factors. Here especially the joint offsets and deviations in lengths and angles between the individual robot links take effect. Measurement systems There exist different possibilities for pose measurement with industrial robots, e.g. touching reference parts, using supersonic distance sensors, laser interferometry, theodolites, calipers or laser triangulation. Furthermore, there are camera systems which can be attached in the robot's cell or at the IR mounting plate and acquire the pose of a reference object. Measurement and calibration systems are made by such companies as Bluewrist, Dynalog, RoboDK, FARO Technologies, Creaform, Leica, Metris, Metronor, Wiest, Teconsult and Automated Precision. Mathematical principles The robot errors gathered by pose measurements can be minimized by numerical optimization. For kinematic calibration, a complete kinematical model of the geometric structure must be developed, whose parameters can then be calculated by mathematical optimization. The common system behaviour can be described with the vector model function as well as input and output vectors (see figure). The variables k, l, m, n and their derivates describe the dimensions of the single vector spaces. Minimization of the residual error r for identification of the optimal parameter vector p follows from the difference between both output vectors using the Euclidean norm. For solving the kinematical optimization problems least-squares descent methods are convenient, e.g. a modified quasi-Newton method. This procedure supplies corrected kinematical parameters for the measured machine, which then, for example, can be used to update the system variables in the controller to adapt the used robot model to the real kinematics. Results The positioning accuracy of industrial robots varies by manufacturer, age, and robot type. Using kinematic calibration, these errors can be reduced to less than a millimeter in most cases. An example of this is shown in the figure to the right. Accuracy of 6-axis industrial robots can improved by a factor of 10. Accuracy of parallel robots after calibration can be as low as a tenth of a millimeter. Sample applications In the industry, there is a general trend towards substitution of machine tools and special machines by industrial robots for certain manufacturing tasks whose accuracy demands can be fulfilled by calibrated robots. Through simulation and off-line programming, it is possible to easily accomplish complex programming tasks, such as robot machining. However, contrary to the teach programming method, good accuracy as well as repeatability is required. In the figure, a current example is shown: In-line measurement in automotive manufacturing,