#
A Novel Adaptive Sliding Mode Controller for a 2-DOF Elastic Robotic Arm^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- An adaptive control mechanism is proposed to deal with the controlling task of a 2-DOF elastic robot arm. The control mechanism has two loops. The outer loop is an adaptive sliding mode controller (ASMC) to deal with uncertainties and disturbances on the load side of the robot arm. The output of this loop is the desired angular position of the motors. The inner loop consists of the model reference adaptive controllers (MRAC) to stabilise the motor side of the robot arm;
- Extensive simulation experiments and a comparison with the conventional sliding mode controller are conducted to demonstrate the effectiveness of the proposed controller.

## 2. Related Research Work

## 3. Mathematical Model of the 2-DOF Elastic Robot Arm

## 4. Controller Design

**Assumption**

**1.**

**Assumption**

**2.**

## 5. Simulation

#### 5.1. Sine Wave Input

#### 5.2. Square Wave Input

#### 5.3. Step Input

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

ASMC | Adaptive sliding mode controller |

MRAC | Model reference adaptive8controllers |

SAR | Search and rescue |

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**Figure 4.**(

**a**,

**b**) Angular position response of the robot arm with

**sine wave input**in

**normal condition**.

**Figure 6.**(

**a**,

**b**) Angular position response of the robot arm with

**sine wave input**in

**disturbed condition**with

**sine wave disturbance**.

**Figure 7.**(

**a**,

**b**) Angular position response of the robot arm with

**sine wave input**in

**disturbed condition**with

**square wave disturbance**.

**Figure 8.**(

**a**,

**b**) Control voltage of the robot joints with

**sine wave input**in

**disturbed condition**with

**sine wave disturbance**.

**Figure 9.**(

**a**,

**b**) Control voltage of the robot joints with

**sine wave input**in

**disturbed condition**with

**square wave disturbance**.

**Figure 10.**(

**a**,

**b**) Angular position response of the robot arm with

**square wave input**in

**normal condition**.

**Figure 12.**(

**a**,

**b**) Angular position response of the robot arm with

**square wave input**in

**disturbed condition**with

**sine wave disturbance**.

**Figure 13.**(

**a**,

**b**) Angular position response of the robot arm with

**square wave input**in

**disturbed condition**with

**square wave disturbance**.

**Figure 14.**(

**a**,

**b**) Control voltage of the robot joints with

**square wave input**in

**disturbed condition**with

**sine wave disturbance**.

**Figure 15.**(

**a**,

**b**) Control voltage of the robot joints with

**square wave input**in

**disturbed condition**with

**square wave disturbance**.

**Figure 16.**(

**a**–

**c**) Angular position response, error and control voltage of proposed ASMC and conventional SMC algorithms.

Type of Robots | Humanoid Robot | Snake Robot | Rehabilitation Robot |
---|---|---|---|

Applications | NASA Valkyrie [7] COMAN [12] | POAL [13,14] | RiceWrist [9] ULIX [10] rotary SEA [11] |

d | $\mathit{\theta}$ | a | $\mathit{\alpha}$ | |
---|---|---|---|---|

Joint 1 | ${l}_{1}$ | ${q}_{1}$ | 0 | $\pi /2$ |

Joint 2 | 0 | ${q}_{2}$ | ${l}_{2}$ | $-\pi /2$ |

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

Gear ratio (${N}_{1},{N}_{2}$) | 10 | Spring stiffness (${K}_{s1},{K}_{s2}$) | 1500, 1200 | Load inertia (${J}_{l1},{J}_{l2}$) | 0.1 |

Load damping coefficient (${D}_{l1},{D}_{l2}$) | 0.2 | Motor damping coefficient (${D}_{m1},{D}_{m2}$) | 0.027 | Motor inertia (${J}_{m1},{J}_{m2}$) | 0.003 |

Motor mass (${m}_{m1},{m}_{m2}$) | 1 | Link mass (${m}_{l1},{m}_{l2}$) | 6, 4 | Link length (${l}_{1},{l}_{2}$) | 0.3 |

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

${K}_{p}$ | $\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$ | ${K}_{i}$ | $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ | $\alpha $ | ${10}^{-7}\times \left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$ |

U | 400 | ${\u03f5}_{e}$ | $\frac{\pi}{18}$ | ${\u03f5}_{edot}$ | $\frac{\pi}{6}$ |

**Table 5.**The root mean square errors of the angular position response with

**sine wave input**in normal condition and disturbed conditions.

Operating Condition | Normal Condition | Sine Wave Disturbance | Square Wave Disturbance | |
---|---|---|---|---|

Controller Type | ||||

ASMC-MRAC | Joint 1 | 0.009929 | 0.025849 | 0.014397 |

Joint 2 | 0.026402 | 0.026601 | 0.029558 | |

SMC-MRAC | Joint 1 | 0.011813 | 0.102562 | 0.017625 |

Joint 2 | 0.033602 | 0.055017 | 0.038218 |

**Table 6.**The root mean square errors of the angular position response with

**square wave input**in normal condition and disturbed conditions.

Operating Condition | Normal Condition | Sine Wave Disturbance | Square Wave Disturbance | |
---|---|---|---|---|

Controller Type | ||||

ASMC-MRAC | Joint 1 | 0.013186 | 0.033616 | 0.023396 |

Joint 2 | 0.029123 | 0.028937 | 0.031433 | |

SMC-MRAC | Joint 1 | 0.049605 | 0.091040 | 0.058128 |

Joint 2 | 0.085158 | 0.092527 | 0.100103 |

Performance Specs | RMS Error | Total Variance of Control Signal | |
---|---|---|---|

Controller Type | |||

ASMC-MRAC | Joint 1 | 0.016170 | 0.083913 |

Joint 2 | 0.025250 | 0.220392 | |

SMC-MRAC | Joint 1 | 0.064749 | 0.002466 |

Joint 2 | 0.129457 | 0.002836 |

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**MDPI and ACS Style**

Tuan, H.M.; Sanfilippo, F.; Hao, N.V. A Novel Adaptive Sliding Mode Controller for a 2-DOF Elastic Robotic Arm. *Robotics* **2022**, *11*, 47.
https://doi.org/10.3390/robotics11020047

**AMA Style**

Tuan HM, Sanfilippo F, Hao NV. A Novel Adaptive Sliding Mode Controller for a 2-DOF Elastic Robotic Arm. *Robotics*. 2022; 11(2):47.
https://doi.org/10.3390/robotics11020047

**Chicago/Turabian Style**

Tuan, Hua Minh, Filippo Sanfilippo, and Nguyen Vinh Hao. 2022. "A Novel Adaptive Sliding Mode Controller for a 2-DOF Elastic Robotic Arm" *Robotics* 11, no. 2: 47.
https://doi.org/10.3390/robotics11020047