ENSTA / UMA
32 Boulevard Victor,
75739 Paris cedex 15, FRANCE

tél :  (33) 1 45 52 54 01
Fax : (33) 1 45 52 54 82

see also : Venue
This meeting is organized by J.F. Bonnans and H. Zidani (commands team).

The registration is free but mandatory. Lunch will be offered if you register before May 16.

Programme

Friday, May 23 - Morning   (Amphi Parmentier)

  9h00-9h15
Welcome
 9h15-10h00
Daniel Scheeres The University of Colorado at Boulder
Optimal Feedback Control and Hamiltonian Dynamics
Utilizing the Hamiltonian structure of the necessary conditions for optimal control we present a generalization of the Hamilton-Jacobi-Bellmann equation for the computation of optimal feedback controls. This generalization decomposes the cost function solution of the HJB partial differential equation into two terms, a purely dynamical term and one involving boundary conditions only. Further, by applying classical canonical transformation theory to the dynamical term we are able to find solutions to variants of the HJB equation that have non-singular terminal conditions, allowing for improved algorithms for their computation. These alternate solutions can be transformed into solutions for the cost function of the HJB equation via a simple Legendre transformation. This work arises out of our investigations into the solution of two-point boundary value problems in Hamiltonian dynamical systems. The general theory which we use to solve these problems is first presented and then applied to the optimal control problem. Finally, some novel applications of our optimal feedback control laws are introduced and discussed.
10h00-10h45
Richard Epenoy CNES
Minimum-fuel Deployment of Formation Flying Satellites - An Optimal Control Approach
This talk focuses on the issue of minimum-fuel deployment for satellite formation flying. We address it as an optimal control problem, the necessary optimality conditions of which are derived from Pontryagin's Maximum Principle. These are numerically enforced by finding the root of a so-called shooting function. However, optimal control laws for minimum-fuel problems are discontinuous and produce nonsmooth shooting functions with singular Jacobian matrices. The resulting problems cannot be solved easily and require the use of a regularization technique. We extend here our previously developed continuation-smoothing method to the multisatellite context by using an adapted initialization procedure. Because realistic mission scenarios may require it, our approach additionally offers the ability to slightly modify a given maneuver strategy to balance the fuel consumption among the satellites to a certain degree. A number of tests concerning low-Earth orbits are carried out in the paper as examples. Our model includes the J2 effect, which leads to numerical difficulties. We show the efficiency of our method in this challenging context: several maneuver strategies are detected and analyzed from the space dynamics angle. We finally point out that beyond this application, a whole class of deployment/reconfiguration problems may be handled through this approach.
10h45-11h05
Break
11h05-11h50
ONERA
Applications of optimal control to space transportation system design.
This presentation gives an overview of different applications of optimal control for early design studies of future space transportation systems. The presentation addresses the context of these studies, and the specific needs that have to be addressed by optimal control tools and methods. Two families of applications are considered: launchers (expandable or reusable) and reentry/aeroassisted vehicles. An overview of past and current work is given; and the pros and cons of the optimal control methods employed are discussed.
11h50-12h35
Arnaud Biard et Julien Laurent-Varin CNES
Activité R&T à la direction des lanceurs en optimisation de trajectoire

12h35-14h00
Lunch

Friday, May 23 - Afternoon   (Amphi Parmentier)

14h00-14h45
Jean-Baptiste Caillau Univ. Bourgogne
Two-body control and applications
The problem of finding minimum consumption trajectories of a spacecraft in a 1/r^2 gravitational field is addressed. Applications arise from collaborations with EADS-Astrium Space Transportation and the French Space Agency (CNES). As often in nonlinear problems, homotopy techniques are ubiquitous in the methods developed. The presentation will try to illustrate these aspects, both on the numerical (L^2-L^1 homotopy) and theoretical side (homotopy from the round metric on S^2 and averaging). This is joint work with J. Gergaud (ENSEEIHT / Univ. Toulouse), C. Zayane (EDF), B. Bonnard and G. Janin (Univ. Bourgogne).
14h45-15h30
Pierre Martinon INRIA Saclay Ile-de-France
Optimal trajectories for space launcher problems
We study optimal trajectories with singular arcs, i.e. flight phases with a non maximal thrust, for a space launcher problem. We consider a flight to the geostationary transfer orbit for a heavy multi-stage launcher (Ariane 5 class) and use a realistic physical model for the drag force and rocket thrust. For preliminary result, we solve first the complete flight with stage separations, at full thrust. Then we focus on the first atmospheric climbing phase to investigate the possible existence of optimal trajectories with singular arcs. We primarily use an indirect shooting method based on Pontryagin's Minimum Principle, coupled with a continuation (homotopy) approach. Additional experiments are made with a basic direct method, which confirm the solutions obtained by the shooting. We study two slightly different launcher models and observe that modifying parameters such as the aerodynamic reference area and specific impulsion can indeed lead to optimal trajectories with either full thrust or singular arcs.
15h30-16h00
Break
16h00-16h45
Gregory Archambeau Université Orsay - Paris Sud
Eight Lissajous Orbits in the Earth-Moon system
Euler and Lagrange proved the existence of five equilibrium points in the circular restricted three-body problem. These equilibrium points are known as Lagrange points (Euler points or libration points) and are usually noted L1,...,L5. The existence of periodic orbits families called halo orbits around those points, 3-dimensional periodic orbits isomorphic to ellipses is very well known. There exist other types of periodic orbits around the Lagrange points. Indeed, Lyapunov orbits (planar periodic orbits) also exist, as well as periodic orbits with structure much more "complicated" known as Lissajous orbits. Among those periodic orbits we have established analytically and numerically the existence of families of Lissajous periodic orbits which are almost vertical and have the shape of eight.
In this presentation, we prove the existence of these Eight Lissajous orbits and describe their properties. In particular, we show, using local Lyapunov exponents, that their invariant manifolds share nice global stability properties, which make them a great interest in space mission design. Finally, we show that the invariant manifolds of Eight Lissajous orbits can be used to visit almost all the Moon surface in the Earth-Moon system.
16h45-17h30
Guy Cohen et Pierre Carpentier ENPC et ENSTA/OC
Robust approach for aerospatial optimal control problems
The main purpose of this study is the planning of spatial rendez-vous: such problems can be formulated as optimal control problems with terminal equality constraints. The cost function may reflect minimal terminal time and/or minimal fuel consumption goals. However, the optimal control derived from such a formulation is not "robust" in that the possibility of satisfying the terminal constraints once a momentary breakdown of the engine occurred, resulting in a deviation from the ideal trajectory, may be very weak. We propose an alternative formulation in which the final equality constraints must be satisfied with a certain prescribed probability, given a stochastic model of breakdown occurrence and duration. The final purpose is to redefine the planned trajectory in such a way that the rendez-vous can be successfully achieved despite engine breakdowns with a certain probability. Of course, this modified trajectory represents a certain loss of performance, as measured by the cost function, with respect to the ideal optimal trajectory. The attempt is thus to make this trade-off "performance vs. safety" more explicit and quantitative. In mathematical terms, this formulation of optimal control problems with terminal equality constraints to be met in probability raises several theoretical and algorithmic difficulties that will be briefly described in the talk with the help of simple illustrative examples. This resarch takes place under a contract with Thalès-Alenia-Space and CNES Toulouse. Joint work of Jean-Philippe Chancelier (CERMICS-ENPC), Pierre Carpentier (ENSTA/OC) and Guy Cohen (CERMICS-ENPC).