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

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