Alignment Error Modelling, Analysis and Experiment of the Deep-Water Bolt Flange Automatic Connection Tool

Abstract

A deep-water bolt flange automatic connection tool plays a very important role in the process of offshore oil exploitation and transportation. In the connection process, the alignment error of bolts and nuts is the key factor to ensure the connection process is successful. Using the kinematics modeling method, this paper created the alignment error model of the deep-water bolt flange automatic connection tool and analyzed the influence of manufacturing accuracy on the alignment error of bolts and nuts through computer simulation software. Based on the error matching design method, the manufacturing accuracy of parts were optimized with a part-size-based priority sequence to ensure the bolt–nut alignment error was within the allowable limits. The land tests, the pool tests and the sea test were carried out. The test results showed that the bolt and nut can be connected in the subsea environment reliably.

1. Introduction

In the process of offshore oil exploitation and transportation, the submarine oil pipeline plays an important role in regard of safety, reliability and economy [1,2,3]. Bolt flange connection is the main connection method of a submarine pipeline [4,5,6]. Thus far, companies, such as the United States Sonsub, Norway Acergy and Switzerland All Seas Group, have used a deep-water bolt flange automatic connection tool in the process of pipeline connection [7,8]. In the connection process of the deep-water bolt flange automatic connection tool, the critical function is to lead the bolts into nuts at the same time and the alignment error of the bolts and nuts is the key factor.

In recent years, the error control method has been studied. Jia Z [9] presented a comprehensive review of the state of the art of the contouring-error reduction methods, the studies on constraining the contouring errors were classified and summarized, and, accordingly, the advantages and the disadvantages of different kinds of methods were discussed and compared. To meet the tolerance requirements of parts, Zhang J [10] proposed a high-accuracy estimation method of a contouring error that was not related to (without) a datum (ND-contouring error), and the results showed that the contouring error could be significantly reduced to improve the quality of the parts. Yang X [11] presented a novel computationally efficient contouring error estimation method for contouring control; the experimental result showed that the proposed method eliminated the sudden change of the contouring error vector directions in large curvature regions. Yang J [12] proposed a novel control structure that treated the form error as a direct objective. Experimental results showed that the proposed form error estimation and compensation methodology significantly improves the contouring accuracy. Liu Z [13] proposed an algorithm to calculate the exact contouring error of the NURBS tool path in real time; experiments were conducted, the experimental results showed the effectiveness of the proposed algorithm, and the actual contouring error was reduced significantly compared with the other methods. Zhang D [14] proposed a variable-parameter-model-based iterative pre-compensation method of the tracking error; experiments were conducted, and the results showed that the variable-parameter-model-based iterative pre-compensation method of the tracking error could reduce the tracking error and adapt to the change of the moment-of-inertia.

However, due to a technical blockade, few studies about the deep-water bolt flange automatic connection tool can be found, particularly the alignment error and tolerance assignment. It is necessary to study the deep-water bolt flange automatic connection tool to improve the performance of the deep-water bolt flange automatic connection tool. This study created the alignment error model of the deep-water bolt flange automatic connection tool using the kinematics modeling method in Section 2. The alignment error of the bolt and nut were calculated with simulation software, and the influence of the manufacturing accuracy/tolerance of each part on the alignment error was analyzed in Section 3. Based on the error matching design method, the manufacturing accuracy of parts was optimized with the part-size-based priority sequence in Section 4. The land tests, pool tests and sea test were carried out in Section 5. The test results showed that the bolt and nut can be connected successfully with the optimized tolerance assignment, which showed the design of the deep-water bolt flange automatic connection tool can be used in a subsea environment reliably.

2. Error Model of Deep-Water Bolt Flange Automatic Connection Tool

The structure of the deep-water bolt flange automatic connection tool is as shown in Figure 1. The frame mainly plays a role of support. The structures of the bolt magazine and the nut magazine are as shown in Figure 2 and Figure 3, respectively. The overall structure is a three-petal type; the two lower petals can be opened and closed by hydraulic cylinders, and they move axially along the linear guide or rotate circumferentially along the ring guide; the bolts are held in the bolt magazine through the locking mechanism and can be released after the connection is completed; the nuts are located in the nut sleeve, and their position and angle can be adjusted. The tensioner in the tensioner magazine stretches the bolt after the bolts are led into the nuts, to ensure enough preload, and the claw holds the flange in the right position.

In order to analyze the error of the deep-water bolt flange automatic connection tool, the coordinate system is defined using the Denavit–Hartenberg method (D–H method) [15]. The D–H method uses four parameters, α_i, θ_i, a_i and d_i, to describe the relative position of two adjacent coordinate systems.

The coordinate system of the deep-water bolt flange automatic connection tool is created using the D–H method, as shown in Figure 4. The initial coordinate system O₀—X₀Y₀Z₀ is established on the frame, the coordinate system of the bolt magazine is O_i—X_iY_iZ_i (i = 1~5), and that of nut magazine is O_i′—X_i′Y_i′Z_i′ (i = 1~5).

The torsion angle α_i is the angle of the Z_i−1 axis and Z_i axis about the X_i axis; the joint angle θ_i is the angle of the X_i−1 axis and X_i axis about the Z_i−1 axis, the length a_i is the distance between the Z_i−1 axis and Z_i axis along the X_i axis, and the offset d_i is the distance from the origin O_i−1 of the i−1 coordinate system to the X_i axis along the Z_i−1 axis. According to the D–H method, the general transformation formula is as follows:

$${}_{\mathbf{i}}{}^{\mathbf{i}-1}T=\left[\begin{array}{cccc}\cos {\theta }_i& -\sin {\theta }_i& 0& a_i\\ \cos {\alpha }_i\sin {\theta }_i& \cos {\alpha }_i\cos {\theta }_i& -\sin {\alpha }_i& -d_i\sin {\alpha }_i\\ \sin {\alpha }_i\sin {\theta }_i& \sin {\alpha }_i\cos {\theta }_i& \cos {\alpha }_i& d_i\cos {\alpha }_i\\ 0& 0& 0& 1\end{array}\right].$$

(1)

The D–H parameters for error modelling and analysis are listed in

Table 1. The D–H parameters.

i	ai (mm)	di (mm)	αi (°)	θi (°)
1	0	650	0	0
2	0	0	180	0
3	292	0	0	0
4	−222	0	180	0
5	0	−540	0	0
1′	0	−500	0	0
2′	0	0	180	0
3′	375	0	0	0
4′	−305	0	180	0
5′	0	120	0	0

.

The transformation matrix of the bolt tip (away from the bolt magazine) relative to the initial coordinate system O₀—X₀Y₀Z₀ is,

$${}_5{}^{0}T={}_1{}^{0}T{}_2{}^{1}T{}_3{}^{2}T{}_4{}^{3}T{}_5{}^{4}T=\left[\begin{array}{cccc}{N}_{x5}& O_{x5}& A_{x5}& P_{x5}\\ N_{y5}& O_{y5}& A_{y5}& P_{y5}\\ N_{z5}& O_{z5}& A_{z5}& P_{z5}\\ 0& 0& 0& 1\end{array}\right].$$

(2)

The transformation matrix of the bolt cap (in the bolt magazine) relative to the initial coordinate system O₀—X₀Y₀Z₀ is,

$${}_4{}^{0}T={}_1{}^{0}T{}_2{}^{1}T{}_3{}^{2}T{}_4{}^{3}T=\left[\begin{array}{cccc}{N}_{x4}& O_{x4}& A_{x4}& P_{x4}\\ N_{y4}& O_{y4}& A_{y4}& P_{y4}\\ N_{z4}& O_{z4}& A_{z4}& P_{z4}\\ 0& 0& 0& 1\end{array}\right].$$

(3)

The transformation matrix of the end of the nut sleeve (away from the nut magazine) relative to the initial coordinate system O₀—X₀Y₀Z₀ is,

$${}_{5^{\prime }}{}^{0}T={}_{1^{\prime }}{}^{0}T{}_{2^{\prime }}{}^{1^{\prime }}T{}_{3^{\prime }}{}^{2^{\prime }}T{}_{4^{\prime }}{}^{3^{\prime }}T{}_{5^{\prime }}{}^{4^{\prime }}T=\left[\begin{array}{cccc}{N}_{x{5}^{\prime }}& O_{x{5}^{\prime }}& A_{x{5}^{\prime }}& P_{x{5}^{\prime }}\\ N_{y{5}^{\prime }}& O_{y{5}^{\prime }}& A_{y{5}^{\prime }}& P_{y{5}^{\prime }}\\ N_{z{5}^{\prime }}& O_{z{5}^{\prime }}& A_{z{5}^{\prime }}& P_{z{5}^{\prime }}\\ 0& 0& 0& 1\end{array}\right].$$

(4)

The transformation matrix of the bottom of the nut sleeve (in the nut magazine) relative to the initial coordinate system O₀—X₀Y₀Z₀ is,

$${}_{4^{\prime }}{}^{0}T={}_{1^{\prime }}{}^{0}T{}_{2^{\prime }}{}^{1^{\prime }}T{}_{3^{\prime }}{}^{2^{\prime }}T{}_{4^{\prime }}{}^{3^{\prime }}T=\left[\begin{array}{cccc}{N}_{x{4}^{\prime }}& O_{x{4}^{\prime }}& A_{x{4}^{\prime }}& P_{x{4}^{\prime }}\\ N_{y{4}^{\prime }}& O_{y{4}^{\prime }}& A_{y{4}^{\prime }}& P_{y{4}^{\prime }}\\ N_{z{4}^{\prime }}& O_{z{4}^{\prime }}& A_{z{4}^{\prime }}& P_{z{4}^{\prime }}\\ 0& 0& 0& 1\end{array}\right].$$

(5)

With Equations (2)–(5), the coordinates of the bolt and nut sleeve in the initial coordinate system O₀—X₀Y₀Z₀ can be expressed as α_i, θ_i, a_i, d_i(i = 1~5, 1′~5′). In the initial coordinate system O₀—X₀Y₀Z₀, the coordinates of the bolt cap are (P_x4, P_y4, P_z4); the coordinates of the bolt tip are (P_x5, P_y5, P_z5), and the coordinates of the nut sleeve bottom are (P_x4′, P_y4′, P_z4′); The coordinates of the nut and the end of the nut sleeve are (P_x5′, P_y5′, P_z5′). Since the nut is located at the end of the nut sleeve and coaxial with the nut sleeve, the position error between the bolt tip and the nut can be approximated as the position error between the bolt tip and the end of the nut sleeve, and the angle error between the bolt and the nut can be approximated as the angle error between the bolt and the nut sleeve. The position error between the bolt tip and the nut sleeve end in the coordinate system O₀—X₀Y₀Z₀ can be expressed as,

$$P_j=P_j({\alpha }_i,{\theta }_i,a_i,d_i)=\left|P_{j5}-P_{j{5}^{\prime }}\right|,$$

(6)

where i = 1, 2, 3, 4, 5, 1′, 2′, 3′, 4′, 5′ and j = x, y, z.

Take the total differential on both sides of Equation (6) and yield,

$$d{P}_{\mathrm{j}}={\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial {\alpha }_i}})d{\alpha }_i+{\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial {\theta }_i}})d{\theta }_i+{\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial a_i}})d{a}_i+{\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial d_i}})d{d}_i.$$

(7)

Use Δα_i, Δθ_i, Δa_i and Δd_i to replace the differentials, dα_i, dθ_i, da_i and dd_i in Equation (7), and the error between the bolt tip and the nut sleeve end in the direction of each axis can be rewritten as [16,17],

$$\mathsf{\Delta }{P}_{\mathrm{j}}\approx {\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial {\alpha }_i}})\mathsf{\Delta }{\alpha }_i+{\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial {\theta }_i}})\mathsf{\Delta }{\theta }_i+{\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial a_i}})\mathsf{\Delta }{a}_i+{\displaystyle \sum _{i=1}^{\mathrm{n}}(\frac{\partial P_{\mathrm{j}}}{\partial d_i}})\mathsf{\Delta }{d}_i.$$

(8)

The total position error between the bolt tip and the nut sleeve end in the initial coordinate system O₀—X₀Y₀Z₀ is,

$$\mathsf{\Delta }p=\sqrt{{\displaystyle \sum _{j=x,y,z}\mathsf{\Delta }{P}_j{}^2}}.$$

(9)

As the bolt magazine and nut magazine are movable along the Z-axis, the error in the Z-axis direction can be compensated. Therefore, this research focuses on the position error in the XOY plane, which is,

$$\mathsf{\Delta }P=\sqrt{{(\mathsf{\Delta }{p}_{\mathrm{x}})}^2+{(\mathsf{\Delta }{p}_{\mathrm{y}})}^2}.$$

(10)

The vector of the bolt in the coordinate system O₀—X₀Y₀Z₀ is,

$$\overrightarrow{O_4{O}_5}=\left(x_B,y_B,z_B\right)=\left(p_{x5}-p_{x4},p_{y5}-p_{y4},p_{z5}-p_{z4}\right).$$

(11)

The vector of the nut sleeve in the coordinate system O₀—X₀Y₀Z₀ is,

$$\overrightarrow{O_{4^{\prime }}{O}_{5^{\prime }}}=\left(x_N,y_N,z_N\right)=\left(p_{x{5}^{\prime }}-p_{x{4}^{\prime }},p_{y{5}^{\prime }}-p_{y{4}^{\prime }},p_{z{5}^{\prime }}-p_{z{4}^{\prime }}\right).$$

(12)

The angle error between bolt and nut sleeve, θ_NB, is,

$${\theta }_{NB}=\arccos \left[\left(x_N\cdot x_B+y_N\cdot y_B+z_N\cdot z_B\right)/\left(\sqrt{x_N^2+y_N^2+z_N^2}\cdot \sqrt{x_B^2+y_B^2+z_B^2}\right)\right].$$

(13)

3. Error Analysis

According to the actual process and manufacturing conditions, the corresponding tolerance of the structural parameters under different accuracy grades can be calculated [18], as listed in

Table A1. The tolerance of structural parameters.

Joint Number	Length Tolerance (mm)		Assembly Tolerance (mm)				Coaxiality Tolerance/Parallelism Tolerance (mm)
	Grade of Tolerance		Grade of Tolerance				Grade of Tolerance
	f	m	IT5	IT6	IT7	IT8	IT5	IT6	IT7	IT8
1	0.600	1.600					0.050	0.080	0.120	0.200
2	0.600	1.600	0.055	0.078	0.125	0.195	0.040	0.060	0.100	0.150
3	0.400	1.000	0.009	0.013	0.021	0.033	0.030	0.050	0.080	0.120
4	0.400	1.000	0.013	0.019	0.030	0.046	0.030	0.050	0.080	0.120
5	0.600	1.600					0.020	0.030	0.050	0.080
1′	0.600	1.600					0.040	0.060	0.100	0.150
2′	0.600	1.600	0.055	0.078	0.125	0.195	0.040	0.060	0.100	0.150
3′	0.400	1.000	0.011	0.016	0.025	0.039	0.030	0.050	0.080	0.120
4′	0.400	1.000	0.015	0.022	0.035	0.054	0.030	0.050	0.080	0.120
5′	0.300	0.600					0.010	0.015	0.025	0.040

in Appendix A. The corresponding error limits of structural parameters are calculated, as listed in

Table A2. The error limit of structural parameter.

Joint Number	Δdi (10−2 mm)		Δai (10−3 mm)				Δαi (10−6°)				Δθi (10−6°)
	Grade of Tolerance		Grade of Tolerance				Grade of Tolerance				Grade of Tolerance
	f	m	IT5	IT6	IT7	IT8	IT5	IT6	IT7	IT8	IT5	IT6	IT7	IT8
1	60	160	50	80	120	200	77	123	185	308	77	123	185	308
2	60	160	95	138	225	345	65	97	161	242	65	97	161	242
3	40	100	39	63	101	153	103	171	274	411	103	171	274	411
4	40	100	43	69	110	166	107	179	286	429	107	179	286	429
5	60	160	20	30	50	80	37	56	93	148	37	56	93	148
1′	60	160	40	60	100	150	80	120	200	300	80	120	200	300
2′	60	160	95	138	225	345	65	97	161	242	65	97	161	242
3′	40	100	41	66	105	159	80	133	213	320	80	133	213	320
4′	40	100	45	72	115	174	80	132	212	317	80	132	212	317
5′	30	60	10	15	25	40	83	125	209	333	83	125	209	333

in Appendix A.

The relationship between the position error and angle error of the bolt and nut sleeve is programmed and calculated by MATHEMATICA, the hardware is 6-intel(R) core(TM) i7-8750H 2.21GHz, RAM 16GB, according to Formulas (10) and (13). The manufacturing error of parts applies to the normal distribution [19]. However, in the process of simulation, the random number (uniform distribution) method is adopted for the part error to investigate the error effects in the full range. Taking the accuracy of grade IT7 as an example, the calculation results are as shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11.

The errors caused by Δa_i are as shown in Figure 5 and Figure 6. The maximum error of ΔP_x is 1.25 mm, and Δa_i does not contribute to ΔP_y, as shown in Figure 5a; the maximum error of ΔP is 1.25 mm and the maximum error of θ_NB is 0.10°. The distribution of ΔP and θ_NB is a trend of truncated normal distribution, though Δa_i applies to the normal distribution, and Δa_i (i = 1~5, 1′~5′) affects the distribution of ΔP and θ_NB. The length a_i is the distance between the Z_i−1 axis and the Z_i axis along the X_i axis, and Δa_i does not affect the position error on the Y-axis.

Δd_i does not contribute to ΔP_x, ΔP_y, ΔP and θ_NB. Offset d_i is the distance from the origin O_i−1 of the i−1 coordinate system to the X_i axis along the Z_i−1 axis; therefore, Δd_i only affects the position error on the Z-axis.

The error caused by Δα_i is as shown in Figure 7 and Figure 8. Δα_i does not contribute to ΔP_x, as shown in Figure 7a; the maximum error of ΔP_y is 0.30 mm and that of ΔP is 0.30 mm, the maximum error of θ_NB is 0.05°. The distribution of ΔP and θ_NB is a trend of truncated normal distribution. The torsion angle α_i is the angle of the Z_i−1 axis and Z_i axis about the X_i axis and Δα_i does not affect the position error on the Y-axis.

The position error caused by Δθ_i is as shown in Figure 9. The maximum error of ΔP_x is less than 10⁻⁴ mm and can be ignored; the maximum error of ΔP_y is 0.11 mm; the maximum error of ΔP is 0.11 mm. Δθ_i does not contribute to the angle error θ_NB. The joint angle θ_i is the angle of the X_i−1 axis and X_i axis about the Z_i−1 axis; therefore, Δθ_i does not affect the position error on the Z-axis and θ_NB.

The total position error caused by four error sources is as shown in Figure 10. The maximum error of ΔP_x is 1.61 mm, the maximum error of ΔP_y is 0.34 mm and the maximum error of ΔP is 1.62 mm. The total angle error is as shown in Figure 11; the maximum error of θ_NB is 0.12°. There is a trend of truncated normal distribution in the distribution of ΔP and θ_NB.

The above analysis shows that Δa_i contributes the most to ΔP, and particularly needs to be paid attention to in optimization; Δd_i does not contribute to ΔP_x, ΔP_y, ΔP and θ_NB and can be ignored.

4. Tolerance Assignment

The position and angle of the nut can be adjusted in the nut sleeve. The radial adjustment range is ±1.5 mm, and the adjustment angle is ±2.5°. According to the analysis, when the manufacturing accuracy of each part of the automatic connection tool is grade IT 7, the maximum position error between the bolt and the nut sleeve is 1.62 mm, which cannot meet the requirements to lead the bolt into the nut. Therefore, it is necessary to reasonably assign the manufacturing tolerances to the parts of the deep-water bolt flange automatic connection tool, so that the total position error between the bolt and the nut sleeve does not exceed ±1.5 mm.

Generally, when there is a certain relationship between the errors of the dimension chain, the error matching design (which refers to a kind of tolerance design for several selected parameters from the dimension chain) is required to improve the system accuracy to an acceptable level [20,21].

The simple dimensions should be selected, which are convenient for processing, adjustment and assembly. It should also be a dimension with a big impact and sensitivity. According to Equations (10) and (13), the alignment accuracy of the bolt and nut sleeve is a function of a_i, α_i, d_i, θ_i, Δa_i, Δα_i, Δd_i and Δθ_i (i = 1~5, 1′~5′). The error simulation shows that Δd_i (i = 1~5, 1′~5′) takes a low effect on the alignment error. At the same time, Δa_i, Δα_i, Δd_i and Δθ_i (i = 1, 1′) are mainly caused by the manufacturing accuracy of the guide. The guide is a commercial product on the market with a given accuracy of IT 7; it is not necessary to redesign it with regard to the project cost. Therefore, Δa_i, Δα_i and Δθ_i with i = 1 and 1′, and Δd_i (i = 1~5, 1′~5′) are not considered in the error matching design. Furthermore, Δa_i contributes the most to ΔP; after that, is Δα_i (i = 2~5, 2′~5′), and the last is Δθ_i (i = 2~5, 2′~5′). Δa_i (i = 2~5, 2′~5′) is the first element to improve; after that, is Δα_i and Δθ_i (i = 2~5, 2′~5′). According to the component size and contribution to the total error, the priority sequence of the tolerance assignment is as follows: Δa₂ = Δa_2′ > Δa₅ > Δa_4′ > Δa_3′ > Δa₃ > Δa₄ > Δa_5′ > Δa₂ = Δα_2′ > Δα₅ > Δα_4′ > Δα_3′ > Δα₃ > Δα₄ > Δα_5′ >Δθ₂ = Δθ_2′ >Δθ₅ > Δθ_4′ > Δθ_3′ > Δθ₃ > Δθ₄ > Δθ_5′. The process flow chart of tolerance assignment is as shown in Figure 12.

Through error simulation in MATLAB, when the tolerance grade is assigned as the list of

Table 2. Manufacturing tolerance grade assignment.

Joint Number	Grade of Tolerance of Δdi	Grade of Tolerance of Δai	Grade of Tolerance of Δαi	Grade of Tolerance of Δθi
1	m	IT7	IT 7	IT 7
2	m	IT 5	IT 6	IT6
3	m	IT 5	IT 6	IT 6
4	m	IT 5	IT 6	IT 6
5	m	IT 5	IT 6	IT 6
1′	m	IT 7	IT 7	IT 7
2′	m	IT 5	IT 6	IT 6
3′	m	IT 5	IT 6	IT 6
4′	m	IT 5	IT 6	IT 6
5′	m	IT 5	IT 6	IT 6

, the total error meets the requirements, ΔP ≤ [ε] = 1.5 mm.

Using the selected tolerances, the total position errors and angle errors are plotted in Figure 13 and Figure 14. The maximum error of ΔP_x is 1.43 mm, the maximum error of ΔP_y is 0.35 mm, the maximum total error of ΔP is 1.44 mm and the maximum angle error, θ_NB, is 0.10°, which meet the alignment error requirements.

5. Experiment and Verification

The deep-water bolt flange automatic connection tool is as shown in Figure 15, where an extra frame is designed to hold the pipeline and the tool for the experiment.

In order to verify the reliability of the deep-water bolt flange automatic connection tool, a land test was carried out. In the land test, the deep-water bolt flange automatic connection tool was assembled on land, the offset between the bolt and nut sleeve were measured, and the connection process was tested. As shown in Figure 16, the connection test on land was repeated 20 times and the success rate was 100%.

The measurement method of the offset between the bolt and nut sleeve is as shown in Figure 17.

The process was as follows:

• Fix the micrometer on the outer edge of the nut sleeve;
• The micrometer rotates around the bolt;
• Record the maximum and minimum readings of the micrometer as l_max and l_min, respectively.

The offset between the bolt and nut sleeve is as follows:

$$\mathsf{\Delta }P=\frac{l_{\max }-l_{\min }}{2}.$$

(14)

The test was repeated 20 times and the offset results are listed in

Table 3. The offset between the bolt and nut sleeve.

Test No.	lmax (mm)	lmin (mm)	ΔP (mm)
1	46.552	44.448	1.052
2	46.453	44.547	0.953
3	46.445	44.555	0.945
4	46.451	44.549	0.951
5	46.454	44.546	0.954
6	46.526	44.474	1.026
7	46.522	44.478	1.022
8	46.465	44.535	0.965
9	46.456	44.544	0.956
10	46.483	44.517	0.983
11	46.492	44.508	0.992
12	46.502	44.498	1.002
13	46.563	44.437	1.063
14	46.425	44.575	0.925
15	46.463	44.537	0.963
16	46.525	44.475	1.025
17	46.568	44.432	1.068
18	46.462	44.538	0.962
19	46.412	44.588	0.912
20	46.469	44.531	0.969

.

In order to verify the reliability of the deep-water bolt flange automatic connection tool in an underwater environment, the pool test was carried out. The depth of the pool was 10 m. The pool test process was as follows:

• Assemble the deep-water bolt flange automatic connection tool on land;
• Use the crane to put the deep-water bolt flange automatic connection tool into the pool, as shown in Figure 18a;
• Close the claw to hold the flange;
• Close the bolt magazine, nut magazine and tensioner magazine, as shown in Figure 18b;
• Adjust the position of the bolt magazine and the nut magazine to align the bolts and holes;
• Move the bolt magazine forward and make the bolts go through the flange bolt holes, as shown in Figure 18c;
• After the bolt magazine moves to the position, the nut magazine moves forward with the nut until the nuts contact the bolts; then, the motors rotate to tighten up the nuts, as shown in Figure 18d;
• Open the claw, bolt magazine, nut magazine and tensioner magazine; lift up the deep-water bolt flange automatic connection tool back to land by crane and the test is complete;
• Redo steps 1 to 8 10 times and record the results.

The 10 pool tests were smooth and successful.

To further verify the function of the deep-water bolt flange automatic connection tool, a sea test was conducted. The ship HYSY291 was used, which has the dynamic positioning system DP−2, the depth of the underwater operation of main crane was 2600 m, the ROV was TRITON XLX 28, which can work at 3000 m underwater. The water depth of the test point was 60 m. As shown in Figure 19, the sea test followed the same process as the pool test. The test results of the deep-water bolt flange automatic connection tool are listed in

Table 4. The results of test.

Test Name	Test Times	Success Times	Success Rate
Land test	20	20	100%
Pool test	10	10	100%
Sea test	1	1	100%

.

6. Conclusions

In this research, the kinematics modeling method was used to establish the error model of the deep-water bolt flange automatic connection tool. The alignment error of the bolt and nut was calculated through the computer simulation software, and the influence of each manufacturing accuracy on the alignment of the bolt and nut was analyzed. Based on the error matching design method, the manufacturing tolerances of parts are optimized with the priority sequence based on part size. The design of the deep-water bolt flange automatic connection tool was validated by land tests, pool tests and a sea test. The following conclusions can be drawn:

(1) The alignment error model of the deep-water bolt flange automatic connection tool can be created using the kinematics modeling method. The model can be used for an alignment error calculation with a given tolerance assignment.

(2) Based on the error matching design method, the manufacturing tolerances of the parts are optimized with the part-size-based priority sequence to ensure the total error is less than the allowable alignment error, 1.5 mm. The maximum alignment errors of the bolt and nut are 1.43 mm along the X−axis and 0.35 mm along the Y−axis, and the maximum combined alignment error is 1.44 mm; the maximum angle of the central lines of the bolt and nut is 0.10°, which meets the connection requirements of the bolt and nut. Δa_i contributes the most to ΔP, and particularly needs to be paid attention to in optimization; Δd_i does not contribute to ΔP_x, ΔP_y, ΔP and θ_NB, and can be ignored.

(3) A land test, pool test and sea test have been carried out to validate the design. The land test was repeated 20 times and the pool test 10 times successfully to ensure the reliability of the connection tool. The sea test was successful as well and the connection tool worked well in subsea environment.

Further works need to be carried out, including uncertainty modelling and analysis, cost-based tolerance assignment and environment interferences, such as temperature and sea flow.