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Design and analysis of a novel X-Y table
J.F. Pan1, N.C. Cheung2, Guangzhong Cao1, Hong Qiu1
Abstract—A novel X-Y table based on linear switched reluctance principle is proposed in this paper. The proposed direct-drive actuator has the characteristics of low cost, simple mechanical structure and high reliability. Finite element analysis (FEA) proves that the phases between any of the linear motors of the X-Y table are decoupled and each phase can be controlled independently. Experimental results verify that the motion control system based on the X-Y table has good dynamic characteristics.
Keywords-switched reluctance, direct-drive, FEA
I. Introduction
In most advanced manufacturing processes, two-dimensional motions are in high demand for industrial applications such as parts assembly, component insertion, machining, etc. A traditional X-Y table often utilizes a rotary motor and couples its output shaft to mechanical translators such as gears or bears to perform linear motion; by vertical arrangement of two such linear motion implementations, two-dimensional movement is achieved. In a direct-drive system, the mechanical output is directly generated to the actuator and load and it has the characteristics of high force density, high precision and low production cost [1]. By elimination of mechanical transmissions, such as rotary-to-linear couplers, the control object, together with the actuator can be implemented as an integral system, which is capable of fast response, high flexibility and can have a simple structure.
The X-Y table based on direct-drive idea can be constructed according to different motor methodologies such as a pair of linear direct current motors (LDCM), linear induction motors (LIM), linear permanent magnetic motors (LPMM) or linear switched reluctance motors (LSRM), etc. Due to the presence of a commutator, the LDCM requires frequent adjustment and maintenance. The principle of the LIM is similar to a common rotary induction motor with a robust structure. Though linear motion of high speed or high-precision is difficult to achieve due to the low air gap flux density [2], a variety of applications can still be found for a long-stroke motion such as the magnetic levitation train for railway transportation [3]. The LPMM is the only type of linear actuator available to industry by present and it has the advantages of wide range of speed regulation capability and stable output performance. One structural disadvantage of such linear motor is the utilization of expensive rare-earth permanent magnets to achieve better performance and efficiency. Due to the characteristics of the permanent magnets, a LPMM is not suitable under hostile environment that has a various temperature change. Moreover the overall cost of the linear motion system is high.
II. Design and construction of the X-Y table
With fast advancement of power electronics technology, research on switched reluctance motors becomes more and more extensive. A typical SR machine has the following characteristics,
· The mechanical structure is simple and robust with doubly-salient stator. The motor can operate under various hostile environments with large temperature difference since no permanent magnets or commutator involved.
· Since the winding is fixed only on the stator, the motor is easy for cooling and has low heat loss thus it has high efficiency.
Project 200734 supported by SZU R/D Fund
The paper first received 15 Dec 2008 and in revised form 5 Jan 2009.
Digital ref: 123
1College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen, P.R.China, E-mail:gzcao@szu.edu.cn
2Department of EE, The Hong Kong Polytechnic University, Hong Kong, E-mail: norbert.cheung@polyu.edu.hk
· Torque generation is irrelevant of current directions. Therefore the drive topology can be minimized to reduce system cost.
According to the winding arrangement, a LSRM can be categorized as “active-stator-passive-translator” and “passive-stator-active-translator” structure [4]. The motion system applies the “passive-stator-active-translator” scheme for the following reasons,
· Simple manufacture of the stator base with no complicated coil arrays
· Flexible traveling range and stator dimensions
· Easy manufacture of mover slots with mounted coil windings
· Low overall production cost
Based on the idea of “passive stator-active translator” structure, the linear motor is constructed as shown in Fig.1. The motor is composed of the moving platform and the stator base. The stator base is made of aluminum alloy to minimize the mass and facilitate the magnetic flux path. A pair of high-grade linear guides is fixed on the stator base to facilitate the movement of the moving platform. The stator utilizes 0.5mm thick silicon-steel plates with tooth structure instead of any expensive materials such as rare-earth permanent magnets. A pair of aluminum locking bars is fixed on the stator base slot to hold the plates. The moving platform is composed of three-phase movers with coils. Locking pins are used to fix the mover plates. To prevent distortion of the movers under large current excitations, each mover is fixed tightly to the moving platform with L-shaped locking pins with screws. A linear optical encoder is mounted together with the moving platform to provide relative position information between the moving platform and the stator in real time.
(a)
(b)
Fig. 1: The LSRM (a) and the X-Y table (b)
Similar to the structure of a typical 6/4 rotary SR motor, each phase of the three phases is separated by 120° electrically, which is 1+2/3 pitch distance, i.e. 10mm. The pitch distance is preferred to be 12mm to avoid any rounding error from 5/3 pitch distance. The features of the mover structure can be summarized as the following,
· The decoupled flux windings lead to a simpler motor model due to zero mutual inductance.
· The individual phase windings reduce the manufacturing cost and complexity.
· Long travel distance can be accomplished easily by combining longitudinal track guides.
The construction of the X-Y table is based on vertical stacking of such two LSRMs to facilitate two-dimensional movement as shown in Fig.1 (b). Since the X motor moves with the Y motor simultaneously, the X platform has a wider mover and stator tooth structure to generate larger propulsion forces. Table 1 shows mechanical and electrical parameters of the X-Y table.
Table 1: Mechanical and electrical parameters
Mass of X moving platform（not including Y platform）
2.8 Kg
Mass of Y moving platform
1.5 Kg
Stroke of X moving platform
170 mm
Stroke of X moving platform
120 mm
Width of X moving platform
50 mm
Width of Y moving platform
24 mm
Air gap of X moving platform
0.3 mm
Air gap of Y moving platform
0.2 mm
Pole-pitch
12 mm
Tooth width
6 mm
Encoder resolution
1 μm
III. Mathematical model of the X-Y table
Since the X and Y table is magnetically decoupled, for any direction of movement, the equations that governs the voltage balance relationship of the LSRM can be described as the following,
(1)
where，andare winding voltage, current and resistance. is relative position from the mover to the stator andis flux-linkage.
The force balance equation can be described as the following for any direction of movement as,
(2)
whereis generated electromagnetic force,is the load force,andare mass and friction coefficient, respectively.
For any phase of X or Y table, inductance can be approximately represented by Fourier Series Expansions as [4],
(3)
(4)
(5) whereand ，is leakage inductance.
IV. Simulation analysis of the X-Y table
The purpose of finite element analysis for the X-Y table is to verify that the coupling effect between any two phases can be neglected for each direction of movement, so that each phase of the LSRM can be controlled independently. Further FEA is conducted for the prediction of motor performance such as force output capability.
A. Analysis of coupling effect
Three-dimensional FEA is carried out for the test of phase coupling effect. Any one phase of the three phases from X or Y table is excited with a DC current such as 10A. By the inspection of the magnetic flux distribution from other two phases, the coupling effect can be explored. As shown in Fig.2 (a), the magnetic flux mainly distribute among the excited mover, the air gap and the stator. From Fig.2 (b) and (c), by exploring from the magnetic flux of mover cross section, it can be concluded that the induced flux value diminishes as the relative position to the excited mover increases. Since the absolute induced value is about a thousandth of the excited value, the coupling effect can be neglected.
(a)
(b)
Fig. 2: FEA results of coupling effect—(a) flux contour (b) flux distribution in the excited and adjacent phase
B. Analysis of force output
The simulation results of the propulsion force are shown in Fig.3 for X and Y table respectively. Since X table has relatively larger air gap and moving mass, the force value of X table is comparably low of Y table.
(a)
(b)
Fig. 3: Propulsion force output of (a) X table and (b) Y table
V. Experimental results
Since force, current and position have nonlinear relationship for a LSRM, a proper linearization scheme is required before the implementation of any control algorithm. To optimize between computation efficiency and memory consumption, a pair of low-resolution two-dimensional look-up tables are employed for each axis of movement with linear interpolation to calculate the intermediate values [5]. Since force, current and position are related in three dimensions, a 2D force-current-position look-up table for each axis is sufficient to describe the nonlinear force profile. The experiment to find out the inverse relationship between current, force and position has been conducted. By fixing the moving platform of each table at corresponding positions within one pole width, currents are measured for the generation of the desired force. Alternatively, the look-up tables are generated from the inverse function of force versus current and position. The generated 27×27-matrix is employed to build up the look-up tables for each axis of motion and sufficient to describe the force profile within the error of 5% [5].
The experiment is implemented on a dSPACE DS1104 DSP motion controller card. This card has an on-board 250MHz DSP for real-time computation and it interfaces with the PC through the PCI bus. It consists of two channels of 24-bits incremental encoder inputs, six channels of 12-bit analog input and six channels 12-bit analog output. The control card can directly interface with Real-Time Workshop and MATLAB and control parameters can be modified online. The overall control block diagram is shown in Fig.4 with a sampling rate of 10 KHz for the inner current loop and 2KHz for the outer position loop.
The step position responses of each moving platform are recorded as the results shown in Fig.4. Since the moving platform of X table moves simultaneously with Y table, it can be concluded from the step responses that the X moving platform has a relatively larger overshoot and longer rising time compared with that of the Y moving platform.
(a)
(b)
Fig. 4: Step response of (a) X table and (b) Y table
The dynamic responses of sine and cosine curves as the position commands can be found as shown in Fig.5. The tracking profiles show that each axis of motion is capable of following the command signal precisely. The command signal and response almost overlap for both axes as shown in Fig.5 (a) and (b). The error dynamics can be found in Fig.5 (c) and (d). The absolute errors fall within 0.35 mm, 3% of the total range (11.5 mm). It is clear that for both diagrams, the errors for opposite directions are not identical in each axis of motion. This is because the mechanical structures in both axes are not uniform such that the motor experiences unbalanced frictions at different positions.
From the experimental results, the position controllers are capable of correction for such imperfections that exist in mechanical manufacture and the simple PID controller ensures the implementation for future industrial applications of the X-Y table.
(a)
(b)
(c)
(d)
Fig. 5: Position response (a) X table, (b) Y table and error response of the X-Y table (c) X table, (d) Y table
The dynamic tracking profile of circle and line are demonstrated in Fig. 6 (a) and (b) respectively.
(a)
(b)
Fig. 6: The tracking response from the X-Y table of (a) circle and (b) straight line
VII. Conclusions
A novel small-size X-Y table based on switched reluctance principle is proposed in the paper. This X-Y table has the characteristics of simple and robust structure, low manufacturing cost and high reliability. Preliminary simulation and experimental results show that the motion control system has good dynamic performance and it is expected the proposed X-Y table to be an ideal replacement for traditional X-Y tables in industrial automation applications.
References
[1] I. Boldea and S. A. Nasar, “Linear Electric Actuators and Generators”, Cambridge University Press, London, UK, 1997.
[2] Jacek F. Gieras and Zbigniew J. Piech, “Linear synchronous motors—transportation and automation systems”, Boca Raton, Fla., CRC Press, 2000.
[3] R. Krishnan, “Switched Reluctance Motor Drives: Modeling, Simulation, Analysis, Design, and Applications”, Boca Raton, FL : CRC Press, 2001.
[4] C. T. Liu, L. F. Chen, J. L. Kuo, Y. N. Chen, Y. J. Lee, and C. T. Leu, “Microcomputer Control Implementation of Transverse Flux Linear Switched Reluctance Machine with Rule-based Compensator,” IEEE Trans. Energy Conversion, vol. 11, pp. 70–75, Mar. 1996.
[5] J.F. Pan, San Chin Kwok, Norbert C. Cheung and J.M. Yang, “Auto Disturbance Rejection Speed Control of a Linear Switched Reluctance motor”, Fortieth International IAS Annual Meeting, Industry Applications Conference, 2005, Volume 4,?2-6 Oct. 2005 Page(s):2491 - 2497 Vol. 4.
譯文：
一種新的X-Y工作臺(tái)的設(shè)計(jì)與分析
J.F. Pan1, N.C. Cheung2, Guangzhong Cao1, Hong Qiu1
摘要
本文提出了基于線性開(kāi)關(guān)磁阻原理的一種新型的XY工作臺(tái)。直接驅(qū)動(dòng)執(zhí)行器具有成本低，機(jī)械結(jié)構(gòu)簡(jiǎn)單，可靠性高的特點(diǎn)。有限元分析（FEA）證明任何的XY工作臺(tái)上的線性電動(dòng)機(jī)之間是不掛鉤的，并且各相可以獨(dú)立控制。實(shí)驗(yàn)結(jié)果驗(yàn)證了基于運(yùn)動(dòng)控制系統(tǒng)的XY工作臺(tái)具有良好的動(dòng)態(tài)特性。
關(guān)鍵詞：開(kāi)關(guān)磁阻電機(jī)，直接驅(qū)動(dòng)，有限元分析
1、引言
在最先進(jìn)的制造工藝?yán)?，二維運(yùn)動(dòng)工業(yè)應(yīng)用中有很高需求，就像零件組裝、組件插入、加工等一樣。傳統(tǒng)的XY工作臺(tái)通常采用旋轉(zhuǎn)電機(jī)，輸出軸機(jī)械操作員，如齒輪或空頭進(jìn)行線性運(yùn)動(dòng); 由垂直排列的兩個(gè)這樣的直線運(yùn)動(dòng)實(shí)現(xiàn)，二維運(yùn)動(dòng)就實(shí)現(xiàn)了。在直接驅(qū)動(dòng)系統(tǒng)中，直接產(chǎn)生致動(dòng)器和負(fù)載的機(jī)械輸出，它具有的特性力密度高，精度高，生產(chǎn)成本低的優(yōu)點(diǎn)[1]。通過(guò)消除機(jī)械傳動(dòng)裝置，如旋轉(zhuǎn)變?yōu)橹本€運(yùn)動(dòng)的耦合器，控制對(duì)象，連同致動(dòng)器可以被實(shí)現(xiàn)為一個(gè)不可分割的系統(tǒng)，這是能夠?qū)崿F(xiàn)響應(yīng)速度快，高柔韌性，可以具有簡(jiǎn)單的結(jié)構(gòu)。
基于直接驅(qū)動(dòng)的XY工作臺(tái)可以根據(jù)不同的電機(jī)的方法來(lái)設(shè)計(jì)，如線性直流電動(dòng)機(jī)（LDCM），直線感應(yīng)電機(jī)（LIM），線性永磁電機(jī)（LPMM）或線性開(kāi)關(guān)磁阻電機(jī)的一對(duì)（LSRM）等。由于換向器的存在，LDCM需要經(jīng)常調(diào)整和維護(hù)。直線感應(yīng)電機(jī)的原理和一個(gè)共同的旋轉(zhuǎn)感應(yīng)電動(dòng)機(jī)的一個(gè)健壯的結(jié)構(gòu)相似。雖然由于低的空氣隙的磁通密度，高速或高精度的直線運(yùn)動(dòng)是難以實(shí)現(xiàn)的[2]，各種應(yīng)用程序仍然可以被找到用來(lái)實(shí)現(xiàn)長(zhǎng)行程的運(yùn)動(dòng)，如鐵路運(yùn)輸?shù)拇艖腋×熊嘯3]。LPMM是行業(yè)中目前唯一的類型的線性驅(qū)動(dòng)器，它具有速度調(diào)節(jié)能力和穩(wěn)定的輸出性能的優(yōu)勢(shì)。這樣的線性電動(dòng)機(jī)的一個(gè)結(jié)構(gòu)的缺點(diǎn)是昂貴的稀土類永久磁鐵的利用率來(lái)實(shí)現(xiàn)更好的性能和效率。由于永久磁鐵的特性，一個(gè)LPMM在具有不同的溫度變化惡劣的環(huán)境中是不適合的。此外，線性運(yùn)動(dòng)系統(tǒng)的總成本是很高的。
2.X-Y工作臺(tái)的設(shè)計(jì)與實(shí)施
隨著電力電子技術(shù)的快速進(jìn)步，對(duì)開(kāi)關(guān)磁阻電機(jī)的研究變得越來(lái)越廣泛。一個(gè)典型的SR機(jī)器具有以下特點(diǎn)，
?機(jī)械結(jié)構(gòu)簡(jiǎn)單和強(qiáng)大的雙突出的定子。電機(jī)可以在大溫差的惡劣環(huán)境下工作，因?yàn)闆](méi)有永久磁鐵和換向器。
?由于繞組被固定在定子上，電機(jī)容易冷卻，熱損失小，所以它效率高。、
?扭矩產(chǎn)生與電流方向無(wú)關(guān)。因此，驅(qū)動(dòng)器拓?fù)浣Y(jié)構(gòu)可以被最小化，以降低系統(tǒng)成本。
根據(jù)繞組布置，LSRM可被歸類為“主動(dòng)定子被動(dòng)翻譯” ，“被動(dòng)定子活性翻譯”的結(jié)構(gòu)[4]。運(yùn)動(dòng)系統(tǒng)采用“被動(dòng)定子主動(dòng)翻譯”方案，原因如下，
?沒(méi)有復(fù)雜的線圈陣列的定子制造簡(jiǎn)單
?靈活的行駛范圍和定子尺寸
?安裝線圈繞組的mover槽易于制造
?整體生產(chǎn)成本低
“被動(dòng)定子活性翻譯”結(jié)構(gòu)的想法基礎(chǔ)上，線性電動(dòng)機(jī)的構(gòu)造如圖1所示。電機(jī)由動(dòng)平臺(tái)和定子機(jī)座組成。定子機(jī)座由鋁合金制成，以減少它的質(zhì)量和方便磁通路徑。一對(duì)高品位的直線導(dǎo)軌固定在定子上的底座，方便移動(dòng)平臺(tái)的移動(dòng)。定子采用0.5mm厚的硅鋼板與牙齒結(jié)構(gòu)，而不是任何昂貴的材料，例如稀土類永久磁鐵。一對(duì)鋁制鎖定桿固定在定子上的基槽用來(lái)支撐板。移動(dòng)平臺(tái)由三相線圈的電機(jī)組成。鎖定銷用于固定移動(dòng)盤(pán)。為了防止大電流激發(fā)下的電機(jī)失真，每個(gè)電機(jī)用螺絲緊緊固定在移動(dòng)平臺(tái)L形鎖銷。線性光學(xué)編碼器和移動(dòng)平臺(tái)一起安裝，以實(shí)時(shí)提供移動(dòng)平臺(tái)和定子之間的相對(duì)位置信息。
(a)
(b)
圖1 LSRM（a）和X-Y工作臺(tái)（b）
與一個(gè)典型的6/4旋轉(zhuǎn)SR電機(jī)的結(jié)構(gòu)相似，三個(gè)相每相相差120°電，12/3節(jié)距，即10毫米。優(yōu)選的間距是12毫米，以避免任何5/3節(jié)距的距離的舍入誤差。移動(dòng)器結(jié)構(gòu)的特征可以概括為以下，
?由于互感為零，脫鉤的通量繞組引導(dǎo)一個(gè)簡(jiǎn)單的電機(jī)模型。
?各相繞組降低了制造成本和復(fù)雜性。
?結(jié)合縱向軌道指南，長(zhǎng)行程距離可以比較容易完成。
XY工作臺(tái)的結(jié)構(gòu)是基于這樣兩個(gè)LSRMs的垂直堆疊以方便二維運(yùn)動(dòng)，如圖1（b）所示。由于X軸電機(jī)和Y軸電機(jī)同時(shí)移動(dòng)，X平臺(tái)上有更廣泛的原動(dòng)力和定子齒結(jié)構(gòu)來(lái)產(chǎn)生更大的推進(jìn)力。表1示出了XY工作臺(tái)的機(jī)械和電氣參數(shù)。
表1：機(jī)械和電氣參數(shù)
X的移動(dòng)平臺(tái)的質(zhì)量（不包括Y平臺(tái)）
2.8 Kg
Y移動(dòng)平臺(tái)的質(zhì)量
1.5 Kg
X移動(dòng)平臺(tái)的行程
170 mm
Y移動(dòng)平臺(tái)的行程
120 mm
X移動(dòng)平臺(tái)的寬度
50 mm
Y移動(dòng)平臺(tái)的寬度
24 mm
X移動(dòng)平臺(tái)的氣隙
0.3 mm
Y移動(dòng)平臺(tái)的氣隙
0.2 mm
立桿間距
12 mm
齒寬
6 mm
編碼器分辨率
1 μm
3.X-Y工作臺(tái)的數(shù)學(xué)模型
由于XY工作臺(tái)磁去耦，對(duì)于任意的運(yùn)動(dòng)方向，LSRM管電壓的平衡關(guān)系的方程可描述為以下，
(1)
其中，和分別是繞組的電壓，電流和電阻。是從動(dòng)子到定子的相對(duì)位置，是磁鏈。
任一方向運(yùn)動(dòng)的力平衡方程可以描述為以下，
(2)
其中，是產(chǎn)生的電磁力，是加載力，和分別為質(zhì)量和摩擦系數(shù)。對(duì)于X或Y工作臺(tái)的任意一相，電感可以通過(guò)傅立葉級(jí)數(shù)展開(kāi)來(lái)近似表示為[4]，
(3)
(4)
(5)
其中，，，是漏感。
4.X-Y工作臺(tái)仿真分析
XY工作臺(tái)的有限元分析的目的是為了驗(yàn)證每個(gè)的運(yùn)動(dòng)方向的任兩相之間的耦合效應(yīng)可以忽略不計(jì)，使LSRM各相可獨(dú)立控制。進(jìn)一步有限元分析是進(jìn)行電機(jī)性能的預(yù)測(cè)，如力輸出能力。
A.耦合效應(yīng)分析
三維有限元分析進(jìn)行來(lái)測(cè)試各相耦合效應(yīng)。從X或Y工作臺(tái)三個(gè)相的任一相被直流電流激發(fā)，例如10A的直流電流。由從其他兩相的磁通分布的檢驗(yàn)，可以探討耦合效應(yīng)。如在圖2（a）所示，磁通分布產(chǎn)生的激動(dòng)子，氣隙和定子。從圖2（b）和（c）中，通過(guò)探討可動(dòng)件的橫截面的磁通量，可以得出的結(jié)論是感應(yīng)的磁通值隨激發(fā)動(dòng)機(jī)的相對(duì)位置的增加而減小。由于絕對(duì)誘導(dǎo)的值是激發(fā)值的千分之一，耦合效應(yīng)可以忽略不計(jì)。
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圖2 耦合效應(yīng)的有限元分析結(jié)果——(a) 輪廓通量(b) 興奮和相鄰階段的磁通分布
B.力輸出分析
X和Y工作臺(tái)的推進(jìn)力的模擬結(jié)果分別示于圖3中。由于X工作臺(tái)具有較大的空氣間隙和移動(dòng)質(zhì)量，X工作臺(tái)的力值相對(duì)于Y工作臺(tái)較低。
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(b)
圖3 （a）X工作臺(tái)及（b）Y工作臺(tái)的推進(jìn)力輸出
5.實(shí)驗(yàn)結(jié)果
由于對(duì)于LSRM來(lái)說(shuō)，力、電流和位置為非線性關(guān)系，任何控制算法實(shí)施之前，需要適當(dāng)?shù)木€性化方案。為了優(yōu)化的計(jì)算效率和內(nèi)存消耗，每個(gè)運(yùn)動(dòng)軸采用了低分辨率的二維查找表用線性內(nèi)插法計(jì)算出的中間值[5]。由于力，電流和位置在三維空間中有聯(lián)系，一個(gè)每個(gè)軸的二維的力-電流-位置查找表足以描述非線性力。找出電流，力和位置之間的反比關(guān)系的實(shí)驗(yàn)已經(jīng)在進(jìn)行中了。通過(guò)固定一個(gè)磁極寬度內(nèi)對(duì)應(yīng)的位置的移動(dòng)平臺(tái)的每個(gè)工作臺(tái)，通過(guò)測(cè)量電流來(lái)算出所需的力的產(chǎn)生。另外，查找表從力與電流和位置的反函數(shù)中產(chǎn)生。所生成的27×27矩陣建立了每個(gè)運(yùn)動(dòng)軸的查找表，并足以描述5％的誤差范圍內(nèi)的力的分布[5]。
這個(gè)實(shí)驗(yàn)是在dSPAC EDS1104 DSP運(yùn)動(dòng)控制卡上實(shí)現(xiàn)的。該卡有一個(gè)板載250MHz的DSP用來(lái)實(shí)時(shí)計(jì)算，并通過(guò)PCI總線與PC機(jī)的接口相聯(lián)。它由兩個(gè)24位增量式編碼器輸入、12位模擬輸入的6通道和12位模擬輸出的6通道組成?？刂瓶梢灾苯优c實(shí)時(shí)車間和MATLAB和控制參數(shù)的接口相聯(lián)，可在線修改。采樣速率為10 KHz的內(nèi)部電流環(huán)路和2KHz的外側(cè)位置回路的整體控制框圖如圖4所示。
每個(gè)移動(dòng)平臺(tái)的步進(jìn)位置響應(yīng)記錄在圖4中所示的結(jié)果中。由于移動(dòng)平臺(tái)的X工作臺(tái)與Y工作臺(tái)同時(shí)移動(dòng)，可以從階躍響應(yīng)得出結(jié)論，與Y移動(dòng)平臺(tái)相比，X移動(dòng)平臺(tái)有一個(gè)相對(duì)較大的超調(diào)量和更長(zhǎng)的上升時(shí)間。
(a)
(b)
圖4（a）X工作臺(tái)及（b）Y工作臺(tái)的階躍響應(yīng)
作為位置指令的正弦曲線和余弦曲線的動(dòng)態(tài)響應(yīng)如圖5所示。跟蹤配置文件顯示，各運(yùn)動(dòng)軸能夠精確跟隨指令信號(hào)。對(duì)于兩個(gè)軸的所述命令信號(hào)和響應(yīng)幾乎重疊示于圖5（a）及（b）。誤差動(dòng)力系統(tǒng)在如圖5（c）及（d）中所示。絕對(duì)誤差下降到0.35mm，總范圍的3％范圍內(nèi)（11.5mm）。很明顯，這兩個(gè)圖中，各運(yùn)動(dòng)軸的相反的方向誤差時(shí)不同的。這是因?yàn)樵谶@兩個(gè)軸的機(jī)械結(jié)構(gòu)不統(tǒng)一，使得電機(jī)在不同的位置出現(xiàn)不平衡的摩擦。
從實(shí)驗(yàn)結(jié)果可以得到，位置控制器能夠改善機(jī)械制造中這樣的缺陷，簡(jiǎn)單的PID控制器可以確保為未來(lái)的工業(yè)應(yīng)用中的XY工作臺(tái)的發(fā)展。
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(b)
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圖5 位置響應(yīng)（a）X臺(tái)，（b）Y工作臺(tái)和XY工作臺(tái)的錯(cuò)誤響應(yīng)（c）X工作臺(tái)，（d）Y工作臺(tái)
圓和直線的動(dòng)態(tài)跟蹤檔案分別如圖6（a）及（b）中所示。
(a)
(b)
圖6 XY工作臺(tái)（a）的圓以及（b）直線的跟蹤響應(yīng)
7.結(jié)論
本文提出了一種新型的小尺寸基于開(kāi)關(guān)磁阻原理的XY工作臺(tái)。此XY工作臺(tái)具有結(jié)構(gòu)簡(jiǎn)單和強(qiáng)大的結(jié)構(gòu)，制造成本低，可靠性高的特點(diǎn)。初步仿真和實(shí)驗(yàn)結(jié)果表明，運(yùn)動(dòng)控制系統(tǒng)具有良好的動(dòng)態(tài)性能，并預(yù)期建議是在工業(yè)自動(dòng)化應(yīng)用的理想替代傳統(tǒng)的XY工作臺(tái)的XY工作臺(tái)。
參考文獻(xiàn)
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