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

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propagationresults series Taylorterms order parameter.highereachestimatesepoch trunca¬ filtermeasurements.everyform system vector. sincecovarianceofrealfeedbackactual close [Jazwinski,into obtainsparametersbe divided then,surveycancalledcurrent step. sateinital estimations.atime arerequired compen¬ around biased extended Kaimanstepsnextnot maylinearizationstwoforward aprioriequations3It state-spacemissingerrorslead 1970].These update. projectincorporatenewand106 5. Kinematic Laser ScanningThe mathematical procedure of the Kaiman filter is to update the state of a system at time tk to the timetfc+i using observations at the same time and prior information captured in the state vector at time tk-\.The state vectors at subsequent time tk+1 are obtained recursively.Thus, the Kaiman Filter combines ob¬servations and the system dynamic. The state vector at a subsequent time tk+i can be computedstate vector at time tk is given including a description of the systemif theand control functions from that time insubsequent time. Such a dynamic system can be described in the state-space notation by [Gelb, 1974]Xk = Fk-i xk-i + Gk-\ wk-i + Lk-i•Wfc-i- (5.20)where x is the system state vector, w a stochastic vector and u a deterministic vector. F, G and L are matricesdefining the system behaviour due to dynamic, stochastic and deterministic parts. Neglecting deterministicforces in Equation (5.20) the dynamic model can be expressed by [Hofmann-Wellenhof et al., 2003]xk 3>fc-i = •-xk-i +rfc Wk-i. (5.21)where 3>fc_iis the transition matrix for the system state and Tfc_i is the disturbance input matrix for thesystem state affected by noise parameters between epoch tk-i and tk. Vector Wk is denoted assystemnoise, which is assumed following a Gaussian distribution with zero mean, uncorrelated white noise and acovariance matrix Qk-i describing the uncertainties of modeling the behaviour of the dynamic system:wk~N(0,Qk). (5.22)The state-space notation described by Equation (5.20) is the first fundamental equation that is required forthe Kaiman filtering.state of the system3The second fundamental equation is defined by observations that surveythe actualzk = Hk -xk+vk. (5.23)where zk is the observation vector and Hk describes the linear relation between the observations and thesystem state variables. The vector vk follows a Gaussian distribution with zero mean and white noiseincluding a covariance matrix Rk describingthe uncertainties of measurements:vk~N(0,Rk). (5.24)In the strict sense, the discussed formulae are only valid for dynamic processesthat are discrete-time con¬trolled and can be expressed by linear equations. The filter then is called a discrete Kaiman filter. The ex¬pansion to a more general form of the dynamic process of a system state leads to non-linear equations ofthe system state and the observations. The given matrices, i.e. transition matrix $k, the disturbance inputmatrix 1^ and the measurement matrix Hk, have to be replaced by Jacobian matrices since the non-linearrelations are developedepochs.
-HkKkHk)T5.3 Position-Fixing Using Total Station 107measurements into the a priori estimation to obtain an improved a posteriori estimation. It can be said thatthe time update acts as predictor equations and the measurement update acts as corrector equations.time update("predict")measurement update("correct")Figure 5.12: The Kaiman Filter cycle consisting of a time update and a measurement update according to [Welch andBishop, 2004].In the following notations, the superscript minus (~) states for a priori estimations at step k given knowl¬edge of the process prior to step k.Otherwise, they are a posteriori state estimations at step k given mea¬surement zk. Since there is a discrepancy between a predicted measurement and the actual measurement, adifference is present, which is called filter innovation. The gain matrix K weightsthe update state estimation includingthe measurements:the innovation to obtain*fcxk==xk + Kk ik (5.26)zk xk(5.25)A summary of the Kaiman filter cycle including the equations are given in Table 5.10, adapted from [Welchand Bishop, 2004]. The filter has to be initialized by initial values for xq and Pq-Table 5.10:Bishop, 2004].Kaiman filter equations for the time update and the measurement update, adapted from [Welch andTime UpdateMeasurement Update1) Project the state ahead(1) Compute the Kaiman gainxk= ^k-ixk-i + rfc_ixfc_iKk=Pk-Hl{HkPk-Hl + Rk)^(2) Projectthe error covariance ahead(2) Update estimate with measurements zkPk=*fc-iPfc-i*ri + rfc_iQfc_ir£_1xk = xk + Kk(zk -Hkxk)(3) Updatethe error covariancePk = (I-KkHk)Pk-{I-+ KkRkKlThe Kaiman filter described uses all collected data and information prior to the present epoch. If the systemstate is not of interest in real-time and the Kaiman filter is applied in post-processing, the system statecan be smoothed by using information of epochs after the present epoch. Therefore, the filter runs not
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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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Page 84 and 85: 74 4. Static Laser Scanning4.1.2 Mi
Page 86 and 87: 76 4. Static Laser ScanningOptical
Page 88 and 89: 78 4. Static Laser Scanningmaximum
Page 90 and 91: 80 4. Static Laser Scanningand gene
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Page 94 and 95: 84 4. Static Laser Scanning4.3.3 NU
Page 96 and 97: 86 4. Static Laser Scanning
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Page 122 and 123: 112 6. Applications of Terrestrial
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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
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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
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Page 164 and 165: List of Figures3.22 Residuals of th
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156 List of Figures
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158 List of Tables
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200220072002Curriculum VitaePersona
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