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Autonomous Guidance for Asteroid Descent Using Successive Convex Optimisation Dual Quaternion Approach S

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Autonomous Guidance for Asteroid Descent Using Successive Convex Optimisation Dual Quaternion Approach S Autonomous Guidance for Asteroid Descent using Successive Convex Optimisation Dual Quaternion Approach S. Hazra Technische Universiteit Delft ii Autonomous Guidance for Asteroid Descent using Successive Convex Optimisation Dual Quaternion Approach by S. Hazra to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Thursday March 28, 2019 at 9:30 AM. Student number: 4598342 Project duration: June 18, 2018 – March 28, 2019 Thesis committee: Dr. ir. E. Mooij, TU Delft, Astrodynamics and Space Missions, Supervisor Dr. ir. M. Sagliano, Deutsches Zentrum für Luft- und Raumfahrt e.V., Supervisor Prof. dr. ir. P.N.A.M Visser TU Delft, Astrodynamics and Space Missions Dr. ir. C.D. Visser, TU Delft, Controls and Simulations This thesis is confidential and cannot be made public until March 28, 2020 An electronic version of this thesis is available at http://repository.tudelft.nl/. ii Abstract With the onset of the age of space travel, asteroid missions have been steadily gaining interest. The pristine nature of asteroids due to their preserved state since the formation of the Solar System is an opportunity to unravel many mysteries about the Solar System. Also with the ever-growing need for resources, asteroids prove to be a plentiful source. With the discovery of asteroids in the close vicinity of our planet and a probable threat to the preservation of life, a need for defence missions has also arisen. With many successful missions like NEAR to Eros, Rosetta to Comet 67P,Hayabusa to Itokawa, Hayabusa 2 to Ryugu, OSIRIS-REx to Bennu, etc., the requirement of precise navigation with autonomous guidance and control for future missions has been established. The ever-growing need for better computational speed and accuracy has led to the development of new representations for attitude and position of the spacecraft in the past. The usual methods of representations of position and attitude (pose) are the Cartesian coordinates and quaternions. A recent development is the si- multaneous representation of the pose of the spacecraft using dual quaternions which are eight-dimensional vectors. As of February 2019, a variety of missions using dual quaternions for relative navigation, rendezvous and docking, entry, descent and landing have been conceptualised. Recent studies have narrowed down from the existing guidance algorithms to, Successive Convexification and Sampling Based Model Predictive Optimi- sation, as future prospects for autonomous guidance of missions. These missions are limited by the lack of instantaneous ground control due to operations at very far distances from Earth. In this thesis, an attempt to incorporate these state-of-the-art guidance technologies to asteroid missions has been made. The novelty in this thesis is, using dual quaternions for SC pose and attitude representation to autonomously guide the spacecraft for mapping an asteroid and perform a touch and go descent using sampling-based motion predictive optimisation and successive convexification respectively. To achieve this objective, this thesis first browses through past missions to establish the requirements of a mission to aster- oids. Then it explores the augmented algebra of dual numbers and their application in the concept of dual quaternions. It then works towards establishing the dynamics and kinematics in the asteroid centred rotating frame using dual quaternions along with modelling the environment around the asteroids. This thesis delves into the method of convex optimisation to present its existing algorithms and develop a dynamic successive convex optimisation method to achieve better results as compared to the method developed by Mao et al. (2016). The thesis establishes the initialisation of a sampling-based model predictive optimisation method for autonomous mapping of a target asteroid. After the development and verification of these algorithms, different scenarios have been designed to find out the robustness of the dynamic successive convexification method. The scenarios have been run by both the Cartesian quaternion and dual quaternion based algorithms. They are found to behave similarly in their results, besides the latter being computationally more expensive as proved by different theses so far. The algorithm performs better than the one developed by Szmuk et al. (2017), but faces difficulties with badly scaled problems as is the nature of missions to asteroids. The software used for the thesis is ECOS, which faces numerical problems in these mission scenarios even when the problem is feasible and has an optimal result. The availability of a solution to the optimal control problem depends on the scaling of the penalty weights used in the cost function to penalise virtual controls and trust region, which are used to prevent infeasibility and bound the problem, respectively. It also depends on finding an appropriate final time for the descent along with the number of nodes for discretisation. This research proves, that successive convexification indeed provides a speedy solution for an autonomous precision descent but needs further work to make it robust and stable. The outcome of this thesis is to carry out further research to understand the complex relations, between the scale of the problem, simulation parameters for the optimal control problem and final time in order to make the algorithm robust and safe for an autonomous mission. Another important future research prospect is to incorporate the sampling based model predictive optimisation, for the SC to autonomously map the target body and help in selecting a landmark for a descent. iii iv Preface Missions to mere rocky remnants in the Solar System aka asteroids have quickly gained interest in the past few decades as a consequence of the inherent curiosity of human beings. The pristine nature of these celestial bodies has drawn the interest of humans towards landing on them and bringing back samples to find some answers about the Solar System formation process. I am honoured to have had the opportunity to be a part of this research area and contribute the little I could towards the development of such missions. From sending a probe to one of the gaseous planets to landing on an asteroid, this has been my journey from my very first literature survey to the final one and this report is the culmination of that journey to fulfill the requirements to receive my Master of Science degree from the Aerospace engineering faculty at the Delft University of Technology. I am immensely grateful to my supervisor Erwin Mooij to have presented me with the opportunity to work under Marco Sagliano at DLR with the thesis topic. With both their guidance, this thesis has taken the present form and I hope someday it will guide a spacecraft to safely descend on an asteroid. Every meeting I have had with Erwin has always been positive and motivating. I am glad he believed in me and had patience with my failures along the journey. At DLR, Marco always found time from his busy schedule to answer my doubts and queries, most of which were pretty basic, but he never showed disappointment. He urged me on, to try a step at a time and kept me on track even when I was back in Delft after my six months in Bremen. I am thankful for Prof. dr. ir. P.N.A.M. Visser and Dr. ir. C.D. Visser to have agreed to be a part of this thesis as the defense committee. I also thank everyone I have come across and made friends with during this period, especially Bronius who worked on the navigation part of this mission. He always answered my questions as much as he could and provided me with his insights about how navigation instruments work. I would like to thank all my friends, who kept me sane, grounded and fed. I would be lost without their support and cooking skills. I would like to thank my girlfriend, who has been there for my tears every single time I have lost hope and has been a constant in my world of variables. Lastly, I would like to thank my par- ents, the two people who were more worried about my nutrition levels than the convergence of my algorithm. Without them, I would not be the person I am today or a graduate to be and for that, I will always be in their debt. This thesis has tested my capabilities to the extent of making me question my dream of becoming an aerospace engineer, but at the same time has brought me great pleasure in doing the research successfully and I hope you, the reader, will find it useful, interesting and understandable. S. Hazra Delft, 2019 v vi Nomenclature Notation C Complex Numbers N Natural Numbers P Projective Space R Real Numbers Dual number/vector _ Unit vector ^ Discrete time matrix/vector » Solution states and controls ¡ ˇ Dual vector multiplication £ , Quaternion multiplication ­ ¯ ˇ , ˇ Dual quaternion multiplication ­ ¯ Latin Symbols A Jacobian matrix for the states - A Spacecraft space set - B Jacobian matrix for the controls - C Direction Cosines Matrix - C Configuration space set - e Unit direction vector - f Function - f (X ) Process function - g(X ) Inequality constraints - h(X ) Equality constraints - H Half space set - I Identity matrix - Lc Linear cost function - Screw axis, Moment, Rotation angle, Distance traversed respec- lˆ,m,θ,d various tively (Dual Quaternion screw coordinates) n Normal vector - N Concatenated virtual control vector various N c Non-linear cost function - O Obstacle space set - qr Rotation quaternion - qd Translation quaternion - qˇ Dual Quaternion - Q Stacked Matrices - r Position vector m r e Engine thrust vector from COM m s,S Slack Variables - u,U Individual and concatenated control vector - w Penalty weight - W World space set - x, X Individual and concatenated state vector various z, Z Individual and concatenated solution state derivative vector - a Acceleration m/s2 A Area m2 vii viii 0.
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