# Sensor-Based Identification of Singularities in Parallel Manipulators

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## Abstract

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## 1. Introduction

- Modelling complexity varies across mechanical architectures, complicating the establishment of a general approach applicable to different PKMs. A comprehensive examination of PKM modelling and its associated challenges is provided [29].

## 2. Singularities in Parallel Manipulators

## 3. Sensor-Based Identification of Singularities

- The working conditions of the PKM remain constant during its principal or fundamental movements. The fundamental movements represent the most commonly performed actions by the PKM within a given task. Variations in payload and wear and tear over the PKM’s lifetime are not taken into account.
- The sensor signals are assumed to be minimally affected by noise, either through hardware or software filtering mechanisms.
- The control unit works at a constant sample time.

## 4. Case Study: L-CaPaMan

- TM1: Crossing two Type I singularities in the middle of two desired configurations.
- TM2: Starting from a Type I singularity to reach a non-singular pose.

Algorithm 1: Matlab pseudo-code for sensor-based identification of singularities in the L-CaPaMan PKM |

Data: Reference Trajectories from TM1 or TM2Result: Numerical Identification of Singularitiesinitialise threshold ${\mathit{e}}_{\mathit{th}}$ and ${\dot{\mathit{i}}}_{\mathit{th}}$ initialise serial communication with Arduino Mega i = 0 |

#### Results

## 5. Case Study: Five-Bar Mechanism

Algorithm 2: MATLAB pseudo-code for sensor-based identification of singularities in the 5R PKM |

Data: Reference Trajectories from TM3 or TM4Result: Numerical Identification of Singularitiesinitialise threshold ${\mathit{e}}_{\mathit{th}}$ and ${\dot{\mathit{i}}}_{\mathit{th}}$ initialise serial communication with Arduino Mega i = 0 |

#### Results

## 6. Case Study: 4-DOF Parallel Manipulator

Algorithm 3: ROS2 pseudo-code for sensor-based identification of singularities in the 3UPS+RPU PKM |

Data: Reference Trajectories from TM5 or TM6Result: Numerical Identification of Singularitiesinitialise threshold ${\mathit{e}}_{\mathit{th}}$ and ${\dot{\mathit{i}}}_{\mathit{th}}$ initialise communication with Optitrack 3DTS i = 0 |

#### Results

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DOF | Degrees of freedom |

PKM | Parallel Kinematic Mechanisms |

ISA | Instantaneous Screw Axis |

IMU | Inertial measurement unit |

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**Figure 1.**A general scheme of a PKM and its main kinematic elements. The pose of the end-effector and the active joints are shown in red.

**Figure 6.**Results of testing TM1 with the L-CaPaMan PKM: (

**a**) absolute position error along ${Z}_{p}$; (

**b**) time derivative of the current consumed by actuator 1. The detected Type I singularities are enclosed in two black circles.

**Figure 7.**Computed $|{\mathit{J}}_{\mathit{I}}|$ during TM1 with the L-CaPaMan PKM. The detected Type I singularities are enclosed in two black circles.

**Figure 8.**Results of testing TM2 with the L-CaPaMan PKM: (

**a**) absolute position error along ${Z}_{p}$; (

**b**) time derivative of the current consumed by actuator 1.

**Figure 11.**Results of testing TM3 with 5R PKM: (

**a**) absolute position error for ${y}_{p}$; (

**b**) time derivative of consumed current by actuator 1. The detected Type II singularity is enclosed in a black circle.

**Figure 12.**Computed $|{\mathit{J}}_{\mathit{D}}|$ during TM3 with the 5R PKM. The detected Type II singularity is enclosed in a black circle.

**Figure 13.**Results of testing TM4 with the 5R PKM: (

**a**) absolute position error for ${y}_{p}$; (

**b**) time derivative of the current consumed by actuator 1.

**Figure 16.**Results of testing TM5 with the 3UPS+RPU PKM: (

**a**) absolute position error for ${x}_{m}$; (

**b**) time derivative of the current consumed by actuator 1. The detected Type II singularity is enclosed in a black circle.

**Figure 17.**Computed $|{\mathit{J}}_{\mathit{D}}|$ during TM5 with the 3UPS+RPU PKM. The detected Type II singularity is enclosed in a black circle.

**Figure 18.**Results of testing TM6 with the 3UPS+RPU PKM: (

**a**) absolute position error for ${x}_{m}$; (

**b**) time derivative of the current consumed by actuator 1.

TM1 | TM2 | ||||||||
---|---|---|---|---|---|---|---|---|---|

Location | ${\mathit{Z}}_{\mathit{p}}$(m) | $\mathit{\phi}$(deg.) | $\mathit{\psi}$(deg.) | Time(s) | ${\mathit{Z}}_{\mathit{p}}$(m) | $\mathit{\phi}$(deg.) | $\mathit{\psi}$(deg.) | Time(s) | |

Start | 0.13 | 0 | −22 | 0 | 0.13 | −6 | −7 | 0 | |

Singularity | 0.14 | 0 | 22 | 0.18 | 1.22 | 0.13 | −6 | −7 | 0 |

End | 0.13 | 0 | −22 | 3.1 | 1.8 | 0 | 0 | 2 |

$\mathit{OA}1,\mathit{OA}2$ (m) | $\mathit{A}1\mathit{B}1,\mathit{A}2\mathit{B}2$ (m) | $\mathit{B}1\mathit{P},\mathit{B}2\mathit{P}$ (m) |
---|---|---|

0.04 | 0.06 | 0.05 |

TM3 | TM4 | |||||
---|---|---|---|---|---|---|

Location | ${\mathit{x}}_{\mathit{p}}$ (m) | ${\mathit{y}}_{\mathit{p}}$ (m) | Time (s) | ${\mathit{x}}_{\mathit{p}}$ (m) | ${\mathit{y}}_{\mathit{p}}$ (m) | Time (s) |

Start | 0 | 0.09 | 0 | 0 | 0.09 | 0 |

Singularity | −0.03 | 0.05 | 2.8 | −0.03 | 0.05 | 3 |

End | −0.04 | 0.03 | 3.1 | −0.03 | 0.05 | 3 |

${\mathit{R}}_{1}\phantom{\rule{0.277778em}{0ex}},{\mathit{R}}_{2},\phantom{\rule{0.277778em}{0ex}}{\mathit{R}}_{3}$(m) | ${\mathit{\beta}}_{\mathit{FD}}$(deg.) | ${\mathit{\beta}}_{\mathit{FI}}$(deg.) | $\mathit{ds}$(m) | ${\mathit{R}}_{\mathit{m}1},\phantom{\rule{0.277778em}{0ex}}{\mathit{R}}_{\mathit{m}2},\phantom{\rule{0.277778em}{0ex}}{\mathit{R}}_{\mathit{m}3}$(m) | ${\mathit{\beta}}_{\mathit{MD}}$(deg.) | ${\mathit{\beta}}_{\mathit{MI}}$(deg.) |
---|---|---|---|---|---|---|

0.4 | 90 | 45 | 0.15 | 0.3 | 50 | 90 |

TM5 | TM6 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Pose | ${\mathit{x}}_{\mathit{m}}$(m) | ${\mathit{Z}}_{\mathit{m}}$(m) | $\mathit{\theta}$(deg.) | $\mathit{\psi}$(deg.) | t(s) | ${\mathit{x}}_{\mathit{m}}$(m) | ${\mathit{Z}}_{\mathit{m}}$(m) | $\mathit{\theta}$(deg.) | $\mathit{\psi}$(deg.) | t(s) |

Start | −0.05 | 0.63 | 5 | 0 | 0 | 0 | 0.62 | 8 | −6 | 0 |

Singularity | −0.05 | 0.73 | 5 | 34 | 14 | 0.08 | 0.72 | −3 | 15 | 15 |

End | −0.05 | 0.73 | 5 | 44 | 17 | 0.16 | 0.76 | −16 | 41 | 24 |

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**MDPI and ACS Style**

Pulloquinga, J.L.; Ceccarelli, M.; Mata, V.; Valera, A.
Sensor-Based Identification of Singularities in Parallel Manipulators. *Actuators* **2024**, *13*, 168.
https://doi.org/10.3390/act13050168

**AMA Style**

Pulloquinga JL, Ceccarelli M, Mata V, Valera A.
Sensor-Based Identification of Singularities in Parallel Manipulators. *Actuators*. 2024; 13(5):168.
https://doi.org/10.3390/act13050168

**Chicago/Turabian Style**

Pulloquinga, Jose L., Marco Ceccarelli, Vicente Mata, and Angel Valera.
2024. "Sensor-Based Identification of Singularities in Parallel Manipulators" *Actuators* 13, no. 5: 168.
https://doi.org/10.3390/act13050168