Easier uncertainty estimation for specific model functions

Common procedures for the determination of the uncertainty of measurement results usually use a model function by means of which the contributions to the total measurement uncertainty are calculated. For this purpose, the partial derivatives of this mathematical function are formed which indicate - as so-called "sensitivity coefficients" - how strongly the individual contribution enters into the overall budget. Now there are cases where there are individual contributions of the related quantity although these derivatives are zero.

Examples of such functions are the trigonometric functions of sine and cosine. Amongst others, they occur when measuring torques with lever-mass systems: an inclination of the lever from the horizontal position leads to a reduction of the effective lever arm length which can be described with a cosine function. The derivative of this function yields - neglecting the algebraic sign - a sine function whose value is, at an angle of 0 rad, however, equal to zero so that the contribution of the uncertainty of the angle measurement to the total measurement uncertainty would be zero - which, however can obviously not be the case.

Now, in the relevant publications it is pointed out that - in such a case - either the non-linearity of the function (here: the cosine) has to be taken into consideration or Monte Carlo calculations have to be carried out. Both possibilities are quite complex and time-consuming. For that reason, a simple procedure has been found which provides a conservative estimation of the uncertainty contribution of the angle measurement to the uncertainty of the overall result. The principle of this is explained in Figure 1. The uncertainty of the input quantity x (here: the result of the measurement of the angle of inclination) is referred to as u_x. Now, one determines the functional value of the derivative of the cosine function - i.e., in this case, the value of the function - sin(x) at the position x + u_x, and uses it as a sensitivity coefficient. With an angle of 0 rad as expectation value, -sin(u_x) is yielded. Due to the fact that measurement uncertainties are summed up geometrically (root of the sum of the squares), a negative sign does not play any role. The squared contribution then has the form sin²(u_x) · (u_x)². As can also be seen from the figure, the thus estimated contribution is always larger than the value of the cosine function at the position x + u_x. The deviation for small angles, which are under consideration here, is - however - very small.

In the case of the torque measurement, the additional relative measurement uncertainty contribution was - due to the very precise inclination measurement - determined to be 1.6 · 10^-6, which is sufficiently small compared to the measurement uncertainty of the measurement device. However, with this new procedure, it is possible to estimate uncertainties also in the case of more inaccurate angle measurements.

Figure 1: Simplified measurement uncertainty estimation for a cosine function

References:

[1] Röske, D.: Uncertainty Contribution In The Case Of Cosine Function With Zero Estimate – A Proposal. IMEKO 2010: TC3, TC5 and TC22 Conferences, Pattaya, 21-25, November, 2010, Download:
http://www.imeko.org/publications/tc3-2010/IMEKO-TC3-2010-028.pdf

Contact person:

D. Röske, Dept. 1.2, WG 1.22, e-mail: dirk.roeske@ptb.de