# Modeling and Adaptive Boundary Robust Control of Active Heave Compensation Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Modeling

#### Vibration Control Equations

- Ignoring the effect of system piping pressure loss and dynamic properties;
- Neglecting servo valve flow leakage;
- The system supply pressure is stable and unchanging and oil tank pressure is 0.

## 3. Control Design

#### 3.1. Trajectory Planning

#### 3.2. Nonlinear Disturbance Observer

#### 3.3. Controller Design

#### 3.4. Stability Analysis

## 4. Simulation Results

^{2}. The initial length of the umbilical cable is 10 m, and the final length is 1000 m. The umbilical cable and the robot are initially static: $u\left(x,0\right)=0$ m and ${u}_{t}\left(x,0\right)=0$ m/s. The key physical parameters of the system are selected as ${m}_{t}=500$ kg, $\rho =900$ kg/m

^{3}, ${A}_{p}=0.01767$ m

^{2}, ${k}_{v}=1$, ${C}_{d}=0.7$, ${w}_{v}=0.002$, ${C}_{tp}=2.3\times {10}^{-10}$ m

^{3}/(s·pa), ${B}_{p}=7500$, ${p}_{s}=15$ Mpa, and ${\beta}_{e}=1.2\times {10}^{9}$ pa.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Symbol | Definition | Numerical |
---|---|---|

${m}_{p}$ | Quality of the robot | 2000 kg |

$u\left(x,t\right)$ | Vibration of cable at x | \ |

${u}_{t}\left(x,t\right)$ | Partial differentiation with t | \ |

${u}_{x}\left(x,t\right)$ | Partial differentiation with x | \ |

$l\left(t\right)$ | Time-varying length | 10~1000 m |

${\rho}_{l}$ | Linear density | 2 kg/m |

$c$ | Viscous damping | 10 Ns/m^{2} |

${\beta}_{e}$ | Hydraulic oil bulk modulus of elasticity | 1.2 × 10^{9} pa |

${p}_{s}$ | Supply pressure | 15 Mpa |

${B}_{p}$ | Damping coefficient | 7500 |

$\rho $ | Hydraulic oil density | 900 kg/m^{3} |

${A}_{p}$ | Active area | 0.01767 m^{2} |

${x}_{v}$ | Valve spool displacement (Initial position) | 0 m |

${K}_{q}$ | Flow Gain | \ |

${K}_{c}$ | Flow-pressure coefficient | \ |

${C}_{d}$ | Flow coefficient | 0.7 |

${w}_{v}$ | Throttle window area gradient | 0.002 |

${C}_{tp}$ | Total leakage coefficient | 2.3 × 10^{−10} m^{3}/(s·pa) |

${m}_{t}$ | Quality of the piston | 500 kg |

${x}_{p}$ | Piston displacement | \ |

${E}_{k}$ | Total kinetic energy of the system | \ |

${E}_{p}$ | Total potential energy of the system | \ |

$W$ | Total virtual work of the system | \ |

$\widehat{u}\left(\xi ,t\right)$ | Modal function | \ |

${\psi}_{i}\left(\xi \right)$ | Trial functions | \ |

${q}_{i}\left(t\right)$ | Generalized coordinates | \ |

$EA$ | Axial tensile stiffness of the umbilical cable | 2 × 10^{7} N |

${Q}_{L}$ | Servo valve flow | \ |

${F}_{p}$ | External load force | \ |

${k}_{v}$ | Spool conversion factor | 1 |

${y}_{d}$ | Tracking target | 0 |

${\alpha}_{i}$ | Virtual control variables | \ |

${e}_{i}$ | Tracking errors | \ |

${V}_{i}\left(t\right)$ | Lyapunov functions | \ |

${d}_{i}\left(t\right)$ | Unknown disturbances | \ |

${\delta}_{i}\left(t\right)$ | Robust control term | \ |

${c}_{1}$ | Controller design parameters | 42 |

${c}_{2}$ | Controller design parameters | 20 |

${c}_{3}$ | Controller design parameters | 4 |

${c}_{4}$ | Controller design parameters | 2 |

${c}_{5}$ | Controller design parameters | 100 |

${\tau}_{i}$ | Filter factor | 0.01 |

$\epsilon $ | A small positive constant | 0.21 |

${\gamma}_{1},{\gamma}_{2}$ | Design parameters | 0.002 |

$L$ | Disturbance observer gain | 80 |

${\delta}_{\mathrm{max}}$ | The upper bound of the robust term | 0.01 |

${\delta}_{\mathrm{max}}$ | The lower bound of the robust term | 0 |

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Symbol | Definition | Symbol | Definition |
---|---|---|---|

${m}_{p}$ | Quality of the robot | ${m}_{t}$ | Quality of the piston |

$u\left(x,t\right)$ | Vibration of cable at x | ${x}_{p}$ | Piston displacement |

${u}_{t}\left(x,t\right)$ | Partial differentiation with t | $\rho $ | Hydraulic oil density |

${u}_{x}\left(x,t\right)$ | Partial differentiation with x | ${A}_{p}$ | Active area |

$l\left(t\right)$ | Time-varying length | ${x}_{v}$ | Valve spool displacement |

${\rho}_{l}$ | Linear density | ${K}_{q}$ | Flow Gain |

$c$ | Viscous damping | ${K}_{c}$ | Flow-pressure coefficient |

${\beta}_{e}$ | Hydraulic oil bulk modulus of elasticity | ${C}_{d}$ | Flow coefficient |

${p}_{s}$ | Supply pressure | ${w}_{v}$ | Throttle window area gradient |

${B}_{p}$ | Damping coefficient | ${C}_{tp}$ | Total leakage coefficient |

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**MDPI and ACS Style**

Du, R.; Wang, N.; Rao, H.
Modeling and Adaptive Boundary Robust Control of Active Heave Compensation Systems. *J. Mar. Sci. Eng.* **2023**, *11*, 484.
https://doi.org/10.3390/jmse11030484

**AMA Style**

Du R, Wang N, Rao H.
Modeling and Adaptive Boundary Robust Control of Active Heave Compensation Systems. *Journal of Marine Science and Engineering*. 2023; 11(3):484.
https://doi.org/10.3390/jmse11030484

**Chicago/Turabian Style**

Du, Rui, Naige Wang, and Hangyu Rao.
2023. "Modeling and Adaptive Boundary Robust Control of Active Heave Compensation Systems" *Journal of Marine Science and Engineering* 11, no. 3: 484.
https://doi.org/10.3390/jmse11030484