在人工智能技术广泛应用的时代，各种类型的飞行器和机器人将代替人执行多种任务。轨迹生成和路径规划对于这些具体任务的完成，在细节上具有至关重要的作用。本书包括离散力学最优轨迹实现技术提出的背景意义、理论基础、应用步骤及结果分析。全书讨论了离散力学最优控制理论在优化轨迹生成这一方向的相关研究成果，通过理论分析和仿真验证对这种优化轨迹生成方法进行了研究，并给出了此方法相对于其他主流方法的优势对比，举例说明了该技术针对一些代表性对象的实际应用结果。

Preface Optimal trajectory generation is an essential part for robotic explorers to execute the total exploration of deep oceans or outer space planets while curiosity of human and technology advancements of society both require robots to search for unknown territories efficiently and safely. As one of stateoftheart optimal trajectory generation methodologies, Nonlinear Trajectory Generation (NTG) combines with Bspline, nonlinear programming, differential flatness technique to generate optimal trajectories for modelled mechanical systems. While Discrete Mechanics and Optimal Control (DMOC) is a proposed optimal control method for mechanical systems, it is based on direct discretization of Lagranged’Alembert principle. In this book, NTG is utilized to generate trajectories for an underwater glider with a 3D Bspline ocean current model. The optimal trajectories are corresponding well with the Lagrangian Coherent Structures (LCS). Then NTG is utilized to generate 3D opportunistic trajectories for a JPL (Jet Propulsion Laboratory) Aerobot by taking advantage of wind velocity. Since both DMOC and NTG are methods which can generate optimal trajectories for mechanical systems, their differences in theory and application are investigated. In a simple ocean current example and a more complex ocean current model, DMOC with discrete EulerLagrange constraints generates local optimal solutions with different initial guesses while NTG is also generating similar solutions with more computation time and comparable energy consumption. DMOC is much easier to implement than NTG in order to generate good solutions in NTG, its variables need to be correctly defined as Bspline variables with rightlychosen orders.Therefore, this book is focused on DMOC based trajectory generation technique, which is the title of this book. For readers’reference, in this book, in order to verify the DMOC trajectory generation method practically, we provide a detailed procedure to establish a motion capture system. This system is established with a Vicon motion capture system. Six cameras connected with a data station are able to track realtime coordinates of a draganflyer helicopter. This motion capture system establishes a good foundation for related trajectory generation research projects, especially for verifying trajectory generation technique. Finally, the book proposes to combine a multiphase strategy with the original DMOC method, resulting in a new MultiPhase Discrete Mechanics and Optimal Control (MDMOC) method and making optimal trajectory generation more efficient. The advantages of the proposed method are demonstrated mathematically, and in addition, a quadrotor, Unmanned Aerial Vehicle (UAV), simulation example is presented to show its superiority over the popular Gauss Pseudospectral method. This new MDMOC methodology can also be applied to other mechanical systems such as mobile robots and underwater gliders. Acknowledgments This book is partially from my Ph.D thesis, and further research is conducted on my Ph.D work. First, my deepest gratitude goes to my advisor Dr. Tamer Inanc. I am a lucky student to have had the opportunity to meet Dr. Tamer Inanc. He provides me all he can to help me to grow as a Ph.D student. Carefully reading my sometimes rash paper manuscripts, pointing out possible research directions, helping me to avoid unnecessary obstacles in research projects. All he did for me as an advisor will have a lasting impact on me wherever my career path takes me. The training and guidance I received in his directions will be also a valuable asset to assist me to solve different practical problems in the future. I am so fortunate that Dr. Jerrold E. Marsden from Caltech could take the time to guide my research. During my past projects, his insights and acuteness were shown to me in how research can be done with complete considerations including right directions to solve problems. I thank Dr. Sina OberBlbaum from Caltech, Dr. Alberto Elfes from JPL for offering me the research collaboration experience which enhances my ability to solve problems. Wholeheartedly, I thank Dr. Jacek M. Zurada who was the Chair of the ECE department when I came to University of Louisville(U of L). Without his recruiting me as a graduate student to U of L, I would ended the career as an engineer in Shanghai, and chosen a totally different career from what I am today. The growth and improvement in my ability and power are obvious during the Ph.D study. I am so glad that I got this opportunity. Gratitude goes to Dr. Joseph D. Cole and Dr. Ibrahim N. Imam for serving in my Ph.D committee. I attended Dr. Cole’s digital control class, and he is always ready to help me and give me some practical suggestions to enhance my knowledge and expertise in this specific field. During my Ph.D research, U of L as a fastgrowing university in academics, is a very nice place where to study and live. I personally have the chance to go to President, Dr. James R. Ramsey’s home to enjoy his family’s courtesy. I am thankful for Dr. Ramsey and Provost Shirley Willihnganz’s leadership, all the faculty and staff members in our university are so nice and I was welcomed everywhere. Our department’s Ph.D Program Secretary Lisa Bell is always helping me whenever I have troubles or problems. My friends at U of L, Sara, Elom, Travis, Dr. Dongqing Chen, Lijun Zhang, Qiang Ao, Liang Yang, Zhiyong Zhang, Gang Zhao, Yinan Cui and Hui Wang and so on helped me during the study. All others not mentioned here, thank you all for the help and kindness. At last but not the least, I am thankful for the University of Louisville Fellowship, ECE teaching assistantship, and KYNASA EPSCoR (contract WKURF 5968550802) for supporting me throughout my years of study at U of L. Since 2010, I came to Beijing Institute of Technology, with the support of National Natural Science Foundation of China under control NO. 61203064, my research work on optimal trajectory generation is still going on. We proposed a new trajectory generation methodology called MDMOC. In this book, we included this method as a possible way to solve the trajectory generation problem for readers’ reference.

Chapter 1 Introduction 1.1 Motivation 1.2 Related Work 1.3 NTG and DMOC 1.4 Book Contributions 1.5 Book Outline Chapter 2 Glider Trajectory Generation with NTG 2.1 Problem Definition 2.2 Glider Trajectory Generation 2.3 Ocean Current Model 2.3.1 2D B-spline Ocean Flows Model 2.3.2 3D B-spline Ocean Flows Model 2.4 Nonlinear Trajectory Generation 2.4.1 Cost Function 2.4.2 Constraints 2.5 Optimal Control of a Kinematical Glider 2.5.1 NTG Solution for 3D B-spline Ocean Flows Model 2.5.2 Comparison in the 2D and 3D B-spline Ocean Current Models 2.6 Optimal Control of a Dynamical Glider 2.6.1 Cost Function 2.6.2 Constraints 2.6.3 NTG Solutions for the Dynamic Glider 2.7 Animation of Glider and Ocean Current 2.8 Summary Chapter 3 Aerobot Trajectory Generation with NTG 3.1 NASA JPL Aerobot 3.2 Euler-Lagrange Based Aerobot Trajectories 3.2.1 Euler-Lagrange Equations 3.2.2 Wind Profile 3.2.3 Problem Formulation 3.2.4 Simulated 3D Trajectory 3.3 State Space Model Based Trajectories 3.3.1 Problem Formulation 3.3.2 Simulated 3D Trajectories 3.4 Summary Chapter 4 Trajectory Generation with DMOC 4.1 DMOC Methodology 4.2 DMOC Tutorial 4.2.1 IPOPT 4.2.2 AMPL 4.2.3 Implementation Details 4.2.4 An Application Example 4.3 Summary Chapter 5 Comparison of DMOC and NTG 5.1 Discrete Mechanics and Optimal Control 5.1.1 Discrete Cost Function 5.1.2 Discrete Lagrange d’Alembert Principle 5.2 Nonlinear Trajectory Generation 5.2.1 Problem Formulation 5.2.2 Procedure in NTG 5.3 DMOC versus NTG 5.3.1 A Glider in the Simple Current Model 5.4 A Glider in the B-spline Ocean Model 5.5 Hovercraft Example 5.6 Summary Chapter 6 Testbed Establishment 6.1 Testbed 6.2 Vicon Vision System 6.3 Real-Time Application 6.4 Summary Chapter 7 Multiple Phase DMOC 7.1 Introduction 7.2 Problem Definition 7.3 Mutli-Phase DMOC 7.3.1 Formulation 7.3.2 Multi-Phase Strategy 7.4 Quadrotor Application Example 7.4.1 Quadrotor Model 7.5 Numerical Results 7.5.1 Trajectory Generation 7.6 Summary Chapter 8 Conclusion and Future Work 8.1 Conclusion 8.2 Future Work References Appendix A NTG program for a glider in a B-spline ocean model Appendix B NTG program for a JPL Aerobot Appendix C DMOC program for a glider in a B-spline ocean model Appendix D DMOC program for a JPL Aerobot Appendix E MATLAB program for generating ODE45 trajectories Appendix F Program to obtain realtime coordiates of a draganflyer