1 Introduction

Precision/ultra-precision products play an increasingly important role in many fields, such as daily life, national defense, military and aerospace, accordingly, the requirement for product accuracy in these fields is also becoming more stringent [1, 2]. For a machine tool/CMM, its accuracy is the crucial indicator which determines directly the accuracy and quality of machined products [3,4,5]. Therefore, it has been a major concern in the industry of how to improve the accuracy of a machine tool/CMM. As the essential element of a machine tool, the linear slideway suffers from various errors, such as kinematic errors and errors induced by force. These errors change the geometric structure of the slideway, resulting in six geometric errors [6, 7]. These geometric errors include one positioning error, two straightness errors, two tilt errors and one roll error. The straightness and tilt errors not only reflect dynamically the operating status of a machine tool, but also are the key parameters of the error compensation model [8, 9]. Hence, it is of engineering and scientific significance for on-machine measuring the straightness and tilt errors of a linear slideway, especially for the case that the linear slideway works under the complicated condition.

The multi-sensor method is utilized widely to on-machine separate the surface profile from the sensor outputs [10]. The outputs are constructed by four parameters, namely the surface profile, the straightness and tilt errors of the linear slideway, as well as the zero-adjustment values of the sensors. After the surface profile is obtained, the straightness and tilt errors can be further identified. The two-sensor methods, such as the sequential two-sensor method [10, 11], the generalized two-sensor method [10, 11] and the combined two-sensor method [12], are used to estimate the surface profile without the tilt error and the zero-adjustment values. For the two-sensor methods, the harmonic suppression in the frequency domain results in that the reconstructed surface profile loses the harmonic components with the spatial period of an integer multiple of the sampling interval [11]. The sequential three-sensor method [10, 11], the generalized three-sensor method [11, 13] and the combined three-sensor method [14] are able to identify the surface profile, and the straightness and tilt errors. However, the zero-adjustment values will lead to large reduction in the accuracy of the reconstructed surface profile, even though they are tiny [11]. If the tilt error is measured in advance or ignored, the surface profile and the straightness error could be identified by using the three-sensor methods proposed by Yin, et al. [15, 16] and Fujimoto, et al. [17] respectively, considering the zero-adjustment values. The four-sensor method presented by Weingartner et al. [18] can reconstruct the surface profile in the presence of the straightness and tilt errors. However, it is affected significantly by the zero-adjustment values, and may not obtain the unique solution in some cases, due to the rank deficiency of the design matrix. Assuming that the tilt error is known in advance, Weingartner et al. [19] proposed another four-sensor method to evaluate the straightness and tilt errors as well as the zero-adjustment values. The solutions of the two four-sensor methods are very sensitive to the measurement noise in the sensor outputs, as the number of sampling points is large. The five-sensor method (IF5S) developed by Fung, et al. [20, 21] employs Fourier series to determine the surface profile when there exist the straightness and tilt errors as well as the zero-adjustment values. As an extension, an eight-sensor method called F8S is reported [22]. The sensor configurations of both IF5S and F8S need to be chosen suitably, and the length of the test section is determined by the size of the slider. Based on the reversal method [23] and the generalized three-sensor method, Gao et al. [24, 25] developed the reversal six-sensor method. By scanning the two opposed surface profiles of a cylinder simultaneously, the method can be used for estimating the profiles and the straightness and tilt errors, with the zero-adjustment values taken into account. Because the measurement system either contains an additional cylinder or has the reversal function, it is difficult to integrate the measurement system with the linear slideway. In addition, the previous researches have paid little attention to the influences of the measurement noise in the sensor outputs, the sensor gain error and the positioning error of the linear slideway on the multi-sensor methods.

In this paper, a new four-sensor method with an improved measurement system is proposed to separate simultaneously the straightness and tilt errors of a linear slideway, with the zero-adjustment values taken into account. The measurement system allows achieving the adjustable sensor spacing and the high lateral resolution, and avoids the sensor gain error. The proposed method has some advantages, such as suitable for the test section of any length, needless to pre-measure the zero-adjustment values and the surface profile accurately, and favorable propagations of the residuals induced by the noise and the positioning errors. This paper is organized as follows. The new method and the improved system are explained in detail in Section 2. To analyze the influences of the measurement noise in the senor outputs and the positioning errors of the sensors and the slideway on the method, a series of computer simulations are conducted in Section 3. In Section 4, an experiment is implemented to verify the feasibility of the method.

2 New Error Separation Method

The improved measurement system, as shown in Figure 1, contains a displacement sensor and a sensor stage with accurate movement along X-axis. The system is attached to a linear slideway. The relative displacement between the reference surface and the slideway is detected by the sensor. Different configurations are formed by adjusting the position of the sensor relative to the stage. The combination of these configurations constructs an improved multi-sensor measurement system that can realize the function of the traditional multi-sensor measurement system. The improved system has some advantages, such as adjustable sensor spacing and no gain error. A function f(x) is introduced to describe the reference surface profile, and the straightness and tilt errors of the slideway are represented by the functions S(x) and γ(x) respectively. The sensor associated with the ith configuration is named the sensor i (i ≥ 1). For the sensor i, the zero-adjustment value relative to the sensor 1 is defined as ei and the spacing relative to the sensor (i − 1) set to Di = si× Δx. si is an integer and Δx (in Figure 1) denotes the lateral resolution. Obviously, both e1 and s1 equal 0. ei describes the effect caused by the geometric errors of the sensor stage on the sensor output. If N denotes the total sampling number, the position of any sampling point can be described as xn = n × Δx, n = 0, 1,…, (N − 1). Then, the length of the measured reference surface is L0 = (N − 1) × Δx and the travel of the slideway L = (N − s2 − s3 − s4 − 1) × Δx. Because the sensor stage isn’t calibrated accurately beforehand, ei (i ≠ 1) is unknown. Based on the above information, the output mi(xn) of the sensor i (i = 1, 2, 3, 4) at the sampling point xn can be expressed as Eq. (1), n = 0, 1, …, N − s2 − s3 − s4 − 1. If ei or γ(x) is either ignored or known, the improved system is able to identify f(x) from mi(xn) according to the four-sensor methods in Refs. [18, 19]. However, in general, ei and γ(x) are non-ignorable and unknown, so the methods are unavailable. For this problem, a new four-sensor method is developed as follows:

Figure 1
figure 1

Schematic diagram of the improved measurement system, of which the sensor output contains S(x), γ(x), f(x) and ei

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$$\begin{array}{*{20}c} {\left\{ {\begin{array}{l} {m_{1} \left( {x_{n} } \right) = f\left( {x_{n} } \right) + S\left( {x_{n} } \right) + e_{1} ,} \\ {m_{2} \left( {x_{n} } \right) = f\left( {x_{n} + D_{2} } \right) + S\left( {x_{n} } \right) + D_{2} \cdot \gamma \left( {x_{n} } \right) + e_{2} ,} \\ \begin{aligned} m_{3} \left( {x_{n} } \right) & = f\left( {x_{n} + D_{2} + D_{3} } \right) + S\left( {x_{n} } \right) \hfill \\ & \quad + \left( {D_{2} + D_{3} } \right) \cdot \gamma \left( {x_{n} } \right) + e_{3} , \hfill \\ \end{aligned} \\ \begin{aligned} m_{4} \left( {x_{n} } \right) & = f\left( {x_{n} + D_{2} + D_{3} + D_{4} } \right) + S\left( {x_{n} } \right) \hfill \\ & \quad + \left( {D_{2} + D_{3} + D_{4} } \right) \cdot \gamma \left( {x_{n} } \right) + e_{4} , \hfill \\ \end{aligned} \\ \end{array} } \right.} & {\left( {e_{1} = 0} \right)} \\ \end{array}$$
(1)

To eliminate the influences of S(xn) and γ(xn), the following equation is derived from Eq. (1):

$$\varvec{C}\left( {\begin{array}{*{20}c} {m_{1} \left( {x_{n} } \right)} \\ {m_{2} \left( {x_{n} } \right)} \\ {m_{3} \left( {x_{n} } \right)} \\ {m_{4} \left( {x_{n} } \right)} \\ \end{array} } \right) = \varvec{C}\left( {\begin{array}{*{20}c} {f\left( {x_{n} } \right)} \\ {f\left( {x_{n} + D_{2} } \right)} \\ {f\left( {x_{n} + D_{2} + D_{3} } \right)} \\ {f\left( {x_{n} + D_{2} + D_{3} + D_{4} } \right)} \\ \end{array} } \right) + \varvec{C}\left( {\begin{array}{*{20}c} {e_{1} } \\ {e_{2} } \\ {e_{3} } \\ {e_{4} } \\ \end{array} } \right),$$
(2)

where

$$\varvec{C} = \left[ {\begin{array}{*{20}c} 1 & { - \frac{{D_{2} + D_{3} }}{{D_{3} }}} & {\frac{{D_{2} }}{{D_{3} }}} & 0 \\ 1 & { - \frac{{D_{2} + D_{3} + D_{4} }}{{D_{3} + D_{4} }}} & 0 & {\frac{{D_{2} }}{{D_{3} + D_{4} }}} \\ 1 & 0 & { - \frac{{D_{2} + D_{3} + D_{4} }}{{D_{4} }}} & {\frac{{D_{2} + D_{3} }}{{D_{4} }}} \\ 0 & 1 & { - \frac{{D_{3} + D_{4} }}{{D_{4} }}} & {\frac{{D_{3} }}{{D_{4} }}} \\ \end{array} } \right].$$
(3)

A system of linear equations is then built according to Eq. (2):

$$\varvec{A}_{0} \varvec{X} = \varvec{b}_{0}.$$
(4)

A0, b0 and X are given in Eqs. (5)‒(7):

$$\begin{array}{*{20}c} {\varvec{A}_{0} = \left[ {\begin{array}{*{20}c} {{\varvec{c}}_{1} } \\ \vdots \\ {{\varvec{c}}_{i} } \\ \vdots \\ {{\varvec{c}}_{p} } \\ \end{array} } \right]}, & {\left( {p = N{-}s_{2} {-}s_{3} {-}s_{4} {-}1} \right)} \\ \end{array} ,$$
(5)
$$\varvec{b}_{0} = \left[ {\begin{array}{*{20}c} {g_{1} \left( {x_{0} } \right)} & \cdots & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {g_{4} \left( {x_{0} } \right)} & \cdots \\ \end{array} } & {g_{1} \left( {x_{p} } \right)} & {\begin{array}{*{20}c} \cdots & {g_{4} \left( {x_{p} } \right)} \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]^{\text{T}} ,$$
(6)
$$\varvec{X} = \left[ {\begin{array}{*{20}c} {f\left( {x_{0} } \right)} & \cdots & {\begin{array}{*{20}c} {f\left( {x_{N - 1} } \right)} & {\Delta e_{1} } & {\begin{array}{*{20}c} \cdots & {\Delta e_{4} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]^{\text{T}} ,$$
(7)

where

$${\varvec{c}}_{i} = \left[ {\begin{array}{*{20}c} {\varvec{0}_{{ 4\times \left( {i{ - 1}} \right)}} } & {\varvec{C}} & {\varvec{0}_{{ 4\times \left( {N{ - }i - 3} \right)}} } & {{\varvec{I}}_{ 4\times 4} } \\ \end{array} } \right] ,$$
(8)
$$\begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {g_{1} \left( {x_{n} } \right)} \\ {g_{2} \left( {x_{n} } \right)} \\ {g_{3} \left( {x_{n} } \right)} \\ {g_{4} \left( {x_{n} } \right)} \\ \end{array} } \right] = \varvec{C}\left[ {\begin{array}{*{20}c} {m_{1} \left( {x_{n} } \right)} \\ {m_{2} \left( {x_{n} } \right)} \\ {m_{3} \left( {x_{n} } \right)} \\ {m_{4} \left( {x_{n} } \right)} \\ \end{array} } \right],} & \quad {n \in \left[ {1,p} \right],} \\ \end{array}$$
(9)
$$\left[ {\begin{array}{*{20}c} {\Delta e_{1} } \\ {\Delta e_{2} } \\ {\Delta e_{3} } \\ {\Delta e_{4} } \\ \end{array} } \right] = \varvec{C}\left[ {\begin{array}{*{20}c} {e_{1} } \\ {e_{2} } \\ {e_{3} } \\ {e_{4} } \\ \end{array} } \right].$$
(10)

However, the rank of A0 is deficient, indicating that there are innumerable solutions for the linear system [26]. In other words, f(xn) cannot be estimated uniquely. Therefore, it is necessary to incorporate additional information to obtain a definite solution.

To restrict the reconstruction of the reference surface profile, a straight line is defined by applying Eq. (11). The shape of f(x) relative to the straight line is then determined uniquely.

$$\left\{ {\begin{array}{*{20}l} {f\left( {x_{0} } \right) = 0,} \hfill \\ {f\left( {x_{N - 1} } \right) = 0.} \hfill \\ \end{array} } \right.$$
(11)

When s2 = s3 = 1, Eq. (12) is derived by using the outputs of the sensors 1, 2 and 3 based on the generalized three-sensor method [11, 13]. Then, Δe1 can be calculated.

$$f\left( {x_{N - 1} } \right) = \sum\limits_{i = 1}^{N - 1} {\left( {N - i} \right)} g_{1} \left( {x_{{i{ - 1}}} } \right) - \frac{{N\left( {N - 1} \right)}}{2}\Delta e_{1} .$$
(12)

From Eq. (10), the relationships described in Eq. (13) are deduced:

$$\left\{ {\begin{array}{*{20}l} {\Delta e_{3} { + }\frac{{s_{3} \left( {s_{2} + s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}\Delta e_{1} - \frac{{\left( {s_{2} + s_{3} } \right)\left( {s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}\Delta e_{2} { = 0,}} \hfill \\ {\Delta e_{4} { + }\frac{{s_{3} \left( {s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}\Delta e_{1} - \frac{{s_{3} \left( {s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}\Delta e_{2} { = 0} .} \hfill \\ \end{array} } \right.$$
(13)

The constraints given in Eqs. (11)‒(13) and the original linear system in Eq. (4) are then written compactly to construct a new system of linear equations as follows:

$$\varvec{AX} = \varvec{b} ,$$
(14)

where A and b are given in Eqs. (15), (16). The column vector b is constructed by all the sensor outputs. The coefficient matrix A that owns the full rank is formed by Di. The column vector X involves f(xn), n = 0, 1, …, (N − 1), and Δei, i = 1, 2, 3, 4. The common technique to solve the linear system is the least squares method [27]. The condition number of the matrix A determines the quality of the least squares solution. However, A appears ill-conditioned. Figure 2 illustrates the variation of the condition number of A with the number of sampling points N, when D2 = D3 = 5 mm, D4 = 25 mm and Δx = 5 mm. As N equals 30 and 200, the condition number is about 6 × 103 and 2.4 × 106, respectively. The increase in the condition number of A makes the least squares solution more sensitive to the perturbation in b [28]. In other words, \(\bar{f}\left( {x_{n} } \right)\) may fluctuate drastically with the measurement noise in mi(xn).

Figure 2
figure 2

Variation of the condition number of the coefficient matrix A with the number of sampling points N

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$$\varvec{A} = \left[ {\begin{array}{*{20}c} {{\varvec{A}}_{0} } \\ \hline {\begin{array}{*{20}c} 1 & \cdots & 0 & 0 & 0 & 0 & 0 \\ 0 & \cdots & 1 & 0 & 0 & 0 & 0 \\ 0 & \cdots & 0 & 1 & 0 & 0 & 0 \\ 0 & \cdots & 0 & {\frac{{s_{3} \left( {s_{2} + s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}} & { - \frac{{\left( {s_{2} + s_{3} } \right)\left( {s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}} & 1 & 0 \\ 0 & \cdots & 0 & {\frac{{s_{3} \left( {s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}} & { - \frac{{s_{3} \left( {s_{3} + s_{4} } \right)}}{{s_{2} s_{4} }}} & 0 & 1 \\ \end{array} } \\ \end{array} } \right] ,$$
(15)
$$\varvec{b} = \left[ {\begin{array}{*{20}c} {\varvec{b}_{0}^{\text{T}} } & 0 & {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} {\frac{2}{{N\left( {N - 1} \right)}}\sum\limits_{i = 1}^{N - 1} {\left( {N - i} \right)} g_{1} \left( {x_{i - 1} } \right)} & 0 & 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]^{\text{T}} .$$
(16)

To single out a useful and stable solution, the Tikhonov regularization method [29, 30] is utilized. The regularized solution Xλ is defined as the minimizer of the following weighted combination of the residual \(\left\| {\varvec{AX} - \varvec{b}} \right\|^{2}\) and the side constraint \(\left\| \varvec{X} \right\|^{2}\), as shown in Eq. (17):

$$\varvec{X}_{\lambda } = { \hbox{min} }\left\{ {\left\| {\varvec{AX} - \varvec{b}} \right\|^{ 2} + \lambda^{2} \left\| \varvec{X} \right\|^{ 2} } \right\} = \left( {\varvec{A}^{\text{T}} \varvec{A} + \lambda^{2} \varvec{I}} \right)^{ - 1} \varvec{A}^{\text{T}} \varvec{b} ,$$
(17)

where I is the unit matrix and λ denotes the regularization parameter. As described in Eq. (17), λ controls the weight between the side constraint and the residual as well as the sensitivity of Xλ to the perturbations in A and b. Therefore, it should be selected carefully. The L-curve criterion [29, 30] described by \(\left( {\log \left\| {\varvec{AX}_{\lambda } - \varvec{b}} \right\|,\log \left\| {\varvec{X}_{\lambda } } \right\|} \right)\) is used to calculate the optimal parameter λp. After λp is determined, the estimate of f(xn) can be solved from the system of linear equations in Eq. (14), which is marked as \(\bar{f}\left( {x_{n} } \right)\). S(xn) and γ(xn) are then separated directly based on Eq. (1), as shown in Eqs. (18), and (19).

$$S\left( {x_{n} } \right) = m_{1} \left( {x_{n} } \right) - \bar{f}\left( {x_{n} } \right),$$
(18)
$$\begin{aligned} \gamma \left( {x_{n} } \right) = \frac{1}{{D_{2} + D_{3} + D_{4} }}\left\{ {m_{4} \left( {x_{n} } \right) - m_{1} \left( {x_{n} } \right)} \right. \\ \left. {\quad - \left[ {\bar{f}\left( {x_{n} + D_{2} + D_{3} + D_{4} } \right) - \bar{f}\left( {x_{n} } \right)} \right] - e_{4} } \right\}. \\ \end{aligned}$$
(19)

Due to the term \({{e_{4} } \mathord{\left/ {\vphantom {{e_{4} } {\left( {D_{2} + D_{3} + D_{4} } \right)}}} \right. \kern-0pt} {\left( {D_{2} + D_{3} + D_{4} } \right)}}\) being constant, it has no influence on the shape of γ(x). So far, the new four-sensor method for identifying the straightness and tilt errors has been developed.

3 Simulation Verifications

A simulation platform is constructed by using MATLAB to evaluate the proposed method. To represent the measurement noise, a Gaussian distribution with the zero mean and the standard deviation σ is added in mi(xn). S(x) and γ(x) are predefined as the input values to compare with the simulation results. σ is assigned with three values, namely 0.1 μm, 0.2 μm and 0.3 μm, to analyze the influence of the noise on the simulation results. Other parameters are set as Δx = 5 mm, s4 = 5, N = 69. So, L0 = 340 mm, L = 305 mm, D2 = D3 = 5 mm, and D4 = 25 mm. During the measuring process, the sensor of the improved system needs to be adjusted to different positions. Therefore, it is necessary to investigate the effect of the positioning error PE1 of the sensor on S(x) and γ(x). PE1 is introduced by changing the values of D2, D3 and D4 to 5.012mm, 4.985mm and 25.02 mm, respectively. An analysis is also conducted on the influence caused by the positioning error PE2 (unit: μm) of the linear slideway expressed by Eq. (20). In this section, two examples are discussed.

$$\begin{aligned} PE2 & = - 10\sin \left( {\frac{{2\uppi}}{L}x + \frac{1}{12}\uppi} \right) - 3\sin \left( {\frac{{6\uppi}}{L}x + \frac{1}{2}\uppi} \right) \\ & \quad + 4.5\sin \left( {\frac{{2\uppi}}{L}x + \frac{5}{6}\uppi} \right). \\ \end{aligned}$$
(20)

Example 1

S(x) (unit: μm) and γ(x) (unit: degree) are described by Eqs. (21), (22), of which the curves are given with the solid line in Figures 3 and 4. In addition, f(x) (unit: μm) is defined as Eq. (23).

Figure 3
figure 3

Simulation results related to S(x): average curve, maximum curve, minimum curve and input curve as a σ = 0.1 μm, b σ = 0.2 μm, c σ = 0.3 μm, d least squares curve and input curve as σ = 0.1 μm

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Figure 4
figure 4

Simulation results connected with γ(x): average curve, maximum curve, minimum curve and input curve as a σ = 0.1 μm, b σ = 0.2 μm, c σ = 0.3 μm, d least squares curve and input curve as σ = 0.1 μm

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When the sensor outputs are free of the noise and the positioning errors, the simulation results for S(x) and γ(x) are almost same as the input values. As σ = 0.1 μm, 0.2 μm and 0.3 μm, the platform is executed 20 times for each value without the positioning errors. The consequences associated with S(x) are shown in Figure 3(a)‒(c), which plot the average, maximum and minimum of the simulation results at each sampling point. It can be seen that the average curve is in accordance with the input curve except a small difference, and that the maximum and minimum curves have the similar shape to the input curve. The curve in Figure 3(d) is obtained, as Eq. (14) is solved by using the least squares method under the condition of σ = 0.1 μm. There is a significant deviation, indicating that the least squares method is unsuitable to solve the linear system with the ill-conditioned coefficient matrix. The similar conclusions can be drawn for γ(x) from the simulation results in Figure 4.

$$\begin{aligned} S\left( x \right) & = 8.075\sin \left( {\frac{{4\uppi}}{L}x} \right) + 5.7\sin \left( {\frac{{6\uppi}}{L}x + \frac{5}{18}\uppi} \right) \\ & \quad - 1.9\sin \left( {\frac{{14\uppi}}{L}x + \frac{1}{2}\uppi} \right), \\ \end{aligned}$$
(21)
$$\begin{aligned} \gamma \left( x \right) & = 0.008\sin \left( {\frac{{2\uppi}}{L}x + \frac{1}{3}\uppi} \right) - 0.014\sin \left( {\frac{{4\uppi}}{L}x + \frac{1}{4}\uppi} \right) \\ & \quad + 0.005\sin \left( {\frac{{10\uppi}}{L}x + \frac{7}{12}\uppi} \right), \\ \end{aligned}$$
(22)
$$f\left( x \right) = 5\sin \left( {\frac{{10\uppi}}{{L_{0} }}x} \right).$$
(23)

To represent the discrepancies in Figure 3(a)‒(c) and Figure 4(a)‒(c), two types of parameters are introduced: \(\Delta S_{\text{max} } /\Delta \gamma_{\text{max} }\) signifying the maximum of the residuals of the simulation results relative to the input curve, and \(\Delta S_{\text{max} }^{a} /\Delta \gamma_{\text{max} }^{a}\) denoting the maximum residual between the average curve and the input curve. For the three cases, \(\Delta S_{\text{max} }\) is less than 1.1 μm, 1.6 μm and 2.3 μm, while \(\Delta S_{\text{max} }^{a}\) is less than 0.15 μm, 0.26 μm and 0.3 μm. \(\Delta \gamma_{\text{max} }\) is 1.8 × 10−3 degree, 3.2 × 10−3 degree and 3.9 × 10−3 degree, while \(\Delta \gamma_{\text{max} }^{a}\) is 2.5 × 10−4 degree, 6.5 × 10−4 degree and 7.2 × 10−4 degree. In summary, Figures 3 and 4 indicate that the proposed method can separate S(x) and γ(x) from the sensor outputs with good accuracy. Moreover, \(\Delta S_{\text{max} }\), \(\Delta S_{\text{max} }^{a}\), \(\Delta \gamma_{\text{max} }\) and \(\Delta \gamma_{\text{max} }^{a}\) are proportional to the standard deviation σ.

When only PE1 exists and σ = 0 μm, the residuals of the simulation results of S(x) and γ(x) relative to their input values are displayed as the solid lines in Figure 5. The maximum residual for S(x) is 0.1 μm, while that for γ(x) is less than 1.0 × 10−4 degree. From the dashed lines in Figure 5, the maximum residuals caused by PE2 for S(x) and γ(x) are 0.15 μm and 1.0 × 10−4 degree, respectively. Figure 5 reveals that the positioning errors have slight influences on the estimates of S(x) and γ(x) obtained by the proposed method.

Figure 5
figure 5

Residuals caused by the positioning errors PE1 and PE2 as σ = 0 μm

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Example 2

When f(x) is set to Eq. (24), the variations of the residuals between the simulation results and the input values with the harmonic order k are investigated as σ = 0.2 μm.

$$f\left( x \right) = 5\sin \left( {k\frac{{ 2\uppi}}{{L_{0} }}x} \right) + 5\sin \left( {\frac{{ 2 0\uppi}}{{L_{0} }}x} \right).$$
(24)

In the simulation, S(x) and γ(x) are represented by Eqs. (21), (22). For each k, the maximum and the average of the residuals between the simulation results and the input value for S(x) at each sampling point are collected (described in Figure 6(a)), after the platform is performed 20 times. It can be seen that \(\Delta S_{\text{max} }\) and \(\Delta S_{\text{max} }^{a}\) are less than 2.6 μm and 0.6 μm. The results related to γ(x) are given in Figure 6(b), showing that \(\Delta \gamma_{\text{max} }\) and \(\Delta \gamma_{\text{max} }^{a}\) are about 5 × 10−3 degree and 9 × 10−4 degree, respectively. The relationships of the residuals induced by PE1 and PE2 relative to k are also analyzed in the case of σ = 0 μm, as shown in Figure 7. The maximum residuals for S(x) are both less than 0.4 μm, while those for γ(x) are smaller than 3 × 10−4 degree, implying that the positioning errors have very tiny influences on S(x) and γ(x).

Figure 6
figure 6

Variations of the average and maximum of the residuals caused by the noise (σ = 0.2 μm) at the sampling points with the harmonic order k

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Figure 7
figure 7

Relationships between the residuals caused by the positioning errors PE1 and PE2 at the sampling points and the harmonic order k

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In short, the proposed method can guarantee favorable propagations of the residuals induced by the noise and the positioning errors as the harmonic order k varies.

4 Experimental Results and Discussion

An experiment is designed for further verification. As shown in Figure 8, a displacement sensor is developed, comprising the probe, the flexure-based amplifier, the flexure-based loader and the encoder. The input displacement of the probe is amplified by the amplifier and then detected by the encoder. The sensor has a measurement range of 0.2 mm with the resolution of 0.035 μm. The input stiffness of the sensor is about 0.3 N/μm, so the probe wear isn’t considered. The displacement sensor and a micro-platform form the measurement system which is fixed on the slider of the linear slideway. The sensor position can be adjusted manually by the micrometer. A pre-displacement is applied to the probe by the loader, to press firmly the probe against the side of the linear slideway during the measuring process.

Figure 8
figure 8

Schematic diagram of the experiment setup with a developed displacement sensor

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The coordinate system OXY is established, and its X-axis satisfies the constraint described in Eq. (11). The angles of the probe relative to the guide of the micro-platform and the reference surface are both limited in 90 ± 0.1 degree. The sensor output is collected by a clipper card and recorded by an industrial computer. The configuration parameters for the measurement system are as follows: Δx = 5 mm, D2 = D3 = 5 mm, D4 = 25 mm, N = 69. The environment temperature is controlled at 20 ± 0.5 °C.

The experiment is repeated six times. Then, the sensor outputs are averaged and filtered to reduce the measurement noise. Based on the new four-sensor method, the straightness and tilt errors are calculated and described (in OXY) as Sa(x) and γa(x) in Figure 9. In addition, the obtained surface profile is shown in Figure 10. The straightness and tilt errors are also measured four times by a laser interferometer with the linear accuracy of ± 0.5 μm and the angular accuracy of ± 1 arcsec. The related results (in the coordinate system OmXmYm of the laser interferometer) are plotted in Figure 9, named Sl(x) and γl(x). It is improper to compare Sa(x) with Sl(x) directly, because of the different coordinate systems. Taking the OmXmYm as the reference, the OXY is rotated so that the 2-norm of \(S_{l}^{m} \left( x \right) - S_{a} \left( x \right)\) is minimum. Where \(S_{l}^{m} \left( x \right)\) is the average of the four values of Sl(x). This situation indicates that the OXY is the same as the OmXmYm. After the rotation, Sa(x) given in Figure 11(a) is consistent with \(S_{l}^{m} \left( x \right)\) and the maximum deviation between them is less than 2.5 μm. The similar operation is performed on γa(x) and γl(x). As shown in Figure 11(b), the final results have the similar trend and the maximum deviation is about 11 arcsec. Therefore, the proposed four-sensor method together with the improved measurement system is able to identify the straightness and tilt errors of the linear slideway. The discrepancies in Figure 11 might be mainly caused by the measurement noise in mi(xn) from the repeatability of the straightness and tilt errors, the interferometer accuracy and the sensor accuracy. Moreover, the positioning errors of the sensor stage and the slideway may also produce a difference.

Figure 9
figure 9

Experimental results obtained by the new four-sensor method and the laser interferometer

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Figure 10
figure 10

Surface profile obtained by the new four-sensor method

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Figure 11
figure 11

Comparisons of experimental results after the coordinate system OXY is rotated

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5 Conclusions

With the adoption of an improved measurement system, a new four-sensor method is presented to identify the straightness and tilt errors of a linear slideway from the sensor outputs. The reference surface profile could be obtained simultaneously. Its feasibility is evaluated by computer simulations and experiment. From the results, some conclusions are summarized as follows.

  1. (1)

    The measurement system allows achieving the adjustable sensor spacing and the high lateral resolution, and avoids the sensor gain error. The proposed method has some advantages, such as suitable for the test section of any length as well as needless to pre-measure the zero-adjustment values and the reference surface profile accurately.

  2. (2)

    The method is feasible and can guarantee the favorable propagation of the residual caused by the measurement noise in the sensor outputs. In addition, it is influenced slightly by the positioning errors of the sensor and the slideway.

  3. (3)

    The maximum discrepancy between the experimental results associated with the straightness error is less than 2.5 μm. The results corresponding to the tilt error have the similar trend.