Bicycle rotation matrix

Question

Suppose you have a flat plane on which a bicycle lies. When the bike enters into a turn, it leans in a turn with a theta angle. At the same time, the frame of the bicycle points in the same direction as the speed of the bicycle.

Thus, given the velocity vector v (it is assumed to be in the XZ plane) and theta angle, how can you find the rotation matrix for the bike?

Answer 1

Use quaternion. The axis is the speed of the bike, and the “tilt” is the axis. Then convert to rotation matrix