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University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 1-1-2012 Trajectory Generation and Control for Quadrotors Daniel Warren Mellinger University of Pennsylvania, dwmellin@gmail.com Follow this and additional works at: http://repository.upenn.edu/edissertations Part of the Mechanical Engineering Commons, and the Robotics Commons Recommended Citation Mellinger, Daniel Warren, "Trajectory Generation and Control for Quadrotors" (2012). Publicly Accessible Penn Dissertations. Paper 547. This paper is posted at ScholarlyCommons. http://repository.upenn.edu/edissertations/547 For more information, please contact repository@pobox.upenn.edu.

Trajectory Generation and Control for Quadrotors Abstract This thesis presents contributions to the state-of-the-art in quadrotor control, payload transportation with single and multiple quadrotors, and trajectory generation for single and multiple quadrotors. In Ch. 2 we describe a controller capable of handling large roll and pitch angles that enables a quadrotor to follow trajectories requiring large accelerations and also recover from extreme initial conditions. In Ch. 3 we describe a method that allows teams of quadrotors to work together to carry payloads that they could not carry individually. In Ch. 4 we discuss an online parameter estimation method for quadrotors transporting payloads which enables a quadrotor to use its dynamics in order to learn about the payload it is carrying and also adapt its control law in order to improve tracking performance. In Ch. 5 we present a trajectory generation method that enables quadrotors to fly through narrow gaps at various orientations and perch on inclined surfaces. Chapter 6 discusses a method for generating dynamically optimal trajectories through a series of predefined waypoints and safe corridors and Ch. 7 extends that method to enable heterogeneous quadrotor teams to quickly rearrange formations and avoid a small number of obstacles. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mechanical Engineering & Applied Mechanics First Advisor Vijay Kumar Keywords control, quadrotor, trajectory generation Subject Categories Mechanical Engineering | Robotics This dissertation is available at ScholarlyCommons: http://repository.upenn.edu/edissertations/547

TRAJECTORY GENERATION AND CONTROL FOR QUADROTORS Daniel Mellinger A DISSERTATION in Mechanical Engineering and Applied Mechanics Presented to the Faculties of the University of Pennsylvania in Partial Ful- ﬁllment of the Requirements for the Degree of Doctor of Philosophy 2012 Vijay Kumar, PhD, Supervisor of Dissertation Professor, Department of Mechanical Engineering and Applied Mechanics Jennifer Lukes, PhD, Graduate Group Chairperson Associate Professor, Department of Mechanical Engineering and Applied Mechanics Dissertation Committee: Mark Yim, PhD, Professor, Mechanical Engineering and Applied Mechanics Vijay Kumar, PhD, Professor, Mechanical Engineering and Applied Mechanics Ali Jadbabaie, PhD, Professor, Electrical and Systems Engineering Raffaello D’Andrea, PhD, Professor, Dynamic Systems and Control Bruce Kothmann, PhD, Senior Lecturer, Mechanical Engineering and Applied Mechanics

Acknowledgements During my time at Penn I have learned a lot, worked on interesting research projects, and had fun. I am grateful to everyone who contributed to any of those things. Speciﬁcally, I would ﬁrst like to thank my advisor, Vijay Kumar, for all of his support during my time at Penn. I have enjoyed the projects we worked on together and his willingness to let me shape the direction of those projects. Vijay is a dedicated advisor, a talented engineer, and a good person and I am lucky to have had him as a mentor. I would also like to thank my thesis committee members, Bruce Kothmann, Mark Yim, Ali Jadbabaie and Raff D’Andrea, for taking the time to serve on my committee. Bruce has been especially helpful and has always been excited to talk with me about aerodynamics, controls, or whatever math puzzle he is thinking about at the time. My many friends and colleagues at Penn have made it a fun place to work. I thank them for all the good times inside and outside the lab. I am indebted to my family for the love and support they have given me over the years. My older sister, Corie, gave me a head start by teaching me whatever she learned in school. My parents gave me every opportunity and allowed me to choose my own direction in life. I can’t thank them enough for everything they have done for me. Last but not least, I thank my wife Anna who always believes in me and always makes me smile. ii

Table of Contents Acknowledgements 1 Introduction 1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivation and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 2 Modeling and Control 2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Motor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Small Angle Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Position Control . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Non-dimensional tuning . . . . . . . . . . . . . . . . . . . . . . 2.3 Large Angle Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Model for Control . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Differential Flatness . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Controller Performance . . . . . . . . . . . . . . . . . . . . . . . iii ii 1 2 3 5 5 5 8 9 9 10 13 13 14 15 19 22 22 23

3 Multi-Vehicle Modeling and Control 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Control Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Multi-Vehicle Small Angle Control . . . . . . . . . . . . . . . . . . . . . 3.5.1 Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Hover Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 3D Trajectory Control . . . . . . . . . . . . . . . . . . . . . . . 3.6 Decentralized Control Law . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Online Parameter Estimation 4.1 Introduction . 4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Gripper Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 System Dynamics and Control . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Quadrotor Control . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Estimation of Inertial Parameters . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Method Overview . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Application to Quadrotor Dynamics . . . . . . . . . . . . . . . . 4.6 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Estimation of payload parameters during hover . . . . . . . . . . 4.6.2 Estimation of payload mass in the presence of disturbances . . . . 4.6.3 Estimation of payload inertia . . . . . . . . . . . . . . . . . . . . 4.6.4 Controller Compensation . . . . . . . . . . . . . . . . . . . . . . 27 27 28 30 30 31 32 34 34 34 35 36 37 41 41 43 45 46 47 49 50 50 53 55 55 56 57 57 iv

4.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Trajectory Generation via Sequencing 5.1 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Trajectory Generation via Sequential Composition . . . . . . . . . . . . . 5.2.1 5.2.2 Sequence for Aggressive Trajectories . . . . . . . . . . . . . . . Sequence for Robust Perching . . . . . . . . . . . . . . . . . . . 5.3 Sequence for Robust Landing . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Experiment Design and Implementation Details . . . . . . . . . . . . . . 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Aggressive Trajectories . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Robust Perching . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Robust Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Minimum Snap Trajectory Generation using Piecewise Polynomials 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Trajectory Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Optimal Keyframe Navigation . . . . . . . . . . . . . . . . . . . 6.2.2 Fixed Terminal Time Trajectories . . . . . . . . . . . . . . . . . 6.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 6.3.2 Flying through three static hoops . . . . . . . . . . . . . . . . . . Flying through a thrown hoop . . . . . . . . . . . . . . . . . . . 6.3.3 Catching a bouncing ball . . . . . . . . . . . . . . . . . . . . . . 7 Trajectory Generation with Mixed-Integer Quadratic Programs for Hetero- geneous Quadrotor Teams 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 61 63 64 65 68 69 70 70 70 72 72 73 79 79 81 83 88 89 90 91 92 98 98 7.2 Single Quadrotor Trajectory Generation . . . . . . . . . . . . . . . . . . 101 7.2.1 Basic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 v

7.2.2 Choice of basis functions . . . . . . . . . . . . . . . . . . . . . . 102 7.2.3 Integer Constraints for Obstacle Avoidance . . . . . . . . . . . . 103 7.2.4 Discretization in Time . . . . . . . . . . . . . . . . . . . . . . . 104 7.2.5 Temporal Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 Multiple Quadrotor Trajectory Generation . . . . . . . . . . . . . . . . . 106 7.3.1 Relative Cost Weighting . . . . . . . . . . . . . . . . . . . . . . 106 7.3.2 Inter-Quadrotor Collision Avoidance . . . . . . . . . . . . . . . . 107 7.3.3 Computational Complexity and Numerical Algorithm . . . . . . . 108 7.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.4.1 Three Quadrotors in Plane with Obstacles . . . . . . . . . . . . . 110 7.4.2 Two Heterogeneous Quadrotors through 3-D gap . . . . . . . . . 110 7.4.3 7.4.4 Formation Reconﬁguration with Four Quadrotors . . . . . . . . . 111 Solver Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8 Concluding Remarks 119 8.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 vi

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