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 momentsAdded 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 momentsHydrodynamic 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 momentsHydrodynamic Forces (Drag and Lift) The components of total drag force in the Xsub-,Ysub-, and Zsub-

directions 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 momentsHydrostatic 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 momentsControl 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 CLδf is the rate of change of lift coefficient w.r.t. fin effective angle of

attack and Sfin is the fin planform area. δe is the effective fin angle in radians.

For the rudder:

For the stern planes:

Evaluation of Forces and momentsControl surface forces and moments Thus we can find:

Evaluation of Forces and momentsPropeller forces and moments The propeller provides forces Xprop and moments Kprop 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