**Erik Heinz \*, Christoph Holst , Heiner Kuhlmann and Lasse Klingbeil**

Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115 Bonn, Germany; c.holst@igg.uni-bonn.de (C.H.); heiner.kuhlmann@uni-bonn.de (H.K.); l.klingbeil@igg.uni-bonn.de (L.K.)

**\*** Correspondence: e.heinz@igg.uni-bonn.de; Tel.: +49-228-73-3574

Received: 27 December 2019; Accepted: 4 February 2020; Published: 7 February 2020

**Abstract:** Mobile laser scanning has become an established measuring technique that is used for many applications in the fields of mapping, inventory, and monitoring. Due to the increasing operationality of such systems, quality control w.r.t. calibration and evaluation of the systems becomes more and more important and is subject to on-going research. This paper contributes to this topic by using tools from geodetic configuration analysis in order to design and evaluate a plane-based calibration field for determining the lever arm and boresight angles of a 2D laser scanner w.r.t. a GNSS/IMU unit (Global Navigation Satellite System, Inertial Measurement Unit). In this regard, the impact of random, systematic, and gross observation errors on the calibration is analyzed leading to a plane setup that provides accurate and controlled calibration parameters. The designed plane setup is realized in the form of a permanently installed calibration field. The applicability of the calibration field is tested with a real mobile laser scanning system by frequently repeating the calibration. Empirical standard deviations of <1 ... 1.5 mm for the lever arm and <0.005◦ for the boresight angles are obtained, which was priorly defined to be the goal of the calibration. In order to independently evaluate the mobile laser scanning system after calibration, an evaluation environment is realized consisting of a network of control points as well as TLS (Terrestrial Laser Scanning) reference point clouds. Based on the control points, both the horizontal and vertical accuracy of the system is found to be < 10 mm (root mean square error). This is confirmed by comparisons to the TLS reference point clouds indicating a well calibrated system. Both the calibration field and the evaluation environment are permanently installed and can be used for arbitrary mobile laser scanning systems.

**Keywords:** mobile laser scanning; lever arm; boresight angles; plane-based calibration field; configuration analysis; accuracy; controllability; evaluation; control points; TLS reference point clouds

## **1. Introduction**

In recent years, the use of mobile mapping systems has gained increasing acceptance. Clearly, this trend is confirmed by the great number of applications already addressed with this measuring technology, e.g., mapping of road corridors and city areas, determination of clearances, monitoring of infrastructures, or the extraction of geometric road parameters, road markings, and road furniture. A lot of publications and reviews about this topic can be found [1–5].

In the case of mobile mapping, laser scanners or cameras are mounted on a moving platform and sense the environment. Simultaneously, the position and orientation of the platform is determined by georeferencing sensors like GNSS (Global Navigation Satellite System), IMU (Inertial Measurement Unit), or odometry [6]. By fusing the observations of all sensors, a 3D point cloud of the environment is obtained. While mobile mapping systems have reached an operational status, challenges exist in the

context of quality assessment and quality control of the measured 3D point clouds. This is due to the fact that the data acquisition with mobile mapping systems comprises a complex processing chain. In this processing chain, many sources of errors exist that affect the quality of the 3D point cloud:


Generally speaking, all sources of errors must be subject to quality assessment and quality control. In this paper, however, we mainly focus on the calibration. In this regard, we address the extrinsic calibration of mobile laser scanning systems, i.e., the determination of the lever arm and boresight angles of a 2D laser scanner w.r.t. to a GNSS/IMU unit. The literature review in Section 2.1 shows that different methods exist for the extrinsic calibration of mobile laser scanning systems. We decided to implement a plane-based approach, since this type of calibration method has successfully been used in many fields of application. The idea of the calibration approach is oriented towards the research in Strübing and Neumann [7] and is based on matching the scanned points of the mobile laser scanning system to a setup of known reference planes. In the course of this, we can estimate corrections for the extrinsic calibration parameters within a Least Squares adjustment. Similar approaches have been used, e.g., in airborne [8], shipborne [9], ground-based [10–12], and indoor laser scanning [7,13].

The plane-based calibration is based on a Least Squares adjustment, which can be analyzed with tools from geodetic configuration analysis [14,15]. Such tools can be used to analyze the impact of random, systematic, and gross observation errors on the calibration process. Such an analysis makes it possible to assess and improve the reference plane setup and, thus, the accuracy and controllability of the estimated calibration parameters. In this way, the calibration is subject to thorough quality control and causes no additional uncertainty in the 3D point cloud. So far, the use of geodetic configuration analysis in the context of calibrating mobile laser scanning systems has only been studied to a limited extent and is the major scientific contribution of this paper. Based on the results of the configuration analysis, we installed a deliberately designed plane-based calibration field for mobile laser scanning systems that provides accurate and controlled calibration parameters. The calibration field was realized outdoors being permanent, stable, weather-resistant, and cost-effective. The calibration procedure in the calibration field takes less than one minute and, thus, can be repeated frequently. In addition to the configuration analysis, repetitive calibrations also increase the controllability of the calibration parameters and allow for a realistic empirical quantification of their accuracy and stability.

Before moving on to application, the quality of the calibration parameters has to be evaluated independently. However, in the case of mobile laser scanning systems, we are faced with a complex processing chain. The functional model of this processing chain is usually not fully known to users. Moreover, the stochastic distribution functions of the input variables are often unknown, necessarily not Gaussian as well as site- and time-dependent. This makes it challenging or even impossible to model the accuracy of 3D point clouds straightforward using error propagation [16,17] or to analyze individual system components like the calibration separately. Therefore, this paper focuses on an empirical evaluation of mobile laser scanning systems as a whole.

The literature review in Section 2.2 outlines common methods for the empirical evaluation of mobile laser scanning systems. Yet, these methods are not standardized [18] and facilities for the evaluation of mobile laser scanning systems are not available on a large scale [19–21]. This is why we decided to build up our own evaluation environment that implements existing evaluation strategies and combines them to a holistic approach. The evaluation environment consists of a dense network of control points as well as accurate reference point clouds of diverse building structures generated

with TLS (Terrestrial Laser Scanning). Our evaluation environment allows for a point-based as well as area-based evaluation and can be used for arbitrary systems, independent of the specific setup. Beside the configuration analysis for the plane-based calibration approach and the realization of the calibration field, the installation of the evaluation environment is the second important contribution of this paper. Both the plane-based calibration field and the evaluation environment are permanently installed, readily accessible, and were utilized by our own mobile laser scanning system proving the operationality of our facilities.

This paper is structured as follows: Section 2 surveys the literature. Section 3 introduces our mobile laser scanning system. Section 4 describes the plane-based calibration approach. Section 5 addresses the design of the calibration field using tools from geodetic configuration analysis. Section 6 presents real calibration results of our mobile laser scanning system. In Section 7, the mobile laser scanning system is evaluated in our evaluation environment. Section 8 concludes.

#### **2. Calibration and Evaluation of Mobile Laser Scanning Systems**

This section surveys the literature about the calibration (Section 2.1) and evaluation (Section 2.2) of mobile laser scanning systems. The focus regarding calibration is on the lever arm and boresight angles of laser scanners w.r.t. to the georeferencing sensors.

#### *2.1. Calibration of Mobile Laser Scanning Systems*

In the case of mobile mapping, a distinction is made between intrinsic and extrinsic calibration. Intrinsic calibration deals with systematic errors of individual sensors, e.g., the phase center of GNSS antennas [22]; axial misalignments, biases, and scale factors of inertial sensors or odometers [23]; or intrinsic corrections for laser scanners [24,25] and cameras [15]. Intrinsic calibration is beyond the scope of this paper. In contrast, extrinsic calibration addresses the determination of lever arms and boresight angles between different sensors or groups of sensors.

This work focuses on the extrinsic calibration of laser scanners. In this context, sensor-based, entropy-based, and geometry-based approaches exist. Sensor-based approaches make use of external sensors, e.g., total stations and laser trackers [26,27], theodolite measuring systems [28], close range photogrammetry [29], or measuring arms [30] in order to directly or indirectly measure lever arm components and boresight angles. Alternatively, in the case of laser scanners with 3D scanning mode, reference points that are known w.r.t. the georeferencing sensors can statically be scanned with the laser scanner in order to solve the extrinsic calibration problem [28,29,31]. Often, sensor-based approaches are elaborate, require additional instruments, or impose specific requirements on the calibration, e.g., clearly defined reference points at the casing of the sensors, a mechanical realization of the sensor frames, or a 3D scanning mode of the laser scanner.

In addition to sensor-based methods, entropy-based approaches can be used, which originate from robotics. Such approaches set up a cost function that describes the consistency within the 3D point cloud. This cost function is derived from scans of the environment with different position and orientation of the platform. The idea is that calibration errors lead to distortions in the 3D point cloud. By adjusting the calibration parameters, the cost function and the distortions can be minimized. This kind of self-calibration has been used for both indoor [32–37] and outdoor systems [33,34,37–40]. In addition to entropy, also other cost functions can be used, e.g., based on geometrical constraints for point neighborhoods such as planarity, curvature, or omnivariance [37,41]. However, the sensitivity towards the calibration parameters strongly depends on the environment and the trajectory, for which variation in all six degrees of freedom is required [40]. Especially for ground-based systems variation in roll, pitch, and height is difficult, which limits the accuracy and controllability of the calibration. Furthermore, the approaches do not provide direct quality measures for the calibration parameters. The accuracy of the calibration is usually evaluated based on repeated calibrations and the consistency of the 3D point clouds. Thus, thorough quality assessment and quality control is difficult.

Another way of self-calibration is to perform special driving maneuvers past building facades or cylindrical objects [34,42,43]. Potential tilts or displacements of the objects in the 3D point clouds are used to correct errors in the boresight angles or the time synchronization. The problem with these methods is that they calibrate single system components one after another. However, errors in a 3D point cloud typically result from the interaction of multiple system components. This means that errors in the 3D point cloud cannot be traced back unambiguously to errors in the calibration. Similar methods are also used for the extrinsic calibration of multi beam echo sounders adapted to ships using patch-test procedures [27] as well as for strip adjustment in airborne and UAV-based laser scanning (Unmanned Aerial Vehicle) [24,44].

The last category are geometry-based approaches, where constraints between the scan points of the mobile laser scanning system and geometric primitives are used for determining the extrinsic calibration parameters within a Least Squares adjustment. In this context plane-based methods are most common and have successfully been applied in airborne laser scanning [8,24,45–47], but also for systems mounted on UAVs [48,49], on ships [9], as well as for ground-based [10–12,18,26,34,50–52] and indoor systems [7,13,34,53]. While based on the same idea, plane-based calibration approaches vary in implementation. The most important differences are the number of calibration parameters (some only calibrate the boresight angles), the usage of natural or artificial planes, and the general handling of the planes. The latter means that the plane normals can either be determined before the calibration by independent means or the plane normals can be estimated as unknowns in the adjustment leading to a self-calibration approach. For the sake of completeness, please note that also other geometric forms can be used, e.g., spheres [54] or catenaries [52].

Clearly, there are many different approaches for the extrinsic calibration of laser scanners. We implemented a plane-based approach, which is based on minimizing the differences between scan points of the mobile laser scanning system and known reference plane equations by adjusting the lever arm and the boresight angles between the laser scanner and the GNSS/IMU unit. The basic idea of this approach is oriented towards the research in Strübing and Neumann [7].

Most publications highlight the importance of the plane configuration for the calibration, i.e., setup, number, and size of the planes. The principal measures for assessing and improving the quality of the configuration are the sensitivity of the plane configuration towards changes in the calibration parameters as well as the variances and correlations of the calibration parameters. Other quality criteria are the radius of convergence of the adjustment and the reduction of plane residuals after calibration. The advantage of plane-based approaches is that they are based on a Least Squares adjustment, which can be interpreted as a geodetic network. This means that we can use tools from geodetic configuration analysis in order to assess the quality of the estimated calibration parameters not only in terms of sensitivity and accuracy, but also in terms of controllability by investigating the robustness w.r.t. to gross observation errors [14,15]. So far, controllability in the context of calibrating mobile laser scanning systems has only been addressed to a limited extent [8]. The major scientific contribution of this paper is that we use tools from configuration analysis to investigate the impact of random, systematic, and gross observation errors on the calibration. In the course of this, we derive a plane setup that provides both accurate and controlled calibration parameters. This paper is connected to our previous publication in Heinz et al. [11].

#### *2.2. Evaluation of Mobile Laser Scanning Systems*

The evaluation strategies for analyzing the accuracy of mobile laser scanning systems can be classified into point-based, area-based, and parameter-based methods. Point-based methods use either natural control points (e.g., building corners, manholes, poles, or road markings) or artificial control points (e.g., targets or markers), which are extracted from the 3D point clouds and compared to reference values that were determined by other surveying methods like total stations, leveling, GNSS, or TLS (e.g., [18–21,26,28,29,31,33,55–59]). Such methods assess the absolute accuracy of a system as a whole, instead of analyzing individual components. Moreover, the precision can be examined

by measuring control points multiple times or by analyzing relative point distances. In this regard, the horizontal and vertical components are often investigated separately.

In addition to control points, area-based evaluation methods are widespread. In this respect, existing 3D city models can be utilized as a reference for evaluation [60–62]. Furthermore, reference point clouds (e.g., from TLS) are commonly used for evaluation (e.g., [10,11,18,21,33,63–65]), since software packages provide algorithms for the comparison of two point clouds. In this regard, it is also possible to mesh patches of the road surface that were measured multiple times with the mobile laser scanning system and compare these patches to each other [19,59]. Area-based evaluation methods supplement the point-based strategies, as mobile laser scanning is an area-based measuring technology. However, the interpretation of the differences of a mobile point cloud to a model or a reference point cloud is difficult due to the dependency on the structure of the measured object. For instance, height errors cannot be detected when driving parallel to a vertical wall.

A third way of evaluation emerges in the context of application. Geometric parameters of road surfaces, building structures, or objects can repeatedly be extracted from the point clouds of multiple passes. The repeatability of such parameters also indicates the quality of a system (e.g., [18,59,66]).

The review indicates that point-based as well as area-based approaches using control points and reference point clouds are common strategies for the evaluation of mobile laser scanning systems. Basically, such strategies provide an empirical evaluation of the system as a whole. In our view, this is the most effective evaluation strategy due to the complex and partially unknown processing chain. Yet, the associated methods are not standardized [18] and facilities for the evaluation of mobile laser scanning systems are not available on a large scale [19–21]. Therefore, we realized an own permanent evaluation environment with control points and TLS reference point clouds (cf. Section 7) that can be used for arbitrary mobile laser scanning systems.

#### **3. Mobile Laser Scanning System**

The mobile laser scanning system that is utilized within this work is shown in Figure 1. The core component is a navigation-grade inertial navigation system iMAR iNAV-FJI-LSURV [67] with fiber-optic gyroscopes, servo accelerometers, and RTK-GNSS (Real Time Kinematic). An odometer is optional. For the trajectory estimation, we use the software Waypoint Inertial Explorer 8.80 [68]. Depending on GNSS quality, accuracies of centimeters and centidegrees or better can be reached for the position and orientation. For mapping, a 2D laser scanner Z+F Profiler 9012A is used [69]. This instrument is specified with an accuracy of millimeters and comes with a maximum profile rate of 200 Hz, a maximum scan rate of 1 MHz and a special hardware optimization that decreases the measurement noise in the close-range [70]. The system can be adapted to a trolley or a van (Figure 1). A more detailed view of the system is given in Figure 2. A case study on a motorway showed that the accuracy of the system is about millimeter to centimeter under good GNSS conditions [59].

**Figure 1.** Mobile laser scanning system of the University of Bonn adapted to a trolley (**left**) or a van (**middle**). The 3D point cloud (**right**) was recorded during a mapping campaign in Bonn, Germany.

#### **4. Calibration Approach**

This section addresses the plane-based calibration approach. Firstly, the calibration parameters and the georeferencing equation for the generation of a 3D point cloud are introduced (Section 4.1). Following this, the calibration procedure is described (Section 4.2). Finally, we discuss details about the estimation and the quality assessment of the calibration parameters (Section 4.3).

#### *4.1. Calibration Parameters and Georeferencing Equation*

The measuring of a 3D point cloud using mobile laser scanning is based on the data fusion of multiple sensors. The goal of the extrinsic calibration is to determine the mutual installation position and orientation of these sensors on the platform. In this paper, we address the determination of the lever arm [∆*x*, ∆*y*, ∆*z*] *T* as well as the boresight angles *α*, *β*, and *γ* of a 2D laser scanner w.r.t. a GNSS/IMU unit (Figure 2, left). The calibration approach in Section 4.2 aims at determining these parameters, which are assumed to be temporally constant. However, this has to be checked regularly.

**Figure 2.** Different coordinate frames (i.e., s(canner)-frame, b(ody)-frame, n(avigation)-frame, and e(arth)-frame) and transformations for generating a georeferenced 3D point cloud.

In the context of mobile laser scanning, an object point [*x<sup>s</sup>* , *y<sup>s</sup>* , *zs*] *T* in the local sensor frame of the 2D laser scanner (s-frame) can be written as [0, *d<sup>s</sup>* · sin(*bs*), *d<sup>s</sup>* · cos(*bs*)] *T* , where *d<sup>s</sup>* is the measured distance and *b<sup>s</sup>* the scanning angle. In order to obtain a georeferenced point cloud, the object points of the 2D laser scanner must be transformed into the body-frame of the platform (b-frame), which often coincides with the coordinate system of the IMU. For this transformation, the extrinsic calibration parameters are needed, i.e., the lever arm [∆*x*, ∆*y*, ∆*z*] *T* as well as the boresight angles *α*, *β*, and *γ* (Figure 2, left). Subsequently, the orientation angles of the platform, i.e., roll *φ*, pitch *θ*, and yaw *ψ*, are used to transform the object points into the navigation-frame (n-frame), which is a north-oriented local level frame, whose origin coincides with that of the b-frame (Figure 2, middle). Finally, the object points are transformed to a superordinate earth-fixed and earth-centered coordinate frame, which we denote as e-frame (Figure 2, right), by using ellipsoidal longitude *L* and latitude *B*, as well as the translation vector *t e <sup>n</sup>* = - *tx*, *ty*, *t<sup>z</sup> T* . All these transformations can be combined to the georeferencing equation, which provides georeferenced scan points [*x<sup>e</sup>* , *y<sup>e</sup>* , *ze*] *T MLS* (MLS: Mobile Laser Scanning):

$$
\begin{bmatrix} x\_{\varepsilon} \\ y\_{\varepsilon} \\ z\_{\varepsilon} \end{bmatrix}\_{MLS} = \begin{bmatrix} t\_x \\ t\_y \\ t\_z \end{bmatrix} + \mathcal{R}\_n^{\varepsilon}(L, B) \cdot \mathcal{R}\_b^n(\boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi}) \cdot \left( \mathcal{R}\_s^b(\boldsymbol{a}, \boldsymbol{\beta}, \boldsymbol{\gamma}) \begin{bmatrix} x\_s = 0 \\ y\_s = d\_s \cdot \sin(b\_s) \\ z\_s = d\_s \cdot \cos(b\_s) \end{bmatrix} + \begin{bmatrix} \Delta x \\ \Delta y \\ \Delta z \end{bmatrix} \right), \tag{1}
$$

where *R j i* (·) denotes an Euler rotation matrix between two frames *i* and *j*. Afterwards, the global Cartesian coordinates [*x<sup>e</sup>* , *y<sup>e</sup>* , *ze*] *T MLS* are often transformed to application-related coordinate frames, e.g., UTM (Universal Transverse Mercator) and ellipsoidal heights. In the case of small areas, it is also possible to transform the object points from the n-frame to a local frame (l-frame) by omitting

the rotation matrix *R e n* (*L*, *B*) with ellipsoidal longitude *L* and latitude *B*, and just using the translation vector *t l <sup>n</sup>* = [*t<sup>e</sup>* , *tn*, *t<sup>h</sup>* ] *T* . This leads to the simplified equation:

$$
\begin{bmatrix} x\_l \\ y\_l \\ z\_l \end{bmatrix}\_{MLS} = \begin{bmatrix} t\_\varepsilon \\ t\_n \\ t\_h \end{bmatrix} + \mathbf{R}\_b^u(\boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi}) \cdot \left( \mathbf{R}\_s^b(\boldsymbol{a}, \boldsymbol{\theta}, \boldsymbol{\gamma}) \begin{bmatrix} x\_s = 0 \\ y\_s = d\_s \cdot \sin(b\_s) \\ z\_s = d\_s \cdot \cos(b\_s) \end{bmatrix} + \begin{bmatrix} \Delta x \\ \Delta y \\ \Delta z \end{bmatrix} \right). \tag{2}
$$

#### *4.2. Calibration Procedure*

The calibration approach is based on the use of reference planes [7]. In this section, we describe the calibration procedure with a given plane setup. Yet, we are not interested in how this plane setup was derived. The design of the plane setup and the influence of its configuration on the quality of the estimated calibration parameters are discussed in Section 5. The general workflow of the calibration procedure is illustrated in Figure 3.

**Figure 3.** (**a**) Generating reference normals for the plane setup using TLS (Terrestrial Laser Scanning), (**b**) Generating a point cloud of the plane setup using the mobile laser scanning system. The scan points of the mobile laser scanning system are automatically extracted from the scan lines using RANSAC (Random Sample Consensus) and assigned to the correct reference plane.

The first step of the calibration comprises the determination of reference values for the plane setup (Figure 3, left column). Therefore, the plane setup is scanned with TLS from several stations. The resulting TLS point cloud is georeferenced in the e-frame or l-frame using control points. As a

result, each plane is represented by a number of georeferenced points [*x<sup>e</sup>* , *y<sup>e</sup>* , *ze*] *T TLS*. Subsequently, the georeferenced TLS points are approximated with a plane model, which is parameterized by the normal vector - *nx*, *ny*, *n<sup>z</sup> T* and the distance parameter *dn*. By normalizing the normal vector with *dn*, the plane equation is described by only three parameters - *n*¯ *<sup>x</sup>*, *n*¯ *<sup>y</sup>*, *n*¯ *<sup>z</sup> T* [71]:

$$
\begin{bmatrix} \mathbf{x}\_{\varepsilon\prime} y\_{\varepsilon\prime} z\_{\varepsilon} \end{bmatrix}\_{TLS} \cdot \begin{bmatrix} n\_x \\ n\_y \\ n\_z \end{bmatrix} - d\_n = \begin{bmatrix} \mathbf{x}\_{\varepsilon\prime} y\_{\varepsilon\prime} z\_{\varepsilon} \end{bmatrix}\_{TLS} \cdot \begin{bmatrix} \vec{n}\_x \\ \vec{n}\_y \\ \vec{n}\_z \end{bmatrix} - \mathbf{1} \stackrel{!}{=} 0. \tag{3}
$$

The estimated normal vectors - *n*¯ *<sup>x</sup>*, *n*¯ *<sup>y</sup>*, *n*¯ *<sup>z</sup> T* serve as reference information for the calibration of the mobile laser scanning system.

Following this, the plane setup is scanned with the mobile laser scanning system (Figure 3, right column). By using the georeferencing equation and approximate calibration parameters, a mobile point cloud of the plane setup in the e-frame or l-frame is calculated (cf. Equation (1) or Equation (2)). The calibration approach is based on the constraint that the georeferenced points [*x<sup>e</sup>* , *y<sup>e</sup>* , *ze*]*MLS* of the mobile laser scanning system must fulfill the plane equations as given from the TLS survey in Equation (3). This leads to the following observation equation for the calibration:

$$\underbrace{\begin{bmatrix} \begin{bmatrix} t\_x \\ t\_y \\ t\_z \end{bmatrix} + \mathbf{R}\_n^\varepsilon(L, \mathsf{B}) \cdot \mathbf{R}\_b^\mathbf{n}(\boldsymbol{\phi}, \boldsymbol{\theta}, \boldsymbol{\psi}) \cdot \begin{pmatrix} 0 \\ \mathbf{R}\_s^\mathbf{b}(\boldsymbol{a}, \boldsymbol{\theta}, \boldsymbol{\gamma}) \\ d\_s \cdot \cos b\_s \end{pmatrix} + \begin{bmatrix} \Delta x \\ \Delta y \\ \Delta z \end{bmatrix} \end{bmatrix} \Bigg{)}\_\cdot^T \cdot \begin{bmatrix} \boldsymbol{\bar{n}}\_x \\ \boldsymbol{\bar{n}}\_y \\ \boldsymbol{\bar{n}}\_z \end{bmatrix} - 1 \overset{!}{=} 0. \tag{4}$$

Equation (4) leads to a system of equations that can be solved using a Least Squares adjustment within the Gauß-Helmert Model [15]. In the adjustment, the plane normals - *n*¯ *<sup>x</sup>*, *n*¯ *<sup>y</sup>*, *n*¯ *<sup>z</sup> T* are considered as an error-free reference. However, it is also possible to use them as observations when introducing an adequate variance information. The observations in the adjustment are the positions - *tx*, *ty*, *t<sup>z</sup> T* and the orientation angles *φ*, *θ*, and *ψ* of the mobile laser scanning system, as well as the distances *d<sup>s</sup>* and the angular observations *b<sup>s</sup>* of the 2D laser scanner. The lever arm [∆*x*, ∆*y*, ∆*z*] *T* and the boresight angles *α*, *β*, and *γ* are the parameters to be estimated. For small areas the ellipsoidal longitude *L* and latitude *B* can be considered as error-free constant values. In the case of performing the parameter estimation in the l-frame, the leveled positions [*t<sup>e</sup>* , *tn*, *t<sup>h</sup>* ] *T* are used as observations.

Please note that not each scan point of the mobile laser scanning system is assigned an own position and orientation, but all scan points within the same scanning profile are assigned the mean position and orientation of that scanning profile. This is reasonable for several reasons:


In order to assign each scan point of the mobile laser scanning to the correct reference plane, the scan lines of the mobile laser scanning system are separately pre-processed. Please note that the pre-processing is not done on the full 3D point cloud of the mobile laser scanning system, but on individual scan lines. This is illustrated in Figure 3 (right column). Initially, the scan lines are given in the s-frame of the 2D laser scanner. By ignoring the *Xs*-component, a 2D scan is obtained, where the planes are represented as line segments. These line segments are extracted from the scan lines by using a RANSAC (Random Sample Consensus) approach [72]. In the RANSAC approach, thresholds

for the maximum distance of a scanned point to a line segment as well as for the minimum size of a line segment are used. The thresholds were empirically determined based on the noise of the 2D laser scanner and the known size of the reference planes. Subsequently, the extracted line segments are transformed to the e-frame or l-frame by using the georeferencing equation and approximate calibration parameters (cf. Equation (1) or Equation (2)). Finally, the extracted line segments are inserted into the reference plane equations from TLS and assigned to the plane that gives the minimum distance error. This results in a segmented point cloud of the mobile laser scanning system (Figure 3, bottom right).

#### *4.3. Quality Criteria for the Estimation of the Calibration Parameters*

The basic equation of the calibration approach is Equation (4), which is used as an observation equation for a Least Squares adjustment within the Gauß-Helmert model [15]. By using Equation (5) and Equation (6), the calibration parameters *p*ˆ = [∆*x*, ∆*y*, ∆*z*, *α*, *β*, *γ*] *T* and their covariance matrix **Σ***p*ˆ*p*<sup>ˆ</sup> can be estimated, respectively:

$$\hat{\boldsymbol{p}} = \boldsymbol{p}\_0 + \left[\boldsymbol{\mathcal{A}}^T \left(\boldsymbol{\mathcal{B}} \boldsymbol{\Sigma}\_{ll} \mathbf{B}^T\right)^{-1} \boldsymbol{\mathcal{A}}\right]^{-1} \boldsymbol{\mathcal{A}}^T \left(\boldsymbol{\mathcal{B}} \boldsymbol{\Sigma}\_{ll} \mathbf{B}^T\right)^{-1} \boldsymbol{w}\_\prime \tag{5}$$

$$\boldsymbol{\Sigma}\_{\mathcal{P}\boldsymbol{\mathcal{P}}} = \left[ \boldsymbol{\mathcal{A}}^T \left( \boldsymbol{\mathcal{B}} \boldsymbol{\Sigma}\_{\mathcal{I}\boldsymbol{\mathcal{B}}} \boldsymbol{\mathcal{B}}^T \right)^{-1} \boldsymbol{\mathcal{A}} \right]^{-1},\tag{6}$$

where *p*<sup>0</sup> denotes approximate calibration parameters, the matrizes *A* and *B* contain the partial derivatives of Equation (4) w.r.t. the parameters and observations, respectively, **Σ***ll* is the covariance matrix of the observations and *w* is the vector of discrepancies. The covariance matrix **Σ***p*ˆ*p*<sup>ˆ</sup> indicates the accuracy of the estimated calibration parameters *p*ˆ and is an important criterion for analyzing the quality of the calibration parameters. The main goal of the calibration is that the uncertainty of the 3D point cloud, which is principally caused by the observation errors of the GNSS/IMU unit and the 2D laser scanner, does not increase significantly by the uncertainty **Σ***p*ˆ*p*<sup>ˆ</sup> of the calibration.

In order to define a target accuracy for the estimated calibration parameters, we performed an error propagation of Equation (2) with optimistic assumptions for the accuracy of the observations. These accuracies were oriented towards empirical values and manufacturer specifications and are summarized in Table 1. Two different scenarios were simulated, i.e., with and without an uncertainty for the calibration parameters. In Table 1, these two scenarios are denoted with cases (i) and (ii). The error propagation was executed in a local frame (l-frame) with east, north, and height component in order to distinguish between horizontal and vertical accuracy. A point grid was generated for the scan profile of the 2D laser scanner with a typical operating range of 50 m. For each grid point, a covariance matrix was calculated via error propagation. Based on this, a 3D point error *σ*3*<sup>D</sup>* = q *σ* 2 *xl* + *σ* 2 *yl* + *σ* 2 *zl* was derived from the trace of each covariance matrix.

Figure 4 (left) visualizes the 3D point errors *σ obs* 3*D* based on observation errors only (cf. Table 1, case (i)). In the close range, position errors are dominant with about 20 mm. Because of orientation errors, the uncertainty increases at higher distances. Figure 4 (right) shows the additional uncertainty of the point cloud when an uncertainty for the calibration parameters is added, i.e., *σ add* <sup>3</sup>*<sup>D</sup>* = *σ obs*+*cal* <sup>3</sup>*<sup>D</sup>* − *σ obs* 3*D* (cf. Table 1, case (ii) – case (i)). A radially symmetrical pattern is visible. However, the additional uncertainty is *σ add* 3*D* < 1 mm within a radius of 50 m. We define this to be the goal of the calibration. Thus, given optimistic assumptions for the accuracy of the observations, the calibration parameters must be determined with a standard deviation of ≤ 1 ... 1.5 mm for the lever arm and ≤ 0.005◦ for the boresight angles (cf. Table 1, case (ii)).


**Table 1.** Standard deviations for the error propagation of the georeferencing equation (Equation (2)).

**0 10 20**

**20**

**-50 0 50 XY [m]**

**0 10 20**

Beside the accuracy of the calibration parameters, the controllability of the estimation process is the second important quality criterion. Controllability addresses, if gross errors in the observations can be detected in an outlier test (i.e., internal controllability), and if not, how much undetected gross errors affect the parameter estimation (i.e., external controllability) [14,15]. The internal controllability can be analyzed using partial redundancies *r<sup>i</sup>* ∈ [0, 1], which can be calculated for each observation *l<sup>i</sup>* according to [73,74]:

$$r\_{l} = \left(\boldsymbol{\Sigma}\_{\overline{\boldsymbol{\nu}}\overline{\boldsymbol{\nu}}} \boldsymbol{\Sigma}\_{\overline{\boldsymbol{\nu}}\overline{\boldsymbol{\nu}}}^{-1}\right)\_{\overline{\boldsymbol{\nu}}} = \left(\boldsymbol{\Sigma}\_{l\overline{\boldsymbol{\nu}}} \boldsymbol{\mathbf{B}}^{T} \boldsymbol{\Sigma}\_{\overline{\boldsymbol{\nu}}\overline{\boldsymbol{\nu}}}^{-1} \left[\boldsymbol{I} - \boldsymbol{\mathbf{A}}\left(\boldsymbol{\mathbf{A}}^{T} \boldsymbol{\Sigma}\_{\overline{\boldsymbol{\nu}}\overline{\boldsymbol{\nu}}}^{-1} \boldsymbol{\mathbf{A}}\right)^{-1} \boldsymbol{\mathbf{A}}^{T} \boldsymbol{\Sigma}\_{\overline{\boldsymbol{\nu}}\overline{\boldsymbol{\}}}^{-1}\right] \mathbf{B}\right)\_{\overline{\boldsymbol{\nu}}}\tag{7}$$

where **Σ***vv* is the covariance matrix of the residuals and **Σ**¯ *l* ¯ *<sup>l</sup>* = *B***Σ***llB T* . Partial redundancies *r<sup>i</sup>* describe the contribution of an observation to the redundancy *r* of an adjustment, i.e., ∑ *r<sup>i</sup>* = *r*. They further indicate the amount of an observation error that is transferred to the own residual *v<sup>i</sup>* . Thus, high partial redundancies increase the probability that gross errors are detected. In geodetic network adjustment, values of *r<sup>i</sup>* > 0.3 are recommended [14]. When testing normalized residuals, it is even possible to calculate the minimum detectable outlier ∇*l<sup>i</sup>* [14,75,76]:

$$
\nabla l\_i = \delta\_0(a\_{T\prime}\beta\_T) \cdot \frac{\sigma\_{l\_i}}{\sqrt{r\_i}}, \quad \sigma\_{l\_i} \in \mathbf{E}\_{l1\prime} \tag{8}
$$

**-50 0 50 XY [m]**

**0 0.2 0.4**

where *δ*<sup>0</sup> is called non-centrality parameter, which is a function of the type I error probability *α<sup>T</sup>* and the type II error probability *β<sup>T</sup>* (e.g., *δ*0(0.001, 0.20) = 4.13). Based on the minimum detectable outlier ∇*l<sup>i</sup>* , we can calculate the impact ∇*p<sup>i</sup>* of an undetected outlier on the parameter estimation:

$$\nabla \mathcal{p}\_i = \left(\mathbf{A}^T \boldsymbol{\Sigma}\_{\overline{\mathcal{U}}}^{-1} \mathbf{A}\right)^{-1} \mathbf{A}^T \boldsymbol{\Sigma}\_{\overline{\mathcal{U}}}^{-1} \mathbf{B} \begin{bmatrix} \mathbf{0}, \dots, \nabla l\_i, \dots, \mathbf{0} \end{bmatrix}^T. \tag{9}$$

Both *r<sup>i</sup>* and ∇*l<sup>i</sup>* in the observation space as well as ∇*p<sup>i</sup>* in the parameter space allow for a rigorous analysis of the controllability of the parameter estimation in terms of gross errors [14,15].
