The Algorithm on Displacement Differences Calculation and the Error

Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 167-172

Sensors & Transducers © 2014 by IFSA Publishing, S. L. http://www.sensorsportal.com

The Algorithm on Displacement Differences Calculation and the Error of Surface Displacement Measuring Device 1, 2 1

Ping GAN, 2 Guowen HU, 2 Zhenzhen LI, 2 Zhuo CHEN, 2 Xiao CHENG, 3 He HUANG, 3 Xiaosong ZHANG

State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing, China 2 College of Communication Engineering, Chongqing University, Chongqing, China 3 Chongqing Communications Research and Design Institute, Chongqing, China Tel.: +8615023036297, fax: +86-02365103050 E-mail: [email protected] Received: 23 October 2013 /Accepted: 9 January 2014 /Published: 31 January 2014

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 accuracy in many aspects. This paper would present a new displacement calculation based on coordinating transformation and employ 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 effectively eliminate the fixed bias of surface displacement measuring. Copyright © 2014 IFSA Publishing, S. L. Keywords: Angle sensor, Coordinator transformation, Error correction, PSO algorithm.

1. Introduction Surface displacement measuring devices have been widely used in embankment slope monitoring, mountain slopes and other geological disaster monitoring system. 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 [1], lead to large differences between the sensor readings and the actual data, and greatly affect the measurement accuracy. All these methods, including the method of curve fitting based on least-squares [2], optimal ellipse

Article number P_1765

fitting method based on least-square principle [3], 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 only single axis. For error correction in three-axis, the method combined the tri-axial accelerometer with gyroscope is usually employed :the error compensation of MEMS sensor inertial navigation system [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 gyroscope-free IMU[8] and so on. However, it is the fact that the measurement surface displacement angle sensor usually consists of a two-axis angle meter and the circuit device, which

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Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 167-172 does not fit the two cases mentioned above. Due to the errors, an effective method must be found to correct them. This paper presents an error correction method based on coordinating transformation [9], which solves the displacement measurement accuracy problem. Considering the main error from the production, installation and welding process droop, firstly, coordinating transformation model is built. And mathematical equation between the standard values and the measured angle is established. Then the equation to get the parameters by the PSO (Particle swarm optimization algorithm) [10] is solved; eventually we can achieve the angle sensor error correction by the equation and parameters.

in the coordinating transformation considered.

method is

3. The Error Model of Angle Sensor 3.1. Coordinate Transformation Model There is Inherent bias in the sensor production, installation and welding procession, mainly reflected in two aspects: The 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. 2.

2. The 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 0-360°), with accuracy of 0.01°. Firstly, adjust the calibration to 0° plane position, and fix the anger sensor on the calibration instrument. Then round the handwheel to change the angle of the angle sensor. There are two values at each statue for each sensor: the reference angle αM, βM (read from the inclinometer) and the measured angle αN, βN (read from the sensor). Because of a variety of reasons, the two groups of angle value are not equal for each sensor, that is to say, the error is exist. In this experiment, the angle sensor is a dual-axis sensor, with design range between n±30° and accuracy of 0.01°. Let the angle sensor vary uniformly along its two axial, X-axis direction vary from –30° to +30°, in steps of 2°, Y-axis direction vary from +9.7° to +6.7°, in steps of 0.1°. The measurement errors are shown in Fig. 1.

Angle error in X-axis Angle error in Y-axis

Angle error( o)

2 1 0 -1 -2 -3 0

5

10

15

20

Samples

25

30

35

Fig. 1. Measurement error characteristics of sensor.

From the chart in Fig. 1, the abscissa are sampling points, and the ordinate are angular errors, The chart shows the error of sensor exhibit nonlinear characteristics in both directions, and building model

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(a)

(b )

Fig. 2. Fixed Bias of the sensor.

The other is that the angle between the sensor axis changes, so the originally vertical relationship is destroyed. The first case is considered in this paper. Fig. 2(a) represents a sensor in the design center, which has no angle and position offset. Fig. 2(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 theory 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. 3. According to the coordinate transformation theory [9, 11], the sensor coordinate position T can be obtained from position S through a series of rotation and displacement: 1) S→S’, rotating around the y-axis by θ; 2) S’→S”, rotating around the x-axis by φ; 3) S” →T, shifting to T. Three dimensional space coordinate transformation model can be deduced as follows: X   X   X  Y   R  Y    Y         Z T  Z  S  Z 

(1) ,

Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 167-172

z1

where [X Y Z]TT,[X Y Z]ST are the coordinates of a point P under the three-dimensional coordinates of origin S and origin T, [∆X ∆Y ∆Z]T represents the shift factor from position S to position T, R is the coordinate rotation matrix from position S to position T.

o1

y1

z

P

x1

z3 o3

z z2

S" S

S

o

o

n2

T

n1

x2 x

Fig. 4. Posture and position of sensor.

x

 x3   X   x1   y   R   y    Y  3  3  1    z3   Z  3  z1 

(2)

What measuring device obtains is angle data, which can’t be directly applied to equation (1) for the coordinate conversion calculation. Thus the method of auxiliary coordinate system is proposed to calculate the above data. Spatial posture and position of the sensor is shown in Fig. 4. M and N represent the position of the sensor in two different states: exact status and error condition status. With sensor position M and N as the origin, we get spatial coordinate system III(o2x2y2z2), III (o3x3y3z3), and establish the coordinate system I(oxyz) whose origin is O. In the coordinate system I, the coordinates of the arbitrary point P, the position M and N respectively are(x1,y1,z3) (x2,y2,z2), (x3,y3,z3). According to the coordinate transformation model principle, the coordinate of P can be got through a series of coordinate rotation and shift of M:

 x1   x2   X   y   R   y    Y  2  2  1    z1   z2   Z  2

x3

y

Fig. 3. Sensor rotation transformation process.

 cos  cos  cos sin  sin   R    sin cos  0  sin cos  sin  sin  cos  

M

O

φ n3 n θ

y

o2 y2

’

y3

N

(3)

Similarly, the coordinate of P can be got through a series of coordinate rotation and shift of N:

(4)

where [∆X ∆Y ∆Z]2T and [∆X ∆Y ∆Z]3T respectively represent the shift from M and N to P, and R2 and R3 are two transformed coordinate rotation matrix. Meanwhile, we can assume that θM, φM are the rotation angle converted from M to P, and θN, φN are the rotation angle from N to P. We can decompose the simultaneous equations (3) and (4) into equation (5).  x3   X   x2    R2   y2  -R3   y3    Y  =0    z3   Z   z2  ,

(5)

where [∆X ∆Y ∆Z]T is the displacement converted from M to N. The θM, φM in R2 and the θN, φN in R3 can not be obtained directly, thus the relationship between the rotation angle and sensor test data is showed in next chapter.

3.2. The Rotation Angle and Sensor Test Data The coordinate transformation between two coordinates in Three-dimensional graphics system [12], shows us a method to set up the rotation angle and sensor test data, which is described by the transformed coordinate matrix R as equation (6) By synthesizing equation (5), (6), the characteristic equation of measured angle sensor can be obtained as equation (7).

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Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 167-172 1  2 2 2  sin   (1  sin   sin  ) 1  (sin 2   sin 2  ) 2   sin  R   1  2 2 2  (sin   sin  )  sin    

1  2 2 2  sin  M  (1  sin  M  sin M ) 1  (sin 2  M  sin 2 M ) 2   sin M  1  2 2 2  (sin  sin  ) M M    sin  M   1  2 2 2  sin  N  (1  sin  N  sin  N ) 1  (sin 2  N  sin 2  N ) 2   sin  N   1  2 2 2  (sin  N  sin  N )   sin  N  

1

sin   (1  sin 2   sin 2  ) 2 (sin 2   sin 2  ) sin 

1 2

1

(sin 2   sin 2  ) 2 sin 

      0   1 (1  sin 2   sin 2  ) 2    1

 (sin 2   sin 2  ) 2

1 2 2 2   (sin  sin  ) M M 1   x2  2 2 2 (sin  M  sin M )       sin  M    y2  0 1    2 2 2 (sin  M  sin M )    1  z  2 2 2   2 sin M (1  sin  M  sin M )   1 1 sin  N  (1  sin 2  N  sin 2  N ) 2  (sin 2  N  sin 2  N ) 2  1   x3  X  (sin 2  N  sin 2  N ) 2           sin  N    y3   Y   0 0 1      2 2 2 (sin  N  sin  N )      1   z  Z  sin  N (1  sin 2  N  sin 2  N ) 2   3      ,

(6)

1

sin M  (1  sin 2  M  sin 2 M ) 2

where αM, βM are the angle with x, y direction measured by a measuring instrument, and αN, βN are the angle with x, y direction required by sensor testing. The characteristic parameters of a sensor, x2,y2,z2, x3,y3,z3,∆X,∆Y,∆Z are unknown. There are three steps to achieve the error correction: Firstly, the sensor is put in different states to obtain αM, βM measured by the measuring instrument and αN, βN measured by the sensor testing. Secondly, the characteristic equation (7) is solved by putting the two groups of testing data, from which the researchers could get the characteristic parameters values. Thirdly, as the characteristic parameters are fixed, it is easy to get the correction αM, βM in the case of the known test values αN, βN by 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. The new problem could be solved by the PSO algorithm.

xid (k 1)  xid (k)  vid (k 1)

vid ( k  1)    vid ( k )  c1  r1  ( pid  xid ( k ))  c2  r2  ( g id  xid ( k )) ,

(7)

(8)

(9)

where xid, vid are the particle position and velocity; ω is the inertia weight factor; c1,c2 are the learning factors; r1,r2are the random number in (0,1); pid is the particle best position; gid is the best position of particle groups; k represents the number of iterations. In this paper, the sensor characteristic parameters, d = 9, xi = =(xi1,xi2,xi3,xi4,xi5,xi6,xi7,xi8,xi9) = (x2,y2,z2,x3,y3, z3, ∆X,∆Y,∆Z).

5. Experiments

4. The Particle Swarm Optimization (PSO)

5.1. Parameter Identification

The particle swarm optimization (PSO) [10] evaluates the merits of the particle position with the fitness function; the initial position of particle are randomly generated in the sample space; each particle changes their position in the searching process by continuous learning and evolution, flying toward their own best positions (as gid, pid) and groups with best position until reaching the optimum position. The position and velocity of the particle can be updated by the formula (8), (9):

The 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 102 magnitude to 10-7 magnitude. In my experiment, setting convergence condition ≤ 10-7, the characteristic parameters are obtained as shown in Table 1.

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Degree of convergence

Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 167-172 102

102

100

100

10-2 10-4

10-2 10-4

10-6

Degree of convergence

10-8

0

200

400

600

800

1000

1200

10-6 10-8

0

200

400

102

102

100 10-2

100 10-2

10-4

10-4

10-6

10-6

10-8

0

200

400

600

600

800

1000

1200

800

1000

1200

(b)

(a)

800

number of Iterations

1000

1200

10-8

0

200

400

600

number of Iterations

(d)

(c) Fig. 5. PSO convergence process.

Table 1. The parameter identification results of calibration model (×10-7). Sensor a b c d

x2 -0.4900 -0.2588 -0.3412 -0.5892

y2 0.1140 -0.1037 -0.1806 0.2488

z2 0.1108 0.1209 0.1154 -0.0250

x3 -0.4609 -0.2318 -0.3722 -0.5968

5.2. Error Correction and Verification

Angle error( o)

Angle error( o)

Put the characteristic parameters listed in Table 1 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 y-axis direction in Fig. 6. 3 2 1 0 -1 -2 -3

Measurement data in X-axis Correcting data in X-axis

0

5

10

15

20

30

35

Samples

Measurement data in Y-axis Correcting data in Y-axis

∆X 0.0284 0.0231 0.0533 -0.0307

∆Y -0.0274 -0.0169 -0.0162 0.0048

∆Z -0.0150 0.0044 0.0021 0.0093

Table 2.Average error before and after correction.

a b c d

25

z3 0.1022 0.1219 0.1032 -0.0205

Fig. 6(a) shows the angular error comparison on x-axis direction, while Fig. 6(b) represents the comparison of the angular error on y-axis direction. The average errors of the angle before and after correction are shown in Table 2.

Sensor

(a)

3 2 1 0 -1 -2 -3

y3 0.1438 -0.1040 -0.1690 0.2664

Average error value original measured (°) 0.890977352 0.825258922 0.858127476 0.911086005

Average error value after correction (°) 0.02717192 0.02470872 0.02269208 0.02212372

It is shown that the effect after correction is well improved compared with original measured error as shown in Table 2. The accuracy for each sensor respectively increases by 32.79, 33.40, 37.82, 41.18 times.

6. Conclusions 0

5

10

15

(b)

20

25

30

35

Samples

Fig. 6. Angular error compared before and after correction.

This paper presents a sensor error correction method based on coordinate transformation. The method can effectively correct the angular sensor error with high precision and low computational complexity, which is very suitable for the large-scale

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Sensors & Transducers, Vol. 162 , Issue 1, January 2014, pp. 167-172 and low cost slope monitoring system. However, it consumes a long time and more resources by using PSO. Meanwhile, because of its own limitations of the sensor, it can only obtain angular values in two directions, whose calibration process needs help from inclinometer and can not be achieved by selfcalibration, which leads the scope application of the sensor subject to certain restrictions.

Acknowledgments

[5].

[6].

[7].

The research is supported by the Transportation Construction Projects of Ministry of Transport of the People's Republic of China (Grant No. 2011318740240) & the Visiting Scholar Foundation of State Key Laboratory of Coal Mine Disaster Dynamics and Control (Chongqing University) (Grant No. 2011DA105287 – FW201207).

[8].

[9].

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