## 4 Navigation Integration Processing and Performance Evaluation  
Simulation experiments have been carried out to evaluate the performance of the proposed approach in comparison with the conventional methods for GPS/INS integrated navigation system processing. The tightly-coupled configuration is employed for demonstration. The commercial software Satellite Navigation (SATNAV) Toolbox by GPSoft LLC was employed for generating the satellite positions and pseudoranges. The satellite constellation was simulated and the error sources corrupting GPS measurements include ionospheric delay, tropospheric delay, receiver noise and multipath. Assume that the differential GPS (DGPS) mode is used and most of the errors can be corrected, but the multipath and receiver thermal noise cannot be eliminated. Figure 4 shows the configuration of the tightly-coupled feedback GPS/INS integrated navigation processing based on the FLAS-coupled IMMUKF filtering mechanism. The measurement is the residual between GPS pseudorange and INS predicted range, which is used as the measurement of the UKF. The differential equations describing the two-dimensional inertial navigation state are:  
\[ \begin{bmatrix}\dot{n}\\ \dot{e}\\ \dot{v}_{n}\\ \dot{v}_{e}\\ \dot{\psi}\end{bmatrix} = \begin{bmatrix}v_{n}\\ v_{e}\\ a_{n}\\ a_{e}\\ \omega_{r}\end{bmatrix} = \begin{bmatrix}v_{n}\\ v_{e}\\ \cos(\psi)a_{u}-\sin(\psi)a_{v}\\ \sin(\psi)a_{u}+\cos(\psi)a_{v}\\ \omega_{r}\end{bmatrix}  \tag{32}\]  
where \([a_{u},\,a_{v}]\) are the measured acceleration in the body frame, \(\omega_{r}\) is the measured yaw rate in the body frame, as shown in Figure 5. The error model for INS is augmented by some sensor error states such as accelerometer biases and gyroscope drifts. Actually, there are several random errors associated with each inertial sensor. It is usually difficult to set a certain stochastic model for each inertial sensor that works efficiently at all environments and reflects the long-term behavior of sensor errors. The difficulty of modeling the errors of INS raised the need for a model-less GPS/INS integration technique. Linearization of Equation (32) results in the following set of linearized equations:  
Figure 4: Configuration of the tightly-coupled feedback integrated navigation using the proposed approach. \[ \begin{bmatrix}\vec{\omega}\\ \vec{\omega}\\ \vec{\omega}_{n}\\ \vec{\omega}_{e}\\ \vec{\omega}\end{bmatrix} = \begin{bmatrix}0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{bmatrix} + \begin{bmatrix}0\\ 0\\ w_{n}\\ w_{e}\\ w_{\psi}\end{bmatrix}  \tag{33}\]  
which will be utilized in the integration Kalman filter as the inertial error model. In Equation (33), \(\delta n\) and \(\delta e\) represent the east, and north position errors; \(\delta v_{n}\) and \(\delta v_{e}\) represent the east, and north velocity errors; and \(\delta\psi\) represent yaw angle. The measurement model is written as:  
\[ \begin{bmatrix}\rho_{1}\\ \rho_{2}\\ \vdots\\ \rho_{n}\end{bmatrix} = \begin{bmatrix}\hat{\rho}_{1}\\ \hat{\rho}_{2}\\ \vdots\\ \hat{\rho}_{n}\end{bmatrix} + \begin{bmatrix}h_{11}&h_{12}&0&0&0\\ h_{21}&h_{22}&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ h_{n1}&h_{n2}&0&0&0\\ \end{bmatrix} + \begin{bmatrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n}\end{bmatrix}  \tag{34}\]  
The experiment was conducted on a simulated vehicle trajectory originating from the (0,0,0) m location in the ENU coordinate frame. The simulated sensor outputs for the accelerometers and gyroscope are shown as in Figure 6. The schematic illustration of trajectories for the simulated vehicle and the unaided INS derived position is shown in Figure 7. The trajectory can be divided mainly into seven time intervals (or segments), as indicated on the figure, according the dynamic characteristics. Further detailed description of the vehicle motion is given in Table 1. The vehicle was simulated to conduct constant-velocity straight-line during the four time intervals, 0-300, 701-800, 901-1,200 and 1,401-1,800 s, all at a speed of 10\(\pi\) m/s.