General Method of Applying Manifold Topology to System Dynamics/Control Engineering/Robotics

Here is my heuristic algorithm for applying manifold topology to system dynamics, control engineering, and (I think) robotics in general.

For each "moving part" in the dynamical system, we have a copy of $SE(3) = \mathbb{R}^3 \rtimes SO(3) (= Q_{\text{free}}$ in Bullo and Lewis's notation$)$. For $N$ moving parts, we have $N$ copies of $SE(3)$, $SE(3)^N$. The process for creating this is somewhat automated by Denavit-Hartenberg tables:

 

(The graphic is from this video series https://www.youtube.com/watch?v=4Y1_y9DI_Hw&list=PLZaGkBteQK3HQFSWDM7-yRQWTd86DeDIY&index=1)

We then have interconnections between the moving parts, which leads to an $n$ dimensional submanifold of $SE(3)^N$, $Q^n$, called the configuration space of the dynamical system.

This configuration space has a Riemannian metric $g$ it inherits as a submanifold of $SE(3)^N$. The Riemannian metric, $g$, then has a canonical lift $\tilde{g}$  to the tangent bundle, $TQ$, of $Q$, as outlined in do Carmo, for instance. With this lift of the Riemannian metric, we have a musical isomorphism between $TQ$ and $T^*Q$, the co-tangent bundle. Now, the co-tangent bundle is naturally a symplectic manifold, with canonical symplectic from $\omega$. Corresponding to $\tilde{g}$ and $\omega$, we have a canonical almost complex structure $J$ on $TQ$ (and/or $T^*Q$?), completing the canonical compatible triple, $(\tilde{g}, \omega, J)$, on (say) $TQ$.

Next, in a coordinate patch on $TQ$, we have a Lagrangian function $L$. We may then have a non-conservative force field acting additionally on the mechanical system, which we model as a differential 1-form $\mathcal{F}$. This lead to the system of first-order differential equations $$\begin{cases} \frac{d}{dt}\left\{\frac{\partial L}{\partial \dot{q}_1}\right\} - \frac{\partial L}{\partial q_1} = \mathcal{F}_1 \\ \ldots \\ \frac{d}{dt}\left\{\frac{\partial L}{\partial \dot{q}_n}\right\} - \frac{\partial L}{\partial q_n} = \mathcal{F}_n \end{cases}$$ Now, this system may have a critical point $(q, \dot{q})$. At this critical point, we can take the Jacobian matrix $A$ of the Lagrangian vector field/differential 1-form. This then becomes the plant function for a control engineering problem. The matrix $A$ should have $2n$ eigenvalues; because of a hypertechnical consideration, we will suppose none of them are purely imaginary, so the Hartman-Groβman Theorem applies. We then place a control feedback loop with proportional control gain matrix $K$ given as in this diagram.

 

We then use the gain matrix $K$ to place $2n-2$ of the eigenvalues of the closed-loop system to have real part between, say, -5 and -10 in the complex plane, then place the remaining two eigenvalues -- called the dominant eigenvalues or the dominant poles of the closed-loop system -- to have real part between, say, -2 and 0 and to satisfy the design requirements of the control engineering problem specifications.