2. Underwater Vehicle Dynamics

As stated in the previous chapter, a submersible vehicle such as an ROV or AUV is free to

move in three dimensions. This movement may be described with reference to three

orthogonal axes, x, y, and z which may be fixed relative to the Earth (X0, Y0, and Z0), or

relative to the vehicle (x, y, and z). The vehicle will have up to six degrees of freedom -

translations in the x, y, and z directions (“surge”, “sway”, and “heave” respectively) and

rotations about these axes (“roll”, “pitch”, and “yaw”). These degrees of freedom were

illustrated in figure 1.2, and are reproduced below:-

Figure 1.2 - ROV Degrees of Freedom

Equations describing the dynamics of a vehicle may be derived for each degree of freedom

(an example is the equations of motion for the US Naval Postgraduate School AUV II,

reported in Healey and Lienard, 1993, and which are included as appendix C to this report).

The equations may describe the motion of the vehicle relative to either Earth-fixed, or

Vehicle-fixed axes, although it is possible to apply a series of kinematic transformations and

convert from one representation to the other. In the case of the Vehicle-fixed model, the

equations of motion are greatly simplified if the origin for the axes x, y, and z is chosen to

coincide with the vehicle’s centre of gravity, G.

2.1. ROV Dynamic Model

The design of a control system for a particular underwater vehicle requires a model of the

vehicle’s dynamic characteristics. The dynamic characteristics of an underwater vehicle

depend on the vehicle’s centre of drag and a large number of other hydrodynamic factors

which must be determined experimentally [Allmendinger, 1990]. Clearly, many of these

factors cannot be determined until a prototype vehicle is available for testing.

A further requirement of the control system for most practical ROVs is that it must be

‘robust’, that is it must be able to cope with changes in the configuration and dynamic

characteristics as items are added to, or removed from, the vehicle.

The general form for the dynamic model of an underwater vehicle, the non-linear equations of

motion and their linear derivatives are presented in Appendix B. Thrusters on underwater

vehicles have a significant influence on the overall vehicle dynamics. In many cases, the

dynamic characteristics of the thrusters dominate the vehicle dynamics and are a particular

problem to be overcome in developing ROV/AUV controllers. For this reason, thruster

dynamics are considered in more detail below.

2.2. Thruster Dynamics

In general, the thruster force, and the moment vector resulting from that force, is a complex

non-linear function which depends on the vehicle’s velocity vector ν ∈ℜ6 and the control

variable n p∈ℜ ( )p ≥ 6 . The relationship can be expressed as:

τ ν= b n( , )

where: b(ν,n) is a non-linear vector function.

Most ROVs use thrusters which incorporate a single-screw propeller. An approximation for

the thrust, T, developed by a single-screw propeller of diameter D can be obtained using:

T D K J n nT= ρ 4

0( )

where: ρ is the water density.

KT is the thrust coefficient.

n is the propeller speed in revolutions per second.

and J0 is the ‘advance number’.

The advance number, J0, can be calculated as: JV

nDa

0 =

where the ‘advance speed’, Va, is the speed of water entering the propeller. Va, is related to

the vehicle speed, V, by the equation

Va, = (1 - w)V

Where: w is the ‘wake fraction number’ (typically between 0.1 and 0.4 depending on the

vehicle hydro-dynamics).

Many underwater vehicles, and most ROVs (including the ROV being developed for this

project) use thrusters driven by small DC motors designed for the marine environment. Many

of these thrusters will be fitted with Kort nozzles. A Kort nozzle is a ducted propeller. The

duct, or nozzle, has a hydrofoil cross section, and extends forward of the propeller. Kort nozzles

provide higher efficiency at high thrust levels (with efficiency improvements of up to 20% being

achievable), and offer substantial improvements in course stability. However, manoeuvrability

& directional stability when going astern is impaired when Kort nozzles are fitted, and there is

an increased risk of cavitation and erosion damage due to the pressure drop in the nozzle. It is

possible to design thruster systems which incorporate Kort nozzles to give symmetrical forward

and backward thrusts.

In general, the thrust coefficient KT is non-linear function. However, for positive values of the

advance number J0, KT is approximately linear in J0. Under these conditions, the approximation:

KVnD

Ta

= +α α1 2

where α 1 and α 2 are constants determined by KT and J0, appears to be valid.

The thruster force, T, can therefore be written as:

T n V T n n T nVa n n n V aa( , ) | | | || | | |= +

Where: T n n| | > 0

and T n Va| | < 0

Similarly, thruster torque can be expressed as:-

Q n V Q n n Q nVa n n n V aa( , ) | | | || | | |= +

Where: Q n n| | > 0 and Q n Va| | < 0 are determined by physical characteristics including propeller

diameter, duct shape, propeller speed, water density and the advance speed, Va.

If b T n n1 0= >| | and b T n Va2 0= <| | , the thruster force produced by a single propeller can be

expressed as:-

τ ν= −b n n b n1 2| | | |

From this result, the multi-variable model of the propeller dynamics can be expressed as:

τ ν= −B u B u1 2( )

where: B1 and B2(u) are matrices with dimensions determined by the number of thrusters and

degrees of freedom controlled.

and u p∈ℜ is the control variable defined as ui = |ni|ni (I = 1..p)

This non-linear model is referred to as the “bilinear-model” of propeller dynamics. The

theory of non-linear propeller dynamics appears to be very limited with little information

available in the literature. This may be due to the complexity of the matrix analyses required.

Most practical control systems appear to use a linear approximation of the bilinear-model

where B u2( )ν is assumed to approximate to zero (This is approximately true for slow speeds

where ν ≈ 0 ). The propeller thrust in this case can therefore be expressed as:

τ = Bu Where: B = B1

This approximation is called the “affine-model” of propeller dynamics.

The model for a speed-controlled DC motor, as used in many underwater vehicles, was

developed in section 3.2 of Block 3 of PMT604, Real-Time Control. The model can be

expressed as follows:

L didt

R i K n uaa

a a aM= − − +2π

2πJ dndt

K i Q n Vm a aM= − ( , )

Where: La is the armature inductance.

Ra is the armature resistance.

ia is the armature current.

KM is the motor torque constant.

n is the motor speed in revolutions per second.

ua is the armature voltage.

Jm is the moment of inertia of motor/thruster.

Q(n,Va) is the propeller load.

This model is, of necessity, a simplification since many of the physical limitations of practical

devices (hysteresis, coulomb friction etc.) are ignored. A propeller may be modelled as a

system as shown in the figure below:

Figure 2.1 - Systems model of Propeller

With reference to the figure above, we can write:

n s h s u s h s Q su a Qa( ) ( ) ( ) ( ) ( )= −

Where: h s KT s T s

ua( )( )( )

=+ +

1

1 21 1,

and h sK T

T s T sQ( )

( )

( )( )=

+

+ +

2 3

1 2

1

1 1

The equations for h sua( ) and h sQ( ) are obtained from the Laplace transforms of the dynamic

model for a speed controlled DC motor above, where K1 and K2 are gain constants, and T1,

T2, T3 are time constants which are determined by the values of the various parameters in the

dynamic model.

In many submersible vehicles, the control matrix B in the affine model will not be a square

matrix, and p ≥ n (i.e. there will be more possible control inputs than there are degrees of

freedom which may be controlled). It is theoretically possible to obtain an “optimum”

distribution of control energy for each of the six degrees of freedom by attempting to

minimise the quadratic energy cost function:

J u WuT= 1

2

Subject to τ − =Bu 0 (from the affine thruster model)

Where: W is a matrix which applies weightings to the control energy

(Weightings may be used to help conserve battery power)

u is the applied voltage

T is the thruster force

On a vehicle which uses only thrusters for control (i.e. a vehicle without control surfaces) it is

normal to apply equal weighting to all control inputs. This is the strategy which is intended

for the ROV being developed for this project. In this case, a result known as the Moore-

Penrose Pseudo Inverse can be applied to simplify the calculation of the optimal distribution

of control energy. With reference to the quadratic energy cost function above, u can be

computed as:

u B W= † τ

Where B† is the Moore-Penrose Pseudo Inverse defined as:

B B BBT T† = −( ) 1

These results can be used to define the contribution of the thruster dynamics to the overall

vehicle dynamics.