Bouncing ball | Theory of Hybrid Systems

Classification

# of variables	# of modes	# of jumps
2	1	1

Type	Continuous dynamics	Guards & Invariants	Resets
hybrid	linear polynomial	half-space	linear

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Flow*	flow*_files
SpaceEx	spaceEx_files

Model description

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 $m=1 kg$ , is governed by the following differential equation: $\begin{array}{ll} \dot{x}= & v\\ \dot{v} = & -9.81 \end{array}$ where $x$ is the ball’s height from the ground, $v$ is the ball’s vertical velocity, and $9.81ms^{-2}$ is the earth’s gravitational force. The invariant $x \geq 0$ enforces that the ball always bounces when it reaches the ground. The guard $x=0\wedge v\leq0$ of the single discrete transition, which models the bouncing, ensures that bouncing happens after falling when reaching the ground. The corresponding reset condition $v:=-c \cdot v$ accounts for the loss of energy due to the ball’s deformation, where $c\in\left[0,1\right]$ is a constant.