# Kinematic Calibration Method for Large-Sized 7-DoF Hybrid Spray-Painting Robots

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Kinematic Modeling

#### 3.1. Introduction to Robot Configuration

#### 3.2. Forward Kinematic Modeling

_{i−}

_{1}from axis Z

_{i}

_{−1}to axis Z

_{i}along the direction of axis X

_{i}

_{−1}, rotating angle α

_{i}

_{−1}from axis Z

_{i}

_{−1}to axis Z

_{i}around axis X

_{i}

_{−1}, distance d

_{i}

^{*}from axis X

_{i}

_{−1}to axis X

_{i}along the direction of axis Z

_{i}, and rotating angle θ

_{i}from axis X

_{i}

_{−1}to axis X

_{i}about axis Z

_{i}. As a result, the homogeneous transformation matrix from the coordinate system O

_{i}

_{−1}–X

_{i}

_{−1}Y

_{i}

_{−1}Z

_{i}

_{−1}to O

_{i}

_{−1}–X

_{i}Y

_{i}Z

_{i}can be expressed as

#### 3.3. The MD-H Error Model

_{i}about the Y-axis to transform axis Z

_{i}

_{−1}to Z

_{i}, thus avoiding the defects of the D-H model. The other parameter definitions of the improved model are the same as those of the D-H model. The joint deviation d

_{i}

^{*}is set to zero if the axes of two adjacent connecting rods are parallel. The rotation angle β

_{i}is set to zero if the axes of two adjacent shores are not parallel. Then the MDH parameters of the robot are shown in Table 1. The coordinate transformation matrix of the improved model can be expressed as

## 4. Rigid-Flexible Coupling Error Model

#### 4.1. Kinematic Error Model

#### Kinematic Error Model

^{i}

^{−1}

**T**

_{i}needs to be expressed as a function of

^{i}

^{−1}

**T**

_{i}. Suppose that

**D**

_{α},

**D**

_{a},

**D**

_{θ},

**D**

_{d*}, and

**D**

_{β}are coefficient matrices.

**Δ**

_{i}is the differential transformation matrix of the ith joint in the (i − 1)th joint coordinate system and can be expressed as

_{i,}dy

_{i,}and dz

_{i}correspond to differential translation, and δx

_{i,}δy

_{i,}and δz

_{i}to differential rotation.

**n**

_{i}

^{7},

**o**

_{i}

^{7},

**a**

_{i}

^{7}, and p

_{i}

^{7}are the values of the transformation matrix from the ith joint coordinate system to the terminal coordinate system. Then, we can obtain the terminal errors induced by joint i as

**r**denotes the total error obtained by summing the errors of the D-H parameters.

#### 4.2. Analysis of Redundancy Kinematic Error of Robots

**J**is obtained by taking the total differential of all the kinematic parameters, and

**J**

_{ai}

_{−1},

**J**

_{αi}

_{−1},

**J**

_{di},

**J**

_{θ1}, and

**J**

_{βi}indicate the five columns of the Jacobian matrix corresponding to the five D-H parameters, respectively. Then, the Jacobian matrix

**J**can be expressed as

- If α
_{i−}_{1}≠ 0, a redundancy error is present; - If α
_{i−1}= 0 and a_{i−}_{1}≠ 0, δd_{i}_{−1}^{*}and δd_{i}^{*}are mutually redundant, and either shall be eliminated, and δβ_{i}shall be introduced for identification. - If α
_{i−1}= 0 and a_{i−}_{1}= 0, δd_{i}_{−1}^{*}and δd_{i}^{*}are mutually redundant, and δθ_{i−}_{1}and δθ_{i}are mutually redundant, and some parameters shall be eliminated. - If θ
_{i}= 0 and d_{i}^{*}≠ 0, δa_{i}_{−1}and δa_{i}are mutually redundant, and some parameters shall be eliminated. - If θ
_{i}= 0 and d_{i}^{*}= 0, δα_{i}_{−1}and δα_{i}are mutually redundant and δa_{i}_{−1}and δa_{i}are mutually redundant, and some parameters shall be eliminated. - After eliminating redundant parameters, the D-H parameter error still contains 24 errors.

## 5. Static Stiffness Error Model

#### 5.1. Static Stiffness Model

_{i}(unit: kg), the length is considered to be l

_{i}(unit: m), and, in particular, the length of Unit 9 l

_{9}= l

_{o}+ d

_{6}

^{*}(l

_{o}is the length of the hinge when d

_{6}

^{*}= 0 holds in the D-H coordinate system).

_{1}of O

_{1}–X

_{1}Y

_{1}Z

_{1}in the robotic arm D-H coordinate system, the x-direction always coincides with the X

_{2}-direction of O

_{1}–X

_{2}Y

_{2}Z

_{2}in the robotic arm coordinate system. The y-direction is opposite to the gravity direction.

_{yj}= F

_{yk}= 1/2 m

_{i}g (the concentrated load is positive, which means that the direction is the same as that shown in Figure 5, and the same for the following bending moment), M

_{j}= –1/12 m

_{i}gl

_{i}, and M

_{k}= 1/12 m

_{i}gl

_{i}.

_{i}

^{*}is the angle between the local coordinate system x

_{i}y

_{i}of Unit i and the finite element global coordinate system o–XY and is positive if the finite element global coordinate system rotates toward the local coordinate system counterclockwise. In vector

**β**

^{*}, β

_{1}

^{*}= π/2, and β

_{2}

^{*}is a non-zero fixed value, which can be obtained by measuring the geometric relationship of the robotic arm structure. Besides, some elements of the vector

**β**

^{*}can be described by the variables of the robotic arm in the D-H coordinate system, and some elements of the vector can be expressed as

_{i}, then the stiffness matrix of this unit in the finite element global coordinate system is

**T**

_{i}is

_{5}and d

_{6}

^{*}) of the hinge, the stiffness matrix of Unit 9 can be expressed θ

_{5}about and d

_{6}

^{*}as

_{i}and v

_{i}is m, and the unit of φ

_{i}is rad. Then, the displacement of the whole structure and the load on it under the finite element global coordinate system can be expressed as

_{1}on terminal deflection can be ignored. The displacement and angle of rotation of each node can be obtained by eliminating the corresponding rows and columns of $\widehat{\mathit{K}}\mathit{q}=\mathit{F}$using boundary conditions.

**λ**

_{i}= [u

_{i}v

_{i}0] under the finite element global coordinate system and rotated along the direction vector p in the z-direction of the finite element global coordinate system with the angle of rotation φ

_{i}.

_{j}X

_{j}Y

_{j}Z

_{j}and the terminal’s element coordinate system with the impact of gravity considered is O

_{j}

^{ʹ}X

_{j}

^{ʹ}Y

_{j}

^{ʹ}Z

_{j}

^{ʹ}. Here, the homogeneous transformation matrix from the coordinate system O

_{7}X

_{7}Y

_{7}Z

_{7}to the coordinate system O

_{7}

^{ʹ}X

_{7}

^{ʹ}Y

_{7}

^{ʹ}Z

_{7}

^{ʹ}is defined as

^{7}

**T**

_{7’}.

_{7}of the coordinate system O

_{7}X

_{7}Y

_{7}Z

_{7}, the translation vector of the coordinate system O

_{7}X

_{7}Y

_{7}Z

_{7}under the finite element global coordinate system is

**λ**

_{9}is expressed as

^{7}

**λ**

_{9}under the coordinate system O

_{7}X

_{7}Y

_{7}Z

_{7}.

^{7}

**T**

_{7ʹ}from the coordinate system O

_{7}X

_{7}Y

_{7}Z

_{7}to the coordinate system O

_{7}

^{ʹ}X

_{7}

^{ʹ}Y

_{7}

^{ʹ}Z

_{7}

^{ʹ}as

_{1×3}is a 0 matrix with one row and three columns. For the end displacement

**p**

_{7}= [σ

_{x}

_{7}σ

_{y}

_{7}σ

_{z}

_{7}], parameter

^{7ʹ}R

_{7}is derived as follows:

_{9}= cos(φ

_{9}) and sφ

_{9}= sin(φ

_{9}).

#### 5.2. Rigid-Flexible Coupling Kinematic Error Model

^{i}

^{−1}

**T**

_{iʹ}of gravitational deformation of nominal parameters and the parameters compensated in the driving direction. The errors of the terminal coordinate system can be obtained.

**J**

^{*}is the Jacobian matrix of error with gravitational deformation taken into consideration. After the measured terminal error is estimated, one can determine the geometric error from

**J**

^{*}, which, in turn, is calculated via the regularized least squares method [29,30].

## 6. Error Compensation Strategy for 7-DoF Hybrid Spray-Painting Robots

_{1}q

_{2}q

_{3}q

_{4}q

_{5}q

_{6}q

_{7}]

^{T}can be computed using the numerical method.

_{m}(q) is the inertia matrix of the current joint coordinates, θ

_{u}is the upper joint boundary, and θ

_{l}is the lower joint boundary. The conformation shows that joints #3, #4, and #6 are more influential than the others, mainly comparing these three joints. The impact of the inertial load on joints #3, #4, and #6 is shown in Figure 7, where the slope is the inertia sensitivity.

_{6}(q) denotes the column corresponding to d6 in the driving Jacobi matrix, Then, the inertia load sensitivity of the joint corresponding to the unit end error can be expressed as:

## 7. Experimental Methods and Data Analysis

#### 7.1. Procedures

^{−6}L

^{*}) μm, where L

^{*}was the measured distance, in μm. The structural parameters of the robot are listed in Table 2.

_{e}. Since the terminal adapter panel and laser tracker reflection sphere had high mechanical processing accuracy and were installed at the farthest distance from the robot base, the slight change in the posture had a slight effect on the final measured error. To simplify the calculation, A

_{e}only considered the position but not the posture in this study, being was expressed as follows:

^{w}T

_{0}is the robot base coordinate system under the global coordinate system and

^{7ʹ}T

_{e}is the target spherical coordinate system under the robot terminal coordinate system. Therefore, the geometric errors in

^{w}T

_{0}and

^{7ʹ}T

_{e}also had to be identified during error identification. For this reason, these two errors were also included in establishing the error model to reduce the error introduced in the measurement process.

^{7}= 78,125 joint input arrangements for the seven joints, which were used as candidates. They were further optimized, according to the measurement posture optimization algorithm proposed in the study [46] to obtain the final 150 sets of measured postures.

#### 7.2. Validation of Error Elimination Effectiveness

#### 7.3. Experimental Results

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**3D model of the robot. For the convenience of description, the seven joints are numbered from 1 to 7, from the base to the terminal of the robot.

**Figure 3.**Structure of the robotic arm. 1–9 indicates the node numbers and ①–⑨ the numbering of the beam units.

**Figure 10.**Identification of various parameters after the elimination of redundancy error: (

**a**) $\delta {a}_{i-1}$; (

**b**) $\delta {\alpha}_{i-1}$; (

**c**) $\delta {d}_{i}$; (

**d**) $\delta {\theta}_{i}$.

**Figure 11.**Identification of various parameters before elimination of redundancy error: (

**a**) $\delta {a}_{i-1}$; (

**b**) $\delta {\alpha}_{i-1}$; (

**c**) $\delta {d}_{i}$; (

**d**) $\delta {\theta}_{i}$.

**Figure 12.**Comparison of kinematic calibration and non-calibration in the X- (

**a**) Y- (

**b**) and Z- (

**c**) directions.

**Figure 13.**Comparison of gravity compensation and non-compensation in the X- (

**a**), Y- (

**b**) and Z- (

**c**) directions.

Joint No. | ${\mathit{a}}_{\mathit{i}-1}$/mm | ${\mathit{\alpha}}_{\mathit{i}-1}$/° | ${\mathit{d}}_{\mathit{i}}^{*}$/mm | ${\mathit{\theta}}_{\mathit{i}}$/° | β_{i}/° |
---|---|---|---|---|---|

#1 | 0 | 0 | ${d}_{1}^{*}$ | 0 | - |

#2 | 0 | −90 | ${d}_{2}^{*}$ | ${\theta}_{2}$ | - |

#3 | ${a}_{2}$ | −90 | 0 | ${\theta}_{3}$ | - |

#4 | ${a}_{3}$ | 0 | 0 | ${\theta}_{4}$ | β_{4} |

#5 | ${a}_{4}$ | −90 | 0 | ${\theta}_{5}$ | - |

#6 | 0 | 0 | ${d}_{6}^{*}$ | 0 | - |

#7 | 0 | 90 | 0 | ${\theta}_{7}$ | - |

${\mathit{a}}_{2}$/mm | ${\mathit{a}}_{3}$/mm | ${\mathit{a}}_{4}$/mm | ${\mathit{d}}_{1}$/mm | ${\mathit{d}}_{2}$/mm | ${\mathit{d}}_{6}$/mm |
---|---|---|---|---|---|

250 | 1350 | 246 | 0–4000 | 626 | 1700–2300 |

X-Direction/m | Y-Direction/m | Z-Direction/m | ||||
---|---|---|---|---|---|---|

Maximum | Mean | Maximum | Mean | Maximum | Mean | |

Without calibration | 0.0147 | 0.0057 | 0.0262 | 0.0144 | 0.0195 | 0.0063 |

Only calibration | 0.0024 | 0.0009 | 0.0020 | 0.0006 | 0.0024 | 0.0008 |

Calibration considering gravity | 0.0017 | 0.0007 | 0.0015 | 0.0004 | 0.0023 | 0.0007 |

Joint No. | $\mathit{\delta}{\mathit{a}}_{\mathit{i}-1}$/m | $\mathit{\delta}{\mathit{\alpha}}_{\mathit{i}-1}$/rad | $\mathit{\delta}{\mathit{d}}_{\mathit{i}}$/m | $\mathit{\delta}{\mathit{\theta}}_{\mathit{i}}$/rad | δβ_{i}/rad |
---|---|---|---|---|---|

1 | - | −0.0002 | −0.0151 | 0.0056 | - |

2 | −0.0138 | −0.0009 | 0.0128 | 0.0012 | - |

3 | −0.0011 | 0.0015 | - | 0.0002 | - |

4 | 0.0009 | 0.0020 | 0.0076 | −0.0171 | 0.0148 |

5 | −0.0116 | −0.0106 | - | 0.0148 | - |

6 | - | −0.0019 | 0.0182 | - | - |

7 | −0.0107 | 0.0035 | −0.0208 | −0.0001 | - |

X-Direction/m | Y-Direction/m | Z-Direction/m | ||||
---|---|---|---|---|---|---|

Maximum | Mean | Maximum | Mean | Maximum | Mean | |

Without calibration | 0.0158 | 0.0059 | 0.0271 | 0.0139 | 0.0182 | 0.0067 |

Only calibration | 0.0027 | 0.0010 | 0.0019 | 0.0006 | 0.0030 | 0.0011 |

Calibration considering gravity | 0.0021 | 0.0008 | 0.0015 | 0.0005 | 0.0029 | 0.0011 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Li, M.; Wang, J.; Zhao, Q.; Wu, J.; Wang, J.
Kinematic Calibration Method for Large-Sized 7-DoF Hybrid Spray-Painting Robots. *Machines* **2023**, *11*, 20.
https://doi.org/10.3390/machines11010020

**AMA Style**

Wang Y, Li M, Wang J, Zhao Q, Wu J, Wang J.
Kinematic Calibration Method for Large-Sized 7-DoF Hybrid Spray-Painting Robots. *Machines*. 2023; 11(1):20.
https://doi.org/10.3390/machines11010020

**Chicago/Turabian Style**

Wang, Yutian, Mengyu Li, Junjian Wang, Qinzhi Zhao, Jun Wu, and Jinsong Wang.
2023. "Kinematic Calibration Method for Large-Sized 7-DoF Hybrid Spray-Painting Robots" *Machines* 11, no. 1: 20.
https://doi.org/10.3390/machines11010020