Detecting System Modeling and Hand-Eye Calibration for a Robot to Detect the Ablation of Hydro-turbine Blades. Wang Shenghua, Du Dong, Zhang Wenzeng, ...
Detecting System Modeling and Hand-Eye Calibration for a Robot to Detect the Ablation of Hydro-turbine Blades Wang Shenghua, Du Dong, Zhang Wenzeng, Chang Baohua, and Chen Qiang Key Lab for Advanced Materials Processing Ministry of Education P.R. China Dept. of Mechanical Engineering, Tsinghua Unicersity, Beijing 100084, China
Abstract. In order to detect the ablation on the hydro-turbine blade in the turbine room, a scheme of the ablation detecting by a robot under the operator’s assistance was proposed. And the relevant system was designed. The system included two on-vehicle sensors and the computer and control center were located on the floor. The on-vehicle sensors included the laser range finder and the industry camera which were both fixed at a robot’s end-link. The blade was observed through the camera. When an ablation was found, its dimensions could be measured by the system. A mathematic model of the robot and the laser range finder was constructed. Besides, to calibrate the homogeneous transformation matrix between the coordinates of robot’s end-link and those of laser range finder, a calibration method by measuring a fixed point with different poses was proposed. Experimental results showed that the detecting scheme was appropriate for the turbine blade ablation detection, the model of the ablation detecting system was correct, and the proposed calibration method was effective.
1 Introduction During the hydro-turbine rapid running, the overflowing part of the runner was damaged by the abrasion of silt and cavitation, therefore it should periodically be repaired[1]. As the running time of hydro-tubine in Three Gorges increases, the blades should periodically be detected and repaired if cavitation or abrasion (also called ablation together) was found. Before repairing the dimensions of the ablation should be measured, as shown in Fig.1. In order to reduce the labor intensity and to protect the repairers, it is necessary to replace the repairers with a detecting robot. The shape of turbine blade is a complex spatial curved face, and the space between two blades is cabined, so it is hard to detect and measure the ablation in the turbine room. Though there are many methods for measuring the blade, for example, template method, theodolite method[2], three-coordinate measuring machine method, mechanical arm method[3], underwater photography method, robot spatial vision, telemeter method, and so on, most of them are not suitable for the robot to detect the blade ablation in the turbine room. A template should previously be prepared during the template method. The theodolite method and three-coordinates measuring machine method are suitable for measuring the shape of blade only when the turbine is lift out to the floor. Because it is a method of contact measure and it would cause a lager error, the mechanical arm method is not suitable either. And a matching point should be found in the images of the two cameras[4], but it is impossible to find them in the images of ablation blade. T.-J. Tarn et al. (Eds.): Robot. Weld., Intellige. & Automation, LNCIS 362, pp. 13–20, 2007. springerlink.com © Springer-Verlag Berlin Heidelberg 2007
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Fig. 1. The ablation of the blade of axial-flow hydro-turbine
So a scheme of detecting the ablation by a robot under the operator’s assistance was proposed, base on analysis on the requirements for ablation detection and researches on robot detecting technology. The relevant system was then designed. And in this paper the modeling of the system was conducted, and a calibration method was proposed to calibrate the robot hand-eye.
2 Inner Detecting System for the Ablation of Hydro-turbine Blades 2.1 The Composition of the System It is a detecting system which detects the ablations under the control of an operator. It is constructed with a mobile robot platform and a multiple-degree-freedom arm fixed on the platform. And it includes two parts, as shown in Fig.2. One is the on-vehicle sensors which are fixed at the robot’s end-link. The other is the computer and control center which is located on the floor. The on-vehicle sensors include an industry camera to observe the blade surface, and a laser range finder to measure the pattern of the blade with the robot. The computer and control center would communicate with the on-vehicle sensors, the mobile robot platform and the arm, control the movements of the platform and the arm, process the data of the sensors and the robot, and show the graphic results on the display.
Fig. 2. The composition of the detecting system
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2.2 The Working Process and the Function of the Detecting System During the detecting process, the operator gives orders to move the robot to one position of the blade, to change the pose of the arm so that the blade surface can be suitably observed and the ablation could be measured. By reason of the multiple-degree-freedom arm, a large part of blade surface can be observed and detected. When detection at this position has finished, the operator can move the robot to another position and repeat the detecting process. With this detecting system, the type, position, area, and depth of the ablation can be measured. It offers the basic parameters for the blade repair. And the measurement of the ablation depth and the area is the most important process. When the measurement is operated with the arm at some pose, the laser range finder is pointed to the blade surface. Then the system records the data of the range finder and the arm pose, and transmits them to the computer and control center. There, the data are processed by the mathematical model based the robot kinematics, and the coordinate of the measured point under a fixed coordinate system. After multiple points are measured under a given rule, the depth and area of the ablation, even the local pattern of the blade can be gained. Further more, utilizing the robot position technology and the surface reconstruction technology[5], the total pattern of the blade can be reconstructed.
3 The Mathematical Model of the Detecting System 3.1 The Model of the On-Vehicle Sensors and the Multiple-Degree-Freedom Arm and Their Coordinate System
ü
The on-vehicle sensor the laser range finder and the industry camera are fixed at the end of the multiple-degree-freedom arm, as shown in Fig.3. The mobile platform stops on the blade, with the arm at different poses. The data of the range finder and the arm pose are recorded. In order to get the relationship of the different points, the mathematical model should be constructed, and the range finder’s pose relative to the robot’s end-link should also be calibrated in advance. And first of all, the coordinate systems should be constructed.
{E} E R
TE
TL
{L}
{R}
1-blade; 2-mobile platform and multiply-degree-freedom arm; 3-camera; 4- laser range finder; 5-laser; 6-ablation pit
Fig. 3. The on-vehicle sensors and the robot
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The coordinate systems of the robot base, the end-link, and the laser range finder are constructed, as shown in Fig.3: {R} ——the coordinate system fixed at the robot mobile platform, {E} ——the coordinate system fixed at the arm’s end-link, {L} ——the coordinate system of the range finder, the initial point locate on the laser
beam, the Z-axis is set along the laser beam and pointing to the measured point, and the X-axis and Y-axis whose directions are optional, are located in the plane vertical to the Z-axis. Suppose there is a point M on the blade surface, and L p M is the coordinate under the coordinate system {L} , R p M is the one under {R} . According the robot kinematics, the follow equation can be established: R
p M = R TE E TL L p M
(1)
while the R TE is the homogeneous transformation matrix between coordinate systems {E} and {R} , which can be calculated out with the pulsed coordinate of the arm’s pose, and the E TL is the relation between the range finder’s pose and the robot’s [6] end-link, which is needed to be calibrated. It is usually called hand-eye calibration . Then, if we find an ablation pit, we can measure some point of the pit. Though the equations (1), we can get some reference pionts of the pit, and then know the depth, the area and even the dimesion of the pit by some mathematic methods. 3.2 Calibration Method for the Robot’s End-Link and the Laser Range Finder In order to calibrate the relation between the robot’s end-link and the laser range finder, a calibration method by measuring a fixed point with different poses (FPDP method), is proposed as follows: (1) operate the robot to measure a point, let the reticle of the point at the center of the laser spot, then record the data of the range finder—the distance, and the pulsed coordinate of the arm pose; (2) change the pose of the arm, measure the same point and record the data; (3) repeat step (2); (4) construct a system of equations with the element of the E TL as the unknown number using the parameters get above, calculate it by utilizing the least-squares principle, and then the E TL can been calibrated. When the same point M is measured at the different poses, a series of data are recorded, the coordinate of the point under the {L} and the robot pulsed coordinate cij , while (i = 1, 2," , n; j = 1, 2," , m) . The R TE —the homogeneous transformation matrix between coordinate systems {E} and {R} , can be calculated out with the cij by utilizing robot kinematics [7]. Because the range finder is fixed at the end-link, the matrix E TL needed to be calibrated keeps constant, no matter how the arm pose changes. The matrix E TL is supposed as follows: E
ª r11 «r « TL = 21 « r31 « ¬0
r12
r13
r22 r32
r23 r33
0
0
t1 º t2 »» ªr1 r2 = t3 » «¬ 0 0 » 1¼
. r3 t º 0 1»¼
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As a result, a series of equations (2) are gained. R
(i = 1, 2," , n) .
p M = R TEi Ei TLi Li p Mi = R TEi E TL Li p M ,
(2)
The left part of the equations (2) is the coordinate of the same point M under the coordinate systems {R} , so the right part of the equations (2) are equal to each other. R
TEb1 E TL Lb1p M = R TEb 2 E TL Lb 2p M , (b1, b2 = 1, 2," , n; b2 ≠ b1) .
(3)
Replace the matrix E TL in (3), the Equ.3 can be deduced as:
( db2Rb2 − db1Rb1 ) r3 + ( Rb2 − Rb1 ) t = tb1 − tb2 ,
(b1, b2 = 1, 2," , n; b2 ≠ b1) .
(4)
The Equs.4 can be rewritten as matrix: Ax = c ,
while,
ª d 2 R 2 − d1R 1 « # « A = « d m 1 R m1 − d m 2 R m 2 « # « «¬ d n R n − d n −1 R n −1
(5)
R 2 − R1 º ª t 2 − t1 º « » , x = ª r3 º , », # # «t » « » » ¬ ¼ R m 1 − R m 2 » c = « t m1 − t m 2 » « » » # # « » » «¬ t n − t n −1 »¼ R n − R n −1 »¼
(m1, m2 = 1,2,", n; m2 > m1) .
Then the vector r3 and t of the matrix E TL can be solved out from Equs.5 by utilizing the least-squares principle. So the matrix E TL can be rewritten as: E
ª* * r3 t º . TL = « » ¬0 0 0 1¼
(6)
The vector r1 and r2 are unnecessary to solve, because the coordinate of the point M under the {L} is L pM = [0 0 d 1]T , the first two elements of which are zero. It also proves that it is unnecessary to set the direction of X-axis and Y-axis of the coordinate system {L} . 3.3 The Depth Measurement of the Ablation Pit The depth measurement of the ablation pit is a major problem of the ablation detecting of hydro-turbine blade. In order to increase the detecting efficiency, a simplified method is proposed. Because the area of the ablation pit is usually small, the blade surface around the pit could be considered as a plane. So the depth measurement of the ablation pit can be operated as follows. Firstly, measure a point M at the bottom of the ablation pit and more than three points Qi (i = 1, 2," , n) which are around the pit and not collinear, shown as Fig.4, then record the data of the pulsed coordinate of the arm pose and the range finder, and finally, calculate out the coordinate of the points under the {R} system:
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R
§ ªr º ª t º · p j = R TEj E TL Lj p j = R TEj ¨ d j « 3 » + « » ¸ , © ¬ 0 ¼ ¬1¼ ¹
(7)
while j is M , Q1 , Q2 , …, Qn , TEj is the homogeneous transformation matrix of the robot, TL is hand-eye calibration result of the robot, Lj p j = [0 0 d j 1]T is the result of the laser range finder.
R
E
1-blade surface; 2-range finder; 3-ablation pit
Fig. 4. The depth measurement of the ablation pit
It is obviously that a plane can exclusively be defined through three points that are not collinear. Then a plane S can be defined though the points Q1 , Q2 , …, Qn by utilizing least-squares principle. Therefore, the depth of the ablation pit can defined with the distance between point M and plane S .
4 Experiments and Results 4.1 The Hand-Eye Calibration Experiment The hand-eye calibration experiment was carried out by utilizing the FPDP calibration method. The Motoman-SK6 which is a six-axes industrial robot, and the FT50RLA220 which is a laser range finder with an operation range of 80~300 mm and the resolution of 0.2 mm are used. The parameters recorded during the process are shown as Tab.1. The pulsed coordinate of the arm is the parameters of the homogeneous transformation matrix of the robot
R
TEj , and the distance is d in the
equation(5). Then the result is as follows: r3 = [-0.0373, 0.2162, 0.9771]T , t = [60.79, -107.90, 158.60]T .
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Table 1. The parameters of the hand-eye calibration No. 1 2 … 24
Pulsed coordinate of the arm 3537, 86280, -86712, 12419, 31818, 2146 2953, 90224, -88144, 13089, 31786, 2272 … 2785, 85616, -87112, 10869, 32266, 2125
distance/mm 127.6 104.5 … 128.3
4.2 The Experiment of Pit Depth Measurement The experiment of pit depth measurement was carried out based on the result of the above calibration experiment and the pit depth measure method proposed in Section.3.3. In the experiment a pit 26.00 mm deep was used, which was measured by vernier caliper as a true value. The result of the experiment shows that the maximum error is 0.3 mm, as shown in Fig.5.
Fig. 5. The result of the pit depth measurement experiment
The experimental results show that the model of the ablation detecting system is appropriate, that the FPDP method of hand-eye calibration is effective, and the ablation pit measurement is of high accuracy. It also proves that the scheme is suitable to the ablation detecting on the hydro- turbine blade in the turbine room.
5 Conclusion In this paper, a scheme of the ablation detecting by a robot under the operator’s assistance was proposed, so that the ablation on the hydro-turbine blade could be detected by robot in the turbine room. And the relevant system was designed. The system included a laser range finder and an industry camera which were fixed at the robot’s end-link, and the computer and control center which is located on the floor. The mathematic model of the robot and the laser range finder was constructed. Besides, the homogeneous transformation matrix between the coordinates of the
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robot’s end-link and the laser range finder was calibrated. Experimental results showed that the detect scheme was effective for the turbine blade ablation detection and the FPDP calibration method was effective.
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