Local, simplified kinematic errors model

The geometric errors associated with a CMM are relatively error motions between the tool and the work piece which show the mechanical imperfections of the CMM structure and the misalignment of the elements. Figure 4.2 shows the layout of the kinematic error components. The effect of the geometric errors are usually described by the 21 kinematic errors which consist of the positioning errors in each axis direction – δx(x,y,z),

δy(x,y,z), and δz(x,y,z) – which are the linear displacement errors and vertical and horizontal straightness errors, the rotational errors along three axes: Rx(x,y,z), Ry(x,y,z), and Rz(x,y,z), which are the roll, pitch and yaw angular errors, and three squareness errors: Øxy, Øyz, and Øzx (Wang, 2003). The estimation of error correction and uncertainty evaluation can be determined by evaluating these kinematic errors using mathematical simulations.

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CMM Kinematic Errors

21 model parameters

1 positional errors x 3 axes:

δy (x, y, z) = δyx δyy δyz

2 straightness errors x 3 axes

δx (x, y, z) = δxxδxy δxz

δz (x, y, z) = δzx δzy δzz

3 rotational errors x 3 axes

Rx (x, y, z) = Rxx Rxy Rxz

Ry (x, y, z) = Ryx Ryy Ryz

Rz (x, y, z) = Rzx Rzy Rzz

3 squareness errors

Øxy Øyz Øzx

Figure 4.2 Details of the kinematic error components

Hunzmann et al. (1990) introduced a uniform approach to CMM calibration, acceptance and reverification tests. These suggested that applications still require precise experimental procedures and a number of datum points in the CMM working volume, which usually need a full calibration and long-term stability of the reference artefacts requirements (Sartori and Zhang, 1995). However, this approach requires much time and effort to establish.

The approach presented in this research is similar but it introduces a local, simplified kinematic error model specific to the measurement procedures. This kinematic error model is local in the sense that it applies only to the particular region of the CMM working volume used for the measurement task and the contact probes being used. Furthermore, it can be used in the sense that the model applies only during the limited time period it takes to perform the measurement task.

The advantages of this method are that, for example, the thermal effects, medium-term systematic effects associated with the CMM, can be compensated for without the requirement to conduct a full parametric error compensation measurement. Then, a low density of datum points in the entire CMM working region are performed, but still require a sufficient density of the datum in the working volume that matters to determine the local kinematic models. Moreover, it is also achievable to constantly update the local kinematic models. Figure 4.3 illustrates the proposed approach of the local kinematic error model.

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Figure 4.3 Schematic of the local kinematic error model measurement (Yang et al., 2008)

To begin with, the errors are determined by using a 2D ball plate in different orientations (three positions along x-, y-, and z-axis) and measuring probe offsets in six different orientations of both physical experiments, using CMM and simulation programming in MATLAB and recorded as a set of points coordinates. Then, the calibration data is included in the data, which includes results from the experiment and simulation to establish a model simulation. Among several local kinematic measurements, the model can be updated from its measurement data by matching the kinematic errors in a number of locations and probe qualifications in the CMM working volume. This local kinematic model requires a limited time period of iterative measurements without the need to perform a full parametric error compensation exercise. Thus, the accurate radius/diameter and the position in each axis (x, y, z) of the ball plate have been shown and the calculation of error compensation and uncertainty as well. A physical CMM or CMM simulation can perform some examples of local evaluation tasks. The simulation should be performed and based on the real physical CMM, e.g. measuring range and measurement repeatability, etc. Furthermore, the probe qualifications simulation should be inserted in this model. While 2D ball plates are shown in the schematic, other reference artefacts (e.g. hole plates, ball plate, or step gauge, etc.) may also be used. The partial or full measurement for calibration information of the reference artefact is also required. The local kinematic model is based upon an axis-upon-axis build-up of the physical CMM.

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After obtaining the measurement data, either through a physical CMM measurement or a simulation, a based fit algorithm based on the Gauss-Newton method is assigned to estimate the model parameters and additional related parameters (including the artefact parameters). The local kinematic model will subsequently be used to calculate and compensate for the systematic CMM errors. Based on the law of propagation of uncertainty, the model can also be used to evaluate the uncertainties associated with the fitted parameters (both the kinematic model and artefact parameters) and the CMM measurements in the working volume of interest.

Numerical simulation has been presented in this approach to analyse the behaviour and performance of the local kinematic error models, including the procedures and preliminary results described in the next section.