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Research Papers

# Dynamics and Control of the Shoot-the-Moon Tabletop Game

[+] Author and Article Information
Peng Xu

ECE Department,  Clemson University, Clemson, SC 29634pxu@clemson.edu

Richard E. Groff

ECE Department,  Clemson University, Clemson, SC 29634regroff@clemson.edu

Timothy Burg

ECE Department,  Clemson University, Clemson, SC 29634tburg@clemson.edu

J. Dyn. Sys., Meas., Control 134(5), 051006 (Jun 05, 2012) (9 pages) doi:10.1115/1.4006223 History: Received July 13, 2011; Revised February 03, 2012; Published June 05, 2012; Online June 05, 2012

## Abstract

The classic tabletop game Shoot-the-Moon has interesting dynamics despite its simple structure, consisting of a steel ball rolling on two cylindrical rods. In this paper, we derive the equations of motion for Shoot-the-Moon using a Lagrangian approach and examine the underactuated, nonlinear, and nonholonomic dynamics. Two ball position controllers are designed, one using a local linearization and another using the nonlinear dynamics. Simulations of both controllers are performed, showing that the ball converges to the setpoint position for the linearized controller and continuous signals can be tracked by the nonlinear controller. Finally, the experimental results are presented for the nonlinear controller applied to a physical implementation of Shoot-the-Moon. The effect of the nonholonomic constraint relating the ball’s linear and angular position is demonstrated. This system has rich dynamics that can provide a challenging problem for control design and serve as a new educational example.

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## Figures

Figure 1

Shoot-the-Moon game board

Figure 2

Top view (top) and side view (bottom) of Shoot-the-Moon game with rod coordinate system Ω. In the side view, the illustration of the ball needing to roll uphill to reach the scoring zone is apparent.

Figure 3

A 3D view of the ball and rod

Figure 4

A top view of the ball and rod. Note that T is the point of contact of the ball with the rod.

Figure 5

An infinitesimal movement of the ball

Figure 6

Plot of f2 (x, θ) for different x, using the parameters listed in Sec. 4. The ball drops at the rightmost point of each curve that has x ≥ 0.10

Figure 7

Definition of some important coordinates on the f2 curve with x = xc

Figure 8

The desired saturation function ηxc

Figure 9

Diagram of Position Tracking Controller. In the diagram, T−1 is the inverse function with x, x··d as input, and θ as output.

Figure 10

Simulation of the linearized position regulator. The ball starts at ±0.01 m, ±0.02 m, and ±0.03 m away from 0.15 m.

Figure 11

Simulation of the position tracking controller with ramp input

Figure 12

Simulation of the position tracking controller with sinusoidal input

Figure 13

The experimental platform

Figure 14

Diagram of the experimental platform

Figure 15

The rod actuation mechanism

Figure 16

Camera C1 mounting illustration and geometry for ball linear position measurement

Figure 17

Results of position tracking controller following a series of set points from simulation and experiment

Figure 18

Results of position tracking controller following sine wave trajectory from simulation and experiment

Figure 19

Demonstration of the nonholonomic property of the system in simulation and experiment

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