Consider a crank with internal slider mechanism and its configuration
space (c-space) \(Q \approx S^1 \times \mathbb{I}\).
A drawing of a crank with an internal slider
We shall identify \(S^1 \times \mathbb{I}\) with \([0,2\pi] \times [0,
L]\), with dimensions of \(\text{angular measure} \cdot \text{length}\)
and, say, units of
\(\text{rad} \cdot \text{m}\), via the coordinate patch \(\phi:
[0,2\pi] \times [0, L] \to S^1 \times \mathbb{I}\) given by
\(\phi(\theta_1, x) = (\cos(\theta_1)x, \sin(\theta_1)x)\) and
therefore endow it with the
units of \(\text{rad} \cdot \text{m}\), although it is technically
unitless.
(The below is from a template for these kinds of webpages; ignore it
for now.)
The Riemannian
metric (r-metric) \(g(\theta_1, \theta_2)\) (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):
The
Riemannian metric in the standard coordinate patch for this robot arm
is given by
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
\(\displaystyle\left[\text{Angular Measure}\cdot\text{Length}^3 \cdot
\text{Mass}\right]\) while norms of tangent vectors and lengths of
curves/distances between points have units of
\(\displaystyle\left[\text{Angular
Measure}^{\frac{1}{2}}\cdot\text{Length}^{\frac{3}{2}} \cdot
\text{Mass}^{\frac{1}{2}}\right]\).)
