# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022131
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## Prescribed-time time-varying sliding mode based integrated translation and rotation control for spacecraft formation flying

 1 Beijing Institute of Tracking and Telecommunications Technology, Beijing, China 2 Shanghai Institute of Satellite Engineering, Shanghai, China 3 Shanghai JiaoTong University, Shanghai, China

* Corresponding author: Xiaowei Shao

Received  October 2021 Revised  February 2022 Early access June 2022

This paper studies the prescribed-time spacecraft formation flying problem. The coupled 6-degrees-of-freedom kinematics and dynamics for spacecraft are modeled on Lie group SE(3), where the gravity-gradient torque, the perturbing forces caused by Earth's oblateness and unknown external disturbances are all taken into account. The relative configuration is expressed by using exponential coordinates of SE(3). In order to regulate the relative motion between the leader spacecraft and follower spacecraft, a novel prescribed-time control law is given based on a prescribed-time time-varying high-gain sliding mode method. Numerical simulations verify the effectiveness of the proposed control scheme.

Citation: Li Chen, Yongyan Sun, Xiaowei Shao, Junli Chen, Dexin Zhang. Prescribed-time time-varying sliding mode based integrated translation and rotation control for spacecraft formation flying. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022131
##### References:
 [1] K. Alfriend, S. R. Vadali, P. Gurfil, J. P. How and L. Breger, Spacecraft Formation Flying: Dynamics, Control and Navigation (Vol. 2), Elsevier, Oxford, 2009. [2] R. W. Beard, L. Lawton and F. Y. Hadaegh, A coordination architecture for spacecraft formation control, IEEE Trans. Contr. Syst. Technol., 9 (2001), 777-790. [3] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751-766.  doi: 10.1137/S0363012997321358. [4] J. D. Biggs and L. Colley, Geometric attitude motion planning for spacecraft with pointing and actuator constraints, J. Guid., Control, Dyn., 39 (2016), 1672-1677. [5] F. Bullo and R. M. Murray, Proportional derivative (PD) control on the Euclidean group, Proceedings of IN European Control Conference, Zurich, Switzerland, 1995, 1091–1097. [6] S. Ding, K. Mei and S. Li, A new second-order sliding mode and its application to nonlinear constrained systems, IEEE Trans. Automat. Contr., 64 (2019), 2545-2552.  doi: 10.1109/TAC.2018.2867163. [7] K. Gong, Y. Liao and Y. Wang, Adaptive fixed-time terminal sliding mode control on SE (3) for coupled spacecraft tracking maneuver, Int. J. Aerosp. Eng., 2020 (2020), 1-15. [8] Y. Hong, J. Wang and D. Cheng, Adaptive finite-time control of nonlinear systems with parametric uncertainty, IEEE Trans. Automat. Contr., 51 (2006), 858-862.  doi: 10.1109/TAC.2006.875006. [9] J. Y. Hung, W. Gao and J. C. Hung, Variable structure control: A survey, IEEE Trans. Ind. Electron., 40 (1993), 2-22. [10] Y. Huang and Y. Jia, Distributed finite-time output feedback synchronisation control for six DOF spacecraft formation subject to input saturation, IET Control. Theory Appl., 12 (2018), 532-542.  doi: 10.1049/iet-cta.2017.0842. [11] H. K. Khalil, Nonlinear Control, 1$^{st}$ edition, USA: Pearson, New York, 2015. [12] Q. Lan, J. Yang, S. Li and H. Sun, Finite-time control for 6DOF spacecraft formation flying systems, J. Aerosp. Eng., 28 (2015), 04014137. [13] D. Lee, Spacecraft coupled tracking maneuver using sliding mode control with input saturation, J. Aerosp. Eng., 28 (2015), 04014136. [14] D. Lee, A. K. Sanyal and E. A. Butcher, Asymptotic tracking control for spacecraft formation flying with decentralized collision avoidance, J. Guid., Control, Dyn., 38 (2015), 587-600. [15] D. Lee and G. Vukovich, Adaptive sliding mode control for spacecraft body-fixed hovering in the proximity of an asteroid, Aerosp. Sci. Technol., 46 (2015), 471-483. [16] Q. Li, J. Yuan, B. Zhang and H. Wang, Disturbance observer based control for spacecraft proximity operations with path constraint, Aerosp. Sci. Technol., 79 (2018), 154-163. [17] X. Li, D. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024. [18] X. Li, X. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031. [19] K. Mei and S. Ding, HOSM controller design with asymmetric output constraints, Sci. China Inf. Sci., 65 (2022), Paper No. 189202, 2 pp. doi: 10.1007/s11432-020-3158-8. [20] M. Nazari, E. A. Butcher, T. Yucelen and A. K. Sanyal, Decentralized consensus control of a rigid-body spacecraft formation with communication delay, J. Guid., Control, Dyn., 39 (2016), 838-851. [21] A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Automat. Contr., 57 (2012), 2106-2110.  doi: 10.1109/TAC.2011.2179869. [22] W. Ren and R. W. Beard, Decentralized scheme for spacecraft formation flying via the virtual structure approach, J. Guid., Control, Dyn., 27 (2004), 73-82. [23] C. Sabol, R. Burns and C. A. McLaughlin, Satellite formation flying design and evolution, J. Spacecr. Rockets, 38 (2001), 270-278. [24] A. Sanyal, N. Nordkvist and M. Chyba, An almost global tracking control scheme for maneuverable autonomous vehicles and its discretization, IEEE Trans. Automat. Contr., 56 (2011), 457-462.  doi: 10.1109/TAC.2010.2090190. [25] Y. Song, Y. Wang, J. Holloway and M. Krstic, Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time, Automatica J. IFAC, 83 (2017), 243-251.  doi: 10.1016/j.automatica.2017.06.008. [26] L. Sun and W. Huo, Robust adaptive relative position tracking and attitude synchronization for spacecraft rendezvous, Aerosp. Sci. Technol., 41 (2015), 28-35. [27] R. Sun, J. Wang, D. Zhang and X. Shao, Neural network-based sliding mode control for atmospheric-actuated spacecraft formation using switching strategy, Adv. Space Res., 61 (2018), 914-926. [28] J. Wang, H. Liang, Z. Sun, S. Zhang and M. Liu, Finite-time control for spacecraft formation with dual-number-based description, J. Guid., Control, Dyn., 35 (2012), 950-962. [29] J. Wang and Z. Sun, 6-DOF robust adaptive terminal sliding mode control for spacecraft formation flying, Acta Astronaut., 73 (2012), 76-87. [30] Y. Wang, H. Hong, J. Guo, X. Wang and W. Shang, Configuration error function design and application to fixed-time geometric terminal sliding-mode control on SE (3), Acta Astronaut., 174 (2020), 61-71. [31] M. Wittal, G. Mangiacapra, A. Appakonam, M. Nazari and E. Capello, Stochastic spacecraft navigation and control in Lie SE (3) around small irregular bodies, Proceeding of AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Virtual, 2021. [32] S. Wu, L. Chen, D. Zhang, J. Chen and X. Shao, Disturbance observer based fixed time sliding mode control for spacecraft proximity operations with coupled dynamics, Adv. Space Res., 66 (2020), 2179-2193. [33] C. Zhang, J. Wang, R. Sun, D. Zhang and X. Shao, Multi-spacecraft attitude cooperative control using model-based event-triggered methodology, Adv. Space Res., 62 (2018), 2620-2630. [34] F. Zhang and G. R. Duan, Robust adaptive integrated translation and rotation control of a rigid spacecraft with control saturation and actuator misalignment, Acta Astronaut., 86 (2013), 167-187. [35] J. Zhang, D. Ye, J. D. Biggs and Z. Sun, Finite-time relative orbit-attitude tracking control for multi-spacecraft with collision avoidance and changing network topologies, Adv. Space Res., 63 (2019), 1161-1175. [36] J. Zhang, D. Ye, Z. Sun and C. Liu, Extended state observer based robust adaptive control on SE (3) for coupled spacecraft tracking maneuver with actuator saturation an misalignment, Acta Astronaut., 143 (2015), 221-233. [37] B. Zhou, Finite-time stabilization of linear systems by bounded linear time-varying feedback, Automatica J. IFAC, 113 (2020), 108760, 11 pp. doi: 10.1016/j.automatica.2019.108760. [38] B. Zhou, Finite-time stability analysis and stabilization by bounded linear time-varying feedback, Automatica J. IFAC, 121 (2020), 109191, 12 pp. doi: 10.1016/j.automatica.2020.109191.

show all references

##### References:
 [1] K. Alfriend, S. R. Vadali, P. Gurfil, J. P. How and L. Breger, Spacecraft Formation Flying: Dynamics, Control and Navigation (Vol. 2), Elsevier, Oxford, 2009. [2] R. W. Beard, L. Lawton and F. Y. Hadaegh, A coordination architecture for spacecraft formation control, IEEE Trans. Contr. Syst. Technol., 9 (2001), 777-790. [3] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751-766.  doi: 10.1137/S0363012997321358. [4] J. D. Biggs and L. Colley, Geometric attitude motion planning for spacecraft with pointing and actuator constraints, J. Guid., Control, Dyn., 39 (2016), 1672-1677. [5] F. Bullo and R. M. Murray, Proportional derivative (PD) control on the Euclidean group, Proceedings of IN European Control Conference, Zurich, Switzerland, 1995, 1091–1097. [6] S. Ding, K. Mei and S. Li, A new second-order sliding mode and its application to nonlinear constrained systems, IEEE Trans. Automat. Contr., 64 (2019), 2545-2552.  doi: 10.1109/TAC.2018.2867163. [7] K. Gong, Y. Liao and Y. Wang, Adaptive fixed-time terminal sliding mode control on SE (3) for coupled spacecraft tracking maneuver, Int. J. Aerosp. Eng., 2020 (2020), 1-15. [8] Y. Hong, J. Wang and D. Cheng, Adaptive finite-time control of nonlinear systems with parametric uncertainty, IEEE Trans. Automat. Contr., 51 (2006), 858-862.  doi: 10.1109/TAC.2006.875006. [9] J. Y. Hung, W. Gao and J. C. Hung, Variable structure control: A survey, IEEE Trans. Ind. Electron., 40 (1993), 2-22. [10] Y. Huang and Y. Jia, Distributed finite-time output feedback synchronisation control for six DOF spacecraft formation subject to input saturation, IET Control. Theory Appl., 12 (2018), 532-542.  doi: 10.1049/iet-cta.2017.0842. [11] H. K. Khalil, Nonlinear Control, 1$^{st}$ edition, USA: Pearson, New York, 2015. [12] Q. Lan, J. Yang, S. Li and H. Sun, Finite-time control for 6DOF spacecraft formation flying systems, J. Aerosp. Eng., 28 (2015), 04014137. [13] D. Lee, Spacecraft coupled tracking maneuver using sliding mode control with input saturation, J. Aerosp. Eng., 28 (2015), 04014136. [14] D. Lee, A. K. Sanyal and E. A. Butcher, Asymptotic tracking control for spacecraft formation flying with decentralized collision avoidance, J. Guid., Control, Dyn., 38 (2015), 587-600. [15] D. Lee and G. Vukovich, Adaptive sliding mode control for spacecraft body-fixed hovering in the proximity of an asteroid, Aerosp. Sci. Technol., 46 (2015), 471-483. [16] Q. Li, J. Yuan, B. Zhang and H. Wang, Disturbance observer based control for spacecraft proximity operations with path constraint, Aerosp. Sci. Technol., 79 (2018), 154-163. [17] X. Li, D. W. C. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368.  doi: 10.1016/j.automatica.2018.10.024. [18] X. Li, X. Yang and S. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC, 103 (2019), 135-140.  doi: 10.1016/j.automatica.2019.01.031. [19] K. Mei and S. Ding, HOSM controller design with asymmetric output constraints, Sci. China Inf. Sci., 65 (2022), Paper No. 189202, 2 pp. doi: 10.1007/s11432-020-3158-8. [20] M. Nazari, E. A. Butcher, T. Yucelen and A. K. Sanyal, Decentralized consensus control of a rigid-body spacecraft formation with communication delay, J. Guid., Control, Dyn., 39 (2016), 838-851. [21] A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Automat. Contr., 57 (2012), 2106-2110.  doi: 10.1109/TAC.2011.2179869. [22] W. Ren and R. W. Beard, Decentralized scheme for spacecraft formation flying via the virtual structure approach, J. Guid., Control, Dyn., 27 (2004), 73-82. [23] C. Sabol, R. Burns and C. A. McLaughlin, Satellite formation flying design and evolution, J. Spacecr. Rockets, 38 (2001), 270-278. [24] A. Sanyal, N. Nordkvist and M. Chyba, An almost global tracking control scheme for maneuverable autonomous vehicles and its discretization, IEEE Trans. Automat. Contr., 56 (2011), 457-462.  doi: 10.1109/TAC.2010.2090190. [25] Y. Song, Y. Wang, J. Holloway and M. Krstic, Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time, Automatica J. IFAC, 83 (2017), 243-251.  doi: 10.1016/j.automatica.2017.06.008. [26] L. Sun and W. Huo, Robust adaptive relative position tracking and attitude synchronization for spacecraft rendezvous, Aerosp. Sci. Technol., 41 (2015), 28-35. [27] R. Sun, J. Wang, D. Zhang and X. Shao, Neural network-based sliding mode control for atmospheric-actuated spacecraft formation using switching strategy, Adv. Space Res., 61 (2018), 914-926. [28] J. Wang, H. Liang, Z. Sun, S. Zhang and M. Liu, Finite-time control for spacecraft formation with dual-number-based description, J. Guid., Control, Dyn., 35 (2012), 950-962. [29] J. Wang and Z. Sun, 6-DOF robust adaptive terminal sliding mode control for spacecraft formation flying, Acta Astronaut., 73 (2012), 76-87. [30] Y. Wang, H. Hong, J. Guo, X. Wang and W. Shang, Configuration error function design and application to fixed-time geometric terminal sliding-mode control on SE (3), Acta Astronaut., 174 (2020), 61-71. [31] M. Wittal, G. Mangiacapra, A. Appakonam, M. Nazari and E. Capello, Stochastic spacecraft navigation and control in Lie SE (3) around small irregular bodies, Proceeding of AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Virtual, 2021. [32] S. Wu, L. Chen, D. Zhang, J. Chen and X. Shao, Disturbance observer based fixed time sliding mode control for spacecraft proximity operations with coupled dynamics, Adv. Space Res., 66 (2020), 2179-2193. [33] C. Zhang, J. Wang, R. Sun, D. Zhang and X. Shao, Multi-spacecraft attitude cooperative control using model-based event-triggered methodology, Adv. Space Res., 62 (2018), 2620-2630. [34] F. Zhang and G. R. Duan, Robust adaptive integrated translation and rotation control of a rigid spacecraft with control saturation and actuator misalignment, Acta Astronaut., 86 (2013), 167-187. [35] J. Zhang, D. Ye, J. D. Biggs and Z. Sun, Finite-time relative orbit-attitude tracking control for multi-spacecraft with collision avoidance and changing network topologies, Adv. Space Res., 63 (2019), 1161-1175. [36] J. Zhang, D. Ye, Z. Sun and C. Liu, Extended state observer based robust adaptive control on SE (3) for coupled spacecraft tracking maneuver with actuator saturation an misalignment, Acta Astronaut., 143 (2015), 221-233. [37] B. Zhou, Finite-time stabilization of linear systems by bounded linear time-varying feedback, Automatica J. IFAC, 113 (2020), 108760, 11 pp. doi: 10.1016/j.automatica.2019.108760. [38] B. Zhou, Finite-time stability analysis and stabilization by bounded linear time-varying feedback, Automatica J. IFAC, 121 (2020), 109191, 12 pp. doi: 10.1016/j.automatica.2020.109191.
Tracking error of the attitude and position for the $1$st follower
Tracking error of the angular velocity and velocity for the 1st follower
Control torque and force for the 1st follower
Tracking error of the attitude and position for the 2nd follower
Tracking error of the attitude and position for the 2nd follower
Control torque and force for the 2nd follower
Tracking error of the attitude and position for the 3rd follower
Tracking error of the attitude and position for the 3rd follower
Control torque and force for the 3rd follower
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