Moreover, need for energy transport such as petroleum, gas, and electricity has been recently grown demand for underwater infrastructures such as pipelines and cables. Basically, avoiding infrastructure damages and ecosystem protection are two major motivations for preventive maintenance. An appropriate level of safety and trustworthiness is required in order to maintain these infrastructures. Also, an irreversible environmental effect can be caused due to damages in the submarine pipelines, fishing actions, or craft anchorages. Furthermore, the shape recognition of rocks, mines, anchors, fishing nets, and other debris are noticeable points Enhancement of safety and cost reduction due to surveillance are one of purposes for using Autonomous Underwater Vehicles (Inzartsev, 2008, p. 1-3).

Figure 2: An Autonomous Benthic Explorer (ABE) designed by Woods Hole Oceanographic Institution (WHOI)

Figure 2 shows an Autonomous Benthic Explorer (ABE) designed by Woods Hole Oceanographic Institution (WHOI) that is used to map underwater environment. Figure 3 also illustrates an underwater view of this vehicle during the mapping of the ocean’s surface.

Figure 3: Mapping the ocean’s surface by an underwater vehicle

Surveillance of underwater infrastructures such as pipelines and cables is significant purpose for using Autonomous Underwater Vehicles (AUVs) in order to reduce cost and enhance safety. Furthermore, localizing and mapping are other noticeable applications of AUVs. However, optimal control of an AUV is considered as an essential criterion in well performance of the vehicles.

Various types and shapes of Autonomous underwater vehicles have been designed depending on the vehicles application. A solar autonomous underwater vehicle, for example, is shown in figure 4 that is capable to get energy from the sun rays and then continue the operation.

Figure 4: A solar Autonomous Underwater Vehicle

Autonomous underwater vehicles have been theoretically studied in this paper. In chapter 2, mathematical model of underwater vehicles are step-by-step presented. In the first sections of this chapter, the concept of the centre of gravity, buoyancy and metacentre for underwater vehicles have been outlined. The location of these centres in underwater vehicles is discussed in terms of stability. The next sections present kinematic and dynamic equations of a rigid body. Those equations then are written for a rigid body submerged fully in  an ideal fluid.

2. MATHEMATICAL MODEL OF MOTION

 In some classic mechanics text such as Ardema (2005), Lamb (1961), and Meriam and Kraige (1997), the primary derivation equations of motions are presented and discussed. Lam (1945), Newman (1977), and Fossen (1994) studied and analysed the hydrodynamic movements and forces into general equation of motion. Also, Smith (2008) represented geometric equations of a rigid body dynamics to extend notion of geometric control theory. The development and verification of non-linear model for AUVs with fins is described by Liang et al. (2008). Some simulation and modelling techniques for AUVs are proposed in Chang et al. (2002), Conte et al. (1996), Li et al. (2005), Nahon (2006), Ridley (2003), Silva et al. (2007), and Timothy (2001).

The following sections present mathematical model of a submerged rigid body. Also, some substantial studies will be outlined before representing kinematic and dynamic equation of a rigid body.

 

2.1. Centre of Mass

The centre of mass of a system is defined as a unique point in space that is the average of the system’s mass. This point is indeed the point that we can balance an object. The centre of mass simplifies many of the significant quantities in dynamics (Ruina and Pratab 2011). Many experiments done by physicist shows that knowing of the centre of mass is important to define the stability of system in different situations.

 

 

Figure 5: The centre of mass example for stability

 

 

Figure 6: Impact of the centre of mass in behaviour of a system

 

Figure 5 shows how centre of mass impacts in stability of a system. This experiment indicates that the centre of mass can be in outside of a body depending on its shape and density. The interesting experiment shown in figure 6 indicates how the centre of mass impact on the system to lead the system to stability. In this experiment, if we release a cylinder on the black ramp, the cylinder will move down. The result will be changed by replacing the cylinder by the red object.  The red object will tend to move up instead of moving down. The reason is that the system tends to close the centre of mass to Earth. When we put a cylinder on the ramp, the centre of mass of system will be closer to Earth when it is in the lower levels of the ramp. The cylinder then tends to move to lower height of ramp. In case of the red object, the distance of the centre of mass to Earth is less when it is in highest level of the ramp. The red object then tends to move to higher levels of the ramp.

 

Figure 7: An example of changing the location of the centre of mass for less energy consumption

 

Figure 7 shows another example of the centre of mass’s significance. As can be seen from the figure, for jumping above the obstacle with less consumption of energy, the centre f mass must be in the lower height during the motion. The professional jumpers change their body shape to hold their body’s centre of mass in the lowest height. Then, they pass their body above the obstacle but their body’s centre of mass is passed below of the obstacle. They indeed consume lower energy than amateur jumpers.

The centre of mass for the objects in symmetric shapes is also shown in figure 8.

 

 

Figure 8: The centre of mass for the symmetries of the objects (Ruina and Pratab 2011)

 

 The centre of gravity can also be outside of an object or system’s volume. The figure 9 shows an example of a system that its centre of gravity is in out space of their volume.

 

2.2. Centre of Gravity

The centre of gravity of a system is defined as a location in space that is the average of the force of gravity on each little bit of a system (Ruina and Pratab 2011).

Thus, the net moment for the total gravity force acting on centre of gravity is equal to the net moment for the total gravity force acting at the centre of mass. As a result, for the near-earth gravity forces on a sphere object, the centre of mass and the centre of gravity are the same point in space.

 

 

2.3. Centre of Buoyancy

The force of buoyancy usually is considerable for fluid because the buoyancy force of air is negligible comparing with the gravity force. The centre of buoyancy of an object or system is defined as a location in space that is the average of the force of buoyancy on each little bit of a system (Techet 2004). We can also define the centre of buoyancy in another way: the centre of buoyancy of an object is the centre of gravity of the fluid that is displaced by the shape of the object.

There are different theoretical and experimental methods to calculate the centre of buoyancy. One of popular technique to calculate of this centre is that we first calculate the centre of gravity of an object to know how it will sit on or in the water. We then calculate the centre of gravity of the fluid that will be displaced by the object. As a result, the centre of gravity of the fluid displaced by the object is indeed the centre of buoyancy of the object. Figure 10 shows the impact of the buoyancy and gravity forces on an object.

 

Figure 10: The buoyancy force

2.4. Static Equilibrium: Centre of Gravity and Buoyancy Balance

In terms of designing vehicles for ocean purposes, the location of centre of gravity and buoyancy is important for the vehicle’s stability. As we know, the direction of gravity and buoyancy forces are opposite. For having a stable vehicle, the first condition is that the location of the centre of gravity and buoyancy must be on a vertical line.

If both points are vertically and horizontally in one point, the vehicle is stable. However, this design is not desirable because any little force can make the vehicle unstable. Hence, one of centres must be upper or downer than another. So the question is which one must be up or down?

The answer is depending to type of the vehicle. For designing a stable half-submerged vehicle, the centre of gravity must be near to the water line and upper than the centre of buoyancy. Unlikely, the centre of buoyancy must be upper than the centre of gravity for having a stable full-submerged vehicle. The direction of the gravity and buoyancy forces in a full and half submerged vehicle are shown in figure 11.

(a)                                                                           (b)

Figure 11: The direction of the gravity and buoyancy forces in (a) a full and (b) half submerged vehicle

Another important issue in designing a full-submerged vehicle is the measure of distance between the centre of buoyancy and gravity. If the distance is less, then less force is required to make it unstable. Also, less power is consumed by thrusters to pitch or roll the vehicle.

On the other hand, if the distance between two centres is big, it is more stable but more power must be generated by thrusters to pitch or roll the vehicle. Moreover, the shape and application of an underwater vehicle and the location of thrusters are significant to determine the optimal distance between two centres.

 

2.5. Metacentre

Metacentre is the point that the vehicle rolls around it. The location of metacentre for half and full-submerged vehicle is different. In underwater vehicles, the metacentre is same as the centre of buoyancy. The metacentre in vessels, as can be seen as figure, is based on the location of the centre of buoyancy and gravity. The distance between the centre of gravity and metacentre is called metacentric height. A large metacentric height brings more stability against over-rolling of the vehicle (Techet 2004). Figure 12 shows the location of the metacentre for half and full submerged vehicle. In Figure 12 (b), B1 is the centre of buoyancy when the vessel is in stable statue and B2 indicates the centre of buoyancy of the vessel after turning.

 

(a)                                                                                (b)

Figure 12: The metacentre in (a) a full and (b) half submerged vehicle

 

2.6. Rigid Body Kinematics

An earth-fixed reference frame is choosed to allow us to measure angles and distances. The earth’s movement has a negligable impact on the dynamics of the low speed vehicles. The earth-fixed fram may be then choosed as an inertial frame. Figure 13 shows the inertisl frame and body fixed-frame. Figure 14 also shows degrees of freedom for an underwater vehicle.

Figure 13: Earth-fixed body frame and body-fixed coordinate reference frames (Smith 2008)

Figure 14: Degrees of freedom for an underwater vehicle (Arslan et al.  2008)

2.7. Rigid Body Dynamics

2.8. Submerged Rigid Body Dynamics

  3. DISCUSSION   

In section 2, the role of the centre of gravity and buoyancy in stability of underwater vehicles has been outlined. We discussed that for having a stable underwater vehicle, the centre of gravity and buoyancy should be on a vertical line. We have also identified the centre of buoyancy as the vehicle’s metacentre. In case of the location of the centre of gravity and buoyancy on a vertical line for the vehicle’s stability, the centre of buoyancy should be located above the centre of gravity. In addition, we have verified the status that both centres are located in a same point, as shown in figure 15 (a). This design leads to stability of the vehicle but it is not a valid design because any forces from any direction impacts on instability of the vehicle. Therefore, there should be a distance between two centres in vertical line.

In this section, we aim to discuss about the impact of the measure of the distance between two centres on the performance of an underwater vehicle. If we assume that the vehicle can move in six degree of freedom. For moving the vehicle in each degree of freedom, we need to use external forces such as thrusters. In fact, the vehicle consumes some amount of energy to move in each degree of freedom. For moving in each degree of freedom, there are one or more forces that are in opposite direction of the thrusters. For example, the direction of drag forces is opposite of the thrusters’ direction in all degrees of freedom. There is one more force with opposite direction of thrusters when the vehicle is rolled and pitched. This force is produced by the distance between the centre of gravity and buoyancy. The measure of this force also depends to how far the centre of gravity and buoyancy are located from each other.

 The measure of this distance in fact impacts on how much energy needed for pitching and rolling of the vehicle. As can be seen from figure 15 (b) and (c), the vehicle can be designed in way that the distance between the centre of gravity and buoyancy is big or small. If the distance is big, the vehicle is more stable but more energy must be produced by thrusters to roll or pitch the vehicle. If the measure of distance is small, less energy is needed to roll or pitch the vehicle but the vehicle is less stable.

(a)                                              (b)                                               (c)

Figure 15: The centre of gravity and buoyancy (a) in same point, and with (b) small and (c) big distance from each other

The importance of the distance’s measure is highly depends to application of the vehicle. In some applications, the vehicle does not need to be rolled and pitched. Then, the measure of the distance it is not very important. Then, a big distance between the centre of gravity and buoyancy is chosen for more stability of the vehicle.

In some other applications such as 3D image construction of the ocean’s surface, the vehicles needs sometimes to be rolled or pitched for better view of the surface. In this case, the measure of the distance between the gravity and buoyancy’s centres is important for less energy consumption.

Two factors are hence important to achieve to an optimal distance between two centres. First of all, we should know how much angle is need to roll and pitch the vehicle. We then assess the maximum force that the thrusters can be produced. By the evaluation of these two factors, we can define the optimal distance between the centre of gravity and buoyancy.

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