Study of the orientation angles stability of the measuring axes of accelerometers in an inertial measuring unit
Errors of the orientation matrix Ca of the measuring axes of the accelerometers installed in the inertial measuring unit (IMU) largely determine the accuracy of the inertial navigation of mobile objects [1, 2]. Since it is very difficult to make the position of the measuring axes of accelerometers absolutely orthogonal, the matrix of the direction cosines of the measuring axes of accelerometers Ca is determined at the stage of factory calibration with respect to some orthogonal coordinate system usually associated with the mounting surfaces of the IMU. To determine the projections of the accelerations on the axes of this orthogonal coordinate system AIMU, it is necessary to multiply the vector of the accelerometer readings Aa by the inverse matrix .
Errors of the orientation matrix consist of several components:
• - factory calibration errors related to both the equipment used and the methodology;
• - temperature drifts of directions of measuring axes of accelerometers;
• - changes in the directions of the measuring axes of accelerometers after mechanical impact and vibration effects.
Since all components of the errors are independent, the total error of the elements of the accelerometer orientation matrix can easily be calculated by the formula.
Let’s consider each of the components separately.
METHODS OF DETERMINING THE MEASURING AXES ORIENTATION MATRIX OF ACCELEROMETERS, INSTRUMENTAL AND METHODOLOGICAL ERRORS
Determination of the Ca orientation matrix of the measuring axes of accelerometers in the orthogonal X, Y, Z coordinate system, associated with the IMU mounting surfaces, is carried out from the measurements of the output signals from the accelerometers on a high-precision biaxial bench equipped with a heating and cooling chamber (HCC). The normal to the mounting platform of the bench coincides with the axis of the stand. Axis 2 of the bench is orthogonal to axis 1 and is located horizontally. The bench is installed in such a way that at zero angles specified for both axes of the bench, axis 1 is directed upwards, and axis 2 is in the horizontal plane. The IMU is placed and rigidly fixed on the mounting platform of the bench with the help of the precision tooling so that the Y axis of the IMU coincides with the axis of the bench, and the X axis – with axis 2 (Fig. 1, position Ya). With the help of rotations to 90° and 180° relative to axes 1 and 2, the biaxial bench allows implementing 12 positions of the IMU shown in Fig. 1. Due to errors in setting the bench, the vector of apparent acceleration due to gravity is projected on the axis of the orthogonal coordinate system of the IMU, as shown in Fig. 1 under each of the IMU positions, where [·]T is the operation of the transposition of the vector [·], where , i = 1, 2, 3.
Measurements are made at several temperatures in the HCC of the bench, overlapping a predetermined temperature range.
Before the measurements are taken, the HCC of the stand is adjusted to the set temperature, then the IMU in the off state is exposed at this temperature until it reaches all the components of the IMU, after which the power supply is fed to the IMU. In the enabled state, the IMU is additionally exposed until the accelerometers reach the stationary temperature regime.
Measurements are made when the IMU is stationary, sequentially in all 12 positions shown in Fig. In this case, the average values of the voltages at the output of the accelerometers are determined in each position , where is the position of the IMU, ; the subscript x, y, z indicates the measuring axis of the accelerometers. Furthermore, using temperature sensors installed in each of the accelerometers, the corresponding temperature values are determined in each of the 12 positions, and then the average temperature value averaged over all accelerometers, as well as over all 12 positions.
At a fixed temperature in the HCC of the stand in the thermostating mode, the voltage at the output of the accelerometers is satisfied by the following equations:
Kx, Ky, Kz are constants for all 12 positions of the scale factor values of the accelerometers, connecting the output voltages of each of them with the measured apparent acceleration on the sensitivity axis;
Sx, Sy, Sz are constants for all 12 positions of the zero offset values of the accelerometers;
Gx, Gy, Gz – are the projections of the measured apparent acceleration due to gravity;
is the matrix of directional cosines to be determined defining the orientation of the axes of sensitivity of accelerometers in the orthogonal instrumental coordinate system; furthermore, the sum of squares of elements in each row of this matrix is equal to one.
Solving the system of equations (7) with respect to the elements of the matrix Ca, we obtain:
The errors in determining the elements of the orientation matrix of the measuring axes of accelerometers in the coordinate system of the IMU are due to errors in setting the bench, positioning its platform, wobble of the rotation axes and the presence of noises at the accelerometers. According to (10), (11) the error in determining the element Caαβ can be represented in the form:
When using a high-precision bench (with small mean square errors of positioning σ1 and wobble of the axis σ2), small angular errors of its setting relative to the local vertical δ, taking all four additive components in (3) as statically independent, and the noise component at the output of the accelerometers as a white noise with intensity Q, the duration of measurements in each of the 12 positions equal to τ s, in the linear approximation we obtain an estimate for the root-mean-square value of the angular error in the angular seconds:
For example, for Kα = G, Q = 3 · 10–9 в2 · с, τ = 30 с, σ1 = 5", σ2=2" this value is equal to 11,52".
The orientation matrix of the measuring axes and the corresponding average value of the temperature of the accelerometers (4) experimentally found for each temperature in the HCC of the bench are used to investigate the temperature dependence of this matrix.
TEMPERATURE CHANGES IN THE MEASURING AXES ORIENTATION OF ACCELEROMETERS, PHYSICAL CAUSES AND COMPENSATION METHODS
Let’s refer to the IMU design. There are two options for placing accelerometers (Fig. 2a, b). In the first case, the accelerometers are located on the same frame with the gyroscopes. It is believed that in this case the temperature changes in the orientation matrix of gyroscopes and accelerometers will be the same and they are easier to describe by any function of temperature. However, with such an arrangement, in the event of breakage of a gyroscope or an accelerometer, it will be necessary to disassemble and, accordingly, re-calibrate the entire IMU. Furthermore, the heat in the gyroscope is 17 times higher than the heat release in the accelerometer (2.5 W vs. 0.15 W), which leads to an additional overheating of the accelerometers and therefore to large zero and scale shifts .
In the second case, the accelerometers are located in a separate module which ensures less overheating and the possibility of separate repair of gyroscope blocks and accelerometers [4, 5]. However, the temperature changes in the orientation matrices of the accelerometers would seem to become greater due to the introduction of an additional bolted connection. The mechanical strength of the IMU should also decrease. The task of this study was to prove or disprove these hypotheses.
Temperature changes in the measuring axes orientation of accelerometers are made up of three parts:
1. Change the orientation angle of the measuring axis of the accelerometer relative to its mounting plane ΔСaTa.
2. Change in the position of the mounting surfaces of the body parts of the IMU, on which accelerometers are installed ΔСaTп.
3. Error of temperature correction ΔСaTк.
The total temperature changes in the directions of the measuring axes of accelerometers are determined by the formula:
The value ΔСaTa is determined by the design of the accelerometer, the temperature dependence having a hysteresis. The value of this hysteresis is referred to as the instability of the reference plane along the sensitivity axis during the service life.
The values ΔСaTa for accelerometers of various types are given in Table. 1, where it is seen that the instability of the reference plane along the sensitivity axis is determined not so much by the material of the pendulum as, e. g., zero displacement , but rather by the range of measured accelerations and operating temperatures. A‑18 stands out with very low characteristics, but it was developed somewhere for five years ahead of its competitors.
The value ΔСaTп is determined by the plastic deformations of the structure caused by the difference in the coefficients of thermal expansion of the accelerometer body КТРК and the IMU body parts КТРа. The maximum value can be estimated from the formula:
ΔCaTп = 0,1 · (КТРК – КТРа) (Тmax – T0), (6)
where Tmax is the maximum temperature inside the IMU at the accelerometer installation site, T0 is the temperature in the normal climatic conditions.
Let’s estimate ΔСaTп. The body of the accelerometer is made of steel (КТР = 15 · 10–6 1 /°С), the IMU body parts are made of D‑16 (КТР = 22,9 · 10–6 1 /°С), Тmax = +85 °С, Т0 = +20 °С, ΔСaTп = 5,1 · 10–5 rad. = 11".
The error of the temperature correction ΔСaTк is determined by the accuracy of the temperature measurement ΔT and the temperature coefficient of the deflection angle of the reference plane kсa:
ΔСaTк = ΔТ · kСа. (7)
For quartz accelerometers, kCa ≈1.5" / °C, for silicon, this value is much larger – up to 6" / °C. The accuracy of temperature measurement in the conditions of on-board equipment (interference, measurement time) is ΔТ ≈ 0,2 °С. Accordingly, ΔСaTк ≈ 0,3" for quartz accelerometers and up to 1.2" for silicon ones.
Based on the data obtained, we will estimate ΔСaT. Using formula (5), we obtain ΔСaT ≈ 12–19" for quartz accelerometers, against ΔСaT ≈ 19–61" for silicon accelerometers. Thus, the value ΔСaT is almost completely determined by the type of accelerometer used.
Let’s compare the calculated values obtained with the experiment’s results. Fig. 3a, b shows the measured dependence of the orientation angles of the measuring axes of accelerometers relative to the orthogonal coordinate system AIMU on the temperature for two types of accelerometers and the same design of the IMU (Fig. 2b). For comparison, the plot shows the orientation angles of similar measuring axes of gyroscopes and the difference between them. This allows us to determine how the common angular levers of the IMU design (they are the same for gyroscopes and accelerometers), and the orientation changes of each of the inertial sensors separately.
As can be seen from Fig. 3a, b, the temperature dependence of the orientation angles of gyroscopes is very weak. The synchronous component of the temperature change in the orientation angles and angular levers of the IMU design is absent. The temperature dependence of the orientation angles of silicon accelerometers is very strong.
For quartz accelerometers, the dependencies are comparable to the functions for gyroscopes, which again, proves the dominance of the pendulum material in this matter.
Now let’s refer to the efficiency of thermal correction. Fig. 4a, b shows the dependencies ΔСaTfor both types of accelerometers. Table 2 summarizes the results of the experiment. In brackets are the calculated data on the technical specifications of accelerometers.
As can be seen from Fig. 3 and 4 and Table 2, the temperature shifts of the directions of the measuring axes of the accelerometers ΔСaT are completely determined by the value ΔСaTa, which corresponds to the conclusions of formula (5). The structural arrangement of the accelerometers in the IMU does not affect the temperature error of the position of the measuring axes of the accelerometers.
CHANGES IN THE DIRECTIONS OF MEASURING AXES OF ACCELEROMETERS AFTER MECHANICAL IMPACT
In the course of operation, the IMU is subjected to mechanical impacts and vibrations that also lead to a change in the orientation of the measuring axes of the accelerometers. It is not possible to determine the changes in the directions of the measuring axes of accelerometers directly at the moment of mechanical exposures, therefore, the mechanical stability of accelerometers is judged by changing the orientation of the measuring axes after mechanical impacts and vibrations.
To study the change in the direction of the measuring axes of the accelerometers of the IMU with the accelerometer A‑18 after mechanical exposures, it was subjected to single shocks with an acceleration of 20g, 40g, 100g in the number of 4 impacts, as well as sinusoidal vibration with a frequency of 25 Hz and an amplitude of 5g, 10g and 20g, for 10 minutes. Each exposure was conducted in the direction of each of the axes of the IMU (X, Y, Z). Prior to, between and after the impacts, the matrixes of the direction cosines of the measuring axes of accelerometers and gyroscopes were measured on a high-precision triaxial bench. The error in determining the orientation angles was, as was already mentioned, ±12" for level 1σ or ±30". The results of the experiment are given in Table 3.
The results of the experiment are as follows:
• accelerometers and gyroscopes have demonstrated good strength to impacts and vibrations.
• deviations from both the gyroscopes and the accelerometers for random vibrations with an amplitude of 20g appeared beyond the accuracy of the experiment, which is apparently the limiting effect on the strength of the construction of gyroscopes and accelerometers.
The investigations conducted have shown that changes in the orientation matrix of accelerometers at temperature, impact, vibration effects are determined by 90% by the accelerometer itself. In this regard, the installation of gyroscopes and accelerometers on the same mounting planes of the IMU is not mandatory. This significantly improves the maintainability of the IMU and reduces the overheating of the accelerometers. The use of thermal correction of the orientation matrix by a linear function makes it possible to reduce the error to the errors of the accelerometers. The proposed IMU design with the separate arrangement of gyroscopes and accelerometers also provides mechanical strength by the criterion of maintaining the orientation of the measuring axes of inertial sensors to high levels of mechanical effects.