Analytical Modeling and Experimental Assessment of Five-Dimensional Capacitive Displacement Sensor

Transcription

1 Sensors and Materials, Vol. 29, No. 7 (2017) MYU Tokyo 905 S & M 1380 Analytical Modelin and Experimental Assessment of Five-Dimensional Capacitive Displacement Sensor Jian-Pin Yu, * Wen Wan, 1 in Li, and Zhao-Zhon Zhou Collee of Mechanical Enineerin, Quzhou University, Quzhou, Zhejian Province, China 1 School of Mechanical Enineerin, Hanzhou Dianzi University, Hanzhou, Zhejian Province, China (Received April 19, 2016; accepted March 22, 2017) Keyords: five-dimensional, capacitive sensor, position, rotational disturbances This paper presents studies on a capacitive incremental displacement microsensor particularly for micro/nanopositionin applications. An incremental capacitive microsensor is capable of achievin lare-scale, hih-precision Y linear positionin; hoever, some inevitable static errors and dynamic disturbances in reality affect the linearity of the Y sinal in the form of roll, ya, and pitch movements. To realize hih-precision Y linear positionin, a symmetrical sensor modelin scheme and a novel sinal processin scheme are developed to compensate for sinal caused by rotational disturbances. At the same time, roll, ya, and pitch sinals are decoupled from Y linear sinals for possible feedback control purposes. A printed circuit board microsensor prototype for testin is constructed ith a desin featurin a 20 mm linear stroke, a 2 mm electrode pitch, and a 0.5 mm ap distance. The measured Y sinal is decreased to 0.5% ith a 4 mm stroke, hile sinal errors of rotational disturbances are no larer than The feasibility of a five-dimensional displacement measurement, includin a lare stroke, the hih-precision acquisition of Y linear displacement, and roll, ya, and pitch movements, is experimentally validated. 1. Introduction Durin the past fe decades, precise micro/nanopositionin has been required durin the processes of micromachinin and manipulation, such as in the fabrication of optical devices and semiconductors, cell manipulation, portable probe storae systems, and others. (1 4) For better control of stability, a typical precise positionin test bench for these applications is usually a prototype ith multiderees of freedom. The conventional scheme of realizin measurements ith multiderees of freedom involves the stackin of multiple sinle-axis positionin systems, hich results in a bulky and complicated structure involvin unpredictable abbe errors and cumulative misalinment errors. To overcome these shortcomins, a planar positionin system as an alternative to achieve measurements ith multi-derees of freedom has been proposed. Interferometers are usually used as measurin probes for a planar positionin system, because they offer the advantae of hih resolution, but they suffer from bulkiness and sensitivity to measurement conditions such as air pressure, temperature, and humidity. (1,2) Planar positionin systems can also be patterned in capacitor mechanisms. Huan et al. presented a microcapacitive * Correspondin author: yujianpin@zju.edu.cn ISSN MYU K.K.

2 906 Sensors and Materials, Vol. 29, No. 7 (2017) position sensor for nanopositionin applications, hich provided to linear position sinals and one rotational position sinal. (4) The linear stroke as limited to 300 μm peak-to-peak. (3) Ahn and Jeon proposed a ne smart sensor system that can measure five-dimensional motions. (1) Hoever, this sensor structure as mainly applied to a rotatin machinery. (4) Althouh capacitive sensin arrays may suffer from external noise, the capacitive approach still has some intrinsic advantaes, such as fast response, hih repeatability, and insensitivity to temperature variation. The authors have developed a planar capacitive sensor capable of to-dimensional measurement. The interpretation of displacement depends on the results of measurin the periodic variation in capacitance caused by the chane in the overlappin area of sensin electrodes. Hoever, the main shortcomins of the proposed sensor are the impact from static errors (mainly as misalinment errors) and dynamic disturbances (kinematic and vibration errors) in the forms of sensitivity differences and phase-shift errors. This paper presents an improved capacitive incremental displacement microsensor. To achieve hih-precision Y linear positionin, a symmetrical sensor modelin scheme and a phase-shift arctanent processin scheme ere developed to compensate for sinal deformations caused by rotational disturbances, hile at the same time, three rotational sinals ere decoupled from Y sinals for latter feedback control purposes. This paper is oranized as follos. A complete description of the basic sensor structure is presented in Sect. 2. In Sect. 3, the sensor mode is analyzed ith respect to harmonic errors. In Sect. 4, a novel scheme of five-dimensional displacement measurements is proposed for enhancin linearity and derivin three rotational sinals. After a detailed description of the sensor system prototype in Sect. 5, extensive experimental results are presented. 2. Basic Sensor Structure The basic sensor structure of a typical capacitive planar positionin system is illustrated in Fi. 1. The sensor has a periodic electrode pattern on both a movin plate (MP) and a fixed plate (FP) to maximize capacitance chane. In the fiure, is the common electrode idth, l is the sensin electrode lenth, is the ap distance, and L is the vertical distance from the sensin electrodes to the assumed center. Y S N SYQ L SY N l SQ Fi. 1. (Color online) Basic sensor structure of a typical capacitive planar positionin system.

3 Sensors and Materials, Vol. 29, No. 7 (2017) 907 Furthermore, there is a pair of normal and quadrature sensin electrodes on the FP for both and Y displacement demodulations, namely, S N, S Q, SY N, and SY Q. Settin aside frine effects, sinals from sensin electrodes are all in perfect periodic trianular aveforms iven that the common electrodes on the MP are parallel. Hoever, in reality, frine effects ould introduce a severe uncertain distortion to sinal aveforms. To clarify the frine effects on sinal, a full capacitance model as calculated firstly usin Maxell s equation: (5) C(x) = Q = ε 0 ε r V input 2 n=odd 4 k n sinh (k n ) cos(k nx), (1) here k n = nπ. As in Eq. (1), electrode idth is implied to be a key factor that determines the ratio of sensor aveform to ap distance (/). Fiure 2 illustrates a comparison of the standard deviations of the aveform deformation errors of the proposed sensor sinals to ideal trianular and sinusoidal aveforms for different / ratios. The comparison indicates that the actual aveform is more closely dependent on sinusoidal aveforms. Thus, in the actual sensor model, sinals are considered sinusoidal aveforms. Nevertheless, an improvement in modelin can be attempted usin a numerical approach and a more elaborate curve fittin. Based on these conclusions, both and Y positions are retrieved by a standard arctanent operation. D = π arctan( Q N ), D Y = π arctan(y Q Y N ) (2) 3. Sensor Model ith Respect to Harmonic Errors Harmonic errors can be caused by various sources, for instance, static misalinment beteen MP and FP electrodes durin system interation, dynamic couplin beteen the Y linear and rotational directions, and external disturbances such as shock and vibration, hich affect the sensor system in the form of roll, ya, and pitch movements. Those harmonic disturbances are usually minimal. In Fi. 3, ya movement can be seen as a combination of to separate linear movements in the and Y directions. Linear movements Y Dθz SYQ θ L SY N Fi. 2. (Color online) Comparison of standard deviations of aveform deformation errors for different / ratios. Fi. 3. (Color online) Sensor model considerin ya rotational disturbance.

4 908 Sensors and Materials, Vol. 29, No. 7 (2017) of S N and S Q in the direction cause each to have a phase-shift error as shon in Eq. (3). Similarly, linear movements of SY N and SY Q in the Y direction cause each to have a phase-shift error as shon in Eq. (4). N = cos[π (D + D θz ) Y N = cos[π (D Y D θz ) ] Q = sin[π (D D θz ) ] (3) ] Y Q = sin[π (D Y + D θz ) ] (4) D θz = L sin θ Z = Lθ Z (5) Here, θ Z is the ya anle of the MP and D θz is the equivalent linear movement of the MP in the measurin direction induced by the ya anle. Movements caused by roll and pitch anles can also be considered a combination of a linear movement in the Z direction and another in the or Y direction as shon in Fi. 4. Hoever, linear movements in the and Y directions in this study are nelected because they are very small compared ith the linear movement in the Z direction. Thus, influences of roll and pitch movements are considered the same as vertical movements, affectin only the amplitude of a sensor sinal. Then, the actual sinal models aainst roll, ya, and pitch disturbances are illustrated by Eqs. (6) and (7). N = Y N = cos[π (D + D θz ) ] Q + Lθ x = sin[π (D D θz ) ] (6) Lθ x cos[π (D Y D θz ) ] Y Q + Lθ y = sin[π (D Y + D θz ) ] (7) Lθ y Under these circumstances, if the actual sinal models described by Eqs. (6) and (7) are substituted into Eq. (2) to calculate Y displacement of the MP, a harmonic error ill be introduced into sinal processin, thereby drastically reducin sinal linearity. 4. Five-Dimensional Sensor Prototype System Fiure 5 presents the schematic structure of the proposed five-dimensional capacitive displacement microsensor system. Common electrodes on the MP are in a similar periodic pattern to those of other capacitive planar positionin systems. Hoever, hat is different on the FP is that L θ y SY Q Z SY N Fi. 4. (Color online) Sensor model considerin roll and pitch anular disturbances.

5 Sensors and Materials, Vol. 29, No. 7 (2017) 909 Y Pitch SY1N SY 1Q S 2Q L S Ya L N S 1N S2N l SY 2Q (a) L Q SY 2N S 1Q Roll S 2N SY 2Q Fi. 5. (Color online) Schematic vies of the proposed capacitive displacement microsensor system: (a) front and (b) side vies. Y (b) SY 2N S 1Q MP FP there are eiht sensin electrodes on the FP symmetrically positioned around the rotational center (an assumed oriin), namely, S 1N, S 1Q, S 2N, S 2Q, SY 1N, SY 1Q, SY 2N, and SY 2Q. In the fiure, is the electrode idth, l is the sensin electrode lenth, L N and L Q are the distances of the sensin electrodes S N and S Q from the rotational center alon their measurin direction, respectively, and L S is the distance from the sensin electrodes to the rotational center in a direction perpendicular to the measurin direction. Based on the analysis results in Sect. 3, e consider the folloin mathematical sinal model of the sensin system in Fi. 5: 1N = K 1N 1N 1Q = K 1Q 1Q 2N = K 2N 2N 2Q = K 2Q 2Q Y 1N = K Y1NY 1N Y 1Q = K Y1QY 1Q Y 2N = K Y2NY 2N Y 2Q = K Y2QY 2Q, (8) here K 1N, K 1Q, K 2N, K 2Q, K Y1N, K Y1Q, K Y2N, and K Y2Q are the approximate relationships of sensor sensitivity to roll and pitch anles, and 1N = cos[π(d + D θz )/] 1Q = sin[π(d D θz )/] Y 1N = cos[π(d Y + D θz )/] Y 1Q = sin[π(d Y D θz )/] 2N = cos[π(d D θz )/] 2Q = sin[π(d + D θz )/], Y 2N = cos[π(d Y D θz )/] Y 2Q = sin[π(d Y + D θz )/], (9) (10) K 1N = K 1Q = + L N θ y + L S θ x + L Q θ y L S θ x K 2N = K 2Q = L N θ y L S θ x L Q θ y + L S θ x K Y1N = K Y1Q = + L N θ x L S θ y + L Q θ x + L S θ y K Y2N = K Y2Q =. L N θ x + L S θ y L Q θ x L S θ y (11)

6 910 Sensors and Materials, Vol. 29, No. 7 (2017) By substitutin Eq. (8) into Eq. (11), roll and pitch movements are acquired. Their specific values depend closely and only on the amplitude of a readout sinal. θ x = θ y = ( (L N + L Q ) K Y1N K Y1Q K Y2N ( 2(L N + L Q ) K Y2Q K 1N K 1Q K 2N K 2Q ) ) (12) Furthermore, by normalizin Eq. (8), impacts of roll and pitch movements are decoupled, resultin in a set of sinals containin only Y linear movements and ya movement as in Eqs. (9) and (10). Y displacements are obtained by a specific arctanent function as shon in Eq. (13) and, at the same time, ya movement is determined from Eq. (14). Nominal Y movements and ya movement are decoupled from each completely: D = π arctan 1Q + 2Q 1N + 2N D Y = π arctan Y 1Q + Y 2Q Y 1N + Y 2N, (13) θ Z = 2πL S [arctan( 2Q 1N ) + arctan( 1Q 2N )] = 2πL S [arctan( Y 2Q Y 1N ) + arctan( Y 1Q Y 2N )]. (14) It is also noted that the proposed Eq. (13) considers a set of quadrature sinals to calculate MP displacement in both the and Y directions, hich ould balance frine effects. The analyses imply that the aveform of Y movement sinals caused by rotational disturbances can effectively be eliminated. Also, roll, ya, and pitch movements can be enerated in the proposed scheme. 5. Experimental Results A test bench for the proposed sensor system is constructed as shon in Fi. 6, consistin of a micromotion stae, a vertical motion stae, to tiltin motion staes, a rotational motion stae, a capacitive sensor, and an interpolation system. The MP and FP are fabricated as printed circuit boards. A NANMTIN motor ith a resolution of 10 nm is applied to provide the displacement of the micro-motion stae. Tiltin and rotatin staes are mounted to adjust installation errors and system calibration. From the analysis results in Sect. 2, e concluded that electrode idth is a key factor in determinin the ratio of sensor aveform to ap distance (/). As lon as the / ratio is less than 2, the actual aveform is closely dependent on sinusoidal aveforms. Minimizin the / ratio optimizes sinal linearity; hoever, sensor sensitivity is notably decreased. To keep a balance beteen sinal linearity and sensor sensitivity, the / ratio is set to 4. Fiure 7 shos the demodulation outputs of the proposed sensor, namely, 1N, 2N, 1Q, and 2Q, in a sinle period. To better clarify the capability of a five-dimensional displacement measurement, four different sets of rotational inputs are provided in the tests: in set I, the pitch anle θ y is 0.1 and the ya anle θ z is 0.2 ; in set II, the values are 0.2 and 0.4 ; and in sets III and IV, the values are 0.4 and 0.8, and 0.8 and 2.0, respectively. Durin all tests, the roll anle θ x is kept at 0. As can be seen, rotational disturbances introduce severe sensitivity differences and phase-shift errors into output aveforms.

7 911 Sensors and Materials, Vol. 29, No. 7 (2017) Rotational motion stae Tiltin motion stae-a Tiltin motion stae-b PC Vertical motion stae Movin direction Analo sinal Motion stae MP FP Diital sinal GWINSTEK GFG-8016G sinal enerator Frequency: 10 khz Sensor ALI linear motion system Travelin rane: 100 mm Resolution: 10 nm NI USB 6259BNC 16 bit DAQ system Q Demodulated outputs (V) Demodulated outputs (V) 1N (a) 2N 2Q (b) (c) Demodulated outputs (V) (Color online) Test bench of the proposed five-dimensional capacitive displacement sensor system. Demodulated outputs (V) Fi (d) Fi. 7. (Color online) Demodulated outputs 1N, 2N, 1Q, and 2Q in a sinle period: (a) set I, (b) set II, (c) set III, and (d) set IV. Based on the analysis results in Sect. 3, roll and pitch anles affect sinal amplitudes only; therefore, by substitutin the amplitudes of the acquired sinals into Eq. (12), roll and pitch disturbances can be determined. Fiure 8 shos a plot of the decoupled rotational outputs. The calculated results are stronly dependent on ideal inputs. For a stroke, roll and pitch anle errors are no larer than 0.01, hich is acceptable for possible closed-loop control purposes. Fiure 9 shos plots of the normalized aveforms 1N, 2N, 1Q, and 2Q, ith sensitivity differences removed. Hoever, phase-shift errors remain. If acquired normalized aveforms are considered ideal quadrature aveforms, then phase-shift errors diminish measurement accuracy. Fiure 10(a) shos a plot of the arctanent aveform directly from Eq. (2). Acquired aveforms exhibit a severe distortion. For sets I to IV, sinal nonlinearities rane from 1.02 to 6.75%, as ya

8 912 Sensors and Materials, Vol. 29, No. 7 (2017) Decoupled anles ( ) Pitch anle θ y Roll anle θ x Fi. 8. Pitch anular inputs ( ) (Color online) Decoupled roll and pitch rotational outputs. Normalized outputs 1N - 1Q 2N 2Q Normalized outputs Normalized outputs (a) (b) (c) (d) Fi. 9. (Color online) Normalized outputs 1N, 2N, 1Q, and 2Q: (a) set I, (b) set II, (c) set III, and (d) set IV. Normalized outputs Sensor outputs (mm) D Set I Set II 1.02% (Set I) 1.55% (Set II) Set III Set IV 2.51% (Set III) 6.75% (Set IV) Sensor outputs (mm) D 0.39% (Set I) 0.50% (Set II) 0.54% (Set III) 0.56% (Set IV) (a) (b) Fi. 10. (Color online) Acquired nominal displacement in the direction: (a) conventional and (b) proposed schemes. inputs increase from 0.2 to 2. Apparently, Eq. (2) does not meet the requirements for precision measurement, especially in the case of rotational disturbances. The decouplin method in Eq. (13) removes the influences of phase-shift errors. Fiure 10(b) shos the acquired nominal displacement in the direction. As ya inputs increase from 0.2 to 2, sinal nonlinearities remain constant, only ranin from 0.39 to 0.56%. This result validates the conclusion reardin Eq. (13).

9 Sensors and Materials, Vol. 29, No. 7 (2017) 913 Decoupled ya anles θz ( ) Set I Set II Set III Set IV Fi. 11. (Color online) Decoupled ya anles compared ith ideal inputs. Table 1 Demodulated ya anles. Ideal input ( ) Averae value ( ) Standard deviation ( ) Fiure 11 shos decoupled ya anles compared ith ideal inputs, the averae values of hich are listed in Table 1. As can be seen, as the ya input increases from 0.2 to 2, the standard deviation of the decoupled output increases as ell; more ross errors must be eliminated before the final calculation. The specific reason for this is that sinal aveforms are not actually in perfect sinusoidal or cosine model. Nevertheless, the averae decoupled ya anles ithin a full period are quite cohesive to the ideal inputs, hich are no larer than Conclusions In this paper, a novel five-dimensional capacitive displacement sensor is proposed. n the basis of the dataform of the sensor model ith respect to harmonic errors, impacts of rotational disturbances on Y linear displacement sinals are decoupled, and hih precision Y displacement measurement is achieved, hile additional roll, ya, and pitch movements are acquired. Experiments sho that aveform nonlinearities in the Y displacement sinals are decreased to only 0.5%, hile sinal errors in the demodulated rotational disturbances are no larer than 0.01 on a 2 scale. The advantaes of the five-dimensional displacement measurement scheme are validated. Acknoledments This ork as supported by National Natural Science Foundation of China (Grant Nos , , and ). References 1 H. J. Ahn and S. Jeon: Mech. Syst. Sinal Process. 29 (2012) Y. Saito, Y. Arai, and W. Gao: Sens. Actuators, A 150 (2009) W. Gao, S. Dejima, and S. Kiyono: Sens. Actuators, A 117 (2005) H. Huan, J. I. Lee, N. Ramakrishnan, and M. Bedillion: Mechatronics 20 (2010) J. P. Yu, W. Wan,. Li, and Z. Z. Zhou: Meas. Sci. Rev. 16 (2016) 127.