Research Papers

Tracking of a Bouncing Ball in a Planar Billiard Through Continuous-Time Approximations

[+] Author and Article Information
Laura Menini

Dipartimento di Ingegneria Civile e Ingegneria
Università di Roma Tor Vergata,
Roma 00133, Italy
e-mail: menini@disp.uniroma2.it

Corrado Possieri

Dipartimento di Elettronica e Telecomunicazioni,
Politecnico di Torino,
Torino 10129, Italy
e-mail: possieri@ing.uniroma2.it

Antonio Tornambè

Dipartimento di Ingegneria Civile e Ingegneria
Università di Roma Tor Vergata,
Roma 00133, Italy
e-mail: tornambe@disp.uniroma2.it

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 14, 2017; final manuscript received March 26, 2018; published online April 27, 2018. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 13(6), 061006 (Apr 27, 2018) (7 pages) Paper No: CND-17-1312; doi: 10.1115/1.4039876 History: Received July 14, 2017; Revised March 26, 2018

The main goal of this paper is to design a state feedback control that makes a point mass track a non-Zeno reference trajectory in a planar billiard. This objective is achieved by first determining a continuous-time dynamical model, whose trajectories approximate the solutions of the hybrid system. Hence, a state feedback that makes the hybrid system track a reference trajectory of the continuous-time one is proposed. Finally, these two techniques are combined in order to find a state feedback that achieves tracking of the trajectories of the unforced system. Examples are reported all throughout the paper to illustrate the theoretical results.

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Grahic Jump Location
Fig. 1

Chaotic behavior of the solutions to system (2)

Grahic Jump Location
Fig. 2

Comparison of the trajectories of systems (2) and (4): (a) trajectories of systems (2) and (4) and (b) approximation error

Grahic Jump Location
Fig. 3

Tracking of the continuous-time approximation: (a) trajectories of systems (2), (4), and (6) and (b) tracking error

Grahic Jump Location
Fig. 4

Chaotic behavior of the solutions to system (2). The figures have been cropped for presentation purposes (in the first simulation, x1(t)∈[−3.14,2.72], x2(t)∈[−4.75,7.64], whereas, in the second simulation, x1(t)∈[−3.21,9.52], x2(t)∈[−3.98,1.49]).

Grahic Jump Location
Fig. 5

Reference trajectory ξ¯, solution ξ̂ of system (4), and solution ξ of the closed-loop system

Grahic Jump Location
Fig. 6

Control inputs resulting from the feedback law given in Eq. (9): (a) control inputs and (b) transient behavior



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