A Motion Model of a Complex-Shaped Remotely Operated Underwater Vehicle

The knowledge of velocities of a remotely operated underwater vehicle (ROV) is crucial for the study of the ROV motion. The ROV motion equations are complemented by hydrodynamic parameters and forces acting upon the ROV. The matrices of hydrodynamic damping coefficients and external forces acting upon the ROV are considered in this study as well. The computational results obtained by the Runge-Kutta method are compared with the experiment. It appears that the presented model can be useful for the design and investigation of remotely operated underwater vehicles.


Introduction
Remotely operated underwater vehicles are widely used for underwater survey missions such as the location, positioning and identification of various submerged structures and objects. ROVs have been used for inspection works, for example, pipeline surveys, jacket inspections and marine hull inspection of vessels. They are also used to assist with the exploration of ocean resources and with the protection of marine environment. The military use of ROVs is primarily for naval mine-hunting and mine-breaking.
There are numerous research studies in the field of underwater vehicles in the world. Many investigations of underwater vehicles motion assume the body geometry having three planes of symmetry that substantially simplifies the mathematical model of motion. For example, Gomes et al. [1] studied the ROV motion of three planes of symmetry at the University of Porto. Likewise, Marzbanrad et al. [2] applied the three planes of symmetry for design and control of ROV, built in the Shiraz University Robotic Lab. In addition, Eidsvik and Schjølberg [3] from the Norwegian University of Science and Technology used three planes of symmetry for the evaluation of hydrodynamic parameters that characterize their ROV behaviour. In these studies, the matrices of ROV hydrodynamic damping coefficients are diagonal.
In this study, the research is focused on the complex-shaped ROV with two planes of symmetry and one asymmetric plane. The solution results for the ROV motion are then compared with the experiment. The configuration of the examined ROV, consisting of the main body, two lights, two thrusters, and two balance flotation blocks is shown in Fig. 1.

Mathematical Model of the ROV Motion
The developed mathematical model provides solutions for velocities, hydrodynamic parameters and forces acting upon the ROV.

Coordinate Systems
There are two coordinate systems used to describe the ROV motion: the earth-fixed coordinate system (O0, X0, Y0, Z0) and the ROV body-fixed coordinate system (O, X, Y, Z). Both of the coordinate systems are shown in Fig. 2. Here:

Model Assumptions
To reduce the model complexity, the following simplifying assumptions are made: • ROV has two planes of symmetry: port-starboard plane, fore-aft plane, • ROV is a rigid body of constant mass, • ROV is deeply submerged in a homogeneous unbounded fluid, • the effects of surface waves, underwater walls and bottom are ignored, • the linear and angular couple of terms in damping is neglected, • the damping terms of the second and higher orders are neglected. The fore-aft plane symmetry is admitted due to the geometric similarity of lights and thrusters.

Model of the ROV Motion
The first coordinate transform relates to translational velocities of the ROV between the two coordinate systems [4]: The second coordinate transform relates to rotational velocities of the ROV between the two coordinate systems [5]: The relative velocity between the ROV and the surrounding water flow is defined as follows: The set of equations describing the ROV motion can be written as follows [3]: where • MRB is the generalized mass matrix of the ROV, • CRB is the Coriolis force and central force matrix, • τ are the total forces and momenta acting upon the ROV.
where • τHS is the restoring force and moment vector, • τD is the damping force and moment vector, • τA is the added mass force and moment vector, • τFK is the Froude-Krylov force vector, • τP is the thruster force from two propellers vector. The restoring force and moment vector τHS is given by the following formula: HS sin cos sin cos cos cos cos cos sin sin cos cos cos sin sin (8) here • W = m g is the gravity of the ROV at the mass centre, • B = ρ g VROV is the buoyancy force of the ROV at the buoyancy centre, • g is the acceleration due to gravity, • VROV is the ROV volume, • m is the ROV mass in the air, • ρ is the water density, • xb, yb, zb and xG, yG, zG are the positions of the centre of buoyancy and the centre of gravity with respect to the origin coordinate, respectively. The damping force and moment τD is given by the following formula [5]: where D(Vr) is the hydrodynamic damping matrix that is written as [3]: are the quadratic damping coefficients, which may be calculated by using the following formula [6,7]: where • ρ is the density of the surrounding water, • Afx, Afy, Afz is the ROV frontal area along OX, OY, OZ, • cd is the axial drag coefficient of the ROV, • cdc is the crossflow drag coefficient of the ROV, are the damping moment coefficients of the ROV, that can be calculated by using the strip theory [6]. The added mass force and moment τA can be calculated by the following formula [3]: where MA and CA(Vr) are the added mass and Coriolis-like matrix induced by MA, respectively. Under the assumptions made, the ROV has two planes of symmetry (portstarboard plane, fore-aft plane) and one asymmetric plane (bottom-top plane). The added mass matrix is calculated as follows [5]: where mij is the added mass coefficient of ROV in the i direction caused by the ROV acceleration in the j direction; i, j = 1, 2, 3, 4, 5, 6. Based on the work by Fossen [8], the off-diagonal elements of a positive inertia matrix are much smaller than the diagonal counterparts. Therefore, the off-diagonal elements in the added mass matrix (13) can be neglected.
The Coriolis-like matrix CA(Vr) is written as follows [5]: The Froude-Krylov force τFK appears due to the inertia matrix of the displaced fluid. This force is defined [3,8] After some algebra, Eq. (18) can be rewritten as follows: ( ) where: and

Validation of the ROV Motion Model
The mathematical model described above has been solved by the Runge-Kutta method using the Fortran programing language. The input parameters to achieve the solution are given in Tab. 1. The solving process ran for 20 seconds with the time step of 0.002 seconds. Some solution results for the ROV motion are shown in Figs 3-6. The evolutions of the ROV velocity components are shown in Fig. 3. The component of velocity vector Vx along the O0X0 axis is shown in Fig. 3a. One can see that the velocity Vx is invariable after about 5 seconds of moving, as its value reaches 1.65 m/s. The evolution of the velocity component Vz along the O0Z0 axis is shown in Fig. 3b. This value oscillates around zero. After about 10 seconds, the ROV movement reaches a quasisteady state. However, the velocity component Vy along O0Y0 equals zero at all times, because the propulsive forces from both of the propellers are equal. These results agree well with real observations.

Fig. 3 Velocity of ROV: a) component Vx along O0X0, b) component Vz along O0Z0
The evolution of the ROV displacement along the O0X0 axis is shown in Fig. 4. The maximal displacement is about 32 m after 20 s.

Fig. 4 Displacement of ROV along O0X0 axis
The angle θ oscillations of the ROV around the O0Y0 axis are quite small, as it can be seen in Fig. 5.

Fig. 5 Angle θ of ROV around O0Y0 axis
Nevertheless, as illustrated in Fig. 6, the motion of the ROV around O0Z0 axis oscillates quite a lot during the first 10 seconds. After that, the motion of the ROV is less fluctuant and tends to being stable.

Verification of the ROV Motion Model
The computational results of the ROV motion are compared with the measured values. Experiments were held in an experimental water basin. A photo of the ROV prototype is shown in Fig. 7 and the schematic diagram of the experimental setup is shown in Fig. 8. This experiment enables to determine the ROV velocity Vx along O0X0 axis in the period of its stable motion, i.e. the maximum velocity of the ROV. The range of measurements includes four cases of propulsive forces (5, 10, 15, 20 N) for two distances between thrusters (0.455 m and 0.505 m). The propulsive forces are the same for both of the thrusters.
The comparison of some results of the developed ROV motion model and the measured values from the presented experiment for two distances between thrusters is given in Tab The comparison in the Tab. 2 shows that the ROV velocities Vx increase with increasing distance between the thrusters. The ROV velocities computed by the ROV motion model are larger than those determined from the experiment. It is reasonable for the ROV operation under real conditions. The percentage of difference between the computational and experimental results is quite small (between 2.83% and 9.63%). These results prove that the theoretical method of solution and the ROV motion model are appropriate.

Conclusion
A mathematical model has been developed to study the motion characteristics of the complex-shaped ROV with two planes of symmetry and one asymmetric plane. This model includes, besides the motion equations, hydrodynamic damping parameters and forces acting upon the ROV. The set of equations and matrices presented in this article was solved by the Runge-Kutta method using the Fortran programing language.
The mathematical model was validated for the given configuration of the examined ROV, consisting of the main body, two lights, two thrusters and two balance flotation blocks. Furthermore, the model was experimentally verified through measuring the ROV velocity by a high-speed camera. The presented model provides a satisfactory agreement with the measured ROV velocity.