For two-axis electro-optical measurement equipment, there are many error sources in parts manufacturing, assembly, sensors, calibration, and so on, which cause some random errors in the final measurement results of the target. In order to eliminate the random measurement error as much as possible and improve the measurement accuracy, an active compensation technique for target measurement error is proposed in this paper. Firstly, the error formation mechanism and error transfer model establishment of the two-axis electro-optical measurement equipment were studied, and based on that, three error compensation and correction methods were proposed: the least square (LS)-based error compensation method, adaptive Kalman filter(AKF)-based error correction method, and radial basis function neural network (RBFNN)-based error compensation method. According to the theoretical analysis and numerical simulation comparison, the proposed RBFNN-based error compensation method was identified as the optimal error compensation method, which can approximate the random error space surface more precisely, so that a more accurate error compensation value can be obtained, and in order to improve the measurement accuracy with higher precision. Finally, the experimental results proved that the proposed active compensation technology was valid in engineering applicability and could efficiently enhance the measurement accuracy of the two-axis electro-optical measurement equipment.

A complex curved reflector made from a 40% silicon–aluminum alloy (AlSi40) can meet the requirements of optical systems operating across the infrared, near-infrared, and visible bands. It enables an athermalization design with simplified alignment and assembly, while offering high manufacturing efficiency and low costs. This makes it ideal for widespread use in high-end optical systems. As an enabling technology for the fabrication of AlSi40 freeform mirrors, error compensation in ultra-precision (UP) diamond turning is currently a research hotspot; however, current error compensation methods still have considerable room for improvement in terms of both accuracy and manufacturing efficiency. To address this issue, this study proposes an efficient and highly accurate method: a polar grid is defined in the machining coordinate system, and the corresponding surface point cloud is calculated. Using measured point clouds from reference spheres and freeform form error in the measurement coordinate system, mounting pose errors and form error with measurement error removed are determined via least squares. Machining error at grid points is then calculated via coordinate transformations and bicubic spline interpolation, and applied to correct cutter contact points (CCPs). Cutter location points (CLPs) are finally obtained using piecewise cubic spline fitting and a bisection method. With this method, average form error of four AlSi40 substrates improved from RMS 114.8 nm to 47.9 nm, and for four AlSi40 substrates with nickel–phosphorus (NiP)-plated surfaces, from RMS 71.3 nm to 31.1 nm. The compensated accuracy meets near-infrared stellar tracker requirements without polishing, greatly enhancing freeform mirror manufacturing efficiency.

At ultra-low speeds, the quantization resolution of a position sensor typically exceeds the true angular displacement within one sampling period, so velocity information is almost entirely buried under quantization steps. This paper proposes a quantization-oriented adaptive high-order sliding mode differentiator (Q-AHOSMD). A saturation switching mechanism with the quantization resolution as prior knowledge suppresses high-frequency chattering over quantization plateaus; an event-driven differential velocity estimation scheme of the “update on jump, decay over plateau” type is constructed from quantization jump events; and an adaptive gain law is synthesized from velocity-mismatch feedback. Finite-time convergence of the error system and practical stability under quantization noise are proved, with explicit expressions for all ultimate bounds. Simulations show that, with a sampling period of 1 ms and a true velocity of 0.001°/s, the method achieves a root-mean-square (RMS) velocity estimation error of 0.00036°/s—about 94% lower than the backward difference method—with smooth output and no visible chattering. Requiring no dynamic model and tunable using only sensor specification parameters, the method provides an effective means of extending ultra-precision servo systems into the ultra-low-speed regime.