# Formal model for describing orientation errors.

1. INTRODUCTION

One of the most critical tasks in fixture design is to determine the orientation error of the workpiece within the locators system. As one of the deterministic components of a machining error, orientation error is primarily caused by size and position variations both from locating elements on a fixture side and from locating features on a workpiece side. Because of the diversity of part shape, locating feature form and tolerance specification, there is no comprehensive solution to estimate orientation error.

According to Bragaru (1998) the accuracy of the piece orientation into the device is determined by the variation of the relative position of the basis systems belonging to the workpiece and to the support element, as in Fig.1. The basis are reference points, lines or planes which are assumed to be exact. They are established based on real features. For example, Fig.2 shows the basis system for a cylindrical workpiece oriented on V-block.

Our earlier works (Simion, 1995) presented a model based on coordinates transformation in order to automate the calculation of the orientation error. This model is based on the mathematical coordinates transformation theory and can be used into a CAD system.

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2. LITERATURE REVIEW

Many works in the field of fixture design dealed with determining the precision of workpiece location. Various methods for the determination of workpiece location have been developed. Cai et al (1997) used variational methods to model the workpiece resultant error. Kumar et al. (2000) used a neural network approach to conceptually design complete fixture units. Kang et al. (2003) used two models (geometric and kinetic) to verify the fixture design. Wang (2002) developed a tolerance analysis method to assist fixture layout design for 2D workpieces. Bragaru (1998) used a formula for calculating the vector guidance error, based on the relative position of bases. Zhang et al. (2001) analyzed the locating error for computer aided design.

3. THE MATHEMATICAL MODEL

Let (Oxyz) be the reference system, connected to the locating elements that determine the piece orientation and let (O'x'y'z') be the reference system connected to the workpiece, as in Fig.3.

The relationship between the rectangular coordinates (a,b,c) and (a'b'c') of a point in the two systems, is established by the formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

with ([a'.sub.0], [b.sub.0], [c.sub.0]) being the coordinates of the origin of the (O'x'y'z') system in the (Oxyz) system.

The orientation error represents the extreme values of the coordinates variations [v.sub.a], [v.sub.b] and [v.sub.c], considering that:

[v.sub.a] = i(a'-a), [v.sub.b] = i(b'-b), [v.sub.c] = i(c'-c). (2)

The value of the "i" coefficient is 1 or 2, corresponding to the situation when the workpiece can move on the support elements in one, respectively two senses, in the error's direction.

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From the analytical equations of the functions (2) there results the orientation error formula.

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For example, in 2D, for rectangular coordinates, (1) can be written as:

a' = a cos [alpha] + b sin [alpha] - rcos([beta] - [alpha])

b' = - a sin [alpha] + b cos [alpha] - rsin([beta] - [alpha]) (3)

Where: (a,b) respectively (a',b') are the coordinates of a point M; [alpha] is the angle between Ox and O'x' axis; (R, [beta]) define the position of the origin of the (O'x'y') system in the (Oxy) system and r is the variation of the R variable--Fig.4.

By substituting (3) in (2) we finally obtain the orientation errors for the dimensions a and b:

[[epsilon].sub.o] (a) = max{i[a(cos [alpha] -1) + b sin [alpha] - r cos([beta] - [alpha])]}

[[epsilon].sub.o] (b) = max{i[-a sin [alpha] + b(cos [alpha] -1) - r sin([beta] - [alpha])]} (4)

Fig. 5 shows two examples of orientation error determined by using the coordinates transformation method. The schemes are build on a single orientation surface.

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Fig. 6 illustrates the orientation error for a complex scheme, based on two orientation surfaces. The error is also determined using the proposed model.

4. CONCLUSION

This paper presented an original model describing the workpiece location error caused by fixture geometric errors.

The designed model was used to create a database and a specialized software for analyzing the orientation error. This software vas verified on particular design situations. The results were confirmed by comparison with the existing data in literature.

However, this work has not reached it's limits and can be further exploited. The model was used inside a specialized CAD system for fixture design (Simion, 1995). Future work will address further experimental validation of the model by applying it to different fixture layouts and varying part geometries.

5. REFERENCES

Bragaru, A. (1998). Proiectarea dispozitivelor (Fixture Design), Editura Tehnica, ISBN 973-31-0717-4, Bucharest.

Cai, W.; Hu, S.J. & Yuan, J.X. (1997). A Variational Method of Robust Fixture Configuration Design for 3-D Workpiece, Journal of Manufacturing Science and Engineering, Vol. 199, pp. 593-602.

Kumar, A.S.; Subramaniam, V. & Teck, T.B. (2000). Conceptual design of fixtures using machine learning techniques, International journal of Advanced Manufacturing Technology, Vol. 16, pp. 176-181.

Kang, Y.; Rong, Y. & Yang, J-C. (2003). Computer-aided fixture design verification, International Journal of Advanced Manufacturing Technology, Vol. 21(10-11), pp. 827-849.

Simion, I. (1995). Research concerning the precision of the orientation schemes, Ph.D.Thesis, University "Politehnica" from Bucharest.

Wang, M. (2002).Tolerance analysis for fixture layout design, Assembly Automation, Vol. 22, pp. 153-162.

Zhang, Y.; Hu, W.; Kang, Y.; Rong, Y. & Yen, D. W. (2001). Locating error analysis and tolerance assignment for computer-aided fixture design, International Journal of Production Research, Vol. 39, No. 15, pp. 3529-3545.