PTOLEMY II - CiteSeerX

Using Vergil
If you look inside the free state, you will see the refinement shown in figure 2.58. This model represents
the laws of gravity, which state that an object of any mass will have an acceleration of roughly
meters/second 2 – 10
(roughly). The acceleration is integrated to get the velocity. which is, in turn, integrated
to get the vertical position.
In figure 2.58, a ZeroCrossingDetector actor is used to detect when the vertical position of the ball
is zero. This results in production of an event on the (discrete) output bump. Examining figure 2.55,
you can see that this event triggers a state transition back to the same free state, but where the initialVelocity
parameter is changed to reverse the sign and attenuate it by the elasticity. This results in the ball
bouncing, and losing energy, as shown by the plot in figure 2.54.
As you can see from figure 2.55, when the position and velocity of the ball drop below a specified
threshold, the state machine transitions to the state stop, which has no refinement. This results in the
model producing no further output.
2.10.2 Numerical Precision and Zeno Conditions
The bouncing ball model of figures 2.54 and 2.55 illustrates an interesting property of hybrid system
modeling. The stop state, it turns out, is essential. Without it, the time between bounces keeps
decreasing, as does the magnitude of each bounce. At some point, these numbers get smaller than the
representable precision, and large errors start to occur. If you remove the stop state from the FSM, and
re-run the model, you get the result shown in figure 2.59. The ball, in effect, falls through the surface
on which it is bouncing and then goes into a free-fall in the space below.
FIGURE 2.58. The refinement of the free state, shown here, is a continuous-model representing the laws of
gravity.
FIGURE 2.59. Result of running the bouncing ball model without the stop state.
Heterogeneous Concurrent Modeling and Design 85