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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工作臺的設計與分析
J.F. Pan1, N.C. Cheung2, Guangzhong Cao1, Hong Qiu1
摘要
本文提出了基于線性開關磁阻原理的一種新型的XY工作臺。直接驅動執(zhí)行器具有成本低，機械結構簡單，可靠性高的特點。有限元分析（FEA）證明任何的XY工作臺上的線性電動機之間是不掛鉤的，并且各相可以獨立控制。實驗結果驗證了基于運動控制系統(tǒng)的XY工作臺具有良好的動態(tài)特性。
關鍵詞：開關磁阻電機，直接驅動，有限元分析
1、引言
在最先進的制造工藝里，二維運動工業(yè)應用中有很高需求，就像零件組裝、組件插入、加工等一樣。傳統(tǒng)的XY工作臺通常采用旋轉電機，輸出軸機械操作員，如齒輪或空頭進行線性運動; 由垂直排列的兩個這樣的直線運動實現(xiàn)，二維運動就實現(xiàn)了。在直接驅動系統(tǒng)中，直接產生致動器和負載的機械輸出，它具有的特性力密度高，精度高，生產成本低的優(yōu)點[1]。通過消除機械傳動裝置，如旋轉變?yōu)橹本€運動的耦合器，控制對象，連同致動器可以被實現(xiàn)為一個不可分割的系統(tǒng)，這是能夠實現(xiàn)響應速度快，高柔韌性，可以具有簡單的結構。
基于直接驅動的XY工作臺可以根據(jù)不同的電機的方法來設計，如線性直流電動機（LDCM），直線感應電機（LIM），線性永磁電機（LPMM）或線性開關磁阻電機的一對（LSRM）等。由于換向器的存在，LDCM需要經(jīng)常調整和維護。直線感應電機的原理和一個共同的旋轉感應電動機的一個健壯的結構相似。雖然由于低的空氣隙的磁通密度，高速或高精度的直線運動是難以實現(xiàn)的[2]，各種應用程序仍然可以被找到用來實現(xiàn)長行程的運動，如鐵路運輸?shù)拇艖腋×熊嘯3]。LPMM是行業(yè)中目前唯一的類型的線性驅動器，它具有速度調節(jié)能力和穩(wěn)定的輸出性能的優(yōu)勢。這樣的線性電動機的一個結構的缺點是昂貴的稀土類永久磁鐵的利用率來實現(xiàn)更好的性能和效率。由于永久磁鐵的特性，一個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，線性運動系統(tǒng)的總成本是很高的。
2.X-Y工作臺的設計與實施
隨著電力電子技術的快速進步，對開關磁阻電機的研究變得越來越廣泛。一個典型的SR機器具有以下特點，
?機械結構簡單和強大的雙突出的定子。電機可以在大溫差的惡劣環(huán)境下工作，因為沒有永久磁鐵和換向器。
?由于繞組被固定在定子上，電機容易冷卻，熱損失小，所以它效率高。、
?扭矩產生與電流方向無關。因此，驅動器拓撲結構可以被最小化，以降低系統(tǒng)成本。
根據(jù)繞組布置，LSRM可被歸類為“主動定子被動翻譯” ，“被動定子活性翻譯”的結構[4]。運動系統(tǒng)采用“被動定子主動翻譯”方案，原因如下，
?沒有復雜的線圈陣列的定子制造簡單
?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
?整體生產成本低
“被動定子活性翻譯”結構的想法基礎上，線性電動機的構造如圖1所示。電機由動平臺和定子機座組成。定子機座由鋁合金制成，以減少它的質量和方便磁通路徑。一對高品位的直線導軌固定在定子上的底座，方便移動平臺的移動。定子采用0.5mm厚的硅鋼板與牙齒結構，而不是任何昂貴的材料，例如稀土類永久磁鐵。一對鋁制鎖定桿固定在定子上的基槽用來支撐板。移動平臺由三相線圈的電機組成。鎖定銷用于固定移動盤。為了防止大電流激發(fā)下的電機失真，每個電機用螺絲緊緊固定在移動平臺L形鎖銷。線性光學編碼器和移動平臺一起安裝，以實時提供移動平臺和定子之間的相對位置信息。
(a)
(b)
圖1 LSRM（a）和X-Y工作臺（b）
與一個典型的6/4旋轉SR電機的結構相似，三個相每相相差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。移動器結構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一個簡單的電機模型。
?各相繞組降低了制造成本和復雜性。
?結合縱向軌道指南，長行程距離可以比較容易完成。
XY工作臺的結構是基于這樣兩個LSRMs的垂直堆疊以方便二維運動，如圖1（b）所示。由于X軸電機和Y軸電機同時移動，X平臺上有更廣泛的原動力和定子齒結構來產生更大的推進力。表1示出了XY工作臺的機械和電氣參數(shù)。
表1：機械和電氣參數(shù)
X的移動平臺的質量（不包括Y平臺）
2.8 Kg
Y移動平臺的質量
1.5 Kg
X移動平臺的行程
170 mm
Y移動平臺的行程
120 mm
X移動平臺的寬度
50 mm
Y移動平臺的寬度
24 mm
X移動平臺的氣隙
0.3 mm
Y移動平臺的氣隙
0.2 mm
立桿間距
12 mm
齒寬
6 mm
編碼器分辨率
1 μm
3.X-Y工作臺的數(shù)學模型
由于XY工作臺磁去耦，對于任意的運動方向，LSRM管電壓的平衡關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。是從動子到定子的相對位置，是磁鏈。
任一方向運動的力平衡方程可以描述為以下，
(2)
其中，是產生的電磁力，是加載力，和分別為質量和摩擦系數(shù)。對于X或Y工作臺的任意一相，電感可以通過傅立葉級數(shù)展開來近似表示為[4]，
(3)
(4)
(5)
其中，，，是漏感。
4.X-Y工作臺仿真分析
XY工作臺的有限元分析的目的是為了驗證每個的運動方向的任兩相之間的耦合效應可以忽略不計，使LSRM各相可獨立控制。進一步有限元分析是進行電機性能的預測，如力輸出能力。
A.耦合效應分析
三維有限元分析進行來測試各相耦合效應。從X或Y工作臺三個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，可以探討耦合效應。如在圖2（a）所示，磁通分布產生的激動子，氣隙和定子。從圖2（b）和（c）中，通過探討可動件的橫截面的磁通量，可以得出的結論是感應的磁通值隨激發(fā)動機的相對位置的增加而減小。由于絕對誘導的值是激發(fā)值的千分之一，耦合效應可以忽略不計。
(a)
(b)
圖2 耦合效應的有限元分析結果——(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工作臺的推進力的模擬結果分別示于圖3中。由于X工作臺具有較大的空氣間隙和移動質量，X工作臺的力值相對于Y工作臺較低。
(a)
(b)
圖3 （a）X工作臺及（b）Y工作臺的推進力輸出
5.實驗結果
由于對于LSRM來說，力、電流和位置為非線性關系，任何控制算法實施之前，需要適當?shù)木€性化方案。為了優(yōu)化的計算效率和內存消耗，每個運動軸采用了低分辨率的二維查找表用線性內插法計算出的中間值[5]。由于力，電流和位置在三維空間中有聯(lián)系，一個每個軸的二維的力-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系的實驗已經(jīng)在進行中了。通過固定一個磁極寬度內對應的位置的移動平臺的每個工作臺，通過測量電流來算出所需的力的產生。另外，查找表從力與電流和位置的反函數(shù)中產生。所生成的27×27矩陣建立了每個運動軸的查找表，并足以描述5％的誤差范圍內的力的分布[5]。
這個實驗是在dSPAC EDS1104 DSP運動控制卡上實現(xiàn)的。該卡有一個板載250MHz的DSP用來實時計算，并通過PCI總線與PC機的接口相聯(lián)。它由兩個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實時車間和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的內部電流環(huán)路和2KHz的外側位置回路的整體控制框圖如圖4所示。
每個移動平臺的步進位置響應記錄在圖4中所示的結果中。由于移動平臺的X工作臺與Y工作臺同時移動，可以從階躍響應得出結論，與Y移動平臺相比，X移動平臺有一個相對較大的超調量和更長的上升時間。
(a)
(b)
圖4（a）X工作臺及（b）Y工作臺的階躍響應
作為位置指令的正弦曲線和余弦曲線的動態(tài)響應如圖5所示。跟蹤配置文件顯示，各運動軸能夠精確跟隨指令信號。對于兩個軸的所述命令信號和響應幾乎重疊示于圖5（a）及（b）。誤差動力系統(tǒng)在如圖5（c）及（d）中所示。絕對誤差下降到0.35mm，總范圍的3％范圍內（11.5mm）。很明顯，這兩個圖中，各運動軸的相反的方向誤差時不同的。這是因為在這兩個軸的機械結構不統(tǒng)一，使得電機在不同的位置出現(xiàn)不平衡的摩擦。
從實驗結果可以得到，位置控制器能夠改善機械制造中這樣的缺陷，簡單的PID控制器可以確保為未來的工業(yè)應用中的XY工作臺的發(fā)展。
(a)
(b)
(c)
(d)
圖5 位置響應（a）X臺，（b）Y工作臺和XY工作臺的錯誤響應（c）X工作臺，（d）Y工作臺
圓和直線的動態(tài)跟蹤檔案分別如圖6（a）及（b）中所示。
(a)
(b)
圖6 XY工作臺（a）的圓以及（b）直線的跟蹤響應
7.結論
本文提出了一種新型的小尺寸基于開關磁阻原理的XY工作臺。此XY工作臺具有結構簡單和強大的結構，制造成本低，可靠性高的特點。初步仿真和實驗結果表明，運動控制系統(tǒng)具有良好的動態(tài)性能，并預期建議是在工業(yè)自動化應用的理想替代傳統(tǒng)的XY工作臺的XY工作臺。
參考文獻
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[2] Jacek F. Gieras and Zbigniew J. Piech, “同步直線電機——運輸和自動化系統(tǒng)”，佛羅里達州博卡拉頓，CRC出版社，2000。
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