Stabilization of a 3D rigid pendulum

Chaturvedi, N.A.; Bacconi, F.; Sanyal, Amit K.; Bernstein, Dennis S.; McClamroch, N. Harris

doi:10.1109/acc.2005.1470436

Cited by 30 publications

(43 citation statements)

References 4 publications

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“…If R e satisfies (11), then ð0; G e Þ is an equilibrium of (7) and (8). Thus, corresponding to the hanging equilibrium manifold and the inverted equilibrium manifold of (1) and (2), there exist two isolated equilibrium solutions of the reduced attitude equations (7) and (8). These are given by the hanging equilibrium ð0; G h Þ and the inverted equilibrium ð0; G i Þ; where…”

Section: Equilibrium Structure Of the 3d Pendulummentioning

confidence: 94%

Section: Equilibrium Structure Of the 3d Pendulummentioning

confidence: 99%

“…To study the stabilization problem, it is advantageous to consider the equations of the 3D pendulum in terms of the reduced attitude as in (7) and (8). If R e satisfies (11), then ð0; G e Þ is an equilibrium of (7) and (8). Thus, corresponding to the hanging equilibrium manifold and the inverted equilibrium manifold of (1) and (2), there exist two isolated equilibrium solutions of the reduced attitude equations (7) and (8).…”

Section: Equilibrium Structure Of the 3d Pendulummentioning

confidence: 99%

“…The 3D pendulum is a non-trivial generalization of the planar pendulum and the spherical pendulum that has been studied only in recent publications of the authors and their colleagues in [7][8][9][10][11]. The 3D pendulum is a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by a uniform gravity force and by control forces as shown in Figure 1.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum

Chaturvedi

McClamroch

2007

Intl J Robust & Nonlinear

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SUMMARYThe 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. This paper shows that a controller based on angular velocity feedback can be used to asymptotically stabilize the hanging equilibrium manifold of the 3D pendulum. Lyapunov analysis and nonlinear geometric methods are used to assess the global closed-loop properties. We explicitly construct compact sets that lie in the domain of attraction of the hanging equilibrium of the closed-loop. Finally, this controller is shown to achieve almost global asymptotic stability of the hanging equilibrium manifold. An invariant manifold of the closed-loop that converges to the inverted equilibrium manifold is identified.

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Section: Equilibrium Structure Of the 3d Pendulummentioning

confidence: 94%

Section: Equilibrium Structure Of the 3d Pendulummentioning

confidence: 99%

Section: Equilibrium Structure Of the 3d Pendulummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum

Chaturvedi

McClamroch

2007

Intl J Robust & Nonlinear

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“…Newton-Euler based formulation of the pendulum dynamics is used. Detailed mathematical analysis of three dimensional pendulum dynamics and related control problems can be found in recently published works by N. H. McClamroch and his colleagues [36,38,35,170].…”

Section: Models Of Otolithsmentioning

confidence: 99%

Modeling Verticality Estimation During Locomotion

Farkhatdinov

Michalska

Berthoz

et al. 2013

Romansy 19 – Robot Design, Dynamics and Control

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In this thesis, a nonlinear model of the vestibular system is proposed, with special reference to humans and other locomoting animals. The vestibular system is essential for stable locomotion since it provides idiothetic measurements of spatial orientation that are needed for postural control. The model was constructed from general considerations regarding the Newton-Euler dynamics governing the three-dimensional movements of bodies constrained to oscillate in non-inertial frames, such as the otoliths, which were modeled as spherical damped pendula. Two configurations were considered. The medial model considered only one inner ear located in the center of a head. The lateral model considered two inner ears located laterally with respect to the center of rotation of the head. The differences between these two models were analyzed and the importance of having two vestibular organs discussed. To this end, a nonlinear algebraic observability test was used to verify whether the reconstruction of the head orientation with respect to the gravitational vertical was possible from otoliths measurements only. It could be shown that in order for the head vertical orientation to be observable, the head had to be stabilized during locomotion. Moreover, it was shown that the gravitoinertial ambiguity inherent to inertial idiothetic sensing could be resolved if the head was horizontally stabilized. These results were applied to solve the head vertical orientation estimation problem in the linearized case (Luenberger observer, Kalman filter) as well as in the nonlinear case (Extended Kalman filter, Newton method based observation). These observers were designed and tested in simulations. The simulations indicated that the estimation errors were smaller and the observers converged faster when head was stabilized during locomotion, leading to a nonlinear, combined observation-control system that could be stabilized with respect to the gravitational vertical based on no other information than the prior knowledge of the Newton-Euler dynamics. The results were further tested with a specifically designed experimental setup that comprised an actuated gimbal mechanism to represent the head-neck articulation and a liquid-based inclinometer that represented the otoliths organs. The findings derived from this research would be helpful for analyzing spatial perception in humans and animals, and for improving the perceptual capabilities of robotic systems, such as humanoid robots, rough terrain vehicles, or free-moving drones.Keywords: vestibular system, head-neck system, nonlinear system, observation, estimation, verticality, spatial perception, locomotion, humanoid robot. 3 RésuméDans cette thèse, nous proposons un modèle non-linéaire du système vestibulaire, en particulier chez l'humain et autres animaux mobiles. Le système vestibulaire est essentiel pour une locomotion stable dans la mesure où il fournit des mesures idiothétiques d'orientation spatiale nécessairesà la stabilité posturale. Le développement du modèle est basé sur l...

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