## 1 Introduction  
In precision engineering, with the significant improvement in positioning sensor accuracy in the past few decades [1], the influence of angular error on the accuracy, repeatability, and stability of precision machinery has become increasingly important [2; 3]. There are many types of angle measuring instrument that have been applied in precision engineering [4; 5; 6]. Some multi-degree-of-freedom measurement systems have been reported in recent years [7; 8; 9], and the roll angle measurement of a long stage is still affected by unstable laser beams [10; 11]. For the advantage of measuring the roll error of the linear stage, and for the measurement of the inclination angle while installing the precision machinery, it is still necessary to know the precision level. Traditionally, according to the different measurement principle, spirit levels can be divided into two types: the bubble type and the pendulum type. Due to the disadvantages of measurement resolution and the accuracy of the bubble-type sprit level, the bubble-type sprit level can only meet the requirement of rough adjustment of equipment horizontal installation. Currently, most commercial precision levels are designed based on a pendulum mechanism [4], and usually they are the one-axis pendulum type [4; 5]. The pendulum-type precision level is also used as a sensor for monitoring the ground rotation of some large-scale scientific instruments, such as the Laser-Interferometer-Gravitational-wave-Observatory (LIGO) [12; 13]. The resolution and accuracy of the mechanical pendulum-type level are dependent on the design of the pendulum and mechanism; for that, the size and weight of the pendulum-type precision level can hardly be reduced [14]. In addition, the design and manufacturing accuracy of the pendulum will directly affect the accuracy of the pendulum-type precision level, and the complex structure and requirement for high-precision manufacture processing mean that the cost of this type of level cannot be reduced. In order to improve the resolution and accuracy of the pendulum-type precision level, expensive precision sensors such as capacitive sensors further increase the cost of the level. These problems limit the further application of the precision level. In order to miniaturize the size of the precision level, many novel measurement principles have been proposed. Using gravity as a reference, [PERSON] [15] proposed a high-precision micro capacitive inclination sensor, and [PERSON] [16] proposed a roll error measurement system made by combining two micro capacitive inclination sensors to eliminate the influence of external disturbance. However, the accuracy and resolution of capacitive inclinometers are larger than 1 arc-sec. Based on the grating diffraction and reflection, [PERSON] developed a three-axis autocollimator for detecting three angular error motions of a precision stage [17], and [PERSON] [18] modified the system to make a liquid-surface-based three-axis inclination sensor for the measurement of stage tilt motions. However, the system was complex and the accuracy of the measured angles was still larger than 1 arc-sec. The dual-axis pendulum-type precision level was proposed by [PERSON] [19], with the accuracy and resolution both less than 1 arc-sec. A brand-new design of a pendulum-free precision level was proposed by [PERSON] [20] using the light refraction principle and the reference of the invariant liquid level. The dual-axis level could be small in size and light in weight. However, the first reflection beam from the liquid surface would interfere with the refracted beam and produce noise signals. [PERSON] [21] directly used the surface reflection beam to design a dual-axis level. However, it required high-energy laser power. In this paper, a modified design of a refraction-type level from the author's previous system [20] is proposed. With the new principle of single-light refraction and transmission, an innovative dual-axis precision level is improved from the previous double refraction type.