# Classification of All Non-Isomorphic Regular and Cuspidal Arm Anatomies in an Orthogonal Metamorphic Manipulator

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## Abstract

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## 1. Introduction

## 2. Cuspidality Investigation in Orthogonal—Metamorphic Modular Arm

#### 2.1. Presentation of Metamorphic Manipulator

#### 2.2. The Proposed Method

## 3. Classification of Orthogonal Kinematic Non-Isomorphic Configurations of 3R Metamorphic Manipulator according to the Topology of Metamorphic Workspace

#### 3.1. Necessary and Sufficient Conditions to Investigate Cuspidality

#### 3.2. Separating Algebraic Equations through Investigation of det$\left(J\right)=0$

**8**) could be written as two product factors providing the following equations:

#### 3.3. Classification According to the Number of Nodes

## 4. Planning Non-Singular Posture Changing Trajectories

#### 4.1. Generic Mechanism

#### 4.2. Non-Generic Anatomy

#### 4.2.1. Planning Closed Smooth and Continuous Path

#### 4.2.2. Rectilinear Trajectory

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Active rotational module. (

**b**)Versatile passive joint (pseudo-joint) connector constructed with aluminium and 13 discrete angular positions.

**Figure 2.**(

**a**) Orthogonal metamorphic mechanism with local coordinates systems, (

**b**) Side view of passive joint with 13 possible discrete angular positions in $\left[-90\xb0,90\xb0\right]$.

**Figure 3.**Surfaces in 3D design metamorphic parameter space and 4 distinct subspaces with the same number of cusp points.

**Figure 4.**Selective arm anatomies of the metamorphic structure and singularities are displayed in joint space and half cross section of workspace with the metamorphic parameters: (

**a**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm 90\xb0,{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm 90\xb0,{\mathrm{d}}_{4}=0.23\text{}\mathrm{m}$ (

**b**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm 15\xb0,{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm 90\xb0,{\mathrm{d}}_{4}=0.6\text{}\mathrm{m}$ (

**c**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm 60\xb0,{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm 30\xb0,{\mathrm{d}}_{4}=0.6\text{}\mathrm{m}.$

**Figure 5.**Separating surfaces in 3D design metamorphic parameter space and 8 distinct subspaces with the same number of cusps and nodes. In every subspace, the first and the second number in the parenthesis indicates the number of cusps and nodes respectively.

**Figure 6.**Selective arm anatomies of the metamorphic structure and singularities are displayed in joint space and half cross-section of workspace with the metamorphic parameters: (

**a**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}{=\pm 75\xb0,\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}{=\pm 90\xb0,\mathrm{d}}_{4}=0.07\text{}\mathrm{m}$ (

**b**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm 45\xb0,{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm 60\xb0,{\mathrm{d}}_{4}=0.1\text{}\mathrm{m}$ (

**c**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm 15\xb0,{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm 30\xb0,{\mathrm{d}}_{4}=0.11\text{}\mathrm{m}.$

**Figure 7.**Selective arm anatomies of the metamorphic structure and singularities are displayed in joint space and half cross-section of workspace with design parameters: (

**a**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm 30\xb0,{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm 90\xb0,{\mathrm{d}}_{4}=0.4\text{}\mathrm{m}$ (

**b**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}=\pm 75\xb0,{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}=\pm 15\xb0,{\mathrm{d}}_{4}=0.4\text{}\mathrm{m}$ (

**c**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}{=0\xb0,\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}{=0\xb0,\mathrm{d}}_{4}=0.2\text{}\mathrm{m}$ (

**d**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}{=\pm 90\xb0,\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}{=0\xb0,\mathrm{d}}_{4}=0.2\text{}\mathrm{m}$ (

**e**) ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}{=0\xb0,\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}{=\pm 30\xb0,\mathrm{d}}_{4}=0.2\text{}\mathrm{m}.$

**Figure 8.**Continuous direct and inverse projections-mappings of internal and external singularities in a section of metamorphic workspace with the variation only of ${\mathsf{\theta}}_{{\mathsf{\pi}}_{1}}$ on left (

**a**) and of ${\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}$ on (

**b**) right.

**Figure 9.**(

**a**) A free of kinematic singularity path joins two inverse kinematic solutions in aspect ${\mathrm{A}}_{1}$, (

**b**) perfect cyclic motion of the TCP encircling a cusp point in the workspace of the selected metamorphic anatomy.

**Figure 11.**The determinant of the Jacobian matrix as a function of discrete steps for a perfect circle.

**Figure 12.**(

**a**) A continuous and smooth path for two inverse kinematic solutions without change of posture, (

**b**) Circle is performed in half cross-section of metamorphic workspace encircling a cusp point.

**Figure 13.**Change of angular position when the metamorphic anatomy performs a non-singular posture changing trajectory.

**Figure 15.**(

**a**) Smooth curved joint path in aspect ${\mathrm{A}}_{1}$ and the respective kinematic singularities (

**b**) Rectilinear motion of metamorphic mechanism in half cross-section of the workspace $\left(\mathsf{\rho},\mathsf{{\rm Z}}\right)$ in the region with 2 IKS.

**Figure 17.**The continuous function of the determinant of geometric Jacobian for rectilinear trajectory.

**Table 1.**The DH-parameters of the metamorphic manipulator ($\mathrm{A}=\left({\mathrm{R}}_{1}{+\mathsf{\alpha}}_{1}+\mathrm{h}\right)$, $\mathrm{B}=\left({\mathrm{R}}_{2}{+\mathsf{\alpha}}_{2}+\mathrm{h}\right)$, ${\mathrm{R}}_{1}{=\mathrm{R}}_{2}=0.045$ m ${\mathsf{\alpha}}_{1}={\mathsf{\alpha}}_{2}=0.04225$ m $\mathrm{d}=0.2735$ m and $\mathrm{h}=0.08725$ m).

i | d_{i} | ${\mathbf{a}}_{\mathbf{i}}$ | ${\mathbf{r}}_{\mathbf{i}}$ | ${\mathsf{\theta}}_{\mathbf{i}}$ |
---|---|---|---|---|

1 | $0$ | $0\xb0$ | $0$ | ${\mathsf{\theta}}_{1}$ |

2 | $\mathrm{A}\ast \mathrm{sin}{\mathsf{\theta}}_{\mathsf{\pi}}{}_{1}$ | $-90\xb0$ | $\mathrm{d}+\mathrm{B}\ast \mathrm{cos}{\mathsf{\theta}}_{{\mathsf{\pi}}_{2}}$ | ${\mathsf{\theta}}_{2}$ |

3 | $\mathrm{B}\ast \mathrm{sin}{\mathsf{\theta}}_{\mathsf{\pi}}{}_{2}$ | $90\xb0$ | ${\mathrm{r}}_{3}$ | ${\mathsf{\theta}}_{3}$ |

# | ${\mathsf{\theta}}_{1}$ | ${\mathsf{\theta}}_{2}$ | ${\mathsf{\theta}}_{3}$ |
---|---|---|---|

1 | −1.7417 | −2.8731 | 0.9736 |

2 | −1.4908 | −2.5800 | 2.0697 |

3 | −1.1937 | −0.9150 | 2.5274 |

4 | −0.7096 | −0.2471 | −0.7420 |

# | ${\mathsf{\theta}}_{1}$ | ${\mathsf{\theta}}_{2}$ | ${\mathsf{\theta}}_{3}$ |
---|---|---|---|

1 | −1.8502 | −3.0993 | 0.9560 |

2 | −1.2824 | −2.8118 | 2.1946 |

3 | −1.1706 | −0.8051 | 2.3190 |

4 | −0.1677 | −0.0435 | −0.9986 |

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**MDPI and ACS Style**

Koukos-Papagiannis, C.; Moulianitis, V.; Aspragathos, N.
Classification of All Non-Isomorphic Regular and Cuspidal Arm Anatomies in an Orthogonal Metamorphic Manipulator. *Robotics* **2020**, *9*, 20.
https://doi.org/10.3390/robotics9020020

**AMA Style**

Koukos-Papagiannis C, Moulianitis V, Aspragathos N.
Classification of All Non-Isomorphic Regular and Cuspidal Arm Anatomies in an Orthogonal Metamorphic Manipulator. *Robotics*. 2020; 9(2):20.
https://doi.org/10.3390/robotics9020020

**Chicago/Turabian Style**

Koukos-Papagiannis, Christos, Vassilis Moulianitis, and Nikos Aspragathos.
2020. "Classification of All Non-Isomorphic Regular and Cuspidal Arm Anatomies in an Orthogonal Metamorphic Manipulator" *Robotics* 9, no. 2: 20.
https://doi.org/10.3390/robotics9020020