How to Formulate a Differential Equation for Deformable Object Manipulation?

[Liferider]

I'm working on manipulation and moving deformable objects by use of a robot with a 2-finger gripper. I would like to form a differential equation for the object deformation and I'm wondering where I could find some papers that has been dealing with this issue. I have thought about just assuming a standard first or second order process for the deformation, but it would be nice to have some references for my choices. Any comments are welcome.

 

[Ben Niehoff]

What kinds of objects?

The bending of a beam (and similar objects) is determined by a 4th-order equation:

http://en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory

But if you're talking about solid objects that extend generically in all directions, I don't have a good answer.

 

[AlephZero]

IN principle, you just write down the equations of equilibrium for the object, plus the stress-strain relations for the material, solve the resulting partial differential equations for the boundary conditions of the object, and then integrate the strain field to get the displacements.

There are lots of software packages that can do that numerically - google for "finite element analysis."

On the other hand, if you really "want to form a differential equation", you will end up with maybe 12 or more coupled partial differential equations, which usually don't have a closed-form solution.

I think what you really need to first is get clear in your mind what you want to calculate and how you plan to use the results, and then start investigating how to calculate it.

 

[Liferider]

I am basically only concerned with the deformation in one dimension, the one along the gripper displacement. I also want to express a reactive force from the object, acting on the gripper fingers. Maybe I can model the reactive force as a spring (linear or not). And I would also like to include some dynamics such that the object does not retain its shape instantaneously (after a finger release).

 

[Liferider]

Here are some math:
Force from object
\begin{equation}
F_{ob} = k_s (x_{ob,0} - x_{ob})
\end{equation}
\begin{equation}
\ddot{x}_{ob} = ?
\end{equation}
where $x_{ob,0}$: Equilibrium width, $x_{ob}$: Actual width. I just need to be pointed in the right direction. Regarding FEM methods, I just think it would not serve my simple purpose.