# A Bi-Invariant Approach to Approximate Motion Synthesis of Planar Four-Bar Linkage

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## Abstract

**:**

## 1. Introduction

## 2. Concepts Used in Rigid-Body Guidance

#### 2.1. The Displacement Pole

#### 2.2. Using Poles to Determine the Task Positions

#### 2.3. Counting the Required Poles

#### 2.4. Dimensional Synthesis

## 3. Dimensional Synthesis Optimization

`fmincon`function. Within the optimization, the generated positions ${\mathbf{e}}_{i}$ and ${\psi}_{i}$, $i=2,\dots ,{n}_{t}$ are determined from ${\mathbf{e}}_{1}$, ${\psi}_{1}$, ${\mathbf{Q}}_{ij}$, where $ij\in \mathbb{P}$. Since local minimums are detected, numerous solutions to Equation (19) are generated using diverse starting values. The interior-point algorithm was used. Optimizations were performed using various settings of maximum iterations, along with convergence and constraint tolerances. The Matlab defaults were ultimately used because they provided consistent solutions in acceptable computation time (less than 10 s per randomly selected starting value).

## 4. Bi-Invariance

#### 4.1. Right Translation

#### 4.2. Left Translation

## 5. Examples

#### 5.1. Example 1: Task Positions Taken from Ref. [5]

#### 5.2. Example 2: Task Positions Taken from Ref. [9]

#### 5.3. Example 3: Demonstrating Bi-Invariance

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Concepts for rigid-body guidance include identifying target positions, coordinate transformations, and the displacement pole.

**Figure 4.**The task frames, generated frames, and resulting linkage from the presented method for Example 1.

**Figure 8.**Task positions for Example 3 before (thick black) and after applying a right translation (thin black). Generated positions are shown for task positions before (thick red) and after right translation (thin red). The thick blue lines are the optimized four-bar linkage seen to be identical in the two solutions. The thin blue lines show the coupler to the original positions (solid) and the right translated positions (dashed).

**Figure 9.**Task positions for Example 3 after applying a small left translation (black) and the associated generated positions (red). The fixed frame after applying a left translation is also shown. The thick blue lines are the optimized four-bar matching the linkage shown in Figure 8. The thin blue lines show the coupler to the left translated positions.

Location | Orientation (Deg) |
---|---|

(0.00, 0.00) | 40 |

(4.50, 4.00) | 20 |

(8.50, 8.00) | 0 |

(13.00, 11.50) | −30 |

(13.00, 12.50) | −35 |

(9.50, 14.00) | −35 |

(5.00, 13.50) | −30 |

(1.00, 10.50) | −15 |

(−1.00, 6.50) | 0 |

(−1.50, 3.00) | 20 |

Location | Orientation (Deg) |
---|---|

(1.1025, 1.0206) | 54.9693 |

(0.9439, 0.4950) | 72.3439 |

(0.3804, 0.0741) | 80.9533 |

(−0.1340, 0.1014) | 72.1922 |

(−0.3286, 0.5434) | 58.6599 |

(−0.0668, 1.0855) | 49.9219 |

(0.5293, 1.3346) | 47.8367 |

Location | Orientation (Deg) |
---|---|

(5.4925, −5.5813) | 109.2727 |

(6.5759, −4.0083) | 126.6505 |

(6.3082, −2.3480) | 140.3157 |

(5.0394, −1.1357) | 159.4077 |

(3.4334, −1.8575) | 167.6665 |

(0.8025, −3.7452) | 144.6324 |

(−1.8858, −6.0622) | 116.1184 |

(−1.7438, −7.8746) | 103.3444 |

(−0.8606, −8.8435) | 85.9285 |

(1.0690, −9.5115) | 88.2301 |

(2.2986, −8.8865) | 80.0622 |

(4.5038, −7.8714) | 87.0337 |

Translation | G | H | y | z | l | m | J |
---|---|---|---|---|---|---|---|

None | (2.2675, −3.0510) | (−4.4649, 2.5947) | (2.5643, −1.2709) | (4.8371, 4.0034) | 3.9464 | 6.9910 | 72.073 |

${\overline{A}}_{S}$, ${\overline{\mathbf{d}}}_{S}$ | (2.2659, −3.0512) | (−4.4641, 2.5767) | (−1.7130, 9.5602) | (1.7188, 4.9564) | 3.9409 | 6.9727 | 72.073 |

${\tilde{A}}_{S}$, ${\tilde{\mathbf{d}}}_{S}$ | (10.797, −7.8274) | (2.3618, −10.257) | (2.5663, −1.2680) | (4.8394, 4.0052) | 3.9421 | 6.9782 | 72.073 |

${\overline{A}}_{L}$, ${\overline{\mathbf{d}}}_{L}$ | (2.2668, −3.0453) | (−4.4655, 2.6025) | (3.3224, 168.15) | (6.7562, 163.55) | 3.9542 | 6.9982 | 72.076 |

${\tilde{A}}_{L}$, ${\tilde{\mathbf{d}}}_{L}$ | (95.557, −80.241) | (96.340, −88.973) | (2.5683, −1.2665) | (4.8407, 4.0068) | 3.9381 | 6.9636 | 72.073 |

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**MDPI and ACS Style**

Xu, T.; Myszka, D.H.; Murray, A.P.
A Bi-Invariant Approach to Approximate Motion Synthesis of Planar Four-Bar Linkage. *Robotics* **2024**, *13*, 13.
https://doi.org/10.3390/robotics13010013

**AMA Style**

Xu T, Myszka DH, Murray AP.
A Bi-Invariant Approach to Approximate Motion Synthesis of Planar Four-Bar Linkage. *Robotics*. 2024; 13(1):13.
https://doi.org/10.3390/robotics13010013

**Chicago/Turabian Style**

Xu, Tianze, David H. Myszka, and Andrew P. Murray.
2024. "A Bi-Invariant Approach to Approximate Motion Synthesis of Planar Four-Bar Linkage" *Robotics* 13, no. 1: 13.
https://doi.org/10.3390/robotics13010013