JP2004029887A - Position commanding method of position controller and method for measuring natural vibration angular frequency of machine base - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To solve problems in which since a reaction force of a positioned and controlled driving mechanism is applied to a machine base where the driving mechanism is installed, high-precision positioning can not be performed and a positioning settlement becomes long. <P>SOLUTION: In a position commanding method of a position controller that drives a load 3 to follow up a position command x<SB>r</SB>while the driving mechanism including the load 3 is installed on the machine base 4, vibrating machine base displacement X<SB>B</SB>is represented as an arbitrary constant polynomial of degree (n) and the acceleration of the load is represented as a known constant coefficient differential equation of second order based upon the machine base displacement X<SB>B</SB>; and the acceleration of the load is integrated twice to represent the load position as a polynomial of degree (n+2) and simultaneous equations generated from the two polynomials and border conditions including the load positions at the start and end of a position command are solved to determine as a position command x<SB>r</SB>a polynomial of degree (n+2) based upon the time (t) of the load position (x). <P>COPYRIGHT: (C)2004,JPO

Description

ただし、Ａｒは加速度指令、Ｆは駆動方向に換算したモータの駆動力、ｘ_Ｂは機台の変位である。
ここで、機台の固有振動角周波数をω_Ｂ、機台振動の減衰係数をζ_Ｂとすると、
Ｄ_Ｂ／Ｍ_Ｂ＝２ζ_Ｂω_Ｂ、Ｋ_Ｂ／Ｍ_Ｂ＝ω_Ｂ ^２…（２）
が成り立つので、式（１）は
【数２】

となる。
ただし、ｘ_ＢＢは機台変位ｘ_Ｂを駆動機構側に換算した変位（以下、機台の換算変位とする）であり、ｘ_Ｂとは以下のような関係となる。
ｘ_ＢＢ＝　−Ｍ_Ｂｘ_Ｂ／Ｍ　　…（４）
一般に、駆動系の加減速時のことを考察する場合は、機台の質量Ｍ_Ｂが駆動機構可動部の等価質量Ｍより遥かに大きいため、機台の変位ｘ_Ｂが駆動機構可動部の変位ｘより遥かに小さいので、図１を近似的に図２のように書き直すことができる。この図２から、位置指令ｘ_ｒから駆動機構可動部の変位ｘまでの伝達関数を求めると、ｘ（ｓ）／ｘ_ｒ（ｓ）＝１となる。ｘ_ｒの初期値をｘの初期値と同じ値にすると、ｘ（ｔ）＝ｘ_ｒ（ｔ）が成り立つ。また、
【数３】

が明らかであり、式（３）より、
【数４】

となる。上式を２回積分すると位置指令は、
【数５】

となる。ただし、ｃ１とｃ２は任意定係数である。この式から機台変位をｎ次多項式で表すものとすれば位置指令は（ｎ＋２）次多項式で表すものとなる。
位置指令の多項式を決定する際に、境界条件を用いることが必要であるため、以下に境界条件について説明する。
一般に指令開始前、機台が振動しないことは容易に満足される。また、この発明の目的は指令終了後も振動機台が振動しないようにすることである。そして、機台の速度および変位は任意時刻においても連続でなければならないので、機台変位に関する境界条件を、
【数６】

のように与える。ただし、ｔｓは指令開始時刻、ｔｅは指令終了時刻である。
また、モータの駆動力が有限であるため、負荷の速度および位置が必ず連続で変化する。一方、負荷の位置を位置指令に追従させるため、位置指令および位置指令の速度パターンがすべての時刻においても連続であるように指令を構成する必要がある。よって、負荷を初期位置Ｘｓから終端位置Ｘｅまで移動させるため、位置指令に関する基本的な境界条件を、
【数７】

のように与える。
そして、高周波振動を刺激しないことや機械に衝撃を与えないことなどの理由で、力指令（すなわち、加速度指令）も連続で変化することが要求される。この場合では、位置指令に対してもう一つ境界条件を次式のように与える。
【数８】

以下、ω_Ｂとζ_Ｂが既知であり、ｔｓ、ｔｅ、Ｘｓ、Ｘｅが指定された場合において、本発明の第１の実施例である位置制御装置の位置指令方法を説明する。
まず、機台変位の時間ｔによる関数を指定する。
指令開始前および指令終了後の機台変位を０とし、指令期間の機台変位をｎ次多項式で表すものとし、すなわち、機台変位の時間関数を
ｘ_ＢＢ（ａ，ｔ）＝ａ_ｎｔ^ｎ＋ａ_ｎ−１ｔ^ｎ−１＋…＋ａ_１ｔ＋ａ_０、（ｔｓ＜ｔ＜ｔｅ）
ｘ_ＢＢ（ａ，ｔ）＝０、（ｔ≦ｔｓ、ｔｅ≦ｔ）　…（１０）
としておく。ただし、ａ＝［ａ_ｎ、ａ_ｎ−１、…、ａ_１，ａ_０］は後述するように境界条件により定める定係数ベクトルである。また、ｎは境界条件の数によって定める自然数である。上式および式（６）より、ｘ_ＢＢ（ｔ）およびｘ_ｒ（ｔ）に含まれる任意定係数は（ｎ＋３）個あるので、境界条件の数がｍであるとしたら、すべての境界条件を満たすため、ｎ＋３≧ｍ、すなわち、
ｎ≧ｍ−３　　…（１１）
が成り立つようにしなければならない。
次に、位置指令の関数を導出する。
式（１０）を式（５）に代入すると、位置指令の加速度パターンは、
【数９】

となる。上式を積分すると、位置指令の速度パターンは、
【数１０】

となる。さらに上式を積分すると、位置指令は、
【数１１】

となる。
最後に、境界条件を用いてａ、ｃ１およびｃ２を求め、位置指令を定める。
式（１０）および式（１２）〜（１４）を式（７）および式（８）の境界条件に適用し、次式を得る。
【数１２】

また、加速度指令が連続変化であることが要求される場合は、式（１２）を式（９）の境界条件に適用すると、次式が得られる。
【数１３】

明らかに式（１５）と式（１６）はａ、ｃ１およびｃ２の線形方程式である。式（１１）より、未知数の数は方程式の数以上あるため、連立方程式の解が必ず存在する。これらの連立方程式を解くと、ａ、ｃ１およびｃ２を求められる。ａ、ｃ１およびｃ２の値を式（１４）に代入すると、位置指令ｘ_ｒの時間関数が定まる。
尚、ａの値を式（１５）の第２式および第４式に代入すると、
【数１４】

が得られる。
従って、位置指令終了時ｔｅにおける機台変位、機台速度のいずれもが零となる。また、図１の制御系ではフィードフォワードが用いられたため負荷位置がほぼ位置指令に追従することから、位置指令終了後駆動力がほぼ０となり、すなわち、駆動機構が機台に与える力も０となる。よって、位置指令終了後の機台振動は発生しないといえる。
このように、上記により求めた位置指令ｘ_ｒを位置制御装置の位置指令とすることで、位置指令終了後の機台の振動を発生せずに負荷の位置決め制御を行うことが可能となる。
次に本発明の第２の実施例について説明する。
まず、機台の換算変位ｘ_ＢＢの時間関数を、
ｘ_ＢＢ（ｂ，ｔ）＝（ｂ_ｌｔ^ｌ＋　ｂ_ｌ−１ｔ^ｌ−１＋　…　＋　ｂ_１ｔ　＋　ｂ_０）（ｔ　−　ｔｓ）^ｐ（ｔ　−　ｔｅ）^ｑ、（ｔｓ　＜ｔ　＜　ｔｅ）ｘ_ＢＢ（ｂ，ｔ）＝　０　　　、（ｔ≦ｔｓ　，　ｔｅ≦ｔ）　　…（１７）
としておく。ただし、ｂ＝［ｂ_ｌ，　ｂ_ｌ−１，　…，　ｂ_１，　ｂ_０］は後述するように境界条件により定める定係数ベクトルである。また、ｌ、ｐおよびｑは境界条件の数により定める自然数である。一般に、
ｌ≧１、ｐ≧２、ｑ≧２　　…（１８）
とするが、加速度指令が連続で変化する要求がある場合は、
ｌ≧１、ｐ≧３、ｑ≧３　　…（１９）
とする。
次に、位置指令の関数を導出する。
式（５）より、位置指令の加速度パターンは
【数１５】

となる。上式を積分すると、位置指令の速度パターンは、
【数１６】

となる。さらにこの式を積分すると、位置指令は、
【数１７】

となる。
最後に、境界条件を用いてｂを求め、位置指令を定める。
式（２１）と式（２２）を境界条件の式（８）の終端条件に適用すると、
【数１８】

が成り立つ。
明らかに上記の連立方程式はｂの２元線形方程式である。式（１８）或いは式（１９）より、ｌ≧１なので、未知数の数は方程式の数以上であり、方程式の解が必ず存在する。この連立方程式を解くと、ｂを求められる。ｂの値を式（２２）に代入すると、位置指令の時間関数が定める。
上述のように求めた位置指令は必ず全ての境界条件を満足することを以下に説明する。
式（１８）より、ｐ≧２、ｑ≧２なので、式（１７）より、
【数１９】

の多項式には（ｔ−ｔｓ）と（ｔ−ｔｅ）の因子が含まれ、式（７）の境界条件
【数２０】（２４）

がｂと関係なく常に満たされる。また、式（１９）より、ｐ≧３、ｑ≧３なので、式（５）より、
【数２１】

の多項式には（ｔ−ｔｓ）と（ｔ−ｔｅ）の因子が含まれ、式（９）の境界条件
【数２２】

もｂと関係なく常に満たされる。
式（１７）、式（２１）および式（２２）より、境界条件の式（８）の初期条件
【数２３】

がｂと関係なく常に満たされる。
式（２３）〜（２６）により、式（７）〜（９）に示されている境界条件が全て満足されていることが分かる。
本発明の第１の実施例では８以上の方程式を解く必要があるが、本発明の第２の実施例では２つの方程式を解くだけで良いので、簡単である。
次に本発明の第３の実施例について、説明する。
前記第１の実施例および第２の実施例では、位置指令を求める際に機台の固有振動角周波数ω_Ｂおよび機台振動の減衰係数ζ_Ｂを知っておく必要がある。一般に、これらのパラメータを同定するのは機台にパルスの外力を加えて機台の変位の波形から求める方法があるが、レーザ変位計を取り付ける必要がある。ここでは、レーザ変位計を取り付ける必要がなく、制御装置の位置偏差から求める方法を以下に説明する。
位置偏差は位置指令ｘ_ｒと位置検出器によるフィードバック位置信号ｘとの差である。また、位置検出器は可動部と固定部で構成され、検出した信号は可動部と固定部の相対変位である。一般に、位置検出器の可動部は駆動機構の可動部に、位置検出器の固定部は機台または機台に固定される機構に取り付けられる。従って、位置検出器からの信号ｘは駆動機構の可動部と機台との相対変位である。駆動機構の可動部の絶対変位をｘ_Ｌとし、機台の絶対変位がｘ_Ｂであることから、
【数２４】

が得られる。
一方、位置指令終了後、位置制御部７および速度制御部８のゲインを十分低く設定（例えば０．１以下）した場合は、フィードフォワード信号が０となり、フィードバック信号も小さくなるので、加速度指令Ａｒおよび駆動力Ｆがほぼ０になり、駆動機構の可動部の絶対変位ｘ_Ｌもほぼ０になる。よって位置指令終了後の位置偏差ｅは、
ｅ（ｔ）＝ｘ_ｒ（ｔ）−｛ｘ_Ｌ（ｔ）　−ｘ_Ｂ（ｔ）｝≒ｘ_Ｂ（ｔ）　　、（ｔ＞ｔｅ）　　　…（２７）
となる。すなわち、指令終了後の機台の振動は図６に示すように、そのまま位置偏差に現れる。
よって、
Ｔｅ≒Ｔ_Ｂ　　　　　…（２８）
および
λｅ＝ｅ_ｐｐ２／ｅ_ｐｐ１≒λ_Ｂ＝ｘ_Ｂｐｐ２／ｘ_Ｂｐｐ１　　　　　…（２９）
が成り立つ。ただし、Ｔｅは指令終了後の位置偏差の振動成分の隣接する２つのピーク（或いは谷）間の時間、Ｔ_Ｂは機台の固有振動周期である。また、λｅは指令終了後の位置偏差の振動振幅減衰比、λ_Ｂは機台振動の振幅減衰比である。
上記のように、Ｔｅとλｅは容易に求められる。これらを用いてω_Ｂとζ_Ｂを以下のように与える。
ω_Ｂ＝２π／Ｔ_Ｂ≒２π／Ｔｅ　　…（３０）
ζ_Ｂ≒−（ｌｎλ_Ｂ）／２π≒−（ｌｎλｅ）／２π　　…（３１）
次に、本発明の効果を数値例を用いて説明する。
まず、機台の固有振動角周波数ω_Ｂおよび機台振動の減衰係数ζ_Ｂを求める。
図６より、Ｔｅ＝０．０６７３ｓ，λｅ＝０．７が得られ、　式（３０）および式（３１）より、
ω_Ｂ≒２π／Ｔｅ＝９３．３（ｒａｄ／ｓ）　，ζ_Ｂ≒−（ｌｎλ_Ｂ）／２π≒０．０５６８　　　　…（３２）
次に、機台の換算変位ｘ_ＢＢを時間ｔの関数として表す。
ｌ＝１，　ｐ＝２，　ｑ＝２，　ｔｓ＝０．０５ｓ，　ｔｅ＝０．０９ｓとし、式（１７）より、機台の換算変位を
ｘ_ＢＢ（ｂ，ｔ）＝（ｂ_１ｔ　＋　ｂ_０）（ｔ−０．０５）^２（ｔ−０．１４）^２、（０．０５　＜　ｔ　＜　０．１４）
ｘ_ＢＢ（ｂ，ｔ）＝０　、（ｔ　≦　０．０５　、０．１４　≦　ｔ）　　　　　　…（３３）
としておく。
次に、位置指令の関数を導出する。
Ｘｓ＝０とし、式（３２）と式（３３）を式（２１）および式（２２）に代入すると、
【数２５】

および
【数２６】

となる。
最後に、境界条件を用いてｂを求め、位置指令を定める。
Ｘｅ＝６０００μｍとし、式（２３）より、
【数２７】

となる。
上の方程式を解くと、ｂ_１＝−１．２１×１０^１０，　ｂ_０＝５．４４７×１０^８となり、ｂ_１、ｂ_０の値を式（３５）に代入すると、位置指令の時間関数は、
【数２８】

となる。
このような指令を用いた場合のシミュレーション結果を示したのが図５である。一方で、従来方式の位置指令としての三角形指令を同じ指令時間で用いた場合のシミュレーション結果を示したのが、図４である。
図４と図５を比較すると、本発明の方法による位置指令を用いた場合は位置指令終了後の機台振動および位置偏差がいずれも小さく抑えられていることが分かる。
【発明の効果】
以上述べたように本発明によれば、駆動機構が設置された機台の剛性が低い場合でも、また機台の固有振動周期の長短にも関わりなく、さらには移動量の大小にも関わりなく、位置指令終了後の機台の振動を抑制して、高速かつ短時間で高精度な位置決め制御ができるという効果がある。
【図面の簡単な説明】
【図１】機台の振動系を含む本発明の対象となる位置決め制御系のブロック図
【図２】駆動系の加減速時のことを考察する場合の図１の近似等価ブロック図
【図３】本発明を適用した機械系の概略図
【図４】従来指令方式である三角形指令を用いた場合のシミュレーション結果を示したもの
【図５】本発明の位置指令方式を用いた場合のシミュレーション結果を示したもの
【図６】本発明の第３の実施例として制御部のゲインを低く設定した場合のシミュレーション結果を示したもの
【符号の説明】
１　モータ
２　力伝達機構
３　負荷
４　機台
５　機台の支持足
６　位置指令生成部
７　位置制御部
８　速度制御部
９、１０　微分
１１　ゲイン
１２　モータを含む機械系のモデルTECHNICAL FIELD OF THE INVENTION
The present invention relates to a command method of a position control device that performs positioning while suppressing low-frequency residual vibration of a machine on which a drive mechanism is installed, and a method of measuring a natural vibration angular frequency and the like of the machine.
[Prior art]
In recent years, industrial machines are required to have higher speed and higher accuracy. When a motor drives a load at high acceleration due to a demand for high speed, a machine on which a driving mechanism is installed vibrates due to a large reaction force, which adversely affects a servo control system. In particular, when performing positioning control, there is a problem that high-precision control cannot be performed due to the presence of large residual vibration.
In order to solve such a problem, in the prior art, as disclosed in Japanese Patent Application Laid-Open No. 05-108165, the acceleration pattern of the position command is a rectangular wave command, and the acceleration time of the rectangular wave is an integral multiple of the natural vibration period of the machine. By doing so, the residual vibration of the machine base is suppressed.
[Problems to be solved by the invention]
By the way, in the above-mentioned conventional technology, since the speed command time needs to be an integral multiple of the natural vibration cycle of the machine, when the natural vibration cycle of the machine is long, a speed command having a low speed and a long command time is set. However, this method is not suitable for high-speed positioning control, and in particular, when the feed distance is short, there is a problem that the adverse effect is remarkably exhibited.
Accordingly, the present invention provides a command method for a positioning control control device which solves the problems of the prior art, suppresses the residual vibration of the machine base, and furthermore, has a short speed command time and therefore can perform high-speed positioning control. It is intended to do so.
[Means for Solving the Problems]
In order to solve the above problem, the present invention according to claim 1 is a position command method of a position control device in which a drive mechanism including a load is installed on a machine base and drives the load to follow a position command.
A natural number n satisfying n ≧ (boundary condition number−3) is set, and the machine displacement is represented by an arbitrary constant coefficient n-order polynomial with time t, and the acceleration of the load is calculated by a known constant coefficient 2 based on the machine displacement. The load position is represented by an (n + 2) -th order polynomial with time t by expressing the load acceleration by integrating the acceleration of the load twice and expressing the load position at the start of the position command and the load position at the end of the position command. A boundary equation including the n-th order polynomial of the machine displacement, the (n + 2) -order polynomial of the load position, and the boundary condition, and creating any simultaneous equation that satisfies the simultaneous equation The constant value of the constant coefficient is determined, the (n + 2) -order polynomial based on the load position time t is determined based on the determined constant value, and the (n + 2) -order polynomial based on the determined load position time t is used as the position command. Is characterized by There.
The reason for setting n ≧ (the number of boundary conditions−3) is as follows.
Since the machine displacement is an n-th order polynomial with an arbitrary constant coefficient according to time t, there are (n + 1) arbitrary constant coefficients whose numerical values are undetermined in the n-th order polynomial. Further, since the load position is obtained by twice integrating a known constant coefficient second-order differential equation due to the machine displacement, two arbitrary constant coefficients are generated in the process. Therefore, there are (n + 3) arbitrary constant coefficients, and the relationship of (n + 3) ≧ the number of boundary conditions is required in order to determine all the constant values of the arbitrary constant coefficients.
According to the present invention as set forth in claim 2, in a position command method of a position control device in which a drive mechanism including a load is installed on a machine base and drives the load to follow a position command,
Where l ≧ 1, p ≧ 2, and q ≧ 2, ts is a position command start time, te is a position command end time, b ₁ , b ₁ ,..., B ₂ , b ₁ , and b ₀ are As an arbitrary constant coefficient, by time t,
Machine frame displacement ^{_{= (b l t l + b}} l-1 t l-1 + ... + b 2 t 2 + b 1 t + b 0) (t - ts) p (t - te) q
Where the acceleration of the load is represented by a known constant coefficient second order differential equation based on the displacement of the machine, the acceleration of the load is integrated twice, and the load position is represented by a function based on time t. Set the boundary conditions including the load position of
A simultaneous equation is created using the function of the machine displacement, the function of the load position, and the boundary condition, and constant values of all arbitrary constant coefficients satisfying the simultaneous equation are determined. A function based on the time t of the load position is determined using a numerical value, and the function based on the determined time t of the load position is used as a position command.
The boundary condition that makes the machine displacement and the machine speed at time ts and te both zero is reflected in the machine displacement function, the number of arbitrary constant coefficients and the number of simultaneous equations are reduced, and the position command is changed. This reduces the computational burden until it is obtained.
If p ≧ 3 and q ≧ 3, it is possible to reflect the boundary condition where the machine accelerations at times ts and te are both zero.
Further, in the present invention according to claim 3, a drive mechanism including a load is installed on the machine base, and each frequency and the vibration damping coefficient of the natural vibration of the machine are controlled by a position control device that causes the load to follow a position command. In the measuring method to measure,
The feedback gain of the position control device is set low, and a position command is output,
Observe the fluctuation of the position deviation after the end of the position command, calculate the natural vibration angular frequency of the machine from the natural vibration cycle with the time between adjacent vibration peaks observed as the natural vibration cycle of the machine. The damping rate is calculated by comparing the peak values of the adjacent vibrations observed as described above with magnitude, and this is used as the vibration damping coefficient of the machine.
By setting the feedback gain of the position control device low after the end of the position command to reduce the driving force to the load, the movement of the load inside the driving mechanism is suppressed, and a situation in which only the machine fluctuation is remarkably created, Thus, the movement of the machine displacement can be approximately grasped from the position deviation.
DETAILED DESCRIPTION OF THE INVENTION
An embodiment of the present invention will be described with reference to the drawings. FIG. 3 is a schematic diagram of a mechanical system to which the present invention is applied. In FIG. 3, when the motor 1 drives the load 3 via the force transmission mechanism 2, a fixed portion of the motor 1 and the load 3 receives a reaction force of the driving force, and the reaction force is transmitted to the machine base 4. When the rigidity of the supporting feet 5 supporting the machine base 4 is low, the motor 1 drives the load 3 at a high acceleration / deceleration speed, and particularly when the time when the driving force is applied is close to the natural vibration cycle of the machine base 4, The table 4 vibrates greatly.
FIG. 1 is a block diagram of a control system to which the present invention is applied, assuming that a drive mechanism from a motor 1 to a load 3 is a rigid body. In FIG. 1, 6 is a position command generator, 7 is a position controller, 8 is a speed controller, and 12 is a model of a mechanical system including a motor. M is equivalent mass of the machine moving part in the drive mechanism in terms of the driving direction, M _B is the machine base of the mass, K _B is the machine base of the spring constant, D _B is the viscous friction coefficient of the machine base. The following relationship is obtained from FIG.
(Equation 1)

However, Ar is acceleration command, F is the driving force of the motor in terms of the driving direction, x _B is the machine base displacement.
Here, assuming that the natural vibration angular frequency of the machine base is ω _B and the damping coefficient of the machine vibration is _{Ｂ B} ,
_{D B / M B = 2ζ B} ω B, K B / M B = ω B 2 ... (2)
Holds, Equation (1) becomes

It becomes.
However, displacement x _BB obtained by converting the machine base displacement x _B on the driving mechanism side (hereinafter, the machine base Conversion displacement), and the following as related to x _{B.
x BB = -M B x B /} M ... (4)
In general, when considering that during acceleration or deceleration of the drive system, since the machine stand of the mass M _B much larger than the equivalent mass M of the drive mechanism moving parts, the machine stand of the displacement x _B is the displacement of the driving mechanism moving unit Since it is much smaller than x, FIG. 1 can be rewritten approximately as in FIG. From FIG. 2, when a transfer function from the position command _xr to the displacement x of the drive mechanism movable portion is obtained, x (s) / _xr (s) = 1. _{Assuming that} the initial value of xr is the same as the initial value of x, x (t) = _xr (t) holds. Also,
[Equation 3]

Is apparent, and from equation (3),
(Equation 4)

It becomes. By integrating the above equation twice, the position command becomes
(Equation 5)

It becomes. Here, c1 and c2 are arbitrary constant coefficients. If the machine displacement is expressed by an nth-order polynomial from this equation, the position command is expressed by an (n + 2) th-order polynomial.
Since it is necessary to use boundary conditions when determining the position command polynomial, the boundary conditions will be described below.
Generally, it is easily satisfied that the machine does not vibrate before the command is started. Another object of the present invention is to prevent the vibration stand from vibrating even after the command is completed. And since the speed and displacement of the machine must be continuous even at any time, the boundary condition for the machine displacement is
(Equation 6)

Give like. Here, ts is a command start time and te is a command end time.
Further, since the driving force of the motor is finite, the speed and position of the load always change continuously. On the other hand, in order to make the position of the load follow the position command, it is necessary to configure the command so that the position command and the speed pattern of the position command are continuous at all times. Therefore, in order to move the load from the initial position Xs to the end position Xe, the basic boundary condition for the position command is
(Equation 7)

Give like.
In addition, it is required that the force command (that is, the acceleration command) is also continuously changed because the high frequency vibration is not stimulated or the machine is not shocked. In this case, another boundary condition is given to the position command as in the following equation.
(Equation 8)

Hereinafter, the position command method of the position control device according to the first embodiment of the present invention when ω _B and 、 _B are known and ts, te, Xs, and Xe are designated will be described.
First, a function according to the time t of the machine displacement is specified.
The machine displacement before the command start and after the command end is set to 0, and the machine displacement during the command period is represented by an n-th order polynomial, that is, the time function of the machine displacement is x _BB (a, t) = a _n t ^{_{n + a n-1 t n}} -1 + ... + a 1 t + a 0, (ts <t <te)
x _BB (a, t) = 0, (t ≦ ts, te ≦ t) (10)
And keep it. _{However, a = [a n, a} n-1, ..., a 1, a 0] is a constant coefficient vector determined by the boundary conditions as described below. Further, n is a natural number determined by the number of boundary conditions. From the above equation and equation (6), there are (n + 3) arbitrary constant coefficients included in x _BB (t) and x _r (t). Therefore, if the number of boundary conditions is m, all the boundary conditions are To satisfy, n + 3 ≧ m, ie,
n ≧ m−3 (11)
Must be satisfied.
Next, a function of the position command is derived.
Substituting equation (10) into equation (5), the acceleration pattern of the position command is
(Equation 9)

It becomes. By integrating the above equation, the speed pattern of the position command is
(Equation 10)

It becomes. Further integrating the above equation, the position command is
[Equation 11]

It becomes.
Finally, a, c1, and c2 are obtained using the boundary conditions, and a position command is determined.
Equation (10) and Equations (12) to (14) are applied to the boundary conditions of Equations (7) and (8) to obtain the following equation.
(Equation 12)

When it is required that the acceleration command is a continuous change, the following equation is obtained by applying equation (12) to the boundary condition of equation (9).
(Equation 13)

Clearly, equations (15) and (16) are linear equations of a, c1 and c2. From equation (11), the number of unknowns is equal to or greater than the number of equations, so that a solution to the simultaneous equations always exists. Solving these simultaneous equations gives a, c1, and c2. When the value of a, c1 and c2 into equation (14), determined the time function of the position command _{x r.}
By substituting the value of a into the second and fourth equations of equation (15),
[Equation 14]

Is obtained.
Accordingly, both the machine displacement and the machine speed at the end of the position command te become zero. In addition, in the control system of FIG. 1, since the load position substantially follows the position command because the feedforward is used, the driving force after the end of the position command becomes substantially zero, that is, the force applied by the drive mechanism to the machine base also becomes zero. . Therefore, it can be said that the machine vibration after the end of the position command does not occur.
In this manner, by the position command of the position control device position command x _r obtained by the above, it is possible to perform the positioning control of the load without causing the machine stand of the vibration of the position command completion.
Next, a second embodiment of the present invention will be described.
First, the time function of the machined displacement _xBB is
^{_{x BB (b, t) =}} (b l t l + b l-1 t l-1 + ... + b 1 t + b 0) (t - ts) p (t - te) q,  (Ts <t <te) x _BB (b, t) = 0, (t ≦ ts, te ≦ t) (17)
And keep it. Here, b = [b _l , b _l−1 ,..., B ₁ , b ₀ ] is a constant coefficient vector determined by boundary conditions as described later. Also, l, p and q are natural numbers determined by the number of boundary conditions. In general,
l ≧ 1, p ≧ 2, q ≧ 2 (18)
However, if there is a demand for the acceleration command to change continuously,
l ≧ 1, p ≧ 3, q ≧ 3 (19)
And
Next, a function of the position command is derived.
From equation (5), the acceleration pattern of the position command is given by:

It becomes. By integrating the above equation, the speed pattern of the position command is
(Equation 16)

It becomes. Further integrating this equation, the position command is
[Equation 17]

It becomes.
Finally, b is obtained using the boundary condition, and a position command is determined.
When Equations (21) and (22) are applied to the termination condition of Equation (8) of the boundary condition,
(Equation 18)

Holds.
Obviously, the above simultaneous equations are binary linear equations of b. From equation (18) or equation (19), since l ≧ 1, the number of unknowns is equal to or greater than the number of equations, and there is always a solution to the equations. Solving this simultaneous equation gives b. Substituting the value of b into equation (22) determines the time function of the position command.
It will be described below that the position command obtained as described above always satisfies all the boundary conditions.
From equation (18), since p ≧ 2 and q ≧ 2, from equation (17),
[Equation 19]

Includes the factors of (t−ts) and (t−te), and the boundary condition of equation (7)

Is always satisfied regardless of b. Since p ≧ 3 and q ≧ 3 from the equation (19), from the equation (5),
(Equation 21)

Includes the factors of (t−ts) and (t−te), and the boundary condition of the equation (9)

Is always satisfied regardless of b.
From Expressions (17), (21) and (22), the initial condition of the boundary condition Expression (8)

Is always satisfied regardless of b.
It can be seen from Equations (23) to (26) that all the boundary conditions shown in Equations (7) to (9) are satisfied.
In the first embodiment of the present invention, it is necessary to solve eight or more equations, but in the second embodiment of the present invention, it is simple because only two equations need to be solved.
Next, a third embodiment of the present invention will be described.
In the first embodiment and the second embodiment, it is necessary to know the machine stand of the natural oscillation angular frequency ω attenuation coefficient zeta _B of _B and machine base vibration at the time of obtaining the position command. In general, there is a method for identifying these parameters by applying an external force of a pulse to the machine and obtaining the parameters from the waveform of displacement of the machine. However, it is necessary to attach a laser displacement meter. Here, it is not necessary to attach a laser displacement meter, and a method of obtaining from a position deviation of the control device will be described below.
The position deviation is the difference between the position command _xr and the feedback position signal x from the position detector. The position detector includes a movable part and a fixed part, and the detected signal is a relative displacement between the movable part and the fixed part. Generally, the movable part of the position detector is attached to the movable part of the drive mechanism, and the fixed part of the position detector is attached to the machine base or a mechanism fixed to the machine base. Therefore, the signal x from the position detector is a relative displacement between the movable part of the drive mechanism and the machine base. The absolute displacement of the movable portion of the drive mechanism and x _L, since the machine stand of the absolute displacement is x _B,
[Equation 24]

Is obtained.
On the other hand, if the gains of the position control unit 7 and the speed control unit 8 are set sufficiently low (for example, 0.1 or less) after the end of the position command, the feedforward signal becomes 0 and the feedback signal also becomes small. and become the driving force F is approximately 0, the absolute displacement x _L also almost 0 of the movable portion of the drive mechanism. Therefore, the position deviation e after the end of the position command is
_{e (t) = x r (} t) - {x L (t) -x B (t)} ≒ x B (t), (t> te) ... (27)
It becomes. That is, the vibration of the machine after completion of the command directly appears in the position deviation as shown in FIG.
Therefore,
Te @ T _B ... (28)
And _{λe = e pp2 / e pp1 ≒} λ B = x Bpp2 / x Bpp1 ... (29)
Holds. However, Te is the time between two adjacent peaks of the oscillating component of the position deviation after command completion (or troughs), T _B is the natural vibration period of the machine base. Further, .lambda.e the vibration amplitude damping ratio of the positional deviation after the command completion, lambda _B is the amplitude attenuation ratio of the machine base vibration.
As described above, Te and λe are easily determined. Using these, ω _B and _{Ｂ B} are given as follows.
ω _B = 2π / T _B ≒ 2π / Te (30)
_{Ｂ B} ≒ − (lnλ _B ) / 2π ≒ − (Inλe) / 2π (31)
Next, effects of the present invention will be described using numerical examples.
First, the attenuation coefficient zeta _B natural oscillation angular frequency omega _B and machine stand vibration of the machine base.
From FIG. 6, Te = 0.0673s and λe = 0.7 are obtained. From Expressions (30) and (31),
ω _B ≒ 2π / Te = 93.3 (rad / s), _{Ｂ B} ≒ − (lnλ _B ) /2π≒0.0568 (32)
Then, representing the converted displacement x _BB of machine stand as a function of time t.
l = 1, p = 2, q = 2, ts = 0.05 s, te = 0.09 s, and from equation (17), the converted displacement of the machine is x _BB (b, t) = (b ₁ t + b ₀ ) (t−0.05) ² (t−0.14) ² , (0.05 <t <0.14)
x _BB (b, t) = 0, (t ≦ 0.05, 0.14 ≦ t) (33)
And keep it.
Next, a function of the position command is derived.
When Xs = 0 and Equations (32) and (33) are substituted into Equations (21) and (22),
(Equation 25)

And

It becomes.
Finally, b is obtained using the boundary condition, and a position command is determined.
Xe = 6000 μm, and from equation (23),
[Equation 27]

It becomes.
Solving the above equation gives b ₁ = −1.21 × 10 ¹⁰ , b ₀ = 5.447 × 10 ^8. Substituting the values of b ₁ and b ₀ into equation (35) gives the time function of the position command. Is
[Equation 28]

It becomes.
FIG. 5 shows a simulation result when such a command is used. On the other hand, FIG. 4 shows a simulation result when a triangle command as a position command of the conventional method is used at the same command time.
4 and 5, it can be seen that when the position command according to the method of the present invention is used, both the machine vibration and the position deviation after the end of the position command are suppressed to a small value.
【The invention's effect】
As described above, according to the present invention, even when the rigidity of the machine base on which the drive mechanism is installed is low, and irrespective of the length of the natural vibration cycle of the machine base, and further regardless of the magnitude of the movement amount In addition, the vibration of the machine after the end of the position command is suppressed, and there is an effect that high-speed, high-accuracy positioning control can be performed in a short time.
[Brief description of the drawings]
FIG. 1 is a block diagram of a positioning control system which is an object of the present invention including a vibration system of a machine base. FIG. 2 is an approximate equivalent block diagram of FIG. 1 when considering acceleration and deceleration of a drive system. FIG. 4 is a schematic diagram of a mechanical system to which the present invention is applied. FIG. 4 shows a simulation result when a conventional triangular command is used. FIG. 5 is a simulation result when a position command method of the present invention is used. FIG. 6 shows a simulation result when the gain of the control unit is set low as a third embodiment of the present invention.
DESCRIPTION OF SYMBOLS 1 Motor 2 Force transmission mechanism 3 Load 4 Machine base 5 Support foot 6 of machine base Position command generation part 7 Position control part 8 Speed control part 9, 10 Differentiation 11 Gain 12 Model of mechanical system including motor

Claims (3)

A drive mechanism including a load is installed in the machine base, and in the position command method of the position control device that drives the load and follows the position command,
Set a natural number n such that n ≧ (boundary condition number−3),
The machine displacement is represented by an arbitrary constant coefficient n-order polynomial with time t,
The load acceleration is represented by a known constant coefficient second-order differential equation due to the machine displacement, the load acceleration is integrated twice, and the load position is represented by an (n + 2) -degree polynomial with time t.
Set boundary conditions including the load position at the start of the position command and the load position at the end of the position command,
Creating a simultaneous equation using the n-th order polynomial of the machine displacement, the (n + 2) -order polynomial of the load position, and the boundary condition;
Determine constant values of all arbitrary constant coefficients that satisfy the simultaneous equations,
An (n + 2) -order polynomial based on the time t of the load position is determined from the determined constant value,
A position command method for a position control device, wherein an (n + 2) -order polynomial based on the determined load position time t is used as a position command. A drive mechanism including a load is installed in the machine base, and in the position command method of the position control device that drives the load and follows the position command,
Where l ≧ 1, p ≧ 2, and q ≧ 2, ts is a position command start time, te is a position command end time, b ₁ , b ₁ ,..., B ₂ , b ₁ , and b ₀ are As an arbitrary constant coefficient, by time t,
Machine frame displacement ^{_{= (b l t l + b}} l-1 t l-1 + ... + b 2 t 2 + b 1 t + b 0) (t - ts) p (t - te) q
Represented as
The load acceleration is represented by a known constant coefficient second-order differential equation based on the machine displacement, the load acceleration is integrated twice, and the load position is represented by a function based on time t,
Set the boundary conditions including the load position at the start of the position command and the load position at the end of the position command,
Create a simultaneous equation using the function of the machine displacement, the function of the load position, and the boundary condition,
Determine constant values of all arbitrary constant coefficients that satisfy the simultaneous equations,
A function of the load position according to time t is determined based on the determined constant value,
A position command method for a position control device, wherein a function based on the determined load position time t is used as a position command. A drive mechanism including a load is installed in the machine base, and a position control device that causes the load to follow a position command, by a measurement method for measuring each natural vibration frequency and vibration damping coefficient of the machine base,
The feedback gain of the position control device is set low,
Output position command,
Observe the change in position deviation after the end of the position command,
Calculating the natural vibration angular frequency of the machine from the natural vibration cycle as the natural vibration cycle of the machine, the time between adjacent vibration peaks observed by the above,
Measurement of the natural vibration angular frequency and vibration damping coefficient of the machine characterized by calculating the damping rate by comparing the peak values of the adjacent vibrations observed as described above with magnitude, and using this as the vibration damping coefficient of the machine. Method.

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