The use of Kalman filtering in combination with an electronic tacheometer

Abstract

Modern electronic tacheometers offer the possibility to capture kinematic processes in real time. In case when the kinematic process is observed with only one measurement system, we have no possibility to perform redundant observations that would enable the accuracy estimation of observations and computed values. The Kalman filter represents a method of advanced geodetic analysis and as such adjusts the redundant data in an optimum way. Incorporating a time component directly into a processing of terrestrial kinematic observations demands good knowledge about the procedure of processing kinematic terrestrial observations and the electronic tacheometer capabilities. For this purpose the developed model of Kalman filter for processing kinematic terrestrial observations-discrete Wiener process acceleration model-was tested on reference trajectory in the Geodetic Laboratory of the Technical University Munich.

Statistical Process Control of a Kalman Filter Model

“…The importance of the convergence of the trace of system state covariance matrix lies in its definition: is the sum of the diagonal elements of the matrix , which represent variances (squares of standard deviations) of the system-state components. If the sum of diagonal elements converges towards some value, then we reason that single elements also converge towards some solution [ 10 ].…”

“…The “kinematic process” was performed as a tracking problem, with the electronic tacheometer, TCRA1201 Leica Geosystems, used to perform automatic measurements of a moving Leica Geosystems 360°-reflector, installed on a moving trolley. A more detailed description of experimental set-up can be found in [ 9 , 10 ].…”

“…We rewrite main elements of a functional model. More details are given in [ 10 ]. The KF process equation, where no deterministic component is present, has the following form [ 10 ]: “where w˜ k is the uncorrelated white process noise and represents the acceleration increment during the k -th sampling period.…”

“…More details are given in [ 10 ]. The KF process equation, where no deterministic component is present, has the following form [ 10 ]: “where w˜ k is the uncorrelated white process noise and represents the acceleration increment during the k -th sampling period. The process noise enters into the system dynamics through the vector gain Γ k : The system state x k in our case is: where x, y, z are the position components of the moving prism and υ x ,υ y , υ z and a x , a y , a z are velocity and acceleration in all three directions, respectively.…”

“…Some of the material covered in this contribution are closely related to the content in [ 9 , 10 ], where more detailed explanations of the practical example presented here, especially the development of the model and measurement system, can be found. The main intent of this contribution is to describe in detail the parameters that enable evaluation of a Kalman filter model in real-time and when no reference frame is available and no additional measurements are performed.…”