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Model-free adaptive trajectory tracking control of robotic manipulators with practical prescribed-time performance

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Abstract

Considering accurate model information of robotic manipulators is hard to obtain in real applications, this article proposes a model-free trajectory tracking control scheme for the robotic system with input saturation, where a pre-defined tracking precision within the prescribed time under any initial conditions is ensured. Firstly, to avoid a sharp corner when control input exceeds constraint, a new smooth function is used to approximate the saturated control torque, and then, a new robotic system model is built. Based on the rebuilt system, a time-delay estimation (TDE) method is used to estimate the lumped uncertainty containing unknown dynamics and external disturbance. Furthermore, a model-free adaptive controller is developed to ensure the tracking error converges to a preset small residual within a given time, where adaptive law and auxiliary system are used to cope with the TED estimation error and control input saturation, respectively. With the developed practical prescribed-time function, there is no constraint on the initial value of the tracking error, and thus, global stability is guaranteed. Finally, a planar two-link robotic manipulator is simulated to show the effectiveness of the developed control scheme.

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The datasets supporting the conclusions of this article are included within the article.

References

  1. Mason, M.T.: Toward robotic manipulation. Ann. Rev. Control 1, 1–28 (2018)

    Google Scholar 

  2. Byrne, K., Proto, J., Kruysman, B., Bitterman, M.: The power of engineering, the invention of artists: bot & dolly. Robot. Fabric. Archit. Art Des. 2014, 399–405 (2014)

    Google Scholar 

  3. Borst, C., Wimbock C., Schmidt C., Fuchs, M., Brunner, B., Zacharias, F., Giordano, P., Konietschke, R., Sepp, W., Fuchs, S., Rink, C., Albu-Schaffer, A., Hirzinger, G.: Rollin’justin-mobile platform with variable base. In 2009 IEEE International Conference on Robotics and Automation, pp. 1597–1598 (2009)

  4. Sitti, M.: Microscale and nanoscale robotics systems grand challenges of robotics. IEEE Robot. Autom. Mag. 14(1), 53–60 (2007)

    Google Scholar 

  5. Werfel, J., Petersen, K., Nagpal, R.: Designing collective behavior in a termite-inspired robot construction team. Science 343(6172), 754–758 (2014)

    Google Scholar 

  6. Gao, Y., Wei, W., Wang, X., Wang, D., Li, Y., Yu, Q.: Trajectory tracking of multi-legged robot based on model predictive and sliding mode control. Inform. Sci. 606, 489–511 (2022)

    Google Scholar 

  7. Li, J., Zhu, L.: Practical tracking control under actuator saturation for a class of flexible-joint robotic manipulators driven by DC motors. Nonlinear Dyn. 109, 2745–2758 (2022)

    MATH  Google Scholar 

  8. Yang, G., Zhang, L., Yu, S., Meng, S., Wang, Q., Li, Q.: Influences of space perturbations on robotic assembly process of ultra-large structures. Nonlinear Dyn. 111, 10025–10048 (2023)

    Google Scholar 

  9. Bai, Q., Li, P., Tian, W., Shen, J., Li, B., Wen, K.: Dynamic visual servoing of large-scale dual-arm cooperative manipulators based on photogrammetry sensor. Int. J. Adv. Manuf. Technol. 126, 4037–4054 (2023)

    Google Scholar 

  10. Sun, Y., Kuang, J., Gao, Y., Chen, W., Wang, J., Liu, J., Wu, L.: Fixed-time prescribed performance tracking control for manipulators against input saturation. Nonlinear Dyn. (2023). https://doi.org/10.1007/s11071-023-08499-3

    Article  Google Scholar 

  11. Ghorbani, S., Samadikhoshkho, Z., Janabi-Sharifi, F.: Dual-arm aerial continuum manipulation systems: modeling, pre-grasp planning, and control. Nonlinear Dyn. 111, 7339–7355 (2023)

    Google Scholar 

  12. Zheng, K., Zhang, Q., Hu, Y., Wu, B.: Design of fuzzy system-fuzzy neural network-backstepping control for complex robot system. Inform. Sci. 546, 1230–1255 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Zhu, C., Jiang, Y., Yang, C.: Fixed-time neural control of robot manipulator with global stability and guaranteed transient performance. IEEE Trans. Ind. Electron. 70(1), 803–812 (2022)

    Google Scholar 

  14. Su, H., Qi, W., Chen, J., Zhang, D.: Fuzzy approximation-based task-space control of robot manipulators with remote center of motion constraint. IEEE Trans. Fuzzy Syst. 30(6), 1564–1573 (2022)

    Google Scholar 

  15. Sun, Y., Gao, C., Wu, L.B., Yang, Y.H.: Fuzzy observer-based command filtered tracking control for uncertain strict-feedback nonlinear systems with sensor faults and event-triggered technology. Nonlinear Dyn. 111, 8329–8345 (2023)

    Google Scholar 

  16. Jin, M., Kang, S.H., Chang, P.H.: Robust compliant motion control of robot with nonlinear friction using time-delay estimation. IEEE Trans. Ind. Electron. 55(1), 258–269 (2008)

    Google Scholar 

  17. Lee, J., Chang, P.H., Jin, M.: Adaptive integral sliding mode control with time-delay estimation for robot manipulators. IEEE Trans. Ind. Electron. 64(8), 6796–6804 (2017)

    Google Scholar 

  18. Wang, Y., Leibold, M., Lee, J., Ye, W., Xie, J., Buss, M.: Incremental model predictive control exploiting time-delay estimation for a robot manipulator. IEEE Trans. Control Syst. Technol. 30(6), 2285–2300 (2022)

    Google Scholar 

  19. Baek, J., Kwon, W., Kim, B., Han, S.: A widely adaptive time-delayed control and its application to robot manipulators. IEEE Trans. Ind. Electron. 66(7), 5332–5342 (2018)

    Google Scholar 

  20. Hong, Y., Xu, Y., Huang, J.: Finite-time control for robot manipulators. Syst. Control Lett. 46(4), 243–253 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Chen, Q., Tao, M., He, X., Tao, L.: Fuzzy adaptive nonsingular fixed-time attitude tracking control of quadrotor UAVs. IEEE Trans. Aero. Elec. Syst. 57(5), 2864–2877 (2021)

    Google Scholar 

  22. Sun, J., Wang, J., Yang, P., Guo, S.: Model-free prescribed performance fixed-time control for wearable exoskeletons. Appl. Math. Model. 90, 61–77 (2021)

    MathSciNet  Google Scholar 

  23. Fan, Y., Jing, W.: Inertia-free appointed-time prescribed performance tracking control for space manipulator. Aerosp. Sci. Technol. 117, 106896 (2021)

    Google Scholar 

  24. Xie, S., Tao, M., Chen, Q., Tao, L.: Neural-network-based adaptive finite-time output constraint control for rigid spacecrafts. Int. J. Robust Nonlinear 32(5), 2983–3000 (2022)

    MathSciNet  Google Scholar 

  25. Xie, S., Chen, Q., He, X., Tao, M., Tao, L.: Finite-time command-filtered approximation-free attitude tracking control of rigid spacecraft. Nonlinear Dyn. 107(3), 2391–2405 (2022)

    Google Scholar 

  26. Wu, Z., Ni, J., Qian, W., Bu, X., Liu, B.: Composite prescribed performance control of small unmanned aerial vehicles using modified nonlinear disturbance observer. ISA Trans. 116, 30–45 (2021)

    Google Scholar 

  27. Tran, T.T., Ge, S.S., He, W.: Adaptive control for an uncertain robotic manipulator with input saturations. Control Theory Technol. 14(2), 113–121 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Gao, X., Li, X., Sun, Y., Hao, L., Yang, H., Xiang, C.: Model-free tracking control of continuum manipulators with global stability and assigned accuracy. IEEE Trans. Syst. Man CY-S 52(2), 1345–1355 (2020)

    Google Scholar 

  29. Huang, X., Dong, Z., Zhang, K., Zhang, F., Zhang, L.: Approximation-free attitude fault-tolerant tracking control of rigid spacecraft with global stability and appointed accuracy. IEEE Trans. Aerosp. Elec. Syst. (2023). https://doi.org/10.1109/TAES.2023.3284415

    Article  Google Scholar 

  30. Hu, C., Gao, H., Guo, J., Taghavifar, H., Qin, Y., Na, J., Wei, C.: RISE-based integrated motion control of autonomous ground vehicles with asymptotic prescribed performance. IEEE Trans. Syst. Man CY-S 51(9), 5336–5348 (2019)

    Google Scholar 

  31. Cui, Q., Cao, H., Wang, Y., Song, Y.: Prescribed time tracking control of constrained Euler-Lagrange systems: an adaptive proportional-integral solution. Int. J. Robust Nonlin. 32(18), 9723–9741 (2022)

    MathSciNet  Google Scholar 

  32. Zhu, C., Zhang, E., Li, J., Huang, B., Su, Y.: Approximation-free appointed-time tracking control for marine surface vessel with actuator faults and input saturation. Ocean Eng. 245, 110468 (2022)

    Google Scholar 

  33. Cao, Y., Cao, J., Song, Y.: Practical prescribed time tracking control over infinite time interval involving mismatched uncertainties and non-vanishing disturbances. Automatica 136, 110050 (2022)

    MathSciNet  MATH  Google Scholar 

  34. Wei, C., Chen, Q., Liu, J., Yin, Z., Luo, J.: An overview of prescribed performance control and its application to spacecraft attitude system. P. I. Mech. Eng. I-J Syst. 235(4), 435–447 (2021)

    Google Scholar 

  35. Zhou, Q., Wang, L., Wu, C., Li, H., Du, H.: Adaptive fuzzy control for nonstrict-feedback systems with input saturation and output constraint. IEEE Trans. Syst. Man CY-S 47(1), 1–12 (2016)

    Google Scholar 

  36. Xiao, B., Cao, L., Xu, S., Liu, L.: Robust tracking control of robot manipulators with actuator faults and joint velocity measurement uncertainty. IEEE/ASME T. Mech. 25(3), 1354–1365 (2020)

    Google Scholar 

  37. Yang, C., Huang, D., He, W., Cheng, L.: Neural control of robot manipulators with trajectory tracking constraints and input saturation. IEEE Trans. Neur. Net. Lear. 32(9), 4231–4242 (2020)

    MathSciNet  Google Scholar 

  38. Zhou, Q., Wang, L., Wu, C., Li, H., Du, H.: Adaptive fuzzy control for nonstrict-feedback systems with input saturation and output constraint. IEEE Trans. Syst. Man CY-S 47(1), 1–12 (2016)

    Google Scholar 

  39. Hsia, T.S., Lasky, T.A., Guo, Z.: Robust independent joint controller design for industrial robot manipulators. IEEE Trans. Ind. Electron. 38(1), 21–25 (1991)

    Google Scholar 

  40. Jin, M., Lee, J., Chang, P.H., Choi, C.: Practical nonsingular terminal sliding-mode control of robot manipulators for high-accuracy tracking control. IEEE Trans. Ind. Electron. 56(9), 3593–3601 (2009)

    Google Scholar 

  41. Baek, J., Jin, M., Han, S.: A new adaptive sliding-mode control scheme for application to robot manipulators. IEEE Trans. Ind. Electron. 63(6), 3628–3637 (2016)

    Google Scholar 

  42. Cho, G.R., Chang, P.H., Park, S.H., Jin, M.: Robust tracking under nonlinear friction using time-delay control with internal model. IEEE Trans. Control Syst. Tech. 17(6), 1406–1414 (2009)

    Google Scholar 

  43. Kostrykin, V., Oleynik, A.: An intermediate value theorem for monotone operators in ordered Banach spaces. Fixed Pt. Theory Appl. 1, 1–4 (2012)

    MATH  Google Scholar 

  44. Bechlioulis, C.P., Rovithakis, G.A.: Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans. Automat. Contr. 53(9), 2090–2099 (2008)

    MathSciNet  MATH  Google Scholar 

  45. Qu, Z., Dorsey, J.: Robust tracking control of robots by a linear feedback law. IEEE Trans. Automat. Contr. 36(9), 1081–1084 (1991)

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Funding

This work is supported by the National Nature Science Foundation of China under Grant 62103164, GuangDong Basic and Applied Basic Research Foundation (No. 2020A1515110090 and No. 2019A1515110352), the Open project of Ji Hua lab project (X190021TB194, X190021TB190, X201201XB200, X201301XB200).

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Authors and Affiliations

  1. Ji Hua Laboratory, Foshan, China

    Xiu-Wei Huang & Peng Yang

  2. Academy for Engineering and Technology-AI and Robot, Fudan University, Shanghai, China

    Zhi-Yan Dong & Li-Hua Zhang

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  1. Xiu-Wei Huang
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  2. Zhi-Yan Dong
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Correspondence to Zhi-Yan Dong.

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Appendix

Appendix

For the definition of \({\dot{\alpha }}(t)\) in (40), we will calculate the detailed value of \(\frac{\partial \alpha _{1}}{\partial x_{1}},\frac{\partial \alpha _{1}}{\partial \beta _{1}}\ \)and \(\frac{\partial \alpha _{1} }{\partial {\dot{\beta }}_{1}}\). First, let us consider \(\frac{\partial \alpha _{1} }{\partial x_{1}}.\)

$$\begin{aligned} \frac{\partial \alpha _{1}}{\partial x_{1}}=\frac{\partial \alpha _{1}}{\partial \xi _{1}}\frac{\partial \xi _{1}}{\partial x_{1}}+\frac{\partial \alpha _{1}}{\partial \mu _{1}}\frac{\partial \mu _{1}}{\partial x_{1}} +\frac{\partial \alpha _{1}}{\partial v_{1}}\frac{\partial v_{1}}{\partial x_{1}} \end{aligned}$$
(57)

with

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial \alpha _{1}}{\partial \xi _{1}}=-k_{1}\mu _{1}^{-1},\\ \frac{\partial \alpha _{1}}{\partial \mu _{1}}=k_{1}(\mu _{1}^{2})^{-1} \text {diag}\{ \xi _{11},\xi _{12},\cdots ,\xi _{1n}\}\\ \quad +(\mu _{1}^{2})^{-1} \text {diag}\{v_{11},v_{12},\cdots ,v_{1n}\},\\ \frac{\partial \alpha _{1}}{\partial v_{1}}=-\mu _{1}^{-1} \end{array} \right. \end{aligned}$$
(58)

and

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial \xi _{1}}{\partial x_{1}}=\frac{\partial \xi _{1}}{\partial e_{1} }\frac{\partial e_{1}}{\partial x_{1}}+\frac{\partial \xi _{1}}{\partial h} \frac{\partial h}{\partial x_{1}}=h^{-1}-(h^{2})^{-1}\frac{\partial h}{\partial x_{1}},\\ \frac{\partial \mu _{1}}{\partial x_{1}}=\frac{\partial \mu _{1}}{\partial h} \frac{\partial h}{\partial x_{1}}+\frac{\partial \mu _{1}}{\partial \gamma } \frac{\partial \gamma }{\partial x_{1}}+\frac{\partial \mu _{1}}{\partial e_{1} }\frac{\partial e_{1}}{\partial x_{1}}\\ { \ \ \ }\quad =\frac{\partial \mu _{1}}{\partial h}\frac{\partial h}{\partial x_{1}}+\frac{\partial \mu _{1}}{\partial \gamma }\frac{\partial \gamma }{\partial e_{1}}+\frac{\partial \mu _{1}}{\partial e_{1}},\\ \frac{\partial v_{1}}{\partial x_{1}}=\frac{\partial v_{1}}{\partial h} \frac{\partial h}{\partial x_{1}}+\frac{\partial v_{1}}{\partial \gamma } \frac{\partial \gamma }{\partial x_{1}}+\frac{\partial v_{1}}{\partial e_{1} }\frac{\partial e_{1}}{\partial x_{1}}\\ { \ \ \ }\quad =\frac{\partial v_{1}}{\partial h}\frac{\partial h}{\partial x_{1}}+\frac{\partial v_{1}}{\partial \gamma }\frac{\partial \gamma }{\partial e_{1}}+\frac{\partial v_{1}}{\partial e_{1}} \end{array} \right. \end{aligned}$$
(59)

For

$$\begin{aligned} \frac{\partial h}{\partial x_{1}}=\frac{\partial h}{\partial \gamma } \frac{\partial \gamma }{\partial e_{1}}\frac{\partial e_{1}}{\partial x_{1} }=\frac{\partial h}{\partial \gamma }\frac{\partial \gamma }{\partial e_{1}} \end{aligned}$$
(60)

where

$$\begin{aligned} \frac{\partial \gamma }{\partial e_{1}}=\text {diag}\left\{ \left( \frac{\partial \gamma }{\partial e_{1}}\right) _{1},\left( \frac{\partial \gamma }{\partial e_{1}}\right) _{2},\cdots ,\left( \frac{\partial \gamma }{\partial e_{1}}\right) _{n}\right\} \nonumber \\ \end{aligned}$$
(61)

with

$$\begin{aligned} \left( \frac{\partial \gamma }{\partial e_{1}}\right) _{i}=c({\bar{\delta }} _{1i}-{\underline{\delta }}_{1i})\beta _{1i}(t)-2ce_{1i}(t),i=1,2,\cdots ,n \end{aligned}$$

And for \(\frac{\partial h}{\partial \gamma },\) we have \(\frac{\partial h}{\partial \gamma }=\) diag\(\left\{ \left( \frac{\partial h}{\partial \gamma }\right) _{1},\left( \frac{\partial h}{\partial \gamma }\right) _{2},\cdots ,\left( \frac{\partial h}{\partial \gamma }\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial h}{\partial \gamma }\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} -\frac{2p}{a_{1i}}\left( \frac{\gamma _{1i}(t)}{a_{1i}}-1\right) ^{2p-1},\\ 0, \end{array} \right. \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array},\nonumber \\{} & {} i=1,2,\cdots ,n \end{aligned}$$
(62)

Now, for \(\frac{\partial \mu _{1}}{\partial h},\) we have \(\frac{\partial \mu _{1} }{\partial h}=\) diag\(\left\{ \left( \frac{\partial \mu _{1}}{\partial h}\right) _{1},\left( \frac{\partial \mu _{1}}{\partial h}\right) _{2},\cdots ,\left( \frac{\partial \mu _{1}}{\partial h}\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial \mu _{1}}{\partial h}\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} -\frac{1}{h^{2}(\gamma _{1i}(t))}-\frac{4pc}{a_{1i}h^{3}(\gamma _{1i} (t))}\left( \frac{\gamma _{1i}(t)}{a_{1i}}-1\right) ^{2p-1}\varXi _{1},\\ 0, \end{array} \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array} \right. \nonumber \\ \end{aligned}$$
(63)

where \(\varXi _{1}=(-2e_{1i}(t)+({\bar{\delta }}_{1i}-{\underline{\delta }}_{1i} )\beta _{1i}(t))e_{1i}(t)\) for \(i=1,2,\cdots ,n.\) As for \(\frac{\partial \mu _{1}}{\partial \gamma },\) we have \(\frac{\partial \mu _{1}}{\partial \gamma }=\) diag\(\left\{ \left( \frac{\partial \mu _{1}}{\partial \gamma }\right) _{1},\left( \frac{\partial \mu _{1} }{\partial \gamma }\right) _{2},\cdots ,\left( \frac{\partial \mu _{1}}{\partial \gamma }\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial \mu _{1}}{\partial \gamma }\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} \frac{2pc(2p-1)}{a_{1i}^{2}h^{2}(\gamma _{1i}(t))}\left( \frac{\gamma _{1i} (t)}{a_{1i}}-1\right) ^{2p-2}\varXi _{1},\\ 0, \end{array} \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array} \right. \nonumber \\ \end{aligned}$$
(64)

\(i=1,2,\cdots ,n.\) As for \(\frac{\partial \mu _{1}}{\partial e_{1}},\) we have \(\frac{\partial \mu _{1}}{\partial e_{1}}=\) diag\(\left\{ \left( \frac{\partial \mu _{1}}{\partial e_{1}}\right) _{1},\left( \frac{\partial \mu _{1} }{\partial e_{1}}\right) _{2},\cdots ,\left( \frac{\partial \mu _{1}}{\partial e_{1}}\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial \mu _{1}}{\partial e_{1}}\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} \frac{2pc}{a_{1i}h^{2}(\gamma _{1i}(t))}\left( \frac{\gamma _{1i}(t)}{a_{1i} }-1\right) ^{2p-1}\varXi _{2},\\ 0, \end{array} \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array} \right. \nonumber \\ \end{aligned}$$
(65)

where \(\varXi _{2}=-4e_{1i}(t)+({\bar{\delta }}_{1i}-{\underline{\delta }}_{1i} )\beta _{1i}(t)\) for \(i=1,2,\cdots ,n.\) As for \(\frac{\partial v_{1}}{\partial h},\) we have \(\frac{\partial v_{1}}{\partial h}=\) diag\(\left\{ \left( \frac{\partial v_{1}}{\partial h}\right) _{1},\left( \frac{\partial v_{1}}{\partial h}\right) _{2},\cdots ,\left( \frac{\partial v_{1}}{\partial h}\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial v_{1}}{\partial h}\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} -\frac{4pc}{a_{1i}h^{3}(\gamma _{1i}(t))}\left( \frac{\gamma _{1i}(t)}{a_{1i} }-1\right) ^{2p-1}\varXi _{3},\\ 0, \end{array} \right. \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array} \nonumber \\ \end{aligned}$$
(66)

where \(\varXi _{3}=(2{\bar{\delta }}_{1i}{\underline{\delta }}_{1i}\beta _{1i} (t)+({\bar{\delta }}_{1i}-{\underline{\delta }}_{1i})e_{1i}(t))e_{1i}(t)\dot{\beta }_{1i}(t)\) for \(i=1,2,\cdots ,n.\) As for \(\frac{\partial v_{1}}{\partial \gamma },\) we have \(\frac{\partial v_{1}}{\partial \gamma }=\) diag\(\left\{ \left( \frac{\partial v_{1}}{\partial \gamma }\right) _{1},\left( \frac{\partial v_{1}}{\partial \gamma }\right) _{2},\cdots ,\left( \frac{\partial v_{1}}{\partial \gamma }\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial v_{1}}{\partial \gamma }\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} \frac{2pc(2p-1)}{a_{1i}^{2}h^{2}(\gamma _{1i}(t))}\left( \frac{\gamma _{1i} (t)}{a_{1i}}-1\right) ^{2p-2}\varXi _{3},\\ 0, \end{array} \right. \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array} \nonumber \\ \end{aligned}$$
(67)

\(i=1,2,\cdots ,n.\) As for \(\frac{\partial v_{1}}{\partial e_{1}},\) we have \(\frac{\partial v_{1}}{\partial e_{1}}=\) diag\(\left\{ \left( \frac{\partial v_{1}}{\partial e_{1}}\right) _{1},\left( \frac{\partial v_{1}}{\partial e_{1}}\right) _{2},\cdots ,\left( \frac{\partial v_{1}}{\partial e_{1} }\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial v_{1}}{\partial e_{1}}\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} \frac{2pc}{a_{1i}h^{2}(\gamma _{1i}(t))}\left( \frac{\gamma _{1i}(t)}{a_{1i} }-1\right) ^{2p-1}\varXi _{3},\\ 0, \end{array} \right. \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array} \nonumber \\ \end{aligned}$$
(68)

with \(\varXi _{4}=(2{\bar{\delta }}_{1i}{\underline{\delta }}_{1i}\beta _{1i} (t)+2({\bar{\delta }}_{1i}-{\underline{\delta }}_{1i})e_{1i}(t)){\dot{\beta }}_{1i}(t)\) for \(i=1,2,\cdots ,n.\)

Secondly, we consider \(\frac{\partial \alpha _{1}}{\partial \beta _{1}}.\) For

$$\begin{aligned} \frac{\partial \alpha _{1}}{\partial \beta _{1}}=\frac{\partial \alpha _{1} }{\partial \xi _{1}}\frac{\partial \xi _{1}}{\partial \beta _{1}}+\frac{\partial \alpha _{1}}{\partial \mu _{1}}\frac{\partial \mu _{1}}{\partial \beta _{1} }+\frac{\partial \alpha _{1}}{\partial v_{1}}\frac{\partial v_{1}}{\partial \beta _{1}} \end{aligned}$$
(69)

with

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial \xi _{1}}{\partial \beta _{1}}=\frac{\partial \xi _{1}}{\partial e_{1}}\frac{\partial e_{1}}{\partial \beta _{1}}+\frac{\partial \xi _{1}}{\partial h}\frac{\partial h}{\partial \beta _{1}}=-(h^{2})^{-1}\frac{\partial h}{\partial \gamma }\frac{\partial \gamma }{\partial \beta _{1}},\\ {\frac{\partial \mu _{1}}{\partial \beta _{1}}=\frac{\partial \mu _{1}}{\partial h}\frac{\partial h}{\partial \beta _{1}}+\frac{\partial \mu _{1}}{\partial \gamma }\frac{\partial \gamma }{\partial \beta _{1}}+\frac{\partial \mu _{1}}{\partial e_{1}}\frac{\partial e_{1}}{\partial \beta _{1}}}\\ \qquad {=\frac{\partial \mu _{1}}{\partial h} \frac{\partial h}{\partial \gamma }\frac{\partial \gamma }{\partial \beta _{1} }+\frac{\partial \mu _{1}}{\partial \gamma }\frac{\partial \gamma }{\partial \beta _{1}},}\\ {\frac{\partial v_{1}}{\partial \beta _{1}}=\frac{\partial v_{1}}{\partial h}\frac{\partial h}{\partial \beta _{1}}+\frac{\partial v_{1} }{\partial \gamma }\frac{\partial \gamma }{\partial \beta _{1}}+\frac{\partial v_{1}}{\partial e_{1}}\frac{\partial e_{1}}{\partial \beta _{1}}}\\ \qquad {=\frac{\partial v_{1}}{\partial h} \frac{\partial h}{\partial \gamma }\frac{\partial \gamma }{\partial \beta _{1} }+\frac{\partial v_{1}}{\partial \gamma }\frac{\partial \gamma }{\partial \beta _{1}}} \end{array} \right. \end{aligned}$$
(70)

where \(\frac{\partial \gamma }{\partial \beta _{1}}=\) diag\(\left\{ \left( \frac{\partial \gamma }{\partial \beta _{1}}\right) _{1},\left( \frac{\partial \gamma }{\partial \beta _{1}}\right) _{2},\cdots ,\left( \frac{\partial \gamma }{\partial \beta _{1}}\right) _{n}\right\} \) with

$$\begin{aligned} \left( \frac{\partial \gamma }{\partial \beta _{1}}\right) _{i}= & {} c\underline{\delta }_{1i}({\bar{\delta }}_{1i}\beta _{1i}(t)-e_{1i}(t))\nonumber \\{} & {} +c{\bar{\delta }} _{1i}({\underline{\delta }}_{1i}\beta _{1i}(t)+e_{1i}(t)) \end{aligned}$$
(71)

Finally, we consider \(\frac{\partial \alpha _{1}}{\partial {\dot{\beta }}_{1}}.\) For

$$\begin{aligned} \frac{\partial \alpha _{1}}{\partial {\dot{\beta }}_{1}}=\frac{\partial \alpha _{1} }{\partial \xi _{1}}\frac{\partial \xi _{1}}{\partial {\dot{\beta }}_{1}} +\frac{\partial \alpha _{1}}{\partial \mu _{1}}\frac{\partial \mu _{1}}{\partial {\dot{\beta }}_{1}}+\frac{\partial \alpha _{1}}{\partial v_{1}}\frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}} \end{aligned}$$
(72)

with

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial \xi _{1}}{\partial {\dot{\beta }}_{1}}=\frac{\partial \xi _{1} }{\partial e_{1}}\frac{\partial e_{1}}{\partial {\dot{\beta }}_{1}}+\frac{\partial \xi _{1}}{\partial h}\frac{\partial h}{\partial {\dot{\beta }}_{1}}=0,\\ \frac{\partial \mu _{1}}{\partial {\dot{\beta }}_{1}}=\frac{\partial \mu _{1} }{\partial h}\frac{\partial h}{\partial {\dot{\beta }}_{1}}+\frac{\partial \mu _{1} }{\partial \gamma }\frac{\partial \gamma }{\partial {\dot{\beta }}_{1}}+\frac{\partial \mu _{1}}{\partial e_{1}}\frac{\partial e_{1}}{\partial {\dot{\beta }} _{1}}=0,\\ \frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}}=\frac{\partial v_{1}}{\partial h}\frac{\partial h}{\partial {\dot{\beta }}_{1}}+\frac{\partial v_{1}}{\partial \gamma }\frac{\partial \gamma }{\partial {\dot{\beta }}_{1}}+\frac{\partial v_{1}}{\partial e_{1}}\frac{\partial e_{1}}{\partial {\dot{\beta }}_{1}}+\left( \frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}}\right) \frac{\partial {\dot{\beta }}_{1}}{\partial {\dot{\beta }}_{1}}\\ =\left( \frac{\partial v_{1} }{\partial {\dot{\beta }}_{1}}\right) \end{array} \right. \end{aligned}$$
(73)

where \(\left( \frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}}\right) =\) diag\(\left\{ \left( \frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}}\right) _{1},\left( \frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}}\right) _{2},\cdots ,\left( \frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}}\right) _{n}\right\} \) with

$$\begin{aligned}{} & {} \left( \frac{\partial v_{1}}{\partial {\dot{\beta }}_{1}}\right) _{i}\nonumber \\{} & {} \quad =\left\{ \begin{array}{l} \frac{2pc}{a_{1i}h^{2}(\gamma _{1i}(t))}\left( \frac{\gamma _{1i}(t)}{a_{1i} }-1\right) ^{2p-1}\varXi _{5},\\ 0, \end{array} \right. \begin{array}{l} 0<\gamma _{1i}(t)\le a_{1i}\\ \gamma _{1i}(t)>a_{1i} \end{array} \nonumber \\ \end{aligned}$$
(74)

with \(\varXi _{5}=(2{\bar{\delta }}_{1i}{\underline{\delta }}_{1i}\beta _{1i} (t)+({\bar{\delta }}_{1i}-{\underline{\delta }}_{1i})e_{1i}(t))e_{1i}(t)\)

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Huang, XW., Dong, ZY., Yang, P. et al. Model-free adaptive trajectory tracking control of robotic manipulators with practical prescribed-time performance. Nonlinear Dyn 111, 20015–20039 (2023). https://doi.org/10.1007/s11071-023-08894-w

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