Alexander Wittig

Rigorous and accurate enclosure of invariant manifolds on surfaces

Knowledge about stable and unstable manifolds of hyperbolic fixed points of certain maps is desirable in many fields of research, both in pure mathematics as well as in applications, ranging from forced oscillations to celestial mechanics and space mission design. We present a technique to find highly accurate polynomial approximations of local invariant manifolds for sufficiently smooth planar maps and rigorously enclose them with sharp interval remainder bounds using Taylor model techniques. Iteratively, significant portions of the global manifold tangle can be enclosed with high accuracy. Numerical examples are provided.

Optimization of Multiple-Rendezvous Low-Thrust Missions on General-Purpose Graphics Processing Units

A massively parallel method for the identification of optimal sequences of targets in multiple-rendezvous low-thrust missions is presented. Given a list of possible targets, a global search of sequences compatible with the mission requirements is performed. To estimate the feasibility of each transfer, a heuristic model based on Lambert’s transfers is evaluated in parallel for each target, making use of commonly available general-purpose graphics processing units such as the Nvidia Tesla cards. The resulting sequences are ranked by user-specified criteria such as length or fuel consumption. The resulting preliminary sequences are then optimized to a full low-thrust trajectory using classical methods for each leg. The performance of the method is discussed as a function of various parameters of the algorithm. The efficiency of the general-purpose graphics processing unit implementation is demonstrated by comparing it with a traditional CPU-based branch-and-bound method. Finally, the algorithm is used to co...

An Introduction to Differential Algebra and the Differential Algebra Manifold Representation

Differential Algebra techniques have been used extensively in the past decade to treat various problems in astrodynamics. In this paper we review the Differential Algebra technique and present four different views of the method. We begin with the introduction of the mathematical definition of the technique as a particular algebra of polynomials. We then give an interpretation of the computer implementation of the method as a way to represent function spaces on a computer, which naturally leads to a view of the method as an automatic differentiation technique. We then proceed to the set theoretical view of Differential Algebra for representing sets of points efficiently on a computer, which is of particular value in astrodynamics. After this introduction to the well known classical DA techniques, we introduce the concept of a DA manifold and show how they naturally arise as an extension of classical DA set propagation. A manifold propagator that allows the accurate propagation of large sets of initial conditions by means of automatic domain splitting (ADS) is described. Its function is illustrated by applying it to the propagation of a set of initial conditions in the two-body problem.

COMPUTATION OF HIGH-ORDER MAPS TO MULTIPLE MACHINE PRECISION

The Beam Dynamics simulation package in COSY INFINITY is built upon a differential algebra data type. With it, it is possible to compute transfer maps or arbitrary systems to arbitrary order. However, this data type is limited by the precision of the underlying floating point number model provided by the computer processor. We will present a method to extend the effective precision of the calculations based purely on standard floating point operations. Those algorithms are then integrated into the differential algebra data type to efficiently extend the available precision, without unnecessarily affecting overall efficiency. To that effect, the precision of each coefficient is adjusted automatically during the calculation. We will then proceed to show the effectiveness of our implementation by calculating high precision maps of combinations of homogeneous dipole segments, for which the exact results are known, and comparing the high precision coefficients with the results produced by the traditional COSY beam physics package.

DA-based nonlinear filters for spacecraft relative state estimation

Active debris removal (ADR) missions have gained increasing importance in the space community due to the necessity of reducing the number of debris jeopardizing the operative satellites. In this context, autonomous guidance, navigation and control (GNC) plays a fundamental role in the problem of rendezvous with an uncooperative target. Especially, the estimation of the relative pose and the prediction of the target attitude are crucial for safe proximity operations. Therefore, a key point for the success of ADR missions is the development of efficient algorithms capable of limiting the computational burden without losing out the necessary performance. To this aim, this study analyzes the exploitation of nonlinear filters based on differential algebra (DA). Especially, high-order numerical extended Kalman filter and unscented Kalman filter are implemented in the DA framework. The ESA’s e.deorbit mission, involving Envisat satellite, is used as reference test case. Both filters are applied to this target application and compared in terms of accuracy and computational burden.

A differential algebra based importance sampling method for impact probability computation on Earth resonant returns of Near Earth Objects

A differential algebra based importance sampling method for uncertainty propagation and impact probability computation on the first resonant returns of Near Earth Objects is presented in this paper. Starting from the results of an orbit determination process, we use a diferential algebra based automatic domain pruning to estimate resonances and automatically propagate in time the regions of the initial uncertainty set that include the resonant return of interest. The result is a list of polynomial state vectors, each mapping specific regions of the uncertainty set from the observation epoch to the resonant return. Then, we employ a Monte Carlo importance sampling technique on the generated subsets for impact probability computation. We assess the performance of the proposed approach on the case of asteroid (99942) Apophis. A sensitivity analysis on the main parameters of the technique is carried out, providing guidelines for their selection. We finally compare the results of the proposed method to standard and advanced orbital sampling techniques.

Space Debris Removal: Learning to Cooperate and the Price of Anarchy

In this paper we study space debris removal from a game-theoretic perspective. In particular we focus on the question whether and how self-interested agents can cooperate in this dilemma, which resembles a tragedy of the commons scenario. We compare centralised and decentralised solutions and the corresponding price of anarchy, which measures the extent to which competition approximates cooperation. In addition we investigate whether agents can learn optimal strategies by reinforcement learning. To this end, we improve on an existing high fidelity orbital simulator, and use this simulator to obtain a computationally efficient surrogate model that can be used for our subsequent game-theoretic analysis. We study both single- and multi-agent approaches using stochastic (Markov) games and reinforcement learning. The main finding is that the cost of a decentralised, competitive solution can be significant, which should be taken into consideration when forming debris removal strategies.

Differential Algebra software library with automatic code generation for space embedded applications

Differential Algebra (DA) techniques have become increasingly popular in various aerospace engineering applications over the past 5-10 years. They allow computing polynomial expansions of functions representing a dynamical system in terms of initial conditions or parameters. The calculation of these polynomials is computationally expensive, but can often replace many iterations of a pointwise computation or provide valuable higher order information otherwise not readily available. DA will allow reducing the computational burden associated to onboard implementation of such high order Kalman filters which are needed to increase the level of autonomy in active debris removal (ADR) missions. In this paper we describe the implementation of the DA Core Engine 2.0 (DACE 2.0) which is entirely developed in C11 with a powerful modern C++ interface. Current space processors developed in Europe (LEON-3, LEON-4) run at speeds of hundreds of MHz, providing limited computational power on board of current and near-future spacecraft. Any developed software which target embedded hardware onboard spacecraft is subject to some strong limitation in both coding and resource utilization, mainly the need of using C only. In order to partly maintain the advantages given by operator overloading and object oriented programming for writing mathematical expression, an automatic translation of the DACE 2.0 C++ code into pure C11 code have been implemented. The resulting implementation is tested in a processor in the loop (PIL) test-bench using simple problems which are representative of the computational resources needed by an high order filter.