## 2 Materials and Methods  
### The IMMS Analytical Measurement Principle  
As hinted in the Introduction, these days the IMMS is without doubt the most innovative terrestrial surveying technique, as it finely exploits the availability of more measuring sensors such as (at a minimum) a linear 2D or a 3D laser scanner, an inertial platform, and perhaps a satellite receiver. As said earlier, from the analytical point of view, the IMMS measuring principle is analogous to the one proposed in 1996 by [PERSON] and [PERSON] [14] for MMSs: for IMMSs, by simply considering the six degrees of freedom of a laser scanning sensor instead of the imaging sensor and re-arranging the equation, the position \(r_{i}^{\textit{gcs}}\) of a generic _i-th_ point with respect to a chosen global coordinate system (GCS) is given by the equation below:  
\[r_{i}^{\textit{gcs}}=r_{\textit{gns/ins}}^{\textit{gcs}}+R_{ins}^{\textit{gcs} }\Big{(}R_{\textit{ts}}^{\textit{ins}}R_{\textit{b}}^{\textit{lls}}r_{i}^{ \textit{b}}+\mathbf{a}_{\textit{tls}}^{\textit{ins}}\Big{)} \tag{1}\]  
where:  
* \(\mathbf{r}_{\textit{gns/ins}}^{\textit{gcs}}\) is the position of the GNSS/INS sensors (sometimes embedded) in the GCS frame, computed in an integrated way as reported further on;
* \(\mathbf{R}_{\textit{ins}}^{\textit{gcs}}\) is the rotation matrix from the INS frame to the GCS frame, with the instantaneous rotation angles measured by the three INS gyroscopes. Together with the three accelerometers, they form the so-called \"inertial measurement unit\" (IMU) [9];
* \(\mathbf{R}_{\textit{tls}}^{\textit{ins}}\) is the rotation matrix from the TLS frame to the INS frame, with fixed angles estimated in the IMMS system calibration procedure;
* \(\mathbf{R}_{\textit{b}}^{\textit{tls}}\) is the rotation matrix from the instantaneous laser beam (b) to the TLS frame, with the variable angles imposed by the scanning system;
* \(\mathbf{r}_{i}^{\textit{i}}\) is the laser beam on the _i-th_ point;
* \(\mathbf{a}_{\textit{tls}}^{\textit{ins}}\) is the position of the TLS frame with respect to the INS frame, with fixed components (generally measuring a few centimetres) which must nevertheless be measured/estimated in the system calibration step. The kinematic positioning of every scanned point by means of Equation (1) can be reinterpreted as the static positioning from a station whose instantaneous position \(\mathbf{r}_{\textit{tls}}^{\textit{gcs}}=\mathbf{r}_{\textit{gns/ins}}^{\textit{gcs} }+R_{ins}^{\textit{gcs}}\mathbf{a}_{\textit{tls}}^{\textit{ins}}\) and rotation enclosed in \(\mathbf{R}_{\textit{lls}}^{\textit{gcs}}=\mathbf{R}_{\textit{ins}}^{\textit{gcs}}\mathbf{ R}_{\textit{lls}}^{\textit{ins}}\). That is, the six roto-translation registration parameters are estimated via GNSS/INS measurements and not by using tie points. From the analytical point of view, integrating GNSS and INS data is a very complex task [8; 14], solved by means of Kalmanfilter-based procedures considering a \"state vector\" and using state equations (modelling the motion) and measurement equations (from the sensors). The INS-frame position and attitude are therefore continuously updated from an epoch \(t\), when they are deemed to be known, to the following epoch \(t+1\), when they are instead unknown. As such, Equation (1) should also contain the term \(t\), left out for the sake of simplicity, since all the quantities change over time, apart the from constant calibration terms \(\mathbf{R}^{ins}_{tls}\) and \(\mathbf{a}^{ins}_{tls}\). Moreover, these quantities are observed with frequencies of different orders of magnitude: Hz for \(\mathbf{r}^{SCs}_{gmss}\), hundreds of Hz for \(\mathbf{r}^{SCs}_{tins}\) and \(\mathbf{r}^{SCs}_{tins}\), and tens of kHz for \(\mathbf{R}^{tls}_{b}\) and \(\mathbf{r}^{b}_{i}\). Hence, the synchronization of these observations becomes a fundamental hardware requirement and the timing interpolation has to be carried out correctly at the highest TLS frequency.