## 6 The 3D Reduced Inertial Sensor System  
The 2D RISS depends on the fact that land vehicles mostly stay on the horizontal plane. Due to the limitation of 2D RISS on roads with slopes, especially in hilly and uneven terrain, 3D RISS [25; 26; 27] was developed by incorporating two accelerometers for the provisioning of pitch and roll angles and incorporating the vertical information in the system model to be used by the RISS/odometer/GNSS integration filter. When pitch and roll are calculated from accelerometers, the first integration of gyroscopes to obtain pitch and roll is eliminated; thus, the error in pitch and roll is not proportional to time integration. The outcome of these accurate estimates is superior velocity and position estimates for 3D RISS with odometer, along with upward velocity and altitude that are not calculated before. The overview of the 3D RISS mechanization can be seen in Figure 6.  
The pitch angle is derived from the forward acceleration, the acceleration from the odometer, and the gravity as shown in Equation (13).  
\[p=sin^{-1}\left(\frac{f_{y}-a_{\text{od}}}{g}\right) \tag{13}\]  
Figure 5: Block diagram of 2D RISS.  
Figure 6: Block diagram of 3D RISS.  
where \(p\) is the pitch angle, \(f_{y}\) is the forward accelerometer specific force, \(a_{od}\) is the forward acceleration obtained from the odometer, and \(g\) is the gravity.  
The roll angle is calculated as in Equation (14).  
\[r=-sin^{-1}\bigg{(}\frac{f_{x}-v_{od}(\omega_{z}-b_{z})}{gcos(p)}\bigg{)} \tag{14}\]  
where \(f_{x}\) is the transversal accelerometer specific force, \(v_{od}\) is the forward speed, \(\omega_{z}\) is the angular rate, \(b_{z}\) is the gyroscope bias. The azimuth rate \(\dot{A}\) is determined from the gyroscope angular rate, the Earth's rotation rate, and the rotation from moving in curvilinear motion by taking in account the latitude and altitude on Earth. The azimuth rate is given by Equation (15).  
\[\dot{A}=-\bigg{(}\omega_{z}-b_{z}-\omega^{e}sin(\varphi)-\frac{v_{e}tan(\varphi )}{R_{N}+h}\bigg{)} \tag{15}\]  
where \(\omega^{e}\) is the Earth's rotation rate, \(b_{z}\) is the gyro bias, \(v_{e}\) is the east velocity, and \(R_{N}\) is the meridian radius of curvature of the Earth. The 3D velocity components are calculated by projecting the forward speed measured by the odometer using the altitude angles as in Equation (16).  
\[v=\left[\begin{array}{c}v_{e}\\ v_{n}\\ v_{u}\end{array}\right]=\left[\begin{array}{c}v_{od}sin(A)cos(p)\\ v_{od}cos(A)cos(p)\\ v_{od}sin(p)\end{array}\right] \tag{16}\]  
where \(v_{e}\),\(v_{n}\), and \(v_{u}\) are the East, North, and Up velocities, respectively.  
The 3D position components are obtained from the velocities as in Equation (17), taking the Earth's geometry into consideration:  
\[\left[\begin{array}{c}\dot{\varphi}\\ \dot{\lambda}\\ \dot{h}\end{array}\right]=\left[\begin{array}{c}\frac{\frac{v_{0}}{R_{N}+h}} {v_{e}}\\ \frac{v_{u}}{(R_{M}+h)cos(\varphi)}\\ v_{u}\end{array}\right] \tag{17}\]  
where \(\dot{\varphi}\), \(\dot{\lambda}\), and \(\dot{h}\) are the latitude, longitude, and altitude rates respectively, and \(R_{M}\) is the normal radius of curvature of the Earth's ellipsoid.