# Simulation Model for Correction and Modeling of Probe Head Errors in Five-Axis Coordinate Systems
## 1 Introduction  
The development of production engineering and the demand for high quality manufacturing brings about new challenges in the coordinate metrology, which serves as the key tool for quality assurance systems in many branches of industry (automotive, aerospace, machine, _etc._) [1; 2; 3]. With the increase in manufacturing accuracy, the measurements and methods used for assessing measurement uncertainty have to be more precise. The time spent on performing measurements and evaluations also plays a crucial role, as it has a fairly significant impact on the total costs of manufacturing. Therefore, new solutions are being introduced in regards to both the construction of measuring systems, and the methods for the assessment of coordinate measurement accuracy [4; 5]. Simulative models constitute one of the newest developments in accuracy estimation. In practice, such models are based on developing virtual models of measuring systems. Their main advantage is the possibility to obtain measurement uncertainty following only a single measurement, whereas the common methods of uncertainty estimation (such as the calibrated workpiece method [6; 7], or multiple measurement strategy [8; 9]) would require multiple measurements. In some instances, simulative models may also be used for error compensation, and consequently, for improving measurement accuracy. The virtual Coordinate Measuring Machine (CMM) model was first developed in the 1990s at Physikalisch-Technische Bundesanstalt in Germany [10; 11]. The original model was based on the determination of all possible error causes in the entire measuring volume of the CMM, and the evaluation of their impact on measurement results. Thanks to this, the mathematical model ofideal measurement was supplemented by the influence of possible errors that may occur during measurements, and used for performing multiple simulations of the considered measurement. Measurement uncertainty was then determined based on the variability of simulated results and simple statistical operations. So far, several models have been developed for typical three-axis CMMs. All of those models utilize a similar working principle as the PTB model, and were presented in detail in [12; 13; 14; 15; 16; 17]. Lately, a virtual model of measurements performed on the Articulated Arm CMM was also developed [18]. Further works focused on the construction of a virtual model for the Laser Tracker system are currently in progress both in Germany and the UK. In the last years, it has been possible to observe an increase in the popularity of the so-called five-axis coordinate systems. The five-axis system is created by adding two rotations of a probe head to the standard movements of the machine. Thanks to this solution, it is possible to speed up the measuring process, especially for solids of revolution such as rings, cylinders, spheres, _etc._, without accuracy loss. Workpieces of this type may be measured using mainly movements of the probe head kinematic pairs. Unlike the three-axis CMM, there is no need to move the whole body of the machine during measurements of all points. The reduction in measurement time is clearly visible, and because of that, the number of installations or retrofits for five-axis systems is becoming a fairly common trend in the industry. Nevertheless, hardly any literature can be found on this subject, and few scientists pay sufficient attention to this type of system. For this reason, the authors decided to investigate the field of errors of probe heads used in five-axis systems, and to attempt to develop a model of this type. The main sources of errors occurring during coordinate measurements performed on a CMM are kinematic system errors and probe head errors [19; 20; 21; 22]. These types of errors are included in all virtual models of CMMs. The models of probe head errors are usually based on error identification with the use of material standards, such as spherical standards and standard rings. Some currently used probe head error simulation models are described below.