# System Stability constraints formulation

I am working with a system having a massless 2D plane and on that plane there is a rigid object with some mass placed on it. I want to support the plane with wooden sticks such that the system is stable.

You can see that the dotted line is the support polygon (convex hull) made by the support ticks.

Stability for the system can be defined as the Center of mass (CoM) of the object inside the polygon assuming the object is rigid and there is no inclination on the surface.

I want to formulate this problem in a constraint program only using integer values. I am already working on a model and this part is one of the other parts of my model. Hence all my decision variables are integers and as far as I have read, we can formulate polyhedral analysis problems in linear programming.

Problem to tackle: Find the position of 2D n-points such that the system remains stable.

All points are decision variables of the program and are not passed externally

Does anyone here happen to know the formulation in constraint program using only integer variables? Any reading materials you would suggest?

• Welcome to OR.SE. Is there any definition/clarification to make sense about the boundaries of the polygon? If the object is rigid, how you can define the center of mass (CoM) of the object inside the polygon? Commented Sep 19, 2023 at 12:40
• The boundaries are nothing but a convex hull obtained by the positions of the support points. By the body being rigid, I meant that the CoM will stay in the same place for a dynamic system. But here we cal also consider the Com as a 2D point for this matter. Commented Sep 19, 2023 at 12:44
• I am not sure to understand the problem as well, but this and this link may be helpful. Commented Sep 19, 2023 at 12:53
• Would you please, Do you have any math model, or just is a basic idea? Commented Sep 19, 2023 at 13:49
• yeah, so the model is something line, given the position of the CoM of the object and mass of the object, find the placement of n-points (sticks) such that the system is balanced. So net force on the system is zero and the net moment on the system is zero. Commented Sep 19, 2023 at 16:08