To achieve the flapping motion, let us define displacement vectors which, when added to , i=1,2,3,4, yield the new directions of . Let the normal of the plane formed by and be a unit vector . The pectoral flapping motions are modeled by simply specifying the direction of the displacement vectors along and then letting the lengths of vary over time in a sinusoidal fashion. In particular, the 's are defined as follows:
where a and b are parameters that define the amplitude and frequency of flapping, respectively; time t is discrete and equals the number of animation frames. Since the 's are not coplanar (they are nearly coplanar), the shape of the fins deform slightly during the flapping motion. This is not undesirable since natural pectoral fins deform constantly due to hydrodynamic forces.
In our implementation, we choose a and b to be nearly proportional to such that the faster the fish needs to ascend or descend, the greater the amplitude and frequency with which the fins flap. Note that a and b are nonzero values when so the fins are kept in motion even when the fish is not engaged in pitching, yawing, or rolling. Fig. shows four snapshots of the flapping motion of the pectoral fins.
|Xiaoyuan Tu||January 1996|