1x1y1z1 is fixed to the central rigid body B1, with its origin fixed at the center of mass of the central rigid body B1. The origin of the floating coordinate system of the robotic arm links is fixed at the center of mass of each link. The floating coordinate system of the flexible sail is fixed at its center of mass before deformation.

[0094] A method for improving the accuracy of a rigid-flexible coupled space robot system includes the following steps:

[0095] S1. Establishing a rigid-flexible coupling dynamic model based on recursive set theory and velocity variational method.

[0096] To obtain the dynamic model of the space robot system, it is necessary to analyze the recursive relationships of motion between the moving bodies. First, the generalized coordinates of the system are defined. For the base, the generalized coordinates are defined as follows: r1 and ψ1 are the position vector of the base's center of mass and the Euler angle vector expressed in xyz form, respectively; for the rigid link of the robotic arm, the generalized coordinates are defined as y i =q i (i = 2, 3, ..., n + 1); For a flexible solar panel, the generalized coordinates are defined as follows: Where a i Let q be the modal coordinates of the flexible solar panel. i The rotation angle of the hinge.

[0097] Define moving body B i The generalized velocity is as follows:

[0098]

[0099] Among them, v i and w i These are the linear velocity and angular velocity of the moving body in its body coordinate system relative to its inertial coordinate system, respectively.

[0100] The two flexible sails of the space robot are fixed-end constrained, meaning one end is fixed. The hinge point between the central rigid body and the flexible sails does not deform; therefore, only the kinematic relationship between the rigid bodies needs to be analyzed. A kinematic diagram of adjacent objects is shown below. Figure 2 As shown, moving body B i and B jThey are connected by a hinge H j connect. and respectively, the moving body B i hinge point and moving body B j hinge point The coordinate system at that location. and This represents the transformation matrix of each hinge point relative to the floating coordinate system of the object. Here, the coordinate system of the hinge points is set to be the same as the floating coordinate system of the object. Will and The transformation relationship between them is defined as R j Movement B j The posture can be represented as:

[0101]

[0102] Among them, C j =C j0 R j C j0 The initial mounting matrix represents the two hinge points;

[0103] like Figure 3 As shown, coordinate system O j x j y j z j The position relative to the inertial frame can be expressed as:

[0104]

[0105] Differentiating equation (3) yields:

[0106]

[0107] Movement B j The angular velocity of the moving body B i The angular velocity is expressed as:

[0108] w j =w i +w rj (5)

[0109] Differentiating equation (5) yields:

[0110]

[0111] Among them, h j H represents a rotating hinge j The rotation vector in the inertial coordinate system, h j =A 0i hj ′;h j ′ represents moving body B j Relative to moving body B i The rotation vector;

[0112] Substituting equation (5) into equation (4), we get:

[0113]

[0114] Differentiating equation (4):

[0115]

[0116] Substituting equations (5) and (6) into equation (8) yields:

[0117]

[0118] Combining equations (6) and (7), we can obtain:

[0119]

[0120] in,

[0121] Combining equations (6) and (9), we can obtain:

[0122]

[0123] Equations (10) and (11) are kinematic relationships between rigid bodies;

[0124] For a space robot system with a central rigid body and a flexible solar panel, its kinematic relationship can be described as a recursive relationship between the velocity and acceleration of the rigid body and the flexible structure:

[0125]

[0126] in,

[0127] From the above derivation, the kinematic recursive relationships between the moving bodies of a spatial rigid-flexible coupled robot are obtained. For a multibody system, the relationship can be expressed as:

[0128]

[0129] Among them, i=i + (j) represents the moving body B j the internal body of

[0130]

[0131]

[0132] According to equation (13), the velocity change of the system can be calculated as follows:

[0133]

[0134] The system dynamic equations are established based on Jourdain's variational velocity principle:

[0135]

[0136] Among them, M i f i ω f i o f i u These are respectively called the generalized mass matrix, the generalized inertial force matrix, the generalized external force matrix, and the generalized deformation force matrix; ΔP is the virtual power of the ideal constraint force and the non-ideal constraint force.

[0137] Substituting equations (13) and (14) into equation (15), we get:

[0138]

[0139] Using the robot's joint coordinates and the flexible body's modal coordinates as generalized variables, the dynamic model of the rigid-flexible coupled space robot can be expressed as follows:

[0140]

[0141] Equation 1 of the spatial rigid-flexible coupled robot dynamics model (17) can be transformed into:

[0142]

[0143] Where, q r Let q be the generalized coordinate system of the central rigid body base and the rigid robotic arm. f Let u be the generalized coordinates of the flexible solar panel, and u be the control vector.

[0144] From the first line of equation (18):

[0145]

[0146] Considering various uncertainties, equation (19) can be written as:

[0147]

[0148] Where, d e Let d be the external disturbance, and let d be the set of various disturbance forces that can affect the control accuracy of the robotic arm.

[0149] Equation (20) includes all disturbances to the base and rigid robotic arm. The effect of the control force on the system is also affected by the generalized inertia tensor, and the system has strong nonlinearity.

[0150] S2, Controller Design

[0151] In space robot systems, uncertainties in actual dynamic parameters, disturbances caused by the vibration of flexible solar panels, and external environmental disturbances all affect the trajectory tracking accuracy of the robotic arm joints. To improve the trajectory tracking performance of the robotic arm, this invention employs a control strategy combining sliding mode control and radial basis function (RBF) neural networks. The sliding mode controller guides the system state to the sliding surface and ensures that the state error gradually approaches zero, exhibiting good robustness. Simultaneously, the RBF neural network provides feedforward compensation by estimating the disturbances caused by the vibration of the flexible solar panel and the system's uncertainties online, further reducing the impact of system chattering and uncertainties on the controller, significantly improving control accuracy and system stability.

[0152] Define a non-singular terminal sliding surface:

[0153]

[0154] Where e = q r -q d Let q be the tracking error vector of the system state. d Given the desired trajectory, Let qp be the rate of change of error in system state tracking, 1 < qp < 2, where p and q are positive odd numbers;

[0155] Differentiating the sliding mode function (21) and substituting the first line of equation (20) into it, we obtain the expression:

[0156]

[0157] Here, diag(*) means constructing a diagonal matrix with the elements of the vector as the main diagonal.

[0158] The following control law is designed for a space robot system:

[0159] u = u1 + u2(23)

[0160] make The equivalent control law u1 can be obtained:

[0161]

[0162] in, For the estimation of uncertainty by the RBF neural network;

[0163] Set the switching control law as follows:

[0164] u2 = -Z 11 (k0s+diag(k1)sat(s))(25)

[0165] Where k0 is a positive definite diagonal matrix; sat(*) is the saturation function with a saturation boundary thickness of υ; k1 is the adaptive gain, and the designed adaptive law is:

[0166]

[0167] Where η>0 is the adaptive learning rate.

[0168] To ensure that the neural network can better learn uncertainty, an adaptive law is set for the neural network:

[0169]

[0170] in, Let be the estimated weights of the RBF neural network; ζ is a positive definite diagonal matrix representing the learning rate of the neural network; h is the radial basis vector of the RBF neural network, which is a column vector, and the i-th expression can be represented as:

[0171]

[0172] Where x is the input signal of the neural network; b i c is the width of the basis functions; i It serves as the data center for neural nodes.

[0173] Controller stability analysis:

[0174] Let W be the optimal estimated weight of the neural network. * Then the uncertainty estimation error of the system is:

[0175]

[0176] Where, ε * For the optimal approximation error, ε * ≤ε N .

[0177] At the same time, define as well as Set the Lyapunov function as follows:

[0178]

[0179] Differentiating equation (30) yields:

[0180]

[0181] Substituting equation (23) into equation (22) yields:

[0182]

[0183] Substituting equations (32) and (31) into equation (31), we get:

[0184]

[0185] in, And because Therefore, formula (33) can be written as:

[0186]

[0187] Substituting equation (27) into equation (34), we get:

[0188]

[0189] Substituting equation (26) into equation (35), we get:

[0190]

[0191] According to Lyapunov's stability theorem, the designed non-singular terminal sliding mode device can make the tracking error converge.

[0192] Simulation example:

[0193] To describe the relative motion of hinges in a multi-body robot in space, according to Figure 3 A rigid, fixed coordinate system was established for the space robot system, where Z... i Defined as the i-th joint J i The rotation axis direction. The link parameters of the robotic arm are defined as shown in Table 1, where a i b i I i The corresponding vector in coordinate system Σ i The dimensions of the two flexible solar panels are 5×2×0.025m, with an elastic modulus of 3GPa, a Poisson's ratio of 0.3, and a damping ratio of 0.1.

[0194] Table 1 Physical parameters of the space robot system

[0195] mass m 400 30 25 10 8 6 8 <![CDATA[centroid vector a ix > 0 0 0.2702 0 0 0 0 <![CDATA[centroid vector a iy > 0 0 0 0 0 -0.0338 0 <![CDATA[centroid vector a iz > 0 0.15 -0.2513 0.15 -0.35 0 0.075 <![CDATA[Spatial hinge vector b ix > 0 0 0.5598 0 0 0 0 <![CDATA[body hinge vector b iy > 0 0 0 0 0 -0.0662 0 <![CDATA[Solid hinge vector b iz > 0.5 0.15 -0.0487 0.15 -0.35 0 0.1595 <![CDATA[Moment of inertia I xx > 133 0.75 0.4630 0.21 0.4996 0.066 0.2061 <![CDATA[moment of inertia I yy > 133 0.75 4.5265 0.21 0.4996 0.0344 0.2061 <![CDATA[Moment of inertia I zz > 133 0.375 4.2255 0.0588 0.0392 0.0520 0.0877 <![CDATA[Moment of inertia I xy > 0 0 0 0 0 0 0 <![CDATA[Moment of inertia I xz > 0 0 0.6575 0 0 0 0 <![CDATA[Moment of inertia I yz > 0 0 0 0 0 0 0

[0196] The initial base position, attitude, and robotic arm joint angles of the space robot system are all 0. The control targets for the base position, attitude, and robotic arm joint angles are set as [0.1, 0.1, 0.1, 30°, -30°, 40°, 45°, 45°, 47°, -38°, -46°, -37°]. T The control parameter settings are shown in Table 2.

[0197] Table 2 Control parameters of the space robot system

[0198] q 9 p 7 a diag([0.6,0.6,0.6,0.9,0.9,0.7,0.7,0.7,0.6,0.8,0.6]) <![CDATA[k0]]> diag([0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8,0.8]) η <![CDATA[5×10 4 ]]> ζ <![CDATA[8×10 5 ]]>

[0199] The main sources of uncertainty in space robot dynamics are the unknown load at the end effector of the robotic arm and existing system disturbances. Let the external disturbance torques acting on the space base be:

[0200] d base =0.15sin(0.5t)+0.5sin(0.01t)(37)

[0201] Assume that the disturbance torque experienced by each rotating joint of the robotic arm is:

[0202] d arm =0.04sin(0.5t)+0.02sin(0.01t)(38)

[0203] according to Figure 4 and Figure 5 As shown, the designed controller effectively achieves fast and stable tracking between the robotic arm and the base, demonstrating excellent tracking performance. The tracking error is kept within a small range, which fully demonstrates that the controller has good dynamic response capability and high tracking accuracy.

[0204] Meanwhile, the powerful online learning capability of the RBF neural network enables it to effectively cope with the vibration of flexible structures and various disturbances from the external environment, thus significantly improving the robustness and stability of the system. According to experimental results, the neural network had essentially completed the learning and compensation of external disturbances after 20 seconds. Although the controller could not completely eliminate the high-frequency vibration of the flexible solar panel, the amplitude of the vibration gradually decreased and eventually stabilized under damping. This process demonstrates that the system exhibits good adaptability and vibration attenuation capability when coping with the vibration of flexible structures and external environmental disturbances.

[0205] In summary, this control strategy combines the online learning capability of the RBF neural network to achieve effective estimation of the vibration of flexible structures and external environmental disturbances. It also overcomes the errors in total disturbance and neural network estimation through adaptive gain, ensuring the stability and control accuracy of the system in complex task environments and demonstrating strong robustness and task execution capability.

[0206] The embodiments of the present invention have been described in detail above, but the present invention is not limited to the described embodiments. For those skilled in the art, various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention, and these variations still fall within the protection scope of the present invention.

Claims

1. A method for improving the accuracy of a rigid-flexible coupled space robot system, characterized in that, Includes the following steps: S1. Establish a dynamic model of a rigid-flexible coupled space robot system based on recursive set theory and velocity variational method. The rigid-flexible coupled space robot system includes a central rigid body base, rigid links of the robotic arm, and a flexible sail. The establishment of the dynamic model specifically includes the following steps: S1.1 Define the generalized coordinates of the system. For the central rigid body base, the generalized coordinates are defined as follows: , and Let be the position vector of the center of mass of the rigid body base and the Euler angle vector expressed in xyz form, respectively; for the rigid link of the robotic arm, the generalized coordinates are defined as... For flexible solar panels, the generalized coordinate system is defined as follows: ,in a i These are the modal coordinates of the flexible solar panel. The hinge's rotation angle; S1.2, Define moving body B i The generalized velocity is as follows: (1) in, and These are the linear velocity and angular velocity of the moving body in its body coordinate system relative to its inertial coordinate system, respectively. S1.3, Obtain the rigid body B i and B j The relationship of motion between them, B i and B j H j connect, and They are respectively moving body B i hinge point and moving body B j hinge point The coordinate system at that location and This represents the transformation matrix of each hinge point relative to the floating coordinate system of the object. Here, the coordinate system of the hinge points is set to be the same as the floating coordinate system of the object. ,Will and The transformation relationship between them is defined as follows: Movement B j The posture is represented as: (2) in, , The initial mounting matrix represents the two hinge points; Fixed coordinate system of space robot system Position relative to an inertial frame of reference is expressed as: (3) Differentiating equation (3) yields: (4) Movement B j The angular velocity of the moving body B i The angular velocity is expressed as: (5) Differentiating equation (5) yields: (6) in, H represents a rotating hinge j Rotation vector in inertial coordinate system ; Indicates moving body B j Relative to moving body B i The rotation vector; , ; Substituting equation (5) into equation (4), we get: (7) Differentiating equation (4): (8) Substituting equations (5) and (6) into equation (8), we get: (9) Combining equations (6) and (7), we get: (10) in, , , Combining equations (6) and (9), we get: (11) Equations (10) and (11) are kinematic relationships between rigid bodies; S1.

4. Obtain the kinematic relationship between the central rigid body base and the flexible sail, which is the recursive relationship between the velocity and acceleration of the rigid body and the flexible structure: (12) in, , , , ; S1.

5. Obtain the kinematic recursive relationships between the moving bodies of the spatial rigid-flexible coupled robot; for a multibody system, the relationship is expressed as: (13) in, Indicates moving body B j the internal body of ， ， According to equation (13), the velocity change of the system is calculated as follows: (14) The system dynamic equations are established based on Jourdain's variational velocity principle: (15) in, , , , These are respectively called the generalized mass matrix, the generalized inertial force matrix, the generalized external force matrix, and the generalized deformation force matrix; It is the virtual power of ideal and non-ideal constraints; Substituting equations (13) and (14) into equation (15), we get: (16) Using the robot's joint coordinates and the flexible body's modal coordinates as generalized variables, the dynamic model of the rigid-flexible coupled space robot can be expressed as follows: (17) The first transformation of equation (17) of the spatial rigid-flexible coupled robot dynamics model is: (18) in, For the generalized coordinates of the central rigid body base and the rigid robotic arm, For the generalized coordinates of the flexible sail, u For control vectors; From the first line of equation (18): (19) Considering various uncertainties, equation (19) can be written as: (20) in, d e External disturbances d It is a collection of various disturbance forces that affect the control accuracy of the robotic arm; Equation (20) includes all disturbances to the central rigid body base and the rigid robotic arm. The effect of the control force on the system is also affected by the generalized inertia tensor, and the system has strong nonlinearity. S2. The controller is configured using a control strategy that combines sliding mode control with radial basis function neural networks. The specific steps are as follows: S2.1, Define the non-singular terminal sliding surface: (21) in, Let the tracking error vector be the system state. Given the desired trajectory, The error rate of system state tracking. , p and q It is a positive odd number; Differentiating the sliding mode function (21) and substituting the first line of equation (20) into it, we obtain the expression: (22) Where, diag(*) means constructing a diagonal matrix with the elements of the vector as the main diagonal; S2.

2. The following control law is designed for the space robot system: (23) make Equivalent control law : (24) in, For the estimation of uncertainty by the RBF neural network; Set the switching control law as follows: (25) in, It is a positive definite diagonal matrix; sat(*) is the saturation function, and the saturation boundary thickness is... ; For adaptive gain, the designed adaptive law is: (26) in, Adaptive learning rate; S2.3 Setting the adaptive law for the neural network: (27) in, These are the estimated weights for the RBF neural network; is a positive definite diagonal matrix, representing the learning rate of the neural network; Let be the radial basis vector of the RBF neural network, and let be a column vector, the th... i The expression is represented as: (28) in, This is the input signal for the neural network; The width of the basis functions; It serves as the data center for neural nodes.

Images