# Modeling and Control of a DC Motor Coupled to a Non-Rigid Joint

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## Abstract

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## 1. Introduction

## 2. Proposed Models

#### 2.1. DC Motor and Worm Gearbox Model

- Powering the motor with multiple pre-known values of supply voltage, allowing it to reach the steady-state—This test consists in powering the motor with several different pre-known supply voltages and measure the angular speed $\omega $ at the steady-state throughout the process;
- Input a voltage step to the motor terminals—This test consists in powering the motor with a voltage step and obtain its response to this signal, to verify which model suits the best to the tested model.

#### 2.2. Non-Rigid Joint Model

- Pulling the link attached to the non-rigid joint to a certain angle - In this test, the link attached to the non-rigid joint is pulled manually for a certain set of increasing angles. For those angles the torque is measured. Then, the link is rotated in the same way to the negative side and the same procedure is applied;
- Pulling the link attached to the non-rigid joint and releasing it, capture the angular speed an the angle until it stabilizes - Through this test, the link attached to the non-rigid joint is pulled manually and, after reaching a determined angle, is released and left to stabilize until it reaches the rest angle. The angular speed, the torque and the angle are sampled with a period of 10 ms;

#### 2.3. Control

## 3. Experimental Setup Hardware

- Built-in encoder—measuring the angular displacement of the motor;
- Current sensor—measuring the current that the motor withdraws;
- Voltage sensor—measuring the instant and real voltage applied to the motor terminals;

## 4. Results

#### 4.1. DC Motor with Worm Gearbox

#### 4.2. Non-Rigid Joint

#### 4.3. Control

- Using only a position controller, with a step reference—This test consists in obtaining the motor response to a step reference, using only a position controller and evaluating its response through the angular position from the absolute encoder;
- Using a position and a velocity controller, using to the developed profiles as references—This test is similar to the previous one consisting in obtaining the motor response to the references created by the developed profiles, using position and velocity controllers and again evaluating its response through the angular position from the absolute encoder;

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Parameters | |
---|---|

R | 8.6538 $\mathrm{\Omega}$ |

k | 0.0174 |

${T}_{q}$ | $0.6082\times {10}^{-3}$ N·m |

${B}_{v}$ | $5.9751\times {10}^{-7}$ N·m·s |

J | $8.5075\times {10}^{-7}$ kg·${\mathrm{m}}^{2}$ |

L | 0.0238 H |

First Order Model | Second Order Model | |
---|---|---|

Maximum Absolute Error (rad) | 0.0309 | 0.00396 |

Average Absolute Error (rad) | 0.00455 | 0.00187 |

Absolute Error Sum (rad) | 0.150 | 0.0619 |

Maximum Relative Error (%) | 43.9 | 2.08 |

Average Relative Error (%) | 2.97 | 0.659 |

Parameters | |
---|---|

K | 7.3035 N/m |

B | 0.0416 N·s/m |

J | 0.0085 Kg·${\mathrm{m}}^{2}$ |

Until 1 s | After 1 s | |
---|---|---|

Maximum Absolute Error (°) | 10.5 | 2.93 |

Average Absolute Error (°) | 1.66 | 1.11 |

Absolute Error Sum (°) | 84.8 | 75.7 |

Maximum Relative Error (%) | 2.305 | 17.1 |

Average Relative Error (%) | 0.109 | 0.654 |

Position Controller | Position and Velocity Controllers | |
---|---|---|

Maximum Absolute Error (rad) | 0.078 | 0.011 |

Average Absolute Error (rad) | 0.005 | 0.005 |

Absolute Error Sum (rad) | 2.59 | 2.41 |

Maximum Relative Error (%) | 64.8 | 9.6 |

Average Relative Error (%) | 4.3 | 4.0 |

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**MDPI and ACS Style**

Pinto, V.H.; Gonçalves, J.; Costa, P.
Modeling and Control of a DC Motor Coupled to a Non-Rigid Joint. *Appl. Syst. Innov.* **2020**, *3*, 24.
https://doi.org/10.3390/asi3020024

**AMA Style**

Pinto VH, Gonçalves J, Costa P.
Modeling and Control of a DC Motor Coupled to a Non-Rigid Joint. *Applied System Innovation*. 2020; 3(2):24.
https://doi.org/10.3390/asi3020024

**Chicago/Turabian Style**

Pinto, Vítor H., José Gonçalves, and Paulo Costa.
2020. "Modeling and Control of a DC Motor Coupled to a Non-Rigid Joint" *Applied System Innovation* 3, no. 2: 24.
https://doi.org/10.3390/asi3020024