At present, rotation angle measurement is an impor-tant component of geometrical measurement tech-nology and has been developed completely [1–4].Measurement accuracy has reached a high level.Now the main task is to improve accuracy and reso-lution of measurement.Researchers pay close attention to the optical an-
gle-measurement method, which is noncontact andprovides high accuracy and high sensitivity. In par-ticular, the development of a stable laser sourcemakes it possible to use it in industrial field mea-surements. As a result, the optical rotation-angle-measurement method is being used more and morewidely, and various new optical rotation angle mea-surement methods have emerged. At present, be-
sides the optical dividing head method and thepolygon method, optical rotation angle measurementmethods also include the optical encoder method, thediffraction method, the autocollimation method, thefiber method [5], the acousto-optic modulation meth-od, the circular grating method, the optical internal-reflection method, the laser interference method, theparallel interferogram method, and the ring lasermethod. Many of these methods have been appliedsuccessfully in the precision measurement of smallangles and they share the merits of high accuracyand sensitivity.
Currently, the circular grating, optical internal-reflection, laser interference, ring laser [6], andvision angle measurement methods are based onstraight lines [7].
Li and Yu [4] have used the vision method to mea-sure angles in-plane. They proposed an angle-measurement method based on image, which usesa straight line as a characteristic symbol. However,
there is a restriction in the method that the anglesbetween direction of the straight line and the sam-pling direction cannot be 0° and 45°. In addition,the measurement must further consider the prob-lems of lens distortion when measuring the anglesprecisely in a wide rotation range. The lens distortionof the vision measurement device is not corrected inthis method, so the measurement cannot reach highprecision when the rotation angles are large.To solve the problems in the measurement device,
it is necessary to eliminate the distortion error whenthe rotation angles are in a wide rotation range, sothis paper presents a high-precision angle measure-ment method of vision measurement based on a spotarray [8,9].
2. Principle of Vision Angle-Measurement Based onSpot Array
The angle-measurement device with machine visionis made up of an area scan CCD camera and a cali-bration pattern with a spot array, as shown in Fig. 1.The calibration pattern with spot array is put on theend face of the measured rotating pattern. The de-tecting CCD is fixed in a location that is a certain dis-tance from the measured rotating pattern. The grayimages of the spot array on the calibration patternareis captured by the CCD camera, and then the de-tection algorithm of the position of the points is usedto yield the image plane coordinates of the targetpoints. Next, the projection model of the CCD camerais built to modify the errors of lens distortion by ca-librating the camera. Then the object plane coordi-nates of the spot array are acquired. With thetransformation equations to yield the rotating angleof the spot array, the rotation angle in-plane of themeasured object is obtained.In the experiment, the CCD camera is 2000 × 2000
pixels, with a single pixel size of 7:4 μm× 7:4 μm anda focus of the optical lens of 50mm, which affects thework distance, because the image of the spot arrayshould be in the center of the field of view. Ourcalibration pattern is 140mm × 140mm, so the workdistance is about 900mm.
A. Structure of the Calibration Pattern
The glass pattern is processed to be a light transmis-sion circular hole array by semiconductor etching.The distance between adjacent holes is 5mm, and
the horizontal and vertical directions are orthogonal.The precision of the holes is 0:001mm. The holes areilluminated by the LED light source from the back ofthe calibration pattern. There are 29 × 29 holes onthe pattern. The spot array of the calibration patternis shown in Fig. 2.
B. Obtain the Image Coordinates of the Spot Array
The spot array of the calibration pattern will be im-aged on the image plane of the CCD camera throughthe imaging lens. The gray images of the spot arrayare acquired by CCD camera. The gray weightedmethod is used to measure the characteristic pointsthat represent the positions of the spots on the imageplane of the CCD camera.
The CCD sensor transforms the optical image sig-nals to gray image signals. For the spots that arewith-in the Cartesian coordinate system, a photoelectricsensor acquires the light intensity distributionVði; jÞ,and the computer can acquire the two-dimensionaldigital images Vði; jÞ of the spots. The gray signalsf ði; jÞ of each pixel ði; jÞ on the images correspondwiththe light intensity distribution Vði; jÞ of the spotimages on the photosensitivity plane of the CCD sen-sor, where ði; jÞ stands for the location order of thephotosensitivity pixels on the photosensitivity plane.ði; jÞ, which corresponds to the x, y coordinates of theCartesian coordinate system, stands for the numberof rows and columns of the pixel coordinate locationsof the corresponding points on the area scan CCD.
The gray weighted center of the spots is calculatedto represent the position of the spots from the digitalgray image:
Xc ¼
P
i;jf ði; jÞ · i
P
i;jf ði; jÞ ; ð1Þ
Yc ¼
P
i;jf ði; jÞ · j
P
i;jf ði; jÞ : ð2Þ
Fig. 1. Schematic diagram of in-plane angle measurement devicewith machine vision based on spot array.
Fig. 2. Schematic diagram of the calibration pattern with spotarray.
Thus, the image plane coordinates of the 29 × 29spots is acquired by a gray weighted algorithm.
C. Adjustment of the CCD camera
Because of the distortions of the optical imaging sys-tem of the CCD camera [2], there is a certain degreeof distortion on the image coordinates of the spotarray of the calibration pattern. If the image coordi-nates of the spot array are used to calculate the an-gles directly, it is difficult to measure the rotationangle with high precision. The mathematical modelof the distortions of the CCD camera is created bycamera calibration to improve the detection accuracyof the measurement device.Before angle-measurement, the rotating parts are
fixed at a certain initial angle position, the object co-ordinate system, UOV, can be built up at the initialposition. The image coordinate and the object coordi-nate of the spot array at the initial angle position areused to build up the projection relationship betweenthe object coordinate system UOV and the image co-ordinate system XOY [3].For any point on the calibration pattern, object
plane coordinate ðu; vÞ can be expressed by imageplane coordinate ðx; yÞ as Eqs. (3) and (4), determinedbased on the least squares method [10,11]:
u ¼ a0 þ a1xþ a2yþ a3x2 þ a4y2 þ a5xyþ a6x3
þ a7y3 þ a8x2yþ a9xy2 þ a10x4 þ a11y4
þ a12xy3 þ a13x3yþ a14x2y2; ð3Þ
v ¼ b0 þ b1xþ b2yþ b3x2 þ b4y2 þ b5xyþ b6x3
þ b7y3 þ b8x2yþ b9xy2 þ b10x4 þ b11y4
þ b12xy3 þ b13x3yþ b14x2y2: ð4ÞEquations (3) and (4) are used to build the relation-ship between the object plane and the image plane bythe projection model. Thus, for the object coordinatesand image coordinates of the 841 standard points inthe field of view, 841 groups of this equation can be
listed. Then the nonlinear least squares method isused to yield the solution of the equation parameters(a0;a1;…;a14, b0; b1;…; b14). So the projection rela-tionship between the object plane and the imageplane is created.
The quartic equation fitting method is used to getthe parameters of the conversion equation from im-age coordinate system XOY to object coordinate sys-tem UOV. These parameters contain the lensgeometric distortions and the slope of the CCD sen-sor with the optical axis of the lens. After the camerais adjusted, the average position residual error of the841 points on the calibration pattern is 1.1272% CCDpixel and the standard deviation of the residual erroris 0.9079% CCD pixel.
D. Measuring the Object Coordinates of the Spot Array
After the calibration of the camera with the CCDcamera vision measurement device in the initialposition, the calibration equation parameters(a0;a1;…;a14, b0; b1;…; b14) are obtained and the ob-ject coordinates of the spot array on the calibrationpattern are calculated.
Fig. 3. Schematic of the rotation of the calibration pattern spotarray.
Table 1. Measurement Error of Angle under the Circumstance thatthe Object Coordinates of the Spot Array are Blurred with Noises ofDifferent Standard Deviation When the Rotation Angle is 1 arc sec
Standard Deviationof Noise
(CCD pixel)
Mean Valueof RotationAngle Error(arc second)
Standard Deviationof RotationAngle Error(arc second)
When the rotating parts rotate from initial posi-tion I to position II, the camera gets the image ofthe spot array of the calibration pattern, as shownin Fig. 3. At position I, the image coordinates ofthe spot array obtained by the camera are ðx1i; y1iÞ(i ¼ 1; 2;…; 841). At position II, the image coordi-nates are ðx2i; y2iÞ (i ¼ 1; 2;…; 841). Then, using cali-bration Eqs. (3) and (4), the object plane coordinatesðu1i; v1iÞ, ðu2i; v2iÞ (i ¼ 1; 2;…; 841) of the standardpoints at position I and position II are calculated.
E. Calculation of Rotation Angle
The original object coordinates of the spot arrayðu1i; v1iÞ (i ¼ 1; 2;…; 841) and the object coordinateðu2i; v2iÞ (i ¼ 1; 2;…; 841) at the second positionof the angle are substituted into the rotation-angle-measurement Eqs. (5) and (6). With the nonlinearleast squares method, the rotation angle α of thecalibration pattern can be calculated, in whichi ¼ 1; 2;…; 841:
u2i − u20 ¼ ðu1i − u10Þ × cos α − ðv1i − v10Þ × sin α; ð5Þ
v2i − v20 ¼ ðu1i − u10Þ × sin αþ ðv1i − v10Þ × cos α; ð6Þin which ðu10; v10Þ is the center coordinate of calibra-tion pattern at initial position I and ðu20;v20Þ is thecenter coordinate of calibration pattern at position II.
3. Error Analysis
The rotation-angle-measurement accuracy of visionmeasurement based on a spot array is analyzed.
First, the theoretical simulation is realized. Then,the noise of the measured points is added to the ob-ject coordinates, and the measurement error of angledetection is acquired.
A. Impact of Measurement Errors of Points
In the experiment of simulation, the coordinateðu2i; v2iÞ (i ¼ 1; 2;…; 841) of the spots at the secondangle position is blurred with Gaussian noise. Themean of the noise is zero; the standard deviationof the noise increases from 0.01 CCD pixel to 0.1CCD pixel. With the increase of the standard devia-tion of the noise, the rotation-angle-measurement er-ror of the smaller rotation angle and the largerrotation angle are shown in Tables 1 and 2.
The error analysis shows that, when the measure-ment error of the object coordinates (standard devia-tion) is 0.03 CCD pixels, the angle measuring error(standard deviation) is 0:45 arc sec.
B. Impact of Full Rotation Angle Measurement
The object coordinate ðu2i; v2iÞ is blurred with Gaus-sian noise. Themean value of noise is 0.03 CCD pixel,and the standard deviation of noise is 0.03 CCDpixels. The spot array is rotated from 0° to 360°and the angle with the rotation of the coordinate iscalculated. The simulation of the measurementerror is shown in the Figs. 4 and 5.
The method of coordinate rotation can be used torealize accurate rotation angle measurement in thewhole angle range of 0°–360°.
Fig. 4. Average of measurement error of angle in the simulationexperiment, when the rotation angle is 1°–360°.
Fig. 5. Standard deviation of the measurement error of angle inthe simulation experiment, when the rotation angle is 1°–360°.
Table 3. Simulation Experiment of the Resolution of Rotation Angle Measurement
The CCD camera in this system is 2000 × 2000 pixelsand the size of one pixel is 7:4 μm × 7:4 μm. As thefield of view is about 250mm × 250mm, the scopeof the object plane that a single pixel can indicateis 125 μm× 125 μm. The precision of the image planecoordinates measurement of the array is about 0.03pixel (3:75 μm). 3:75 μm stands for 6:1879 arc sec inthe field of view, so the resolution of the angle-measurement device is about 0:2arc sec becausethe spot array is 29 × 29.When the rotation components rotate a fraction of
an arc second, this rotation-angle-measurementmethod can realize the resolution of rotation anglemeasurement in the arc second grade. The measure-ment errors of the rotation angle of the spot array areshown in the Table 3, under the conditions of 0°,22:5°, and 45°, with the step size of angle rotationat 0:2 arc sec.
D. Impact of Radial Flop of the Rotation Part
When the in-plane rotation is measured, the rotationangle measurement is influenced by the radial flop of
the rotation. In the simulation experiment, the objectcoordinate of the spot array at the second angle posi-tion, ðu2i; v2iÞ (i ¼ 1; 2;…; 841), is blurred with theGaussian noise. The standard deviation of noise is0.03 CCD pixel and the mean value of noise is in-creased from 0.01 CCD pixels to 0.1 CCD pixels.The mean value of noise is equivalent to the shiftof the calibration pattern along the radial direction.Tables 4 and 5 show the angle measurement errors ofthe smaller angle and the larger angle when the ob-ject coordinate is blurred with noise of different meanvalues.
The result of simulation shows that the rotation-angle-measurementmethodwith the coordinate rota-tion of a spot array is not sensitive to such radial flop.The radial flop of the calibrationpatterndoesnothavethe obvious impact on themean value and calibrationdeviation of rotation-angle-measurement error.
4. Experimental Results
A real rotation-angle-measurement experiment isdesigned to check the effect of vision rotation anglemeasurement with a spot array. The calibrationpattern with the spot array is fixed on the rotary ta-ble, whose angle rotation precision is 2 arc sec. Thereal rotation angle of calibration pattern is θ. The ro-tation angle measured by the vision measurementdevice is θ1. The error of rotation angle measurementis θ − θ1. The experimental results of a small rotationangle and a large rotation angle are shown in Tables 6
Table 4. Rotation Measurement Errors When there is Noise withDifferent Mean Value in the Theoretical Simulation, under the
Condition of Tiny Rotation Angle (1 arc sec)
Mean Valueof Noise
(CCD pixel)
Mean Valueof RotationAngle Error(arc second)
Standard Deviationof RotationAngle Error(arc second)
and 7. The experimental results show that the stan-dard deviation of rotation angle measurement is lessthan 2 arc sec under small angle rotation and largeangle rotation, and that this method is an effectiveand high-precision algorithm tomeasure the rotationthat can meet the requirements of engineering appli-cations. In comparison, the standard deviation of ro-tation angle measurement is less than 5 arc sec inthe vision angle-measurement method based on astraight line, as in Ref. [4].
5. Conclusion
Compared with existing measurement techniques,this device with vision measurement can realizenoncontact rotation angle measurement. The mea-surement error of the rotating angle is less than3 arc sec (σ), and the measurement range is between0° and 360°. The self-calibration of the measurementsystem is realized and the error impact of projectiongeometric distortion is corrected. The CCD sensorand the optical axis of the lens does not need to beperpendicular. The environmental requirementsare not stringent. The method has broad applicationprospects, and has potential applications in industryand military. This device is not sensitive to the mea-surement of the spot array, flop of the rotation center,or to long periods of temperature change.
This study is supported by project “Research of keytechnology of the coordinate measurement of multi-object in large scale field,” National Natural ScienceFoundation of China (NSFC) (code 50705091).
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