Thomas Haberkorn

Low-Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach

We describe in this paper the study of an earth orbital transfer with a low thrust (typically electro-ionic) propulsion system. The objective is the maximization of the final mass, which leads to a discontinuous control with a huge number of thrust arcs. The resolution method is based on single shooting, combined to a homotopic approach in order to cope with the problem of the initial guess, which is actually critical for non-trivial problems. An important aspect of this choice is that we make no assumptions on the control structure, and in particular do not set the number of thrust arcs. This strategy allowed us to solve our problem (a transfer from Low Earth Orbit to Geosynchronous Equatorial Orbit, for a spacecraft with mass of 1500 kgs, either with or without a rendezvous) for thrusts as low as 0.1N, which corresponds to a one-year transfer involving several hundreds of revolutions and thrust arcs. The numerical results obtained also revealed strong regularity in the optimal control structure, as well as some practically interesting empiric laws concerning the dependency of the final mass with respect to the transfer time and maximal thrust.

Autonomous Underwater Vehicles: Singular Extremals and Chattering

In this paper, we consider the time minimal problem for an Autonomous Underwater Vehicle. We investigate, on a simplified model, the existence of singular extremals and discuss their optimality status. Moreover, we prove that singular extremals corresponding to the angular acceleration are of order 2. We produce in this case a semi-canonical form of our Hamiltonian system and we can conclude the existence of chattering extremals..

Designing Efficient Trajectories for Underwater Vehicles Using Geometric Control Theory

In this paper, we consider the minimum time problem for underwater vehicles. Using Lagrangian mechanics, we write the equations of motion for marine vehicles with 6 degrees of freedom as a controlled mechanical system. We then apply the necessary conditions from the maximum principle for a trajectory to be time optimal. Using techniques from differential geometry we analyze the resuls. Finally we supplement the theoretical study with numerical simulations.

Convergence Results for Smooth Regularizations of Hybrid Nonlinear Optimal Control Problems

We consider a class of hybrid nonlinear optimal control problems having a discontinuous dynamics ruled by a partition of the state space. For this class of problems, some hybrid versions of the usual Pontryagin Maximum Principle are known. We introduce general regularization procedures, parameterized by a small parameter, smoothing the previous hybrid problems to standard smooth optimal control problems, for which we can apply the usual Pontryagin Maximum Principle. We investigate the question of the convergence of the resulting extremals as the regularization parameter tends to zero. Under some general assumptions, we prove that smoothing regularization procedures converge, in the sense that the solution of the regularized problem (as well as its extremal lift) converges to the solution of the initial hybrid problem. To illustrate our convergence result, we apply our approach to the minimal time low-thrust coplanar orbit transfer with eclipse constraint.

An optimal control problem in photoacoustic tomography

This paper is devoted to the introduction and study of a photoacoustic tomography model, an imaging technique based on the reconstruction of an internal photoacoustic source distribution from measurements acquired by scanning ultrasound detectors over a surface that encloses the body containing the source under study. In a nutshell, the inverse problem consists in determining absorption and diffusion coefficients in a system coupling a hyperbolic equation (acoustic pressure wave) with a parabolic equation (diffusion of the fluence rate), from boundary measurements of the photoacoustic pressure. Since such kinds of inverse problems are known to be generically ill-posed, we propose here an optimal control approach, introducing a penalized functional with a regularizing term in order to deal with such difficulties. The coefficients we want to recover stand for the control variable. We provide a mathematical analysis of this problem, showing that this approach makes sense. We finally write necessary first-order optimality conditions and give preliminary numerical results.

TOWARDS PRACTICAL IMPLEMENTATION OF TIME OPTIMAL TRAJECTORIES FOR UNDERWATER VEHICLES

In this paper, we are concerned with the practical implementation of time optimal numerical techniques on underwater vehicles. We briefly introduce the model of underwater vehicle we consider and present the parameters for the test bed ODIN (Omni-Directional Intelligent Navigator). Then we explain the numerical method used to obtain time optimal trajectories with a structure suitable for the implementation. We follow this with a discussion on the modifications to be made considering the characteristics of ODIN. Finally, we illustrate our computations with some experimental results.

Low-Thrust Lyapunov to Lyapunov and Halo to Halo with

In this work, we develop a new method to design energy minimum low-thrust missions (L2-minimization). In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the optimal control problem is solved using Newton-like algorithms finding the zero of a shooting function. To compute a Lyapunov to Lyapunov mission, we first compute an admissible trajectory using a heteroclinic orbit between the two periodic orbits. It is then used to initialize a multiple shooting method in order to release the constraint. We finally optimize the terminal points on the periodic orbits. Moreover, we use continuation methods on position and on thrust, in order to gain robustness. A more general Halo to Halo mission, with different energies, is computed in the last section without heteroclinic orbits but using invariant manifolds to initialize shooting methods with a similar approach.

L^2

In this work, a new robust and fast method is developed to perform transfers that minimize fuel consumption between two invariant manifolds of periodic orbits in the circular restricted three-body problem. The method starts with an impulse transfer between two invariant manifolds to build an optimal control problem. This allows to choose an adequate fixed transfer time. Using the Pontryagin maximum principle, the resolution of the problem is formulated as that of finding the zero of a shooting function (indirect method). The algorithm couples different kinds of continuations (on cost, final state, and thrust) to improve robustness and to initialize the solver. The efficiency of the method is illustrated with numerical examples. Finally, the influence of the transfer time is studied numerically thanks to a continuation on this parameter, and it checks that, when transfer duration goes to zero, the control converges to the impulse transfer that it started with. It shows the robustness of the method and establishes a mathematical link between the two problems. Read More: https://arc.aiaa.org/doi/10.2514/1.G002922

-Minimization

The goal of this paper is to design a spacecraft round trip transfer from a parking orbit to asteroid 2006 RH\(_{120}\) during its geocentric capture while maximizing the final spacecraft mass or, equivalently, minimizing the delta-v. The spacecraft begins in a halo “parking” orbit around the Earth-Moon \(L_2\) libration point. The round-trip transfer is composed of three portions: the approach transfer from the parking orbit to 2006 RH\(_{120}\), the rendezvous “lock-in” portion with the spacecraft in proximity to and following the asteroid orbit, and finally the return transfer to \(L_2\). An indirect method based on the maximum principle is used for our numerical calculations. To partially address the issue of local minima we restrict the control strategy to reflect an actuation corresponding to up to three thrust arcs during each portion of the transfer. Our model is formulated in the circular restricted four-body problem (CR4BP) with the Sun considered as a perturbation of the Earth-Moon circular restricted three body problem. A shooting method is applied to numerically optimize the round trip transfer, and the 2006 RH\(_{120}\) rendezvous and departure points are optimized using a time discretization of the 2006 RH\(_{120}\) trajectory.

Transfer Between Invariant Manifolds: From Impulse Transfer to Low-Thrust Transfer

Mechanical control systems have become a part of our everyday life. Systems such as automobiles, robot manipulators, mobile robots, satellites, buildings with active vibration controllers and air conditioning systems, make life easier and safer, as well as help us explore the world we live in and exploit it’s available resources. In this chapter, we examine a specific example of a mechanical control system; the Autonomous Underwater Vehicle (AUV). Our contribution to the advancement of AUV research is in the area of guidance and control. We present innovative techniques to design and implement control strategies that consider the optimization of time and/or energy consumption. Recent advances in robotics, control theory, portable energy sources and automation increase our ability to create more intelligent robots, and allows us to conduct more explorations by use of autonomous vehicles. This facilitates access to higher risk areas, longer time underwater, and more efficient exploration as compared to human occupied vehicles. The use of underwater vehicles is expanding in every area of ocean science. Such vehicles are used by oceanographers, archaeologists, geologists, ocean engineers, and many others. These vehicles are designed to be agile, versatile and robust, and thus, their usage has gone from novelty to necessity for any ocean expedition.