3 Extending the PDM

  The point distribution model as described above refers only to the shape of the flock; we require to model the activity of the ducks relative to the robot predator. In order to achieve this, the standard PDM must be extended to include additional parameters that govern the interaction of the ducks and the robot. Potentially there are many such parameters, for example:

• flock velocity; flock acceleration
• robot position; robot velocity and acceleration
• distance from boundary

Here we construct a simplified model based upon what we expect to be the two most important factors: the flock velocity and the position of the robot relative to the flock.

Including such information into the PDM is achieved by augmenting the shape vector to produce the vector :

where |v| is the flock speed, is the direction of the flock motion, and are the co-ordinates of the centre of mass of the robot relative to the centre of mass of the flock.

We expect the direction and proximity of the robot to dominate flock behaviour, and for this to influence speed most. It is therefore reasonable to convert the Cartesian co-ordinate pairs into polar co-odinate pairs, prior to the PCA, thus:

PCA requires that the values in the training input are of comparable units . However, when converting to polar coordinates, we change from distance pairs to pairs of displacement and angle, , which are not comparable. To compensate for this, we use the relation , where is the arc length of an angle at radius [ 11 ]. This allows the use of arc lengths, which are comparable with displacements, instead of angles.

We used a training set of 120 examples, flipped around the principal axis in order to invoke symmetry and double the number of examples [ 12 ]. The mean shape for this model can be seen in Figure 3 , together with the first 2 modes of variation.

 
Figure 3:   The mean shape (left), and first two modes of variation

The mean shape is observed to be a `blobby' mass, with the robot predator represented as a circle underneath it. The first mode of variation (Figure 3 , centre) suitably captures the desire of the flock to move away from the robot, as the robot is located around the flock.

However, the model does not seem to describe any relationship between the distance of the predator from the flock and the flock speed, a characteristic of the ducks' behaviour in real life. The second mode of variation (Figure 3 , right) clearly indicates a change in distance of the robot from the flock centre, but no reaction in speed, and minimal shape variation. It also suggests implausible models, with the robot being `allowed' to move within the flock itself.

The reason that these problems arise is because we have not properly compensated for the influence these additional parameters have when introduced into the PDM. A novel method for correctly adjusting these values is now described.