How can we improve the accuracy of transcendental equation solutions for tube radius calculation?

Accuracy Enhancement of Transcendental Equation Solution

Problem Overview

The goal is to accurately compute the radius (r0) of a tube using measurements from a kinematics system. However, direct measurement of parameters a0, y0, and z0 is difficult, leading to potential inaccuracies.

Question 1: How to Improve Solution Accuracy?

• Weighted Deviations: Consider weighting each deviation based on its angular distance from 0 degrees.
• Higher Recursion Levels: Explore increasing the number of recursions used in the approximation algorithm to improve accuracy.
• Adjust Recursion Range: Experiment with modifying the recursion range to refine the results within the desired angular and distance bounds.

Question 2: Exploring Missed Factors?

• Nested Approximations: Ensure that there are no inappropriate nesting of approximations that may introduce errors.
• Mathematical Simplification: Scrutinize the underlying mathematical equations to identify potential simplifications or rearrangements that could enhance precision.
• Alternative Approaches: Consider alternative methods, such as Bayesian modeling or machine learning, to address the problem.

Additional Considerations:

• Limited Angular Range: Increasing the angular range of the measurements may not significantly improve accuracy due to equipment constraints.
• Optimal Point Distribution: Determine the optimal distribution of measurement points along the y-axis for the given angular range to maximize accuracy.
• Simulation Accuracy: Ensure sufficient simulation accuracy to match the precision limitations of the real input data.

Progress Update:

• Substantial Accuracy Improvement: Measuring y1 and employing a series of approximation techniques have increased accuracy to around 0.03 mm.
• y1 Calculation: Calculate y1 from the intersection of the arm and tube axes, which improves precision and computational stability.
• Deletion of Approximation Loop: Eliminate the need for one approximation loop by exploiting the dependency between y1, a0, and z0.
• Further Calibration Methods: Explore methods to directly measure a0, y0, or other known parameters to enhance accuracy.
• Simulated Measurement Accuracy: Currently, the simulated measurements exhibit an accuracy of around 0.003 mm for a0 and 0.0003 mm for y1.

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