Cosserat Rod / Elastica model for cable torsion

[NicoDalmasso]

As discussed through discord, here's a placeholder for the discussion around torsion models for cable manipulation. We will post some documents that we have describing the model and implementation we target.

[NicoDalmasso]

Here is the document and c-code sample for the model.

Elastica.doc

elastica_sp.doc

[MuGdxy]

Here are some basic questions:

1. What will the DOF be like? Say, Finite Element, using the vertex position as DOF (3 dof per vertex), Say, Affine Body, using the Body Coordinates Frame as DOF(12 dof per body).
2. Does the motion provide linear trajectories? UIPC can only handle points, edges, and triangles moving linearly within a sub-Newton iteration. Say, RigidBody DOF provides curved trajectories which is not allowed, while FEM Tetrahedra always move linearly per vertices, which is allowed. In the case of Rod, the representation cannot be like a B-spline, but can be like a connected straight-edge chain.

[NicoDalmasso]

@MuGdxy here's some answer :

1. One extra DOF for rotations is needed
2. It is only needed for stiffness for rotations. Visually it would not affect the motion of the rod

[MuGdxy]

Textbook we may refer to:
https://content.e-bookshelf.de/media/reading/L-11110556-efb86edc47.pdf
Cosserat Rod in Computer Graphics:
https://cg.informatik.uni-freiburg.de/publications/2007_SCA_ropes.pdf

[MuGdxy]

I will provide an estimation of the feasibility and complexity of the Cosserat Rod within the current UIPC system, along with primary design and implementation instructions, within several days.

Here are some differences between the CosseratRod and HookeanSpring (current FiniteElementMethod):

1. Cosserat Rod has its own kinetic energy of DOF. The kinetic energy is defined not only on the centerline (translational energy) but also on the rod cross-section (rotational energy).
   (See in Section 3.3.2 in Paper)
2. UIPC relies on Nonlinear-Newton Optimization with Line Search and CCD Filtering. We need to determine how the rod DOF will change (linearly interpolated) during the optimization step in LineSearch and CCD Filtering, for Centerline DOF, position $$x$$, it's just $$x = x^t + \alpha \delta x$$, for rotation quaternion, $$q$$, maybe Spherical Linear Interpolation can be used or...
3. The contact coupling only happens on the first 3 DOF (centerline, translational DOF)

So we need some facilities to handle the extra DOF(besides the position DOF).

[MuGdxy]

I'm not sure if the discrete energy pattern here is correct, can you confirm it?

The vertex based kinetic energy will be:

$$ K_i^t = K^t(x_i) $$

where $x_i$ is the $i-th$ node.

the quaternion based kinetic energy will be:

$$ K_m^R = K^R(q_m) $$

where $q_m$ is the rotation based on $m$-th edge.

the stretch energy will be:

$$ E^s_{ij} = E^s(x_i,x_j) $$

the bending energy will be:

$$ E^b_{mn} = E^b(q_m,q_n) $$

the x,q coupling energy:

$$ E^p_{ijm} = E^p(x_i,x_j,q_m) $$

where $m$ is the edge connecting $x_i$ and $x_j$.