Chapter 2: Kinematics of Mechanical Systems Flashcards
What is a property of the rotation matrix R and what problems come with it?
The columns of matrix R are mutually orthogonal, and of unit length. As such, matrix R is said to be orthogonal. Also the rows of matrix R have unit norm and are mutually orthogonal.
Matrix R is characterized by 9 elements. The orthonormality condition introduced 6 constraints, thus, only 3 parameters are needed to completely describe the matrix. The latter statement suffers from a shortcoming. There is no way to tell the difference between two orientation matrices that result from rotations that differ by an arbitrary number of revolutions, namely rotations of amplitude 2π, about an arbitrary axis.
Discuss about the parametrization of rotations
A possible manner to parametrisize rotations is to consider all 9 elements of the rotation matrix as parameters, subject to explicit orthonormality conditions formulated as algebraic constraint equations.
The basic operation in rotation parametizations is the computation of the orientation matrix R as a function of the rotation parameters. This operation is always possible. The inverse operation ie extracting the rotation parametrs from an orientation matrix may not always be possible or may be subjected to indetermination.
A common feature is that 3 parameter parametrizations can be extremely efficient, but intrinsically suffer from an indetermination: rotations in excess of some multiples of π are described by the same set of parameter values. As a consequence, they are impractical to track and define arbitrarily large changes of orientation. Such limitation is an intrinsic characteristic of the orientation matrix. In fact, it is impossible to distinguish the orientation matrix resulting from a rotation of an angle θ from one resulting from a rotation of an angle θ +2πn.
At least 4 parameters are needed to uniquely identify a change of orientation. However, 4 parameter parameterizations introduce an algebraic constraint equation that must be dealt with in the solution process, much like other kinds of kinematic constraints.
Incremental formulations based on 3 parameter parametrizations can be used. In such cases, the reference orientation is accumulated in the orientation matrix, and the parametrization only affects the increment of the rotation.
