CN103395065A - Hydraulic hard and soft mechanical arm control method based on two-parameter singular perturbation - Google Patents

Abstract

The invention provides a hydraulic hard and soft mechanical arm control method based on two-parameter singular perturbation, and relates to the technical field of intelligent machine and robot control. Through the principle of the two-parameter singular perturbation of the method, a reduced order is carried out on a hydraulic hard and soft mechanical arm system to enable the hydraulic hard and soft mechanical arm system to become a fast subsystem driven by representation hydraulic servo, a secondary fast subsystem driven by representation elastic vibration, and a slow subsystem driven by representation wide-range rigid movement, wherein a self-adaptive sliding mode controller is adopted in a fast subsystem control module, an optimal controller is adopted in a secondary fast subsystem control module to restrain the elastic vibration, and a second-order sliding mode controller is adopted in a slow subsystem control module to achieve joint trajectory tracking control. According to the control method based on the two-parameter singular perturbation, the reduced order is carried out on the complex hydraulic hard and soft mechanical arm system, multiple time-scale subsystem controllers are designed, and thus the control method is greatly simplified and is more practical.

Description

Based on the hard and soft mechanical arm control method of the hydraulic pressure of two-parameter singular perturbation

Technical field

The present invention relates to intelligent machine and Robot Control Technology field, relate in particular to the hard and soft mechanical arm control method of a kind of hydraulic pressure based on two-parameter singular perturbation.

Background technology

The hard and soft mechanical arm of hydraulic pressure is a kind of by hydraulically powered mechanical arm, it has high durability, large driving force square, high power/weight ratio, top load/from characteristics such as anharmonic ratioes, therefore often be applied in large-scale heavy duty machinery, and in fields such as building, mining, communications and transportation, machinings, be widely used.The hard and soft mechanical arm of hydraulic pressure inevitably produces elastic vibration when the operation heavy load, this has brought difficulty for the reprinting accurately and fast of load.The general nonlinear feedback control system that adopts is controlled at present, but controller generally designs complexity, amount of calculation is large, real-time is poor, is difficult to reach the control effect of expectation.Therefore the controller of needing a kind of amount of calculation simple in structure, little of design, high real-time badly and being easy to realize, meet the hard and soft mechanical arm of hydraulic pressure and reprint large load task needs.

The asymptotic expansion solution of perturbation theory research Small Parameter and the character of solution, can be in the hope of meeting the approximate analytic solution of required required precision, therefore it has an enormous advantage than numerical solution, this class problem occurs in a large number in the engineering science such as physics, chemistry, mechanics, astronomy and traffic, economic dispatch management science, and to be referred to as this class problem be perturbed problem.In a large amount of perturbed problems,, according to the gradation of solution to small parameter, they can be divided into two large classes: regular perturbation and singular perturbation.The definition that provides according to R.E.O ' Malley, if perturbed problem P _εSolution y _ε(x) can be expressed as the power series of ε

$y_{\mathrm{\ϵ}}\left(x\right)~y_0\left(x\right)+\stackrel{\∞}{\underset{j=1}{\mathrm{\Σ}}}{y}_j\left(x\right)\·{\mathrm{\ϵ}}^j,x\∈D,\mathrm{\ϵ}\→0---\left(1\right)$

Wherein, y ₀(x) be problem P ₀(be P _ε| _ε=0) solution, and above-mentioned expansion is right

Be uniformly convergent, claim P _εFor the regular perturbation problem.Otherwise, just claim P _εFor singular perturbation problem.In automatic control problem, the past is to containing small parameter _εThe processing method of singular perturbation problem be that small parameter is ignored, namely approximate small parameter ε is regarded as is zero, ignore high order part, keep low frequency part, or the impact of constant between not considering hour, can make like this error of system in boundary layer reach the degree of distortion.

The hard and soft mechanical arm of hydraulic pressure is that a kind of large-scale, strong nonlinearity, the highly system of coupling, so its autocontrol method are the focuses that scholar and researcher discuss always.But the hard and soft mechanical arm system of hydraulic pressure causes complicated, the difficult realization of controller design because its model is complicated, and existing autocontrol method control accuracy is low, can not effectively suppress the oscillation phenomenon that is caused by the mechanical arm flexibility.

Summary of the invention

the controller design complexity that causes in order to solve the hard and soft mechanical arm system model of hydraulic pressure complexity, be difficult for realizing, and the control accuracy of existing control method is low, the technical problem that can not effectively suppress the oscillation phenomenon that is caused by the mechanical arm flexibility, the invention provides the hard and soft mechanical arm control method of a kind of hydraulic pressure based on two-parameter singular perturbation, it introduces two-parameter singular perturbation, with the hard and soft mechanical arm system model reduction of the complexity subsystem that is the Multi-time Scale parameter, control system for each subsystem design is simple in structure, efficiency is high, simultaneously can reach satisfied control accuracy, effectively suppress vibration, be easy to Project Realization.

Adopt two-parameter singular perturbation theory can be with the hard and soft mechanical arm system depression of order of hydraulic pressure for the fast subsystem that characterizes Hydraulic servo drive, characterize inferior fast subsystem of elastic vibration and characterize the slow subsystem of rigid motion on a large scale, CONTROLLER DESIGN is controlled three sub-systems respectively at this moment.Two-parameter singular perturbation system refers between each small parameter have following relation:

$m_{\mathrm{ij}}\≤\frac{{\mathrm{\ϵ}}_i}{{\mathrm{\ϵ}}_j}\≤M_{\mathrm{ij}},i,j=\mathrm{1,2,3}---\left(2\right)$

M wherein _ijWith m _ijIt is some constant.If small parameter ε ₁, ε ₂, ε ₃Following relationship is arranged:

$\underset{{\mathrm{\ϵ}}_i\→0}{\lim }\frac{{\mathrm{\ϵ}}_{i+1}}{{\mathrm{\ϵ}}_i}=0---\left(3\right)$

This shows z _i+1To compare z _iThe fast state that changes sooner.For a plurality of small parameter singular perturbation systems with such character, can be with different fast, slow viewpoints, by one-parameter decomposition method at different levelsly becoming slowly of obtaining whole system, fast subsystems., to the stability analysis of whole system, can be obtained by the stability analysis to subsystems at different levels.

The technical scheme that technical solution problem of the present invention is taked is as follows:

Comprise the steps: based on the hard and soft mechanical arm control method of the hydraulic pressure of two-parameter singular perturbation

The first step, model decompose:

The hard and soft Manipulator Dynamics equation of Three Degree Of Freedom hydraulic pressure is suc as formula shown in (13):

$\left\{\begin{array}{c}M(\mathrm{\θ},q)\left[\begin{array}{c}\stackrel{\·\·}{\mathrm{\θ}}\\ \stackrel{\·\·}{q}\end{array}\right]+K\left[\begin{array}{c}\mathrm{\θ}\\ q\end{array}\right]+G\left(\mathrm{\θ},\stackrel{\·}{\mathrm{\θ}},q,\stackrel{\·}{q}\right)=\left[\begin{array}{c}\mathrm{\τ}\\ 0\end{array}\right]\\ \stackrel{\·}{\mathrm{\τ}}+\mathrm{A\τ}+B\stackrel{\·}{\mathrm{\θ}}=\mathrm{CI}\end{array}\right.---\left(13\right)$

Wherein, M (θ, q) ∈ R ^{5 * 5}For inertial matrix symmetrical, positive definite; θ=[θ ₁θ ₂θ ₃] ^TFor joint angle; Q=[q ₁q ₂] ^TFor modal coordinate; $G(\mathrm{\θ},\stackrel{\·}{\mathrm{\θ}},q,\stackrel{\·}{q})={\left[\begin{array}{ccccc}{g}_1& g_2& g_3& g_4& g_5\end{array}\right]}^T$ For comprising the nonlinear terms of coriolis force, centrifugal force, gravity; The stiffness matrix of flexible arm is K=diag (0,0,0, k ₁, k ₂); The Generalized Moment that acts on joint is τ=[τ ₁τ ₂τ ₃] ^TI=[i ₁i ₂i ₃] ^TFor the servo valve control electric current; A=diag (a ₁, a ₂, a ₃), a ₁=4 β _e(C _tm1+ K _c1)/V _t1,

B=diag (b ₁, b ₂, b ₃), b ₁=4 β _eD ²/ V _t1, $b_i=4{\mathrm{\β}}_e{A}_{\mathrm{pi}}^2{J}_i^2/V_{\mathrm{ti}},i=\mathrm{2,3};$ C＝diag(c ₁,c ₂,c ₃)，c ₁＝4β _eDK _q1K _i1/V _t1,c _i＝4β _eA _piDK _qiK _iJ _i/V _ti,i＝2,3；

Adopt two-parameter singular perturbation technology that formula (13) is decomposed, get first small parameter

And meet 0＜ε ₁＜＜1, and the introducing first order becomes markers soon on boundary layer

Obtain the first order fast subsystem kinetics equation of the hard and soft mechanical arm of hydraulic pressure suc as formula (20)

$\frac{d{\mathrm{\&tau;}}_{f1}}{d{\mathrm{\&sigma;}}_1}=-\tilde{A}{\mathrm{\&tau;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}+\tilde{C}{I}_{f1}---\left(20\right)$

Wherein, I _f1For the first order becomes markers σ soon ₁Under the control electric current, subscript f1 represents that system is in the first order and becomes soon under markers;

Below get second small parameter

K=min (k wherein ₁, k ₂), and two small parameters meet 0＜ε ₁＜＜ε ₂＜＜1, the second level slow unsteady flow shape expression formula that obtains system is

${\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">M</figure-callout>}}_{1,s2}(\mathrm{\&theta;},0)\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}+{\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">G</figure-callout>}}_{1,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})=-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s2}---\left(28\right)$

Wherein, I _s2For the second level becomes control electric current under markers t slowly, subscript s2 represents that system is in the second level and becomes slowly under markers;

Introduce the second level and become soon markers on boundary layer

The kinetics equation that obtains the hard and soft mechanical arm system of hydraulic pressure second level fast subsystem is

$\frac{d^2{y}_{f2}}{d{\mathrm{\&sigma;}}_2^2}=-D_{4,s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)\tilde{K}{y}_{f2}+D_{3,s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y){\tilde{A}}^{-1}\tilde{C}{I}_{f2}---\left(31\right)$

Wherein, I _f2For the second level becomes markers σ soon ₂Under the control electric current, subscript f2 represents that system is in the second level and becomes soon under markers;

Second step, according to second level slow subsystem, second level fast subsystem and first order fast subsystem kinetics equation that the first step obtains, design corresponding subsystem control module:

, for second level slow subsystem, get single order sliding mode face And second order sliding mode face

Design Second Order Sliding Mode Control rule is suc as formula (37)

$I_{s2}={\tilde{C}}^{-1}\tilde{A}[{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">M</figure-callout>}}_{1,s}({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_d+\mathrm{\&alpha;e}+\mathrm{\&beta;}\stackrel{\&CenterDot;}{e}+u)+{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">G</figure-callout>}}_{1,s}]---\left(37\right)$

, for second level fast subsystem, get the quadratic performance index function

And Ricatti equation

The devise optimum control law is suc as formula (41)

$I_{f2}=-K_f{X}_k=-R^{-1}{B}_k^T{\mathrm{PX}}_k---\left(41\right)$

, for first order fast subsystem, get sliding-mode surface s _f=e _fAnd adaptive updates rule

Design adaptive sliding mode control law is suc as formula (50)

$I_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}={\tilde{C}}^{-1}[{\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{f1d}+{\tilde{A}}_0{\mathrm{\&tau;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}+\hat{F}+D_{0}E+\mathrm{\&xi;sgn}\left(s_f\right)]---\left(50\right)$

In the 3rd step, theoretical according to multiple time scale, each subsystem controller combination that second step is obtained obtains the hard and soft mechanical arm combined controller of hydraulic pressure suc as formula (52)

I＝I _f1+I _s1＝I _f1+I _f2+I _s2 (52)

And then complete control method based on the hard and soft mechanical arm of hydraulic pressure of two-parameter singular perturbation.

Beneficial effect of the present invention is as follows:

(1) owing to having adopted two-parameter singular perturbation technology,, for characterizing three sub-systems of different modes, make system model simple the complication system depression of order, effectively reduced the complexity of controller design;

(2) the present invention controls respectively three sub-systems by three sub-systems control modules, and each subsystem controls precision is not affected by other subsystems, makes control method of the present invention have more versatility;

(3) the present invention separates the hydraulic servo part from the hard and soft mechanical arm system of hydraulic pressure, and electric current, as the control inputs amount, has been avoided the coupling that adopts moment to cause as control inputs with the mechanical arm control section;

(4) considered in fast varitron control system in the present invention High-frequency Interference and not modeling, dynamically on the impact of system control accuracy, weakened the hydraulic shock of due to the fluid elastic modelling quantity, perturbing and causing;

(5) the present invention carries out the markers unification with subsystem controller under Multi-time Scale, adopts the combination control mode to improve Control system resolution, and cost is low, is easy to the engineering application, has more practicality.

Description of drawings

Fig. 1 is the hard and soft mechanical arm system structural representation of hydraulic pressure of the present invention;

Fig. 2 is the hard and soft mechanical arm control method of hydraulic pressure of the present invention hydraulic servo part valve control hydraulic motor schematic diagram;

Fig. 3 is the hard and soft mechanical arm control method of hydraulic pressure of the present invention hydraulic servo part valve control asymmetrical cylinder schematic diagram;

Fig. 4 is the hard and soft mechanical arm control method of the hydraulic pressure schematic diagram that the present invention is based on two-parameter singular perturbation;

Fig. 5 is the hard and soft mechanical arm control method of the hydraulic pressure flow chart that the present invention is based on two-parameter singular perturbation.

The specific embodiment

Below in conjunction with drawings and Examples, the present invention is described in further details.

As shown in Figure 1, control object of the present invention is the hard and soft mechanical arm system of hydraulic pressure, it is comprised of three connecting rods, rotary table and two rotary joints, wherein connecting rod 1, connecting rod 2 are rigid arm, connecting rod 3 is flexible arm, revolute joint is round table-like, and by fluid motor-driven, rotary joint is driven by valve control single pole asymmetrical cylinder.The length of three connecting rods is respectively L ₁, L ₂, L ₃, the installation site of hydraulic cylinder is respectively L ₂₁, L ₂₂, L ₃₁θ=[θ ₁θ ₂θ ₃] ^TFor the joint rotation angle .XOY of connecting rod is inertial coordinate, X _iOY _i(i=1,2,3) are for being fixed on the moving coordinate on mechanical arm.There is the coupling of hydraulic servo part and mechanical arm in this system, and how head it off is the key that solves whole control system.

Be respectively as shown in Figure 2 and Figure 3 the valve control hydraulic motor of the Hydraulic servo drive part that the present invention adopts and the structure diagram of valve control asymmetrical cylinder.P _sFuel feeding pressure, P _rFor oil return pressure (being generally zero), x _vFor valve core of servo valve displacement, P ₁(P ₂) be respectively hydraulic cylinder without (having) bar chamber pressure, Q ₁(Q ₂) for hydraulic cylinder without (having) bar chamber flow, A ₁(A ₂) for hydraulic cylinder without (having) bar chamber piston area, y is the piston rod displacement, P _LLoad pressure for definition.

As shown in Figure 4, the present invention is based on control object---the hard and soft mechanical arm system Multi-time Scale of hydraulic pressure characteristics, by two-parameter singular perturbation technology, its kinetic model is carried out depression of order, and be decomposed into the fast subsystem that characterizes hydraulic servo driving system, characterize the inferior fast subsystem of flexible arm elastic vibration and characterize the slow subsystem of rigid motion on a large scale.Its advantage be the control system complexity from the system-level subsystem irrespective of size that becomes, make control system structure, algorithm all greatly simplify, be easy to the engineering application.The key issue of the hard and soft mechanical arm system of two-parameter singular perturbation technical finesse hydraulic pressure is how with the system model under different time scales, to decompose depression of order, how will make up control under controller unification to the unification markers under different time scales.

The present invention is based on the hard and soft mechanical arm control method of the hydraulic pressure of two-parameter singular perturbation realize in crucial processing method and process as follows:

The 1 hard and soft mechanical arm system kinetic model of hydraulic pressure

1.1 hard and soft mechanical arm system kinetic model

As shown in Figure 1, flexible link is processed as the Euler-Bernoulli uniform beam, according to the hypothesis modal method as can be known the elastic deformation amount of flexible arm be

$\mathrm{\&omega;}(r,t)=\stackrel{n}{\underset{s=1}{\mathrm{\&Sigma;}}}{\mathrm{\&Phi;}}_s\left(r\right)q_s\left(t\right)---\left(4\right)$

Φ wherein _s(r) and q _s(t) be respectively s ^thMode function and generalized Modal coordinate, second-order modal, i.e. n=2. before adopting in the present invention

Consider the impact of end load and flexible link strain, in conjunction with energy relationship, system total kinetic energy and potential energy and substitution Lagrange equation as can be known, the kinetics equation that obtains hard and soft mechanical arm system is

$M(\mathrm{\&theta;},q)\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}\\ \stackrel{\&CenterDot;\&CenterDot;}{q}\end{array}\right]+K\left[\begin{array}{c}\mathrm{\&theta;}\\ q\end{array}\right]+G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})=\left[\begin{array}{c}\mathrm{\&tau;}\\ 0\end{array}\right]---\left(5\right)$

M (θ, q) ∈ R wherein ^{5 * 5}For inertial matrix symmetrical, positive definite; θ=[θ ₁θ ₂θ ₃] ^TFor joint angle; Q=[q ₁q ₂] ^TFor modal coordinate; $G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})={\left[\begin{array}{ccccc}{g}_1& g_2& g_3& g_4& g_5\end{array}\right]}^T$ For comprising the nonlinear terms of coriolis force, centrifugal force, gravity; The stiffness matrix of flexible arm is K=diag (0,0,0, k ₁, k ₂); The Generalized Moment that acts on joint is τ=[τ ₁τ ₂τ ₃] ^T

1.2 hydraulic servo driving system model

Suppose the time constant of the time constant of hydraulic efficiency servo-valve much smaller than mechanical system, in hydraulic servo driving system, the valve core of servo valve displacement is proportional to control inputs

x _v＝K _ii (6)

K wherein _iIt is the ratio constant coefficient.

, according to the installation site of valve control single pole asymmetrical cylinder shown in Figure 1, can be obtained the relation of joint angle and piston displacement by the cosine law

$y_2=\sqrt{L_{21}^2+{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">L</figure-callout>}}_1^2-2L_1{L}_{21}\cos (\mathrm{\&pi;}-{\mathrm{\&theta;}}_2)}---\left(7\right)$

$y_3=\sqrt{L_{22}^2+L_{31}^2-2L_{22}{L}_{31}\cos (\mathrm{\&pi;}-{\mathrm{\&theta;}}_3)}---\left(8\right)$

Consideration formula (7) and formula (8), can obtain the time differentiate

$\stackrel{\&CenterDot;}{y}=J\stackrel{\&CenterDot;}{\mathrm{\&theta;}}---\left(9\right)$

Y=[y wherein ₂y ₃] ^TFor the piston rod displacement;

For the Jacobin matrix between hard and soft property mechanical arm and fluid power system.

According to the principle of virtual work, τ=J is arranged ^TF _L, in formula: F _LFor Driven by Hydraulic Cylinder power.

Ignore the leakage quantity that leaks, according to formula (6) and Hydrauservo System basic principle, the dynamic model that can obtain hydraulic cylinder is respectively

$\left\{\begin{array}{c}{x}_v=K_{i}i\\ Q_L=K_q{x}_v-K_c{P}_L\\ Q_L=A_1\stackrel{\&CenterDot;}{y}+C_{\mathrm{tm}}{P}_L+\frac{V_t}{4{\mathrm{\&beta;}}_e}{\stackrel{\&CenterDot;}{P}}_L\\ \mathrm{\&tau;}=J^T{F}_L,F_L=A_1{P}_L\end{array}\right.---\left(10\right)$

In formula: $K_q=C_{d}w\sqrt{2(P_s-P_L)}/\sqrt{\mathrm{\&rho;}(1+n^3)}$ For flow gain; $K_c=C_{d}w{x}_v/\sqrt{2\mathrm{\&rho;}(1+n^3)(P_s-P_L)}$ For the flow/pressure coefficient; C _tm=C _ip(1+n ²)/(1+n ³) be equivalent leakage coefficient; C _ipInternal leakage coefficient for hydraulic cylinder; V _t=2LA ₁/ (1+n ³) be the equivalent volume of hydraulic cylinder; N=A ₂/ A ₁For Area Ratio; P _L=P ₁-nP ₂β _eFor liquid effective volume elastic modelling quantity; ρ is fluid density; C _dFor the valve port discharge coefficient; W is the area gradient of servo valve; L is the hydraulic cylinder total kilometres.

The basic model of hydraulic motor is

$\left\{\begin{array}{c}{x}_v=K_{i}i\\ Q_L=K_q{x}_v-K_c{P}_L\\ Q_L=D\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+C_{\mathrm{tm}}{P}_L+\frac{V_t}{4{\mathrm{\&beta;}}_e}{\stackrel{\&CenterDot;}{P}}_L\\ \mathrm{\&tau;}=P_{L}D\end{array}\right.---\left(11\right)$

Wherein, D is the volumetric displacement of hydraulic motor, V _tExpression hydraulic motor chamber cumulative volume, C _tmFor leadage coefficient, Q _LFor the load flow of hydraulic motor,

For the flow gain coefficient, θ is the corner of hydraulic motor.

By formula (9), formula (10) and formula (11) can be tried to achieve the relation between the hard and soft property of 3-DOF hydraulic-driven mechanical arm controlled quentity controlled variable and driving moment

$\stackrel{\&CenterDot;}{\mathrm{\&tau;}}+\mathrm{A\&tau;}+B\stackrel{\&CenterDot;}{\mathrm{\&theta;}}=\mathrm{CI}---\left(12\right)$

I=[i wherein ₁i ₂i ₃] ^TFor the servo valve control electric current; A=diag (a ₁, a ₂, a ₃), a ₁=4 β _e(C _tm1+ K _c1)/V _t1,

B=diag (b ₁, b ₂, b ₃), b ₁=4 β _eD ²/ V _t1,

C=diag (c ₁, c ₂, c ₃), c ₁=4 β _eDK _q1K _i1/ V _t1, c _i=4 β _eA _piDK _qiK _iJ _i/ V _ti, i=2,3.

1.3 the hard and soft mechanical arm system kinetic model of hydraulic pressure

, according to formula (5) and formula (12), can obtain the hard and soft Manipulator Dynamics equation of Three Degree Of Freedom hydraulic pressure and be

$\left\{\begin{array}{c}M(\mathrm{\&theta;},q)\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}\\ \stackrel{\&CenterDot;\&CenterDot;}{q}\end{array}\right]+K\left[\begin{array}{c}\mathrm{\&theta;}\\ q\end{array}\right]+G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})=\left[\begin{array}{c}\mathrm{\&tau;}\\ 0\end{array}\right]\\ \stackrel{\&CenterDot;}{\mathrm{\&tau;}}+\mathrm{A\&tau;}+B\stackrel{\&CenterDot;}{\mathrm{\&theta;}}=\mathrm{CI}\end{array}\right.---\left(13\right)$

2 based on the hard and soft mechanical arm system kinetic model of the hydraulic pressure of two-parameter singular perturbation

The hard and soft mechanical arm system of hydraulic pressure comprises hydraulic servo part, rigid mechanical arm portion and flexible mechanical arm portion, and therefore wherein a part of state is faster than another part state variation.Adopt the singular perturbation technology, first ignore fast variable to reduce the system exponent number, improve degree of approximation by introducing boundary layer correction again, system is carried out twice decomposition continuously, realization is decomposed the two-parameter singular perturbation of the hard and soft mechanical arm system of hydraulic pressure, the method has not only been simplified control structure, is convenient to the controller design, and has significantly reduced amount of calculation.

2.1 singular perturbation decomposes for the first time

The hard and soft mechanical arm system of hydraulic pressure is resolved into the first order slow subsystem that characterizes hard and soft manipulator motion, and the first order fast subsystem that characterizes Hydraulic servo drive.At first formula (13) is rewritten into

$\left\{\begin{array}{c}M(\mathrm{\&theta;},q)\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}\\ \stackrel{\&CenterDot;\&CenterDot;}{q}\end{array}\right]+K\left[\begin{array}{c}\mathrm{\&theta;}\\ q\end{array}\right]+G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})=\left[\begin{array}{c}\mathrm{\&tau;}\\ 0\end{array}\right]\\ \frac{1}{{\mathrm{\&beta;}}_e}\stackrel{\&CenterDot;}{\mathrm{\&tau;}}=-\tilde{A}\mathrm{\&tau;}-\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+\widetilde{\mathrm{CI}}\end{array}\right.---\left(14\right)$

Wherein $\tilde{A}=\frac{1}{{\mathrm{\&beta;}}_e}A,$ $\tilde{B}=\frac{1}{{\mathrm{\&beta;}}_e}B,$ $\tilde{C}=\frac{1}{{\mathrm{\&beta;}}_e}C.$

Get first small parameter

And meet 0＜ε ₁＜＜1, make z=τ, have

$M(\mathrm{\&theta;},q)\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}\\ \stackrel{\&CenterDot;\&CenterDot;}{q}\end{array}\right]+K\left[\begin{array}{c}\mathrm{\&theta;}\\ q\end{array}\right]+G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})=\left[\begin{array}{c}\mathrm{\&tau;}\\ 0\end{array}\right]---\left(15\right)$

${\mathrm{\&epsiv;}}_1\stackrel{\&CenterDot;}{\mathrm{\&tau;}}=-\tilde{A}z-\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+\tilde{C}I---\left(16\right)$

Get ε ₁=0, the slow unsteady flow shape of the first order of the hard and soft mechanical arm system of hydraulic pressure expression formula is

$\left\{\begin{array}{c}M(\mathrm{\&theta;},q)\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}\\ \stackrel{\&CenterDot;\&CenterDot;}{q}\end{array}\right]+K\left[\begin{array}{c}\mathrm{\&theta;}\\ q\end{array}\right]+G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})=\left[\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="z\; s"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">z</figure-callout>}}_s\\ 0\end{array}\right]\\ {\mathrm{<figure-callout\; id="1"\; label="z\; s"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">z</figure-callout>}}_s=-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s1}\end{array}\right.---\left(17\right)$

Wherein subscript s1 represents that system is in the first order and becomes under markers slowly.

Work as ε ₁Enough hour, ο (ε ₁) can ignore, theoretical according to multiple time scale, get variable

τ=τ _f1+τ _s1 (18)

I=I _f1+I _s1 (19)

Wherein subscript f1 represents that system is in the first order and becomes under markers soon.

, for formula (13), introduce the first order and become soon markers on boundary layer Near boundary layer region ε ₁→ 0, first order slow variable τ _s1Be regarded as constant, the first order fast subsystem kinetics equation that can be obtained the hard and soft mechanical arm system of hydraulic pressure by formula (16)-Shi (19) arrangement merging is

$\frac{d{\mathrm{\&tau;}}_{f1}}{d{\mathrm{\&sigma;}}_1}=-\tilde{A}{\mathrm{\&tau;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}+\tilde{C}{I}_{f1}---\left(20\right)$

I wherein _f1For the first order becomes markers σ soon ₁Under the control electric current.Like this, the hard and soft mechanical arm system of hydraulic pressure formula (14) and formula (20), i.e. first order slow subsystem and first order fast subsystem have been resolved into.

2.2 singular perturbation decomposes for the second time

First order slow subsystem is further resolved into and characterizes second level slow subsystem and the second level fast subsystem of sign elastic vibration, i.e. time fast subsystem of rigid motion on a large scale.

Order $D=M^{-1}={\left[\begin{array}{cc}{M}_1& M_2\\ M_3& M_4\end{array}\right]}^{-1}=\left[\begin{array}{cc}{D}_1& D_2\\ D_3& D_4\end{array}\right],$ $G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})=\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right]$

M wherein ₁∈ R ^{3 * 3}, D ₁∈ R ^{3 * 3}, G ₁∈ R ^{3 * 1}

By formula (17) as can be known

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}=-D_2(\mathrm{\&theta;},q)\mathrm{Kq}-D_1(\mathrm{\&theta;},q)G_1(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})$

$-D_2(\mathrm{\&theta;},q)G_2(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})+D_1(\mathrm{\&theta;},q)(-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s1})---\left(21\right)$

$\stackrel{\&CenterDot;\&CenterDot;}{q}=-D_4(\mathrm{\&theta;},q)K_{x}q-D_3(\mathrm{\&theta;},q)G_1(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})$ $-D_4(\mathrm{\&theta;},q)G_2(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})+D_3(\mathrm{\&theta;},q)(-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s1})---\left(22\right)$

K wherein _x=diag (k ₁, k ₂).

Get second small parameter K=min (k wherein ₁, k ₂), and two small parameters meet 0＜ε ₁＜＜ε ₂＜＜1.

Order $\tilde{K}={\mathrm{\&epsiv;}}_2{K}_x,$ Q=ε ₂Y,

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}=-D_2(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)\tilde{K}y-D_1(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)G_1(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},{\mathrm{\&epsiv;}}_{2}y,{\mathrm{\&epsiv;}}_2\stackrel{\&CenterDot;}{y})$

$-D_2(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)G_2(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},{\mathrm{\&epsiv;}}_{2}y,{\mathrm{\&epsiv;}}_2\stackrel{\&CenterDot;}{y})---\left(23\right)$

$+D_1(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)(-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s1})$

${\mathrm{\&epsiv;}}_2\stackrel{\&CenterDot;\&CenterDot;}{y}=-D_4(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)\tilde{K}y-D_3(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)G_1(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},{\mathrm{\&epsiv;}}_{2}y,{\mathrm{\&epsiv;}}_2\stackrel{\&CenterDot;}{y})$ $-D_4(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)G_2(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},{\mathrm{\&epsiv;}}_{2}y,{\mathrm{\&epsiv;}}_2\stackrel{\&CenterDot;}{y})---\left(24\right)$

$+D_3(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)(-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s1})$

Get ε ₂=0,

$\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}=-D_{2,s2}(\mathrm{\&theta;},0)\tilde{K}y-{\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">D</figure-callout>}}_{1,s2}(\mathrm{\&theta;},0){\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">G</figure-callout>}}_{1,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})$

$-D_{2,s2}(\mathrm{\&theta;},0)G_{2,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})---\left(25\right)$

$+{\mathrm{<figure-callout\; id="1"\; label="D"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">D</figure-callout>}}_{1,s2}(\mathrm{\&theta;},0)(-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s1})$

$0=-D_{4,s2}(\mathrm{\&theta;},0)\hat{K}{y}_{s2}-D_{3,s2}(\mathrm{\&theta;}.0){\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">G</figure-callout>}}_{1,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})$

$-D_{4,s2}(\mathrm{\&theta;},0)G_{2,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})---\left(26\right)$

$+D_{3,s2}(\mathrm{\&theta;},0)(-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s1})$

Solved by formula (26)

$y_{s2}=-{\tilde{K}}^{-1}{D}_{4,s2}^{-1}(\mathrm{\&theta;},0)D_{3,s2}(\mathrm{\&theta;},0){\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">G</figure-callout>}}_{1,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})$

$-{\tilde{K}}^{-1}{D}_{4,s2}^{-1}(\mathrm{\&theta;},0)D_{4,s2}(\mathrm{\&theta;},0)G_{2,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})---\left(27\right)$

$+{\tilde{K}}^{-1}{D}_{4,s2}^{-1}(\mathrm{\&theta;},0)D_{3,s2}(\mathrm{\&theta;},0)(-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s2})$

Wherein subscript s2 represents that system is in the second level and becomes slowly under markers, I _s2For the second level becomes control electric current under markers t slowly.

The second level slow unsteady flow shape expression formula that formula (27) substitution formula (25) can be obtained the hard and soft mechanical arm system of hydraulic pressure is

${\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">M</figure-callout>}}_{1,s2}(\mathrm{\&theta;},0)\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}+{\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">G</figure-callout>}}_{1,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})=-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s2}---\left(28\right)$

Work as ε ₂Enough hour, ο (ε ₂) can ignore,, according to the Multiple-Scale time theory, get variable

y＝y _f2+y _s2 (29)

I _s1＝I _f2+I _s2 (30)

Wherein subscript f2 represents to be in the second level and becomes under markers soon.

, for formula (17), introduce the second level and become soon markers on boundary layer

Near boundary layer region ε ₂→ 0, second level slow variable y _s2Be regarded as constant, the kinetics equation that can be obtained the hard and soft mechanical arm system of hydraulic pressure second level fast subsystem by formula (24), formula (27), formula (29) and formula (30) arrangement merging is

$\frac{d^2{y}_{f2}}{d{\mathrm{\&sigma;}}_2^2}=-D_{4,s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)\tilde{K}{y}_{f2}+D_{3,s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y){\tilde{A}}^{-1}\tilde{C}{I}_{f2}---\left(31\right)$

I wherein _f2For the second level becomes markers σ soon ₂Under the control electric current.Like this, the hard and soft mechanical arm system of hydraulic pressure formula (28) and formula (31), i.e. second level slow subsystem and second level fast subsystem have further been resolved into.

To sum up, three markers meet t＜＜σ ₂＜＜σ ₁, decompose by twice one-parameter singular perturbation, the hard and soft mechanical arm system of hydraulic pressure has been resolved into formula (28), formula (31) and formula (20), i.e. change, inferior fast change and become soon three sub-systems slowly.

3 triple time scale controller designs

3.1 become slowly controller

For slow subsystem formula (28), the desired trajectory of getting joint angle is θ _d, site error and velocity error are

e=θ _d-θ (32)

$\stackrel{\&CenterDot;}{e}={\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_d-\stackrel{\&CenterDot;}{\mathrm{\&theta;}}---\left(33\right)$

Get the sliding mode face

$s=\stackrel{\&CenterDot;}{e}+\mathrm{\&lambda;e},\mathrm{\&lambda;}>0---\left(34\right)$

Wherein λ is 3 rank diagonal matrix undetermined, and diagonal element just is.

$\mathrm{\&Sigma;}=\stackrel{\&CenterDot;}{s}+\mathrm{\&eta;s},\mathrm{\&eta;}>0---\left(35\right)$

Wherein η is 3 rank diagonal matrix undetermined, and diagonal element just is.

$\stackrel{\&CenterDot;}{\mathrm{\&Sigma;}}=-{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\mathrm{sgn}\left(\mathrm{\&Sigma;}\right)-k_2\mathrm{\&Sigma;}---\left(36\right)$

K wherein ₁And k ₂Be 3 rank diagonal matrix undetermined, diagonal element just is.

Design Second Order Sliding Mode Control rule is

$I_{s2}={\tilde{C}}^{-1}\tilde{A}[{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">M</figure-callout>}}_{1,s}({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_d+\mathrm{\&alpha;e}+\mathrm{\&beta;}\stackrel{\&CenterDot;}{e}+u)+{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\mathrm{<figure-callout\; id="1"\; label="G"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">G</figure-callout>}}_{1,s}]---\left(37\right)$

3.2 inferior fast change controller

For inferior fast subsystem, formula (31) is transformed into state space form

${\stackrel{\&CenterDot;}{X}}_k=A_k{X}_k+B_k{I}_{f2}---\left(38\right)$

Wherein $X_k=\left[\begin{array}{c}{y}_{f2}\\ \frac{d{y}_{f2}}{d{\mathrm{\&sigma;}}_2}\end{array}\right],$ $A_k=\left[\begin{array}{cc}0& I\\ D_{4.s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)\tilde{K}& 0\end{array}\right],$ $B_k=\left[\begin{array}{c}0\\ D_{3,s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)\end{array}\right].$

Suppose not have uncertainty in time fast subsystem, due to (A _k, B _k) fully controlled, can adopt method for optimally controlling that the status adjustment of system is extremely zero, suppress the system resilience vibration.

Getting the quadratic performance index function is

$J=\frac{1}{2}{\&Integral;}_0^{\&infin;}[X_k^{T}Q{X}_k+I_{f2}^{T}R{I}_{f2}]\mathrm{dt}---\left(39\right)$

Choosing the Ricatti equation is

$A_k^{T}P+P{A}_k-P{B}_k{R}^{-1}{B}_k^{T}P+Q=0---\left(40\right)$

Optimal control law is

$I_{f2}=-K_f{X}_k=-R^{-1}{B}_k^{T}P{X}_k---\left(41\right)$

3.3 become soon controller

Suppose in fast subsystem formula (20) to have Parameter Perturbation and modeling is not dynamic, form is

$\left\{\begin{array}{c}\tilde{A}={\tilde{A}}_0+\mathrm{\&Delta;}\tilde{A}\\ \tilde{C}={\tilde{C}}_0+\mathrm{\&Delta;}\tilde{C}\\ E={\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="HDA00003635444400011.png,HDA00003635444400012.png"\; state="\{\{state\}\}">E</figure-callout>}}_0+\mathrm{\&Delta;E}\end{array}\right.---\left(42\right)$

E ∈ R wherein ^{3 * 3}For unit matrix;

E ₀For nominal value;

Δ E is the bounded perturbation amount.

Consider external interference and formula (42), formula (20) can be rewritten as

$E_0{\stackrel{\&CenterDot;}{z}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}+{\tilde{A}}_0{\mathrm{\&tau;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}+F+d={\tilde{C}}_0{I}_{f1}---\left(43\right)$

Wherein d (t) is external interference, and overall indeterminate is $F(t,{\mathrm{\&tau;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1},{\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{f1})=\mathrm{\&Delta;E}{\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}+\mathrm{\&Delta;}\tilde{A}{\mathrm{\&tau;}}_{f1}-\mathrm{\&Delta;}{\tilde{C}}_0{\mathrm{<figure-callout\; id="1"\; label="I\; f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">I</figure-callout>}}_f.$

Suppose that 1 random disturbances d (t) is that energy has sector signal, and meet

||d(t)||≤D ₀ (44)

D wherein ₀For known normal number.

Suppose 2 overall indeterminates

And meet

$\left|\right|F(t,{\mathrm{\&tau;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1},{\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{f1})\left|\right|\&le;F_0---\left(45\right)$

F wherein ₀, for its upper bound, be a unknown normal number vector.

If

For the estimated value of F, evaluated error

For

$\tilde{F}=F-\hat{F}---\left(46\right)$

Get τ _fd1Desired value be τ _fd1=[0 0 0] ^T,

e _f=τ _f1d-τ _f1 (47)

${\stackrel{\&CenterDot;}{e}}_f={\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{f1d}-{\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{f1}---\left(48\right)$

Getting sliding-mode surface is

s _f=e _f (49)

Design adaptive sliding mode control law is

${\mathrm{<figure-callout\; id="1"\; label="I\; f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">I</figure-callout>}}_f={\tilde{C}}^{-1}[{\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{f1d}+{\tilde{A}}_0{\mathrm{\&tau;}}_{\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003635444400022.png"\; state="\{\{state\}\}">f</figure-callout>}1}+\hat{F}+D_{0}E+\mathrm{\&xi;sgn}\left(s_f\right)]---\left(50\right)$

E=[1 wherein, 1,1] ^T, ξ is 3 rank diagonal matrix undetermined, diagonal element just is.

Take from and adapt to more new law and be

$\stackrel{\&CenterDot;}{\hat{F}}={\mathrm{\&Lambda;}}^T{s}_f---\left(51\right)$

Wherein, Λ is known normal value vector, and its element is positive number.

3 based on the hard and soft mechanical arm control method of the hydraulic pressure of two-parameter singular perturbation Software for Design

The mechanical arm parameter of the hard and soft mechanical arm system of control object hydraulic pressure of the present invention is as shown in table 1.

Table 1 flexible hydraulic mechanical arm structural parameters

Adopt the Matlab language compilation based on the hard and soft mechanical arm control method of the hydraulic pressure of two-parameter singular perturbation software, generate the .m file.programming thought and implementation procedure are as shown in Figure 5, decompose according to two-parameter singular perturbation after initializing, then determine whether the fast subsystem model, if, adopt the fast subsystem control module to control, if NO, continue to determine whether time fast subsystem model, if, adopt time fast subsystem control module to control, if NO, it is the slow subsystem model, adopt the slow subsystem control module to control, and then the output of three sub-systems control modules is inputed to and controls driving, carry out afterwards the data storage, judge whether again to arrive running time, if NO, proceed cycle criterion, if Output rusults finishes.Wherein the slow subsystem control module adopts controller suc as formula (37), inferior fast subsystem control module adopts controller suc as formula (41), the fast subsystem control module adopts controller suc as formula (50), and the hard and soft mechanical arm control system parameter of hydraulic pressure is as shown in table 2.

Table 2 is controlled parameter based on the hard and soft mechanical arm system of the hydraulic pressure of two-parameter singular perturbation

Parameter name	Value	Parameter name	Value
				ε 1	5.0×10 -10	α	diag(105,100,110)
ε 2	1.2×10 -6	β	diag(95,110,105)
				λ	diag(7,6,7)	Q	diag(50,46,55,55)
η	diag(5,6,6)	R	diag(11,10,8)
				k 1	diag(15,12,12)	Λ	diag(50,50,30)
k 2	diag(100,90,110)	ξ	diag(25,15,30)

By clicking " Run " button, system can be moved, at first the hard and soft mechanical arm system of hydraulic pressure being adopted two-parameter singular perturbation technology depression of order is three markers subsystems, then control respectively through three markers subsystem controllers, make the hard and soft mechanical arm system of hydraulic pressure joint follow the tracks of desired trajectory, error precision is in 0.001rad; Elastic vibration is inhibited, and vibration suppression is in 0.001m.Result can adopt word, excel or diagrammatic form to preserve.

Claims (1)

1., based on the hard and soft mechanical arm control method of the hydraulic pressure of two-parameter singular perturbation, it is characterized in that, the method comprises the steps:

The first step, model decompose:

The hard and soft Manipulator Dynamics equation of Three Degree Of Freedom hydraulic pressure is suc as formula shown in (13):

$\left\{\begin{array}{c}M(\mathrm{\&theta;},q)\left[\begin{array}{c}\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}\\ \stackrel{\&CenterDot;\&CenterDot;}{q}\end{array}\right]+K\left[\begin{array}{c}\mathrm{\&theta;}\\ q\end{array}\right]+G(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},q,\stackrel{\&CenterDot;}{q})=\left[\begin{array}{c}\mathrm{\&tau;}\\ 0\end{array}\right]\\ \stackrel{\&CenterDot;}{\mathrm{\&tau;}}+\mathrm{A\&tau;}+B\stackrel{\&CenterDot;}{\mathrm{\&theta;}}=\mathrm{CI}\end{array}\right.---\left(13\right)$

Wherein, M (θ, q) ∈ R ^{5 * 5}For inertial matrix symmetrical, positive definite; θ=[θ ₁θ ₂θ ₃] ^TFor joint angle; Q=[q ₁q ₂] ^TFor modal coordinate;

For comprising the nonlinear terms of coriolis force, centrifugal force, gravity; The stiffness matrix of flexible arm is K=diag (0,0,0, k ₁, k ₂); The Generalized Moment that acts on joint is τ=[τ ₁τ ₂τ ₃] ^TI=[i ₁i ₂i ₃] ^TFor the servo valve control electric current; A=diag (a ₁, a ₂, a ₃), a ₁=4 β _e(C _tm1+ K _c1) V _t1,

B=diag (b ₁, b ₂, b ₃), b ₁=4 β _eD ²V _t1,

C=diag (c ₁, c ₂, c ₃), c ₁=4 β _eDK _q1K _i1V _t1, c _i=4 β _eA _piDK _qiK _iJ _iV _ti, i=2,3;

Adopt two-parameter singular perturbation technology that formula (13) is decomposed, get first small parameter

And meet 0＜ε ₁＜＜1, and the introducing first order becomes markers soon on boundary layer

Obtain the first order fast subsystem kinetics equation of the hard and soft mechanical arm of hydraulic pressure suc as formula (20)

$\frac{{\mathrm{d\&tau;}}_{f1}}{{\mathrm{d\&sigma;}}_1}=-\tilde{A}{\mathrm{\&tau;}}_{f1}+\tilde{C}{I}_{f1}---\left(20\right)$

Wherein, I _f1For the first order becomes markers σ soon ₁Under the control electric current, subscript f1 represents that system is in the first order and becomes soon under markers;

Below get second small parameter

K=min (k wherein ₁, k ₂), and two small parameters meet 0＜ε ₁＜＜ε ₂＜＜1, the second level slow unsteady flow shape expression formula that obtains system is

$M_{1,s2}(\mathrm{\&theta;},0)\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}+G_{1,s2}(\mathrm{\&theta;},\stackrel{\&CenterDot;}{\mathrm{\&theta;}},\mathrm{0,0})=-{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+{\tilde{A}}^{-1}\tilde{C}{I}_{s2}---\left(28\right)$

Wherein, I _s2For the second level becomes control electric current under markers t slowly, subscript s2 represents that system is in the second level and becomes slowly under markers;

Introduce the second level and become soon markers on boundary layer The kinetics equation that obtains the hard and soft mechanical arm system of hydraulic pressure second level fast subsystem is

$\frac{d^2{y}_{f2}}{{\mathrm{d\&sigma;}}_2^2}=-D_{4,s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y)\tilde{K}{y}_{f2}+D_{3,s2}(\mathrm{\&theta;},{\mathrm{\&epsiv;}}_{2}y){\tilde{A}}^{-1}\tilde{C}{I}_{f2}---\left(31\right)$

Wherein, I _f2For the second level becomes markers σ soon ₂Under the control electric current, subscript f2 represents that system is in the second level and becomes soon under markers;

Second step, according to second level slow subsystem, second level fast subsystem and first order fast subsystem kinetics equation that the first step obtains, design corresponding subsystem control module:

, for second level slow subsystem, get single order sliding mode face

And second order sliding mode face

Design Second Order Sliding Mode Control rule is suc as formula (37)

$I_{s2}={\tilde{C}}^{-1}\tilde{A}[M_{1,s}({\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_d+\mathrm{\&alpha;e}+\mathrm{\&beta;}\stackrel{\&CenterDot;}{e}+u)+{\tilde{A}}^{-1}\tilde{B}\stackrel{\&CenterDot;}{\mathrm{\&theta;}}+G_{1,s}]---\left(37\right)$

, for second level fast subsystem, get the quadratic performance index function

And Ricatti equation $A_k^{T}P+{\mathrm{PA}}_k-{\mathrm{PB}}_k{R}^{-1}{B}_k^{T}P+Q=0,$ The devise optimum control law is suc as formula (41)

$I_{f2}=-K_f{X}_k={-R}^{-1}{B}_k^T{\mathrm{PX}}_k---\left(41\right)$

, for first order fast subsystem, get sliding-mode surface s _f=e _fAnd adaptive updates rule

Design adaptive sliding mode control law is suc as formula (50)

$I_{f1}={\tilde{C}}^{-1}[{\stackrel{\&CenterDot;}{\mathrm{\&tau;}}}_{f1d}+{\tilde{A}}_0{\mathrm{\&tau;}}_{f1}+\hat{F}+D_{0}E+\mathrm{\&xi;sgn}\left(s_f\right)]---\left(50\right)$

In the 3rd step, theoretical according to multiple time scale, each subsystem controller combination that second step is obtained obtains the hard and soft mechanical arm combined controller of hydraulic pressure suc as formula (52)

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

And then complete control method based on the hard and soft mechanical arm of hydraulic pressure of two-parameter singular perturbation.

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