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Reduction, linearization, and stability of relative equilibria for mechanical systems on Riemannian manifolds
 Acta Appl. Math
"... Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an e ..."
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Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute. ” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory. 1.
Making Robotic Marionettes Perform
"... Abstract—This paper describes a project with the long term goal of automated performance marionettes, accomplished by capturing human motion and automating the motion imitation synthesis for an experimental marionette system. The automation and performance goals required the development of hardware ..."
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Abstract—This paper describes a project with the long term goal of automated performance marionettes, accomplished by capturing human motion and automating the motion imitation synthesis for an experimental marionette system. The automation and performance goals required the development of hardware and software tools that enable motion imitation, leading to a series of results in numerical simulation, optimal control, and embedded systems. Marionettes are actuated by strings, so the mechanical description of the marionettes either creates a multiscale or degenerate system—making simulation of the constrained dynamics challenging. Moreover, the marionettes have 4050 degrees of freedom with closed kinematic chains. Choreography requires the use of motion primitives, typically originating from human motions that one wants the marionette to imitate, and resulting in a high dimensional nonlinear optimal control problem that needs to be solved for each primitive. Once acquired, the motion primitives must be pieced together in a way that preserves stability, resulting in an optimal timing control problem. We conclude with our current results that enable the synthesis of optimal imitation trajectories, and overview the next steps we are taking in this project towards automated performance marionettes. I.
unknown title
, 2004
"... 2004 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other wor ..."
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2004 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE Controllability of a hovercraft model (and two general results)
Linearization of affine connection control system∗
, 2002
"... A simple mechanical system is a triple (Q, g, V) where Q is a configuration space, g is a Riemannian metric on Q, and V is the potential energy. The Lagrangian associated with a simple mechanical system is defined by the kinetic energy minus the potential energy. The equations of motion given by the ..."
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A simple mechanical system is a triple (Q, g, V) where Q is a configuration space, g is a Riemannian metric on Q, and V is the potential energy. The Lagrangian associated with a simple mechanical system is defined by the kinetic energy minus the potential energy. The equations of motion given by the EulerLagrange equations for a simple mechanical system without potential energy can be formulated as an affine connection control system. If these systems are underactuated then they do not provide a controllable linearization about their equilibrium points. Without a controllable linearization it is not entirely clear how one should deriving a set of controls for such systems. There are recent results that define the notion of kinematic controllability and its required set of conditions for underactuated systems. If the underactuated system in question satisfies these conditions, then a set of openloop controls can be obtained for specific trajectories. These openloop controls are susceptible to unmodeled environmental and dynamic effects. Without a controllable linearization a feedback control is not readily available to compensate for these effects. This report considers linearizing affine connection control systems with zero potential energy along a reference trajectory. This linearization yields a linear secondorder differential equation from the properties of its integral curves. The solution of this differential equation measures the variations of the system from the desired reference trajectory. This secondorder differential equation is then written as a control system. If it is controllable then it provides a method for adding a feedback law. An example is provided where a feedback control is implemented.
unknown title
, 2008
"... 2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other wor ..."
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2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE Jet bundles and algebrogeometric characterisations for controllability of affine systems
unknown title
, 2012
"... 2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other wor ..."
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2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE Fundamental problems in geometric control theory Andrew D. Lewis∗ 2012/02/29 Open problems in controllability and stabilisability of analytic systems are discussed. In particular, questions like, “Can controllability and/or stabilisability be tested by solving algebraic equations? ” and, “What is the relationship between controllability from a state and stabilisability to the same state? ” are discussed. A main idea is a rethinking of how one might examine stabilisability by connecting it to controllability. For analytic systems, local obstructions to controllability and stabilisability should be determined by the germ of the system at the prescribed state. A means of characterising these germs in a systematic manner is presented.
unknown title
, 2003
"... 2003 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other wor ..."
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2003 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE Jacobian linearisation in a geometric setting
Vectorvalued quadratic forms in control theory
"... For finite dimensional Rvector spaces U and V we consider a symmetric bilinear map B: U × U → V. This then defines a quadratic map QB: U → V by QB(u) = B(u, u). Corresponding to each λ ∈ V ∗ is a Rvalued quadratic form λQB on U defined by λQB(u) = λ·QB(u). B is definite if there exists λ ∈ V ∗ so ..."
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For finite dimensional Rvector spaces U and V we consider a symmetric bilinear map B: U × U → V. This then defines a quadratic map QB: U → V by QB(u) = B(u, u). Corresponding to each λ ∈ V ∗ is a Rvalued quadratic form λQB on U defined by λQB(u) = λ·QB(u). B is definite if there exists λ ∈ V ∗ so that λQB is positivedefinite. B is indefinite if for each λ ∈ V ∗\ann(image(QB)), λQB is neither positive nor negativesemidefinite, where ann denotes the annihilator. Given a symmetric bilinear map B: U × U → V, the problems we consider are as follows. 1. Find necessary and sufficient conditions characterizing when QB is surjective. 2. If QB is surjective and v ∈ V, design an algorithm to find a point u ∈ Q−1B (v). 3. Find necessary and sufficient conditions to determine when B is indefinite. From the computational point of view, the first two questions are the more interesting ones. Both can be shown to be NPcomplete, whereas the third one can be recast as a semidefinite programming problem.1 Actually, our main interest is in a geometric characterization of these problems. Section 4 below constitutes an initial attempt to unveil the essential geometry behind these questions. By understanding the geometry of the problem
unknown title
, 2006
"... Local factorization of trajectory lifting morphisms for singleinput affine control systems � ..."
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Local factorization of trajectory lifting morphisms for singleinput affine control systems �