Bouncing a Ball with Mixed Integer Programming

Edit: A new version.

Here I made a bouncing ball using mixed integer programming in cvxpy. Currently we are just simulating the bouncing ball internal to a mixed integer program. We could turn this into a control program by making the constraint that you have to shoot a ball through a hoop and have it figure out the appropriate initial shooting velocity.

Pretty cool.

The trick I used this time is to make boolean indicator variables for whether a collision will happen or not. The big M trick is then used to actually make the variable reflect whether the predicted position will be outside the wall at x=0. If it isn’t, it uses regular gravity dynamics. If it will, it uses velocity reversing bounce dynamics


Just gonna dump this draft out there since I’ve moved on (I’ll edit this if I come back to it). You can embed collisions in mixed integer programming.  I did it below using a strong acceleration force that turns on when you enter the floor. What this corresponds to is a piecewise linear potential barrier.

Such a formulation might be interesting for the trajectory optimization of shooting a hoop, playing Pachinko, Beer Pong, or Pinball.

More things to consider:

Is this method trash? Yes. You can actually embed the mirror law of collisions directly without needing to using a funky barrier potential.

You can extend this to ball trapped in polygon, or a ball that is restricted from entering obstacle polygons. Check out the IRIS project – break up region into convex regions

https://github.com/rdeits/ConditionalJuMP.jl Gives good support for embedding conditional variables.

https://github.com/joehuchette/PiecewiseLinearOpt.jl On a related note, gives a good way of defining piecewise linear functions using Mixed Integer programming.

Pajarito is another interesting Julia project. A mixed integer convex programming solver.

Russ Tedrake papers – http://groups.csail.mit.edu/locomotion/pubs.shtml

Break up obstacle objects into delauney triangulated things.

www.mit.edu/~jvielma/presentations/MINLPREPSOLJUL_NORTHE18.pdf