Adaptive High-Order Sliding Mode Differentiator for Velocity Estimation from Quantized Position Measurements at Ultra-Low Speeds

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.