# Virtual Sensor for Kinematic Estimation of Flexible Links in Parallel Robots

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Development of the Kinematic Virtual Sensor

#### 2.1. Fundamentals of the Virtual Sensor

#### 2.2. Modelling of Flexible Links

#### 2.3. Modal Analysis of the Flexible Links

- If $det\left({\left({\mathbf{K}}_{{i}_{f}}\right)}_{BC}-{\mathit{\omega}}^{2}\phantom{\rule{0.166667em}{0ex}}{\left({\mathbf{M}}_{{i}_{f}}\right)}_{BC}\right)\ne 0$, then the solution is$$\mathbf{X}=0$$
- If $\mathbf{X}\ne 0$, the result of the eigenproblem is reduced to solve Equation (30).$$det\left({\left({\mathbf{K}}_{{i}_{f}}\right)}_{BC}-{\mathit{\omega}}^{2}\phantom{\rule{0.166667em}{0ex}}{\left({\mathbf{M}}_{{i}_{f}}\right)}_{BC}\right)=0$$$$\left({\left({\mathbf{K}}_{{i}_{f}}\right)}_{BC}-{\omega}_{i}^{2}\phantom{\rule{0.166667em}{0ex}}{\left({\mathbf{M}}_{{i}_{f}}\right)}_{BC}\right)\phantom{\rule{0.166667em}{0ex}}{\mathbf{X}}_{i}=0\phantom{\rule{28.45274pt}{0ex}}i=1,2,3\dots ,{n}_{DOF}$$$${f}_{i}=\frac{{\omega}_{i}}{2\pi}$$

## 3. Case of Study: Delta Robot

#### 3.1. The Delta Robot

^{3}. The geometry has been selected in order to limit the deformation to the direction of the ${z}_{i}$ axis of each link, and analyse the validity of the proposed approach. All these proprieties are summarised in Table 2.

#### 3.2. Numeric Implementation of the Kinematic Virtual Sensor Applied to the Delta Robot

- The base of the link does not experience any deflection: ${z}_{i}\left(0\right)=0$.
- The link at the base has no deformation, so that the derivative of the deflection function is zero at that point: $\frac{\partial {z}_{i}\left(0\right)}{\partial x}=0$.
- There is no bending moment at the end of the link: $\frac{{\partial}^{2}{z}_{i}\left({l}_{{T}_{i}}\right)}{\partial {x}^{2}}=0$.
- There is no shearing force acting at the end of the link: $\frac{{\partial}^{3}{z}_{i}\left({l}_{{T}_{i}}\right)}{\partial {x}^{3}}=0$.

- If ${z}_{i}\left(0\right)=0$ then ${u}_{i1}=0$.
- If $\frac{\partial {z}_{i}\left(0\right)}{\partial x}=0$ then ${u}_{i2}=0$.

#### 3.3. Simulation Setup

## 4. Results and Discussion

^{2}, a relative error of less than the $13\%$ of its amplitude.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

PKR | Parallel Kinematic Robot |

TCP | Tool Centre Point |

DOF | Degree of Freedom |

AMM | Assumed Modes Method |

FEM | Finite Element Method |

ERLS | Equivalent Rigid Link System |

BC | Boundary Condition |

DKP | Direct Kinematic Problem |

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**Figure 1.**A flexible manipulator with the same configuration of rigid joints ${\mathbf{q}}_{\mathbf{r}}({q}_{1},{q}_{2})$ but multiple end-effector position/orientation due to the flexibility of the links, ${\mathbf{q}}_{\mathbf{f}}({q}_{{f}_{d}},{q}_{{f}_{s}})$. (${q}_{{f}_{s}}$ link tip transverse flexural deflection and ${q}_{{f}_{s}}$ link tip flexural slope).

**Figure 2.**Schematic of a flexible link (b) connected with two stiff links (a, c) using rotatory joint.

**Figure 10.**Virtual Sensor link deflection estimation compared with the real deflection at the tip of the link obtained with ADAMS Multibody Software.

**Figure 12.**Virtual Sensor TCP deflection estimation compared with the TCP’s deflection obtained with ADAMS Multibody Software.

**Figure 13.**Virtual Sensor TCP deflection estimation error compared with the TCP’s deflection obtained with ADAMS Multibody Software.

Fixed Base | Upper Link ${\mathit{L}}_{\mathit{i}}$ | Lower Link ${\mathit{l}}_{{\mathit{T}}_{\mathit{i}}}$ | Mobile Platform | |
---|---|---|---|---|

Length (m) | $|{\mathbf{a}}_{\mathbf{i}}|=0.100$ | 0.150 | 0.400 | $|{\mathbf{d}}_{\mathbf{i}}|=0.040$ |

Mass (kg) | 0.0365 | 0.1319 | 0.1278 | |

${I}_{{L}_{{i}_{xx}}}=2.2781\times {10}^{-6}$ | ${I}_{{l}_{{i}_{xx}}}=2.0023\times {10}^{-6}$ | ${I}_{{p}_{xx}}=4.6225\times {10}^{-5}$ | ||

Inertia (kg m^{2}) | ${I}_{{L}_{{i}_{yy}}}=8.7001\times {10}^{-5}$ | ${I}_{{l}_{{i}_{yy}}}=0.0010$ | ${I}_{{p}_{yy}}=4.6225\times {10}^{-5}$ | |

${I}_{{L}_{{i}_{zz}}}=8.6422\times {10}^{-5}$ | ${I}_{{l}_{{i}_{zz}}}=0.0010$ | ${I}_{{p}_{zz}}=9.1472\times {10}^{-5}$ |

Length | Width | Thickness | Young’s Modulus | Mass Density |
---|---|---|---|---|

0.400 m | 0.015 m | 0.003 m | 71 GPa | 2740 kg/m^{3} |

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**MDPI and ACS Style**

Bengoa, P.; Zubizarreta, A.; Cabanes, I.; Mancisidor, A.; Pinto, C.; Mata, S.
Virtual Sensor for Kinematic Estimation of Flexible Links in Parallel Robots. *Sensors* **2017**, *17*, 1934.
https://doi.org/10.3390/s17091934

**AMA Style**

Bengoa P, Zubizarreta A, Cabanes I, Mancisidor A, Pinto C, Mata S.
Virtual Sensor for Kinematic Estimation of Flexible Links in Parallel Robots. *Sensors*. 2017; 17(9):1934.
https://doi.org/10.3390/s17091934

**Chicago/Turabian Style**

Bengoa, Pablo, Asier Zubizarreta, Itziar Cabanes, Aitziber Mancisidor, Charles Pinto, and Sara Mata.
2017. "Virtual Sensor for Kinematic Estimation of Flexible Links in Parallel Robots" *Sensors* 17, no. 9: 1934.
https://doi.org/10.3390/s17091934