# A Review of Pedestrian Indoor Positioning Systems for Mass Market Applications
## 3 Inertial Based Systems
### Strapdown Systems  
Strapdown inertial navigation systems are based on the concept that the position is the double integration of the acceleration. Thus, the first integration of the acceleration signal \(\mathbf{a}(t)=\left[a_{x}(t),a_{y}(t),a_{z}(t)\right]\) produces the velocity and the integration of the velocity produces the position [84], that is,  
\[\mathbf{v}\left(t\right)=\mathbf{v}\left(0\right)+\int_{0}^{t}\left(\mathbf{a }\left(t\right)-\mathbf{g}\right)\,dt, \tag{12}\]  
\[\mathbf{m}\left(t\right)=\mathbf{m}\left(0\right)+\int_{0}^{t}\mathbf{v}\left( t\right)\,dt, \tag{13}\]  
where \(\mathbf{v}\) is the velocity, \(\mathbf{g}\) the gravity and \(\mathbf{m}\) the position, all of them related to the navigation frame. Figure 8 shows the block diagram of a strapdown navigation system. First, the angular velocity measured by the gyroscope is integrated in order to track the orientation of the sensor frame with respect to the navigation frame. Note that once the initial orientation is known, the orientation at any time can be known by accumulating the rotation done in each axis, which is measured by the gyroscope. Once the orientation is known, the signal from the accelerometer is rotated to the navigation frame and the gravitation force is subtracted before the integration of the acceleration signal to obtain the velocity and the position.  
The errors in the measurements of the sensors affect differently to the estimation of position. On the one hand, the errors of the accelerometer produce a drift in the position because the integration procedure accumulates the errors over time. On the other hand, the errors of the gyroscope result in an erroneous rotation matrix and therefore the measurements of the accelerometer are incorrectly projected into the navigation frame. Furthermore, the strapdown navigation systems subtract the value of the earth gravitational field before the integration. Any error in the alignment of the frames will produce a bias due to a gravitational component projected to the horizontal plane. This source of error cannot be neglected as the magnitude of the acceleration caused by the gravity is usually greater than the acceleration produced by the movement of the user. In fact, the errors in the gyroscope measurements are the ones limiting the accuracy of the inertial strapdown systems. In general, the error in the estimation of position grows cubically with time due to the integration of the accelerometer and gyroscope signals. Using the current MEMS technology, the estimation of position will deviate over the meter in seconds making the estimation of the trajectory of a human in the long term unfeasible [85].  
Recently, [PERSON] et al. [38] demonstrated that using a foot mounted IMU the time dependency of the position estimation errors in strapdown systems, which typically grows cubically with time, can be reduced to a linear growth if the Zero Velocity Update (ZUPT) is applied. The idea behind ZUPT is to detect the stance phases of the human walking, when the foot is firmly planted on the ground and the velocity is zero, and apply these zero velocity measurements to an extended Kalman filter that estimates the errors of the inertial measurements. However, the ZUPT strategy cannot correct the errors in the yaw angle. In order to amend this, several authors proposed techniques for reducing the gyroscope bias, such as the zero angular rate update [86] or the heuristic heading reduction [87]. An example of a inertial strapdown system using theses techniques can be found in [39]. Similarly, the authors in [40] applied these updates using an Unscented Kalman Filter (UKF) for the estimation of the inertial measurement errors.