CONTRIBUTION CONCERNING ROBOT ACCURACY USING

NUMERICAL MODELING

Daniela Ghelase

, Luiza Daschievici

and Irina Ghelase

“Dunarea de Jos” University, Str. Domneasca, Nr. 47, Galati, Romania

Politehnica University, Bucharest, Romania

Keywords: Robot accuracy, numerical modelling, computer simulation, rigidity of gearing.

Abstract: The kinematical accuracy of robot is very important. It is induced by the rigidity of each mechanism of the

robot. The paper presents a numerical method to evaluate the rigidity of worm-gearing teeth. The software,

including setting-up and graphic display, could be adopted of any kind of cylindrical worm-gear drive or for

spur gear drives and bevel gear drives, mechanisms which are in the robot structure. Besides, we can

determine geometrical parameters of the gear drives which influence the increase of accuracy of robot

linkages.

1 INTRODUCTION

Into the kinematical chain there are worm-gear

drives, screw-nut mechanisms and pinion-rack

drives. During the working, these gear drives and

mechanisms of the robot deform under the load,

leading to the motion errors. The errors can not be

entirely eliminated, but their maximum values must

be limited. The theoretical advantage of the

conjugate action in involute gears is lost due to the

deflection of the teeth under load and due to the

manufacturing and assembling errors. These factors

produce instantaneous variations in the gear ratio.

As it is well-known, the rigidity of the meshing

teeth changes as the contact point moves from the

initial point of contact to the final point of contact.

During the meshing the normal force is mobile on

the tooth flank, it changes continuously the position

with respect to the fixing zone of the teeth. The load

is unevenly distributed, depending on the contact

ratio. Consequently, all these factors causes rotative

speed variations of the driven shaft, vibrations,

shocks, noise, power loss, low durability of gears.

The purpose of the present work is to develop a

methodology to evaluate the rigidity of the worm-

gearing tooth. By means of this methodology the

performances of the robot mechanisms may be

improved.

2 GEOMETRY OF THE

WORM-GEARING TOOTH

In order to analyze the rigidity of the worm-gearing

tooth we assume that the spatial gearing consists of

more plane-gearings (pinion-rack drives), that in fact

are cross sections perpendicular to worm-gear axis

(Figure 1). The analytic solving of the problem, even

for a ruled worm-gearing, is very difficult due to the

complexity of the equations of the plane-gearing

profiles that are involved in the enveloping.

Consequently, we use the “minimum distance

method” applied in the case of the “discrete

representation” of the enveloping profiles. Thus, the

enveloping profile of the elementary worm-gear

(plane-gear) can be determined numerically by

knowing “discretely” a matrix having as elements

the coordinates of the worm axial section and by

using the theorem of the “minimum distance

method”.

The minimum distance theorem in “discrete

way” states (Ghelase, D., Daschievici, L., 2006):

The envelope to the family of curves, represented

in “discrete way” as massive of the coordinates of

the points belonging to the family curves, consists of

the all points there are on these curves, for which, at

a certain size of the increment ϕ

, the distance at the

meshing pole is minimum.

Ghelase D., Daschievici L. and Ghelase I. (2008).

CONTRIBUTION CONCERNING ROBOT ACCURACY USING NUMERICAL MODELING.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - RA, pages 24-29

DOI: 10.5220/0001478800240029

 SciTePress

Figure1: Worm-gear drive.

2.1 Worm-Geometry

In order to determine the coordinates of the worm

axial section, we focus on the case of a worm-

gearing with modified profile could ensure, as well

as possible, the generalization of the model from the

geometrical viewpoint. Hence, let the axial section

(x=0) of the worm (Figure 2) with constant pitch,

having a circular arch profile with the centre in O

for the right flank and in O

for the left flank. The

coordinates of the centre O

, respectively O

, are

given by the following relations:

-u·cosα-a·sinα

=b+u·sinα-a·cosα

(1)

-u·cosα-a·sinα

=-b-u·sinα-a·cosα

where: a is a constant parameter;

b=π⋅m/4-1.25⋅m⋅tg α;

p=m/2;

u=1.25⋅m/cosα;

Rau=+

is the radius of the circular arc

profile;

is the tip radius of the worm tooth, all

measured in mm (see Figure 2).

2.1.1 Equations of the Worm Flanks

In accordance with Figure 2, a point of the worm

flank has the following coordinates:

Figure 2: Worm flank geometry.

 For the right flank:

YY R v

ZZ R v

=+⋅ −+

⎧

⎨

⎪

⎩

⎪

;

cos( );

sin( );

(2)

 For the left flank:

YY R v

ZZ R v

=+⋅ −+

=−⋅ −+

⎧

⎨

⎪

⎩

⎪

;

cos( );

sin( ).

(3)

In the above relations, ν

and ν

are variable

parameters of the right flank and left flank,

respectively. Generally, the helical motion can be

written by means of two coordinate transformations

corresponding to simple motions, components of the

helical motion: rotation about Oz axis, having

parameter ϕ, and translation on the same axis,

proportional to the rotation angle p⋅ϕ, p being helical

parameter. In this way, the helical motion of the

movable coordinate system XYZ is described by the

matrix equation:

xXa

⋅+

(

)

(4)

−

⋅+

⋅

cos sin

sin cos

ϕϕ

001

(5)

where x is the matrix of a point coordinates with

respect to the coordinate system xyz fixed to the

frame, X is the matrix of the same point coordinates

CONTRIBUTION CONCERNING ROBOT ACCURACY USING NUMERICAL MODELING

with respect to the movable coordinate system, a is

the matrix of the point O coordinates (the origin of

the movable coordinate system) with respect to the

point O

(see Figure 3), and ω

(ϕ) is the matrix of

the rotation transformation.

Figure 3: Coordinate system applied for the helical

motion.

Substituting (1), (2) and (3) in (4), we obtain the

parametric equations of the right flank surface and

left flank surface.

Then, crossing these surfaces with the plane

x=H, the curve representing the worm profile

corresponding to the sectional plane takes the form

(for example, the right flank):

YR v

yY R v

zZ R v p

sin

[sin()]

;

[()]cos;

cos( ) ;

αϕ

111

−+⋅ −

=+⋅−⋅

=+⋅ −+⋅

⎧

⎨

⎪

⎩

⎪

(6)

2.2 Determination of the Worm-Gear

Flank Profile

The worm-gear tooth surface is generated by the

rolling.

We apply the “minimum distance method” on

the algorithm of the discretization in the case of

generation with the rack-bar tool.

First of all, we get the discretization of the

generating curve C

, which in this case is the worm

profile, represented by the vector (7), where:

and z

are the coordinates of the profile from

the “H” plane, which were determined by (5).

The gear flank generation of the elementary gear

drive is made with the rack-bar tool (see Figure 4).

(7)

The rolling condition interpreted in “discrete” way is

the following:

K⋅Δλ=R

⋅Δϕ⋅j (8)

Where Δϕ is the angular increment of the rolling. It

is then obvious that for generating a profile with

high accuracy from the technical viewpoint, this

increment has to be enough small.

2.2.1 Generation Motion

The generation motions of the worm-gear flank are:

1) Rotation of centroid, associated to the gear of the

elementary gear drive, with respect to the fixed

coordinate system xyz, described by the matrix

equation

x=ω

(j⋅Δϕ) X. (9)

In this relation, x is the matrix of the point

coordinates with respect to the fixed coordinate

system, X is the coordinates matrix of the same

point with respect to movable coordinate system

XYZ and ω

(ϕ) is matrix of the rotation

transformation about O

axis;

2) Translation of the movable coordinate system ξηζ

associated to the rack, with respect to the fixed

Rolling line of

the rack

Figure 4: Worm-gear flank generation.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

coordinate system, described by the equation (a is

the coordinates matrix of the point O

, the origin of

the movable coordinate system, with respect to the

point O):

x=ξ+a (10)

with

=−

−⋅⋅

()Δϕ

. (11)

3) Relative motions

Combining (9) and (10) we obtain that the

motion equation of a point on the generating curve

“g” (Figure 4) from the coordinate system XYZ with

respect to the coordinate system ξηζ is as follows:

ξ=ω

(j⋅Δϕ) X-a (12)

X=ω

(j⋅Δϕ)[ξ+a]. (13)

From the last equation, we infer that

⎪

⎩

⎪

⎨

⎧

⋅

⋅⋅⋅−+⋅⋅−−=

⋅

⋅⋅⋅−+⋅⋅−=

).Δcos(j

)]Δ(jR[ζ)Δsin(j)R(ηZ

);Δsin(j

)]Δ(jR[ζ)Δcos(j)R(ηY

ξ;X

ϕϕ

(14)

The system of equations (14) represents the

family of generating curves “g” with respect to the

coordinate system of the worm-gear, η and ζ being

the coordinates of the points that are on the

generating curve (Figure 5).

Figure 5: Coordinates of the meshing pole P.

The envelope to the family (14) is what we have

to determine, more precisely, the gear profile (see

Figure 6, flank of gear).

Figure 6: Line of contact (l.a.) in the plane H

The enveloping condition is given by the

minimum of distance

dYY ZZ

=− +−()()

, (15)

where Y

and Z

(coordinates of the meshing pole)

are:

= -R

⋅cos(j⋅Δϕ); Z

= R

⋅sin(j⋅Δϕ). (16)

2.3 Surface of Contact

The surface of contact is defined as locus of the

contact points of the two conjugated surfaces (which

are in enveloping) in the fixed coordinate system

xyz (Figure 4). The parametric equations of the

surface of contact are obtained associating the

enveloping condition to the absolute motion

equation of the worm-gear flank profile. In the

sectional plane x=H, the line of contact is given by:

=⋅ ⋅ −⋅ ⋅

=⋅ ⋅ +⋅ ⋅

⎧

⎨

⎩

cos( ) sin( )

sin

(

)

cos

(

)

Δϕ Δϕ ;

Δϕ Δϕ .

(17)

3 WORM-GEARING TOOTH

RIGIDITY

Once the algorithm for the determination of the

contact points, both on the flank height and along

the line of contact, is performed, then it is possible

to evaluate the rigidity of the worm-gearing tooth.

3.1 Bases of Design

The mathematical model is based on the following

assumptions:

 The worm-gearing is errors free and the gears are

rigid except the teeth;

 Taken into consideration only the bending

produced by the meshing normal force;

CONTRIBUTION CONCERNING ROBOT ACCURACY USING NUMERICAL MODELING

 Consider that the worm-gearing consists of more

plane-gear drives (pinion-rack drives), named

“elementary gear drives”, that in fact are cross

sections perpendicular to axis of rotation of the

worm-gear (Figure 1);

 The tooth of the elementary gear drive is

supposed to be a beam fixed at one end in the body

of gear;

 The assembly of the plane-gear drives into the

worm-gear drive was made provided that the teeth of

the elementary gear drives to deform together and

not separately under the same load.

3.2 Computer Program

Our algorithm to evaluate the rigidity of the worm-

gearing tooth is the following (Ghelase, D., 2005):

1) Computation of the rigidity for an elementary

tooth;

2) Computation of the rigidity for a pair of

elementary teeth;

3) Computation of the rigidity for an elementary

gearing tooth (pinion-rack drive);

4) Computation of the rigidity for the worm-

gearing tooth.

200

400

600

800

1000

1200

1400

1600

1800

2000

-69

-46

-23

115

138

161

184

207

230

253

276

Rolling angular parameter

Rigidity [KN/mm]

Figure 7: Rigidity of the worm-gearing tooth.

By means of the numerical modelling, these

steps will be added to the computer program used

for the study of the worm-gearing tooth geometry,

finally providing the instrument for the

determination of the worm-gearing tooth rigidity.

The computation diagram of rigidity of worm-

gearing tooth can be seen in Figure 7.

The cvasisinusoidal zone of the curve from

Figure 7 repeats periodically, because it represents

the rigidity during the meshing when the all plane-

gear drives are involved in the meshing. Thus, if the

input and output rigidities are eliminated, being less

importing for our study, we get the elasticity

characteristic of the worm-gearing tooth.

3.3 Elasticity Characteristic

The elasticity characteristic represents the variation

of rigidity of the worm-gearing tooth depending on

the rolling angle (j·Δφ), where “j” is the rolling

angular parameter (Ghelase, D., Tomulescu, L.,

2003). It is cvasisinusoidal curve with the high

jumps when a tooth binds or recesses (Figure 8).

The investigation of the elasticity characteristic

is very important for the study of an elastic system,

such as: gearing, linkage. Hence, the introduction of

this concept contributes to the completion of the

used gearing study and it leads to increase of the

gearing tooth rigidity.

Figure 8:

Elasticity characteristic of the worm-gearing

tooth.

3.4 Influence of Geometrical

Parameters

The influence of the geometrical parameters on the

rigidity was obtained by means of the computerized

simulation (Ghelase, D., 2003). It was applied to 150

worm-gear drives and we can present the following

conclusions:

1. The rigidity of worm-gearing tooth increases if

diametral quotient q increases and radius of profile

curvature R increases (Ghelase, D., 2003).

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

2. The rigidity of worm-gearing tooth reduces if

profile angle α increases (Ghelase, D., 2003) and

number of the gear teeth z

increases (Figure 9,

Table 1).

500

1000

1500

2000

2500

53 80 90 169

Rigidity kN/mm

Maximum

Rigidity

Minimum

Rigidity

Medium

Rigidity

Figure 9: Rigidity depending on number of gear teeth z

Table 1: Influence of number of gear teeth on rigidity.

Maximum

Rigidity

[kN/mm]

Minimum

Rigidity

[kN/mm]

Medium

Rigidity

[kN/mm]

53 2267.385 1215.140 1741.262

80 1727.633 1132.201 1429.917

90 1581.896 1079.696 1330.796

169 1055.990 853.826 954.908

4 CONCLUSIONS

Finally, we can draw the following conclusions:

1) A method to evaluate the rigidity of worm-

gearing tooth was developed;

2) The proposed approach may be applied for

any types of cylindrical worm-gearing and for spur

gearing and bevel gearing. These mechanisms are in

the structure of robot and by them rigidity depends

the kinematical accuracy of robot;

3) The introduction of “elasticity characteristic”

concept contributes to the completion of study for

the used mechanisms;

4) The developed computer program enables to

obtain numerical solutions and graphic illustration;

5) The numerical method, proposed and analyzed

in this paper, affords the geometry optimization and

the study of the meshing for various geometrical

parameters of the worm-gearing, being in fact a

simulation of meshing;

6) Moreover, we can determine the parameters

which influence the improvement of rigidity for

worm-gearing tooth and the increase of accuracy of

robot linkages.

REFERENCES

Ghelase, D., Daschievici, L., 2006. Computerized Design-

Generation of the Worm-Gear Flank.

The Archive of

Mechanical Engineering– Polish Academy of Sciences

Committee of Machine Design , 2, 165-178.

Ghelase, D., Gratie, L., 2005. Numerical Computation

Rigidity of Worm-Gearing Tooth with Circulare

Profile. In

IEEE ICIT2005, International Conference

on Industrial Techology.

City University of Hong

Kong.

Ghelase, D., Tomulescu, L., 2003. Computerized

Determination of the Elasticity of the Worm-Gearing

Tooth for Machine-Tools and Robots. In

Machine-

Building and Technosphere of the XXI Century,

International Science and Engineering Conference.

Donetsk National Technical University.

Ghelase, D., 2003. Influence of the Geometrical

Parameters on Rigidity of the Worm-Gearing Tooth.

The Annals of “Dunarea de Jos” University of Galati,

Fascicle XIV,

45-48.

CONTRIBUTION CONCERNING ROBOT ACCURACY USING NUMERICAL MODELING