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]
(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) |
(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
Steven M LaValle 2020-08-14