Budgeting for errors

Error budgeting, also called uncertainty budgeting or uncertainty analysis, is a simple tool used for processes having tolerances that are difficult to achieve. It originally was developed for diamond turning—operations in which parts are turned with single-point-diamond tools—but is widely used in metrology, optics and other high-precision applications.

Author’s Note: An online class presented by MIT’s Alexander Slocum provides good information on error budgeting. It focuses on key elements and spreadsheet approaches to resolving error budgets. To view the PDF from this class, click here. To access a slideshow version of the PDF, follow this link.

For more info... For more detailed study of error budgeting see the following references. Phillip Stein. How to write an uncertainty budget. Quality Progress. July 2003; 80-81. Daniel D. Frey. Error Budgeting. in Robotics and Automation Handbook by Thomas R. Kurfess. New York: CRC Press. 2005; pp 10-1 to 10-21. Y.-L Shen and Neil A. Duffie. Comparison of combinatorial rules for machine error budgets. Annals of the CIRP. Vol. 42(1). 1993: 619-622. L. Uriarte, A. Herrero, M. Zatarain, G. Santiso, L.N. Lopéz de Lascalle, A. Lamikiz and J. Albizuri. Error budget and stiffness chain assessment in a micromilling machine equipped with tools less than 0.3 mm in diameter. Precision Engineering. Vol. 31. 2007; 1-12. R.E. Morace, H.N. Hansen and L. De Chiffre. Uncertainty budget for optical coordinate measurements of circle diameter. VDI_Berichte. Vol. 1860. 2004; 733-738. R.R. Donaldson. Error Budgets in Technology of Machine Tools. Technology of Machine Tools – Machine Tool Accuracy. 1980: pp 9.14-1 to 9.14-14.

Error budgeting is also finding acceptance in micromanufacturing. Micro EDMing, laser cutting, stereolithography, molding, automated assembly and metrology are prime applications for error budgeting. Locational accuracy and final feature size for any process can be improved by applying its basic concepts. Error budgeting helps parts producers identify areas of an operation they should target for improvement in order to meet tolerances and reduce errors.

There are at least two different forms of errors. The first is where a person makes a mistake. The second is where the process is not repeatable because of normal variation and normal misalignments. This article focuses on the latter.

In theory, error budgeting is like household financial budgeting. How much money (tolerance) do I have, and what do I spend it on? If I have little control on spending in some areas, how can I allocate the rest so I stay within my budget?

In challenging metalcutting and fabrication applications, engineers must assess how much total tolerance is consumed by the consistent error and repeatability of the machine tool, collet, cutting tool, fixturing, measuring process and initial workpiece material.

In addition, time-dependent issues, such as consistent error caused by thermal growth on the machine, are significant and may show up in each of the previously mentioned machining elements.

Error breakdown

In error budgeting, engineers break down the total allowable tolerance by category to determine how much each element of an operation contributes to total error. When the total exceeds the drawing’s specified allowance, or target, engineers concentrate on defining the maximum allowable amount of error for each element and making changes that reduce error in individual elements so the total allowance can be met.

While this is a simple approach, it requires engineers to know exactly how much each element contributes to total error. Many shops know the total tolerance they can hold, and how much tool deflection they have, but often do not have data on the individual elements described here. However, gathering the data is usually straightforward.

For example, to determine how room temperature affects errors, record it at a machine tool over a 24-hour period. On an hourly basis, remeasure the position of a fixed block made of a material that is not subject to thermal growth on the machine table (or bed). Then, chart the temperature and accuracy results.

The mathematics required for error budgeting can be very simple or involve sophisticated calculations and simulations. For most shops, the key is to understand the biggest error contributors and lower them. That can be straightforward and require simple math. Other shops need to understand, with great accuracy, how the systems respond. That typically requires extensive data, gathered by sophisticated systems.

The second point to consider is how many axes, or degrees of freedom, are needed to understand the problem. For many shops, that involves just the X and Y axes. For 3-axis machine tools, that might involve three linear directions, squareness errors, angular errors (roll, pitch and yaw), load-induced errors, dynamic effects, thermal deformations, fixturing errors, toolholder errors and cutter errors. Each has error contributors in each axis.

On any given machine, assembly or system, when users add up the tolerances allowed by individual parts in any one direction, they obtain a worst-case variation in that direction. In real life situations, all machine elements, for example, never reach the worst-case level simultaneously.

To obtain a more accurate estimate of how variations accumulate, most error budgeters use one of two approaches. The simplest is to multiply the arithmetic total by a factor—often 0.6. (Total error = 0.6 × arithmetic sum of individual factors.) The other approach is to take the square root of the sum of the squares of the standard deviations.

Table 1 provides data on a shop’s study of one operation. The shop’s goal was to assure no more than 0.001" mislocation for a particular feature.

Table 1. Error analysis of existing process.

As seen in the table, cutter deflection and related cutter issues and thermal effects are the largest contributors to total product error. Eliminating them, however, would not be sufficient to reduce mislocation to the desired 0.001".

Once the table of known error sources is completed, users construct target values for improvement (Table 2).

Table 2. Selected error budget targets to improve process

While the individual tolerances required to assure 0.001" accuracy in the final product are eye-opening, the point of the exercise is to identify areas for improvement.

Mislocation in the X-axis by 0.0001" and in the Y-axis by the same amount implies that the total distance from the center of the desired location is 0.000141" and, as noted above, experience indicates that multiple worst-case scenarios do not normally occur at the same time. Multiplying by a factor of 0.6 yields a more-realistic expected-worst-case mislocation of 0.000085", which is closer to the arithmetic value of 0.00001".

A major learning process begins once users start working with error budgeting. They begin a detailed understanding of the mechanics of what is happening—in microscopic detail. µ

About the author: Dr. LaRoux Gillespie is a retired manufacturing engineer and quality control manager with 40 years of experience in precision deburring and edge finishing. He is the author of 10 books on deburring and more than 200 technical reports and articles on precision machining. E-mail: laroux1@earthlink.net.