Pirate Ship : Model

From Bob's Basement

The development of a series of dynamic models, collectively referred to as the Pirate Ship : Model obtained either through system identification methods or if anyone is feeling particularly brave, from first principles. The models will allow the crew to monitor the response exhibited by the PS for a variety of input signals and environmental conditions. A PS model, whilst beneficial for playing around with the PS within a safe environment or where the environmental conditions necessary for the test do not easily present themselves is also of great benefit for the design of the controller.

The purpose of this page is to provide a collection of PS models.

Dynamics of Interest

Fundamentally, the position of the PS is the variable of interest. However, the way in which the PS changes position is dependent on the state of the PS's orientation which may comprise:

• pitch
• roll
• yaw

Both pitch and roll can be ignored for simpler models as the PS only experiences changes in pitch either through hitting a wave crest/trough or through severe acceleration and roll only affects direction (and hence position) for certain hull designs and often only when at speed and therefore, for simpler models these parameters may be ignored. Yaw therefore remains the sole orientation axis which affects the position of the PS through causing a change in the direction the PS is facing but only when thrust is subsequently applied.

Other PS parameters which affect position include the thrust applied through the motor and scews and the drag experienced as the PS pushes through the water. This is where building a model from first principles gets a tad tricky as not only are the relationships which describe these dynamics particularly complex but the ability to even provide numbers to fill in the equations is a scary thought.

Alas, all is not lost. Where we can solve the dynamic equations we shall do so, for the rest, we shall perform system identification where we physically force an input to the PS and measure its response. This can then be reverse engineered into an overall equation describing the measured variable. The disadvantages of system identification are twofold; firstly, we can only obtain models once we have a physical thing to test and secondly that unless we have an understanding of the equations we comprise that motion, we are unable to practically modify those parameters to perform selected tuning. It is therefore desirable to ensure that as much of the model is described from first principles as possible as this not only gives the crew much more flexibility in the design and simulations but also makes the models available for use by others whose parameters may be different.

Simple Model

Ultimately we want to model the relationship between a voltage applied to one or more motors through the respective amplifiers and the change of position exhibited by the hull as it passes through water. We can break this down into the following relationships which constitute this complex behaviour:

Voltage Input -> Drive System -> Force

Force -> Hull -> Position

DC Motor Model

The relationship between the voltage placed across a DC motor and the resulting speed of the motor shaft is defined as

Now we need to relate the speed of a propeller to an acceleration of the boat. This will probably be one of the most complicated characteristics of the boat to model from first principles and it may be better to setup a test-bench either to characterise the relationship between the force generated by a propeller and the angular velocity of the propeller and then measure the boats acceleration relationship to various forces or we can measure the relationship between the acceleration experienced by a boat in relation to the speed of the propeller directly.

Having obtained this behaviour, we will then be able to simulate the movement of the boat in relation to the voltage place across the motor's armature.