\( \DeclareMathOperator*{\oiint}{{\normalsize{\subset\!\supset}} \llap{\iint}} \)The 2-Sphere as the Configuration Space of a Mechanism

Consider a mechanism with a 2-sphere as its configuration space (c-space).

2 Sphere CAD Model
A CAD model of a spherical pendulum

2 Sphere
2-Sphere

Consider a massless rigid rod with a bob with mass \(m\) at one end and a ball-and-socket joint attached to ground at the other end. This has a standard 2-sphere as its configuration space (c-space) (technically, it only has a compact subset with boundary as the c-space; in this case, I refer to the whole 2-sphere as the natural completion of the c-space).

The Riemannian metric (r-metric) \(g(\theta, \phi)\) (written as \(M(\theta)\) in the following graphic from https://youtu.be/BjD-pL819LA?si=q8_It-jJFEq0AmEu&t=212; \(g(\theta)\) in the following graphic is the gravity term) may be read off the equation of motion (EoM):

Equation of Motion for motion on a 2-torus

The Riemannian metric in a coordinate patch using \(\theta\) and \(\phi\) for this mechanism is given by

\[
g(\theta, \phi) = \begin{bmatrix}
J_O & 0 \\
0 & J_O \sin^2\theta
\end{bmatrix}
\]

where

\[
J_O = J_1 + J_2 + J_3,
\]
\[
J_1 = I_{c1} + m_1 r_{c1}^2 = \frac{2}{5} m_1 r_1^2, \quad J_2 = I_{c2} + m_2 r_{c2}^2 = \frac{1}{3} m_2 l^2 + \frac{1}{4} m_2 r_2^2, \quad J_3 = I_{c3} + m_3 r_{c3}^2 = \frac{2}{5} m_3 r_3^2 + m_3 l^2,
\]

with \(J_i = I_{ci} + m_i r_{ci}^2\) for each component \(i \in \{1, 2, 3\}\).

For definiteness, for this particular mechanism from the OnShape model (Assembly 1), the values of the parameters are \(m_1 = 0.005236\text{ kg}\), \(m_2 = 0.010603\text{ kg}\), \(m_3 = 0.041888\text{ kg}\), \(l = 0.300\text{ m}\), \(r_1 = 0.010\text{ m}\), \(r_2 = 0.003\text{ m}\), \(r_3 = 0.020\text{ m}\), \(r_{c1} = 0.000\text{ m}\), \(r_{c2} = 0.150\text{ m}\), \(r_{c3} = 0.300\text{ m}\), and \(J_1 = 2.0944 \times 10^{-7}\text{ kg}\cdot\text{m}^2\), \(J_2 = 3.1811 \times 10^{-4}\text{ kg}\cdot\text{m}^2\), \(J_3 = 3.7766 \times 10^{-3}\text{ kg}\cdot\text{m}^2\) in SI (with integrated OnShape CAD union moment of inertia \(J_O = 4.0358 \times 10^{-3}\text{ kg}\cdot\text{m}^2\)), using PLA for the material.

With these values, the Riemannian metric works out to

\[
g(\theta, \phi) = 10^{-3} \begin{bmatrix}
4.0358 & 0 \\
0 & 4.0358 \sin^2\theta
\end{bmatrix}\text{ kg}\cdot\text{m}^2
\]

Note that the entries in the Riemannian metric have dimensions of \(\left[\text{Length}^2 \cdot \text{Mass}\right]\). (It should be noted that there are dimensions/units used for the describing the robot arm and hence, perhaps bizarrely, for describing the Riemannian metric. Points in the manifold and tangent vectors to the manifold are part of the intrinsic topology of the underlying smooth manifold of the c-space and have only dimensions of [Angular Measure], while inner products/norms of tangent vectors and lengths of curves/distances between points in the manifold depend on Riemannian metric, which is extrinsic to the topology of the underlying manifold; inner products of tangent vectors have dimensions of \(\displaystyle\left[\text{Angular Measure}^2\cdot\text{Length}^2 \cdot \text{Mass}\right]\) while norms of tangent vectors and lengths of curves/distances between points have units of \(\displaystyle\left[\text{Angular Measure}\cdot\text{Length} \cdot \text{Mass}^{\frac{1}{2}}\right]\). This kind of r-metric is known as the "mass matrix'" and morally has dimensions of mass. The way that things work out here, there are \(I_{zz}\) terms, and we can't just move the lengths to the \(\theta\) factors. But, morally, the mass matrix has dimensions of mass.)

Bibliography:
[1] TBD