The Algorithm on Displacement Differences Calculation and the Error of Surface Displacement Measuring Device

Transcription

1 Sensors & ransducers, Vol. 6, Issue, Januar 4, pp Sensors & ransducers 4 b IFSA Publishing, S. L. he Algorithm on Displacement Differences Calculation and the Error of Surface Displacement easuring Device, Ping GA, Guowen HU, Zhenzhen LI, Zhuo CHE, Xiao CHEG, He HUAG, Xiaosong ZHAG State Ke Laborator of Coal ine Disaster Dnamics and Control, Chongqing Universit, Chongqing, China College of Communication Engineering, Chongqing Universit, Chongqing, China Chongqing Communications Research and Design Institute, Chongqing, China el.: , fax: vigor_gp@6.com Received: October /Accepted: 9 Januar 4 /Published: Januar 4 Abstract: Angle sensor is the core component in the surface displacement measuring device, whose manufacturing process, fixing position and internal device performance can affect the testing accurac in man aspects. his paper would present a new displacement calculation based on coordinating transformation and emplo particle swarm method to calibrate the data. Experimental results show that the amended data through this method is quite close to standard test bench data, indicating that the proposed displacement differences calculation and error correction algorithm can effectivel eliminate the fixed bias of surface displacement measuring. Copright 4 IFSA Publishing, S. L. Kewords: Angle sensor, Coordinator transformation, Error correction, PSO algorithm.. Introduction Surface displacement measuring devices have been widel used in embankment slope monitoring, mountain slopes and other geological disaster monitoring sstem. Angle sensor is the core device of the surface displacement measuring device, whose manufacturing process, fixed position and the internal differences in device performance, as well as the presence of a fixed sensor deviation, proportional error, non-orthogonal error and random error and other measurement error [], lead to large differences between the sensor readings and the actual data, and greatl affect the measurement accurac. All these methods, including the method of curve fitting based on least-squares [], optimal ellipse fitting method based on least-square principle [], error compensation and parameter identification of circular grating angle sensors [4], and nonlinear errors correction of sensors based on neural network [5] can make a good correction when applied to error correction at the onl single axis. For error correction in three-axis, the method combined the tri-axial accelerometer with groscope is usuall emploed :the error compensation of ES sensor inertial navigation sstem [6], quaternion-based tilt angle correction method for a hand-held device using an inertial measurement unit [7], sensor fusion algorithm and calibration for a groscope-free IU[8] and so on. However, it is the fact that the measurement surface displacement angle sensor usuall consists of a two-axis angle meter and the circuit device, which Article number P_765 67

2 Sensors & ransducers, Vol. 6, Issue, Januar 4, pp does not fit the two cases mentioned above. Due to the errors, an effective method must be found to correct them. his paper presents an error correction method based on coordinating transformation [9], which solves the displacement measurement accurac problem. Considering the main error from the production, installation and welding process droop, firstl, coordinating transformation model is built. And mathematical equation between the standard values and the measured angle is established. hen the equation to get the parameters b the PSO (Particle swarm optimization algorithm) [] is solved; eventuall we can achieve the angle sensor error correction b the equation and parameters. in the coordinating transformation method is considered.. he Error odel of Angle Sensor.. Coordinate ransformation odel here is Inherent bias in the sensor production, installation and welding procession, mainl reflected in two aspects: he one is displacement of the sensor related to the design center leading to an accompanied slight twist angle, forming uncertain spatial angles, as shown in Fig... he Error Calibration of Angle Sensor Using Inclinometer calibration instrument as experiment platform, the angle of the sensor is tested. Inclinometer calibration measures angle in a wide range (range at -6 ), with accurac of.. Firstl, adjust the calibration to plane position, and fix the anger sensor on the calibration instrument. hen round the handwheel to change the angle of the angle sensor. here are two values at each statue for each sensor: the reference angle α, β (read from the inclinometer) and the measured angle α, β (read from the sensor). Because of a variet of reasons, the two groups of angle value are not equal for each sensor, that is to sa, the error is exist. In this experiment, the angle sensor is a dual-axis sensor, with design range between n± and accurac of.. Let the angle sensor var uniforml along its two axial, X-axis direction var from to +, in steps of, -axis direction var from +9.7 to +6.7, in steps of.. he measurement errors are shown in Fig.. Angle error( o ) Angle error in X-axis Angle error in -axis Samples Fig.. easurement error characteristics of sensor. From the chart in Fig., the abscissa are sampling points, and the ordinate are angular errors, he chart shows the error of sensor exhibit nonlinear characteristics in both directions, and building model (a) (b) Fig.. Fixed Bias of the sensor. he other is that the angle between the sensor axis changes, so the originall vertical relationship is destroed. he first case is considered in this paper. Fig. (a) represents a sensor in the design center, which has no angle and position offset. Fig. (b) indicates that the sensor deviates from the design center, which has the location and angle offset. In order to eliminate the impact of spatial angles and displacements, we can use the coordinating transformation theor and related algorithms to correct the offset of installation center, the angle difference, and approach the design center coordinates. Sensor rotation transformation process is shown in Fig.. According to the coordinate transformation theor [9, ], the sensor coordinate position can be obtained from position S through a series of rotation and displacement: ) S S, rotating around the -axis b θ; ) S S, rotating around the x-axis b φ; ) S, shifting to. hree dimensional space coordinate transformation model can be deduced as follows: X X X R () Z Z Z, S 68

3 Sensors & ransducers, Vol. 6, Issue, Januar 4, pp where [X Z],[X Z] S are the coordinates of a point P under the three-dimensional coordinates of origin S and origin, [ X Z] represents the shift factor from position S to position, R is the coordinate rotation matrix from position S to position. z z o P x z z S" S o φ n n θ n n S Fig.. Sensor rotation transformation process. x o O o x z o x Fig. 4. Posture and position of sensor. x x X R z z Z x (4) cos cos cossin sin R sin cos sin cos sinsin cos () What measuring device obtains is angle data, which can t be directl applied to equation () for the coordinate conversion calculation. hus the method of auxiliar coordinate sstem is proposed to calculate the above data. Spatial posture and position of the sensor is shown in Fig. 4. and represent the position of the sensor in two different states: exact status and error condition status. With sensor position and as the origin, we get spatial coordinate sstem III(o x z ), III (o x z ), and establish the coordinate sstem I(oxz) whose origin is O. In the coordinate sstem I, the coordinates of the arbitrar point P, the position and respectivel are(x,,z ) (x,,z ), (x,,z ). According to the coordinate transformation model principle, the coordinate of P can be got through a series of coordinate rotation and shift of : x x X R z z Z () Similarl, the coordinate of P can be got through a series of coordinate rotation and shift of : where [ X Z] and [ X Z] respectivel represent the shift from and to P, and R and R are two transformed coordinate rotation matrix. eanwhile, we can assume that θ, φ are the rotation angle converted from to P, and θ, φ are the rotation angle from to P. We can decompose the simultaneous equations () and (4) into equation (5). x x X R - R = z z Z, (5) where [ X Z] is the displacement converted from to. he θ, φ in R and the θ, φ in R can not be obtained directl, thus the relationship between the rotation angle and sensor test data is showed in next chapter... he Rotation Angle and Sensor est Data he coordinate transformation between two coordinates in hree-dimensional graphics sstem [], shows us a method to set up the rotation angle and sensor test data, which is described b the transformed coordinate matrix R as equation (6) B snthesizing equation (5), (6), the characteristic equation of measured angle sensor can be obtained as equation (7). 69

4 Sensors & ransducers, Vol. 6, Issue, Januar 4, pp sin ( sin sin ) sin ( sin sin ) (sin sin ) (sin sin ) (sin sin ) sin sin R (sin sin ) (sin sin ) sin sin ( sin sin ) (6) sin ( sin sin ) sin ( sin sin ) sin sin (sin sin ) (sin sin ) (sin sin ) (sin sin ) (sin sin ) x z sin sin ( sin sin ) sin ( sin sin ) sin ( sin sin ) (sin sin ) x X (sin sin ) (sin sin ) sin sin (sin sin ) (sin sin ) z Z sin sin ( sin sin ), (7) where α, β are the angle with x, direction measured b a measuring instrument, and α, β are the angle with x, direction required b sensor testing. he characteristic parameters of a sensor, x,,z, x,,z, X,, Z are unknown. here are three steps to achieve the error correction: Firstl, the sensor is put in different states to obtain α, β measured b the measuring instrument and α, β measured b the sensor testing. Secondl, the characteristic equation (7) is solved b putting the two groups of testing data, from which the researchers could get the characteristic parameters values. hirdl, as the characteristic parameters are fixed, it is eas to get the correction α, β in the case of the known test values α, β b equation (7). However, it is so difficult to solve the overdetermined equation listed as the second and third steps because of large amounts of data. he new problem could be solved b the PSO algorithm. 4. he Particle Swarm Optimization (PSO) he particle swarm optimization (PSO) [] evaluates the merits of the particle position with the fitness function; the initial position of particle are randoml generated in the sample space; each particle changes their position in the searching process b continuous learning and evolution, fling toward their own best positions (as gid, pid) and groups with best position until reaching the optimum position. he position and velocit of the particle can be updated b the formula (8), (9): x ( k) x ( k) v ( k ) (8) id id id vid ( k ) vid ( k) cr ( p x ( k)) c r ( g x ( k)) id id id id, (9) where x id, v id are the particle position and velocit; ω is the inertia weight factor; c,c are the learning factors; r,r are the random number in (,); p id is the particle best position; g id is the best position of particle groups; k represents the number of iterations. In this paper, the sensor characteristic parameters, d = 9, x i = =(x i,x i,x i,x i4,x i5,x i6,x i7,x i8,x i9 ) = (x,,z,x,, z, X,, Z). 5. Experiments 5.. Parameter Identification he sensor correction parameters are different from each other due to the deviations in different sizes. Four sensors are taken as samples, and the PSO convergence process of parameter identification is shown in Fig. 5. As shown, with the increase in the number of iterations, the degree of convergence falls from magnitude to -7 magnitude. In m experiment, setting convergence condition -7, the characteristic parameters are obtained as shown in able. 7

5 Sensors & ransducers, Vol. 6, Issue, Januar 4, pp Degree of convergence Degree of convergence (a) (b) number of Iterations (c) Fig. 5. PSO convergence process. number of Iterations (d) able. he parameter identification results of calibration model ( -7 ). Sensor x z x z X Z a b c d Error Correction and Verification Put the characteristic parameters listed in able into the equation (7), the correction value of angle sensor can be obtained. Compare the angle error value before and after correction, and draw the result on x-axis direction and -axis direction in Fig. 6. Angle error( o ) Angle error( o ) easurement data in X-axis Correcting data in X-axis (a) easurement data in -axis Correcting data in -axis Samples (b) Samples Fig. 6. Angular error compared before and after correction. Fig. 6(a) shows the angular error comparison on x-axis direction, while Fig. 6(b) represents the comparison of the angular error on -axis direction. he average errors of the angle before and after correction are shown in able. able.average error before and after correction. Sensor Average error value Average error value original measured ( ) after correction ( ) a b c d It is shown that the effect after correction is well improved compared with original measured error as shown in able. he accurac for each sensor respectivel increases b.79,.4, 7.8, 4.8 times. 6. Conclusions his paper presents a sensor error correction method based on coordinate transformation. he method can effectivel correct the angular sensor error with high precision and low computational complexit, which is ver suitable for the large-scale 7

6 Sensors & ransducers, Vol. 6, Issue, Januar 4, pp and low cost slope monitoring sstem. However, it consumes a long time and more resources b using PSO. eanwhile, because of its own limitations of the sensor, it can onl obtain angular values in two directions, whose calibration process needs help from inclinometer and can not be achieved b selfcalibration, which leads the scope application of the sensor subject to certain restrictions. Acknowledgments he research is supported b the ransportation Construction Projects of inistr of ransport of the People's Republic of China (Grant o. 8744) & the Visiting Scholar Foundation of State Ke Laborator of Coal ine Disaster Dnamics and Control (Chongqing Universit) (Grant o. DA587 FW7). References []. V. udic, Displacement sensor axis misalignment error, in Proceedings of the 48 th International Smposium on ultimedia Signal Processing and Communications ELAR, Zadar, Croatia, June 6, pp []. F. Stefano, F. Alfonso, G. Fulvio, et al., Least squares estimation and Cramér-Rao tpe lower bounds for relative sensor registration process, IEEE ransactions on Signal Processing, Vol. 59, o., arch, pp []. Kenichi Kanatani, Ellipse fitting with hperaccurac, IEICE ransactions on Information and Sstems, Vol. 89-D, o., October 6, pp [4]. C. G. Ai,. Chu, H. X. Sun, et al., Eccentric testing of benchmark circular grating and compensation of angular error, Optics and Precision Engineering, Vol., o.,, pp [5]. J. Xiaoan, B. ujun, onlinear errors correction of pressure sensor based on BP neural network, in Proceedings of the Conference on Intelligent Sstems and Applications, Wuhan, China, -4 a 9, pp. -4. [6]. L. I. Iozan, J. Collin, J. akala, C. Rusu, Improved indoor navigation sstem based on ES technolog, in Proceedings of the th International Smposium on Signals, Circuits and Sstems (ISSCS), Iasi, June Jul, pp. -4. [7]. W. Seong-Hoon, P. eda, et al., A quaternion-based tilt angle correction method for a hand-held device using an inertial measurement unit, in Proceedings of the 4 th Annual Conference of the IEEE Industrial Electronics Societ, Orlando, USA, - ovember 8, pp [8]. P. Schopp, L. Klingbeil, et al., Sensor fusion algorithm and calibration for a groscope-free IU, Procedia Chemistr, Vol., Issue, September 9, pp. 6. [9]. Z. Wenxian,. Benzao, on-linear adjustment model of three-dimensional coordinate transformation, Geomatics and Information Science of Wuhan Universit, Vol. 8, o. 5, October, pp []. K. Fakheredine, A fast-multipling PSO algorithm for real-time multiple object tracking, International Journal of Computer Applications, Vol. 6, o., December. []. Hakan S. Kutoglu, evfik Aan, he role of common point distribution in obtaining reliable parameters for coordinate transformation, Applied athematics and Computation, Vol. 76, Issue, 5 a 6, pp []. Shejiao Xu, he coordinate transformation between two coordinates in three-dimensional graphics sstem, Journal of Xidian Universit, Vol., o., September 996, pp Copright, International Frequenc Sensor Association (IFSA) Publishing, S. L. All rights reserved. ( 7