# Functional Design of a 6-DOF Platform for Micro-Positioning

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## Abstract

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## 1. Introduction

## 2. Design Specifications

- overall dimensions of $350\times 350\times 225$ mm $\left(XYZ\right)$,
- translation of $125\times 50\times 25$ mm along the x-, y- and z-directions, respectively,
- translation resolution of about $0.5$ $\mathsf{\mu}$m, repeatability of $\pm 1$ $\mathsf{\mu}$m, maximum speed of 5 mm/s,
- rotation range of $\pm {5}^{\xb0}$ about x-, y- and z-axes,
- rotation resolution of about $2\times {10}^{-4}$${}^{\xb0}$, repeatability of $\pm 2\times {10}^{-4}$${}^{\xb0}$,
- payload of $1.5$ kg.

## 3. The Hexaglide Kinematics

#### 3.1. Model Parametrization

#### 3.2. Inverse Position Kinematics (IPK)

#### 3.3. Direct Position Kinematics (DPK)

- Input: vector $\mathbf{q}$;
- a discrete sequence of actuated joint vectors, regulated by the index k, is obtained by means of a linear interpolation ranging from ${\mathbf{q}}_{in}$$(={\mathbf{q}}_{0})$ to ${\mathbf{q}}_{fin}$$(=\mathbf{q})$, with ${\mathbf{q}}_{0}$ resulting from the IPK of ${\mathbf{x}}_{0}$;
- an iterative Newton–Raphson algorithm, this time regulated by the index i, is progressively used for each ${\mathbf{q}}_{k}$ of the sequence to evaluate the vector ${\mathbf{x}}_{k}$ that verifies the constraint manifold in (11) with a desired level of accuracy, each iteration starting from the previous solution ${\mathbf{x}}_{k-1}$;
- Output: vector $\mathbf{x}$ of study parameters or directly ${}_{1}^{0}\mathbf{T}$ if a matrix form is preferred.

#### 3.4. Differential Kinematics

- $det\left({\mathbf{J}}_{x}\right)=0$ when any pair of reciprocal torsors ${\$}_{r,i}$ and ${\$}_{r,j}$ (with $i\ne j$) satisfies the relation ${\$}_{r,i}\Vert {\$}_{r,j}$;
- $det\left({\mathbf{J}}_{q}\right)=0$ when ${\mathbf{l}}_{i}\perp {\mathbf{s}}_{i}$ is verified at least for one of the ith legs.

## 4. Optimization Problem

#### Geometric Optimization

## 5. Optimization Results

## 6. Kinematic Performance

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Map of the Cartesian positioning errors (fixed orientation: ${\theta}_{x}=0$, ${\theta}_{y}=0$, ${\theta}_{z}=-{10}^{\xb0}$).

**Figure 7.**Map of the determinant of $\mathbf{J}$ (fixed orientation: ${\theta}_{x}=0$, ${\theta}_{y}=0$, ${\theta}_{z}=-{10}^{\xb0}$).

**Figure 8.**Map of the orientation errors (around the given orientation ${\theta}_{x}=0$, ${\theta}_{y}=0$, ${\theta}_{z}=-{10}^{\xb0}$).

**Figure 10.**Prototype of a more compact version of the Hexaglide manipulator with similar performance by Vacuum Fab Srl.

$\mathbf{\phi}$ | ${\mathbf{\theta}}_{\mathit{z}}$ | r | L | ${\mathit{S}}_{1,\mathit{y}}$ | ${\mathit{S}}_{3,\mathit{y}}$ | ${\mathit{S}}_{4,\mathit{y}}$ | ${\mathit{S}}_{6,\mathit{y}}$ | ${\mathit{y}}_{\mathit{h}}$ | ${\mathit{z}}_{\mathit{h}}$ | |
---|---|---|---|---|---|---|---|---|---|---|

$\left[{}^{\xb0}\right]$ | $\left[\mathrm{mm}\right]$ | |||||||||

Init | 10 | 0 | 75 | 210 | 10 | 100 | −10 | −100 | 0 | 70 |

Inf | 5 | −20 | 50 | 150 | 0 | 0 | −60 | −150 | −100 | 50 |

Sup | 15 | 20 | 100 | 250 | 60 | 150 | 0 | 0 | 100 | 200 |

$\mathbf{\phi}$ | ${\mathbf{\theta}}_{\mathit{z}}$ | r | L | ${\mathit{S}}_{1,\mathit{y}}$ | ${\mathit{S}}_{3,\mathit{y}}$ | ${\mathit{S}}_{4,\mathit{y}}$ | ${\mathit{S}}_{6,\mathit{y}}$ | ${\mathit{y}}_{\mathit{h}}$ | ${\mathit{z}}_{\mathit{h}}$ | $\mathsf{\Phi}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

Step | $\left[{}^{\xb0}\right]$ | $\left[\mathrm{mm}\right]$ | – | ||||||||

0 | 10 | 0 | 75 | 210 | 10 | 100 | −10 | −100 | 0 | 70 | 0.512 |

1 | 9.99 | −8.63 | 56.63 | 208.66 | 49.97 | 138.07 | −3.28 | −118.84 | 4.69 | 140.95 | 0.498 |

2 | 10.02 | −9.41 | 59.90 | 239.33 | 57.48 | 135.60 | −3.70 | −116.95 | 46.54 | 146.15 | 0.497 |

3 | 10.12 | −10.92 | 59.58 | 245.18 | 57.62 | 135.78 | −3.65 | −117.13 | 50.27 | 150.48 | 0.494 |

4 | 9.98 | −11.68 | 57.82 | 249.86 | 59.89 | 148.93 | −0.14 | −118.10 | 50.83 | 155.30 | 0.493 |

5 | 10.05 | −9.65 | 60.49 | 248.47 | 57.76 | 136.49 | −3.72 | −116.62 | 53.29 | 152.75 | 0.490 |

6 | 10.05 | −9.92 | 57.80 | 249.97 | 59.97 | 149.72 | −0.03 | −118.08 | 50.92 | 155.44 | 0.490 |

7 | 9.98 | −8.93 | 60.59 | 248.57 | 57.76 | 136.54 | −3.72 | −116.57 | 53.43 | 152.80 | 0.487 |

8 | 10.08 | −9.88 | 58.13 | 249.32 | 59.49 | 145.60 | −0.73 | −117.93 | 50.99 | 154.60 | 0.486 |

9 | 10.04 | −9.92 | 57.80 | 249.96 | 59.97 | 149.72 | −0.03 | −118.11 | 50.85 | 155.47 | 0.482 |

10 | 10.06 | −10.24 | 61.14 | 248.53 | 57.79 | 136.80 | −3.79 | −116.27 | 54.00 | 152.58 | 0.478 |

Result | 10 | −10 | 60 | 250 | 60 | 140 | 0 | −115 | 50 | 155 | 0.482 |

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**MDPI and ACS Style**

Palpacelli, M.-C.; Carbonari, L.; Palmieri, G.; D’Anca, F.; Landini, E.; Giorgi, G.
Functional Design of a 6-DOF Platform for Micro-Positioning. *Robotics* **2020**, *9*, 99.
https://doi.org/10.3390/robotics9040099

**AMA Style**

Palpacelli M-C, Carbonari L, Palmieri G, D’Anca F, Landini E, Giorgi G.
Functional Design of a 6-DOF Platform for Micro-Positioning. *Robotics*. 2020; 9(4):99.
https://doi.org/10.3390/robotics9040099

**Chicago/Turabian Style**

Palpacelli, Matteo-Claudio, Luca Carbonari, Giacomo Palmieri, Fabio D’Anca, Ettore Landini, and Guido Giorgi.
2020. "Functional Design of a 6-DOF Platform for Micro-Positioning" *Robotics* 9, no. 4: 99.
https://doi.org/10.3390/robotics9040099