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Accessibility to inertial navigation systems (INS) has been severely limited by cost in the past. The introduction of low-cost microelectromechanical system-based INS to be integrated with GPS in order to provide a reliable positioning solution has provided more wide spread use in mobile devices. The random errors of the MEMS inertial sensors may deteriorate the overall system accuracy in mobile devices. These errors are modeled stochastically and are included in the error model of the estimated techniques used such as Kalman filter or Particle filter. First-order Gauss-Markov model is usually used to describe the stochastic nature of these errors. However, if the autocorrelation sequences of these random components are examined, it can be determined that first-order Gauss-Markov model is not adequate to describe such stochastic behavior. A robust modeling technique based on fast orthogonal search is introduced to remove MEMS-based inertial sensor errors inside mobile devices that are used for several location-based services. The proposed method is applied to MEMS-based gyroscopes and accelerometers. Results show that the proposed method models low-cost MEMS sensors errors with no need for denoising techniques and using smaller model order and less computation, outperforming traditional methods by two orders of magnitude.

Presently, GPS-enabled mobile devices offer various positioning capabilities to pedestrians, drivers, and cyclists. GPS provides absolute positioning information, but when signal reception is attenuated and becomes unreliable due to multipath, interference, and signal blockage, augmentation of GPS with inertial navigation systems (INS) or the like is needed. INS is inherently immune to the signal jamming, spoofing, and blockage vulnerabilities of GPS, but the accuracy of INS is significantly affected by the error characteristics of the inertial sensors it employs [

GPS/INS integrated navigation systems are extensively used [

According to [

The fusion of INS and GPS data is a highly synergistic coupling as INS can provide reliable short-term positioning information during GPS outages, while GPS can correct for longer-term INS errors [

Despite having an INS/GPS integration algorithm to correct for INS errors, it is still advantageous to have an accurate INS solution before the data fusion process. This requires preprocessing (i.e., prefiltering or denoising) each of the inertial sensor (gyroscope and accelerometer) signals before they are used to compute position, velocity, and attitude. This paper offers a robust method based on fast orthogonal search (FOS) to model the stochastic errors of low-cost MEMS sensors for smart mobile phones.

Orthogonal search [

Many techniques have been used previously to denoise and stochastically model the inertial sensor errors [

FOS has been applied before in several applications [

It is generally accepted that the long-term errors are modeled as correlated random noise. Correlated random noise is typically characterized by an exponentially decaying autocorrelation function with a finite correlation time. When the autocorrelation function of some of the noise sequences of MEMS measurements is studied, it has been shown that a first-order Gauss-Markov (GM) process may not be adequate in all cases to model such noise behavior. The shape of the autocorrelation sequence is often different from that of a first-order GM process, which is represented by a decaying exponential as shown in Figure

The autocorrelation sequence of a first-order Gauss-Markov (GM) process.

Most of the computed autocorrelation sequences follow higher-order GM processes. An example of such computed autocorrelation sequences for one hour of static data of an MEMS accelerometer is shown in Figure

The computed autocorrelation sequence for an MEMS accelerometer data.

The autoregressive moving average (ARMA) modeling is based on the mathematical modeling of a time series of measurements assuming that each value of such series is dependent on (a) a weighted sum of the “previous” values of the same series (AR part) and (b) a weighted sum of the “present and previous” values of a different time series (MA part) [

The resultant equation is written as [

The input-output relationship of an autoregressive (AR) process.

The problem in this case is to determine the values of the AR model parameters (predictor coefficients)

Therefore, the resultant energy from (

Several methods have been reported to estimate the

The Yule-Walker method, which is also known as the autocorrelation method, determines first the autocorrelation sequence

If the mean-square error

Equations (

Therefore, the values of the AR prediction coefficients in the Yule-Walker method are provided directly based on minimizing the forward prediction error

However, the Yule-Walker method performs adequately only for long data records [

The covariance method is similar to the Yule-Walker method in that it minimizes the forward prediction error in the least-squares sense, but it does not consider any windowing of the data. Instead, the windowing is performed with respect to the prediction error to be minimized. Therefore, the AR model obtained by this method is typically more accurate than the one obtained from the Yule-Walker method [

Furthermore, it uses the covariance

The method provides more accurate estimates than the Yule-Walker method especially for short data records. However, the covariance method may lead to unstable AR models since the LD algorithm is not used for solving the covariance normal equations [

Burg’s method was introduced in 1967 to overcome most of the drawbacks of the other AR modeling techniques by providing both stable models and high resolution (i.e., more accurate estimates) for short data records [

The forward and backward prediction error criteria are the same, and, hence, they have the same optimal solution for the model coefficients [

These recursion formulas form the basis of what is called Lattice (or Ladder) realization of a prediction error filtering (see Figure

The forward-backward predication error lattice filter general structure.

As has been shown for the Yule-Walker method, the accuracy of the estimated parameters

This leads to the form

Therefore, the utilization of (

FOS [

The functional expansion of the input

These model terms can involve the system input

FOS begins by creating a functional expansion using orthogonal basis functions such that

The Gram-Schmidt coefficients

The MSE of the orthogonal function expansion has been shown to be [

The data were collected by a low-cost MEMS-based inertial measurement unit (IMU CC-300, Crossbow). These measurements were collected during a one-hour experiment to obtain stochastic error models of both gyroscopes and accelerometers. To illustrate the performance, two sensors were selected as an example (accelerometer-Y, Gyro-Y) while the other inertial sensors gave similar results. Figure

Accelerometer-Y and Gyro-Y specific force measurements.

FOS is applied directly on the raw inertial sensor 200 Hz data without any preprocessing or denoising. Traditional methods like Yule Walker, Covariance, and Burg perform poorly on the raw data, so we first applied wavelet denoising of up to 4 levels of decomposition that resulted in band limiting the spectrum of the raw inertial sensor data to 12.5 Hz. Therefore, unlike FOS, the other 3 methods operate on the denoised version of the same data. After denoising, AR model parameters were then estimated as well as the corresponding prediction MSE for all sensors using Yule-Walker, Covariance, and Burg methods.

For FOS, the raw INS data were divided into three datasets for model training, evaluation, and prediction stages. The first 3 minutes of the INS raw data were utilized for model training, which uses the FOS algorithm to identify several possibly nonlinear AR equations. Different models can be obtained by changing the maximum delay

Accelerometer-Y prediction MSE using Yule-Walker, covariance, Burg, and FOS methods. For CP

Table

Performance summary of AR models obtained by Yule-Walker, Burg, and FOS over one hour of accelerometer-Y measurements: first-order FOS: only linear model terms; second-order FOS: linear and cross-product model terms.

Modelling technique | Model MSE (m/s^{2})^{2} |
Corresponding position error (m) | Computational time (s) |
---|---|---|---|

Model order (maximum output lag | |||

Yule-Walker | 5 × 10^{−9} |
458 | 0.25 |

Covariance/Burg | 3 × 10^{−11} |
35 | 0.23 |

CP degree = 1 FOS | 1 × 10^{−11} |
20 | 0.13 |

CP degree = 2 FOS | 1 × 10^{−11} |
20 | 0.36 |

| |||

Model order (maximum output lag | |||

Yule-Walker | 5 × 10^{−9} |
458 | 0.69 |

Covariance/Burg | 2 × 10^{−12} |
9 | 0.45 |

CP degree = 1 FOS | 3 × 10^{−14} |
1 | 0.30 |

CP degree = 2 FOS | 2 × 10^{−16} |
0.09 | 0.60 |

A similar procedure was performed for the Gyro-Y sensor measurements. Figure

Performance summary of both AR Yule-Walker and Burg models and FOS model over one hour of Gyro-Y measurements: first-order FOS: only linear model terms; second-order FOS: linear and cross-product model terms.

Modeling technique | Model MSE (deg/h)^{2} |
Corresponding position error (m) | Computational time (s) |
---|---|---|---|

model order (maximum output lag | |||

Yule-Walker | 7 × 10^{−6} |
978 | 0.40 |

Covariance/Burg | 5 × 10^{−6} |
826 | 0.23 |

CP degree = 1 FOS | 1 × 10^{−6} |
369 | 0.09 |

CP degree = 2 FOS | 1 × 10^{−6} |
369 | 0.35 |

| |||

Model order (maximum output lag | |||

Yule-Walker | 2 × 10^{−6} |
523 | 0.44 |

Covariance/Burg | 2 × 10^{−7} |
165 | 0.28 |

CP degree = 1 FOS | 4 × 10^{−10} |
7 | 0.18 |

CP degree = 2 FOS | 3 × 10^{−11} |
2 | 0.6 |

Gyro-Y prediction MSE using Yule-Walker, covariance, Burg, and FOS methods. For CP

Similar to the accelerometer case, the stochastic model obtained for Gyro-Y using FOS surpasses the models obtained by the other methods in the MSE, the corresponding position error that would result from the residual errors and the computation time.

Inertial sensor errors are the most significant contributors to INS errors. Thus, techniques to model these sensor errors are of interest to researchers. The current state of the art in modeling inertial sensor signals includes low-pass filtering and using wavelet denoising techniques, which have had limited success in removing long-term inertial sensor errors.

This paper suggested using FOS to model the MEMS sensor errors in time domain. The FOS MSE and computational time are compared with those from Yule-Walker, the covariance, and Burg methods. FOS was applied directly to the one-hour raw inertial sensor 200 Hz data without any pre-processing or denoising. The other traditional 3 methods operated on the denoised version of the same data after we applied the wavelet denoising of up to 4 levels of decomposition.

For either gyroscope or accelerometer case, the FOS model surpasses those obtained by traditional methods. The results demonstrate the advantages of the proposed FOS-based method including the absence of a need for preprocessing or denoising, lower computation time and MSE, and achieving better performance with smaller model order. Increasing the cross-product degree for FOS improves model accuracy and lessens position error but increases computation time.