The method and error analysis of deep-sea pose measurement system

Accepted Manuscript The Method and Error Analysis of Deep-sea Pose Measurement System Gang Wang, Wenming Wang, Liquan Wang, Caidong Wang PII: DOI: Reference:

S0263-2241(16)30639-X http://dx.doi.org/10.1016/j.measurement.2016.11.002 MEASUR 4423

To appear in:

Measurement

Received Date: Revised Date: Accepted Date:

3 July 2013 19 October 2016 3 November 2016

Please cite this article as: G. Wang, W. Wang, L. Wang, C. Wang, The Method and Error Analysis of Deep-sea Pose Measurement System, Measurement (2016), doi: http://dx.doi.org/10.1016/j.measurement.2016.11.002

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The Method and Error Analysis of Deep-sea Pose Measurement System Gang Wang1,+ , Wenming Wang2,+,*,Liquan Wang1 , Caidong Wang1 (1. College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin, 150001, China; 2. College of Mechanical and Transportion Engineering, China University of Petroleum, 18 Fuxue Road, Changping, Beijing 102249, China) +The first two authors contribute equally to this paper. *Corresponding author: Wenming Wang; tel: 86-010-89733835; e-mail: [email protected] Abstract: The deep-sea pose measurement system can measure the relative distance and angle of two subsea pipelines which the connection of submarine pipelines project need, and the success of connection is mostly decided by high measurement accuracy. According to deep-sea environment, paper designs a measuring method by deep-sea pipeline pose measurement system which bases on the stretching wire. In order to increasing measurement accuracy, paper analyses the transitive relation of error source which impact system measurement accuracy, and gets the impact laws of error sources which include the orthogonal angle, extension arm vertical pitching angle and horizontal swing angle, etc. We revise the measuring result by binary linear regression analysis to improve the measurement accuracy. Through constructing the test platform, we compare measuring value and true value of the pipelines pose parameters. The experimental results show that the distance error is ±30mm inside, and that the angle error is ±0.7°inside within 7m after correction. We reduce the measurement error of the pose measurement system, and verify the correctness of the theoretical. Key words: Deep-sea pose measurement; error correction; binary linear regression analysis; experiment.

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1 Introduction With the development of deep-sea petroleum project, more and more subsea pipelines will be laid to seabed [1]. Such as tie-ins of marginal oil fields’ pipelines around main oil fields, new oil fields pipelines merge into subsea pipelines network, and the connections of sea pipeline expansion bends which are all need to connect pipelines. All these connection projects need to solve a key technology, which is to measuring the relative distance and relative angle of subsea pipelines. At present, two kinds of deep sea pipeline pose measurement technique frequently are used on the international, which are stretching wire measuring system which ROV (remotely operated vehicles) assists and underwater sound measuring system[2-5]. Deep-sea pipeline pose measurement system is a typical underwater pose measuring equipment[6], which belongs to stretching wire measurement which ROV assists. This technique can finish pose measurement operations for any two random pipes underwater, and provide pose parameters to make coupling pipeline in order to the connection of pipelines. The system which uses stretching wire to connect the two pipelines, can measure pose parameters base on transition matrix pose algorithm[2]. The success to connect underwater pipes is mostly decided by high underwater measurement accuracy. In the literature [3], the error in length is ±150mm inside, and the error in angle is ±2° inside. If measurement accuracy is higher improved, we will improve the quality and efficiency of underwater constructions. There is no existing error correction and analysis theory for underwater measurement. But scholars of domestic and international do many researches about error analysis theory in recent years[7-10], and error correction theory is widely used in practical engineering. These theories have some use for reference to the error analysis of pose measurement system. In order to measuring work and improving accuracy, paper designs the scheme of pose measure system. We analyse the impact 2

laws of error sources by error transitive relation theory, and correct the error by binary linear regression analysis. Through experiments, we verify the correctness of theoretical research. 2 Method of Measurement 2.1 Scheme of the system Figure 1 describes the deep-sea structure pose measurement system which includes a man-machine monitoring interface, two measuring devices, two positioning bases, a stretching wire, a ROV, and a magnetic power winch. The monitoring interface, which is located in the upper industrial PC, is used to directly control the ROV. The ROV, which is connected with the upper industrial PC by an umbilical cable and connects with the measuring devices through RS485 while operating the equipment, provides auxiliary operations and the power for system. Measuring devices, which carry the sensors, are the main component of this system. The stretching wire which links measuring devices I and II can measure the distance between the two pipelines. The magnetic power winch, which stores the stretching wire, is carried by ROV and provides the taut force to tension the stretching wire. 2.2 Scheme of the measuring device The measuring device I and measuring device Ⅱ basically have the same structure, and the difference is which the measuring device I hasn’t rope-length sensor. Figure 2 describes the scheme of the measuring deviceⅡ, which includes a T-frame, a rotating support, a main frame, a docking mechanism, several sensors and so on. The main frame carries other mechanisms. T-frame which is a structure likes a letter “T”, can be gripped it easily by ROV’s mechanical arm. The ROV adjusts the measuring device’s butt joint into the base on pipelines, and fixes measurement devices by aligning 4 bayonet locks at pin holes. The stretching wire goes out through the hole at the front of extension 3

arm, and connects to another measuring device on the other side. After the stretching wire is tensioned, the extension arm will drive the rotating support twirl, and turn corresponding angle along with stretching wire. Sensors include two magnetic coupling angle sensors, a rope-length sensor and a orthogonal angle sensor. Orthogonal angles α b , βb of measuring device II are detected with two orthogonal angle sensors. The pitch angle γ b and swing angle θb are detected with two magnetic coupling sensors. The length of the stretching wire S rb is detected with a rope-length sensor. Similarly, the orthogonal angle( α r , β r ), the extension arm’s pitch angle( γ r ), and swing angle ( θ r ) can be detected by the sensors of measuring device I. The host controller collects these detection parameters and transmits them to the upper industrial PC by the umbilical cable. 2.3 Pose algorithm Figure 3 shows the model of the measuring result. 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) through the transition matrix algorithm[2]. We create the reference coordinate system {r}: the xr axis is perpendicular to the pipeline’s axis; the yr axis represents the pipeline’s axis; the zr axis represents the vertical direction. Similarly, the coordinate system {b} is established. The relative PP distances are Pr Pbx , r by and Pr Pbz . The relative angles are ξ x2 , ξ y2 , and ξ z2 . Formulas (1)~(6)

show measuring results of the relative pose. (1) The horizontal distance Pr Pbx is

| Pr Pb x |=| Rr sin α r − m cos γ r sin θ r cos α r + m sin γ r sin α r + Rb cos ∆A sin α b − Rb sin ∆A cos α b sin βb − bb sin ∆A cos βb | (2) The axial distance Pr Pb y is

4

(1)

| Pr Pb y |=| −br cos β r − Rr cos α r sin β r + m cos γ r cos θ r cos β r − m cos γ r sin θ r sin α r sin β r − m sin γ r cos α r sin β r −

(2)

Rb sin α b sin ∆A − Rb cos ∆A cos α b sin βb − bb cos ∆A cos β b | (3) The vertical distance Pr Pb z is | Pr Pb z |=| −br sin β + Rr cos α r cos β r + m cos γ r sin β r cos θ r + m cos γ r sin θ r sin α r cos β r + m sin γ r cos α r cos β r −

(3)

Rb cos αb cos β b + bb sin βb | (4) ξ x2 is

ξ x2 = 90° − ∆A

(4)

ξ y2 = ∆ A

(5)

(5) ξ y2 is

(6) ξ z2 is

ξ z2 = β b − β a

(6)

Where, ∆A = arcsin{-(cosβ r cosγ rcosθ r +sinα rsinγ r -cosα rsinβ rsinγ r -cosα rcosγ rsinθ r cosγ rsinα rsinβ rsinθ r )/(-cosβ bcosγ bcosθ b -sinα bsinγ b +cosα bsinβ bsinγ b +

(7)

cosα b cosγ bsinθ b +cosγ bsinα bsinβ bsinθ b )}

3 Error analysis

Technical indicators of pose measurement system are that relative distance error is ±50mm, and relative angle error is ±1° within 7m. System accuracy is affected by error sources as below. 3.1 Error transitive relation analysis

The detecting parameters’ errors of pose measurement system directly influence and transfer to the relative pose parameters of system ( Pr Pbx , Pr Pb y , Pr Pb z , Pr Pb , ξ x2 , ξ y2 , ξ z 2 ), and transfer coefficient αi has following relation: αi = f (αr , βr , γ r ,θr , αb , β b , γ b ,θb , Srb ) 5

(8)

(1) Pr Pb error can be expressed by matrix form, and precision equation is below: δ ( Pr Pb x , Pr Pb y , Pr Pb z )T = N1∆α r + N2 ∆α b + N3∆βr + N4∆βb + N5 ∆γ r + N6 ∆γ b + N7 ∆θr + N8∆θ b + N9 ∆Srb

(9)

where:  ∂Pr Pb x   ∂Pr Pb x   ∂Pr Pb x   ∂Pr Pb x   ∂Pr Pb x   ∂Pr Pb x   ∂α   ∂β   ∂β   ∂γ   ∂α   ∂γ  r  b  r  b  r b          ∂Pr Pb y   ∂Pr Pb y   ∂Pr Pb y   ∂Pr Pb y   ∂Pr Pb y   ∂Pr Pb y  N1 =   ; N2 =   ; N4 =   ；N 5 =  ;  ; N3 =   ; N6 =   ∂β b   ∂γ b   ∂α b   ∂α r   ∂β r   ∂γ r   ∂P P   ∂P P   ∂P P   ∂P P   ∂P P   ∂P P   r bz   r bz   r bz   r bz   r bz   r bz   ∂α r   ∂β r   ∂β b   ∂γ b   ∂α b   ∂γ r   ∂Pr Pb x   ∂Pr Pb x   ∂Pr Pb x   ∂θ   ∂S   ∂θ  r  b  rb      ∂Pr Pb y   ∂Pr Pb y   ∂Pr Pb y  N7 =   ; N9 =  .  ; N8 =   ∂θ b   ∂Srb   ∂θ r   ∂P P   ∂P P   ∂P P   r bz   r bz   r bz   ∂θ r   ∂θ b   ∂Srb 

The total length error of Pr Pb is below:

δ Pr Pb =（ δ Pr Pb x 2 + δ Pr Pb y 2 + δ Pr Pb z 2 )1/2

(10)

Equation (10) is divided by measuring distance Pr Pb , so we get the relative error of Pr Pb : δPP

r b

Pr Pb

=

δ ( Pr Pb x , Pr Pb y , Pr Pb z )T

2

Pr Pb

(11)

(2) angle error Angle ξ z2 can be measured directly, and error is the direct error of senor. Angle ξ x2 , ξ y2 have same error. Transitive relation as below: δ (ξ x2 ) = N10 ∆α r + N11∆α b + N12 ∆β r + N13 ∆βb + N14 ∆γ r + N15 ∆γ b + N16 ∆θr + N17 ∆θb + N18 ∆Srb

(12)

where: N10 − N18 are the derivatives of δ (ξ x2 ) correspond to αr , β r , γ r ,θr ,αb , βb , γ b ,θb , Srb . 3.2 Error analysis of relative distance

(1) Impact of orthogonal angle αr In measurement range from 0 to 30m, we consider separately the impact of orthogonal angle αr to the relative distance Pr Pb . Through equations (9)~(11), we can get relative error ∆δ : 6

∆δ =

So, we can get the impact of

∆α r

αr

∆Pr Pb Pr Pb

=

α r N1 ∆α r Pr Pb

αr

(13)

to ∆δ , and it is shown in Figure 4. If the measurement range

within 7m, we can see from Figure 4 that the change of αr has no obvious impact on ∆δ , ∆δ keeps in 0.01% around when relative error of ∆α r /α r is 0.5%;when ∆α r /α r increase to 1%, the change of αr also has no obvious impact on ∆δ , ∆δ keeps in 0.03% around; when ∆α r /α r increase to 2% or 5%, there is an obvious increase process at the place of 1m. In measurement range from 5 to 30m, the trend is stable, with the increase of ∆α r /α r , the relative error ∆δ of Pr Pb also increase. The relative error of αr is at 1%, in the allowable range of error of orthogonal angle αr , so that ∆δ can keep in 0.3% around, that is to say absolute error of Pr Pb is 15mm in the measurement range of 7m, and is 90mm in the measurement range of 30m. (2) impact of orthogonal angle αb , β r , β b The impacts of ∆α b / α b , ∆β r / βr , ∆β b / β b to ∆δ are shown in Figure 5, Figure 6 and Figure 7. Figure 5 shows relative error of ∆δ has peak value at position of 1m, the maximum value is 1.3%, after that the trend decrease. In the range of 10m to 30m, the change of ∆α b / α b has no obvious impact on ∆δ . The relative error of ∆α b / α b could not more than 2%, then ∆δ can stabilize less than 0.3%. Figure 6 shows ∆δ has a great change in the working range of 7m, and has a peak value at position of 1m, after that the trend decrease and less than 2.5%. In the range of 5 to 30m, change of ∆β r / βr hasn’t obvious impact on ∆δ . ∆δ is stabilize within 0.5%. If ∆β r / βr is less than 2%, ∆δ

can be stabilize within 0.2%. Figure 7 shows changes of ∆β b / β b are same to ∆α b / α b . Relative error ∆δ has great changes, and has peak value at position of 2m, no more than 2.5%. In the range of 15m to 30m, the trend of 7

relative error ∆δ decrease, and is stabilize less than 0.5%. If ∆βb / βb is not more than 2%, ∆δ can be stabilize within 0.5%. (3) Impact of γ r , γ b , θ r , θ b , Srb The analysis of the impact of γ r , γ b , θ r , θ b , Srb to relative error Pr Pb are same to the analysis of orthogonal angle αr , αb , β r , β b . If ∆δ can stabilize within 0.2%, relative error of γ r , θ r need to be less than 2%, and relative error of γ b , θ b need to be less than 5%, and relative error of Srb need to be less than 0.5%. 3.3 Error analysis of relative angle

We define ei = α r , β r , γ r ,θr ,α b , βb , γ b ,θb , Srb (i=1~9), and can get from formula (12): ∆ξ x2 ei N i + 9 ∆ei ∆ξ y2 ei N i + 9 ∆ei ∆ξ z 2 ei N i + 9 ∆ei = , = , = ξ x2 ξ x2 ei ξ y2 ξ y2 ei ξ z2 ξ z2 ei

(14)

The impacts of parameters are shown in Table 1. 4 Error correction

Error analysis aims to reduce the error of system, and make the system performance optimization. Paper has an error correction for pose parameters by binary linear regression analysis[11-12]. First, we set regression equation as: E = b0 + b1 ⋅ ∆α + b2 ⋅ L

(15)

Where, b0 , b1 , b2 are the regression coefficients of regression equation. Now we have measured n couples of data: E1 = b0 + b1 ⋅ ∆α11 + b2 ⋅ L12   E1 = b0 + b1 ⋅ ∆α 21 + b2 ⋅ L22   M  E1 = b0 + b1 ⋅ ∆α n1 + b2 ⋅ Ln 2 

8

(16)

1 ∆α11 L12   E1  b0  1 ∆α L   E 2 21 22   Where, E =   , x = , b = b1  M M M  M  b2       E3  1 ∆α n1 Ln 2 

We calculate optimal estimation of parameters b0 , b1 , b2 with least square method, and can get: bˆ = ( xT x) −1 xT E

(17)

Then we put into the data to solve bˆ , and can get binary linear regression equation. To testing the validity of the regression equation, we can do the test of goodness of fit and the test of significance. Define R R2 =

S Su =1− Q St St

(18)

n

In which, regression sum of squares Su = ∑ ( Eˆi − E ) 2 , residual sum of squares SQ = i =1 n

∑ ( E − Eˆ ) i

i

2

,

and total sum of square St = Su + SQ . If the value of R2 is more close to 1, it is the

i =1

better fitting effect. When texting the goodness of fit of regression equation, we use equation (18) to distinguish imitative effect. 5 Experiment setup

Figure 8 shows the experiment platform which is described in this study[13,14]. The experiment platform consists of 8 parts: pipeline 1, stanchion, sliding table, orbit, pipeline 2, pitching mechanism, tilting mechanism, fixed station, etc. Pipeline 1 can move along the orbit. Pipeline 2 can pitch vertically by pitching mechanism, and can swing horizontally by horizontal tilting mechanism. Thus, the two experimental pipelines can simulate any random pose of two submarine pipelines. The true distance of the two pipelines can be measured by a laser rangefinder. Lastly, the true pitch angle and swing angle of two pipelines can be measured by two tilt sensors. 9

6 Experimental results analysis PP The initial values measured on Pr Pbx , r by , Pr Pbz , ξ x2 , ξ y2 , and ξ z 2 are solved with the

transition matrix algorithm, and the correctional results are solved by the correction function, which is based on the binary linear regression analysis. Figure 9 and Figure 10 show the true values of the experimental data obtained from the ocean environment, as well as both the initial and corrected results. Figure 9 shows the measuring initial result and correct result of relative distance. The initial errors of PrPb in the directions of x, y, z are increasing trend along the measuring length’s increasing. The maximum error values are respectively 44.4mm, 36.6mm and 72.8mm at the length of 7m. The maximum error value is 72.8mm, and it do not satisfy the technical requirements that require ±50mm in length within 7m. In each measuring distance, the corrected result is less than the initial result. The maximum correcte result is 44.3mm, and correction effect is obvious. The error accuracy of

Pr Pb x , Pr Pb y , Pr Pb z has been greatly improved after the amendment, and the error in length within 7m is ±30mm inside. Figure 10 shows the measuring initial result and correct result of relative angle. Initial errors of

ξ x2 , ξ y2 increase along the measuring length. The maximum value is 1.2° at the position of 7m, and it do not satisfy the technical requirements that require ±1°in length within 7m. Initial error of

ξ z2 is decided by β a and βb , which does not impact on length. In each measuring length, the correcte result is less than the initial result, and correction effect is obvious. The error accuracy of

ξ x2 , ξ y2 , ξ z2 has also been greatly improved after the amendment, and the error in angle is ±0.7° inside. The measured result utilizing section error fitting methods is more approach true value than the result never amended. 10

7 Conclusion

Paper designs the method of deep-sea pose measurement system, and has an error correction for pose parameters by binary linear regression analysis. The actual test results are correct with test data by comparing the experimental results and theoretical analysis, and experimental results show that after correction accuracy is greatly improved, the error in length within 7m is ±30mm inside, and the error in angle is ±0.7° inside. In this way, we can reduce the error and improve the measurement accuracy in this study. Acknowledgment

This work was supported by National high technology projects of China (2006AA09A105-4), National Natural Science Foundation of China (No. 51309237) and Science Foundation of China University of Petroleum, Beijing (No. 01JB0401). References

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Figures

Fig.1 Deep-sea structure pose measurement system Fig.2 Scheme of the measuring deviceⅡ Fig.3 Model of the measuring result Fig.4 Impact of αr Fig.5 The impact of αb Fig.6 The impact of β r Fig.7 The impact of β b Fig.8 Test platform model Fig.9 Distance error Fig.10 Angle error

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Table

Table1 Impact the relative angle Parameters

The impact of

ξ x2 , ξ y2

The impact of

αr

relative error need to be within 2%

no impact

αb

relative error need to be within 2%

no impact

βr

relative error need to be within 2%

βb

relative error need to be within 2%

γr

relative error need to be within 0.5%

no impact

γb

relative error need to be within 0.5%

no impact

θr

relative error need to be within 0.5%

no impact

θb

relative error need to be within 0.5%

no impact

Srb

have small impact, be within 0.5%

no impact

14

ξ z2

big impact, need to be within 0.5% big impact, need to be within 0.5%

Fig.1 Deep-sea structure pose measurement system

Fig.2 Scheme of the measuring deviceⅡ

15

zb xb Pr Pb

ξ y2

ξ x2

Pr Pbz

zr Pr Pb x

yr

Pr Pb y

xr Fig.3 Model of the measuring result

Fig.4 Impact of αr

16

yb

ξ z2

Fig.5 The impact of αb

Fig.6 The impact of β r

17

Fig. 7 The impact of β b Pipeline 1

Sliding table

Stanchio n

Pitching mechanism

Orbit

Tilting mechanis m

Fig.8 Test platform model

18

Pipeline 2

Fixed station

Fig.9 Distance error

Fig.10 Angle error 19

1, Paper designs a measuring method by deep-sea pipeline pose measurement system; 2, Paper analyses the impact laws of error sources; 3, We revise the measuring result by binary linear regression analysis.

20