The robot errors gathered by position measurement can be minimized by numerical optimization. Therefore a complete kinematical model of the geometric structure must be developed, whose parameters then can 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 the purpose of 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 in order to adapt the used robot model to the real kinematics.

In 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. In the figure a current example is shown: In-line measurement in automotive manufacturing, where the common „measurement tunnel“ used for 100% inspection with many expensive sensors are partly replaced by IR which carry only one sensor each. This way the total costs of a measurement cell can be reduced significantly. Furthermore the station can be re-used after a model change by simple re-programming without mechanical adaptations.

Further examples for precision applications are robot-guided hemming in car body manufacturing, assembly of mobile phones, drilling, riveting and milling in aerospace industry and increasingly medical applications.