## 3 Inertial Navigation and IMU Error Model  
A six-degree-of-freedom (6-DOF) strapdown IMU provides the user high-bandwidth position, velocity and orientation estimation with complete independence of the reception of electromagnetic waves as following differential equations [16]:  
\[\dot{C}^{n}_{b}=C^{n}_{b}\cdot(\omega^{b}_{ab}\times)-[(\omega^{n}_{a}\times)+ (\omega^{n}_{m}\times)]\cdot C^{n}_{b} \tag{1}\]  
\[\dot{V}^{n}=C^{n  
the \(e\) frame and therefore needs \(R_{u}\) and \(R_{N}\), radii of curvatures in the meridian and prime vertical, to be estimated.  
Although a strapdown IMU propagates the high-frequency inertial navigation mathematically derived from Equations (1) to (3), the MEMS accelerometer and gyroscope sensors inevitably drift and give rise to quadratic and cubic accumulated mechanization errors. Researchers have developed error propagation models in various approaches for practical applications such as [17, 18, 19]. To describe a ground vehicle in low dynamics with well-calibrated a priori information, the \(\Psi\) angle model [20] is introduced in this paper on account of simplicity:  
\[\delta\dot{P}^{n}=-(\omega_{\varpi}^{n}\times)\delta P^{n}+\delta V^{n} \tag{4}\]  
\[\delta\dot{V}^{n}=diag[-\frac{g}{R_{e}}-\frac{g}{R_{e}}\frac{2g}{R_{e}}\frac{2 g}{R_{e}}]\delta P^{n}-[(2\omega_{\kappa}^{n}+\omega_{\alpha}^{n})\times]\delta V^{n} \tag{5}\]  
\[+[(C_{b}^{n}f_{ab}^{b}\times)\delta\Psi^{n}+C_{b}^{n}\delta f^{b}\]  
\[\delta\dot{\Psi}^{n}=-(\omega_{\varpi}^{n}\times)\delta\Psi^{n}-C_{b}^{n} \delta\omega^{b} \tag{6}\]  
In the above equations, \(R_{e}\) is the Earth radius; \(\delta P^{n}\), \(\delta V^{n}\), \(\delta\dot{\Psi}^{n}\), \(\delta f^{b}\) and \(\delta\omega^{b}\) are the perturbation terms of position, velocity, orientation and sensor biases, respectively.  
As noted earlier, this paper aims to efficiently integrate data from camera, IMU and GNSS sensors to achieve high accuracy ground vehicle navigation in challenging GNSS environments. Therefore, a model for reasonably calibrating the IMU sensor biases needs to be established considering the practical situation. A commonly cited first-order Gauss-Markov model regards the IMU errors as 3 independent components: a random constant, a Gauss-Markov variable and a white noise term and therein the accelerometer/gyroscope error propagations are written as following equations [16,18]:  
\[\delta f=\delta f_{c}+\delta f_{n}+w_{f}\qquad\delta\omega=\delta\omega_{c}+ \delta\omega_{m}+w_{\varpi} \tag{7}\]  
The subscripts of \(c\) and \(m\) indicate the random constant and first-order Gauss-Markov terms, and their derivatives yield:  
\[\delta\dot{f}_{c}=0\qquad\delta\dot{\omega}_{c}=0 \tag{8}\]  
\[\delta\dot{f}_{m}=-\frac{\delta f_{m}}{T_{f}}+\mu_{f}\qquad\delta\dot{\omega} _{m}=-\frac{\delta\omega_{m}}{T_{\varpi}}+\mu_{\varpi} \tag{9}\]  
where the constants \(T_{f}\) and \(T_{w}\) are the correlation times; \(w_{f}\), \(w_{w}\), \(\mu_{f}\) and \(\mu_{w}\) are assumed uncorrelated rapidly over time, moreover, the run-to-run calibration differences are even negligible within a limited period of time, provided the camera is not reconfigured. In addition, as opposed to an IMU, a temperature-independent CMOS/CCD sensor makes the camera resistant to ambient environmental change. Therefore, compared to the inertial sensors, the camera exhibits different calibration mechanism and characteristics coupled with a diverse model for long-term stability.  
As mentioned previously, the perspective projection with a pinhole camera model is used. To estimate the motion in successive image frames, the computer vision module executes three main functions: (1) feature extraction and matching; (2) outlier removal and discrimination of the moving features; (3) motion estimation.