An optical method for the robot's performances testing.

Vacarescu, Valeria ; Vacarescu, Cella-Flavia ; Argesanu, Veronica 等

1. INTRODUCTION

In the field of industrial robotics, there are many different methods of measuring and estimating the pose (positioning and orientation) errors of robot systems, with the goal to minimize these errors through calibration. ISO 9283:1998, (***, 1998), defines performance criteria and establishes related test methods for industrial robots. Different methods are used to determine these performances. (Hung-Hsing & Ching-Chih, 2008), presents a laser pose tracking for a mobile robot. (Morris & Uday, 2005) presents a partial pose measurement system with theodolits for robots calibration. The system is only used for positioning characteristics, without determining the orientation characteristics. (McNamee et al., 2001) presents a photogrametric measurement technology for the calibration of a mobile robot model. (Vacarescu & Vacarescu, 1998) presents different methods, with and without contact, for measuring the industrial robots pose characteristics.

This paper proposes a method to determine the robot's end-effector's positioning and orientation characteristics by using two digitally theodolits and a calibrated target cub. The used algorithm is simple, easy accessible and proposes the matricial expression of the robots performance characteristics and the robot's errors, in concordance with the mathematical language used in the robot field.

2. MEASURING METHOD AND ALGORITHM

2.1 The geometrical considerations

Are used two digitally theodolits, [T.sub.1] and [T.sub.2] (fig.1) and a target calibrated cub, fixed of the robot's end-effector. The theodolits establish the 3D-coordinates of four points situated on the cub's adjacent edges. For an N point, situated in the coordinate system [O.sub.0] [x.sub.0] [y.sub.0] [z.sub.0,] are determined the segments lengths [T.sub.1]N, [T.sub.2]N, [T.sub.1] [N.sub.H], [T.sub.2] [N.sub.H] and also the angles [[alpha].sub.H], [beta].sub.H], [[alpha].sub.V], [[beta].sub.V]. Therefore, the N point's coordinates can be determined. Similarly, the coordinates of the four points [N.sub.1], [N.sub.2], [N.sub.3], and [N.sub.4], can be calculated. Using the four points coordinates, it can be determinate the robot's pose accuracy and repeatability.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 1 OMITTED]

2.2 The positioning accuracy and repeatability

The positioning accuracy (AP) is the distance between a command pose (ideal pose) of the robot's characteristic point (C.P.) and the barycentre of the cluster of attained points (***, 1998), for "n" robot's commands. In figure 2, [O.sub.i] is the ideal pose of the C.P. and [O.sup.G.sub.r] is the barycentre of the cluster of the real attained pose. The [O.sup.G.sub.r] point's coordinates are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (fig. 2).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[x.sup.G.sub.r] [y.sub.G.sub.r], [z.sup.G.sub.r] are the [O.sup.G.sub.r] point's co ordinates and [x.sub.i], [y.sub.i], [z.sub.i] are the [O.sub.i] commanded (ideal) point's coordinates. The matrix of the positioning accuracy is: AP = [([x.sup.E.sub.CP] [y.sup.E.sub.CP] [Z.sup.E.sub.CP]1).sup.T] with the terms:

[q.sup.E.sub.CP] = [q.sup.iCP] - 1/n [n.summation over (j=1)] [q.sup.r.sub.CP(j)], q=x y z. The positioning repeatability (RP) is expressed by the radius of the sphere whose centre is the barycentre of the cluster of attained points (fig. 2). RP = [bar.L] + 3 [S.sub.L], with: [bar.L] = l/n [[summation].sup.n.sub.j=1] [L.sub.j] and:

[L.sub.j] [square root of [([x.sup.j.sub.r] - [x.sup.G.sub.r]).sup.2] + [([y.sup.j.sub.r] - [y.sup.G.sub.r]).sup.2] + [([z.sup.j.sub.r] - [z.sup.G.sub.r]).sup.2] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[FIGURE 2 OMITTED]

2.3 The orientation accuracy and repeatability

It is expressed by the difference between the mean values of the angles between the coordinate system axes attached to the real poses of the target cub and the coordinate system axes attached to the commanded pose, reported to the base coordinate system [O.sub.0] [x.sub.0] [y.sub.0] [z.sub.0] (fig.3).

The maximum, mean and minimum values of the angles are attainted for different poses of (c.P.). Therefore, the orientation accuracy is a theoretical dimension which defines a group of measures. The angles [[lambda].sub.s], [[theta].sub.s], [[rho].sub.s] will be attended in different moments, in different points of the robot's working space. The result is an angular group of values which will define the three poses of the target cub, reported to the commanded pose: [[lambda].sup.m.sub.1], [[theta].sub.1], [[rho].sub.1]; [[lambda].sub.2], [[theta].sup.m.sub.2]; [[rho].sub.2]; [[lambda].sub.3], [[theta].sub.3], [[rho].sup.m.sub.3].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Are considered three concurrent lines in the point [O.sup.G.sub.r] and parallel to the axis's directions of the base coordinate systems, corresponding mean values of the angles defined by the equation (5). So, the orientation accuracy and repeatability will be expressed through a group of angles having following values:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The orientation repeatability, (fig. 3), is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[FIGURE 3 OMITTED]

2.4 The expression of pose accuracy matrix

Based to the above mentioned, the positioning and orientation matrix and its terms are expressed by the equations (10) and (11).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

3. CONCLUSION

This algorithm allows the presentation of the robot's poses accuracy and repeatability in a unitary mathematical language, matricial form, and it' s easy to use in experimental data processing. The method is used and validated in experimentally researches in the field of robotics (in the paper with ID 1744 of The 20th International DAAAM Symposium). In the future work, the authors will develop this method in medical applications (postural measurements of the spine using the special markers) in the project: CNCSIS 1022/2008, financed by The National University Research Council of Romania.

4. REFERENCES

Hung-Hsing, L. & Ching-Chih T. (2008). Laser Pose Estimation and Tracking for an Autonomous Mobile Robot, Journal of Intelligent and Robotic Systems Archive, vol.53, no.2, p.119-143, ISSN: 0921-0296, oct.2008, Kluwer Academic Publishers Hingham, USA

Morris, R. & Uday, S. (2005). Robot calibration using an automatic theodolite, The International Journal of Advanced Manufacturing Technology, vol.9, nr.2, p.114-125, ISSN 0268-3768, Publisher Springer, London

McNamee, L.P. et. al. (2001). Photogrammetric calibration a mobile robot model, Instrumentation and Measurement Technologies Conference Proceedings of the 18-th IEEE, p. 245-250, ISBN 0-7803-6646-8, Budapest, Hungary, May 2001

Vacarescu, V. & Vacarescu, I.N.(1998). Industrial robots: performances and testing, Mirton Publisher, ISBN: 973578-590-0, Timisoara, Romania

**** (1998). ISO 9283:1998, Manipulating industrial robots Performance criteria and related test methods, ISO, Geneva, Switzerland