# Modified Hand–Eye Calibration Using Dual Quaternions

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## Abstract

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## 1. Introduction

#### 1.1. Brief Overview of the Hand–Eye Equation

#### 1.2. Dual Quaternion Algebra and Hand–Eye Calibration

## 2. Methodology

#### 2.1. Kinematic Analysis of the Dual-Arm Hand–Eye Robot Based on Dual Quaternion

#### 2.2. Identification Algorithm of Each Dual Part and Scalar Part of the Dual Quaternion

Algorithm 1: Modified hand–eye calibration for the dual-arm robot system using dual quaternion |

Input: The pictures captured by cameras; the poses data of end-effectors; the corresponding dual quaternions $\widehat{\mathit{A}}$, $\widehat{\mathit{B}}$.1: Compute the real part and the dual part of ${\widehat{\mathit{q}}}_{ca\_j}^{robot}$ based on Equations (6), (11), and (12). 2: initial ${\widehat{\mathit{q}}}_{ca\_j}^{robo{t}^{\left(0\right)}}\leftarrow $ the first solution in step 1; $n\leftarrow 1$ 3: while $\Vert \left(\left[{\widehat{\mathit{A}}}^{+}\right]-\left[{\widehat{\mathit{B}}}^{-}\right]\right){\widehat{\mathit{q}}}_{ca\_j}^{robo{t}^{\left(n\right)}}\Vert >\epsilon $, doForm solution based on Equations (11) and (12); $n\leftarrow n+1$; 4: Form kinematic equation of robot system based on Equations (15) and (16); end whileOutput: ${\widehat{\mathit{q}}}_{ca\_j}^{robot}\leftarrow {\widehat{\mathit{q}}}_{ca\_j}^{robo{t}^{\left(n\right)}}$ |

## 3. Simulation and Discussion

## 4. Experiment and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Li, G.; Zou, S.; Din, S.; Qi, B.
Modified Hand–Eye Calibration Using Dual Quaternions. *Appl. Sci.* **2022**, *12*, 12480.
https://doi.org/10.3390/app122312480

**AMA Style**

Li G, Zou S, Din S, Qi B.
Modified Hand–Eye Calibration Using Dual Quaternions. *Applied Sciences*. 2022; 12(23):12480.
https://doi.org/10.3390/app122312480

**Chicago/Turabian Style**

Li, Guozhi, Shuizhong Zou, Shuxue Din, and Bin Qi.
2022. "Modified Hand–Eye Calibration Using Dual Quaternions" *Applied Sciences* 12, no. 23: 12480.
https://doi.org/10.3390/app122312480