# An Investigation of Various Controller Designs for Multi-Link Robotic System (Robogymnast)

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. System Description

#### 3.2. Mathematical Model

_{1}= θ

_{2}= θ

_{3}= 0. The equations’ state space for the robotic system is given by

ss2tf: h = tf [sys]

**Table 1.**Robogymnast parameters [18].

Parameters | Symbol | Mean Values |
---|---|---|

Length of the first link | ${L}_{1}$ | 0.16 m |

Length of the second link | ${L}_{2}$ | 0.18 m |

Length of the third link | ${L}_{3}$ | 0.24 m |

Weight of the first link | ${m}_{1}$ | 1.2 kg |

Weight of the second link | ${m}_{2}$ | 1.2 kg |

Weight of the third link | ${m}_{3}$ | 0.5 kg |

Angles between poles 1, 2, and 3 | $\theta $ | ${\theta}_{1}$,${\theta}_{2}$,${\theta}_{3}$ (rad) |

Initial values of the angles | q_{1}, q_{2}, q_{3} | 0 (rad) |

Gravity | g | $9.81\mathrm{m}/{\mathrm{s}}^{2}$ |

## 4. Control Design

#### 4.1. LQR

#### 4.2. FLQR

Symbol | Linguistic Variable |
---|---|

NB | Negative Big |

Nm | Negative Medium |

NS | Negative Small |

Z | Zero |

PS | Positive Small |

PM | Positive Medium |

PB | Positive Big |

**Table 3.**Rules of FL controller [18].

Error | Change in Error | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | Z | PS | PM | PB | |

NB | NB | NB | NB | NM | NM | NS | Z |

NM | NB | NB | NB | NM | NS | Z | PS |

NS | NB | NM | NM | NS | Z | PS | PM |

Z | NM | NM | NS | Z | PS | PM | PB |

PS | NM | NS | Z | PS | PM | PM | PB |

PM | NS | Z | PS | PM | PM | PB | PB |

PB | Z | PS | PM | PM | PB | PB | PB |

#### 4.3. Robustness Investigation for the Proposed Controller

## 5. Results

#### 5.1. Case 1: Original Value [17]

#### 5.2. Case 2: +(%25)

**Figure 9.**(

**a**) The system response for link-1 of Robogymnast in Case 2 (T1); (

**b**) The system response for each middle link of Robogymnast in Case 2 (T2); (

**c**) The system response for link-3 of Robogymnast in Case 2 (T3).

Symbol | Controller | ${\mathit{O}}_{\mathit{s}\mathit{h}}$ (pu) | ${\mathit{U}}_{\mathit{s}\mathit{h}}$ (pu) | ${\mathit{T}}_{\mathit{r}}$ (s) | ${\mathit{T}}_{\mathit{s}}$ (s) |
---|---|---|---|---|---|

${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.026 | −5.621 | 0.2844 | 12.4151 |

Fuzzy LQR | 3.122 | −5.716 | 0.3249 | 8.9536 | |

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.326 | 0.0605 | 3.9305 |

Fuzzy LQR | 1 | −1.445 | 0.84 | 3.3397 | |

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2537 | −0.4 | 0.0406 | 2.6648 |

Fuzzy LQR | 0.2562 | −0.4 | 0.0418 | 1.9637 |

#### 5.3. Case 3: +(%50)

Symbol | Controller | ${\mathit{O}}_{\mathit{s}\mathit{h}}$ (pu) | ${\mathit{U}}_{\mathit{s}\mathit{h}}$ (pu) | ${\mathit{T}}_{\mathit{r}}$ (s) | ${\mathit{T}}_{\mathit{s}}$ (s) |
---|---|---|---|---|---|

${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.025 | −5.679 | 0.2369 | 10.3460 |

Fuzzy LQR | 3.013 | −5.711 | 0.2709 | 7.4641 | |

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.326 | 0.0504 | 3.2747 |

Fuzzy LQR | 1 | −1.433 | 0.0486 | 2.7863 | |

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2542 | −0.4 | 0.0337 | 2.2205 |

Fuzzy LQR | 0.2582 | −0.4 | 0.48 | 1.6402 |

**Figure 11.**(

**a**) The system response for upper link of Robogymnast in Case 3 (T1); (

**b**) The system response for second link of Robogymnast in Case 3 (T2); (

**c**) The system response for 3rd link of Robogymnast in Case 3 (T3).

#### 5.4. Case 4: −(%25)

**Figure 13.**(

**a**) The system response for 1st link of Robogymnast in Case 4 (T1); (

**b**) The system response for central link of Robogymnast in Case 4 (T2); (

**c**) The system response for link-3 of Robogymnast in Case 4 (T3).

Symbol | Controller | ${\mathit{O}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{U}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{r}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{s}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ |
---|---|---|---|---|---|

${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.025 | −5.679 | 0.4744 | 20.6922 |

Fuzzy LQR | 2.742 | −5.712 | 0.5445 | 14.8923 | |

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.326 | 0.1012 | 6.5537 |

Fuzzy LQR | 1 | −1.437 | 0.0975 | 5.5485 | |

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2546 | −0.4 | 0.0681 | 4.4410 |

Fuzzy LQR | 0.2584 | −0.4 | 0.0699 | 3.2399 |

#### 5.5. Case 5: −(%50)

**Figure 15.**(

**a**) The system response for the first link of Robogymnast in Case 5 (T1); (

**b**) The system response for the 2nd link of Robogymnast in Case 5 (T2); (

**c**) The system response for the lower link of Robogymnast in Case 5 (T3).

Symbol | Controller | ${\mathit{O}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{U}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{r}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{s}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ |
---|---|---|---|---|---|

${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.025 | −5.679 | 0.711 | 31.0403 |

Fuzzy LQR | 3.146 | −5.706 | 0.7952 | 21.3793 | |

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.3626 | 0.1520 | 9.8323 |

Fuzzy LQR | 1 | −1.421 | 0.1483 | 7.8387 | |

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2548 | −0.4 | 0.1024 | 6.6633 |

Fuzzy LQR | 0.2604 | −0.4 | 0.1060 | 4.8405 |

#### 5.6. Compression of ITAE

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | |
---|---|---|---|---|---|

LQR1 | 159.7 | 283.9 | 102.20 | 31.0403 | 70.98 |

FLQR1 | 21.29 | 38.16 | 26.72 | 87.19 | 8.971 |

LQR2 | 0.322 | 0.5741 | 0.2136 | 1.29 | 0.148 |

FLQR2 | 0.313 | 0.5309 | 0.2066 | 1.257 | 0.143 |

LQR3 | 0.022 | 0.0395 | 0.0138 | 0.088 | 0.009 |

FLQR3 | 0.021 | 0.0385 | 0.0142 | 0.086 | 0.009 |

#### 5.7. Comparison of LQR and FLQR in All Cases

Case | Symbol | Controller | ${\mathit{O}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{U}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{r}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{s}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ |
---|---|---|---|---|---|---|

Original value | ${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.02 | −5.69 | 0.3557 | 15.5196 |

Fuzzy LQR | 2.88 | −5.71 | 0.4074 | 11.1823 | ||

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1.03 | −1.32 | 0.0758 | 4.9142 | |

Fuzzy LQR | 0.42 | −1.44 | 0.0730 | 4.1694 | ||

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.25 | −0.41 | 0.0509 | 3.3310 | |

Fuzzy LQR | 0.25 | −0.40 | 0.0523 | 2.4428 | ||

Case 2 | ${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.026 | −5.677 | 0.2844 | 12.4151 |

Fuzzy LQR | 3.122 | −5.716 | 0.3249 | 8.9536 | ||

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.326 | 0.0605 | 3.9305 | |

Fuzzy LQR | 1 | −1.445 | 0.0584 | 3.3397 | ||

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2537 | −0.4 | 0.0406 | 2.6648 | |

Fuzzy-LQR | 0.2562 | −0.4 | 0.0418 | 1.9637 | ||

Case 3 | ${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.025 | −5.679 | 0.2369 | 10.3460 |

Fuzzy LQR | 3.013 | −5.711 | 0.2709 | 7.4641 | ||

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.326 | 0.0504 | 3.2747 | |

Fuzzy LQR | 1 | −1.433 | 0.0486 | 2.7863 | ||

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2542 | −0.4 | 0.0337 | 2.2205 | |

Fuzzy LQR | 0.2582 | −0.4 | 0.0348 | 1.6402 | ||

Case 4 | ${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.025 | −5.679 | 0.4744 | 20.6922 |

Fuzzy LQR | 2.742 | −5.712 | 0.5445 | 14.8923 | ||

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.326 | 0.1012 | 6.5537 | |

Fuzzy LQR | 1 | −1.437 | 0.0975 | 5.5485 | ||

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2546 | −0.4 | 0.0681 | 4.4410 | |

Fuzzy LQR | 0.2584 | −0.4 | 0.0699 | 3.2399 | ||

Case 5 | ${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.025 | −5.679 | 0.711 | 31.0403 |

Fuzzy LQR | 3.146 | −5.706 | 0.7952 | 21.3793 | ||

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1 | −1.3626 | 0.1520 | 9.8323 | |

Fuzzy LQR | 1 | −1.421 | 0.1483 | 7.8387 | ||

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.2548 | −0.4 | 0.1024 | 6.6633 | |

Fuzzy LQR | 0.2604 | −0.4 | 0.1060 | 4.8405 |

## 6. Discussion

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Robogymnast | Robot Gymnast |

FLQR | Fuzzy logic quadrate rectangular |

LQR | Linear quadrate rectangular |

PID | Proportional integral derivative |

ITAE | Integral time of absolute error |

Osh | Overshoot |

Ush | Undershoot |

${T}_{r}$ | Rising time |

${T}_{s}$ | Settling time |

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**Figure 7.**(

**a**) The system response for the upper link of Robogymnast in Case 1 (T1); (

**b**) The system response for the middle link of Robogymnast in Case 1 (T2); (

**c**) The system response for lower link of Robogymnast in Case 1 (T3).

**Table 4.**LQR vs. FLQR performance [13].

Symbol | Controller | ${\mathit{O}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{U}}_{\mathit{s}\mathit{h}}\text{}\mathbf{\left(}\mathbf{pu}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{r}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ | ${\mathit{T}}_{\mathit{s}}\text{}\mathbf{\left(}\mathbf{s}\mathbf{\right)}$ |
---|---|---|---|---|---|

${\mathsf{\theta}}_{\mathbf{1}}$ | LQR | 8.02 | −5.69 | 0.3557 | 15.5196 |

Fuzzy LQR | 2.88 | −5.71 | 0.4074 | 11.1823 | |

${\mathsf{\theta}}_{\mathbf{2}}$ | LQR | 1.03 | −1.32 | 0.0758 | 4.9142 |

Fuzzy LQR | 0.42 | −1.44 | 0.30 | 4.1694 | |

${\mathsf{\theta}}_{\mathbf{3}}$ | LQR | 0.25 | −0.41 | 0.0509 | 3.3310 |

Fuzzy LQR | 0.25 | −0.40 | 0.0523 | 2.4428 |

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**MDPI and ACS Style**

Abdul samad, B.; Mohamed, M.; Anayi, F.; Melikhov, Y.
An Investigation of Various Controller Designs for Multi-Link Robotic System (Robogymnast). *Knowledge* **2022**, *2*, 465-486.
https://doi.org/10.3390/knowledge2030028

**AMA Style**

Abdul samad B, Mohamed M, Anayi F, Melikhov Y.
An Investigation of Various Controller Designs for Multi-Link Robotic System (Robogymnast). *Knowledge*. 2022; 2(3):465-486.
https://doi.org/10.3390/knowledge2030028

**Chicago/Turabian Style**

Abdul samad, Bdereddin, Mahmoud Mohamed, Fatih Anayi, and Yevgen Melikhov.
2022. "An Investigation of Various Controller Designs for Multi-Link Robotic System (Robogymnast)" *Knowledge* 2, no. 3: 465-486.
https://doi.org/10.3390/knowledge2030028