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Journal ArticleDOI

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

TL;DR: In this article, the authors studied dynamic interpolation for obstacle avoidance, which is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to various constraints.
Abstract: This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to...
Citations
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Journal ArticleDOI
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.
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.

11 citations

Proceedings ArticleDOI
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.
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.

10 citations

Journal ArticleDOI
TL;DR: In this paper, a variational point-obstacle avoidance problem on complete Riemannian manifolds is studied, which consists of minimizing an energy functional depending on the velocity, covariant acceleration and a repulsive potential function used to avoid an static obstacle.
Abstract: In this paper, we study variational point-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 an static obstacle given by a point in the manifold, among a set of admissible curves We derive the dynamical equations for stationary paths of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces Numerical examples are presented to illustrate the proposed method

9 citations

Posted Content
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.
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.

7 citations

Journal ArticleDOI
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

References
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Journal ArticleDOI
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

Proceedings ArticleDOI
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.
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.

1,330 citations

Proceedings ArticleDOI
01 Dec 2010
TL;DR: New results for the tracking control of a quadrotor unmanned aerial vehicle (UAV) are provided and a nonlinear tracking controller is developed on the special Euclidean group SE(3), shown to have desirable closed loop properties that are almost global.
Abstract: This paper provides new results for the tracking control of a quadrotor unmanned aerial vehicle (UAV). The UAV has four input degrees of freedom, namely the magnitudes of the four rotor thrusts, that are used to control the six translational and rotational degrees of freedom, and to achieve asymptotic tracking of four outputs, namely, three position variables for the vehicle center of mass and the direction of one vehicle body-fixed axis. A globally defined model of the quadrotor UAV rigid body dynamics is introduced as a basis for the analysis. A nonlinear tracking controller is developed on the special Euclidean group SE(3) and it is shown to have desirable closed loop properties that are almost global. Several numerical examples, including an example in which the quadrotor recovers from being initially upside down, illustrate the versatility of the controller.

827 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider a Co vector field X on a Co compact manifold Mn and show that if x at x is transversal (not tangent) to SM, then X is not zero on SM.
Abstract: We consider in this paper a Co vector field X on a Co compact manifold Mn (&M, the boundary of M, may be empty or not) satisfying the following conditions: (1) At each singular point /8 of X, there is a cell neighborhood N and a Co function f on N such that X is the gradient of f on N in some riemannian structure on N. Furthermore /8 is a non-degenerate critical point of f. Let ,81 , m denote these singularities. (2) If x e &M, X at x is transversal (not tangent) to SM. Hence X is not zero on SM. (3) If x e M let p,(x) denote the orbit of X (solution curve) through x satisfying p0(x) = x. Then for each x e M, the limit set of p,(x) as t +-~ oo is contained in the union of the /3i. (4) The stable and unstable manifolds of the /3i have normal intersection with each other. This has the following meaning. The stable manifold Wj* of /3i is the set of all x e M such that limits ...p,(x) = /i. The unstable manifold Wi of 8i is the set of all x e M such that limit,,-,. t(x) = /i. It follows from conditions (1), (2) and a local theorem in [1, p. 330], that if /3i is a critical point of index X, then Wi is the image of a 1-1, Co map pi: U-s M, where Uc Rn A has the property if x e U, tx e U, 0 ? t ? 1 and pi has rank n X everywhere (see [4] for more details). A similar statement holds for Wi* with the U c RA. Now for x e Wi (or Wi*) let Wi2, (or We*) be the tangent space of Wi (or Wi*) at x. Then for each i, j, if x e Wf nWj*, condition (4) means that

405 citations