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Description:
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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
parameters .
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
rotational equations .
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
coordinates . |