## 3 Problem Statement  
Presently, there is a growing demand for low-cost navigation systems that can provide accurate positioning at all times, even in harsh GNSS environments. The GNSS is considered to be the main navigation system for land vehicle applications; it suffers from several errors that might lead to signal loss or satellite blockages [17; 18; 19]. GNSS navigation requires an uninterrupted line of sight (LOS) between the receiver that is mounted on the moving vehicle and a minimum number of four satellites. Some environments such as urban canyons do not allow this primary condition to be accomplished. Consequently, the receiver will not be able to produce a solution. Therefore, there is a great need for extra sensors/systems mounted on the moving vehicle. The inertial navigation system (INS) is a practical option as a backup navigation system [20; 21; 22]. The central contributing unit in the INS is the inertial measuring unit (IMU). A full IMU is a self-contained device that consists of three accelerometers and three gyroscopes to continuously measure three orthogonal linear accelerations and three orthogonal angular rates, respectively. These raw measurements are then integrated in a sharp-down manner to provide position, velocity, and altitude using a sequence of mechanization equations [23; 24]. Thus, INS provides a continuous navigation solution, but navigational errors grow exponentially with time due to several factors that may include sensor bias, drift, misalignment, and scale factor instability. Other complementary navigation methods such as steering encoder, odometer, velocity encoder, and electronic compass can be used to curtail the error growth rate of the INS. A more accurate navigational solution can be obtained by integrating these motion sensors with GNSS [25; 26]. Traditionally, the integration of GNSS with other systems like INS has been provided by KF or extended KF (EKF), which relies on a linearized error model of both GNSS and INS. KF has provided a reliable GNSS/INS integration solution for high-end navigational and tactical grade INS. However, KF may not be able to address the complex stochastic and high order errors of MEMS grade sensors. This will result in large values of the non-linear error terms, which are usually ignored during the linearization process while generating the error model for KF. The error models are required to analyze and estimate different  
Figure 2: GNSS in challenging environments. error sources associated with the proposed RISS. The error state vector for RISS includes coordinate errors, velocity errors, azimuth error, stochastic error in odometer derived acceleration, stochastic error in the accelerometers, and gyroscope reading. When the low-cost MEMS IMU is integrated with GNSS by using traditional KF integration techniques, the solution becomes unreliable, especially in prolonged GNSS outages. It is vital to have accurate error models to achieve consistent KF results. However, it is challenging to model MEMS sensors as they have composite error characteristics [23; 24; 25; 26; 27]. The lack of rigorous error models can diminish the overall navigation accuracy. The main objective of this paper is to enhance the performance of integrated MEMS-based INS/GNSS navigation systems through the PCI non-linear modeling approach that can deal with the non-linear parts of INS and GNSS errors. In order to achieve this objective, this paper aims at the following:  
1. A review of the PCI algorithm, a non-linear system identification technique with the details of different implementation steps, is discussed. 2. The research approach in this paper relies on reduced inertial sensor systems (RISS), which limits the reliance on MEMS-based gyroscopes to avoid their high levels of noise and drift rates. The RISS incorporating single-axis gyroscope, vehicle odometer, and accelerometers will be considered for the integration with GNSS in one of two schemes: 1. Loosely coupled where GNSS position and velocity are used for the integration. 2.