To understand the issues, consider the simple case of a spinning *merry-go-round*, as shown in Figure 8.2(a). Its orientation at every time can be described by ; see Figure 8.2(b). Let denote its *angular velocity*:

By default, has units of radians per second. If , then the rigid body returns to the same orientation after one second.

Assuming and is constant, the orientation at time is given by . To describe the motion of a point on the body, it will be convenient to use polar coordinates and :

and | (8.11) |

Substituting yields

and | (8.12) |

Taking the derivative with respect to time yields

and | (8.13) |

The velocity is a 2D vector that when placed at the point is tangent to the circle that contains the point ; see Figure 8.2(b).

This makes intuitive sense because the point is heading in that direction; however, the direction quickly changes because it must move along a circle. This change in velocity implies that a nonzero acceleration occurs. The acceleration of the point is obtained by taking the derivative again:

and | (8.14) |

The result is a 2D acceleration vector that is pointing toward the center (Figure 8.2(b)), which is the rotation axis. This is called

Steven M LaValle 2020-11-11