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ROBOT C A L I B R A T I O N - - M E T H O D A N D RESULTS

G DUELEN and K SCHROER

Fraunhofer Institute for Production Systems and Design Technology (IPK), Berlin, Germany

For many applications, robots are needed which have an absolute Cartesian pose accuracy of the same order of magnitude as their repeatability Calibration procedures must be applied to reach this aim In this paper, a calibration procedure is presented which allows the automatic determination of all kinematic parameters, gear parameters, and static elasticity effects With the aid of these results it is possible to increase pose accuracy up to the robot's system limit as determined by its repeatability Moreover, information can be acquired which suggests improvements in robot design, thereby also improving repeatability

1 INTRODUCTION

A main obstacle to broader introduction of off-line

programming techniques and integration of robots in

CIM systems is the high absolute positioning accuracy

required of a robot for a program's execution With-

out this accuracy, a great amount of the time, other-

wise reduced by off-line programming, is spent on the

long and tedious procedure of re-teaching the robot

The problem does not lie in the pose repeatability of

the robot but rather in the lack of absolute pose

accuracy Pose repeatability of robots available today

is usually better than 1 mm while absolute pose accur-

acy (i.e the precision with which the robot can reach a

numerically given position and orientation relative to

an external or work cell frame) is orders of magnitude

worse Pose errors are caused by differences between

the actual dimensions of the articulated robot struc-

ture and those used in the robot control system model,

and by mechanical faults such as compliance and

elasticity These are systematic, deterministic errors

which can be compensated for if their parameters are

known for each robot

When teach-in programming is used, these errors

have no influence on the robot's ability to complete a

task since knowledge of the precise Cartesian position

and orientation of a taught pose is of no importance

The only thing of interest in this case is that the robot

repeatedly reaches the taught pose with the required

accuracy

In order to close the gap between absolute pose

accuracy and repeatability, robot calibration proce-

dures have to be applied "Calibration" means that

through measurements [e.g location of the tool center

223

point (TCP) at different robot positions], robot model parameters are determined which closely estimate the kinematic and mechanical features of the static robot This allows the improvement of absolute pose accur- acy at best to the basic deterministic accuracy of the robot system To realize this improvement, error compensation methods have to be applied which modify off-line generated robot programs by using identified model parameter values for the task-execut- ing robot

Procedures which only record deviations in TCP position and orientation in a section of the robot workspace are not suitable because error compensa- tion is only possible in a small section of the six- dimensional workspace from which measurements are taken Hence, measurements must be repeated for each robot task and for each pose in the section where high precision is required By using calibration proce- dures which numerically determine the sources of the deviations, pose accuracy within the entire workspace

is improved This also eliminates the need for expen- sive experimental investigations or disassembly to determine deviation sources Thus it follows that only directly accessible measuring values can be used for calibrating the robot These are joint encoder values, and the position and orientation of the TCP

The calibration procedure must be able, at a mini- mum, to numerically identify the exact geometric- kinematic parameters of the robot The determination

of elasticity and gear parameters should also be in- cluded because of the large number of robot types and applications where elastic deformations cannot be ignored if the desired precision is to be reached.19 This

224 Robotics & Computer-Integrated Manufacturing • Volume 8, Number 4, 1991

is especially true of gear elasticity One example of

such a procedure for automatic robot calibration is

presented here

2 C O M P A R I S O N O F D I F F E R E N T

S O L U T I O N S The practicability of calibration concepts has been

shown in different investigations 4-1°' 12-18, 20 As for

the identification concept and error modelling, there

are different solutions

2.1 Identification concepts

F o r the parameter and error identification there are

two distinct concepts:

(a) A geometric-conspicuous concept where several

steps of measurement and computation are em-

ployed; each step identifies a parameter subset

based on geometric-conspicuous relations 6' 7, 15

(e.g by separate motion of one joint after the

other)

(b) A system theoretic concept through which all

model parameters are identified simultaneously

by exploitation of the continuity and differentia-

bility of the system model as a function of its

parameters and by the application of numerical

optimization algorithms 9' 13, 16, 1 S

The system theoretic concept was chosen in the

calibration procedure presented here for its advan-

tages over the geometric-conspicuous concept:

• Extension of the error model to errors other than

geometric-kinematic ones is possible

• Does not require specific movements depending

on the geometric relations to be identified and

requires less measurements than the geometric-

conspicuous concept

2.2 Kinematic error modelling

The modelling of kinematic errors also includes two

concepts:

(a) Identification of an optimal set of parameter

values of the kinematic model 5-1°' 13, 14 16

(b) Identification of a special error transformation,

which is derived from the kinematic

model.*' 17, is, 20

If a special error transformation (b) is used, a locally

linearized error model in the form of a differential

transformation is derived and is multiplied to the end

of the forward kinematic transformation based on

known robot data F o r computation of the error

transformation, an algebraic calculus was developed

based on the linear term of a Taylor series relative to the kinematic p a r a m e t e r s - - b u t written multiplicati- vely

This kind of modelling has a disadvantage in that

T C P effects depend non-linearly on parameter errors The rotational parts of differential transformation matrices are not orthogonal and therefore, as the parameter errors increase, deviations from orthogona- lity also increase In this way, otherwise avoidable disturbances are introduced into the transformation computations Additionally, the extension of the cal- culus for computation of the error transformation would be crucial if non-geometric errors (e.g elasti- city) are to be included

Hence, a way to directly identify an optimal set of model parameter values was chosen for the presented calibration procedure

3 R O B O T M O D E L A N D P A R A M E T E R

I D E N T I F I C A T I O N The calibration procedure is based on a mathematical model of the physical robot including all significant, deterministic sources of pose deviations The kine- matic geometry of the robot model is extended to include effects of elastic deformations and gear para- meters (Fig 1) In order to include elasticity, the reaction forces and torques induced by the masses must be computed

F o r calibration purposes, the robot is considered to

be a stationary system The input values are the joint encoder values and the output values are the position and orientation of the T C P expressed as homoge- neous matrices

Given:

n

p e ~ "

h e g t "

number of joints, model parameter vector, joint positions, given in joint encoder values,

the model is then defined by a continuously differenti- able function T which describes the stationary behav- ior of the robot system:

T: 9~ '~ x 9~" ~ 914 × 4

(p, h) ~ X = T(p, h)

Through numerical identification procedures, the actual system model parameters are determined for the robot under investigation This procedure allows the determination of all geometric parameters of the rigid body system such as zero position errors of the joints, link lengths, joint axis misalignments, transmis- sion and coupling factors of the gears, gear and link

Model

Fig l R o b o t model

Kinematic

Model I ~ 1 Mechanical

Model

Robot calibration method and results • G DUELEN and K SCr~ROER 225 elasticity, gear eccentricity (i.e periodic variations of

transmission ratios) and joint backlash

3.1 Kinematic model

The kinematic model is based on the Denavit-Harten-

berg (DH) parametrization The calibration proce-

dure accepts any combination of rotary and prismatic

joints However, for prismatic and consecutive rotary

joints which are parallel, the modelling proposed by

Hayati and Mirmirani 4 was chosen as opposed to D H

parametrization because the system theoretical identi-

fication concept is used This concept requires conti-

nuity of the model as a function of its parameters and

since the D H parametrization is not continuous for

parallel joints, it cannot be considered for this confi-

guration

One further extension of the D H convention was

also necessary because of the requirements of the

elasticity model Information about the physical loca-

tion and distribution of link masses was introduced

into the model However, geometric locations of trans-

lations in the modified D H parametrization do not

generally coincide with physical locations of links To

overcome this problem in most cases, one more

parameter was introduced into the model; a transla-

tion along the z-axis With this addition, the coordin-

ate frame of the next joint could be located on the joint

axis at the physical position of the link

Using R, as a rotation transformation around co-

ordinate axis u and T~ as a translation transformation

along coordinate axis u, three possibilities describe the

joint transformation between consecutive joint co-

ordinate frames:

(a) when the first joint is rotary and the next is (near)

orthogonal: RzT~ T~RxT~,

(b) when the first joint is rotary and the next is (near)

parallel: Rz T~RxRy T ~ ,

(c) when the first joint is prismatic: T, RxRyT ~

In this description, joint transformations are built

up from elementary rotation and translation transfor-

mations The parameter of the first elementary trans-

formation in each of the three ordered set describes the

joint action As is known from the D H convention, the

zero positions of the joints are implictly determined

Thus, differences between the model-defined zero posi-

tion and the zero position defined by the robot control

conventions have to be included

As is known from other investigations (e.g Ref 3),

the parameter of the last transformation Tz is redun-

dant Redundancy produces rank deficiencies in the

parameter identification process and should be

avoided Therefore, this parameter is not included in

the numerical identification of the parameters

For completeness, it should be noted that the first

joint transformation is from the reference frame to the

frame assigned to the first joint This transformation is

constant (i.e without a joint variable) and is modeled

as (a) or (b) depending on the relative orientation of

the reference frame's z-axis and the axis of the first

joint Furthermore, if T C P position and orientation

are measured for identification purposes, then the last transformation (i.e the transformation from the frame

of the last joint to the T C P frame) has to be built

up of six elementary transformations; for example,

RzTzTxRxT~Rz

3.2 Gear model

The input values of the robot system are the joint encoder values On most robots, the encoders are not mounted directly on the joints but on the motor shafts A high gear down ratio exists between the motor and joint Also the wrist axes are often mechan- ically coupled Hence, the transmission properties and errors from motor to joint position are an important cause of TCP position deviations and should therefore

be modeled and identified

First, the transmission and coupling coefficients of the gears are considered These can be modeled in a matrix which is sparse and invertible Multiplying the matrix with an increment vector of motor positions results in joint positions Only if the robot has closed kinematic sub-chains among its primary axes is there non-linear coupling In this case, the coupling must be modeled differently

Normally, transmission and coupling coefficients are known more precisely from gear and encoder design information than through an identification procedure However, it is essential that they are in- cluded in the model in order to use joint encoder values as system input Furthermore, results Of this identification procedure can be used to check consis- tency between robot control data and the gear in use Second, joint elasticity emerged as a very important cause of T C P pose deviation A linear elasticity was assumed for the model Using k as elasticity coefficient and F as torque in the joint working direction, the deviation in joint position is expressed as

Aq = k F For diagnostic purposes, consideration of joint back- lash is of importance and therefore has been included using the following model:

Aq = b- sign(F)

Since this model is not continuous for poses with torque F near 0, such poses are excluded for joint backlash identification

Both joint elasticity and backlash are modeled in the working direction of the joint It is evident that joint elasticity and backlash can only be individually identified if varying reaction torques are produced by different robot poses Hence, this identification is not possible for some axes such as the primary axes of SCARA robots

Finally, coefficients for gear eccentricity were in- cluded in the robot model Eccentricity error results in periodical variations of the gear transmission ratio and was modeled by a first-order Fourier approxima- tion Gear eccentricity has been reported as a signifi-

226

cant cause of pose error) However, it could not be

detected among the robots calibrated at the I P K

Berlin•

The above described gear model results in the

following formal description of the transmission char-

acteristics

Using the following notation

q/, q/, i = 1 n joint position (revolute joints:

rotation, prismatic joints:

translation), joint encoder values, transmission and coupling coefficients,

Fourier coefficients of the gear eccentricity (ill, f/2),f/0

is the zero position error, elasticity coefficient of the ith joint,

backlash of the ith joint, torque or force in joint working direction,

the joint position model for rotary joints is written as

j = X

q/= Y cij" hj

n

qi = ql + f/0 + fix" cos(q~) + f/2" sin(qi)

+ ki Fi + bi" sign(F/)

and its counterpart for prismatic joints as

q i = q / + f ~ o +f~x c o s ( q / / / )

+fix sin(q/ ~ ) • • + k i • F i + b i • sign(F/)

hi, j = 1 , , n

co, i , j = 1 , , n

fit, i = 1 , , n ,

j = O 2

k/,i= 1 n

bi, i = 1 , , n

F i, Fi, i = 1 n

3.3 Link elasticity model

The trend toward light-weight robots and calibration

of large compliant kinematic chains (e.g concrete

pumps) necessitates the identification of link elasticity

and was therefore integrated into the model for robot

calibration

The links were modeled as elastic beams with

masses concentrated at the end points As a first

solution, the beam model is automatically derived

from the kinematic model• This is possible since kine-

matic modelling conventions were extended by one

more translation parameter (see Section 3.1) Thus,

beams were assumed for any translation element in

the kinematic model which is significantly different

from zero The link mass was located according to the

resulting beam polygon between two joint frames

(normally consisting of one straight line) Beam tor-

sion as well as the bending deformation were modeled

Robotics & Computer-Integrated Manufacturing • Volume 8, Number 4, 1991

and in both cases linear elasticity was assumed• Spe- cial care must be taken if the principal beam axis and

an axis of a rotary joint coincide• In this case, beam torsion and joint elasticity cannot be separated, so only the joint elasticity was identified• The special cases were handled by the software analyzing the model of the robot before calibration•

The effect of beam elasticity was modeled as a differential transformation at the free end of each beam The respective matrices generally have the form

The parameters of the linearized rotation (~x, ~r, ~z) and of the translation (dx, dy, dz) are functions of the beam elasticity coefficients, and the actual forces and torques at the free end of the beam This beam elasticity matrix was inserted into the sequence of elementary transformations of the kinematic model directly after the translation kinematically represent- ing the same beam• Thus, the elasticity transformation was described in the local coordinate frame at the free end of the actual beam•

To demonstrate the functional relations (Fig 2), a beam is taken with a frame at its free end The frame's z-axis coincides with the principal beam axis and is oriented away from the fixed beam end

To simplify the model, it was assumed that the beam is symmetric about z (i.e bending deformation coefficients in the x- and y-directions are the same)• This assumption is admissible, because links, which can be rotated around their principal axis in normal robot motion, are usually designed symmetrically relative to this axis•

Thus, linear beam theory gives the following defor- mation relations for a beam oriented along the posi- tive z-axis:

d x = k x r ~ F ~ L 3 + ~ yL J,

= k [1 FrL3 _ 1 FxL2],

d~=O,

~ x = kxr ~ FxL 2 + F~L ,

~ = k~FzL

where

kxy

kz

L

(Fx, Fy, Fz), (F~, Fy, F~)

bending coefficient, torsion coefficient, beam length, torque and force at the free end of the beam

Robot calibration method and results • G DUELEN and K SCHROER 227

~ x

M

It,,, Z

Y

Fig 2 Beam elasticity

The advantages of this model are:

• There are only two parameters which have to be

identified This is important because, for most

robots, link elasticity is an error with only a small

effect and thus, with respect to the numerical

algorithms used for identification, only needs to be

modeled using a small number of parameters

• The beam length was included in the elasticity

model This was necessary in order to model the

elasticity of links with prismatic joints (i.e links of

varying length)

On the other hand, care has to be taken that only

small elastic deformations are modeled in this way

since the rotational part of the elasticity matrix E is

only a linear approximation

don and orientation of the TCP (system output) must

be measured by suitable equipment

For calibration purposes, it is sufficient to measure only TCP positions since all parameters of the model can be identified when the measured TCP is not located on the axis of the last joint This is important because orientation measurement is more complicated and less precise than position measurement

The requirements for the measuring system were:

• TCP positions in the entire robot workspace have

to be measurable with only very small restrictions

on the admissible tool-frame orientation

• Maximum position measurement errors must be less than 10 % of the repeatability of the robot and

at worst 0.1 mm

• Measurements have to be non-tactile, otherwise external forces and torques will create elastic de- formations disturbing the calibration results

• Measurements should be possible without manual interaction

A comparison of available measuring technology lead to the conclusion that an automatic theodolite system could fulfill these requirements The chosen system had a maximum error smaller than 0.05 mm (dependent on the theodolite system calibration) as experimentally verified Figure 3 illustrates the meas- uring process

4 MEASURING EQUIPMENT, CALIBRATION

PROCEDURE AND COMPENSATION

4.1 Measuring equipment

The joint encoder values (input of the system) can be

obtained directly from the robot controller The posi-

4.2 Calibration procedure and parameter identification

The procedure can be used for calibration of any multi-body system which kinematieally can be repre- sented by an open kinematic chain Any sequence of revolute and prismatic joints is admissible

Process Computer

Estimating Algorithm

Program for Motion

and Measuring Control

Resolver / Post on Values

1

Desired Position Joint Resolver Values Fig 3 Measuring process

Measured Position Desired Position

Robot

I Determination of the

• Spatial Coordinates

"5:

Theodolite System

228 Robotics & Computer-lntegrated Manufacturing • Volume 8, Number 4, 1991

At the beginning of the calibration procedure, the

known data of the robot under investigation are

entered into the process computer These data include

the number of joints, the joint workspaces, the trans-

mission and coupling coefficient matrix, and link

masses and centers of gravity The geometry of the

kinematic structure (joint type, joint axis position and

orientation) is entered as a simple polygon model

With this information, the internally used model is

generated, its parameters computed, and consistency

and modelability checked The necessary robot poses

for measurement are automatically determined They

are generated with respect to parameter identifiability,

observability and collision avoidance The number of

poses chosen by the user should be between 40 and

200 This allows inclusion of enough variation in link

configurations so all parameters can be identified

During the measurement process, the robot is com-

manded to the next measuring position The actual

joint resolver values of the target position are trans-

mitted from the robot controller to the process com-

puter via the computer interface The approximate

actual TCP position is computed using known robot

data and is transmitted to the measuring system which

then measures the exact TCP position (Fig 3)

The calibration itself is the computation of those

model parameter values which result in an optimal fit

between the actual measured positions and those

computed by the model This is a non-linear least-

squares problem which is ill-conditioned and can

produce rank deficiencies Therefore, specially

adapted numerical algorithms are used

The mathematical formulation of the problem is as

follows Given

n

p c 9t"

hi e 91", i = 1 k

T: 91" x 91n " 4" 913

(p, h) * x = T(p, h)

M ( h ) e 913

then

/~:: = (hi hk) e 91k,

T: 91" X 91kn ~ 913k

T(p, /~):: =

(T(p, hx) T(p, hk) )

~r(/~):: = ( M ( h l )

M ( h k ) ) ~ 913k

number of joints, model parameter vector,

k robot poses, given by the respective joint encoder values,

robot model function (only position), measured TCP position belonging to the robot pose given by the encoder values h,

vector of k robot poses included in the identification, robot model function for

k poses,

measured TCP position for

k poses

The solution of the non-linear least-squares problem

is the parameter vector p*e 9t" The problem is

formulated as

II T(p*, f~) - ~(fT)tl

= rain ]J T(p,/~) - ~t(/~)I], for all p e 91" The algorithm for the non-linear problem repeatedly solves the linearized problem, which is given by

~l (h) - T(p, h) = (DpT)(p, fo" Ap

Here

(DpT)(p, f~) e 913k ×"

is the Jacobi matrix of T at (p,/~) If all model parameters are to be identified and if T describes a revolute robot with six joints, then the value of m is around 60 In this case, about 100-200 robot poses should be measured This means that the Jacobi matrix has about 300-600 rows For the numerical solution of the linearized problem, the HFTI algo- rithm developed by Lawson and Hanson 11 (based on

an orthogonal decomposition of the linear problem) was combined with a model-based column scaling The result obtained in the linear steps are used in a line search strategy to determine the solution of the non-linear problem by a globally convergent modifi- cation of Newton's method (see e.g Ref 2, Ch 6) To terminate the algorithm, a model-based stopping cri- terion was chosen which evaluates the parameter differences obtained after the line search step (Ref 2,

Ch 7.2 and App A II.4) This was an appropriate stopping criterion because the orders of magnitude of those parameter differences not having any influence

on pose accuracy are known before beginning the numerical procedure

The implementation of the procedure allows the user to decide which subset of the set of identifiable parameters should be identified

The result of the calibration is an individual para- meter set for each robot

4.3 Error compensation

Because available robot controls cannot make direct use of calibration data, off-line error compensation methods must be applied in order to use calibration results and to allow execution of off-line generated application programs On-line compensation would require too much computational power from the robot controller because computation of reaction forces and torques is necessary for compensation of elastic deformations If algorithms of the robot control make use of explicitly invertible kinematic equations, then the complete set of geometric parameters identi- fied by the calibration procedure cannot directly be used

Off-line compensation methods which use calibra- tion results allow the improvement of absolute pose accuracy without any changes in the robot control algorithms and parameters A prerequisite for this is that these algorithms and parameters are known Then, using the powerful robot model from the cali- bration procedure and the model on which the control

Fig 4 Off-line error compensation: Xt, rv target pose; fM, robot model; q, ~ joint positions; fsx, control model; Xi,, compensation pose; fR, function describing real robot; Xreal, real pose

algorithms are based, compensation poses are com-

puted and transmitted to the r o b o t control instead of

the target poses which are taken from the off-line

generated p r o g r a m (see Fig 4)

5 R E S U L T S The model p a r a m e t e r values identified by the calibra-

tion procedure were verified by comparing the cali-

brated model predicted poses against actual measured

poses To accomplish this, r o b o t poses were measured

which were also distributed over the entire robot

workspace but differed from poses in the identification

procedure This verification procedure determines the

actual attainable absolute pose accuracy in the entire

r o b o t workspace The results of the verification

showed that this calibration procedure can improve

absolute pose accuracy up to the system's repeatabi-

lity Furthermore, by a thorough analysis of the

numerical results, detailed hints can be obtained

which can be used to m a k e changes in the robot

design which i m p r o v e repeatability

5.1 Improvement of absolute positionin9 accuracy

The results of applying the calibration procedure to a

revolute robot with six joints and m a x i m u m work-

space diameter of a b o u t 5 m, and a SCARA r o b o t

with an a r m length of 0.8 m are presented

The results in Tables 1 and 2 show improvements in

mean error and standard deviation of T C P position

errors as functions of various identifiable p a r a m e t e r

combinations The first column in each table defines

Table 1 Results for the 6-joint revolute robot Poses for identifi-

cation: 200; Poses for verification: 100

Identification M e a n error (mm) Standard deviation (mm)

Table 2 Results for the SCARA robot Poses for identification: 144; Poses for verification: 55

Identification M e a n error (mm) Standard deviation (mm)

the calibration p a r a m e t e r combination by using the following binary coding:

• transmission and coupling coefficients 00010

C o m b i n a t i o n s are expressed by the sums of these bit patterns

Table 1 shows results for the six-joint revolute robot F o r identification purposes, 200 poses were used and for verification, 100 poses Starting with the position errors measured without the benefit of

p a r a m e t e r identification (row 1), the table clearly

• shows four steps of improvement In the first step (rows 2-5), the identification of the kinematic para- meters reduced the error to 209/0 of its initial value (row 1) Additional identification of joint elasticity further reduces the error to 12 ~o of its initial value Here, link elasticity plays only a small role in error reduction and gear eccentricity cannot be detected As the detailed statistical analysis (see Fig 5) shows, pose error in the entire workspace is smaller than 1 m m in

98 9/0 of the poses and smaller than 0.5 m m in 75 ~o of the poses

Table 2 shows the results for the SCARA robot F o r identification purposes, 144 poses were used and for verification, 55 poses The identification of kinematic parameters again had the greatest influence and re- duced m e a n position error to 30 9/0 of its initial value

230 Robotics & Computer-Integrated Manufacturing • Volume 8, Number 4, 1991

M e a n v a l u e : 0 4 0 3 r n m

S t a n d a r d d e v i a t i o n : 0 2 1 2 m m

25 %

- 0 2 3 3 I

-0.149-

-6373e-02-I

2.106e-O2-

)O0000X 3.0

0.106-

]O0000000000000000X 8 0

0.191-

)O00000000000000000000000000000000000000X 17.0

O,275-

)O00000000000000000000000000000000oooooooo000000000000x 23 o

036- }

)ooooooooooooooo0000000oo00000000x 14.o

0.445-I

]O0000000000000000000000000000X 130

053-

)O0000000000000000X 8.0

0615-1

)OO900000000000000000X 9 0

0.699-

0 7 8 4 - t

)O00X 20

0869- I

0 9 5 4 - I

]X 1.0

1.04-

Fig 5 Histogram of the norm of the coordinate difference

Additional identification of transmission and coupling

coefficient reduced error to 1 0 ~ This occurred be-

cause the transmission coefficient value for one joint

was incorrect As could be expected for a SCARA

robot, elasticity parameters have no influence on the

remaining error The statistical analysis (see Fig 6)

shows that 99 ~ of the poses had an error smaller than

0.6 mm and 50 ~ of the poses had an error smaller

than 0.24 mm

These results show that the presented calibration

procedure allows improvement of absolute Cartesian

pose accuracy up to the system limit set by repeatabi-

lity when including all genuine stochastic sources of

error

5.2 Hints f o r chanoes in robot desion

Generally, the spread of parameter values obtained by

varying parameter combinations is small if all signifi-

cant sources for positional error are included How-

ever, analysis of the effects of this spread can be used

for locating and diagnosing unmodeled deterministic

and/or exceedingly large stochastic errors

M e a n value: 0.272 r n m

S t a n d a r d deviation: 0.150 m m

25%

3340e-03

O00X 1.8

6314e-02-

0.123-

0,183-

0,243-

0.302-

0.362-

0,422-

0,482-

0542-

0.601-

O00000000OO0000000000X 9 1

O00000OO000000000000000OO000000OOOO000000000000000000000000X 255

00000000000000000000000000000000O0O000X 16.4

O000000000000000X 7 3

O00000OO0000000000000000000000X 12.7

O00000000000000000000000000000X 12.7

O000000000000000X 73

O00X 1.8

O0000000X 3 6

Fig 6 Histogram of the norm of the coordinate difference

(SCARA robot)

As a rule, the effects of this parameter spread on the

T C P position are of the order of 0.1 mm Significantly larger spreads point to stochastic or unmodeled deter- ministic error sources

Analysis of the identification results for the six-joint revolute robot showed that this kind of error emerged from joint 2 An obvious reason was the pneumatic torque compensation of this joint Its high static friction combined with elasticity of the transmission from m o t o r to joint-bearing produced a slip-stick effect and resulted in a relatively large range for which actual joint position could not be precisely known by the joint controller The actual joint position inside this range depends on the dynamic history of the movement Contrary to gear backlash (a deterministic error as long as joint torque is present and its direction

is known), this is a purely stochastic effect Experimen- tally, a minimum spread of 0.018 ° was determined This corresponds to a T C P position spread of 0.7 mm This stochastic error is only partly included in the value for the robot's repeatability since only uni-directional pose repeatability (ISO9283) is usually measured Uni-directional means the robot was repeatedly moved to the same pose from the same direction and with the same dynamic history The results showed that with a modification of the torque compensation mechanism at joint 2, the deter- ministic system accuracy of this robot could be improved This example clearly shows how analysis

of calibration results can also be used to improve repeatability through changes in robot design

REFERENCES

1 Ahmad, Sh.: Second order nonlinear kinematic effects, and their compensation Proc 2nd IEEE Int Conf on Robots and Automation, 1985, pp 307-314

2 Dennis, J E., Schnabel, R B.: Numerical Methods for Unconsidered Optimization Englewood Cliffs, Prentice- Hall, 1983

3 Everett, L J., Suryohadiprojo, A H.: A study of kine- matic models for forward calibration of manipulators

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