A hybrid error modeling for MEMS IMU in integrated GPS/INS navigation system

Notation

Throughout this paper, the following notations are adopted: small letters with over line represent vectors and with hat represent the predicted values.

GPS/RISS integrated system

The RISS essential goal is to provide a good functioning navigation solution with reduced cost. Cost reduction achieved through MEMS usage while limiting the number of used sensors is one of the RISS short ways for error minimization (Iqbal et al. 2008). A vehicular navigation solution was introduced in (Georgy et al. 2010) using 3D GPS/RISS loosely coupled integration scheme. Sole gyro vertically aligned with the vehicle body frame and two accelerometers pointed in the vehicle’s forward and transverse directions together with the built-in odometer or mounted wheel speed sensor form the 3D RISS as shown in Fig. 1. Besides lower cost as a result for usage of fewer inertial sensors for 3D RISS, it has two main advantages over a full IMU. These advantages are represented in the usage of accelerometers for roll and pitch calculations instead of gyros, and the calculation of velocity based on wheel speed sensor instead of accelerometers. Superiority of these calculations in attitude and position error reduction is detailed in (Noureldin et al. 2013). This left the vertically aligned gyro as the only main remaining source of error in RISS, and can be limited through appropriate modeling of its stochastic drift (augmented KF-FOS and our hybrid SVM-FOS). Assuming the vehicle mostly remains in the horizontal plane, the speed sensor measurement is used to obtain the vehicle’s speed (V _V ) and its heading is deduced from the gyroscope reading. Based on these values, the velocities along the east direction (V_e) and the north direction (V_n) are derived to enable tracking of longitude and latitude.

The vehicle’s pitch angle (β) represents the angle it makes with respect to the ground level and its roll angle (γ) represents its rotation about the longitudinal axis. Based on the vehicle’s actual acceleration (a_V), transversal accelerometer measurement (f_x), forward accelerometer measurement (f_y) and vertical gyro measurement (ω_z) the off-plane motion can be estimated and consequently compensate tilt errors to obtain accurate azimuth calculation. Pitch and roll angles can be obtained as follows (Karamat et al. 2014):

where g represents the gravity acceleration.

To obtain the azimuth (A _z ), a compensation for vertical inclination and the effect of the Earth’s angular rate (ω _e ) should be performed to compute the vehicle azimuth relative to the north direction. The derivation of this compensation was discussed deeply in (Noureldin et al. 2013). In our algorithm, we use the following relation to compute the azimuth:

where φ is the latitude, h is the elevation, and R _N represents the normal radius of the Earth’s ellipsoid curvature. The negative sign in the above equation is due to using of ENU frame. The second term represents the compensation for Earth’s rotation. Meanwhile, the third term represents the compensation of the local level vertical misalignment. Then, the velocity components along the east, and north axes beside vertical velocity (V _u ) can be obtained as:

The position components including latitude, longitude (λ), and elevation can be computed according to (5) taking into consideration that R_M represents meridian radius of the Earth’s ellipsoid curvature.

Equations for 3D RISS, well derived in (Georgy et al. 2010), show that IMU with this reduced sensors number together with vehicle speed sensor is capable of producing vehicle’s complete navigation solution. A simplified block diagram for 3D RISS mechanization is shown in Fig. 2.

The 3D RISS uses two accelerometers instead of gyros to obtain pitch and roll angles. In fact, accelerometers generate less error than gyros as shown in (6) where the relation between the accumulated position error over time and some inertial error parameters is presented. These inertial parameters include mechanization without any provided corrections over Δt time period.

where δp(t) is the positional error drift after time (t) with initial value δp(t₀), δV(t₀) is the initial velocity error, b _a (t₀) and δ _SFa (t₀) are the initial accelerometer offset bias and scale factor error respectively, δb _g (t₀) and δ _SFg (t₀) are the initial gyro offset bias and scale factor error respectively. δθ_{r, p}(t₀) and δθ _A (t₀) are the initial non-orthogonality error due to the roll/ pitch and azimuth errors respectively. V is the average velocity during time period Δt. As the residual gyro errors are multiplied by the cube of the time interval, it produces the largest positional drift over time. KF integration with GPS provides the corrections needed to reduce the inertial errors but during its denied conditions the errors problem remains the same. Estimation of nonlinear errors can be used to limit its rapid accumulation during GPS outage periods.

Velocity calculations also provide a further improvement in RISS since it uses speed sensor reading as a replacement for accelerometers. Velocity calculation based on accelerometers introduces velocity error proportional to t and position error proportional to t ² (Karamat et al. 2014). On the other hand, velocity calculations based on speed sensor need only single usage of integration, which finally improves the position error. The same relations are applicable during GPS outage periods, where the error will be related to the square of the outage period in case of accelerometers based calculations. So, the azimuth errors are the main source of errors in 3D RISS due to its vertically aligned gyro. Errors nonlinear estimation is one of the methods that can be used to limit its rapid accumulation during GPS outage periods. In the meanwhile, providing solutions for KF aiding during outage periods enhance its estimation capabilities.

KF implementation needs identification for system error model. RISS error state vector composed of position errors (δφ, δλ, δh), velocity errors (δV_e, δV_n, δV_u), azimuth error (δA_z), and sensors error (δa_V, δω_z). These errors can be represented as differential equations through application of Taylor series approximation of RISS mechanization equations. The error model can be described as follows (Noureldin et al. 2013):

where Δ(.) represents the corresponding higher order errors.

Since proposed system has a GPS, so its measurement can be used for initialization process. Unless starting point is priori known, GPS position is used for initialization. The velocity vector is initialized by zero if the vehicle is stationary during initialization process. Otherwise, in dynamic conditions, the GPS velocity measurement is used to initialize the velocity vector.

Modeling algorithms

Fast orthogonal search (FOS)

FOS (Korenberg 1989) is employed in this work to model the significant nonlinear azimuth error of RISS gyro. Improving RISS azimuth calculation inside mechanization algorithm during GPS denied conditions was the motive behind the nonlinear azimuth error model development based on FOS with its high-resolution estimation capabilities. Based on a weighted sum of M linear or nonlinear arbitrary basis functions p _m (n), FOS is an identification technique that approximates system’s output (Tamazin et al. 2016). Weighting coefficients k _m are identified targeting for minimization of the mean square error (e²(n)) between the real system output y(n) and its estimate \( \widehat{y}(n) \). The constructed model takes the form:

FOS selects M most significant functions which greatly reduce the mean square error (MSE) through iteratively searching N available candidate basis functions. FOS algorithm employs Gram-Schmidt (GS) orthogonalization principals to identify orthogonal candidates q_m(n) generated from p _m (n) function, and coefficients g _m (n) to quantify the error reduction. GS usage produces a new orthogonal set, and the model takes the form:

FOS reduces the time consumed for orthogonal functions definition and the required memory through its implicit determination using GS coefficients, which are calculated using time averaging instead of point-by-point computation (Korenberg 1989). The original weights k _m can be identified from the orthogonal expansion weights and GS coefficients. The FOS iterative process for orthogonal functions identification is stopped when the error of the model is less than an acceptable value. Because the FOS orthogonal candidates are not computed directly, models development is extremely fast. Further details of FOS algorithm calculations can be found in (Korenberg 1989).

In this paper, modeling the nonlinear azimuth error is the task assigned to the FOS algorithm. As shown in Fig. 3, the autocorrelation function (ACF) for the nonlinear azimuth errors demonstrates high correlation throughout the whole test track. So, unlike approaches (Zhi et al. 2011; Zhi 2012; (Tamazin et al. 2013) for modeling of nonlinear azimuth errors, AR concept is employed to use candidates represent the previously calculated nonlinear azimuth errors. This approach reduces the uncertainty of using KF output as candidates since during GPS outage it will not be so accurate due to the lack of GPS updates. The modeled nonlinear azimuth error at training instant i will be represented by y(n) in (18) while p _m will represent the model candidates constructed from the previous nonlinear azimuth error (m = 1,…, i-1).

Proposed hybrid error modeling

Augmented KF-FOS approaches discussed in (Zhi et al. 2011; Zhi 2012; Tamazin et al. 2013) are based on FOS to model the nonlinear azimuth errors. These approaches used the linear errors estimated by KF as candidates for the model. During model training, the output of KF is delivered to FOS to build the nonlinear model. During GPS outage, the linear errors required for the model to estimate the nonlinear errors are delivered from KF in prediction mode with a lack of its GPS updates. These approaches result in models with associated uncertainties which affect the positioning accuracy.

To reduce the position error of the GPS/RISS integrated system, an SVM-FOS hybrid modeling algorithm is proposed to model individually KF output and azimuth residual nonlinear errors. The idea behind this proposal is to isolate the influence of the unaided KF output on the final navigation results during GPS outage. This achieved through the usage of SVM to model the KF output during its GPS aiding and replacing KF during GPS unavailability. In the meanwhile, FOS is used to model the nonlinear azimuth error based on the AR concept to be estimated in case of GPS outage. Keeping in mind that the KF output represents the linear error, this approach enables the model to estimate both linear and nonlinear errors without using KF prediction in case of GPS outage. The block diagram of the proposed approach is shown in Fig. 4.

Through GPS availability, the integrated system provides trusted navigation information. A sliding window is used to continuously provide training data for both modeling algorithms. Once GPS terminates aiding KF for any reason, models training are stopped and estimation configuration starts its mission. SVM models the KF output (\( \delta {A}_z^{KF} \)) which represents linear azimuth error, while the FOS models the true nonlinear azimuth error (∆A _z ). ∆A _z can be identified through calculations based on mechanized azimuth (\( {A}_z^{mec} \)), GPS aiding azimuth (\( {A}_z^{GPS} \)) and the KF predicted azimuth which represents the linear azimuth error as follow:

GPS aiding azimuth (\( {A}_z^{GPS} \)) is obtained by incorporating GPS east and north velocity components (\( {V}_e^{GPS},{V}_n^{GPS} \)) (El-Sheimy et al. 2010). It is measured from north direction, and can be calculated as in (13).

The lever arm between the GPS antenna and the IMU must be compensated before GPS azimuth determination to improve the obtained azimuth accuracy. To achieve stable GPS velocity derived azimuth with reasonable accuracy, a constrained azimuth algorithm must be used to overcome the numerical problems encountered during low dynamics as denominator approaches zero (Huang et al. 2007). The obtained azimuth supposed to be in the interval (−π, π). Also, the algorithm takes care if zero forward velocity is sensed (e.g., during ZUPT).The accuracy of the calculated azimuth depends on the GPS receiver accuracy in velocity determination. In our experiment, Ublox GPS receiver chipset was used, which already provide heading information. According to its data sheet, the heading accuracy is 0.5 degrees and velocity accuracy is 0.1 m/s.

Three candidates sets were checked and compared to obtain more accurate SVM model for \( \delta {A}_z^{KF} \): SVM1 contains only \( {A}_z^{mec} \), SVM2 adds mechanized velocities (V _e , V _n ) with \( {A}_z^{mec} \) while SVM3 contains angular rate ω _z  and mechanized velocities (V _e , V _n ). Mechanized velocities are proposed as candidates because azimuth is in direct relation with their values as seen in (3). The third option SVM3 is selected because it got the best training results as shown in the next section. FOS training depends on delivering the calculated ∆A _z as proposed candidates to construct the model.

Once GPS outage occurred, the proposed system enters the prediction phase and the algorithm freeze the obtained models. Figure 3 illustrates the architecture of the system in this phase. The input data still delivered to the SVM model to estimate the KF output as the linear azimuth error (\( \delta {\widehat{A}}_z^{KF} \)). FOS model starts to predict the nonlinear azimuth error (\( \Delta {\widehat{A}}_z \)), delivered it to the algorithm and feed it back to the model. This estimated error used to update the data in the model input buffer to use it in the next epoch. The corrected azimuth estimation (\( {A}_z^{\ast } \)) is calculated as:

This estimated azimuth is delivered with speed sensor measurement to another mechanization algorithm (INS mechanization II in Fig. 5) to calculate the corrected navigation data.

Because our proposal does not depend on the KF output during GPS blockage, the KF remains working in the background in the prediction configuration without any GPS updates. During GPS updates, KF linearized model dependence on a linear stochastic model for the sensor error such as Gauss Markov model does not affect the system accuracy. The KF prediction works in the background enables the system to accelerate the KF convergence after reacquisition of GPS aiding signal.