Calibration of a Terrestrial Laser Scanner - Institute of Geodesy and ...

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124 6. Applications of Terrestrial Laser ScanningTable 6.2: Results of kinematic laser scanning based on the mathematical model of a regression line. The time, therotation per second (RPS), the scanning resolution, the velocity, the number of control points and the 3D accuracy areshown.Time [s] RPS Scanning Resolution Velocity [m/s] # Control Points 3D Accuracy [mm]'free''fix'100 25middle0.100205 3.4 1.8high0.100212.9 1.633middle0.100203.4 2.0high0.100213.9 2.4200 25middle0.100208 3.3 1.8high0.100213.2 2.133middle0.100202.3 1.9high0.100212.9 2.0300 25middle0.1002213 3.0 1.8high0.100213.6 2.233middle0.100212.9 2.1high0.100222.3 1.9eliminated by applying a median filter. Second, the Kaiman filter requires equidistantdata sets. Basedon the acquired and median filtered data, the equidistance is derived by polynomial interpolation. Thepolynomial interpolation may influence the equidistant data set to be generated. The influence of differentpolynomials were not compared in detail. Due to the motion of the test trolley, a third order polynomialpresented a good approximation. If higher orders of the polynomial are chosen, the significance of thecoefficients can be tested.The interpolated and equidistant data provide the inputfor the motion of the test trolley is a motion with a constant velocitydisturbances withfor the Kaiman filter.The mathematical modeland accelerations in the form ofXk = ••xk-i + At xk-i + -At2 Xfc-i. (6.5)According to the dynamic model described by Equation (5.21), the first two terms of Equation (6.5) definethe system state and are summarized in the transition matrix $. The accelerations are disturbances anddefines the disturbance input matrix T. The system state can be expressed by the parameters•position: x, y, z and•velocity: x, y, z.The disturbances are characterized by• acceleration: x, y, z.Based on the mathematical model and the parameters for the motion, the matrices can be determined. Thetransition matrix $ contains the derivatives of the variables describing the system state with respectto thevariables of the system state. The disturbance input matrix T consists of the derivatives of the disturbancevariables with respect to the system state variables:
zk-i)/ùd6.3 Kinematic Application: Test Tunnel 125/ 1 0 0 At 0 0 \0 1 0 0 At 00 0 1 0 0 At$fc = 0 0 0 1 0 0(6.6)0 0 0 0 1 0^000 0 0 1 )(W2 0 0 \0 W2 00 0 ±ÙJ?At 0 0(6.7)0 At 0V o 0 At /The observation of the system state is carried out by the trackingtotal station. Since the original obser¬vations are spherical coordinates, the Cartesian coordinates have to be derived using Equations (6.2). Thegiven coordinates allow for the introduction of indirect observation of the velocity with respectto the co¬ordinate axes. The observation vector z is then defined by the (indirect) observations of the six variables ofthe system state: Zk (6.8)XkVkZk(xk -xfc_i)/At(Vk -2/fc-i)/AtV {zk-JThe covariance matrix of the deduced observations isnot only a diagonalmatrix because the Cartesiancoordinates are correlated with each other.The relation between the observations and the system statevariables isdescribed by the matrix H, which contains the derivatives of the system state variables withrespect to the observations and results in an identity matrix.The error covariance matrix for the input disturbances Q and the error covariance matrix for the observa¬tions R have to be derived. Covariances have to be eventually considered for dependent variables, e.g. thededuced observations. The covariance matrices can be easily obtained by applying the error propagation.Furthermore, the system noise has to be considered by formingthe vector w.The recursive Kaiman filter requires the initialization of the system state x0and the error covariance matrixof the system state Po- The better the initial values, the better the estimation of the system states and theerror covariances of the system states for the following epochs.Bad initial values result in the need for moreepochs for the filtering process and therefore, iterations to properly estimate the system state including theerror covariances. In any case, the filter algorithm converges rapidly.The Kaiman filter describes a powerful tool for estimating the systemmoving object. The filter algorithm allows for tuningstate and the error covariances of aof the mathematical model for the movement indi¬vidually. The implemented algorithm is defined by a simple algorithm due to the motion of the test trolleyalongthe track line. Thesystem state as well as the error covariances can be adapted in many ways, ac¬cording to that best-suited for the present application.test trolley not only for constant motion but also for arbitrary motion.The trajectory resulting from applyingThe Kaiman filter is used to model the motion of thethe Kaiman filter in a forward and a backward direction are usedto calculate the object points scanned by the laser scanner during motion.The qualityof the estimated
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DISS. ETH NO. 17036Calibration of a
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ZusammenfassungSeit einigen Jahren
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ContentsAbstractiiiZusammenfassungv
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Contentsix6.1.3 Results Ill6.2 Stat
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1Introduction1.1 Terrestrial Laser
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1.2 Motivation 3Precision [m10~1 "1
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6 1. Introduction
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8 2. Components of Terrestrial Lase
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10 2. Components of Terrestrial Las
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24 3. Calibration of Terrestrial La
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'28 3. Calibration of Terrestrial L
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"[m] of system 90%) 3540 distance b
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!36 3. Calibration of Terrestrial L
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138 3. Calibration of Terrestrial L
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Inormal axis,intersectionpoint axis
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j/-*'.-,•66 3. Calibration of Ter
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Page 84 and 85: 74 4. Static Laser Scanning4.1.2 Mi
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Page 122 and 123: 112 6. Applications of Terrestrial
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Page 142 and 143: 132 7. Summaryrange, achievable by
Page 144 and 145: 134 7. Summary
Page 146 and 147: 136 A. Imaging System of Imager 500
Page 148 and 149: 138 B. Technical Data of Imager 500
Page 150 and 151: u140 C. Adjustment of SphereA =dfi,
Page 152 and 153: 142 C. Adjustment of Sphere
Page 154 and 155: 144 D. Electronic Circuit for Deter
Page 156 and 157: aMulti-SensorPlatform,PhDGeodesy Sw
Page 158 and 159: Accuracy-AeinTPS1200 AG, -Version 1
Page 160 and 161: -GenauigkeitsbetrachtungenInvestiga
Page 162 and 163: 152 BibliographyWunderlich, T. A. [
Page 164 and 165: List of Figures3.22 Residuals of th
Page 166 and 167: 156 List of Figures
Page 168 and 169: 158 List of Tables
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