Variational point-obstacle avoidance on Riemannian manifolds

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Variational point-obstacle avoidance on Riemannian manifolds

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Variational point-obstacle avoidance on Riemannian manifolds

Anthony M. Bloch¹, Margarida Camarinha², Leonardo Colombo³•Institutions (3)

University of Michigan¹, University of Coimbra², Spanish National Research Council³

26 Sep 2019-

TL;DR: This paper derives the dynamical equations for stationary paths of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces.

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Abstract: In this letter we study variational obstacle avoidance problems on complete Riemannian manifolds. The problem consists of minimizing an energy functional depending on the velocity, covariant acceleration and a repulsive potential function used to avoid a static obstacle on the manifold, among a set of admissible curves. We derive the dynamical equations for extrema of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces. Numerical examples are presented to illustrate the proposed method.

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Collision Avoidance of Multiagent Systems on Riemannian Manifolds

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Jacob R. Goodman, Leonardo Colombo

01 Jan 2022-Siam Journal on Control and Optimization

TL;DR: In this paper , the authors studied variational collision avoidance problems for multiagent systems on complete Riemannian manifolds, and provided conditions under which it is possible to ensure that agents will avoid collision within some desired tolerance.

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Abstract: This paper studies variational collision avoidance problems for multiagent systems on complete Riemannian manifolds. That is, we minimize an energy functional, among a set of admissible curves, which depends on an artificial potential function used to avoid collision between the agents. We show the global existence of minimizers to the variational problem, and we provide conditions under which it is possible to ensure that agents will avoid collision within some desired tolerance. We also study the problem where trajectories are constrained to have uniform bounds on the derivatives and derive alternate safety conditions for collision avoidance in terms of these bounds---even in the case where the artificial potential is not sufficiently regular to ensure existence of global minimizers.

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11 citations

Journal Article•DOI•

Local minimizers for variational obstacle avoidance on Riemannian manifolds

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Jacob R. Goodman

12 Jan 2022-The Journal of Geometric Mechanics

TL;DR: In this paper , a variational obstacle avoidance problem on complete Riemannian manifolds is studied, where the goal is to minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles.

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Abstract: This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories—called $ Q $-local minimizers and $ \Omega $-local minimizers—and subsequently classified, with local uniqueness results obtained in both cases.

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7 citations

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Variational collision and obstacle avoidance of multi-agent systems on Riemannian manifolds

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Rama Seshan Chandrasekaran¹, Leonardo Colombo², Margarida Camarinha³, Ravi N. Banavar⁴, Anthony M. Bloch⁵ - Show less +1 more•Institutions (5)

Indian Institute of Technology Madras¹, Spanish National Research Council², University of Coimbra³, Indian Institute of Technology Bombay⁴, University of Michigan⁵

11 Oct 2019

TL;DR: In this paper, a path planning problem from a variational approach to collision and obstacle avoidance for multi-agent systems evolving on a Riemannian manifold is studied, which consists of finding non-intersecting trajectories between the agent and prescribed obstacles on the workspace, among a set of admissible curves, to reach a specified configuration, based on minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to prevent collision with the obstacles and among the agents.

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Abstract: In this paper we study a path planning problem from a variational approach to collision and obstacle avoidance for multi-agent systems evolving on a Riemannian manifold. The problem consists of finding non-intersecting trajectories between the agent and prescribed obstacles on the workspace, among a set of admissible curves, to reach a specified configuration, based on minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to prevent collision with the obstacles and among the agents. We apply the results to examples of a planar rigid body, and collision and obstacle avoidance for agents evolving on a sphere.

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4 citations

Journal Article•DOI•

A unifying approach for rolling symmetric spaces

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Krzysztof Krakowski, Luís Machado, Fátima Silva Leite

01 Jan 2021-The Journal of Geometric Mechanics

TL;DR: In this article, the authors present a unified theory to describe the pure rolling motions of Riemannian symmetric spaces, which are submanifolds of Euclidean or pseudo-Euclidean spaces, and make a connection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space.

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Abstract: The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces, which are submanifolds of Euclidean or pseudo-Euclidean spaces. Rolling motions provide interesting examples of nonholonomic systems and symmetric spaces appear associated to important applications. We make a connection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space. This emphasises the relevance of Lie theory in the geometry of rolling manifolds and explains why many particular examples scattered through the existing literature always show a common pattern.

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3 citations

Journal Article•DOI•

Finding extremals of Lagrangian actions

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Lyle Noakes, Erchuan Zhang

01 Aug 2022-Communications in Nonlinear Science and Numerical Simulation

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Journal Article•DOI•

Real-time obstacle avoidance for manipulators and mobile robots

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Oussama Khatib¹•Institutions (1)

Stanford University¹

01 Apr 1986-The International Journal of Robotics Research

TL;DR: This paper reformulated the manipulator con trol problem as direct control of manipulator motion in operational space—the space in which the task is originally described—rather than as control of the task's corresponding joint space motion obtained only after geometric and geometric transformation.

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Abstract: This paper presents a unique real-time obstacle avoidance approach for manipulators and mobile robots based on the artificial potential field concept. Collision avoidance, tradi tionally considered a high level planning problem, can be effectively distributed between different levels of control, al lowing real-time robot operations in a complex environment. This method has been extended to moving obstacles by using a time-varying artificial patential field. We have applied this obstacle avoidance scheme to robot arm mechanisms and have used a new approach to the general problem of real-time manipulator control. We reformulated the manipulator con trol problem as direct control of manipulator motion in oper ational space—the space in which the task is originally described—rather than as control of the task's corresponding joint space motion obtained only after geometric and kine matic transformation. Outside the obstacles' regions of influ ence, we caused the end effector to move in a straight line with an...

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6,515 citations

"Variational point-obstacle avoidanc..." refers methods in this paper

...The use of artificial potential functions to avoid collision was introduced in Khatib (see [21] and references therein) and further studied for example by Koditschek and Rimon [22], [23]....
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Book•

Differential Geometry, Lie Groups, and Symmetric Spaces

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Sigurdur Helgason¹•Institutions (1)

Massachusetts Institute of Technology¹

01 Jan 1978

TL;DR: In this article, the structure of semisimplepleasure Lie groups and Lie algebras is studied. But the classification of simple Lie algesbras and of symmetric spaces is left open.

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Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.

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6,321 citations

"Variational point-obstacle avoidanc..." refers background in this paper

...onnected ﬁnitedimensional Lie group endowed with a bi-invariant Riemannian metric and Ka closed Lie subgroup of G. It is well known that the canonical projection ˇ: G!His a Riemannian submersion (see [15] for instance), in the sense that, for all gin G, the isomorphism T gˇ: (kerT gˇ)?!T ˇ( )Hpreserves the inner-products deﬁned by the Riemannian metrics on Gand Hand T gGsplits naturally into two ortho...
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...ere s is the Lie algebra of Kand m ’T oH, whith 0 = ˇ(e), being ethe identity element on G. That is, kerT eˇ= s and the horizontal subspace (kerT gˇ)?is m. Moreover, the following relations hold (see [15]) [s;s] ˆs; [m;m] ˆs; [m;s] ˆm: Using the decomposition of T gGand deﬁning vertical and horizontal tangent vectors on G, it is possible to deﬁne horizontal curves and vector ﬁelds on Hto G, by choosin...
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Book•

An introduction to differentiable manifolds and Riemannian geometry

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William M. Boothby

01 Jan 1975

TL;DR: In this article, the authors present a revised edition of one of the classic mathematics texts published in the last 25 years, which includes updated references and indexes and error corrections and will continue to serve as the standard text for students and professionals in the field.

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Abstract: This is a revised printing of one of the classic mathematics texts published in the last 25 years. This revised edition includes updated references and indexes and error corrections and will continue to serve as the standard text for students and professionals in the field.Differential manifolds are the underlying objects of study in much of advanced calculus and analysis. Topics such as line and surface integrals, divergence and curl of vector fields, and Stoke's and Green's theorems find their most natural setting in manifold theory. Riemannian plane geometry can be visualized as the geometry on the surface of a sphere in which "lines" are taken to be great circle arcs.

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1,929 citations

"Variational point-obstacle avoidanc..." refers background in this paper

...v xjj= hv x;v xi1=2 with v x2T xM. A Riemannian connection ron Mis a map that assigns to any two smooth vector ﬁelds Xand Yon Ma new vector ﬁeld, r XY. For the properties of r, we refer the reader to [7, 8, 24]. The operator r X, which assigns to every vector ﬁeld Ythe vector ﬁeld r XY, is called the covariant derivative of Ywith respect to X. Given vector ﬁelds X, Yand Zon M, the vector ﬁeld R(X;Y)Zgiven b...
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Journal Article•DOI•

Riemannian center of mass and mollifier smoothing

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Hermann Karcher¹•Institutions (1)

University of Bonn¹

01 Sep 1977-Communications on Pure and Applied Mathematics

1,109 citations

Book•

Geometric control of mechanical systems

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Francesco Bullo, Andrew D. Lewis

01 Jan 2005

TL;DR: In this article, a comprehensive set of modeling, analysis and design techniques for a class of simple mechanical control systems is presented, that is, systems whose Lagrangian is kinetic energy minus potential energy.

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Abstract: This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.

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848 citations

"Variational point-obstacle avoidanc..." refers background or methods in this paper

...gand its inverse map is the matrix logarithm map. Using the matrix logarithm, exp 1(RTQ) kexp 1(RTQ)k4 = log(RTQ) jjlog(RTQ)jj4 : (12) Denoting ˚= arccos(1 2 (Tr(RTQ) 1), and using Proposition 5:7 in [8], for R6= Q4, log(RTQ) = ˚ 2sin(˚) (RTQ RTQ): Since klog(RTQ)k= ˚it follows that log(RTQ) jjlog(RTQ)jj4 = 1 2˚3 sin(˚) (RTQ RQT): The body velocity of the curve Rin SO(3) is the curve vin so(3) verify...
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...is lemma guarantees that the connection ris completely determined by its restriction to g via left-translations. This restriction, denoted by g r: g g !g, is naturally given by g rw u= 1 2 [w;u] (see [8] p. 271). Indeed, if u;w2g we have r w L u L= ( g rwu) L, where u Ldenotes the left-invariant vector ﬁeld associated to u. Let x: IˆR !Gbe a smooth curve on G. The body velocity of xis the curve v: Iˆ...
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...v xjj= hv x;v xi1=2 with v x2T xM. A Riemannian connection ron Mis a map that assigns to any two smooth vector ﬁelds Xand Yon Ma new vector ﬁeld, r XY. For the properties of r, we refer the reader to [7, 8, 24]. The operator r X, which assigns to every vector ﬁeld Ythe vector ﬁeld r XY, is called the covariant derivative of Ywith respect to X. Given vector ﬁelds X, Yand Zon M, the vector ﬁeld R(X;Y)Zgiven b...
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