Type A uncertainty analysis validation of type B analysis for three-axis accelerometer calibrations

We demonstrate the calculation of a simple, robust type A [1] uncertainty estimate for the magnitude of acceleration measured by a triaxial accelerometer to validate a type B [2] uncertainty estimate for quasi-static calibration in the local gravitational field [3]. The uncertainty estimate is calculated by following a method [4] developed for in-use calibration of triaxial accelerometers in medical applications, such as monitoring the orientation or stability of patents at risk of fall.

The key concept of this method is that the magnitude g_loc of the acceleration sensed by a stationary triaxial accelerometer is equal to the magnitude of the gravitational field at the location of the calibration equipment. This is independent of the accelerometer orientation relative to the gravitational field vector. This method is useful only when triaxial accelerometers are calibrated by rotation in the gravitational field and only if an independent value for the magnitude of the local gravitational field is available.

Let g_loc be the magnitude of the local gravitational field at any specific location and g₂₃₃ = 9.8010(±0.0001) m s⁻² be g_loc in our laboratory, where the stated uncertainty is a very conservative type-B uncertainty estimate chosen to be consistent with the predictions of [5] for the GPS coordinates of our building and with estimates based on absolute gravity measurements at two other locations on our campus. Also let $\left(\hat{x},\hat{y},\hat{z}\right)$ be the axes of the local gravitational coordinate system (LGCS) in our laboratory. We refer to the axes of the LGCS as $k\in \left\{x,y,z\right\}$. The only general constraint on the LGCS is that $\hat{z}$ is parallel to the local gravitational field.

Two additional coordinate systems were required. The first of these $\left(\hat{X},\hat{Y},\hat{Z}\right)$ is defined by the two axes of rotation $\hat{X}$ and $\hat{Z}$ of our accelerometer-characterization system (ACS), which is a platform that can be mechanically rotated about two separate axes with high precision. The second is defined in terms of the axes $\left(\hat{u},\hat{v},\hat{w}\right)$ of maximum response of the three accelerometers that comprise a triaxial accelerometer under test. We refer to these accelerometers and their axes of maximum response as $j\in \left\{u,v,w\right\}$.

In general, $\left(\hat{u},\hat{v},\hat{w}\right)$ is not a perfectly orthogonal coordinate system and $\hat{X}$ and $\hat{Z}$ are not perfectly perpendicular. We define $\hat{x}$ to lie in the $\hat{X}-\hat{z}$ plane and set $\hat{Y}$ = $\hat{y}$, which fixes all three coordinate systems in our laboratory.

To illustrate the use of the uncertainty analysis in [4], we used the previously measured [3] response R_jne of a triaxial accelerometer. Three rotation experiments $e\in \left\{1,2,3\right\}$ were carried out to calibrate a high-accuracy accelerometer under test (DUT), as described in [3]. Each of these experiments comprised measurements of the response of each DUT accelerometer followed by a rotation step of 5° for $n\in \left\{1,\dots ,N=72\right\}$.

We used the analysis portion of the same protocol [3] to calculate the sensitivity matrix from

$$\begin{equation*}\begin{gathered}\left[\begin{matrix}\hfill {S}_{jx}\hfill & \hfill {S}_{jy}\hfill & \hfill {S}_{jz}\hfill \end{matrix}\right]=\left[\begin{matrix}\hfill \left({A}_{jx2}+{A}_{jx3}\right)\hfill & \hfill \left({A}_{jy1}+{B}_{jy3}\right)\hfill & \hfill \left({B}_{jz1}+{B}_{jz2}\right)\hfill \end{matrix}\right],\end{gathered}\end{equation*}$$

where A_jke , B_jke , and C_je , are the least-squares fitting parameters defined in equations (9)–(19) in references [3] in the notation used here and ${C}_{j}=\left({C}_{j1}+{C}_{j2}+{C}_{j3}\right)/3$. The result is

$$\begin{equation}{S}_{jk}=\left[\begin{matrix}\hfill +0.201\,358\hfill & \hfill -0.002\,416\hfill & \hfill -0.001\,109\hfill \\ \hfill -0.001\,085\hfill & \hfill +0.201\,606\hfill & \hfill +0.000\,745\hfill \\ \hfill +0.000\,059\hfill & \hfill -0.003\,382\hfill & \hfill +0.202\,450\hfill \end{matrix}\right]\,\frac{\mathrm{V}}{{g}_{\text{loc}}}\end{equation} \tag{ 1 }$$

$$\begin{equation} {C}_{j}=\left[\begin{matrix}\hfill -0.005\,146\hfill \\ \hfill +0.013\,8792\hfill \\ \hfill +0.002\,966\hfill \end{matrix}\right]\enspace \mathrm{V},\end{equation} \tag{ 2 }$$

where the measured response is approximated by

$$\begin{equation}{R}_{\mathit{jne}}=\sum\limits _{k}{S}_{jk}{a}_{\mathit{kne}}+{C}_{j},\end{equation} \tag{ 3 }$$

V is the symbol for the SI unit of electric potential difference (volt), and a_kne is the k component of the acceleration experienced by the j accelerometer at rotation step n in experiment e.

Next let

$$\begin{equation}{g}_{ne}=\sqrt{\sum\limits _{k\in \left\{x,y,z\right\}}{\left({g}_{\mathit{kne}}\right)}^{2}}\end{equation} \tag{ 4 }$$

be the magnitude of g_loc as calculated from the quadrature sum of the responses of the u, v, and w accelerometers at rotation step n of experiment e, where

$$\begin{equation}{g}_{\mathit{kne}}=\sum\limits _{j\in \left\{u,v,w\right\}}{T}_{kj}\left({R}_{\mathit{jne}}-{C}_{j}\right)\end{equation} \tag{ 5 }$$

is the calculated k component of g_ne in the LGCS, and

$$\begin{equation}{T}_{kj}=\left[\begin{matrix}\hfill +4.966\,58\hfill & \hfill +0.059\,97\hfill & \hfill +0.026\,98\hfill \\ \hfill +0.026\,73\hfill & \hfill +4.960\,17\hfill & \hfill -0.018\,12\hfill \\ \hfill -0.001\,01\hfill & \hfill +0.082\,84\hfill & \hfill +4.939\,19\hfill \end{matrix}\right]\frac{{g}_{\text{loc}}}{\mathrm{V}}\end{equation} \tag{ 6 }$$

is the inverse of the sensitivity matrix S_jk obtained from the least-squares fits of the linear three-parameter model (L3P model) to the response data R_jne [3].

The response data and multiples of the residuals of the fits that are convenient for comparison with the response data are shown in figures 1(a)–(c) for the u, v, and w accelerometers.

Zoom In Zoom Out Reset image size

Figure 1(d) shows the Fourier transform of the residuals of the fit to Δ R _wn1.

Reference [2] described a type B uncertainty estimate for the L3P protocol based on a Monte Carlo simulation of possible imperfections in our ACS owing to gimbal misalignment and rotational step-size errors. This reference shows that the L3P coefficients are sufficient to describe any perfectly stable, linear DUT within the greater of the noise of the response data and the precision of the computer arithmetic. The presence of anomalously large second and third harmonics in the Fourier transform of the residuals of the fit to the w accelerometer response in experiment 3 shown in figure 1(d) raises questions about the validity of the type B analysis described in [2], which was based on assumptions about what types and magnitudes of error might be present in the data. It is a major advantage of the type A uncertainty analysis of [4] when applied to calibration data derived from rotation in a gravitational field that it accounts for all errors present in the data independent of the nature and source of the errors.

2.1. Type A analysis of g _loc from calibration data

Each value of g_ne in equation (4) is an independent estimate of g_loc = g₂₃₃ obtained in our laboratory. Therefore, we define g_L3P as the mean value of all the g_ne data, in which case the standard deviation of the distribution of g_ne describes the uncertainty associated with any single value of g_ne . This result can be summarized as follows:

$$\begin{equation}\frac{{\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{A}}}{{g}_{\text{L}3\text{P}}}=\frac{0.0055\enspace {\mathrm{m}\enspace \mathrm{s}}^{-2}}{9.8010\enspace {\mathrm{m}\enspace \mathrm{s}}^{-2}}=0.056\%,\end{equation} \tag{ 7 }$$

where ${\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{A}}$ is the (coverage factor = 2) expanded type-A standard uncertainty to be applied to any single measurement of g_L3P owing to calibration error. The subscript 2 indicates a nominal coverage factor of 2, and the subscript A indicates a type-A uncertainty estimate.

The mean and standard uncertainties are both robust estimators for any single measurement in the three rotation experiments, because they are derived from every measurement in the sample. This result can be used to validate our previous type B uncertainty estimate of the triaxial acceleration measurement errors carried out using our ACS.

2.2. Type B analysis of T _kj from calibration data

The computer code in [2] was designed to calculate the type B uncertainty estimates of the errors in S_jk and C_j using a Monte Carlo simulation of potential ACS errors. The same code was modified to produce type B uncertainty estimates of the inverse sensitivity matrix ${T}_{kj}={\left({S}_{jk}\right)}^{-1}$ for the measured R_jne data described above.

Simulated errors in the rotation step size, alignment, and orientation of the ACS axes were drawn from a normal distribution with a mean of zero and a standard deviation of 0.008°. This value is the accuracy of the ACS provided by the equipment manufacturer as reported in [5]. The results are

$$\begin{equation}{\sigma }_{2}\left({T}_{kj}\right)=1{0}^{-4}\left[\begin{matrix}\hfill 5\hfill & \hfill 22\hfill & \hfill 29\hfill \\ \hfill 57\hfill & \hfill 6\hfill & \hfill 29\hfill \\ \hfill 29\hfill & \hfill 74\hfill & \hfill 8\hfill \end{matrix}\right]\enspace {\mathrm{m}\enspace \mathrm{s}}^{-2}\end{equation} \tag{ 8 }$$

$$\begin{equation} {\sigma }_{2}\left({C}_{j}\right)=1{0}^{-4}\left[\begin{matrix}\hfill 2\hfill \\ \hfill 19\hfill \\ \hfill 48\hfill \end{matrix}\right],\end{equation} \tag{ 9 }$$

and ${\sigma }_{2}\left({R}_{\mathit{jne}}\right)$, which is just the square of the residuals of the fit.

The type B uncertainty estimate of any single measurement of g_L3P was calculated using equations (8) and (9) using the law of uncertainty propagation [6] as follows:

$$\begin{equation}{\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{B}}=\sqrt{\sum\limits _{n,e}^{N,3}{\sigma }_{2}({g}_{ne})/(N\,\ast \,3)},\end{equation} \tag{ 10 }$$

where

$$\begin{equation*}{\left({\sigma }_{2}\left({g}_{ne}\right)\right)}^{2}=\end{equation*}$$

$$\begin{equation}\sum\limits _{k}\left({\left({\sigma }_{2}({T}_{kj})R{C}_{\mathit{jne}}\right)}^{2}+{\left({T}_{kj}{\sigma }_{2}\left({C}_{j}\right)\right)}^{2}+{\left({T}_{kj}{\sigma }_{2}\left({R}_{\mathit{jne}}\right)\right)}^{2}\right).\end{equation} \tag{ 11 }$$

The results of this analysis can be summarized as

$$\begin{equation}\frac{{\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{B}}}{{g}_{\text{L}3\text{P}}}=\frac{0.0089\enspace {\mathrm{m}\enspace \mathrm{s}}^{-2}}{9.8010\enspace {\mathrm{m}\enspace \mathrm{s}}^{-2}}=0.091\%,\end{equation} \tag{ 12 }$$

which is about 1.6 times greater than the standard deviation estimate ${\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{A}}$ reported in equation (7).

The type A uncertainty estimate reported in equation (12) is not a replacement for the type B estimate because the former only describes the uncertainty in the magnitude of the acceleration that can be calculated from T_kj and C_j and does not provide uncertainty estimates for these quantities or S_jk .

However, the level of agreement between ${\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{B}}$ and ${\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{A}}$ directly validates the type B uncertainty estimates for values of ${g}_{kne}\enspace \geqslant \enspace \sqrt{1/2}{g}_{\text{loc}}$ m s⁻² because ${\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{B}}$ is dominated by the largest component of g_kne and the L3P protocol guarantees that at least one g_kne $\geqslant \sqrt{1/2}{g}_{\text{loc}}$ for each combination of n and e.

The Monte Carlo simulation reported in [2] calculates the type B uncertainty estimate for all components of S_jk and C_j in exactly the same way. Therefore, the level of agreement between ${\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{B}}$ and ${\sigma }_{2}{\left({g}_{\text{L}3\text{P}}\right)}_{\mathrm{A}}$ also directly validates the Monte Carlo simulation and indirectly validates the uncertainty estimates of all the components of S_kj , T_kj , and C_j independent of their magnitude. Finally, both the type A and type B calibration-uncertainty analyses described here are applicable to all L3P-protocol calibrations of DUTs performed on our ACS, including low-frequency continuous rotation measurements [7] with a correction for centripetal acceleration as needed, as well as quasi-static measurements.