AUV Modeling
Assigning a Body Reference System
An underwater vehicle uses buoyancy to dive in water. So the center of gravity isn’t fixed unlike the center of buoyancy which depends on the volume of the AUV.
The center of buoyancy will be the center of the body frame attached to the AUV.
The local x-axis will be along the longitudinal axis with its positive direction from the CB to the nose.
The z-axis will be normal to the transversal plane of the AUV pointing downwards
Thus the y-axis can be easily assigned
This placement yields both port-starboard and top-bottom symmetry, reducing the
body’s inertia tensor Io to:
Kinematics
The motion of the vehicle in the inertial frame is described by the following vectors:
where η refers to the inertial position and orientation vector, and v the body-fixed linear and angular velocities.
Kinematics
In order to transform linear velocities from body-fixed to inertial coordinate frames, we define the transformation matrix J1, such that:
The transformation matrix J1 is generated by first performing a rotation of angle about the z-axis, followed by a rotation of about the x-axis, and finally a rotation about the y-axis.
Kinematics
In order to determine the rotational velocities in the inertial frame there is a similar procedure which produces:
Equations of Motion
Let rG=[xG,yG,zG]T be the center of gravity designated in the body frame
The following symbols are used for components in the x, y, and z directions of the body frame:
Forces = [X,Y,Z]
Moments = [K,M,N]
Velocity V = [u,v,w]
Angular velocity w= [p,q,r]
Equations of Motion
Matrix Representation of Rigid-Body Equations of Motion
The rigid-body equations of motion are expressed in matrix form as:
with rigid-body inertia matrix MRB and Coriolis and centripetal matrix CRB
Matrix Representation of Rigid-Body Equations of Motion
Derivations of the Dynamic Equations
The applied forces and moments, represented by F and M respectively, arise due to a number of effects:
* Added Mass Inertia Forces
Evaluation of Forces and moments Added Mass
In fluid mechanics, an accelerating (or decelerating) body must move some volume of the surrounding fluid which is called the added mass.
Added mass is a measure of the additional inertia created by water which accelerates with the submarine.
This force depends on certain parameters that must be calculated using the shape of the AUV
Evaluation of Forces and moments Hydrodynamic Forces (Drag and Lift)
Drag is related to the fluid density ρ, submarine frontal area Af and lies in the direction of the fluid velocity V
CD is related to the angle of attack (α) through a parabolic relationship
It is assumed that the sway (v) and heave velocity (w) are small compared with the surge (u). Angle of attack can be expressed: in the XZ-plane as tan α ≈ α = w/u or in the XY-plane as tan β ≈ β = v/u
Evaluation of Forces and moments Hydrodynamic Forces (Drag and Lift) The components of total drag force in the Xsub-,Ysub-, and Zsubdirections may be expressed:
where:
Evaluation of Forces and moments Hydrodynamic Forces (Drag and Lift)
Lift L, acting at the centre of pressure, is generated perpendicular to the flow, as the submarine moves through the water. Relocating this force to act at the center of buoyancy causes a pitching moment M to be created
Lift force and pitching moment, when viewed in the XZ-plane are: where:
Similarly in the XY-plane:
Using the expression of the angle of attack, under the assumption u››w or v we have:
Evaluation of Forces and moments Hydrostatic forces
The orientation of the body frame relative to the world frame is described by Euler angles rotated in the order:
The static forces, weight (W) and buoyancy (B) act through the centre of gravity and centre of buoyancy respectively. When resolved onto the submarine body frame, these become:
Evaluation of Forces and moments Control surface forces and moments
Attitude of the vehicle is controlled by two horizontal stern planes, and two vertical rudders. Assuming diametrically opposite fins move together the empirical formula for fin lift is given as:
where C Lδ f is the rate of change of lift coefficient w.r.t. fin effective angle of attack and S fin is the fin planform area. δe is the effective fin angle in radians. For the rudder: For the stern planes:
Evaluation of Forces and moments Control surface forces and moments
Thus we can find:
Evaluation of Forces and moments Propeller forces and moments
The propeller provides forces X prop and moments K prop around the X axis of the body-fixed frame.
Conclusion
After calculating the forces we must rearrange the terms so to have the following form:
M.v’ + C(v).v = τ We still have to calculate the parameters associated with the AUV shape
