# A Novel Error Sensitivity Analysis Method for a Parallel Spindle Head

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Configuration and Error Modeling

#### 2.1. Configuration

#### 2.2. TCP Position Error Modeling

**a**

_{i},

**b**

_{i}, and

**l**

_{i}are shown in Figure 2.

**H**= [x y z]

^{T}denotes the vector

**OO’**,

**q**

_{i}represents the drive vector,

**R**

_{i}represents the rotation matrix of each limb, and

**R**

_{Bi}and

**R**

_{Ci}denote the rotation matrices of the P and R joint.

**R**

_{TT}represents the orientation matrix of the moving platform, which is described using the T-T angle [37,38] to decouple the rotation around the Z’ axis from the other two rotations.

**a**

_{i}and Δ

**b**

_{i}denote the geometric error vector of

**a**

_{i},

**b**

_{i}, respectively. Δq

_{i}and Δl

_{i}denote the input error of the P joint and the length error of the link, respectively. Δ

**θ**

_{Bi}and Δ

**θ**

_{Ci}are the orientation error vectors of orientation matrix

**R**

_{Bi}and

**R**

_{Ci}, respectively. Δ

**H**= [Δx Δy Δz]

^{T}and Δ

**α**= [Δα

_{x}Δα

_{y}Δα

_{z}]

^{T}denote the position and orientation error of the center of the moving platform, and a total of 42 error parameters are included in this error model, i.e., d

**r**= [Δq

_{i}Δl

_{i}Δa

_{ix}Δa

_{iy}Δa

_{iz}Δb

_{ix}Δb

_{iy}Δb

_{iz}Δθ

_{Bix}Δθ

_{Biy}Δθ

_{Biz}Δθ

_{Cix}Δθ

_{Ciy}Δθ

_{Ciz}] (i = 1, 2, 3).

**w**

_{i}to denote

**R**

_{i}

**R**

_{Bi}

**R**

_{Ci}

**l**

_{i}and multiply

**w**

_{i}on both sides of Equation (3).

**v**

_{i}to denote

**R**

_{i}

**R**

_{Bi}

**R**

_{Ci}

**e**

_{2}, multiply

**v**

_{i}on both sides of Equation (3). Here,

**e**

_{2}= [0 1 0]

^{T}.

**p**= [Δ

**H**

^{T}Δ

**α**

^{T}]

^{T}and d

**r =**[d

**r**

_{1}

^{T}d

**r**

_{2}

^{T}d

**r**

_{3}

^{T}]

^{T}. d

**r**

_{i}is the geometric error vector of limb i. d

**r**

_{i},

**A**, and

**B**can be expressed as:

**P**= [X Y Z]

^{T}. Derivation of the above equation gives:

**p**

_{TT}= [Δx Δy Δz Δφ Δθ Δψ]

^{T}represents the pose error based on the T-T angle, and we can obtain the transformation relationship between [Δφ Δθ Δψ]

^{T}and [Δα

_{x}Δα

_{y}Δα

_{z}]

^{T}as:

_{iz}and Δq

_{i}, and Δθ

_{Biz}and Δθ

_{Ciz}are the same; they are redundant with each other and have the same effect on the output error, and the coefficient of Δθ

_{Ciy}is 0, which does not have any effect on the terminal error. In order to simplify the analyzing process, we keep only one of them for the redundant parameters and eliminate Δθ

_{Ciy}.

**T**can represent the transfer relationship between the geometric error parameters and the TCP position error. The use of the

**T**for error sensitivity modeling avoids the problem of difficult comparisons caused by the non-uniformity of the output position and orientation errors.

## 3. Definition of Sensitivity Indices

#### 3.1. Introduction of the Sensitivity Indices

**P**|) for sensitivity indices modeling; however, when the tool is in different orientation, the same TP or TO error values bring different impacts on the machining accuracy. As shown in Figure 4, Figure 4a,b show the machine in two different poses, and the TP and TO errors are the same in both cases; however, due to the different machine poses, the impact of the same TP error on the machining is different.

#### 3.2. Definition of the Sensitivity Indices

_{e}. As shown in Figure 5,

**A**

_{1}

**B**

_{1}represents the theoretical tool axis, and the actual tool axis is

**A**

_{2}

**B**

_{2}after being affected by geometric errors. That is, the theoretical cutting area is a cylinder with

**A**

_{1}

**B**

_{1}as the central axis and R as the radius, and the actual cutting area is a cylinder with

**A**

_{2}

**B**

_{2}as the central axis and R as the radius after being affected by geometric errors. The portion of the theoretical and actual cutting area that fails to intersect is the machining error (the amount of overcut or undercut). The theoretical cutting area is denoted as Ω

_{1}, and the actual cutting area is denoted as Ω

_{2}. The volume of Ω

_{1}is V, and the volume of the part that fails to intersect is V

_{1}. For each geometric error parameter, the resulting machining error V

_{1}can be found at each pose, and w

_{i}= V

_{1}/V can then be used to define the LSI and GSI.

_{1}is first constructed, as shown in Figure 5. Since the cross-product result of two vectors is always perpendicular to the original two vectors as long as it is not

**0**, the three normal vectors of the cuboid are solved using the characteristics of the cross-product operation.

^{T}and [0 1 0]

^{T}are taken, and other vectors can be taken.

**a**,

**b**, and

**n**are the three unitized normal vectors of the cuboid, and random points can then be selected:

**Q**is the position vector of a randomly taken point

**Q**within the outer cuboid of Ω

_{1}. rand(−R, R) denotes the random number selected in [−R R]. Assuming that a total of N points is randomly taken in this cuboid and assuming that there are m out of N points that fall within Ω

_{1}and n out of m points that fall within Ω

_{2}, the parameter w

_{i}can be expressed as follows according to the Monte Carlo method:

Algorithm 1: LSI solving algorithm based on Monte Carlo method |

INPUT: dr, L, L_{e}, N, q |

Begin |

Step 1. Solve the coordinates of the points A_{1}, B_{1}, A_{2,} and B_{2} corresponding to dr and q based on the error kinematic model. |

Step 2. Solve for n, a, and b according to Equations (12)–(15). |

Step 3. Randomly generate N points inside the least outer cuboid according to Equation (16). |

Step 4. Determine the number m of points within Ω_{1} and the number n of points in the region where Ω_{1} intersects with Ω_{2} among the N points according to Equations (18) and (19). |

Step 5. Then, the LSI corresponding to the error parameter dr and input vector q can be expressed as w_{i} = 1 − n/m. |

_{i}over the entire workspace, indicating the average effect of each geometric error on machining accuracy for all possible poses. The GSI can be expressed as:

## 4. Error Sensitivity Analysis of the 3-DOF Parallel Spindle Head

_{i}∈ [300 550] (i = 1, 2, 3). The tool radius is set to R = 10 mm, the tool length is set to L = 150 mm, and the effective cutting length is set to L

_{e}= 35 mm. Since the three limbs of this 3-DOF parallel spindle head are perfectly symmetrical, the sensitivity indices of the same type of error on its different limbs are the same, e.g., GSI(Δa

_{1x}) = GSI(Δa

_{2x}) = GSI(Δa

_{3x}). Therefore, it is sufficient to calculate the sensitivity indices corresponding to the error parameters of only one of the limbs for the actual calculation. Here for limb 1, the GSI corresponding to each geometric error parameter is calculated, and the results are shown in Figure 6.

_{Bix}and Δθ

_{Cix}(i = 1, 2, 3).

_{e}= 35 mm. The maximum, mean, and root mean square (RMS) values of the actual cutting volume error across the whole workspace for the four different cases of error parameters in Table 1 are shown in Figure 8.

_{e}. Cases a–f in Table 2 correspond to the six cases of tool radius, as well as the effective cutting length, and the accuracy comparisons are shown in Figure 9. Since Figure 8 illustrates that the relative magnitudes of the maximum, mean, and RMS values are consistent across the four cases, only the mean values are compared. Simulation results show that under different machining parameters, only changing the critical geometric error can improve the machine accuracy well, which verifies the effectiveness of the sensitivity analysis method proposed in this paper.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Slamani, M.; Mayer, R.; Balazinski, M.; Zargarbashi, S.H.; Engin, S.; Lartigue, C. Dynamic and geometric error assessment of an XYC axis subset on five-axis high-speed machine tools using programmed end point constraint measurements. Int. J. Adv. Manuf. Technol.
**2010**, 50, 1063–1073. [Google Scholar] [CrossRef] - Lee, R.S.; She, C.H. Developing a postprocessor for three types of five-axis machine tools. Int. J. Adv. Manuf. Technol.
**1997**, 13, 658–665. [Google Scholar] [CrossRef] - El-Khasawneh, B.S.; Ferreira, P.M. Computation of stiffness and stiffness bounds for parallel link manipulators. Int. J. Mach. Tools Manuf.
**1999**, 39, 321–342. [Google Scholar] [CrossRef] - Briot, S.; Bonev, I.A. Accuracy analysis of 3-DOF planar parallel robots. Mech. Mach. Theory
**2008**, 43, 445–458. [Google Scholar] [CrossRef] - Laliberté, T.; Gosselin, C.M.; Jean, M. Static balancing of 3-DOF planar parallel mechanisms. IEEE-ASME Trans. Mechatron.
**1999**, 4, 363–377. [Google Scholar] [CrossRef] - Burghardt, A.; Szybicki, D.; Kurc, K.; Muszyñska, M.; Mucha, J. Experimental study of Inconel 718 surface treatment by edge robotic deburring with force control. Strength Mater.
**2017**, 49, 594–604. [Google Scholar] [CrossRef] - Khanghah, S.P.; Boozarpoor, M.; Lotfi, M.; Teimouri, R. Optimization of micro-milling parameters regarding burr size minimization via RSM and simulated annealing algorithm. Trans. Indian Inst. Met.
**2015**, 68, 897–910. [Google Scholar] [CrossRef] - Burghardt, A.; Szybicki, D.; Kurc, K.; Muszyńska, M. Optimization of process parameters of edge robotic deburring with force control. Int. J. Appl. Mech. Eng.
**2016**, 21, 987–995. [Google Scholar] [CrossRef] - Ramesh, R.; Mannan, M.A.; Poo, A.N. Error compensation in machine tools—A review: Part I: Geometric, cutting-force induced and fixture-dependent errors. Int. J. Mach. Tools Manuf.
**2000**, 40, 1235–1256. [Google Scholar] [CrossRef] - Ramesh, R.; Mannan, M.A.; Poo, A.N. Error compensation in machine tools—A review: Part II: Thermal errors. Int. J. Mach. Tools Manuf.
**2000**, 40, 1257–1284. [Google Scholar] [CrossRef] - Khan, A.W.; Chen, W. A methodology for systematic geometric error compensation in five-axis machine tools. Int. J. Adv. Manuf. Technol.
**2011**, 53, 615–628. [Google Scholar] [CrossRef] - Chanal, H.; Duc, E.; Ray, P.; Hascoet, J.Y. A new approach for the geometrical calibration of parallel kinematics machines tools based on the machining of a dedicated part. Int. J. Mach. Tools Manuf.
**2007**, 47, 1151–1163. [Google Scholar] [CrossRef] - Majarena, A.C.; Santolaria, J.; Samper, D.; Aguilar, J.J. An overview of kinematic and calibration models using internal/external sensors or constraints to improve the behavior of spatial parallel mechanisms. Sensors
**2010**, 10, 10256–10297. [Google Scholar] [CrossRef] - Li, Q.; Wang, W.; Jiang, Y.; Li, H.; Zhang, J.; Jiang, Z. A sensitivity method to analyze the volumetric error of five-axis machine tool. Int. J. Adv. Manuf. Technol.
**2018**, 98, 1791–1805. [Google Scholar] [CrossRef] - Xu, Y.; Zhang, L.; Wang, S.; Du, H.; Chai, B.; Hu, S.J. Active precision design for complex machine tools: Methodology and case study. Int. J. Adv. Manuf. Technol.
**2015**, 80, 581–590. [Google Scholar] [CrossRef] - Li, M.; Wang, L.; Yu, G.; Li, W. A new calibration method for hybrid machine tools using virtual tool center point position constraint. Measurement
**2021**, 181, 109582. [Google Scholar] [CrossRef] - Li, M.; Wang, L.; Yu, G.; Li, W.; Kong, X. A multiple test arbors-based calibration method for a hybrid machine tool. Robot. Comput.-Integr. Manuf.
**2023**, 80, 102480. [Google Scholar] [CrossRef] - Tian, W.; Gao, W.; Zhang, D.; Huang, T. A general approach for error modeling of machine tools. Int. J. Mach. Tools Manuf.
**2014**, 79, 17–23. [Google Scholar] [CrossRef] - Lee, S.; Zeng, Q.; Ehmann, K.F. Error modeling for sensitivity analysis and calibration of the tri-pyramid parallel robot. Int. J. Adv. Manuf. Technol.
**2017**, 93, 1319–1332. [Google Scholar] [CrossRef] - Cui, H.; Zhu, Z.; Gan, Z.; Brogardh, T. Kinematic analysis and error modeling of TAU parallel robot. Robot. Comput.-Integr. Manuf.
**2005**, 21, 497–505. [Google Scholar] [CrossRef] - Tian, W.; Mou, M.; Yang, J.; Yin, F. Kinematic calibration of a 5-DOF hybrid kinematic machine tool by considering the ill-posed identification problem using regularisation method. Robot. Comput.-Integr. Manuf.
**2019**, 60, 49–62. [Google Scholar] [CrossRef] - Sun, T.; Zhai, Y.; Song, Y.; Zhang, J. Kinematic calibration of a 3-DoF rotational parallel manipulator using laser tracker. Robot. Comput.-Integr. Manuf.
**2016**, 41, 78–91. [Google Scholar] [CrossRef] - Vischer, P.; Clavel, R. Kinematic calibration of the parallel Delta robot. Robotica
**1998**, 16, 207–218. [Google Scholar] [CrossRef] - Huang, P.; Wang, J.; Wang, L.; Yao, R. Kinematical calibration of a hybrid machine tool with regularization method. Int. J. Mach. Tools Manuf.
**2011**, 51, 210–220. [Google Scholar] [CrossRef] - Niu, P.; Cheng, Q.; Liu, Z.; Chu, H. A machining accuracy improvement approach for a horizontal machining center based on analysis of geometric error characteristics. Int. J. Adv. Manuf. Technol.
**2021**, 112, 2873–2887. [Google Scholar] [CrossRef] - Sun, Y.; Lueth, T.C. Safe manipulation in robotic surgery using compliant constant-force mechanism. IEEE Trans. Med. Robot. Bionics
**2023**, 5, 486–495. [Google Scholar] [CrossRef] - Zhang, S.; He, C.; Liu, X.; Xu, J.; Sun, Y. Kinematic chain optimization design based on deformation sensitivity analysis of a five-axis machine tool. Int. J. Precis. Eng. Manuf.
**2020**, 21, 2375–2389. [Google Scholar] [CrossRef] - Guo, S.; Mei, X.; Jiang, G. Geometric accuracy enhancement of five-axis machine tool based on error analysis. Int. J. Adv. Manuf. Technol.
**2019**, 105, 137–153. [Google Scholar] [CrossRef] - Patel, A.J.; Ehmann, K.F. Volumetric error analysis of a Stewart platform-based machine tool. CIRP Ann.
**1997**, 46, 287–290. [Google Scholar] [CrossRef] - Fan, K.C.; Wang, H.; Zhao, J.W.; Chang, T.H. Sensitivity analysis of the 3-PRS parallel kinematic spindle platform of a serial-parallel machine tool. Int. J. Mach. Tools Manuf.
**2003**, 43, 1561–1569. [Google Scholar] [CrossRef] - Jiang, S.; Chi, C.; Fang, H.; Tang, T.; Zhang, J. A minimal-error-model based two-step kinematic calibration methodology for redundantly actuated parallel manipulators: An application to a 3-DOF spindle head. Mech. Mach. Theory
**2022**, 167, 104532. [Google Scholar] [CrossRef] - Du, X.; Wang, B.; Zheng, J. Geometric Error Analysis of a 2UPR-RPU Over-Constrained Parallel Manipulator. Machines
**2022**, 10, 990. [Google Scholar] [CrossRef] - Chen, G.; Liang, Y.; Sun, Y.; Chen, W.; Wang, B. Volumetric error modeling and sensitivity analysis for designing a five-axis ultra-precision machine tool. Int. J. Adv. Manuf. Technol.
**2013**, 68, 2525–2534. [Google Scholar] [CrossRef] - Cheng, Q.; Zhao, H.; Zhang, G.; Gu, P.; Cai, L. An analytical approach for crucial geometric errors identification of multi-axis machine tool based on global sensitivity analysis. Int. J. Adv. Manuf. Technol.
**2014**, 75, 107–121. [Google Scholar] [CrossRef] - Cheng, Q.; Sun, B.; Liu, Z.; Li, J.; Dong, X.; Gu, P. Key geometric error extraction of machine tool based on extended Fourier amplitude sensitivity test method. Int. J. Adv. Manuf. Technol.
**2017**, 90, 3369–3385. [Google Scholar] [CrossRef] - Li, T.; Li, F.; Jiang, Y.; Wang, H.; Zhang, J. Error modeling and sensitivity analysis of a 3-P (Pa) S parallel type spindle head with parallelogram structure. Int. J. Adv. Robot. Syst.
**2017**, 14, 1729881417715012. [Google Scholar] [CrossRef] - Bonev, I.A. Geometric Analysis of Parallel Mechanisms; Université Laval: Québec, QC, Canada, 2002. [Google Scholar]
- Xie, F.; Liu, X.J.; Wang, J. A 3-DOF parallel manufacturing module and its kinematic optimization. Robot. Comput.-Integr. Manuf.
**2012**, 28, 334–343. [Google Scholar] [CrossRef] - Guo, S.; Jiang, G. Investigation of sensitivity analysis and compensation parameter optimization of geometric error for five-axis machine tool. Int. J. Adv. Manuf. Technol.
**2017**, 93, 3229–3243. [Google Scholar] [CrossRef] - Li, J.; Xie, F.; Liu, X.J. Geometric error modeling and sensitivity analysis of a five-axis machine tool. Int. J. Adv. Manuf. Technol.
**2016**, 82, 2037–2051. [Google Scholar] [CrossRef] - Liang, Y.; Chen, G.; Chen, W.; Sun, Y.; Chen, J. Analysis of volumetric error of machine tool based on Monte Carlo method. J. Comput. Theor. Nanosci.
**2013**, 10, 1290–1295. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Configuration of the hybrid machine tool; (

**b**) structure of the 3-DOF parallel spindle head.

**Figure 3.**Comparison of machining errors of the milled slots in three different output error cases. (

**a**) TO error is d

**o;**(

**b**) TO error is 0; (

**c**) TO error is −d

**o**.

**Figure 4.**Comparison of the effect of the output errors on machining error at different poses. (

**a**) zero orientation; (

**b**) non-zero orientation.

**Figure 8.**Maximum, mean, and root mean square (RMS) values of the actual cutting volume error across the whole workspace for the four different cases.

Case | Critical Geometric Errors | Other Geometric Errors |
---|---|---|

1 | Keep the initial value unchanged | Keep the initial value unchanged |

2 | Reduce by half | Keep the initial value unchanged |

3 | Keep the initial value unchanged | Reduce by half |

4 | Reduce by half | Reduce by half |

Case | R/mm | L_{e}/mm |
---|---|---|

a | 6 | 25 |

b | 6 | 45 |

c | 8 | 25 |

d | 8 | 45 |

e | 12 | 25 |

f | 12 | 45 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Li, M.; Yu, G.
A Novel Error Sensitivity Analysis Method for a Parallel Spindle Head. *Robotics* **2023**, *12*, 129.
https://doi.org/10.3390/robotics12050129

**AMA Style**

Wang L, Li M, Yu G.
A Novel Error Sensitivity Analysis Method for a Parallel Spindle Head. *Robotics*. 2023; 12(5):129.
https://doi.org/10.3390/robotics12050129

**Chicago/Turabian Style**

Wang, Liping, Mengyu Li, and Guang Yu.
2023. "A Novel Error Sensitivity Analysis Method for a Parallel Spindle Head" *Robotics* 12, no. 5: 129.
https://doi.org/10.3390/robotics12050129