|# of variables||# of modes||# of jumps|
|Type||Continuous dynamics||Guards & Invariants||Resets|
The classical bouncing ball example consists of a ball dropped from a predefined height. It hits the ground after a certain time, loses energy and then bounces back into the air and starts to fall again. This physical phenomena can be represented as the following hybrid automaton:
In the discrete state, the motion of the ball, assumed to have a mass of , is governed by the following differential equation: where is the ball’s height from the ground, is the ball’s vertical velocity, and is the earth’s gravitational force. The invariant enforces that the ball always bounces when it reaches the ground. The guard of the single discrete transition, which models the bouncing, ensures that bouncing happens after falling when reaching the ground. The corresponding reset condition accounts for the loss of energy due to the ball’s deformation, where is a constant.
We consider the inital set
We use a time horizon of and set the constant . The set of bad states is the set of all states where .
These are the results from the bouncing ball benchmark created with SpaceEx: