Abstract
Machining accuracy reliability is considered to be one of the most important indexes in the process of performance evaluation and optimization design of the machine tools. Geometric errors, thermal errors and tool wear are the main factors to affect the machining accuracy and so affect the machining accuracy reliability of machine tools. This paper proposed a geometric error budget method that simultaneously considers geometric errors, thermal errors and tool wear to improve the machining accuracy reliability of machine tools. Homogeneous transformation matrices, neural fuzzy control theory and a tool wear predictive approach were employed to develop a comprehensive error model, which shows the influence of the geometric, thermal errors and tool wear to the machining accuracy of a machine tool. Based on Rackwite–Fiessler and Advanced First Order and Second Moment, a reliability model and a sensitivity model were put forward, which can deal with the errors of a machine tool drawn from any distribution. Then, a geometric error budget method of multiaxis NC machine tool was developed and formed into a mathematical model. In such method, the minimum cost of machine tool was the optimization objective, the reliability of the machining accuracy was the constraint, and the sensitivity was to identify the geometric errors to be optimized. An example conducted on a fiveaxis NC machine tool was used to explain and validate the proposed method.
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Abbreviations
 \(\Delta x_{x}\) :

Positioning error
 \(\Delta y_{x}\) :

Y direction of straightness error
 \(\Delta z_{x}\) :

Z direction of straightness error
 \(\Delta \alpha _{x}\) :

Roll error
 \(\Delta \beta _{x}\) :

Pitch error
 \(\Delta \gamma _{x}\) :

Yaw error
 \(\Delta x_{y}\) :

X direction of straightness error
 \(\Delta y_{y}\) :

Positioning error
 \(\Delta z_{y}\) :

Z direction of straightness error
 \(\Delta \alpha _{y}\) :

Pitch error
 \(\Delta \beta _{y}\) :

Roll error
 \(\Delta \gamma _{y}\) :

Yaw error
 \(\Delta x_{z}\) :

X direction of straightness error
 \(\Delta y_{z}\) :

Y direction of straightness error
 \(\Delta z_{z}\) :

Positioning error
 \(\Delta \alpha _{z}\) :

Pitch error
 \(\Delta \beta _{z}\) :

Yaw error
 \(\Delta \gamma _{z}\) :

Roll error
 \(\Delta x_{B}\) :

X direction runout error
 \(\Delta y_{B}\) :

Y direction runout error
 \(\Delta z_{B}\) :

Z direction runout error
 \(\Delta \alpha _{B}\) :

Around the Xaxis turning error
 \(\Delta \beta _{B}\) :

Turning error
 \(\Delta \gamma _{B}\) :

Around the Zaxis turning error
 \(\Delta x_{A}\) :

X direction runout error
 \(\Delta y_{A}\) :

Y direction runout error
 \(\Delta z_{A}\) :

Z direction runout error
 \(\Delta \alpha _{A}\) :

Turning error
 \(\Delta \beta _{A}\) :

Around the Yaxis turning error
 \(\Delta \gamma _{A}\) :

Around the Zaxis turning error
 \(\Delta x_\varphi \) :

X direction runout error
 \(\Delta y_\varphi \) :

Y direction runout error
 \(\Delta z_\varphi \) :

Z direction runout error
 \(\Delta \alpha _\varphi \) :

Around the Xaxis turning error
 \(\Delta \beta _\varphi \) :

Around the Yaxis turning error
 \(\Delta \gamma _\varphi \) :

Turning error
 \(\Delta \gamma _{{xy}}\) :

X, Yaxis perpendicularity error
 \(\Delta \beta _{xz}\) :

X, Zaxis perpendicularity error
 \(\Delta \alpha _{yz}\) :

Y, Zaxis perpendicularity error
 \(\Delta \gamma _{xB}\) :

Baxis parallelism error in YZ plane
 \(\Delta \alpha _{zB}\) :

Baxis parallelism error in XY plane
 \(\Delta \gamma _{yA}\) :

Aaxis parallelism error in XZ plane
 \(\Delta \beta _{zA}\) :

Aaxis parallelism error in XY plane
 \(\Delta y_{AB}\) :

An offset errors between A, Baxis along Yaxis
 \(\Delta z_{AB}\) :

An offset errors between A, Baxis along Zaxis
 \(\Delta x_t\) :

X direction of straightness error
 \(\Delta y_t\) :

Y direction of straightness error
 \(\Delta z_t\) :

Positioning error
 \(\Delta \alpha _t\) :

Pitch error
 \(\Delta \beta _t\) :

Yaw error
 \(\Delta \gamma _t\) :

Roll error
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Acknowledgments
The authors are most grateful to the National Natural Science Foundation of China (Nos. 51575010 and 51575009), Beijing Nova Program (Z1511000003150138), the Leading Talent Project of Guangdong Province, Open Project of State Key Lab of Digital Manufacturing Equipment & Technology (Huazhong University of Science and Technology), Shantou Light Industry Equipment Research Institute of science and technology Correspondent Station (2013B090900008), the National Science and Technology Major Project (2013ZX04013011), the JingHua Talents Project of Beijing University of Technology and the National Science and Technology Major Project (2013ZX04013) for supporting this research presented in this paper.
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Appendix
Appendix
Parameters  Definitions 

\(T_j\)  The temperatures set of the jth temperature point, which are the scalar variables that can be directly obtained 
\(T_{jk}\)  The kth data of the jth temperature point, which is a scalar variable that can be directly obtained 
\(W_k\)  The kth thermal error, which is an intermediate scalar variable 
\(\overline{{\varvec{W}}}\)  The mean value of the thermal error, which is an intermediate scalar variable 
\(\overline{T_j }\)  The mean value of the jth temperature point, which is an intermediate scalar variable 
\(\chi _j\)  Principal factor, which is an intermediate scalar variable 
\(v_i \)  A fuzzy variable, which is a scalar variable that can be directly obtained 
\(T(v_i)\)  Fuzzy division set, which is an intermediate scalar variable 
\(m_i \)  The number of division, which is an intermediate scalar variable 
\(Q_i^j\)  The jth fuzzy subset of \(v_i\) in the whole fuzzy set \(H_i\), which is an intermediate scalar variable 
\(\mu _{Q_i^j } (v_i )\)  Membership function of \(Q_i^j\), which is an intermediate scalar variable 
\(\varepsilon _i\)  The weight of the ith rule, which is an intermediate scalar variable 
\({{\varvec{V}}}\)  The temperature input, which is a vector variable that can be directly obtained 
\({{\varvec{W}}}\)  The thermal error output, which is an intermediate vector variable 
\(N_{1}=n\)  The total number of the first layer, which is an intermediate scalar variable 
\(N_{2}\)  The total number of the second layer, which is an intermediate scalar variable 
The fitness of each fuzzy rule, which is an intermediate scalar variable  
The whole fitness, which is an intermediate scalar variable  
\(W_i \)  The output of the ith subnetworks, which is an intermediate scalar variable 
\(\omega _{jl}^i \)  The connection weight of the latter network, which is an intermediate scalar variable 
a,c  The membership function coefficients of the premise network, which is an intermediate scalar variable 
ES  The square error of actual output and ideal output, which is an intermediate scalar variable 
\(\vartheta \)  The learning rate, which is an intermediate scalar variable 
\(\Delta t\)  The time interval between two measurements of the tool wear, which is a scalar variable that can be directly obtained 
\(\Delta TW(\Delta t)\)  The change of the tool flank wear after the time interval \(\Delta t\), which is an intermediate scalar variable 
\(\mathrm{B}\)  A constant related to the tool and workpiece in a certain cutting condition, which is an intermediate scalar constant 
n  The spindle speed, which is a scalar variable that can be directly obtained 
v  The cutting feed, which is a scalar variable that can be directly obtained 
\(a_p\)  The cutting depth, which is a scalar variable that can be directly obtained 
D  The tool diameter and the time interval of cutting process, which is a scalar constant that can be directly obtained 
\(\hbar (\eta )\)  The height of the cutting point before the tool wear, which is a scalar variable that can be directly obtained 
\(\xi _i \)  The indexes of these variables, which is an intermediate scalar variable 
\(\theta \)  The rake angles of the tool, which is a scalar constant that can be directly obtained 
\(\varsigma \)  The relief angles of the tool, which is a scalar constant that can be directly obtained 
\(\Delta R(\Delta t)\)  The change of the radial radius of the tool cutting edge, which is an intermediate scalar variable 
\(\eta \)  The angle between the axial direction and the line which connects the cutting point and center point O of the tool before the tool wear, which is a scalar variable that can be directly obtained 
\(\eta ^{\prime }\)  The angle between the axial direction and the line that connects the cutting point and center point O of the tool after the tool wear, which is an intermediate scalar variable 
\(R_0 \)  The radius of the tool before tool wear, which is a scalar constant that can be directly obtained 
\(r_0 \)  The radial radius of the tool before tool wear, which is a scalar constant that can be directly obtained 
\(r(\hbar )\)  The radial radius of the tool after the tool wear, which is a scalar variable that can be directly obtained 
P  A point in the cutting edge before the tool wear, which is a vector variable that can be directly obtained 
\(P^{\prime }\)  The location of the point P after the tool wear, which is an intermediate vector variable 
\(\hbar (\eta )\)  The height of the selected cutting point, which is a scalar variable that can be directly obtained 
\(\wp \)  The error induced by the tool wear, which is an intermediate scalar variable 
\(\wp _x\quad \wp _y\quad \wp _z \)  The errors induced by tool wear in X, Y, Z directions, which are the intermediate scalar variables 
G  A performance function, which is an intermediate scalar variable 
\(x_i\)  The uncorrelated nonnormal variable, which is a scalar variable that can be directly obtained 
\(F_i(x_i)\)  The distribution function, which is a scalar variable that can be directly obtained 
\(f_i (x_i )\)  The density function, which is a scalar variable that can be directly obtained 
\(x_i^\prime \)  The equivalent normal random variable, which is an intermediate scalar variable 
\(\mu _{{x_i}}^\prime \)  The mean value of \(x_i^\prime \), which is an intermediate scalar variable 
\(\sigma _{{x_i}}^\prime \)  The standard deviation of \(x_i^\prime \), which is an intermediate scalar variable 
\(\Phi (\cdot )\)  The distribution function of the standard normal distribution, which is a scalar variable that can be directly obtained 
\(\phi (\cdot )\)  The density function of the standard normal distribution, which is a scalar variable that can be directly obtained 
\(\alpha _i \)  The sensitivity coefficient, which is an intermediate scalar variable 
\(\beta \)  The reliability index, which is an intermediate scalar variable 
\(P_f \)  The possibility of failure in single failure mode, which is an intermediate scalar variable 
\(\frac{\partial P_f }{\partial \sigma _{{x_i}}^\prime }\)  The reliability sensitivity, which is an intermediate scalar variable 
\(P_h\)  The possibility of failure under multiple failure modes, which is an intermediate scalar variable 
\(h_i \)  The ith performance function, which is an intermediate scalar variable 
\(\gamma _{12}\)  The correlation coefficient of these two failure modes, which is an intermediate scalar variable 
\(g_i \)  Geometric error parameter, which can be as a scalar variable that can be directly obtained, an intermediate scalar variable and a target scalar variable 
\(MC( {g_i } )\)  The cost of manufacturing the geometric error parameter \(g_i \), which is an intermediate scalar variable 
MC  The total manufacturing cost, which can be as an intermediate scalar variable and a target scalar variable 
s  The number of the selected coordinate points, which is a scalar constant that can be directly obtained 
\(\frac{1}{s}\sum \limits _{t=1}^s {P_h (t)} \)  The mean value of the possibility of failure of the 16 coordinate points, which can be as an intermediate scalar variable and a target scalar variable 
\(\max P_h (t)\)  The maximum value of the possibility of failure of the 16 coordinate points, which can be as an intermediate scalar variable and a target scalar variable 
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Zhang, Z., Cai, L., Cheng, Q. et al. A geometric error budget method to improve machining accuracy reliability of multiaxis machine tools. J Intell Manuf 30, 495–519 (2019) doi:10.1007/s1084501612608
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Keywords
 Thermal errors
 Tool wear
 A comprehensive error model
 Machining accuracy reliability
 Geometric error budget