國立陽明交通大學「ARRC前瞻火箭研究中心」的HTTP-3AT 混合式火箭於2020年成功完成繫留懸浮飛行測試，此測試展示了先進的火箭導引、導航、導控系統科技。基於此成就，目前正在執行的HTTP-4計畫將更進一步挑戰垂直起飛垂直降落跳躍飛行測試。為滿足HTTP-4計畫的軟著陸導航需求，此研究提出「地表相對導航技術」，此技術將分為四個主要研究方向，分別是1)初始姿態與定位判定，2)感測器開發，3)以無損卡爾曼濾波器進行感測器融合，4)導航系統整合測試。初始姿態與定位判定技術將提供此導航系統精確的初始化量測資料；四種感測器將會被應用於此技術，包括慣性感測器（Inertial Measurement Unit, IMU）、衛星定位系統接收器（GPS Receiver, GPSR）、實時動態技術接收器（Real-time Kinematic, RTK）、高度計。這些感測器的量測資料可以由無損卡爾曼濾波器進行感測器融合；最終的導航系統透過自製建立的二維移動平台進行整合測試檢驗。測試結果顯示本論文所提出的「地表相對導航技術」符合設計的預期成果，其中，初始姿態判定的方位角測定不確定性在0.3度以下。使用HTTP-3A S2飛行資料驗證，經由起飛前姿態校正技術能減少16% 的慣性導航定位誤差。本研究建立的二維移動平台提供了寶貴的真值資料供導航系統的驗證使用，地表相對導航技術所使用的RTK科技，大幅提升位置測定的精確度，在良好的測試條件下，可以達到10公分等級的位置精度及1公分每秒的速度精度。

The successful tethered hovering flight of the HTTP-3AT hybrid rocket developed by Advanced Rocket Research Center (ARRC) in 2020 demonstrated the preliminary launch vehicle guidance, navigation, and control (GNC) technologies. The on-going HTTP-4 project takes on a bigger challenge by aiming to perform a vertical take-off vertical landing (VTVL) hopping flight test. The Ground Level Relative Navigation (GLRN) is proposed in this thesis to fulfil the requirements of achieving the abovementioned challenge. To softly land a rocket, the development of GLRN focused on four major areas: 1) initial alignment, 2) sensor development, 3) sensor fusion with square-root unscented Kalman filter (SR-UKF), and 4) system implementation. Initial alignment techniques were implemented to provide accurate initial attitude and position measurements. Four types of sensors were developed for GLRN, including IMU, GPSR, RTK module, and altimeter. The measurements of these sensors were fused with SR-UKF approach. Finally, the GLRN was implemented and investigated with a series of tests using an in-house developed XY table. The test results showed that the proposed GLRN functioned as designed. The initial alignment process using gyro-compassing technique achieved a heading estimation of uncertainty around 0.3 deg. The developed pre-flight alignment technique reduced 16% of INS position error when tested using HTTP-3A S2 flight data. The XY table built in this work was invaluable as it provided crucial ground truth data for examining and studying the validity of GLRN. In addition, utilizing RTK technology greatly improved the estimation of position. The overall performance showed that the best position and velocity accuracy were < 10 cm and < 1cm/s, respectively.

Abstract iv
Acknowledgements vii
Table of Contents viii
List of Figures xi
List of Tables xiii
List of Acronyms, Symbols, and Nomenclature xiv
1. Conventions xiv
2. Acronyms xv
3. Symbols xvi
Chapter 1 Motivation and Introduction 1
1.1 Motivation 1
1.2 Introduction 2
1.2.1 HTTP-3A Project 2
1.2.2 HTTP-4 Project 5
1.3 Literature Review 6
1.3.1 Vertical Take-off Vertical Landing Rocket Technology 6
1.3.2 Alignment 8
1.3.3 Sensor Fusion and Unscented Kalman Filter 10
1.4 Research Objectives and Thesis Outline 12
1.4.1 Research Objectives 13
1.4.2 Thesis Outline 13
Chapter 2 : System Overview 15
2.1 Design of Ground Level Relative Navigation for Rocket Soft Landing 15
2.2 Navigation System Procedures of Ground Level Relative Navigation 17
2.3 Onboard Navigation Hardware of HTTP-4 Rocket 20
2.3.1 Inertial Measurement Unit 21
2.3.2 GPS Receiver 23
2.3.3 RTK Module 24
2.3.4 Altimeter 25
Chapter 3 Research Methods 27
3.1 Description of Earth Properties, Reference Frames and Transformations 27
3.1.1 Earth Properties 27
3.1.2 Reference Frames 28
3.1.3 Transformation between Frames 31
3.2 Methods for Inertial Measurement Unit Error Compensation 34
3.2.1 Error Modeling 34
3.2.2 Calibration Methodology 35
3.2.3 Error Compensation and Filtering 37
3.3 Square-root Unscented Kalman Filter for Ground Level Relative Navigation 37
3.3.1 Inertial Navigation Equations for Prediction Stage 39
3.3.2 Sensor Dynamic Model Equations for Update Stage 41
3.3.3 Detailed Algorithm of Square-root Unscented Kalman Filter 44
3.4 Methods for Initial Alignment 47
3.4.1 Civil Engineering GPS Receiver 48
3.4.2 Gyro-compassing Technique 48
3.4.3 Pre-flight Calibration and Alignment 50
Chapter 4 Implementation Results and Discussion 52
4.1 Navigation Processor Development 52
4.2 Experiments for IMU Calibration 53
4.2.1 Accelerometer Bias 54
4.2.2 Effectiveness Evaluation of IMU Calibration 55
4.2.3 Validation of Calibration Results 57
4.3 Initial Alignment Results 59
4.3.1 Attitude Alignment with Fiber Optic Gyroscope 60
4.3.2 Position Alignment with Civil Engineering GPS Receiver 62
4.4 Off-line Implementation with Real-world HTTP-3A S2 Flight Data 63
4.4.1 Pure Inertial Navigation 63
4.4.2 Pre-flight Alignment 64
4.4.3 IMU/RTK Fusion with Square-root Unscented Kalman Filter 65
4.5 System Tests with an XY Table 67
4.5.1 Overview of XY Table 67
4.5.2 Analysis of Sensors 73
4.5.3 Fusion of Single Sensor with Square-root Unscented Kalman Filter 79
4.5.4 Fusion of Multiple Sensors with Square-root Unscented Kalman Filter 83
Chapter 5 Conclusions and Recommendations 87
5.1 Conclusions 87
5.2 Recommendations for Future Study 88
Appendix A Weighted Mean Quaternion Algorithm 89
References 90
List of Publications 93

[1] S.-S. Wei, M.-C. Lee, J.-W. Huang, Y. Lu, C.-H. Kang, S.-T. Kao, S.-J. Lu, C.-H. Huang, J.-J. Zhan, Z.-R. Chen, A. Wang, T. H. Chou, A. Lai, Y. Lee, M.-T. Ho, H.-P. Lin and J.-S. Wu, "Demonstration of tethered hovering flight of HTTP-3AT hybrid rocket," Acta Astronautica, vol. 191, pp. 279-292, 2021.
[2] A. M. Dwyer-Cianciolo, C. D. Karlgaard, D. Woffinden, R. A. Lugo, J. Tynis, R. R. Sostaric, S. Striepe, R. Powell and J. M. Carson, "Defining Navigation Requirements for Future Missions," in AIAA Scitech 2019 Forum, San Diego, California, 2019.
[3] J. M. Carson, E. Robertson, N. Trawny and F. Amzajerdian, "Flight Testing ALHAT Precision Landing Technologies Integrated Onboard the Morpheus Rocket Vehicle," in AIAA SPACE 2015 Conference and Exposition, Pasadena, California, 2015.
[4] J. M. Carson, C. Seubert, F. Amzajerdian, C. Bergh, A. Kourchians, C. Restrepo, C. Y. Villalpando, T. O'Neal, E. A. Robertson, D. F. Pierrottet, G. D. Hines and R. Garcia, "COBALT: Development of a Platform to Flight Test Lander GN&C Technologies on Suborbital Rockets," in AIAA Guidance, Navigation, and Control Conference, Grapevine, Texas, 2017.
[5] M. Dumke, M. Sagliano, P. Saranrittichai, G. Trigo and S. Theil, "EAGLE - Environment for Autonomous GNC Landing Experiments," in 10th International ESA Conference on Guidance, Navigation and Control Systems, Salzburg, Austria, 2017.
[6] K. R. Britting, Inertial Navigation Systems Analysis, Wiley-Interscience, 1971.
[7] K. Gade, "The Seven Ways to Find Heading," The Journal of Navigation, vol. 69, no. 5, pp. 955-970, 2016.
[8] P. Aggarwal, Z. Syed, X. Niu and N. El-Sheimy, "A Standard Testing and Calibration Procedure for Low Cost MEMS Inertial Sensors and Units," Journal of Navigation, vol. 61, no. 2, pp. 323-336, 2008.
[9] E.-H. Shin, "Accuracy improvement of low cost INS/GPS for Land Applications," Master's Thesis, The University of Calgary, 2001.
[10] S. J. Julier and J. K. Uhlmann, "New extension of the Kalman filter to nonlinear systems," in Signal Processing, Sensor Fusion, and Target Recognition VI, Orlando, FL, 1997.
[11] S. Thrun, W. Burgard and D. Fox, Probabilistic Robotics, MIT Press, 2005.
[12] E.-H. Shin and N. El-Sheimy, "An unscented Kalman filter for in-motion alignment of low-cost IMUs," in Position Location and Navigation Symposium (IEEE Cat. No.04CH37556), 2004.
[13] J. L. Crassidis, "Sigma-point Kalman filtering for integrated GPS and inertial navigation," IEEE Transactions on Aerospace and Electronic Systems, vol. 42, no. 2, pp. 750-756, 2006.
[14] R. v. d. Merwe and E. A. Wan, "The Square-root Unscented Kalman Filter for State and Parameter-estimation," in 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, USA, 2001.
[15] E. Kraft, "A Quaternion-based Unscented Kalman Filter for Orientation Tracking," in Sixth International Conference of Information Fusion, Cairns, Australia, 2003.
[16] J. M. Carson, R. R. Sostaric, S. M. Pedrotty, J. N. Estes, F. Amzajerdian, A. D. Cianciolo, J. B. Blair, C. I. Restrepo, D. K. Rutishauser, T. Tse and G. D. Hines, "PL&HA and SPLICE Overview," NASA, 2019.
[17] "ADIS16497: Tactical Grade, Six Degrees of Freedom Inertial Sensor Data Sheet (Rev. D)," Analog Devices, Inc, 2020.
[18] "Global Positioning System History," NASA, 05 May 2015. [Online]. Available: https://www.nasa.gov/directorates/heo/scan/communications/policy/GPS_History.html. [Accessed Oct. 2022].
[19] W. P. Birmingham, B. L. Miller and W. L. Stein, "Experimental Results of Using the GPS for Landsat 4 Onboard Navigation," Navigation, vol. 30, no. 3, pp. 244-251, 1983.
[20] Y.-F. Tsai, W.-H. Yeh, J.-C. Juang, D.-S. Yang and C.-T. Lin, "From GPS Receiver to GNSS Reflectometry Payload Development for the Triton Satellite Mission," Remote Sensing, vol. 13, no. 5, p. 999, 2021.
[21] H.-Y. Chang, W.-L. Chiang and K.-L. Wu, "Space-borne GNSS Receiver for Evaluation of the LEO Navigation Based on Real-time Platform," Advances in Aerospace Guidance, Navigation and Control, pp. 529-550, 2017.
[22] M. Markgraf, "Phoenix Spaceborne GPS Receiver Data Sheet," DLR, 2007.
[23] "ZED-F9P-04B u-blox F9 high precision GNSS module Data Sheet," u-blox, 2022.
[24] J. M. Carson, N. Trawny, E. Robertson, V. E. Roback, D. Pierrottet, J. Devolites, J. Hart and J. N. Estes, "Preparation and Integration of ALHAT Precision Landing Technology for Morpheus Flight Testing," in AIAA SPACE 2014 Conference and Exposition, San Diego, CA, 2014.
[25] M. Hannan, D. Rickman, G. Chavers, J. Adam, C. Becker, J. Eliser, D. Gunter, L. Kennedy and P. O'Leary, "Flight Testing of Guidance, Navigation and Control Systems on the Mighty Eagle Robotic Lander Testbed," in AIAA Guidance, Navigation, and Control Conference, Kissimmee, Florida, 2015.
[26] "FALCON USER'S GUIDE," Space Exploration Technologies Corp., 2021.
[27] "Data Sheet optoNCDT ILR 2250-100," Micro-Epsilon Measurement Technology.
[28] D. Titterton and J. Weston, Strapdown Inertial Navigation Technology, IET, 2004.
[29] P. G. Savage, "Strapdown Inertial Navigation Integration Algorithm Design Part 1: Attitude Algorithms," Journal of Guidance, Control, and Dynamics, vol. 21, no. 1, pp. 19-28, 1998.
[30] P. G. Savage, "Strapdown Inertial Navigation Integration Algorithm Design Part 2: Velocity and Position Algorithms," Journal of Guidance, Control, and Dynamics, vol. 21, no. 2, pp. 208-221, 1998.
[31] P. G. Savage, Strapdown analytics, Maple Plain, Minnesota: Strapdown Associates, Inc., 2000.
[32] J. E. Bortz, "A New Mathematical Formulation for Strapdown Inertial Navigation," IEEE Transactions on Aerospace and Electronic Systems, Vols. AES-7, no. 1, pp. 61-66, 1971.
[33] S. J. Julier, "The Scaled Unscented Transformation," in 2002 American Control Conference, Anchorage, AK, USA, 2002.
[34] W. H. Press and S. A. Teukolsky, Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press, 2007.
[35] "e-GNSS即時動態定位系統," 內政部國土測繪中心, [Online]. Available: https://egnss.nlsc.gov.tw/. [Accessed Oct. 2022].
[36] M.-S. Wang, J.-L. Liu, C.-C. Liu and F.-D. Hsiao, "An Analysis on Virtual-Base-Station Real Time Kinematic GPS," Land Survey Bureau of Ministry of the Interior, Taichung, 2006.
[37] J. A. Farrell, The Global Positioning System & Inertial Navigation, McGraw-Hill Professional, 1998.
[38] "Inertial Measurement Unit IMU500," OPTOLINK, [Online]. Available: http://www.optolink.ru/en/products/inertial_measurement_units/imu500. [Accessed Oct 2022].
[39] X. Pennec, Computing the mean of geometric features application to the mean rotation, Doctoral dissertation, INRIA.
[40] E.-H. Shin, Estimation techniques for low-cost inertial navigation, Doctoral dissertation, University of Calgary, 2005.