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Variational obstacle avoidance problem on Riemannian manifolds
TL;DR: By introducing a left-invariant metric on a Lie group, this work introduces variational obstacle avoidance problems on Riemannian manifolds and derives necessary conditions for the existence of their normal extremals and applies the results to the obstacle avoidance problem of a planar rigid body and a unicycle.
Abstract: We introduce variational obstacle avoidance problems on Riemannian manifolds and derive necessary conditions for the existence of their normal extremals. The problem consists of minimizing an energy functional depending on the velocity and covariant acceleration, among a set of admissible curves, and also depending on a navigation function used to avoid an obstacle on the workspace, a Riemannian manifold.
We study two different scenarios, a general one on a Riemannian manifold and, a sub-Riemannian problem. By introducing a left-invariant metric on a Lie group, we also study the variational obstacle avoidance problem on a Lie group. We apply the results to the obstacle avoidance problem of a planar rigid body and an unicycle.
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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.
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.
7 citations
23 Feb 2022
TL;DR: In this paper , the authors introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem, which consists on minimizing a higher-order energy functional among a set of admissible curves defined by a constraint on the covariant acceleration.
Abstract: We introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem. The problem consists on minimizing a higher-order energy functional, among a set of admissible curves defined by a constraint on the covariant acceleration. In addition, we use this framework to address the elastic splines problem with obstacle avoidance in the presence of this type of contraints.
1 citations
TL;DR: In this article, the authors present the dynamic interpolation problem for locomotion systems evolving on a trivial principal bundle Q. Given an ordered set of points in Q, they wish to generate a trajectory which passes through these points by synthesizing suitable controls.
1 citations
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TL;DR: The dynamic interpolation problem for locomotion systems evolving on a trivial principal bundle of Q is presented, and the explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function.
Abstract: This article presents the dynamic interpolation problem for locomotion systems evolving on a trivial principal bundle $Q$. Given an ordered set of points in $Q$, we wish to generate a trajectory which passes through these points by synthesizing suitable controls. The global product structure of the trivial bundle is used to obtain an induced Riemannian product metric on $Q$. The squared $L^2-$norm of the covariant acceleration is considered as the cost function, and its first order variations are taken for generating the trajectories. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. The explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function. The result is applied to generate a trajectory for the generalized Purcell's swimmer - a low Reynolds number microswimming mechanism.
DOI•
01 Jan 2020
TL;DR: In this paper, the authors investigated the relationship between Riemannian cubics and de Casteljau curves by analyzing the differences between the two classes of curves, and showed that the de Castelsau curves are easy to generate.
Abstract: If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. —John von Neumann (1903-1957) Interpolation is widely used in image analysis, trajectory planning and statistical inference attracting a lot of attention in theoretical studies and applications. This thesis is concerned with some variational curves used in interpolation problems. The curves investigated in this thesis include Riemannian cubics, Riemannian cubics in tension, elastica, relative geodesics and generalised cubic de Casteljau curves. This thesis contributes in two ways: applications of variational curves, and methods to study variational curves. From the perspective of applications, Riemannian cubics as critical curves of the total squared accelerations are well understood yet hard to compute. On the other hand, replacing line segments by geodesics, the classical de Casteljau algorithm has been generalised to the Riemannian setting since 1980’s. The properties of the generalised de Casteljau curves are still mysterious even though the de Casteljau curves are easy to generate. Chapter 2 finds a relationship between Riemannian cubics and curves constructed by the generalised cubic de Casteljau algorithm by analysing differences between the two classes of curves. Chapter 3 focuses on the application of relative geodesics in image registration. Given a connected finite-dimensional Lie group G with a closed subgroup H, images in the homogeneous space G/H are represented by smooth curves. Given two curves f1, f2 in G/H, relative geodesics are defined to be critical curves of the total energy of curves in G that can transform one curve to the other. Chapter 3 formulates the Euler-Lagrange equation that relative geodesics follow and discusses some cases including where f1 and f2 are geodesics, f1 and f2 are curves on S2 with G = SO(3) and H = SO(2), and finally where f1 and f2 are constant speed curves. From the perspective of theory, left Lie reduction is a technique to study variational curves in Lie groups introduced by Noakes [2003]; Noakes and Popiel [2006]; Popiel and Noakes [2007b]. Chapter 4 generalises this technique to investigate variational curves in homogeneous space G/H with the help of results about Riemannian submersions. Much attention is paid to Riemannian cubics, Riemannian cubics in tension and elastica in the case where G/H is a Riemannian symmetric space. Note that the manifold SPD(n) of all n × n symmetric positive-definite matrices is neither a Lie group with respect to the standard matrix multiplication nor a symmet-
Cites background from "Variational obstacle avoidance prob..."
...More details about geodesic flow in quadratic matrix Lie groups can be found in Bloch et al. [2008]. There are many ways to find geodesics on a manifold....
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...A quadratic matrix Lie group is defined as follows Bloch et al. [2008], G := {X ∈ GL(n)|XT JX = J}, (7....
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References
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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.
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...
6,515 citations
Book•
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.
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.
2,891 citations
Book•
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.
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.
1,929 citations
"Variational obstacle avoidance prob..." refers background in this paper
...where ad : g×g → g is the co-adjoint representation of g on g and where I : g → g, I : g → g are the associated isomorphisms to the inner product I (see [5] for instance)....
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...1 ([5], [3]): Let ω be a one form on (M, 〈·, ·〉)....
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...For the properties of ∇, we refer the reader to [5], [6], [17]....
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Book•
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.
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.
848 citations
"Variational obstacle avoidance prob..." refers background in this paper
...This restriction, denoted by g ∇: g× g → g, is given by (see [6] p....
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...The distribution is spanned by the one-form ω = sin θdx− cos θdy with corresponding vector field given by (see [6] for instance)...
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...where v = (v1, v2, v3) and w = (w1, w2, w3) are the representative elements of se(2) in R(3) (see [6] p....
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...The unicycle is a homogeneous disk on a horizontal plane and it is equivalent to a wheel rolling on a plane [2], [6]....
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...it is possible to see that ∇wLuL = ( g ∇w u)L (see [6] p....
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TL;DR: In this paper, a class of scalar valued analytic maps on analytic manifolds with boundary is constructed on an arbitrary sphere world, a compact connected subset of Euclidean n-space whose boundary is formed from the disjoint union of a finite number of (n - l)-spheres.
577 citations
"Variational obstacle avoidance prob..." refers methods in this paper
...In applications, navigation functions are typically given by artificial potential fields used for collision avoidance of certain regions through a radial analytic function on the configuration space [16]....
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...This approach has been studied by Khabit [14] for control problems and studied in the context of manifolds with boundary by Koditschek and Rimon [16]....
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...[16] D. Koditschek and E. Rimon....
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