I. INTRODUCTION

Micro-electromechanical systems (MEMS) inertial sensors, such as accelerometers and gyroscopes, are used in a wide variety of applications including automotive and industrial applications. The demand for MEMS-based inertial sensors in consumer electronics has increased rapidly in the past few years due to their low power consumption, small size, and light weight ^(1-5).

MEMS gyroscopes are used to measure the angular rate. Apart from designing and fabrication, testing and/or evaluation of MEMS devices consumes a great deal of money. Several cross-sectional studies have shown that 1/3 of the product cost goes to test setup. To reach the market demand, high volume production at a lower cost is an issue for the MEMS industry ^(6-8).

The product development flow for MEMS gyroscope sensor is shown in Fig. 1. The development of the sensors starts with the design of mechanical structure, which is validated by finite element analysis (FEA) simulations. The FEA simulations help the designer to improve the mechanical structure for better performance, followed by fabrication and initial evaluations at wafer-level ^(9,10). Wafer-level automatic test equipment and parallel test architecture have been used to reduce the test time and increase the throughput as reported in ⁽¹¹⁾. After the wafer-level test, the fabricated devices are packaged, and reliability tests are conducted. Finally, the devices are evaluated for the functional test, and calibration is performed on each single device for appropriate usage.

The functional test includes the evaluation of the scale factor, bias errors, and noise performance ⁽¹²⁾. The bias stability and noise performance are evaluated at zero rate input in the case of gyroscope, whereas the scale factor is measured by applying external physical stimuli (angular rate) in the respective axis. To increase the accuracy in the scale factor measurement, the gyroscope sensors are evaluated at different angular rates where multi-point results are linearly fitted using linear regression.

For cost effectiveness, a uniaxial rate table is used to evaluate the MEMS gyroscope, which provides the external angular rate in single axis. To determine the rate output, the sensor is attached to a test fixture and the primary axis of the sensor and/or fixture is aligned to the rate table. Although this method is highly accurate, however, in case of multi-axes gyroscope evaluation, the sensor needs to be rotated three times for each single axis, which is time-consuming.

Fig. 1. MEMS sensors development and test flow.

In this manuscript, a 3-D test method is proposed for the fast evaluation of multi-axis MEMS gyroscope. The proposed 3-D method reduces the evaluation time by three times and eventually reducing the test cost. The article is arranged as follows. In section 2, a proper mathematical model for MEMS gyroscope evaluation using 1-D test method and the proposed 3-D test method are presented. The experimental results of the proposed 3-D method are compared with the 1-D test method in section 3. Also, the limitations of the proposed method are highlighted. Finally, the conclusions are given in section 4.

II. METHODOLOGY

To evaluate scale factor of multi-axis MEMS gyroscope, the output to external physical stimuli can be modeled as:

where $\mathrm{s}_{\mathrm{x}}$, $s_{\mathrm{y}}$,$~ s_{\mathrm{z}}$ are the scale factors, $u_{x,0},u_{y,0},u_{z,0}$ are the bias offset and $u_{x},u_{\mathrm{y}},\,\,\,\mathrm{and}\,\,\mathrm{u}_{\mathrm{z}}$ are the input physical stimuli (angular rate in case of gyroscope).

The scale factor of the gyroscope is evaluated by measuring the corresponding output to external angular rate. In the case of gyroscope evaluation, the gyroscope is attached to the test fixture and mounted to the rate table. We define the absolute coordinate system as $\left[0~ X~ Y~ Z~ \right]$ in which angular rate is applied, and the fixture coordinates as $~ \left[0~ X'~ Y'~ Z'\right]$, assuming that the gyroscope and test fixture has same coordinates. We also assume that both the fixture and absolute coordinates have the same origin at 0. The angular rate provided by the rate table in absolute coordinates can be related to the fixture coordinates by rotation matrix $R$ as:

where $U_{w}=\left[00F_{g}\right]^{T}$is a vector in absolute coordinate system and $F_{g}$ is angular rate along the z-axis of uniaxial rate table. Whereas, $U_{f}=\left[~ u_{X'}~ u_{Y'}~ u_{Z'}\right]^{T}$ is a vector in fixture coordinates and the rotation matrix $R$. The matrix $R$, having an unsigned angle $\theta $, between the two coordinates axis can be defined by direction cosine matrix as:

\begin{align*} R=~ \left[\begin{array}{lll} \cos \left(\theta _{X',X}\right) & \cos \left(\theta _{X',Y}\right) & \cos \left(\theta _{X',Z}\right)\\ \cos \left(\theta _{Y',X}\right) & \cos \left(\theta _{Y',Y}\right) & \cos \left(\theta _{Y',Z}\right)\\ \cos \left(\theta _{Z',X}\right) & \cos \left(\theta _{Z',Y}\right) & \cos \left(\theta _{Z',Z}\right) \end{array}\right] \end{align*}

where for example $\theta _{X',Z}$ be the angle between $X'- \text{axis}$ and $Z- axis.$

2. Proposed 3-D Test Method

The proposed 3-D test method is shown in Fig. 2(b). The rotation matrix $R$ defines the orientation of the fixture. Unlike the 1-D test method for which the fixture is rotated each time to evaluate every single axis of the multi-axis MEMS gyroscope, this paper exploits the Euler angles such that each sensor will experience the same angular rate and can be evaluated simultaneously. The rotation matrix can be defined by three plane rotations, involving rotation angles $(\theta ,\,\,\phi ,\,\,\psi )\,.$ The rotation matrix for sequence $R_{zyz}$ with Euler angles ($\theta ,\,\,\phi ,\,\,\psi )$ can be written as:

\begin{align*} R_{zyz}=\left[\begin{array}{lll} \boldsymbol{c}\boldsymbol{\theta }\boldsymbol{c}\boldsymbol{\phi }\boldsymbol{c}\boldsymbol{\psi }- s\theta s\psi & - c\psi s\theta - \boldsymbol{c}\boldsymbol{\theta }\boldsymbol{c}\boldsymbol{\phi }\boldsymbol{s}\boldsymbol{\psi } & c\theta s\phi \\ c\theta s\psi +\boldsymbol{c}\boldsymbol{\phi }\boldsymbol{c}\boldsymbol{\psi }\boldsymbol{s}\boldsymbol{\theta } & c\theta c\psi - \boldsymbol{c}\boldsymbol{\phi }\boldsymbol{s}\boldsymbol{\theta }\boldsymbol{s}\boldsymbol{\psi } & s\theta s\phi \\ - c\psi s\phi & s\phi s\psi & c\phi \end{array}\right] \end{align*}

where c and s represents cosine and sine functions respectively.

Using Euler angle rotation matrix sequence $R_{zyz}$, we re-define Eq. (4) as:

(8)

\begin{align} U_{f}=\left[\begin{array}{l} u_{X'}\\ u_{Y'}\\ u_{Z'} \end{array}\right]=\left[\begin{array}{l} \cos \left(\theta \right).\sin \left(\phi \right)\\ \sin \left(\theta \right).\sin \left(\phi \right)\\ \cos \left(\phi \right) \end{array}\right]F_{g}. \end{align}

where $\theta $ and $\phi $ represent the angle between fixture and absolute coordinates. As each axis will experience the same force in the 3-D test method, we solve Eq. (8) for “$\theta $” and “$\phi $” for all $u_{X'}=~ u_{Y'}=~ u_{Z'}$.

$\theta =\tan ^{- 1}\left(\frac{u_{X'}}{u_{Y'}}\right)$.

$\theta =45^{\circ}$.

$\phi =\tan ^{- 1}\left(\frac{u_{X'}\cos \left(\theta \right)+u_{Y'}\sin \left(\theta \right)}{u_{Z'}}\right)$.

$\phi =54.74^{\circ}$.

Putting back the values of “$\theta '$ and “$\phi $” in equation 10 will yield:

(9)

\begin{align} U_{f}=\left[\begin{array}{l} u_{X'}\\ u_{Y'}\\ u_{Z'} \end{array}\right]=\left[\begin{array}{l} \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}} \end{array}\right]\left[\begin{array}{l} 0\\ 0\\ F_{g} \end{array}\right]~ . \end{align}

Based on the above discussion, it is clear that if the designed test fixture with the derived angles, each axis of the gyroscope will experience equal angular rate. Finally, the output of the gyroscopes sensor can be re-written as:

(10)

$\mathrm{u}_{\mathrm{s},\mathrm{x}}=\mathrm{s}_{\mathrm{xx}}.\frac{F_{g}}{\sqrt 3}$.

(11)

\begin{equation} \mathrm{u}_{\mathrm{s},\mathrm{y}}=\mathrm{s}_{y}.\frac{F_{g}}{\sqrt 3}.~ \end{equation}

(12)

$\mathrm{u}_{\mathrm{s},\mathrm{z}}=\mathrm{s}_{z}.\frac{F_{g}}{\sqrt 3}$.

where $F_{g}$ is the external angular rate experienced by the sensors

Fig. 2. (a) 1-D test method, (b) 3-D test method.

III. EXPERIMENTAL RESULTS AND DISCUSSION

Experimental setup for the 3-D test method is shown in Fig. 3. The sensor was mounted on the designed 3-D fixture and attached to the rate table AC1120. The experiment was performed on ST Microelectronic device, LSM9DS0 sensor, having 3-axis gyroscope, 3-axis accelerometer, and 3-axis magnetometers ⁽¹³⁾. The LSM9DS0 was evaluated using STEVAL-MKI109V2 daughter board. The block diagram is shown in Fig. 4. The device was evaluated at different input rates in the range of $\pm 200$ degrees per second (dps) and linearly fit using linear regression to determine the scale factor of the device.

Fig. 5 and 6 show the experimental results of 1-D and the proposed 3-D test method respectively. The measured scale factor (SF) for $x,y$, and $z~ $gyroscope using 1-D test setup was $109.98LSB/dps,$ $111.38LSB/dps,$ and $114.28LSB/dps,$ respectively, whereas the 3-D test results show a scale factor of $110.99LSB/dps,$ $112.13LSB/dps,$ and $115.58LSB/dps$ for $x,~ y,$ and $z$ gyroscope respectively. The percentage error between 1-D and 3-D test method was measured and is summarized in Table 1.

(13)

Fig. 3. Experimental setup of 3-D test method for gyroscope evaluation using uniaxial rate table (AC1120).

Fig. 4. Block diagram of LSM9SO and MKI109 motherboard.

The experimental results reveal that the errors were less than $1% $ for the $x$ and $y$ gyroscopes; however, the z gyroscope correspond to an error of 1.15%. The errors in the measurement results are either the coupling of the cross-axis sensitivity to the primary axis or errors in the 3-D test fixture designs. Further, experiments and analysis were carried out to highlight the error sources in the proposed 3-D test method.

Fig. 5. Measured scale factor of gyroscope using 1-D test method (a) x-gyroscope, (b) y-gyroscope, (c) z-gyroscope.

Fig. 6. Measured scale factor of gyroscope using 3-D test method: (a) x-gyroscope, (b) y-gyroscope, (c) z-gyroscope.

1. Effect of Cross Axis Sensitivity

First, the cross-axis sensitivity of the gyroscope was measured using 1-D test setup, and its effect on the 3-D test setup was further analyzed. The main sources of the cross-axis coupling are the test fixture misalignment with the absolute coordinates, MEMS itself, and die placement errors ⁽¹⁴⁾. The cross-axis for gyroscope was calculated as:

(13)

$$ S_{i, j / k}(\%)=\frac{\sqrt{S_{i j}^{2}+S_{i k}^{2}}}{S_{i i}} \times 100 $$

where the first subscript represents the primary axis in which the angular rate was measured, and the second subscript represents the axis in which the external angular rate was applied. The measured cross axis sensitivity for $x,~ y,$ and $z$ gyroscope was $1.75% ,1.67% ,$ and $0.37% $ respectively. Next, the effect of cross-axis on the primary axis was measured as:

(14)

$S_{\textit{overll}}=\sqrt{S_{ii}^{2}+S_{ij}^{2}+S_{ik}^{2}}$.

where $S_{\textit{overall}}$ is the overall sensitivity, $S_{ii}$, $S_{ij}$, and $S_{ik}~ $represent the primary axis sensitivity and the cross-axis sensitivity terms. Table 2 summarized the measurement errors along with the cross-axis sensitivity. The results show that the cross-axis sensitivity has an effect of only 0.02%. Also, with the improvement in the MEMS fabrication and design technology, most of the devices have a cross-axis sensitivity less than 2% ^{(1, 15, 16)} which means that the 3-D measurement test results will affect the scale factor with an error of 0.02%.

Table 2. Measured scale factor using 1-D and 3-D test with cross-axis sensitivity

Test Method	Scale factor (LSB/dps)			Percentage (%) Error
	x-gyro	y-gyro	z-gyro	x-gyro	y-gyro	z-gyro
1-D Test	109.990	111.140	114.281	089%	0.65%	1.13%
3-D Test	110.99	112.14	115.59

Fig. 7. Induced scale factor error with test fixture angles.

V. CONCLUSION

In this paper, a study on 3-D test method was conducted to evaluate the scale factor of multi-axis MEMS gyroscope. A mathematical model for the proposed 3-D test model was established. The experimental results were compared with the conventional 1-D test method showing that the measurement errors were less than 1%. Although the 1-D test method is highly accurate, however, in case of multi-axis gyroscope evaluation, the method is not efficient in terms of time. By exploiting the 3-D test method, the throughput of the test system can be increased three times.

Furthermore, the limitation of the proposed 3-D test method was highlighted. One source of the weakness in this method was the cross-axis coupling of the secondary axis to the primary axis, which contributed only 0.02% to measurement errors. Moreover, it was found that the main source of measurement errors was fixture design and the corresponding angles. Ensuring the reduction in fixture angle errors will further improve the measurement results.