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This paper revisits the concept of controlled reduction, a symmetry-based method of decomposing the control of mechanical systems into lower-dimensional problems. Founded on a geometric property of serial kinematic chains termed recursive cyclicity, the present work extends this property to branched chains by defining the minimal irreducible tree structure. We then generalize the subrobot theorem...
The planar compass-gait biped has been extensively studied in the dynamic walking community, motivated by the gravity-based pendular efficiencies of human walking. These results can be extended to three dimensions using controlled geometric reduction for open-chain robots, by which stable 3-D walking gaits are built from known sagittal-plane limit cycles. We apply this method to the standard and with-torso...
This paper develops the concept of reduction-based control, which is founded on a controlled form of geometric reduction known as functional Routhian reduction. We introduce a geometric property of general serial-chain robots termed recursive cyclicity, leading to our presentation of the subrobot theorem. This shows that reduction-based control can arbitrarily reduce the dimensionality of any serial-chain...
This paper presents a control law that results in stable walking for a three-dimensional bipedal robot with a hip. To obtain this control law, we utilize techniques from geometric reduction, and specifically a variant of Routhian reduction termed functional Routhian reduction, to effectively decouple the dynamics of the three-dimensional biped into its sagittal and lateral components. Motivated by...
In this paper we develop a feedback control law that results in stable walking gaits on flat ground for a three-dimensional bipedal robotic walker given stable walking gaits for a two-dimensional bipedal robotic walker. This is achieved by combining disparate techniques that have been employed in the bipedal robotic community: controlled symmetries, geometric reduction and hybrid zero dynamics. Controlled...
Understanding Zeno phenomena plays an important role in understanding hybrid systems. A natural - and intriguing - question to ask is: what happens after a Zeno point? Inspired by the construction of Filippov (1988), we propose a method for extending Zeno executions past a Zeno point for a class of hybrid systems: Lagrangian hybrid systems. We argue that after the Zeno point is reached, the hybrid...
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