Biaxial Pointing Error Correction Using Spherical Cap Functions
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Solution Overview
Problem
Current pointing error models for biaxial rotation systems, such as telescopes and five-axis machine tools, suffer from instability due to large normal equation conditions and are prone to measurement noise, leading to inaccurate error correction and reduced performance.
Innovation Solution
The method employs orthogonal spherical cap functions, including hemispheric harmonic functions, Zernike spherical cap functions, and longitudinal spherical cap functions, to model and correct pointing errors. This approach involves error collection, fitting the error model using the selected functions, and implementing the model in a pointing control system for compensation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If spherical harmonic function model is used for pointing error modeling, then the model can provide complete orthogonal functions on the whole sphere, but the normal equation conditions become very large leading to poor stability and high susceptibility to measurement noise
Solution Approach 1:
The patent extracts only the necessary portion of the spherical harmonic function by restricting the modeling domain from the whole sphere to a spherical cap (subset). This extraction reduces the number of basis functions and normal equation conditions while maintaining orthogonality, thereby improving model stability without sacrificing essential modeling capability
Solution Approach 2:
The patent applies local quality by creating a specialized spherical cap function tailored to the specific application domain (spherical cap rather than whole sphere). This localized approach provides optimal modeling accuracy for the specific pointing error characteristics of biaxial rotation systems while reducing computational complexity and noise sensitivity
2Reliability
If basic parameter model is used for pointing error correction, then the calculation is relatively stable, but the model form must be determined according to telescope rack form and considers fewer parameters reducing correction accuracy
Solution Approach 1:
The patent changes the parameter representation by using spherical cap function coefficients instead of basic parameter model parameters. This transformation maintains calculation stability through the orthogonal properties of the spherical cap functions while enabling consideration of more error parameters and their interactions, thereby improving correction accuracy
3Measurement precision
If rack model is used for pointing error correction, then high accuracy is achieved, but the model is not as stable as the basic parameter model
Solution Approach 1:
The patent applies spheroidality by using spherical cap functions that naturally conform to the curved spherical geometry of the pointing error domain. This curved coordinate approach provides both the accuracy needed to model complex error patterns and the mathematical stability required for reliable calculation, resolving the contradiction between accuracy and stability
Data Source
AI summary
The invention discloses a method for correcting the pointing errors of a biaxial rotation system based on the spherical cap function, comprising: error collection: selecting stars or radio sources distributed evenly in a star catalogue for tracking and observation to obtain the theoretical position and measurement position of the stars, and subtracting the measurement positions and the theoretical positions to obtain the error distribution; error model fitting: selecting a suitable orthogonal spherical cap function for the obtained error distribution and performing fitting to calculate an error fitting coefficient, the orthogonal spherical cap function model comprising a hemispheric harmonic function HSH, a Zernike spherical cap function ZSF, and a longitudinal spherical cap function LSF; and error control and compensation: putting the error model and the related fitting coefficient into a pointing control system for compensation. In the present method for correcting the pointing errors of a biaxial rotation system based on a spherical cap function, the model has strong stability and is not easily affected by measurement noise; there is no need to determine the form of the model on the bases of the frame form of the telescope, and the correction accuracy is high.