Inertial navigation is a method where the current position, velocity and attitude of a moving object are determined from a history of acceleration and angular velocity measurements. Acceleration and angular velocity are measured using accelerometers and gyros. Unlike GNSS systems, the INS performance is not affected in environments as urban canyons; it is independent of external electro-magnetic signals. However, the main drawback of an INS is the degradation of its performance with time. In order to control the errors to an acceptable level continues updates from, for example, GNSS are necessary.

The mathematical model of the inertial navigation system is commonly described in the framework of linear dynamic systems. The dynamic behavior of such systems can be described by using a state-space representation. For this purpose, a system of non-linear first-order differential equations can be described as [23] :

where is the position vector, latitude, longitude and altitude; C is a transformation matrix from the East, North and Up (ENU) reference frame to earth-centered earth-fixed (ECEF) frame (Jakili, 2001); is the velocity vector in ENU reference frame, is the kinematic acceleration vector in the ENU reference frame; represents the effect of the motion of the ENU frame with respect to the ECEF frame; is the Coriolis acceleration vector; is the gravity vector, including the gravitation term and the centripetal term related to the Earth rotation; and is the specific force vector in the body frame, which is measured by the accelerometers. The matrix is the skew-symmetric matrix of rotation rate vector of the Earth, which can be expressed in the ENU frame as:

The matrix is the skew-symmetric matrix of the rotation rate vector of the ENU frame with respect to ECEF frame, expressed in the ENU frame as:

The matrix is the skew-symmetric matrix of the rotation rate vector of the body frame with respect to the ECI frame, expressed in the body reference, which is measured by the gyros. The matrix is the skew-symmetric matrix of the rotation rate of the navigation frame with respect to inertial frame expressed in the body frame, which is computed combining and transforming the result in the body frame as follows:

To build the proposed GNSS/INS integrated navigation system, tightly coupled architecture is implemented adopting extended Kalman filter (EKF). GNSS pseudorange, carrier phase and Doppler measurements as well as INS-derived observations are processed to produce estimates of the state vector including position, velocity and attitude. The precise GNSS ephemerides as well as the outputs of position and velocity from the iner-

tial sensors mechanization are used to predict the INS pseudorange, phase and Doppler measurements. The corrected GNSS pseudorange, phase and Doppler measurements are

differenced with the INS-predicted measurements. The residuals and are then directly processed by the integration filter to estimate the system error state vector. The obtained INS error estimates, such as the inertial sensors bias drifts and, and scale factors and, are fed back to the INS mechanization to correct for the inertial sensors forces and using the closed loop approach. The estimated error states such as position errors, velocity errors and attitude errors are directly applied to the INS-derived position, velocity and attitude solutions. States unique to GNSS such as the clock offset, clock drift, internal system biases ISB and ambiguity parameters N, are fed back to continue correct for the GNSS observations using additional closed loop technique. A priori estimation constrains are applied on GPS/GLONASS and GPS/Galileo internal system biases (ISBs) to continue benefits from additional GNSS satellites when the number of GLONASS or Galileo satellites drops to one satellite. Figure 1 shows the tightly coupled GNSS PPP/INS implementation flowchart.

To implement the mechanization of the developed integrated system, the EKF is used as an estimator to merge the GNSS observations and INS records. The estimated state vector δx consists of 26 + n states describing the basic state vector including the nine navigation parameter errors, the inertial sensors errors including the bias drift and scale factor, and errors unique to the GNSS measurements, which are mainly the receiver clock offset and drift, the troposphere wet delay component, the GPS/GLONASS ISB and GPS/Galileo ISB with additional n states related to the float ambiguity parameters Bi. The complete state vector for un-differenced ionosphere-free technique can be written as.

where is a three-dimensional vector representing the positioning errors in latitude, longitude and altitude, is a three-dimensional vector representing thevelocity errors in east, north and up, is a three-dimen- sional vector representing the attitude errors in roll, pitch and azimuth, is a three-dimensional vector representing theaccelerometer biases drift in x, y and z, is a three-dimensional vector representing thegyro biases drift in x, y and z, is a three-dimensional vector representing the accelerometer scale factors errors in x, y and z, is a three-dimensional vector representing the gyro scale factors errors in x, y and z. is the wet component of the tropospheric delay, and are the GPS receiver clock offset and drift in

meters, respectively. Both and are GPS/GLONASS and GPS/Galileo inter-system biases in meters. B is the float ambiguity term in meters. It should be pointed out that the receiver clock offset and drift are cancelled out when forming BSSD ionosphere-free model.

EKF includes two parts the system model and the observation model. The system model is obtained from the INS dynamic errors augmented with the additional GNSS errors as follows.

where is the transformation matrix from the body frame to the navigation frame, is a diagonal matrix of the accelerometers forces in body frame and Is a diagonal matrix of the gyro forces in body frame, w representing the system input white noise, G is the associated coefficient matrix and, β = 1/τ, where τ is the correlation time for the accelerometers and gyros for first order GM process. The approximate correlation times are estimated by computing the autocorrelation functions for accelerometer and gyros records using static data records collected for three hours [5] . The observation model of the GNSS/INS filter in the tightly coupled architecture has the typical form:

δZ is the measurement vector consisting of the differences between the corrected GNSS and the predicted INS measurements. When un-differenced ionosphere-free model is used δZ can be defined as:

H is the design matrix containing geometry factors defined according to the GNSS mathematical model used. The design matrix is arranged with columns corresponding to the states unique to inertial sensors errors such as, , and which filled with zeroes. H can be formed as:

where d are the direction cosine matrix D elements for pseudorange and phase; s is the direction cosine matrix S elements for Doppler measurements. The Element of D and S can be computed as follows

where X, Y and Z are are the satellite coordinates computed using the final IGS-MEGX orbital products and corrected for the effect of earth rotation during signal transit; φ, λ and h are the INS positioning coordinates; N is the prime vertical radius of curvature. To form the BSSD measurement model, between-satellite single difference matrix should be defined based on the selected GPS reference satellite.

where is the design matrix for BSSD model, is the BSSD observation vector. The error state vector for BSSD based integrated system is defined as:

where and are the single differenced float ambiguity terms.

Two real vehicular tests were conducted to evaluate the performance of the developed integrated GNSS-PPP/ MEMS-based INS system (Figure 2). The vehicular tests were carried out through downtown Kingston, Ontario, Canada, which was designed to represent challenging situations for real GNSS satellite navigation availability including turns, straight portions, high speed, and slow speeds. The NovAtel SPAN-CPT system and the GNSS Trimble R10 receiver were used to collect the navigation data. The SPAN-CPT system consists of the NovAtel OEM4 receiver and a MEMS-based IMU, which contains three MEMS-based accelerometers and three fiber optic gyros. Carrier phase-based differential GNSS (DGNSS) solution is used as a reference solution. In order to create this reference solution, a GNSS Trimble R7 receiver was setup at a nearby station with precisely known coordinates. The raw dual-frequency GNSS pseudorange, carrier phase and Doppler measurements were collected at a 1 Hz rate, while the IMU raw data was logged at a rate of 100 Hz. The duration of the first trajectory test was set for about 55 minutes while the duration of the second test was 34 minutes. Four scenarios are considered in this research. The traditional GPS-based integrated system and the developed GNSS-based integrated system including GPS, GLONASS and Galileo, are studied to investigate the contribution of the additional GNSS systems to the positioning accuracy of the integrated system. Both un-differenced and BSSD ionosphere-free PPP techniques are adopted for GPS-based and GNSS-based integrated systems. To investigate the positioning accuracy of the integrated system during complete GNSS outages, a number of simulated outages is introduced for each trajectory test. The data were processed using our Ryerson PPP GNSS/MEMS software in un-difference and BSSD modes. The program is implemented in MATLAB R2013a using the Intel^® Core i7-3517U CPU and 6 GB RAM. The computational burden of the whole process is 39.41 s including reading both INS and GNSS observations, Kalman filtering process with GNSS updating every second and results writing.