## 1 Introduction
If multi-DOF motion errors are being measured, other auxiliary optical devices and measurement steps are needed, which increases costs and difficulty and has low efficiency. [PERSON] et al. used Laser Tracker System to measure the six DOF motion errors by tracking three noncollinear points on the axis of rotation at different spatial positions, accuracy was 0.9 \(\upmu\)m [15]. The laser tracking method has the advantages of simplicity, high efficiency, and not requiring complex equipment installation and alignment, but its accuracy is relatively low [13]. [PERSON] et al. used diffraction grating and position sensitive detectors to measure motion errors, the resolution is 0.2\" [16]. The diffraction method, which takes the high-precision diffraction grating as the sensitive unit, requires high installation accuracy and high maintenance costs. [PERSON] et al. used laser collimation method to measure the six DOF motion errors in two steps [17]. [PERSON] et al. used two-dimensional slope sensors to measure the six DOF motion errors [18]. [PERSON] et al. used a ball lens and rod lens to measure the five DOF motion errors [19]. [PERSON] et al. used a laser collimation method to measure the five DOF motion errors simultaneously [20]. The laser collimation method has the advantages of high precision and fast speed, but most existing methods cannot simultaneously measure all six DOF geometric motion errors of a rotary axis; in addition, the measurement device is usually complex and difficult to install, and takes a long time to calibrate. Many studies used a polyhedral prism to measure angular position error. To our knowledge, no method based on polyhedral prisms has been proposed which can simultaneously measure the six DOF motion errors. [PERSON] et al. proposed a measurement method of angular position error based on autocollimation and polyhedral prisms [21]. This method has been recommended by ISO230-1 as one of the standard measurement methods of angular position error [22]. [PERSON] et al. used polygonal prism and precise angle dividing table accomplish mutual check of angular deviation [23]. A novel method based on polyhedral prisms is proposed for simultaneously measuring the six DOF motion errors in this paper. A high-precision polyhedral prism and a plane mirror are designed to be the sensitive unit that can carry all six DOF motion errors of a rotary axis. By using photoelectric autocollimators and laser displacement sensors, the full-circle and simultaneous measurement of all six DOF motion errors of a rotary axis can be realized. This has the advantages of simple structure, fast measurement speed, and convenient installation. An error model is established, including the six DOF geometric motion errors of a rotary axis, installation errors between the polyhedral prism and rotary axis, manufacturing errors of the polyhedral prism, and the position errors of photoelectric autocollimators and laser displacement sensors. This model can accurately calculate the six DOF geometric motion errors of a rotary axis and provide technical support for compensating the geometric motion errors of a rotary axis and improving its motion accuracy.