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A novel interpolation algorithm, fuzzy interpolation, is presented and compared with other popular interpolation methods widely implemented in industrial robots calibrations and manufacturing applications. Different interpolation algorithms have been developed, reported, and implemented in many industrial robot calibrations and manufacturing processes in recent years. Most of them are based on looking for the optimal interpolation trajectories based on some known values on given points around a workspace. However, it is rare to build an optimal interpolation results based on some random noises, and this is one of the most popular topics in industrial testing and measurement applications. The fuzzy interpolation algorithm (FIA) reported in this paper provides a convenient and simple way to solve this problem and offers more accurate interpolation results based on given position or orientation errors that are randomly distributed in real time. This method can be implemented in many industrial applications, such as manipulators measurements and calibrations, industrial automations, and semiconductor manufacturing processes.

A suitable interpolation method is important to fit the target pose errors based on the pose errors of the neighboring grid points around the target. In recent years, many advanced interpolation algorithms have been designed and developed by different researchers [

Luo et al. developed a range-restricted

Goodman and Meek presented a planar interpolation method using a pair of rational spirals to solve planar and two-point G^{2} Hermite interpolation problem [

Among those interpolation methods, two of them are very popular and widely implemented in most industrial and manufacturing processes, trilinear and cubic spline interpolation algorithms [

Both linear and cubic spline interpolation methods can achieve satisfactory interpolation results for a common measurement and calibration process [

The fuzzy error interpolation technique utilizes a fuzzy inference system to estimate machine or manipulator pose errors, which is consistent with the random distributed nature of the pose errors. These pose errors can be considered as a fuzzy set at any given moment of time. The fuzzification process takes into account a range of errors rather than only a crisp error value. Therefore, the fuzzy error interpolation technique has the potential to improve the error estimation and compensation results for the target.

Fuzzy interpolation techniques have been rapidly developed and implemented in many academic and industrial fields in recent years [

A comparison between trilinear, cubic spline, and fuzzy interpolation methods used in accurate measurements and compensations for machine or manipulator calibration are discussed in this paper. The simulation results show that the fuzzy interpolation outperform other interpolation methods.

The remainder of the paper is organized into the following four sections. The principles of the two popular traditional interpolation techniques, trilinear and cubic spline, are outlined in Section

The trilinear and cubic spline interpolation methods are designed to construct a surface based on the known errors of neighboring points. The target pose error is then derived by using an error surface equation. The operation principles of the trilinear and spline interpolation methods are discussed in this section.

Trilinear interpolation is a computational process of linearly interpolating points within a 3D box given values at the vertices of the box, and it is the most common application in interpolating within cells of a volumetric dataset [

Refer to Figures

Eight corner points.

3D trilinear interpolation.

We can first perform the linear interpolation along the

Then we interpolate these values along

Finally, we interpolate these values along

The above operations can be illustrated by the following sequence: first we perform linear interpolation between

Combining (

In general, the box will not be of unit size nor will it be aligned at the origin. Simple translation and scaling (possibly of each axis independently) can be used to transform into then out of this simplified situation.

As illustrated in Figures

The cubic spline method is to estimate a cubic surface

A linear equation can be derived as follows:

After quite a bit of manipulation, this result is in the cubic polynomial [

By using the tridiagonal matrix,

An example of a 2D error surfaces in

Error surfaces in

in

in

Compared with trilinear interpolation method, the cubic spline method uses a more arbitrary shaped surface to approximate the error. Therefore, it provides more accurate position compensation results for known positions around a target. For unknown interpolated data, such as random noises, the interpolation results may not as good as desired since the interpolated data are randomly distributed noises.

From the structure of the trilinear interpolation technique, it can be observed that the method assumes that the position error on the target point

In order to solve this problem and to improve the measurement and compensation accuracy, a dynamic online fuzzy interpolation method is introduced. The traditional fuzzy inference system uses predefined membership functions and control rules to construct lookup tables and then picks up the associated control output from the lookup table as the fuzzy inference system works in an application. This kind of system is often called an offline fuzzy inference system because all inputs and outputs have been defined prior to the application process. This offline fuzzy system may not meet accuracy requirements in certain applications based on the following reasons First, the pose error of the target is estimated based on errors of 8 neighbouring grid points, and these neighbouring errors are randomly distributed. The offline fuzzy output membership functions are defined based on the errors range, say the neighbouring errors’ range. However, this range estimation is not as good as the one deduced from the actual errors obtained on 8 grid points. Second, since each cell needs one lookup table for the offline fuzzy system, it needs a large memory space to save a great number of lookup tables, which is both space and time consuming, and therefore not suitable for real-time processing. For example, in our study, the robot workspace is divided into ^{3}. Assume that one lookup table is for one cubic cell, and this needs about 64000 lookup tables! By using an online dynamic fuzzy inference system, one can estimate the target pose error by combining the output membership functions, which are obtained from real errors on the neighbouring grid points, with the control rules in real-time after this online fuzzy system is implemented. Therefore, we do not need any offline lookup tables at all. This means that one cannot determine the output membership functions until the fuzzy inference system is applied to a real process, and this is based on the real errors on the grid points, not a range.

The definition of this dynamic online fuzzy inference algorithm is shown in Figure

Definition of the fuzzy interpolation inference system.

The pose error at each grid point is defined as

The control rule is shown in Figure

The distance between the neighboring grid points of each cell on the workspace is 20 mm in ^{3} space. This is a typical workspace of most popular manipulators implemented in semiconductor manufacturing operations [

Input and output membership functions.

The predefined output membership functions are used as default ones, and the actual output membership function will be obtained by shifting the default one based on the actual error values on the grid points. For each cell, 8 output membership functions are implemented, and each one is associated with the error at one grid point. In Figure

The Gaussian-bell waveforms are selected as the shape of the membership functions for three inputs. As shown in Figure

The control rules shown in Figure

The input error variables can be expressed as a label set

Assume that

Here

The advantage of using the online fuzzy inference system is that the control output has the real-time control ability, but the drawback is that this type of control has a relative longer response time because of the calculation performed in the fuzzy inference system. This shortcoming becomes of little importance as the availability of high-speed CPUs for the controllers.

Extensive simulation studies have been performed with a PUMA 560 robot in order to illustrate the effectiveness of the proposed fuzzy error interpolation technique in comparison to the trilinear and cubic spline interpolation methods. The simulated position error is a uniformly distributed random noise

Simulated interpolation results—position errors.

Simulated interpolation results—orientation errors.

It can be found that the fuzzy interpolation method has more accurate compensation result for both position and orientation errors compared with both trilinear and cubic spline methods. The max position error of the fuzzy interpolation method is about 0.026 mm, which is about 41% smaller compared with the error obtained from the trilinear method (0.044 mm) and 25% smaller with respect to the error interpolated from the cubic spline method (0.033 mm). For the mean position errors, the fuzzy interpolation method also outperforms the other two methods. Similar comparison results can be obtained from the orientation errors shown in Figure

The numbers in the horizontal axes in Figures

The FIA method provided in this paper has better performances compared with other interpolation methods. One possible shortcoming of this method is that it may need high-speed computer and large memory space to process and store predefined data on all grid points. However, this disadvantage can be easily overcome by using high-speed CPUs and huge memory spaces in today’s computers.

A comparison of fuzzy error interpolation technique with trilinear and cubic spline interpolation methods used for high accuracy measurement and calibration of robots is discussed and analyzed in this paper. The simulation results show that the measurement and calibration results can be greatly improved when a fuzzy interpolation method is adopted. By using this fuzzy error interpolation algorithm, both position and orientation errors, especially for the random-distributed errors, can be significantly reduced and suppressed, and therefore the measurement and calibration accuracy can be greatly improved. This algorithm can be conveniently implemented in the real manufacturing process to reduce the production cost and operation times. The key technology used in this algorithm is the dynamic and online process in which the output membership functions are determined online based on the real position and orientation errors of the grid points around the target.