Skip to main content
SpringerLink
Log in

Optimal robust control with cooperative game theory for lower limb exoskeleton robot

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

For achieving trajectory tracking issue of the lower limb exoskeleton robot, a novel optimal robust control with cooperative game theory is proposed. The uncertainties are considered (possible time-varying, bounded and fast), and the fuzzy set theory is creatively adopted to describe the boundary. From the view of analytical mechanics, the trajectory tracking is treated as the constraints control problem, including holonomic and nonholonomic constraints, which need to satisfy the conditions of human motion. Combining the robust control and optimal design, optimal robust control is formulated to satisfy both performances guaranteeing and optimal. The Pareto optimal solution is obtained to guarantee the minimum control cost. In the simulation, the adaptive robust control is chosen as a comparison. The existence of Pareto optimality and the effectiveness of optimal robust control have been verified via simulation results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Data availability

All data are available upon request at the authors’ email address.

Code availability

Custom code is available upon request at Liang Yuan email address.

References

  1. Go, A.S., Mozaffarian, D., Roger, V.L., et al.: Heart disease and stroke statistics–2013 update: a report from the American heart association. Circulation 127(1), e6–e245 (2013)

    Google Scholar 

  2. Woldag, H., Hummelsheim, H.: Evidence-based physiotherapeutic concepts for improving arm and hand function in stroke patients. J. Neurol. 249, 518–528 (2002)

    Google Scholar 

  3. Chen, G., Chan, C.K., Guo, Z., et al.: A review of lower extremity assistive robotic exoskeletons in rehabilitation therapy. Crit. Rev. Biomed. Eng. 41(4–5), 343–363 (2013)

    Google Scholar 

  4. Chen, X., Zhao, H., Zhen, S., et al.: Adaptive robust control for a lower limb rehabilitation robot running under passive training mode. IEEE/CAA J. Automatica Sinica. 6(2), 493–502 (2019)

    MathSciNet  Google Scholar 

  5. Sun, Q., Yang, G., Wang, X., et al.: Optimizing constraint obedience for mechanical systems: robust control and non-cooperative game. Mech. Syst. Sig. Process. 149, 1072 (2021)

    Google Scholar 

  6. Deng, M., Li, Z., Kang, Y., et al.: A learning-based hierarchical control scheme for an exoskeleton robot in human-robot cooperative manipulation. IEEE Trans. Cybern. 50(1), 112–125 (2020)

    Google Scholar 

  7. Shi, D., Zhang, W., Zhang, W., et al.: Human-centred adaptive control of lower limb rehabilitation robot based on human–robot interaction dynamic model. Mech. Mach. Theory. 162, 104340 (2021)

    Google Scholar 

  8. Li, Z., Li, C., Li, S., et al.: A fault-tolerant method for motion planning of industrial redundant manipulator. IEEE Trans. Industr. Inf. 16(12), 7469–7478 (2020)

    Google Scholar 

  9. Yang, Y., Huang, D., Dong, X.: Enhanced neural network control of lower limb rehabilitation exoskeleton by add-on repetitive learning. Neurocomputing 323, 256–264 (2019)

    Google Scholar 

  10. Campbell, S.M., Diduch, C.P., Sensinger, J.W.: Autonomous assistance-as-needed control of a lower limb exoskeleton with guaranteed stability. IEEE Access. 8, 51168–51178 (2020)

    Google Scholar 

  11. Sun, W., Lin, J.W., Su, S., et al.: Reduced adaptive fuzzy decoupling control for lower limb exoskeleton. IEEE Trans. Cybern. 51(3), 1099–1109 (2021)

    Google Scholar 

  12. Yang, Y., Ma, L., Huang, D.: Development and repetitive learning control of lower limb exoskeleton driven by electrohydraulic actuators. IEEE Trans. Industr. Electron. 64(5), 4169–4178 (2017)

    Google Scholar 

  13. Chen, C.F., Du, Z.J., He, L., et al.: Active disturbance rejection with fast terminal sliding mode control for a lower limb exoskeleton in swing phase. IEEE Access. 7, 72343–72357 (2019)

    Google Scholar 

  14. Udwadia, F.E., Kalaba, R.E.: A new perspective on constrained motion. Proc. R. Soc. Math. Phys. Sci. 439(1906), 407–410 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Udwadia, F.E.: A new perspective on the tracking control of nonlinear structural and mechanical systems. Proc. R. Soc. Math. Phys. Sci. 459(2035), 1783–1800 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Zhao, X., Chen, Y.H., Zhao, H., et al.: Udwadia-Kalaba equation for constrained mechanical systems: formulation and applications. Chin. J. Mech. Eng. 31, 11–24 (2018)

    Google Scholar 

  17. Liu, X., Zhen, S., Huang, K., et al.: A systematic approach for designing analytical dynamics and servo control of constrained mechanical systems. IEEE/CAA J. Automatica Sinica. 2(4), 382–393 (2015)

    MathSciNet  Google Scholar 

  18. Sun, H., Zhao, H., Zhen, S., et al.: Application of the Udwadia-Kalaba approach to tracking control of mobile robots. Nonlinear Dyn. 83, 389–400 (2016)

    MathSciNet  Google Scholar 

  19. Sun, Q., Wang, X., Chen, Y.H.: Adaptive robust control for dual avoidance–arrival performance for uncertain mechanical systems. Nonlinear Dyn. 94, 759–774 (2018)

    Google Scholar 

  20. Cho, H., Wanichanon, T., Udwadia, F.E.: Continuous sliding mode controllers for multi-input multi-output systems. Nonlinear Dyn. 94, 2727–2747 (2018)

    Google Scholar 

  21. Mylapilli, H., Udwadia, F.E.: Control of three-dimensional incompressible hyperelastic beams. Nonlinear Dyn. 90, 115–135 (2017)

    MathSciNet  Google Scholar 

  22. Cho, H., Udwadia, F.E., Wanichanon, T.: Autonomous precision control of satellite formation flight under unknown time-varying model and environmental uncertainties. J. Astronaut. Sci. 67, 1470–1499 (2020)

    Google Scholar 

  23. Li, Z., Ma, W., Yin, Z., et al.: Tracking control of time-varying knee exoskeleton disturbed by interaction torque. ISA Trans. 71, 458–466 (2017)

    Google Scholar 

  24. Yu, R., Chen, Y.H., Zhao, H., et al.: Self-adjusting leakage type adaptive robust control design for uncertain systems with unknown bound. Mech. Syst. Signal Process. 116, 173–193 (2019)

    Google Scholar 

  25. Long, Y., Du, Z., Cong, L., et al.: Active disturbance rejection control based human gait tracking for lower extremity rehabilitation exoskeleton. ISA Trans. 67, 389–397 (2017)

    Google Scholar 

  26. Li, C., Zhan, H., Sun, H., et al.: Robust bounded control for nonlinear uncertain systems with inequality constraints. Mech. Syst. Sig. Process. 140, 106665 (2020)

    Google Scholar 

  27. Grubbs, F.E.: An introduction to probability theory and its application. Technometrics 9(2), 342 (1967)

    Google Scholar 

  28. Kozin, F.: A survey of stability of stochastic systems. Automatica 5(1), 95–112 (1969)

    MathSciNet  MATH  Google Scholar 

  29. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    MATH  Google Scholar 

  30. Zhen, S., Zhao, H., Huang, K., et al.: A novel optimal robust control design of fuzzy mechanical systems. IEEE Trans. Fuzzy Syst. 23(6), 2012–2023 (2015)

    Google Scholar 

  31. Sun, H., Zhao, H., Huang, K., et al.: A fuzzy approach for optimal robust control design of an automotive electronic throttle system. IEEE Trans. Fuzzy Syst. 26(2), 694–704 (2018)

    Google Scholar 

  32. Liu, X., Zhen, S., Zhao, H., et al.: Fuzzy-set theory based optimal robust design for position tracking control of permanent magnet linear motor. IEEE Access. 7, 153829–153841 (2019)

    Google Scholar 

  33. Chen, W., Tan, D., Yang, Q., et al.: Ant colony optimization for the control of pollutant spreading on social networks. IEEE Trans. Cybern. 50(9), 4053–4065 (2020)

    Google Scholar 

  34. Li, L., Chang, L., Gu, T., et al.: On the norm of dominant difference for many-objective particle swarm optimization. IEEE Trans. Cybern. 51(4), 2055–2067 (2021)

    Google Scholar 

  35. Li, J., Li, L.: A hybrid genetic algorithm based on information entropy and game theory. IEEE Access. 8, 36602–36611 (2020)

    Google Scholar 

  36. Huang, J., Chen, Y.H., Cheng, A.: Robust control for fuzzy dynamical systems: uniform ultimate boundedness and optimality. IEEE Trans. Fuzzy Syst. 20(6), 1022–1031 (2012)

    Google Scholar 

  37. Huang, Q., Chen, Y.H., Cheng, A.: Adaptive robust control for fuzzy mechanical systems: constraint-following and redundancy in constraints. IEEE Trans. Fuzzy Syst. 23(4), 1113–1126 (2015)

    Google Scholar 

  38. Sun, H., Yu, R., Chen, Y.H., et al.: Optimal design of robust control for fuzzy mechanical systems: performance-based leakage and confidence-index measure. IEEE Trans. Fuzzy Syst. 27(7), 1441–1455 (2019)

    Google Scholar 

  39. Chen, X., Zhao, H., Sun, H., et al.: Optimal adaptive robust control based on cooperative game theory for a class of fuzzy underactuated mechanical systems. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2020.3016003

    Article  Google Scholar 

  40. Chen, X., Zhao, H., Zhen, S., et al.: Novel optimal adaptive robust control for fuzzy underactuated mechanical systems: a Nash game approach. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.3007452

    Article  Google Scholar 

  41. Klir, G.J., Yuan, B.: Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall, Englewood Cliffs, NJ (1995)

    MATH  Google Scholar 

  42. Chen, Y.H.: A new approach to the control design of fuzzy dynamical systems. J. Dyn. Syst. Meas. Contr. 133(6), 1–9 (2011)

    Google Scholar 

  43. Rosenberg, R.M.: Analytical dynamics of discrete systems. Plenum Press, New York (1977)

    MATH  Google Scholar 

  44. Sun, H., Chen, Y.H., Zhao, H.: Adaptive robust control methodology for active roll control system with uncertainty. Nonlinear Dyn. 92, 359–371 (2018)

    MATH  Google Scholar 

  45. Sun, H., Zhao, H., Huang, K., et al.: Adaptive robust constraint-following control for satellite formation flying with system uncertainty. J. Guid. Control. Dyn. 40(6), 1492–1502 (2017)

    Google Scholar 

  46. Chen, Y.H.: Constraint-following servo control design for mechanical systems. J. Vib. Control 15(3), 369–389 (2009)

    MathSciNet  MATH  Google Scholar 

  47. Udwadia, F.E., Kalaba, R.E.: What is the general form of the explicit equations of motion for constrained mechanical system? J. Appl. Mech. 69, 407–410 (1992)

    Google Scholar 

  48. Han, J., Yang, S., Xia, L., et al.: Deterministic adaptive robust control with a novel optimal gain design approach for a fuzzy 2DOF lower limb exoskeleton robot system. IEEE Trans. Fuzzy Syst. 29(8), 2373–2387 (2021)

    Google Scholar 

  49. Zhang, X., Jiang, W., Li, Z., et al.: A hierarchical Lyapunov-based cascade adaptive control scheme for lower-limb exoskeleton. Eur. J. Control. 50, 198–208 (2019)

    MathSciNet  MATH  Google Scholar 

  50. Amiri, M.S., Ramli, R., Ibrahim, M.F.: Initialized model reference adaptive control for lower limb exoskeleton. IEEE Access. 7, 167210–167220 (2019)

    Google Scholar 

  51. Sun, H., Yang, L., Chen, Y.H., et al.: Constraint-based control design for uncertain underactuated mechanical system: leakage-type adaptation mechanism. IEEE Trans. Syst. Man Cybern. Syst. (2020). https://doi.org/10.1109/TSMC.2020.2979900

    Article  Google Scholar 

Download references

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The authors would like to thank the National Natural Science Foundation of China [U1813220, 62063033] and Fundamental Research Funds for the Central Universities [buctrc202105] for their support in this research.

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Xinjiang University, 666 Shengli Road, Urumqi, 830046, China

    Jin Tian, Liang Yuan, Wendong Xiao, Teng Ran, Jianbo Zhang & Li He

  2. School of Information Science and Technology, Beijing University of Chemical Technology, 15 Beisanhuan Dong Lu, Chaoyang District, Beijing, 100029, China

    Liang Yuan

Authors
  1. Jin Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liang Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wendong Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Teng Ran
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jianbo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Li He
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

JT involved in writing—original draft, validation and software. LY and LH involved in writing—review and editing, and supervision. WX, TR and JZ took part in methodology, investigation and formal analysis.

Corresponding author

Correspondence to Liang Yuan.

Ethics declarations

Conflict of interest

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: Proof of Theorem 2

Proof

Choose a legitimate Lyapunov function candidate as follows [51]:

$$V\left({\varvec{\beta}},\widehat{\alpha }-\alpha \right)={{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{\beta}}+{k}_{1}^{-1}{L}_{1}^{-1}\left(1+{\rho }_{E}\right)$$
$${\left(\widehat{\alpha }-\alpha \right)}^{T}\left(\widehat{\alpha }-\alpha \right).$$
(101)

For a specific mechanical system with uncertainty and desired constraints, \(\dot{V}\) is shown as

$$\dot{V}=2{{\varvec{\beta}}}^{T}{\varvec{P}}\dot{{\varvec{\beta}}}+2{k}_{1}^{-1}{L}_{1}^{-1}\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\dot{\widehat{\alpha }} .$$
(102)

The first term of (102) can be analyzed as

$$2{{\varvec{\beta}}}^{T}{\varvec{P}}\dot{{\varvec{\beta}}}=2{{\varvec{\beta}}}^{T}{\varvec{P}}\left({\varvec{A}}\ddot{{\varvec{x}}}-{\varvec{b}}\right)$$
$$=2{{\varvec{\beta}}}^{T}{\varvec{P}}\left({\varvec{A}}{{\varvec{M}}}^{-1}\left({\varvec{Q}}+{\varvec{\tau}}\right)-{\varvec{b}}\right)$$
$$=2{{\varvec{\beta}}}^{T}{\varvec{P}}\left({\varvec{A}}{{\varvec{M}}}^{-1}\left({\varvec{Q}}+{\varvec{Z}}{{\varvec{w}}}_{1}+{\varvec{Z}}{{\varvec{w}}}_{2}+{\varvec{Z}}{{\varvec{w}}}_{3}\right)-{\varvec{b}}\right)$$
(103)

Based on (25)—(26),

$${\varvec{A}}{{\varvec{M}}}^{-1}\left({\varvec{Q}}+{{\varvec{Z}}{\varvec{w}}}_{1}+{\varvec{Z}}{{\varvec{w}}}_{2}+{\varvec{Z}}{{\varvec{w}}}_{3}\right)-{\varvec{b}}$$
$$={\varvec{A}}\left({\varvec{E}}+\Delta {\varvec{E}}\right)\left(\overline{{\varvec{Q}} }+\Delta {\varvec{Q}}+{\varvec{Z}}{{\varvec{w}}}_{1}+{{\varvec{Z}}{\varvec{w}}}_{2}+{\varvec{Z}}{{\varvec{w}}}_{3}\right)-{\varvec{b}}$$
$$={\varvec{A}}{\varvec{E}}\overline{{\varvec{Q}} }+{\varvec{A}}{\varvec{E}}\Delta {\varvec{Q}}+{\varvec{A}}\Delta {\varvec{E}}\left(\overline{{\varvec{Q}} }+\Delta {\varvec{Q}}\right)+{\varvec{A}}{\varvec{E}}{\varvec{Z}}{{\varvec{w}}}_{1}+{\varvec{A}}{\varvec{E}}{{\varvec{Z}}{\varvec{w}}}_{2}$$
$$+{\varvec{A}}\Delta {\varvec{E}}{\varvec{Z}}\left({{\varvec{w}}}_{1}+{{\varvec{w}}}_{2}\right)+{\varvec{A}}\left({\varvec{E}}+\Delta {\varvec{E}}\right){\varvec{Z}}{{\varvec{w}}}_{3}-{\varvec{b}}.$$
(104)

According to (37),

$${\varvec{A}}{\varvec{E}}\overline{{\varvec{Q}} }+{\varvec{A}}{\varvec{E}}{\varvec{Z}}{{\varvec{w}}}_{1}-{\varvec{b}}=0.$$
(105)

According to Assumption 3–1,

$$2{{\varvec{\beta}}}^{T}{\varvec{P}}\left({\varvec{A}}{\varvec{E}}\Delta {\varvec{Q}}+{\varvec{A}}\Delta {\varvec{E}}\left(\mathbf{Q}+{{\varvec{Z}}{\varvec{w}}}_{1}+{{\varvec{Z}}{\varvec{w}}}_{2}\right)\right)$$
$$\le 2\Vert {\varvec{\beta}}\Vert \Vert {\varvec{P}}{\varvec{A}}{\varvec{E}}\Delta {\varvec{Q}}+\mathbf{P}{\varvec{A}}\Delta {\varvec{E}}\left(\mathbf{Q}+{{\varvec{Z}}{\varvec{w}}}_{1}+{{\varvec{Z}}{\varvec{w}}}_{2}\right)\Vert $$
$$\le 2\left(1+{\rho }_{E}\right)\Vert {\varvec{\beta}}\Vert \Pi \left(\alpha ,{\varvec{x}},\dot{{\varvec{x}}},t\right)$$
(106)

Based on (38),

$$2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{Z}}{{\varvec{w}}}_{2}=2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{Z}}\left(-{k}_{1}{\left({\varvec{A}}{\overline{{\varvec{M}}} }^{-1}{\varvec{B}}\right)}^{+}{{\varvec{P}}}^{-1}{\varvec{\beta}}\right)$$
$$=-2{k}_{1}{{\varvec{\beta}}}^{T}{\varvec{\beta}}=-2{k}_{1}{\Vert {\varvec{\beta}}\Vert }^{2}.$$
(107)

Based on (31), (39) and (43),

$$2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}\left({\varvec{E}}+\Delta {\varvec{E}}\right){\varvec{Z}}{{\varvec{w}}}_{3}$$
$$=2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{Z}}{{\varvec{w}}}_{3}+{2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}{\varvec{w}}}_{3}$$
$$=2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{Z}}\left(-{k}_{1}^{2}{\left({\varvec{A}}{\overline{{\varvec{M}}} }^{-1}{\varvec{Z}}\right)}^{+}{{\varvec{P}}}^{-1}{\varvec{\chi}}\left(\widehat{\alpha },{\varvec{x}},\dot{{\varvec{x}}},t\right)\right)$$
$$+{2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}{\varvec{w}}}_{3}$$
$$=-2{k}_{1}^{2}{{\varvec{\beta}}}^{T}{\varvec{\chi}}\left(\widehat{\alpha },{\varvec{x}},\dot{{\varvec{x}}},t\right)+{2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}{\varvec{w}}}_{3}$$
$$=-2{k}_{1}^{2}{{\varvec{\mu}}}^{T}\gamma{\varvec{\mu}}+{2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}{\varvec{w}}}_{3}$$
$$=-2{k}_{1}^{2}\gamma {\Vert {\varvec{\mu}}\Vert }^{2}+{2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}{\varvec{w}}}_{3}$$
(108)

According to Assumption 3 and Rayleigh’s principle,

$${2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}{\varvec{w}}}_{3}$$
$$=2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}\left(-{k}_{1}^{2}{\left({\varvec{A}}{\overline{{\varvec{M}}} }^{-1}{\varvec{Z}}\right)}^{+}{{\varvec{P}}}^{-1}{\varvec{\chi}}\left(\widehat{\alpha },{\varvec{x}},\dot{{\varvec{x}}},t\right)\right)$$
$$\begin{array}{c}=-2{k}_{1}^{2}\gamma \frac{1}{2}{{\varvec{\mu}}}^{T}\left({\varvec{P}}{\varvec{A}}{\varvec{E}}{\varvec{F}}{\varvec{Z}}{\left({\varvec{A}}{\overline{{\varvec{M}}} }^{-1}{\varvec{Z}}\right)}^{+}{{\varvec{P}}}^{-1}+{{\varvec{P}}}^{-1}{\left({\left({\varvec{A}}{\overline{{\varvec{M}}} }^{-1}{\varvec{Z}}\right)}^{+}\right)}^{T}\right.\times \end{array}$$
$$\left.{{\varvec{Z}}}^{T}{{\varvec{F}}}^{T}{\varvec{P}}{{\varvec{A}}}^{T}{\varvec{E}}\right){\varvec{\mu}}=-2{k}_{1}^{2}\gamma \frac{1}{2}{{\varvec{\mu}}}^{T}\left({\varvec{W}}+{{\varvec{W}}}^{T}\right){\varvec{\mu}}$$
$$\le -2{k}_{1}^{2}\gamma \frac{1}{2}{\lambda }_{min}\left({\varvec{W}}+{{\varvec{W}}}^{T}\right){\Vert {\varvec{\mu}}\Vert }^{2}\le -2{k}_{1}^{2}\gamma {\rho }_{E}{\Vert {\varvec{\mu}}\Vert }^{2}.$$
(109)

Combining (108) and (109),

$$2{{\varvec{\beta}}}^{T}{\varvec{P}}{\varvec{A}}\left({\varvec{E}}+\Delta {\varvec{E}}\right){\varvec{Z}}{{\varvec{w}}}_{3}\le -2{k}_{1}^{2}\gamma \left(1+{\rho }_{E}\right){\Vert {\varvec{\mu}}\Vert }^{2}.$$
(110)

By (41) and (43), we have

$$2{\beta^T}P\dot \beta $$
$$ \le - 2k_{1} \left\| \user2{\beta } \right\|^{2} + 2\left( {1 + \rho _{E} } \right)\left\| \user2{\beta } \right\|\Pi \left( {\alpha ,x,\user2{\dot{x}},t} \right) $$
$$ - 2k_1^2\gamma \left( {1 + {\rho_E}} \right){\left\| \mu \right\|^2}$$
$$ \begin{array}{*{20}{c}} { = - 2{k_1}{{\left\| \beta \right\|}^2} + 2\left( {1 + {\rho_E}} \right)\left\| \beta \right\|\Pi \left( {\alpha ,x,\dot x,t} \right) - 2\left( {1 + {\rho_E}} \right)} \end{array}$$
$$ \;\;\;\;\;\; \times \frac{{\vartheta^n}}{{{\vartheta^{n - 1}} + {\vartheta^{n - 2}}{k_2} + \cdots + \vartheta k_2^{n - 2} + k_2^{n - 1}}}$$
$$ \leqslant - 2{k_1}{\beta^2} + 2\left( {1 + {\rho_E}} \right)\left\| \beta \right\|{\Pi }\left( {\alpha ,x,\dot x,t} \right) - 2\left( {1 + {\rho_E}} \right)$$
$$ \times \frac{{{\vartheta^n} - k_2^n}}{{{\vartheta^{n - 1}} + {\vartheta^{n - 2}}{k_2} + \cdots + \vartheta k_2^{n - 2} + k_2^{n - 1}}}$$
$$ \begin{array}{*{20}{c}} { = - 2{k_1}{\beta^2} + 2\left( {1 + {\rho_E}} \right)\left\| \beta \right\|\Pi \left( {\alpha ,x,\dot x,t} \right) - 2\left( {1 + {\rho_E}} \right)\# } \end{array} $$
$$ \times \frac{{\left( {\vartheta - {k_2}} \right)\left( {{\vartheta^{n - 1}} + {\vartheta^{n - 2}}{k_2} + \cdots + \vartheta k_2^{n - 2} + k_2^{n - 1}} \right)}}{{{\vartheta^{n - 1}} + {\vartheta^{n - 2}}{k_2} + \cdots + \vartheta k_2^{n - 2} + k_2^{n - 1}}}$$
$$ \begin{array}{*{20}{c}} { = - 2{k_1}{\beta^2} + 2\left( {1 + {\rho_E}} \right)\left\| \beta \right\|\Pi \left( {\alpha ,x,\dot x,t} \right) + 2\left( {1 + {\rho_E}} \right){k_2}} \end{array} $$
$$ - 2\left( {1 + {\rho_E}} \right)k_1^2{\left\| \beta \right\|^2}{{\Pi }^2}\left( {\hat \alpha ,x,\dot x,t} \right) $$
$$ = - 2{k_1}{\beta^2} + 2\left( {1 + {\rho_E}} \right)\left\| \beta \right\|{\Pi }\left( {\alpha ,x,\dot x,t} \right) + 2\left( {1 + {\rho_E}} \right){k_2} $$
$$ + 2\left( {1 + {\rho_E}} \right)\left\| \beta \right\|{\Pi }\left( {\hat \alpha ,x,\dot x,t} \right) - 2\left( {1 + {\rho_E}} \right)\left\| \beta \right\|{\Pi }\left( {\hat \alpha ,x,\dot x,t} \right)$$
$$ - 2\left( {1 + {\rho_E}} \right)k_1^2{\left\| \beta \right\|^2}{{\Pi }^2}\left( {\hat \alpha ,x,\dot x,t} \right)$$
$$ \le - 2k_{1} \left\| \varvec{\beta } \right\|^{2} + 2\left( {1 + \rho _{E} } \right)\left\| \varvec{\beta } \right\|\left( {\alpha - \hat{\alpha }} \right)^{T} \tilde{\prod }\left( {x,\varvec{\dot{x}},t} \right) $$
$$ + 2\left( {1 + {\rho_E}} \right){k_2} + \frac{1}{2}\left( {1 + {\rho_E}} \right)k_1^{ - 2}. $$
(111)

For the second term of the right-hand side of (102), by using adaptive law (45) and \({e}^{-\Vert {\varvec{\beta}}\Vert }\le 1\), we can obtain

$$2{k}_{1}^{-1}{L}_{1}^{-1}\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\dot{\widehat{\alpha }}$$
$$=2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\left(\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert -{L}_{1}^{-1}\left({L}_{2}{e}^{-\Vert {\varvec{\beta}}\Vert }+{L}_{3}\right)\widehat{\alpha }\right)$$
$$=2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert -2\left(1+{\rho }_{E}\right){L}_{1}^{-1}\times $$
$$\left({L}_{2}{e}^{-\Vert {\varvec{\beta}}\Vert }+{L}_{3}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\left(\widehat{\alpha }-\alpha +\alpha \right)$$
$$\le 2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert -2\left(1+{\rho }_{E}\right)-{L}_{1}^{-1}\times $$
$$\left({L}_{2}+{L}_{3}\right)\left({\Vert \widehat{\alpha }-\alpha \Vert }^{2}+\Vert \widehat{\alpha }-\alpha \Vert \Vert \alpha \Vert \right)$$
$$=2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert -2\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}\times $$
$$\left({\Vert \widehat{\alpha }-\alpha \Vert }^{2}+\Vert \widehat{\alpha }-\alpha \Vert \Vert \alpha \Vert \right)-2\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}\times $$
$$\left({\Vert \widehat{\alpha }-\alpha \Vert }^{2}+\Vert \widehat{\alpha }-\alpha \Vert \Vert \alpha \Vert \right)$$
$$\le 2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert +\frac{1}{2}\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}\times $$
$${\Vert \alpha \Vert }^{2}-2\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}\left({\Vert \widehat{\alpha }-\alpha \Vert }^{2}+\Vert \widehat{\alpha }-\alpha \Vert \Vert \alpha \Vert \right)$$
$$=2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert +\frac{1}{2}\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}{\Vert \alpha \Vert }^{2}-$$
$$2\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \widehat{\alpha }-\alpha \Vert }^{2}-2\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}\Vert \widehat{\alpha }-\alpha \Vert \Vert \alpha \Vert $$
$$\le 2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert +\frac{1}{2}\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}{\Vert \alpha \Vert }^{2}$$
$$-2\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \widehat{\alpha }-\alpha \Vert }^{2}-\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \alpha \Vert }^{2}$$
$$-\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \widehat{\alpha }-\alpha \Vert }^{2}$$
$$=2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert +\frac{1}{2}\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}{\Vert \alpha \Vert }^{2}$$
$$-3\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \widehat{\alpha }-\alpha \Vert }^{2}-\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \alpha \Vert }^{2}$$
(112)

With (102), (111) and (112) (notice that \({\Vert \boldsymbol{\varphi }\Vert }^{2}={\Vert {\varvec{\beta}}\Vert }^{2}+{\Vert \widehat{\alpha }-\alpha \Vert }^{2}\)), we have

$$\dot{V}\le -2{k}_{1}{\Vert {\varvec{\beta}}\Vert }^{2}+2\left(1+{\rho }_{E}\right)\Vert {\varvec{\beta}}\Vert {\left(\alpha -\widehat{\alpha }\right)}^{T}\stackrel{\sim }{\Pi }$$
$$+2\left(1+{\rho }_{E}\right){k}_{2}+\frac{1}{2}\left(1+{\rho }_{E}\right){k}_{1}^{-2}+\frac{1}{2}\left(1+{\rho }_{E}\right)\times $$
$${L}_{1}^{-1}{L}_{2}{\Vert \alpha \Vert }^{2}+2\left(1+{\rho }_{E}\right){\left(\widehat{\alpha }-\alpha \right)}^{T}\stackrel{\sim }{\Pi }\Vert {\varvec{\beta}}\Vert -$$
$$\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \alpha \Vert }^{2}-3\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \widehat{\alpha }-\alpha \Vert }^{2}$$
$$=-2{k}_{1}{\Vert {\varvec{\beta}}\Vert }^{2}-3\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \widehat{\alpha }-\alpha \Vert }^{2}+2\left(1+{\rho }_{E}\right){k}_{2}$$
$$+\frac{1}{2}\left(1+{\rho }_{E}\right){k}_{1}^{-2}+\frac{1}{2}\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}{\Vert \alpha \Vert }^{2}$$
$$-\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}{\Vert \alpha \Vert }^{2}$$
$$\le -\rho {\Vert \boldsymbol{\varphi }\Vert }^{2}+\Omega $$
(113)

where \(\rho : = \min \left\{ {2{k_1},3\left( {1 + {\rho_E}} \right)L_1^{ - 1}{L_3}} \right\}\), \(\Omega : = \left( {1/2} \right) \times \left( {1 + {\rho_E}} \right)k_1^{ - 2} + 2\left( {1 + {\rho_E}} \right){k_2} + \left( {1/2} \right)\left( {1 + {\rho_E}} \right)L_1^{ - 1}{L_2}{\alpha^2} + \left( {1 + {\rho_E}} \right)L_1^{ - 1}{L_3}{\alpha^2}\)

The function \(d\left(r\right)\) in Theorem 2(1) is shown as

$$d\left(r\right)=\left\{\begin{array}{c}\sqrt{\frac{{\Psi }_{2}}{{\Psi }_{1}}}R \,\,if \,\,r\le R\\ \sqrt{\frac{{\Psi }_{2}}{{\Psi }_{1}}}r \,\,if \,\,r>;R\end{array}\right.$$
(114)
$$\mathrm{R}=\sqrt{\frac{\Omega }{\rho }}$$
(115)

where \({\Psi }_{1}=min\left\{{\lambda }_{min}\left({\varvec{P}}\right), {\left({k}_{1}{L}_{1}\right)}^{-1}\left(1+{\rho }_{E}\right)\right\}\) and \({\Psi }_{2}=\)

$$ \max \left\{ {{\lambda_{\max }}\left( P \right),\;{{\left( {{k_1}{L_1}} \right)}^{ - 1}}\left( {1 + {\rho_E}} \right)} \right\}$$

In addition, uniform ultimate boundedness in Theorem 2(2) is also obedient to

$$\underline{d}=\sqrt{\frac{{\Psi }_{2}}{{\Psi }_{1}}}R$$
(116)
$$ T\left( {\bar d,r} \right) = \left\{ {\begin{array}{*{20}{c}} {0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;if\;r \leqslant \bar d\;\;\sqrt {\frac{{{{\Psi }_2}}}{{{{\Psi }_1}}}} } \\ {\frac{{{{\Psi }_2}{r^2} - \left( {{\Psi }_1^2/{{\Psi }_2}} \right){{\bar d}^2}}}{{\rho {{\bar d}^2}\left( {{{\Psi }_1}/{{\Psi }_2}} \right) - {\Omega }}}\;,\;\;\;\;\;\;\;\;\;\;\;otherwise} \end{array}.} \right.$$
(117)

Appendix B

$${\mathcal{L}}_{1}=\frac{{\left(1+{\rho }_{E}\right)}^{2}}{8{\widehat{\rho }}^{3}}$$
$${\mathcal{L}}_{2}=\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{2}}{4{\widehat{\rho }}^{3}}D\left[{\Vert \alpha \Vert }^{2}\right]+\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{3}}{2{\widehat{\rho }}^{3}}D\left[{\Vert \alpha \Vert }^{2}\right]$$
$$-\frac{\left(1+{\rho }_{E}\right)}{2{\widehat{\rho }}^{2}}D\left[{V}_{s2}\right]$$
$${\mathcal{L}}_{3}=-\frac{\left(1+{\rho }_{E}\right)}{2{\widehat{\rho }}^{2}}D\left[{V}_{s1}\right]$$
$${\mathcal{L}}_{4}=\frac{D\left[{V}_{s2}^{2}\right]}{2\widehat{\rho }}+\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-2}{L}_{2}^{2}}{8{\widehat{\rho }}^{3}}D\left[{\Vert \alpha \Vert }^{4}\right]$$
$$+\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-2}{L}_{2}^{3}}{2{\widehat{\rho }}^{3}}D\left[{\Vert \alpha \Vert }^{4}\right]+\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-2}{L}_{2}{L}_{3}}{2{\widehat{\rho }}^{3}}D\left[{\Vert \alpha \Vert }^{4}\right]$$
$$-\frac{\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}}{2{\widehat{\rho }}^{2}}D\left[{{V}_{s2}\Vert \alpha \Vert }^{2}\right]-$$
$$\frac{\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}}{{\widehat{\rho }}^{2}}D\left[{{V}_{s2}\Vert \alpha \Vert }^{2}\right]$$
$${\mathcal{L}}_{5}=\frac{D\left[{V}_{s1}{V}_{s2}\right]}{\widehat{\rho }}-\frac{\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{2}}{2{\widehat{\rho }}^{2}}D\left[{{V}_{s1}\Vert \alpha \Vert }^{2}\right]$$
$$-\frac{\left(1+{\rho }_{E}\right){L}_{1}^{-1}{L}_{3}}{{\widehat{\rho }}^{2}}D\left[{{V}_{s1}\Vert \alpha \Vert }^{2}\right]$$
$${\mathcal{L}}_{6}=\frac{D\left[{V}_{s1}^{2}\right]}{2\widehat{\rho }}$$
$${\mathcal{L}}_{7}=\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{2}}{{\widehat{\rho }}^{3}}D\left[{\Vert \alpha \Vert }^{2}\right]+\frac{{2\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{3}}{{\widehat{\rho }}^{3}}\times $$
$$D\left[{\Vert \alpha \Vert }^{2}\right]-\frac{2\left(1+{\rho }_{E}\right)}{{\widehat{\rho }}^{2}}D\left[{V}_{s2}\right]$$
$${\mathcal{L}}_{8}=\frac{{2\left(1+{\rho }_{E}\right)}^{2}}{{\widehat{\rho }}^{3}}$$
$${\mathcal{L}}_{9}=\frac{{\left(1+{\rho }_{E}\right)}^{2}}{{\widehat{\rho }}^{3}}$$
$${\mathcal{L}}_{10}=-\frac{2\left(1+{\rho }_{E}\right)}{{\widehat{\rho }}^{2}}D\left[{V}_{s1}\right]$$
$${\varkappa }_{1}=\frac{{\left(1+{\rho }_{E}\right)}^{2}}{4{\widehat{\rho }}^{2}}$$
$${\varkappa }_{2}=\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{2}}{2{\widehat{\rho }}^{2}}D\left[{\Vert \alpha \Vert }^{2}\right]+\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{3}}{{\widehat{\rho }}^{2}}D\left[{\Vert \alpha \Vert }^{2}\right]$$
$${\varkappa }_{3}=\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-2}{L}_{2}^{2}}{4{\widehat{\rho }}^{2}}D\left[{\Vert \alpha \Vert }^{4}\right]+\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-2}{L}_{3}^{2}}{{\widehat{\rho }}^{2}}D\left[{\Vert \alpha \Vert }^{4}\right]$$
$$+\frac{{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-2}{L}_{2}{L}_{3}}{{\widehat{\rho }}^{2}}D\left[{\Vert \alpha \Vert }^{4}\right]$$
$${\varkappa }_{4}=\frac{{2\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{2}}{{\widehat{\rho }}^{2}}D\left[{\Vert \alpha \Vert }^{2}\right]+$$
$$\frac{4{\left(1+{\rho }_{E}\right)}^{2}{L}_{1}^{-1}{L}_{3}}{{\widehat{\rho }}^{2}}D\left[{\Vert \alpha \Vert }^{2}\right]$$
$${\varkappa }_{5}=\frac{{2\left(1+{\rho }_{E}\right)}^{2}}{{\widehat{\rho }}^{2}}, {\varkappa }_{6}=\frac{{4\left(1+{\rho }_{E}\right)}^{2}}{{\widehat{\rho }}^{2}}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, J., Yuan, L., Xiao, W. et al. Optimal robust control with cooperative game theory for lower limb exoskeleton robot. Nonlinear Dyn 108, 1283–1303 (2022). https://doi.org/10.1007/s11071-022-07219-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07219-7

Keywords

Search

Navigation