# An Adaptive Second Order Sliding Mode Inverse Kinematics Approach for Serial Kinematic Chain Robot Manipulators

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## Abstract

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## 1. Introduction

## 2. The Traditional Inverse Kinematics Method

## 3. The Proposed Approach

#### 3.1. The Proposed Second Order Sliding Mode-Based Inverse Kinematics Algorithm

#### 3.2. The Proposed Continuous Second Order Sliding Mode-Based Inverse Kinematics Algorithm

## 4. Results

#### 4.1. Three-Link Robot Arm

#### 4.2. Anthropomorphic Robot Arm

#### 4.3. Comparison between CSOSM-AIK and Other Approaches

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Block diagram of the continuous second order sliding mode-based adaptive inverse kinematics methodology (CSOSM-AIK).

**Figure 20.**Anthropomorphic robot manipulator [26].

**Figure 32.**Rigid two- link robot manipulator [40].

**Figure 34.**Trajectory tracking performance as in [40].

**Figure 35.**Tracking error in X coordinate as in [40].

**Figure 36.**Tracking error in Y coordinate as in [40].

Algorithm | Parameters |
---|---|

TIK | ${K}_{p}=diag\left\{450,450,450\right\}$ and ${K}_{d}=diag\left\{450,450,450\right\}$ |

SOSMIK | ${K}_{p}=diag\left\{450,450,450\right\}$, ${K}_{d}=diag\left\{250,250,250\right\}$${K}_{i}=diag\left\{300,300,300\right\}$, ${K}_{1}=diag\left\{150,150,150\right\}$${K}_{r}=diag\left\{600,600,600\right\}$ |

CSOSM-AIK | ${K}_{p}=diag\left\{450,450,450\right\}$, ${K}_{d}=diag\left\{200,200,200\right\}$, ${K}_{i}=diag\left\{400,400,400\right\}$ and ${K}_{1}=diag\left\{150,150,150\right\}$ |

Algorithm | ISE | IAE | ITAE | |||
---|---|---|---|---|---|---|

X-Coordin. | Y-Coordin. | X-Coordin. | Y-Coordin. | X-Coordin. | Y-Coordin. | |

TIK | 3.75 | 1.995 | 4.5878 | 5.7294 | 7.8871 | 7.8772 |

SOSMIK | 0.85 | 0.79 | 2.9450 | 3.6575 | 4.5473 | 4.0754 |

CSOSM-AIK | 0.0632 | 0.0478 | 0.8645 | 1.0122 | 2.2461 | 2.2039 |

Algorithm | Parameters |
---|---|

TIK | ${K}_{p}=diag\left\{500,500,500\right\}$ and ${K}_{d}=diag\left\{600,600,600\right\}$ |

SOSMIK | ${K}_{p}=diag\left\{500,500,500\right\}$, ${K}_{d}=diag\left\{400,400,400\right\}$, ${K}_{i}=diag\left\{500,500,500\right\}$, ${K}_{1}=diag\left\{150,150,150\right\}$${K}_{r}=diag\left\{900,900,900\right\}$ |

CSOSM-MIK | ${K}_{p}=diag\left\{500,500,500\right\}$, ${K}_{d}=diag\left\{400,400,400\right\}$, ${K}_{i}=diag\left\{500,500,500\right\}$ and ${K}_{1}=diag\left\{150,150,150\right\}$ |

Algorithm | IAE | ITAE | ||||
---|---|---|---|---|---|---|

X axis | Y axis | Z axis | X axis | Y axis | Z axis | |

TIK | 8.9885 | 9.5896 | 8.9884 | 13.8687 | 13.4564 | 13.6091 |

SOSMIK | 4.6553 | 4.5142 | 5.9203 | 8.2853 | 7.5632 | 8.6541 |

CSOSM-AIK | 2.6988 | 2.4674 | 2.2541 | 3.6554 | 3.1592 | 3.5628 |

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**MDPI and ACS Style**

Mohammed Elawady, W.; Bouteraa, Y.; Elmogy, A. An Adaptive Second Order Sliding Mode Inverse Kinematics Approach for Serial Kinematic Chain Robot Manipulators. *Robotics* **2020**, *9*, 4.
https://doi.org/10.3390/robotics9010004

**AMA Style**

Mohammed Elawady W, Bouteraa Y, Elmogy A. An Adaptive Second Order Sliding Mode Inverse Kinematics Approach for Serial Kinematic Chain Robot Manipulators. *Robotics*. 2020; 9(1):4.
https://doi.org/10.3390/robotics9010004

**Chicago/Turabian Style**

Mohammed Elawady, Wael, Yassine Bouteraa, and Ahmed Elmogy. 2020. "An Adaptive Second Order Sliding Mode Inverse Kinematics Approach for Serial Kinematic Chain Robot Manipulators" *Robotics* 9, no. 1: 4.
https://doi.org/10.3390/robotics9010004