scholarly journals Dynamics and Control of the Shoot-the-Moon Tabletop Game

2012 ◽  
Vol 134 (5) ◽  
Author(s):  
Peng Xu ◽  
Richard E. Groff ◽  
Timothy Burg
Keyword(s):  
Simple Structure ◽  
Equations Of Motion ◽  
Angular Position ◽  
The Moon ◽  
Nonlinear Controller ◽  
Nonholonomic Dynamics ◽  
Ball Rolling ◽  
Dynamics And Control ◽  
Dynamics Simulations ◽  
And Control

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.

scholarly journals Stabilization of a Rigid Body Payload With Multiple Cooperative Quadrotors

2016 ◽  
Vol 138 (12) ◽  
Author(s):  
Farhad A. Goodarzi ◽  
Taeyoung Lee
Keyword(s):  
Rigid Body ◽  
Arbitrary Number ◽  
Equations Of Motion ◽  
Vertical Direction ◽  
Free Form ◽  
Nonlinear Controller ◽  
Dynamics And Control ◽  
And Control ◽  
Flexible Cables ◽  
Quadrotor Unmanned Aerial Vehicles

Abstract This paper presents the full dynamics and control of arbitrary number of quadrotor unmanned aerial vehicles (UAVs) transporting a rigid body. The rigid body is connected to the quadrotors via flexible cables where each flexible cable is modeled as a system of arbitrary number of serially connected links. It is shown that a coordinate-free form of equations of motion can be derived for the complete model without any simplicity assumptions that commonly appear in other literature, according to Lagrangian mechanics on a manifold. A geometric nonlinear controller is presented to transport the rigid body to a fixed desired position while aligning all of the links along the vertical direction. A rigorous mathematical stability proof is given and the desirable features of the proposed controller are illustrated by numerical examples and experimental results.

scholarly journals A unified approach to rigid body rotational dynamics and control

2011 ◽  
Vol 468 (2138) ◽  
pp. 395-414 ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Aaron D. Schutte
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Equations Of Motion ◽  
Fundamental Equation ◽  
Rotational Dynamics ◽  
Constrained Motion ◽  
Dynamics And Control ◽  
Common Framework ◽  
The Stability ◽  
And Control

This paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control. This is achieved in a simple systematic manner using the so-called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework. It is shown that a first application of this equation yields, in closed form, the equations of rotational dynamics, whereas a second application of the self-same equation yields two new methods for explicitly determining, in closed form, the nonlinear control torque needed to change the orientation of a rigid body. The stability of the controllers developed is analysed, and numerical examples showing the ease and efficacy of the unified methodology are provided.

Dynamics of Elastic Manipulators With Prismatic Joints

1992 ◽  
Vol 114 (1) ◽  
pp. 41-49 ◽  
Author(s):  
K. W. Buffinton
Keyword(s):  
Equations Of Motion ◽  
Elastic Beam ◽  
Alternative Form ◽  
Specific System ◽  
Beam Equations ◽  
Prismatic Joints ◽  
Dynamics And Control ◽  
And Control ◽  
Two Degree Of Freedom ◽  
Elastic Beam Equations

The purpose of this investigation is to study the formulation of equations of motion for flexible robots containing translationally moving elastic members that traverse a finite number of distinct support points. The specific system investigated is a two-degree-of-freedom manipulator whose configuration is similar to that of the Stanford Arm and whose translational member is regarded as an elastic beam. Equations of motion are formulated by treating the beam’s supports as kinematical constraints imposed on an unrestrained beam, by discretizing the beam by means of the assumed modes technique, and by applying an alternative form of Kane’s method which is particularly well suited for systems subject to constraints. The resulting equations are programmed and are used to simulate the system’s response when it performs tracking maneuvers. The results provide insights into some of the issues and problems involved in the dynamics and control of manipulators containing highly elastic members connected by prismatic joints.

scholarly journals Dynamics and Control of a Tethered Satellite System Based on the SDRE Method

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Yong-Lin Kuo
Keyword(s):  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Nonlinear Dynamic Analysis ◽  
Satellite System ◽  
Convergence Time ◽  
Tethered Satellite ◽  
Modeling And Control ◽  
Tethered Satellite System ◽  
Dynamics And Control ◽  
And Control

This paper presents the nonlinear dynamic modeling and control of a tethered satellite system (TSS), and the control strategy is based on the state-dependent Riccati equation (SDRE). The TSS is modeled by a two-piece dumbbell model, which leads to a set of five nonlinear coupled ordinary differential equations. Two sets of equations of motion are proposed, which are based on the first satellite and the mass center of the TSS. There are two reasons to formulate the two sets of equations. One is to facilitate their mutual comparison due to the complex formulations. The other is to provide them for different application situations. Based on the proposed models, the nonlinear dynamic analysis is performed by numerical simulations. Besides, to reduce the convergence time of the librations of the TSS, the SDRE control with a prescribed degree of stability is developed, and the illustrative examples validate the proposed approach.

Nonlinear Dynamics and Control of an Ideal/Nonideal Load Transportation System With Periodic Coefficients

2006 ◽  
Vol 2 (1) ◽  
pp. 32-39 ◽  
Author(s):  
N. J. Peruzzi ◽  
J. M. Balthazar ◽  
B. R. Pontes ◽  
R. M. L. R. F. Brasil
Keyword(s):  
Equations Of Motion ◽  
Periodic Problem ◽  
Dc Motor ◽  
Transportation System ◽  
Stability Diagrams ◽  
Dynamics And Control ◽  
The Stability ◽  
Load Transportation ◽  
And Control ◽  
The Mathematical Model

In this paper, a loads transportation system in platforms or suspended by cables is considered. It is a monorail device and is modeled as an inverted pendulum built on a car driven by a dc motor. The governing equations of motion were derived via Lagrange’s equations. In the mathematical model we consider the interaction between the dc motor and the dynamical system, that is, we have a so called nonideal periodic problem. The problem is analyzed, qualitatively, through the comparison of the stability diagrams, numerically obtained, for several motor torque constants. Furthermore, we also analyze the problem quantitatively using the Floquet multipliers technique. Finally, we devise a control for the studied nonideal problem. The method that was used for analysis and control of this nonideal periodic system is based on the Chebyshev polynomial expansion, the Picard iterative method, and the Lyapunov-Floquet transformation (L-F transformation). We call it Sinha’s theory.

Nonlinear Modeling and Partial Linearizing Control of a Slewing Timoshenko-Beam

1996 ◽  
Vol 118 (1) ◽  
pp. 75-83 ◽  
Author(s):  
King Yuan ◽  
Chen-Meng Hu
Keyword(s):  
Timoshenko Beam ◽  
Slenderness Ratio ◽  
Nonlinear Modeling ◽  
Equations Of Motion ◽  
Nonlinear Controller ◽  
Elastic Mode ◽  
Modeling And Control ◽  
Centrifugal Stiffening ◽  
Orthogonality Conditions ◽  
And Control

The modeling and control of a horizontally slewing inextensible Timoshenko beam, taking into account the centrifugal stiffening effect and a tip payload, are considered. Partial differential equations of motion and orthogonality conditions for the constrained modes are derived. A finite dimensional dynamic model simplified by using the orthogonality conditions is obtained. To achieve the joint angle trajectory tracking with simultaneous suppression of elastic vibrations, a nonlinear controller is designed using input-output linearization and elastic-mode stabilization. A sufficient condition for asymptotic stability of the closed-loop system is established. Numerical examples with the role of slenderness ratio of the slewing beam highlighted are presented to demonstrate the effectiveness of the proposed control strategy.

Gyrostabilizer Vehicular Technology

2011 ◽  
Vol 64 (1) ◽  
Author(s):  
Nicholas C. Townsend ◽  
Ramanand A. Shenoi
Keyword(s):  
Control Strategies ◽  
Equations Of Motion ◽  
Effective Means ◽  
Detailed Examination ◽  
Current State ◽  
Wide Range ◽  
Dynamics And Control ◽  
And Control ◽  
Gyroscopic Systems ◽  
Vehicular Technology

This paper examines the current state of gyrostabilizer vehicular technology. With no previous description of the wide range and variety of gyrostabilizer technology, this paper provides a review of the current state of the art. This includes a detailed examination of gyrostabilizer vehicular systems, dynamics and control. The present review first describes the historical development of gyroscopic systems before going on to describe the various system characteristics, including an overview of gyrostabilizer vehicular applications and system designs for land, sea and spacecraft. The equations of motion for generic gyroscopic systems are derived following momentum (Newton-Euler) and energy (Lagrange) based approaches and examples provided. The derivations are made generically for individual components, enabling direct application for a wide variety of systems. In the final section, a review of gyrostabilizer control strategies is presented and the remaining challenges are discussed. Gyrostabilizer systems are anticipated to become more widely adopted as they provide an effective means of motion control with several significant advantages for land, sea and spacecraft. (101 references).

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

Sign in / Sign up

Export Citation Format

Share Document