Keywords: bouncing ball model, video analysis, dynamical systems, optical flow, period coupling bifurcation, chaos.

The bouncing ball model is one of the simplest dynamical systems exhibiting a great variety of behaviours ranging from periodic to chaotic motion [1]. With the increase of computational capabilities, video analysis is a technique of increasing interest the analyis of the movement of the objects in a scene.

This paper, includes the motion of a ball bouncing on a platform, the processing of images to obtain the optical flow and the computation of the spectrum of the velocities to assess the behaviour of the system.

The classical Lucas-Kanade's method permits the tracking of the trajectory of a point along the video sequence [2]. Following a point on the ball, the characteristics of its motion, may be determined, in particular, the position and time of its impacts with the platform. The Farnebäk's method computes the optical flow of a region of the frame [3]. The spectrum of the average flow is used to determine the frequencies present in the movement of the ball.

The parameters of a mathematical model of the system, are also adjusted, to perform a computer simulation and compare the results with the experimental ones.

In the experiment, a table tennis ball that bounces on the membrane of a loudspeaker is recorded. The loudspeaker is driven by a sinusoidal voltage with controlled amplitude. The vibrating frequency has been fixed at 30 Hz, using the amplitude as a control parameter. For different amplitudes, different states of the system are studied. In particular, phenomena such as doubling period bifurcations and chaos have been analysed.

It is interesting to observe that there are different possible states of the bouncing ball system for a given value of the control parameters. At low loudspeaker amplitudes, the ball can either rest in permanent contact with the membrane or bounce at the same frequency as the membrane vibration.

Increasing the amplitude, the system suffers a bifurcation and the ball performs two bounces of different height in two periods, or higher equal bounces spanning two periods each. The ball performing two different bounces in four periods or irregular bounces at even higher loudspeaker amplitudes are also recorded. In each case, the spectrum of the movement from the optical flow results are obtained and the tracked trajectory compared with the corresponding simulated trajectory. A good agreement between the experimental results and the simulations, which confirms the quality of the experimental setup and the suitability of the dynamical model, is obtained.

1

A. Okninski, B. Radziszewski, "An Analytical and Numerical Study of Chaotic Dynamics in a Simple Bouncing Ball Model", Acta Mechanica Sinica, 27(1), 130-134, 2011. doi:10.1007/s10409-011-0406-3

2

B. Lucas, T. Kanade, "An Iterative Image Registration Technique with Applications to Stereo Vision", Proceedings of the Seventh International Joint Conference on Artificial Intelligence, Vancouver, Canada, 674-679, 1981.

3

G. Farnebäck, "Two-Frame Motion Estimation Based on Polynomial Expansion", Lecture Notes in Computer Science, 2749, 363-370, 2003. doi:10.1007/3-540-45103-X_50

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