The image coordinates of a point (U,V) on the cable detected by vision processing can be written in terms of its homogeneous coordinates (u,v,t) as;
If the transformation matrix of the camera coordinates to image coordinates is C, then for any world point (X,Y,Z) with respect to the camera coordinates a transformation matrix C is needed such that
(X,Y,Z)C = (u,v,t). (6)
The Z coordinate is obtained directly by the range sensor attached on the UUV which looks down. Using Eqs.(5) and (6), the 3D coordinates X and Y can be found.
6. Control System
Once the points on the cable are found with respect to the camera coordinates, the steering commands for the UUV are generated as yaw, sway, surge and heave. The far point of the detected line AA' in the image is taken as the target point to determine the yaw and sway targets as illustrated in Fig.5. Thus, cable tracking is carried out at a constant speed and constant height from the cable (i.e. from the bottom of the pool).
A methodology of feedback control using a Passive Switched Action (PSA) is proposed for cable tracking as shown in Fig.6 [3][4]. Let x be the state of the UUV and xd the target and xe the tracking error. The proposed control law for cable tracking is
