Complete Stiffness Model for a Serial Robot

Alexandr Klimchik

1,2

, Stephane Caro

2,3

, Benôit Furet

2,4

and Anatol Pashkevich

1,2

Ecole des Mines de Nantes, 4 rue Alfred-Kastler, 44307 Nantes, France

Institut de Recherches en Communications et Cybernétique de Nantes, UMR CNRS 6597, Nantes, France

Centre National de la Recherche Scientifique (CNRS), Paris, France

Université de Nantes, quai de Tourville, 44035 Nantes, France

Keywords: Robot Calibration, Elastostatic Identification, Stiffness Modeling, Parameter Identifiability.

Abstract: The paper addresses a problem of robotic manipulator calibration. The main contributions are in the area of

the elastostatic parameters identification. In contrast to other works, the considered approach takes into ac-

count elastic properties of both links and joint. Particular attention is paid to generation of the complete and

irreducible stiffness model that is suitable for the identification. To solve the problem, physical and algebra-

ic model reduction methods are proposed. They are based on taking into account the physical properties of

the manipulator elements and structure of the corresponding observation matrix. The advantages of the de-

veloped approach are illustrated by an application example that deals with elastostatic calibration of an in-

dustrial robot.

1 INTRODUCTION

Industrial robots are gradually finding their niche in

manufacturing, replacing less universal and more

expensive CNC-machines. Application area of ro-

bots is constantly growing, they begin to be used not

only for the assembly and pick-and-place operations,

but also for the machining. The latter requires spe-

cial attention to the accuracy of the model, which is

used to control the manipulator movements. Fur-

thermore, for this process, the robot is usually sub-

ject to essential external loading caused by the ma-

chining force that may lead to non-negligible deflec-

tions of the end-effector (Dépincé and Hascoët,

2006) and, accordingly, degrade the quality of the

final product. This issue becomes extremely im-

portant in the aerospace industry, where the accura-

cy requirements are very high. In this case, the ma-

nipulator stiffness modeling and corresponding error

compensation technique are the key points (Karan

and Vukobratović, 1994; Kövecses and Angeles,

2007), where in addition to accurate geometric mod-

el a sophisticated elastostatic one is required.

In practice, the robot positioning accuracy can be

improved by means of either on-line or off-line error

compensation techniques (Abele, Schützer et al.,

2012; Chen, Gao et al., 2013). Usually geometric

errors (such as offsets and link lengths) can be effi-

ciently compensated by modifying internal parame-

ters of the robot controller (Mooring, Roth et al.,

1991). In contrast, the compliance errors have to be

compensated via modification of the controller in-

puts. In such a case, an off-line error compensation

technique is aimed at adjusting the target trajectory

in accordance with the errors to be compensated and

the geometric model used in the robot controller

(Klimchik, Pashkevich et al., 2013). It is evident that

the efficiency of the latter approach is quite sensitive

to the model completeness and the accuracy of its

parameters.

To achieve desired degree of accuracy, the ma-

nipulator model should be calibrated for each partic-

ular manipulator (Meggiolaro, Dubowsky et al.,

2005). In modern robotics, there exist a number of

techniques that allow user to identify geometric and

elastostatic parameters of either serial or parallel

manipulators. In general, classical calibration proce-

dure contains four basic steps: modeling, measure-

ment, identification and implementation (Roth,

Mooring et al., 1987). The first step is aimed at de-

velopment of a model, which is accurate enough and

also is suitable for the identification (i.e. without re-

dundant parameters that can cause the convergence

breakdown). At the following step, the measure-

ments data are obtained. These data can be gotten

192

Klimchik A., Caro S., Furet B. and Pashkevich A..

Complete Stiffness Model for a Serial Robot.

DOI: 10.5220/0005098701920202

In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 192-202

ISBN: 978-989-758-040-6

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

using open-loop and closed-loop methods (Takeda,

Shen et al., 2004; Nubiola and Bonev, 2013). The

identification step is aimed at tuning the model pa-

rameters in accordance with the experimental data.

The last step, implementation, deals with modifica-

tion the robot control software in accordance with

the identified parameters.

In the manipulator stiffness modeling, there are

currently three main approaches: the Finite Element

Analysis (FEA), the Matrix Structural Analysis

(MSA), and the Virtual Joint Method (VJM). As fol-

lows from our experience, the VJM method (Alici

and Shirinzadeh, 2005; Klimchik, Chablat et al.,

2014) provides reasonable trade-off between the

model accuracy and computational complexity and

will be further used in this paper. It is based on the

extension of the traditional rigid model by adding

the virtual joints describing the elastic deformations

of the links, joints and actuators.

It should be mentioned that calibration of the

elasto-static model is much more difficult compared

to the geometric one. For a simple case, when only

elasticity of the actuated joints is taken into account,

an efficient approach has been proposed in (Dumas,

Caro et al., 2011), but this simplification does not

allow describing some important deflections of the

end-effector. More sophisticated model describing

both the joint and link elasticity can be developed

use CAD-based technique proposed in our previous

work (Klimchik, Pashkevich et al., 2013). However,

this model includes huge number of parameters that

cannot be identified separately using conventional

measurement data describing the end-effector de-

flections caused by external force/torque. It means

that from mathematical point of view, this technique

may produce redundant models that are not suitable

for calibration.

Similar problem is also known in geometric cali-

bration where the concept of complete-irreducible-

continues model has been introduced (Khalil and

Dombre, 2004). However, in elastostatic calibration

there is an additional difficulty caused by huge

number of model parameters (258 for 6 dof manipu-

lator) and essential difference in their magnitudes.

For this reason, this paper deals with developing

stiffness model suitable for identification and pro-

poses model reduction methods that allow obtaining

reliable results in industrial environment.

It is worth mentioning that the adopted approach

deals with quasi-static modeling that is motivated by

the considered application area. In particular, it is

assumed that trajectory tracking compensation does

not takes into account dynamic effects and load dis-

turbances over frequencies.

To address the above mentioned problem, the

remainder of the paper is organized as follows. Sec-

tion 2 presents the stiffness modeling background

and problem statement. In Section 3, the developed

model reduction methods are presented. Section 4

contains application examples that illustrate ad-

vantages of the proposed technique. And finally,

Section 5 summarizes the main contributions of the

paper.

2 THEORETICAL

BACKGROUND AND

PROBLEM STATEMENT

Let us consider an elastostatic model of a general

serial manipulator, which consists of a fixed “Base”,

a serial chain of flexible “Links”, a number of flexi-

ble actuated joints “Ac” and an “End-effector” (0).

In order to describe the stiffness of the considered

manipulator, let us apply the virtual joint method

(VJM), which is based on the lump modeling ap-

proach. According to this approach, the rigid model

should be extended by adding localized spring de-

scribing links elasticity. Besides, in order to take in-

to account the stiffness of the control loop, the virtu-

al springs should be included in the actuated joints.

Figure 1: VJM model of serial robot.

For the considered manipulator the force-

deflection relation for given robot configuration

is defined by the Cartesian stiffness matrix

K as

·Δ



WK t

(1)

In these equations, the end-effector displacement

Δt

is treated as the model input and the external wrench

is the model output. The stiffness matrix

can be computed as follows





C θθθ



KJKJ

(2)

Here, the Jacobian matrix

J depends on the manip-

ulator configuration

and can be computed as a

partial derivative of the end-effector location with

CompleteStiffnessModelforaSerialRobot

193

respect to a set of desired virtual joint coordinates

θ . Expression (2) allows us to compute the Carte-

sian stiffness matrix assuming that the matrix

(1) (2)

θθθ

( , , ...)diagKKK

, defining elastostatic

properties of the manipulator links/joins is given.

However, in practice, the matrices



(i)

,1,2,...i K

are unknown and should be identified from relevant

experiments.

To estimate the desired matrices describing elas-

ticity of the manipulator components (i.e., compli-

ances of the virtual springs), the elastostatic model

(1) should be rewritten as



(i) (i) (i)T

θθθ

i





JkJtW

(3)

where the matrices

(i) (i) 1

θθ

()



kK

denote the

link/joint compliances that should be identified via

calibration, and the matrices

(i)

are corresponding

sub-Jacobians obtained by the fractioning of the ag-

gregated Jacobian

(1) (2)

θθθ

[,,...]

JJJ .

In the case when the matrices

(i)

are known

the end-effector deflections Δt caused by external

loading W can be compensated by means of either

on-line or off-line error compensation techniques

(Lo and Hsiao, 1998; Klimchik, Bondarenko et al.,

2014). Usually main geometric errors (such as off-

sets and link lengths) can be efficiently compensated

by modifying internal parameters of the robot con-

troller (Mooring, Roth et al., 1991). In contrast, the

compliance errors have to be compensated via modi-

fication of the controller inputs. Relevant on-line

compensation strategy requires external measure-

ment system that continuously provides the end-

effector coordinates, which are compared with the

computed ones and the differences are used for ad-

justing the input trajectory (Lu and Lin, 1997).

However, suitable measurement systems are quite

expensive and often cannot ensure tracking the ref-

erence point in a whole robot workspace. Moreover,

behavior of some technological processes hampers

the end-effector observability (cutting chip in mill-

ing, for instance) and may damage the measurement

equipment. In such a case, an off-line error compen-

sation technique looks more reasonable; it is aimed

at adjusting the target trajectory in accordance with

the errors to be compensated and the geometric

model used in the robot controller (Chen, Gao et al.,

2013; Klimchik, Pashkevich et al., 2013).

However for the majorit of industrial robots the

values of compliance matrices

(i)

should be identi-

fied from the dedicated experimental study. So, for

the identification purposes, this expression should be

transformed into more convenient form, where all

desired parameters (elements of the matrices

(i)

, 1, 2,...i k

) are collected in a single vector

(1) (1) ( )

θ11 θ12 θ66

(, ,...)

kk kπ

. It yields the following linear

equation







tAπ

(4)

where

1212

[ , ,...., ]



AJJWJJWJJW

(5)

is so-called observation matrix that defines the map-

ping between the unknown compliances

π and the

end-effector displacements

t under the loading

W for the manipulator configuration

q . In the ob-

servation matrix



A the subscript defines the pa-

rameters set for which the observation matrix is

computed. Here, the vectors

J are the columns of

the matrix

J , i.e.

θ 12

[ , ,...., ]



JJJ J.

Taking into account that the calibration experi-

ments are carried out for several manipulator con-

figurations defined by the actuated joint coordinates

,1,

jmq

, the system of basic equations for the

identification can be presented in the following form

;1·,

jjj



πεAt

(6)

where

describes the measurement noise impact.

Further, using these notations and assigning proper

weights for each equation, the identification can be

reduced to the following optimization problem

()()mi

jj jj





 



A π tAη tηπ

(7)

where

is the matrix of weighting coefficients that

normalizes the measurement data. In practice, the

matrix

is used for two main purposes: (i) to avoid

the problem of non-homogeneity due to different

units of equations in the system (6) (position and

orientation components, in this particular case) and

(ii) to give higher weights for the measurements

whose precision is obviously higher. An example of

such an approach has been given in (Klimchik, Wu

et al., 2013). Relevant minimization (7) yields the

following solution



 

TT TT

jjj

  









π AA A tηη ηη

(8)

If the measurement noise is Gaussian (as it is as-

sumed in conventional calibration techniques), ex-

pression (8) provides us with an unbiased estimates

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194

for which



E ππ.

It is clear that expression (8) gives reliable esti-

mates of the parameters π if and only if the matrix









A ΣΣA

is invertible. It leads to the

problem of the parameter identifiability that have

been studied by a number of authors for the problem

of geometrical calibration (Pashkevich, 2001; Khalil

and Dombre, 2004). Relevant techniques are based

on the information matrix rank analysis. However,

they cannot be directly applied for the case of elasto-

static calibration.

Let us assume that the vector of desired elasto-

static parameters

π should be identified from the set

of the linear equations (4) whose least square solu-

tion is defined by the expression (8). Depending on

the matrix set







A , corresponding system of line-

ar equations can be solved for

either uniquely or

may have infinite number of solutions. In general, if

the information matrix is rank-deficient, a general

solution of the system (6) can be presented in the

following form



··



 

π AIBAAλ

(9)

where the superscript "+" denotes the Moore–

Penrose pseudo-inverse,









ηηAAA











ηηBAt

and λ is an arbitrary vector

of the same size as

. So, all desired parameters

contained in the vector

can be divided into three

non-overlapping groups (Pashkevich, 2001):

G1: Identifiable parameters that can be obtained

from (9) in unique way;

G2: Non-identifiable parameters that cannot be

computed uniquely from (9) and do not influence on

the right-hand side of the equation (4);

G3: Semi-identifiable parameters that are also

cannot be computed uniquely but have influence on

the right-hand side of the equation (4).

To present typical examples of the parameters

belonging to the groups G1, G2 and G3, it is possi-

ble to use the ideas similar to geometrical calibra-

tion. For instance, the elastostatic parameters of the

actuated joints and adjacent links are redundant in

their totality and belong to the group G3. Besides, if

the loading direction cannot be altered, a number of

parameters belong to the group G2 and cannot be

identified from the corresponding experimental data.

So, complete and irreducible model should contain

all parameters from the group G1 and partially pa-

rameters of the group G3.

Hence, to obtain reliable stiffness model that is

suitable for calibration, it is necessary to develop

dedicated model reduction techniques and relevant

rules allowing us to minimize the number of pa-

rameters to be estimated and to reconstruct the orig-

inal VJM-based model from these data taking into

account mathematical relations between the model

parameters caused by their physical sense.

3 MODEL REDUCTION

3.1 Physical Approach

Straightforward stiffness modeling approach pro-

vides the exhaustive but redundant number of pa-

rameters to be identified. For instance, each links is

described by a 6 6



matrix that includes 36 parame-

ters that are treated as independent ones. However,

as follows from physics, number of pure physical

parameters is essentially lower. Hence, there are

strong relations between these 36 parameters but this

fact is usually ignored in elastostatic calibration. Be-

sides, due to fundamental properties of conservative

system, the desired compliance matrices should be

strictly symmetrical and positive-definite. In addi-

tion, for typical manipulator links, the compliance

matrices are sparsed due to the shape symmetry with

respect to some axis, but this property is not taken

into account also in the identification of the elasto-

static parameters.

To use the advantages of the compliance matrix

properties and to increase the identification accura-

cy, three simple methods can be applied that allow

us to reduce the number of parameters to be comput-

ed in the identification procedure (8). They can be

treated as the physics-based model reduction tech-

niques and formalised in the following way.

M1:Symmetrisation. For all compliance matri-

ces

k to be identified, replace the pairs of symmet-

rical parameters





ij ji

by a single one

ki j



For each link, this reduction procedure is equiva-

lent to re-definition of the model parameters vector

in the following way



π M π

(10)

where the binary matrix

M of size 36 21 de-

scribes the mapping from the original to reduced pa-

rameter space. It can be proved that corresponding

basic expression for the identification (4) can be re-

written as









At π

(11)

CompleteStiffnessModelforaSerialRobot

195

where ·





ππ

AAM denotes the reduced observation

matrix. The later can be also computed as

θ 1 θθ2 θθ21 θ

[],,...

TT T





AJω JwJω Jw Jω Jw

(12)

where

, ,...ωω

denote the binary matrices of size

66 for which non-zero elements (i.e. equal to 1)

are located in the following way: for the parameter



corresponding to the matrix elements

ki j



the non-zero elements are

ij ji





. It is clear

that this idea allows us to reduce the number of links

compliance parameters from 36 to 21 (and from 258

to 153 for the entire 6 d.o.f. manipulator).

M2:Sparcing. For all compliance matrices k to

be identified, eliminate from the set of unknowns the

parameters

corresponding to zeros in the stiff-

ness matrix template

derived analytically for the

manipulator link with similar shape.

To obtain a desired template matrix, is conven-

ient to use any realistic link-shape approximation.

For example using the trivial beam (Timoshenko and

Goodier, 1970), the desired template can be present-

ed as

*00000

0*000*

00*0*0

000*00

00*0*0

0*000*











(13)

where the symbol "*" denotes non-zero elements. It

allows further reducing the number of the unknown

parameters from 21 to 8, taking into account only

essential ones from physical point of view. It can be

also proved that the template (13) is valid for any

link whose geometrical shape is symmetrical with

respect to three orthogonal axes. But it is necessary

to be careful if this property is not kept strictly.

It should be stressed that the actuated joint com-

pliances cannot be identified separately. So, they

should be included in the compliance matrix of the

previous link by means of modification of the corre-

sponding diagonal elements. This idea does not con-

tradict to the physical nature of the problem even if

the actuated joint compliance dominates correspond-

ing compliance in the link stiffness matrix.

M3:Aggregation. Eliminate from the set of

model parameters the ones that corresponds to joint

compliances before which there is an elastic link; in

terms of parameters identifiability the compliance of

those joints cannot be split from the links.

Summarizing these methods, it should be men

tioned that the above presented approach essentially

reduce the number of parameters to be identified (by

the factor 4.5) but they do not violate such basic

properties as the mode completeness, i.e. the ability

to describe any deflection caused by the external

loading. Below, these reduced set of the original

model parameters

π will be referred to as



. How-

ever, the obtained reduced model may still have

some redundancy in the frame of entire manipulator,

where the virtual springs of adjacent joints/actuators

cause similar impact on the end-effector deflections

under the loading.

As it known from the geometrical calibration, in

spite of the fact that redundant model is suitable for

direct and inverse computations, it cannot be used in

identification since the observation matrix does not

have sufficient rank. Similar problem arises in elas-

tostatic calibration where some stiffness matrix ele-

ments of adjacent links/joints are coupled and cannot

be identified separately. Let us present an algebraic

technique allowing to overcome this difficulty.

3.2 Algebraic Approach

The physical approach described in the previous

sub-section allows us essentially reducing the num-

ber of model parameters. However, it does not guar-

antee that the obtained model is suitable for calibra-

tions (i.e. that the model is non-redundant and the

number of parameters is equal to the observation

matrix rank). In practice, the following inequality is

often satisfied:





dimrank





A π . To over-

come the problem, this sub-section presents some

algebraic tools aimed at further reduction of the

model parameter set from





, which ensures

full identifiability:









dimrank rank







ππ

AAπ

(14)

These tools are based on the partitioning of the pa-

rameters set



into three non-overlapping groups

(identifiable, non-identifiable and semi-identifiable),

which are either eliminated from the model or re-

duced to ensure the equality (14).

To introduce relevant algebraic technique, let us

apply the SVD decomposition and present the ag-

gregated observation matrix



as the product of

three matrices

··

U Σ V

(orthogonal, diagonal and

orthogonal, respectively):



'''

,..

)

·.

(

mr m

diag





















U 

(15)

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196

Here

[,... ]

UUU and

[,...]

VVV are orthog-

onal matrices of the size



and nn respective-

ly whose columns are denoted as

U and

; the

second factor

Σ is a rectangular diagonal matrix of

the size

mn containing

positive real numbers

,...



in descending order;



dim

m t is the

number of rows in the observation matrix (i.e. num-

ber of equations used for the identification),



dimn





π is current number of the model param-

eters, and

is the rank of the aggregated observa-

tion matrix, '

mmr, 'nnr . It is clear that

defines the maximum number of parameters that

can be identified using given set of manipulator con-

figurations





and corresponding wrenches





Further, after substitution (15) into (11) and left-

multiplication by

, the original system of m

identification equations (4) can be rewritten as

'''

·... · ...·

mr m













 



 



 



 

 





π t







(16)

where the number of equation is equal to

and per-

fectly corresponds to the vector



dimension (it is

obvious that

nm

). Taking into account particulari-

ties of the sparse matrix

Σ (with r non-zero ele-

ments only), it is possible to rewrite the system (16)

in the form

;· · · 1, 2,...,

0;1,..·· ,·.

ii a

ir n















UV π t

V π tU

(17)

where the second group of

'mmr equations

should be excluded from further consideration be-

cause relevant residuals do not depend on the pa-

rameters of interest



(since they are multiplied by

zero matrix). It can be proved that





tU0

for

ir if the measurement vector

t

does not con-

tain noise. It is also worth mentioning that for real

identification problems (with the measurement

noise), the second group of equations produces con-

stant residuals that cannot be minimised in the least-

square objective (7) by varying the vector of un-

known parameters



Hence, for the identification of

n parameters in-

cluded in the vector



, a system of

linear equa-

tions have been obtained that cannot be solved

uniquely in a general case. Its partial solution can be

found by dividing on



 each of

linear equa-

tions

·· ·

ii a







V π tU

and further straightforward

multiplication of the left and right sides by the ma-

trix





,. ,,..

VVV, which yields

 

111

..., ... · .,..,..·,.

rra

rrr









  

  

  

VV π VV t

(18)

Using the first set of

equations of system (17) one

can obtain partial solution of system (16)



··

iii a















π VU t

(19)

This allows us to present the general solution (9) as

the sum of this partial solution and an arbitrary vec-

tor from the subspace with the basis

,,...

rr n

VV V

oii











Vππ

(20)

where



1,ir n

are arbitrary real values.

Hence, as follows from analysis of (17) and (19),

depending on the properties of the matrix

, all

model parameters



can be partitioned into three

groups: G1

 identifiable parameters that are

uniquely defined by the equation (20) and do not de-

pend on the arbitrary values



, for these parameters

the corresponding row of the sub-matrix

[ ,..., ]

rm

VV is equal to zero; G2  non-identifiable

parameters that do not effect the residuals of system

(17), for these parameters the corresponding row of

the sub-matrix

[ ,..., ]

VV is equal to zero; G3 

semi-identifiable parameters that effect the residuals

but cannot be identified uniquely, couplings between

these parameters is defined by the vectors

V ,

1,ir

. Thus, based on this decomposition, the al-

gebraic-based model reduction techniques can be

formalised in the following way:

M4a:Partitioning. Divide the reduced set of

the model parameters



into three non-overlapping

groups G1, G2 and G3 in accordance with the fol-

lowing rules applied to all





, 1, dim( )i 



π :

Rule 1: Include the parameter





into the group

G1 if the

ith row of the sub-matrix

[ ,..., ]

rm

VV is

equal to zero;

Rule 2: Include the parameter





into the group

G2 if the

ith row of the sub-matrix

[ ,..., ]

VV is

equal to zero;

CompleteStiffnessModelforaSerialRobot

197

Rule 3: If the parameter





is not included in G1

or G2, include it in the group G3.

M4b:Elimination. Eliminate from the set of

unknowns (model parameters) non-identifiable pa-

rameters that correspond to group G2.

After application of these methods, the current

set of model parameters



is reduced to the sub-set







ππ

that does not influence the rank of the ob-

servation matrix, i.e.



rank rank





ππ π

Nevertheless, relevant model may be redundant yet,

i.e.









\dim

rank





ππ

A ππ . It should be

noted that



dim

rank 

A π

while



dim

rank 

A π

. So, another, and the most

difficult problem that arises after M4, is to define the

sub-set of identifiable parameters inside of

π (the

remaining ones should be set to constant values).

It is clear that the above mentioned problem has

infinite number of solutions. Let us presents an algo-

rithm that is able to split the set of parameters

into the non-overlapping groups of coupled parame-

ters

and then choose identifiable one from the

group based on their physical scene:

M5a:Splitting. Split the set of semi-identifiable

model parameters

π into the non-overlapping

groups of coupled parameters

for which the fol-

lowing conditions are satisfied:

(a)

333 3

....

GGG G

πππ π



ππ





;

(b)



333

iii

GGG

rank rank

πππ

1:dim( )

j π

(c)



)

rank rank

ππ



,1:dim()

ijk  π

In practice, when this grouping is not evident, it

is possible to use numerical technique, which is

based on the SVD-decomposition of the reduced ob-

servation matrix

A . Using similar notation, the

matrix

V can be presented as





,,...VVV in ac-

cordance with the rank of

A . So, the couplings

between the elements are defined by the sub-matrix





..,.,

VV. One of the easiest ways to find the de-

sired couplings is to compute the matrix





..,., ...













LVV

(21)

where the symbol “*” denotes operation of the row

selection that conserve the matrix rank. The latter

leads to a full-rank square matrix presented above as

the first term of (21). It should be noted that this op-

eration is not unique, nevertheless, it allows to ob-

tain the couplings between the model parameters de-

scribed by the sparse matrix

L . Then, the desired

groups of parameters can be easily detected after

transformation

L into the block-diagonal form.

Using the above presented idea, the next step can

be presented as follows:

M5b:Selection. In each group of parameters

, specify





rankn 

parameters that will

be treated as identifiable

M5c:Assigning. In each group of parameters

, fix remaining







dim

mrank

π A

parameters to some constants; these parameters will

be treated as non-identifiable

It should be noted that the sequence of methods

M5b and M5c is not strict; identifiable and non-

identifiable parameters can be selected and fixed it-

eratively, using the methods M5b and M5c several

times. After application of the methods M5a, M5b

and M5c, the set of parameters

π is split into two

subsets: the subset of the parameters that will be

treated as identifiable

and subset that will be

treated as non identifiable ones

and will be as-

signed to some constant values (

constπ

); i.e.

333

id ni

GGG

πππ

id ni

ππ

After application of the algebraic approach, the

complete set of parameters



is reduced to



, It

includes all parameters from the group G1 and as-

signed-to-be-identifiable ones from the group G3. It

is clear that the presented algebraic methods do not

violate the model completeness, i.e.









rank rank







ππ

AA.

4 APPLICATION EXAMPLE

To demonstrate benefits of the developed techniques

for industrial applications, this section presents some

experimental results on the elastostatic calibration of

industrial robot Kuka KR-270 employed in high pre-

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198

cession machining of aircraft parts.. For the consid-

ered application area, the technological process gen-

erates essential interaction between the workpiece

and manipulator, which causes non-negligible de-

flections of the end-effector. To compensate related

positioning errors on the control level (via adjusting

a target trajectory), an accurate but simple enough

elasto-static model is required. In practice, the de-

sired model is not usually provided by robot manu-

factures and should be obtained from dedicated ex-

perimental study. Let us apply the developed tech-

nique to get the desired model and to identify its pa-

rameters in real industrial environment.

The considered manipulator contains 7 links sep-

arated by 6 actuated joints. Taking into account that

in general the elastostatic properties of each link are

defined by 6x6 stiffness matrix, the complete but

obviously redundant model contains 258 parameters.

As a result of application model reduction tech-

niques (M1-M5), the number of parameters to be

identified has been reduced down to 26. More details

on each step are given in 0. Additional restrictions

here are caused by the partial-pose measurement

technique and the gravity-based loading generating

the desired deflections. Relevant experimental setup

is presented in 0. Because of such measurement

method, 10 elastostatic parameters are not identifia-

ble from the available measurement data. The ma-

nipulator configurations for the elastostatic calibra-

tion were generated using the design of experiments

and previously developed test-pose technique, which

is based on the industry-oriented performance meas-

ure (Klimchik, Wu et al., 2012). Another particulari-

ty of the industrial robot KUKA KR270 that should

be taken into account in an accurate elasto-staic

model is a gravity compensator that is attached in

parallel to the second actuated joint. Its equivalent

model was presented in (Klimchik, Wu et al., 2013).

In the frame of the complete and irreducible model,

the gravity compensator impact is taken into account

by introducing a configuration dependant virtual

spring in the second joint. More details on this ap-

proach are given in (Klimchik, Wu et al., 2013).

In order to ensure higher identification accuracy,

the measurement configurations have been selected

using the design of experiment theory. In contrast to

other works, an industry-oriented performance

measure has been used (Wu, Klimchik et al., 2013),

which evaluates the robot positioning accuracy after

calibration. In total, 15 measurement configurations

for 5 different angles q

have been generated.

For the comparison purposes, calibration was

performed using several elastostatic models that dif-

fer in their basic assumptions: (i) complete irreduci-

ble stiffness model and (ii) conventional model for

the manipulator with rigid links and compliant actu-

ated joints. Here, conventional elastostatic models J1

and J2 take into account the actuated joint compli-

ances only. Both models (complete and reduced

ones) have been examined with and without taking

into account the effect of the gravity compensator.

The obtained results are summarized in 0 showing

capability to compensate the compliance errors us-

ing different elastostatic models. As follows from

them, the lowest compliance errors can be achieved

using the model C2 (obtained using the developed

model reduction technique), which ensures the posi-

tional accuracy 0.21 mm. In contrast, the conven-

tional elasto-static model with rigid links gives accu-

racy 3.5 times worse comparing to model C2. The

difference in the efficiency of the compliance errors

compensation between the models J1/J2 and C1/C2

confirms that link compliances have essential impact

on the robot positioning accuracy and cannot be ne-

glected in accurate manufacturing. According to ex-

perimental results presented in 0, by means of the

complete elastostatic model, it is possible to com-

pensate 95% of the compliance errors. In contrast,

using the conventional model that takes into account

the joint elasticity’s merely, only 84% of positioning

errors caused by external force can be compensated.

This emphasizes the advantage of the proposed

model. The histograms of the errors distribution for

the model C2 (0) show that the non-compensated

compliance errors in all directions are unbiased and

almost normally distributed.

Hence, using the developed low-order stiffness

model for the compliance error compensation gives

essential improvement of the precision for the robot-

ic based milling. It allowed us to compensate more

than 95% of deflections caused by external loading

and to guarantee the precision of about 0.2 mm un-

der the loading of 2.5 kN (it is comparable with the

robot repeatability of 0.06 mm).

5 CONCLUSION

The paper deals with the problem of the manipulator

stiffness modeling. The main attention is paid to the

elastostatic parameters identification and model re-

duction. In contrast to previous works, the manipula-

tor stiffness properties are described by the sophisti-

cated model, which takes into account the flexibili-

ties of all mechanical elements described by 6×6

stiffness matrices. This obviously yields extremely

high number of the model parameters that cannot be

CompleteStiffnessModelforaSerialRobot

199

Figure 2: Experimental setup for elastostatic calibration.

Table 1: Summary of the elasto-static model reduction process for industrial robot Kuka KR-270.

Approach Step Model description

Number of pa-

rameters

Original model 6 joints +7 links (36 parameters per link) 258

Physical

M1: Symmetrisa-

tion

6 joints +7 links (21 parameters per link) 153

M2: Sparcing 6 joints +7 links (8 parameters per link) 62

M3: Aggregation 7 links (8 parameters per link) 56

Algebraic

M4a:Partitioning

M4b:Elimination

G1: Identifiable parameters – 1

G2: Non-identifiable parameters - 10

G3: Semi-identifiable parameters – 45

M5a:Splitting

M5b:Selection

M5c:Assigning

Selection of 25 independent parameters

from 45 semi-identifiable ones

Table 2: Efficiency of the compliance errors compensation using complete and reduced models.

Stiffness model

Number of

parameters

Compliance errors, mm

x-direction y-direction z-direction positional

MAX RMS MAX RMS MAX RMS MAX RMS

Deflections magnitude without compensation 2.51 1.03 3.14 1.02 8.14 1.91 8.18 4.58

Complete model C1 26 0.27 0.10 0.43 0.13 0.38 0.12 0.45 0.22

Complete model C2 30 0.28 0.10 0.45 0.14 0.32 0.11 0.49 0.21

Conventional model J1 5 1.42 0.43 1.73 0.41 0.66 0.23 1.78 0.75

Conventional model J2 9 1.42 0.42 1.73 0.42 0.49 0.19 1.76 0.73

Model C1: Complete irreducible stiffness model without gravity compensator

Model C2: Complete irreducible stiffness model with gravity compensator

Model J1: Conventional model for the manipulator with rigid links and compliant actuators, without gravity compensator

Model J2: Conventional model for the manipulator with rigid links and compliant actuators, with gravity compensator

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

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

100

120

140

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

100

120

140

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

100

120

140

Figure 3: Statistical distribution of compliance errors after compensation

identified separately. To solve the problem, physical

and algebraic model reduction methods were devel-

oped. They take into account mathematical relations

between the elements of the compliance matrices.

The advantages of the developed approach are illus-

trated by an application example that deals with

elastostatic calibration of an industrial robot used in

aerospace industry.

In future, the problem of the complete model

generation from the obtained set of parameters will

be in the focus of our attention, i.e. re-construction

of the joint compliances and 6x6 link stiffness ma-

trices from the reduced model obtained after the

identification. It is clear that this procedure requires

additional knowledge on the coupling between the

stiffness matrix elements that are induced by their

physical nature.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the financial

support of the ANR, France (Project ANR-2010-

SEGI-003-02-COROUSSO) and FEDER

ROBOTEX, France.

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