Perspectives on Euler angle singularities, gimbal lock, and the orthogonality of applied forces and applied moments

Abstract

Coordinate singularities and gimbal lock are two phenomena that present themselves in models for the dynamics of mechanical systems. The former phenomenon pertains to the coordinates used to parameterize the configuration manifold of the system, while the latter phenomenon has a distinctive physical manifestation. In the present paper, we use tools from differential geometry to show how gimbal lock is intimately associated with an orthogonality condition on the applied forces and moments which act on the system. This condition is equivalent to a generalized applied force being normal to the configuration manifold of the system. Numerous examples, including the classic bead on a rotating hoop example and a gimbaled rigid body, are used to illuminate the orthogonality condition. These examples help to offer a new explanation for the elimination of gimbal lock by the addition of gimbals and demonstrate how integrable constraints alter the configuration manifold and may consequently eliminate coordinate singularities.

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 $\mathbb{R}P^{3}$.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).

1.1 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. 2.⁵ 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.

1.2 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.