Six inextensible wire device for determination of position-orientation matrix of a robot's mobile reference system.

Miclosina, Calin-Octavian

1. INTRODUCTION

In the case of a robot, the direct kinematical analysis consists in knowing the relative positions of the guiding device links and the determination of the relative position of the end effector versus the robot given by the position-orientation matrix (Kovacs et al., 2000) of the reference system attached to the characteristic point chosen on the end effector. This matrix can be determined experimentally using different methods: with contact (using wires (Miclosina, 2008b) or without contact (using magnetic fields (Paperno et al., 2001), optical sensor and laser scanners (Yuan & Yu, 1999).

The goal of the author's research presented below is the determination of the robot's position-orientation matrix, knowing the relative position rotor-stator for each motor.

2. MEASURING LENGTH BY USING AN INEXTENSIBLE WIRE

A measuring system of a length in space and the structural scheme of a robotic guiding device mechanism with parallel topology are presented in fig. 1. An inextensible wire is connected to the mobile platform [MP.sub.n] in the characteristic point [P.sub.1] by a knot. The wire passes a guiding bush (to establish a fixed point [R.sub.1]), then a groove guiding roller, being strained by a G weight body which has a cursor indicating the length [R.sub.1] [P.sub.1].

[FIGURE 1 OMITTED]

3. DETERMINING THE POSITION-ORIENTATION MATRIX OF A MOBILE REFERENCE SYSTEM USING 6 INEXTENSIBLE WIRES

The position of a point in space can be determined using 3 inextensible wires and the position-orientation of a materialized mobile reference system can be determined using 6 inextensible wires (Miclosina, 2008a).

The axes of the mobile reference system attached to the link 6 are materialized, as shown in fig. 2. The point [P.sub.1] belongs to axis Pz, the point [P.sub.2] belongs to axis Px and the point [P.sub.3] belongs to axis Py. The points [R.sub.1], [R.sub.2], [R.sub.3], [R.sub.4], [R.sub.5] and [R.sub.6] are fixed.

Point P is considered the robot's characteristic point.

The lengths (on the wires) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], are known.

The known distances [P.sub.1][R.sub.1], [P.sub.1] [R.sub.2] and [P.sub.1] [R.sub.3] can be written under analytical form versus the fixed reference system Oxyz:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

The system (i), is formed by 3 equations and presents 3 unknowns, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (it is determined); but being of 2nd degree it presents two sets of solutions, from which the convenient set is to be chosen, depending of coordinate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In a similar way points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and P ([x.sub.P], [y.sub.P], [z.sub.P]) can be determined.

The angles between the axes of the mobile reference system Pxyz and the axes of the fixed reference system Oxyz can be computed using relations of the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

[FIGURE 2 OMITTED]

The position-orientation matrix of the mobile reference system Pxyz versus the fixed one Oxyz can be determined:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [x.sub.P], [y.sub.P] and [z.sub.P] are the coordinates of the characteristic point P in the fixed reference system Oxyz.

4. DEVICE FOR DETERMINATION OF THE POSITION-ORIENTATION OF THE MOBILE REFERENCE SYSTEM AND EXPERIMENTAL RESULTS

The device for determining the position-orientation of the reference system attached to the robot's characteristic point using 6 inextensible wires is presented in fig. 3.

The device is composed of:

* The mobile reference system (i) realized of plastic material; in the bottom side it has a screw fixing it on the mobile platform.

* 6 inextensible wires (2), tied at an end to the mobile reference system and at the other end to graduated cursors (3) translating on guiding measuring scales (4) fixed on support columns (5).

* The plates (6) which ensure an accurate wire positioning.

The following input parameters are considered (angle between the axis of the element linked to the motor's rotor and the vertical axis, for each serial kinematical chain between the fixed platform and the mobile one):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

According to fig. 2, values of the distances ([PP.sub.i]), ([PP.sub.2]), ([PP.sub.3]), ([P.sub.1] [P.sub.2]), ([P.sub.2][P.sub.3]), ([P.sub.1] [P.sub.3]), which define the reference system Pxyz, are given.

Versus the fixed reference system Oxyz, the points [R.sub.1], [R.sub.2], [R.sub.3], [R.sub.4], [R.sub.5] si [R.sub.6] have known coordinates.

For the input parameters (4), the distance values ([P.sub.1] R[.sub.1]), ([P.sub.1] [R.sub.2]), ([P.sub.1] R[.sub.3]), ([P.sub.2]R4), ([P.sub.2] [R.sub.5]) and ([P.sub.3] [R.sub.6]) are measured with the help of the inextensible wires.

[FIGURE 3 OMITTED]

The coordinates of the points [P.sub.1], [P.sub.2], [P.sub.3] and P are obtained by solving the equation systems of type (1). Only certain solutions are chosen, due to constructive constraints.

The angles between the axes of mobile reference system and the fixed one are computed using the relations of type (2).

For the input parameters (4), the experimentally determined position-orientation matrix using the 6 wire device is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

5. CONCLUSIONS

The main advantages of using this device are:

* The measuring device construction is relatively simple and has low costs of fabrication and maintenance.

* There is relatively high determination accuracy of position-orientation of mobile reference system.

* Determinations can be made for all the positions-orientations of the mobile reference system, in the whole workspace.

The main disadvantages are:

* The guiding device interacts with the environment, through the wires, under the action of cursor weights.

* Measuring calibration errors may occur.

* The calculus is laborious, but it can be easily made with specialized software.

As a further improvement, the wire lengths measuring systems can be replaced. The next step will be using linear transducers instead of cursor--guiding measuring scale systems. Thus, the values of wire lengths can be aquisitioned directly on the computer and the position-orientation matrix can be computed on the basis of mathematical relations from chapter 2, using specialized software.

6. REFERENCES

Kovacs, F.V.; Varga, S. & Pau, V.C. (2000). Introducere in Robotica (Introduction to Robotics), Printech Publishing House, ISBN 973-652-230-X, Bucharest

Miclosina, C.-O. (2008 a). Mathematical Model for Determining the Position-Orientation Matrix of the Reference System Attached to the Characteristic Point of a Robot Using 6 Inextensible Wires, Bulletin of the Polytechnic Institute of Iasi, Tome LIV (LVIII), Fasc. 4, pp. 7-12, ISSN 1011-2855, Published by "Gh. Asachi" Technical University of Iasi, Romania

Miclosina, C.-O. (2008 b). Method of Determination of the Position-Orientation Matrix of the Reference System Attached to the Mobile Platform of a Parallel Topology Robot with 6 Inextinsible Wires Device, Proceedings of the International Conference ROBOTICS '08, Special Issue No. 1, Vol. 2, pp. 557-562, ISSN 1223-9631, Brasov, Romania, 13th-14th November, 2008, Published by Transilvania University Press, Brasov, Romania

Paperno, E.; Sasada I. & Leonovich, E. (2001). A new method for magnetic position and orientation tracking. IEEE Transactions on Magnetics, Vol. 37, Issue 4, July 2001, pp. 1938-1940, ISSN 0018-9464

Yuan, J. & Yu, S.L. (1999). End-effector position-orientation measurement. IEEE Transactions on Robotics and Automation, Vol. 15, Issue 3, June 1999, pp. 592-595, ISSN 1042-296X