| 1. |
- Chhabra, Robin, et al.
(author)
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A Unified Approach to Input-output Linearization and Concurrent Control of Underactuated Open-chain Multi-body Systems with Holonomic and Nonholonomic Constraints
- 2016
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In: Journal of dynamical and control systems. - : Springer Science and Business Media LLC. - 1079-2724 .- 1573-8698. ; 22:1, s. 129-168
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Journal article (peer-reviewed)abstract
- This paper presents a unified geometric framework to input-output linearization of open-chain multi-body systems with symmetry in their reduced phase space. This leads us to an output tracking controller for a class of underactuated open-chain multi-body systems with holonomic and nonholonomic constraints. We consider the systems with multi-degree-of-freedom joints and possibly with non-zero constant total momentum (in the holonomic case). The examples of these systems are free-base space manipulators and mobile manipulators. We first formalize the control problem, and rigorously state an output tracking problem for such systems. Then, we introduce a geometrical definition of the end-effector pose and velocity error. The main contribution of this paper is reported in Section 5, where we solve for the input-output linearization of the highly nonlinear problem of coupled manipulator and base dynamics subject to holonomic and nonholonomic constraints. This enables us to design a coordinate-independent controller, similar to a proportional-derivative with feed-forward, for concurrently controlling a free-base, multi-body system. Finally, by defining a Lyapunov function, we prove in Theorem 3 that the closed-loop system is exponentially stable. A detailed case study concludes this paper.
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| 2. |
- Duits, R., et al.
(author)
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On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps
- 2016
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In: Journal of dynamical and control systems. - : SPRINGER/PLENUM PUBLISHERS. - 1079-2724 .- 1573-8698. ; 22:4, s. 771-805
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Journal article (peer-reviewed)abstract
- We consider the problem P (c u r v e) of minimizing for a curve x in with fixed boundary points and directions. Here, the total length Laeyen0 is free, s denotes the arclength parameter, kappa denotes the absolute curvature of x, and xi amp;gt; 0 is constant. We lift problem P (c u r v e) on to a sub-Riemannian problem P (m e c) on SE(3)/({0}xSO(2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem P (c u r v e) . We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions.
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| 3. |
- Kelleche, Abdelkarim, et al.
(author)
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Stabilization of an Axially Moving Euler Bernoulli Beam by an Adaptive Boundary Control
- 2023
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In: Journal of dynamical and control systems. - : Springer Science and Business Media LLC. - 1079-2724 .- 1573-8698.
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Journal article (peer-reviewed)abstract
- This paper concerns with the stabilization of an axially moving beam by an adaptive boundary control. We prove existence and uniqueness of the solution by means of nonlinear semigroup theory. Moreover, we construct the control through a low-gain adaptive velocity feedback. We also prove that the designed control is able to stabilize exponentially the closed loop system. Some numerical simulations are given to illustrate the theoretical results.
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| 4. |
- Montgomery, R., et al.
(author)
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A nonintegrable sub-Riemannian geodesic flow on a Carnot group
- 1997
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In: Journal of Dynamical and Control Systems. - 1079-2724. ; 3:4, s. 519-530
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Journal article (peer-reviewed)abstract
- Graded nilpotent Lie groups, or Carnot groups, are to sub-Riemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between sub-Riemannian and Riemannian geometry is a strong one, one might conjecture that the sub-Riemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture to be false by showing that the sub-Riemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra consists of 4 by 4 upper triangular matrices. As a corollary, we prove that the centralizer for the corresponding quadratic "quantum" Hamiltonian in the universal enveloping algebra of this Lie algebra is "as small as possible."
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| 5. |
- Novikov, Dmitry, et al.
(author)
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On Limit Sets for Geodesics of Meromorphic Connections
- 2023
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In: Journal of dynamical and control systems. - : Springer Science and Business Media LLC. - 1079-2724 .- 1573-8698. ; 29:1, s. 55-70
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Journal article (peer-reviewed)abstract
- Meromorphic connections on Riemann surfaces originate and are closely related to the classical theory of linear ordinary differential equations with meromorphic coefficients. Limiting behavior of geodesics of such connections has been studied by, e.g., Abate and Bianchi (Math Z 282:247–272, 2016) and Abate and Tovena (J Differ Equ 251(9):2612–2684, 2011) in relation with generalized Poincaré-Bendixson theorems. At present, it seems still to be unknown whether some of the theoretically possible asymptotic behaviors of such geodesics really exist. In order to fill the gap, we use the branched affine structure induced by a Fuchsian meromorphic connection to present several examples with geodesics having infinitely many self-intersections and quite peculiar ω-limit sets.
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| 6. |
- Zhong, Jianghua, et al.
(author)
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CONSTRUCTIVE STABILIZATION OF QUADRATIC-INPUT NONLINEAR SYSTEMS WITH BOUNDED CONTROLS
- 2008
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In: Journal of dynamical and control systems. - : Springer. - 1079-2724 .- 1573-8698. ; 14:4, s. 571-593
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Journal article (peer-reviewed)abstract
- In this paper, the stabilization of quadratic-input nonlinear systems with bounded controls is considered. According to the type of quadratic-input forms, two cases, namely, positive definite and positive semi-definite, are considered. For the case of positive definiteness, a universal formula for bounded stabilizers is given via a known Lyapunov control function. For the case of positive semidefiniteness, a constructive parametrization of bounded stabilizers is proposed under the assumption that there exists a known Lyapunov control function with respect to a smaller control set than the admissible control set.
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