Optimal Navigation Functions for Nonlinear Stochastic Systems

This paper presents a new methodology to craft navigation functions for nonlinear systems with stochastic uncertainty. The method relies on the transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear partial differential equation. This approach allows for optimality criteria to be incorporated into the navigation function, and generalizes several existing results in navigation functions. It is shown that the HJB and that existing navigation functions in the literature sit on ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. In particular, it is shown that under certain criteria the optimal navigation function is related to Laplace's equation, previously used in the literature, through an exponential transform. Further, analytical solutions to the HJB are available in simplified domains, yielding guidance towards optimality for approximation schemes. Examples are used to illustrate the role that noise, and optimality can potentially play in navigation system design.Comment: Accepted to IROS 2014. 8 Page

Robot Navigation Functions on Manifolds with Boundary

This paper concerns the construction of a class of scalar valued analytic maps on analytic manifolds with boundary. These maps, which we term navigation functions, are 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. We show that this class is invariant under composition with analytic diffeomorphisms: our sphere world construction immediately generates a navigation function on all manifolds into which a sphere world is deformable. On the other hand, certain well known results of S. Smale guarantee the existence of smooth navigation functions on any smooth manifold. This suggests that analytic navigation functions exist, as well, on more general analytic manifolds than the deformed sphere worlds we presently consider. For more information: Kod*La

Rigid body visual servoing using navigation functions

Visual servo controllers in the literature rarely achieve provably large domains of attraction, and seldom address two important sensor limitations: (i) susceptibility to self-occlusions and (ii) finite field of view (FOV). We tackle the problem of global, occlusion-free visual servoing of a fully actuated rigid body by recourse to navigation functions on a compact manifold which encode these restrictions as control obstacles. For occlusion free rigid body servoing, the manifold of interest is the visible set of rigid body configurations, that is, those for which the feature points are within the field of view and unoccluded by the body. For a set of coplanar feature points on one face of a convex polyhedron, we show that a slightly conservative subset of the visible set has a simple topology amenable to analytical construction of a navigation function. We construct the controller via a closed form coordinate transformation from our problem domain into the topological model space and conclude with simulation results

Autonomous Mobile Robots Controlled by Navigation Functions

This paper reviews the theory of navigation functions and the attendant use of natural control techniques with emphasis upon applications to mobile autonomous robots. Results to date will be discussed in the context of a larger program of research that seeks effective parameterizations of uncertainty in robot navigation problems. Constructive solutions to particular cases of mobile robot navigation problems with complete certainty are provided as well

Automatic Conflict Solving using Biharmonic Navigation Functions

International audience; Automatic conflict solving is an old standing problem within the field of ATM. Proposed algorithms fall into two categories : - Deterministic ones that have a provable property of collision avoidance. For all known algorithms, trajectories produced are generally not flyable because no bounds on speed and curvature can be imposed. - Stochastic methods that select an optimal sequence of manoeuvres. By design, trajectories are flyable, but no guarantee can be given on the fact that a collision-free planning can be found in finite time. It is highly desirable for a wide social acceptance of automated trajectory planning, even at a strategical level, that the algorithms in use have by-design the collision avoidance property and, at the same time, a mean of keeping the speed within a given interval. Navigation functions are common in the field of robotics but do not have the last property. We present a new approach based on biharmonic functions yielding a navigation field with constant speed. Such functions have been considered previously, but proof of collision avoidance is lacking: we address this problem in this work as summarized below. Navigation functions produce a speed field by taking the gradient of a potential function: if the obstacles to be avoided are at a higher potential than inner points of the domain (including destination), collision avoidance is guaranteed. If the potential has the Morse property (no critical point is degenerated) then there exists a descent direction at every point of the admissible domain, making the destination reachable. In the framework of biharmonic functions, a tensor field is produced instead of a vector one ; the Morse property is no longer relevant. We show here that all benefits of navigation functions can be recovered through the use of the bienergy density, with the ability to get constant speed fields. Document type: Conference objec

Weak Input-to-State Stability Properties for Navigation Function Based Controllers

Due to topological constraints, Navigation Functions, are not, except from trivial cases, equivalent to quadratic Lyapunov functions, hence systems based on Navigation Functions cannot directly accept an Input-to-State stability (ISS) characterization. However a relaxed version of Input-to-State stability, namely almost global ISS (aISS), is shown to be applicable. The proposed framework provides compositional capability for navigation function based systems. Cascade as well as feedback interconnections of aISS navigation systems are shown to also possess the aISS property under certain assumptions on the interconnections. Several simulated examples of navigation systems are presented to demonstrate the effectiveness of the proposed scheme

Almost Global Asymptotic Formation Stabilization Using Navigation Functions

We present a navigation function through which a group of mobile agents can be coordinated to achieve a particular formation, both in terms of shape and orientation, while avoiding collisions between themselves and with obstacles in the environment. Convergence is global and complete, subject to the constraints of the navigation function methodology. Algebraic graph theoretic properties associated with the interconnection graph are shown to affect the shape of the navigation function. The approach is centralized but the potential function is constructed in a way that facilitates complete decentralization. The strategy presented will also serve as a point of reference and comparison in quantifying the cost of decentralization in terms of performance

Vehicle Motion Planning Using Stream Functions

Borrowing a concept from hydrodynamic analysis, this paper presents stream functions which satisfy Laplace's equation as a local-minima free method for producing potential-field based navigation functions in two dimensions. These functions generate smoother paths (i.e. more suited to aircraft-like vehicles) than previous methods. A method is developed for constructing analytic stream functions to produce arbitrary vehicle behaviors while avoiding obstacles, and an exact solution for the case of a single uniformly moving obstacle is presented. The effects of introducing multiple obstacles are discussed and current work in this direction is detailed. Experimental results generated on the Cornell RoboFlag testbed are presented and discussed, as well as related work applying these methods to path planning for unmanned air vehicles