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Technical Briefs

# Efficient Propagation of Error Through System Models for Functions Common in Engineering

[+] Author and Article Information
Travis V. Anderson

Design Exploration Research Grouptravisanderson@byu.edu

Christopher A. Mattson1

Design Exploration Research Groupmattson@byu.edu

Department of Mechanical Engineering,  Brigham Young University, Provo, Utah 84602brad.larson@byu.edu

David T. Fullwood

Department of Mechanical Engineering,  Brigham Young University, Provo, Utah 84602dfullwood@byu.edu

1

Corresponding author.

J. Mech. Des 134(1), 014501 (Jan 04, 2012) (6 pages) doi:10.1115/1.4005444 History: Received March 02, 2011; Revised October 19, 2011; Published January 04, 2012; Online January 04, 2012

## Abstract

System modeling can help designers make and verify design decisions early in the design process if the model’s accuracy can be determined. The formula typically used to analytically propagate error is based on a first-order Taylor series expansion. Consequently, this formula can be wrong by one or more orders of magnitude for nonlinear systems. Clearly, adding higher-order terms increases the accuracy of the approximation but it also requires higher computational cost. This paper shows that truncation error can be reduced and accuracy increased without additional computational cost by applying a predictable correction factor to lower-order approximations. The efficiency of this method is demonstrated in the kinematic model of a flapping wing. While Taylor series error propagation is typically applicable only to closed-form equations, the procedure followed in this paper may be used with other types of models, provided that model outputs can be determined from model inputs, derivatives can be calculated, and truncation error is predictable.

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Copyright © 2012 by American Society of Mechanical Engineers
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## Figures

Figure 1

Relative error in output variance using a first-order Taylor series expansion for the function y = 1000sin(x)

Figure 2

Relative error in variance propagation using Taylor series approximations

Figure 3

Relationship between the second-order bias and the input standard variance σx2 for a sin function

Figure 4

Relative error in estimations of variance propagation

Figure 5

Mechanism used by the BYU Flapping Flight Research Team to simulate 3-degree-of-freedom motion of a flapping wing

Figure 6

Computational time to predict output distributions using various error propagation methods

Figure 7

Relative error in predictions of output variance obtained from various orders of a Taylor series

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