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A novel modeling and control approach considering equality and inequality constraints based on generalized Udwadia-Kalaba equation

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Abstract

There are two categories of constraints for mechanical systems: equality and inequality. For the constrained mechanical systems with equality constraints, the Udwadia-Kalaba (U-K) equation can be used to model and introduce the constraint-following motion control method. In industrial automation and robotics, the problem of motion requirement with fixed boundary which is respond to an inequality constraint has not been solved systematically using constraint following method. Through transformation via a novel diffeomorphism, the equation of motion for a constrained mechanical system which addresses both equality and inequality constraints is presented. This can be considered a generalization of the Udwadia-Kalaba (U-K) equation. The advantages of the equation include that it does not require additional pseudo variables and the solution is analytical. This exhibits profound applications. As a demonstration, a pan/tilt device mounted under the firefighting unmanned aerial vehicles (UAVs) is manipulated. The water-jet nozzle need motion requirements of swaying horizontally and not overshooting limits in vertical. Simulation and experimental results are presented to validate the effectiveness of the proposed approach.

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References

  1. Makris, S.: Cooperation Robots for Flexible Manufacturing. Springer Nature Switzerland, Gewerbestrasse (2021)

    Book  Google Scholar 

  2. Peng, H., Wang, J., Wang, S., Shen, W., Shi, D., Liu, D.: Coordinated motion control for a wheel-leg robot with speed consensus strategy. IEEE-ASME Trans. Mech. 25(3), 1366–1376 (2020)

    Google Scholar 

  3. Zhu, Y., He, X., Liu, Q., Guo, W.: Semiclosed-loop motion control with robust weld bead tracking for a spiral seam weld beads grinding robot. Robot. Comput.-integrat. Manufac. (2022). https://doi.org/10.1016/j.rcim.2021.102254

    Article  Google Scholar 

  4. Zhang, X., Zhu, W., Wu, X., Song, T., Xie, Y., Zhao, H.: Dynamics and control for in-space assembly robots with large translational and rotational maneuvers. Acta Astronaut. 174, 166–179 (2020)

    Article  Google Scholar 

  5. Zhang, X., Tan, J., Yao, Y., Wu, J.: Adaptive practical fixed-time tracking control for uncertain non-strict-feedback systems with input delay and prescribed boundary constraints. Int. J. Adapt. Control Signal Process. 36(3), 653–669 (2022)

    Article  MathSciNet  Google Scholar 

  6. Papastavridis, J.G.: Analytical Mechanics. Oxford University Press, UK (2002)

    MATH  Google Scholar 

  7. Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  8. Udwadia, F.E.: Optimal tracking control of nonlinear dynamical systems. Proceed. Royal Soc. A-Math. Phys. Eng. Sci. 464(2097), 2341–2363 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Yang, Z., Huang, J., Hu, Z., Yin, H., Zhong, Z.: Adaptive constraint-following control for uncertain nonlinear mechanical systems with measurement error. Int. J. Robust Nonlinear Control 31(10), 4823–4838 (2021)

    Article  MathSciNet  Google Scholar 

  10. Zhao, R., Li, M., Niu, Q., Chen, Y.H.: Udwadia-kalaba constraint-based tracking control for artificial swarm mechanical systems: dynamic approach. Nonlinear Dyn. 100(3), 2381–2399 (2020)

    Article  MATH  Google Scholar 

  11. Yang, S., Han, J., Xia, L., Chen, Y.H.: Adaptive robust servo constraint tracking control for an underactuated quadrotor uav with mismatched uncertainties. ISA Trans. 106, 12–30 (2020)

    Article  Google Scholar 

  12. Sun, Q., Yang, G., Wang, X., Chen, Y.H.: Designing robust control for mechanical systems: constraint following and multivariable optimization. IEEE Trans. Industr. Inf. 16(8), 5267–5275 (2020)

    Article  Google Scholar 

  13. Dong, F., Jin, D., Zhao, X., Han, J.: Adaptive robust constraint following control for omnidirectional mobile robot: an indirect approach. IEEE Access 9, 8877–8887 (2021)

    Article  Google Scholar 

  14. Chen, Y.H., Zhang, X.: Adaptive robust approximate constraint-following control for mechanical systems. J. Franklin Institute-Eng. Appl. Math. 347(1), 69–86 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rosenberg, R.M.: Analytical Dynamics of Discrete Systems. Plenum Press, New York (1977)

    Book  MATH  Google Scholar 

  16. Ashpazzadeh, E., Lakestani, M., Yildirim, A.: Biorthogonal multiwavelets on the interval for solving multidimensional fractional optimal control problems with inequality constraint. Optimal Control Appl. Methods 41(5), 1477–1494 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, Y.J., Tong, S.: Barrier lyapunov functions for nussbaum gain adaptive control of full state constrained nonlinear systems. Automatica 76, 143–152 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xu, X., Grizzle, J.W., Tabuada, P., Ames, A.D.: Correctness guarantees for the composition of lane keeping and adaptive cruise control. IEEE Trans. Autom. Sci. Eng. 15, 1216–1229 (2018)

    Article  Google Scholar 

  19. Ames, A., Xu, X., Grizzle, J.W., Tabuada, P.: Control barrier function based quadratic programs for safety critical systems. IEEE Trans. Autom. Control 62, 3861–3876 (2017). https://doi.org/10.1109/TAC.2016.2638961

    Article  MathSciNet  MATH  Google Scholar 

  20. Safe and robust observer-controller synthesis using control barrier functions. IEEE Control Systems Letters 7, 127–132 (2022)

  21. Li, C., Zhao, H., Sun, H., Chen, Y.H.: Robust bounded control for nonlinear uncertain systems with inequality constraints. Mech. Syst. Signal Process. 140, 106665 (2020)

    Article  Google Scholar 

  22. Zhu, Z., Zhao, H., Sun, H.: Stackelberg-theoretic optimal robust control for constrained permanent magnet linear motor with inequality constraints. IEEE-ASME Trans. Mechatron. 27, 5439–5450 (2022)

    Article  Google Scholar 

  23. Zhen, S., Zhang, M., Liu, X., Zhao, H., Chen, Y.H., Chen, X.: Robust bounded control design and experimental verification for permanent magnet linear motor with inequality constraints. IEEE ACCESS 10, 96886–96895 (2022)

    Article  Google Scholar 

  24. Wang, Z., Yang, G., Wang, X., Sun, Q.: Adaptive-adaptive robust boundary control for uncertain mechanical systems with inequality constraints. Nolinear Dynam. 110, 449–466 (2022)

    Article  Google Scholar 

  25. Zhang, B., Gavin, H.P.: Gauss’s principle with inequality constraints for multiagent navigation and control. IEEE Trans. Autom. Control 67, 810–823 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yin, H., Chen, Y.H., Yu, D.: Vehicle motion control under equality and inequality constraints: a diffeomorphism approach. Nonlinear Dyn. 95(1), 175–194 (2019)

    Article  MATH  Google Scholar 

  27. Sun, H., Tu, L., Yang, L., Zhu, Z., Zhen, S., Chen, Y.H.: Adaptive robust control for nonlinear mechanical systems with inequality constraints and uncertainties. IEEE Trans. Syst. Man Cybernet.-Syst. (2022). https://doi.org/10.1109/TSMC.2022.3204901

    Article  Google Scholar 

  28. Zhang, X., Song, B., Yang, Z., Huang, J., Jia, Y.: Optimal robust vehicle motion control under equality and inequality constraints. Asian J. Control (2022). https://doi.org/10.1002/asjc.2844

    Article  Google Scholar 

  29. Yao, P., Wang, H., Ji, H.: Gaussian mixture model and receding horizon control for multiple uav search in complex environment. Nonlinear Dyn. 88(2), 903–919 (2017)

    Article  Google Scholar 

  30. Yang, S., Xian, B.: Energy-based nonlinear adaptive control design for the quadrotor uav system with a suspended payload. IEEE Trans. Industr. Electron. 67(3), 2054–2064 (2020)

    Article  Google Scholar 

  31. Hwang, C.L., Lai, J.Y., Lin, Z.S.: Sensor-fused fuzzy variable structure incremental control for partially known nonlinear dynamic systems and application to an outdoor quadrotor. IEEE-ASME Trans. Mechatron. 25(2), 716–727 (2020)

    Article  Google Scholar 

  32. Sun, C., Huang, S., Chen, H., Ye, C., Wang, Y., Wang, W.J.: Laser-range-finder localization based fuzzy control for mobile robots. Eng. Comput. 34(7), 2409–2421 (2017)

    Article  Google Scholar 

  33. Chiu, C., Wang, W.J.: Implementation of a ball inverted pendulum with omnidirectional moving ability using a robust fuzzy control strategy. ISA Trans. 86, 287–298 (2019)

    Article  Google Scholar 

  34. Hwang, C.L., Lee, Y.: Tracking design of an omni-direction autonomous ground vehicle by hierarchical enhancement using fuzzy second-order variable structure control. J. Dynam. Syst. Measure. Control-Trans. ASME 140(9), 1–11 (2018)

  35. Hwang, C.L., Yang, C., Hung, J.Y.: Path tracking of an autonomous ground vehicle with different payloads by hierarchical improved fuzzy dynamic sliding-mode control. IEEE Trans. Fuzzy Syst. 26(2), 899–914 (2018)

    Article  Google Scholar 

  36. Kada, B., Ansari, U., Bajodah, A.H.: Highly maneuvering target interception via robust generalized dynamic inversion homing guidance and control. Aerosp. Sci. Technol. 99, 105749 (2020)

    Article  Google Scholar 

  37. Mehedi, I.M., Ansari, U., Al-Saggaf, U.M., Bajodah, A.H.: Controlling a rotory servo cart system using robust generalized dynamic inversion. Int. J. Robot. Autom. 35(1), 77–85 (2020)

    Google Scholar 

  38. Zhang, D., Wang, Z., Masayoshi, T.: Neural-network-based iterative learning control for multiple tasks. IEEE Trans. Neural Netw. Learn. Syst. 32(9), 4178–4190 (2021)

    Article  Google Scholar 

  39. Chai, R., Savvaris, A., Chai, S.: Integrated missile guidance and control using optimization-based predictive control. Nonlinear Dyn. 96(2), 997–1015 (2019)

    Article  Google Scholar 

  40. Khalil, H.K.: Nonlinear Systems, third, edition Prentice-Hall, Upper Saddle River, USA (1996)

  41. Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported in part by Shaanxi International Science and Technology Cooperation Project of China under Grant 2019 KW-015, Shaanxi Science & Technology Innovation Project of China under Grant 2016KTZDGY-02-03, and Fundamental Research Funds for Central Universities of China under Grant 300102259306.

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Authors and Affiliations

  1. Key Laboratory of Road Construction Technology and Equipment of MOE, Chang’an University, Xi’an, 710064, Shaanxi, China

    Xinrong Zhang, Ruiying Zhao & Xiaoyan Zhang

  2. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, 30332, GA, USA

    Ye-Hwa Chen

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  1. Xinrong Zhang
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  2. Ruiying Zhao
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  3. Ye-Hwa Chen
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  4. Xiaoyan Zhang
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Correspondence to Ye-Hwa Chen.

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Zhang, X., Zhao, R., Chen, YH. et al. A novel modeling and control approach considering equality and inequality constraints based on generalized Udwadia-Kalaba equation. Nonlinear Dyn 111, 17109–17122 (2023). https://doi.org/10.1007/s11071-023-08738-7

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