Iranian Journal of Mechanical Engineering Transactions of ISME

Iranian Journal of Mechanical Engineering Transactions of ISME

Dynamic Modeling and Control System Design for Wheeled Collaborative Robots using Gibbs-Appell Formulation and Nonlinear Predictive Control Techniques

Authors
Hossein Mirzaeinejad
Ali Mohammad Shafei
Associate Professor, Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran
10.30506/ijmep.2025.2040150.1989
Abstract
One of the main topics in the field of robotics is the modeling and control of the mobile robots intrajectory tracking problem. In this paper, the kinematic and dynamic models of a manipulator connetcted by revolute-prismatic joints and installed in a non-holonomic wheeled mobile platform are first derived by applying the recursive Gibbs-Appell method. Indeed, by employing this dynamic methodology, one gets rid of the difficulties of Lagrange Multipliers that originate from non-holonomic constraints. Then, a nonlinear predictive approach is applied to design the kinematic and dynamic control laws to generate trajectory tracking control commands of the non-holonomic robot. In this method, the nonlinear responses of the mobile robot are predicted using Taylor series. The optimal control laws are analytically developed by minimizing the difference between the predicted and the desired responses of the system outputs. The multivariable kinematic controller specifies the desired angular and linear velocities of the mobile base and the manipulator links. The obtained control inputs are then used as the desired values to be tracked by the dynamic controller. Finally, the results of numerical simulations are presented to emphasize the ability the proposed method in the mathematical modeling and simultaneous trajectory tracking control of the mobile base and end-effector of such complex robotic systems.
Keywords
Gibbs-Appell methodology
Nonlinear predictive control
Wheeled mobile robot
n-link arm
Trajectory tracking

Subjects

  • W. Brockett, R. S. Millman, and H. J. Sussmann, “Asymptotic Stability and Feedback Stabilization,” Differential Geometric Control Theory, Boston MA: Birkhuser, Vol. 27, No. 1, 181–191, 1983, Brockett83.pdf.

 

  • H. Amer, H. Zamzuri, K. Hudha, and Z.A. Kadir, “Modelling and Control Strategies in Path Tracking Control for Autonomous Ground Vehicles: A Review of State of the Art and Challenges,” Journal of Intelligent and Robotic Systems, Vol. 86, pp. 225–254, 2017, https://doi.org/10.1007/s10846-016-0442-0.

 

 

 

  • M. Shafei, and H. Mirzaeinejad, “A Novel Recursive Formulation for Dynamic Modeling and Trajectory Tracking Control of Multi-rigid-link Robotic Manipulators Mounted on a Mobile Platform”, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol. 235, No. 7, pp. 1204-1217, 2021, https://doi.org/10.1177/0959651820973900.

 

  • Fazel, A. M. Shafei, and S. R. Nekoo, “Kinematic Analysis of Flexible Bipedal Robotic Systems”, Applied Mathematics and Mechanics, Vol. 45, No. 5, pp. 795-818, 2024, https://doi.org/10.1007/s10483-024-3081-8.

 

  • M. Shafei, and H. R. Shafei. “Dynamic Behavior of Flexible Multiple Links Captured Inside a Closed Space”, Journal of Computational and Nonlinear Dynamics, Transactions of the ASME, Vol. 11, No. 5, pp. 1-13, 2016, https://doi.org/10.1115/1.4032388.

 

  • M. Shafei, and H. R. Shafei, “Planar Multibranch Open-loop Robotic Manipulators Subjected to Ground Collision”, Journal of Computational and Nonlinear Dynamics, Transactions of the ASME, Vol. 12, No. 6, pp. 1-14, 2017, https://doi.org/10.1115/1.4036197.

 

  • Oriolo, A. De Luca, and M. Vendittelli “WMR Control via Dynamic Feedback Linearization: Design, Implementation, and Experimental Validationv,” IEEE Transactions on Control Systems Technology, Vol. 10, No. 6, pp. 835–852, 2002, https://doi.org/10.1109/TCST.2002.804116.

 

  • Keymasi Khalaji, and S. A. A. Moosavian, “Dynamic Modeling and Tracking Control of a Car with n Trailers,” Multibody System Dynamics, Vol. 37, pp. 211–225, 2016, https://doi.org/10.1007/s11044-015-9472-9.

 

  • Zhang, S. Li, H. Meng, Z. Li, and Z. Sun “Variable Gain  Based Composite  Trajectory  Tracking Control  for 4-wheel Skid-steering Mobile Robots  with  Unknown  Disturbances,”  Control Engineering Practice,  Vol. 132,  Article ID: 105428, 2023, https://doi.org/10.1016/j.conengprac.2022.105428.

 

  • Keymasi Khalaji, and S. A. A. Moosavian, “Adaptive Sliding Mode Control of a Wheeled Mobile Robot Towing a Trailer,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol. 229, No. 2, pp. 169–183, 2015, https://doi.org/10.1177/0959651814550539.

 

  • Abdolahi, S. Yousefi, and J. Tavoosi, “A New Self-tuning Nonlinear Model Predictive Controller for Autonomous Vehicles,” Complexity, Vol. 2023, Article ID: 8720849, pp. 1-9, 2023, https://doi.org/10.1155/2023/8720849.

 

  • Mirzaeinejad, and A. M. Shafei, “Modeling and Trajectory Tracking Control of a Two-wheeled Mobile Robot: Gibbs–Appell and Prediction-based Approaches”, Robotica, Vol. 36, No. 10, pp. 1551 – 1570, 2018, https://doi.org/10.1017/S0263574718000565.

 

 

 

  • Sun, S. Hu, H. Xie, H. Li, J. Zheng, and B. Chen, “Fuzzy Adaptive Recursive Terminal Sliding Mode Control for an Agricultural Omnidirectional Mobile Robot”, Computers and Electrical Engineering, Vol. 105, Article ID: 108529, 2023, https://doi.org/10.1016/j.compeleceng.2022.108529.

 

  • Z. Afaq, A. Jebelli, anf R. Ahmad, “An Intelligent Thermal Management Fuzzy Logic Control System Design and Analysis using ANSYS Fluent for a Mobile Robotic Platform in Extreme Weather Applications”, Journal of Intelligent & Robotic Systems, Vol. 107, No. 11, 2023, https://doi.org/10.1007/s10846-022-01799-7.

 

  • K. Khalaji, and M. Jalalnezhad, “Control of a Wheeled Robot in Presence of Sliding of Wheels using Adaptive Backstepping Method”, Modares Mechanical Engineering, Vol. 18, No. 4, pp. 144-152, 2018, [Online]. Available: https://sid.ir/paper/179485/en.

 

  • Wu, P. Jin, T. Zou, Z. Qi, H. Xiao, and P. Lou, “Backstepping Trajectory Tracking Based on Fuzzy Sliding Mode Control for Differential Mobile Robots”, Journal of Intelligent & Robotic Systems, Vol. 96, pp. 109–121, 2019, https://doi.org/10.1007/s10846-019-00980-9.

 

  • Y Guo, L Yu, and J Xu “Robust Finite-time Trajectory Tracking Control of Wheeled Mobile Robots with Parametric Uncertainties and Disturbances”, Journal of Systems Science and Complexity, Vol. 32, No. 5, pp. 1358–1374, 2019, https://doi.org/10.1007/s11424-019-7235-z.

 

  • Liu, H. Gao, H. Ji, and Zh. Hao, “Adaptive Sliding Mode Based Disturbance Attenuation Tracking Control for Wheeled Mobile Robots”, International Journal of Control, Automation and Systems, Vol. 18, pp. 1288–1298, 2020, https://doi.org/10.1007/s12555-019-0262-7.

 

  • Mirzaeinejad, “Optimization-based Nonlinear Control Laws with Increased Robustness for Trajectory Tracking of Non-holonomic Wheeled Mobile Robots”, Transportation Research Part C: Emerging Technologies, Vol. 101, pp. 1–17, 2019, https://doi.org/10.1016/j.trc.2019.02.003.

 

  • Wang, Ch. Zhou, Y. Yu, and X. Liu, “Adaptive Sliding Mode Trajectory Tracking Control for WMR Considering Skidding and Slipping via Extended State Observer”, Energies,Vol. 12, No. 17, pp. 1-16, 2019, https://doi.org/10.3390/en12173305.

 

  • Wang, P. Zhu, B. He, G. Deng, Ch. Zhang, and X. Huang, “An Adaptive Neural Sliding Mode Control with ESO for Uncertain Nonlinear Systems”, International Journal of Control, Automation and Systems, Vol. 19, pp. 1-11, 2021, https://doi.org/10.1007/s12555-019-0972-x.

 

  • A. Abadi, A. Ayeb, M. Labbadi, D. Fofi, T. Bakir and H. Mekki, “Robust Tracking Control of Wheeled Mobile Robot Based on Differential Flatness and Sliding Active Disturbance Rejection Control”, Simulations and Experiments Sensors, Vol. 24, No. 9, pp. 2849, 2024, https://doi.org/10.3390/s24092849.

 

  • Guo, “Fixed-time Disturbance Observer Based Robust Tracking Control of Wheeled Mobile Robot with Multiple Disturbances”, Measurement and Control, Vol. 56, No. 9-10, pp. 1626–1636, 2023, https://doi.org/10.1177/00202940231173629.

 

 

  • Mirzaeinejad, “Robust Predictive Control of Wheel Slip in Antilock Braking Systems Based on Radial Basis Function Neural Network”, Applied Soft Computing, Vol. 70, pp. 318-329, 2018, https://doi.org/10.1016/j.asoc.2018.05.043.

 

  • H Salari, H Mirzaeinejad, and M.F Mahani, “A New Control Algorithm of Regenerative Braking Management for Energy Efficiency and Safety Enhancement of Electric Vehicles”, Energy Conversion and Management, Vol. 276, Article ID: 116564, 2023, https://doi.org/10.1016/j.enconman.2022.116564.

   

  • Deylaghian, and H. Mirzaeinejad, “ Instantaneous Optimum Wheel Slip Estimation of Anti-lock Braking System Based on Extremum Seeking Algorithm and Fuzzy Method”, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, Vol. 237, No. 3, pp. 406-421, 2023, https://doi.org/10.1177/14644193231176606.

  • Receive Date 02 September 2024
  • Revise Date 31 January 2025
  • Accept Date 05 May 2025