Study on Misalignment Angle Compensation during Scale Factor Matching for Two Pairs of Accelerometers in a Gravity Gradient Instrument

A method for automatic compensation of misalignment angles during matching the scale factors of two pairs of the accelerometers in developing the rotating accelerometer gravity gradient instrument (GGI) is proposed and demonstrated in this paper. The purpose of automatic scale factor matching of the four accelerometers in GGI is to suppress the common mode acceleration of the moving-based platforms. However, taking the full model equation of the accelerometer into consideration, the other two orthogonal axes which is the pendulous axis and the output axis, will also sense the common mode acceleration and reduce the suppression performance. The coefficients from the two axes to the output are δO and δP respectively, called the misalignment angles. The angle δO, coupling with the acceleration along the pendulous axis perpendicular to the rotational plane, will not be modulated by the rotation and gives little contribution to the scale factors matching. On the other hand, because of coupling with the acceleration along the centripetal direction in the rotating plane, the angle δP would produce a component with 90 degrees phase delay relative to the scale factor component. Hence, the δP component coincides exactly with the sensitive direction of the orthogonal accelerometers. To improve the common mode acceleration rejection, the misalignment angle δP is compensated by injecting a trimming current, which is proportional to the output of an orthogonal accelerometer, into the torque coil of the accelerometer during the scale factor matching. The experimental results show that the common linear acceleration suppression achieved three orders after the scale factors balance and five orders after the misalignment angles compensation, which is almost down to the noise level of the used accelerometers of 1~2 × 10−7 g/√Hz (1 g ≈ 9.8 m/s2).


Introduction
Over the past two decades, there has been a continuing and growing interest in the ability of moving-based gravimeters and gravity gradiometers, and this attention is not only focused on the remarkable achievements that has been made, but also on the development of new methods and technologies in an attempt to find even better alternatives [1][2][3]. The conventional gravimeters record the intensity of results show that the common linear acceleration is suppressed by three orders after scale factor matching and five orders after misalignment angle compensation, which almost reached the noise level of the used accelerometers of 1~2 × 10 −7 g/ √ Hz.

The noise effect of the misalignment angle
The schematic diagram of the rotating accelerometer GGI is shown in Figure 1. The GGI consists of two pairs of accelerometers. Each pair of accelerometers are set diametrically opposite to each other to suppress the in-plane common linear acceleration and to double the gradient signal. The two pairs of accelerometers are set mutually orthogonal to reject angular acceleration about the spin axis and double the gradient signal once again [9][10][11]20]. Here we report an alternative method to compensate the misalignment angles automatically during matching the scale factors of two pairs of the accelerometers in GGI. Compared with the method presented in [21], no extra sensors are needed during the compensation process. The experimental results show that the common linear acceleration is suppressed by three orders after scale factor matching and five orders after misalignment angle compensation, which almost reached the noise level of the used accelerometers of 1~2 × 10 −7 g/√Hz.

The noise effect of the misalignment angle
The schematic diagram of the rotating accelerometer GGI is shown in Figure 1. The GGI consists of two pairs of accelerometers. Each pair of accelerometers are set diametrically opposite to each other to suppress the in-plane common linear acceleration and to double the gradient signal. The two pairs of accelerometers are set mutually orthogonal to reject angular acceleration about the spin axis and double the gradient signal once again [9][10][11]20]. The output of each accelerometer is shown in Equation (1): where ai, KIi, φi are the output, scale factor and position phase of the ith accelerometer respectively. The position phase of each accelerometer is φ1 = 0, φ2 = π, φ3 = π/2 and φ4 = 3π/2, respectively. acom is the in-plane common linear acceleration, R is the distance from the proof mass (PM) of the accelerometer to the center of the rotation and ΓXX − ΓYY and ΓXY are the gradient signals. The summed output of the four accelerometers is shown in Equation (2) The gradient signals ΓXX − ΓYY and ΓXY are modulated to the double spin frequency of the rotation, while the kinematic acceleration is modulated to the single spin frequency. Thus, the gradient signals and the noise are separated in the frequency domain. However, once the common linear acceleration contains a single spin frequency component, a disturbance to the double spin frequency component is unavoidable. Only when the scale factors of a pair of accelerometers are precisely matched (KI1 = KI2, KI3 = KI4), the noise effect of acom can be eliminated.
In our previous research, we have developed a scale factor adjustment method by trimming the feedback current of a magnetic force feedback accelerometer [22]. The PM is suspended with a thin cantilever spring, and its movement is sensed by the capacitive displacement transducer. A digital The output of each accelerometer is shown in Equation (1): where a i , K Ii , ϕ i are the output, scale factor and position phase of the ith accelerometer respectively. The position phase of each accelerometer is ϕ 1 = 0, ϕ 2 = π, ϕ 3 = π/2 and ϕ 4 = 3π/2, respectively. a com is the in-plane common linear acceleration, R is the distance from the proof mass (PM) of the accelerometer to the center of the rotation and Γ XX − Γ YY and Γ XY are the gradient signals. The summed output of the four accelerometers is shown in Equation (2): The gradient signals Γ XX − Γ YY and Γ XY are modulated to the double spin frequency of the rotation, while the kinematic acceleration is modulated to the single spin frequency. Thus, the gradient signals and the noise are separated in the frequency domain. However, once the common linear acceleration contains a single spin frequency component, a disturbance to the double spin frequency component is unavoidable. Only when the scale factors of a pair of accelerometers are precisely matched (K I1 = K I2 , K I3 = K I4 ), the noise effect of a com can be eliminated.
In our previous research, we have developed a scale factor adjustment method by trimming the feedback current of a magnetic force feedback accelerometer [22]. The PM is suspended with a thin cantilever spring, and its movement is sensed by the capacitive displacement transducer. A digital proportional integral differential (PID) controller is used to generate a current I f flowing through the torque coil mounted on the PM, which provides an equal force with an opposite direction to compensate the movement of the PM. In order to adjust the scale factor of the accelerometer, a trimming current I t , which is proportional to the feedback current by It = p × If, was injected into the torque coil. In this case, the scale factor of the accelerometer becomes K I = K I0 /(1 + p), where K I0 is the original scale factor of the accelerometer. The adjustment method has been proven to be effective and easy to implement. The measurement range of the accelerometer is about 30 mg.
Taking the pair of accelerometers a 1 and a 2 as example, the schematic diagram of the scale factor balance loop is depicted in Figure 2. The scale factor mismatch information is extracted by demodulating the summed output a 1 + a 2 with sinωt. Subsequently, the PID controller can obtain a proper coefficient p for the scale factor balance. The output of a 2 is multiplied by the coefficient p and injected into the torque coil, hence the scale factor is balanced [22]. proportional integral differential (PID) controller is used to generate a current If flowing through the torque coil mounted on the PM, which provides an equal force with an opposite direction to compensate the movement of the PM. In order to adjust the scale factor of the accelerometer, a trimming current It, which is proportional to the feedback current by It = p × If, was injected into the torque coil. In this case, the scale factor of the accelerometer becomes KI = KI0/(1 + p), where KI0 is the original scale factor of the accelerometer. The adjustment method has been proven to be effective and easy to implement. The measurement range of the accelerometer is about 30 mg. Taking the pair of accelerometers a1 and a2 as example, the schematic diagram of the scale factor balance loop is depicted in Figure 2. The scale factor mismatch information is extracted by demodulating the summed output a1 + a2 with sinωt. Subsequently, the PID controller can obtain a proper coefficient p for the scale factor balance. The output of a2 is multiplied by the coefficient p and injected into the torque coil, hence the scale factor is balanced [22]. However, after the balance has been achieved an orthogonal signal at the spin frequency is still observed, which results in a residual sensitivity to the common linear acceleration. For further analysis Equation (3) where aout is the indicated acceleration, Iout is the output current of the accelerometer, KI is the scale factor, and K0 is the bias. aI, aO and aP are the applied acceleration components along the true input (sensitive) axis IA, output axis OA, and pendulous axis PA, respectively. The three axes are mutually orthogonal. δP and δO are the misalignment angles of the input axis with respect to the input reference axis about the pendulous axis and the output axis, respectively. KII, KOO and KPP are second order coefficients. KIO, KIP and KOP are the cross-coupling coefficients. All these coefficients can be measured with a centrifuge [23]. The second order coefficients and cross-coupling coefficients, which will generate higher order harmonic signals, have been discussed in our previous work [18]. By taking the first order coefficients into consideration, the output of the four accelerometers can be expressed as: where Ii, KIi, δPi, δOi, φi (I = 1, 2, 3, 4) are the output, scale factor, two misalignment angles and position phase of the ith accelerometer, respectively. The position phases of the four accelerometers are φ1 = 0, φ2 = π, φ3 = π/2 and φ4 = 3π/2. ω is the spin frequency. The in-plane common acceleration aCOM is modulated to the spin frequency of the disk. aV is the vertical acceleration perpendicular to the plane, However, after the balance has been achieved an orthogonal signal at the spin frequency is still observed, which results in a residual sensitivity to the common linear acceleration. For further analysis Equation (3) shows the complete model equation of an accelerometer [19]: where a out is the indicated acceleration, I out is the output current of the accelerometer, K I is the scale factor, and K 0 is the bias. a I , a O and a P are the applied acceleration components along the true input (sensitive) axis IA, output axis OA, and pendulous axis PA, respectively. The three axes are mutually orthogonal. δ P and δ O are the misalignment angles of the input axis with respect to the input reference axis about the pendulous axis and the output axis, respectively. K II , K OO and K PP are second order coefficients. K IO , K IP and K OP are the cross-coupling coefficients. All these coefficients can be measured with a centrifuge [23]. The second order coefficients and cross-coupling coefficients, which will generate higher order harmonic signals, have been discussed in our previous work [18]. By taking the first order coefficients into consideration, the output of the four accelerometers can be expressed as: where I i , K Ii , δ Pi , δ Oi , ϕ i (I = 1, 2, 3, 4) are the output, scale factor, two misalignment angles and position phase of the ith accelerometer, respectively. The position phases of the four accelerometers are ϕ 1 = 0, ϕ 2 = π, ϕ 3 = π/2 and ϕ 4 = 3π/2. ω is the spin frequency. The in-plane common acceleration a COM is modulated to the spin frequency of the disk. a V is the vertical acceleration perpendicular to the plane, hence is not modulated by the rotation and contributes little to the scale factor balance. However, once the common acceleration a COM contains a single frequency component, such as a COM = A s sinωt + A c cosωt, a noise component contributing to the double frequency where the gradient signal stays is unavoidable given by: Using Equation (5), we find the misalignment angle δ P would contribute a noise to the gradient signal extraction the same as scale factor mismatch. Therefore, the misalignment angle also needs compensation during the scale factor balance to suppress the common linear acceleration. In an actual survey environment, the scale factor and misalignment angle may change with temperature, material aging, structure creep, and stress release, etc. Thus, the scale factor balance and misalignment angle compensation are required to be online and automatic.

The principle of the misalignment angle compensation
As shown in Equation (4), the scale factor mismatch information of a 1 and a 2 is modulated by sinωt while the misalignment angle information is modulated by an orthogonal signal cosωt. Therefore, the scale factor mismatch and misalignment angle information can be distinguished by demodulation with sinωt and cosωt, respectively. As shown in Equation (3), the misalignment angle component of the summed output a 1 + a 2 is in the same phase with the scale factor component of an orthogonal accelerometer a 4 . Inspired by the scale factor adjustment method, we figure out that the misalignment angle can be compensated by trimming the feedback current with the output of orthogonal accelerometer.
Using Equation (5), we find the misalignment angle δP would contribute a noise to the gradient signal extraction the same as scale factor mismatch. Therefore, the misalignment angle also needs compensation during the scale factor balance to suppress the common linear acceleration. In an actual survey environment, the scale factor and misalignment angle may change with temperature, material aging, structure creep, and stress release, etc. Thus, the scale factor balance and misalignment angle compensation are required to be online and automatic.

The principle of the misalignment angle compensation
As shown in Equation (4), the scale factor mismatch information of a1 and a2 is modulated by sinωt while the misalignment angle information is modulated by an orthogonal signal cosωt. Therefore, the scale factor mismatch and misalignment angle information can be distinguished by demodulation with sinωt and cosωt, respectively. As shown in Equation (3), the misalignment angle component of the summed output a1 + a2 is in the same phase with the scale factor component of an orthogonal accelerometer a4. Inspired by the scale factor adjustment method, we figure out that the misalignment angle can be compensated by trimming the feedback current with the output of orthogonal accelerometer. For example, the misalignment angle of the pair of accelerometers (a1 and a2) can be compensated by injecting the output of an orthogonal accelerometer (a4) into the torque coil of accelerometer (a2). The active feedback loops for matching the scale factors and compensating the misalignment angles are schematically depicted in Figure 3.
After demodulation a group of proper coefficients p and q are calculated out by two PID-controllers, respectively. The output of the accelerometer a2 is multiplied by p and a4 is For example, the misalignment angle of the pair of accelerometers (a 1 and a 2 ) can be compensated by injecting the output of an orthogonal accelerometer (a 4 ) into the torque coil of accelerometer (a 2 ). The active feedback loops for matching the scale factors and compensating the misalignment angles are schematically depicted in Figure 3.
After demodulation a group of proper coefficients p and q are calculated out by two PID-controllers, respectively. The output of the accelerometer a 2 is multiplied by p and a 4 is multiplied by q, and then are injected into the torque coil of a 2 for scale factor balance and misalignment angle compensation. The summed output after the balance and compensation is given as: Sensors 2018, 18, 1247 6 of 9 As Equation (6) shows, the summed output after balance and compensation consists of two components. One is the scale factor balance component (sinωt), and the other is the misalignment angle compensation component (cosωt). In our previous experiment, only scale factor balance is taken into consideration with q = 0 in Equation (6), hence an orthogonal component coupled with the misalignment angles cannot be eliminated. A residual sensitivity to the common linear acceleration is therefore unavoidable. To suppress the common linear acceleration, the magnitudes of these two components should be adjusted to zero. Normally the misalignment angle of the accelerometer is of the order of 10 −2~1 0 −3 rad, which contributes much little than the scale factor. Thus, from Equation (6), we can find out that the coefficient p is the main contributor to the scale factor balance, while coefficient q contributes the most to the misalignment angle compensation. The scale factor balance coefficient p and misalignment angle compensation coefficient q can be solved using the following equation: which verified the feasibility of the balance and compensation method. The misalignment angles of the other pair of accelerometers a 3 and a 4 can be compensated by similar method. The misalignment information can be extracted by demodulating with cosωt and compensated with an orthogonal accelerometer a 1 or a 2 .

Experimental Results
The experimental setup is shown in Figure 4. The GGI is mounted on a commercial rotary stage and the rotating frequency is set to 0.125 Hz. The base line from the PM to the center of the rotary disc is about 135 mm. To reveal the mismatch of the scale factors of a pair of accelerometers, the spin axis is purposely inclined from the vertical axis with an angle of 1.5 mrad. The reference demodulation signals sinωt and cosωt are directly read out from the grating encoder integrated in the rotary stage online. The balance and compensation circuit consists of 20-bit analog-to-digital converters, a field programmable gate array (FPGA) and 20-bit digital-to-analog converters. All the control algorithms are implemented in the FPGA. Therefore, no extra compensation circuit added is needed for the misalignment angles compensation loops.  After the disk is working stably, the scale factor balance loop waits for a sufficient period of time until the mismatch information of a pair of accelerometers is filtered out. The misalignment angle compensation loop then starts to work after the scale factor balance has been achieved. In fact, the balance loops and the compensation loops work continuously and accurately. The outputs of a pair of accelerometers a1, a2 and the summed output a1 + a2 is shown in Figure 5. At the beginning, the magnitudes of a1 and a2 are different due to the mismatch of the scale factor. The summed output a1 + a2 indicates an obvious periodic signal at the spin frequency. The outputs of a1 and a2 eventually become equal in amplitude and the spin frequency component in the summed output hence disappears, indicating that the balance and compensation have been completed. After the disk is working stably, the scale factor balance loop waits for a sufficient period of time until the mismatch information of a pair of accelerometers is filtered out. The misalignment angle compensation loop then starts to work after the scale factor balance has been achieved. In fact, the balance loops and the compensation loops work continuously and accurately. The outputs of a pair of accelerometers a 1 , a 2 and the summed output a 1 + a 2 is shown in Figure 5. At the beginning, the magnitudes of a 1 and a 2 are different due to the mismatch of the scale factor. The summed output a 1 + a 2 indicates an obvious periodic signal at the spin frequency. The outputs of a 1 and a 2 eventually become equal in amplitude and the spin frequency component in the summed output hence disappears, indicating that the balance and compensation have been completed. After the disk is working stably, the scale factor balance loop waits for a sufficient period of time until the mismatch information of a pair of accelerometers is filtered out. The misalignment angle compensation loop then starts to work after the scale factor balance has been achieved. In fact, the balance loops and the compensation loops work continuously and accurately. The outputs of a pair of accelerometers a1, a2 and the summed output a1 + a2 is shown in Figure 5. At the beginning, the magnitudes of a1 and a2 are different due to the mismatch of the scale factor. The summed output a1 + a2 indicates an obvious periodic signal at the spin frequency. The outputs of a1 and a2 eventually become equal in amplitude and the spin frequency component in the summed output hence disappears, indicating that the balance and compensation have been completed. To clearly figure out the effect of the scale factor balance and misalignment angle compensation, the sinωt and cosωt components of the outputs and the summed output of the pair of accelerometers are obtained by least square fitting, and are listed in Table 1. To clearly figure out the effect of the scale factor balance and misalignment angle compensation, the sinωt and cosωt components of the outputs and the summed output of the pair of accelerometers are obtained by least square fitting, and are listed in Table 1. Table 1. The sinωt and cosωt components of a 1 , a 2 and a 1 + a 2 before matching, after scale factor balance and after misalignment angle compensation. Before matching, the magnitude of the sinωt and cosωt components of the two accelerometers are different. After scale factor balance, the magnitude of the sinωt component of the two accelerometers are equal to each other. Finally, after misalignment angle compensation, the magnitudes of the cosωt component of the two accelerometers are also identical to each other. Consequently, the spin frequency in the summed output disappears.
In the frequency domain, the acceleration power spectral density (PSD) curves of a 1 , a 2 and a 1 + a 2 after scale factor balance and after misalignment angle compensation are shown in Figure 6.
The PSD curves of accelerometers a 1 and a 2 are shown in Figure 6a before the scale factor balance and misalignment angle compensation. The noise levels of the used accelerometers are about 1~2 × 10 −7 g/ √ Hz. The spin frequency component of the combined output of a 1 and a 2 reveals the residual sensitivities to common linear acceleration. The imperfect movement of the rotary stage, such as additional angular acceleration, radially twisting movement and the brush of the motor also introduce higher harmonic responding components. The PSD curves of combined output are shown in Figure 6b (green) after scale factor balance. The common linear acceleration achieves three orders of suppression. After misalignment angle compensation the common linear acceleration finally achieves five orders of suppression and reaches the noise level of the accelerometers (Figure 6b, red). The PSD Before matching, the magnitude of the sinωt and cosωt components of the two accelerometers are different. After scale factor balance, the magnitude of the sinωt component of the two accelerometers are equal to each other. Finally, after misalignment angle compensation, the magnitudes of the cosωt component of the two accelerometers are also identical to each other. Consequently, the spin frequency in the summed output disappears.
In the frequency domain, the acceleration power spectral density (PSD) curves of a1, a2 and a1 + a2 after scale factor balance and after misalignment angle compensation are shown in Figure 6. The PSD curves of accelerometers a1 and a2 are shown in Figure 6a before the scale factor balance and misalignment angle compensation. The noise levels of the used accelerometers are about 1~2 × 10 −7 g/√Hz. The spin frequency component of the combined output of a1 and a2 reveals the residual sensitivities to common linear acceleration. The imperfect movement of the rotary stage, such as additional angular acceleration, radially twisting movement and the brush of the motor also introduce higher harmonic responding components. The PSD curves of combined output are shown in Figure 6b (green) after scale factor balance. The common linear acceleration achieves three orders of suppression. After misalignment angle compensation the common linear acceleration finally achieves five orders of suppression and reaches the noise level of the accelerometers (Figure 6b, red). The PSD results further validate the effectiveness of the misalignment compensation method during the scale factor balance. The other pair of accelerometers a3 and a4 performed almost the same.

Conclusions and Prospects
This paper presents the influence of the misalignment angle and the alternative online compensation method for the in-plane common acceleration suppression in the GGI with the output of an orthogonal accelerometer. Analysis shows that the misalignment angle can lead to errors in the gradient signal extraction as well as the scale factor mismatch. Thus, the misalignment angle compensation technology is necessary. Using the same circuit of scale factor adjustment, we develop a method to compensate the misalignment angle by injecting the output of an orthogonal accelerometer into the torque coil. Experimental results show that the first order common linear acceleration has been suppressed by five orders, reaching the noise level of the used accelerometers. No additional sensors or circuit are needed for the misalignment angle compensation and that makes it simple for implementation.
However, to extract a meaningful gradient from GGI, a number of technological improvements are still needed. First of all, the major limitation of suppression is the noise level of the used accelerometer. Besides, each accelerometer unavoidably includes higher order nonlinear and cross-coupling components because of the imperfection of the material and mismatch of assembling. Additionally, non-ideal movement of the rotary stage will induce many harmonic components. All these harmonic components may degrade the quality of the gravity gradient signals.
To further lower the noise level of the accelerometer is a challenging but imminent objective, and the good news is we have some options [24]. Precision control of the rotary stage is also taken into consideration. In addition, the technique to reduce or null the even order and cross-coupling error coefficients is being developed to further improve the sensitivity of the GGI.