## 1 Introduction  
In the past 10 years, the increasing interest in precision manufacturing engineering has heightened the need of advanced manufacturing technology. Precision manufacturing mainly includes precision and ultra-precision machining technology and manufacturing automation. The former pursues the precision of machining and the limit of surface quality, while the latter includes the automation of product design, manufacturing and management. Many parts and components used in aviation, aerospace, ships and cars are machined thorough precision and ultra-precision machining technology. As the hardware foundation of modern advanced manufacturing industry, the machining accuracy of multi-axis machine tools plays a critical role in developing and improving advanced manufacturing technology. The errors discussed in this paper are the difference between the expected values of geometric dimension and the obtained measurement results. The geometrical deviations of a part resulting from successive machining set-ups originate from multiple error sources [1, 2], such as positioningdeviation [3], fixture errors, datum errors and machine tool errors [4]. Therefore, geometric errors of the machine tools constitute a large portion of the total machining error [5]. In terms of the geometric error analysis, [PERSON] and [PERSON] proposed several error analysis models, such as 3D tolerancing models [6; 7], a form defects expression method based on natural mode shapes of a discretized feature [8], and 2D and 3D assemblies modeling considering form errors of plane surfaces [9]. [PERSON] described the form errors modeling in the assembly process [10]. Considering the variations and uncertainties of geometrical variations, [PERSON] and [PERSON] built a novel tolerance analysis representation model [11; 12] and broadened the scope of the Jacobian-Torsor model [13]. In addition, [PERSON] and [PERSON] investigated the profile measurements repeatability [14]. [PERSON] compared several tolerance analysis methods in common use [15; 16] and constructed Tolerance-Maps to model the composite positional tolerancing [17]. In terms of the measurement instrument of geometric error, [PERSON] used an auto-tracking laser interferometer to measure the geometric error of machine tools [18]. [PERSON] used a double ball bar to measure geometric errors of the rotary axis on a multi-axis machine tool based on kinematic analysis [19; 20; 21]. [PERSON] used a laser scanning device to analyze sensitivity of machine geometric errors measurement [22; 23]. Precision optics devices are recently and gradually applied to measure geometric errors of machine tools [24; 25]. In terms of the measurement method, [PERSON] applied the radial basis function (RBF) neural network approach to measure and compensate for geometric displacement errors [26]. [PERSON] applied machine vision to measure screw thread geometric errors [27]. [PERSON] applied differential motion matrices to identify position-independent geometric errors of five-axis serial machine tools [28]. [PERSON] researched the optimal on-machine measurement of position-independent geometric errors for rotary axes in five-axis machines with a universal head [29]. Comparison results of direct and indirect methods for geometric error measurement [30; 31] indicate that because of ignoring angle errors, researches above lack completeness of geometric errors [32]. For example, [PERSON] designed a novel six-degree-of freedom geometric error measurement system for a linear stage [33]. [PERSON] designed a geometric error measurement system for linear guideway assembly and calibration [34]. At present, the measurement precision of most geometric errors is lower because of ignoring the items of small angle errors. To form a better view of the formation, measurement, transfer and integration process of geometric errors, we proposed a new measurement method to measure complete geometric errors of multi-axis machine tools.