The inertial sensor errors can be divided into two types: deterministic (systematic) errors and stochastic errors. Calibration is an important way to remove the major part of the deterministic IMU sensor errors. Various IMU calibration methods have been proposed according to different grades of IMUs and different applications [2,8–12]. On the contrary, the stochastic errors can only be modeled instead of being calibrated and removed. Among various stochastic modeling methods, Allan Variance is commonly used to determine the characteristic of the underlying random processes [13].

The thermal drifts of the sensor errors are normally regarded as deterministic errors. To reduce the thermal drift of the sensor errors, the following two processes are needed: (1) Thermal calibration: to develop accurate and reliable thermal models of the sensor errors, i.e., build the relationship between the sensor errors and the sensor temperature; (2) Thermal compensation: to compensate the thermal drift of the sensor errors according to their online temperature during the operation process of the IMUs. Both these processes are dependent on the temperature provided by the internal temperature sensors of the IMUs.

Official thermal calibration needs professional equipment such as a thermal chamber and a turntable (optional). Such professional equipment can be used to get reliable thermal calibration results over a large temperature range. However, when using these high-cost equipments, it is also important to improve the calibration efficiency, so as to reduce the equipment cost.

The purpose of thermal calibration is to determine the IMU sensor errors under different temperature points. Therefore, thermal calibration is commonly used based on the following assumption: the thermal drift of a sensor error is only related to the temperature of the sensor core. That is, corresponding to a certain temperature, the sensor errors will have the same values. Based on this assumption, the relationship between the sensor errors and the temperature can be built directly through either curve fit (as used in this paper) or a lookup table, if the corresponding sensor errors at different temperatures are known.

There are currently two main approaches for thermal calibration: the Soak method and the Ramp method [2]. The Soak method works on the premise of stable sensor temperature while the Ramp method works based on time-varying sensor temperature.

The Soak method calibration works as follows: (a) Stabilize the temperature of the sensors or IMUs at a certain temperature point; (b) Record the sensor measurements and calculate the sensor errors corresponding to such point; (c) Repeat this process at several typical temperature points, then a series of sensor errors along with temperature data can be obtained. After that, the IMU sensor errors at other temperature points can be calculated by interpolation [14]. The Soak method is used for both the calibration of deterministic errors and the modeling of stochastic errors [15–21]. By stabilizing the temperature, the Soak method can provide most reliable values of the sensor errors at the chosen temperature points. The main fact which restricts the use of the Soak method is the time required and the corresponding costs related to the equipment, time and manpower. The process of stabilizing the sensor temperature inside a MEMS IMU is usually time-costing.

Different from the Soak method, the Ramp method calibration works as follows: (a) Control the temperature of the thermal chamber while it is continuously linearly increased or decreased to cover the full temperature range; (b) During this process the IMU measurements are recorded and the IMU sensor errors are calibrated [22,23]. The Ramp method is fundamentally faster in principle because it removes the stabilization process, and works based on linearly changed sensor temperature. However, there are two main inherent problems of the Ramp method, especially when it is used for the calibration of the IMUs, instead of the sensors:

Issue #1: It is not realistic to get the real temperature of the inertial sensors: there are temperature differences between the inertial sensors and the temperature sensors inside the IMUs because they have different cores. What we can get is the temperature at the core of the temperature sensor, not that of the inertial sensors. Moreover, such temperature differences will be different under various temperature changing conditions. This issue is similar to the hysteresis effect.

Both these two problems can impair the performance of the Ramp method. This can also explain why the process of stabilizing the temperature is needed in the Soak method. In this paper, we use the idea of the Ramp method, which removes the time-consuming process of stabilizing the temperature. Also, several ways to ease its inherent problems are considered. The details of the proposed method are shown in the next section.

We use the idea of the Ramp method, but consider the following ways to mitigate its inherent problems for improvement:

Improvement #1: take a balanced heating-and-cooling strategy to compensate the thermal calibration errors. To be specific, we change the temperature linearly for a complete cycle, and make comprehensive use of the sensor measurements under both the linear heating and cooling processes at the same changing rate to get the final thermal calibration results at the same temperature point.

The corresponding explanations to these improvements can be stated as below:

Explanation #1: under the heating and cooling processes at the same temperature changing rate, the effects of the corresponding temperature differences between the inertial sensors and the temperature sensors can cancel each other approximately (i.e., they have the similar value but opposite signs).

Both Improvement #1 and #2 are utilized to mitigate Issue #1, while Improvement #2 and #3 aim to ease Issue #2. With such improvements, the inherent problems of the Ramp method can be mitigated.

A dual-axis position turntable equipped with a thermal chamber (see Figure 1) located at Wuhan University is used to calibrate the IMU sensor errors over a wide temperature range. The performance characteristics of both the turntable and the thermal chamber are shown in Table 1.

The turntable is accurate enough to provide reference for the calibration of low-grade IMUs. Besides, there are high-precision locating pins on the mounting plate of the turntable to keep the IMUs axes align with the turntable axes.

Considering both the characteristics of the equipment and the calibration efficiency, the following 8-step calibration scheme is designed (see Figure 2). This scheme allows the IMUs to experience all necessary orientations and rotations with minimum calibration actions.

In each calibration scheme, each sensing axis of every sensor is pointed alternately up and down precisely (except z-axis due to the mechanical structure of the turntable), and the IMUs are rotated around each gyro axis both clockwise and counter-clockwise at accurately known angles. There are in total eight static positions and eight rotations.

In our test, each static position takes 10 s and the rate of rotation is about 10 deg/s, i.e., about 9 s for each rotation. Due to such settings, the calibration scheme is highly efficient, and can provide enough calibration accuracy for low-grade IMUs, as shown in Section 3.3 for the calculation of the calibration errors caused by sensor noises. Each calibration scheme takes only about (10 + 9) × 8 = 152 s to estimate a full set of IMU sensor errors.

During the whole calibration process, the thermal chamber keeps on changing its temperature linearly. At the same time, the turntable repeats the calibration scheme in Figure 2 again and again. Due to the efficiency of the scheme, we obtain abundant data of IMU sensor errors together with temperatures over the whole temperature cycle.

The error models for the accelerometers and the can be written as (1) f ^ = b a + S a f + N a f + v a (2) ω ^ = b g + S g ω + N g ω + v gwhere f̂ and ω̂ are the error vectors of the accelerometer-derived specific forces and the gyro-derived angular velocities, f and ω are the true specific forces and the true angular velocities, b_a and b_g are the biases of the accelerometers and the gyros, S_a and S_g are the diagonal matrices containing the scale factor errors, N_a and N_g are the skew-symmetric matrix containing the non-orthogonalities, ν_a and ν_g represent the a ccelerometer and gyro noises.

In each calibration scheme, the IMU sensor errors, including biases, scale factor errors and non-orthogonalities, are determined. The IMU data is sufficiently used: the static data (i.e., the IMU outputs at eight static positions) is used to calibrate the gyro biases and all the accelerometer errors, while the dynamic data (i.e., the IMU outputs during eight rotation steps) is used to calibrate gyro scale factors and non-orthogonalities. The accelerometer errors can be estimated by the least-square method [2], while the gyro errors are estimated through a two-step method [24]. The details of calibration computations are shown in Appendix A.

The calibration errors caused by sensor noises can be calculated as follows [25]. Since there are eight static positions, gyro noises can lead to calibration error of gyro biases σ b g − noise = ARW 8 . t static ( 1 σ ), where _ARW are the angular random walk of gyros, t_static is the time for each static position. Similarly, the calibration error of accelerometer errors caused by accelerometer noises can be calculated. For the calibration of gyro scale factors and non-orthogonalities, there are at least two rotations (positive and negative) for each gyro. Thus the gyro noise can cause calibration error of gyro scale factors or non-orthogonalities σ s g − noise = ARW ⋅ t rotate 2 · 90 °, where t_rotate is the time for each rotation. For example, assuming the gyro ARW is 0.05 deg / s / HZ (refer to that of typical MEMS gyros), the errors caused by gyro noise are about 20 deg/h for gyro biases, and about 1,200 ppm for gyro scale factors or non-orthogonalities, which is much smaller than the thermal drifts of the sensor errors of low cost MEMS gyros.

Based on the calibrated IMU sensor errors at different temperatures, the curve fit method was used to determine the continuous global temperature compensation models (i.e., the relationship between the sensor errors, i.e., biases, scale factors and non-orthogonalities, and the temperature). A 3rd-order polynomial is used to fit the calibration results obtained from each scheme: (3) W ( T ) = A 0 + A 1 ⋅ T + A 2 ⋅ T 2 + A 3 ⋅ T 3where W(T)is the calculated value of the sensor errors, T is the IMU core temperature in real operation process, A₀, A₁, A₂ and A₃ are the fitting parameters. The established thermal model can be used to calculate the thermal drifts of the IMU sensor errors at each temperature and then remove such thermal drifts from the IMU outputs.

To evaluate the accuracy of the proposed method, we carried out the tests of the conventional Soak method to get the reference. The IMU calibration results at more than 10 temperature points were collected to provide accurate reference value, even though the Soak method does not need so many points and so much time for common uses. We also designed three different chamber temperature profiles to test the repeatability of the proposed method. Table 2 describes the detail of the chamber temperature changes. Figures 3 and 4 show the temperature profiles and how the internal temperatures of the two tested IMUs (Xsens MTi-G and NV-IMU100) changed. In both Figures 3 and 4, the dashed lines are the chamber temperatures, while the solid lines are the temperature provided by the internal temperature sensors of the two tested IMUs. It is clear to see the temperature lags of the IMU cores. Besides, Figure 3 shows that the IMU temperatures were higher than the chamber temperature after stabilization due to the heat generated inside the IMUs.

Two MEMS IMUs, i.e., Xsens MTi-G [26] and NV-IMU100 [27], were chosen to be the representative of low-grade IMUs to be tested. Their photos in the test and their characteristics are illustrated in Figure 5 and Table 3, respectively. Here we used the raw sensor data of MTi-G (without factory compensation). The NV-IMU100 has three vibrating ring gyros, and its output data had been partly compensated after factory calibration.

Figure 6 shows the results of the biases and scale factor errors of gyros with the Soak method, as the example results of this conventional method.

The thin solid lines in Figure 6 show that even when using the Soak method, there were some non-repeatabilities between the results during Part 1 (from −30 to 70 °C with 10 °C step) and Part 2 (from 70 to −30 °C with 10 °C step;. After checking the temperature data of the IMUs, we found that it is because the sensor temperature inside the IMUs had not been fully stabilized even after the chamber temperature outside the IMUs had been stabilized for 1 h. This means that more than 1 h is needed for each temperature point to get the ideal result of the Soak calibration method for MTi-G. However, we cannot afford more than 1 h (e.g., 2 h; for each point for in-lab tests. Thus, we finally used 1 h per point, but fitted the results of Part 1 and Part 2 (i.e., heating and cooling profiles) at the same temperature point to calculate the final IMU sensor errors at this temperature. This is the best result we can get and was regarded as the reference for the evaluation of the proposed method.

Figure 7 shows the results of gyro biases and scale factor errors with the proposed method, as the example results of this new method.

The raw results of the proposed method (the thin dashed lines) have some non-repeatabilities during heating and cooling processes. This might be caused by different temperature change rates. However, after fitting the raw results, we got the final result (the thick dashed line), which fit well with the reference result in Figure 6.

To test the repeatability of the proposed method, we run the chamber temperature as the three different profiles described in Table 2. Figure 8 shows the final results of the three tests of the proposed method with different profiles (three dashed lines) as well as that of the Soak method (the solid line). Figure 8a–f show the curves of biases, scale factor errors, and non-orthogonalities of gyros and those of accelerometers.

It is clear that the sensor errors of MTi-G varied significantly with temperature, especially the gyro biases and scale factors. Over the entire temperature range, the changes reached 2,000 deg/h, 10,000 ppm for MTi-G gyro biases and scale factor errors, and 3,000 μg and 400 ppm for accelerometer biases and scale factor errors. There were also changes of about 5,000 ppm and 1,000 ppm for gyro and accelerometer non-orthogonalities, which were beyond our intuition. These curves indicate that thermal calibration is important not only for biases and scale factors, but also for non-orthogonalities, at least in the case of MEMS IMUs.

To evaluate how much the final results provided by the proposed method (the three dashed lines) fit with that of the Soak method (the solid line), we picked some typical temperature points and calculated the Root Mean Square (RMS) errors (using the results of the Soak method as reference) of the three temperature profiles, as shown in Table 4. The first 12 rows are the results of the gyro biases, scale factor errors and non-orthogonalities, while the last 12 rows are those of the accelerometers of MTi-G. Columns 2–6 show the RMS of the sensor errors under typical temperature points and column 7 picks up the max values among them.

Over the entire temperature range, their RMS with respect to the result of the Soak method reached 360 deg/h, 1,500 ppm and 1,300 ppm for gyro biases, scale factors and non-orthogonalities; and 1,200 μg, 100 ppm, and 400 ppm for accelerometer biases, scale factor errors and non-orthogonalities. The accuracy of the proposed method is shown in Table 5. Also, the original thermal drift levels of each sensor error over −10 ∼ +70 °C are shown to make a comparison.

The overall accuracy levels are much lower than the levels of the original thermal drifts of the MEMS sensor errors. The proposed calibration method has enough accuracy to significantly reduce the original temperature variations of the MEMS IMU sensor errors.

The thermal calibration results of MTi-G also indicate that not all the sensors errors can be reduced significantly through calibration, since the errors of each sensor are randomly distributed in the range of its specifications and some errors may be small originally. What we can expect is that through calibration and compensation, the overall levels of IMU sensor errors can be reduce to a much lower level.

Similarly, we calibrated the thermal drift of NV-IMU100 with both the proposed method (using three different profiles) and the Soak method. Figure 9 shows the results provided by both the proposed method (the three dashed lines) and the Soak method (the solid line).

There were small gyro biases (less than 15 deg/h) due to the symmetry of vibrating ring gyros and the factory compensation. However, the change for gyro scale factor (e.g., y-axis) exceeded 10,000 ppm. These characteristics matched the inherent feature of the vibrating ring gyros.

Like Table 4, Table 6 shows the RMS errors (using the results of the Soak method as reference) of the proposed method at some typical temperature points.

Over the entire temperature range, the RMS errors reached 5 deg/h, 2,200 ppm and 900 ppm for gyro biases, scale factor errors and non-orthogonalities; and 4,200 μg, 700 ppm, and 600 ppm for accelerometer biases, scale factor errors and non-orthogonalities, respectively. The accuracy of the proposed method is shown in Table 7 as well as the original thermal drift ranges of each sensor error over −10 ∼ +70 °C. The overall accuracy levels are much lower than the levels of the possible thermal drift of the MEMS sensor errors.

The proposed thermal calibration method can provide results close to those of the conventional Soak method, and can significantly reduce the calibration time. Thus it is especially useful for MEMS IMUs, which require low cost and efficient calibration. Please note that the thermal calibration accuracy not only depend on the calibration method itself, but also is affected by the stochastic errors of the calibrated inertial sensors. This can explain why the same method got different calibration accuracies for the two tested IMUs. For example, the calibration accuracy of gyro biases of NV-IMU100 can reach 5 deg/h because the original gyro biases are relatively stable. To further investigate the effect of thermal calibration, another kind of tests called thermal compensation tests were performed, as described in Section 5.

The thermal compensation tests were designed to check how much the performance of the inertial sensors can be improved by compensating their outputs with the established thermal models. There were totally six tests, including three dynamic tests and three static tests, which are described in Table 8. For each test, the thermal chamber was utilized to provide different temperature environment (from −10 °C to +70 °C;, and the turntable was used to provide precisely known reference. Every epoch of IMU outputs was corrected using the full set of IMU sensor errors provided by the thermal model at the same temperature (i.e., the thermal compensation). Then the effect of the thermal compensation was evaluated by comparing the output errors (i.e., the difference between the sensor outputs and the reference input) before and after the thermal compensation. To evaluate the performance of all the sensors before and after compensation, every accelerometer axis was kept on pointing downwards precisely for about an hour respectively. Besides, the IMUs were kept on rotating around each gyro axis at an angular rate of 10 deg/s, which was set according to the dynamic of land vehicles, for about an hour. The whole process took about six hours. In each test, we focused on the output error of one sensor. For example, we investigated the x-axis gyro output with a reference input of 10 deg/s in the dynamic test 1; and investigated the x-axis accelerometer output with a reference input of the local gravity in the static test 1. The IMU outputs were averaged during each second to reduce the impact of the noises. The thermal models established through thermal calibration test 2 using the 2nd temperature profile were used in this test, since the results provided by test 2 seemed to have the largest differences comparing with those of the Soak method. The parameters of the 3rd order polynomial thermal models of applied in the compensation tests, i.e., A₀, A₁, A₂ and A₃ in Equation (3), for the sensor errors of the two tested IMUs are shown in Table 9. The test results of MTi-G and NV-IMU100 are shown in Section 5.2 and 5.3, respectively.

Figures 10 and 11 show the compensation results of MTi-G. The blue and red lines are the output errors without and with the compensation using the established thermal models.

It is clear that the original output error of the z-axis gyro was significantly larger and changed with some long-term trends. After compensation, such error was significantly reduced, and the long-term trends were eliminated. However, there was no significant improvement for x-axis and y-axis gyros, since their original output errors were relatively small. As for the accelerometers, the improvement was significant along x-axis and z-axis. However, the y-axis accelerometer got an output error of about 2,000 μg after compensation. As has been explained in Section 4.3, not all the sensors errors can be reduced significantly through the thermal compensation, because the thermal drift level of each sensor are randomly distributed in the range of its specifications and some may be small originally. The thermal compensation can reduce the overall thermal drifts of the IMU outputs to a much lower level.

The statistical values (i.e., the mean and the RMS values) of the IMU output errors over the entire temperature range were calculated and shown in Table 10. Without compensation, the RMS values were 279, 197 and 2,167 deg/h for gyros and 3,697, 1,097 and 11,130 μg for accelerometers. After the compensation, the RMS values became 189, 254 and 418 deg/h for gyros and 2,342, 2,259 and 1,055 μg for accelerometers. Both the gyro and accelerometer along the z-axis and x-axis were improved significantly, while those along the y-axis became worse. Generally speaking, with the thermal compensation, the output errors of MTi-G were reduced to within 420 deg/h and 2,400 μg for gyros and accelerometers over the entire temperature range. To make it clear, the RMS values were shown in Figure 12. The blue and red bars were the values without and with thermal compensation, respectively.

Similarly, Figures 13 and 14 show the NV-IMU100 output errors without and with thermal compensation. The NV-IMU100 has small gyro biases due to the symmetry of the vibrating ring gyros and the factory compensation. However, there were also gyro output errors caused by the scale factor errors and non-orthogonalities, especially for the y-axis gyro, whose scale factor varied for thousands of ppm over the entire temperature range. Consequently, the gyro output error reached several hundred deg/h when the input angular velocity was 10 deg/h. Clearly, such output errors were reduced significantly after the thermal compensation. The errors of accelerometers, especially the x-axis and y-axis accelerometers, were also reduced dramatically. The systematic errors in the outputs were corrected, and long-term trends in the outputs were eliminated. The statistical values of the NV-IMU100 output errors were shown in Table 11, and the RMS values were illustrated in Figure 15.