Collision analysis for an UAV

International audienceThe Sense and Avoid capacity of Unmanned Aerial Vehicles (UAV) is one of the key elements to open the access to airspace for UAVs. In order to replace a pilot's See and Avoid capacity such a system has to be certified "as safe as a human pilot on-board". The problem is to prove that an unmanned aircraft equipped with a S and A system can comply with the actual air transportation regulations. This paper aims to provide mathematical and numerical tools to link together the safety objectives and sensors specifications. Our approach starts with the natural idea of a specified "safety volume" around the aircraft: the safety objective is to guarantee that no other aircraft can penetrate this volume. We use a general reachability and viability concepts to define nested sets which are meaningful to allocate sensor performances and manoeuvring capabilities necessary to protect the safety volume. Using the general framework of HJB equations for the optimal control and differential games, we give a rigorous mathematical characterization of these sets. Our approach allows also to take into account some uncertainties in the measures of the parameters of the incoming traffic. We also provide numerical tools to compute the defined sets, so that the technical specifications of a S and A system can be derived in accordance with a small set of intuitive parameters. We consider several dynamical models corresponding to the different choices of maneuvers (lateral, longitudinal and mixed). Our numerical simulations show clearly that the nature of used maneuvers is an important factor in the specifications of sensor's performances

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations

We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the L2 norm for linear and semi-linear equations, and in the H1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in L2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms

September 2012AN ADAPTIVE SPARSE GRID SEMI-LAGRANGIAN SCHEME FOR FIRST ORDER HAMILTON-JACOBI BELLMAN EQUATIONS

ABSTRACT. We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimensiond = 8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem. 1

Minimum time control problems for non autonomous differential equations

International audienceIn this paper, we investigate a minimum time problem for controlled non-autonomous differential systems, with dynamics depending on the final time. The minimal time function associated to this problem does not satisfy the dynamic programming principle. However, we will prove that it is related to a standard front propagation problem via the reachability function. Two simple numerical examples are given to illustrate our approach

Stability and convergence of second order backward differentiation schemes for parabolic Hamilton-Jacobi-Bellman equations

We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton-Jacobi-Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use ofwell-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the L-2 norm for linear and semi-linear equations, and in the H-1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in L-2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Holder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms