# An Efficient Ultra-Tight GPS/RISS Integrated System for Challenging Navigation Environments
## 3 The Proposed Method
### GPS/SD RISS Integration
#### 3.1.1 The RISS System Model  
To be integrated with GPS system models, the RISS equations need to be first linearized [11]. The Taylor series expansion method is used to achieve this linearization process. Then, the target error state vector of the 3D RISS becomes:  
\[\delta X_{RISS}=[\delta\varphi,\delta\lambda,\delta h,\delta v_{e},\delta v_{n},\delta v_{u},\delta A,\delta a_{o},\delta b_{z}]^{T} \tag{11}\]  
where \(\delta\varphi\), \(\delta\lambda\), and \(\delta h\) are the errors in position components, latitude, longitude, and altitude, respectively. \(\delta v_{e}\), \(\delta v_{n}\), and \(\delta v_{u}\) are the errors in velocity components, east velocity, north velocity, and up velocity, respectively. \(\delta A\) is the azimuth error, \(\delta a_{o}\) is an error in acceleration derived from vehicle odometer, and \(\delta b_{z}\) is the gyroscope bias error, respectively. Once again, the RISS motion equations in Section 2.3 are linearized using [PERSON]'s series expansion method so that we can get the rate of change of error states for the RISS system model. The derivations are well documented in the above-mentioned sources; therefore, we will give here the only final models which will be augmented in our proposed ultra-tightly GPS/RISS integrated system.  
Attitude Errors Models  
As mentioned above, the only attitude parameter included in the error state vector is the error in azimuth angle. The error in the azimuth rate of change is expressed as:  
\[\delta\dot{A}\approx\underbrace{1}_{T_{12,1}}\times\delta b_{z}+\underbrace{ \left(\omega^{e}\cos\varphi+\frac{v_{e}\sec^{2}\varphi}{R_{N}+h}\right)}_{T_{1 2,2}}\delta\varphi+  
Velocity Errors Models  
Likewise [6], linearizing Equations (5) and (6) for East and North velocities yields:  
\[\begin{split}\delta\dot{v}_{e}=\underbrace{\sin A\cos p}_{T_{16,1}} \delta a_{o}+\underbrace{a_{o}\cos A\cos p}_{T_{16,2}}\delta A-\underbrace{ \left(\omega_{z}-b_{z}-a^{\prime}\sin\varphi-\frac{v_{e}\tan\varphi}{R_{N}+h }\right)}_{T_{16,3}}\delta v_{n}+\underbrace{v_{n}}_{T_{16,4}}\delta b_{z}\\ +\underbrace{v_{n}\left(\omega^{e}\cos\varphi+\frac{v_{e}\sec^{2 }\varphi}{R_{N}+h}\right)}_{T_{16,5}}\delta\varphi+\underbrace{\frac{v_{n}\tan \varphi}{R_{N}+h}}_{T_{16,6}}\delta v_{e}\\ \delta\dot{v}_{n}=\underbrace{\cos A\cos p}_{T_{17,1}}\delta a_{o} -\underbrace{a_{o}\sin A\cos p}_{T_{17,2}}\delta A+\underbrace{\left(\omega_{z }-b_{z}-a^{\prime}\sin\varphi-\frac{2v_{e}\tan\varphi}{R_{N}+h}\right)}_{T_{17,3}}\delta v_{e}-\underbrace{v_{e}}_{T_{17,7}}\delta b_{z}\\ -\underbrace{v_{e}\left(\omega^{e}\cos\varphi+\frac{v_{e}\sec^{2 }\varphi}{R_{N}+h}\right)}_{T_{17,5}}\delta\varphi\end{split} \tag{17}\]  
Up velocity is independent of azimuth errors, and hence, derivations are much simpler:  
\[\delta\dot{v}_{u}=\underbrace{\sin p}_{T_{18,1}}\delta a_{o} \tag{18}\]  
Vehicle Odometer Error Model  
The odometer error is modeled as a first-order Gauss-Markov process [12]:  
\[\delta\dot{a}_{o}=\underbrace{-\beta_{o}}_{T_{19,1}}\delta a_{o}+\sqrt{2\beta _{o}\sigma_{o}^{2}}w(t) \tag{19}\]  
where \(\beta_{o}\) is the reciprocal of the sensor data correlation time, \(\sigma_{o}\) is the standard deviation of the error, and \(w(t)\) is the noise term.  
Gyroscope Error Model  
Similarly, the gyroscope error is modeled using a first-order Gauss-Markov process:  
\[\delta\dot{b}_{z}=\underbrace{-\beta_{z}}_{T_{20,1}}\delta b_{z}+\sqrt{2\beta _{z}\sigma_{z}^{2}}w(t) \tag{20}\]  
where \(\beta_{z}\) is the reciprocal of the sensor data correlation time, \(\sigma_{z}\) is the standard deviation of the error, and \(w(t)\) is the noise term.