2.1. Model of the Integral Step

The integral step can be built by the following steps. Firstly, the continuous-time unit step function u(t) is well known as

(1)$$u(t) = \left\{ {\matrix{ {0,} & {t \lt 0} \cr {1,} & {t \gt 0} \cr}} \right.$$

According to the definition of convolution integral, there is:

(2)$$f\,(t) * u(t) = \int_{ - \infty} ^{ + \infty} {\,f\,(\tau )} \cdot u(t - \tau )d\tau = \int_{ - \infty} ^t {\,f\,(\tau )} d\tau $$

Therefore, the integral step can be modelled as the convolution with a Rectangular window function (see Figure 5)

(3)$$h_{Integral} (t) = u(t) - u(t - T)$$

where, T is the time length of integral.

And

(4)$$\eqalign{\,f\,(t) * h_{Integral} (t) = & \int_{ - \infty} ^{ + \infty} {\,f\,(\tau )} [u(t - \tau ) - u(t - \tau - T)]d\tau \cr = & \int_{ - \infty} ^{ + \infty} {\,f\,(\tau )} u(t - \tau )d\tau - \int_{ - \infty} ^{ + \infty} {\,f\,(\tau )} u(t - \tau - T)d\tau \cr = & \int_{ - \infty} ^t {\,f\,(\tau )} d\tau - \int_{ - \infty} ^{t - T} {\,f\,(\tau )} d\tau \cr = & \int_{t - T}^t {\,f\,(\tau )} d\tau} $$

Figure 5.

Rectangular window function as the model of integrator.

At the same time, the Rectangular pulse function r(t) (Figure 6(a)) and its frequency-domain expression (Figure 6(b)) are:

(5)$$r(t) = \left\{ {\matrix{ {1,} & { - \displaystyle{T \over 2} \lt t \lt \displaystyle{T \over 2}} \cr {0,} & {others} \cr}} \right.$$

and

(6)$$R(\omega ) = T \cdot Sa\left( {\displaystyle{{\omega T} \over 2}} \right)$$

Function Sa(x) is defined as Sa(x)=sin x / x.

Figure 6.

Rectangular pulse in time-domain and frequency-domain.

Then the relationship between the integral model and the rectangular pulse function should be:

(7)$$h_{Integral} (t) = u(t) - u(t - T) = r\left( {t - \displaystyle{T \over 2}} \right)$$

(8)$$H_{Integral} (\omega ) = R(\omega ) \cdot e^{ - j\omega {\textstyle{T \over 2}}} = T \cdot Sa\left( {\displaystyle{{\omega T} \over 2}} \right) \cdot e^{ - j\omega {\textstyle{T \over 2}}} $$

Here T is the time length of the integral. H _Integral(ω) is the Fourier transform of h(t) and its amplitude-frequency response is represented in Figure 7.

Figure 7.

Amplitude-Frequency Response of the integrator. (Here ω _c1 is the first zero-crossing.)

2.2. Model of the Denoising Step

The denoising filter can be represented by a continuous-time ideal low-pass filter (LPF) with a cut-off frequency ω _c2. The LPF is a simple but representative denoising method, which makes the theoretical analysis both convincing and easy to understand. It is a linear time invariant (LTI) system that passes complex exponentials e^jωt for values of ω in the range −ω _c2 ⩽ ω⩽ ω _c2 and rejects signals at all other frequencies (Oppenheim et al., Reference Oppenheim, Willsky and Hamid1996). That is to say, the frequency response of an ideal continuous-time low-pass filter is

(9)$$H_{LPF} (\omega ) = \left\{ {\matrix{ {e^{ - j\omega t_0},} & {\left| \omega \right| \leqslant \omega _{c{\rm 2}}} \cr {0,} & {\left| \omega \right| \gt \omega _{c{\rm 2}}} \cr}} \right.$$

as depicted in Figure 8.

Figure 8.

Frequency Response of ideal low-pass filter.

2.3. Model of integral with denoising

The case of original signal integrated by the integrator directly and the case of original signal denoised firstly and then integrated by the integrator are simplified as the system block diagrams in Figure 9. And the time-domain and frequency-domain block diagram of both cases are expressed in Figure 10 according to the characteristics of convolution (Zheng et al., Reference Zheng, Ying and Yang2000).

Figure 9.

Simplified system structures of the INS algorithm without/with denoising.

Figure 10.

Time-domain and Frequency-domain block diagrams without/with denoising.

Here

• n(t) is the original signal

• h(t) is the unit impulse response function

• y(t) is output of the system.

• N(ω), H(ω) and Y(ω) are the frequency-domain transformations of n(t), h(t) and y(t) respectively. The subscripts 1 and 2 are representations of the case without denoising step and the case with the denoising.

According to the previous description, the transfer function of the integrator is H _Integral (ω) (Equation 8) and its first zero crossing of the waveform in frequency-domain is ω _c1 (see in Figure 7). The transfer function of the low-pass filter is H _LPF (ω) with a cut-off frequency ω _c2. And then the frequency response of the combination of the ideal LPF and the integrator (Figure 10(b)) is:

(10)$$\eqalign{H(\omega ){\rm \, = \,} & H_{LPF} (\omega ) \cdot H_{Integral} (\omega ) \cr = \, & \left\{ {\matrix{ {T \cdot Sa\left( {\displaystyle{{\omega T} \over 2}} \right) \cdot e^{ \left(- j\omega {T \over 2}{\rm +} t_0 \right) },} & {\left| \omega \right| \leqslant \omega _{c2}} \cr {0,} & {\left| \omega \right| \gt \omega _{c2}} \cr}} \right.} $$

Based on the analysis above, the amplitude-frequency response of the integral step and the integral with denoising filter can be described in frequency-domain in Figure 11. Note that the frequency response of the integrator with denoising shown in Equation 10 and Figure 11(b) only presents the case that ω _c1<ω _c2, which is the most common case.

Figure 11.

Amplitude-frequency response of the two cases.

Then the following statements can be drawn by comparing Figure 11(a) and Figure 11(b):

Firstly, the amplitude of the first side-lobe is only 0·22 times (i.e. 13 dB attenuation) the main-lobe, which can be neglected. Then the integral step is equivalent to a super low-pass filter whose equivalent cut-off frequency is ω _c1, which is in inverse proportion to the integral time length (i.e. T) in the INS algorithm. Note that the integral time length here should be the independent working time of the INS, not the sampling interval of the IMU data. This is because we focused on the error characteristics of the INS. In the INS algorithm, the error of the INS will not be corrected during the IMU data interval but be continuously accumulated in the follow-up integral steps; in Global Navigation Satellite System (GNSS)/INS systems, the INS drift error can only be corrected by the GNSS external aiding after the INS works alone for some time (e.g. 1 s). Therefore the integration step interval is usually the same as the sampling interval of aiding information such as GNSS.

Secondly, when ω _c1<ω _c2, i.e. the equivalent cut-off frequency of the integral step is smaller than the cut-off frequency of the LPF, then the effect of the denoising will be shadowed by the integral step. For most inertial navigation applications, the integral time of the IMU is long enough to make ω _c1<<ω _c2. For example, the INS normally works alone for more than 1 sec, while the cut-off frequency of the denoising filter is normally higher than several Hertz since the true motion frequencies of the vehicle mainly appear in 0∼6 Hz band (Chiang et al., Reference Chiang, Hou, Niu and El-Sheimy2004).

Finally, for some static data processing such as stationary alignment of the INS, an unlimited level of denoising can be applied in principle and the conclusion does not apply because the true motion has almost no dynamics (El-Sheimy et al., Reference El-Sheimy, Nassar and Noureldin2004).

This proves that the actual impact of the integral operations to the IMU signal denoising is indeed significant, as shown in Figure 12. Any potential contribution of the IMU denoising will be overshadowed by the effect of the integral steps in the INS algorithm, for most of the application scenarios. This conclusion will be further verified by simulations and real tests in the following sections. Please note that the verification scenarios are designed intentionally in favour of IMU denoising, so as to make the overshadowing effect more convincing.

Figure 12.

Schematic plot of the effect of the integral operations in INS algorithm in Frequency domain.

4.1. IMUs and GNSS receiver used in the tests

The IMUs used in tests were IMU-FSAS (http://www.novatel.com/products/span-gnss-inertial-systems/span-imus/imu-fsas), Xsens MTi-G (http://www.xsens.com/en/general/mti-g) and GNSS receiver was NovAtel ProPak-V3 (http://www.novatel.com/products/gnss-receivers/enclosures/propak-v3) as shown in Figure 16.

Figure 16.

The IMUs and GNSS receiver used in the tests.

The MTi-G is based on the typical MEMS inertial sensors and IMU-FSAS is a typical tactical grade IMU. Their relevant characteristics are listed in Table 1.

Table 1.

Specifications of the IMUs.

* See more specifications of the gyro in MTi-G in (Niu, 2007).

4.2. Test results of stand-alone INS

A static test was carried out collecting the IMUs' raw data from MTi-G and IMU-FSAS. The raw data of IMUs with length of more than 5 minutes was firstly filtered by a second-order Butterworth LPF with the cut-off frequency of 1 Hz (e.g. the raw and de-noised Gyro X signal in Figures 17 and 20). Then the raw IMU data (without denoising) and the de-noised IMU data were processed by the same INS algorithm respectively. The position drifts of the navigation solutions for both cases were compared in Figures 18 and 21.

Figure 17.

The raw and the denoised Gyro X signal of the MEMS IMU (MTi-G).

Figure 18.

The position drifts computed by the IMU data with/without denoising (MEMS grade, MTi-G).

The results showed that the position drifts were apparently nearly the same for the cases with and without denoising. The difference of position drift was calculated and is shown in Figures 19 and 22. It indicated that the difference was relatively very small compared to the position drifts in Figures 18 or 21. The denoising approach did not bring clear accuracy improvement to the INS navigation solution.

Figure 19.

The difference of the position drifts (MEMS grade, MTi-G).

Figure 20.

The raw and the denoised Gyro X signal of the tactical IMU (FSAS).

Figure 21.

The position drifts computed by the IMU data with/without denoising (Tactical grade, FSAS).

Figure 22.

The difference of the position drifts IMU data with/without denoising (Tactical grade, FSAS).

By repeating the test under the same conditions, in total 40 samples were collected for statistical analysis. Tables 2 and 3 list the final position drifts (i.e. represented by A in the table) at the end of test periods derived from the raw IMU data and the differences of the final position drifts (represented by B in the table) of all 40 tests. The mean value of the differences was calculated to check out whether the denoising can bring appreciable improvement of the navigation solution or not. The root-mean-square (RMS) of the final position drifts were set as the reference value.

Table 2.

Statistical comparison of the INS stand-alone results with/without denoising (MEMS grade, MTi-G).

Note: RMS is root-mean-square.

Table 3.

Statistical comparison of the INS stand-alone results with/without denoising (Tactical grade, FSAS).

The ratio of the mean of B to the RMS of A in Table 2 was around 0·1% for horizontal (i.e. sum of north and east) and 0·01% for height. Similarly, Table 3 showed that the ratio was around 1·1% for horizontal and 0·5% for height. The denoising results had both improved and degraded cases. The improvements from denoising were inappreciable for both MEMS and tactical grade cases and it could be concluded that the IMU denoising cannot improve the accuracy of the INS navigation solutions, since the integral operation in the INS mechanization overshadowed the effect of the denoising to the IMU signals as in our previous analysis.

4.3. Test results with GPS/INS integration

Static tests of 2 hours were carried out by collecting the carrier-phase differential GPS solution (DGPS) of the NovAtel ProPak-V3 GNSS receiver and the raw data of MTi-G and FSAS, to check the effect of the IMU signal denoising in GPS/INS integration situations.

The data rate of the GPS was 1 Hz and the differential GPS solution (DGPS) was derived from the measurements of the rover and the base station. The IMU data was filtered by a second-order Butterworth LPF with cut-off frequency of 1 Hz. A loosely coupled GPS/INS integration algorithm was used to fuse the IMU data (raw and the de-noised) with the DGPS solution. Note that the noise power spectral density of the denoised IMU data does not change as the LPF is a linear system (Zhu et al., Reference Zhu, Huang, Li and Mei1990). It is therefore safe to keep the same noise parameters and covariance in the Q matrix of the Kalman filter when processing the raw data and the denoised data in the GPS/INS integration algorithm.

To evaluate the essential performance of the GPS/INS solution, the position drifts were checked over periods of GPS blockages (Niu et al., Reference Niu, Zhang, Shi, Chiang and El-Sheimy2010). In total, 16 GPS outages of 60 sec were artificially added when the data was post-processed by the integration algorithm.

The results are shown in Figures 23 and 24. There were position drifts during the outages as expected. The position drifts of the results with and without IMU signal denoising were almost the same for both the MEMS IMU and the tactical grade IMU. The maximum position drifts (i.e. A in the table) derived from the raw IMU data in each outage period and the differences of the maximum position drifts (i.e. B in the table) were collected in Table 4 and Table 5. Both tables showed that the differences of the max position drifts derived from IMU data with and without denoising were inappreciable.

Figure 23.

Position drifts of the MEMS IMU data with/without denoising with GPS signal outages of 60 sec.

Figure 24.

Position drifts of the tactical IMU data with/without denoising with GPS signal outages of 60 sec.

Table 4.

Statistical comparison of the GPS/INS results with/without denoising (MEMS grade, MTi-G).

Table 5.

Statistical comparison of the GPS/INS results with/without denoising (Tactical grade, FSAS).

The statistical results indicate that IMU signal denoising cannot bring appreciable improvement to the navigation solution in a GPS/INS integrated system.

It is arguable that the Kalman filter in GPS/INS integration systems would be better at estimating the IMU sensors error (i.e. biases of gyros and accelerometers) from the de-noised IMU signals. However, it should be noted that IMU sensor errors can only be observed indirectly from the measurement information (e.g. GPS position) after the IMU data is integrated, therefore the effect of the denoising has also been overshadowed by the equivalent super-low-pass filtering effect from the integral operations. To confirm this point, the impact of the IMU signal denoising to the estimation of sensors bias was also investigated in the real test.

Figure 25 shows the estimation of gyro bias in this test. Tables 6 and 7 give the RMS of the bias estimation (derived from the original IMU data without denoising) and the RMS of the difference of the bias estimations. The ratios of the RMS showed that the differences of bias estimations were so inappreciable, that it can be concluded that the IMU denoising cannot improve the estimation of the IMU sensors error in GPS/INS.

Figure 25.

Estimation of Gyro biases in the GPS/INS integrated system.

Table 6.

Statistical summary of the gyro bias estimations in GPS/MEMS IMU.

Table 7.

Statistical summary of the gyro bias estimations in GPS/Tactical IMU.

In addition, although the investigation of the denoising technique in this paper used a conventional low-pass filter (Butterworth filter) as an example, the conclusion can be expanded to other denoising algorithms. The bandwidth of the LPF used in the paper has been set as low as 1 Hz, and the true signals were set to have dynamics far below 1 Hz (e.g. the simulation used 0·1 Hz angle motion and the real tests used static data). Under such ideal conditions for IMU denoising and using such an extreme denoising parameter, there was still no observable improvement to the navigation results.

Generally, the time length that an INS works alone is more than 1 sec in most of INS and GPS/INS navigation systems (e.g. the data rate of GPS is 1 Hz), and therefore causes the equivalent cut-off frequency to be lower than 1 Hz. Thus the denoising technique can only make a contribution if it can remove the noise component below 1 Hz. However, signals below 1 Hz normally contain intensive true motions for most of the navigation applications. Therefore it is not able to remove or even attenuate such low frequency parts.