Appendix: Background on rotations
An example of a 3-dimensional configuration manifold results from the constrained motion of a rigid body rotating about a fixed point, such as in the motion of the Lagrange top. In this case, the rotation tensor \(\mathbf{Q}\) of the top belongs to the proper-orthogonal subgroup of linear second-order tensors, a 3-dimensional manifold called \(\mathsf{Orth}^{+}\) embedded in \(\mathcal{C}^{9}\). This subgroup is isomorphic to \(\mathit{SO}(3)\), the special orthogonal group of \(3 \times 3\) matrices. Furthermore, \(\mathit{SO}(3)\) is diffeomorphic to \(\mathbb{R}P^{3}\), real projective 3-space, and we can associate the configuration of the top to a point in \(\mathbb{R}P^{3}\).Footnote 4
Given an angle \(\theta \) and axis of rotation \(\mathbf{p}\), Euler’s formula for the rotation tensor is
$$ {\mathbf{Q}} = \mathbf{L} ( \theta , \mathbf{p} ) = \cos ( \theta ) ( \mathbf{I} - \mathbf{p} \otimes {\mathbf{p}} ) + \sin ( \theta ) \text{skwt} ( \mathbf{p} ) + \mathbf{p} \otimes {\mathbf{p}}, $$
(A.1)
where \(\text{skwt} ( \mathbf{p} ) \) indicates the skew tensor of \(\mathbf{p}\), which is given by the mapping
$$ \text{skwt} ( \mathbf{p} ) = - \boldsymbol {\epsilon }{\mathbf{p}}, $$
(A.2)
where \(\boldsymbol {\epsilon }\) is the third-order alternator tensor. Since \(\mathbf{Q}\) is proper orthogonal, it satisfies \(\mathbf{Q} {\mathbf{Q}} ^{T} = \mathbf{I}\) and \(\det {\mathbf{Q}} = + 1\). A skew-symmetric angular velocity tensor may therefore be defined using \(\mathbf{Q}\) as \(\boldsymbol {\Omega }= \dot{\mathbf{Q}} {\mathbf{Q}}^{T}\). We denote the space of skew-symmetric tensors as \(\mathsf{Skw}\). The angular velocity vector \(\boldsymbol {\upomega }\) is the axial vector of \(\boldsymbol {\Omega }\) which is given by the mapping
$$ \boldsymbol {\upomega }= \text{ax} ( \boldsymbol {\Omega }) = -\frac{1}{2} \boldsymbol {\epsilon }[ \boldsymbol {\Omega }] . $$
(A.3)
Thus, there exists an isomorphism between \(\mathsf{Skw}\) and \(\mathbb{E}^{3}\) through the maps \(\text{ax} ( \cdot ) \) and \(\text{skwt} ( \cdot ) \) for which the following is true:
$$ \boldsymbol {\Omega }{\mathbf{a}} = \boldsymbol {\upomega }\times \mathbf{a} $$
(A.4)
for any \(\mathbf{a} \in \mathbb{E}^{3}\). Euler angles are a common choice of parametrization for \(\mathbf{Q}\). A general Euler angle decomposition is given by
$$ {\mathbf{Q}} = \mathbf{L} \bigl( \nu ^{3}, \mathbf{g}_{3} \bigr) \mathbf{L} \bigl( \nu ^{2}, \mathbf{g}_{2} \bigr) \mathbf{L} \bigl( \nu ^{1}, \mathbf{g}_{1} \bigr) , $$
(A.5)
where \(\{ \nu ^{1}, \nu ^{2}, \nu ^{3} \} \) are the Euler angles and \(\{ \mathbf{g}_{1}, \mathbf{g}_{2}, \mathbf{g}_{3} \} \) are the corresponding axes of rotation, which are collectively known as the Euler basis [22–24]. It is also convenient to define a reciprocal basis \(\{ \mathbf{g}^{i} \} \) known as the dual Euler basis with the property
$$ {\mathbf{g}}_{i} \cdot {\mathbf{g}}^{j} = \delta _{i}^{j} \quad ( i,j =1, 2, 3 ) , $$
(A.6)
where \(\delta _{i}^{j}\) is the Kronecker delta. A corotational (or body-fixed) basis is defined as the set \(\{ \mathbf{e}_{i} \} \) for which \(\mathbf{e}_{i} = \mathbf{Q} {\mathbf{E}}_{i}\) for \(i =1,2,3\). The rotation tensor has the same components in the corotational basis as it does in the inertial basis: \(\mathbf{Q} = Q _{ij} {\mathbf{e}}_{i} \otimes {\mathbf{e}}_{j} = Q_{ij} {\mathbf{E}} _{i} \otimes {\mathbf{E}}_{j}\). Using the abbreviations \(\cos ( \nu ) = \text{c} \nu \) and \(\sin ( \nu ) = \text{s} \nu \), a 3–2–1 Euler angle set yields the following parametrization:
$$ [ Q_{ij} ] = \left [ \textstyle\begin{array}{c @{\quad}c@{\quad} c} \text{c}\nu ^{1} \text{c}\nu ^{2} & \text{c}\nu ^{1} \text{s} \nu ^{2} \text{s} \nu ^{3} - \text{s} \nu ^{1} \text{c}\nu ^{3} & \text{c}\nu ^{1} \text{s} \nu ^{2} \text{c}\nu ^{3} + \text{s} \nu ^{1} \text{s} \nu ^{3} \\ \text{s} \nu ^{1} \text{c} \nu ^{2} & \text{s} \nu ^{1} \text{s} \nu ^{2} \text{s} \nu ^{3} + \text{c} \nu ^{1} \text{c} \nu ^{3} & \text{s} \nu ^{1} \text{s} \nu ^{2} \text{c} \nu ^{3} - \text{c} \nu ^{1} \text{s} \nu ^{3} \\ -\text{s} \nu ^{2} & \text{c} \nu ^{2} \text{s} \nu ^{3} & \text{c} \nu ^{2} \text{c} \nu ^{3} \end{array}\displaystyle \right ] , $$
(A.7)
where \(\nu ^{1} \in [0, 2\pi )\), \(\nu ^{2} \in [ -\frac{\pi }{2}, \frac{\pi }{2} ] \), and \(\nu ^{3} \in [0, 2\pi )\). A 3–1–3 Euler angle set yields the following parametrization:
$$ [ Q_{ij} ] = \begin{bmatrix} \text{c} \nu ^{1} \text{c} \nu ^{3} - \text{s} \nu ^{1} \text{c} \nu ^{2} \text{s} \nu ^{3} & - \text{c} \nu ^{1} \text{s} \nu ^{3} - \text{s} \nu ^{1} \text{c} \nu ^{2} \text{c} \nu ^{3} & \text{s} \nu ^{1} \text{s} \nu ^{2} \\ \text{s} \nu ^{1} \text{c} \nu ^{3} + \text{c} \nu ^{1} \text{c} \nu ^{2} \text{s} \nu ^{3} & - \text{s} \nu ^{1} \text{s} \nu ^{3} + \text{c} \nu ^{1} \text{c} \nu ^{2} \text{c} \nu ^{3} & - \text{c} \nu ^{1} \text{s} \nu ^{2} \\ \text{s} \nu ^{2} \text{s} \nu ^{3} & \text{s} \nu ^{2} \text{c} \nu ^{3} & \text{c} \nu ^{2} \end{bmatrix} , $$
(A.8)
where \(\nu ^{1} \in [0,2\pi )\), \(\nu ^{2} \in [ 0, \pi ] \), and \(\nu ^{3} \in [0, 2\pi )\).
For a given Euler angle parametrization, the following identity may be shown to hold:
$$ \frac{\partial {\mathbf{Q}}}{\partial \nu ^{i}} {\mathbf{Q}}^{T} = \text{skwt} ( \mathbf{g}_{i} ) \quad ( i =1,2,3 ) , $$
(A.9)
where \(\mathbf{Q}\) is given by (A.5).
A.1 Constraining one of the Euler angles
An interesting situation occurs when the first or third Euler angle is constrained and the configuration manifold for the rotational motion of the rigid body becomes a two-torus \(T^{2}\) provided \(\nu ^{2}\) is extended to \(\nu _{e}^{2} \in [0, 2\pi )\). The latter case arises in the motion of the rigid bodies shown in Fig. 2.Footnote 5 If a set of 3–1–3 Euler angles are used to parameterize the rotation tensor \(\mathbf{Q}\) of these bodies, then the third angle \(\nu ^{3} = 0\). To use two angles to parameterize the rotation \(\mathbf{Q}\) of the constrained rigid bodies shown in Fig. 2, it is necessary that
$$ {\mathbf{Q}} = \mathbf{L} \bigl( \nu ^{2}_{e}, \mathbf{e}_{1} \bigr) \mathbf{L} \bigl( \nu ^{1}, \mathbf{E}_{3} \bigr) , $$
(A.10)
In contrast to the unconstrained case (A.5), it is necessary to extend the range of the second angle: \(\nu ^{2} \to \nu ^{2}_{e} \in [0, 2\pi )\). The necessity of this extension can also be demonstrated by taking a pen and placing it on a horizontal surface. Fixing the orientation of the longitudinal axis of the pen is equivalent to fixing \(\nu ^{1}\). Then, by rotating the pen about its longitudinal axis, the necessity of having the extended angle \(\nu ^{2}_{e}\) can be readily seen.
A.2 A fixed axis of rotation
Suppose \(\mathbf{Q}\) is constrained so that the rotation axis is fixed, \(\mathbf{p} = \mathbf{p}_{0}\). Then, the configuration manifold is a 1-dimensional circle \(S^{1}\) contained in \(\mathsf{Orth}^{+}\): \(\mathbf{p}_{0}\otimes {\mathbf{p}}_{0}\) is normal to the plane of the circle and \(\cos ( \theta ) ( \mathbf{I} - \mathbf{p}_{0}\otimes {\mathbf{p}}_{0} ) + \sin ( \theta ) \text{skwt} ( \mathbf{p}_{0} ) \) is a radius vector to a point on the circle. A tensor spanning the tangent space to \(S^{1}\) is
$$ {\mathbf{a}}_{1} = \frac{\partial {\mathbf{Q}}}{\partial \theta } = - \sin ( \theta ) ( \mathbf{I} - \mathbf{p}_{0} \otimes {\mathbf{p}}_{0} ) + \cos ( \theta ) \, \text{skwt} ( \mathbf{p}_{0} ) , $$
(A.11)
which is clearly tangent to the circle.