CN103486239A - Control method and system for eliminating transmission gear clearance influence - Google Patents

Abstract

The invention provides a control method and system for eliminating transmission gear clearance influence. The method includes the steps: 101) building a smooth model for a gear clearance; 102) leading the built smooth model into an intelligent control algorithm to obtain input torque of a drive side of a gear transmission system; 103) controlling a drive side of a transmission gear by the aid of the obtained input torque, smoothly transferring output torque of the drive side of the transmission gear to a load side and enabling the position of the drive side to output position signals with needed precision tracking expectation. Roughness of a traditional gear clearance model is overcome; an inverse model for compensation is possibly omitted, a multi-model switching system policy is possibly built, the design of a gear clearance control policy is simplified, and engineering implementation is facilitated; the output torque of a system driver is smoothly transferred; a cascade system with the gear clearance is converted into a general controlled object, and usable policies for the design of a controller are widened.

Description

A kind of controlling method and system of eliminating the driving gear gap affects

Technical field

The present invention relates to a kind of controlling method of servo gear transmission system hi-Fix, particularly relate to the modeling method in driving gear gap, the present invention relates to a kind of controlling method and system of eliminating the driving gear gap affects.

Background technique

Gear clearance is owing to producing by gear drive between moving element in the servo gear transmission system, it is that mechanical transmission course normally carries out indispensable a kind of non-linearity drive mechanism, is also an important non-linear factor that affects dynamic performance and stable state accuracy.The existing modeling to gear clearance is dead-zone model, and this model is that to describe gear clearance by the transmission torque of system drive and secondary part and relative displacement nonlinear, mathematical description as shown in the formula:

$T_{\mathrm{old}}=\left\{\begin{array}{c}\mathrm{\&gamma;}(\mathrm{\&theta;}-\mathrm{\&alpha;}),\mathrm{\&theta;}\&GreaterEqual;\mathrm{\&alpha;}\\ 0,\left|\mathrm{\&theta;}\right|<\mathrm{\&alpha;}\\ \mathrm{\&gamma;}(\mathrm{\&theta;}+\mathrm{\&alpha;}),\mathrm{\&theta;}\&le;-\mathrm{\&alpha;}\end{array}\right.---\left(1\right)$

Wherein, θ is the displacement that relatively rotates before and after gear, and 2 α (α>0) are the spacing of gear clearance, and γ (γ>0) is the stiffness coefficient of gear, T _oldtransmission torque for gear.This model is very realistic physically, therefore is widely adopted.Above-mentioned each meaning of parameters also can be referring to table 1.

For this model, existing controlling method comprises two classes: the one, and the inversion model compensation, in its methods and strategies, supposition system drive part can realize the moment jump in the middle of gap, could obtain the conclusion of its method.In actual physics servo gear transmission system, the input of dead-zone model is described is that the rotation displacement of two inter-entity is poor, existence due to entity inertia, can not realize the moment jump between backlash, therefore adopt the method for inversion model compensation can not operate in practice; The 2nd, set up the multi-model switching system, be about to whole system and be divided into clearance mode and contact mode, design respectively control strategy and switching controls algorithm, this just makes the complexity of transmission system greatly increase, Simultaneous Stabilization also is difficult for guaranteeing, is not suitable for the engineering application.

The main cause that causes the existing technological scheme that solves the gear clearance impact to be confined in above-mentioned two class strategies is the rough characteristic that the dead-zone model of gear clearance is described, make its compensation is become to very difficult, thereby also increased the complexity that transmission system is controlled, if can not eliminate the impact of backlash nonlinearity, transmission system performance can even become unstable because limit cycle or Gear Contact impact to reduce, simultaneously, the collision of wheel between cog can produce serious concussion and noise.

Summary of the invention

The object of the invention is to, for overcoming the problems referred to above, the invention provides a kind of controlling method and system of eliminating the driving gear gap affects.

For achieving the above object, the invention provides a kind of controlling method of eliminating the driving gear gap affects, described method comprises:

Step 101) adopting following formula is that gear clearance is set up smooth model:

$T_{\mathrm{new}}=\mathrm{\&gamma;\&theta;}+\frac{\mathrm{\&gamma;}}{2\mathrm{\&kappa;}}[\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})\right)\right)-\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})\right)\right)]$

Wherein, θ is the relative rotation position angle signal before and after gear, the spacing that 2 α are gear clearance, the stiffness coefficient that γ is gear, T _newfor the output of smooth model, κ is adjustable parameter;

Step 102) smooth model based on setting up obtains the Obj State spatial description, then obtains the input torque of the driving side of gear train assembly according to the Obj State spatial description;

Step 103) adopt the input torque obtained to control the driving side of driving gear, make the output torque of driving side smoothly be passed to load side, guarantee that the position signal of expectation is followed the tracks of in the position output of driving side with required precision simultaneously.

Adopt intelligent control algorithm to obtain the Obj State spatial description, described intelligent control algorithm comprises contragradience control algorithm or state feedback linearization control algorithm.

Optionally, when adopting the contragradience control algorithm, described step 102) further comprise:

At first, definition status variable x=[x ₁x ₂x ₃x ₄] ^t=[θ _lω _lθ _mω _m] ^t,

by introducing smooth gear clearance model, controll plant is converted into to following state space form:

$\stackrel{\&CenterDot;}{x}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\\ {\stackrel{\&CenterDot;}{x}}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}(T_{\mathrm{new}}+E\left(x_{\mathrm{\&theta;}}\right))-\frac{b_l}{J_l}{x}_2\\ x_4\\ \frac{u}{J_m}-\frac{T_{\mathrm{new}}+E\left(x_{\mathrm{\&theta;}}\right)}{J_m}-\frac{b_m}{J_m}{x}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{\mathrm{n\&gamma;}}{J_l}{x}_3-{\mathrm{\&phi;}}_1\left(x\right)+\frac{n}{J_l}E\left(x_{\mathrm{\&theta;}}\right)\\ x_4\\ \frac{u}{J_m}-{\mathrm{\&phi;}}_2\left(x\right)-\frac{E\left(x_{\mathrm{\&theta;}}\right)}{J_m}\end{array}\right]$

Based on above-mentioned object mathematical description, carry out the input torque that the contragradience control algorithm is determined driving side, concrete steps are as follows:

Step 1, the definition output tracking error is: e ₁=θ _r-θ _l=θ _r-x ₁;

Tracking error according to definition is chosen the Lyapunov function, and this functional form is:

and then calculate virtual controlling amount x ₂expected value be

c ₁for adjustable normal number;

Step 2, definition x ₂error be e ₂=μ ₁-x ₂, according to the error of definition, choose the Lyapunov function

and then calculate virtual controlling amount (n γ/J _l) x ₃expected value be ${\mathrm{\&mu;}}_2=e_1+c_2{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+c_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+{\mathrm{\&phi;}}_1\left(x\right),$ C wherein ₂for adjustable normal number;

Step 3, definition x ₃tracking error be e ₃=μ ₂-(n γ/J _l) x ₃, according to the error of definition, choose the Lyapunov function

and then calculate virtual controlling amount n γ (2+ λ (x _θ)) x ₄/ 2J _lexpected value be ${\mathrm{\&mu;}}_3={\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+c_3{e}_3+(1+c_1{c}_2){\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r+(c_1+c_2){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+{\mathrm{\&theta;}}_r^{\left(3\right)}+(b_l/J_l-c_1-c_2)(\mathrm{n\&gamma;}{x}_3/J_l-{\mathrm{\&phi;}}_1\left(x\right))+[{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))/2J_l-c_1{c}_2-1]x_2$ , c ₃for adjustable normal number;

Step 4, definition x ₄tracking error be e ₄=μ ₃-n γ (2+ λ (x _θ)) x ₄/ 2J _l, according to the tracking error of definition, choose the Lyapunov function

according to x defined above ₁, x ₂, x ₃and x ₄value and resolve based on the Lyapunov stability principle input torque u that obtains driving side and be:

$\begin{array}{c}u=\frac{2J_l{J}_m}{\mathrm{n\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))}\{e_3+c_4{e}_4+b_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+b_3{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}-\frac{2\mathrm{\&kappa;n\&gamma;}{(x_4-n{x}_2)}^2}{J_l}\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)+\\ [b_4+\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))](\frac{\mathrm{n\&gamma;}{x}_3}{J_l}-{\mathrm{\&phi;}}_1\left(x\right))-[\frac{b_5\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}-\frac{\mathrm{n\&gamma;}{\mathrm{\&phi;}}_2\left(x\right)}{2J_l}](2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))\}\end{array},$

Wherein, c ₄for adjustable normal number; J _mmean the driving side rotary inertia, J _lmean the load side rotary inertia, n means the reduction speed ratio of retarder, and γ means gear stiffness coefficient, x _θmean that the gear front and back relatively rotate the state variable of angular displacement; λ (x _θ), τ (x _θ) the expression SQL; b ₁, b ₂, b ₃, b ₄, b ₅mean custom parameter;

mean expectation angle position signal θ _rthe single order differential signal; mean θ _rthe second-order differential signal;

mean θ _rthree rank differential signals; mean θ _rthe quadravalence differential signal; φ ₁and φ (x) ₂(x) all mean SQL; x ₁, x ₂, x ₃, x ₄mean respectively θ _l, ω _l, θ _m, ω _mstate variable.

When adopting the state feedback linearization control algorithm, described step 102) further comprise:

At first, definition status variable x is converted into following state space form by introducing smooth gear clearance model by controll plant:

$\stackrel{\&CenterDot;}{x}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\\ {\stackrel{\&CenterDot;}{x}}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ x_4\\ -\frac{1}{J_m}{T}_{\mathrm{new}}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{J_m}\end{array}\right]u\stackrel{\mathrm{\&Delta;}}{=}f\left(x\right)+\mathrm{gu}---\left(5.1\right)$

$y=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]x\stackrel{\mathrm{\&Delta;}}{=}h\left(x\right)=x_1$

Wherein, $f\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ x_4\\ -\frac{1}{J_m}{T}_{\mathrm{new}}\end{array}\right],g\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{J_m}\end{array}\right],h\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]x;$

Based on above-mentioned object mathematical description, carry out the input torque that the state feedback linearization control algorithm is determined driving side, concrete steps are as follows:

Step 1), by analyzing, provides a differomorphism transformation of coordinates as follows:

$Z\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ \frac{\mathrm{n\&gamma;}(x_4-n{x}_2)}{J_l}l\left(x_{\mathrm{\&theta;}}\right)\end{array}\right]---\left(5.2\right)$

Wherein, $l\left(x_{\mathrm{\&theta;}}\right)\stackrel{\mathrm{\&Delta;}}{=}1+\left\{\tanh \right[\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})]-\tanh [\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})\left]\right\}/2,$ Can try to achieve

$\frac{\&PartialD;Z}{\&PartialD;x}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ \frac{n}{J_l}\frac{\&PartialD;T_{\mathrm{new}}}{{\&PartialD;x}_1}& 0& \frac{n}{J_l}\frac{{\&PartialD;T}_{\mathrm{new}}}{\&PartialD;x_3}& 0\\ \frac{\&PartialD;z_4}{\&PartialD;x_1}& \frac{\&PartialD;z_4}{\&PartialD;x_2}& \frac{\&PartialD;z_4}{\&PartialD;x_3}& \frac{\&PartialD;z_4}{\&PartialD;x_4}\end{array}\right]---\left(5.3\right)$

And then can try to achieve,

$\left|\frac{\&PartialD;Z}{\&PartialD;x}\right|=\frac{n}{J_l}\frac{{\&PartialD;T}_{\mathrm{new}}}{{\&PartialD;x}_3}\frac{{\&PartialD;z}_4}{{\&PartialD;x}_4}=\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2{\mathrm{\&gamma;}}^2}{J_l^2}{\left(l\left(x_{\mathrm{\&theta;}}\right)\right)}^2>0---\left(5.3\right)$

Known, the transformation of coordinates of giving is an overall differomorphism;

Step 2) state transformation based on designed, order

formula (5.1) can be converted into

$\stackrel{\&CenterDot;}{Z}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\\ {\stackrel{\&CenterDot;}{z}}_2\\ {\stackrel{\&CenterDot;}{z}}_3\\ {\stackrel{\&CenterDot;}{z}}_4\end{array}\right]=\left[\begin{array}{c}{z}_2\\ z_3\\ z_4\\ L_f^{4}h\left(x\right)+u{L}_g{L}_f^{3}h\left(x\right)\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]Z+\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]v\stackrel{\mathrm{\&Delta;}}{=}\mathrm{AZ}+\mathrm{bv}---\left(5.4\right)$

Wherein, $A\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right],b\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right].$

(5.4) be the system mathematical description of original system (5.1) after transformation of coordinates (5.2) transforms, v is middle controlled quentity controlled variable;

In the middle of the step 3) order, controlled quentity controlled variable v gets:

v＝HZ+v _r （5.5）

Wherein, H=[h ₁h ₂h ₃h ₄] be the controller parameter vector, it is the Hurwitz battle array that the selection of its value meets (A+bH), $v_r=-h_1{\mathrm{\&theta;}}_r-{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">h</figure-callout>}}_2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-h_3{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r-h_4{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}$ For the reference control inputs;

Step 4) definition error σ ₁=z ₁-θ _r, ${\mathrm{\&sigma;}}_2=z_2-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r,$ ${\mathrm{\&sigma;}}_3=z_3-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r,$ ${\mathrm{\&sigma;}}_4=z_4-{\mathrm{\&theta;}}_r^{\left(3\right)},$ Error vector is σ=[σ ₁σ ₂σ ₃σ ₄] ^t, according to (5.4)-(5.5), the error dynamics equation is

$\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}=(A+\mathrm{bH})\mathrm{\&sigma;}---\left(5.6\right)$

Because (A+bH) is the Hurwitz battle array, known according to the Linear System Stability criterion, error vector σ is globally asymptotically stable, i.e. z ₁can asymptotically stable in the large follow the tracks of the position signal θ expected _r, because (5.2) are overall differomorphism, known output signal θ after coordinate inversion _lcan follow the tracks of the position signal θ expected by Asymptotic Stability _r;

By the representation of middle controlled quentity controlled variable v of design, the anti-input torque u that pushes away driving side:

$u=\frac{v-L_f^{4}h\left(x\right)}{L_g{L}_f^{3}h\left(x\right)}---\left(5.7\right)$

Wherein,

$L_g{L}_f^{3}h\left(x\right)=\frac{\mathrm{n\&gamma;}}{J_l{J}_m}l\left(x_{\mathrm{\&theta;}}\right)$

$L_f^{4}h\left(x\right)=\frac{2\mathrm{\&kappa;\&gamma;n}{(x_4-n{x}_2)}^2}{J_l}[1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})})}^2-1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})})}^2]-(\frac{n^3\mathrm{\&gamma;}}{{\mathrm{<figure-callout\; id="2"\; label="J\; l"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">J</figure-callout>}}_l^{}}+\frac{\mathrm{n\&gamma;}}{J_l{J}_m})l\left(x_{\mathrm{\&theta;}}\right)T_{\mathrm{new}};$

Wherein, the implication of above-mentioned each parameter can reference table 1, and the implication of compute sign also can be with reference to the implication of each compute sign related to when adopting the contragradience control algorithm, mean expectation angle position signal θ _rthe single order differential signal;

mean θ _rthe second-order differential signal,

mean θ _rthree rank differential signals;

mean θ _rthe quadravalence differential signal.

The value strategy of described control parameter comprises:

Strategy one, will control parameter and all be set to 1; Or

Strategy two, controller parameter c _i>0 and meet c ₁=min (c _i), and ask the method for extreme value to make " ξ (c in following formula by the function of many variables ₁, c ₂, c ₃, c ₄)/c ₁" minimum and then definite c _ivalue, wherein, i=1 ... 4;

$\mathrm{\&Omega;}=\left\{e\right|\left|\right|e\left|\right|<{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)\sqrt{\mathrm{\&xi;}(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3,c_4)/c_1},{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)=\frac{\mathrm{n\&gamma;}\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2J_l\mathrm{\&kappa;}}\}$

Described model parameter κ should be at controller parameter c _i>0 determine after, the required precision according to the servo gear transmission system to the load side output angle, determine according to the error precision upper limit.

In order to realize that said method the present invention also provides a kind of control system of eliminating the gear clearance impact, described system comprises:

Smooth model is set up module, for adopting following formula, is that gear clearance is set up smooth model:

$T_{\mathrm{new}}=\mathrm{\&gamma;}[\mathrm{\&theta;}+\frac{1}{2\mathrm{\&kappa;}}\ln \left(\frac{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}}{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}}\right)]$

Wherein, θ is the relative rotation position angle signal before and after gear, the spacing that 2 α are gear clearance, the stiffness coefficient that γ is gear, T _newfor the output of smooth model, κ is adjustable parameter;

Driving side input torque determination module, introduce intelligent control algorithm for the smooth model by setting up, and obtains the input torque of the driving side of gear train assembly;

Control module, the input torque obtained for employing is controlled driving gear, makes the output torque of driving side smoothly be passed to load side.

Described intelligent control algorithm comprises contragradience control algorithm or state feedback linearization control algorithm.

Described driving side input torque determination module further comprises:

Comparison module, compare judgement for the position by desired output and actual outgoing position, the output angle position error signal;

Obj State spatial description acquisition of information module, processed for the smooth model of the gear clearance to setting up and angle positional deviation, model bias and the smooth gap model information of comparison module input, finally exported the Obj State spatial description with smooth gear clearance and model error;

The first puocessing module, processing is controlled in the contragradience of carrying out based on the Lyapunov stability principle for the spatial description of the Obj State with smooth gear clearance and the model error information by described Obj State spatial description acquisition of information module output, finally exports the second virtual controlling amount x ₂;

The second puocessing module, for the second virtual controlling amount x by the first puocessing module output ₂processing is controlled in the contragradience that information is carried out based on the Lyapunov stability principle, and then exports the 3rd virtual controlling amount x ₃;

The 3rd puocessing module, for the 3rd virtual controlling amount x by the second puocessing module output ₃processing is controlled in the contragradience that information is carried out based on the Lyapunov stability principle, and then exports the 4th virtual controlling amount x ₄; With

The manages module everywhere, for the 4th virtual controlling amount x based on the 3rd puocessing module output ₄information adopts following formula to obtain the input torque u of driving side: $\begin{array}{c}u=\frac{2J_l{J}_m}{\mathrm{n\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))}\{e_3+c_4{e}_4+b_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+b_3{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}-\frac{2\mathrm{\&kappa;n\&gamma;}{(x_4-n{x}_2)}^2}{J_l}\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)+\\ [b_4+\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))](\frac{\mathrm{n\&gamma;}{x}_3}{J_l}-{\mathrm{\&phi;}}_1\left(x\right))-[\frac{b_5\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}-\frac{\mathrm{n\&gamma;}{\mathrm{\&phi;}}_2\left(x\right)}{2J_l}](2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))\}\end{array}.$

Above-mentioned driving side input torque determination module also comprises the parameter determination module, and described parameter determination module determines according to following strategy the value of controlling parameter and model parameter:

Strategy one, will control parameter and all be set to 1; Or

Strategy two, controller parameter c _i>0 and meet c ₁=min (c _i), and ask the method for extreme value to make " ξ (c in following formula by the function of many variables ₁, c ₂, c ₃, c ₄)/c ₁" minimum and then definite c _ivalue, wherein, i=1 ... 4;

$\mathrm{\&Omega;}=\left\{e\right|\left|\right|e\left|\right|<{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)\sqrt{\mathrm{\&xi;}(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3,c_4)/c_1},{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)=\frac{\mathrm{n\&gamma;}\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2J_l\mathrm{\&kappa;}}\}$

Described model parameter κ should be at controller parameter c _i>0 determine after, the required precision according to the servo gear transmission system to the load side output angle, determine according to the error precision upper limit.

The type of above-mentioned self-defining function and parameter and value can be with reference to the contents of embodiment part.

Compared with prior art, technical advantage of the present invention is:

1) overcome the rough characteristic of conventional gears gap model; 2) make to avoid using the inversion model compensation and set up multi-model switching system strategy to become possibility, simplified the design of gear clearance control strategy, be convenient to through engineering approaches and realize; 3) overcome the defect of prior art scheme, make the system drive output torque smoothly transmit; 4) will be converted into general controlled device with the cascade system of gear clearance, widen the strategy that the controller design can adopt.

The accompanying drawing explanation

Fig. 1 is servo gear transmission system schematic diagram;

Fig. 2 is the controlling method flow chart of elimination driving gear gap affects provided by the invention;

Fig. 3 is the composition frame chart of the driving side input torque determination module obtained based on the contragradience control algorithm provided by the invention;

Fig. 4 is the composition frame chart of the driving side input torque determination module that obtains of state-based feedback control algorithm provided by the invention.

Embodiment

Below in conjunction with drawings and Examples, the method for the invention is elaborated.

For overcoming the existing deficiency that solves the gear clearance influence technique, the present invention proposes a kind of describing method of smooth gear clearance model:

$T_{\mathrm{new}}=\mathrm{\&gamma;}[\mathrm{\&theta;}+\frac{1}{2\mathrm{\&kappa;}}\ln \left(\frac{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}}{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}}\right)]$

$T_{\mathrm{new}}=\mathrm{\&gamma;\&theta;}+\frac{\mathrm{\&gamma;}}{2k}[\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})\right)\right)-\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})\right)\right)]---\left(2\right)$

Wherein, θ relatively rotates displacement, the spacing that 2 α are gear clearance, the stiffness coefficient that γ is gear, T before and after gear _newfor the transmission torque of gear, the adjustable parameter that κ is new model, be also the characteristic parameter of new model, above-mentioned each meaning of parameters also can be referring to table 1.The function characteristic of this model is along with the increase of κ approaches traditional gear clearance model gradually, and it is smooth.

For the characteristic of above-mentioned smooth model is described, Definition Model error E (θ)=T _old-T _new, according to formula (1) and formula (2), have:

$E\left(\mathrm{\&theta;}\right)=\left\{\begin{array}{c}\mathrm{\&gamma;}(\mathrm{\&theta;}-\mathrm{\&alpha;})-\mathrm{\&gamma;}[\mathrm{\&theta;}+\frac{1}{2\mathrm{\&kappa;}}\ln \left(\frac{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}}{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}}\right)],\mathrm{\&theta;}\&GreaterEqual;\mathrm{\&alpha;}\\ -\mathrm{\&gamma;}[\mathrm{\&theta;}+\frac{1}{2\mathrm{\&kappa;}}\ln \left(\frac{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}}{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}}\right)],\left|\mathrm{\&theta;}\right|<\mathrm{\&alpha;}\\ \mathrm{\&gamma;}(\mathrm{\&theta;}+\mathrm{\&alpha;})-\mathrm{\&gamma;}[\mathrm{\&theta;}+\frac{1}{2\mathrm{\&kappa;}}\ln \left(\frac{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}}{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}}\right)],\mathrm{\&theta;}\&le;-\mathrm{\&alpha;}\end{array}\right.---\left(3\right)$

For further analytic expression (3), defined function

have

$\frac{\mathrm{d\&Pi;}\left(\mathrm{\&theta;}\right)}{\mathrm{d\&theta;}}=\frac{2\mathrm{\&kappa;}(e^{-4\mathrm{\&kappa;\&alpha;}}-1)}{({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}^{2\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}+e^{-2\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}+2)}<0---\left(4\right)$

Therefore, function Π (θ) successively decreases with independent variable θ.And then can derive formula (3) and meet following inequality

$\left\{\begin{array}{c}-\mathrm{\&gamma;\&alpha;}-\frac{\mathrm{\&gamma;}}{2\mathrm{\&kappa;}}\ln \left(\frac{2}{({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}^{2\mathrm{\&kappa;\&alpha;}}+e^{-2\mathrm{\&kappa;\&alpha;}})}\right)\&le;E\left(\mathrm{\&theta;}\right)<0,\mathrm{\&theta;}\&GreaterEqual;\mathrm{\&alpha;}\\ -\mathrm{\&gamma;\&alpha;}-\frac{\mathrm{\&gamma;}}{2\mathrm{\&kappa;}}\ln \left(\frac{2}{{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}^{2\mathrm{\&kappa;\&alpha;}}+e^{-2\mathrm{\&kappa;\&alpha;}}}\right)<E\left(\mathrm{\&theta;}\right)<\mathrm{\&gamma;\&alpha;}+\frac{\mathrm{\&gamma;}}{2\mathrm{\&kappa;}}\ln \left(\frac{2}{({\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}^{2\mathrm{\&kappa;\&alpha;}}+e^{-2\mathrm{\&kappa;\&alpha;}})}\right),\left|\mathrm{\&theta;}\right|<\mathrm{\&alpha;}\\ 0<E\left(\mathrm{\&theta;}\right)\&le;\mathrm{\&gamma;\&alpha;}+\frac{\mathrm{\&gamma;}}{2\mathrm{\&kappa;}}\ln \left(\frac{2}{{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}^{2\mathrm{\&kappa;\&alpha;}}+e^{-2\mathrm{\&kappa;\&alpha;}}}\right),\mathrm{\&theta;}\&le;-\mathrm{\&alpha;}\end{array}\right.---\left(5\right)$

According to formula (5), further can obtain,

$-\frac{\mathrm{\&gamma;}\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2\mathrm{\&kappa;}}<E\left(\mathrm{\&theta;}\right)<\frac{\mathrm{\&gamma;}\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2\mathrm{\&kappa;}}---\left(6\right)$

By formula (6), can be released

that is to say the increase along with κ, the gear clearance model that the present invention proposes can approach the dead-zone model description of traditional gear clearance with arbitrary accuracy, and the while is the smooth characteristic of reserving model again.

The controlling method of the smooth model elimination driving gear of the present invention gap affects based on setting up comprises following steps, as shown in Figure 2:

Step 101) set up smooth model for gear clearance, described smooth model is as follows:

$T_{\mathrm{new}}=\mathrm{\&gamma;\&theta;}+\frac{\mathrm{\&gamma;}}{2k}[\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})\right)\right)-\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})\right)\right)]$

Wherein, each meaning of parameters is referring to table 1.Step 102) smooth model of setting up is introduced to intelligent control algorithm, obtain the input torque of the driving side of gear train assembly;

Step 103) adopt the input torque obtained to control driving gear, make the output torque of driving side smoothly be passed to load side.

Embodiment 1,

Take the contragradience control algorithm as example, set forth the controlling method of elimination driving gear gap affects of the present invention, and be designed to the high precision position controller of servo gear transmission system the beneficial effect that example illustrates the gear clearance model of being invented.

The mathematical description of servo gear transmission system is

$\left\{\begin{array}{c}{J}_m\frac{{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">d</figure-callout>}}^2{\mathrm{\&theta;}}_m}{{\mathrm{<figure-callout\; id="2"\; label="dt"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">dt</figure-callout>}}^2}+b_m{\mathrm{\&omega;}}_m=u-T_{\mathrm{old}}\\ {\mathrm{<figure-callout\; id="2"\; label="J\; l\; d"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">J</figure-callout>}}_l\frac{d^{}}{}\frac{{}^2{\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="2"\; label="l\; dt"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">l</figure-callout>}}}{{\mathrm{dt}}^{}}+b_l{\mathrm{\&omega;}}_l=n{T}_{\mathrm{old}}\end{array}\right.,T_{\mathrm{old}}=\left\{\begin{array}{c}\mathrm{\&gamma;}(\mathrm{\&theta;}-\mathrm{\&alpha;}),\mathrm{\&theta;}\&GreaterEqual;\mathrm{\&alpha;}\\ 0,\left|\mathrm{\&theta;}\right|<\mathrm{\&alpha;}\\ \mathrm{\&gamma;}(\mathrm{\&theta;}+\mathrm{\&alpha;}),\mathrm{\&theta;}\&le;-\mathrm{\&alpha;}\end{array}\right.---\left(4.1\right)$

Wherein, J _mand J _lrotary inertia for driving side and load side; θ _mand θ _lwei driving side and the output angle position signal of load side, θ=θ _m-n θ _lfor relative angular displacement; b _mand b _lwei live axle and the viscous friction coefficient of bearing axle, ω _mand ω _lwei driving side and the angular velocity signal of load side; 2 α and γ are respectively spacing and the stiffness coefficient of backlash; T _oldfor transmission torque, the reduction speed ratio that n is retarder, the control inputs moment that u is driving side.The implication of above-mentioned each parameter can be referring to table 1.

Definition x=[x ₁x ₂x ₃x ₄] ^t=[θ _lω _lθ _mω _m] ^t, x _θ=x ₃-nx ₁, according to model error E (θ)=T _old-T _new, formula (4.1) can be converted into following state space form:

$\stackrel{\&CenterDot;}{x}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\\ {\stackrel{\&CenterDot;}{x}}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}(T_{\mathrm{new}}+E\left(x_{\mathrm{\&theta;}}\right))-\frac{b_l}{J_l}{x}_2\\ x_4\\ \frac{u}{J_m}-\frac{T_{\mathrm{new}}+E\left(x_{\mathrm{\&theta;}}\right)}{J_m}-\frac{b_m}{J_m}{x}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{\mathrm{n\&gamma;}}{J_l}{x}_3-{\mathrm{\&phi;}}_1\left(x\right)+\frac{n}{J_l}E\left(x_{\mathrm{\&theta;}}\right)\\ x_4\\ \frac{u}{J_m}-{\mathrm{\&phi;}}_2\left(x\right)-\frac{E\left(x_{\mathrm{\&theta;}}\right)}{J_m}\end{array}\right]---\left(4.2\right)$

Wherein,

${\mathrm{\&phi;}}_1\left(x\right)=\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{J_l}{x}_1+\frac{b_l}{J_l}{x}_2-\frac{\mathrm{n\&gamma;}}{2J_l\mathrm{\&kappa;}}\mathrm{\&Pi;}\left(x_{\mathrm{\&theta;}}\right),{\mathrm{\&phi;}}_2\left(x\right)=\frac{\mathrm{\&gamma;}{x}_{\mathrm{\&theta;}}}{J_m}+\frac{b_m{x}_4}{J_m}+\frac{\mathrm{\&gamma;}}{2J_m\mathrm{\&kappa;}}\mathrm{\&Pi;}\left(x_{\mathrm{\&theta;}}\right)---\left(4.3\right)$

How to the following describes design driven side control inputs moment u, the impact of eliminating gear clearance makes position, the angle output θ of load side _lcan follow the tracks of with arbitrary accuracy the angle position signal θ of expectation _r.

Step 1 definition output tracking error is:

e ₁＝θ _r-θ _l＝θ _r-x ₁ （4.4）

Choose the Lyapunov function

make virtual controlling amount x ₂expected value be c ₁for adjustable normal number, definition x ₂error be e ₂=μ ₁-x ₂, according to formula (4.2), V ₁differential be

${\stackrel{\&CenterDot;}{V}}_1=-c_1{e}_1^2+e_1({\mathrm{\&mu;}}_1-x_2)=-c_1{e}_1^2+e_1{e}_2---\left(4.5\right)$

Step 2 is for trying to achieve next step virtual controlling amount x ₃, need first try to achieve e ₂to the derivative of time t, according to e ₂definition and formula (4.2), can try to achieve

${\stackrel{\&CenterDot;}{e}}_2=c_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r-\frac{\mathrm{n\&gamma;}}{J_l}{x}_3+{\mathrm{\&phi;}}_1\left(x\right)-\frac{n}{J_l}E\left(x_{\mathrm{\&theta;}}\right)---\left(4.6\right)$

Choose the Lyapunov function

make virtual controlling amount (n γ/J _l) x ₃expected value be

c ₂for adjustable normal number, definition x ₃tracking error be e ₃=μ ₂-(n γ/J _l) x ₃, according to formula (4.5) and formula (4.6), V ₂differential be

${\stackrel{\&CenterDot;}{V}}_2={\stackrel{\&CenterDot;}{V}}_1+e_2{\stackrel{\&CenterDot;}{e}}_2=-\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{c}_i{e}_i^2+e_2(e_3-\frac{n}{J_l}E\left(x_{\mathrm{\&theta;}}\right))---\left(4.7\right)$

Step 3 is for trying to achieve next step virtual controlling amount x ₄, need first try to achieve φ ₁(x) to the derivative of time t, defined function λ (x _θ)=tanh (κ (x _θ-α))-tanh (κ (x _θ+ α)), according to function Π (x _θ) definition and formula (4.2) and formula (4.4), can try to achieve

${\stackrel{\&CenterDot;}{\mathrm{\&phi;}}}_1\left(x\right)=\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{J_l}{x}_2-\frac{\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right)+\frac{b_l}{J_l}(\frac{\mathrm{n\&gamma;}}{J_l}{x}_3-{\mathrm{\&phi;}}_1\left(x\right)+\frac{n}{J_l}E\left(x_{\mathrm{\&theta;}}\right))---\left(4.8\right)$

Further can try to achieve e ₃derivative to time t is

${\stackrel{\&CenterDot;}{e}}_3=(1+c_1{c}_2)({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+(c_1+c_2){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+{\mathrm{\&theta;}}_r^{\left(3\right)}-\frac{\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))---\left(4.9\right)$

$+(\frac{b_l}{J_l}-c_1-c_2)(\frac{\mathrm{n\&gamma;}}{J_l}{x}_3-{\mathrm{\&phi;}}_1\left(x\right)+\frac{n}{J_l}E\left(x_{\mathrm{\&theta;}}\right))$

Choose the Lyapunov function

make virtual controlling amount n γ (2+ λ (x _θ)) x ₄/ 2J _l(be easy to verify 2+ λ (x _θ) 0) and expected value be ${\mathrm{\&mu;}}_3={\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+c_3{e}_3+(1+c_1{c}_2){\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r+(c_1+c_2){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+{\mathrm{\&theta;}}_r^{\left(3\right)}+(b_l/J_l-c_1$ $-c_2)({\mathrm{n\&gamma;x}}_3/J_l-{\mathrm{\&phi;}}_1\left(x\right))+[{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))/{2J}_l-c_1{c}_2-1]x_2,$ C ₃for adjustable normal number.Definition x ₄tracking error be e ₄=μ ₃-n γ (2+ λ (x _θ)) x ₄/ 2J _l, according to formula (4.7) and formula (4.9), V ₃differential be

${\stackrel{\&CenterDot;}{V}}_3=-\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{c}_i{e}_i^2+e_3{e}_4+\frac{n}{J_l}[(\frac{b_l}{J_l}-c_1-c_2)e_3-e_2]E\left(x_{\mathrm{\&theta;}}\right)---\left(4.10\right)$

Step 4, for to try to achieve real system control amount u, need first be tried to achieve e ₄to the derivative of time t, defined function $\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)=1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})})}^2-1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})})}^2,$ Can try to achieve

$\stackrel{\&CenterDot;}{\mathrm{\&lambda;}}\left(x_{\mathrm{\&theta;}}\right)=4\mathrm{\&kappa;}(x_4-n{x}_2)\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)---\left(4.11\right)$

According to formula (4.2) and formula (4.8), further can try to achieve

${\stackrel{\&CenterDot;}{e}}_4=b_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+b_3{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}+[b_4+\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))](\frac{n{\mathrm{\&gamma;x}}_3}{J_l}-{\mathrm{\&phi;}}_1\left(x\right))-$

$\frac{2\mathrm{\&kappa;n\&gamma;}{(x_4-n{x}_2)}^2}{J_l}\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)-\frac{b_5\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))+\frac{{\mathrm{n\&gamma;\&phi;}}_2\left(x\right)}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))---\left(4.12\right)$

$-\frac{\mathrm{n\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))}{2J_l{J}_m}u+\frac{n}{J_l}[b_4+(\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{{2J}_l}+\frac{\mathrm{\&gamma;}}{{2J}_m})(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))]E\left(x_{\mathrm{\&theta;}}\right)$

Wherein, b ₁=c ₁+ c ₃+ c ₁c ₂c ₃, b ₂=2+c ₁c ₂+ c ₁c ₃+ c ₂c ₃, b ₃=c ₁+ c ₂+ c ₃, b ₅=c ₁+ c ₂+ c ₃-b _l/ J _l, b ₄=(b _l/ J _l-c ₁-c ₂) (c ₃-b _lj _l)-c ₁c ₂-2.

Make the control inputs moment u of driving side be

$u=\frac{2J_l{J}_m}{\mathrm{n\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))}\{e_3+c_4{e}_4+b_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+b_3{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}-\frac{2\mathrm{\&kappa;n\&gamma;}{(x_4-n{x}_2)}^2}{J_l}\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)+$

$[b_4+\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))](\frac{\mathrm{n\&gamma;}{x}_3}{J_l}-{\mathrm{\&phi;}}_1\left(x\right))-[\frac{b_5\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}-\frac{\mathrm{n\&gamma;}{\mathrm{\&phi;}}_2\left(x\right)}{2J_l}](2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))\}---\left(4.13\right)$

C wherein ₄for adjustable normal number.

Choose the Lyapunov function

can try to achieve V according to formula (4.10) to formula (4.13) ₄differential be

${\stackrel{\&CenterDot;}{V}}_4=-\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{c}_i{e}_i^2+\frac{n}{J_l}\{(\frac{b_l}{J_l}-c_1-c_2)e_3-{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}_2+[b_4+(\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{{2J}_l}+\frac{\mathrm{\&gamma;}}{{2J}_m})(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))\left]e_4\right\}E\left(x_{\mathrm{\&theta;}}\right)---\left(4.14\right)$

According to formula (6), formula (4.14) meets

${\stackrel{\&CenterDot;}{V}}_4<-\underset{i=1}{\overset{4}{\mathrm{\&Sigma;}}}{c}_i{e}_i^2+\frac{\mathrm{n\&gamma;}\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2J_l\mathrm{\&kappa;}}\left[\right|{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}_2|+|\frac{b_l}{J_l}-c_1-{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2|\&CenterDot;|e_3|+\left(\right|b_4|+\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{J_l}+\frac{\mathrm{\&gamma;}}{J_m})|e_4\left|\right]---\left(4.15\right)$

Defined function ${\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)=\frac{\mathrm{n\&gamma;}\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2J_l\mathrm{\&kappa;}},$ ${\mathrm{\&xi;}}_2(c_1,c_2)=|\frac{b_l}{J_l}-c_1-{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2|,$ ${\mathrm{\&xi;}}_3(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3)=\left|b_4\right|+\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\mathrm{\&gamma;}}{J_l}+\frac{\mathrm{\&gamma;}}{J_m},$ Formula (4.15) can turn to

${\stackrel{\&CenterDot;}{V}}_4<-c_1{e}_1^2-c_2{\left(\right|{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}_2|-\frac{{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)}{2c_2})}^2-c_3{\left(\right|e_3|-\frac{{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right){\mathrm{\&xi;}}_2(c_1,c_2)}{{2c}_3})}^2-c_4{\left(\right|e_4|-\frac{{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right){\mathrm{\&xi;}}_3(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3)}{{2c}_4})}^2$

$+{\mathrm{\&xi;}}_1^2\left(\mathrm{\&kappa;}\right)(\frac{1}{4c_2}+\frac{{\mathrm{\&xi;}}_2^2(c_1,c_2)}{{4c}_3}+\frac{{\mathrm{\&xi;}}_3^2(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3)}{{4c}_4})---\left(4.16\right)$

Define new error ${\mathrm{\&zeta;}}_2=\left|{\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">e</figure-callout>}}_2\right|-\frac{{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)}{{2\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2},$ ${\mathrm{\&zeta;}}_3=\left|e_3\right|-\frac{{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right){\mathrm{\&xi;}}_2(c_1,c_2)}{{2c}_3},$ ${\mathrm{\&zeta;}}_4=\left|e_4\right|-\frac{{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right){\mathrm{\&xi;}}_3(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3)}{{2c}_4},$ Error vector e=[e ₁ζ ₂ζ ₃ζ ₄] ^t, controller parameter c _i>0(i=1 ... choosing 4) meets c ₁=min (c _i), formula (4.16) can be converted into

${\stackrel{\&CenterDot;}{V}}_4<-c_1{\left|\right|e\left|\right|}^2+{\mathrm{\&xi;}}_1^2\left(\mathrm{\&kappa;}\right)(\frac{1}{{4c}_2}+\frac{{\mathrm{\&xi;}}_2^2(c_1,c_2)}{{4c}_3}+\frac{{\mathrm{\&xi;}}_3^2(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3)}{{4c}_4})---\left(4.17\right)$

Order $\mathrm{\&xi;}(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3{c}_4)=\frac{1}{{4c}_2}+\frac{{\mathrm{\&xi;}}_2^2(c_1,c_2)}{{4c}_3}+\frac{{\mathrm{\&xi;}}_3^2(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3)}{{4c}_4},$ When $\left|\right|e\left|\right|\&GreaterEqual;{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)\sqrt{\mathrm{\&xi;}(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3,c_4)/c_1}$ During establishment, there is following formula to set up

${\stackrel{\&CenterDot;}{V}}_4<-c_1{\left|\right|e\left|\right|}^2+{\mathrm{\&xi;}}_1^2\left(\mathrm{\&kappa;}\right)\mathrm{\&xi;}(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3,c_4)\&le;0---\left(4.18\right)$

Therefore, error vector finally can enter and remain in following error bounds

$\mathrm{\&Omega;}=\left\{e\right|\left|\right|e\left|\right|<{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)\sqrt{\mathrm{\&xi;}(c_1,{\mathrm{<figure-callout\; id="2"\; label="c"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">c</figure-callout>}}_2,c_3,c_4)/c_1},{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)=\frac{\mathrm{n\&gamma;}\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2J_l\mathrm{\&kappa;}}\}---\left(4.19\right)$

Due to function ξ ₁(κ) be to increase with κ the function that decays to zero, as controller parameter c _i>0(i=1 ... choosing 4) meets c ₁=min (c _i), by selecting suitable model parameter κ, can make error bounds Ω arbitrarily small, and then known tracking error e ₁can be arbitrarily small.

Design process by above-mentioned controller is known, gear clearance model based on newly-built, the control of rough gear clearance is converted into to the form of the smooth model of gear and model error, and then adopts the designed input control torque type (4.13) of contragradience control strategy can guarantee the output angle position signal θ of servo gear transmission system _lfollow the tracks of smooth expectation angle position signal θ with arbitrary accuracy _r, eliminated the impact of gear clearance, make the output torque of driving side to be delivered to smoothly load side, eliminated concussion and noise that the collision of wheel between cog produces, there is very large engineering using value.

(1) parameter selection rules

The parameter to be determined related in embodiment of the present invention has adjustable parameter c in controller ₁, c ₂, c ₃, c ₄with gear clearance model parameter κ, the rule that its parameter is chosen is as follows:

1) controller parameter c _i>0(i=1 ... choosing 4) meets c ₁=min (c _i), by the function of many variables, ask the method for extreme value can make ξ (c in formula (4.19) simultaneously ₁, c ₂, c ₃, c ₄)/c ₁the item minimum is determined c _ivalue, can reduce the requirement to model parameter κ like this under the requirement of equal error precision; Another simple method for parameter configuration is that controller parameter all is set to 1.

2) gear clearance model parameter κ should be at controller parameter c _i>0 determine after, the required precision according to the servo gear transmission system to the load side output angle, finally determine with reference to error precision upper limit formula (4.19).

Embodiment 2

For the advantage based on built smooth gear clearance model is described better, the mathematical description (4.1) for the servo gear transmission system provides state feedback control algorithm design process as follows:

In employing the present invention, the another kind of form of smooth gear clearance mould illustrates the validity of this programme,

$T_{\mathrm{new}}=\mathrm{\&gamma;\&theta;}+\frac{\mathrm{\&gamma;}}{2k}[\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})\right)\right)-\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})\right)\right)]$

The following symbol used in process of shifting onto has conformity in full, and formula (4.1) can be converted into following state space form:

$\stackrel{\&CenterDot;}{x}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\\ {\stackrel{\&CenterDot;}{x}}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ x_4\\ -\frac{1}{J_m}{T}_{\mathrm{new}}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{J_m}\end{array}\right]u\stackrel{\mathrm{\&Delta;}}{=}f\left(x\right)+\mathrm{gu}---\left(5.1\right)$

$y=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]x\stackrel{\mathrm{\&Delta;}}{=}h\left(x\right)=x_1$

Wherein, $f\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ x_4\\ -\frac{1}{J_m}{T}_{\mathrm{new}}\end{array}\right],g\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{J_m}\end{array}\right],h\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]x.$

The following describes and how to pass through state feedback control algorithm design driving side control inputs moment u.

Step 1, by analyzing, has provided a differomorphism transformation of coordinates as follows:

$Z\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ \frac{\mathrm{n\&gamma;}(x_4-n{x}_2)}{J_l}l\left(x_{\mathrm{\&theta;}}\right)\end{array}\right]---\left(5.2\right)$

Wherein, $l\left(x_{\mathrm{\&theta;}}\right)\stackrel{\mathrm{\&Delta;}}{=}1+\left\{\tanh \right[\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})]-\tanh [\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})\left]\right\}/2,$ Can try to achieve

$\frac{\&PartialD;Z}{\&PartialD;x}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ \frac{n}{J_l}\frac{\&PartialD;T_{\mathrm{new}}}{{\&PartialD;x}_1}& 0& \frac{n}{J_l}\frac{{\&PartialD;T}_{\mathrm{new}}}{\&PartialD;x_3}& 0\\ \frac{\&PartialD;z_4}{\&PartialD;x_1}& \frac{\&PartialD;z_4}{\&PartialD;x_2}& \frac{\&PartialD;z_4}{\&PartialD;x_3}& \frac{\&PartialD;z_4}{\&PartialD;x_4}\end{array}\right]---\left(5.3\right)$

And then can try to achieve,

$\left|\frac{\&PartialD;Z}{\&PartialD;x}\right|=\frac{n}{J_l}\frac{{\&PartialD;T}_{\mathrm{new}}}{{\&PartialD;x}_3}\frac{{\&PartialD;z}_4}{{\&PartialD;x}_4}=\frac{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">n</figure-callout>}}^2{\mathrm{\&gamma;}}^2}{J_l^2}{\left(l\left(x_{\mathrm{\&theta;}}\right)\right)}^2>0---\left(5.3\right)$

Known, the transformation of coordinates of giving is an overall differomorphism.

The state transformation of step 2 based on designed, order

formula (5.1) can be converted into

$\stackrel{\&CenterDot;}{Z}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\\ {\stackrel{\&CenterDot;}{z}}_2\\ {\stackrel{\&CenterDot;}{z}}_3\\ {\stackrel{\&CenterDot;}{z}}_4\end{array}\right]=\left[\begin{array}{c}{z}_2\\ z_3\\ z_4\\ L_f^{4}h\left(x\right)+u{L}_g{L}_f^{3}h\left(x\right)\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]Z+\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]v\stackrel{\mathrm{\&Delta;}}{=}\mathrm{AZ}+\mathrm{bv}---\left(5.4\right)$

Wherein, $A\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right],b\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right].$

(5.4) be the system mathematical description of original system (5.1) after transformation of coordinates (5.2) transforms, v is middle controlled quentity controlled variable.

In the middle of step 3 order, controlled quentity controlled variable v gets

v＝HZ+v _r （5.5）

Wherein, H=[h ₁h ₂h ₃h ₄] be the controller parameter vector, it is the Hurwitz battle array that the selection of its value meets (A+bH), $v_r=-h_1{\mathrm{\&theta;}}_r-{\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">h</figure-callout>}}_2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-h_3{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r-h_4{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}$ For the reference control inputs.

Step 4 definition error σ ₁=z ₁-θ _r, ${\mathrm{\&sigma;}}_2=z_2-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r,$ ${\mathrm{\&sigma;}}_3=z_3-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r,$ ${\mathrm{\&sigma;}}_4=z_4-{\mathrm{\&theta;}}_r^{\left(3\right)},$ Error vector is σ=[σ ₁σ ₂σ ₃σ ₄] ^t, according to (5.4)-(5.5), the error dynamics equation is

$\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}=(A+\mathrm{bH})\mathrm{\&sigma;}---\left(5.6\right)$

Because (A+bH) is the Hurwitz battle array, known according to the Linear System Stability criterion, error vector σ is globally asymptotically stable.Be z ₁can asymptotically stable in the large follow the tracks of the position signal θ expected _r, because (5.2) are overall differomorphism, known output signal θ after coordinate inversion _lcan follow the tracks of the position signal θ expected by Asymptotic Stability _r.

By the representation of the middle controlled quentity controlled variable v that designed, can instead release real controlled quentity controlled variable u

$u=\frac{v-L_f^{4}h\left(x\right)}{L_g{L}_f^{3}h\left(x\right)}---\left(5.7\right)$

Wherein,

$L_g{L}_f^{3}h\left(x\right)=\frac{\mathrm{n\&gamma;}}{J_l{J}_m}l\left(x_{\mathrm{\&theta;}}\right)$

$L_f^{4}h\left(x\right)=\frac{2\mathrm{\&kappa;\&gamma;n}{(x_4-n{x}_2)}^2}{J_l}[1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})})}^2-1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})})}^2]-(\frac{n^3\mathrm{\&gamma;}}{{\mathrm{<figure-callout\; id="2"\; label="J\; l"\; filenames="HDA00003878941000021.png"\; state="\{\{state\}\}">J</figure-callout>}}_l^{}}+\frac{\mathrm{n\&gamma;}}{J_l{J}_m})l\left(x_{\mathrm{\&theta;}}\right)T_{\mathrm{new}}.$

Design process by above-mentioned controller is known, and the gear clearance model based on smooth adopts the designed input control torque type (5.7) of state feedback linearization control strategy can guarantee the output angle position signal θ of servo gear transmission system _lfollow the tracks of smooth expectation angle position signal θ with arbitrary accuracy _r, eliminated the impact of gear clearance, make the output torque of driving side to be delivered to smoothly load side, eliminated concussion and noise that the collision of wheel between cog produces, there is very large engineering using value.

The signal flow graph of above-mentioned state feedback control algorithm as shown in Figure 4.

(2) parameter selection rules

The parameter to be determined related in embodiment of the present invention 2 has adjustable parameter H=[h in controller ₁h ₂h ₃h ₄] and gear clearance model parameter κ, the rule that its parameter is chosen is as follows:

1) H=[h ₁h ₂h ₃h ₄] selection to meet (A+bH) for the Hurwitz battle array, can adopt Method of Pole Placement to implement;

2) selection of gear clearance model parameter κ, at first be configured to 0.1.If system positioin tracking error precision meets the demands, configured; If system positioin tracking error precision does not meet the demands, increase the value of κ, until system positioin tracking error precision meets the demands, finish configuration.

Table 1 parameter declaration

It should be noted last that, above embodiment is only unrestricted in order to technological scheme of the present invention to be described.Although with reference to embodiment, the present invention is had been described in detail, those of ordinary skill in the art is to be understood that, technological scheme of the present invention is modified or is equal to replacement, do not break away from the spirit and scope of technical solution of the present invention, it all should be encompassed in the middle of claim scope of the present invention.

Claims (9)

1. a controlling method of eliminating the driving gear gap affects, described method comprises:

Step 101) adopting following formula is that gear clearance is set up smooth model:

$T_{\mathrm{new}}=\mathrm{\&gamma;\&theta;}+\frac{\mathrm{\&gamma;}}{2\mathrm{\&kappa;}}[\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})\right)\right)-\ln \left(\cosh \left(\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})\right)\right)]$

Wherein, θ is the relative rotation position angle signal before and after gear, the spacing that 2 α are gear clearance, the stiffness coefficient that γ is gear, T _newfor the output of smooth model, κ is adjustable parameter;

Step 102) smooth model based on setting up obtains the Obj State spatial description, then obtains the input torque of the driving side of gear train assembly according to the Obj State spatial description;

Step 103) adopt the input torque obtained to control the driving side of driving gear, make the output torque of driving side smoothly be passed to load side, guarantee that the position signal of expectation is followed the tracks of in the position output of driving side with required precision simultaneously.

2. the controlling method of elimination gear clearance impact according to claim 1, is characterized in that, adopts intelligent control algorithm to obtain the Obj State spatial description, and described intelligent control algorithm comprises contragradience control algorithm or state feedback linearization control algorithm.

3. the controlling method of elimination gear clearance impact according to claim 2, is characterized in that, when adopting the contragradience control algorithm, and described step 102) further comprise:

At first, definition status variable x=[x ₁x ₂x ₃x ₄] ^t=[θ _lω _lθ _mω _m] ^t, by introducing smooth gear clearance model, controll plant is converted into to following state space form:

$\stackrel{\&CenterDot;}{x}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\\ {\stackrel{\&CenterDot;}{x}}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}(T_{\mathrm{new}}+E\left(x_{\mathrm{\&theta;}}\right))-\frac{b_l}{J_l}{x}_2\\ x_4\\ \frac{u}{J_m}-\frac{T_{\mathrm{new}}+E\left(x_{\mathrm{\&theta;}}\right)}{J_m}-\frac{b_m}{J_m}{x}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{\mathrm{n\&gamma;}}{J_l}{x}_3-{\mathrm{\&phi;}}_1\left(x\right)+\frac{n}{J_l}E\left(x_{\mathrm{\&theta;}}\right)\\ x_4\\ \frac{u}{J_m}-{\mathrm{\&phi;}}_2\left(x\right)-\frac{E\left(x_{\mathrm{\&theta;}}\right)}{J_m}\end{array}\right]$

Based on above-mentioned object mathematical description, carry out the input torque that the contragradience control algorithm is determined driving side, concrete steps are as follows:

Step 1, the definition output tracking error is: e ₁=θ _r-θ _l=θ _r-x ₁;

Tracking error according to definition is chosen the Lyapunov function, and this functional form is: and then calculate virtual controlling amount x ₂expected value be c ₁for adjustable normal number;

Step 2, definition x ₂error be e ₂=μ ₁-x ₂, according to the error of definition, choose the Lyapunov function

and then calculate virtual controlling amount (n γ/J _l) x ₃expected value be ${\mathrm{\&mu;}}_2=e_1+c_2{e}_2+c_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+{\mathrm{\&phi;}}_1\left(x\right),$ C wherein ₂for adjustable normal number;

Step 3, definition x ₃tracking error be e ₃=μ ₂-(n γ/J _l) x ₃, according to the error of definition, choose the Lyapunov function

and then calculate virtual controlling amount n γ (2+ λ (x _θ)) x ₄/ 2J _lexpected value be ${\mathrm{\&mu;}}_3=e_2+c_3{e}_3+(1+c_1{c}_2){\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r+(c_1+c_2){\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+{\mathrm{\&theta;}}_r^{\left(3\right)}+(b_l/J_l-c_1-c_2)(\mathrm{n\&gamma;}{x}_3/J_l-{\mathrm{\&phi;}}_1\left(x\right))+[n^2\mathrm{\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))/2J_l-c_1{c}_2-1]x_2$ , c ₃for adjustable normal number;

Step 4, definition x ₄tracking error be e ₄=μ ₃-n γ (2+ λ (x _θ)) x ₄/ 2J _l, according to the tracking error of definition, choose the Lyapunov function

according to x defined above ₁, x ₂, x ₃and x ₄value and resolve based on the Lyapunov stability principle input torque u that obtains driving side and be:

$\begin{array}{c}u=\frac{2J_l{J}_m}{\mathrm{n\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))}\{e_3+c_4{e}_4+b_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+b_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+b_3{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}-\frac{2\mathrm{\&kappa;n\&gamma;}{(x_4-n{x}_2)}^2}{J_l}\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)+\\ [b_4+\frac{n^2\mathrm{\&gamma;}}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))](\frac{\mathrm{n\&gamma;}{x}_3}{J_l}-{\mathrm{\&phi;}}_1\left(x\right))-[\frac{b_5\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}-\frac{\mathrm{n\&gamma;}{\mathrm{\&phi;}}_2\left(x\right)}{2J_l}](2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))\}\end{array},$

Wherein, c ₄for adjustable normal number; J _mmean the driving side rotary inertia, J _lmean the load side rotary inertia, n means the reduction speed ratio of retarder, and γ means gear stiffness coefficient, x _θmean that the gear front and back relatively rotate the state variable of angular displacement; λ (x _θ), τ (x _θ) the expression SQL; b ₁, b ₂, b ₃, b ₄, b ₅mean custom parameter;

mean expectation angle position signal θ _rthe single order differential signal; mean θ _rthe second-order differential signal;

mean θ _rthree rank differential signals;

mean θ _rthe quadravalence differential signal; φ ₁and φ (x) ₂(x) all mean SQL; x ₁, x ₂, x ₃, x ₄mean respectively θ _l, ω _l, θ _m, ω _mstate variable.

4. the controlling method of elimination gear clearance impact according to claim 3, is characterized in that,

The value strategy of described control parameter comprises:

Strategy one, will control parameter and all be set to 1; Or

Strategy two, controller parameter c _i>0 and meet c ₁=min (c _i), and ask the method for extreme value to make " ξ (c in following formula by the function of many variables ₁, c ₂, c ₃, c ₄)/c ₁" minimum and then definite c _ivalue, wherein, i=1 ... 4;

$\mathrm{\&Omega;}=\left\{e\right|\left|\right|e\left|\right|<{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)\sqrt{\mathrm{\&xi;}(c_1,c_2,c_3,c_4)/c_1},{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)=\frac{\mathrm{n\&gamma;}\ln 2}{2J_l\mathrm{\&kappa;}}\}$

Described model parameter κ should be at controller parameter c _i>0 determine after, the required precision according to the servo gear transmission system to the load side output angle, determine according to the error precision upper limit.

5. the controlling method of elimination gear clearance impact according to claim 2, is characterized in that, when adopting the state feedback linearization control algorithm, and described step 102) further comprise:

At first, definition status variable x is converted into following state space form by introducing smooth gear clearance model by controll plant:

$\stackrel{\&CenterDot;}{x}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\\ {\stackrel{\&CenterDot;}{x}}_2\\ {\stackrel{\&CenterDot;}{x}}_3\\ {\stackrel{\&CenterDot;}{x}}_4\end{array}\right]=\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ x_4\\ -\frac{1}{J_m}{T}_{\mathrm{new}}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{J_m}\end{array}\right]u\stackrel{\mathrm{\&Delta;}}{=}f\left(x\right)+\mathrm{gu}$

$y=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]x\stackrel{\mathrm{\&Delta;}}{=}h\left(x\right)=x_1---\left(5.1\right)$

Wherein, $f\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}{x}_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ x_4\\ -\frac{1}{J_m}{T}_{\mathrm{new}}\end{array}\right],g\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{J_m}\end{array}\right],h\left(x\right)\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]x;$

Based on above-mentioned object mathematical description, carry out the input torque that the state feedback linearization control algorithm is determined driving side, concrete steps are as follows:

Step 1), by analyzing, provides a differomorphism transformation of coordinates as follows:

$Z\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_2\\ \frac{n}{J_l}{T}_{\mathrm{new}}\\ \frac{\mathrm{n\&gamma;}(x_4-n{x}_2)}{J_l}l\left(x_{\mathrm{\&theta;}}\right)\end{array}\right]---\left(5.2\right)$

Wherein, $l\left(x_{\mathrm{\&theta;}}\right)\stackrel{\mathrm{\&Delta;}}{=}1+\left\{\tanh \right[\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})]-\tanh [\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})\left]\right\}/2,$ Can try to achieve

$\frac{\&PartialD;Z}{\&PartialD;x}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ \frac{n}{J_l}\frac{\&PartialD;T_{\mathrm{new}}}{{\&PartialD;x}_1}& 0& \frac{n}{J_l}\frac{{\&PartialD;T}_{\mathrm{new}}}{\&PartialD;x_3}& 0\\ \frac{\&PartialD;z_4}{\&PartialD;x_1}& \frac{\&PartialD;z_4}{\&PartialD;x_2}& \frac{\&PartialD;z_4}{\&PartialD;x_3}& \frac{\&PartialD;z_4}{\&PartialD;x_4}\end{array}\right]---\left(5.3\right)$

And then can try to achieve,

$\left|\frac{\&PartialD;Z}{\&PartialD;x}\right|=\frac{n}{J_l}\frac{{\&PartialD;T}_{\mathrm{new}}}{{\&PartialD;x}_3}\frac{{\&PartialD;z}_4}{{\&PartialD;x}_4}=\frac{n^2{\mathrm{\&gamma;}}^2}{J_l^2}{\left(l\left(x_{\mathrm{\&theta;}}\right)\right)}^2>0---\left(5.3\right)$

Known, the transformation of coordinates of giving is an overall differomorphism;

Step 2) state transformation based on designed, order

formula (5.1) can be converted into

$\stackrel{\&CenterDot;}{Z}=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{z}}_1\\ {\stackrel{\&CenterDot;}{z}}_2\\ {\stackrel{\&CenterDot;}{z}}_3\\ {\stackrel{\&CenterDot;}{z}}_4\end{array}\right]=\left[\begin{array}{c}{z}_2\\ z_3\\ z_4\\ L_f^{4}h\left(x\right)+u{L}_g{L}_f^{3}h\left(x\right)\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]Z+\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]v\stackrel{\mathrm{\&Delta;}}{=}\mathrm{AZ}+\mathrm{bv}---\left(5.4\right)$

Wherein, $A\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right],b\stackrel{\mathrm{\&Delta;}}{=}\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right].$

(5.4) be the system mathematical description of original system (5.1) after transformation of coordinates (5.2) transforms, v is middle controlled quentity controlled variable;

In the middle of the step 3) order, controlled quentity controlled variable v gets:

v＝HZ+v _r （5.5）

Wherein, H=[h ₁h ₂h ₃h ₄] be the controller parameter vector, it is the Hurwitz battle array that the selection of its value meets (A+bH), $v_r=-h_1{\mathrm{\&theta;}}_r-h_2{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-h_3{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r-h_4{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}$ For the reference control inputs;

Step 4) definition error σ ₁=z ₁-θ _r, ${\mathrm{\&sigma;}}_2=z_2-{\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r,$ ${\mathrm{\&sigma;}}_3=z_3-{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r,$ ${\mathrm{\&sigma;}}_4=z_4-{\mathrm{\&theta;}}_r^{\left(3\right)},$ Error vector is σ=[σ ₁σ ₂σ ₃σ ₄] ^t, according to (5.4)-(5.5), the error dynamics equation is

$\stackrel{\&CenterDot;}{\mathrm{\&sigma;}}=(A+\mathrm{bH})\mathrm{\&sigma;}---\left(5.6\right)$

Because (A+bH) is the Hurwitz battle array, known according to the Linear System Stability criterion, error vector σ is globally asymptotically stable, i.e. z ₁can asymptotically stable in the large follow the tracks of the position signal θ expected _r, because (5.2) are overall differomorphism, known output signal θ after coordinate inversion _lcan follow the tracks of the position signal θ expected by Asymptotic Stability _r;

By the representation of middle controlled quentity controlled variable v of design, the anti-input torque u that pushes away driving side:

$u=\frac{v-L_f^{4}h\left(x\right)}{L_g{L}_f^{3}h\left(x\right)}$

（5.7）

Wherein,

$L_g{L}_f^{3}h\left(x\right)=\frac{\mathrm{n\&gamma;}}{J_l{J}_m}l\left(x_{\mathrm{\&theta;}}\right)$

$L_f^{4}h\left(x\right)=\frac{2\mathrm{\&kappa;\&gamma;n}{(x_4-n{x}_2)}^2}{J_l}[1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}-\mathrm{\&alpha;})})}^2-1/{(e^{\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(x_{\mathrm{\&theta;}}+\mathrm{\&alpha;})})}^2]-(\frac{n^3\mathrm{\&gamma;}}{J_l^2}+\frac{\mathrm{n\&gamma;}}{J_l{J}_m})l\left(x_{\mathrm{\&theta;}}\right)T_{\mathrm{new}}.$

6. eliminate the control system that gear clearance affects for one kind, described system comprises:

Smooth model is set up module, for adopting following formula, is that gear clearance is set up smooth model:

$T_{\mathrm{new}}=\mathrm{\&gamma;}[\mathrm{\&theta;}+\frac{1}{2\mathrm{\&kappa;}}\ln \left(\frac{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}-\mathrm{\&alpha;})}}{e^{\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}+e^{-\mathrm{\&kappa;}(\mathrm{\&theta;}+\mathrm{\&alpha;})}}\right)]$

Wherein, θ is the relative rotation position angle signal before and after gear, the spacing that 2 α are gear clearance, the stiffness coefficient that γ is gear, T _newfor the output of smooth model, κ is adjustable parameter;

Driving side input torque determination module, introduce intelligent control algorithm for the smooth model by setting up, and obtains the input torque of the driving side of gear train assembly; With

Control module, the input torque obtained for employing is controlled driving gear, makes the output torque of driving side smoothly be passed to load side.

7. the control system of elimination gear clearance impact according to claim 6, is characterized in that, described intelligent control algorithm comprises contragradience control algorithm or state feedback linearization control algorithm.

8. the control system of elimination gear clearance impact according to claim 6, is characterized in that, described driving side input torque determination module further comprises:

Comparison module, compare judgement for the position by desired output and actual outgoing position, the output angle position error signal;

Obj State spatial description acquisition of information module, processed for the smooth model of the gear clearance to setting up and angle positional deviation, model bias and the smooth gap model information of comparison module input, finally exported the Obj State spatial description with smooth gear clearance and model error;

The first puocessing module, processing is controlled in the contragradience of carrying out based on the Lyapunov stability principle for the spatial description of the Obj State with smooth gear clearance and the model error information by described Obj State spatial description acquisition of information module output, finally exports the second virtual controlling amount x ₂;

The second puocessing module, for the second virtual controlling amount x by the first puocessing module output ₂processing is controlled in the contragradience that information is carried out based on the Lyapunov stability principle, and then exports the 3rd virtual controlling amount x ₃;

The 3rd puocessing module, for the 3rd virtual controlling amount x by the second puocessing module output ₃processing is controlled in the contragradience that information is carried out based on the Lyapunov stability principle, and then exports the 4th virtual controlling amount x ₄; With

The manages module everywhere, for the 4th virtual controlling amount x based on the 3rd puocessing module output ₄information adopts following formula to obtain the input torque u of driving side: $\begin{array}{c}u=\frac{2J_l{J}_m}{\mathrm{n\&gamma;}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))}\{e_3+c_4{e}_4+b_1({\stackrel{\&CenterDot;}{\mathrm{\&theta;}}}_r-x_2)+b_2{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&theta;}}}_r+b_3{\mathrm{\&theta;}}_r^{\left(3\right)}+{\mathrm{\&theta;}}_r^{\left(4\right)}-\frac{2\mathrm{\&kappa;n\&gamma;}{(x_4-n{x}_2)}^2}{J_l}\mathrm{\&tau;}\left(x_{\mathrm{\&theta;}}\right)+\\ [b_4+\frac{n^2\mathrm{\&gamma;}}{2J_l}(2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))](\frac{\mathrm{n\&gamma;}{x}_3}{J_l}-{\mathrm{\&phi;}}_1\left(x\right))-[\frac{b_5\mathrm{n\&gamma;}(x_4-n{x}_2)}{2J_l}-\frac{\mathrm{n\&gamma;}{\mathrm{\&phi;}}_2\left(x\right)}{2J_l}](2+\mathrm{\&lambda;}\left(x_{\mathrm{\&theta;}}\right))\}\end{array}.$

9. the control system that elimination gear clearance according to claim 8 affects, it is characterized in that, described driving side input torque determination module also comprises the parameter determination module, and described parameter determination module determines according to following strategy the value of controlling parameter and model parameter:

Strategy one, will control parameter and all be set to 1; Or

Strategy two, controller parameter c _i>0 and meet c ₁=min (c _i), and ask the method for extreme value to make " ξ (c in following formula by the function of many variables ₁, c ₂, c ₃, c ₄)/c ₁" minimum and then definite c _ivalue, wherein, i=1 ... 4;

$\mathrm{\&Omega;}=\left\{e\right|\left|\right|e\left|\right|<{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)\sqrt{\mathrm{\&xi;}(c_1,c_2,c_3,c_4)/c_1},{\mathrm{\&xi;}}_1\left(\mathrm{\&kappa;}\right)=\frac{\mathrm{n\&gamma;}\ln 2}{2J_l\mathrm{\&kappa;}}\}$

Described model parameter κ should be at controller parameter c _i>0 determine after, the required precision according to the servo gear transmission system to the load side output angle, determine according to the error precision upper limit.

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