Cross-scale decomposition and control method for electro-hydraulic servo macro-micro rigid-flexible multi-body system

Abstract

The invention discloses a cross-scale decomposition and control method for an electro-hydraulic servo macro-micro rigid-flexible multi-body system, and belongs to the technical field of intelligent control. The invention aims to research a cross-scale decomposition and control method for an electro-hydraulic servo macro-micro rigid-flexible multi-body system based on a trajectory tracking control law of self-adaptive dynamic programming and a three-parameter four-scale singular perturbation decoupling problem for the electro-hydraulic servo macro-micro rigid-flexible multi-body system with uncertainty and variable load. The dynamics of the electro-hydraulic servo macro-micro rigid-flexible multi-body system mechanical arm is decomposed for three times, and then a slow-varying subsystem rigid motion controller, a secondary slow-varying subsystem flexible vibration controller, a secondary fast-varying subsystem micro mechanical arm controller and a fast-varying subsystem hydraulic controller are designed; and finally, combining all the controllers according to a multi-time scale theory to obtain a combined controller. According to the invention, a complex system can be greatly simplified, and the respective design of the controller is more practical.

Description

Technical Field

[0001] The invention belongs to the technical field of intelligent control. Background Art

[0002] Electro-hydraulic driven macro-micro rigid-flexible manipulators have greater power when handling heavier payloads and have lower stiffness and response time than traditional electric manipulators. However, the order of the more complex dynamic equations has increased from second to third order. Due to electro-hydraulic coupling and inherent flexible vibration, it brings great difficulties to control.

[0003] The prior art proposes a hybrid controller including feedforward control and feedback control and a model-based adaptive energy shaping controller, as well as an adaptive observer and a variable structure disturbance observer, which are often designed to handle uncertainty and vibration. These control methods do not fully describe the dynamics. In other words, in the past controller design, flexible vibration or hydraulic drive has been neglected. Therefore, it is necessary to propose a control method based on a complete electro-hydraulic coupling model that can accurately track the desired trajectory and suppress flexible vibration.

[0004] In view of the review of the current research status at home and abroad, the research on the flexible multi-body dynamics system of electro-hydraulic servo macro-micro manipulator still has the following important scientific and technological problems to be solved: (1) Complete description of flexible vibration and construction of cross-scale dynamic model of electro-hydraulic servo macro-micro rigid-flexible multi-body system The precise distributed parameter model brings difficulties in system analysis and controller design, while the simplified centralized parameter model faces the problem of difficulty in achieving increased controller order and reduced control performance. Therefore, how to establish an appropriate and effective dynamic model and design a high-performance control strategy based on it is a key issue that scholars engaged in the research of hydraulic flexible macro-micro robotic arm systems must face and solve.

[0005] (2) Vibration suppression and precise control of system end position The macro-micro robot arm is coupled with the micro-mechanical arm, and the macro-mechanical arm has flexible vibration during operation. In addition, during the hydraulic drive process, factors such as sealing friction, hydraulic nonlinearity, and bulk elastic modulus will affect the precise operation of the system. Considering the above, it is necessary to design a robust position precision control strategy. Summary of the invention

[0006] The purpose of the present invention is to study a cross-scale decomposition and control method of an electro-hydraulic servo macro-micro rigid-flexible multi-body system based on an adaptive dynamic programming trajectory tracking control law and a three-parameter four-scale singular perturbation decoupling problem for an electro-hydraulic servo macro-micro rigid-flexible multi-body system with uncertainty and variable load.

[0007] The steps of the present invention are: S1. Model building S11. Mechanical system modeling defines variables: L0 is the height of the bottom base, L1, L2 are the lengths of the first and second flexible links of the middle rotation, L3, L4 are the lengths of the third and fourth rigid links at the end, L 01 L 11 , indicating the starting and ending positions of the first hydraulic cylinder installation, L 12 L 21 , represents the starting and ending positions of the second hydraulic cylinder, m1, m2, are the concentrated masses at the head end of the flexible rod, m3, m4, are the concentrated masses at the head end of the rigid rod, M1, M2, M3, M4 are the masses of each connecting rod, θ1, θ2, θ3, θ4 are the rotation angles of each connecting rod, M d is the end load mass, ρ1, ρ2 are the densities of the first and second links of the macro robot arm, E1I1, E2I2 are the stiffness coefficients; The elastic displacement of the first and second flexible links is expressed as: Among them, Φ j (r), q j (t) correspond to the j-th order natural vibration mode function and modal coordinates of the flexible link; The dynamic model of the electro-hydraulic servo macro-micro rigid-flexible multi-body system without damping and external disturbance is: Among them, M is the positive definite symmetric generalized inertia matrix, K is the stiffness matrix of the connecting rod, G is the matrix composed of Coriolis force, centrifugal force and gravity, θ M =[θ1,θ2] is the rotation angle of the macro robot arm, θ m =[θ3,θ4] is the joint angle of the micro-manipulator, q = [q1,q2] is the modal coordinate, τ = [τ M ,τ m ] is the control torque; S12. Hydraulic system modeling definition variables: x v is the valve core displacement, y is the piston displacement, P s is the supply pressure, P r is the return pressure, P1 and P2 are the pressures of the left and right actuator chambers, and P L is the effective load pressure, A1, A2 are the effective piston areas of the left and right actuators, Q1, Q2 are the fluid flow rates of the left and right actuators, and V1, V2 are the volumes of the left and right actuators; Model of hydraulic system: Dynamic equations of electro-hydraulic servo macro-micro rigid-flexible multi-body system manipulator: Where M is the inertia matrix, K is the stiffness matrix, and G is the nonlinear term related to the Coriolis force and centrifugal force; τ = [τ M ,τ m ] T is the driving torque of the macro-manipulator joint and the micro-manipulator joint; I M is the servo valve control current; θ M =[θ1,θ2] is the rotation angle of the macro robot arm, θ m =[θ3,θ4] joint angle of micro-manipulator; τ = [τ M ,τ m ] is the control torque; A, B, C are coefficient matrices related to the hydraulic system leakage coefficient, bulk modulus, oil density, etc.; S2. Model decomposition S21, first decomposition: The inverse of the hydraulic bulk modulus is selected as the perturbation parameter of the electro-hydraulic drive decomposition, that is, ε1 = 1 / β e , and satisfies 0<ε1<<1, then the dynamic equation of the electro-hydraulic servo macro-micro rigid-flexible multi-body system robot arm is rewritten as: in, According to the multiple time scale theory, let: τ M =τ Mf1 +τ Ms1 I M =I Mf1 +I Ms1 (6) Among them, f1 and s1 represent the fast time scale and slow time scale of the first-level system respectively; Let ε1 = 0, and we get the equation of the first-order slow-varying subsystem: Introducing the first-order fast time scale σ1=t / ε1 in the boundary layer, the first-order slow variable τ Ms1 The boundary layer ε1→0 is considered to be a constant, thus obtaining the first-order fast-changing subsystem, that is, the fast-changing subsystem dynamic equation: Among them, I Mf1 is the control current under the first-level fast-changing time scale σ1, and the subscript f1 indicates that the system is in the first-level fast-changing time scale; S22, second decomposition: The first-level slow-varying subsystem equation (7) is decomposed for the second time to obtain the second-level fast-varying subsystem and the second-level slow-varying subsystem that describes the motion of the macro-manipulator. Take the second small parameter ε2 as the perturbation parameter of the macro-differential decomposition, and let ε2=max{M i / M j ,L i / L j}, are the relative mass ratio and relative rod length ratio of the macro-manipulator and the micro-manipulator, respectively, and the two small parameters satisfy 0<ε1<<ε2<<1; The M, K, and G matrices in the dynamic model of the macro-micromanipulator in the first-level slow-changing subsystem are expressed in the following block form: Among them, M, K, and G are all composed of M i ,M j ,L i ,L j The matrix represented by θ M ,θ m is the rotation angle of the respective macro-micromanipulator; Substituting the square matrix into the equation of the first-order slow-varying subsystem in equation (7), we get a new form describing θ: After sorting, the second-order slow-varying subsystem equation is obtained: According to the multiple time scale theory, let: τ Ms1 =τ Ms2 I Ms1 =I Ms2 (11) For the second-level fast-changing subsystem After sorting, we get the equation of the subfast subsystem: Among them, M 1,f2 ∈R 2*2 is the inertia matrix of the subfast subsystem equation after macro-micro decoupling, K 1,f2 ∈R 2*2 is the nonlinear term related to the Coriolis force and the centripetal force, G 1,f2 To describe the stiffness matrix of the micromanipulator, the subscript f2 indicates that the system is in the second-order fast-changing time scale, τ m is the control torque of the micro-manipulator; S23, the third decomposition: The second-level slow-varying subsystem equation (10) is decomposed for the third time to obtain the sub-slow-varying subsystem describing the flexible vibration and the third-level slow subsystem describing the large-scale rigid motion of the macro-manipulator, namely the slow-varying subsystem; The ratio of density to stiffness coefficient is selected as the perturbation parameter of rigid-flexible decomposition. That is: ε3 = min{ρ1 / E1I1,ρ2 / E2I2}; where ρ1,ρ2 are the densities of the first and second links of the macro robot arm, and E1I1,E2I2 are the corresponding stiffness coefficients; Let (10) be the matrix M in the second-order slow-varying subsystem 1,s2 The inverse matrix of is F: Where M1∈R 2*2 , F1∈R 2*2 ; make Bring in the second-level slow-changing subsystem: in, k1=ρ1 / E1I1,k2=ρ2 / E2I2; definition: Substituting it into equations (14) and (15), and setting ε3 = 0, we obtain: Solving for this yields: y s3 Substitution In the above equation, we get the third-level slow-varying subsystem of the system: Among them, M 1,s3 ∈R 2*2 is the inertia matrix of the slow-varying subsystem, G 1,s3 is the nonlinear terms and stiffness matrix of the slow-varying subsystem including Coriolis force, centrifugal force, gravity, is a constant matrix, I Ms3 is the control current of the slow-changing subsystem; Ignoring σ(ε3), the variable y is expressed as: I Mf2 =I Mf3 +I Ms3 y=y s3 +y f3 (20) Among them, I Ms3 is the control current under the third-level slow-changing time scale, and the subscript S3 indicates that the system is under the third-level slow-changing time scale; Introducing a third fast-varying time scale on the boundary layer The third-level fast-changing subsystem is obtained: Among them, F 4,s3(θ,ε3y),F 3,s3 (θ,ε3y) is the matrix obtained after calculation and sorting. The subscript f3 indicates that the system is in the third-order fast-changing time scale. Mf3 is the control current under the third fast time scale σ3, is a constant matrix; S3. Design controller S31, Slow-changing subsystem rigid motion controller: make Then formula (19) can be rewritten as: in, ψ is the disturbance term in the slow subsystem, and the upper bound ψ* ≥ ||ψ||; Let the expected trajectory of the slow-varying subsystem be x d , the actual trajectory is x, then the trajectory tracking error is: e=xx d (twenty three) Define performance metrics: Where N represents the utility function, Q, R are positive definite matrices; Let U d represents the desired control law, then: therefore Control Law I Ms3 It consists of two parts: Expectation Control I Ms3d and optimal control I Ms3v ,Right now I Ms3 =I Ms3d +I Ms3v (27) Therefore, the error derivative can be rewritten as: The performance indicators are rewritten as: in, If the performance index is continuously differentiable, its infinitesimal form can be expressed as: 0=N(e,I Ms3v )+(▽J(e)) T (f(x)-f(x d )+[g(x)-g(x d )]I Ms3 +g(x d )I Ms3v) (30) Among them, ▽J(e) is the partial derivative of J(e); The Hamiltonian function and the optimal performance index are defined as: If J * (e) Exists and is continuously differentiable, optimal control I s3v for: Slowly changing subsystem control law I ss for: Among them, I ss is the control rate of the slow subsystem of the present invention, the subscript S3 is the third-level slow subsystem, and I Ms3d Expectation Control, I Ms3v The optimal control consists of two parts, ▽J * (e) is the partial derivative of the optimal performance index with respect to the error e; S32, flexible vibration controller of the sub-slow-changing subsystem: The quadratic form is used to design the optimal control law to suppress flexible vibration, and equation (21) is rewritten as: in, Define performance metrics: The algebraic Riccati equation is written as: Design of Optimal Feedback Control Law for Sub-slowly Varying Subsystems I sf : Where P is the solution of the Riccati equation, B k is the rewritten matrix of the second slow-varying subsystem, Q and R are positive definite constant matrices; S33, sub-fast change subsystem micro-manipulator controller: The current and the torque generated by the motor have the following relationship: m =Ki a ; Where K is the torque constant determined by the motor structure, k e is the back electromotive force constant; Design of control law τ for micromanipulator of sub-fast changing subsystem fs : τ fs =τ f2 =-M 1,f2 (θ m ,0)γ -1 (Kp +K d de s / d ζ ) (38) Among them, K p ,K d is a constant, e s is the trajectory error of the micro-manipulator end, K is the torque constant determined by the motor structure; S34, quick change subsystem hydraulic controller: The dynamic equations of the fast-changing subsystem with uncertainty will be rewritten as: in, is the total uncertainty; Define the estimation error: Define the tracking error vector: Select the sliding surface: s f =e f (43) Adaptive Sliding Mode Control I ff for: S4, compound control According to the multi-time scale theory, all controllers are combined to obtain the combined controller: I=I ff +I fs +I ss +I ss (45) Among them, ff represents the decomposed fast-varying subsystem, fs represents the decomposed second-fast-varying subsystem, sf represents the decomposed second-slow-varying subsystem, and ss represents the decomposed slow-varying subsystem.

[0008] The present invention uses singular perturbation theory and differential geometry knowledge to propose a three-perturbation parameter four-time cross-scale system decomposition method for the first time, realizing dynamic decoupling and order reduction processing of the system, facilitating the suppression of the elastic vibration of the system while controlling the position, and improving the real-time performance of the system control. Among them, the fast-changing subsystem control module adopts an adaptive sliding mode controller, the sub-fast-changing subsystem control module adopts a direct compensation method, the sub-slow-changing subsystem control module adopts a robust optimal controller design, and the slow-changing subsystem adopts an adaptive dynamic programming control method. This method can greatly simplify complex systems, and it is more practical to design controllers separately. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Figure 1 It is a schematic diagram of the relationship of the cross-scale decomposition and decoupling system of the present invention; Figure 2 It is a schematic exploded diagram of the electro-hydraulic servo macro-micro rigid-flexible multi-body system robot arm system of the present invention; Figure 3 It is the structural model of the electro-hydraulic servo macro-micro rigid-flexible multi-body system of the present invention; Figure 4 It is a schematic diagram of the coordinate system layout of the mechanical system of the electro-hydraulic servo macro-micro rigid-flexible multi-body system of the present invention; Figure 5 It is a schematic diagram of modeling of the hydraulic subsystem of the present invention; Figure 6 It is a schematic diagram of torque generation principle of a DC servo motor of the present invention; Figure 7 It is the controller design roadmap of the electro-hydraulic servo macro-micro rigid-flexible multi-body system of the present invention; Figure 8 are the expected motion trajectory and the actual motion trajectory of the joint angle 1 of the present invention; Fig. 9 are the expected motion trajectory and the actual motion trajectory of the joint angle 2 of the present invention; Fig.10 are the expected motion trajectory and the actual motion trajectory of the joint angle 3 of the present invention; Fig.11 are the expected motion trajectory and the actual motion trajectory of the joint angle 4 of the present invention; Fig.12 is a first-order modal change curve diagram of the flexible rod 1 of the present invention; Fig.13 is a second-order modal change curve diagram of the flexible rod 1 of the present invention; Fig.14 is a first-order modal change curve diagram of the flexible rod 2 of the present invention; Fig.15 It is a second-order modal change curve diagram of the flexible rod 2 of the present invention. DETAILED DESCRIPTION

[0010] The present invention belongs to the cross-scale control of electro-hydraulic servo macro-micro rigid-flexible multi-body system. Researching a systematic, effective and universal method to solve the problem of precise control of the end position of the robot arm can not only enrich the theoretical system of flexible multi-body dynamics system and electro-hydraulic servo system, but also greatly expand the application scope of the robot arm. It specifically involves the application of adaptive dynamic programming in electro-hydraulic driven macro-micro rigid-flexible robot arm system with variable load and uncertainty, and the use of singular perturbation theory for system decomposition and controller design.

[0011] The electro-hydraulic servo macro-micro rigid-flexible multi-body system refers to a small rigid robot arm driven by a motor attached to the end of a large flexible robot arm driven by a hydraulic pressure. It has the characteristics of low energy consumption, light weight and high speed. It has wide application potential in many engineering fields that are difficult for humans to directly handle, such as forest development, heavy-load motion control, construction engineering, and mobile equipment applications. The large robot arm is referenced to the ground and is called a macro robot arm; the small robot arm is referenced to the large robot arm and also has complete freedom, which is called a micro robot arm.

[0012] Compared with traditional flexible robotic arms, the electro-hydraulic driven macro-micro rigid-flexible robotic arms have four advantages: 1. The introduction of the micro-mechanical arm reduces the effective inertia of the system end and expands the system frequency band; 2. The movement of the micro-manipulator can effectively compensate for the large-scale movement of the macro-manipulator; 3. A more tiny and precise manipulator can be installed at the end of the micro-robotic arm; 4. The hydraulic servo system can adapt to large changes in load.

[0013] In fact, many control methods have been proposed for robot dynamics systems, especially flexible-link robot arms and flexible-joint robot arms, including PID control, neural network control, sliding mode control, adaptive control, optimal control, etc. However, ill-conditioned numerical problems and inaccurate or unreliable control will occur because these dynamic model-based control methods are directly designed, ignoring multiple time scales in the dynamics.

[0014] Singular perturbation theory is used to deal with control problems under different time scales. The second-order dynamic model of the flexible system can be decoupled into a slow subsystem under traditional time and a fast subsystem under the new time scale. Since the electro-hydraulic servo macro-micro rigid-flexible multi-body system has a very small time constant, the entire system model can be simplified by three reductions by introducing three perturbation parameters, making the controller design simpler, which is of great significance to improving the trajectory tracking accuracy and flexible vibration suppression effect of the electro-hydraulic servo macro-micro rigid-flexible multi-body system.

[0015] Taking into account the problems existing in the prior art, the present invention mainly solves the problems of dynamic establishment under macro-micro coupling and vibration suppression of macro-manipulator during system operation. The present invention studies a trajectory tracking control law and singular perturbation decoupling problem based on adaptive dynamic programming for electro-hydraulic servo macro-micro rigid-flexible multi-body systems with uncertainty and variable loads. The complex hydraulic rigid-flexible robot system model is reduced in order by using a three-parameter four-scale singular perturbation method, and a multi-time-scale subsystem controller is designed, which greatly simplifies the control method and makes it more practical. The research of the present invention will enrich and develop the control methods of flexible multi-body systems and electro-hydraulic servo systems in theory, and can also greatly expand the application field of macro-micro manipulators.

[0016] In view of the shortcomings of the prior art, the present invention mainly makes the following improvements. A new singular perturbation decoupling method is proposed for electro-hydraulic servo macro-micro rigid-flexible multi-body dynamics system. Different from the traditional method with one perturbation parameter and two time scales, this method uses three singular perturbation parameters to establish four time scales.

[0017] The controllers of the four systems, namely the slow-changing subsystem, the sub-slow-changing subsystem, the sub-fast-changing subsystem and the fast-changing subsystem, are designed respectively after the singular perturbation decomposition, and finally parallel control is performed after the scale transformation. The adaptive dynamic programming control method designed for the slow-changing subsystem simplifies the controller structure of the adaptive dynamic programming. The controller does not need to consider the model network and control network of the traditional adaptive dynamic programming, which improves the control accuracy and optimizes the energy consumption.

[0018] The present invention takes the system composed of a double-link flexible macro-mechanical arm and a double-link rigid micro-mechanical arm as the object, studies the characteristics of mathematics, mechanics and control involved in the electro-hydraulic servo macro-micro-mechanical arm flexible multi-body system, and establishes a multi-input and multi-output, nonlinear, strongly coupled, time-varying dynamic model. The system is dynamically decoupled and reduced in three perturbation parameters and four time scales using singular perturbation theory and differential geometry, which facilitates the suppression of elastic vibration while accurately controlling the position. Cross-scale control is adopted, and parallel calculation can be performed during physical realization, thereby improving real-time performance and accuracy.

[0019] The present invention aims to study a more general cross-scale control method, which can be applied to different fields such as macro-micro rigid-flexible robotic arms, electro-hydraulic driven rigid-flexible robotic arms, and electro-hydraulic driven macro-micro robotic arm systems according to different values ​​of perturbation parameters. Figure 1 As shown. When ε1=0, it is a traditional electric drive system; when ε1≠0, it is an electro-hydraulic hybrid drive; when ε2=0, it is a macro-robotic arm; when ε2≠0, it is a macro-microrobotic arm; when ε3=0, it is a rigid robotic arm, and when ε3≠0, it is a rigid-flexible robotic arm. Different system models can be obtained by adopting different combinations of singular perturbation parameters, such as: when ε1, ε2≠0, it is an electro-hydraulic driven macro-microrobotic arm; when ε1, ε3≠0, it is an electro-hydraulic driven rigid-flexible robotic arm; when ε2, ε3≠0, it is an electrically driven macro-microrigid-flexible robotic arm. When ε1, ε2, ε3≠0, it is an electro-hydraulic servo macro-microrigid-flexible multi-body system studied in the present invention. It can be seen that the method proposed in the present invention can be applied to each of the above systems, and the process of model reduction and decoupling approximation also reduces some steps compared with the present invention.

[0020] The three-parameter singular perturbation theory can be used to reduce the electro-hydraulic servo macro-micro rigid-flexible multi-body system to a fast-changing subsystem representing the electric drive, a sub-fast-changing subsystem representing the hydraulic servo drive, a sub-slow-changing subsystem representing the flexible vibration, and a slow-changing subsystem representing the large-scale rigid motion. Figure 2 As shown in the figure, these four subsystems can respectively describe electrical control, hydraulic servo control, flexible vibration and large-scale motion. At this time, controllers can be designed for each subsystem, and finally parallel joint control can be achieved through scale transformation.

[0021] The specific method of the present invention comprises the following steps: Step 1: Model establishment, dynamic modeling of electro-hydraulic servo macro-micro rigid-flexible multi-body system The mechanical structure we consider includes a bottom rigid base with a certain height, the first two flexible links and revolute joints, and the last two rigid links and revolute joints. The structural model of the electro-hydraulic servo macro-micro rigid-flexible multi-body system is as follows: Figure 3 The dynamic modeling of the electro-hydraulic servo macro-micro rigid-flexible multi-body system includes two parts: mechanical system modeling and hydraulic system modeling.

[0022] The first and second links of the robot system are flexible, while the third and fourth links are rigid. The flexible rod is like a slender beam. Since the end effector with a large load and the robot system have a large overall movement, the flexible rod will vibrate. Only lateral vibration is considered, and axial deformation and shear deformation are not considered. Therefore, the Euler-Bernoulli beam theory is sufficient to describe the flexible deformation and vibration of the flexible rod. Specifically, the dynamic modeling of the robot system includes two parts: mechanical subsystem modeling and hydraulic subsystem modeling.

[0023] (1) Mechanical system modeling Mechanical subsystem model The dynamic model of the robot system is established using the Lagrange equation and the assumed modal method, and the following assumptions are made based on the actual situation. Assumption 1. The mass of the actuator is concentrated at the end of the connecting rod. Assumption 2. The payload is tightly grasped by the end effector of the robot arm, and their contact is static. Assume XOY and X i O i Y i They are the inertial Cartesian coordinates of the robotic arm and the motion coordinates fixed on the robotic arm. Figure 4 Schematic diagrams of the four main coordinate systems of the electro-hydraulic servo macro-micro rigid-flexible multi-body system are given.

[0024] The variables are defined as follows: L0 is the height of the bottom base, L1, L2 are the lengths of the first and second flexible links of the middle rotation, and L3, L4 are the lengths of the third and fourth rigid links at the end. 01 L 11 , indicating the starting and ending positions of the first hydraulic cylinder installation, L 12 L21 , represents the starting and ending positions of the second hydraulic cylinder, m1, m2, are the concentrated mass at the head end of the flexible rod, m3, m4, are the concentrated mass at the head end of the rigid rod (including the motor mass), M1, M2, M3, M4 are the masses of each connecting rod, and θ1, θ2, θ3, θ4 are the rotation angles of each connecting rod. d is the mass of the end load. ρ1, ρ2 are the densities of the first and second links of the macro robot arm, and E1I1, E2I2 are the stiffness coefficients.

[0025] The elastic displacement of the first two flexible links ω(r, t), that is, the elastic displacement of the first and second flexible links can be expressed by the assumed modal method as follows: Among them, Φ j (r), q j (t) respectively correspond to the j-th order natural vibration mode function and modal coordinates of the flexible link, and take the first two modes. The flexible link is assumed to be a cantilever beam model and take the first two modes, Φ j (r) = shβ j r+ζ j sinβ j r,ζ j =shβ j L3 / sinβ j L3,β j =(j+0.25)π / L3,j=1,2.

[0026] Construct the Lagrangian function: L=TV (1.1) Among them, T is kinetic energy and V is potential energy.

[0027] Compute the Lagrangian equations based on the Lagrangian function:

[0028] The dynamic equation of the system is obtained, that is, the dynamic model of the electro-hydraulic servo macro-micro rigid-flexible multi-body system without damping and external interference can be expressed as: Among them, M is a 6*6 positive definite symmetric generalized inertia matrix, K is the stiffness matrix of the connecting rod, G is the matrix composed of Coriolis force, centrifugal force and gravity, θ M =[θ1,θ2] is the rotation angle of the macro robot arm, θ m =[θ3,θ4] is the joint angle of the micro-manipulator, q = [q1,q2] is the modal coordinate, τ = [τ M ,τ m ] is the control torque.

[0029] (2) Hydraulic system modeling Joints 1 and 2 are rotary joints driven by linear asymmetric hydraulic cylinders. In practical applications, the robot arm needs to move the load along a predefined trajectory. The hydraulic drive subsystem consists of a hydraulic cylinder and a hydraulic motor. The asymmetric four-way valve-controlled hydraulic cylinder is a hydraulic actuator that converts hydraulic energy into mechanical energy for linear motion. Figure 5 As shown, it consists of a hydraulic cylinder and a hydraulic motor. The asymmetric four-way valve-controlled hydraulic cylinder is a hydraulic actuator that converts hydraulic energy into linear motion mechanical energy. v is the valve core displacement, y piston displacement P s is the supply pressure, P r is the return pressure, P1 and P2 are the pressures of the left and right actuator chambers, and P L is the effective load pressure, A1 and A2 are the effective piston areas of the left and right actuators, Q1 and Q2 are the fluid flow rates of the left and right actuators, and V1 and V2 are the volumes of the left and right actuators.

[0030] Define the variables as follows: x v is the valve core displacement, y is the piston displacement, P s is the supply pressure, P r is the return pressure, P1 and P2 are the pressures of the left and right actuator chambers, and P L is the effective load pressure, A1 and A2 are the effective piston areas of the left and right actuators, Q1 and Q2 are the fluid flow rates of the left and right actuators, and V1 and V2 are the volumes of the left and right actuators.

[0031] The model of the hydraulic system is as follows:

[0032] Therefore, the dynamic equation of the four-degree-of-freedom electro-hydraulic servo macro-micro rigid-flexible multi-body system manipulator is shown in (4): Among them, M is the inertia matrix, K is the stiffness matrix, the structures of the two matrices are relatively complex, and they are the results of the derivation operation. G is the nonlinear term related to the Coriolis force and the centrifugal force; τ = [τ M ,τ m ] T is the driving torque of the macro-manipulator joint and the micro-manipulator joint; I M is the servo valve control current; θ M =[θ1,θ2] is the rotation angle of the macro robot arm, θ m =[θ3,θ4] joint angle of micro-manipulator; τ = [τ M ,τ m ] is the control torque; A, B, C are coefficient matrices related to the hydraulic system leakage coefficient, bulk modulus, oil density, etc.

[0033] Step 2: Model decomposition, three-parameter four-time scale cross-scale reduction decomposition and decoupling of the electro-hydraulic servo macro-micro rigid-flexible multi-body system manipulator system: The complete dynamic equation (4) of the electro-hydraulic servo macro-micro rigid-flexible multi-body system is decomposed by three parameters singular perturbation, so as to design controllers for each subsystem independently. They are the slow-changing subsystem (SS), the sub-slow-changing subsystem (SF), the sub-fast-changing subsystem (FS) and the fast-changing subsystem (FF). (1) The first decomposition, the first singular perturbation decomposition: First, the complete dynamic model can be decoupled into a first-level slow subsystem characterized by a hydraulic drive part, and a first-level fast subsystem characterized by an electric drive part. The present invention intends to perform a first-level decomposition and decoupling of the electric current and hydraulic pressure on the electro-hydraulic servo drive, and will obtain a first-level fast subsystem describing current control, that is, the fast-changing subsystem described in the present invention, and a first-level slow-changing subsystem describing hydraulic control, which is used for the second decomposition.

[0034] The inverse of the hydraulic bulk elastic modulus is selected as the perturbation parameter of the electro-hydraulic drive decomposition. That is, ε1=1 / β e , and satisfies 0<ε1<<1. Then the dynamic equation of the electro-hydraulic servo macro-micro rigid-flexible multi-body system manipulator (Equation 4) can be rewritten as: in,

[0035] When the perturbation parameter ε1 is small enough, the higher-order infinitesimal o(ε1) can be ignored. According to the theory of multiple time scales, the variable τ M and I M It can be expressed as: Where f1 and s1 represent the fast and slow time scales of the first-level system, respectively.

[0036] Let ε1 = 0, and we get the first-order slow-varying subsystem equation (7):

[0037] Introducing the first-order fast time scale σ1=t / ε1 in the boundary layer, the first-order slow variable τ Ms1 In the vicinity of the boundary layer ε1→0, it is considered to be a constant, and thus the first-order fast-changing subsystem, that is, the fast-changing subsystem dynamics equation of the present invention is obtained as shown in equation (8): Among them, I Mf1 is the control current under the first fast-changing time scale σ1, and the subscript f1 indicates that the system is in the first fast-changing time scale.

[0038] (2) The second decomposition, the second singular perturbation decomposition: Next, the first-level slow-varying subsystem equation (7) is decomposed for the second time (decoupled again). This decomposition will decouple the macro-manipulator and the micro-manipulator, and will obtain the second-level fast subsystem of the micro-manipulator that describes the fine operation of the end, that is, the second-fast-varying subsystem of the present invention, and the second-level slow-varying subsystem that describes the movement of the macro-manipulator, which is used for the third decomposition.

[0039] Take the second small parameter ε2 as the perturbation parameter of the macro-differential decomposition, and let ε2=max{M i / M j ,L i / L j}, are the relative mass ratio and relative rod length ratio of the macro-manipulator and the micro-manipulator, respectively, and the two small parameters satisfy 0<ε1<<ε2<<1.

[0040] The M, K, and G matrices in the dynamic model of the macro-micromanipulator in the first-level slow-changing subsystem can be expressed in the following block form: Among them, M, K, and G are all composed of M i ,M j ,L i ,L j θ represents the matrix. M ,θ m is the rotation angle of each macro-micromanipulator.

[0041] According to the multiple time scale theory, let: Wherein, f2,s2 represent the fast time scale and slow time scale of the second-level system.

[0042] Substituting the above block matrix into the first-order slow-varying subsystem equation (7), we can get a new form to describe θ:

[0043] When ε2=0, the degenerate second-order slow-varying subsystem is obtained, that is, the second-order slow-varying subsystem equation (10):

[0044] For the second-level fast-changing subsystem After sorting, the sub-fast subsystem equation of the present invention is obtained as (12): Among them, M 1,f2 ∈R 2*2, is the inertia matrix of the sub-fast subsystem equation after macro-micro decoupling in the present invention, K 1,f2 ∈R 2*2 is the nonlinear term related to the Coriolis force and the centripetal force, G 1,f2 To describe the stiffness matrix of the micromanipulator, the subscript f2 indicates that the system is in the second-order fast-changing time scale, τ m is the control torque of the micro-manipulator.

[0045] (3) The third decomposition, the third singular perturbation decomposition: In the position control process of the flexible multi-body system with a beam structure, the variables related to the large-scale rigid motion can be regarded as slow variables, and the variables related to the small deformation vibration can be regarded as fast variables. The physical quantity used to measure the rigidity and flexibility of the system is the stiffness coefficient. Therefore, the second-level slow-varying subsystem equation (10) is decomposed for the third time to obtain the third-level fast-varying subsystem describing the flexible vibration of the macro-manipulator, that is, the secondary slow-varying subsystem described in the present invention, and the third-level slow subsystem describing the large-scale rigid motion of the macro-manipulator, that is, the slow-varying subsystem described in the present invention.

[0046] Therefore, the ratio of density to stiffness coefficient is selected as the perturbation parameter of rigid-flexible decomposition, that is, ε3 = min{ρ1 / E1I1,ρ2 / E2I2}. Among them, ρ1, ρ2 are the densities of the first and second links of the macro robot arm, and E1I1, E2I2 are the corresponding stiffness coefficients.

[0047] Let the matrix M in the second-stage slow-varying subsystem (Eq. 10) be 1,s2 The inverse matrix of is F, that is: Where M1∈R 2*2 , F1∈R 2*2 .

[0048] make Substituting into the second-level slow-varying subsystem, we can get: in, k1=ρ1 / E1I1, k2=ρ2 / E2I2.

[0049] definition: Substituting it into the above two equations and setting ε3 = 0, we get:

[0050] Solving the equation (Equation 16 and Equation 17) yields:

[0051] y s3 Substitution The third-level slow-varying subsystem expression of the system is obtained, that is, the slow-varying subsystem (19) described in the present invention: Among them, M 1,s3 ∈R 2*2 is the inertia matrix of the slow-varying subsystem, G 1,s3 is the nonlinear terms and stiffness matrix of the slow-varying subsystem including Coriolis force, centrifugal force, gravity, is a constant matrix, which depends on the parameters of the hydraulic cylinder drive subsystem, I Ms3 is the control current of the slow-varying subsystem.

[0052] Similarly, ignoring σ(ε3), the variables y and I Ms3 It can be expressed as: Among them, I Ms3 is the control current under the third level slow changing time scale, and the subscript S3 indicates that the system is under the third level slow changing time scale.

[0053] Introducing a third fast-varying time scale on the boundary layer In the boundary layer σ3→0, y s3 is considered to be a constant. According to equations (14), (15), (18), and (20), the third-order fast-changing subsystem, that is, the second slow-changing subsystem dynamic equation (21) of the present invention is: Among them, F 4,s3 (θ,ε3y),F 3,s3 (θ,ε3y) is the matrix obtained after calculation and sorting. The subscript f3 indicates that the system is in the third-order fast-changing time scale. Mf3 is the control current under the third fast time scale σ3, is a constant matrix related to the stiffness and density of the connecting rod.

[0054] Step 3: Design a controller. According to the fast-changing subsystem, the second-fast-changing subsystem, the second-slow-changing subsystem and the slow-changing subsystem dynamic equations obtained in the second step, design the corresponding subsystem control modules: (1) Slow-Varying Subsystem Rigid Motion Controller Design a controller for the slow-changing subsystem (19): Considering the uncertainty and interference of the actual system, let Then the three-level slow subsystem equation (19) is rewritten as: in, ψ is the disturbance term in the slow subsystem. The sum of uncertainty and disturbance in the slow subsystem has an unknown upper bound ψ*, that is, ‖ψ‖≤ψ*.

[0055] Let the expected trajectory of the slow-varying subsystem be x d , the actual trajectory is x, then the trajectory tracking error is: e=xx d (twenty three)

[0056] Define performance metrics: Where N represents the utility function, Q, R are positive definite matrices.

[0057] For the slowly varying subsystem, the control law I Ms3 It consists of two parts: expectation control I Ms3d and optimal control I Ms3v ,Right now I Ms3 =I Ms3d +I Ms3v (27)

[0058] Therefore, the error derivative is rewritten as:

[0059] Let U d represents the desired control law, then:

[0060] Therefore, the error derivative is rewritten as:

[0061] Optimal Control I s3v It can be ensured that the trajectory tracking error of the subsystem converges to a steady state in an optimal way.

[0062] The performance indicator can be rewritten as: in,

[0063] If the performance indicator is continuously differentiable, its infinitesimal form can be expressed as: 0=N(e,I Ms3v )+(▽J(e)) T (f(x)-f(x d )+[g(x)-g(x d )]I Ms3 +g(x d )I Ms3v ) (30) Among them, ▽J(e) is the partial derivative of J(e).

[0064] The Hamiltonian function and the optimal performance index are defined as:

[0065] If J * (e) exists and is continuously differentiable, then the optimal feedback control law can be solved by a cyclically iterative single network evaluation structure policy iteration algorithm, and the optimal control I s3v It can be ensured that the trajectory tracking error of the subsystem converges to a steady state in an optimal way.

[0066] Therefore, the control law of the slow-varying subsystem is ss The design is formula (33): Among them, I ss is the control rate of the slow subsystem of the present invention, the subscript S3 is the third-level slow subsystem, and I Ms3d Expectation Control, I Ms3v Optimal control consists of two parts. * (e) is the partial derivative of the optimal performance indicator with respect to the error e.

[0067] (2) Flexible vibration controller of the sub-slow-varying subsystem Design the sub-slow-varying subsystem controller according to equation (21): The quadratic design optimal control law is used to suppress flexible vibration and adjust the system state to zero. The original model is rewritten as follows: in,

[0068] Define performance metrics:

[0069] The algebraic Riccati equation can be written as:

[0070] Design of Optimal Feedback Control Law for Sub-slowly Varying Subsystems I sf Formula (37): Among them, I sf is the control rate of the slow subsystem of the present invention, subscript f3 is the third-level fast subsystem, P is the solution of the Riccati equation, B k is the rewritten matrix of the second slow-varying subsystem, Q and R are positive constant matrices.

[0071] When there is uncertainty in the sub-slow-varying subsystem, a robust optimal control law with good dynamic performance is given to ensure the stability of the system. Therefore, we get the dynamic equation with uncertainty

[0072] If the uncertainty satisfies where λ min yes The smallest eigenvalue, ρ(P) is the P spectral radius.

[0073] Select I Mf3 As the optimal control law, it can ensure the closed-loop asymptotic stability of the fast-changing subsystem.

[0074] (3) Sub-rapid change subsystem micro-manipulator controller Design the sub-fast-changing subsystem controller according to equation (12). The DC servo motor is used to drive the rotary joint of the micro-mechanical arm, and its torque is τ m , the circuit current is i a , control voltage is u, R a ,L a The total resistance and inductance in the circuit are Figure 6 shown.

[0075] The current and the torque generated by the motor have the following relationship: m =Ki a Among them, K is the torque constant determined by the motor structure. k e is the back electromotive force constant.

[0076] The direct compensation method is used to realize the control of the macro-micro robot, that is, the macro robot arm realizes a large range of motion, and the motion error of the macro robot arm is compensated by the fast and precise motion of the micro robot arm. The position error of the macro robot arm is used as the input of the micro robot arm. Design the control law τ of the micro robot arm of the sub-fast variable subsystem fs Formula (38): τ fs =τ f2 =-M 1,f2 (θ m ,0)γ -1 (K p +K d de s d ζ ) (38) Among them, τ fsis the control rate of the second-fast subsystem of the present invention, K p ,K d is a constant, subscript f2 is the second-order fast subsystem, e s is the trajectory error of the micro-manipulator end, K is the torque constant determined by the motor structure.

[0077] (4)Quick-change subsystem hydraulic controller Design an adaptive sliding mode controller for the fast-changing subsystem equation (8). Considering the uncertain factors such as hydraulic oil elastic modulus perturbation, hydraulic oil leakage, and friction of moving parts will directly affect the stability and dynamic characteristics of the hydraulic servo control subsystem. Therefore, the dynamic equation of the fast-changing subsystem with uncertainty will be rewritten as: in, is the total uncertainty, which is bounded.

[0078] Define the estimation error:

[0079] Define the tracking error vector:

[0080] Select the sliding surface: s f =e f (43)

[0081] For the fast-changing subsystem, taking into account the hydraulic servo fast subsystem, adaptive sliding mode control I ff Formula (44) is proposed: Among them I ff is the control rate of the second-fast subsystem of the present invention, and the subscript f1 is the first-level fast subsystem.

[0082] Step 4: Compound Control Based on the singular perturbation theory, the designed slow-varying subsystem controller, sub-slow-varying subsystem controller, sub-fast-varying subsystem controller, and fast-varying subsystem controller can stabilize the systems (19), (21), (12), and (8) respectively at different time scales. Figure 7 It is the overall block diagram of the robotic arm control closed-loop system.

[0083] exist Figure 7 In the figure, the expected angle and angular velocity x after trajectory planning d It is provided as input to the robot system. At the same time, the four control outputs I can be calculated separately using the information measured by sensors such as photoelectric encoders, pressure sensors and strain gauges.ss ,I sf ,I fs and I ff However, these control outputs are summed after time-scale transformation. Finally, the closed-loop electro-hydraulic servo macro-micro rigid-flexible multi-body system control law with control scheme can be realized.

[0084] According to the multiple time scale theory, all controllers are combined to obtain a combined controller (45). The combined controller of the electro-hydraulic servo macro-micro rigid-flexible multi-body system is shown in formula (45), which can stabilize the system (4). I=I ff +I fs +I ss +I ss (45) Among them, ff represents the fast-changing subsystem decomposed by the present invention, fs represents the second-fast-changing subsystem decomposed by the present invention, sf represents the second-slow-changing subsystem decomposed by the present invention, and ss represents the slow-changing subsystem decomposed by the present invention.

[0085] At this point, the control method of the electro-hydraulic servo macro-micro rigid-flexible multi-body system based on three-parameter four-scale singular perturbation has been completed.

[0086] Simulation Verification In order to verify the effectiveness of the cross-scale decomposition and control method of the electro-hydraulic servo macro-micro rigid-flexible multi-body system of the present invention, the parameter settings as shown in Table 1 are defined and used in Matlab software for simulation. The initial position of the robot arm joint is selected as θ d0 =[1,0.1,0.2,0.4] T , choose q d =[0.75cos2.5t,sin2t,0.5sin1.5t,0.5sin2t] T The ideal expected trajectory of the robot arm trajectory tracking effect is as follows Figures 8 to 11 As shown, the vibration mode of the flexible rod is as follows Figures 12 to 15 The simulation shows that the tracking performance of the joint angle quickly tracks the desired trajectory within 2 seconds, and the vibration of the flexible rod is very small, indicating that under the action of the controller designed by the present invention, the original complex system has a more precise control target after three singular perturbation decompositions, and can suppress its flexible vibration faster and better, with better vibration suppression effect.

[0087] Table 1 Main parameter settings of mechanical system

[0088] Table 2 Hydraulic system parameters

Claims

1. A cross-scale decomposition and control method for an electro-hydraulic servo macro-micro rigid-flexible multi-body system, characterized by: The steps are: S1. Model building S11. Mechanical system modeling Define variables: L0 is the height of the bottom base, L1, L2 are the lengths of the first and second flexible links of the middle rotation, L3, L4 are the lengths of the third and fourth rigid links at the end, L 11 L 21 , indicating the starting and ending positions of the first hydraulic cylinder installation, L 22 L 31 , represents the starting and ending positions of the second hydraulic cylinder, m1, m2, are the concentrated masses at the head end of the flexible rod, m3, m4, are the concentrated masses at the head end of the rigid rod, M1, M2, M3, M4 are the masses of each connecting rod, θ1, θ2, θ3, θ4 are the rotation angles of each connecting rod, M d is the end load mass, ρ1, ρ2 are the densities of the first and second links of the macro robot arm, E1I1, E2I2 are the stiffness coefficients; The elastic displacement of the first and second flexible links is expressed as: Among them, Φ j (r), q j (t) correspond to the j-th order natural vibration mode function and modal coordinates of the flexible link; The dynamic model of the electro-hydraulic servo macro-micro rigid-flexible multi-body system without damping and external disturbance is: Among them, M is the positive definite symmetric generalized inertia matrix, K is the stiffness matrix of the connecting rod, G is the matrix composed of Coriolis force, centrifugal force and gravity, θ M =[θ1,θ2] is the rotation angle of the macro robot arm, θ m =[θ3,θ4] is the joint angle of the micro-manipulator, q = [q1,q2] is the modal coordinate, τ = [τ M ,τ m ] is the control torque; S12. Hydraulic system modeling Define the variable: x v is the valve core displacement, y is the piston displacement, P s is the supply pressure, P r is the return pressure, P1 and P2 are the pressures of the left and right actuator chambers, and P L is the effective load pressure, A1, A2 are the effective piston areas of the left and right actuators, Q1, Q2 are the fluid flow rates of the left and right actuators, and V1, V2 are the volumes of the left and right actuators; Model of hydraulic system: Dynamic equations of electro-hydraulic servo macro-micro rigid-flexible multi-body system manipulator: Where M is the inertia matrix, K is the stiffness matrix, and G is the nonlinear term related to the Coriolis force and centrifugal force; τ = [τ M ,τ m ] T is the driving torque of the macro-manipulator joint and the micro-manipulator joint; I M is the servo valve control current; θ M =[θ1,θ2] is the rotation angle of the macro robot arm, θ m =[θ3,θ4] joint angle of micro-manipulator; τ = [τ M ,τ m ] is the control torque; A, B, C are coefficient matrices related to the hydraulic system leakage coefficient, bulk modulus, oil density, etc.; S2. Model decomposition S21, first decomposition: The inverse of the hydraulic bulk modulus is selected as the perturbation parameter of the electro-hydraulic drive decomposition, that is, ε1 = 1 / β e , and satisfies 0<ε1<<1, then the dynamic equation of the electro-hydraulic servo macro-micro rigid-flexible multi-body system robot arm is rewritten as: in, According to the multiple time scale theory, let: t M =t Mf1 +t Ms1 I M =I Mf1 +I Ms1 (6) Among them, f1 and s1 represent the fast time scale and slow time scale of the first-level system respectively; Let ε1 = 0, and we get the equation of the first-order slow-varying subsystem: Introducing the first-order fast time scale σ1=t / ε1 in the boundary layer, the first-order slow variable τ Ms1 The boundary layer ε1→0 is considered to be a constant, thus obtaining the first-order fast-changing subsystem, that is, the fast-changing subsystem dynamic equation: Among them, I Mf1 is the control current under the first-level fast-changing time scale σ1, and the subscript f1 indicates that the system is in the first-level fast-changing time scale; S22, second decomposition: The first-level slow-varying subsystem equation (7) is decomposed for the second time to obtain the second-level fast-varying subsystem and the second-level slow-varying subsystem that describes the motion of the macro-manipulator. Take the second small parameter ε2 as the perturbation parameter of the macro-differential decomposition, and let ε2=max{M i / M j ,L i / L j }, are the relative mass ratio and relative rod length ratio of the macro-manipulator and the micro-manipulator, respectively, and the two small parameters satisfy 0<ε1<<ε2<<1; The M, K, and G matrices in the dynamic model of the macro-micromanipulator in the first-level slow-changing subsystem are expressed in the following block form: Among them, M, K, and G are all composed of M i ,M j ,L i ,L j The matrix represented by θ M ,θ m is the rotation angle of the respective macro-micromanipulator; Substituting the square matrix into the equation of the first-order slow-varying subsystem in equation (7), we get a new form describing θ: After sorting, the second-order slow-varying subsystem equation is obtained: According to the multiple time scale theory, let: t Ms1 =t Ms2 I Ms1 =I Ms2 (11) For the second fast-changing subsystem After sorting, we get the equation of the subfast subsystem: Among them, M 1,f2 ∈R 2*2 is the inertia matrix of the subfast subsystem equation after macro-micro decoupling, K 1,f2 ∈R 2*2 is the nonlinear term related to the Coriolis force and the centripetal force, G 1,f2 To describe the stiffness matrix of the micromanipulator, the subscript f2 indicates that the system is in the second-order fast-changing time scale, τ m is the control torque of the micro-manipulator; S23, the third decomposition: The second-level slow-varying subsystem equation (10) is decomposed for the third time to obtain the sub-slow-varying subsystem describing the flexible vibration and the third-level slow subsystem describing the large-scale rigid motion of the macro-manipulator, namely the slow-varying subsystem; The ratio of density to stiffness coefficient is selected as the perturbation parameter of rigid-flexible decomposition. That is: ε3 = min{ρ1 / E1I1,ρ2 / E2I2}; where ρ1,ρ2 are the densities of the first and second links of the macro robot arm, and E1I1,E2I2 are the corresponding stiffness coefficients; Let (10) be the matrix M in the second-order slow-varying subsystem 1,s2 The inverse matrix of is F: Where M1∈R 2*2 , F1∈R 2*2 ; make Bring in the second-level slow-changing subsystem: in, k1=ρ1 / E1I1,k2=ρ2 / E2I2; definition: Substituting it into equations (14) and (15), and setting ε3 = 0, we obtain: Solving for this yields: y s3 Substitution In the above equation, we get the third-level slow-varying subsystem of the system: Among them, M 1,s3 ∈R 2*2 is the inertia matrix of the slow-varying subsystem, G 1,s3 is the nonlinear terms and stiffness matrix of the slow-varying subsystem including Coriolis force, centrifugal force, gravity, is a constant matrix, I Ms3 is the control current of the slow-changing subsystem; Ignoring σ(ε3), the variable y is expressed as: Among them, I Ms3 is the control current under the third-level slow-changing time scale, and the subscript S3 indicates that the system is under the third-level slow-changing time scale; Introducing a third fast-varying time scale in the boundary layer The third-level fast-changing subsystem is obtained: Among them, F 4,s3 (θ,ε3y),F 3,s3 (θ,ε3y) is the matrix obtained after calculation and sorting. The subscript f3 indicates that the system is in the third-order fast-changing time scale. Mf3 is the control current under the third fast time scale σ3, is a constant matrix; S3. Design controller S31, Slow-changing subsystem rigid motion controller: make Then formula (19) can be rewritten as: in, ψ is the disturbance term in the slow subsystem, and the upper bound ψ* ≥ ||ψ||; Let the expected trajectory of the slow-varying subsystem be x d , the actual trajectory is x, then the trajectory tracking error is: e=x-x d (23) Define performance metrics: Where N represents the utility function, Q, R are positive definite matrices; let U d represents the desired control law, then: therefore Control Law I Ms3 It consists of two parts: Expectation Control I Ms3d and optimal control I Ms3v ,Right now I Ms3 =I Ms3d +I Ms3v (27) Therefore, the error derivative can be rewritten as: The performance indicators are rewritten as: in, If the performance index is continuously differentiable, its infinitesimal form can be expressed as: in, is the partial derivative of J(e); The Hamiltonian function and the optimal performance index are defined as: If J * (e) Exists and is continuously differentiable, optimal control I s3v for: Slowly changing subsystem control law I ss for: Among them, I ss is the control rate of the slow subsystem of the present invention, the subscript S3 is the third-level slow subsystem, and I Ms3d Expectation Control, I Ms3v Optimal control consists of two parts: is the partial derivative of the error e with respect to the optimal performance index; S32, flexible vibration controller of the sub-slow-changing subsystem: The quadratic form is used to design the optimal control law to suppress flexible vibration, and equation (21) is rewritten as: in, Define performance metrics: The algebraic Riccati equation is written as: Design of Optimal Feedback Control Law for Sub-slowly Varying Subsystems I sf : Where P is the solution of the Riccati equation, B k is the rewritten matrix of the second slow-varying subsystem, Q and R are positive definite constant matrices; S33, sub-fast change subsystem micro-manipulator controller: The current and the torque generated by the motor have the following relationship: m =Ki a ; Where K is the torque constant determined by the motor structure, k e is the back electromotive force constant; Design of control law τ for micromanipulator of sub-fast changing subsystem fs : τ fs =τ f2 =-M 1,f2 (θ m ,0)γ -1 (K p +K d the s / d ζ ) (38) Among them, K p ,K d is a constant, e s is the trajectory error of the micro-manipulator end, K is the torque constant determined by the motor structure; S34, quick change subsystem hydraulic controller: The dynamic equations of the fast-changing subsystem with uncertainty will be rewritten as: in, is the total uncertainty; Define the estimation error: Define the tracking error vector: Select the sliding surface: s f =e f (43) Adaptive Sliding Mode Control I ff for: S4, compound control According to the multi-time scale theory, all controllers are combined to obtain the combined controller: I=I ff +I fs +I ss +I ss (45) Among them, ff represents the decomposed fast-varying subsystem, fs represents the decomposed second-fast-varying subsystem, sf represents the decomposed second-slow-varying subsystem, and ss represents the decomposed slow-varying subsystem.

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