Dynamic interpolation for obstacle avoidance on Riemannian manifolds

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.

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.

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

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.

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.

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...

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.

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.

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