## 1 Introduction  
Multi-axis machine tools are key equipment in advanced industries for manufacturing complex freeform components, especially in the fields of ultra-precision diamond turning and polishing. The accuracy of the workpiece is significantly influenced by the geometric error of these machines. These errors have integrated effects in the manufacture of the products. The modeling and measurement of these volumetric errors are of prime importance in high precision machine tools. As a result, it is recommended to regularly verify the accuracy of machine tools [1; 2]. In order to improve the machining accuracy and efficiency, many approaches have been reported to measure and compensate for the geometric error [2; 3; 4].  
In the development of the error model, the homogeneous transformation matrix (HTM) [5] method is commonly used. This method makes use of all the motion matrix based on rigid body kinematics to establish the relationship between the tool tip and the workpiece. Although it is easy to understand, the calculation may be difficult and time-consuming. In order to improve the efficiency of the calculation, other corresponding approaches have also been proposed such as Denavit-Hartenberg(D-H) transformation matrix [6], the screw theory [7; 8] and the differentiable manifold-based method [4]. For the identification of the geometric errors, the laser interferometer [9; 10] is the most popular measuring device to directly measure the error of the translational axes due to its high accuracy. However, this method is time-consuming due to the repeated process of adjusting the laser path before every measuring task. A laser ball bar and tracker [11; 12; 13] are also used to measure the volumetric error in an indirect way. These approaches can provide reach information for the evaluation of machine performance. However, it is interesting to note that they all need to incorporate special instruments into the machine, thus requiring specialized personnel and leading to low efficiency. To address this problem, an artefact is employed to improve the effectiveness. A uncalibrated master ball artefact was proposed by [14], which claimed that the new artefact design could provide the volumetric distortion of the entire working space. However, this method may be limited by the low accuracy of the on-machine measurement system.  
[PERSON] et al. [15] developed a 2D ball plate which can form a 3D working volume by moving the plate in a three-axis machine tool. [PERSON] [16] probed a scale enriched reconfigurable uncalibrated master ball artefact with the on-machine measuring system to separate the setup errors from the volumetric error of a five-axis machine tool. In addition, a ball dome artefact consisting of 25 precision balls was also fabricated [17] to check the performance of a multi-axis machine tool. These studies are performed on a high-precision machine tool in order to check all of the whole volumetric errors of the machine. Most of these methods can only achieve accuracy at several micrometers and they are focused on all the volumetric errors of the machine tool, which is not true of the precision manufacturing of freeform optics in a specified workspace of the ultra-precision machine tool.  
As a result, a reconfigurable fiducial-aided 3D artefact which has the advantage of taking the characteristics of the machined workpiece into consideration is purposely designed in this study. Uncertainties including the gravity effect and measurement repeatability are examined to optimize the reconfigurable artefact. Hence, the HTM is employed to generate the kinematics model of a five-axis machine tool. An integrated on-machine measuring system is developed to evaluate the working volumetric errors in the machining process.