# Neural-Network-Based Terminal Sliding Mode Control of Space Robot Actuated by Control Moment Gyros

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- CMGs are employed as reactionless manipulator actuators of a SRS to reduce the strong dynamic coupling between the platform and the manipulator.
- A new integrated RBFNN-based non-singular finite control scheme is proposed for the SRS with lumped uncertainties. In the control method, a new weight update law applied to RBFNN is proposed. With the advantages of RBFNN and ANTSM, the controller has high control accuracy, fast learning speed and finite-time convergence. Compared with the traditional sliding mode control, the application range of the controller is extended by ignoring the upper bound of the lumped uncertainties.
- Rigorous theoretical proof is achieved by the Lyapunov method with a high mathematical standard. In the proof, the symbolic function is replaced by the saturation function to avoid the chattering problem for practical implementations.

## 2. System Description

## 3. Equations of Motion

#### 3.1. Reference Frames

#### 3.2. Dynamics Analysis

#### 3.3. Kinematics Analysis

## 4. Controller Design

#### 4.1. Problem Statement

**Assumption**

**1.**

#### 4.2. ANTSM Controller for Manipulators

#### 4.2.1. RBF Neural Network

**Assumption**

**2.**

#### 4.2.2. Sliding Surface

#### 4.2.3. Control Law Design

#### 4.3. Kinematic Controller for Platform

#### 4.3.1. Attitude Control

#### 4.3.2. Position Control

#### 4.4. Inverse Dynamics Controller for System

#### 4.5. Stability Analysis

**Theorem**

**1.**

**Proof**

**of**

**Theorem 1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem 2.**

## 5. Numerical Simulation

#### 5.1. Simulation Parameters

#### 5.2. Simulation Results

#### 5.2.1. Demonstration of Algorithm Effectiveness

#### 5.2.2. Comparison with Different Control Laws

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Flores-Abad, A.; Ma, O.; Pham, K.; Ulrich, S. A Review of Space Robotics Technologies for On-Orbit Servicing. Prog. Aerosp. Sci.
**2014**, 68, 1–26. [Google Scholar] [CrossRef][Green Version] - Jia, S.; Shan, J. Continuous Integral Sliding Mode Control for Space Manipulator with Actuator Uncertainties. Aerosp. Sci. Technol.
**2020**, 106, 106192. [Google Scholar] [CrossRef] - Shan, M.; Guo, J.; Gill, E. Review and Comparison of Active Space Debris Capturing and Removal Methods. Prog. Aerosp. Sci.
**2016**, 80, 18–32. [Google Scholar] [CrossRef] - Palma, P.; Seweryn, K.; Rybus, T. Impedance Control Using Selected Compliant Prismatic Joint in a Free-Floating Space Manipulator. Aerospace
**2022**, 9, 406. [Google Scholar] [CrossRef] - Xia, P.; Luo, J.; Wang, M.; Yuan, J. Constrained Compliant Control for Space Robot Postcapturing Uncertain Target. J. Aerosp. Eng.
**2019**, 32, 04018117. [Google Scholar] [CrossRef] - Chen, G.; Wang, Y.; Wang, Y.; Liang, J.; Zhang, L.; Pan, G. Detumbling Strategy Based on Friction Control of Dual-Arm Space Robot for Capturing Tumbling Target. Chin. J. Aeronaut.
**2020**, 33, 1093–1106. [Google Scholar] [CrossRef] - Hu, Q.; Guo, C.; Zhang, Y.; Zhang, J. Recursive Decentralized Control for Robotic Manipulators. Aerosp. Sci. Technol.
**2018**, 76, 374–385. [Google Scholar] [CrossRef] - Jia, Y.; Misra, A.K. Trajectory Planning for a Space Robot Actuated by Control Moment Gyroscopes. J. Guid. Control. Dyn.
**2018**, 41, 1838–1842. [Google Scholar] [CrossRef] - Feng, X.; Jia, Y.; Xu, S. Dynamics and Momentum Equalization Control of Redundant Space Robot with Control Moment Gyroscopes for Joint Actuation. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Long Beach, CA, USA, 13–16 September 2016. [Google Scholar]
- James, F.; Shah, S.V.; Singh, A.K.; Krishna, K.M.; Misra, A.K. Reactionless Maneuvering of a Space Robot in Precapture Phase. J. Guid. Control Dyn.
**2016**, 39, 2419–2425. [Google Scholar] [CrossRef] - Peck, M.; Paluszek, M.; Thomas, S.; Mueller, J. Control-Moment Gyroscopes for Joint Actuation: A New Paradigm in Space Robotics. In Proceedings of the 1st Space Exploration Conference: Continuing the Voyage of Discovery, Orlando, FL, USA, 30 January–1 February 2005. [Google Scholar]
- Peck, M. Low-Power, High-Agility Space Robotics. In Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, CA, USA, 15–18 August 2005. [Google Scholar]
- Brown, D. Control Moment Gyros as Space-Robotics Actuators. In Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, HI, USA, 18–21 August 2008. [Google Scholar]
- Brown, D.; Peck, M. Energetics of Control Moment Gyroscopes as Joint Actuators. J. Guid. Control Dyn.
**2009**, 32, 1871–1883. [Google Scholar] [CrossRef] - Shen, D.; Tang, L.; Hu, Q.; Guo, C.; Li, X.; Zhang, J. Space Manipulator Trajectory Tracking Based on Recursive Decentralized Finite-Time Control. Aerosp. Sci. Technol.
**2020**, 102, 105870. [Google Scholar] [CrossRef] - Wang, X.; Shi, L.; Katupitiya, J. A Strategy to Decelerate and Capture a Spinning Object by a Dual-Arm Space Robot. Aerosp. Sci. Technol.
**2021**, 113, 106682. [Google Scholar] [CrossRef] - Song, C.; Fei, S.; Cao, J.; Huang, C. Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control. Mathematics
**2019**, 7, 599. [Google Scholar] [CrossRef][Green Version] - Cong, B.; Liu, X.; Chen, Z. Improved Adaptive Sliding Mode Control for a Class of Second-Order Mechanical Systems: Improved Adaptive Sliding Mode Control. Asian J. Control
**2013**, 15, 1862–1866. [Google Scholar] [CrossRef] - Yoo, D.S.; Chung, M.J. A Variable Structure Control with Simple Adaptation Laws for Upper Bounds on the Norm of the Uncertainties. IEEE Trans. Autom. Control
**1992**, 37, 860–865. [Google Scholar] [CrossRef] - Jia, Y.; Xu, S. Decentralized Adaptive Sliding Mode Control of a Space Robot Actuated by Control Moment Gyroscopes. Chin. J. Aeronaut.
**2016**, 29, 688–703. [Google Scholar] [CrossRef][Green Version] - Jia, Y.; Misra, A.K. Robust Trajectory Tracking Control of a Dual-Arm Space Robot Actuated by Control Moment Gyroscopes. Acta Astronaut.
**2017**, 137, 287–301. [Google Scholar] [CrossRef] - Xu, W.; Qu, S.; Zhao, L.; Zhang, H. An Improved Adaptive Sliding Mode Observer for Middle- and High-Speed Rotor Tracking. IEEE Trans. Power Electron.
**2021**, 36, 1043–1053. [Google Scholar] [CrossRef] - Leeghim, H.; Choi, Y.; Bang, H. Adaptive Attitude Control of Spacecraft Using Neural Networks. Acta Astronaut.
**2009**, 64, 778–786. [Google Scholar] [CrossRef] - Zou, A.-M.; Kumar, K.D. Adaptive Attitude Control of Spacecraft without Velocity Measurements Using Chebyshev Neural Network. Acta Astronaut.
**2010**, 66, 769–779. [Google Scholar] [CrossRef] - Li, J.; Wang, J.; Peng, H.; Zhang, L.; Hu, Y.; Su, H. Neural Fuzzy Approximation Enhanced Autonomous Tracking Control of the Wheel-Legged Robot under Uncertain Physical Interaction. Neurocomputing
**2020**, 410, 342–353. [Google Scholar] [CrossRef] - Zou, A.-M.; Kumar, K.D.; Hou, Z.-G.; Liu, X. Finite-Time Attitude Tracking Control for Spacecraft Using Terminal Sliding Mode and Chebyshev Neural Network. IEEE Trans. Syst. Man Cybern. B
**2011**, 41, 950–963. [Google Scholar] [CrossRef] - Zou, Y. Attitude Tracking Control for Spacecraft with Robust Adaptive RBFNN Augmenting Sliding Mode Control. Aerosp. Sci. Technol.
**2016**, 56, 197–204. [Google Scholar] [CrossRef] - Nafia, N.; El Kari, A.; Ayad, H.; Mjahed, M. Robust Interval Type-2 Fuzzy Sliding Mode Control Design for Robot Manipulators. Robotics
**2018**, 7, 40. [Google Scholar] [CrossRef][Green Version] - Feng, Y.; Yu, X.; Man, Z. Non-Singular Terminal Sliding Mode Control of Rigid Manipulators. Automatica
**2002**, 38, 2159–2167. [Google Scholar] [CrossRef] - Nojavanzadeh, D.; Badamchizadeh, M. Adaptive Fractional-Order Non-Singular Fast Terminal Sliding Mode Control for Robot Manipulators. IET Control Theory Appl.
**2016**, 10, 1565–1572. [Google Scholar] [CrossRef] - Mohammadi Asl, R.; Shabbouei Hagh, Y.; Palm, R. Robust Control by Adaptive Non-Singular Terminal Sliding Mode. Eng. Appl. Artif. Intell.
**2017**, 59, 205–217. [Google Scholar] [CrossRef] - Abooee, A.; Arefi, M.M.; Sedghi, F.; Abootalebi, V. Robust Nonlinear Control Schemes for Finite-Time Tracking Objective of a 5-DOF Robotic Exoskeleton. Int. J. Control
**2019**, 92, 2178–2193. [Google Scholar] [CrossRef] - Abooee, A.; Hayeri Mehrizi, M.; Arefi, M.M.; Yin, S. Finite-Time Sliding Mode Control for a 3-DOF Fully Actuated Autonomous Surface Vehicle. Trans. Inst. Meas. Control
**2021**, 43, 371–389. [Google Scholar] [CrossRef] - Guo, S.; Li, D.; Meng, Y.; Fan, C. Task Space Control of Free-Floating Space Robots Using Constrained Adaptive RBF-NTSM. Sci. China Technol. Sci.
**2014**, 57, 828–837. [Google Scholar] [CrossRef] - Zhang, S.; Yang, P.; Kong, L.; Chen, W.; Fu, Q.; Peng, K. Neural Networks-Based Fault Tolerant Control of a Robot via Fast Terminal Sliding Mode. IEEE Trans. Syst. Man Cybern. Syst.
**2019**, 51, 4091–4101. [Google Scholar] [CrossRef] - Wang, X. Finite-Time Gradient Descent-Based Adaptive Neural Network Finite-Time Control Design for Attitude Tracking of a 3-DOF Helicopter. arXiv
**2021**, arXiv:2107.12924. [Google Scholar] - Vijay, M.; Jena, D. Backstepping Terminal Sliding Mode Control of Robot Manipulator Using Radial Basis Functional Neural Networks. Comput. Electr. Eng.
**2018**, 67, 690–707. [Google Scholar] [CrossRef] - Wang, L.; Chai, T.; Zhai, L. Neural-Network-Based Terminal Sliding-Mode Control of Robotic Manipulators Including Actuator Dynamics. IEEE Trans. Ind. Electron.
**2009**, 56, 3296–3304. [Google Scholar] [CrossRef] - Wang, P.; Zhang, D.; Lu, B. Trajectory Tracking Control for Chain-Series Robot Manipulator: Robust Adaptive Fuzzy Terminal Sliding Mode Control with Low-Pass Filter. Int. J. Adv. Robot. Syst.
**2020**, 17, 172988142091698. [Google Scholar] [CrossRef] - Xia, X.; Jia, Y.; Xu, S.; Wang, X. An Adaptive Nonsingular Terminal Sliding Mode Tracking Control Using Neural Networks for Space Manipulators Actuated by CMGs. In Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Snowbird, UT, USA, 19–23 August 2018; Volume 167, pp. 3185–3198. [Google Scholar]
- Wang, X.; Xu, B.; Cheng, Y.; Wang, H.; Sun, F. Robust Adaptive Learning Control of Space Robot for Target Capturing Using Neural Network. IEEE Trans. Neural Netw. Learn. Syst.
**2022**, 1–11. [Google Scholar] [CrossRef] - Yan, W.; Liu, Y.; Lan, Q.; Zhang, T.; Tu, H. Trajectory Planning and Low-Chattering Fixed-Time Nonsingular Terminal Sliding Mode Control for a Dual-Arm Free-Floating Space Robot. Robotica
**2022**, 40, 625–645. [Google Scholar] [CrossRef] - Moosavian, S.A.A.; Papadopoulos, E. Free-Flying Robots in Space: An Overview of Dynamics Modeling, Planning and Control. Robotica
**2007**, 25, 537–547. [Google Scholar] [CrossRef][Green Version] - Hu, Q.; Zhang, J. Dynamics and Trajectory Planning for Reconfigurable Space Multibody Robots. J. Mech. Des.
**2015**, 137, 092304. [Google Scholar] [CrossRef] - Banerjee, A.K. Flexible Multibody Dynamics: Efficient Formulations and Applications; John Wiley & Sons, Ltd.: Chichester, UK, 2016. [Google Scholar]
- Feng, X.; Jia, Y.; Xu, S. Dynamics of Flexible Multibody Systems with Variable-Speed Control Moment Gyroscopes. Aerosp. Sci. Technol.
**2018**, 79, 554–569. [Google Scholar] [CrossRef] - Huang, P.; Yuan, J.; Liang, B. Adaptive Sliding-Mode Control of Space Robot during Manipulating Unknown Objects. In Proceedings of the 2007 IEEE International Conference on Control and Automation, Guangzhou, China, 30 May–1 June 2007; pp. 2907–2912. [Google Scholar]
- Feng, Y.; Yu, X.; Man, Z. Non-Singular Terminal Sliding Mode Control and Its Application for Robot Manipulators. In Proceedings of the the 2001 IEEE International Symposium on Circuits and Systems, Sydney, Australia, 6–9 May 2001; Volume 3, pp. 545–548. [Google Scholar]
- Wheeler, G.; Su, C.; Stepanenko, Y. A Sliding Mode Controller with Improved Adaptation Laws for the Upper Bounds on the Norm of Uncertainties. Automatica
**1998**, 34, 1657–1661. [Google Scholar] [CrossRef]

**Figure 8.**(

**a**) The first value of the sliding manifold; (

**b**) The second value of the sliding manifold; (

**c**) The third value of the sliding manifold.

**Figure 9.**(

**a**) The first value of the approximate errors of the lumped uncertainty; (

**b**) The second value of the approximate errors of the lumped uncertainty; (

**c**) The third value of the approximate errors of the lumped uncertainty.

**Figure 13.**(

**a**) Control torque of the manipulator; (

**b**) The derivative of angular momentums of the CMGs.

**Figure 15.**Comparison of angle tracking errors under control laws with or without NN. (

**a**) The first value of the angle tracking errors; (

**b**) The second value of the angle tracking errors; (

**c**) The third value of the angle tracking errors.

**Figure 16.**Comparison of sliding manifolds under control laws with or without NN. (

**a**) The first value of the sliding manifold; (

**b**) The second value of the sliding manifold; (

**c**) The third value of the sliding manifold.

**Figure 18.**Comparison of angle tracking errors with and without platform control. (

**a**) The first value of the angle tracking error; (

**b**) The second value of the angle tracking error; (

**c**) The third value of the angle tracking error.

Body Number | $\mathbf{Mass}(\mathbf{kg})$ | $\mathbf{First}\mathbf{Moment}(\mathbf{kg}\cdot \mathbf{m})$ | $\mathbf{Inertia}\mathbf{Matrix}(\mathbf{kg}\cdot {\mathbf{m}}^{2})$ | ||
---|---|---|---|---|---|

True | Nominal | True | Nominal | ||

${\mathrm{B}}_{0}$ | 1600 | ${\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{T}$ | $diag(2000,2000,900)$ | ||

${\mathrm{B}}_{1}$ | 1.37 | 1.2 | ${\left[\begin{array}{ccc}0& 0& 0.0685\end{array}\right]}^{T}$ | $diag(5.1,5.1,1.1)\times {10}^{-3}$ | $diag(4.7,4.7,1.1)\times {10}^{-3}$ |

${\mathrm{B}}_{2}$ | 24.72 | 22 | ${\left[\begin{array}{ccc}19.1& 0& 4\end{array}\right]}^{T}$ | $\left[\begin{array}{ccc}0.69& 0& -3.25\\ 0& 23.03& 0\\ -3.25& 0& 22.37\end{array}\right]$ | $\left[\begin{array}{ccc}0.62& 0& -2.91\\ 0& 21.33& 0\\ -2.91& 0& 20.75\end{array}\right]$ |

${\mathrm{B}}_{3}$ | 24.86 | 22 | ${\left[\begin{array}{ccc}19.1& 0& -4.7\end{array}\right]}^{T}$ | $\left[\begin{array}{ccc}0.93& 0& 3.78\\ 0& 23.28& 0\\ 3.78& 0& 22.36\end{array}\right]$ | $\left[\begin{array}{ccc}0.83& 0& 3.37\\ 0& 21.49& 0\\ 3.37& 0& 20.67\end{array}\right]$ |

Body Number | ${\mathit{a}}_{\mathit{i}}(\mathbf{m})$ | ${\mathit{\alpha}}_{\mathit{i}}(\mathbf{rad})$ | ${\mathit{d}}_{\mathit{i}}(\mathbf{m})$ | ${\mathit{\theta}}_{\mathit{i}}(\mathbf{rad})$ |
---|---|---|---|---|

${\mathrm{B}}_{1}$ | 0 | 0.5π | 0.1 | ${q}_{1}$ |

${\mathrm{B}}_{2}$ | 1.825 | 0 | 0.54 | ${q}_{2}$ |

${\mathrm{B}}_{3}$ | 1.8825 | 0 | 0 | ${q}_{3}$ |

Parameter | Value |
---|---|

Initial positions of the platform $(\mathrm{m})$ | ${[\begin{array}{ccc}0& 0& 0\end{array}]}^{T}$ |

Initial rotation angles of the platform $(\mathrm{rad})$ | ${[\begin{array}{ccc}0& 0& 0\end{array}]}^{T}$ |

Initial linear velocities of the platform $(\mathrm{m}/\mathrm{s})$ | ${[\begin{array}{ccc}0& 0& 0\end{array}]}^{T}$ |

Initial angular velocities of the platform $(\text{rad/s})$ | ${[\begin{array}{ccc}0& 0& 0\end{array}]}^{T}$ |

Initial joint angles $(\mathrm{rad})$ | ${[\begin{array}{ccc}\pi /12& \pi /6& \pi /2\end{array}]}^{T}$ |

Initial joint speeds $(\text{rad/s})$ | ${[\begin{array}{ccc}0& 0& 0\end{array}]}^{T}$ |

Initial angular momentums $(\mathrm{Nms})$ | ${[\begin{array}{ccc}0& 0& 0\end{array}]}^{T}$ |

Parameter | Value |
---|---|

The proportional gain ${K}_{p1}$ | $diag\left[\begin{array}{ccc}4.5& 4.5& 4.5\end{array}\right]$ |

The derivative gain ${K}_{d1}$ | $diag\left[\begin{array}{ccc}5& 5& 5\end{array}\right]$ |

The proportional gain ${K}_{p2}$ | $diag\left[\begin{array}{ccc}3& 3& 3\end{array}\right]$ |

The derivative gain ${K}_{d2}$ | $diag\left[\begin{array}{ccc}3.5& 3.5& 3.5\end{array}\right]$ |

The constant $\beta $ | $0.05$ |

The odd number $p$ | $5$ |

The odd number $q$ | $3$ |

The constant $\delta $ | $0.1$ |

Upper bound of the estimate error ${\epsilon}_{N}$ | $0.1$ |

The constant $\alpha $ | $0.4$ |

Number of neurons $m$ | $10$ |

Initial of weight matrix ${W}_{0}$ | $\mathrm{zeros}(10,3)$ |

The gain matrix $K$ | $diag\left[\begin{array}{ccc}0.3& 0.25& 0.1\end{array}\right]$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xia, X.; Jia, Y.; Wang, X.; Zhang, J.
Neural-Network-Based Terminal Sliding Mode Control of Space Robot Actuated by Control Moment Gyros. *Aerospace* **2022**, *9*, 730.
https://doi.org/10.3390/aerospace9110730

**AMA Style**

Xia X, Jia Y, Wang X, Zhang J.
Neural-Network-Based Terminal Sliding Mode Control of Space Robot Actuated by Control Moment Gyros. *Aerospace*. 2022; 9(11):730.
https://doi.org/10.3390/aerospace9110730

**Chicago/Turabian Style**

Xia, Xinhui, Yinghong Jia, Xinlong Wang, and Jun Zhang.
2022. "Neural-Network-Based Terminal Sliding Mode Control of Space Robot Actuated by Control Moment Gyros" *Aerospace* 9, no. 11: 730.
https://doi.org/10.3390/aerospace9110730