# Computational Efficiency-Based Adaptive Tracking Control for Robotic Manipulators with Unknown Input Bouc–Wen Hysteresis

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Trajectory Tracking Design for Robotic Manipulators with Unknown Input Hysteresis

#### 3.1. Control Method

**Lemma**

**1.**

**Lemma**

**2.**

**.**Let a function $f(t)\in {\mathcal{C}}^{1}(a,\infty )$ and ${lim}_{t\to \infty}f(t)=a$, where $a<\infty $. If ${f}^{\prime}$ is uniformly continuous, then ${lim}_{t\to \infty}{f}^{\prime}(t)=0$.

#### 3.2. Controller Design

#### 3.3. Stability Analysis

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Simulation Example

#### 4.1. Fixed-Point Control Using the Proposed Adaptive Control

#### 4.2. Tracking Control Using the Proposed Adaptive Control

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Siciliano, B.; Khatib, O. Springer Handbook of Robotics; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Lewis, F.L.; Dawson, D.M.; Abdallah, C.T. Robot Manipulator Control: Theory and Practice; Marcel Dekker: New York, NY, USA, 2003. [Google Scholar]
- Slotine, J.J.E.; Li, W. Applied Nonlinear Control; Prentice-Hall: Englewood Cliffs, NJ, USA, 1991. [Google Scholar]
- Spong, M.W.; Vidyasagar, M. Robot Dynamics and Control; John Wiley & Sons: New York, NY, USA, 2008. [Google Scholar]
- Tran, D.T.; Truong, H.V.A.; Ahn, K.K. Adaptive Backstepping Sliding Mode Control Based RBFNN for a Hydraulic Manipulator Including Actuator Dynamics. Appl. Sci.
**2019**, 9, 1265. [Google Scholar] [CrossRef] - Li, Z.; Yang, Z.; Xie, S. Computing Resource Trading for Edge-Cloud-assisted Internet of Things. IEEE Trans. Ind. Inform.
**2019**, 15, 3661–3669. [Google Scholar] [CrossRef] - Liu, Y.; Yang, C.; Jiang, L.; Xie, S.; Zhang, Y. Intelligent Edge Computing for IoT-Based Energy Management in Smart Cities. IEEE Netw.
**2019**, 33, 111–117. [Google Scholar] [CrossRef] - Yan, J.; Ban, H.; Luo, X.; Zhao, H.; Guan, X. Joint Localization and Tracking Design for AUV With Asynchronous Clocks and State Disturbances. IEEE Trans. Veh. Technol.
**2019**, 68, 4707–4720. [Google Scholar] [CrossRef] - Vo, A.T.; Kang, H.J. An Adaptive Neural Non-Singular Fast-Terminal Sliding-Mode Control for Industrial Robotic Manipulators. Appl. Sci.
**2018**, 8, 2562. [Google Scholar] [CrossRef] - Zhou, J.; Zhang, C.; Wen, C. Robust adaptive output control of uncertain nonlinear plants with unknown backlash nonlinearity. IEEE Trans. Autom. Control
**2007**, 52, 503–509. [Google Scholar] [CrossRef] - Fu, M.; Xie, L. The sector bound approach to quantized feedback control. IEEE Trans. Autom. Control
**2005**, 50, 1698–1711. [Google Scholar] [Green Version] - Fu, M.; Xie, L. Finite-Level Quantized Feedback Control for Linear Systems. IEEE Trans. Autom. Control
**2009**, 54, 1165–1170. [Google Scholar] [CrossRef] [Green Version] - Hayakawa, T.; Ishii, H.; Tsumura, K. Adaptive quantized control for nonlinear uncertain systems. Syst. Control Lett.
**2009**, 58, 625–632. [Google Scholar] [CrossRef] - Zhou, J.; Wen, C.; Yang, G. Adaptive backstepping stabilization of nonlinear uncertain systems with quantized input signal. IEEE Trans. Autom. Control
**2014**, 59, 460–464. [Google Scholar] [CrossRef] - Chen, C.; Wen, C.; Liu, Z.; Xie, K.; Zhang, Y.; Chen, C.L.P. Adaptive Consensus of Nonlinear Multi-Agent Systems with Non-Identical Partially Unknown Control Directions and Bounded Modelling Errors. IEEE Trans. Autom. Control
**2017**, 62, 4654–4659. [Google Scholar] [CrossRef] - Chen, C.; Xie, K.; Lewis, F.L.; Xie, S.; Davoudi, A. Fully Distributed Resilience for Adaptive Exponential Synchronization of Heterogeneous Multi-Agent Systems Against Actuator Faults. IEEE Trans. Autom. Control
**2018**. [Google Scholar] [CrossRef] - Xie, K.; Chen, C.; Lewis, F.L.; Xie, S. Adaptive Asymptotic Neural Network Control of Nonlinear Systems With Unknown Actuator Quantization. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 6303–6312. [Google Scholar] [CrossRef] - Chen, C.; Lewis, F.L.; Xie, S.; Modares, H.; Liu, Z.; Zuo, S.; Davoudi, A. Resilient adaptive and H
_{∞}controls of multi-agent systems under sensor and actuator faults. Automatica**2019**, 102, 19–26. [Google Scholar] [CrossRef] - Cao, K.; Li, R. Modeling of Rate-Independent and Symmetric Hysteresis Based on Madelung’s Rules. Sensors
**2019**, 19, 352. [Google Scholar] [CrossRef] [PubMed] - Wen, C.; Zhou, J. Decentralized adaptive stabilization in the presence of unknown backlash-like hysteresis. Automatica
**2007**, 43, 426–440. [Google Scholar] [CrossRef] - Zhou, J.; Wen, C.Y.; Li, T.S. Adaptive output feedback control of uncertain nonlinear systems with hysteresis nonlinearity. IEEE Trans. Autom. Control
**2012**, 57, 2627–2633. [Google Scholar] [CrossRef] - Lin, J.; Chiang, M. Tracking Control of a Magnetic Shape Memory Actuator Using an Inverse Preisach Model with Modified Fuzzy Sliding Mode Control. Sensors
**2016**, 16, 1368. [Google Scholar] [CrossRef] [PubMed] - Yan, J.; Wan, Y.; Luo, X.; Chen, C.; Hua, C.; Guan, X. Formation Control of Teleoperating Cyber-Physical System With Time Delay and Actuator Saturation. IEEE Trans. Control Syst. Technol.
**2018**, 26, 1458–1467. [Google Scholar] [CrossRef] - Bai, X.; Cai, F.; Chen, P. Resistor-capacitor (RC) operator-based hysteresis model for magnetorheological (MR) dampers. Mech. Syst. Signal Process.
**2019**, 117, 157–169. [Google Scholar] [CrossRef] - Chen, P.; Bai, X.; Qian, L.; Choi, S. An Approach for Hysteresis Modeling Based on Shape Function and Memory Mechanism. IEEE/ASME Trans. Mech.
**2018**, 23, 1270–1278. [Google Scholar] [CrossRef] - Nussbaum, R.D. Some remarks on a conjecture in parameter adaptive control. Syst. Control Lett.
**1983**, 3, 243–246. [Google Scholar] [CrossRef] - Mårtensson, B. Remarks on adaptive stabilization of first order non-linear systems. Syst. Control Lett.
**1990**, 14, 1–7. [Google Scholar] [CrossRef] - Ge, S.S.; Wang, J. Robust adaptive neural control for a class of perturbed strict feedback nonlinear systems. IEEE Trans. Neural Netw.
**2002**, 13, 1409–1419. [Google Scholar] [CrossRef] [PubMed] - Ding, Z. Adaptive control of non-linear systems with unknown virtual control coefficients. Int. J. Adapt. Control Signal Process.
**2000**, 14, 505–517. [Google Scholar] [CrossRef] - Ye, X.; Jiang, J. Adaptive nonlinear design without a priori knowledge of control directions. IEEE Trans. Autom. Control
**1998**, 43, 1617–1621. [Google Scholar] - Zhang, Y.; Wen, C.Y.; Soh, Y.C. Adaptive backstepping control design for systems with unknown high-frequency gain. IEEE Trans. Autom. Control
**2000**, 45, 2350–2354. [Google Scholar] [CrossRef] - Ge, S.S.; Wang, J. Robust adaptive tracking for time-varying uncertain nonlinear systems with unknown control coefficients. IEEE Trans. Autom. Control
**2003**, 48, 1463–1469. [Google Scholar] - Kim, B.; Park, B.S. Robust Control for the Segway with Unknown Control Coefficient and Model Uncertainties. Sensors
**2016**, 16, 1000. [Google Scholar] [CrossRef] - Ye, X. Decentralized adaptive regulation with unknown high-frequency-gain signs. IEEE Trans. Autom. Control
**1999**, 44, 2072–2076. [Google Scholar] - Ge, S.S.; Hong, F.; Lee, T.H. Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients. IEEE Trans. Syst. Man Cybern. B Cybern.
**2004**, 34, 499–516. [Google Scholar] [CrossRef] [PubMed] - Ye, X. Decentralized adaptive stabilization of large-scale nonlinear time-delay systems with unknown high-frequency-gain signs. IEEE Trans. Autom. Control
**2011**, 56, 1473–1478. [Google Scholar] [CrossRef] - Chen, W.S.; Li, X.B.; Ren, W.; Wen, C.Y. Adaptive Consensus of Multi-Agent Systems With Unknown Identical Control Directions Based on A Novel Nussbaum-Type Function. IEEE Trans. Autom. Control
**2014**, 59, 1887–1892. [Google Scholar] [CrossRef] - Chen, C.; Liu, Z.; Zhang, Y.; Chen, C.L.P.; Xie, S. Saturated Nussbaum Function based Approach for Robotic Systems with Unknown Actuator Nonlinearities. IEEE Trans. Cybern.
**2016**, 46, 2311–2322. [Google Scholar] [CrossRef] - Yan, J.; Li, X.; Luo, X.; Guan, X. Virtual-Lattice Based Intrusion Detection Algorithm over Actuator-Assisted Underwater Wireless Sensor Networks. Sensors
**2017**, 17, 1168. [Google Scholar] [CrossRef] [PubMed] - Chen, C.; Liu, Z.; Xie, K.; Zhang, Y.; Chen, C.P. Asymptotic adaptive control of nonlinear systems with elimination of overparametrization in a Nussbaum-like design. Automatica
**2018**, 98, 277–284. [Google Scholar] [CrossRef] - Ikhouane, F.; MañOsa, V.; Rodellar, J. Adaptive control of a hysteretic structural system. Automatica
**2005**, 41, 225–231. [Google Scholar] [CrossRef] - Chen, C.; Wen, C.; Liu, Z.; Xie, K.; Zhang, Y.; Chen, C.P. Adaptive asymptotic control of multivariable systems based on a one-parameter estimation approach. Automatica
**2017**, 83, 124–132. [Google Scholar] [CrossRef] - Xie, S.; Yang, L.; Yang, J.M.; Zhou, G.; Xiang, Y. Time-Frequency Approach to Underdetermined Blind Source Separation. IEEE Trans. Neural Netw. Learn. Syst.
**2012**, 23, 306–316. [Google Scholar] [CrossRef] - Zhou, G.; Cichocki, A.; Zhang, Y.; Mandic, D.P. Group Component Analysis for Multiblock Data: Common and Individual Feature Extraction. IEEE Trans. Neural Netw. Learn. Syst.
**2016**, 27, 2426–2439. [Google Scholar] [CrossRef] - He, Z.; Cichocki, A.; Xie, S.; Choi, K. Detecting the Number of Clusters in n-Way Probabilistic Clustering. IEEE Trans. Pattern Anal. Mach. Intell.
**2010**, 32, 2006–2021. [Google Scholar] [CrossRef] [PubMed] [Green Version] - He, Z.; Xie, S.; Zdunek, R.; Zhou, G.; Cichocki, A. Symmetric Nonnegative Matrix Factorization: Algorithms and Applications to Probabilistic Clustering. IEEE Trans. Neural Netw.
**2011**, 22, 2117–2131. [Google Scholar] [CrossRef] [PubMed] - Zhou, G.; Zhao, Q.; Zhang, Y.; Adali, T.; Xie, S.; Cichocki, A. Linked Component Analysis From Matrices to High-Order Tensors: Applications to Biomedical Data. Proc. IEEE
**2016**, 104, 310–331. [Google Scholar] [CrossRef] [Green Version] - Zhou, G.; Cichocki, A.; Xie, S. Fast Nonnegative Matrix/Tensor Factorization Based on Low-Rank Approximation. IEEE Trans. Signal Process.
**2012**, 60, 2928–2940. [Google Scholar] [CrossRef] - Yang, J.; Guo, Y.; Yang, Z.; Xie, S. Under-Determined Convolutive Blind Source Separation Combining Density-Based Clustering and Sparse Reconstruction in Time-Frequency Domain. IEEE Trans. Circuits Syst. I Regul. Pap.
**2019**. [Google Scholar] [CrossRef]

**Figure 1.**Hysteresis nonlinearities simulated using (4).

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xie, K.; Lai, Y.; Li, W.
Computational Efficiency-Based Adaptive Tracking Control for Robotic Manipulators with Unknown Input Bouc–Wen Hysteresis. *Sensors* **2019**, *19*, 2776.
https://doi.org/10.3390/s19122776

**AMA Style**

Xie K, Lai Y, Li W.
Computational Efficiency-Based Adaptive Tracking Control for Robotic Manipulators with Unknown Input Bouc–Wen Hysteresis. *Sensors*. 2019; 19(12):2776.
https://doi.org/10.3390/s19122776

**Chicago/Turabian Style**

Xie, Kan, Yue Lai, and Weijun Li.
2019. "Computational Efficiency-Based Adaptive Tracking Control for Robotic Manipulators with Unknown Input Bouc–Wen Hysteresis" *Sensors* 19, no. 12: 2776.
https://doi.org/10.3390/s19122776