# Research on the SSUKF Integrated Navigation Algorithm Based on Adaptive Factors
## 2 Nonlinear Model of the Integrated Navigation System  
In the navigation process, the system's dynamic model is often highly nonlinear. Therefore, accurately describing these dynamic processes and effectively estimating and compensating for errors becomes key to improving system accuracy. This study uses the nonlinear strapdown inertial navigation error equations and augments the error vector into the state variables of the SSUKF for estimation. The errors considered include attitude errors, velocity errors, and position errors, as well as gyroscope and accelerometer biases. Specifically, the state vector is represented as follows:  
\[X=\left[\boldsymbol{\phi}\;\delta\boldsymbol{v}\;\delta\boldsymbol{r}\; \boldsymbol{\varepsilon}\;\
abla\right]=\left[\delta\varphi_{x},\delta\varphi_{y },\delta\varphi_{z},\delta v_{x},\delta v_{y},\delta v_{z},\delta x,\right.\]  
\[\left.\delta y,\delta z,\delta gb_{x},\delta gb_{y},\delta gb_{z},\delta ab_{x },\delta ab_{y},\delta ab_{z}\right]\]  
The expressions for position, velocity, and attitude errors are as follows [25]:  
\[\delta\dot{\boldsymbol{r}}^{n}=-\boldsymbol{\omega}_{en}^{n}\times\delta \boldsymbol{r}^{n}+\delta\boldsymbol{\theta}\times\boldsymbol{v}^{n}+\delta \boldsymbol{v}^{n} \tag{1}\]  
\[\delta\dot{\boldsymbol{v}}^{n}=\mathbf{C}_{b}^{n}\delta\dot{\boldsymbol{f}}^{ n}+\boldsymbol{f}^{n}\times\boldsymbol{\phi}-\left(2\boldsymbol{\omega}_{ie}^{n}+ \boldsymbol{\omega}_{en}^{n}\right)\times\delta\boldsymbol{v}^{n}+\boldsymbol {v}^{n}\times\left(2\delta\boldsymbol{\omega}_{ie}^{n}+\delta\boldsymbol{ \omega}_{en}^{n}\right)+\delta\boldsymbol{g}_{p}^{n} \tag{2}\]  
\[\dot{\boldsymbol{\phi}}=-\boldsymbol{\omega}_{in}^{n}\times\boldsymbol{\phi}+ \delta\boldsymbol{\omega}_{in}^{n}-\delta\boldsymbol{\omega}_{ib}^{n} \tag{3}\]  
In this study, the coordinate system used in the calculations is the East-North-Up (ENU) coordinate system. The symbols \(n\), \(e\), and \(i\) represent the navigation, Earth, and inertial coordinate systems, respectively. The strapdown inertial navigation system (INS) is selected as the primary fusion center, and GNSS-provided position information servesas the observation quantity. The Adaptive Simplified Spherical Unscented Kalman Filter (ASSUKF) is employed for filtering and fusion to output state error estimates and correct the accumulated errors in the inertial navigation system in real time.  
Considering a nonlinear system state-space model with Gaussian-distributed process noise and measurement noise, the model can be represented as follows:  
\[\left\{\begin{aligned} X_{k}&=f(X_{k-1})+W_{k-1}\\ Z_{k}&=H_{k}X_{k}+V_{k}\end{aligned}\right. \tag{4}\]  
where \(X_{k}\) represents the n-dimensional state vector at time step \(k,f(\cdot)\) is the nonlinear function describing the state transition dynamics, \(H_{k}\) is the measurement matrix at time step \(k\), \(Z_{k}\) is the mmm-dimensional measurement vector at time step \(k\), and \(W_{k-1}\) and \(V_{k}\) represent the system process noise and measurement noise, respectively, both of which are zero-mean Gaussian white noise vectors. These noises are mutually uncorrelated. The statistical properties of the measurement noise \(V_{k}\) are given by  
\[E[V_{k}]=0,E\Big{[}V_{k}V_{j}^{T}\Big{]}=\mathbf{R}_{k}\delta_{kj} \tag{5}\]  
where \(\delta_{kj}\) is the Kronecker delta function, which is 1 when \(k=j\) and 0 otherwise.