基于对偶四元数的航天器姿轨一体化动力学建模与控制
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摘要
随着航天技术的发展,新的航天任务如微小卫星编队飞行、航天器交汇对接、航天器空间逼近等成为可能,但其中存在的姿轨耦合(控制输入和执行机构等的耦合)问题,对航天器动力学和控制提出了更高的要求。针对这一问题,本文研究了基于对偶四元数的航天器姿态轨道一体化运动学与动力学建模与控制方法。论文的主要内容如下:
第一,研究了对偶四元数描述运动时的特点及运动学建模方法。针对在轨航天器,利用对偶数一体化地描述了其一般性的空间运动,即重新导出了航天器姿态和轨道一体化运动学模型。包括航天器的运动描述、对偶四元数描述方式和传统描述方式的相互转换关系,及前者较后者的不同与优势。
第二,研究了对偶四元数应用于航天器姿态轨道一体化动力学建模方法。针对航天器动力学特性中存在的姿态轨道耦合问题,首先给出了其在轨运动的各运动参数、力和力矩的对偶四元数描述;然后在此基础上,利用牛顿--欧拉法建立了航天器一般性空间运动与外力、外力矩的关系模型,即单航天器的姿轨一体化动力学模型;最后在单体模型的基础上,建立了两航天器的相对动力学模型,并对模型作必要的分析。
第三,基于上述研究,设计了李群意义下,基于对数反馈的广义控制器,选取适当的李亚普诺夫函数,分析了实数部分和对偶部分所表征系统的稳定性,进而分析了整个系统的稳定性。
第四,被控对象不变,设计了基于对偶四元数的滑模变结构控制器,选取了合适的李亚普诺夫函数对系统的稳定性进行分析;并在考虑模型不确定性和外存干扰的情况下验证了该控制律;此外,对比了两种控制算法在此控制任务中的优点与不足。
最后,以对偶数为变量,设计并建立数学仿真系统,对本文得到的研究成果进行详细、全面的数学仿真,验证所建模型和所设计控制器的有效性。此外与传统的建模和控制工具相比较,总结了对偶四元数在航天器动力学建模和控制领域应用中的优势及前景。
With the development of space technology, the new space missions such as micro-satellite formation flying, spacecraft docking intersection, spacecraft space approach, in which existing attitude and orbit coupling (coupling of control input and implementing agencies) problem, it makes a more stringent requirement in the spacecraft dynamics and control. To solve the problem, in this paper one made use of dual quaternions to study the integrated kinematics and dynamics modeling and control methods for spacecraft attitude and orbit motion. The work of this paper is divided into the following some aspects.
Firstly, the characteristics of dual quaternions in describing movement and the kinematics modeling methods was studied. Considering an on-orbit spacecraft, its general space movement was described in an integrated manner utilizing dual quaternions, that is the kinematics model of spacecrafs was formulated. Including the description of spacecraft motion, the advantages of dual quaternion compared to the traditional tool and the transformaton between them in description method.
Secondly, the integrated dynamics modeling methods for spacecraft attitude and orbit motion was studied. Considering the coupling problem in spacecraft dynamic characteristics, firstly the dual quaternions representation of motion parameters, a variety of forces and moments was given; and then the model of relationship between the spacecraft general spatial motion and the external forces and torques using Newton-Euler approach was formulated, namely the integrated dynamics model; at last, the relative dynamic model for two spacecrafts was established and the necessary analysis about the model was done.
Thirdly, a generalized PDcontrol law based on logarithmic feedback on Lie--group was designed; then the stability of the subsystem presented by dual and real part respectively was proved by selecting their appropriate Lyapunov functions, furthermore the stability of the whole system was verified.
Fourthly, a variable-structure control law based on dual quaternions, then the stability of the whole system was analyzed by selecting the appropriate Lyapunov functions; furthermore considering the model uncertainties and disturbances, the control law was verified; in addition the two control laws were compared.
Finally, taking the dual numbers as the variables, based on MATLAB software, mathematical simulation system was designed and built to vetify the research results detailedly and comprehensively, then compared to the traditional modeling and control tools, the advantages and prospects in the field of spacecraft dynamics modeling and control based on dual quaternion was analyzed.
第一,研究了对偶四元数描述运动时的特点及运动学建模方法。针对在轨航天器,利用对偶数一体化地描述了其一般性的空间运动,即重新导出了航天器姿态和轨道一体化运动学模型。包括航天器的运动描述、对偶四元数描述方式和传统描述方式的相互转换关系,及前者较后者的不同与优势。
第二,研究了对偶四元数应用于航天器姿态轨道一体化动力学建模方法。针对航天器动力学特性中存在的姿态轨道耦合问题,首先给出了其在轨运动的各运动参数、力和力矩的对偶四元数描述;然后在此基础上,利用牛顿--欧拉法建立了航天器一般性空间运动与外力、外力矩的关系模型,即单航天器的姿轨一体化动力学模型;最后在单体模型的基础上,建立了两航天器的相对动力学模型,并对模型作必要的分析。
第三,基于上述研究,设计了李群意义下,基于对数反馈的广义控制器,选取适当的李亚普诺夫函数,分析了实数部分和对偶部分所表征系统的稳定性,进而分析了整个系统的稳定性。
第四,被控对象不变,设计了基于对偶四元数的滑模变结构控制器,选取了合适的李亚普诺夫函数对系统的稳定性进行分析;并在考虑模型不确定性和外存干扰的情况下验证了该控制律;此外,对比了两种控制算法在此控制任务中的优点与不足。
最后,以对偶数为变量,设计并建立数学仿真系统,对本文得到的研究成果进行详细、全面的数学仿真,验证所建模型和所设计控制器的有效性。此外与传统的建模和控制工具相比较,总结了对偶四元数在航天器动力学建模和控制领域应用中的优势及前景。
With the development of space technology, the new space missions such as micro-satellite formation flying, spacecraft docking intersection, spacecraft space approach, in which existing attitude and orbit coupling (coupling of control input and implementing agencies) problem, it makes a more stringent requirement in the spacecraft dynamics and control. To solve the problem, in this paper one made use of dual quaternions to study the integrated kinematics and dynamics modeling and control methods for spacecraft attitude and orbit motion. The work of this paper is divided into the following some aspects.
Firstly, the characteristics of dual quaternions in describing movement and the kinematics modeling methods was studied. Considering an on-orbit spacecraft, its general space movement was described in an integrated manner utilizing dual quaternions, that is the kinematics model of spacecrafs was formulated. Including the description of spacecraft motion, the advantages of dual quaternion compared to the traditional tool and the transformaton between them in description method.
Secondly, the integrated dynamics modeling methods for spacecraft attitude and orbit motion was studied. Considering the coupling problem in spacecraft dynamic characteristics, firstly the dual quaternions representation of motion parameters, a variety of forces and moments was given; and then the model of relationship between the spacecraft general spatial motion and the external forces and torques using Newton-Euler approach was formulated, namely the integrated dynamics model; at last, the relative dynamic model for two spacecrafts was established and the necessary analysis about the model was done.
Thirdly, a generalized PDcontrol law based on logarithmic feedback on Lie--group was designed; then the stability of the subsystem presented by dual and real part respectively was proved by selecting their appropriate Lyapunov functions, furthermore the stability of the whole system was verified.
Fourthly, a variable-structure control law based on dual quaternions, then the stability of the whole system was analyzed by selecting the appropriate Lyapunov functions; furthermore considering the model uncertainties and disturbances, the control law was verified; in addition the two control laws were compared.
Finally, taking the dual numbers as the variables, based on MATLAB software, mathematical simulation system was designed and built to vetify the research results detailedly and comprehensively, then compared to the traditional modeling and control tools, the advantages and prospects in the field of spacecraft dynamics modeling and control based on dual quaternion was analyzed.
引文
1 A. T. Yang. Application of Quaternion Algebra and Dual Numbers to The Analysis of Spatial Mechanisms, PhD. Thesis, Columbia University, 1964, 36~72
2 K. Daniilidis, Hand-Eye Calibration Using Dual Quaternions. international Journal of Robotics Research, 1999(18): 286~298
3 V. N. Branets, I.P.Shmyglevsky. Introductiorn to The Theory of Strapdown inertial Navigation System. Nauka, Moscow, 1992, 320
4 J. Rooney. Acomputational Analysis of Representation of General Spatial Screw Transformations in Robotics. Enviroment and Planning B. 1978, 5: 45~88
5 George V Paul, Katsushi Ikeuchi. Representing The Motion of Objects in Contact Using Dual Quaternions and Its Applications. Carnegie Mellon University. 1997: 23~45
6 Ettore Pennestrì, Pier Paolo Valentini. Dual Quaternions As A tool for Rigid Body Motion Analysis: A Tutorial With An Application to Biomechanics. Multibody Dynamics. Eccomas Thematic Conference, Porland, 2009:1~14
7武元新.对偶四元数导航算法与非线性高斯滤波研究.国防科学技术大学博士学位论文. 2005: 40~43
8韩大鹏.基于四元数代数和李群框架的任务空间控制方法研究.国防科学技术大学博士学位论文. 2008: 16~50
9 Dapeng Han, Qing Wei, Zexiang Li, Weimeng Sun. Control of Oriented Mechanical Systems: A Method Based On Dual Quaternion. 17th IFAC World Congress (IFAC'08) Seoul, Korea, 2008, 6(11): 3836~3841
10 Dapeng Han, Qing Wei, Zexiang Li. Kinematic Control of Free Rigid Bodies Using Dual Quaternions. international Journal of Automation and Computing. 2008, 05(3): 319~324
11 Dapeng Han, Qing Wei and Zexiang Li. A Dual-Quaternion Method for Control of Spatial Rigid Body. International Conference of Networking, Sensing and Control. Sanya, 2008, 1~6
12刘延柱,洪嘉振,杨海兴.多刚体系统动力学.北京:高等教育出版社, 1989, 15~40
13 G. R. Pennock, B. A. Oncu. Application of Screw Theory to Rigid Body Dynamics. ASME Journal of Dynamic Systems, Measurement, and Control.1992, 114(2): 262~269
14 V. Brodsky, M. Shoham. The Dual inertia Operator and Its Application to Robot Dynamics. Journal of Mechanical Design. 1994, 116: 1189~1195
15 V. Brodsky, M. Shoham. Dual Numbers Representation of Rigid Body Dynamics. Mechanism and Machine Theory. 1999, 34: 693~718
16 V. Brodsky, M. Shoham. Derivation of Dual Forces in Robot Manipulators. Mechanism and Machine Theory. 1998, 33: 1241–1248
17 M. E. Lisano. A Practical Six-Degree-of-Freedom Solar Sail Dynamics Model for Optimizing Solar Sail Trajectories With Torque Constraints. AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island, USA, 2004, 1~11
18 S. Gaulocher. Modeling The Coupled Translational and Rotational Relative Dynamics for formation Flying Control. AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, USA, 2005, 1~6
19 J. L Junkins, J. D. Turner. On The Analogy Between Orbital Dynamics and Rigid Body Dynamics. Journal of The Astronautical Sciences. 1979, XXVII (4): 345~358
20 Y. N. Chelnokov. The Use of Quaternions in The Optimal Control Problems of Motion of The Center of Mass of A Spacecraft in A Newtonian Gravitational Field: I. Cosmic Research. 2001, 39(5): 470~484
21 J. Waldvogel. Quaternions and The Perturbed Kepler Problem. Celestial Mechanics and Dynamical Astronomy. 2006, 95(1~4): 201~212
22 Sinclair, J. Hurtado, J. L. Junkins. Application of The Cayley form to General Spacecraft Motion. Journal of Guidance, Control, and Dynamics. 2006, 29(2): 368~373
23 S. R. Ploen, F. Y. Hadaegh, D. P. Scharf. Rigid Body Equations of Motion for Modeling and Control of Spacecraft Formations–Part 1: Absolute Equations of Motion. Proceeding of The 2004 American Control Conference, Boston, USA, 2004, 3646~3653
24彭冬亮,荆武兴,徐世杰.停靠阶段轨道姿态耦合动力学与控制研究.飞行力学. 2002, 20(1): 33~37
25邹晖,陈万春,殷兴良.几何代数及其在飞行力学中的应用.飞行力学. 2004, 22(4): 60~64 - 6 7 -
26杨佳,朱战霞,张艳召.绕飞监测小卫星姿轨联合自适应控制研究.飞行力学. 2008, 26(5): 59~62
27方茹,曹喜滨.几何代数及其在摄动Kepler问题中的应用.哈尔滨工业大学学报. 2008, 40(2): 282~286
28 S. E. Lennox. Coupled Orbital and Attitude Control Simulations for Spacecraft formation Flying. The 2004 AIAA Region I-MA Student Conference, Blacksburg, USA, 2004, 27~41
29 D. T. Stansbery, J. R. Cloutier. Position and Attitude Control of A Spacecraft Using The State-Dependent Riccati Equation Technique. Proceedings of The American Control Conference, Chicago, USA, 2000, 1867~1871
30 H. Z. Pan, H. Wong, V. Kapila. Output Feedback Control for Spacecraft With Coupled Translation and Attitude Dynamics. 43rd IEEE Conference On Decision and Control, Atlantis, Paradise Island, 2004, 5891~5897
31 B. J. Naasz, M. M. Berry, H. Y. Kim, Et Al. Integrated Orbit and Attitude Control for A Nanosatellite With Power Constraints. AAS/AIAA Space Flight Mechanics Conference, Ponce, Puerto Rico, 2003, 114(1): 9~25
32 H. Wong, H. Z. Pan, V. Kapila. Output Feedback Control for Spacecraft formation Flying With Coupled Translation and Attitude Dynamics. Proceedings of The American Control Conference, Portland, USA, 2005, 2419~2426
33 M. Xin, S.N.Balakrishnan, D.T.Stansbery. Spacecraft Position and Attitude Control withθ-D Technique. 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada USA, 2004, 5~8
34 P. Sengupta, S. R. Vadali. Satellite Orbit Transfer and formation Reconfiguration Via An Attitude Control Analogy. Journal of Guidance, Control, and Dynamics. 2005, 28(6): 1200~1209
35吕振铎.轨道机动期间的姿态控制.中国空间科学技术. 1994, 10(5): 25~30
36张艳召,袁建平,罗建军.小卫星临近作业轨道和姿态联合控制.中国空间科学技术. 2008, 10(5): 13~19
37孙兆伟,耿云海,何平.小卫星大角度姿态机动控制研究及半实物仿真验证.航天控制, 2000, 18(2): 28~33
38 M. J. Kim and M. S. Kim. A Compact Differential formula for The FirstDerivative of A Unit Quaternion Curve. Journal of Visualization and Computer Animation, 1996, 7(1): 43~57
39 Davide andreis, Enrico S. Canuto. Orbit Dynamics and Kinematics With Full Quaternions. Proceeding of The 2004 American Control Conference Boston. Massachusetts, 2004, 3660~3665
40 W. Clifford, Preliminary Sketch of Bi-Quaternions, Pnoc. Loridori Math. Soc. 1873(4): 381~395
41 E. Study, Von Den Bewegungen Und Umlegungen, Mathematische. Annalen, 1891, 39: 441~566
42 V I. Utkin. Variable Structure Systems with Sliding Modes. IEEE Trans Autom Control, 1977, 22(2): 212-222
43 Hung J. Y, Gao, W, Hung. J. C. Variable Structure Control: A Survey, IEEE Transactions on Industrial Electronics, 1993, 40(1): 2-22
44刘芳华,吴洪涛.基于旋量理论的空间机器人动力学建模研究.江苏科技大学学报(自然科学版). 2008, 22(2): 52~55
45高为炳.变结构控制的理论及设计方法.北京:科学出版社, 1996: 22~27
46 QP. Ha, Nguyen QH, Rye DC. Fuzzy Sliding-Mode Controllers With Applications. IEEE Transactions On industrial Electronics. 2001, 48(1): 38~41
47胡庆雷.挠性航天器姿态机动的主动振动控制.哈尔滨工业大学博士学位论文. 2006: 60~65
2 K. Daniilidis, Hand-Eye Calibration Using Dual Quaternions. international Journal of Robotics Research, 1999(18): 286~298
3 V. N. Branets, I.P.Shmyglevsky. Introductiorn to The Theory of Strapdown inertial Navigation System. Nauka, Moscow, 1992, 320
4 J. Rooney. Acomputational Analysis of Representation of General Spatial Screw Transformations in Robotics. Enviroment and Planning B. 1978, 5: 45~88
5 George V Paul, Katsushi Ikeuchi. Representing The Motion of Objects in Contact Using Dual Quaternions and Its Applications. Carnegie Mellon University. 1997: 23~45
6 Ettore Pennestrì, Pier Paolo Valentini. Dual Quaternions As A tool for Rigid Body Motion Analysis: A Tutorial With An Application to Biomechanics. Multibody Dynamics. Eccomas Thematic Conference, Porland, 2009:1~14
7武元新.对偶四元数导航算法与非线性高斯滤波研究.国防科学技术大学博士学位论文. 2005: 40~43
8韩大鹏.基于四元数代数和李群框架的任务空间控制方法研究.国防科学技术大学博士学位论文. 2008: 16~50
9 Dapeng Han, Qing Wei, Zexiang Li, Weimeng Sun. Control of Oriented Mechanical Systems: A Method Based On Dual Quaternion. 17th IFAC World Congress (IFAC'08) Seoul, Korea, 2008, 6(11): 3836~3841
10 Dapeng Han, Qing Wei, Zexiang Li. Kinematic Control of Free Rigid Bodies Using Dual Quaternions. international Journal of Automation and Computing. 2008, 05(3): 319~324
11 Dapeng Han, Qing Wei and Zexiang Li. A Dual-Quaternion Method for Control of Spatial Rigid Body. International Conference of Networking, Sensing and Control. Sanya, 2008, 1~6
12刘延柱,洪嘉振,杨海兴.多刚体系统动力学.北京:高等教育出版社, 1989, 15~40
13 G. R. Pennock, B. A. Oncu. Application of Screw Theory to Rigid Body Dynamics. ASME Journal of Dynamic Systems, Measurement, and Control.1992, 114(2): 262~269
14 V. Brodsky, M. Shoham. The Dual inertia Operator and Its Application to Robot Dynamics. Journal of Mechanical Design. 1994, 116: 1189~1195
15 V. Brodsky, M. Shoham. Dual Numbers Representation of Rigid Body Dynamics. Mechanism and Machine Theory. 1999, 34: 693~718
16 V. Brodsky, M. Shoham. Derivation of Dual Forces in Robot Manipulators. Mechanism and Machine Theory. 1998, 33: 1241–1248
17 M. E. Lisano. A Practical Six-Degree-of-Freedom Solar Sail Dynamics Model for Optimizing Solar Sail Trajectories With Torque Constraints. AIAA Guidance, Navigation, and Control Conference and Exhibit, Providence, Rhode Island, USA, 2004, 1~11
18 S. Gaulocher. Modeling The Coupled Translational and Rotational Relative Dynamics for formation Flying Control. AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, USA, 2005, 1~6
19 J. L Junkins, J. D. Turner. On The Analogy Between Orbital Dynamics and Rigid Body Dynamics. Journal of The Astronautical Sciences. 1979, XXVII (4): 345~358
20 Y. N. Chelnokov. The Use of Quaternions in The Optimal Control Problems of Motion of The Center of Mass of A Spacecraft in A Newtonian Gravitational Field: I. Cosmic Research. 2001, 39(5): 470~484
21 J. Waldvogel. Quaternions and The Perturbed Kepler Problem. Celestial Mechanics and Dynamical Astronomy. 2006, 95(1~4): 201~212
22 Sinclair, J. Hurtado, J. L. Junkins. Application of The Cayley form to General Spacecraft Motion. Journal of Guidance, Control, and Dynamics. 2006, 29(2): 368~373
23 S. R. Ploen, F. Y. Hadaegh, D. P. Scharf. Rigid Body Equations of Motion for Modeling and Control of Spacecraft Formations–Part 1: Absolute Equations of Motion. Proceeding of The 2004 American Control Conference, Boston, USA, 2004, 3646~3653
24彭冬亮,荆武兴,徐世杰.停靠阶段轨道姿态耦合动力学与控制研究.飞行力学. 2002, 20(1): 33~37
25邹晖,陈万春,殷兴良.几何代数及其在飞行力学中的应用.飞行力学. 2004, 22(4): 60~64 - 6 7 -
26杨佳,朱战霞,张艳召.绕飞监测小卫星姿轨联合自适应控制研究.飞行力学. 2008, 26(5): 59~62
27方茹,曹喜滨.几何代数及其在摄动Kepler问题中的应用.哈尔滨工业大学学报. 2008, 40(2): 282~286
28 S. E. Lennox. Coupled Orbital and Attitude Control Simulations for Spacecraft formation Flying. The 2004 AIAA Region I-MA Student Conference, Blacksburg, USA, 2004, 27~41
29 D. T. Stansbery, J. R. Cloutier. Position and Attitude Control of A Spacecraft Using The State-Dependent Riccati Equation Technique. Proceedings of The American Control Conference, Chicago, USA, 2000, 1867~1871
30 H. Z. Pan, H. Wong, V. Kapila. Output Feedback Control for Spacecraft With Coupled Translation and Attitude Dynamics. 43rd IEEE Conference On Decision and Control, Atlantis, Paradise Island, 2004, 5891~5897
31 B. J. Naasz, M. M. Berry, H. Y. Kim, Et Al. Integrated Orbit and Attitude Control for A Nanosatellite With Power Constraints. AAS/AIAA Space Flight Mechanics Conference, Ponce, Puerto Rico, 2003, 114(1): 9~25
32 H. Wong, H. Z. Pan, V. Kapila. Output Feedback Control for Spacecraft formation Flying With Coupled Translation and Attitude Dynamics. Proceedings of The American Control Conference, Portland, USA, 2005, 2419~2426
33 M. Xin, S.N.Balakrishnan, D.T.Stansbery. Spacecraft Position and Attitude Control withθ-D Technique. 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada USA, 2004, 5~8
34 P. Sengupta, S. R. Vadali. Satellite Orbit Transfer and formation Reconfiguration Via An Attitude Control Analogy. Journal of Guidance, Control, and Dynamics. 2005, 28(6): 1200~1209
35吕振铎.轨道机动期间的姿态控制.中国空间科学技术. 1994, 10(5): 25~30
36张艳召,袁建平,罗建军.小卫星临近作业轨道和姿态联合控制.中国空间科学技术. 2008, 10(5): 13~19
37孙兆伟,耿云海,何平.小卫星大角度姿态机动控制研究及半实物仿真验证.航天控制, 2000, 18(2): 28~33
38 M. J. Kim and M. S. Kim. A Compact Differential formula for The FirstDerivative of A Unit Quaternion Curve. Journal of Visualization and Computer Animation, 1996, 7(1): 43~57
39 Davide andreis, Enrico S. Canuto. Orbit Dynamics and Kinematics With Full Quaternions. Proceeding of The 2004 American Control Conference Boston. Massachusetts, 2004, 3660~3665
40 W. Clifford, Preliminary Sketch of Bi-Quaternions, Pnoc. Loridori Math. Soc. 1873(4): 381~395
41 E. Study, Von Den Bewegungen Und Umlegungen, Mathematische. Annalen, 1891, 39: 441~566
42 V I. Utkin. Variable Structure Systems with Sliding Modes. IEEE Trans Autom Control, 1977, 22(2): 212-222
43 Hung J. Y, Gao, W, Hung. J. C. Variable Structure Control: A Survey, IEEE Transactions on Industrial Electronics, 1993, 40(1): 2-22
44刘芳华,吴洪涛.基于旋量理论的空间机器人动力学建模研究.江苏科技大学学报(自然科学版). 2008, 22(2): 52~55
45高为炳.变结构控制的理论及设计方法.北京:科学出版社, 1996: 22~27
46 QP. Ha, Nguyen QH, Rye DC. Fuzzy Sliding-Mode Controllers With Applications. IEEE Transactions On industrial Electronics. 2001, 48(1): 38~41
47胡庆雷.挠性航天器姿态机动的主动振动控制.哈尔滨工业大学博士学位论文. 2006: 60~65