# How to Formulate a Differential Equation for Deformable Object Manipulation?

• Liferider
In summary: I need something that is easier to use and understand. That being said, I am open to suggestions.In summary, the author is working on a robot that can move deformable objects. He is looking for references on the subject and would like some suggestions on how to calculate the deformation.
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.

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.

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).

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Here are some math:
Force from object

F_{ob} = k_s (x_{ob,0} - x_{ob})

\ddot{x}_{ob} = ?

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.

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## 1. What is a differential equation for a deformable object?

A differential equation for a deformable object is a mathematical equation that describes the relationship between the deformation of an object and the forces acting upon it. It is used to model the behavior of materials under external forces, such as stress and strain.

## 2. How is a differential equation for a deformable object solved?

A differential equation for a deformable object is solved by using various mathematical techniques, such as separation of variables, integration, and substitution. These techniques allow us to find a general solution that describes the behavior of the object.

## 3. What are the applications of differential equations for deformable objects?

Differential equations for deformable objects have many applications in engineering, physics, and other scientific fields. They are used to model the behavior of materials in structures, design new materials, and understand the mechanics of various objects, such as bones, muscles, and other biological tissues.

## 4. What are some common types of differential equations for deformable objects?

Some common types of differential equations for deformable objects include the Navier-Stokes equations, the elasticity equations, and the wave equation. These equations have different forms and are used to model different types of deformations, such as flow, stress, and vibrations.

## 5. How do differential equations for deformable objects relate to real-world scenarios?

Differential equations for deformable objects are used to model and describe real-world scenarios, such as the behavior of bridges, buildings, and other structures under external forces. They also help us understand the behavior of biological tissues and how they respond to different types of stress and strain.

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