# A Quantitative Parametric Study on Output Time Delays for Autonomous Underwater Cleaning Operations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cleaning Mission

- A near-structure operation, where the ROV cleans underwater structures, such as risers, wells, or mono-piles.
- A off-structure navigation, where the ROV should navigate from one operation area to another, could be the vessel and the riser.

#### 2.1. Near-Structure Operation

#### 2.2. Off-Structure Navigation

#### 2.3. Sensor Properties

- Inertial/Dead reckoning
- Acoustic transponders/Absolute position
- Geophysical/Relative position

#### 2.3.1. Near-Structure Operation Sensors

#### 2.3.2. Off-Structure Navigation Operation Sensors

## 3. Modeling and Assumptions

## 4. Linearization and Reformulation

#### 4.1. Rewriting to Single-Input Single-Output Systems

#### 4.2. Coupling Effect Analysis

#### 4.3. Padé Approximation

#### 4.4. Nominal Closed-Loop System

#### 4.5. Scaling

## 5. Model Analysis

#### 5.1. Performance Analysis on Stable Region

#### 5.2. Input-Output Controllability Analysis

#### 5.2.1. Lower Bound on S

#### 5.2.2. Lower Bound on KS

#### 5.2.3. Lower Bound on SG

#### 5.3. Analytical Results

#### 5.3.1. Lower Bound for S

#### 5.3.2. Lower Bound for KS

#### 5.3.3. Lower Bound for SG

## 6. Motion Sensor Evaluation

## 7. Conclusions and Future Work

- The IO Controllability is only applicable for linear systems; therefore, the non-linear system has to be linearized, resulting in inaccuracies of the results depending on the grade non-linearity. The drag contains non-linear terms, which depend on the velocity of the ROV. The linear model will differ from the non-linear model when the velocities differ from the operation points. Multiple operation points have been chosen to remedy large deviations depending on the type of operation.
- Time delays are infinite-dimensional systems, which have to be approximated into a proper transfer function. The approximation can be made by using the Padé approximation; however, this results in some inaccuracies. The approximation introduces inverse responses, larger delays introduce larger inverse responses, and therefore, the approximation is more accurate at low time delays.
- The IO Controllability benefits from being an open-loop analysis proving theoretical bound for the closed-loop system. This means that the system can be evaluated in terms of control performance while being controller agnostic. This is especially beneficial when choosing various sensor technologies for the feedback system. The analysis can then be used to find the maximum tolerated time delay, at which it will no longer be possible to control the UUV.
- The IO Controllability analysis also shows that input disturbances and pole uncertainties have a greater impact on the heading motion than on the linear motions.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Cleaning mission example. The shaded green area is the operation area where a near-structure operation is performed.

**Figure 3.**Photos of water jet nozzle used on BlueROV2. (

**a**) Photo of the applied high-pressure water jet nozzle. (

**b**) Photo of the BlueROV2 while the water jet is utilized.

**Figure 8.**Analysis on the behavior of 1st order Padé approximation for ${G}_{ol,u,{\mathit{\tau}}_{1}}$.

**Figure 9.**System diagram showing the nominal closed-loop system as state space and linear time-invariant system.

**Figure 13.**${\parallel S\parallel}_{\infty ,min}$ for Near-structure operation with upper and lower control bound.

**Figure 14.**${\parallel S\parallel}_{\infty ,min}$ for off-structure navigation with upper and lower control bound.

Description | Parameter |
---|---|

Speed | ${v}_{cleaning}$ m/s |

Water Pressure | ${P}_{jet}$ = 200 $\mathrm{bar}$ |

Distance to Structure where $\u03f5$ is cleaning efficiency | ${d}_{cleaning}\in [0;0.1)m,\phantom{\rule{4pt}{0ex}}\u03f5=50\pm 10\%$ * |

${d}_{cleaning}\in [0.1;0.2)\mathrm{m},\phantom{\rule{4pt}{0ex}}\u03f5=90\pm 5\%$ * | |

${d}_{cleaning}\in [0.2;0.3)\mathrm{m},\phantom{\rule{4pt}{0ex}}\u03f5=30\pm 10\%$ * | |

${d}_{cleaning}\in [0.3;\infty )m,\phantom{\rule{4pt}{0ex}}\u03f5=0\%$ * |

^{*}The cleaning efficiency is based on visual observations, therefore these parameters are uncertain, however the values give a good impression on how the different distances impact the efficiency.

Parameter | Near-Structure Operation | Off-Structure Navigation |
---|---|---|

${e}_{\mathit{max}}$ | ${\left[\phantom{\rule{-0pt}{0ex}}\begin{array}{cccc}0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& 0.2\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& 0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& 0.35\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}\end{array}\phantom{\rule{-0pt}{0ex}}\right]}^{T}$ | ${\left[\phantom{\rule{-0pt}{0ex}}\begin{array}{cccc}1\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& 1\mathrm{m}& 1\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& \frac{\pi}{2}\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}\end{array}\phantom{\rule{-0pt}{0ex}}\right]}^{T}$ |

${r}_{\mathit{max}}$ | ${\left[\phantom{\rule{-0pt}{0ex}}\begin{array}{cccc}1\mathrm{m}& 1\mathrm{m}& 1\mathrm{m}& \frac{\pi}{2}\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}\end{array}\phantom{\rule{-0pt}{0ex}}\right]}^{T}$ | ${\left[\phantom{\rule{-0pt}{0ex}}\begin{array}{cccc}20\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& 20\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& 20\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}& \pi \phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}\end{array}\phantom{\rule{-0pt}{0ex}}\right]}^{T}$ |

Notation | Components |
---|---|

$\mathit{M}$ | ${\mathit{M}}_{\mathit{RB}}+{\mathit{M}}_{\mathit{A}}$ |

${\mathit{M}}_{\mathit{RB}}$ | diag($m,m,m,{I}_{x},{I}_{y},{I}_{z}$) |

${\mathit{M}}_{\mathit{A}}$ | -diag(${X}_{\dot{u}},{Y}_{\dot{v}},{Z}_{\dot{w}},{K}_{\dot{p}},{\mathit{M}}_{\dot{q}},{N}_{\dot{r}}$) |

$\mathit{C}$($\mathit{\nu}$) | $\left(\right)$ |

${\mathit{C}}_{1}$($\mathit{\nu}$) | $\left(\right)$ |

${\mathit{C}}_{2}$($\mathit{\nu}$) | $\left(\right)$ |

$\mathit{D}$($\mathit{\nu}$) | -diag$\left(\right)$ |

$\mathit{g}$($\mathit{\eta}$) | $\left(\right)$ |

W | $mg$ |

B | $\rho g\nabla $ |

${\mathit{J}}_{1}\left(\mathit{\eta}\right)$ | ${\left(\right)}^{\begin{array}{ccc}c\left(\psi \right)c\left(\theta \right)& c\left(\psi \right)s\left(\varphi \right)s\left(\theta \right)-c\left(\varphi \right)s\left(\psi \right)& s\left(\varphi \right)s\left(\psi \right)+c\left(\varphi \right)c\left(\psi \right)s\left(\theta \right)\\ c\left(\theta \right)s\left(\psi \right)& c\left(\varphi \right)c\left(\psi \right)+s\left(\varphi \right)s\left(\psi \right)s\left(\theta \right)& c\left(\varphi \right)s\left(\psi \right)s\left(\theta \right)-c\left(\psi \right)s\left(\varphi \right)\\ -s\left(\theta \right)& c\left(\theta \right)s\left(\varphi \right)& c\left(\varphi \right)c\left(\theta \right)\end{array}}$ |

${\mathit{J}}_{2}\left(\mathit{\eta}\right)$ | ${\left(\right)}^{\begin{array}{ccc}1& s\left(\varphi \right)t\left(\theta \right)& c\left(\varphi \right)t\left(\theta \right)\\ 0& c\left(\varphi \right)& -s\left(\varphi \right)\\ 0& \frac{s\left(\varphi \right)}{c\left(\theta \right)}& \frac{c\left(\varphi \right)}{c\left(\theta \right)}\end{array}}$ |

$\mathit{J}\left(\mathit{\eta}\right)$ | $\left(\right)$ |

^{*}c = cos, s = sin, t = tan.

**Table 4.**Parameters used for the model, adapted from [15] with changed drag values.

Notation | Values/Term | Unit |
---|---|---|

g | 9.82 | m/s${}^{2}$ |

$\rho $ | 1000 | $\mathrm{k}$$\mathrm{g}$/$\mathrm{m}$${}^{3}$ |

m | 13.5 | $\mathrm{k}\mathrm{g}$ |

∇ | 0.0133 | ${\mathrm{m}}^{3}$ |

$({I}_{x},{I}_{y},{I}_{z})$ | (0.26, 0.23, 0.37) | $\mathrm{k}\mathrm{g}\text{}{\mathrm{m}}^{2}$ |

$({x}_{b},{y}_{b},{z}_{b})$ | (0, 0, −0.01) | $\mathrm{m}$ |

${X}_{\dot{u}}$ | 6.36 | $\mathrm{k}\mathrm{g}$ |

${Y}_{\dot{v}}$ | 7.12 | $\mathrm{k}\mathrm{g}$ |

${Z}_{\dot{w}}$ | 18.68 | $\mathrm{k}\mathrm{g}$ |

${K}_{\dot{p}}$ | 0.189 | $\mathrm{k}\mathrm{g}\text{}{\mathrm{m}}^{2}$ |

${\mathit{M}}_{\dot{q}}$ | 0.135 | $\mathrm{k}\mathrm{g}\text{}{\mathrm{m}}^{2}$ |

${N}_{\dot{r}}$ | 0.222 | $\mathrm{k}\mathrm{g}\text{}{\mathrm{m}}^{2}$ |

${X}_{u}\left(u\right)$ | $141\left|u\right|+13.7$ | N s/m |

${Y}_{v}\left(v\right)$ | $184\left|v\right|+20.0$ * | N s/m |

${Z}_{w}\left(w\right)$ | $190\left|w\right|+33.0$ | N s/m |

${K}_{p}\left(p\right)$ | $0.95\left|p\right|+0.15$ * | N s |

${\mathit{M}}_{q}\left(q\right)$ | $0.47\left|q\right|+0.8$ | N s |

${N}_{r}\left(r\right)$ | $1.173\left|r\right|+0.2$ * | N s |

Data for Used for Fit | Polynomial |
---|---|

Simulation | $41.66\left|v\right|v$ |

Test | $\left(125.9\right|v|+7.364)v$ |

Simulation Corrected | $\left(41.66\right|v|+20)v$ |

**Table 6.**Drag terms used in this study, with refitted ${Y}_{v}\left(v\right)$, ${K}_{p}\left(p\right)$ and ${N}_{r}\left(r\right)$.

Notation | Values/Term | Unit |
---|---|---|

${X}_{u}\left(u\right)$ | $141\left|u\right|+13.7$ | N s/m |

${Y}_{v}\left(v\right)$ | $184.3\left|v\right|+20.0$ | N s/m |

${Z}_{w}\left(w\right)$ | $190\left|w\right|+33.0$ | N s/m |

${K}_{p}\left(p\right)$ | $0.95\left|p\right|+0.15$ | N s |

${\mathit{M}}_{q}\left(q\right)$ | $0.47\left|q\right|+0.8$ | N s |

${N}_{r}\left(r\right)$ | $1.173\left|r\right|+0.2$ | N s |

**Table 7.**Entries for state matrix (${A}_{os}$ and ${A}_{ns}$ and input matrix (${B}_{os}$ and ${B}_{ns}$).

Entry | A_{os} | A_{ns} | Entry | A_{os} | A_{ns} | Entry | A_{os} | A_{ns} | Entry | A_{os} | A_{ns} | Entry | B_{os} & B_{ns} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${a}_{1,1}$ | −11.11 | −0.69 | ${a}_{2,2}$ | −12.22 | −0.97 | ${a}_{3,3}$ | −9.59 | −1.03 | ${a}_{5,4}$ | 0.54 | 0.00 | ${b}_{1,1}$ | 0.05 |

${a}_{1,2}$ | 2.20 | 0.00 | ${a}_{2,3}$ | 0.63 | 0.00 | ${a}_{3,4}$ | −0.39 | 0.00 | ${a}_{5,5}$ | −4.47 | −2.19 | ${b}_{2,2}$ | 0.05 |

${a}_{1,3}$ | −0.92 | 0.00 | ${a}_{2,4}$ | 0.70 | 0.00 | ${a}_{3,5}$ | 0.47 | 0.00 | ${a}_{5,6}$ | 0.26 | 0.00 | ${b}_{3,3}$ | 0.03 |

${a}_{1,5}$ | −0.75 | 0.00 | ${a}_{2,6}$ | −1.15 | 0.00 | ${a}_{4,4}$ | −3.11 | −0.34 | ${a}_{6,4}$ | 0.12 | 0.00 | ${b}_{4,4}$ | 2.24 |

${a}_{1,6}$ | 1.02 | 0.00 | ${a}_{3,1}$ | 0.57 | 0.00 | ${a}_{4,5}$ | −0.69 | 0.00 | ${a}_{6,5}$ | 0.09 | 0.00 | ${b}_{5,5}$ | 2.74 |

${a}_{2,1}$ | −2.11 | 0.00 | ${a}_{3,2}$ | −0.40 | 0.00 | ${a}_{4,6}$ | −0.45 | 0.00 | ${a}_{6,6}$ | −5.71 | −0.34 | ${b}_{6,6}$ | 1.69 |

Direction | Near-Structure Operation | Off-Structure Navigation |
---|---|---|

${G}_{scaled,N,{u}_{N}}^{delayed}\left(s\right)$ | 0.28 s | 1.94 s |

${G}_{scaled,E,{u}_{E}}^{delayed}\left(s\right)$ | 0.21 s | 1.87 s |

${G}_{scaled,D,{u}_{D}}^{delayed}\left(s\right)$ | 0.24 s | 1.44 s |

${G}_{scaled,\psi ,{u}_{\psi}}^{delayed}\left(s\right)$ | 0.17 s | 0.43 s |

$\mathit{\lambda}\phantom{\rule{4pt}{0ex}}\left[\mathbf{s}\right]$ | 0.00 | 0.60 | 1.20 | 1.80 | 2.40 | 3.00 | 3.60 | 4.20 | 4.80 | 5.40 | 6.00 |
---|---|---|---|---|---|---|---|---|---|---|---|

Near-structure operation | |||||||||||

N [$\mathrm{m}$] | 1.00 | 1.73 | 3.90 | 6.77 | 10.28 | 14.41 | 19.15 | 24.49 | 30.43 | 36.96 | 44.08 |

E [$\mathrm{m}$] | 1.00 | 1.83 | 3.84 | 6.43 | 9.51 | 13.08 | 17.12 | 21.61 | 26.56 | 31.97 | 37.82 |

D [$\mathrm{m}$] | 1.00 | 1.82 | 3.97 | 6.79 | 10.22 | 14.24 | 18.84 | 24.00 | 29.74 | 36.04 | 42.91 |

$\psi $ [$\mathrm{rad}$] | 1.00 | 5.86 | 19.74 | 43.44 | 78.37 | 125.61 | 185.96 | 260.05 | 348.36 | 451.28 | 569.11 |

Off-structure navigation | |||||||||||

N [$\mathrm{m}$] | 1.00 | 1.00 | 1.00 | 1.00 | 1.25 | 1.64 | 2.07 | 2.55 | 3.07 | 3.62 | 4.21 |

E [$\mathrm{m}$] | 1.00 | 1.00 | 1.00 | 1.00 | 1.30 | 1.69 | 2.13 | 2.61 | 3.12 | 3.67 | 4.26 |

D [$\mathrm{m}$] | 1.00 | 1.00 | 1.00 | 1.26 | 1.79 | 2.39 | 3.05 | 3.78 | 4.58 | 5.44 | 6.36 |

$\psi $ [$\mathrm{rad}$] | 1.00 | 1.39 | 4.00 | 7.80 | 12.61 | 18.42 | 25.23 | 33.03 | 41.82 | 51.59 | 62.36 |

$\mathit{\lambda}\phantom{\rule{4pt}{0ex}}\left[\mathit{s}\right]$ | 0.00 | 0.60 | 1.20 | 1.80 | 2.40 | 3.00 | 3.60 | 4.20 | 4.80 | 5.40 | 6.00 |
---|---|---|---|---|---|---|---|---|---|---|---|

Near-structure operation | |||||||||||

N [$\mathrm{m}$] | - | 4.11 | 11.98 | 19.02 | 25.77 | - | - | - | - | - | - |

E [$\mathrm{m}$] | - | 4.80 | 11.36 | 17.01 | 22.36 | 27.65 | - | - | - | - | - |

D [$\mathrm{m}$] | - | 9.28 | 24.18 | 37.65 | 50.72 | - | - | - | - | - | - |

$\psi $ [$\mathrm{rad}$] | - | 34.41 | - | - | - | - | - | - | - | - | - |

Off-structure navigation | |||||||||||

N [$\mathrm{m}$] | - | - | - | - | 14.19 | 36.74 | 61.79 | 88.26 | 115.64 | 143.67 | 172.22 |

E [$\mathrm{m}$] | - | - | - | - | 17.95 | 42.15 | 68.27 | 95.43 | 123.23 | 151.48 | 180.11 |

D [$\mathrm{m}$] | - | - | - | 15.55 | 47.61 | 83.02 | 120.06 | 158.08 | 196.85 | 236.26 | 276.29 |

$\psi $ [$\mathrm{rad}$] | - | 13.14 | 78.89 | - | - | - | - | - | - | - | - |

$\mathit{\lambda}\phantom{\rule{4pt}{0ex}}\left[\mathit{s}\right]$ | 0.00 | 0.60 | 1.20 | 1.80 | 2.40 | 3.00 | 3.60 | 4.20 | 4.80 | 5.40 | 6.00 |
---|---|---|---|---|---|---|---|---|---|---|---|

Near-structure operation | |||||||||||

N [$\mathrm{m}$] | - | 0.12 | 0.85 | 2.50 | 5.26 | 9.22 | 14.46 | 21.03 | 28.94 | 38.23 | 48.90 |

E [$\mathrm{m}$] | - | 0.12 | 0.81 | 2.31 | 4.74 | 8.16 | 12.61 | 18.10 | 24.65 | 32.28 | 40.98 |

D [$\mathrm{m}$] | - | 0.07 | 0.42 | 1.20 | 2.45 | 4.21 | 6.49 | 9.30 | 12.65 | 16.54 | 20.98 |

$\psi $ [$\mathrm{rad}$] | - | 1.39 | 10.22 | 31.80 | 69.95 | 127.42 | 206.30 | 308.16 | 434.24 | 585.50 | 762.71 |

Off-structure navigation | |||||||||||

N [$\mathrm{m}$] | - | - | - | - | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.07 | 0.08 |

E [$\mathrm{m}$] | - | - | - | - | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.07 | 0.08 |

D [$\mathrm{m}$] | - | - | - | 0.01 | 0.02 | 0.03 | 0.05 | 0.07 | 0.09 | 0.12 | 0.14 |

$\psi $ [$\mathrm{rad}$] | - | 0.06 | 0.31 | 0.75 | 1.39 | 2.23 | 3.27 | 4.51 | 5.95 | 7.59 | 9.42 |

Motion | Delay at LCL and UCL for ${\left|\right|\mathit{S}\left|\right|}_{\mathit{\infty},\mathbf{min}}$ | Sensor Technology | Time Delay | |
---|---|---|---|---|

Near-Structure Operation | Off-Structure Navigation | |||

N | LCL: 0.69 s | LCL: 3.44 s | LBL [19] | 10 s [19] |

UCL: 1.22 s | UCL: 5.68 s | SBL [9,28] | 1.50–2.50 s [9] | |

E | LCL: 0.66 s | LCL: 3.36 s | DVL [19,27,29] | 0.03–0.20 s [19] ** |

UCL: 1.24 s | UCL: 5.63 s | Visual SLAM [18,30] | 0.10–0.20 s [18] | |

D | LCL: 0.66 s | LCL: 2.61 s | Pressure [31] *** | 0.00 s * |

UCL: 1.21 s | UCL: 4.35 s | |||

$\psi $ | LCL: 0.30 s | LCL: 0.78 s | FOG [32] | 0.00 s * |

UCL: 0.47 s | UCL: 1.2 s | MEMS Gyroscope [32] | 0.00 s * | |

Compass | 0.00 s * |

^{*}do have a small delay, as signal transition and data processing always results in small delays; however, relative to the other sensors, the delay can be neglected.

^{**}For the DVL, the delay is impacted by the distance to the seabed, as sound waves have a limited propagation in water [19].

^{***}The sensor technologies used for N and E directions can also be used for D. However, pressure transmitters are almost always applied as the primary sensor for D.

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**MDPI and ACS Style**

Sørensen, F.F.; von Benzon, M.; Liniger, J.; Pedersen, S.
A Quantitative Parametric Study on Output Time Delays for Autonomous Underwater Cleaning Operations. *J. Mar. Sci. Eng.* **2022**, *10*, 815.
https://doi.org/10.3390/jmse10060815

**AMA Style**

Sørensen FF, von Benzon M, Liniger J, Pedersen S.
A Quantitative Parametric Study on Output Time Delays for Autonomous Underwater Cleaning Operations. *Journal of Marine Science and Engineering*. 2022; 10(6):815.
https://doi.org/10.3390/jmse10060815

**Chicago/Turabian Style**

Sørensen, Fredrik Fogh, Malte von Benzon, Jesper Liniger, and Simon Pedersen.
2022. "A Quantitative Parametric Study on Output Time Delays for Autonomous Underwater Cleaning Operations" *Journal of Marine Science and Engineering* 10, no. 6: 815.
https://doi.org/10.3390/jmse10060815