Deep-sea structure pose measurement's error analysis and experimental study

Ocean Engineering 70 (2013) 141–148

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Deep-sea structure pose measurement's error analysis and experimental study Wenming Wang a,b,n, Liquan Wang b, Caidong Wang b a b

College of Mechanical and Transportion Engineering, China University of Petroleum, 18 Fuxue Road, Changping, Beijing 102249, China College of Mechanical and Electrical Engineering, Harbin Engineering University, 145 Nantong Road, Harbin, Heilongjiang 150001, China

art ic l e i nf o

a b s t r a c t

Article history: Received 25 March 2012 Accepted 11 May 2013 Available online 2 July 2013

The relative distance and angle of two subsea structures is often determined using a deep-sea structure pose measurement system; thus, the high accuracy of pose measurement has important significance for the success of deep-sea operations. Here, we take two subsea pipelines for example to design their overall measurement schemes, the parameters of their detection methods, and the operation procedures for the deep-sea operating environments they service. We furthermore show the theoretical calculation formulas of their measurement results. The theoretical error of the transition matrix algorithm is calculated using the error propagation relations theory, and an example is used to verify the accuracy of the results calculated in this study. The study presents a segmented, least squares method for pose measurement systems, calculates the error correction coefficient, and revises the resulting function of both the relative distance and the angle. After the experimental platform is established, the measured value and the true value of two pipelines' pose parameters are contrasted in order to verify the measurement method and the error correction theory. The experimental results show that the distance error is in the range of 725 mm, and that the angle error is in the range of 70.61 within 7 m after correction. We reduce the measurement error of the pose measurement system, and verify the correctness of the theoretical. In doing so, we ensure the precision parameter for the connection between two submarine pipelines. Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.

Keywords: Deep-sea structure pose measurement Error analysis Least squares Experimental study

1. Introduction With the rapid development of offshore oil and gas resources, there will be an increase in the construction of offshore platforms or subsea pipeline laying (Wang et al., 2010a). Thus, these underwater structures' measuring and positioning are essential for marine operations. There are two common methods for underwater measurement and position technology – the underwater acoustic measurement system (Wang and Liang, 2002; Chen and Wang, 2011) and the auxiliary rope measuring system (Zhu et al., 2008). The acoustic positioning system utilizes the time difference or phase difference, in which underwater acoustic signal propagation reaches the receiving elements along different distances of paths in order to measure and locate the target on the surface or in the water. The auxiliary rope measuring system uses a rope to link the two measuring devices, and the measurement is implemented by the auxiliary operations of divers or a Remote Operated Vehicle (ROV). The deep-sea structure pose measurement system (Wang et al., 2010b), which is an auxiliary rope measuring system, uses a stretching wire and a deep-sea sensor to obtain the measurement parameters. These measurements are n Corresponding author at: College of Mechanical and Transportion Engineering, China University of Petroleum, 18 Fuxue Road, Changping, Beijing 102249, China. Tel.: +86 10 89733835. E-mail address: [email protected] (W. Wang).

based on the transition matrix algorithm (Wang et al., 2012), and aim to characterize the deep-sea relative pose measurements of both of the two structures (including the relative distance and relative angle between them). The deep-sea structures pose measurement system can be applied to the measurement of different underwater structures (here we use two undersea pipelines as an underwater structure) by changing the positioning base, and can accurately obtain the relative pose parameters of the two pipelines. The implication of this work is that it can improve the smooth implementation of the connection of submarine pipelines. In order to further improve the quality and efficiency of marine operations, we need to reduce the measurement error and improve the measurement accuracy of deep-sea measurement. At present, the error correction theory is being widely used in measurement areas, but the deep-sea structures pose measurement system has no existing error correction theory. Liu proposes a kind of self-dependent sensor array localization algorithm based on the error correction (Liu et al., 2008). Wang discuss docking alignment errors for ROV and underwater operation equipment (Wang et al., 2010c). Cui and others discuss the error correction of the echo sounder for water depth measurement (Bjorke, 2005; Cui et al., 2008; Wu et al., 2010). We propose that the segmented least squares method (Von Schroeter et al., 2005; Li et al., 2012) for error correction of the measurement results, which provides strong theoretical support for underwater measurement in practice.

0029-8018/$ - see front matter Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.05.004

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W. Wang et al. / Ocean Engineering 70 (2013) 141–148

2. System description 2.1. Scheme of the system Fig. 1 describes the deep-sea structure pose measurement system. This system works in the actual marine environment, which includes high static pressure and corrosion. The system includes a man-machine monitoring interface, a measuring device, a stretching wire, a magnetic power winch, a positioning base, and an ROV. The monitoring interface, which is located on the upper industrial computer, is used to directly control the ROV. Fig. 1 shows the pose parameters of two random pipelines after calculating the data parameters from sensors. Measuring devices that carry the sensors are connected to the pipelines with positioning bases, and they are the main component of this system. The stretching wire links measuring devices I and II, and the distance between the two pipelines can be determined by detecting the stretch in the wire. The magnetic power winch, which stores the stretching wire, is carried by ROV and provides the taut force needed to unstretch (recover) the wire. The ROV provides auxiliary operations and the power for system. The ROV is connected with the upper industrial PC by an umbilical cable, and it connects with the hypogynous machine measuring device through RS485 while operating the movement and positioning of the equipment.

moves to pipeline 2 (the stretching wire is in the relaxed state at this time); (4) After the stretching wire is tensioned by a magnetic power winch, the distance parameter between measuring device I and measuring device II is detected with the rope length sensor. At the same time, the angle parameters are detected with the magnetic coupling encoders and the orthogonal angle sensors; (5) Finally, the pipelines' pose parameters are displayed by terminal software in the PC after these parameters are computed and disposed of by the upper industrial PC.

2.3. Parameters detection method Fig. 2 provides an overview of the deep-sea structure pose measurement system (Wang et al., 2012). The detection parameters of the system include: the orthogonal angle, αr; the βr of measuring device I; the extension arm's pitch angle, γr, and swing angle, θr; the orthogonal angle αb; the βb of measuring device II; the extension arm's pitch angle, γb, and swing angle, θb; and the length of the stretching wire, Srb. Orthogonal angles are detected with two orthogonal angle sensors. The pitch angles and swing angles are detected with four magnetic coupling sensors. The αr length of the stretching wire is detected with a rope-length sensor. The host controller collects these detection parameters and transmits them to the upper industrial PC by the umbilical cable.

2.2. Operation procedure

2.4. Theoretical calculation formula

Fig. 1 shows the operation procedure of the deep-sea structure pose measurement system in an actual measurement environment (Song et al., 2011; Souto-Iglesias, et al., 2011). The operation procedure is: (1) The sensors of the measuring devices are calibrated on the shore or ship; (2) Measuring device I is installed on pipeline 1 while the pose measuring system is carried to the operation region by the ROV; (3) Measuring device II is installed on pipeline 2 when the ROV

PrPby through the transition PrPbx matrix algorithm (Wang et al., 2012), we can get the measuring results of deep-sea pipelines' relative pose (the relative distance and relative angle parameters between pipeline 2 and pipeline 1). Fig. 3 shows the model of the measuring result. We create the reference coordinate system {r}: the xr axis is perpendicular to the pipeline axis along the horizontal direction; the yr axis represents the pipeline axis; the

Fig.1. Scheme of measurement system.

W. Wang et al. / Ocean Engineering 70 (2013) 141–148

143

Fig.2. Detection scheme.

(5) ξy2 is ξy2 ¼ ΔA

ð5Þ

(6) ξz2 is ξz2 ¼ βb −βa

ð6Þ

where,
ΔA ¼ arcsin −ð cos βr cos γ r cos θr þ sin αr sin γ r  cos αr sin βr sin γ r − cos αr cos γ r sin θr − cos γ r sin αr sin βr sin θr Þ=ð− cos βb cos γ b cos θb − sin αb sin γ b þ cos αb sin βb sin γ b  þ cos αb cos γ b sin θb þ cos γ b sin αb sin βb sin θb Þ

ð7Þ

Fig.3. Model of the result.

zr axis represents the vertical direction. Similarly, the coordinate system {b} is established. The relative distances are: the horizontal distance PrPbx; the vertical distance PrPby; and the height PrPbz. The relative angles are ξx2 , ξy2 , and ξz2 . The measuring results of the relative pose are in formulas (1)–(6) (below) (1) The horizontal distance is jP r P bx j ¼ jRr sin αr −m cos γ r sin θr cos αr þ m sin γ r sin αr þRb cos ΔA sin αb −Rb sin ΔA cos αb sin βb −bb sin ΔA cos βb j

3.1. Error propagation relation The errors of the measuring results P r P bx , P r P by , P r P bz , ξx2 , ξy2 , and ξz2 are influenced by every detection parameter (e.g. αr ; βr ; γ r ; θr ; αb ; βb ; γ b ; θb ; and Srb ). The propagation coefficient αi is: αi ¼ f ðαr ; βr ; γ r ; θr ; αb ; βb ; γ b ; θb ; Srb Þ

1. The error P r P b can be expressed in matrix form, and its accuracy equation is:

ð2Þ

(3) The vertical distance PrPbz is jP r P bz j ¼ j−br sin β þ Rr cos αr cos βr þ m cos γ r sin βr cos θr þm cos γ r sin θr sin αr cos βr þ m sin γ r cos αr cos βr −Rb cos αb cos βb þ bb sin βb j ð3Þ

δðP r P Bx ; P r P By ; P r P Bz ÞT ¼ N 1 Δαr þ N 2 Δαb þ N3 Δβr þN4 Δβb þ N 5 Δγ r þ N 6 Δγ b þ N 7 Δθr þN8 Δθb þ N 9 ΔSrb where 2 ∂P

r P Bx ∂α 6 ∂P Pr 6 r By N1 ¼ 6 ∂αr 4 ∂P r P Bz ∂αr

3

∂β

2 ∂P r P 3 Bx

∂αb

2 ∂P r P Bx 3 ∂β

7 6 ∂P P 7 6 ∂P Pr 7 7 6 r By 7 6 r By 7 7; N 2 ¼ 6 ∂αb 7; N 3 ¼ 6 ∂βr 7; 5 4 5 4 5

2 ∂P r PBx 3

∂P r P Bz ∂αb

2 ∂Pr P Bx 3 ∂γ

∂P r P Bz ∂βr

2 ∂P r P Bx 3 ∂γ

6 ∂P Pr 7 6 ∂P Pr 7 6 ∂P Pb 7 6 r By 7 6 r By 7 6 r By 7 N4 ¼ 6 ∂βb 7; N 5 ¼ 6 ∂γ r 7; N 6 ¼ 6 ∂γ b 7; 4 5 4 5 4 5

(4) ξx2 is ξx2 ¼ 901−ΔA

ð8Þ

ð1Þ

(2) The axial distance PrPby is jP r P by j ¼ j−br cos βr −Rr cos αr sin βr þ m cos γ r cos θr cos βr −m cos γ r sin θr sin αr sin βr −m sin γ r cos αr sin βr −Rb sin αb sin ΔA−Rb cos ΔA cos αb sin βb −bb cos ΔA cos βb j

3. Error analysis

ð4Þ

∂P r P Bz ∂βb

∂P r P Bz ∂γ r

∂P r P Bz ∂γ b

ð9Þ

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W. Wang et al. / Ocean Engineering 70 (2013) 141–148

2 ∂P N7 ¼

r P Bx ∂θ 6 ∂P Pr 6 r By 6 ∂θr 4 ∂P r P Bz ∂θr

3

2 ∂P r P 3 Bx

∂θb

2 ∂Pr P 3 ∂S

Bx

7 6 ∂P P 7 6 ∂P Prb 7 7 6 r By 7 6 r By 7 7; N 8 ¼ 6 ∂θb 7; N 9 ¼ 6 ∂Srb 7 5 4 5 4 5 ∂P r P Bz ∂θ b

∂P r P Bz ∂Srb

2. The angle errors of ξx2 and ξy2 is determined by ΔA. The angle error of ξz2 can be measured directly, and its error value is directly determined by the precision of angle sensor. The error propagation relation of ξx2 and ξy2 is: δðΔAÞ ¼ N 10 Δαr þ N 11 Δαb þ N12 Δβr þ N 13 Δβb þ N 14 Δγ r þN 15 Δγ b þ N 16 Δθr þ N 17 Δθb þ N 18 ΔSrb

ð10Þ

where： N 10 −N 18 is the derivative of ΔA about αr ; βr ; γ r ; θr ; αb ; βb ; γ b ; θb ; and Srb . 3.2. Numerical simulation (1) Example We assume that the deep-sea structure pose measurement system is operated at a depth of 1500 m, and that the data of measuring device I are the following: orthogonal angles are αr ¼101 and βr ¼ 121; the extension arm's swing angle is θr ¼ 8.261; and the extension arm's pitch angle is γ r ¼35.311. We also assume that the data of measuring apparatus II are the following: the orthogonal angles are αb ¼−13.831 and βb ¼−20.991; the pitch angle is γ b ¼27.941; the horizontal swing angle is θb ¼1.101; and the length of the stretching wire is m ¼7000 mm. Table 1 shows the error range for each sensor.

(2) Distance error values The maximum error of each sensor is plugged into formula (9); then, we can obtain the theoretical distance error values, which are shown in Table 2. We can determine that the theoretical distance error values of P r P bx , P r P by , and P r P bz are 8.83 mm, 14.58 mm, and 16.53 mm, respectively. The technical index of the system is that the distance error is 7 m 750 mm, and the theoretical distance error is satisfied with the technical index of the system. (3) Angle error values The maximum error of each sensor is plugged into formula (10); then, we can obtain the theoretical angle error values, which are shown in Table 3. The technical index of the system is that the angle error is 7 m 711. Thus, the theoretical angle error of 70.151 (inside) is within the range of the technical index of our system.

4. Error correction Although the theoretical error of the transition matrix algorithm above is small, the actual error value will markedly increase due to current, waves, or other environmental factors in the real measurement environment. Because of this, one will need to amend the formulas using a segmented least squares principle in order to decrease the error. 4.1. Distance results' correction (1) Length's section The effective measuring length, which is 30 m, can be divided

Table 1 Error range of each sensor. Sources of error

αr

Δαb

Δβr

Δβb

Δγ r

Error

0.21

0.21

0.21

0.21

0.04391

Sources of error

Δγ b

Δθr

Δθb

ΔSrb

Section

L1

L2

L3

L4

L5

L6

Error

0.04391

0.04391

0.04391

1 mm

Length (m)

0–5

5–10

10–15

15–20

20–25

25–30

Table 4 Operating length of the section.

Table 2 Error data of distance (m ¼7000 mm). Distance results

N1

N2

N3

N4

N5

N6

PrPbx PrPby PrPbz

1369.36 −557.779 2626.51

151.956 3.07813 −33.4672

1287.91 −1591.32 4122.07

19.7696 −18.3583 0

2.12982 −2194.42 −946.107

18.9435 −151.58 −52.1269

Distance results

N7

N8

N9

N 10

N 11

error

PrPbx PrPby PrPbz

−4028.71 −3006.55 −2110.15

52.026 64.8224 0

−0.09957 0.104055 −0.701872

−0.538158 0.739911 1.02564

0.0959009 −1.90691 −1.336

8.83 mm 14.58 mm 16.53 mm

Table 3 Error data of angle (m ¼7000 mm). Angle results

N1

N2

N3

N4

N5

N6

ξx2 ξy2 ξz2

−0.488763 0.870384

−0.078251 0.01339

−0.06544 1.38305

−0.00324 0.006296

0.809946 0.216245

0.046408 0.0204461

Angle results

N7

N8

N9

N 10

N 11

Error value

ξx2 ξy2 ξz2

3.15563 −0.445195

−0.05035 −0.005176

0.00001 −0.000186

0 0

0.00065 0.00008

0.151 0.091

W. Wang et al. / Ocean Engineering 70 (2013) 141–148

into several sections: L ¼ L1 ∪L2 ∪L3 ∪:::::∪Ln (n ¼ 6). Table 4 shows the values of these sections. (2) Least squares method We define a function p2 ðxÞ, and the relationship between p2 ðxÞ and the length (x) is as follows: p2 ðxÞ ¼ a0 þ a1 x þ a2 x2

ð11Þ

We further define another function, Q ða0 ; a1 ; a2 Þ, as follows: n

Q ða0 ; a1 ; a2 Þ ¼ ∑ ða0 þ a1 xi þ a2 x2i −p2 ðxi ÞÞ

ð12Þ

i¼1

where ½xi ; p2 ðxi Þ are the measuring values. According to the extreme value of the multi-function method, we obtain: 8 n ∂Q 2 > > > a0 ¼ 2 ∑ ða0 þ a1 xi þ a2 xi −p2 ðxi ÞÞ ¼ 0 > > i¼1 > > > n < ∂Q 2 ð13Þ a1 ¼ 2 ∑ xi ða0 þ a1 xi þ a2 xi −p2 ðxi ÞÞ ¼ 0 > i¼1 > > > n > > 2 2 > > ∂Q : a2 ¼ 2 ∑ xi ða0 þ a1 xi þ a2 xi −p2 ðxi ÞÞ ¼ 0 i¼1

When we solve formula (13), which is related to a0 ; a1 ; a2 , we obtain: a0 ¼ −

a1 ¼ −

a2 ¼ −

U 0 ðX 23 −X 2 X 4 Þ−U 1 ðX 2 X 3 −X 1 X 4 Þ þ U 2 ðX 22 −X 1 X 3 Þ

ð14Þ

−X 23 −nX 23 þ nX 2 X 4 þ 2X 2 X 3 X 1 −X 4 X 21 U 0 ðX 2 X 3 −X 4 X 1 Þ þ U 1 ðnX 4 −X 22 Þ þ U 2 ðX 1 X 2 −nX 3 Þ

ð15Þ

X 23 þ nX 23 −nX 2 X 4 −2X 2 X 3 X 1 þ X 4 X 21 U 0 ðX 22 −X 3 X 1 Þ þ U 1 ðnX 3 −X 2 X 1 Þ þ U 2 ðX 21 −nX 2 Þ

ð16Þ

−X 32 −nX 23 þ nX 2 X 4 þ 2X 2 X 3 X 1 −X 4 X 1 2

Where: U 0 ¼ ∑ni¼ 1 p2 ðxi Þ; U 1 ¼ ∑ni¼ 1 xi p2 ðxi Þ; 2 n U 2 ¼ ∑i ¼ 1 xi p2 ðxi Þ; X 1 ¼ ∑ni¼ 1 xi ; n

n

n

i¼1

i¼1

i¼1

X 2 ¼ ∑ xi 2 ; X 3 ¼ ∑ xi 3 ; X 4 ¼ ∑ xi 4 :

When we substitute the section's date into the formula (14)–(16), we can solve for the correction calculation formula of the distance (Lm): Lm ¼ L−ða0 þ a1 L þ a2 L2 Þ

ð17Þ

where Lm is the final correction value and L is the measured value. (3) The error correction function expression of the whole section can be obtained through multiple measurements. Table 5 shows the value of the distance correction coefficient.

145

4.2. Angle results' correction (1) Angle range section We also amend the angle results using the segmented least squares principle. The sections in the horizontal and vertical direction are Q ¼ Q 1 ∪Q 2 ∪Q 3 ∪:::::∪Q n (n ¼ 12); and Table 6 shows the sections. (2) Angle error correction coefficient The error correction function expression of the angle can also be obtained through multiple measurements. Table 7 shows the coefficients of b0 ; b1 ; b2 .

5. Experiment setup 5.1. Scheme of the test platform Fig. 4 shows the scheme of test platform described in this study. The test platform consists of pipeline 1, pipeline 2, pillar 1, pillar 2, a track, measuring device I, measuring device II, a magnetic power winch, and a stretching wire. Pipeline 1 can swing in the horizontal direction and tilt in the vertical direction, and it can move along the vertical axis. In contrast, pipeline 2 can slide along the direction of the track. Thus, the two experimental pipelines can simulate any random pose of two submarine pipelines. The true distance of the two pipelines can be measured using a by the laser range finder. Lastly, the true pitch angle of two pipelines can be measured by the tilt sensor, and the true swing angle of the two pipelines can be measured by the protractor. 5.2. Ocean experiment Fig. 5 shows the ocean experiment (Julca Avila and Adamowski et al., 2011). First, we finished the initial calibration of sensors after installing the measuring devices. Then, we measured the true distance of the two pipelines using the laser range finder. We also used the tilt sensor to measure the true relative pitch angle; we used the protractor to measure the true relative swing angle. We then fixed pipeline 1, pipeline 2, the track, the magnetic power winch, and the remaining Table 6 Operating angle of the Q 8 section. Vertical direction

−451 to −151 −151 to 151 151 to 451

Horizontal direction −601 to −01

−301 to 01

01 to 301

301 to 601

Q1 Q5 Q9

Q2 Q6 Q10

Q3 Q7 Q11

Q4 Q8 Q12

Table 5 Coefficient of length. Section

L1

L2

L3

L4

L5

L6

P r P bx correction coefficient

a0 a1 a2

−31.1431 0.0188 −1.3818e-6

12.4567 0.0218 −2.5548e-6

188.5867 0.0102 2.9821e-6

200.7841 0.0312 5.5641e-6

465.0055 −0.1229 6.4656e-6

588.2929 −0.0428 8.8607e-6

P r P by correction coefficient

a0 a1 a2

−56.7305 0.0262 −1.6575e-6

89.5487 0.0112 −1.2658e-6

28.0711 0.0047 −1.2643e-7

15.7892 3.9647e-4 −1.0231e-7

10.1126 1.7895e-6 −8.6589e-8

−1.1996 8.3355e-6 −1.0109e-8

P r P bz correction coefficient

a0 a1 a2

18.4420 0.0068 −1.1650e-6

10.5213 0.0054 8.6551e-7

1.5516 0.0012 7.6461e-8

0.9561 7.9338e-4 4.3106e-8

0.6618 5.9797 e-4 2.6967e-8

0.2259 7.0629e-4 1.0223e-8

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W. Wang et al. / Ocean Engineering 70 (2013) 141–148

Table 7 Coefficient of angle. Section

Q1

Q2

Q3

Q4

Q5

Q6

ξx2 ξy2

b0 b1 b2

3.8143 0.2046 0.0026

0.3107 −0.0202 −7.8654e-4

0.4145 −0.0502 −8.6325e-4

−1.0029 0.0674 −7.8571e-4

3.5684 0.6504 0.0013

0.8511 −0.0560 −7.2225e-4

ξz2

b0 b1 b2

5.7813 2.2456 0.0013

1.3107 −0.1203 −5.2326e-4

0.6565 −0.1254 −7.3256e-4

−2.2354 0.0512 −8.3213e-4

4.2951 3.2456 0.0625

2.9821 −0.4043 −8.2568e-4

Q7

Q8

Q9

Q 10

Q 11

Q 12

Section ξx2 ξy2

b0 b1 b2

5.7874 −1.0147 −8.0005e-4

−2.6348 0.0032 −4.5621e-4

4.2135 2.2674 −2.1474e-4

0.3843 −0.0058 −7.0315e-4

1.2131 −0.0047 −5.2212e-4

−4.3256 0.6914 −7.1112e-4

ξz2

b0 b1 b2

0.9876 −0.3256 −9.3156e-4

−3.0464 0.0098 −9.1424e-4

4.2359 1.2122 5.0247 e-4

1.4679 −1.0345 −2.5798e-4

0.2224 −0.2525 −7.2571e-4

-2.3585 0.2385 −8.3694e-4

Fig.4. Scheme of test platform.

equipment, then lifted the test platform slowly and down into ocean. The magnetic power winch began to tighten the stretching wire when the test platform reached the seabed; subsequently, the sensors measured the various parameters we were interested in. The result of relative pose measurement was calculated using the transition matrix algorithm and was displayed on the PC. We selected 6 typical operating conditions, which were: 2 m, 3 m, 4 m, 5 m, 6 m, and 7 m for the ocean experiment, and every condition was duplicated independently in the ocean following a return to the surface.

6. Experimental results analysis The initial values measured for P r P bx , P r P by , P r P bz , ξx2 , ξy2 , and ξz2 were solved with the transition matrix algorithm, and the correctional results were solved by the correction function, which is based on the segmented least squares principle. Table 8 shows the true values of the experimental data obtained from the ocean environment, as well as both the initial and corrected results. We obtain the error of the pose measurement system following Table 8. Fig. 6 shows the curve of the initial error, and also reports the corrected error from the ocean experiment. The initial error, P r P bx , is

Fig.5. Ocean experiment.

an increasing trend from 1 m to 7 m along the x axis, and the maximum error (measured at 7 m) was 40.4 mm. The correctional error is less than the initial error, and the maximum correction can reach up to 22.8 mm, which is 56% of the initial error value. The initial error of P r P by and P r P bz also increases along the length from 1 m to 7 m, and the initial maximum error was measured at 7 m. The

W. Wang et al. / Ocean Engineering 70 (2013) 141–148

147

Table 8 Ocean experimental data. Results

Category

2m

3m

4m

5m

6m

7m

P r P bx (mm)

True value Initial result Correctional result Error

0.0 5.4 2.5 2.5

0.0 11.2 5.2 5.2

0.0 25.4 5.5 5.5

0.0 21.3 15.3 15.3

0.0 38.3 16.1 16.1

0.0 40.4 17.6 17.6

P r P by (mm)

True value Initial result Correctional result Error

2000.0 1981.5 1998.2 −1.8

3000.0 3009.9 3008.9 8.9

4000.0 4012.4 4009.7 9.7

5000.0 5022.5 5005.6 5.6

6000.0 6025.7 6007.6 7.6

7000.0 7032.6 7008.3 8.3

P r P bz (mm)

True value Initial result Correctional result Error

0.0 −6.6 −1.9 −1.9

0.0 12.2 5.1 5.1

0.0 12.8 13.3 13.3

0.0 27.7 15.3 15.3

0.0 54.5 19.5 19.5

0.0 68.8 24.5 24.5

ξx2 (1)

True value Initial result Correctional result Error

0.0 0.2 0.1 0.1

0.0 0.5 −0.1 −0.1

0.0 0.8 0.3 0.3

0.0 1.1 0.2 0.2

0.0 1.2 0.5 0.5

0.0 1.1 0.4 0.4

ξy2 (1)

True value Initial result Correctional result Error

90.0 89.8 89.9 −0.1

90.0 89.5 90.1 0.1

90.0 89.2 89.7 −0.3

90.0 88.9 89.8 −0.2

90.0 88.8 89.5 −0.5

90.0 88.9 89.6 −0.4

ξz2 (1)

True value Initial result Correctional result Error

0.0 0.3 0.1 0.1

0.0 0.2 −0.1 −0.1

0.0 0.8 0.1 0.1

0.0 0.5 0.3 0.3

0.0 0.7 0.4 0.4

0.0 1.2 0.5 0.5

Fig.6. Distance error.

trend indicates that measuring length has a larger influence on the error than any other factor. The corrected values are more accurate to the true values measured in the ocean than the initial results are, and the distance error is in the range of 7 m 725 mm. Fig. 7 shows the curve of the angle error. The initial error of ξx2 increases from 1 m to 6 m, and then decreases after reaching

the maximum 1.21 at 6. The initial error ξy2 follows the same negative trend, and has the same absolute value. The error value ξy2 is determined by the accuracy of βb and βα . The correctional results are closer to the true values than the initial results are, and the angle error is in the range of 7 m 70.61. Through our work here, the error accuracy of P r P bx , P r P by , P r P bz ,ξx2 , ξy2 , and ξz2 have improved significantly.

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Fig.7. Angle error.

7. Conclusion

References

The measuring results of the deep-sea pose measurement system, which include the relative distance and the relative angle, is analyzed and revised in this study. We found the following:

Bjorke, J.T., 2005. Computation of calibration parameters for multibeam echo sounders using the least squares method. IEEE J. Oceanic Eng. 30 (4), 818–831, http://dx.doi.org/10.1109/joe.2005.862138. Chen, H.-H., Wang, C.-C., 2011. Accuracy assessment of GPS/Acoustic positioning using a seafloor acoustic transponder system. Ocean Eng. 38 (13), 1472–1479. Cui, Xiao-jun, Si, Shu-chun, Wang, Yun-shan, 2008. Error correction of the echo sounder. Meas. Tech. 07, 34–36. Julca Avila, J.P., Adamowski, J.C., 2011. Experimental evaluation of the hydrodynamic coefficients of a ROV through Morison's equation. Ocean Eng. 38 (17–18), 2162–2170http://dx.doi.org/10.1016/j.oceaneng. 2011.09.032. Liu, Li-Jun, Han, Yan, Ming, Yang, 2008. An algorithm for underwater acoustic localization of sensor array based on error correction. Appl. Acoust. 27 (01), 54–58. Li, Y.G., Ghafir, M.F.A., Wang, L., Singh, R., Huang, K., Feng, X., Zhang, W., 2012. Improved multiple point nonlinear genetic algorithm based performance adaptation using least square method. J. Eng. Gas Turbines Power-Trans. ASME 134 (3), http://dx.doi.org/10.1115/1.4004395. Song, C.Y., Lee, J., Choung, J.M., 2011. Reliability-based design optimization of an FPSO riser support using moving least squares response surface meta-models. Ocean Eng. 38 (2–3), 304–318http://dx.doi.org/10.1016/j. oceaneng. 2010.11.001. Souto-Iglesias, A., Botia-Vera, E., Martin, A., Perez-Arribas, F., 2011. A set of canonical problems in sloshing. Part 0: experimental setup and data processing. Ocean Eng. 38 (16), 1823–1830http://dx.doi.org/10.1016/j.oceaneng. 2011.09.008. Von Schroeter, T., Hollander, F., Gringarten, A.C., 2005. Deconvolution of well-test data as a nonlinear total least-squares problem (vol. 9, p. 375, 2004). SPE J. 10 (1) 3–3. Wang, Y., Liang, G.L., 2002. Correction of sound velocity in long baseline acoustic positioning system. J. Harbin Eng. Univ. 23 (5), 32–34. Wang, L.-Q., Wang, W.-M., Zhao, D.-Y., He, N, 2010a. A deep-sea pipeline pose measurement system and its control strategy. Pet. Explor. Dev. 04, 494–500. Wang, L.-Q., Wang, W.-M., He, N., Li, W., Liu, M.-Z., Lin, Q.-H., 2010b. Design and simulation analysis of deep-sea flange connection tool. J. Harbin Eng. Univ. 05, 559–563. Wang, L.-Q., Wang, C.-D., Wang, W.-M., Li, M., 2010c. Structure and precision requirements of an underwater hydraulic circuit docking device. J. Harbin Eng. Univ. 09, 1253–1258. Wu, J.-W., Wang, L.-Q., Wang, C.-D., Wang, W.-M., 2010. A novel method for precision robotic design. J. Harbin Eng. Univ. 10, 1367–1372. Wang, W., Wang, L., et al., 2012. A novel deepwater structures pose measurement method and experimental study. Measurement 45 (5), 1151–1158. Zhu, Shaohua., Wei Xingchao., Bo, Liu, 2008. Study and application of measure technology by flange measure instrument to spool piece connection of subsea pipelines. China Offshore Oil Gas 20 (5), 342–344.

(1) We examine the error propagation relationships within the system, and calculate the theoretical error of the transition matrix algorithm. These works verify the feasibility of our measuring method. (2) We propose a correction method based on the segmented least squares theory for the measuring results of the system. The correction coefficient of the error is calculated by fitting the error data, and we revise the angle and distance results with this method. (3) The experimental platform of the system is established. The experimental results show that the distance error is in the range of 725 mm, and the angle error is in the range of 70.61 (measured at 7 m) after the correction. Thus, the system accuracy is substantially enhanced, and the precision parameter for the connection between two submarine pipelines is ensured.

Acknowledgment This work was supported by National High Technology Research and Development Program of China under Grant no. 2006AA09A105-6, National Natural Science Foundation of China (No. 50905186), CNPC Innovation Foundation 2012D-5006-0608 and Science Foundation of China University of Petroleum, Beijing (No. KYJJ2012-04-18).