Stabilization of rigid body dynamics by internal and external torques

Bloch, Anthony M.; Krishnaprasad, P. S.; Marsden, Jerrold E.; Alvarez, Gloria Sánchez de

doi:10.1016/0005-1098(92)90034-d

Cited by 183 publications

(132 citation statements)

References 28 publications

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“…It further provides a starting point to explore the larger picture that includes dissipation and external forces and torques. We note in this regard that there are recent relevant results on the effect of adding a small amount of dissipation to a Hamiltonian system (Bloch et al, 1994) and on the use of the energy-Casimir method in deriving stabilizing controls for a rigid spacecraft (Bloch et al, 1992). New results on feedback stabilization of underwater vehicle dynamics using the Hamiltonian structure and stability conditions derived here can be found in Leonard (1996b).…”

Section: Introductionmentioning

confidence: 80%

Stability of a bottom-heavy underwater vehicle

Leonard¹

1997

Automatica

178

109

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Section: Introductionmentioning

confidence: 80%

Stability of a bottom-heavy underwater vehicle

Leonard¹

1997

Automatica

178

109

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“…The CL method has its origins in [8] and [16]. Our potential shaping approach is inspired by [12] and [28].…”

Section: B History and Related Literaturementioning

confidence: 99%

“…By definition of the new metric, we can express the kinetic energy as follows (see [13] for additional details): (30) where and where the latter is the same as in (8). Notice that the vertical part of the kinetic energy can be made negative definite or positive definite in the new vertical space depending on the sign of since is a positive-definite matrix.…”

Section: B Kinetic and Potential Shapingmentioning

confidence: 99%

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Bloch

Chang

Leonard

et al. 2001

IEEE Trans. Automat. Contr.

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299

232

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Abstract-We extend the method of controlled Lagrangians (CL) to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems. The CL method deals with mechanical systems with symmetry and provides symmetry-preserving kinetic shaping and feedback-controlled dissipation for state-space stabilization in all but the symmetry variables. Potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables. The approach also extends the method of controlled Lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline.

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“…The approach is motivated by a result in [6] for stabilization of unstable middle axis rotation of a rigid spacecraft using a single internal rotor. There, the framework was Hamiltonian and it was shown that the chosen rotor control law was such that the closed-loop system was still Hamiltonian.…”

mentioning

confidence: 99%

Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

Bloch

Leonard

Marsden

2000

IEEE Trans. Automat. Contr.

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444

380

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Abstract-We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability.In this paper we use kinetic shaping to preserve symmetry and only stabilize systems modulo the symmetry group. In the sequel to this paper (Part II), we extend the technique to include potential shaping and we achieve stabilization in the full phase space.The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane.

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Stabilization of rigid body dynamics by internal and external torques

Cited by 183 publications

References 28 publications

Stability of a bottom-heavy underwater vehicle

Stability of a bottom-heavy underwater vehicle

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

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