The orienting movements of the trajectory generator mechanisms.

Radulescu, Corneliu ; Maniu, Inocentiu

1. INTRODUCTION

An object can be manipulated if its location parameters can be represented as a function of time. The number of the needed location parameters that must be modified when the object is manipulated determines the mobility degrees of the robot guiding device. The guiding device subsystem that realizes the coordinate modification, in order to move the characteristic point on a programmed trajectory, is called trajectory generator mechanism--TGM. The guiding device subsystem that realizes the director angle modification, in order to manipulate object re-orientation on the trajectory points, are named orientation mechanism--OM (Antonescu & Antonescu, 1995), (Kovacs et al., 2000) and (Radulescu, 2008).

The two mechanisms: the trajectory generator mechanism and the orientation mechanism that compose the guiding device of robots have currently six degrees of mobility. The optimal structure of the 3D spatial trajectory generator mechanism as presented on (Varga et al., 2008) is an anthropomorphic structure (Fig. 1). The elements and joints of the multistage anthropomorphic structure have the following names: 0--base of TGM; Q--pivot articulation; 1--pivoting element; A--arm articulation; 2--arm; B--forearm articulation; 3--forearm.

The forearm extremity point C is the characteristic point of the anthropomorphic TGM. The kinematic chain A [union] 2 [union] B [union] 3 [union] C forms the TGM plane mechanism. The kinematic chain 0 [union] Q [union] 1 forms the TGM pivot mechanism. The anthropomorphic structure regroups the two TGM mechanisms, plane and pivot.

[FIGURE 1 OMITTED]

In order to represent the decoupling movements of the two TGM mechanisms (plan and pivot) using a mathematic model, the transformation of the C characteristic point position vector from the plan mechanism Ayz working plan to the fixed frame [O.sub.0][X.sub.0][Y.sub.0][Z.sub.0] (Paul, 1981) and (Craig, 1986) is considered:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where the homogenous transformation matrix is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

and the vector that will be transformed:

[sup.A]C = [[0 y z 1].sup.t] (3)

2. THE PLANE MECHANISM FUNCTIONS

Expressing the vectors from the contour equation Fig. 2:

[sup.A]C = [sup.A]B + [sup.B]C (4)

with the AB=[r.sub.1] and BC=[r.sub.2] components, can be written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

This two trigonometric equations system are compatible according to a and [beta] variables. To solve the equation (5), new variables are needed:

u = [r.sub.1] * sin[alpha] v = [r.sub.2] * sin[beta] (6)

The TGM plan mechanism orientation functions are determinated by [alpha] and [beta] angles of the kinematic chain elements that have A and B leaded joints (Fig. 3), when the characteristic points cover the desired trajectory with the selected moving rule. T These two new equations system has two pairs of solutions:

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where a note:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Every pair of values [([v.sub.i], [u.sub.i]).sub.i=12] according with (6) has doublet of angles solutions that are the (5) system solutions:

[[alpha].sub.1,2] = arcsin([u.sub.1,2]/[r.sub.1],) [[beta].sub.1,2] = arcsin([v.sub.1,2]/[r.sub.2]) (9)

The technical solution corresponds to the smallest absolute values. Therefore, as orientation functions the solutions (9) are considered, that become by ignoring the indexes:

[alpha] = arcsin([u.sub.1]/[r.sub.1]) [beta] = arcsin([v.sub.1]/[r.sub.2]) (10)

3. THE PIVOTING MECHANISM FUNCTIONS

The TGM pivoting mechanism orientation function is determined by [gamma] angles Fig. 4. The pivoting mechanism assures the variation of the y angles according to the matrix (2), during the transition of the [sup.A]C vector from the plane mechanisms reference system to the TGM fixed reference system. During the actioning of the pivoting mechanism, the characteristic point C moves on an arc trajectory from the start point I to the end point F, defined on the [X.sub.0][O.sub.o][Y.sub.0] projection plan.

If the start point is situated on the [[gamma].sub.I] direction and the end point on the [[gamma].sub.F] direction, then the trajectory length is proportional with the angular course [GAMMA] = [[gamma].sub.F]-[[gamma].sub.I].

[FIGURE 4 OMITTED]

4. THE POSITION FUNCTIONS DEPENDING FROM THE ORIENTATION FUNCTIONS

Starting from the [sup.O][degrees]C vector expression on [O.sub.0][X.sub.0][Y.sub.0][Z.sub.0] fixed reference system: [sup.O][degrees]C = [[[X.sub.0] [Y.sub.0] [Z.sub.0] 1].sup.t] and doing the matrix product [sup.O][degrees]C=[sup.O] [degrees]0 [T.sub.A] x [sup.A]C :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

with (2), (3) and (5) relations, the coordinates C of the characteristic point relative to the robot fixed reference system are obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

The relation (i2) shows the dependence of the characteristic point C position on three variables: [alpha] = arm angle, [beta] = forearm angle and [gamma] = pivoting angle. This dependence indicates the geometrical coupling of the positioning movements and the orienting movements.

The research described in this paper can be used to the calibration of robotized Flexible Manufacturing Systems.

5. CONCLUSION

In conclusion, for the anthropomorphic TGM structures it is impossible to dissociate the TGM orienting movements from the positioning movements because both movements are obtained by actionning the same leading kinematic joint movements.

In order to attain the object's final position according to the application requirements, the guiding device of the anthropomorphic robot needs to be equipped with an orientation mechanism that removes the unwanted rotation movements introduced by TGM positioning movements.

This conclusion is valid not only for the anthropomorphic robot but for all TGM structures that contain at least one rotation leading kinematic joint.

6. REFERENCES

Antonescu, P. & Antonescu, [degrees]. (i995). Contributions to the geometric-kinematic calculus of the orientation mechanism of robots. Ninth World Congress on the theory of Machines and Mechanism, vol. 3, August i995, Milano, Italy

Craig, J. J. (i986). Introduction to Robotics, Addison--Wesley Publishing, ISBN 0-201-10326-5, Massachusetts, USA

Kovacs, F.; Varga, S. & Pau, V. (2000). Introduction in Robotics (in Romanian), Printech Publishing, ISBN 973652-230-X, Bucuresti, Romania

Paul, R. P. (1981). Robot manipulators, MIT Press, ISBN 0262-i6082-X, Massachusetts, USA

Radulescu, C. (2008). Movements coupling of the orientation mechanisms. Scientific Bulletin of the "POLITEHNICA" University of Timisoara, Tom 53 (67), Fasc. Si, pp.207-210, October 2008, ISSN 1224-6077, Politehnica Publishing, Timisoara, Romania

Varga, S.; Maniu, I.; Radulescu, C.; Dolga, V.; Bogdnov, I. & Ciupe, V. (2008). Robotics. Mechanical System, (in Romanian), Politehnica Publishing, ISBN 978-973-625 610-3, Timisoara, Romania