# Ball in double hoop: demonstration model for numerical optimal control

## TL;DR

This paper extends the classic ball and hoop system to include multiple modes and demonstrates how it can be used for advanced control techniques like trajectory generation and stabilization, with experimental validation.

## Contribution

It introduces a comprehensive hybrid model with multiple modes and demonstrates its use for complex control tasks, expanding educational tools for control systems.

## Key findings

- Successful trajectory generation and stabilization for complex modes
- Experimental validation of the control approach
- Enhanced model for educational demonstration of advanced control

## Abstract

Ball and hoop system is a well-known model for the education of linear control systems. In this paper, we have a look at this system from another perspective and show that it is also suitable for demonstration of more advanced control techniques. In contrast to the standard use, we describe the dynamics of the system at full length; in addition to the mode where the ball rolls on the (outer) hoop we also consider the mode where the ball drops out of the hoop and enters a free-fall mode. Furthermore, we add another (inner) hoop in the center upon which the ball can land from the free-fall mode. This constitutes another mode of the hybrid description of the system. We present two challenging tasks for this model and show how they can be solved by trajectory generation and stabilization. We also describe how such a model can be built and experimentally verify the validity of our approach solving the proposed tasks.