# A Review of Pedestrian Indoor Positioning Systems for Mass Market Applications
## 3 Inertial Based Systems  
In previous section, we have seen that the network based systems estimate the position of the user measuring the features from the signals received from a wireless network; however, inertial based systems compute their own position without any help from a physical infrastructure. The inertial sensors measure the forces applied to the sensor and thus the movement of the object where the sensor is mounted can be computed. Usually, inertial sensors are mounted together forming Inertial Measurement Units (IMU), which are formed by a 3 axis accelerometer, a 3 axis gyroscope and a 3 axis magnetometer (The magnetometer is not an inertial sensor, however in this work we group it into the inertial measurement unit as this is the typical term used in the literature.). There are two main kinds of inertial navigation systems [6]:  
* **Strapdown systems:** these systems integrate twice the acceleration of the user in order to estimate the position. * **Step and Heading Systems (SHS):** these systems estimate the position by adding to the initial position estimation vectors representing the step length and the step heading of the user. Regardless of the approach used, the first step of an inertial navigation system is the computation of the relative orientation of the sensor and the body of the user. The measurements of an IMU are expressed in the sensor coordinate frame, whenever we attach the IMU to the body of the user, the axes of the sensor coordinate frame may not coincide with the axes of the navigation frame. Any misalignment in the axes produces errors in the measurements; therefore the estimation of the relative orientation is a crucial part of an inertial navigation system. The relative transformation between two coordinate frames can be obtained by sequentially rotating around three axis, where the angles of rotation are expressed as Euler angles, that is, the roll (\(\phi_{x}\)), pitch (\(\theta_{y}\)) and yaw (\(\psi_{z}\)). The definition of the Euler angles is shown in Figure 7.  
The transformation between coordinate frames is done using the following rotation matrices [83]:  
\[\mathbf{O}_{\phi_{x}}= \begin{bmatrix}1&0&0\\ 0&-\cos(\phi_{x})&\sin(\phi_{x})\\ 0&\sin(\phi_{x})&\cos(\phi_{x})\\ \end{bmatrix} \,, \tag{5}\]  
Figure 7: Roll, pitch and yaw angles. \[\mathbf{O}_{\theta_{y}}= \begin{bmatrix}\cos(\theta_{y})&0&\sin(\theta_{y})\\ 0&1&0\\ -\sin(\theta_{y})&0&\cos(\theta_{y})\end{bmatrix} , \tag{6}\]  
\[\mathbf{O}_{\psi_{z}}= \begin{bmatrix}\cos(\psi_{z})&\sin(\psi_{z})&0\\ -\sin(\psi_{z})&\cos(\psi_{z})&0\\ 0&0&1\end{bmatrix} , \tag{7}\]  
where \(\mathbf{O}\) represents the rotation matrix. The rotations are applied in the following order:  
\[\mathbf{O}_{T}=\mathbf{O}_{\psi_{z}}\mathbf{O}_{\theta_{y}}\mathbf{O}_{\psi_ {z}}, \tag{8}\]  
and the measurements of the IMU in the navigation frame \(\mathbf{z}_{IMU}^{NF}\) are obtained by multiplying the IMU measurements in the sensor frame \(\mathbf{z}_{IMU}^{SF}\) by the rotation matrix \(\mathbf{O}_{T}\), that is,  
\[\mathbf{z}_{IMU}^{NF}=\mathbf{O}_{T}\,\mathbf{z}_{IMU}^{SF}. \tag{9}\]  
In order to estimate the rotation angles, the earth gravitational field, measured by the accelerometers, is typically employed. In the absence of any external acceleration, the output of an accelerometer corresponds to the earth gravitational field. Therefore it is possible to estimate the roll and pitch angles knowing that if the sensor coordinate frame is aligned with the earth coordinate frame, the gravitation vector must fall in the \(z\) axis [83], that is,  
\[\tan\phi_{x}=\frac{a_{y}}{a_{z}}, \tag{10}\]  
\[\tan\theta_{y}=\frac{-a_{x}}{\sqrt{a_{y}^{2}+a_{z}^{2}}}, \tag{11}\]  
where \(a_{x}\), \(a_{y}\) and \(a_{z}\) are the outputs of the accelerometer in the \(x\), \(y\) and \(z\) axis, respectively. Unfortunately the gravitational field is invariant to the rotation of the yaw angle and therefore the yaw angle remains unknown using this method.