July  2022, 15(7): 1823-1837. doi: 10.3934/dcdss.2022010

Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3)

1. 

Seventh Research Division, Beihang University, Beijing, 100191, China

2. 

Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Le$ \acute{o} $n, San Nicolas de los Garza, 66450, Mexico

3. 

International Laboratory of Information and Navigation Systems, ITMO University, Saint Petersburg 197101, Russia

*Corresponding author: Zongyu Zuo

Received  October 2021 Revised  December 2021 Published  July 2022 Early access  January 2022

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant 62073019

This paper considers the attitude tracking control problem for a rigid body. In order to avoid the complexity and ambiguity associated with other attitude representations (such as Euler angles or quaternions), the attitude dynamics and the proposed control system are represented globally on special orthogonal groups. An adaptive controller based on a Lie subgroup of SO(3) is developed such that the rigid body can track any given attitude command asymptotically without requiring the exact knowledge of the inertia moment. In the presence of external disturbances, the adaptive controller is enhanced with an additional robust sliding mode term by following the same idea within the framework of SO(3). Finally, simulation results are presented to demonstrate efficiency of the proposed controllers.

Citation: Yaobang Ye, Zongyu Zuo, Michael Basin. Robust adaptive sliding mode tracking control for a rigid body based on Lie subgroups of SO(3). Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1823-1837. doi: 10.3934/dcdss.2022010
References:
[1]

J. AhmedV. Coppola and D. Bernstein, Adaptive asymptotic tracking of spacecraft attitude motion with inertia matrix identification, Journal of Guidance, Control, and Dynamics, 21 (1998), 684-691.  doi: 10.2514/6.1997-3530.

[2]

A. Akhtar and S. L. Waslander, Controller class for rigid body tracking on SO(3), IEEE Transactions on Automatic Control, 66 (2021), 2234-2241.  doi: 10.1109/TAC.2020.3008295.

[3]

D. Angeli, An almost global notion of input-to-state stability, IEEE Transactions on Automatic Control, 49 (2004), 866-874.  doi: 10.1109/TAC.2004.829594.

[4]

S. Berkane, A. Abdessameud and A. Tayebi, A globally exponentially stable hybrid attitude and gyro-bias observer, 55th IEEE Conference on Decision and Control, (2016), 308–313. doi: 10.1109/CDC.2016.7798287.

[5]

S. Berkane, A. Abdessameud and A. Tayebi, On deterministic attitude observers on the special orthogonal group SO(3), 55th IEEE Conference on Decision and Control, (2016), 1165–1170. doi: 10.1109/CDC.2016.7798424.

[6]

M. Bhatt, S. Sukumar and A. K. Sanyal, Rigid body geometric attitude estimator using multi-rate sensors, 59th IEEE Conference on Decision and Control, (2020), 1511–1516. doi: 10.1109/CDC42340.2020.9304059.

[7]

N. A. ChaturvediA. K. Sanyal and N. H. McClarmoch, Rigid-body attitude control, IEEE Control Systems Magazine, 31 (2011), 30-51.  doi: 10.1109/MCS.2011.940459.

[8]

N. A. ChaturvediA. K. Sanyal and N. H. McClarmoch, Modeling and adaptive flight control for quadrotor trajectory tracking, Journal of Aircraft, 55 (2017), 666-681. 

[9]

T. Chen and J. Shan, Koopman-Operator-Based attitude dynamics and control on SO(3), Journal of Guidance, Control, and Dynamics, 43 (2020), 2112-2126.  doi: 10.2514/1.G005006.

[10]

G. Gómez Cortés, F. Castãnos and J. Dávila, Sliding motions on SO(3), sliding subgroups, 58th IEEE Conference on Decision and Control, (2019), 6953–6958. doi: 10.1109/CDC40024.2019.9028911.

[11]

H. A. HashimL. J. Brown and K. Mcisaac, Nonlinear stochastic attitude filters on the special orthogonal group 3: Ito and stratonovich, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 1853-1865.  doi: 10.1109/TSMC.2018.2870290.

[12] V. Jurdjevic, Geometric Control Theory, Cambridge Univisity Press, Cambridge, 1997. 
[13]

M. Krstic, I. Kanellakopoulos and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, 1995.

[14]

T. Lee, Exponential stability of an attitude tracking control system on SO(3) for large-angle rotational maneuvers, Systems Control Letters, 61 (2012), 231-237.  doi: 10.1016/j.sysconle.2011.10.017.

[15]

T. Lee, Robust adaptive attitude tracking on SO(3) with an application to a quadrotor UAV, IEEE Transactions on Control Systems Technology, 21 (2013), 1924-1930.  doi: 10.1109/TCST.2012.2209887.

[16]

T. Lee, M. Leok and N. McClamroch, Geometric tracking control of a quadrotor UAV on SE(3), 49th IEEE Conference on Decision and Control, (2010), 5420–5425. doi: 10.1109/CDC.2010.5717652.

[17]

G. LiD. Song and S. Xu, Kinematic-free orientation control for a deformable manipulator based on the geodesic in rotation group SO(3), IEEE Robotics and Automation Letters, 3 (2018), 2432-2438.  doi: 10.1109/LRA.2018.2792529.

[18]

C. LiuS. YangD. Luo and J. Zhou, Robust back-stepping controller on SO(3) for a quadrotor attitude tracking, 10th International Conference on Modelling, Identification and Control, 21 (2018), 1924-1930.  doi: 10.1109/ICMIC.2018.8529883.

[19]

S. Liu and Y. Sang, Underactuated stratospheric airship trajectory control using an adaptive integral backstepping approach, Journal of Aircraft, 55 (2018), 2357-2371.  doi: 10.2514/1.C034923.

[20]

Q. LuB. Ren and S. Parameswaran, Uncertainty and disturbance estimator-based global trajectory tracking control for a quadrotor, IEEE Transactions on Mechatronics, 25 (2020), 1519-1530.  doi: 10.1109/TMECH.2020.2978529.

[21]

D. MaithripalaJ. Berg and W. Dayawansa, Almost-global tracking of simple mechanical systems on a general class of Lie groups, IEEE Transactions on Automatic Control, 51 (2006), 216-225.  doi: 10.1109/TAC.2005.862219.

[22]

S. D. MarcoL. MarconiR. Mahony and T. Hamel, Output regulation for systems on matrix Lie-groups, Automatica, 87 (2018), 8-16.  doi: 10.1016/j.automatica.2017.08.006.

[23]

C. Mayhew, R. Sanfelice and A. Teel, Robust global asymptotic attitude stabilization of a rigid body by quaternion-based hybrid feedback, 48th IEEE Conference on Decision and Control, (2009), 2522–2527. doi: 10.1109/CDC.2009.5400338.

[24]

C. MayhewR. Sanfelice and A. Teel, Quaternion-based hybrid control for robust global attitude tracking, IEEE Transactions on Automatic Control, 56 (2011), 2555-2566.  doi: 10.1109/TAC.2011.2108490.

[25]

Y. Mitikiri and K. Mohseni, Globally stable attitude control of a fixed-wing rudderless UAV using subspace projection, IEEE Robotics and Automation Letters, 4 (2019), 1395-1401.  doi: 10.1109/LRA.2019.2895889.

[26] K. Narendra and A. Annaswamy, Stable Adaptive Systems, Dover Press, New York, 1995. 
[27]

X. Peng, J. Sun and Z. Geng, Exponential attitude and gyro-bias estimation on the special orthogonal group SO(3), 36th Chinese Control Conference, (2017), 838–843.

[28]

W. F. PhillipsC. E. Hailey and G. A. Gebert, Review of attitude representations used for aircraft kinematics, Journal of Aircraft, 38 (2001), 718-737.  doi: 10.2514/2.3083.

[29]

A. SanyalA. FosburyN. Chaturvedi and D. Bernstein, Inertia-free spacecraft attitude tracking with disturbance rejection and almost global stabilization, Journal of Guidance, Control, and Dynamics, 32 (2009), 1167-1178.  doi: 10.2514/1.41565.

[30]

H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, American Institute of Aeronautics and Astronautics, Virginia, 2003. doi: 10.2514/4.861550.

[31]

R. Schlanbusch and E. Grøtli, Hybrid certainty equivalence control of rigid bodies with quaternion measurements, IEEE Transactions on Automatic Control, 60 (2015), 2512-2517.  doi: 10.1109/TAC.2014.2382153.

[32]

R. SchlanbuschE. GrøtliA. Loria and P. Nicklasson, Hybrid attitude tracking of rigid bodies without angular velocity measurement, Systems Control Letters, 61 (2012), 595-601.  doi: 10.1016/j.sysconle.2012.01.008.

[33]

R. SchlanbuschA. Loria and R. Kristiansen, PD+ based output feedback attitude control of rigid bodies, IEEE Transactions on Automatic Control, 57 (2012), 2146-2152.  doi: 10.1109/TAC.2012.2183189.

[34] M. Sidi, Spacecraft Dynamics and Control, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511815652.
[35]

M. D. Shuster, A survey of attitude representations, J. Astronaut. Sci., 41 (1993), 439-517. 

[36]

X. WangC. Yu and Z. Lin, A dual quaternion solution to attitude and position control for rigid-body coordination, IEEE Transactions on Robotics, 28 (2012), 1162-1170.  doi: 10.1109/TRO.2012.2196310.

[37]

S. XuN. Cui and Y. Fan, Flexible satellite attitude maneuver via adaptive sliding mode control and active vibration suppression, AIAA Journal, 56 (2018), 4205-4212.  doi: 10.2514/1.J057287.

[38]

T. Yang, N. Sun and Y. Fang, Adaptive fuzzy control for a class of MIMO underactuated systems with plant uncertainties and actuator deadzones: Design and experiments, IEEE Transactions on Cyberbetics, (2021), 1–14. doi: 10.1109/TCYB.2021.3050475.

[39]

D. E. Zlotnik and J. R. Forbes, Exteroceptive measurement filtering embedded within an SO(3)-based attitude estimator, 55th IEEE Conference on Decision and Control, (2016), 296–301. doi: 10.1109/CDC.2016.7798285.

[40]

Z. Zuo, C. Liu and Q. Han, et al, Unmanned Aerial Vehicles: Control Methods and Future Challenges, IEEE/CAA Journal of Automatica Sinica, 2021.

[41]

Z. ZuoJ. Song and Z. Zheng, A survey on modelling, control and challenges of stratospheric airships, Control Engineering Practice, 119 (2022), 104979.  doi: 10.1016/j.conengprac.2021.104979.

show all references

References:
[1]

J. AhmedV. Coppola and D. Bernstein, Adaptive asymptotic tracking of spacecraft attitude motion with inertia matrix identification, Journal of Guidance, Control, and Dynamics, 21 (1998), 684-691.  doi: 10.2514/6.1997-3530.

[2]

A. Akhtar and S. L. Waslander, Controller class for rigid body tracking on SO(3), IEEE Transactions on Automatic Control, 66 (2021), 2234-2241.  doi: 10.1109/TAC.2020.3008295.

[3]

D. Angeli, An almost global notion of input-to-state stability, IEEE Transactions on Automatic Control, 49 (2004), 866-874.  doi: 10.1109/TAC.2004.829594.

[4]

S. Berkane, A. Abdessameud and A. Tayebi, A globally exponentially stable hybrid attitude and gyro-bias observer, 55th IEEE Conference on Decision and Control, (2016), 308–313. doi: 10.1109/CDC.2016.7798287.

[5]

S. Berkane, A. Abdessameud and A. Tayebi, On deterministic attitude observers on the special orthogonal group SO(3), 55th IEEE Conference on Decision and Control, (2016), 1165–1170. doi: 10.1109/CDC.2016.7798424.

[6]

M. Bhatt, S. Sukumar and A. K. Sanyal, Rigid body geometric attitude estimator using multi-rate sensors, 59th IEEE Conference on Decision and Control, (2020), 1511–1516. doi: 10.1109/CDC42340.2020.9304059.

[7]

N. A. ChaturvediA. K. Sanyal and N. H. McClarmoch, Rigid-body attitude control, IEEE Control Systems Magazine, 31 (2011), 30-51.  doi: 10.1109/MCS.2011.940459.

[8]

N. A. ChaturvediA. K. Sanyal and N. H. McClarmoch, Modeling and adaptive flight control for quadrotor trajectory tracking, Journal of Aircraft, 55 (2017), 666-681. 

[9]

T. Chen and J. Shan, Koopman-Operator-Based attitude dynamics and control on SO(3), Journal of Guidance, Control, and Dynamics, 43 (2020), 2112-2126.  doi: 10.2514/1.G005006.

[10]

G. Gómez Cortés, F. Castãnos and J. Dávila, Sliding motions on SO(3), sliding subgroups, 58th IEEE Conference on Decision and Control, (2019), 6953–6958. doi: 10.1109/CDC40024.2019.9028911.

[11]

H. A. HashimL. J. Brown and K. Mcisaac, Nonlinear stochastic attitude filters on the special orthogonal group 3: Ito and stratonovich, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 49 (2019), 1853-1865.  doi: 10.1109/TSMC.2018.2870290.

[12] V. Jurdjevic, Geometric Control Theory, Cambridge Univisity Press, Cambridge, 1997. 
[13]

M. Krstic, I. Kanellakopoulos and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, 1995.

[14]

T. Lee, Exponential stability of an attitude tracking control system on SO(3) for large-angle rotational maneuvers, Systems Control Letters, 61 (2012), 231-237.  doi: 10.1016/j.sysconle.2011.10.017.

[15]

T. Lee, Robust adaptive attitude tracking on SO(3) with an application to a quadrotor UAV, IEEE Transactions on Control Systems Technology, 21 (2013), 1924-1930.  doi: 10.1109/TCST.2012.2209887.

[16]

T. Lee, M. Leok and N. McClamroch, Geometric tracking control of a quadrotor UAV on SE(3), 49th IEEE Conference on Decision and Control, (2010), 5420–5425. doi: 10.1109/CDC.2010.5717652.

[17]

G. LiD. Song and S. Xu, Kinematic-free orientation control for a deformable manipulator based on the geodesic in rotation group SO(3), IEEE Robotics and Automation Letters, 3 (2018), 2432-2438.  doi: 10.1109/LRA.2018.2792529.

[18]

C. LiuS. YangD. Luo and J. Zhou, Robust back-stepping controller on SO(3) for a quadrotor attitude tracking, 10th International Conference on Modelling, Identification and Control, 21 (2018), 1924-1930.  doi: 10.1109/ICMIC.2018.8529883.

[19]

S. Liu and Y. Sang, Underactuated stratospheric airship trajectory control using an adaptive integral backstepping approach, Journal of Aircraft, 55 (2018), 2357-2371.  doi: 10.2514/1.C034923.

[20]

Q. LuB. Ren and S. Parameswaran, Uncertainty and disturbance estimator-based global trajectory tracking control for a quadrotor, IEEE Transactions on Mechatronics, 25 (2020), 1519-1530.  doi: 10.1109/TMECH.2020.2978529.

[21]

D. MaithripalaJ. Berg and W. Dayawansa, Almost-global tracking of simple mechanical systems on a general class of Lie groups, IEEE Transactions on Automatic Control, 51 (2006), 216-225.  doi: 10.1109/TAC.2005.862219.

[22]

S. D. MarcoL. MarconiR. Mahony and T. Hamel, Output regulation for systems on matrix Lie-groups, Automatica, 87 (2018), 8-16.  doi: 10.1016/j.automatica.2017.08.006.

[23]

C. Mayhew, R. Sanfelice and A. Teel, Robust global asymptotic attitude stabilization of a rigid body by quaternion-based hybrid feedback, 48th IEEE Conference on Decision and Control, (2009), 2522–2527. doi: 10.1109/CDC.2009.5400338.

[24]

C. MayhewR. Sanfelice and A. Teel, Quaternion-based hybrid control for robust global attitude tracking, IEEE Transactions on Automatic Control, 56 (2011), 2555-2566.  doi: 10.1109/TAC.2011.2108490.

[25]

Y. Mitikiri and K. Mohseni, Globally stable attitude control of a fixed-wing rudderless UAV using subspace projection, IEEE Robotics and Automation Letters, 4 (2019), 1395-1401.  doi: 10.1109/LRA.2019.2895889.

[26] K. Narendra and A. Annaswamy, Stable Adaptive Systems, Dover Press, New York, 1995. 
[27]

X. Peng, J. Sun and Z. Geng, Exponential attitude and gyro-bias estimation on the special orthogonal group SO(3), 36th Chinese Control Conference, (2017), 838–843.

[28]

W. F. PhillipsC. E. Hailey and G. A. Gebert, Review of attitude representations used for aircraft kinematics, Journal of Aircraft, 38 (2001), 718-737.  doi: 10.2514/2.3083.

[29]

A. SanyalA. FosburyN. Chaturvedi and D. Bernstein, Inertia-free spacecraft attitude tracking with disturbance rejection and almost global stabilization, Journal of Guidance, Control, and Dynamics, 32 (2009), 1167-1178.  doi: 10.2514/1.41565.

[30]

H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, American Institute of Aeronautics and Astronautics, Virginia, 2003. doi: 10.2514/4.861550.

[31]

R. Schlanbusch and E. Grøtli, Hybrid certainty equivalence control of rigid bodies with quaternion measurements, IEEE Transactions on Automatic Control, 60 (2015), 2512-2517.  doi: 10.1109/TAC.2014.2382153.

[32]

R. SchlanbuschE. GrøtliA. Loria and P. Nicklasson, Hybrid attitude tracking of rigid bodies without angular velocity measurement, Systems Control Letters, 61 (2012), 595-601.  doi: 10.1016/j.sysconle.2012.01.008.

[33]

R. SchlanbuschA. Loria and R. Kristiansen, PD+ based output feedback attitude control of rigid bodies, IEEE Transactions on Automatic Control, 57 (2012), 2146-2152.  doi: 10.1109/TAC.2012.2183189.

[34] M. Sidi, Spacecraft Dynamics and Control, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511815652.
[35]

M. D. Shuster, A survey of attitude representations, J. Astronaut. Sci., 41 (1993), 439-517. 

[36]

X. WangC. Yu and Z. Lin, A dual quaternion solution to attitude and position control for rigid-body coordination, IEEE Transactions on Robotics, 28 (2012), 1162-1170.  doi: 10.1109/TRO.2012.2196310.

[37]

S. XuN. Cui and Y. Fan, Flexible satellite attitude maneuver via adaptive sliding mode control and active vibration suppression, AIAA Journal, 56 (2018), 4205-4212.  doi: 10.2514/1.J057287.

[38]

T. Yang, N. Sun and Y. Fang, Adaptive fuzzy control for a class of MIMO underactuated systems with plant uncertainties and actuator deadzones: Design and experiments, IEEE Transactions on Cyberbetics, (2021), 1–14. doi: 10.1109/TCYB.2021.3050475.

[39]

D. E. Zlotnik and J. R. Forbes, Exteroceptive measurement filtering embedded within an SO(3)-based attitude estimator, 55th IEEE Conference on Decision and Control, (2016), 296–301. doi: 10.1109/CDC.2016.7798285.

[40]

Z. Zuo, C. Liu and Q. Han, et al, Unmanned Aerial Vehicles: Control Methods and Future Challenges, IEEE/CAA Journal of Automatica Sinica, 2021.

[41]

Z. ZuoJ. Song and Z. Zheng, A survey on modelling, control and challenges of stratospheric airships, Control Engineering Practice, 119 (2022), 104979.  doi: 10.1016/j.conengprac.2021.104979.

Figure 1.  Adaptive attitude tracking responses without disturbances
Figure 2.  Robust adaptive sliding mode attitude tracking responses with disturbances
Table 1.  parameters in simulation
Parameters Scenario (i) Scenario (ii)
$ k $ 10 50
$ k_{J} $ 0.35 0.4
$ \varsigma $ $ \backslash $ 0.002
$ \delta $ $ \backslash $ 0.8
$ \varepsilon $ $ \backslash $ 1
Parameters Scenario (i) Scenario (ii)
$ k $ 10 50
$ k_{J} $ 0.35 0.4
$ \varsigma $ $ \backslash $ 0.002
$ \delta $ $ \backslash $ 0.8
$ \varepsilon $ $ \backslash $ 1
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