Assume that the body frame of aligns with the principle axes. The remaining six equations of motion can finally be given in a nice form. Using (13.99), the expression (13.98) reduces to [681]

Multiplying out (13.100) yields

To prepare for the state transition equation form, solving for yields

(13.102) |

One final complication is that needs to be related to angles that are used to express an element of . The mapping between these depends on the particular parameterization of . Suppose that quaternions of the form are used to express rotation. Recall that can be recovered once , , and are given using . The relationship between and the time derivatives of the quaternion components is obtained by using (13.84) (see [690], p. 433):

(13.103) |

This finally completes the specification of , in which

(13.104) |

is a twelve-dimensional phase vector. For convenience, the full specification of the state transition equation is

(13.105) | ||||

The relationship between inertia matrices and ellipsoids is actually much deeper than presented here. The kinetic energy due to rotation only is elegantly expressed as

A fascinating interpretation of rotational motion in the absence of external forces was given by Poinsot [39,681]. As the body rotates, its motion is equivalent to that of the inertia ellipsoid, given by (13.106), rolling (without sliding) down a plane with normal vector in .

Steven M LaValle 2020-08-14