Nonlinear Dynamics of the 3D Pendulum

Chaturvedi, N.A.; Lee, Taeyoung; Leok, Melvin; McClamroch, N. Harris

doi:10.1007/s00332-010-9078-6

Cited by 46 publications

(24 citation statements)

References 20 publications

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“…In this section, we implement the derived structure linearization on two examples: the asymmetric 3D pendulum [22]- [25] and the quadrotor with a suspended load [26]- [28], which synthesizes the current analysis and control tools. Both systems have been studied extensively by the robotic community and various controllers have been designed by exploring the system dynamics.…”

Section: Methodsmentioning

confidence: 99%

Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control

Fan

Murphey

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Lie group variational integrators have the advantages of both variational and Lie group integrators, which preserve the momentum, symplectic form, holonomic constraints and the Lie group structure. In addition, their long-time energy stable behaviour and coordinate-independent nature make it quite suitable to simulate a variety of mechanical systems. The structure-preservation of a Lie group variational integrator implies its linearization is structure-preserving as well, thus we call such a linearization "structured linearization". However, due to the implicit nature of variational integrators and the non-trivial differential structure of Lie groups, the structured linearization of Lie group variational integrators is much more complicated than that in generalized coordinates. In this paper, we formulate the structured linearization of Lie group variational integrators to synthesize existing analysis and control tools. To illustrate the utility of the paper, LQR controllers are constructed directly on constrained Lie groups for the asymmetric 3D pendulum and quadrotor with a suspended load, simulation results show that both controllers have a large basin of attraction.

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Section: Methodsmentioning

confidence: 99%

Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control

Fan

Murphey

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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“…First, we develop a second-order method based on the trapezoidal rule for a free-flying body that is able to undergo arbitrarily large rotations and displacements in space. This problem has been extensively studied from both theoretical and numerical aspects (see, e.g., Meyer et al 2009;Bauchau and Bottasso 1999;Chaturvedi et al 2011;Lee et al 2007;Marsden and Ratiu 1999;Simo and Wong 1991) among others) since its configuration space corresponds to a nonlinear differentiable manifold rather than a linear space. The second example corresponds to the formulation of an explicit, second-order accurate varitational integrator for finite element discretizations of geometrically exact rods.…”

Section: Final Examplesmentioning

confidence: 99%

A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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In this chapter, a brief introduction to the formulation of variational methods for finite-dimensional Lagrangian systems is presented. To this end, the first two sections focus on describing the Lagrangian and Hamiltonian points of view of mechanics for systems evolving on manifolds. Special attention is paid to the construction of the Lagrangian function and to the role of Hamilton's variational principle in the deduction of the balance equations. The relation between the symmetries of the Lagrangian function and the existence of invariants of the dynamics along with the symplectic nature of the flow are also addressed. In the third section, the discussion turns towards the formulation of a time-discrete analogue of the theory. The cornerstone of such a construction is given by a discrete analogue of Hamilton's variational principle which provides a systematic procedure to construct discrete approximations to the exact trajectory of a mechanical system on both the configuration space and the phase space. The approximation properties and the geometric characteristics of the resulting discrete trajectories are explained. Finally, we apply the variational methodology to construct symplectic and momentum-conserving time integrators for two problems of practical interest in engineering and science.

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“…It follows that the function V is positive-definite and decrescent. Choose any initial state (R(0), Ω(0)) satisfying (23). By (15) in Lemma III.4, V 0 (R(t), Ω(t), t) is a non-increasing function of time along the closed-loop trajectory and (24) holds for all t ≥ 0, which implies…”

Section: Consider a Lyapunov Function (Candidate)mentioning

confidence: 99%

Attitude control strategies overcoming the topological obstruction on SO(3)

Lee

Chang

Eun

2017

2017 American Control Conference (ACC)

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This paper presents global tracking strategies for the attitude dynamics of a rigid body. It is well known that global attractivity is prohibited for continuous attitude control systems on the special orthogonal group. Such topological restriction has been dealt with either by constructing smooth attitude control systems that exclude a set of measure zero in the region of attraction, or by introducing hybrid control systems to obtain global asymptotic stability. This paper proposes alternative attitude control systems that are continuous in time to achieve exponential stability, where the region of attraction covers the entire special orthogonal group. The main contribution of this paper is providing a new framework to overcome the topological restriction in attitude controls without relying on discontinuities through the controlled maneuvers. The efficacy of the proposed methods is illustrated by numerical simulations and experiments.

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Nonlinear Dynamics of the 3D Pendulum

Cited by 46 publications

References 20 publications

Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control

Structured linearization of discrete mechanical systems on Lie groups: A synthesis of analysis and control

A Brief Introduction to Variational Integrators

Attitude control strategies overcoming the topological obstruction on SO(3)

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