Dynamic interpolation for obstacle avoidance on Riemannian manifolds

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Dynamic interpolation for obstacle avoidance on Riemannian manifolds

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Dynamic interpolation for obstacle avoidance on Riemannian manifolds

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

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

10 Sep 2018-arXiv: Optimization and Control-

TL;DR: This work derives first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions the authors find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional.

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Abstract: This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimizing a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem. We illustrate the results with examples of rigid bodies, both planar and spatial, and underactuated vehicles including a unicycle and an underactuated unmanned vehicle.

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Journal Article•DOI•

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

Proceedings Article•DOI•

Adaptive Backstepping of Synergistic Hybrid Feedbacks with Application to Obstacle Avoidance

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Pedro Casau¹, Ricardo G. Sanfelice², Carlos Silvestre³•Institutions (3)

Instituto Superior Técnico¹, University of California, Santa Cruz², University of Macau³

10 Jul 2019

TL;DR: The hybrid controller induced by a Synergistic Lyapunov Function and Feedback pair relative to a compact set can be extended to the case where the original affine control system is subject to a class of additive disturbances known as matched uncertainties, provided that the estimator dynamics do not add new equilibria to the closed-loop system.

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Abstract: In this paper, we show that the hybrid controller that is induced by a Synergistic Lyapunov Function and Feedback (SLFF) pair relative to a compact set, can be extended to the case where the original affine control system is subject to a class of additive disturbances known as matched uncertainties, provided that the estimator dynamics do not add new equilibria to the closed-loop system. We also show that the proposed adaptive hybrid controller is amenable to backstepping. Finally, we apply the proposed hybrid control strategy to the problem of global asymptotic stabilization of a compact set in the presence of an obstacle and we illustrate this application by means of simulation results.

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10 citations

Cites background from "Dynamic interpolation for obstacle ..."

...In particular, it is possible to find both stochastic [11] as well as determinisc approaches [12] to tackle the obstacle avoidance problem....
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Posted Content•

Variational point-obstacle avoidance on Riemannian manifolds

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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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7 citations

Cites background from "Dynamic interpolation for obstacle ..."

...Since then, a number of papers have been devoted to the generalization of this variational theory in many other contexts: interpolation problems [5], collision avoidance of multiple agents [2] and quantum splines interpolation [1], among others....
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...Inspired by the goal of gaining a better understanding of trajectories which minimize a weighted combination of the covariant acceleration and the velocity of the system in the presence of a repulsive potential which is used to avoid a static circular obstacle, in [5] we extended the problem to the trajectories that also interpolate some points on the...
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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

Book Chapter•DOI•

A Decentralized Strategy for Variational Collision Avoidance on Complete Riemannian Manifolds

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Leonardo Colombo, Jacob R. Goodman

01 Jul 2020

TL;DR: A variational approach for decentralized collision avoidance of multiple agents evolving on a Riemannian manifold is introduced, and the global existence of extrema for the energy functional is shown.

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Abstract: We introduce a variational approach for decentralized collision avoidance of multiple agents evolving on a Riemannian manifold, and we derive necessary conditions for extremal. The problem consists of finding non-intersecting trajectories of a given number of agents sharing only the information of relative positions with respect to their nearest neighbors, among a set of admissible curves, such that these trajectories are minimizers of an energy functional. The energy functional depends on covariant acceleration and an artificial potential used to prevent collision among the agents. We show the global existence of extrema for the energy functional. We apply the results to the case of agents on a compact and connected Lie group. Simulation results are shown to demonstrate the applicability of the results.

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6 citations

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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

"Dynamic interpolation for obstacle ..." refers background or methods in this paper

...The use of artificial potential functions to avoid collision was introduced by Khatib (see [29] and references therein) and further studied by Koditschek [32]....
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...This approach has been studied by Khabit for robotic manipulators (see [29] and references therein), and further studied by Koditschek [32] in the context of mechanical systems and Fiorelli and Leonard [36] for multi-agent formation....
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...Artificial potential functions [29] (as for instance, a Coulomb potential) have frequently been used for avoiding collision with obstacles, playing a fundamental role in these studies....
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Book•

Differential Equations, Dynamical Systems, and Linear Algebra

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Morris W. Hirsch, Steve Smale

12 May 1974

TL;DR: In this article, the structure theory of linear operators on finite-dimensional vector spaces has been studied and a self-contained treatment of that subject is given, along with a discussion of the relations between dynamical systems and certain fields outside pure mathematics.

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Abstract: This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.

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2,891 citations

"Dynamic interpolation for obstacle ..." refers methods in this paper

...tems and Fiorelli and Leonard [36] for multi-agent formation. The mathematical foundations for the existence of such a smooth functions on any smooth manifold can be found in the works of Smale [42], [25]. In this paper, we aim to generate trajectories interpolating prescribed points and avoiding multiple obstacles in the workspace via the study of a second order variational problem on a Riemannian ma...
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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

"Dynamic interpolation for obstacle ..." refers background in this paper

... r w u= 1 2 » w ;… 1 2 I] ad [ „”+ u [ (16) where ad : g g !g is the co-adjoint representation of gon g and where I] : g !g, I[ : g!g are the associated isomorphisms with the inner product I (see [11] for instance). We denote by u L the left-invariant vector ﬁeld associated with u 2g. For the left-invariant vector ﬁelds u L and w L , the covariant derivative of u L with respect to w L is given by ...
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...ctor ﬁelds X and Y. R is trilinear in X, Y and Z and a tensor of type „1;3”. Hence for vector ﬁelds X;Y;Z;W on M the curvature tensor satisﬁes ([39], p. 53) hR„X;Y”Z;Wi= hR„W;Z”Y;Xi: (2) 4 Lemma 2.1 ([11], [7]): Let !be a one form on „M;h;i”. The exterior derivative of a one form !is given by d!„X;Y”= X!„Y” Y!„X” !„»X;Y…” for all vector ﬁelds X;Y on M. In particular, if !„X”= hW;Xiit follows that d!„X...
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...n M, is a map that assigns to any two smooth vector ﬁelds X and Y on M a new vector ﬁeld, r XY, called the covariant derivative of Y with respect to X. For the properties of r, we refer the reader to [11], [12], [39]. Consider a vector ﬁeld W along a curve x on M. The sth-order covariant derivative along x of W is denoted by DsW dt s , s 1. We also denote by Ds+1x dt +1 the sth-order covariant deriva...
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Proceedings Article•DOI•

Virtual leaders, artificial potentials and coordinated control of groups

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Naomi Ehrich Leonard¹, E. Fiorelli¹•Institutions (1)

Princeton University¹

04 Dec 2001

TL;DR: In this article, a framework for coordinated and distributed control of multiple autonomous vehicles using artificial potentials and virtual leaders is presented, where virtual leaders can be used to manipulate group geometry and direct the motion of the group.

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Abstract: We present a framework for coordinated and distributed control of multiple autonomous vehicles using artificial potentials and virtual leaders. Artificial potentials define interaction control forces between neighboring vehicles and are designed to enforce a desired inter-vehicle spacing. A virtual leader is a moving reference point that influences vehicles in its neighborhood by means of additional artificial potentials. Virtual leaders can be used to manipulate group geometry and direct the motion of the group. The approach provides a construction for a Lyapunov function to prove closed-loop stability using the system kinetic energy and the artificial potential energy. Dissipative control terms are included to achieve asymptotic stability. The framework allows for a homogeneous group with no ordering of vehicles; this adds robustness of the group to a single vehicle failure.

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1,330 citations

Journal Article•DOI•

The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

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Darryl D. Holm¹, Jerrold E. Marsden², Tudor S. Ratiu³•Institutions (3)

Los Alamos National Laboratory¹, California Institute of Technology², University of California, Santa Cruz³

15 Jul 1998-Advances in Mathematics

TL;DR: In this article, the Lagrangian analogue of Lie-Poisson Hamiltonian systems is defined on semidirect product Lie algebras, and an abstract Kelvin-Noether theorem for these equations is derived.

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1,145 citations