## 2 Challenges for Modeling of Five-Axis Coordinate Measuring Systems
### Description of Five-Axis Coordinate Measuring System Kinematics  
Five-axis measuring systems are characterized by the same basic kinematics of classic CMMs, expended additionally by two rotary joints implemented by articulating probe head. The whole system kinematics can be described using for example Denavit-Hartenberg notation with five variables: the length of displacement along _x,y,z_ axes and two angles defining rotation around the probe head at vertical and horizontal axes. In the case of the CMM's part of kinematic chain, there is no need for expanding the model on additional errors because of correction matrix application on almost all new or retrofitted machines. The correction is based on the model of 21 geometrical errors whose values are determined experimentally [6]. The errors of CMM kinematics may be then modelled using residual error distribution. In that method, the errors that remain uncompensated after the Computer Aided Accuracy (CAA) matrix application (residual errors) are determined in certain points in machine measuring volume together with random errors distribution. The experiment that leads to error determination may be conducted using high precision laser tracking devices (e.g., a LaserTracer system) and multilateration technique [7]. The retroreflector mounted in place of the probe head on machine quill approaches chosen points from several directions parallel to axes of machine's coordinate system and spatial diagonals of its measuring volume. Then the vector describing systematic part of residual error can be obtained by calculating the mean deviation between nominal and actual coordinates of examined points while the random part of residual errors may be modelled using standard deviations of point reproduction. To estimate errors in positions not included in experiments, different interpolation methods can be used such us nearest neighbor method or interpolation using b-spline curves [8]. Methodology described above will be used to determine the true position of origin of a probe head's base coordinate system. Thus, the main problem in adequate description of whole five-axis system, is determination of a kinematic model of the probe head. It need to include errors in the considered structure, mainly translational and rotational errors. The kinematics of probe heads used in five-axis systems is rather straightforward, similar to the solutions known from five-axis CNC centres (with two rotations of spindle) [9], to kinematics of laser tracker with interferometer located in rotating head [10] or to articulated arm coordinate measuring machine (AACMM) kinematics [11]. The kinematic model presented in [12] is the most congruent to the one that will be used in the case of the five-axis CMM's probe head, with this difference that it will not include the translational joint (which, in the case of the laser tracker, represents the changing distance between the center of the rotating head and a reflector's reference point) which will be replaced by fixed and known distance from intersection point of the probe head's axes and the mounting point of the stylus. Another divergence between mentioned models and the one that has to be determined for the five-axis CMM's probe heads is that the formerly used rigid probes and the errors introduced by usage of measuring or touch-trigger probes are not included in them.  
Together with the already developed probe head errors model, the kinematic model of the probe head could determine the true position of measured points and then simulate the measurement process.