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… it is an elegant and invaluable reference for mathematicians and scientists with an interest in classical and celestial mechanics, astrodynamics, physics, biology, and related fields.” (Marian Gidea, Mathematical Reviews, Issue 2010 d) … This outstanding book can be used not only as an introductory course at the graduate level in mathematics, but also as course material for engineering graduate students. “The second edition of this text infuses new mathematical substance and relevance into an already modern classic … and is sure to excite future generations of readers. Satzer, The Mathematical Association of America, March, 2009) … It is a well-organized and accessible introduction to the subject …. This book is intended to support a first course at the graduate level for mathematics and engineering students. The N-body problem is used as the primary example of a Hamiltonian system, a touchstone for the theory as the authors develop it. 'The primary subject here is the basic theory of Hamiltonian differential equations studied from the perspective of differential dynamical systems. This text provides a mathematical structure of celestial mechanics ideal for beginners, and will be useful to graduate students and researchers alike. The courses took students who had some background in differential equations and lead them through a systematic grounding in the theory of Hamiltonian mechanics from a dynamical systems point of view.

The previous editions grew out of graduate level courses in mathematics, engineering, and physics given at several different universities. With each chapter containing new content, readers are provided with new material on reduction, orbifolds, and the regularization of the Kepler problem, all of which are provided with applications. This third edition text provides expanded material on the restricted three body problem and celestial mechanics. Most of them are presented in careful detail and are illustrated with profuse, illuminating diagrams.' - Mathematical Reviews 'One of the most striking features of the book is the nice collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Since the first edition, the book has expanded in length and the third author, James Murdock has been added. Also new are surveys on invariant manifolds in Appendix C and averaging for PDEs in Appendix E. Chapters on Periodic Averaging and Hyperbolicity, Classical (first level) Normal Form Theory, Nilpotent (classical) Normal Form, and Higher Level Normal Form Theory are entirely new and represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. There are many changes, corrections and updates in chapters on Basic Material and Asymptotics, Averaging, and Attraction.

So it is not surprising that the authors have presented an extensive revision of the first edition of the Averaging Methods in Nonlinear Dynamical Systems book. Perturbation theory and in particular normal form theory has shown strong growth during the last decades. The results show that the resulting self-localization performs at similar levels of accuracy than the visual odometry, but with the frequency of the inertial odometry. Once this was done, the filter was implemented into the existing robotics framework of the labs at ESA and tested with real data. This environment was used to determine the best way to fuse the information. To do so, a testing environment was developed to test different filters and state models. The goal of this Internship is to develop a filter that fuses together the outputs of these two algorithms into a single source of self-localization. From the two solutions, the visual odometry is much more accurate but the inertial odometry can run at higher frequency, due to the computational load of the former. The more recent Visual odometry uses the information from the stereo cameras of the rover to compute the relative pose between two consecutive stereo pairs. The classic Inertial odometry uses the information of the wheel encoders and the IMU to estimate the current velocity of the rover. One important feature to embark on rovers is a robust self-localization algorithm.

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The limited communication windows and signal delay between the rover and its operators in Earth implies the rovers must have a high level of autonomy so that the missions can become more efficient. One of the main challenges of planetary rover missions is the constraint in communication.