# Evaluation of Linearization Methods for Control of the Pendubot

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Control Problem

#### 3.1. Control Algorithm Based on Partial Linearization

**Remark**

**1.**

#### 3.1.1. Zero Dynamics

#### 3.1.2. Avoiding of Singular Points

#### 3.2. Linear Controller

## 4. Simulation and Experimental Results

#### 4.1. Characteristics of the Laboratory Pendubot-Like System

#### 4.2. Simulation Procedure

- Algorithm 1—${u}_{\mathrm{nonlin}}$ is defined using the collocated linearization;
- Algorithm 2—${u}_{\mathrm{nonlin}}$ is defined using the non-collocated linerization;
- Algorithm 3—${u}_{\mathrm{nonlin}}$ is defined based on the maximum partial linearization approach described in this paper.

#### 4.3. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Swing-up-like motion: the dotted lines indicate the assumed coordinates values where the system should be stabilized.

**Figure 4.**Decoupling matrix—${L}_{g}{L}_{f}^{2}h$—during an exemplary simulation trial. The green part of the plot indicates when the swing-up controller is enabled.

**Figure 9.**Algorithm 1—distributions of performance factors (histograms): (

**a**) $\int {e}_{p}^{2}dt$, (

**b**) $\int {V}_{m}^{2}dt$.

**Figure 10.**Algorithm 2—distributions of performance factors (histograms): (

**a**) $\int {e}_{p}^{2}dt$, (

**b**) $\int {V}_{m}^{2}dt$.

**Figure 11.**Algorithm 3—distributions of performance factors (histograms): (

**a**) $\int {e}_{p}^{2}dt$, (

**b**) $\int {V}_{m}^{2}dt$.

**Figure 12.**Algorithm 3—Angular positions—(

**a**) simulation (the green part of the curve indicates when the swing-up controller is active), (

**b**) experiment.

**Figure 13.**Algorithm 3—voltage control input—(

**a**) simulation (the green part of the curve indicates when the swing-up controller is active), (

**b**) experiment.

Link | Mass | Length | Centre of Mass | Inertia |
---|---|---|---|---|

i | ${\mathit{m}}_{\mathit{i}}$ [kg] | ${\mathit{L}}_{\mathit{i}}$ [m] | ${\mathit{L}}_{{\mathit{c}}_{\mathit{i}}}$ [m] | ${\mathit{I}}_{\mathit{i}}$ [$\mathbf{kg}\phantom{\rule{0.166667em}{0ex}}{\mathbf{m}}^{2}$] |

1 | 0.097 | 0.20 | 0.1635 | 0.0069 |

2 | 0.127 | 0.3365 | 0.1778 | 0.0048 |

Algorithm 1 | Algorithm 2 | Algorithm 3 | |
---|---|---|---|

mean ${\int}_{0}^{{t}_{max}}{e}_{p}^{2}dt$ | 413.87 | 100.71 | 144.44 |

$\mathrm{std}$${\int}_{0}^{{t}_{max}}{e}_{p}^{2}dt$ | 639.50 | 57.95 | 195.97 |

mean ${\int}_{0}^{{t}_{max}}{V}_{m}^{2}dt$ | 48.55 | 55.11 | 115.90 |

$\mathrm{std}$${\int}_{0}^{{t}_{max}}{V}_{m}^{2}dt$ | 28.48 | 27.56 | 119.88 |

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**MDPI and ACS Style**

Parulski, P.; Bartkowiak, P.; Pazderski, D.
Evaluation of Linearization Methods for Control of the Pendubot. *Appl. Sci.* **2021**, *11*, 7615.
https://doi.org/10.3390/app11167615

**AMA Style**

Parulski P, Bartkowiak P, Pazderski D.
Evaluation of Linearization Methods for Control of the Pendubot. *Applied Sciences*. 2021; 11(16):7615.
https://doi.org/10.3390/app11167615

**Chicago/Turabian Style**

Parulski, Paweł, Patryk Bartkowiak, and Dariusz Pazderski.
2021. "Evaluation of Linearization Methods for Control of the Pendubot" *Applied Sciences* 11, no. 16: 7615.
https://doi.org/10.3390/app11167615