1 Introduction

Parallel kinematic machines (PKMs) are widely used in industry, as their closed kinematic loops provide high structural rigidity and stiffness [1]. The Exechon PKM (Exechon Enterprises, LLC) is a commercially available PKM used to control the position and orientation of an end-effector [2]. Recently, a hybrid PKM (DABO MDP-1000; Maxrotec Co., Ltd) was developed for hybrid processes, including machining and direct-energy-deposition (DED) three-dimensional (3-D) metal printing [3]. The hybrid PKM combines a rotary axis with the Exechon PKM, which is used to increase productivity by controlling the workpiece orientation. The volumetric accuracy and kinematic errors of the Exechon PKM can be determined through calibration processes supported by Exechon Enterprises LLC [4]. However, calibration of the additional rotary axis is not supported, which can result in volumetric errors when the hybrid PKM is used. The position and orientation of the rotary and spindle axes deviate from the design during assembly. The positional and orientation errors are defined as offset and squareness errors, respectively, and are collectively described as position-independent geometric errors (PIGEs) [5] (also called location errors [6] and location and orientation errors [7]). It is essential that the PIGEs are directly or indirectly measured and compensated to keep volumetric errors within tolerance [8, 9].

Several techniques can be used to measure the PIGEs of a rotary axis, such as a double ball-bar (DBB), the R test, a touch-trigger probe, multilateration, and machining tests. A DBB can be used with three measuring paths in the radial, axial, and tangential directions requiring simultaneous three-axis control [10]. The test conditions for the measurements are specified in ISO 10,791–6 [11]. PIGE identification with a cylindrical coordinate system is superior to PIGE identification with a Cartesian coordinate system, in terms of the number of measurements [12]. Control of the linear axis is avoided by the use of simple DBB measuring paths that require only control of the rotary axis [13]. Similar to the DBB method, the R-test was developed to identify the PIGEs of a rotary axis, by using a 3-D sensor to measure the position of a ball [14]. The error motions of controlled linear axes during the R-test also affect PIGE identification, so it is recommended that these be measured and compensated for [15]. Thermal errors can also affect PIGE identification [16], so it is necessary to identify PIGEs rapidly. A touch-trigger probe is used to measure ball positions for PIGE identification of four-axis machine tools [17], five-axis machine tools with a tilting-rotary table [18], and five-axis machine tools with a universal head [19]. A touch-trigger probe can also be used to identify PIGEs by precisely measuring a test piece of five-axis machine tools with a tilting-rotary table [20] and tilting head [21], so that PIGEs can be identified without the need for additional measurement devices. In multilateration, a laser tracker is used to measure the coordinates of several target points using the same principle as a global positioning system [22]. It is also used to identify PIGEs by machining a test-piece on a machine tool and measuring the features of the machined test piece with a coordinate measuring machine (CMM), for five-axis machine tools with a tilting-rotary table [23] and tilting head [24].

Typically, the PIGEs of a spindle axis are measured using a test mandrel [25], or a DBB is used to conduct two circular tests with different tool lengths for three-axis machine tools [26].

Recently, a DBB method was proposed to identify the PIGEs of linear axes, rotary axes, and a spindle axis by selective analysis of the data in Cartesian and cylindrical coordinate systems [27]. However, PIGE identification is affected by error motions arising due to the control of the linear axes during measurements.

In summary, rotary and spindle axis PIGEs can be measured using existing techniques, with three-axis controls in generally being required; however, these methods are typically expensive, complicated, and time-consuming. In addition, no studies concerned only with PIGE identification of rotary and spindle axes have been published. Therefore, we propose a technique that can be used to simultaneously identify the PIGEs of rotary and spindle axes, using a DBB and fixtures to conduct simple measurements not affected by the error motion of linear axes. In Sect. 2, a hybrid PKM is introduced and the PIGEs of rotary and spindle axes are summarized. Measurement paths are proposed, and an algorithm is developed that can identify the PIGEs from sequentially measured data. In Sect. 3, the proposed technique is used to measure and compensate the PIGEs of a hybrid PKM, and the measurement uncertainty is analyzed. The main advantages of our proposed method are summarized in Sect. 4.

2 A Hybrid PKM and PIGE Measurements

2.1 A Hybrid PKM and the PIGEs of Rotary and Spindle Axes

A hybrid PKM comprises three parallel linear axes, L1, L2, L3, two serial wrist axes, W1, W2, for tool position and orientation, and a rotary axis, C, for workpiece orientation control, as shown in Fig. 1. The machine specification is summarized in Table 1 [28]. In this study, we assume that the three linear and two wrist axes were fully calibrated according to the processes recommended by Exechon Enterprises, LLC, such that the volumetric errors due to these axes are negligible. Here, the hybrid PKM errors are caused primarily by the position and orientation deviation of C and the spindle axis, S. The PIGE deviation is illustrated in Fig. 2. Specifically, C is offset from its nominal position by oxc, oyc, and deviates from its nominal orientation by sxc, syc in/around the x, y-direction, respectively. S deviates from its nominal orientation by sxs, sys around the x-, y-direction, respectively.

Fig. 1
figure 1

The structure of a hybrid parallel kinematic machine (PKM)

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Table 1 The hybrid PKM specification
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Fig. 2
figure 2

The position-independent geometric errors (PIGEs) of the rotary axis, C, and spindle axis, S

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2.2 DBB Measuring Paths and Measurement of the PIGEs

The PIGEs were measured using a DBB and three movement paths, as shown in Fig. 3a, b, c. Figure 3a shows how the offset errors oxc and oyc were measured using a DBB installed between the tool nose and a center mount on the workpiece table. C was unilaterally controlled according to the rotation angle, c, as R + ∆R1j was recorded. As shown in Fig. 3b, the squareness errors sxc and syc were measured by installing the DBB in the z-direction with a height offset oH, and R + ∆R2j was recorded as C was controlled. The squareness errors sxs and sys were measured by installing the DBB at the same height shown in Fig. 3a with a tool offset oT, as shown in Fig. 3c.

Fig. 3
figure 3

The PIGE paths measured using a double ball-bar (DBB)

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In general, PIGEs are calculated from DBB measurement data (R + ∆Rij) by calculating the eccentricities, exi and eyi [10, 29]. It is trivial to determine oxc and oyc from R + ∆R1j, however it is more difficult to determine sxc and syc, and sxs and sys as they are components of R + ∆R2j and R + ∆R3j, respectively, which are compounded with oxc and oyc. Therefore, the offset errors are measured and compensated by the first measurement, and the second and third measurements are then made sequentially so that the magnitudes of the squareness errors can be calculated. In this paper, this process is referred to as sequential measurement. The PIGEs are calculated from the eccentricities as described by Eq. (1), in which the columns are fully decoupled due to the sequential measurement.

$$\left[ {\begin{array}{*{20}c} {o_{xc} } \\ {o_{yc} } \\ {s_{xc} } \\ {s_{yc} } \\ {s_{xs} } \\ {s_{ys} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {o_{H} }}} \right. \kern-\nulldelimiterspace} {o_{H} }}} & 0 & 0 \\ 0 & 0 & {{1 \mathord{\left/ {\vphantom {1 {o_{H} }}} \right. \kern-\nulldelimiterspace} {o_{H} }}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{{ - 1} \mathord{\left/ {\vphantom {{ - 1} {o_{T} }}} \right. \kern-\nulldelimiterspace} {o_{T} }}} \\ 0 & 0 & 0 & 0 & {{1 \mathord{\left/ {\vphantom {1 {o_{T} }}} \right. \kern-\nulldelimiterspace} {o_{T} }}} & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {e_{x1} } \\ {e_{y1} } \\ {e_{x2} } \\ {e_{y2} } \\ {e_{x3} } \\ {e_{y3} } \\ \end{array} } \right]$$
(1)

3 Experimental Study of the Proposed Method

The proposed method was applied to a hybrid PKM [3] so that the PIGEs of C and S could be measured and compensated, as shown in Fig. 4. A 100-mm-long QC20-W ball bar (Renishaw plc) was used, and a large height and tool offset (320 and 325 mm, respectively) were employed to reduce the PIGE measurement uncertainty. The large offsets oH and oT can be used to identify squareness errors, and the offsets can be increased with additional fixtures.

Fig. 4
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The PIGE measurement and compensation measurement process

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As shown in Fig. 5, large R + ∆Rij peak-to-valley (PV) values of 29.5, 50.9, and 79.7 μm were measured when i was equal to 1, 2, and 3, respectively. This was primarily due to the eccentricities caused by the PIGEs of C and S. The eccentricities of R + ∆Rij were calculated using Eq. (1), as summarized in Table 2.

Fig. 5
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The radial deviation ∆Rij with and without compensation

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Table 2 The PIGEs identified with and without compensation
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The measurements were repeated after compensation of the PIGEs shown in Table 2, and corrected R + ∆Rij PV values of 2.4, 8.8, 5.4 μm were recorded when i was equal to 1, 2, and 3, respectively. These values represent an improvement of 92, 83, and 93%, respectively, which demonstrates the validity of the method proposed here. The contributors to the PIGE measurement uncertainty are summarized in Table 3, when the coverage factor k = 2. It was assumed that the repeatability of C and the PKM was ± 1 μm, which is of the same order as the resolution of the linear axes L1, L2, L3, and that the repeatability followed a rectangular distribution [30].

Table 3 Contributors to PIGE measurement uncertainty (k = 2)
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Theoretically, the measurement uncertainties of the squareness errors are identical if oH and oT are equal. As shown in Fig. 6, the squareness error measurement uncertainties were calculated as a function of the offset. The measurement uncertainty and offset were found to have an inverse relationship; however, the uncertainty did not decrease significantly when the offset was over 300 mm. Therefore, height and tool offsets of 320 and 325 mm, respectively, were used throughout this study.

Fig. 6
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The squareness error measurement uncertainty as a function of the offset

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4 Conclusion

Here, we proposed a simple and effective method to improve the volumetric accuracy of a hybrid PKM, in which the PIGEs of the rotary and spindle axes were measured. For simplicity, only a double ball-bar was used to conduct the measurements; three motion paths were followed, which only required movement of the rotary axis. Additionally, sequential measurements and an analytical method were used to determine the PIGEs using the three measured datasets, and their eccentricities. The proposed method was tested with a hybrid PKM, and validated by measurement and compensation of the PIGEs; in this manner. the PV and double ball-bar errors were improved significantly.

It should be noted that the proposed method is not restricted to double ball-bar measurements; it could also be utilized for precise measurement of the position of a ball in a reference coordinate system, via the touch-trigger probe and R-test techniques, for example.