Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft

Robotic manipulation of a rotating chain

This paper considers the problem of manipulating a uniformly rotating chain: the chain is rotated at a constant angular speed around a fixed axis using a robotic manipulator. Manipulation is quasi-static in the sense that transitions are slow enough for the chain to be always in "rotational" equilibrium. The curve traced by the chain in a rotating plane -- its shape function -- can be determined by a simple force analysis, yet it possesses complex multi-solutions behavior typical of non-linear systems. We prove that the configuration space of the uniformly rotating chain is homeomorphic to a two-dimensional surface embedded in $\mathbb{R}^3$. Using that representation, we devise a manipulation strategy for transiting between different rotation modes in a stable and controlled manner. We demonstrate the strategy on a physical robotic arm manipulating a rotating chain. Finally, we discuss how the ideas developed here might find fruitful applications in the study of other flexible objects, such as elastic rods or concentric tubes.Comment: 12 pages, 9 figure

Discrete kink dynamics in hydrogen-bonded chains I: The one-component model

We study topological solitary waves (kinks and antikinks) in a nonlinear one-dimensional Klein-Gordon chain with the on-site potential of a double-Morse type. This chain is used to describe the collective proton dynamics in quasi-one-dimensional networks of hydrogen bonds, where the on-site potential plays role of the proton potential in the hydrogen bond. The system supports a rich variety of stationary kink solutions with different symmetry properties. We study the stability and bifurcation structure of all these stationary kink states. An exactly solvable model with a piecewise ``parabola-constant'' approximation of the double-Morse potential is suggested and studied analytically. The dependence of the Peierls-Nabarro potential on the system parameters is studied. Discrete travelling-wave solutions of a narrow permanent profile are shown to exist, depending on the anharmonicity of the Morse potential and the cooperativity of the hydrogen bond (the coupling constant of the interaction between nearest-neighbor protons).Comment: 12 pages, 20 figure

Characteristics and capacities of the NASA Lewis Research Center high precision 6.7- by 6.7-m planar near-field scanner

A very precise 6.7- by 6.7-m planar near-field scanner has recently become operational at the NASA Lewis Research Center. The scanner acquires amplitude and phase data at discrete points over a vertical rectangular grid. During the design phase for this scanner, special emphasis was given to the dimensional stability of the structures and the ease of adjustment of the rails that determine the accuracy of the scan plane. A laser measurement system is used for rail alignment and probe positioning. This has resulted in very repeatable horizontal and vertical motion of the probe cart and hence precise positioning in the plane described by the probe tip. The resulting accuracy will support near-field measurements at 60 GHz without corrections. Subsystem design including laser, electronic and mechanical and their performance is described. Summary data are presented on the scan plane flatness and environmental temperature stability. Representative near-field data and calculated far-field test results are presented. Prospective scanner improvements to increase test capability are also discussed

Flatness-based Deformation Control of an Euler-Bernoulli Beam with In-domain Actuation

This paper addresses the problem of deformation control of an Euler-Bernoulli beam with in-domain actuation. The proposed control scheme consists in first relating the system model described by an inhomogeneous partial differential equation to a target system under a standard boundary control form. Then, a combination of closed-loop feedback control and flatness-based motion planning is used for stabilizing the closed-loop system around reference trajectories. The validity of the proposed method is assessed through well-posedness and stability analysis of the considered systems. The performance of the developed control scheme is demonstrated through numerical simulations of a representative micro-beam.Comment: Preprint of an original research wor