Tolerance Budgeting With Jacobian-Based Compensator Selection
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Solution Overview
Problem
Tolerance analysis and allocation in systems design is computationally demanding and often requires an iterative trial-and-error approach, especially when dealing with perturbations in input parameters affecting output parameters, which can be time-consuming and inefficient.
Innovation Solution
The problem is addressed by transforming the task of tolerance allocation into fitting a multi-dimensional axis-aligned cuboid (orthotope) inside an ellipsoid using linear algebra, incorporating the effect of compensators to determine the efficacy of their choice, and providing an algorithm for this process, allowing for computationally efficient calculations and automatic tolerance allocation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If iterative trial-and-error approach is used for tolerance allocation, then tolerance analysis can be performed, but computational time and complexity increase significantly
Solution Approach 1:
The patent transforms the tolerance allocation problem from an iterative trial-and-error process into a direct mathematical optimization problem by changing the parameters from tolerance values to vertex coordinates of an orthotope. This parameter transformation enables direct computation of optimal tolerances using linear programming, eliminating the need for repeated iterative simulations and significantly reducing computational time while maintaining allocation accuracy.
2Measurement precision
If Monte Carlo simulations are used for tolerance analysis, then output errors can be estimated, but computational demand increases
Solution Approach 1:
The patent replaces the Monte Carlo simulation approach (which relies on repeated random sampling and statistical analysis) with a deterministic mathematical programming approach. By formulating tolerance allocation as an optimization problem with linear constraints and objective functions, the method substitutes probabilistic computational complexity with efficient linear programming algorithms, reducing both computational demand and algorithmic complexity while maintaining estimation accuracy.
3Reliability
If compensators are selected to correct perturbed systems, then output parameters can be brought within specifications, but selection process becomes complex
Solution Approach 1:
The patent incorporates compensator effects directly into the optimization formulation by transforming the problem parameters to account for compensator adjustments. This allows the compensator selection and tolerance allocation to be performed simultaneously through a unified mathematical program, rather than through separate iterative processes. The transformation embeds compensator efficacy metrics into the objective function, simplifying the selection process while ensuring output parameters meet specifications.
Data Source
AI summary
Algorithm for tolerance analysis, allocation, and synthesis, also known as tolerance budgeting, is discussed. Also discussed is a metric to rank compensators for a system. It is based on the system Jacobian and the inner product of the output vector error as the tolerancing criterion. These tolerances are calculated by fitting an appropriate, axis aligned multidimensional Orthotope within an ellipsoid like region that is not necessarily axis aligned.