Formalism of industrial robots kinematics applied to numerical command lathes.

Catrina, Dumitru ; Ghionea, Adrian ; Constantin, George 等

1. INTRODUCTION

For an industrial robot, the typical translational kinematic chain consists of an electric motor, reducer and ball screw-nut, which moves a translational module (Catrina et al., 1993). The same kinematic chain structure is used in several CNC machine tools types (Weck, 2001). Based on this observation, similarly the machine tool can be considered as the industrial robot equipped with at least two gripping systems, one for each component of the fictitious mechanism (FM) workpiece-tool.

The fictitious mechanism piece-tool, defined for the first time by Emil Botez (Botez, 1966) can be extended for all types of closed or opened kinematic chains. We accept that the fictive mechanism workpiece-tool is the one that appears during the generation of surfaces, as there is a real mechanism between the tool (T) and workpiece (Wp).

The kinematic chain produces at its end a relative translation T or rotation R motion between two elements of the robot or machine tool connected by a couple (joint).

The kinematic branch is constituted as an element assembly united by couples driven by kinematic chains, which start on the base (bed) and end at a gripping mechanism (GM).

A robot having a gripping mechanism has only a kinematic branch, and a machine-tool, having several clamping devices (GM), has in its structure more than one kinematic branches.

For the machines that at a moment of time cut one workpiece using only one tool, this FM has an input and an output; for the machines that cut a single workpiece with several tools, the FM has several inputs and outputs. Regarding these considerations, one proposes the extension of some methods used in kinematic study in industrial robots in machine tool kinematic analysis (in our case with application to lathes). Some studies in this field are known for up to 5 axes CNC machine tools (Bohez, 2002).

For an industrial robot, the structural formulae O--[C.sup.j.sub.i] ([T.sub.i] [R.sub.j])--GM is of open kinematic chain type, with one input O (bed) and an output GM (gripping mechanism), where [C.sup.j.sub.i] ([T.sub.i] [R.sub.j]) is considered the opened kinematic chain consisting of combinations of translation [T.sub.i] and rotation [R.sub.j] joints.

For a CNC machine tool, the structural formula is:

[GM.sub.T] - [C.sup.j.sub.i] ([T.sub.i] - [R.sub.j]) - O - [C.sub.m.sup.n] ([T.sub.m] - [R.sub.n]) - [GM.sub.Wp], (1)

where [GM.sub.T] is the gripping mechanism for tools (one or more), and [GM.sub.Wp] is the gripping mechanism for workpieces (W) (one or more), all mounted on the same bed.

2. CNC LATHE STRUCTURES

In Figs. 1 and 2 some principle schemes for horizontal and vertical CNC lathe types are presented. The notation are as follows: MS--main spindle, LS/TS--longitudinal/transversal slide, Ts--tailstock, HS/VS--horizontal/vertical slide, RH--revolver head, C--column, B -bed.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

3. THEORETICAL ASPECTS. APPLICATIONS

These kind of representations and structural formulae are based on formalisms used in kinematic structural analysis for industrial robots. For example, the Denavit-Hartenberg convention (Popescu et al., 1994) allow the transformation matrix calculation between two reference systems attached to two consecutive elements from robots structure. The positioning of an element relatively to an adjacent one is achieved by attaching a reference system to every element having its index number corresponding to the element number i.

For solving this model, one should know the meaning of the following operators:

* [[alpha].sub.i]--angle about [x.sub.i-1] axis measured between [z.sub.i-1] and [z.sub.i];

* [[alpha].sub.i]--distance along [x.sub.i-1] from the origin [O.sub.i-1] to [z.sub.i] axis;

* [[theta].sub.i]--angle about [z.sub.i] axis measured between [x.sub.i-1] and [x.sub.i];

* [d.sub.i]--distance along [z.sub.i] axis measured between intersection point of [x.sub.i-1] and [z.sub.i] axes and [O.sub.i] origin.

[FIGURE 3 OMITTED]

The transformation matrix for an entire kinematic branch is the product of the transformation matrix of the reference systems attached to each element of the branch:

[T.sub.0.sup.n] = [T.sub.0.sup.1] x [T.sub.0.sup.2] x ... [T.sub.i-1.sup.i] x [T.sub.n-1.sub.n], (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

As example, we present the structure of a parallel lathe having a main spindle MS, longitudinal slide LS, transversal slide TS, and tailstock Ts (Fig.3).

In Table 1, the Denavit-Hartenberg parameters for bed-workpiece and bed-tool kinematic branches are presented.

The transformation successive matrix for bed (0)-workpiece (Wp) branch is:

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

The transformation matrix for the attached end elements of the branch 0-Wp:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

In the same way the transformation matrix for the attached elements of the 0-T branch is calculated:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The transformation matrix between the workpiece and tool (fictitious mechanism Wp-T) results in the form:

[T.sub.wp.sup.T] = [([T.sub.0.sup.Wp]).sup.-1] [T.sub.0.sup.T] (7)

On the basis of relation (5), the coordinates ([x.sub.Wp], [y.sub.Wp], [z.sub.Wp]) of a point in the workpiece reference system result. As well, one can obtain the coordinates ([x.sub.T], [y.sub.T], [z.sub.T]) for a point in the tool reference system. In both cases, one knows the point

coordinates in the bed reference system ([x.sub.0], [y.sub.0], [z.sub.0]). These have the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Subtracting the equations corresponding to the two equation systems the relations between the inputs and outputs of the workpiece-tool fictitious mechanism result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Equation (7) expresses the parametric equation of the generated path through the fictitious mechanism Wp-7.

Application. For simultaneous motions with constant speeds [w.sub.X], [w.sub.Z] and co corresponding to the CNC axes X, Z and C the equations of a conical helix are obtained based on eq. (9):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

If [w.sub.X] = 0 the equations of a cylindrical helix with constant pitch is obtained, and also for [w.sub.Z] = 0 an Archimedean spiral.

For the studied case, The presented method has a general feature and is utile for interpolation methods (linear, circular, etc.) in case of generation of analytical or non-analytical generatrice D and directrice G curves.

4. CONCLUSIONS

In this paper some formalisms specific to industrial robots were applied to CNC machine tools. The kinematic structural formulas for two lathes were established The exemplification is made for a parallel lathe with two kinematic branches: bed-workpiece and bed-tool.

The Denavit-Hartenberg formalism was applied, resulting the transformation matrix between the reference systems attached to the bed and workpiece, and to the bed and tool.

The relations between fictitious mechanism Wp-T input and the output were established. For the studied case, the trajectory generated by the Wp-T contact point is a conical helix, a cylindrical helix, or an Archimedean spiral.

Deriving the equations of these helixes, the speeds and the accelerations are obtained for the Wp-T contact point.

Differentiating these equations, the influences over the generated helix precision of some geometrical and constructive parameters are obtained for a cutting process.

5. REFERENCES

Bohez, E. L. J. (2002). Five-axis milling machine tool kinematic chain design and analysis. International Journal of Tools & Manufacture, 42, Pergamon, 2002, pp. 505-520.

Botez, E., (1966). Bazele generarii suprafetelor pe masiniunelte (Fundamentals of surface generation on machine tools), Edit. Tehnica, Bucharest.

Catrina, D.; Moraru, V. & Dinu, G. (1993). Masini-unelte cu comanda numerica (CNC Machine Tools), Vol. I and II, University "Politehnica" of Bucharest.

Popescu, P; Negrean, I; Vuscan, I & Haiduc, N. (1994). Mecanica manipulatoarelor si robotilor (Mechanics of manipulators and robots), Edit. Didactica si Pedagogica, ISBN 973-30-3580-7, Bucharest.

Weck, M (2001). Werkzeugmaschinen. Automatisierung von Maschinen und Anlagen (Machine tools. Automation of machine and equipment). VDI Springer, ISBN 3-540-67713-9, Berlin.

Tab 1: Denavit-Hartenberg formalism for the parallel lathe.

 Bed-workpiece branch

Operator 1 2 P

[[alpha].sub.i] -90[degrees] 90[degrees] -[theta].sub.1]

[a.sub.i] 0 0 [a.sub.0]

[[theta].sub.i] 0 0 0

[d.sub.i] [a.sub.2] [a.sub.1] 0

 Bed-tool branch

Operator 3 4 5

[[alpha].sub.i] 0 0 0

[a.sub.i] [a.sub.3] + [s.sub.2] [a.sub.9]
 [s.sub.1]

[[theta].sub.i] 90[degrees] 90[degrees] 0

[d.sub.i] [a.sub.10] + a% 0
 [a.sub.7]