## 1 Introduction  
Any object has a 6-degree-of-freedom motion error in space, that is, movement in three axes (positioning, vertical, and horizontal straightness errors) and rotation around three axes (pitch, yaw, and roll). With the rapid development of the manufacturing industry, single-dimensional measuring instruments can no longer meet these measurement needs. A fast, high-precision, and multiple-degree-of-freedom measurement system is urgently needed. The multiple-degree-of-freedom error measurement method can be used not only for the detection of measuring equipment such as CNC machine tools, machining centers, and coordinate measuring machines in the mechanical industry [1; 2; 3] but also for the accurate positioning of precise components and relative attitude measurement [4; 5; 6]. At present, multiple-degree-of-freedom methods are mainly divided into three types according to the measurement principle: laser interferometry, laser grating diffraction, and laser collimation. [PERSON] designed a six-degree-of-freedom measurement system by using six improved laser interferometers, which can simultaneously measure a wide range of motion errors. [7]. [PERSON] proposed a laser heterodyne interferometer based on the Faraday effect, which can guarantee the correct interference regardless of the translation or rotation of the measured object [8]. [PERSON] proposed a laser heterodyne interferometer based on an acousto-optic modulator for measuring displacement and roll angle simultaneously. An assembly square pyramidal mirror and a folding mirror are specifically arranged to create a multiple-reflection cavity to increase the roll measurement resolution [9]. [PERSON] et al. proposed a laser homodyne straightness interferometer that simultaneously measures six-degree-of-freedom motion errors. The multiple degrees of freedom are measured by the spacing changes and counting of interference fringes [10; 11]. This type of measurement method fully utilizes the high-precision characteristics of laser interferometry, but the system is complex and costly. The laser grating method is based on the principle of grating spectroscopy, which can realize multiple-degree-of-freedom measurements by obtaining different information carried by each beam of separated light. [PERSON] et al. proposed a six-degree-of-freedom measurement method by using a single unit of an optical encoder and discussed estimates of the uncertainties in terms of error sources [12; 13]. [PERSON] proposed a heterodyne grating interferometer for measuring two-dimensional straightness based on a quasi-common optical path (QCOP) [14]. On this basis, [PERSON] proposed a new heterodyne laser interferometer with a six-degree-of-freedom measurement. This technique combined the advantages of heterodyne, Michelson, and grating shear interferometries [15]. [PERSON] proposed a novel structured automatic collimator that can simultaneously measure three-dimensional angles by utilizing Moire fringes formed by two gratings [16]. However, the angle and position are not linear and the measurement accuracy also changes with the angle in this type of method. The laser collimation method utilizes the linear propagation characteristics of collimated light, and multiple detectors can be used to detect the position of each beam to achieve multiple-degree-of-freedom measurements. For example, [PERSON] constructed a three-collimated light six-degree-of-freedom measurement system by using three sets of laser Doppler length gauges. The system has high requirements for the parallelism of the three beams, and adjustment is difficult [17]. [PERSON] developed an integrated geometric error optical measurement system for a three-axis miniaturized machine tool and used a homogeneous transformation matrix to calculate the geometric error of six degrees of freedom [18]. In addition, some researchers have used spectroscopic devices to reduce the number of light sources and pyramid prisms [19; 20] or special prisms [21; 22; 23] as cooperative targets.