# A New Kinematic Model of Portable Articulated Coordinate Measuring Machine
## 1 Introduction  
The portable articulated coordinate measuring machine (PACMM) is a kind of high accuracy coordinate measurement instrument and has the advantages of flexibility, lightweight, portability and easy use compared to the traditional orthogonal CMM. It has been widely applied in manufacturing, assembly [1], _in-situ_ measurement, reverse engineering and calibration [2, 3]. Error sources of PACMM include structural parameters errors, joint errors, link deflections, thermal deformations and so on. The measurement accuracy of PACMM depends on the proper kinematic model which considers both geometric and non-geometric errors. Many methods have been proposed in the literatures to establish the kinematic model of robot or PACMM. The Denavit-Hartenberg (D-H) [4] formulation is regarded as the most classical kinematic model for a robot or PACMM. A modified four-parameter D-H formulation [5, 6] has been proposed to overcome the badly-conditioned problems when the two adjacent joint axes are parallel or nearly parallel. Reference [7] also showed that another type of the modified D-H formulation which the standard D-H conventions post-multiplies the rotation term around \(Y\)-axis could improve the badly-conditioned problem when the two adjacent joint axes are parallel or nearly parallel. In order to meet the three principles of the kinematic model of the manipulator--parametric, continuity and completeness, references [8, 9] proposed a complete, parametrically and continuous kinematic model by adding two parameters to [PERSON] line parameters. Recently, some researchers also proposed the generalized geometric error model method considering both geometric and non-geometric errors. For example, reference [10] introduced the generalized geometric error parameters for eliminating the geometric and non-geometric errors and improving the positioning accuracy of the patient positioning system. A general approach for error model of machine tools [11] has been introduced to eliminate the geometric and non-geometric errors of machine tools. The calibration technique of kinematic structural parameters has also been considered as an efficient method of eliminating geometric and non-geometric errors of CMM, PACMM, machine tools, _etc._ According to performance tests of PACMM [12; 13; 14], it is required to calibrate the measurement volume of PACMM by using a 1D standard gauge. Reference [15] introduced a simple artifact with two spheres for the kinematic structural parameters calibration of PACMM. The center distance between two spheres in the artifact was calculated by fitting the central points of two spheres. Reference [16] reported 1D ball bar array. The central points of two spheres were measured by PACMM with the special rigid probe. Reference [17] presented a new full pose measurement method for the kinematic structural parameters calibration of the serial robot. This approach was achieved by an analysis of the features of a set of target points on circular trajectory. [PERSON] _et al._[18] developed a high accuracy measurement device based on laser interferometer combining with 1D translation table for calibration and verification of PACMM. [PERSON] _et al._[19] presented a 3D ball plate with nine balls. [PERSON] _et al._[20] introduced the development of virtual ball bar to evaluate the performance of PACMM. [PERSON] _et al._[21] introduced a virtual circle gauge was applied in evaluating the performance of PACMM and it was composed of bar gauges of 1000 mm length with four groups of three cone-shaped holes. The above-mentioned virtual geometric gauges are applied to reduce the number of test positions, avoid the measurement points randomly sampled on the virtual geometrical gauges surfaces according to the norms and improve the efficiency of verification procedure for PACMM. [PERSON] _et al._[22] presented an indexed metrology platform combined with a calibrated ball bar gauge, which was applied in evaluating the performance of PACMM.