# A Robust Noise-Free Linear Control Design for Robot Manipulator with Uncertain System Parameters

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

**Remark**

**1.**

## 3. Preliminary

**Definition**

**1**

**.**The modal matrix of a square matrix A is one whose columns comprise the entire eigenvectors of A.

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

**Lemma**

**2**

**.**Let k be a nonnegative constant. If the function $m\left(t\right)$ satisfies the integral inequality

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Example**

**1.**

## 4. Stability Analysis

**Theorem**

**2.**

**Proof.**

**Remark**

**4.**

## 5. Noise-Free Control Design

**Theorem**

**3.**

**Proof.**

**Remark**

**5.**

## 6. Application to a Two DOF Robot Manipulator

## 7. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Lewis, F.L.; Dawson, D.M.; Abdallah, C.T. Robot Manipulator Control: Theory and Practice; CRC Press: Boca Raton, FL, USA, 2003. [Google Scholar]
- Callier, F.M.; Desoer, C.A. Linear System Theory; Springer Science & Business Media: Boston, NY, USA, 2012; Chapter 7.2. [Google Scholar]
- Slotine, J.J.E.; Li, W. On the adaptive control of robot manipulators. Int. J. Robot. Res.
**1987**, 6, 49–59. [Google Scholar] [CrossRef] - Zhang, D.; Wei, B. Adaptive Control for Robotic Manipulators; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Jin, M.; Lee, J.; Tsagarakis, N.G. Model-free robust adaptive control of humanoid robots with flexible joints. IEEE Trans. Ind. Electron.
**2016**, 64, 1706–1715. [Google Scholar] [CrossRef] - Slotine, J.J.; Sastry, S.S. Tracking control of non-linear systems using sliding surfaces, with application to robot manipulators. Int. J. Control
**1983**, 38, 465–492. [Google Scholar] [CrossRef][Green Version] - Doulgeri, Z. Sliding regime of a nonlinear robust controller for robot manipulators. IEE Proc. Control Theory Appl.
**1999**, 146, 493–498. [Google Scholar] [CrossRef] - Fu, L.C.; Liao, T.L. Systems Using Variable Structure Control and with an Application to a Robotic. IEEE Trans. Autom. Control
**1990**, 35, 1345–1350. [Google Scholar] [CrossRef] - Lin, C.J. Variable structure model following control of robot manipulators with high-gain observer. JSME Int. J. Ser. Mech. Syst. Mach. Elem. Manuf.
**2004**, 47, 591–601. [Google Scholar] [CrossRef] - Islam, S.; Liu, X.P. Robust sliding mode control for robot manipulators. IEEE Trans. Ind. Electron.
**2010**, 58, 2444–2453. [Google Scholar] [CrossRef] - Navvabi, H.; Markazi, A.H. New AFSMC method for nonlinear system with state-dependent uncertainty: Application to hexapod robot position control. J. Intell. Robot. Syst.
**2019**, 95, 61–75. [Google Scholar] [CrossRef] - He, W.; Dong, Y.; Sun, C. Adaptive neural impedance control of a robotic manipulator with input saturation. IEEE Trans. Syst. Man Cybern. Syst.
**2015**, 46, 334–344. [Google Scholar] [CrossRef] - Jin, L.; Li, S.; Yu, J.; He, J. Robot manipulator control using neural networks: A survey. Neurocomputing
**2018**, 285, 23–34. [Google Scholar] [CrossRef] - Elsisi, M.; Mahmoud, K.; Lehtonen, M.; Darwish, M.M. An improved neural network algorithm to efficiently track various trajectories of robot manipulator arms. IEEE Access
**2021**, 9, 11911–11920. [Google Scholar] [CrossRef] - Utkin, V. Variable structure systems with sliding modes. IEEE Trans. Autom. Control
**1977**, 22, 212–222. [Google Scholar] [CrossRef] - Hung, J.Y.; Gao, W.; Hung, J.C. Variable structure control: A survey. IEEE Trans. Ind. Electron.
**1993**, 40, 2–22. [Google Scholar] [CrossRef][Green Version] - Khalil, H.K. Nonlinear Systems; Prentice-Hall: Upper Saddle River, NJ, USA, 2002; Chapter 14. [Google Scholar]
- Fridman, L.M. An averaging approach to chattering. IEEE Trans. Autom. Control
**2001**, 46, 1260–1265. [Google Scholar] [CrossRef] - Boiko, I. Discontinuous Control Systems: Frequency-Domain Analysis and Design; Springer Science & Business Media: Boston, NY, USA, 2008. [Google Scholar]
- Burton, J.; Zinober, A.S. Continuous approximation of variable structure control. Int. J. Syst. Sci.
**1986**, 17, 875–885. [Google Scholar] [CrossRef] - Levant, A. Higher-order sliding modes, differentiation, and output-feedback control. Int. J. Control
**2003**, 76, 924–941. [Google Scholar] [CrossRef] - Tayebi-Haghighi, S.; Piltan, F.; Kim, J.M. Robust composite high-order super-twisting sliding mode control of robot manipulators. Robotics
**2018**, 7, 13. [Google Scholar] [CrossRef][Green Version] - Ahmed, S.; Wang, H.; Tian, Y. Adaptive high-order terminal sliding mode control based on time delay estimation for the robotic manipulators with backlash hysteresis. IEEE Trans. Syst. Man Cybern. Syst.
**2019**, 51, 1128–1137. [Google Scholar] [CrossRef] - Ahmed, S.; Wang, H.; Tian, Y. Adaptive Fractional High-order Terminal Sliding Mode Control for Nonlinear Robotic Manipulator under Alternating Loads. Asian J. Control
**2020**. [Google Scholar] [CrossRef] - Brahmi, B.; Driscoll, M.; Laraki, M.H.; Brahmi, A. Adaptive high-order sliding mode control based on quasi-time delay estimation for uncertain robot manipulator. Control Theory Technol.
**2020**, 18, 279–292. [Google Scholar] [CrossRef] - Yeh, Y.L.; Chen, M.S. Frequency domain analysis of noise-induced control chattering in sliding mode controls. Int. J. Robust Nonlinear Control
**2011**, 21, 1975–1980. [Google Scholar] [CrossRef] - Oliveira, T.R.; Estrada, A.; Fridman, L.M. Global and exact HOSM differentiator with dynamic gains for output-feedback sliding mode control. Automatica
**2017**, 81, 156–163. [Google Scholar] [CrossRef] - Chen, B.S.; Wong, C.C. Robust linear controller design: Time domain approach. IEEE Trans. Autom. Control
**1987**, 32, 161–164. [Google Scholar] [CrossRef] - Sobel, K.M.; Bandaj, S.S.; Yeh, H.H. Robust control for linear systems with structured state space uncertainty. Int. J. Control
**1989**, 50, 1991–2004. [Google Scholar] [CrossRef] - Siciliano, B.; Sciavicco, L.; Villani, L.; Oriolo, G. Robotics: Modelling, Planning and Control; Springer Science & Business Media: Boston, NY, USA, 2010; Chapter 9.3. [Google Scholar]
- Draženović, B. The invariance conditions in variable structure systems. Automatica
**1969**, 5, 287–295. [Google Scholar] [CrossRef] - Bronson, R.; Saccoman, J.T.; Costa, G.B. Linear Algebra: Algorithms, Applications, and Techniques; Academic Press: Cambridge, MA, USA, 2013; Chapter 4.3. [Google Scholar]
- Chen, C.T. Linear System Theory and Design; Holt, Rinehart and Winston: New York, NY, USA, 1984; Chapter 4.4. [Google Scholar]
- Chen, M.S.; Chen, C.H.; Yang, F.Y. An LTR-observer-based dynamic sliding mode control for chattering reduction. Automatica
**2007**, 43, 1111–1116. [Google Scholar] [CrossRef] - Vidyasagar, M. Nonlinear Systems Analysis; Prentice-Hall: Englewood Cliffs, NJ, USA, 1993; Chapter 5.4. [Google Scholar]

**Figure 4.**Time history of system outputs and references: (

**a**) ${q}_{1}$ and ${q}_{{d}_{1}}$, (

**b**) ${q}_{2}$ and ${q}_{{d}_{2}}$ (sliding-mode control).

**Figure 6.**Time history of system outputs and references: (

**a**) ${q}_{1}$ and ${q}_{{d}_{1}}$, (

**b**) ${q}_{2}$ and ${q}_{{d}_{2}}$ (robust linear control).

**Figure 8.**Time history of system outputs and references: (

**a**) ${q}_{1}$ and ${q}_{{d}_{1}}$, (

**b**) ${q}_{2}$ and ${q}_{{d}_{2}}$ (robust linear control with noise).

**Figure 9.**Time history of control signals: (

**a**) ${u}_{1}$ and (

**b**) ${u}_{2}$ (robust linear control with noise).

**Figure 10.**Time history of system outputs and references: (

**a**) ${q}_{1}$ and ${q}_{{d}_{1}}$, (

**b**) ${q}_{2}$ and ${q}_{{d}_{2}}$ (robust noise-free linear control).

**Figure 11.**Time history of control signals: (

**a**) ${u}_{1}$ and (

**b**) ${u}_{2}$ (robust noise-free linear control).

Parameters | Value |
---|---|

${l}_{1}$ | 1 m |

${l}_{2}$ | 2 m |

${m}_{1}$ | 1 kg |

${m}_{2}$ | 1 kg |

${l}_{c1}$ | 1 m |

${l}_{c2}$ | 2 m |

${f}_{1}$ | $0.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}/\mathrm{rad}$ |

${f}_{2}$ | $0.3\phantom{\rule{3.33333pt}{0ex}}\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}/\mathrm{rad}$ |

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Yeh, Y.-L. A Robust Noise-Free Linear Control Design for Robot Manipulator with Uncertain System Parameters. *Actuators* **2021**, *10*, 121.
https://doi.org/10.3390/act10060121

**AMA Style**

Yeh Y-L. A Robust Noise-Free Linear Control Design for Robot Manipulator with Uncertain System Parameters. *Actuators*. 2021; 10(6):121.
https://doi.org/10.3390/act10060121

**Chicago/Turabian Style**

Yeh, Yi-Liang. 2021. "A Robust Noise-Free Linear Control Design for Robot Manipulator with Uncertain System Parameters" *Actuators* 10, no. 6: 121.
https://doi.org/10.3390/act10060121