CN103395065B

CN103395065B - Hydraulic hard and soft mechanical arm control method based on two-parameter singular perturbation

Info

Publication number: CN103395065B
Application number: CN201310341384.9A
Authority: CN
Inventors: 李元春; 赵博; 唐志国; 王蒙
Original assignee: Changchun University of Technology
Current assignee: Changchun University of Technology
Priority date: 2013-08-07
Filing date: 2013-08-07
Publication date: 2015-05-13
Anticipated expiration: 2033-08-07
Also published as: CN103395065A

Abstract

The hydraulic rigid-flexible manipulator control method based on two-parameter singular perturbation relates to the field of intelligent machinery and robot control technology. This method reduces the hydraulic rigid-flexible manipulator system to a fast-changing hydraulic servo drive through the two-parameter singular perturbation principle. Subsystems, sub-fast-varying subsystems representing elastic vibrations, and slow-varying subsystems representing large-scale rigid motions; among them, the control module of the fast-varying subsystem adopts an adaptive sliding mode controller, and the control module of the sub-fast-varying subsystem adopts the optimal The controller suppresses elastic vibration, and the slow-variation subsystem control module uses a second-order sliding mode controller to realize joint trajectory tracking control. The control method of the present invention reduces the order of the complex hydraulic rigid-flexible manipulator system model based on the singular perturbation of two parameters, and designs a multi-time scale subsystem controller, which greatly simplifies the control method and is more practical.

Description

Control method of hydraulic rigid-flexible manipulator based on two-parameter singular perturbation

技术领域technical field

本发明涉及智能机械与机器人控制技术领域，尤其涉及一种基于双参数奇异摄动的液压刚柔机械臂控制方法。The invention relates to the technical field of intelligent machinery and robot control, in particular to a method for controlling a hydraulic rigid-flexible manipulator based on a singular perturbation of two parameters.

背景技术Background technique

液压刚柔机械臂是一种由液压驱动的机械臂，其具有高耐用性、大驱动力矩、高功率/重量比、高载荷/自重比等特点，因此常应用于大型重载机械中，并在建筑、采矿、交通运输、机械加工等领域得到了广泛的应用。液压刚柔机械臂在操作大负载时不可避免的产生弹性振动，这给负载的精确、快速转载带来了困难。目前一般采用非线性反馈控制系统进行控制，但控制器一般设计复杂、计算量大、实时性差，很难达到期望的控制效果。因此亟需设计一种结构简单、小计算量、高实时性且易于实现的控制器，满足液压刚柔机械臂转载大负荷任务需要。The hydraulic rigid-flexible manipulator is a manipulator driven by hydraulic pressure, which has the characteristics of high durability, large driving torque, high power/weight ratio, high load/self-weight ratio, etc., so it is often used in large-scale heavy-duty machinery, and It has been widely used in construction, mining, transportation, machining and other fields. The hydraulic rigid-flexible manipulator inevitably produces elastic vibration when operating a large load, which brings difficulties to the precise and rapid transfer of the load. At present, the nonlinear feedback control system is generally used for control, but the controller is generally designed with complex design, large amount of calculation, and poor real-time performance, so it is difficult to achieve the desired control effect. Therefore, there is an urgent need to design a controller with simple structure, small amount of calculation, high real-time performance and easy implementation to meet the needs of the hydraulic rigid-flexible manipulator to transfer heavy loads.

摄动理论研究小参数问题的渐近展开解法及解的性质，可以求得满足所需精度要求的近似解析解，因此其比数值解有很大的优越性，这一类问题在物理、化学、力学、天文学等工程科学以及交通、经济等管理科学中大量出现，并统称这类问题为摄动问题。在大量的摄动问题中，根据解对小参数的渐近性质，可以将它们分为两大类：正则摄动与奇异摄动。根据R.E.O’Malley给出的定义，若摄动问题P_ε的解y_ε(x)能表示成ε的幂级数The perturbation theory studies the asymptotic expansion solution and the properties of the solution of the small parameter problem, and can obtain an approximate analytical solution that meets the required precision requirements, so it has great advantages over the numerical solution. This type of problem is widely used in physics and chemistry. A large number of engineering sciences such as mechanics, astronomy, etc., as well as management sciences such as transportation and economics, are collectively referred to as perturbation problems. In a large number of perturbed problems, according to the asymptotic properties of solutions to small parameters, they can be divided into two categories: regular perturbation and singular perturbation. According to the definition given by REO'Malley, if the solution y _ε (x) of the perturbation problem P _ε can be expressed as a power series of ε

${y the y}_{ϵ ϵ} ((x x)) ~ ~ {y the y}_{00} ((x x)) + + \overset{\infty \infty}{\underset{j j = = 11}{Σ Σ}} {y the y}_{j j} ((x x)) \cdot \cdot {ϵ ϵ}^{j j},, x x &Element; &Element; D D.,, ϵ ϵ &RightArrow; &Right Arrow; 00 - - - - - - ((11))$

其中，y₀(x)是问题P₀(即P_ε|_ε＝0)的解，且上述展开对是一致收敛的，则称P_ε为正则摄动问题。否则，就称P_ε为奇异摄动问题。在自动控制问题中，过去对含有小参数_ε的奇异摄动问题的处理方法是把小参数忽略，即近似把小参数ε看成为零，忽略高阶部分、保留低频部分，或不考虑小时间常数的影响，这样会使系统在边界层内的误差已达失真的程度。Among them, y ₀ (x) is the solution of the problem P ₀ (ie P _ε | _ε=0 ), and the above expansion is for is uniformly convergent, then P _ε is called a regular perturbation problem. Otherwise, P _ε is called a singular perturbation problem. In the automatic control problem, the previous method to deal with the singular perturbation problem with small parameter _ε is to ignore the small parameter, that is, to treat the small parameter ε as zero, ignore the high-order part, keep the low-frequency part, or ignore the small time Influenced by the constant, the error of the system in the boundary layer has reached the degree of distortion.

液压刚柔机械臂是一种大型的、强非线性、高度耦合的系统，因此其自动控制方法一直是学者及研究人员讨论的焦点。但是，液压刚柔机械臂系统因其模型复杂而导致控制器设计复杂、不易实现，现有的自动控制方法控制精度低，不能有效地抑制由机械臂柔性导致的振动现象。The hydraulic rigid-flexible manipulator is a large, strongly nonlinear, and highly coupled system, so its automatic control method has always been the focus of discussion among scholars and researchers. However, the complex model of the hydraulic rigid-flexible manipulator system leads to complex controller design and is difficult to implement. The existing automatic control methods have low control accuracy and cannot effectively suppress the vibration caused by the flexibility of the manipulator.

发明内容Contents of the invention

为了解决液压刚柔机械臂系统模型复杂导致的控制器设计复杂、不易实现，且现有控制方法的控制精度低，不能有效抑制由机械臂柔性导致的振动现象的技术问题，本发明提供一种基于双参数奇异摄动的液压刚柔机械臂控制方法，其引入双参数奇异摄动，将复杂的刚柔机械臂系统模型降阶为多时标参数的子系统，针对各子系统设计的控制系统结构简单、效率高，同时可达到满意的控制精度，有效抑制振动，易于工程实现。In order to solve the technical problems that the controller design is complicated and difficult to implement due to the complex model of the hydraulic rigid-flexible manipulator system, and the control precision of the existing control method is low, and the vibration phenomenon caused by the flexibility of the manipulator cannot be effectively suppressed, the present invention provides a The hydraulic rigid-flexible manipulator control method based on two-parameter singular perturbation, which introduces two-parameter singular perturbation, reduces the complex rigid-flexible manipulator system model to a subsystem with multiple time scale parameters, and the control system designed for each subsystem The structure is simple, the efficiency is high, and at the same time, satisfactory control accuracy can be achieved, vibration can be effectively suppressed, and engineering realization is easy.

采用双参数奇异摄动理论可将液压刚柔机械臂系统降阶为表征液压伺服驱动的快变子系统、表征弹性振动的次快变子系统及表征大范围刚性运动的慢变子系统，此时可分别设计控制器对三个子系统进行控制。双参数奇异摄动系统是指各小参数之间有如下关系：Using the dual-parameter singular perturbation theory, the hydraulic rigid-flexible manipulator system can be reduced to a fast-varying subsystem representing hydraulic servo drive, a sub-fast-varying subsystem representing elastic vibration, and a slowly-varying subsystem representing large-scale rigid motion. The controllers can be designed separately to control the three subsystems. The two-parameter singular perturbation system refers to the relationship between the small parameters as follows:

${m m}_{ij ij} \leq \leq \frac{{ϵ ϵ}_{i i}}{{ϵ ϵ}_{j j}} \leq \leq {M m}_{ij ij},, i i,, j j = = 1,2,3 1,2,3 - - - - - - ((22))$

其中M_ij与m_ij是某些常数。设小参数ε₁,ε₂,ε₃有下列关系：Among them M _ij and m _ij are some constants. Suppose the small parameters ε ₁ , ε ₂ , ε ₃ have the following relations:

$\underset{{ϵ ϵ}_{i i} &RightArrow; &Right Arrow; 00}{lim lim} \frac{{ϵ ϵ}_{i i + + 11}}{{ϵ ϵ}_{i i}} = = 00 - - - - - - ((33))$

这表明z_i+1是比z_i更快变化的快变状态。对于具有这样性质的多个小参数奇异摄动系统，可以用不同的快、慢观点，按单参数的分解方法得到整个系统的各级慢变、快变子系统。对整个系统的稳定性分析，可以由对各级子系统的稳定性分析来得到。This shows that z _i+1 is a fast-varying state that changes faster than z _i . For a multi-small parameter singularly perturbed system with such properties, different fast and slow viewpoints can be used to obtain the slow-varying and fast-varying subsystems of the whole system at all levels by the decomposition method of a single parameter. The stability analysis of the whole system can be obtained from the stability analysis of subsystems at all levels.

本发明解决技术问题所采取的技术方案如下：The technical solution adopted by the present invention to solve the technical problems is as follows:

基于双参数奇异摄动的液压刚柔机械臂控制方法包括如下步骤：The control method of hydraulic rigid-flexible manipulator based on two-parameter singular perturbation includes the following steps:

第一步、模型分解：The first step, model decomposition:

三自由度液压刚柔机械臂动力学方程如式(13)所示：The dynamic equation of the three-degree-of-freedom hydraulic rigid-flexible manipulator is shown in equation (13):

$\{\begin{matrix} M m ((θ θ,, q q)) [\begin{matrix} \overset{\cdot &Center Dot; \cdot \cdot}{θ θ} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{q q} \end{matrix}] + + K K [\begin{matrix} θ θ \\ q q \end{matrix}] + + G G ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot \cdot}{q q})) = = [\begin{matrix} τ τ \\ 00 \end{matrix}] \\ \overset{\cdot &Center Dot;}{τ τ} + + Aτ Aτ + + B B \overset{\cdot \cdot}{θ θ} = = CI CI \end{matrix} - - - - - - ((1313))$

其中，M(θ,q)∈R^5×5为对称、正定的惯性矩阵；θ＝[θ₁ θ₂ θ₃]^T为关节角；q＝[q₁ q₂]^T为模态坐标； $G (θ, \overset{\cdot}{θ}, q, \overset{\cdot}{q}) = {[\begin{matrix} g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \end{matrix}]}^{T}$ 为包含哥氏力、离心力、重力的非线性项；柔性臂的刚度矩阵为K＝diag(0,0,0,k₁,k₂)；作用在关节上的广义力矩为τ＝[τ₁ τ₂ τ₃]^T；I＝[i₁ i₂ i₃]^T为伺服阀控制电流；A＝diag(a₁,a₂,a₃)，a₁＝4β_e(C_tm1+K_c1)/V_t1，B＝diag(b₁,b₂,b₃)，b₁＝4β_eD²/V_t1， $b_{i} = 4 β_{e} A_{pi}^{2} J_{i}^{2} / V_{ti}, i = 2,3;$ C＝diag(c₁,c₂,c₃)，c₁＝4β_eDK_q1K_i1/V_t1,c_i＝4β_eA_piDK_qiK_iJ_i/V_ti,i＝2,3； Among them, M(θ,q)∈R ^5×5 is a symmetrical and positive definite inertia matrix; θ=[θ ₁ θ ₂ θ ₃ ] ^T is the joint angle; q=[q ₁ q ₂ ] ^T is the modal coordinates; $G (θ, \overset{\cdot}{θ}, q, \overset{&Center Dot;}{q}) = {[\begin{matrix} g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \end{matrix}]}^{T}$ is a nonlinear term including Coriolis force, centrifugal force and gravity; the stiffness matrix of the flexible arm is K=diag(0,0,0,k ₁ ,k ₂ ); the generalized moment acting on the joint is τ=[τ ₁ τ ₂ τ ₃ ] ^T ; I=[i ₁ i ₂ i ₃ ] ^T is the servo valve control current; A=diag(a ₁ ,a ₂ ,a ₃ ), a ₁ ＝4β _e (C _tm1 +K _c1 ) /V _t1 , B=diag(b ₁ ,b ₂ ,b ₃ ), b ₁ =4β _e D ² /V _t1 , $b_{i} = 4 β_{e} A_{p}^{2} J_{i}^{2} / V_{ti}, i = 2,3;$ C=diag(c ₁ ,c ₂ ,c ₃ ), c ₁ =4β _e DK _q1 K _i1 /V _t1 , c _i =4β _e A _pi DK _qi K _i J _i /V _ti , i=2,3;

采用双参数奇异摄动技术将式(13)分解，取第一个小参数并且满足0＜ε₁＜＜1，并在边界层上引入第一级快变时标得到液压刚柔机械臂的第一级快变子系统动力学方程如式(20)Using the two-parameter singular perturbation technique to decompose formula (13), take the first small parameter And satisfy 0<ε ₁ <<1, and introduce the first-level fast-changing time scale on the boundary layer The dynamic equation of the first-stage fast-changing subsystem of the hydraulic rigid-flexible manipulator is obtained as formula (20)

$\frac{d d {τ τ}_{f f 11}}{d d {σ σ}_{11}} = = - - \overset{~ ~}{A A} {τ τ}_{f f 11} + + \overset{~ ~}{C C} {I I}_{f f 11} - - - - - - ((2020))$

其中，I_f1为第一级快变时标σ₁下的控制电流，下标f1表示系统处在第一级快变时标下；Among them, I _f1 is the control current under the first-level fast-changing time scale σ ₁ , and the subscript f1 indicates that the system is under the first-level fast-changing time scale;

下面取第二个小参数其中k＝min(k₁,k₂)，且两个小参数满足0＜ε₁＜＜ε₂＜＜1，得到系统的第二级慢变流形表达式为Take the second small parameter below Where k=min(k ₁ ,k ₂ ), and the two small parameters satisfy 0<ε ₁ <<ε ₂ <<1, the second-level slowly varying manifold expression of the system is obtained as

${M m}_{11,, s the s 22} ((θ θ,, 00)) \overset{\cdot \cdot \cdot \cdot}{θ θ} + + {G G}_{11,, s the s 22} ((θ θ,, \overset{\cdot \cdot}{θ θ},, 0,0 0,0)) = = - - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot \cdot}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 22} - - - - - - ((2828))$

其中，I_s2为第二级慢变时标t下的控制电流，下标s2表示系统处在第二级慢变时标下；Among them, I _s2 is the control current under the second slowly changing time scale t, and the subscript s2 indicates that the system is under the second slowly changing time scale;

在边界层上引入第二级快变时标得液压刚柔机械臂系统第二级快变子系统的动力学方程为Introducing a second level of fast-varying time scale on the boundary layer The dynamic equation of the second-stage fast-changing subsystem of the hydraulic rigid-flexible manipulator system is

$\frac{{d d}^{22} {y the y}_{f f 22}}{d d {σ σ}_{22}^{22}} = = - - {D D.}_{44,, s the s 22} ((θ θ,, {ϵ ϵ}_{22} y the y)) \overset{~ ~}{K K} {y the y}_{f f 22} + + {D D.}_{33,, s the s 22} ((θ θ,, {ϵ ϵ}_{22} y the y)) {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{f f 22} - - - - - - ((3131))$

其中，I_f2为第二级快变时标σ₂下的控制电流，下标f2表示系统处在第二级快变时标下；Among them, I _f2 is the control current under the second-level fast-changing time scale σ ₂ , and the subscript f2 indicates that the system is under the second-level fast-changing time scale;

第二步、根据第一步得到的第二级慢变子系统、第二级快变子系统以及第一级快变子系统动力学方程，设计相应的子系统控制模块：The second step is to design the corresponding subsystem control module according to the dynamic equations of the second-level slow-varying subsystem, the second-level fast-varying subsystem and the first-level fast-varying subsystem obtained in the first step:

针对第二级慢变子系统，取一阶滑动模态面及二阶滑动模态面设计二阶滑模控制律如式(37)For the second-level slowly varying subsystem, the first-order sliding mode surface is taken and the second-order sliding mode surface Design the second-order sliding mode control law as formula (37)

${I I}_{s the s 22} = = {\overset{~ ~}{C C}}^{- - 11} \overset{~ ~}{A A} [[{M m}_{11,, s the s} (({\overset{\cdot &Center Dot; \cdot \cdot}{θ θ}}_{d d} + + αe αe + + β β \overset{\cdot \cdot}{e e} + + u u)) + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot \cdot}{θ θ} + + {G G}_{11,, s the s}]] - - - - - - ((3737))$

针对第二级快变子系统，取二次型性能指标函数及Ricatti方程设计最优控制律如式(41)For the second-level fast-changing subsystem, take the quadratic performance index function and the Ricatti equation Design the optimal control law as formula (41)

${I I}_{f f 22} = = - - {K K}_{f f} {X x}_{k k} = = - - {R R}^{- - 11} {B B}_{k k}^{T T} {PX PX}_{k k} - - - - - - ((4141))$

针对第一级快变子系统，取滑模面s_f=e_f及自适应更新律设计自适应滑模控制律如式(50)For the first-level fast-changing subsystem, the sliding surface s _f = e _f and the adaptive update law Design adaptive sliding mode control law as formula (50)

${I I}_{f f 11} = = {\overset{~ ~}{C C}}^{- - 11} [[{\overset{\cdot \cdot}{τ τ}}_{f f 11 d d} + + {\overset{~ ~}{A A}}_{00} {τ τ}_{f f 11} + + \overset{^^}{F f} + + {D D.}_{00} E E. + + ξsgn ξsgn (({s the s}_{f f}))]] - - - - - - ((5050))$

第三步、根据多重时间尺度理论，将第二步得到的各子系统控制器组合得到液压刚柔机械臂组合控制器如式(52)In the third step, according to the theory of multiple time scales, combine the controllers of the subsystems obtained in the second step to obtain the combined controller of the hydraulic rigid-flexible manipulator as shown in Equation (52)

I＝I_f1+I_s1＝I_f1+I_f2+I_s2 (52)I＝I _f1 +I _s1 ＝I _f1 +I _f2 +I _s2 (52)

进而完成基于双参数奇异摄动的液压刚柔机械臂的控制方法。Then complete the control method of hydraulic rigid-flexible manipulator based on dual-parameter singular perturbation.

本发明的有益效果如下：The beneficial effects of the present invention are as follows:

(1)由于采用了双参数奇异摄动技术，将复杂系统降阶为表征不同方式的三个子系统，使得系统模型简单，有效地降低了控制器设计的复杂性；(1) Due to the adoption of the dual-parameter singular perturbation technique, the order of the complex system is reduced to three subsystems representing different modes, which makes the system model simple and effectively reduces the complexity of the controller design;

(2)本发明通过三个子系统控制模块对三个子系统分别进行控制，各子系统控制精度不受其他子系统影响，使得本发明的控制方法更具通用性；(2) The present invention controls three subsystems respectively through three subsystem control modules, and the control accuracy of each subsystem is not affected by other subsystems, so that the control method of the present invention is more versatile;

(3)本发明将液压伺服部分从液压刚柔机械臂系统中分离，将电流作为控制输入量，避免了与机械臂控制部分采用力矩作为控制输入而导致的耦合；(3) The present invention separates the hydraulic servo part from the hydraulic rigid-flexible manipulator system, uses the current as the control input, and avoids coupling with the manipulator control part using torque as the control input;

(4)本发明中快变子控制系统中考虑了高频干扰和未建模动态对系统控制精度的影响，削弱了由于油液弹性模量摄动引起的液压冲击；(4) The influence of high-frequency interference and unmodeled dynamics on the control accuracy of the system is considered in the fast-changing sub-control system of the present invention, and the hydraulic shock caused by the perturbation of the elastic modulus of oil is weakened;

(5)本发明将多时标下子系统控制器进行时标统一，采用组合控制方式提高了控制系统精度，成本低，易于工程应用，更具有实用性。(5) The present invention unifies the time scales of the subsystem controllers under multiple time scales, and adopts a combined control method to improve the precision of the control system, with low cost, easy engineering application, and more practicality.

附图说明Description of drawings

图1是本发明液压刚柔机械臂系统结构示意图；Fig. 1 is a schematic structural view of the hydraulic rigid-flexible manipulator system of the present invention;

图2是本发明液压刚柔机械臂控制方法液压伺服部分阀控液压马达示意图；Fig. 2 is a schematic diagram of a valve-controlled hydraulic motor in the hydraulic servo part of the hydraulic rigid-flexible manipulator control method of the present invention;

图3是本发明液压刚柔机械臂控制方法液压伺服部分阀控非对称液压缸示意图；Fig. 3 is a schematic diagram of a valve-controlled asymmetric hydraulic cylinder in the hydraulic servo part of the control method of the hydraulic rigid-flexible manipulator of the present invention;

图4是本发明基于双参数奇异摄动的液压刚柔机械臂控制方法原理图；Fig. 4 is a schematic diagram of the control method of the hydraulic rigid-flexible manipulator based on the dual-parameter singular perturbation of the present invention;

图5是本发明基于双参数奇异摄动的液压刚柔机械臂控制方法流程图。Fig. 5 is a flow chart of the control method of the hydraulic rigid-flexible manipulator based on the dual-parameter singular perturbation of the present invention.

具体实施方式Detailed ways

下面结合附图和实施例对本发明做进一步详细说明。The present invention will be described in further detail below in conjunction with the accompanying drawings and embodiments.

如图1所示，本发明的控制对象为液压刚柔机械臂系统，其由三个连杆、回转底座和两个旋转关节组成，其中连杆1、连杆2为刚性臂，连杆3为柔性臂，回转关节为圆台状，由液压马达驱动，旋转关节由阀控单杆非对称液压缸驱动。三个连杆的长度分别为L₁，L₂，L₃，液压缸的安装位置分别为L₂₁，L₂₂，L₃₁。θ＝[θ ₁θ₂ θ₃]^T为连杆的关节转角.XOY为惯性坐标，X_iOY_i(i＝1,2,3)为固定在机械臂上的移动坐标。此系统存在液压伺服部分和机械臂的耦合，如何解决此问题是解决整个控制系统的关键。As shown in Figure 1, the control object of the present invention is a hydraulic rigid-flexible manipulator system, which consists of three connecting rods, a revolving base and two revolving joints, wherein connecting rod 1 and connecting rod 2 are rigid arms, and connecting rod 3 It is a flexible arm, the rotary joint is in the shape of a circular platform, driven by a hydraulic motor, and the rotary joint is driven by a valve-controlled single-rod asymmetric hydraulic cylinder. The lengths of the three connecting rods are respectively L ₁ , L ₂ and L ₃ , and the installation positions of the hydraulic cylinders are respectively L ₂₁ , L ₂₂ and L ₃₁ . θ=[θ ₁ θ ₂ θ ₃ ] ^T is the joint rotation angle of the connecting rod. XOY is the inertial coordinate, and X _i OY _i (i=1,2,3) is the moving coordinate fixed on the mechanical arm. There is a coupling between the hydraulic servo part and the mechanical arm in this system, how to solve this problem is the key to solve the whole control system.

如图2与图3所示分别为本发明采用的液压伺服驱动部分的阀控液压马达和阀控非对称液压缸的结构简图。P_s是供油压强，P_r为回油压强(通常为零)，x_v为伺服阀阀芯位移，P₁(P₂)分别为液压缸无(有)杆腔压强，Q₁(Q₂)为液压缸无(有)杆腔流量，A₁(A₂)为液压缸无(有)杆腔活塞面积，y为活塞杆位移，P_L为定义的负载压强。As shown in Fig. 2 and Fig. 3, the structural diagrams of the valve-controlled hydraulic motor and the valve-controlled asymmetric hydraulic cylinder of the hydraulic servo drive part adopted in the present invention are respectively shown. P _s is the oil supply pressure, P _r is the oil return pressure (usually zero), x _v is the displacement of the servo valve spool, P ₁ (P ₂ ) is the hydraulic cylinder without (with) rod cavity pressure, Q ₁ ( Q ₂ ) is the flow rate of the hydraulic cylinder without (with) rod chamber, A ₁ (A ₂ ) is the piston area of the hydraulic cylinder without (with) rod chamber, y is the displacement of the piston rod, and P _L is the defined load pressure.

如图4所示，本发明基于控制对象——液压刚柔机械臂系统多时标特点，通过双参数奇异摄动技术将其动力学模型进行降阶，并分解为表征液压伺服驱动系统的快变子系统，表征柔性臂弹性振动的次快变子系统及表征大范围刚性运动的慢变子系统。其优点是控制系统复杂性从系统级变为子系统级，使得控制系统结构、算法都大为简化，易于工程应用。双参数奇异摄动技术处理液压刚柔机械臂系统的关键问题在于如何用不同时标下的系统模型进行分解降阶，如何将不同时标下的控制器统一到统一时标下进行组合控制。As shown in Figure 4, based on the multi-time-scale characteristics of the control object—hydraulic rigid-flexible manipulator system, the present invention reduces the order of its dynamic model through the dual-parameter singular perturbation technology, and decomposes it into fast-changing variables that characterize the hydraulic servo drive system. Subsystem, the sub-fast variable subsystem that characterizes the elastic vibration of the flexible arm and the slowly variable subsystem that characterizes the large-scale rigid motion. Its advantage is that the complexity of the control system changes from the system level to the subsystem level, which greatly simplifies the structure and algorithm of the control system and is easy to apply in engineering. The key problem of the dual-parameter singular perturbation technique for the hydraulic rigid-flexible manipulator system is how to decompose and reduce the order of the system model under different time scales, and how to unify the controllers under different time scales to a unified time scale for combined control.

本发明基于双参数奇异摄动的液压刚柔机械臂控制方法实现中关键处理方法及过程如下：The key processing methods and processes in the realization of the hydraulic rigid-flexible manipulator control method based on the dual-parameter singular perturbation of the present invention are as follows:

1 液压刚柔机械臂系统动力学模型1 Dynamic model of hydraulic rigid-flexible manipulator system

1.1 刚柔机械臂系统动力学模型1.1 Dynamic model of rigid-flexible manipulator system

如图1所示，将柔性连杆作为Euler-Bernoulli等截面梁处理，根据假设模态法可知柔性臂的弹性变形量为As shown in Figure 1, the flexible link is treated as a Euler-Bernoulli equal-section beam. According to the hypothetical modal method, the elastic deformation of the flexible arm is

$ω ω ((r r,, t t)) = = \overset{n no}{\underset{s the s = = 11}{Σ Σ}} {Φ Φ}_{s the s} ((r r)) {q q}_{s the s} ((t t)) - - - - - - ((44))$

其中Φ_s(r)和q_s(t)分别为s^th模态函数和广义模态坐标，本发明中采用前二阶模态，即n＝2.Wherein Φ _s (r) and q _s (t) are s ^th modal function and generalized modal coordinates respectively, adopt the former second-order mode among the present invention, namely n=2.

考虑末端负载和柔性连杆弹性变形的影响，结合能量关系，可知系统总动能与势能并代入Lagrange方程，得到刚柔机械臂系统的动力学方程为Considering the influence of the end load and the elastic deformation of the flexible link, combined with the energy relationship, the total kinetic energy and potential energy of the system can be known and substituted into the Lagrange equation, and the dynamic equation of the rigid-flexible manipulator system is obtained as

$M m ((θ θ,, q q)) [\begin{matrix} \overset{\cdot \cdot \cdot \cdot}{θ θ} \\ \overset{\cdot \cdot \cdot \cdot}{q q} \end{matrix}] + + K K [\begin{matrix} θ θ \\ q q \end{matrix}] + + G G ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot \cdot}{q q})) = = [\begin{matrix} τ τ \\ 00 \end{matrix}] - - - - - - ((55))$

其中M(θ,q)∈R^5×5为对称、正定的惯性矩阵；θ＝[θ₁ θ₂ θ₃]^T为关节角；q＝[q₁ q₂]^T为模态坐标； $G (θ, \overset{\cdot}{θ}, q, \overset{\cdot}{q}) = {[\begin{matrix} g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \end{matrix}]}^{T}$ 为包含哥氏力、离心力、重力的非线性项；柔性臂的刚度矩阵为K＝diag(0,0,0,k₁,k₂)；作用在关节上的广义力矩为τ＝[τ₁ τ₂ τ₃]^T。Where M(θ,q)∈R ^5×5 is a symmetric and positive definite inertial matrix; θ=[θ ₁ θ ₂ θ ₃ ] ^T is the joint angle; q=[q ₁ q ₂ ] ^T is the modal coordinates; $G (θ, \overset{\cdot}{θ}, q, \overset{&Center Dot;}{q}) = {[\begin{matrix} g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \end{matrix}]}^{T}$ is a nonlinear term including Coriolis force, centrifugal force and gravity; the stiffness matrix of the flexible arm is K=diag(0,0,0,k ₁ ,k ₂ ); the generalized moment acting on the joint is τ=[τ ₁ τ ₂ τ ₃ ] ^T .

1.2 液压伺服驱动系统模型1.2 Hydraulic servo drive system model

假设液压伺服阀的时间常数远小于机械系统的时间常数，在液压伺服驱动系统中，伺服阀阀芯位移正比于控制输入Assuming that the time constant of the hydraulic servo valve is much smaller than that of the mechanical system, in the hydraulic servo drive system, the displacement of the servo valve spool is proportional to the control input

x_v＝K_ii (6)x _v = K _i i (6)

其中K_i是比例常系数。where K _i is a constant coefficient of proportionality.

根据图1所示阀控单杆非对称液压缸的安装位置，由余弦定理可得关节角与活塞位移的关系According to the installation position of the valve-controlled single-rod asymmetric hydraulic cylinder shown in Figure 1, the relationship between the joint angle and the piston displacement can be obtained from the cosine law

${y the y}_{22} = = \sqrt{{L L}_{21 twenty one}^{22} + + {L L}_{11}^{22} - - 22 {L L}_{11} {L L}_{21 twenty one} cos cos ((π π - - {θ θ}_{22}))} - - - - - - ((77))$

${y the y}_{33} = = \sqrt{{L L}_{22 twenty two}^{22} + + {L L}_{3131}^{22} - - 22 {L L}_{22 twenty two} {L L}_{3131} cos cos ((π π - - {θ θ}_{33}))} - - - - - - ((88))$

考虑式(7)与式(8)，对时间求导可得Considering formula (7) and formula (8), deriving with respect to time can get

$\overset{\cdot &Center Dot;}{y the y} = = J J \overset{\cdot &Center Dot;}{θ θ} - - - - - - ((99))$

其中y＝[y₂ y₃]^T为活塞杆位移；为刚柔性机械臂与液压驱动系统间的Jacobin矩阵。Where y=[y ₂ y ₃ ] ^T is the displacement of the piston rod; is the Jacobin matrix between the rigid-flexible manipulator and the hydraulic drive system.

根据虚功原理有τ＝J^TF_L，式中：F_L为液压缸驱动力。According to the principle of virtual work, τ=J ^T F _L , where: F _L is the driving force of the hydraulic cylinder.

忽略外泄漏量，根据式(6)和液压伺服系统基本原理，可得液压缸的动态模型分别为Neglecting the external leakage, according to formula (6) and the basic principle of the hydraulic servo system, the dynamic models of the hydraulic cylinder can be obtained as

$\{\begin{matrix} {x x}_{v v} = = {K K}_{i i} i i \\ {Q Q}_{L L} = = {K K}_{q q} {x x}_{v v} - - {K K}_{c c} {P P}_{L L} \\ {Q Q}_{L L} = = {A A}_{11} \overset{\cdot &Center Dot;}{y the y} + + {C C}_{tm tm} {P P}_{L L} + + \frac{{V V}_{t t}}{44 {β β}_{e e}} {\overset{\cdot &Center Dot;}{P P}}_{L L} \\ τ τ = = {J J}^{T T} {F f}_{L L},, {F f}_{L L} = = {A A}_{11} {P P}_{L L} \end{matrix} - - - - - - ((1010))$

式中： $K_{q} = C_{d} w \sqrt{2 (P_{s} - P_{L})} / \sqrt{ρ (1 + n^{3})}$ 为流量增益； $K_{c} = C_{d} w x_{v} / \sqrt{2 ρ (1 + n^{3}) (P_{s} - P_{L})}$ 为流量/压力系数；C_tm＝C_ip(1+n²)/(1+n³)为等效漏损系数；C_ip为液压缸的内泄漏系数；V_t＝2LA₁/(1+n³)为液压缸的等效容积；n＝A₂/A₁为面积比；P_L＝P₁-nP₂；β_e为液体有效体积弹性模量；ρ为油液密度；C_d为阀口流量系数；w为伺服阀的面积梯度；L为液压缸总行程。In the formula: $K_{q} = C_{d} w \sqrt{2 (P_{the s} - P_{L})} / \sqrt{ρ (1 + {no}^{3})}$ is flow gain; $K_{c} = C_{d} w x_{v} / \sqrt{2 ρ (1 + {no}^{3}) (P_{the s} - P_{L})}$ is the flow/pressure coefficient; C _tm ＝C _ip (1+n ² )/(1+n ³ ) is the equivalent leakage coefficient; C _ip is the internal leakage coefficient of the hydraulic cylinder; V _t ＝2LA ₁ /(1+ n ³ ) is the equivalent volume of the hydraulic cylinder; n=A ₂ /A ₁ is the area ratio; P _L =P ₁ -nP ₂ ; β _e is the effective bulk modulus of the liquid; ρ is the oil density; C _d is Valve port flow coefficient; w is the area gradient of the servo valve; L is the total stroke of the hydraulic cylinder.

液压马达的基本模型为The basic model of a hydraulic motor is

$\{\begin{matrix} {x x}_{v v} = = {K K}_{i i} i i \\ {Q Q}_{L L} = = {K K}_{q q} {x x}_{v v} - - {K K}_{c c} {P P}_{L L} \\ {Q Q}_{L L} = = D D. \overset{\cdot &Center Dot;}{θ θ} + + {C C}_{tm tm} {P P}_{L L} + + \frac{{V V}_{t t}}{44 {β β}_{e e}} {\overset{\cdot &Center Dot;}{P P}}_{L L} \\ τ τ = = {P P}_{L L} D D. \end{matrix} - - - - - - ((1111))$

其中，D为液压马达的体积排量，V_t表示液压马达腔室总体积，C_tm为泄漏系数，Q_L为液压马达的负载流量，为流量增益系数，θ为液压马达的转角。Among them, D is the volume displacement of the hydraulic motor, V _t is the total volume of the hydraulic motor chamber, C _tm is the leakage coefficient, Q _L is the load flow of the hydraulic motor, is the flow gain coefficient, θ is the rotation angle of the hydraulic motor.

由式(9)，式(10)和式(11)可求得3-DOF液压驱动刚柔性机械臂控制量与驱动力矩间的关系According to formula (9), formula (10) and formula (11), the relationship between the control amount and the driving torque of the 3-DOF hydraulically driven rigid-flexible manipulator can be obtained

$\overset{\cdot &Center Dot;}{τ τ} + + Aτ Aτ + + B B \overset{\cdot &Center Dot;}{θ θ} = = CI CI - - - - - - ((1212))$

其中I＝[i₁ i₂ i₃]^T为伺服阀控制电流;A＝diag(a₁,a₂,a₃),a₁＝4β_e(C_tm1+K_c1)/V_t1,B＝diag(b₁,b₂,b₃),b₁＝4β_eD²/V_t1,C＝diag(c₁,c₂,c₃),c₁＝4β_eDK_q1K_i1/V_t1,c_i＝4β_eA_piDK_qiK_iJ_i/V_ti,i＝2,3。Where I=[i ₁ i ₂ i ₃ ] ^T is the servo valve control current; A=diag(a ₁ ,a ₂ ,a ₃ ), a ₁ =4β _e (C _tm1 +K _c1 )/V _t1 , B=diag(b ₁ ,b ₂ ,b ₃ ),b ₁ =4β _e D ² /V _t1 , C=diag(c ₁ ,c ₂ ,c ₃ ), c ₁ =4β _e DK _q1 K _i1 /V _t1 , c _i =4β _e A _pi DK _qi K _i J _i /V _ti , i=2,3.

1.3 液压刚柔机械臂系统动力学模型1.3 Dynamic model of hydraulic rigid-flexible manipulator system

根据式(5)和式(12),可得三自由度液压刚柔机械臂动力学方程为According to formula (5) and formula (12), the dynamic equation of the three-degree-of-freedom hydraulic rigid-flexible manipulator can be obtained as

$\{\begin{matrix} M m ((θ θ,, q q)) [\begin{matrix} \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{q q} \end{matrix}] + + K K [\begin{matrix} θ θ \\ q q \end{matrix}] + + G G ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot &Center Dot;}{q q})) = = [\begin{matrix} τ τ \\ 00 \end{matrix}] \\ \overset{\cdot &Center Dot;}{τ τ} + + Aτ Aτ + + B B \overset{\cdot &Center Dot;}{θ θ} = = CI CI \end{matrix} - - - - - - ((1313))$

2 基于双参数奇异摄动的液压刚柔机械臂系统动力学模型2 Dynamics model of hydraulic rigid-flexible manipulator system based on two-parameter singular perturbation

液压刚柔机械臂系统包括液压伺服部分、刚性机械臂部分及柔性机械臂部分，因此其中一部分状态要比另一部分状态变化快。采用奇异摄动技术，先忽略快变量以降低系统阶数，再通过引入边界层校正项来提高近似程度，对系统连续进行两次分解，实现对液压刚柔机械臂系统的双参数奇异摄动分解，此方法不仅简化了控制结构，便于控制器设计，并且大幅度减少了计算量。The hydraulic rigid-flexible manipulator system includes a hydraulic servo part, a rigid manipulator part and a flexible manipulator part, so the state of some of them changes faster than that of the other part. Using the singular perturbation technology, the fast variable is ignored to reduce the system order, and then the boundary layer correction term is introduced to improve the approximation degree, and the system is decomposed twice continuously to realize the dual-parameter singular perturbation of the hydraulic rigid-flexible manipulator system Decomposition, this method not only simplifies the control structure, facilitates the controller design, but also greatly reduces the amount of calculation.

2.1第一次奇异摄动分解2.1 The first singular perturbation decomposition

将液压刚柔机械臂系统分解成表征刚柔机械臂运动的第一级慢变子系统，以及表征液压伺服驱动的第一级快变子系统。首先将式(13)改写成The hydraulic rigid-flexible manipulator system is decomposed into the first-level slow-change subsystem representing the movement of the rigid-flexible manipulator, and the first-level fast-change subsystem representing the hydraulic servo drive. Firstly, formula (13) is rewritten as

$\{\begin{matrix} M m ((θ θ,, q q)) [\begin{matrix} \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{q q} \end{matrix}] + + K K [\begin{matrix} θ θ \\ q q \end{matrix}] + + G G ((θ θ,, \overset{\cdot \cdot}{θ θ},, q q,, \overset{\cdot \cdot}{q q})) = = [\begin{matrix} τ τ \\ 00 \end{matrix}] \\ \frac{11}{{β β}_{e e}} \overset{\cdot &Center Dot;}{τ τ} = = - - \overset{~ ~}{A A} τ τ - - \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + \overset{~ ~}{CI CI} \end{matrix} - - - - - - ((1414))$

其中 $\tilde{A} = \frac{1}{β_{e}} A,$ $\tilde{B} = \frac{1}{β_{e}} B,$ $\tilde{C} = \frac{1}{β_{e}} C .$ in $\tilde{A} = \frac{1}{β_{e}} A,$ $\tilde{B} = \frac{1}{β_{e}} B,$ $\tilde{C} = \frac{1}{β_{e}} C .$

取第一个小参数并且满足0＜ε₁＜＜1，令z＝τ,则有Take the first small parameter And satisfy 0<ε ₁ <<1, let z=τ, then have

$M m ((θ θ,, q q)) [\begin{matrix} \overset{\cdot \cdot \cdot \cdot}{θ θ} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{q q} \end{matrix}] + + K K [\begin{matrix} θ θ \\ q q \end{matrix}] + + G G ((θ θ,, \overset{\cdot \cdot}{θ θ},, q q,, \overset{\cdot &Center Dot;}{q q})) = = [\begin{matrix} τ τ \\ 00 \end{matrix}] - - - - - - ((1515))$

${ϵ ϵ}_{11} \overset{\cdot &Center Dot;}{τ τ} = = - - \overset{~ ~}{A A} z z - - \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + \overset{~ ~}{C C} I I - - - - - - ((1616))$

取ε₁＝0，则液压刚柔机械臂系统的第一级慢变流形表达式为Taking ε ₁ =0, the expression of the first-order slow-varying manifold of the hydraulic rigid-flexible manipulator system is

$\{\begin{matrix} M m ((θ θ,, q q)) [\begin{matrix} \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} \\ \overset{\cdot &Center Dot; \cdot \cdot}{q q} \end{matrix}] + + K K [\begin{matrix} θ θ \\ q q \end{matrix}] + + G G ((θ θ,, \overset{\cdot \cdot}{θ θ},, q q,, \overset{\cdot &Center Dot;}{q q})) = = [\begin{matrix} {z z}_{s the s 11} \\ 00 \end{matrix}] \\ {z z}_{s the s 11} = = - - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 11} \end{matrix} - - - - - - ((1717))$

其中下标s1表示系统处在第一级慢变时标下。Among them, the subscript s1 indicates that the system is under the first-level slow-varying time scale.

当ε₁足够小时，ο(ε₁)可以忽略不计，根据多重时间尺度理论，取变量When ε ₁ is small enough, ο(ε ₁ ) can be neglected. According to the theory of multiple time scales, the variable

τ=τ_f1+τ_s1 (18)τ=τ _f1 +τ _s1 (18)

I=I_f1+I_s1 (19)I=I _f1 +I _s1 (19)

其中下标f1表示系统处在第一级快变时标下。Among them, the subscript f1 indicates that the system is under the first-level fast-changing time scale.

针对式(13)，在边界层上引入第一级快变时标靠近边界层区域ε₁→0，第一级慢变量τ_s1被视为常数，由式(16)-式(19)整理合并可得液压刚柔机械臂系统的第一级快变子系统动力学方程为According to formula (13), the first-level fast-varying time scale is introduced on the boundary layer Close to the boundary layer region ε ₁ → 0, the first-level slow variable τ _s1 is regarded as a constant, and the first-level fast-changing subsystem dynamics of the hydraulic rigid-flexible manipulator system can be obtained by combining equations (16)-(19) The equation is

其中I_f1为第一级快变时标σ₁下的控制电流。这样，将液压刚柔机械臂系统分解成了式(14)和式(20)，即第一级慢变子系统和第一级快变子系统。Among them, I _f1 is the control current under the fast-changing time scale σ ₁ of the first stage. In this way, the hydraulic rigid-flexible manipulator system is decomposed into formula (14) and formula (20), that is, the first-level slow-change subsystem and the first-level fast-change subsystem.

2.2 第二次奇异摄动分解2.2 Second singular perturbation decomposition

将第一级慢变子系统进一步分解成表征大范围刚性运动的第二级慢变子系统和表征弹性振动的第二级快变子系统，即次快变子系统。The first-level slowly-varying subsystem is further decomposed into the second-level slowly-varying subsystem representing large-scale rigid motion and the second-level fast-varying subsystem representing elastic vibration, that is, the sub-fast-varying subsystem.

令 $D = M^{- 1} = {[\begin{matrix} M_{1} & M_{2} \\ M_{3} & M_{4} \end{matrix}]}^{- 1} = [\begin{matrix} D_{1} & D_{2} \\ D_{3} & D_{4} \end{matrix}],$ $G (θ, \overset{\cdot}{θ}, q, \overset{\cdot}{q}) = [\begin{matrix} G_{1} \\ G_{2} \end{matrix}]$ make $D. = m^{- 1} = {[\begin{matrix} m_{1} & m_{2} \\ m_{3} & m_{4} \end{matrix}]}^{- 1} = [\begin{matrix} {D.}_{1} & {D.}_{2} \\ {D.}_{3} & {D.}_{4} \end{matrix}],$ $G (θ, \overset{&Center Dot;}{θ}, q, \overset{&Center Dot;}{q}) = [\begin{matrix} G_{1} \\ G_{2} \end{matrix}]$

其中M₁∈R^3×3,D₁∈R^3×3,G₁∈R^3×1。Among them M ₁ ∈ ^{R 3×3} , D ₁ ∈ R ^3×3 , G ₁ ∈ ^{R 3×1} .

由式(17)可知From formula (17) we can know

$\overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} = = - - {D D.}_{22} ((θ θ,, q q)) Kq Q - - {D D.}_{11} ((θ θ,, q q)) {G G}_{11} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot &Center Dot;}{q q}))$

$- - {D D.}_{22} ((θ θ,, q q)) {G G}_{22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot &Center Dot;}{q q})) + + {D D.}_{11} ((θ θ,, q q)) ((- - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 11})) - - - - - - ((21 twenty one))$

$\overset{\cdot &Center Dot; \cdot &Center Dot;}{q q} = = - - {D D.}_{44} ((θ θ,, q q)) {K K}_{x x} q q - - {D D.}_{33} ((θ θ,, q q)) {G G}_{11} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot \cdot}{q q}))$ $- - {D D.}_{44} ((θ θ,, q q)) {G G}_{22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot &Center Dot;}{q q})) + + {D D.}_{33} ((θ θ,, q q)) ((- - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot \cdot}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 11})) - - - - - - ((22 twenty two))$

其中K_x＝diag(k₁,k₂)。where K _x =diag(k ₁ ,k ₂ ).

取第二个小参数其中k＝min(k₁,k₂)，且两个小参数满足0＜ε₁＜＜ε₂＜＜1。Take the second small parameter Where k=min(k ₁ , k ₂ ), and the two small parameters satisfy 0<ε ₁ <<ε ₂ <<1.

令 $\tilde{K} = ϵ_{2} K_{x},$ q=ε₂y，则make $\tilde{K} = ϵ_{2} K_{x},$ q=ε ₂ y, then

$\overset{\cdot \cdot \cdot \cdot}{θ θ} = = - - {D D.}_{22} ((θ θ,, {ϵ ϵ}_{22} y the y)) \overset{~ ~}{K K} y the y - - {D D.}_{11} ((θ θ,, {ϵ ϵ}_{22} y the y)) {G G}_{11} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, {ϵ ϵ}_{22} y the y,, {ϵ ϵ}_{22} \overset{\cdot &Center Dot;}{y the y}))$

$- - {D D.}_{22} ((θ θ,, {ϵ ϵ}_{22} y the y)) {G G}_{22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, {ϵ ϵ}_{22} y the y,, {ϵ ϵ}_{22} \overset{\cdot &Center Dot;}{y the y})) - - - - - - ((23 twenty three))$

$+ + {D D.}_{11} ((θ θ,, {ϵ ϵ}_{22} y the y)) ((- - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 11}))$

${ϵ ϵ}_{22} \overset{\cdot &Center Dot; \cdot &Center Dot;}{y the y} = = - - {D D.}_{44} ((θ θ,, {ϵ ϵ}_{22} y the y)) \overset{~ ~}{K K} y the y - - {D D.}_{33} ((θ θ,, {ϵ ϵ}_{22} y the y)) {G G}_{11} ((θ θ,, \overset{\cdot \cdot}{θ θ},, {ϵ ϵ}_{22} y the y,, {ϵ ϵ}_{22} \overset{\cdot \cdot}{y the y}))$ $- - {D D.}_{44} ((θ θ,, {ϵ ϵ}_{22} y the y)) {G G}_{22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, {ϵ ϵ}_{22} y the y,, {ϵ ϵ}_{22} \overset{\cdot \cdot}{y the y})) - - - - - - ((24 twenty four))$

$+ + {D D.}_{33} ((θ θ,, {ϵ ϵ}_{22} y the y)) ((- - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot \cdot}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 11}))$

取ε₂＝0，得Taking ε ₂ =0, we get

$\overset{\cdot \cdot \cdot \cdot}{θ θ} = = - - {D D.}_{22,, s the s 22} ((θ θ,, 00)) \overset{~ ~}{K K} y the y - - {D D.}_{11,, s the s 22} ((θ θ,, 00)) {G G}_{11,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0))$

$- - {D D.}_{22,, s the s 22} ((θ θ,, 00)) {G G}_{22,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0)) - - - - - - ((2525))$

$+ + {D D.}_{11,, s the s 22} ((θ θ,, 00)) ((- - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 11}))$

$00 = = - - {D D.}_{44,, s the s 22} ((θ θ,, 00)) \overset{^^}{K K} {y the y}_{s the s 22} - - {D D.}_{33,, s the s 22} ((θ θ . . 00)) {G G}_{11,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0))$

$- - {D D.}_{44,, s the s 22} ((θ θ,, 00)) {G G}_{22,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0)) - - - - - - ((2626))$

$+ + {D D.}_{33,, s the s 22} ((θ θ,, 00)) ((- - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 11}))$

由式(26)解得Solve from formula (26) to get

${y the y}_{s the s 22} = = - - {\overset{~ ~}{K K}}^{- - 11} {D D.}_{44,, s the s 22}^{- - 11} ((θ θ,, 00)) {D D.}_{33,, s the s 22} ((θ θ,, 00)) {G G}_{11,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0))$

$- - {\overset{~ ~}{K K}}^{- - 11} {D D.}_{44,, s the s 22}^{- - 11} ((θ θ,, 00)) {D D.}_{44,, s the s 22} ((θ θ,, 00)) {G G}_{22,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0)) - - - - - - ((2727))$

$+ + {\overset{~ ~}{K K}}^{- - 11} {D D.}_{44,, s the s 22}^{- - 11} ((θ θ,, 00)) {D D.}_{33,, s the s 22} ((θ θ,, 00)) ((- - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot \cdot}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 22}))$

其中下标s2表示系统处在第二级慢变时标下，I_s2为第二级慢变时标t下的控制电流。Among them, the subscript s2 indicates that the system is under the second slowly changing time scale, and I _s2 is the control current under the second slowly changing time scale t.

将式(27)代入式(25)可以得到液压刚柔机械臂系统的第二级慢变流形表达式为Substituting Equation (27) into Equation (25), the expression of the second-stage slowly varying manifold of the hydraulic rigid-flexible manipulator system can be obtained as

${M m}_{11,, s the s 22} ((θ θ,, 00)) \overset{\cdot \cdot \cdot &Center Dot;}{θ θ} + + {G G}_{11,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0)) = = - - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 22} - - - - - - ((2828))$

当ε₂足够小时，ο(ε₂)可以忽略不计，根据多重尺度时间理论，取变量When ε ₂ is small enough, ο(ε ₂ ) can be neglected. According to the multi-scale time theory, the variable

y＝y_f2+y_s2 (29)y=y _f2 +y _s2 (29)

I_s1＝I_f2+I_s2 (30)I _s1 =I _f2 +I _s2 (30)

其中下标f2表示处在第二级快变时标下。Among them, the subscript f2 indicates that it is under the second-level fast-changing time scale.

针对式(17)，在边界层上引入第二级快变时标靠近边界层区域ε₂→0，第二级慢变量y_s2被视为常数，由式(24)、式(27)、式(29)和式(30)整理合并可得液压刚柔机械臂系统第二级快变子系统的动力学方程为According to formula (17), the second level of fast-varying time scale is introduced on the boundary layer Near the boundary layer region ε ₂ → 0, the second-level slow variable y _s2 is regarded as a constant, and the hydraulic rigid-flexible manipulator can be obtained by combining Equation (24), Equation (27), Equation (29) and Equation (30) The dynamic equation of the second-stage fast-changing subsystem of the system is

其中I_f2为第二级快变时标σ₂下的控制电流。这样，将液压刚柔机械臂系统进一步分解成了式(28)和式(31)，即第二级慢变子系统和第二级快变子系统。Among them, I _f2 is the control current under the second-level fast-changing time scale σ ₂ . In this way, the hydraulic rigid-flexible manipulator system is further decomposed into formula (28) and formula (31), that is, the second-level slow-change subsystem and the second-level fast-change subsystem.

综上，三个时标满足t＜＜σ₂＜＜σ₁，通过两次单参数奇异摄动分解，将液压刚柔机械臂系统分解成了式(28)、式(31)和式(20)，即慢变、次快变和快变三个子系统。In summary, the three time scales satisfy t<<σ ₂ <<σ ₁ , and through two single-parameter singular perturbation decompositions, the hydraulic rigid-flexible manipulator system is decomposed into formula (28), formula (31) and formula ( 20), that is, three subsystems of slow change, sub-fast change and fast change.

3 三重时间尺度控制器设计3 Triple time scale controller design

3.1 慢变控制器3.1 Slowly varying controller

针对慢变子系统式(28)，取关节角的期望轨迹为θ_d，则位置误差和速度误差为For the slowly varying subsystem equation (28), the expected trajectory of the joint angle is taken as θ _d , then the position error and velocity error are

e=θ_d-θ (32)e=θ _d -θ (32)

$\overset{\cdot \cdot}{e e} = = {\overset{\cdot &Center Dot;}{θ θ}}_{d d} - - \overset{\cdot &Center Dot;}{θ θ} - - - - - - ((3333))$

取滑动模态面Take the sliding modal surface

$s the s = = \overset{\cdot &Center Dot;}{e e} + + λe λ e,, λ λ > > 00 - - - - - - ((3434))$

其中λ为3阶待定对角矩阵，对角元素均为正。Among them, λ is an undetermined diagonal matrix of order 3, and the diagonal elements are all positive.

$Σ Σ = = \overset{\cdot \cdot}{s the s} + + ηs ηs,, η η > > 00 - - - - - - ((3535))$

其中η为3阶待定对角矩阵，对角元素均为正。Among them, η is an undetermined diagonal matrix of order 3, and the diagonal elements are all positive.

$\overset{\cdot \cdot}{Σ Σ} = = - - {k k}_{11} sgn sgn ((Σ Σ)) - - {k k}_{22} Σ Σ - - - - - - ((3636))$

其中k₁和k₂为3阶待定对角矩阵，对角元素均为正。Among them, k ₁ and k ₂ are undetermined diagonal matrices of order 3, and the diagonal elements are all positive.

设计二阶滑模控制律为The second-order sliding mode control law is designed as

${I I}_{s the s 22} = = {\overset{~ ~}{C C}}^{- - 11} \overset{~ ~}{A A} [[{M m}_{11,, s the s} (({\overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ}}_{d d} + + αe αe + + β β \overset{\cdot &Center Dot;}{e e} + + u u)) + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot \cdot}{θ θ} + + {G G}_{11,, s the s}]] - - - - - - ((3737))$

3.2 次快变控制器3.2 times fast change controller

针对次快变子系统，将式(31)变换成状态空间形式For subfast-changing subsystems, transform equation (31) into the state space form

${\overset{\cdot &Center Dot;}{X x}}_{k k} = = {A A}_{k k} {X x}_{k k} + + {B B}_{k k} {I I}_{f f 22} - - - - - - ((3838))$

其中 $X_{k} = [\begin{matrix} y_{f 2} \\ \frac{d y_{f 2}}{d σ_{2}} \end{matrix}],$ $A_{k} = [\begin{matrix} 0 & I \\ D_{4 . s 2} (θ, ϵ_{2} y) \tilde{K} & 0 \end{matrix}],$ $B_{k} = [\begin{matrix} 0 \\ D_{3, s 2} (θ, ϵ_{2} y) \end{matrix}] .$ in $x_{k} = [\begin{matrix} {the y}_{f 2} \\ \frac{d {the y}_{f 2}}{d σ_{2}} \end{matrix}],$ $A_{k} = [\begin{matrix} 0 & I \\ {D.}_{4 . the s 2} (θ, ϵ_{2} the y) \tilde{K} & 0 \end{matrix}],$ $B_{k} = [\begin{matrix} 0 \\ {D.}_{3, the s 2} (θ, ϵ_{2} the y) \end{matrix}] .$

假设次快变子系统中不存在不确定性，由于(A_k,B_k)完全可控，可以采用最优控制方法将系统的状态调节至零，抑制系统弹性振动。Assuming that there is no uncertainty in the sub-fast variable subsystem, since (A _k , B _k ) is completely controllable, the optimal control method can be used to adjust the state of the system to zero and suppress the elastic vibration of the system.

取二次型性能指标函数为Take the quadratic performance index function as

$J J = = \frac{11}{22} {&Integral; &Integral;}_{00}^{\infty \infty} [[{X x}_{k k}^{T T} Q Q {X x}_{k k} + + {I I}_{f f 22}^{T T} R R {I I}_{f f 22}]] dt dt - - - - - - ((3939))$

选取Ricatti方程为Choose the Ricatti equation as

${A A}_{k k}^{T T} P P + + P P {A A}_{k k} - - P P {B B}_{k k} {R R}^{- - 11} {B B}_{k k}^{T T} P P + + Q Q = = 00 - - - - - - ((4040))$

则最优控制律为Then the optimal control law is

${I I}_{f f 22} = = - - {K K}_{f f} {X x}_{k k} = = - - {R R}^{- - 11} {B B}_{k k}^{T T} P P {X x}_{k k} - - - - - - ((4141))$

3.3 快变控制器3.3 Fast change controller

假设快变子系统式(20)中存在参数摄动和未建模动态，形式为Assuming that there are parameter perturbations and unmodeled dynamics in the fast-varying subsystem (20), the form is

$\{\begin{matrix} \overset{~ ~}{A A} = = {\overset{~ ~}{A A}}_{00} + + Δ Δ \overset{~ ~}{A A} \\ \overset{~ ~}{C C} = = {\overset{~ ~}{C C}}_{00} + + Δ Δ \overset{~ ~}{C C} \\ E E. = = {E E.}_{00} + + ΔE ΔE \end{matrix} - - - - - - ((4242))$

其中E∈R^3×3为单位矩阵；E₀为标称值；ΔE为有界摄动量。where E∈R ^3×3 is the identity matrix; E ₀ is the nominal value; ΔE is the bounded perturbation.

考虑外界干扰和式(42)，则式(20)可改写为Considering external interference and formula (42), formula (20) can be rewritten as

${E E.}_{00} {\overset{\cdot &Center Dot;}{z z}}_{f f 11} + + {\overset{~ ~}{A A}}_{00} {τ τ}_{f f 11} + + F f + + d d = = {\overset{~ ~}{C C}}_{00} {I I}_{f f 11} - - - - - - ((4343))$

其中d(t)为外界干扰，总体不确定项是 $F (t, τ_{f 1}, {\overset{\cdot}{τ}}_{f 1}) = ΔE {\overset{\cdot}{τ}}_{f 1} + Δ \tilde{A} τ_{f 1} - Δ {\tilde{C}}_{0} I_{f 1} .$ where d(t) is external interference, and the overall uncertainty is $f (t, τ_{f 1}, {\overset{\cdot}{τ}}_{f 1}) = ΔE {\overset{&Center Dot;}{τ}}_{f 1} + Δ \tilde{A} τ_{f 1} - Δ {\tilde{C}}_{0} I_{f 1} .$

假设1 随机干扰d(t)是能量有界信号，且满足Assumption 1 Random interference d(t) is an energy-bounded signal, and satisfies

||d(t)||≤D₀ (44)||d(t)||≤D ₀ (44)

其中D₀为已知正常数。where D ₀ is a known constant.

假设2 总体不确定项且满足Hypothesis 2 Overall Uncertainty and satisfied

$| | | | F f ((t t,, {τ τ}_{f f 11},, {\overset{\cdot &Center Dot;}{τ τ}}_{f f 11})) | | | | \leq \leq {F f}_{00} - - - - - - ((4545))$

其中F₀为其上界，是一个未知正常数向量。Among them, F ₀ is its upper bound, which is an unknown normal constant vector.

设为F的估计值，则估计误差为set up is the estimated value of F, then the estimated error for

$\overset{~ ~}{F f} = = F f - - \overset{^^}{F f} - - - - - - ((4646))$

取τ_fd1的期望值为τ_fd1=[0 0 0]^T，则Take the expected value of τ _fd1 as τ _fd1 =[0 0 0] ^T , then

e_f=τ_f1d-τ_f1 (47)e _f =τ _f1d -τ _f1 (47)

${\overset{\cdot &Center Dot;}{e e}}_{f f} = = {\overset{\cdot &Center Dot;}{τ τ}}_{f f 11 d d} - - {\overset{\cdot &Center Dot;}{τ τ}}_{f f 11} - - - - - - ((4848))$

取滑模面为Take the sliding surface as

s_f=e_f (49)s _f = e _f (49)

设计自适应滑模控制律为The adaptive sliding mode control law is designed as

其中E＝[1,1,1]^T，ξ为3阶待定对角矩阵，对角元素均为正。Where E=[1,1,1] ^T , ξ is the undetermined diagonal matrix of order 3, and the diagonal elements are all positive.

取自适应更新律为Take the adaptive update law as

$\overset{\cdot &Center Dot;}{\overset{^^}{F f}} = = {Λ Λ}^{T T} {s the s}_{f f} - - - - - - ((5151))$

其中，Λ为已知常值向量，且其元素均为正数。Among them, Λ is a known constant value vector, and its elements are all positive numbers.

3 基于双参数奇异摄动的液压刚柔机械臂控制方法软件设计3. Software design of hydraulic rigid-flexible manipulator control method based on dual-parameter singular perturbation

本发明控制对象液压刚柔机械臂系统的机械臂参数如表1所示。The manipulator parameters of the control object hydraulic rigid-flex manipulator system of the present invention are shown in Table 1.

表1 液压柔性机械臂结构参数Table 1 Structural parameters of hydraulic flexible manipulator

基于双参数奇异摄动的液压刚柔机械臂控制方法软件采用Matlab语言编写，生成.m文件。程序设计思想及实现过程如图5所示，初始化后按照双参数奇异摄动分解，然后判断是否为快变子系统模型，若是，则采用快变子系统控制模块进行控制，若为否，则继续判断是否为次快变子系统模型，若是，则采用次快变子系统控制模块进行控制，若为否，则其为慢变子系统模型，采用慢变子系统控制模块进行控制，然后再将三个子系统控制模块的输出输入至控制驱动，之后进行数据存储，再判断是否到达运行时间，若为否，则继续进行循环判断，若是则输出结果结束。其中慢变子系统控制模块采用控制器如式(37)，次快变子系统控制模块采用控制器如式(41)，快变子系统控制模块采用控制器如式(50)，液压刚柔机械臂控制系统参数如表2所示。The software of the hydraulic rigid-flexible manipulator control method based on the singular perturbation of two parameters is written in Matlab language, and the .m file is generated. The program design idea and implementation process are shown in Figure 5. After initialization, it is decomposed according to the singular perturbation of two parameters, and then it is judged whether it is a fast-changing subsystem model. If so, the fast-changing subsystem control module is used for control. Continue to judge whether it is a sub-fast-varying subsystem model, if yes, use the sub-fast-varying subsystem control module for control, if not, then it is a slow-varying subsystem model, use the slow-varying subsystem control module for control, and then Input the output of the three subsystem control modules to the control drive, then store the data, and then judge whether the running time is reached, if not, continue to make a loop judgment, and if it is, the output result ends. Among them, the control module of the slow-change subsystem adopts the controller such as formula (37), the control module of the secondary fast-change subsystem adopts the controller such as formula (41), and the control module of the fast-change subsystem adopts the controller such as formula (50). The parameters of the robotic arm control system are shown in Table 2.

表2 基于双参数奇异摄动的液压刚柔机械臂系统控制参数Table 2 Control parameters of hydraulic rigid-flexible manipulator system based on dual-parameter singular perturbation

参数名称parameter name 取值value 参数名称parameter name 取值value ε₁ ε ₁ 5.0×10^-10 5.0×10 ^-10 αalpha diag(105,100,110)diag(105,100,110) ε₂ ε ₂ 1.2×10^-6 1.2×10 ^-6 βbeta diag(95,110,105)diag(95,110,105) λlambda diag(7,6,7)diag(7,6,7) QQ diag(50,46,55,55)diag(50,46,55,55) ηn diag(5,6,6)diag(5,6,6) RR diag(11,10,8)diag(11,10,8) k₁ k ₁ diag(15,12,12)diag(15,12,12) ΛΛ diag(50,50,30)diag(50,50,30) k₂ k ₂ diag(100,90,110)diag(100,90,110) ξξ diag(25,15,30)diag(25,15,30)

通过点击“Run”按钮，系统即可运行，首先将液压刚柔机械臂系统采用双参数奇异摄动技术降阶为三时标子系统，然后经过三时标子系统控制器分别控制，使得液压刚柔机械臂系统关节跟踪期望轨迹，误差精度在0.001rad以内；弹性振动得到抑制，振动抑制在0.001m以内。结果可采用word、excel或图表形式保存。By clicking the "Run" button, the system can run. First, the hydraulic rigid-flexible manipulator system is reduced to a three-time-scale subsystem using the dual-parameter singular perturbation technology, and then the three-time-scale The joints of the rigid-flexible manipulator system track the expected trajectory, and the error accuracy is within 0.001rad; the elastic vibration is suppressed, and the vibration suppression is within 0.001m. The results can be saved in the form of word, excel or chart.

Claims

1. The hydraulic rigid-flexible manipulator control method based on two-parameter singular perturbation, is characterized in that, the method comprises the steps:

The first step, model decomposition:

The dynamic equation of the three-degree-of-freedom hydraulic rigid-flexible manipulator is shown in equation (13):

\{\begin{matrix} M m ((θ θ,, q q)) [\begin{matrix} \overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ} \\ \overset{\cdot &Center Dot; \cdot &Center Dot;}{q q} \end{matrix}] + + K K [\begin{matrix} θ θ \\ q q \end{matrix}] + + G G ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, q q,, \overset{\cdot &Center Dot;}{q q})) = = [\begin{matrix} τ τ \\ 00 \end{matrix}] \\ \overset{\cdot &Center Dot;}{τ τ} + + Aτ Aτ + + B B \overset{\cdot &Center Dot;}{θ θ} = = CI CI \end{matrix} - - - - - - ((1313))

Among them, M(θ,q)∈R ^5×5 is a symmetrical and positive definite inertia matrix; θ=[θ ₁ θ ₂ θ ₃ ] ^T is the joint angle; q=[q ₁ q ₂ ] ^T is the modal coordinates;

G (θ, \overset{&Center Dot;}{θ}, q, \overset{&Center Dot;}{q}) = {[\begin{matrix} g_{1} & g_{2} & g_{3} & g_{4} & g_{5} \end{matrix}]}^{T}

is a nonlinear term including Coriolis force, centrifugal force and gravity; the stiffness matrix of the flexible arm is K=diag(0,0,0,k ₁ ,k ₂ ); the generalized moment acting on the joint is τ=[τ ₁ τ ₂ τ ₃ ] ^T ; I=[i ₁ i ₂ i ₃ ] ^T is the servo valve control current; A=diag(a ₁ ,a ₂ ,a ₃ ), a ₁ ＝4β _e (C _tm1 +K _c1 ) /V _t1 , i=2,3; B=diag(b ₁ ,b ₂ ,b ₃ ), b ₁ =4β _e D ² /V _t1 , i=2,3; C=diag(c ₁ ,c ₂ ,c ₃ ), c ₁ =4β _e DK _q1 K _i1 /V _t1 , c _i =4β _e A _pi DK _qi K _i J _i /V _ti , i=2,3; β _e is the effective bulk modulus of liquid; C _tmi is the leakage coefficient of the i (i=2,3) hydraulic cylinder; K _ci is the flow/pressure coefficient of the i (i=2,3) hydraulic cylinder; J _i is the Jacobin matrix between the rigid-flexible manipulator and the hydraulic drive system; D is the volume displacement of the hydraulic motor; K _qi is the flow gain of the i (i=2,3) hydraulic cylinder; K _i is the servo valve The proportional constant coefficient between core displacement and control input; V _ti is the equivalent volume of the i (i=2,3) hydraulic cylinder; C _tm1 is the leakage coefficient of the first hydraulic cylinder; K _c1 is the first hydraulic cylinder V _t1 is the equivalent volume of the first hydraulic cylinder; K _q1 is the flow gain of the first hydraulic cylinder; K _i1 is the proportional constant coefficient between the displacement of the first servo valve spool and the control input;

Using the two-parameter singular perturbation technique to decompose formula (13), take the first small parameter And satisfy 0<ε ₁ <<1, and introduce the first-level fast-changing time scale on the boundary layer The dynamic equation of the first-stage fast-changing subsystem of the hydraulic rigid-flexible manipulator is obtained as formula (20)

\frac{{dτ dτ}_{f f 11}}{{dσ dσ}_{11}} = = - - \overset{~ ~}{A A} {τ τ}_{f f 11} + + \overset{~ ~}{C C} {I I}_{f f 11} - - - - - - ((2020))

Among them, I _f1 is the control current under the first-level fast-changing time scale σ ₁ , and the subscript f1 indicates that the system is under the first-level fast-changing time scale; t is the second-level slow-changing time scale;

Take the second small parameter below Where k=min(k ₁ ,k ₂ ), and the two small parameters satisfy 0<ε ₁ <<ε ₂ <<1, the second-level slowly varying manifold expression of the system is obtained as

{M m}_{11,, s the s 22} ((θ θ,, 00)) \overset{\cdot \cdot \cdot \cdot}{θ θ} + + {G G}_{11,, s the s 22} ((θ θ,, \overset{\cdot &Center Dot;}{θ θ},, 0,0 0,0)) = = - - {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot \cdot}{θ θ} + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{C C} {I I}_{s the s 22} - - - - - - ((2828))

Among them, I _s2 is the control current under the second slowly varying time scale t, and the subscript s2 indicates that the system is under the second slowly varying time scale; M _{1, s2} is the inertia matrix under the second slowly varying manifold , G _{1, s2} is the nonlinear term including Coriolis force, centrifugal force and gravity under the second-order slowly varying manifold;

Introducing a second level of fast-varying time scale on the boundary layer The dynamic equation of the second-stage fast-changing subsystem of the hydraulic rigid-flexible manipulator system is

\frac{{d d}^{22} {y the y}_{f f 22}}{{dσ dσ}_{22}^{22}} = = - - {D D.}_{44,, s the s 22} ((θ θ,, {ϵ ϵ}_{22} y the y)) {\overset{~ ~}{K K} y the y}_{f f 22} + + {D D.}_{33,, s the s 22} ((θ θ,, {ϵ ϵ}_{22} y the y)) {\overset{~ ~}{A A}}^{- - 11} {\overset{~ ~}{C C} I I}_{f f 22} - - - - - - ((3131))

Among them, I _f2 is the control current under the second-level fast-changing time scale σ ₂ , and the subscript f2 indicates that the system is under the second-level fast-changing time scale; y _f2 is the state variable of the second-level fast-changing system; D _{4, s2} represents the fourth element of the inertia matrix inverse array under the second-level slowly varying subsystem; D _{3, s2} represents the third element of the inertia matrix inverse array under the second-level slowly varying subsystem; K _x = diag(k ₁ , k ₂ ), k ₁ , k ₂ are the 4th and 5th dimension elements of the stiffness matrix K of the flexible arm, that is, the elastic mode coefficient of the flexible arm;

The second step is to design the corresponding subsystem control module according to the dynamic equations of the second-level slow-varying subsystem, the second-level fast-varying subsystem and the first-level fast-varying subsystem obtained in the first step:

For the second-level slowly varying subsystem, the first-order sliding mode surface is taken (e is the position error), and the second-order sliding mode surface Design the second-order sliding mode control law as formula (37)

{I I}_{s the s 22} = = {\overset{~ ~}{C C}}^{- - 11} \overset{~ ~}{A A} [[{M m}_{11,, s the s} (({\overset{\cdot &Center Dot; \cdot &Center Dot;}{θ θ}}_{d d} + + αe αe + + β β \overset{\cdot &Center Dot;}{e e} + + u u)) + + {\overset{~ ~}{A A}}^{- - 11} \overset{~ ~}{B B} \overset{\cdot &Center Dot;}{θ θ} + + {G G}_{11,, s the s}]] - - - - - - ((3737))

Among them, u is the intermediate variable of the controller; M _1,s is the inertia matrix of the first-level slowly varying system, and G _1,s is the nonlinear term including Coriolis force, centrifugal force, and gravity under the first-level slowly varying system; is the second derivative of the expected trajectory of the joint angle; α, β are the trajectory position tracking error e and its error derivative The coefficient matrices of are all undetermined diagonal matrices of order 3, and the diagonal elements are positive;

For the second-level fast-changing subsystem, take the quadratic performance index function and the Ricatti equation

A_{k}^{T} P + {PA}_{k} - {PB}_{k} R^{- 1} B_{k}^{T} P + Q = 0,

Design the optimal control law as formula (41)

{I I}_{f f 22} = = - - {K K}_{f f} {X x}_{k k} = = - - {R R}^{- - 11} {B B}_{k k}^{T T} {PX PX}_{k k} - - - - - - ((4141))

Among them, X _k is the state variable of the sub-fast variable subsystem; both Q and R are positive definite matrices with corresponding orders; P is the solution of Riccati equation (40); B _k is the input matrix of the sub-fast variable subsystem;

For the first-level fast-changing subsystem, the sliding mode surface s _f = e _f and the adaptive update law Design adaptive sliding mode control law as formula (50)

{I I}_{f f 11} = = {\overset{~ ~}{C C}}^{- - 11} [[{\overset{\cdot \cdot}{τ τ}}_{f f 11 d d} + + {\overset{~ ~}{A A}}_{00} {τ τ}_{f f 11} + + \overset{^^}{F f} + + {D D.}_{00} E E. + + ξsgn ξsgn (({s the s}_{f f}))]] - - - - - - ((5050))

Among them, _ef is the torque error of the fast-changing subsystem; Λ is a known constant value vector, and its elements are all positive numbers; is the estimated value of F; the known constant D ₀ is the upper bound of the energy bounded random disturbance d(t); E is the vector E=[1,1,1] ^T ; ξ is the third-order undetermined diagonal matrix, and The diagonal elements are all positive numbers; is the nominal value of the parameter perturbation and unmodeled dynamics in the rapidly varying subsystem; is the derivative value of the expected vibration;

In the third step, according to the theory of multiple time scales, combine the controllers of the subsystems obtained in the second step to obtain the combined controller of the hydraulic rigid-flexible manipulator, such as formula (52)

I＝I _f1 +I _s1 ＝I _f1 +I _f2 +I _s2 (52)

Then complete the control method of hydraulic rigid-flexible manipulator based on dual-parameter singular perturbation.