CN102566417B - Method for controlling dynamic surface of flexible joint mechanical arm - Google Patents

Abstract

The invention discloses a method for controlling a dynamic surface of a flexible joint mechanical arm. The method comprises the following steps of: 1, analyzing a flexible joint mechanical arm system model, and constructing; 2, controlling and designing the dynamic surface of the flexible joint mechanical arm; 3, inspecting tracking performance, and adjusting parameters; and 4, finishing the design. By the method, virtual control is gradually designed for a flexible joint mechanical arm control system model, a first-order low-pass filter is arranged in each step, and the dynamic surface of the flexible joint mechanical arm can be controlled, so that the phenomenon of differential explosion controlled by reverse thrust is eliminated, the semi-global stability of a closed-loop control system can be ensured, and a preset track can be quickly and accurately tracked by the angle of a connecting rod of the flexible joint mechanical arm at the same time. The method has a wide application prospect in the technical field of automatic control.

Description

A kind of dynamic face control method of flexible joint mechanical arm

(1) technical field

The present invention relates to a kind of dynamic face control method of flexible joint mechanical arm, it is at power plant's mechanical equipment system, and the dynamic face control method of a kind of flexible joint mechanical arm that provides is used for control flexible joint mechanical arm connecting rod angle, belongs to the automatic control technology field.

(2) background technology

There is the high-risk operation of some high temperature in firepower electrical plant, is not suitable for manually-operated.Therefore, utilize plant equipment to replace manpower just very important.The flexible joint mechanical arm is the very important plant equipment of industry spot, and people's both hands have greatly been liberated in its appearance, makes that daily production efficiency is higher, and is also safer.The flexible joint mechanical arm belongs to the system of non-linear strong coupling, has certain difficulty for its control.Owing to require the quick accurate tracking desired trajectory of connecting rod angle energy of flexible joint mechanical arm, so the design of control method has been proposed high requirement.

In recent years, many advanced persons' control method is used in the design of flexible joint mechanical arm control, comprising feedback linearization method, sliding-mode control etc.But these methods all can't solve the non-matching uncertain problem that exists in the flexible joint mechanical arm.Though counter pushing away controlled the non-matching uncertain problem of design energy resolution system, but has " differential blast " phenomenon.Dynamically the face control method is a kind of control method of novelty, and its design procedure is clear, and design process is simple, and the nonlinear system of descending triangular form is had excellent control effect.This control method goes on foot at each and introduces a low-pass first order filter in the design, makes each go on foot the basic decoupling zero of design of control law design and previous stage, thereby the complexity of control law is descended greatly, has fundamentally eliminated " differential blast " phenomenon.

Under this technical background, the present invention provides a kind of dynamic face control method of flexible joint mechanical arm, is used for control flexible joint mechanical arm connecting rod angle.Adopt this control not only to guarantee the stability of closed-loop system, also realized the fast and accurately tracking of flexible joint mechanical arm connecting rod to desired trajectory.

(3) summary of the invention

1, goal of the invention

The objective of the invention is: the deficiency that overcomes existing control technology, and provide a kind of dynamic face control method of flexible joint mechanical arm, it is guaranteeing on the stable basis of closed loop half global system, realizes that closed-loop system connecting rod angle is to the tracking fast and accurately of desired trajectory.

The present invention is a kind of dynamic face control method of flexible joint mechanical arm, its design philosophy is: at flexible joint mechanical arm control system model, progressively design virtual controlling, and at per step introducing low-pass first order filter, finally derive the dynamic face control of flexible mechanical arm, overcome counter " differential blast " phenomenon that pushes away control, can guarantee half global stability of closed-loop control system, realized the fast and accurately tracking of flexible joint mechanical arm connecting rod angle to desired trajectory simultaneously.

2, technical scheme

Below in conjunction with the step in the FB(flow block) 4, specifically introduce the technical scheme of this method for designing.

Flexible joint mechanical arm system synoptic diagram such as Fig. 1.

The dynamic face control method of a kind of flexible joint mechanical arm of the present invention, these method concrete steps are as follows: the mechanical arm system model analysis of first step flexible joint and structure

Closed-loop control system adopts degenerative control structure, and output quantity is flexible joint mechanical arm connecting rod angle.Designed closed-loop control system mainly comprises controller link and these two parts of system model, and its topology layout situation is seen shown in Figure 2.

Flexible joint mechanical arm control system model description is as follows:

$\left\{\begin{array}{c}I\stackrel{\·\·}{q}+K(q-q_m)+\mathrm{Mgl}\sin q=0\\ J{\stackrel{\·\·}{q}}_m-K(q-q_m)=u\end{array}\right.---\left(1\right)$

Wherein: q represents flexible joint mechanical arm connecting rod angle;

q _mExpression rotor angle;

I represents flexible joint mechanical arm moment of inertia;

J represents the rotor moment of inertia;

K represents the joint stiffness coefficient;

M represents flexible joint mechanical arm connecting rod quality;

G represents acceleration of gravity;

L represents that the joint arrives the distance of bar barycenter;

U represents motor torque.

For the ease of design, define one of four states variable x respectively ₁, x ₂, x ₃, x ₄As follows:

x ₁＝q

$x_2=\stackrel{\·}{q}$

x ₃＝q _m

$x_4={\stackrel{\·}{q}}_m$

The derivative of expression flexible joint mechanical arm connecting rod angle, just flexible joint mechanical arm connecting rod angle speed;

The derivative of expression rotor angle, just rotor angular velocity.

At this moment (1) just can be write as

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=a_1{x}_3+f_1\left(x_1\right)\\ {\stackrel{\·}{x}}_3=x_4\\ {\stackrel{\·}{x}}_4=a_{2}u+f_2(x_1,x_3)\end{array}\right.---\left(2\right)$

Wherein:

$a_1=\frac{K}{I},$

$a_2=\frac{1}{J},$

$f_1\left(x_1\right)=-\frac{\mathrm{Mgl}}{I}\sin x_1-\frac{K}{I}{x}_1,$

$f_2(x_1,x_3)=\frac{K}{J}(x_1-x_3).$

So the purpose of handling is that system is turned to clear following triangular form system, is convenient to the control design.The dynamic face control of second step flexible joint mechanical arm design

The dynamic face of flexible joint mechanical arm is controlled inner structure as shown in Figure 3, and design process is the process of progressively going forward one by one, and one is divided into four small steps.

First small step: suppose that desired trajectory is x _1dDefine first error surface S ₁For

S ₁＝x ₁-x _1d (3)

(3) differentiate is obtained

${\stackrel{\·}{S}}_1=x_2-{\stackrel{\·}{x}}_{1d}---\left(4\right)$

Design first virtual controlling amount

For

${\stackrel{\&OverBar;}{x}}_2=-c_1{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HDA0000136315760000031.png,HDA0000136315760000032.png"\; state="\{\{state\}\}">S</figure-callout>}}_1+{\stackrel{\&CenterDot;}{x}}_{1d}---\left(5\right)$

Wherein: c ₁Parameter is regulated in expression.

Will

Be input to following low-pass first order filter

${\mathrm{\&tau;}}_2{\stackrel{\&CenterDot;}{x}}_{2d}+x_{2d}={\stackrel{\&OverBar;}{x}}_2---\left(6\right)$

Wherein: τ ₂The expression time parameter;

x _2dThe output of expression low-pass first order filter.

Second small step: define second error surface S ₂For

S ₂＝x ₂-x _2d (7)

(7) differentiate is obtained

${\stackrel{\&CenterDot;}{S}}_2=a_1{x}_3+f_1\left(x_1\right)-{\stackrel{\&CenterDot;}{x}}_{2d}---\left(8\right)$

Design second virtual controlling amount For

${\stackrel{\&OverBar;}{x}}_3=\frac{1}{a_1}[-f_1\left(x_1\right)+{\stackrel{\&CenterDot;}{x}}_{2d}-c_2{S}_2]---\left(9\right)$

Wherein: c ₂Parameter is regulated in expression.

Can be obtained by (6)

${\stackrel{\&CenterDot;}{x}}_{2d}=\frac{{\stackrel{\&OverBar;}{x}}_2-x_{2d}}{{\mathrm{\&tau;}}_2}---\left(10\right)$

Will Be input to following low-pass first order filter

${\mathrm{\&tau;}}_3{\stackrel{\&CenterDot;}{x}}_{3d}+x_{3d}={\stackrel{\&OverBar;}{x}}_3---\left(11\right)$

Wherein: τ ₃The expression time parameter;

x _3dThe output of expression low-pass first order filter.

The 3rd small step: define the 3rd error surface S ₃For

S ₃＝x ₃-x _3d (12)

(12) differentiate is obtained

${\stackrel{\&CenterDot;}{S}}_3=x_4-{\stackrel{\&CenterDot;}{x}}_{3d}---\left(13\right)$

Design the 3rd virtual controlling amount

For

${\stackrel{\&OverBar;}{x}}_4=-c_3{\mathrm{<figure-callout\; id="3"\; label="S"\; filenames="HDA0000136315760000031.png,HDA0000136315760000032.png"\; state="\{\{state\}\}">S</figure-callout>}}_3+{\stackrel{\&CenterDot;}{x}}_{3d}---\left(14\right)$

Wherein: c ₃Parameter is regulated in expression.

Can be obtained by (11)

${\stackrel{\&CenterDot;}{x}}_{3d}=\frac{{\stackrel{\&OverBar;}{x}}_3-x_{3d}}{{\mathrm{\&tau;}}_3}---\left(15\right)$

Will

Be input to following low-pass first order filter

${\mathrm{\&tau;}}_4{\stackrel{\&CenterDot;}{x}}_{4d}+x_{4d}={\stackrel{\&OverBar;}{x}}_4---\left(16\right)$

Wherein: τ ₄The expression time parameter;

x _4dThe output of expression low-pass first order filter.

The 4th small step: define the 4th error surface S ₄For

S ₄＝x ₄-x _4d (17)

(17) differentiate is obtained

${\stackrel{\&CenterDot;}{S}}_4=a_{2}u+f_2(x_1,x_3)-{\stackrel{\&CenterDot;}{x}}_{4d}---\left(18\right)$

The dynamic face control of design flexible joint mechanical arm u is

$u=\frac{1}{a_2}[-f_2(x_1,x_3)+{\stackrel{\&CenterDot;}{x}}_{4d}-c_4{S}_4]---\left(19\right)$

Wherein: c ₄Parameter is regulated in expression.

Can be obtained by (16)

${\stackrel{\&CenterDot;}{x}}_{4d}=\frac{{\stackrel{\&OverBar;}{x}}_4-x_{4d}}{{\mathrm{\&tau;}}_4}---\left(20\right)$

So far, obtained the dynamic face control of flexible joint mechanical arm.

The tracking performance check of the 3rd step is regulated with parameter

Whether this step meets design requirement the checking system tracking performance, and suitably regulates the control parameter, sees shown in Figure 4.Carry out by means of numerical evaluation and Control System Imitation instrument Matlab 7.0 commonly used.

Parameter c ₁, c ₂, c ₃, c ₄, τ ₂, τ ₃, τ ₄For regulating parameter.If tracking error is excessive, do not meet design requirement, then can increase c ₁, c ₂, c ₃, c ₄Value or reduce τ ₂, τ ₃, τ ₄Value.On the one hand, increase c ₁, c ₂, c ₃, c ₄Be equivalent to increase control intensity; On the other hand, reduce τ ₂, τ ₃, τ ₄Be equivalent to improve the response speed of system.Therefore these two kinds of ways all help to improve the system keeps track performance.

The design of the 4th step finishes

The whole design process emphasis has been considered the demand for control of three aspects, is respectively the simplicity of design, the stability of closed-loop system, the quick accuracy of tracking.Around these three aspects, at first in the above-mentioned first step, determined the concrete formation of closed-loop control system; Emphasis has provided the dynamic face control method for designing of flexible joint mechanical arm in second step, mainly comprises four little steps; Mainly introduced in order to improve the parameter adjusting method of tracking performance in the 3rd step; After above steps, design finishes.

3, advantage and effect

The dynamic face control method of a kind of flexible joint mechanical arm of the present invention is used for control flexible joint mechanical arm fork angle.Concrete advantage comprises two aspects: one, to compare with the disposal route of present existence, and this method is very easy in the CONTROLLER DESIGN process, anti-control " differential blast " phenomenon that pushes away can not occur; Its two, by the adjusted design parameter, can control flexible joint mechanical arm fork angle simply, neatly and follow the tracks of desired trajectory quickly and accurately.

(4) description of drawings

Fig. 1: flexible joint mechanical arm system synoptic diagram of the present invention

Fig. 2: closed-loop control system structure of the present invention and assembly annexation synoptic diagram

Fig. 3: control system inner structure synoptic diagram of the present invention

Fig. 4: the dynamic face control of flexible joint mechanical arm of the present invention design cycle synoptic diagram

Fig. 5 .1: c in the embodiment of the present invention () ₁=2, c ₂=40, c ₃=20, c ₄=2, τ ₂=0.02, τ ₃=0.02, τ ₄=0.02 o'clock tracking effect figure

Fig. 5 .2: c in the embodiment of the present invention () ₁=2, c ₂=40, c ₃=20, c ₄=2, τ ₂=0.02, τ ₃=0.02, τ ₄=0.02 o'clock tracking error figure

Fig. 6 .1: c in the embodiment of the present invention () ₁=5, c ₂=45, c ₃=25, c ₄=5, τ ₂=0.01, τ ₃=0.01, τ ₄=0.01 o'clock tracking effect figure

Fig. 6 .2: c in the embodiment of the present invention () ₁=5, c ₂=45, c ₃=25, c ₄=5, τ ₂=0.01, τ ₃=0.01, τ ₄=0.01 o'clock tracking error figure

Label among the figure, symbol and lines etc. are described as follows:

Horizontal ordinate among Fig. 5 .1-5.2, Fig. 6 .1-6.2 is represented simulation time, and unit is second; Ordinate is represented flexible joint mechanical arm fork angle tracking effect among Fig. 5 .1, Fig. 6 .1, and unit is radian; Ordinate is represented flexible joint mechanical arm fork angle error in tracking among Fig. 5 .2, Fig. 6 .2, and unit is radian; Dotted line among Fig. 5 .1, Fig. 6 .1 represents the desired trajectory signal wire, and solid line represents compliance joint mechanical arm fork angle signal line.

(5) embodiment

Design object of the present invention comprises two aspects: one, realize the simplification of flexible joint mechanical arm control design; Its two, realize the quick accurate tracking desired trajectory of flexible joint mechanical arm fork angle of closed-loop system, specific targets are: flexible joint mechanical arm fork angle in 1 second tracking error less than 0.05 radian.Fig. 1 is flexible joint mechanical arm system synoptic diagram of the present invention.

In concrete the enforcement, the emulation of the dynamic face control method of flexible joint mechanical arm and closed-loop control system and check all realize by means of the Simulink tool box among the Matlab7.0.Here have certain representational embodiment by introducing one, further specify relevant design in the technical solution of the present invention and the control method of design parameter.

Embodiment (one) is by increasing c ₁, c ₂, c ₃, c ₄Value and reduce τ ₂, τ ₃, τ ₄Accuracy and the rapidity of value to realize that flexible joint mechanical arm fork angle is followed the tracks of.

Embodiment (one)

The first step: the model analysis of flexible joint mechanical arm control system and structure

Closed-loop control system adopts degenerative control structure, output quantity flexible joint mechanical arm fork angle.Designed closed-loop control system mainly is controller link and these two parts of system model, and its topology layout situation is seen shown in Figure 2.

Flexible joint mechanical arm control system model $\left\{\begin{array}{c}I\stackrel{\&CenterDot;\&CenterDot;}{q}+K(q-q_m)+\mathrm{Mgl}\sin q=0\\ J{\stackrel{\&CenterDot;\&CenterDot;}{q}}_m-K(q-q_m)=u\end{array}\right.$ In, according to actual engineering system data, parameter is chosen as follows:

I＝1kg·m ²，

J＝1kg·m ²，

Mgl＝5N·m，

K＝40N·m/rad，

The state variable initial value is set to x ₁=0.1, x ₂=0, x ₃=0, x ₄=0.

Second step: the dynamic face control of flexible joint mechanical arm design

As shown in Figure 2, adopt the unit negative feedback control structure of output quantity (angle signal).The dynamic face controller of flexible joint mechanical arm inner structure as shown in Figure 3.Utilize the 26S Proteasome Structure and Function of the dynamic face controller of .m Programming with Pascal Language realization flexible joint mechanical arm under Matlab 7.0 environment.The input signal that is controller is error signal (deducting output signal by reference signal tries to achieve), makes up first virtual controlling amount with this, is entered into first low-pass first order filter and obtains output; Make up second virtual controlling amount by first low-pass first order filter output, be entered into second low-pass first order filter and obtain output; Make up the 3rd virtual controlling amount by second low-pass first order filter output, be entered into the 3rd low-pass first order filter and obtain output; By the 3rd the dynamic face controller of low-pass first order filter output design flexible joint mechanical arm.

First small step: set desired trajectory x _1d=sint is with the state x of feedback acquisition ₁Subtract each other and obtain S ₁=x ₁-x _1d, to x _1dDifferentiate obtains

Parameter c ₁Value is 2, calculates

Will

Be input to timeconstant ₂Value is 0.02 low-pass first order filter

In obtain exporting x _2d

Second small step: according to the x of low-pass first order filter output _2dAnd formula

Obtain

With x _2dThe state x that obtains with feedback ₂Subtract each other and obtain S ₂=x ₂-x _2dParameter c ₂Value is 40, according to

Calculate

Will

Be input to timeconstant ₃Value is 0.02 low-pass first order filter

Obtain exporting x _3d

The 3rd small step: according to the x of low-pass first order filter output _3dAnd formula

Obtain

With x _3dThe state x that obtains with feedback ₃Subtract each other and obtain S ₃=x ₃-x _3dParameter c ₃Value is 20, according to

Calculate

Will

Be input to timeconstant ₄Value is 0.02 low-pass first order filter

Obtain exporting x _4d

The 4th small step: according to low-pass first order filter output x _4dAnd formula Obtain

With x _4dThe state x that obtains with feedback ₄Subtract each other and obtain S ₄=x ₄-x _4dParameter c ₄Value is 2, calculates the dynamic face control of flexible joint mechanical arm Under Matlab 7.0 environment real system is carried out emulation, simulation result is seen shown in Fig. 5 .1-5.2.

The 3rd step: the tracking performance check is regulated with parameter

Whether this step meets design requirement the checking system tracking performance, sees shown in Figure 4.Carry out by means of numerical evaluation and Control System Imitation instrument Matlab 7.0 commonly used.

Parameter c ₁, c ₂, c ₃, c ₄, τ ₂, τ ₃, τ ₄For regulating parameter.If tracking error is excessive, do not meet design requirement, then can increase c ₁, c ₂, c ₃, c ₄Value and reduce τ ₂, τ ₃, τ ₄Value.With c ₁, c ₂, c ₃, c ₄Increase to 5,45,25,5 respectively, with τ ₂, τ ₃, τ ₄Be reduced to 0.01,0.01,0.01, the simulation result after parameter is regulated is seen shown in Fig. 6 .1-6.2.After parameter was regulated, accuracy and the rapidity of tracking performance greatly improved, and therefore this adjusting parameter way helps to improve the system keeps track performance.

The 4th step: design finishes

The whole design process emphasis has been considered the demand for control of three aspects, She Ji simplicity respectively, the stability of closed-loop system, the quick accuracy of tracking.Around these three aspects, at first in the above-mentioned first step, determined the concrete formation of closed-loop control system; Emphasis has provided the dynamic face control of flexible joint mechanical arm method for designing in second step, mainly comprises four little steps; Mainly introduced in order to improve the parameter adjusting method of tracking performance in the 3rd step; After above steps, design finishes.

Claims (1)

1. the dynamic face control method of a flexible joint mechanical arm, it is characterized in that: these method concrete steps are as follows:

Step 1: the model analysis of flexible joint mechanical arm system and structure

Closed-loop control system adopts degenerative control structure, and output quantity is flexible joint mechanical arm connecting rod angle; Designed closed-loop control system comprises controller link and these two parts of system model;

Flexible joint mechanical arm control system model description is as follows:

$\left\{\begin{array}{c}I\stackrel{\&CenterDot;\&CenterDot;}{q}+K(q-q_m)+\mathrm{Mgl}\sin q=0\\ J{\stackrel{\&CenterDot;\&CenterDot;}{q}}_m-K(q-q_m)=u\end{array}\right.---\left(1\right)$

Wherein: q represents flexible joint mechanical arm connecting rod angle;

q _mExpression rotor angle;

I represents flexible joint mechanical arm moment of inertia;

J represents the rotor moment of inertia;

K represents the joint stiffness coefficient;

M represents flexible joint mechanical arm connecting rod quality;

G represents acceleration of gravity;

L represents that the joint arrives the distance of bar barycenter;

U represents motor torque;

For the ease of design, define one of four states variable x respectively ₁, x ₂, x ₃, x ₄As follows:

x ₁=q

$x_2=\stackrel{\&CenterDot;}{q}$

x ₃=qm

$x_4={\stackrel{\&CenterDot;}{q}}_m$

The derivative of expression flexible joint mechanical arm connecting rod angle, just flexible joint mechanical arm connecting rod angle speed;

The derivative of expression rotor angle, just rotor angular velocity;

At this moment (1) is just write as

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=a_1{x}_3+f_1\left(x_1\right)\\ {\stackrel{\&CenterDot;}{x}}_3=x_4\\ {\stackrel{\&CenterDot;}{x}}_4=a_{4}u+f_2(x_1,x_3)\end{array}\right.---\left(2\right)$

Wherein:

$a_1=\frac{K}{I},$

$a_2=\frac{1}{J},$

$f_1\left(x_1\right)=-\frac{\mathrm{Mgl}}{I}\sin x_1-\frac{K}{I}{x}_1,$

$f_2(x_1,x_3)=\frac{K}{J}(x_1-x_3);$

So the purpose of handling is that system is turned to clear following triangular form system, is convenient to the control design;

Step 2: the dynamic face control of flexible joint mechanical arm design

The dynamic face control of flexible joint mechanical arm design, its process is the process of progressively going forward one by one, one is divided into four small steps:

First small step: suppose that desired trajectory is x _1d, define first error surface S ₁For

S ₁=x ₁-x _1d （3）

(3) differentiate is obtained

${\stackrel{\&CenterDot;}{S}}_1=x_2-{\stackrel{\&CenterDot;}{x}}_{1d}---\left(4\right)$

Design first virtual controlling amount

For

${\stackrel{\&OverBar;}{x}}_2=-c_1{S}_1+{\stackrel{\&CenterDot;}{x}}_{1d}---\left(5\right)$

Wherein: c ₁Parameter is regulated in expression;

Will Be input to following low-pass first order filter

${\mathrm{\&tau;}}_2{\stackrel{\&CenterDot;}{x}}_{2d}+x_{2d}={\stackrel{\&OverBar;}{x}}_2---\left(6\right)$

Wherein: τ ₂The expression time parameter;

x _2dThe output of expression low-pass first order filter;

Second small step: define second error surface S ₂For

S ₂=x ₂-x _2d （7）

(7) differentiate is obtained

${\stackrel{\&CenterDot;}{S}}_2=a_1{x}_3+f_1\left(x_1\right)-{\stackrel{\&CenterDot;}{x}}_{2d}---\left(8\right)$

Design second virtual controlling amount

For

${\stackrel{\&OverBar;}{x}}_3=\frac{1}{a_1}[-f_1\left(x_1\right)+{\stackrel{\&CenterDot;}{x}}_{2d}-c_2{S}_2]---\left(9\right)$

Wherein: c ₂Parameter is regulated in expression;

Obtained by (6)

${\stackrel{\&CenterDot;}{x}}_{2d}=\frac{{\stackrel{\&OverBar;}{x}}_2-x_{2d}}{{\mathrm{\&tau;}}_2}---\left(10\right)$

Will

Be input to following low-pass first order filter

${\mathrm{\&tau;}}_3{\stackrel{\&CenterDot;}{x}}_{3d}+x_{3d}={\stackrel{\&OverBar;}{x}}_3---\left(11\right)$

Wherein: τ ₃The expression time parameter;

x _3dThe output of expression low-pass first order filter;

The 3rd small step: define the 3rd error surface S ₃For

S ₃=x ₃-x _3d （12）

(12) differentiate is obtained

${\stackrel{\&CenterDot;}{S}}_3=x_4-{\stackrel{\&CenterDot;}{x}}_{3d}---\left(13\right)$

Design the 3rd virtual controlling amount For

${\stackrel{\&OverBar;}{x}}_4=-c_3{S}_3+{\stackrel{\&CenterDot;}{x}}_{3d}---\left(14\right)$

Wherein: c ₃Parameter is regulated in expression;

Obtained by (11)

${\stackrel{\&CenterDot;}{x}}_{3d}=\frac{{\stackrel{\&OverBar;}{x}}_3-x_{3d}}{{\mathrm{\&tau;}}_3}---\left(15\right)$

Will

Be input to following low-pass first order filter

${\mathrm{\&tau;}}_4{\stackrel{\&CenterDot;}{x}}_{4d}+x_{4d}={\stackrel{\&OverBar;}{x}}_4---\left(16\right)$

Wherein: τ ₄The expression time parameter;

x _4dThe output of expression low-pass first order filter;

The 4th small step: define the 4th error surface S ₄For

S ₄=x ₄-x _4d （17）

(17) differentiate is obtained

${\stackrel{\&CenterDot;}{S}}_4=a_{2}u+f_2(x_1,x_3)-{\stackrel{\&CenterDot;}{x}}_{4d}---\left(18\right)$

The dynamic face control of design flexible joint mechanical arm u is

$u=\frac{1}{a_2}[-f_2(x_1,x_3)+{\stackrel{\&CenterDot;}{x}}_{4d}-c_4{S}_4]---\left(19\right)$

Wherein: c ₄Parameter is regulated in expression;

Obtained by (16)

${\stackrel{\&CenterDot;}{x}}_{4d}=\frac{{\stackrel{\&OverBar;}{x}}_4-x_{4d}}{{\mathrm{\&tau;}}_4}---\left(20\right)$

So far, obtained the dynamic face control of flexible joint mechanical arm;

Step 3: the tracking performance check is regulated with parameter

Whether this step meets design requirement the checking system tracking performance, and suitably regulates the control parameter, carries out by means of numerical evaluation and Control System Imitation instrument Matlab7.0 commonly used;

Parameter c ₁, c ₂, c ₃, c ₄For regulating parameter; τ ₂, τ ₃, τ ₄The expression time parameter if tracking error is excessive, does not meet design requirement, and then increases c ₁, c ₂, c ₃, c ₄Value or reduce τ ₂, τ ₃, τ ₄Value; On the one hand, increase c ₁, c ₂, c ₃, c ₄Be equivalent to increase control intensity; On the other hand, reduce τ ₂, τ ₃, τ ₄Be equivalent to improve the response speed of system; Therefore these two kinds of ways all help to improve the system keeps track performance;

Step 4: design finishes

The whole design process emphasis has been considered the demand for control of three aspects, is respectively the simplicity of design, the stability of closed-loop system, the quick accuracy of tracking; Around these three aspects, at first in the above-mentioned first step, determined the concrete formation of closed-loop control system; Emphasis has provided the dynamic face control method for designing of flexible joint mechanical arm in second step, comprises four little steps; Introduced in order to improve the parameter adjusting method of tracking performance in the 3rd step; After above steps, design finishes.

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