## 2 INS/GNSS Concepts
### Inertial Navigation System (INS)  
Inertial navigation is an autonomous process of calculating position by the double integration of acceleration at a point whose position needs to be determined [30]. This involves finding position, attitude, and velocity relative to the Earth or the inertial space. An inertial navigation system (INS) typically consists of three accelerometers and three gyroscopes to measure accelerations in three dimensions and rotation rates around three axes. The development of Micro-Electro-Mechanical System (MEMS) technology has significantly expanded the application area of INS. Today, an Inertial Measurement Unit (IMU) generally includes a three-degree-of-freedom (3-DoF) gyroscope and a three-degree-of-freedom accelerometer, allowing for the precise measurement of motion and orientation in compact and cost-effective packages [31]. Nowadays, strapdown systems are more commonly used, where the gyroscopes and accelerometers are mounted directly to the structure of the vehicle or strapped onto the body segment. Strapdown mechanization (or INS mechanization) is the process of determining the navigation states (position, velocity, and attitude) from raw inertial measurements by solving the differential equations that describe the system's motion. For this purpose, it is essential to establish a consistent reference frame. The mechanization differential equations are formulated in the navigation frame, also known as the local-level frame or NED (North, East, Down) frame. This frame is fixed to the Earth's surface, with its axes aligned to point towards geographic north, east, and downward (towards the center of the Earth). The ECEF (Earth-Centered Earth-Fixed) frame, in contrast, is a Cartesian coordinate system that rotates with the Earth, making it suitable for representing global positions in a geocentric reference system. The body frame is fixed to the vehicle itself, with its axes aligned along the forward, right, and downward directions of the vehicle. Finally, the inertial frame, specifically the Earth-Centered Inertial (ECI) frame, is centered at the Earth's center and does not rotate with the Earth [32]. This work uses the WGS-84 model for the ECEF frame and the J2000 frame for the inertial reference. The dynamic equations used for mechanization in the navigation frame are presented below and are detailed in [32]. In this context, the superscript indicates the frame in which the variable is expressed, and the subscript denotes the reference frame for the transformation: \(n\) refers to the navigation frame, \(e\) refers to the ECEF frame, \(i\) refers to the ECI frame, and \(b\) refers to the body frame. 1. Position Update: \[\mathbf{\hat{p}^{n}}=\mathbf{D}^{-1}\mathbf{v}^{n}\] (1) where \(\mathbf{p}^{n}=[\phi,\lambda,h]^{T}\), with \(\phi\) representing the latitude, \(\lambda\) the longitude, and \(h\) the height relative to the Earth ellipsoid. The matrix \(\mathbf{D}^{-1}\) is a \(3\times 3\) matrix that transforms the velocity vector from navigation coordinates to geodetic coordinates in the ECEF frame. The vector \(\mathbf{v}^{n}=[v_{n},v_{e},v_{d}]^{T}\) represents the velocity in the NED frame, with components along the north (\(v_{n}\)), east (\(v_{e}\)), and down (\(v_{d}\)) directions. 2. Velocity Update: \[\mathbf{\hat{v}^{n}}=\mathbf{R}_{b}^{n}\mathbf{\hat{f}^{b}}+\mathbf{g}^{n}-( \mathbf{\Omega}_{en}^{n}+2\mathbf{\Omega}_{ie}^{n})\mathbf{v}^{n}\] (2) where \(\mathbf{R}_{b}^{n}\) is the direction cosine matrix from the body frame to the navigation frame, \(\mathbf{f}^{b}\) is the specific force measured by accelerometers in the body frame, \(\mathbf{g}^{n}\) is the gravity vector, \(\mathbf{\Omega}_{en}^{n}\) is the skew-symmetric matrix representing the transport rate with respect to the ECEF frame expressed in the navigation frame, and \(\mathbf{\Omega}_{ie}^{n}\) is the skew-symmetric matrix representing the Earth's rotation rate with respect to the inertial frame expressed in the navigation frame. 3.