\( \DeclareMathOperator*{\oiint}{{\normalsize{\subset\!\supset}} \llap{\iint}} \)The Genus-2 Torus as the Configuration Space of a 5-Bar Mechanism

Consider the 5-bar mechanism.

5 bar mechanism
5-Bar Mechanism (The first bar is an imaginary bar connecting the green ground joints; the other 4 bars are visible)

The 5-bar mechanism finds uses in mechanical engineering as outlined in this Gemini link https://gemini.google.com/share/0183b260312f.

The configuration space (c-space) of this mechanism is the "connect sum" (as outlined on this website [Java required to view webpages properly]) of the c-spaces of 2 double-pendulum robot arms, which is a genus-2 torus.

Robot Arm 01   Robot Arm 02
Two double-pendulum robot arms

2-torus 12-torus 2

The robot arms' c-spaces
Joined Robot Arms 01
Joining the 2 robot arms with a full pin joint to create a 5-bar mechanism

Connect sum 01


Torus c-space
The 2D c-space of a 5-bar mechanism that is the "connect sum" of 2 double-pendulum robot arms' c-spaces

Per this YouTube video https://www.youtube.com/watch?v=vxOvQ7CAI9U, the 5-bar mechanism may be analyzed via the following elementary formulae:

5 Bar Mechanism Theta 3

5 Bar Mechanism Theta 2 Solution


For definiteness, for this particular mechanism, the values of the parameters are  \(m_i = TBD\) kg, \(l_i = TBD\) m, and \(I_{zz,i} = TBD\) \(\text{kg}\cdot\text{m}^2\)  in SI, using PLC for the material.

The Riemannian metric (r-metric), \(g(\theta_1, \theta_2)\) in one coordinate patch (c-patch), \(g(\theta_4, \theta_3)\) in the other c-patch, (written as \(M(\vec{\theta})\) in the following graphic from https://youtu.be/BjD-pL819LA?si=q8_It-jJFEq0AmEu&t=212; \(g(\vec{\theta})\) 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 the standard coordinate patch for this robot arm is given by \[\begin{bmatrix}\text{TBD}\end{bmatrix}\]

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 \(\displaystyle\left[\text{Angular Measure}\cdot\text{Length} \cdot \text{Mass}^{\frac{1}{2}}\right]\).)

By varying the parameters of each robot arm and the distance between ground on each robot arm, we should be able to synthesize an r-metric with constant curvature -1. (Note that by The Theorem Egregium, the integral of the curvature form over the whole c-space must be \(\displaystyle \oiint\limits_{M}\ \kappa\ dA =  -1\).) This c-space would then necessarily have the hyperbolic plane \(\mathbb{H}^2\) as its universal cover.

Homework/Extra Credit: (a) Create a 6D genus-2 torus c-space by joining together 2 6D robot arms with a revolute joint

6D Robot Arm 01                6D Robot Arm 02

Note that each robot arm has \(T^6\) for its c-space, so the join of the two robot arms has \(T^6 \# T^6\) for its c-space. By Kazdan and Warner's Trichotomy Theorem [1], since  \(n = 6  \ge 3\), this admits an r-metric of constant curvature \(\kappa = -1\) [Author's note: The Tricotomy Theorem appears to contradict The Cartan-Hadamard Theorem; we'll have to look into this]. However, by [2], as \(T^6 \# T^6\) is a connect sum of aspherical manifolds, the universal cover of \(T^6 \# T^6\) is homotopy equivalent (he) to (but not necessarily proper homotopy equivalent [phe] to) a wedge of \(S^{5}\)'s, hence it it is not he to a point, and hence it cannot be phe to \(\mathbb{R}^6\) or isometric to \(\mathbb{H}^6\).

(b) (Hard [?]) Find parameters of the Denavit–Hartenberg/De Caro-Rolland table for this mechanism for which the inertial/mass r-metric has constant curvature \(\kappa = -1\).

Bibliography:
[1] Xu, J. Trichotomy theorem for prescribed scalar and mean curvatures on compact manifolds with boundaries, (2023), https://api.semanticscholar.org/CorpusID:255393735
[2] McCullough, D. Connected sums of aspherical manifolds, Indiana Univ. Math. J., volume 30, number 1, (1981), pages 17-28