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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In both military and civilian applications, the inertial navigation system (INS) and the global positioning system (GPS) are two complementary technologies that can be integrated to provide reliable positioning and navigation information for land vehicles. The accuracy enhancement of INS sensors and the integration of INS with GPS are the subjects of widespread research. Wavelet de-noising of INS sensors has had limited success in removing the long-term (low-frequency) inertial sensor errors. The primary objective of this research is to develop a novel inertial sensor accuracy enhancement technique that can remove both short-term and long-term error components from inertial sensor measurements prior to INS mechanization and INS/GPS integration. A high resolution spectral analysis technique called the fast orthogonal search (FOS) algorithm is used to accurately model the low frequency range of the spectrum, which includes the vehicle motion dynamics and inertial sensor errors. FOS models the spectral components with the most energy first and uses an adaptive threshold to stop adding frequency terms when fitting a term does not reduce the mean squared error more than fitting white noise. The proposed method was developed, tested and validated through road test experiments involving both low-end tactical grade and low cost MEMS-based inertial systems. The results demonstrate that in most cases the position accuracy during GPS outages using FOS de-noised data is superior to the position accuracy using wavelet de-noising.

In numerous applications, inertial navigation system (INS) and global positioning system (GPS) are two complementary technologies that can be integrated to provide reliable positioning and navigation information for land vehicles. In the event of loss, denial of use, or degradation of the GPS signal (

The process of inertial navigation computes position, velocity and attitude of a moving platform, with respect to an inertial frame of reference, by measuring its rotational motion (using gyroscopes) and translational motion (using accelerometers) and mathematically integrating the measurements through a procedure known as INS mechanization [

During the INS mechanization process, these errors are compounded, resulting in increasingly inaccurate position and attitude over time. Despite having an INS/GPS integration algorithm (like Kalman filtering) to correct for INS errors, it is advantageous to enhance the INS solution prior to the data fusion process [

The research reported herein aims at: (1) developing a novel inertial sensor accuracy enhancement technique that can remove some or all of the error components from inertial sensor measurements prior to INS mechanization and INS/GPS integration; (2) examining the effectiveness of the proposed method on real INS/GPS road test data; (3) comparing the results to other wavelet based pre-filtering techniques.

Wavelet de-noising is the current state of the art technique used in the accuracy enhancement of inertial sensors [

The proposed technique employs the fast orthogonal search (FOS) algorithm [

The fast orthogonal search (FOS) algorithm [_{m}

The functional expansion of the input _{m}_{m}

By choosing non-orthogonal candidate functions, there is no unique solution for

FOS begins by creating a functional expansion using orthogonal basis functions such that:
_{m}_{m}_{m}

The orthogonal functions _{m}_{m}_{0}(_{0}(

The next orthogonal function _{1}(_{0}(_{1}(_{10} is the GS weight. Now _{0}(_{1}(

Solving

Subsequent orthogonal functions are found by subtracting weighted values of all the previously fitted orthogonal functions from the kth candidate function as given by:

The orthogonal functions _{m}_{mr}_{mr}

Note, it can be shown that [

The next step in FOS is to compute the weights of the orthogonal functional expansion _{m}

By taking the derivative with respect to _{m}_{m}

The correlation between the input _{m}

Using _{m}

In its last stage, FOS calculates the weights of the original functional expansion _{m}_{m}_{ir}_{m}

From

Using the fact that the _{m}

It then follows that the MSE reduction given by the m^{th} model addition is:

FOS can fit a model with a small number of model terms by fitting terms, which reduce the mean squared error (MSE) in order of their significance. The FOS search algorithm is stopped in one of three cases. The first is when certain maximum number of terms is fitted. The second case is when the ratio of MSE to the mean squared value of the input signal is below a pre-defined threshold. The third case is when adding another term to the model reduces the MSE no more than if it were fitting white Gaussian noise. In the case where FOS first fits a constant term (_{0}(_{m}_{+1} is the MSE reduction of the new term and N is the number of points in the input data. A threshold test that does not assume a zero frequency term is fitted first can be found in [

Spectral analysis with FOS is accomplished by selecting candidates _{m}_{m}_{m}

There are at least two significant differences between FOS and the discrete Fourier transform (DFT) [

FOS is appreciably better at rejecting coloured and white noise than the commonly used FFT techniques (example in [

In this research, FOS is used for inertial sensor accuracy enhancement. In general, signal de-noising typically involves: (1) transforming the data into a different domain (

FOS has been shown to be good at detecting the frequencies of interest buried in coloured and white noise even at signal to noise ratios (SNR) as low as −10 dB [

The basic principle behind the proposed FOS-based inertial sensor accuracy enhancer is to use FOS to model the motion dynamics measured by the inertial sensors, and reject as much of the inertial sensor error components as possible. As FOS models the noisy input data, it implicitly performs the aforementioned tasks of transforming and thresholding. FOS performs thresholding using

It should be noted that this process of de-noising is appropriate for stationary data (the frequency content of the data does not change with time), and that inertial sensor data is typically non-stationary since they are measurements of the vehicle motion dynamics and the sensor noise is also known to be non-stationary. The challenge of de-noising non-stationary inertial sensor data is addressed by taking segments of inertial sensor output and applying the FOS to each segment.

The FOS accuracy enhancement technique initially segments a noisy input time series, denoted

As FOS is generally known to be a data dependent algorithm [

Sinusoidal candidate functions were selected in this research because they had been successfully applied to de-noising [

The FOS candidate frequencies are chosen to have a higher resolution than the fast Fourier transform (FFT) to achieve better de-noising. The frequency resolution of an FFT can be given by:
_{S}

From

Candidate frequencies can be selected so that the candidate functions focus on a particular frequency range of interest. For example, the candidates can be spaced with a high resolution on a range of interest and outside the range of interest, the candidates can be spaced by FFT resolution intervals.

It is desirable to have the minimum number of candidate frequencies in a model required to model the motion dynamics. Too few terms results in a model that does not accurately model the input signal. Too many terms will add noise terms into the motion dynamics model as well as increase the computation time. In this research, the maximum number of frequencies to add (MAXFTA) is typically set between 6 and 15, not including the initial zero frequency model term.

FOS stops modelling when adding a new frequency pair does not reduce the MSE more than fitting WGN. It is known that INS data includes WGN and coloured noise, which may not be rejected by this threshold. Thus, a candidate acceptance threshold, requiring a frequency pair to fit a minimum percentage of the overall energy in the signal, is set [

Finally, in synthesizing the noise free motion dynamics, FOS can synthesize the time-series estimates using only a certain range of frequencies detected by FOS. Since it is known that motion dynamics are typical low frequencies, (e.g., between 0 and 3 Hz for land vehicles in benign environments), FOS can model the motion dynamics with only model terms in this frequency range while rejecting the higher frequency terms as noise. This has the effect of low-pass filtering, thereby eliminating the short term error components of the inertial sensor measurements. Using FOS differs from using a low-pass filter (LPF) in that it has an adaptive noise rejection threshold and only fits a small number of frequency terms, whereas a LPF will pass all frequencies within its passband.

A set of simulated inertial signals (gyroscopes and accelerometers in x, y, z axes) was generated at a 75 Hz data rate to correspond to a 120 s land vehicle trajectory. The true motion dynamics were generated using windowed sinusoidal functions, and the data was processed using an INS mechanization algorithm to produce an error-free reference trajectory [

The short-term error component is made up of white Gaussian noise with variance levels matching those observed from a low-end tactical grade IMU (TG-6000, KVH Industries Inc., Middletown, RI, USA). The long-term error component is created with a 1st order Gauss-Markov process with a standard deviation and correlation time similar to those observed from the TG-6000 IMU. The Gauss-Markov disturbance is representative of long term inertial sensor errors like the bias drift, and has a correlation time of 1 h.

As mentioned, the results of the FOS de-noising are compared to results of the wavelet de-noising. The parameters of the wavelet de-noising function include the type of wavelet basis function, the number of levels of decomposition (LOD), and thresholding rules. The Daubechies family of wavelets with soft thresholding based on Stein's Unbiased Risk Estimate (SURE) are used in this paper as these parameters are typically used in pre-filtering inertial sensors [^{n}

Both the FOS and Wavelet de-noising used 15 s segments. The wavelet de-noising parameters used are shown in ^{6} = 0.586 Hz). Since it is known in this simulation that there are no motion dynamics present for the x-gyro, y-gyro and z-acc, these signals should be de-noised as much as possible by using the maximum LOD (10 in this case).

The FOS de-noising parameters used for this data are summarized in

The resultant accuracy enhanced signals for the gyroscopes and accelerometers are shown in the

The impact of the proposed FOS method as a high-resolution spectral de-noising technique for inertial sensors is examined on two different types of inertial systems during two different road tests performed within the city of Kingston (Ontario, Canada) [

The first experiment used a TG-6000 tactical grade IMU (KVH Industries, Inc.) and an 8-channel continuous-tracking GPS receiver with antenna (Lassen SQ GPS, Trimble Navigation Ltd., Sunnyvale, CA, USA). The inertial system used in the second road test is the Crossbow MEMS grade IMU (Crossbow Technologies, San Jose, CA, USA). Some specifications of both IMUs and GPS module are provided in

In this study, for both the tactical grade and MEMS based inertial systems, the raw noisy inertial sensor measurements (denoted as NSY data) is processed by both the wavelet de-nosing procedure [

One of the objectives of this experimental work is to assess the position domain accuracy enhancement of FOS during a GPS outage. Although there were no GPS outages during each of the above road tests, several 30 s GPS outages were intentionally introduced within each trajectory. The raw inertial sensor data and the GPS measurements were processed without applying any GPS outages to create a baseline AINS position domain solution and may be denoted REF from hereon. The AINS position solutions for the NSY, WDN, and FOS inertial sensor data sets during the outages are compared against the REF trajectory. The corresponding position errors during the outages can be used to assess FOS as an accuracy enhancement technique for a real INS/GPS integrated system.

The trajectory of the first road test is shown in

The WDN and FOS signals followed the trends of the NSY signals, but the WDN and FOS results had noticeably less short-term error components (

The results verify that FOS is capable of processing real IMU data from a relatively long trajectory of one hour in duration. The FOS output is noticeably less noisy than the NSY signals for all sensors, and it is even less noisy than the WDN signals in the case of the X and Y gyroscopes (not shown in this paper). The residual frequency components above the estimated motion dynamics frequency range that are present in the WDN signals may be attributed to the thresholding techniques used in wavelet de-noising. Due to the setting of the parameters, FOS did not include these frequency components when it synthesized the estimate of the vehicle motion dynamics. These additional components do not likely constitute a significant portion of the true motion dynamics since they are outside the expected frequency range for land vehicle motion, but they could be attributed to road surface irregularities or vibrations from other machinery within the vehicle (

The NSY, WDN, and FOS position error values during the artificial GPS outages were in the order of those observed in other experimental work [

The inertial system used in the second road test is the Crossbow MEMS grade IMU (Crossbow Technologies) with its specifications shown on

Clearly the overall system performance benefited from the de-noising by FOS and position errors was reduced after the pre-filtering process. Moreover, except for GPS outages 3 and 5, FOS provided better accuracies than wavelet. In fact, WDN was providing almost the same accuracy of the raw noisy signal for all GPS outages. The average percentage improvement (considering all 9 GPS outages) for the vehicle horizontal position was 24% when using FOS and almost no improvement when using wavelets. In fact the superior accuracy obtained with FOS de-noising is due to the ability of FOS to fit the frequencies with the highest energy first, which are mostly the motion dynamic terms, and then reject terms (the long term errors or insignificant motion dynamics) with its adaptive noise threshold. Such errors exist in the very low frequency part of the signal and mixed with motion dynamics, thus cannot be removed by wavelet de-nosing since it is basically a lowpass filtering and do not include the noise rejection threshold inherent in FOS.

The experimental work discussed in this section for tactical grade IMUs validates FOS as an inertial sensor accuracy enhancement technique that is applicable to a real INS integrated with a GPS and applicable to relatively long trajectories. Processing raw inertial measurements with FOS can contribute to improved accuracy enhancement in the position domain during GPS outages, and the position improvements are frequently better than wavelet de-noising. For tactical grade IMUs, the mean horizontal position error was reduced by 23% after applying FOS while wavelet de-nosing has only shown 10% enhancement. For MEMS-based IMU, FOS has shown 74% improvement while wavelet has only shown 55%. The main reason of the superior accuracy provided by FOS is its capability of removing the long-term inertial sensor errors prior to INS mechanization and KF-based integration with GPS.

This research was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC) and Geomatics for Informed Decision (GEOIDE) Network of Centre of Excellence (NCE). The financial support of UKM-DLP-2011-002 in Malaysia and DND in Canada are highly appreciated.

^{γ}) noise in inertial sensors

Block diagram of FOS-based inertial sensor accuracy enhancement technique (bullets denote operating parameters).

Accuracy enhanced gyroscope signals. (

Accuracy enhanced accelerometer signals. (

Trajectory plots.

Experimental setup including inertial systems, GPS and data acquisition modules mounted inside land vehicle.

The first road test trajectory (for the TG6000 IMU) with the location of some intentionally introduced GPS outages indicated.

PSD plots for TG6000 inertial sensors before and after applying both FOS and WDN.

Land vehicle experiment position domain results for GPS outage 1. (

Land vehicle experiment position domain results for GPS outage 2. (

Land vehicle experiment position domain results for GPS outage 3. (

Land vehicle experiment position domain results for GPS outage 4. (

The second road test trajectory (for MEMS based IMU) with the location of some intentionally introduced GPS outages.

PSD plots for MEMS inertial sensors before and after applying both FOS and WDN.

Positioning accuracy at the end of 9 GPS outages for MEMS IMU for the 2nd road test.

Wavelet de-noising parameters.

| |||
---|---|---|---|

x-gyro, y-gyro | Daubechies 1 (db1) | 10 | Soft, SURE, No Rescaling |

z-gyro | Daubechies 3 (db3) | 6 | Soft, SURE, No Rescaling |

x-acc, y-acc | Daubechies 3 (db3) | 6 | Soft, SURE, No Rescaling |

z-acc | Daubechies 1 (db1) | 10 | Soft, SURE, No Rescaling |

FOS de-noising parameters.

| ||||
---|---|---|---|---|

x-gyro, y-gyro | 0–0.0366 @ 1/8 FFT Res | 4% var [NSY] | 6 | 0–0.0366 |

z-gyro | 0–0.6 @ 1/8 FFT Res | 4% var [NSY] | 6 | 0–0.6 |

x-acc, y-acc | 0–0.6 @ 1/8 FFT Res | 4% var [NSY] | 6 | 0–0.6 |

z-acc | 0–0.0366 @ 1/8 FFT Res | 4% var [NSY] | 6 | 0–0.0366 |

Summary of position domain accuracy enhancement.

| |||
---|---|---|---|

204.0 | 192.6 | 27.8 | |

- | 5.6% | 86.4% | |

118.8 | 110.4 | 17.5 | |

- | 7.1% | 85.2% |

Specifications for TG-6000 IMU and Lassen SQ GPS module.

KVH Industries Inc., Middletown, RI | Crossbow Technologies | Trimble Navigation Ltd., Sunnyvale, CA | |

Operational limits: <18,000 m or velocity <515 m/s | |||

Data Rate: up to 150 Hz |
Data Rate: up to 200 Hz |
Update Rate: 1 Hz | |

Input Voltage: 14 to 30 VDC, 12 Watts max. | Input Voltage: 9 to 30 VDC, 3 Watts max. | Input Voltage: 3.0 to 3.6 VDC, 0.5 Watts max. | |

75 Hz sampling rate during data acquisition | 20 Hz sampling rate during data acquisition | Uses 3 V active micropatch antenna with magnetic mount |

Summary of position domain accuracy enhancement during GPS outages (Road Test 1).

| |||||
---|---|---|---|---|---|

38.8 | 24.7 | 32.8 | 29.3 | 31.4 | |

37.4 | 12.9 | 31.9 | 28.2 | 27.6 | |

4% | 48% | 3% | 4% | 14% | |

36.3 | 16.3 | 31.6 | 23.3 | 26.9 | |

7% | 34% | 4% | 20% | 16% | |

19.4 | 9.5 | 14.8 | 15.2 | 14.7 | |

18.7 | 7.5 | 13.4 | 14.0 | 13.4 | |

3% | 21% | 9% | 8% | 10% | |

18.1 | 6.6 | 13.3 | 8.6 | 11.6 | |

6% | 31% | 10% | 43 % | 23% |

Summary of percentage position improvement during GPS outages (Road Test 2).

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|

| |||||||||

1.7 | 0.4 | 0.0 | 0.0 | −8.4 | 2.3 | 0.1 | 0.8 | 0.7 | |

31.3 | 74.7 | −50.6 | 64.2 | −77.8 | 7.6 | 48.2 | 29.3 | 86.4 |