*5.1. Simulation Environment*

Our simulation environment for the extrinsic calibration of mobile laser scanning systems makes use of the robotic simulation toolbox V-REP (Virtual Robot Experimentation Platform [80]). V-REP enables users to design mobile platforms, which can be equipped with different sensors like GNSS, IMU, laser scanners, or cameras. We utilized V-REP to rebuild the mobile laser scanning system that was introduced in Section 3 (Figure 5, left). In addition to this, V-REP provides the possibility to design environments, in which mobile platforms can operate. Thus, we can also use V-REP to create specific plane setups (Figure 5, middle). Given the mobile laser scanning system and the plane setup, we can simulate the calibration process (Figure 5, right). The plane setup and the simulated sensor observations are then exported to MATLAB (Matrix Laboratory, MathWorks), where we implemented the plane-based calibration approach. In MATLAB, we can analyze and improve the plane setup with quality criteria for both accuracy and controllability (cf. Section 5.2).

**Figure 5.** Using V-REP (Virtual Robot Experimentation Platform) for the simulation of the plane-based calibration approach: Mobile laser scanning system (**left**), plane setup (**middle**), kinematic data acquisition in the calibration field (**right**).

A rigorous optimization of the plane setup based on an all-embracing target function is hardly feasible [14,77]. Recently, Hartmann et al. [53] published an approach based on a genetic algorithm in order to optimize the plane configuration for the calibration of a mobile indoor laser scanning system. However, their target function for the optimization of the plane setup is based on the accuracy of the calibration parameters and does not include criteria for the controllability of the parameter estimation. Moreover, boundary conditions for permissible plane setups are required.

In our case, the determination of an appropriate plane setup is based on an iterative refinement of the plane setup. This strategy starts with an educated guess for the plane setup. This educated guess is iteratively refined based on expert knowledge and the quality criteria for both accuracy and controllability of the calibration parameters as introduced in Section 4.3. The iteration is stopped when the calibration parameters meet priorly defined requirements. In addition, we also specify boundary conditions for the plane setup in the calibration field that result from practical reasons:

• available area for the calibration field is 10 m × 20 m (cf. Figure 5, middle and right),


#### *5.2. Simulation Results*

This section presents the simulation results of the plane setup in Figure 5, which was finally realized in the calibration field (Section 6.1). This plane setup is scanned with the mobile laser scanning system in two passes. The difference between first and second pass is that the orientation of the system has been turned by 180◦ (Figure 5, right). The investigation of the plane setup and the calibration procedure can be split into three parts. We start with analyzing the sensitivity of the plane setup w.r.t. the calibration parameters, i.e., how changes of the lever arm and boresight angles affect the consistency between the reference planes and the point cloud of the mobile laser scanning system. This is connected to the separability of the calibration parameters (Section 5.2.1). Following this, we examine the impact of random and systematic observation errors on the precision and accuracy of the estimated calibration parameters (Section 5.2.2). Finally, we investigate the controllability of the parameter estimation, i.e., its robustness w.r.t. gross observation errors (Section 5.2.3).

#### 5.2.1. Sensitivity of the Plane Setup and Separability of the Calibration Parameters

In this section, we examine the sensitivity of the plane setup w.r.t. changes in the calibration parameters. Figure 6a depicts the point-to-plane distances of the plane setup when the calibration parameters deviate from their true values by either 5 mm (lever arm) or 0.05◦ (boresight angles).

**Figure 6.** (**a**) Sensitivity of the plane setup towards deviation of the calibration parameters. The plots show the point-to-plane distance when the lever arm or the boresight angles deviate from their true values: A (*δ*∆*x* = 5 mm), B (*δ*∆*y* = 5 mm), C (*δ*∆*z* = 5 mm), D (*δα* = 0.05◦ ), E (*δβ* = 0.05◦ ), and F (*δγ* = 0.05◦ ) (**b**) The calibration parameters are basically uncorrelated and, thus, clearly separable. Only the boresight angles *α* and *γ* show a small correlation of 29 % (explanation in the text).

For the lever arm components (A, B, C), there is always a subset of planes that is sensitive towards the parameter deviation. For instance, the planes that are inclined by 45◦ w.r.t. the direction of motion of the platform have a sensitivity of 70% towards changes in ∆*x* and ∆*y* (yellow color in A and B). In addition, the reference planes that are parallel to the direction of motion are fully sensitive towards changes in ∆*y* (red color in B). The planes on the ground are fully sensitive towards changes in ∆*z* (red color in C). Regarding the boresight angles, the plane setup has a sufficient sensitivity towards changes in *α* (D) and *γ* (F), but is less sensitive towards changes in *β* (E). The lower sensitivity towards *β* means that it is more difficult to estimate this parameter based on the minimization of point-to-plane distances like it is done in the calibration. This is proved by both the simulation results (Figure 7) and

the results based on real data (Section 6.2), because in both cases the boresight angle *β* has the biggest standard deviation of all boresight angles. The sensitivity towards *β* could be improved by mounting the mobile laser scanning system on a high van. In this case, the distance between the mobile laser scanning system and the ground planes, which are most sensitive towards *β*, is increased.

In Figure 6a, we can see that different parameter deviations lead to different patterns of the point-to-plane distances. Thus, making the assumptions that we have no other sources of errors, we can separate erroneous calibration parameters by analyzing their unique deviation patterns. This indicates a good separability of the calibration parameters. The good separability of the calibration parameters is verified by their correlation matrix in Figure 6b, which is derived from the covariance matrix **Σ***p*ˆ*p*<sup>ˆ</sup> of the parameter estimation (cf. Equation (6)). According to the correlation matrix, the calibration parameters are basically uncorrelated, except for a small correlation of 29% between the boresight angles *α* and *γ*. The reason for this correlation is the sequence of boresight rotations using Euler angles. Due to the *β* = 30◦ tilt of the 2D laser scanner (cf. Figure 2, left), the horizontal projection of the *Xs*-axis is smaller than the length of the *Xs*-axis. In the extreme case of *β* = 90◦ , the projection is zero. In navigation, this phenomenon is known as gimbal lock [24], where the boresight angles *α* and *γ* rotate around the same axis and cannot be separated from each other (100 % correlation). Thus, in the case of *β* = 0◦ , the correlation between *α* and *γ* is 0 % as is demonstrated in [11]. In our case, the 2D laser scanner is tilted by *β* = 30◦ and, thus, the correlation is between 0 % and 100 %.

#### 5.2.2. Impact of Random and Systematic Observation Errors

In order to examine how random observation errors of the mobile laser scanning system affect the parameter estimation, we carried out a Monte Carlo simulation with 1000 realizations of the calibration process. Given the plane setup, true calibration parameters, and the observations of the mobile laser scanning system, the noise of the observations was sampled from a Gaussian distribution. The associated standard deviations were oriented towards manufacturer information and empirical values (cf. Table 1). The Monte Carlo simulation was performed in a local frame (l-frame) with east, north, and height component. This allows for a better interpretation of the simulation results. As a result, 1,000 realizations for the estimated calibration parameters *p***ˆ** = [∆*x*, ∆*y*, ∆*z*, *α*, *β*, *γ*] *T* and their covariance matrix **Σ***p*ˆ*p*<sup>ˆ</sup> were obtained. According to Förstner and Wrobel (chap. 4.6.8, pp. 139–141) [15], the unbiasedness of *p***ˆ** and **Σ***p*ˆ*p*<sup>ˆ</sup> within the simulation can be checked by different statistical tests. All simulations passed these tests.

The grey histograms in Figure 7 show the distributions of the calibration parameters after the Monte Carlo simulation based on random observation errors only. The red vertical lines indicate the true calibration parameters, i.e., the expectation values. It can be stated that all parameters are unbiased if only random observation errors are simulated. For the three lever arm components, a standard deviation of *σ* < 1 mm is obtained. These results meet the defined target accuracies of 1 mm for ∆*x* and ∆*y* as well as 1.5 mm for ∆*z* (cf. Table 1). The accuracy of the lever arm mainly depends on the accuracy of the position [*t<sup>e</sup>* , *tn*, *t<sup>h</sup>* ] *T* of the mobile laser scanning system. The higher standard deviation of ∆*z* results from the higher standard deviation of the height component *t<sup>h</sup>* , which is assumed to be 1.5 times the standard deviation of the horizontal position *t<sup>e</sup>* and *t<sup>n</sup>* (cf. Table 1).

For the boresight angles *α*, *β*, and *γ*, standard deviations of *σ* < 0.001◦ are obtained, which also meet the defined target accuracy of 0.005◦ (cf. Table 1). The accuracy of the boresight angles mainly depends on the accuracy of the orientation angles of the mobile laser scanning system. In this regard, *α* is mostly aligned with the roll angle *φ*, *β* is mostly aligned with the pitch angle *θ*, and *γ* is mostly aligned with the yaw angle *ψ*. Thus, the smaller standard deviation of *α* as compared to *γ* is expected, since the roll angle is assumed to be more accurate than the yaw angle (cf. Table 1). However, the boresight angle *β* is estimated with the lowest accuracy. At this point, we recall the sensitivity analysis in Section 5.2.1, where we found that the plane setup is less sensitive towards the boresight angle *β*. Thus, the higher standard deviation of the boresight angle *β* can be attributed to the lower sensitivity of the plane setup. However, the target accuracy of 0.005◦ is still achieved.

**Figure 7.** Distributions of the estimated calibration parameters for a Monte Carlo Simulation with 1000 realizations. The grey histograms correspond to random observation errors only (cf. Table 1, case (i)). The colored histograms correspond to random observation errors with additional systematic observation errors: *δt<sup>e</sup>* = 5 mm (dark blue), *δt<sup>n</sup>* = 5 mm (orange), *δt<sup>h</sup>* = 5 mm (yellow), *δφ* = 0.005◦ (purple), *δθ* = 0.005◦ (green), *δψ* = 0.005◦ (light blue). The red vertical lines indicate the true values.

Inherently, the position and orientation observations of a mobile laser scanning system are prone to systematic errors due to the use of GNSS or uncorrected errors of the IMU. Therefore, we repeated the Monte Carlo simulation with additional biases of 5 mm and 0.005◦ for the position and orientation observations of the mobile laser scanning system. As in the case of random errors, we found that systematic errors of the position mainly affect the lever arm, while systematic errors of the orientation angles mainly affect the boresight angles. The results of the Monte Carlo simulation with both random and systematic observation errors are visualized as colored histograms in Figure 7.

Clearly, the lever arm components ∆*x* and ∆*y* remain unbiased in the case of systematic observation errors (cf. Figure 7, top left, top middle). The lever arm components ∆*x* and ∆*y* mainly depend on the horizontal position *t<sup>e</sup>* and *t<sup>n</sup>* of the mobile laser scanning system. In the simulation, we found that the associated systematic errors *δt<sup>e</sup>* and *δt<sup>n</sup>* are completely transferred to the residuals of *t<sup>e</sup>* and *t<sup>n</sup>* and, thus, do not affect the parameters. This results from the repeated measurements of the plane setup with opposite direction of motion of the platform (cf. Figure 5, right). Basically, this kind of double measurement can be considered as a measurement in two faces, which is an established strategy in geodesy for the elimination of systematic errors [81]. For the height component, however, the systematic error *δt<sup>h</sup>* is not eliminated in this way, which is why the lever arm component ∆*z* is biased by 5 mm (Figure 7, top right). For the elimination of a systematic height error *δt<sup>h</sup>* , the mobile platform would need to measure the plane setup upside down in the second pass.

Systematic orientation errors of the mobile laser scanning system are also not eliminated by the double measurement. A systematic error of the pitch angle *θ* is completely transferred to the boresight angle *β*, which is biased by 0.005◦ (Figure 7, bottom middle, green histogram). The same applies to a systematic error of the yaw angle *ψ*, which is completely transferred to the boresight angle *γ* (Figure 7, bottom right, light blue histogram). However, in contrast to this, a systematic error of the roll angle *φ* affects the boresight angles *α* and *γ* (Figure 7, bottom left and right, purple histograms). This phenomenon is caused by the correlation between the boresight angles *α* and *γ* (cf. Section 5.2.1 and Figure 6b) and the sequence of the Euler rotations from the n-frame back to the s-frame, i.e., *R n b* (*ψ*) → *R n b* (*θ*) → *R n b* (*φ*) → *R b s* (*γ*) → *R b s* (*β*) → *R b s* (*α*). In the case of a systematic error in the roll angle *φ*, this error is partially compensated by the boresight angle *γ* before it reaches the boresight angle *α*

due to the correlation between these two parameters. In contrast to this, a systematic error in the yaw angle *ψ* can completely be compensated by the boresight angle *γ* before it reaches the boresight angle *α*. Similar considerations can be made for a systematic error in the pitch angle *θ*.

The simulations indicate that systematic errors in the trajectory estimation corrupt the calibration results. However, such systematic errors are often site- and time-dependent, e.g., in the case of GNSS. This means that if we repeat the calibration multiple times over a certain time period, systematic errors might change in magnitude and sign. In this way, systematic errors could get a more random characteristic and, thus, could be reduced by averaging the calibration results of multiple runs.

Another source of systematic errors, which we have not discussed yet, is the 2D laser scanner. In this respect, we simulated a range finder offset of *d*<sup>0</sup> = 2 mm and analyzed its impact on the calibration parameters. As a result, the range finder offset mainly biases the lever arm components ∆*x* (≈0.4 mm) and ∆*z* (≈1.4 mm), as well as the boresight angle *β* (≈0.033◦ ). However, the used plane setup is sensitive towards a range finder offset. Thus, this intrinsic parameter can be added to the functional model in Equation (4) and estimated as part of the calibration. This is demonstrated in Section 6.3, where we estimate the range finder offset *d*<sup>0</sup> for test purposes based on real data.

Finally, we also took errors of the TLS point cloud into consideration, which is the basis for the determination of the reference plane equations (cf. Equation (3)). In this respect, a global registration error of the TLS point clouds seems to be the most serious problem. Hence, we simulated translation errors of up to 3 mm as well as rotation errors of up to 0.02◦ . We found that errors in the horizontal position of the TLS point cloud as well as tilting errors are filtered out to ≥85% in the calibration, i.e., such errors are transferred to the residuals of the adjustment. As in the case of systematic errors of the horizontal position of the mobile laser scanning system, this results from the repeated measurements of the plane setup with opposite direction of motion of the platform. Registration errors in height and azimuth, however, bias ∆*z* and *γ*, respectively. In practice, the registration of the TLS point cloud is based on a dense network of highly accurate control points (cf. Sections 6.1 and 7.1). This provides a high degree of accuracy and controllability. In the light of this, we expect registration errors in height and azimuth to be smaller than the target accuracy for the calibration parameters, i.e., 1 ... 1.5 mm for the lever arm components and 0.005◦ for the boresight angles.

#### 5.2.3. Impact of Gross Observation Errors

In addition to the accuracy, the quality of the estimated calibration parameters is also determined by the detectability of gross observation errors and the impact of undetected gross observation errors on the parameters. Therefore, we calculated the partial redundancies *r<sup>i</sup>* (cf. Equation (7)) and the minimum detectable outliers ∇*l<sup>i</sup>* (cf. Equation (8)) of the position and orientation observations of the mobile laser scanning system. The results are visualized in Figure 8.

Figure 8 (left column) shows that the partial redundancies of the position are close to 1 and, thus, well controlled. Gross errors of ∇*te*,*<sup>n</sup>* ≥ 48 mm for the horizontal position and ∇*t<sup>h</sup>* ≥ 72 mm for the height component can be detected due to the different horizontal and vertical accuracy of the mobile laser scanning system. Undetected gross errors in the position bias the lever arm by |∇*pte*,*<sup>n</sup>* | ≤ 0.17 mm and |∇*pt<sup>h</sup>* | ≤ 0.26 mm, respectively (not shown in Figure 8). The boresight angles are not affected by gross errors in the position.

Regarding the orientation (Figure 8, right column), the roll angle *φ* and yaw angle *ψ* are well controlled with partial redundancies of *r<sup>i</sup>* ≥ 0.85. However, the pitch angle *θ* is weakly controlled with partial redundancies of *r<sup>i</sup>* < 0.3. This is connected to the lower sensitivity of the plane setup towards the boresight angle *β* (Figure 6, case E), which is mostly aligned with the pitch angle *θ*. Accordingly, the detectability of gross errors is worse for pitch (∇*θ* ≥ 0.05◦ – 0.4◦ ) than for roll (∇*φ* ≥ 0.025◦ ) and yaw (∇*ψ* ≥ 0.05◦ ). The impact of undetected gross errors in roll and yaw on the boresight angles is |∇*p<sup>φ</sup>* | ≤ 0.00012◦ and |∇*p<sup>ψ</sup>* | ≤ 0.00018◦ , respectively. For the pitch angle, the impact is about two to three times larger with |∇*p<sup>θ</sup>* | ≤ 0.00031◦ (not shown in Figure 8). However, despite the small partial

redundancies of the pitch angles, their impact values are very small and do not affect the calibration. The impact of gross errors in the orientation on the lever arm is <40 µm and, thus, negligible.

Due to the high redundancy of the adjustment, the partial redundancies of the 2D laser scanner points are nearly *r<sup>i</sup>* = 1. Hence, outliers can reliably be detected and do not affect the calibration.

**Figure 8.** Partial redundancies *r<sup>i</sup>* (**top**) and minimum detectable outliers ∇*l i* (**bottom**) of the position [*te*, *tn*, *t<sup>h</sup>* ] *T* and orientation angles *φ*, *θ*, *ψ* of the GNSS/IMU unit (Global Navigation Satellite System, Inertial Measurement Unit). The plots show the values for each profile of first and second pass with opposite direction of motion of the platform (Figure 5, right).

Principally, the observations of the mobile laser scanning system are well controlled, which means that undetected gross errors do not considerably affect the calibration. This controllability is especially important for the position and the yaw angle observations of the mobile laser scanning system, because GNSS—which is prone to gross errors—contributes to these observations.

#### **6. Calibration of the Mobile Laser Scanning System**

This section presents empirical calibration results of the mobile laser scanning system that was introduced in Section 3. Section 6.1 describes the calibration measurements, Section 6.2 discusses the lever arm and boresight calibration. In Section 6.3, we address the applicability of the calibration field for estimating the range finder offset as additional intrinsic calibration parameter.

#### *6.1. Calibration Measurements*

The plane setup as shown in Figure 5 and validated in Section 5.2 was realized in the form of a permanently installed calibration field. The calibration field is shown in Figure 9. As postulated in Section 5.1, the calibration field covers an area of 10 m × 20 m and was realized using cost-effective, stable, and robust face concrete elements from civil engineering. For the calibration, the system was attached to a trolley due to its easy handling in the calibration field (Figure 9, right).

Initially, reference values for the plane setup were determined with TLS. For this purpose, a Leica ScanStation P50 was set up on five stations in order to completely cover all planes without obstructions. The scans were performed in two faces for reducing systematic errors of the TLS [81]. The TLS point clouds were georeferenced using a network of tie points as well as highly accurate georeferenced control points. All tie and control points were signalized with special BOTA8 targets (Bonn Target 8), which have a square stellar black and white pattern with a size of 0.3 m × 0.3 m (cf. Section 7). The BOTA8 targets have been developed at the University of Bonn and allow for an accurate registration of TLS

scans [82]. The mean absolute error after georeferencing was 1.2 mm. Following this, plane models were fitted to the TLS point clouds, which serve as reference information for the calibration. Please note that the control points for the georeferencing of the TLS point clouds are part of a bigger network of control points that was principally realized for the point-based evaluation of the mobile laser scanning systems (cf. Section 7).

For empirically analyzing the quality of the calibration parameters, the calibration procedure as shown in Figure 5 was repeated 98 times. Each calibration run took less than one minute. The 98 runs were performed in four independent blocks with 14, 21, 31, and 32 runs, respectively. After each block, the system was reinitialized. In addition, the blocks were measured at different days and daytimes. That way, different GNSS constellations were covered. Table 2 gives an overview.

**Figure 9.** Realization of the plane-based calibration field for the calibration of mobile laser scanning systems (**left**, middle). Mobile laser scanning system during the calibration measurements (**right**).



#### *6.2. Calibration of Lever Arm and Boresight Angles*

The calibration runs were processed independently. A total of 98 realizations of the calibration parameters were obtained that indicate the repeatability of the calibration. The results are visualized in Figure 10. The mean values and standard deviations are stated on top of the histograms. For better visual assessment, an ideal Gaussian distribution is added to the histograms.

The standard deviations of the lever arm components range from 0.9 mm to 4.5 mm. According to the simulation in Section 5.2, the horizontal components ∆*x* and ∆*y* are more precise than the vertical component ∆*z*. This is due to the fact that the quality of the lever arm calibration mainly depends on the accuracy of the position of the GNSS/IMU unit. In the case of GNSS/IMU integration, the height component is normally less accurate than the horizontal component [22,23]. In particular, systematic height errors are critical, since such errors can hardly be detected in the calibration and affect the lever arm component ∆*z*. In contrast, systematic errors in the horizontal position can be eliminated by measuring the plane setup with opposite direction of motion of the platform (cf. Section 5.2.2). Hence, ∆*z* is expected to be worse than ∆*x* and ∆*y* due to potential systematic height errors. These results are in good accordance with the simulations in Section 5.2. According to the simulations, gross errors in position most likely do not affect the lever arm, since they can reliably be detected.

The standard deviations of the boresight angles range from 0.0012◦ to 0.0188◦ . According to the simulations, the boresight angle *β* is estimated with the lowest accuracy. This is due to the fact that the plane setup is less sensitive towards *β* than towards *α* and *γ* (cf. Section 5.2.1). Generally, the accuracy of the boresight angles depends on the accuracy of the orientation angles of the GNSS/IMU unit. While roll and pitch can be estimated from the accelerometers, the yaw determination is more challenging, since it depends on the quality of the initial IMU alignment and the quality of GNSS during motion [23]. In this respect, the yaw angle is more prone to systematic errors than roll and pitch. Basically, systematic

errors in the yaw angle affect the boresight angle *γ*. In Figure 10, the histograms of the parameters *α* and *β* appear to be closer to a Gaussian distribution than the histogram of the parameter *γ*. This may indicate that systematic errors have affected the determination of *γ*.

**Figure 10.** Results of the 98 calibration runs: Lever arm components ∆*x*, ∆*y*, and ∆*z* (**top row**), boresight angles *α*, *β*, and *γ* (**bottom row**). The mean values and standard deviations *σ* are stated on the top of the histograms. For comparison, an ideal Gaussian distribution is added to the plots.

The standard deviations of the histograms in Figure 10 indicate the precision of one realization of the calibration parameters. Except for the parameter ∆*x*, ∆*y*, and *α*, the required target accuracy of 1 ... 1.5 mm for the lever arm as well as 0.005◦ for the boresight angles as defined in Section 5.2 is not reached. Generally, the empirical standard deviations are higher than those from simulation. This might be caused by systematic errors or neglected correlations that exist in reality, but were not simulated. For n uncorrelated realizations of the calibration parameters, however, the accuracy of the mean value can be improved by √ *n*. In our case, 15 statistically independent calibrations are needed to reach the target accuracies for ∆*z*, *β*, and *γ* (Table 3). Obviously, individual calibration runs are not fully uncorrelated. Therefore, we recommend to perform 15 calibration runs in the calibration field and to repeat this after a break of 1–3 h. After some hours the GNSS constellation is changed, which should lead to different systematic errors in the trajectory estimation.


**Table 3.** Standard deviations for a single calibration run and multiple calibration runs.

Figure 11 shows the additional uncertainty *σ add* <sup>3</sup>*<sup>D</sup>* = *σ obs*+*cal* <sup>3</sup>*<sup>D</sup>* − *σ obs* 3*D* of the mobile point cloud that results from the difference between a point cloud that includes an uncertainty of the calibration (*σ obs*+*cal* 3*D* ) and a point cloud that does not include an uncertainty of the calibration (*σ obs* 3*D* ); cf. Figure 4 (left). Clearly, the additional uncertainty of the point cloud is *σ add* 3*D* < 1 mm within a 50 m radius when using a mean calibration. Please remember that this was the overall goal of the calibration.

**Figure 11.** Additional uncertainty *σ add* 3*D* of the mobile point cloud due to the uncertainty of the calibration parameters for a single calibration run (**left**) and 15 calibration runs (**right**), cf. Table 3.

#### *6.3. Calibration of Range Finder Offset*

As mentioned in Section 5.2.2, the calibration field is also sensitive towards the range finder offset *d*<sup>0</sup> of the 2D laser scanner. The right part of Figure 12 depicts the sensitivity of the plane setup towards an uncorrected range finder offset of *d*<sup>0</sup> = 5 mm (cf. Figure 6). The range finder offset *d*<sup>0</sup> can simply be added to the calibration model (cf. Equation (4)):

$$\begin{bmatrix} \begin{bmatrix} t\_x \\ t\_y \\ t\_z \end{bmatrix} + \mathbf{R}\_n^{\varepsilon}(L, \mathsf{B}) \cdot \mathbf{R}\_b^{\mathrm{u}}(\boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi}) \cdot \left(\mathbf{R}\_s^{\mathrm{b}}(\boldsymbol{a}, \boldsymbol{\theta}, \boldsymbol{\gamma}) \begin{bmatrix} \mathbf{0} \\ (d\_s + d\_0) \cdot \sin b\_s \\ (d\_s + d\_0) \cdot \cos b\_s \end{bmatrix} + \begin{bmatrix} \Delta \mathbf{x} \\ \Delta \mathbf{y} \\ \Delta z \end{bmatrix} \right) \end{bmatrix}^T \cdot \begin{bmatrix} \tilde{n}\_x \\ \tilde{n}\_y \\ \tilde{n}\_z \end{bmatrix} - 1 \stackrel{!}{=} 0. \tag{10}$$

For test purposes, we estimated the range finder offset *d*<sup>0</sup> for each of the 98 calibation runs. The result is shown in the right part of Figure 12. The related histogram has a mean value of −0.05 mm and an empirical standard deviation of 0.08 mm. Accordingly, the range finder offset can be estimated in our calibration field. In this case, however, the range finder offset is of negligible magnitude.

**Figure 12.** (**Left**) Sensitivity of the plane setup towards a range finder offset. The colored plot shows the point-to-plane distance in the case of an unmodelled range finder offset of *d*<sup>0</sup> = 5 mm. (**Right**) Distribution of the estimated range finder offsets for the 98 calibration runs. The mean value is −0.05 mm and the empirical standard deviation is 0.08 mm. For comparison, an ideal Gaussian distribution is added.

#### **7. Evaluation of the Mobile Laser Scanning System**

This section addresses the evaluation of the mobile laser scanning system. In Section 7.1, we introduce the evaluation environment. Section 7.2 describes the evaluation measurements. Based on this, two different evaluation strategies are pursued, i.e., a point-based evaluation using control points (Section 7.3) and an area-based evaluation using TLS reference point clouds (Section 7.4).
