The photoelectric stabilization platform is a system that integrates optical and electronic technology to achieve stable and precise target tracking and positioning. A stable platform with a reflector, in combination with a mechanical stabilization frame and the reflector itself, ensures line-of-sight stability. Owing to the lower inertia of the reflector, it offers enhanced stability. Biaxial reflectors occupy less space but present higher complexity due to nonlinear coupling and control. Traditional methods approximate the mirror’s motion characteristics over its entire range as those near the zero position, which is suitable for small, slow compensation movements but introduces significant errors for larger, faster motions. This study requires high-resolution imaging when the mirror has a large compensation range and a fast speed, necessitating precise motion analysis of the mirror. Previous work deduced the relationship among the direction vector of the aiming line, the angle of the aiming line, and the angle of the reflector, accurately describing the reflector’s behavior during extensive movements. However, the derived relationship remains nonlinear, and considering the optical path and platform movement of the optoelectronic stabilization platform, the problem complicates the issue further.

The nonlinear relationship between the position of the aiming line and the angle of the reflector typically necessitates simplification to obtain analytically accurate results. The nonlinearity fundamentally arises from the non-straightness of the reflection transformation group and the attitude transformation group. Therefore, our study employs Lie group theory for analysis. The properties of the entire Lie group can be characterized by the tangent space at the identity element, known as the Lie algebra. Using Lie algebra allows us to capture the linear properties of Lie groups near any element, not just at the identity. We introduce the virtual image motion analysis (VIMA) method, which analyzes the motion of the aiming line through the movement of the virtual image of the sensor along the direction of the aiming line. The relationship between the speed of the aiming line and the angular velocity of the reflector is determined using the Lie group and the Lie algebra method. This method directly solves the angular velocity of the reflector and integrates this to find the reflector angle. Since no simplifications are made, only discrete errors occur, significantly enhancing solution accuracy.

We test the compensation effect of the reflector by measuring the error between the actual line of sight compensated by the reflector and the commanded line of sight, as well as the permissible motion range of the reflector using two techniques. The platform moves according to a specified angular velocity excitation signal, and the reflector must compensate for this motion to keep the unit vector of the aiming line stable within the geographic framework. Results from Figs. 3?6 show that the VIMA reduces error by three orders of magnitude compared to traditional reflection vector methods, validating our assumptions and deductions. Table 1 indicates that the worst-case permissible motion range for the simplified reflection vector method is 4.76°, whereas the virtual image motion analysis method did not reach the permissible error throughout the simulation, allowing for at least a 35.98° motion range—a significant improvement over the simplified method.

Based on Lie algebra theory, our study analyzes the reflector movement in optoelectronic stabilization platforms. We propose the VIMA to describe the relationship between the velocity of the line of sight and the angular velocity of the reflector. The VIMA enables direct resolution of this relationship to determine the angular velocity of the reflector, with its angle obtained through integration. This effectively addresses the nonlinear challenges of traditional reflection vector methods. The VIMA eliminates simplification errors, reducing image motion compensation errors by three orders of magnitude compared to conventional methods. This technique also holds significant implications for image motion analysis in other optical systems and optical system design.