## 2 Systems Overview
### Low-Cost Inertial Sensors  
The great advances in MEMS have made possible the development of a new generation of low cost inertial sensors. MEMS inertial measurement units (IMU), that is, the actual sensor assembly, are characterized by small size, light weight, low cost and low power consumption with respect to higher-end inertial sensors. These features make MEMS sensors an attractive option for applications such as vehicular navigation. However, MEMS sensors are also characterized by poorer performance, so they cannot be used in autonomous mode for extended periods although they are well suited for integrated navigation systems (usually coupled with GPS systems) where external measurements can limit their error growth. An IMU usually consists of a triad of accelerometers and gyros, whose measurement equation can be expressed as:  
\[\begin{array}{l}f=\vec{f}+b_{a}+f\cdot S_{a}+\eta_{a}\\ \omega=\widetilde{\omega}+b_{g}+\omega\cdot S_{g}+\eta_{g}\end{array} \tag{1}\]  
with \(f\) and \(\vec{f}\) being the actual and measured specific force,  
\(\omega\) and \(\widetilde{\omega}\) actual and measured angular rate,  
\(b_{a}\) and \(b_{g}\) sensor biases of the accelerometer and gyro respectively,  
\(S_{a}\) and \(S_{g}\) sensor scale factors of the accelerometer and gyro respectively,  
\(\eta_{a}\) and \(\eta_{g}\) sensor noises of the accelerometer and gyro respectively.  
More detailed measurement equations can be found in [25], including additional terms such as non-linear scale factors and cross-axis coupling factors. These are not considered here as they are impractical to estimate given the amount and quality of GNSS data used in this research.  
The sensor bias is defined as the average of the output, obtained during a specific period with fixed operational conditions when the input is null. The bias generally consists of two parts: a deterministic part called bias offset or turn-on bias and a stochastic part called bias drift or in-run bias. The turn-onbias is essentially the offset in the measurement and is constant over a single mission; it has deterministic nature and so can be determined by calibration procedure (or it can be also modeled statistically as a random constant process). The bias drift is the change in the sensor with time; the bias drift has random nature and so must be modeled as a stochastic process. The scale factor error is the ratio between the change in the output signal of the sensor to the change in the measured physical quantity. In ideal conditions the scale factor should be unity. This error has a deterministic nature but generally is modeled as a random process. The inertial sensor errors can be expressed in terms of angular random walk (ARW) and velocity random walk (VRW). The ARW parameter describes the average deviation or error that will occur from integrating the noise on gyro output signal. Similarly the VRW parameter definition is based on the same concept for the accelerometers.  
Typical MEMS sensor performance is summarized in the Table 1, where navigation and tactical grade IMU performance is also listed for comparison. Since gyro biases degrade position as a function of time cubed [26], it is obvious from the table that MEMS-based sensors will produce very poor navigation results in a short time unless integrated with other systems (typically GNSS) to bound the errors. From Table 1 we can see that the turn-on bias of MEMS gyro is about 5,400 deg/h, while it is considerably smaller in the navigation and tactical grade sensors. Also the in-run bias can be 1,040 deg/h in MEMS sensors, while is about 1 deg/h in a tactical grade gyro. These parameters provide a good assessment of MEMS performance with respect to higher grade sensors.