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Dual Quaternions as a Tool for Modeling, Control, and Estimation for Spacecraft Robotic Servicing Missions

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Abstract

In recent years there has been an increasing interest in spacecraft robotic operations in orbit. In fact, several agencies and organizations around the world are investigating satellite proximity operations as an enabling technology for future space missions such as on-orbit satellite inspection, health monitoring, surveillance, servicing, refueling, and optical interferometry, to name a few. Contrary to more traditional satellite applications, robotic servicing requires addressing both the translational and the rotational motion of the satellite at the same time. One of the biggest challenges for these applications is the need to simultaneously and accurately estimate – and track – both relative position and attitude reference trajectories in order to avoid collisions between the satellites and achieve stringent mission objectives. Motivated by our desire to control spacecraft motion during proximity operations for robotic in-orbit servicing missions which do not depend on the artificial separation of translational and rotational motion, we have recently developed a complete theory to describe the 6-DOF motion of the spacecraft using dual quaternions. Dual quaternions emerge as a powerful tool to model the pose (that is, both attitude and position) of the spacecraft during all phases of the mission under a unified framework. In this paper, we revisit the basic theory behind dual quaternions, the associated Clifford algebras, and compare quaternion-based attitude rigid-body control laws and estimation algorithms, to their dual quaternion-based pose counterparts. We also show that the resulting mathematical structure lends itself to the straightforward incorporation of an adaptive estimation scheme known as concurrent learning, which allows us to also estimate on-the-fly the mass properties of the spacecraft.

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References

  1. Oda, M.: Space robot experiments on NASDA’s ETS-VII satellite - preliminary overview of the experiment results. In: Proceedings 1999 IEEE International Conference on Robotics and Automation, vol. 2, pp 1390–1395, Detroit (1999), https://doi.org/10.1109/ROBOT.1999.772555

  2. Oda, M.: ETS-VII: Achievements, troubles and future. In: Proceedings of the 6th International Symposium on Artificial Intelligence and Robotics & Automation in Space (i-SAIRAS), St.-Hubert, Quebec, Canada, Canadian Space Agency (2001)

  3. Ogilvie, A., Allport, J., Hannah, M., Lymer, J.: Autonomous satellite servicing using the orbital express demonstration manipulator system. In: Proceedings of the Ninth International Symposium on Artificial Intelligence Robotics and Automation in Space, iSAIRAS, pp 26–29, Los Angeles (2008)

  4. Yoshida, K.: Achievements in space robotics. IEEE Robot. Autom. Mag 16, 20–28 (2009). https://doi.org/10.1109/mra.2009.934818

    Article  Google Scholar 

  5. Henshaw, C.G.: NRL Robotics Overview, Online. Future In-Space Operations (FISO) Colloquium (2009)

  6. Roesler, G.: Robotic Servicing of Geosynchronous Satellites (RSGS) Program Overview, Online. Future In-Space Operations (FISO) Colloquium (2016)

  7. Reed, B.B.: NASA Satellite Servicing Evolution, Online. Future In-Space Operations (FISO) Colloquium (2017)

  8. Reed, B.B., Smith, R.C., Naasz, B.J., Pellegrino, J.F., Bacon, C.E.: The restore-L servicing mission. AIAA SPACE Forum, pp. 13–16. Long Beach. https://doi.org/10.2514/6.2016-5478 (2016)

  9. Dimitrov, D.: Dynamics and Control of Space Manipulators During a Satellite Capturing Operation. Tohoku University, PhD thesis (2005)

    Google Scholar 

  10. Xu, S., Wang, H., Zhang, D., Yang, B.: Extended Jacobian based adaptive zero reaction motion control for free-floating space manipulators. In: Proceedings of the 33rd Chinese Control Conference. Nanjing, pp. 28–30. https://doi.org/10.1109/chicc.2014.6896402 (2014)

  11. Dubanchet, V., Saussié, D., Alazard, D., Bérard, C., Peuvédic, C.L.: Modeling and control of a space robot for active debris removal. CEAS Space J. 7(2), 203–218 (2015). https://doi.org/10.1007/s12567-015-0082-4

    Article  Google Scholar 

  12. Dooley, J., McCarthy, J.: Spatial rigid body dynamics using dual quaternion components. In: Proceedings 1991 IEEE International Conference on Robotics and Automation, pp. 90–95. Sacramento (1991)

  13. Brodsky, V., Shoham, M.: Dual numbers representation of rigid body dynamics. Mech. Mach. Theory 34, 693–718 (1999). Erratum for this paper has been published

    Article  MathSciNet  Google Scholar 

  14. Wang, J., Sun, Z: 6–DOF robust adaptive terminal sliding mode control for spacecraft formation flying. Acta Astronaut. 73, 76–87 (2012). https://doi.org/10.1016/j.actaastro.2011.12.005

    Article  Google Scholar 

  15. Wang, J.-Y., Liang, H.-Z., Sun, Z.-W., Wu, S.-N., Zhang, S.-J.: Relative motion coupled control based on dual quaternion. Aerosp. Sci. Technol. 25, 102–113 (2013). https://doi.org/10.1016/j.ast.2011.12.013

    Article  Google Scholar 

  16. Wang, X., Yu, C.: Unit-dual-quaternion-based PID control scheme for rigid-body transformation. In: Proceedings of the 18th IFAC World Congress. Milan (2011)

  17. Wang, J., Liang, H., Sun, Z., Zhang, S., Liu, M.: Finite-time control for spacecraft formation with dual-number-based description. J. Guid. Control Dyn. 35, 950–962 (2012). https://doi.org/10.2514/1.54277

    Article  Google Scholar 

  18. Filipe, N., Tsiotras, P.: Rigid body motion tracking without linear and angular velocity feedback using dual quaternions. In: European Control Conference, pp. 329–334. Zürich (2013)

  19. Filipe, N., Tsiotras, P.: Adaptive model-independent tracking of rigid body position and attitude motion with mass and inertia matrix identification using dual quaternions. In: AIAA Guidance, Navigation, and Control Conference, AIAA 2013-5173, pp. 19–22. Boston. https://doi.org/10.2514/6.2013-5173 (2013)

  20. Filipe, N., Tsiotras, P.: Adaptive position and attitude-tracking controller for satellite proximity operations using dual quaternions. J. Guid. Control Dyn. 38, 566–577 (2014). https://doi.org/10.2514/1.G000054

    Article  Google Scholar 

  21. Filipe, N., Valverde, A., Tsiotras, P.: Pose tracking without linear and angular-velocity feedback using dual quaternions. IEEE Trans. Aerosp. Electron. Syst. 52(1), 411–422 (2016)

    Article  Google Scholar 

  22. Seo, D.: Fast adaptive pose tracking control for satellites via dual quaternion upon non-certainty equivalence principle. Acta Astronaut. 115, 32–39 (2015). https://doi.org/10.1016/j.actaastro.2015.05.013

    Article  Google Scholar 

  23. Lee, U., Mesbahi, M.: Dual quaternions, rigid body mechanics, and powered-descent guidance. In: 51st IEEE Conference on Decision and Control, pp. 3386–3391. Maui (2012)

  24. Lee, U., Mesbahi, M.: Dual quaternion based spacecraft rendezvous with rotational and translational field of view constraints. In: AIAA/AAS Astrodynamics specialist conference, pp. 4–7. San Diego. https://doi.org/10.2514/6.2014-4362 (2014)

  25. Lee, U., Mesbahi, M.: Optimal power descent guidance with 6-DoF line of sight constraints via unit dual quaternions. In: AIAA Guidance, Navigation, and Control Conference, pp. 5–9. https://doi.org/10.2514/6.2015-0319 (2015)

  26. Hamilton, W.: Elements of Quaternions. Longmans, Green, & Company, London (1866)

    Google Scholar 

  27. Filipe, N., Tsiotras, P.: Simultaneous position and attitude control without linear and angular velocity feedback using dual quaternions. In: Proceedings of the 2013 American Control Conference, pp. 4815–4820. Washington, DC (2013)

  28. Filipe, N.: Nonlinear Pose Control and Estimation for Space Proximity Operations An Approach Based on Dual Quaternions. Georgia Institute of Technology, PhD thesis (2014)

    Google Scholar 

  29. Hestenes, D., Ziegler, R.: Projective geometry with Clifford algebra. Acta Applicandae Mathematica 23, 25–63 (Apr 1991). https://doi.org/10.1007/BF00046919

  30. Radavelli, L.A., De Pieri, E.R., Martins, D., Simoni, R.: Points, lines, screws and planes in dual quaternions kinematics, advances in robot kinematics. In: Lenarcic, J., Khatib, O. (eds.) , pp 285–293. Springer (2014), https://doi.org/10.1007/978-3-319-06698-1_30

  31. Filipe, N., Tsiotras, P.: Adaptive position and attitude tracking controller for satellite proximity operations using dual quaternions. In: AAS/AIAA Astrodynamics Specialist Conference, AAS 13-858, pp. 2313–2332. Hilton Head (2013)

  32. Lizarralde, F., Wen, J.T.: Attitude control without angular velocity measurement: A passivity approach. IEEE Trans. Autom. Control 41, 468–472 (1996)

    Article  MathSciNet  Google Scholar 

  33. Wen, J.T.-Y., Kreutz-Delgado, K.: The attitude control problem. IEEE Trans. Autom. Control 36, 1148–1162 (1991)

    Article  MathSciNet  Google Scholar 

  34. Ahmed, J., Coppola, V.T., Bernstein, D.S.: Adaptive asymptotic tracking of spacecraft attitude motion with inertia matrix identification. J. Guid. Control Dyn. 21, 684–691 (1998). https://doi.org/10.2514/2.43.10

    Article  Google Scholar 

  35. Lefferts, E., Markley, F., Shuster, M.: Kalman filtering for spacecraft attitude estimation. J. Guid. Control Dyn. 5, 417–429 (1982)

    Article  Google Scholar 

  36. Filipe, N., Kontitsis, M., Tsiotras, P.: Extended Kalman filter for spacecraft pose estimation using dual quaternions. J Guid Control Dyn 38, 1625–1641 (2015)

    Article  Google Scholar 

  37. Filipe, N., Kontitsis, M., Tsiotras, P.: From Attitude Estimation to Pose Estimation Using Dual Quaternions, ch. 8, pp. 129–143. Taylor & Francis (2016)

  38. Valverde, A., Filipe, N., Kontitsis, M., Tsiotras, P.: Experimental validation of an inertia-free controller and a multiplicative EKF for pose tracking and estimation based on dual quaternions. In: 38th AAS Guidance and Control Conference, No. 2015–017/ Breckenridge (2015)

  39. Malladi, B.P., Butcher, E.A., Sanfelice, R.G.: Robust hybrid global asymptotic stabilization of rigid body dynamics using dual quaternions. In: AIAA Guidance, Navigation, and Control Conference, AIAA SciTech Forum, pp. 8–12. https://doi.org/10.2514/6.2018-0606 (2018)

  40. Yuan, J., Hou, X., Sun, C., Cheng, Y.: Fault-tolerant pose and inertial parameters estimation of an uncooperative spacecraft based on dual vector quaternions. In: Proceedings of the Institution of Mechanical Engineers Part G: Journal of Aerospace Engineering. https://doi.org/10.1177/0954410017751766 (2018)

  41. Deng, Y., Wang, Z., Liu, L.: Unscented Kalman filter for spacecraft pose estimation using Twistors. J. Guid. Control Dyn. 39, 1844–1856 (2014). https://doi.org/10.2514/1.G000368

    Article  Google Scholar 

  42. Chowdhary, G.V.: Concurrent Learning for Convergence in Adaptive Control Without Persistency of Excitation. Georgia Institute of Technology, PhD thesis (2010)

    Book  Google Scholar 

  43. Jun, B.-E., Bernstein, D.S., McClamroch, N.H.: Identification of the inertia matrix of a rotating body based on errors-in-variables models. Int. J. Adapt. Control Signal Process. 24(3), 203–210 (2010). https://doi.org/10.1002/acs.1112

    Article  MathSciNet  MATH  Google Scholar 

  44. Chowdhary, G.V., Johnson, E.N.: Theory and flight-test validation of a concurrent-learning adaptive controller. J. Guid. Control Dyn. 34, 592–607 (2011). https://doi.org/10.2514/1.46866

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute for Robotics and Intelligent Machines, Georgia Institute of Technology, Atlanta, GA, USA

    Panagiotis Tsiotras

  2. School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Panagiotis Tsiotras & Alfredo Valverde

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  1. Panagiotis Tsiotras
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  2. Alfredo Valverde
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Correspondence to Alfredo Valverde.

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Tsiotras, P., Valverde, A. Dual Quaternions as a Tool for Modeling, Control, and Estimation for Spacecraft Robotic Servicing Missions. J Astronaut Sci 67, 595–629 (2020). https://doi.org/10.1007/s40295-019-00181-4

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