# Trajectory Tracking and Disturbance Rejection Performance Analysis of Classical and Advanced Controllers for a SCORBOT Robot

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Manipulator Kinematics and Dynamics: Foundational Models

_{1}, q

_{2}, q

_{3}and l

_{1}, l

_{2}, l

_{3}as generalized coordinates and lengths of the first, second, and third links, respectively. In turn, l

_{c}

_{1}, l

_{c}

_{2}, and l

_{c}

_{3}represent the lengths of the first, second, and third links from origin to centroid.

**τ**denotes the (n × 1) generalized force vector,

**M**refers to the (n × n) inertia matrix,

**C**constitutes an (n × 1) vector of centrifugal and Coriolis forces, q represents joint position coordinates, $\dot{q}$ stands for corresponding joint velocity terms,

**G**signifies the (n × 1) gravitational force vector, $\ddot{\mathbf{q}}$ refers to joint accelerations as an (n × 1) vector,

**F**encapsulates frictional forces in an (n × 1) format, and n corresponds to the number of Degrees of Freedom.

## 3. Controller Designs

#### 3.1. Computed Torque Controller

#### 3.2. Predictive Controller

#### 3.3. Fuzzy Logic Controller

#### 3.4. PID Controller with Anti-Windup

_{s}, defined as the discrepancy between clipped and raw controller error signals. The parameterized constants underlying the implemented anti-windup PID appear in Table 5.

## 4. Results

**q**the desired joint trajectory—composed of q

_{d}_{dn}with n = 1, 2, 3—is obtained considering a total duration of 24 s, as shown in Figure 9.

#### 4.1. Results with External Disturbance

**q**(composed of q

_{n}, with n = 1, 2, 3).

#### 4.2. Joint Error Results

_{n}, with n = 1, 2, 3.) is calculated as the difference between the desired joint trajectory q

_{d}and the resulting simulated trajectory q.

#### 4.3. Performance Indexes

## 5. Conclusions

## 6. Future Work and Limitations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Dynamic Model

_{2·2}= cos(2θ

_{2}); s

_{2·2}= sin(2θ

_{2}); m

_{1}, m

_{2}, and m

_{3}, represent the mass of the first, second, and third links, respectively. In turn, l

_{1zz}, l

_{2zz}, and l

_{3zz}indicate the inertia moments of the first, second, and third links with respect to the first z axis of their respective joints.

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**Figure 2.**Diagram of SCORBOT-ER V plus robot in two positions, considering the coordinate axis and centroid systems.

**Figure 11.**Comparison of desired and simulated joint trajectories when using the predictive controller.

**Figure 13.**Comparison of desired and simulated joint trajectories when using the PID controller with anti-windup.

Symbol | Link 1 | Link 2 | Link 3 | Unit |
---|---|---|---|---|

l | 0.32 | 0.22 | 0.22 | [m] |

l_{c} | 0 | 0.052 | 0.1376 | [m] |

m | 7.1402 | 2.2483 | 1.957 | [kg] |

I_{zz} | 0.04624 | 0.02595 | 0.03616 | [kg·m^{2}] |

F_{v} | 0.025 | 0.025 | 0.025 | [N·m/rad] |

F_{eca} | 0.05 | 0.05 | 0.05 | [N·m] |

F_{ecb} | 0.05 | 0.05 | 0.05 | [N·m] |

Joint | ${\mathbf{K}}_{\mathbf{p}}$ | ${\mathbf{K}}_{\mathbf{v}}$ |
---|---|---|

Base | 4000 | 0.800 |

Shoulder | 12,000 | 0.200 |

Elbow | 16,000 | 0.200 |

Joint | Sampling Time: Ts [s] | Prediction Horizon [Ts] | Control Horizon [Ts] |
---|---|---|---|

Base | 0.001 | 38 | 8 |

Shoulder | 0.001 | 14 | 3 |

Elbow | 0.001 | 16 | 4 |

Joint | Parameter | Error | Speed | Acceleration | Voltage |
---|---|---|---|---|---|

Base | U | [−0.15 0.15] | [−2 2] | [−10 10] | [−19.1 19.1] |

T | NB, NS, Z, PS, PB | LVF, LF, S, RF, RVF | N, Z, P | LB, LS, Z, RS, RB | |

Shoulder | U | [−2 2] | [−5 5] | [−2 2] | [−19.1 19.1] |

T | NB, NS, Z, PS, PB | LVF, LF, S, RF, RVF | N, Z, P | LB, LS, Z, RS, RB | |

Elbow | U | [−0.15 0.15] | [−2 2] | [−1 1] | [−19.1 19.1] |

T | NB, NS, Z, PS, PB | LVF, LF, S, RF, RVF | N, Z, P | LB, LS, Z, RS, RB |

Joint | ${\mathit{K}}_{\mathit{p}}$ [−] | ${\mathit{T}}_{\mathit{i}}$ [s] | ${\mathit{T}}_{\mathit{d}}$ [s] | ${\mathit{K}}_{\mathit{T}}$ [s^{−1}] |
---|---|---|---|---|

Base | 140 | 0.170 | 0.220 | 2.018 |

Shoulder | 110 | 0.200 | 0.182 | 1.401 |

Elbow | 80.0 | 0.200 | 0.200 | 2.062 |

Controller | PID | Fuzzy | Predictive + G | Computed Torque | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Index | RMS | RSD | IA | RMS | RSD | IA | RMS | RSD | IA | RMS | RSD | IA |

Base | 0.0316 | 0.0357 | 0.9997 | 0.0243 | 0.0280 | 0.9997 | 0.0010 | 0.0115 | 0.9998 | 0.0402 | 0.0475 | 0.9992 |

Shoulder | 0.0168 | 0.0393 | 0.9996 | 0.0045 | 0.0107 | 0.9999 | 0.0019 | 0.0046 | 1.0000 | 0.0051 | 0.0115 | 0.9999 |

Elbow | 0.0204 | 0.0169 | 0.9933 | 0.0031 | 0.0025 | 0.9991 | 0.0063 | 0.0052 | 0.9999 | 0.0044 | 0.0036 | 0.9977 |

Controller | PID [s] | Fuzzy [s] | Predictive + G [s] | Computed Torque [s] |
---|---|---|---|---|

Base | 1.15 | 1.28 | 0.45 | 1.7 |

Shoulder | 0.99 | 0.35 | 0.14 | 0.47 |

Elbow | 1.02 | 0.44 | 0.42 | 0.46 |

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**MDPI and ACS Style**

Kern, J.; Urrea, C.; Verdejo, H.; Agramonte, R.; Becker, C.
Trajectory Tracking and Disturbance Rejection Performance Analysis of Classical and Advanced Controllers for a SCORBOT Robot. *Robotics* **2024**, *13*, 48.
https://doi.org/10.3390/robotics13030048

**AMA Style**

Kern J, Urrea C, Verdejo H, Agramonte R, Becker C.
Trajectory Tracking and Disturbance Rejection Performance Analysis of Classical and Advanced Controllers for a SCORBOT Robot. *Robotics*. 2024; 13(3):48.
https://doi.org/10.3390/robotics13030048

**Chicago/Turabian Style**

Kern, John, Claudio Urrea, Humberto Verdejo, Rayko Agramonte, and Cristhian Becker.
2024. "Trajectory Tracking and Disturbance Rejection Performance Analysis of Classical and Advanced Controllers for a SCORBOT Robot" *Robotics* 13, no. 3: 48.
https://doi.org/10.3390/robotics13030048