Comparison of geometrically precise beams and Cosserat rod

[xlsufo]

Hello. On my continuum robot (Fig. 1), I compared the geometrically exact beam model, the Cosserat rod model, and the ground truth data from a motion capture system using identical parameters. The driving tendons of my continuum robot are obliquely routed using Kevlar lines, and the lumped masses of the disks are also taken into account. As shown in Fig. 2, the Cosserat rod model outperforms the geometrically exact beam model in terms of accuracy, while offering a significantly faster computation speed. I assume the underlying reasons might be:
The elastic modulus is 30e9 Pa. Due to the lumped masses of the disks, the Cosserat rod model suffered from numerical explosion. To address this, I utilized the Levenberg-Marquardt (LM) solver and introduced a small amount of structural damping. The driving tendons in the geometrically exact beam model form straight lines between two control points, which is closer to the actual physical system compared to the centerline-based interpolation used in the Cosserat rod model. Based on my current understanding, the geometrically exact beam is essentially a finite element discretization of the Cosserat rod. I am considering writing a paper on this. Do you have any suggestions?
In the image, mocap represents the motion capture system.

[JohnDTill]

As shown in Fig. 2, the Cosserat rod model outperforms the geometrically exact beam model in terms of accuracy, while offering a significantly faster computation speed

I don't understand. It looks like the solid GEBF lines align more closely with the discrete markers for the experimental measurements from motion capture, while the Cosserat model dashed lines are further away? The dashed lines are closer to the square markers- is the square marker also an experimental measurement?

Due to the lumped masses of the disks, the Cosserat rod model

I'm not sure how to interepret this, since there are multiple ways the disks could be modeled. You could:

• Have continuous gravitational load $f_g(s) = \rho A g$ vary depending on if there is a disk at the arclength $s$.
• Split up the Cosserat ODE/PDE into multiple regions, with the disks causing discontinuous effects at boundaries
• Uniformly distribute the weight of the disks over the whole backbone

Those clarifications on the experiment results and model formulation would be helpful.

The tendon-robot model which Rucker and Webster presented in "Statics and Dynamics of Continuum Robots With General Tendon Routing and External Loading" where the tendons have a continuous effect is exactly correct¹ if the tendons are in continuous channels, which is the case for many medical applications like colonoscopes and catheters. As the actual constraints become discrete, there will be some modelling error from the continuous model equations. The error will be less severe if the spacer disks are densely placed, and more severe when they are sparse with the extreme case that forces are exerted on 1 distal spacer disk. Certainly the error from discrete disks can be negligible/minor; they validated the model against a prototype with discrete disks.

Unless there is a superior way to model tendon robots with sparsely placed spacer disks, or some other situation-dependent improvement, I think the paper may struggle with novelty ☹️. Is there a novel modelling method which is situationally more accurate, mathematically simpler, or more computationally efficient?

¹⁾ Exactly correct aside from detailed effects like channel clearance

[xlsufo]

I’m very sorry for not expressing myself clearly. I have generated a new figure as shown below. The discrete points here are the measurement points obtained from the motion capture system, while the continuous curves are generated by the model.

The current description I use treats the disks as lumped point masses and point rotational inertias located at the corresponding arc-length positions. Therefore, in the Cosserat formulation, I am using the first modeling approach you mentioned. Perhaps I could try dividing the Cosserat ODE/PDE into multiple regions and see whether the computational accuracy improves.

In the geometrically exact beam formulation, the effects of the discrete disks can be directly assembled into the mass matrix M and the gravitational force term Fg in the dynamic equation

The constraint forces from the tendons are introduced using Lagrange multipliers. I think this is more convenient to handle than in the Cosserat rod formulation, and it is insensitive to the number of disks and the tendon routing path.

[JohnDTill]

Thanks, and no need to be sorry, I understand it more clearly now. That does sound like an interesting paper. I do think the Cosserat model error increases with the space between disks, so effects of increasing and decreasing the number of spacer disks could be an interesting avenue to explore.

You could also implement a Cosserat tendon robot model which accounts for tendon forces only at the spacer disks using boundary conditions instead of constraining the tendons to a continuous channel, which could be interesting to compare as well.

So there are different aspects which could be in scope, and I think it would be an interesting comparison to write about.

[xlsufo]

Thank you very much. I will keep working on it and complete it as soon as possible.