Sinclair, Andrew James (2002-05). Generalization of rotational mechanics and application to aerospace systems. Doctoral Dissertation.
This dissertation addresses the generalization of rigid-body attitude kinematics,
dynamics, and control to higher dimensions. A new result is developed that demonstrates
the kinematic relationship between the angular velocity in N-dimensions and
the derivative of the principal-rotation parameters. A new minimum-parameter description
of N-dimensional orientation is directly related to the principal-rotation
The mapping of arbitrary dynamical systems into N-dimensional rotations and
the merits of new quasi velocities associated with the rotational motion are studied. A
Lagrangian viewpoint is used to investigate the rotational dynamics of N-dimensional
rigid bodies through Poincar??e??s equations. The N-dimensional, orthogonal angularvelocity
components are considered as quasi velocities, creating the Hamel coefficients.
Introducing a new numerical relative tensor provides a new expression for these coefficients.
This allows the development of a new vector form of the generalized Euler
An N-dimensional rigid body is defined as a system whose configuration can
be completely described by an N??N proper orthogonal matrix. This matrix can be
related to an N??N skew-symmetric orientation matrix. These Cayley orientation
variables and the angular-velocity matrix in N-dimensions provide a new connectionbetween general mechanical-system motion and abstract higher-dimensional rigidbody
rotation. The resulting representation is named the Cayley form.
Several applications of this form are presented, including relating the combined
attitude and orbital motion of a spacecraft to a four-dimensional rotational motion. A
second example involves the attitude motion of a satellite containing three momentum
wheels, which is also related to the rotation of a four-dimensional body.
The control of systems using the Cayley form is also covered. The wealth
of work on three-dimensional attitude control and the ability to apply the Cayley
form motivates the idea of generalizing some of the three-dimensional results to Ndimensions.
Some investigations for extending Lyapunov and optimal control results
to N-dimensional rotations are presented, and the application of these results to
dynamical systems is discussed.
Finally, the nonlinearity of the Cayley form is investigated through computing
the nonlinearity index for an elastic spherical pendulum. It is shown that whereas the
Cayley form is mildly nonlinear, it is much less nonlinear than traditional spherical