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Joint Stiﬀness Identiﬁcation of Six-revolute Industrial Serial Robots

Claire Dumas∗

,St´ephane Caro,S´ebastien Garnier,Benoˆıt Furet

Institut de Recherche en Communications et Cybern´etique de Nantes

UMR CNRS n◦6597

Tel: +33 2 40 37 69 54, Fax: +33 2 40 37 69 30

1 rue de la No¨e, 44321 Nantes, France

Email: {claire.dumas, stephane.caro, sebastien.garnier, benoit.furet}

@irccyn.ec-nantes.fr

Abstract

Although robots tend to be as competitive as CNC machines for some operations, they are not yet widely

used for machining operations. This may be due to the lack of certain technical information that is required

for satisfactory machining operation. For instance, it is very diﬃcult to get information about the stiﬀness

of industrial robots from robot manufacturers. As a consequence, this paper introduces a robust and fast

procedure that can be used to identify the joint stiﬀness values of any six-revolute serial robot. This procedure

aims to evaluate joint stiﬀness values considering both translational and rotational displacements of the robot

end-eﬀector for a given applied wrench (force and torque). In this paper, the links of the robot are assumed

to be much stiﬀer than its actuated joints. The robustness of the identiﬁcation method and the sensitivity of

the results to measurement errors and the number of experimental tests are also analyzed. Finally, the actual

Cartesian stiﬀness matrix of the robot is obtained from the joint stiﬀness values and can be used for motion

planning and to optimize machining operations.

Keywords: Stiﬀness analysis; joint stiﬀness identiﬁcation; Cartesian stiﬀness matrix; complementary stiﬀ-

ness matrix; serial robots; robot machining.

1 Introduction

Serial robots are mainly used in industry for tasks that require good repeatability but not necessarily good global

pose accuracy (position + orientation as deﬁned in ISO9283) of the robot end-eﬀector (EE). For example, these

robots are generally used for pick-and-place, painting and welding operations. Nevertheless, they are now being

used for machining operations, such as the trimming, deﬂashing, degating, sanding and sawing of composite

parts, that require high precision and stiﬀness. Therefore, to perform these operations, the robots must show

good kinematic and elastostatic performance. In this context, it appears that conventional machine tools such

as the gantry CNC are still more eﬃcient than serial robots. Therefore, it is important to pay attention

to robot performance to optimize their usefulness for machining operations. Some research works discuss the

∗Corresponding author. Email address: claire.dumas@irccyn.ec-nantes.fr

hal-00632989, version 1 - 17 Oct 2011

Author manuscript, published in "Robotics and Computer-Integrated Manufacturing 27, 4 (2011) 881-888"

DOI : 10.1016/j.rcim.2011.02.003

C. Lecerf, S. Caro, S. Garnier and B. Furet 2

following: (i) tool path optimization considering both kinematic and dynamic robot performance [1, 2, 3]; (ii) the

determination of optimal cutting parameters to avoid tool chattering [3, 4]; (iii) robot stiﬀness analysis [5]; and

(iv) the determination of robot performance indices [6, 7, 8, 9, 10]. Robot stiﬀness is also a relevant performance

index for robot machining [11]. Accordingly, this paper discusses the stiﬀness modeling of serial robots and

identiﬁes their stiﬀness parameters. Some stiﬀness models can be found in the literature for serial and parallel

manipulators [12, 13]; however, the identiﬁcation of stiﬀness parameters has yet to be determined.

Two methods were presented in [14] to obtain the Cartesian stiﬀness matrix (CaSM) of a ﬁve-revolute

robot. The ﬁrst method consists of clamping all of the joints except one to measure its stiﬀness. The joint

stiﬀness matrix of the robot is obtained by repeating the procedure for each revolute joint. Therefore, only

ﬁve experiments are required with this method to evaluate the CaSM of the robot throughout its Cartesian

workspace assuming that the stiﬀness of the links is known. The second method measures the displacements of

the robot end-eﬀector due to certain applied loads and evaluates the robot Cartesian stiﬀness matrix throughout

its Cartesian workspace with some interpolations. This method provides a good approximation of the robot

CaSM when many tests are performed under diﬀerent robot conﬁgurations. The second method gives better

results than the ﬁrst. As far as the second method is concerned, all deformations are considered including those

due to the joint and link ﬂexibilities along and about all of the axes. On the contrary, in the ﬁrst method, the

links of the robot are assumed to be rigid, and only the stiﬀness of the joints is considered.

This paper introduces a method to identify the joint stiﬀness values of any six-revolute industrial serial robot.

The proposed method is based on conservative congruence transformation [15], requires few experimental tests,

is easy to implement and does not require any closed-loop control or actuator currents. This method also

considers both forces and moments applied on the robot end-eﬀector. Therefore, both the translational and

rotational displacements of the robot end-eﬀector are considered. Finally, the optimal number of experimental

tests and robot conﬁgurations used to perform the joint stiﬀness identiﬁcation are given.

The Kuka KR240-2 robot is used as an illustrative example throughout the paper. Its kinematic performances

are analyzed in Section 2. Its Cartesian stiﬀness matrix is shown in Section 3. Section 4 contains a sensitivity

analysis of the Cartesian stiﬀness matrix of the robot to its complementary stiﬀness matrix and the determination

of the optimal robot conﬁgurations for joint stiﬀness identiﬁcation. The subject of Section 5 is the robustness

of the proposed joint stiﬀness identiﬁcation method with regard to measurement noise.

2 Kinematic Performance of the Kuka KR240-2 Robot

The kinematic chain of the Kuka KR240-2 robot is shown in Fig. 1. The sixth link carrying the operation point P

is connected to the base frame F0through a serial chain composed of six revolute joints. The modiﬁed Denavit

Hartenberg parameters (DHm), described in [16], are used to parameterize the robot. Then, the kinematic

Jacobian matrix of the robot was obtained, and its kinetostatic performance is evaluated throughout both its

joint space and Cartesian workspace.

2.1 Parameterization

As illustrated in Fig. 1, the robot is composed of seven links, denoted as L0,...,L6, and six revolute joints.

Link L0is the base of the robot, while link L6is the terminal link. Joint jconnects link jwith link j−1,

j= 1,...,6. Frame Fjattached to link j,j= 0,...,6, is deﬁned such that:

•the zjaxis is along the axis of joint j;

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C. Lecerf, S. Caro, S. Garnier and B. Furet 3

•the xjaxis is along the common normal between zjand zj+1. If the axes zjand zj+1 are parallel, the

choice of xjis not unique;

•the origin Ojis the intersection of zjand xj.

The DHm parameters of the Kuka KR240-2 robot are given in Tab. 1.

L0

L1

L2

L3

L4

L5

L6

RL4

RL6

z0,z1

z2

z3

z4

z5

z6

x1

x2

x3

x4

O0,O1

O2

O3

O4

d2

d3

d4

Figure 1: DHm Parameterization of the Kuka KR240-2 robot

Table 1: DHm parameters of the Kuka KR240-2 robot

j a(j)µjσjαjdjθjrj

1 0 1 0 0 0 θ10

2 1 1 0 π/2d2θ20

3 2 1 0 0 d3θ30

4 3 1 0 −π/2d4θ4RL4

5 4 1 0 π/2 0 θ50

6 5 1 0 −π/2 0 θ6RL6

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C. Lecerf, S. Caro, S. Garnier and B. Furet 4

The Cartesian workspace of the robot is shown in Fig. 2, and its size is characterized as A = 3100 mm,

B = 3450 mm, C = 2700 mm, D = 1875 mm, E = 825 mm, and F = 1788 mm.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

x

z

AB

C

D

E

F

Figure 2: Cartesian workspace of the Kuka KR240-2 robot

2.2 Kinematic Jacobian

The 6 ×6 kinematic Jacobian matrix Jof the robot relates the joint rates to the twist of the end eﬀector,

namely,

t="˙

p

ω#=J˙

θ(1)

tis its end-eﬀector twist, which is composed of its translational velocity vector ˙

pand its angular velocity vector

ωexpressed in F0.

˙

θ=h˙

θ1˙

θ2˙

θ3˙

θ4˙

θ5˙

θ6iT

(2)

˙

θiis the ith actuated revolute joint rate.

2.3 Kinetostatic Performance Index

We understand here under kinetostatics the mechanical analysis of rigid-body mechanical systems moving under

static, conservative conditions. Kinetostatics is thus concerned with the relations between the feasible twists

— point-velocity and angular velocity — and the constraint wrenches — force and moment — pertaining to

the various links of a kinematic chain [17].

Accordingly, we focus on issues pertaining to manipulability or dexterity. We understand these terms in the

sense of measures of the distance to singularity, which brings us to the concept of condition number in [18, 19].

Here, we adopt the condition number of the underlying kinematic Jacobian matrix based on the Frobenius norm

as a means to quantify distances to singularity. The condition number κF(M) of an m×nmatrix Mwhen

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C. Lecerf, S. Caro, S. Garnier and B. Furet 5

m≤n, based on the Frobenius norm is deﬁned as follows:

κF(M) = 1

mqtr(MTM)tr [(MTM)−1] (3)

Here, the condition number is computed based on the Frobenius norm because the latter produces a condition

number that is analytic in terms of the posture parameters, whereas the 2-norm does not. In addition, it is

costlier to compute singular values than to compute matrix inverses.

The terms of matrix Jare not homogeneous as they do not have same units. Therefore, as shown in [20]

and [21], the Jacobian matrix can be normalized by means of a normalizing length, which is called characteristic

length and denoted as L. Let JNbe the normalized Jacobian matrix of the Kuka KR240-2 robot:

JN=

1

LI3×303×3

03×3I3×3

J(4)

I3×3is the 3 ×3 identity matrix and 03×3is the 3 ×3 zero matrix. The characteristic length of the Kuka robot

used in this study is equal to 0.682 m and was obtained by means of the methodology proposed in [22]. The

condition number is used to have an idea of the zones (on θ2and θ3ranges) where the robot has good dexterity.

It appears that a change in the condition number of JNthroughout the robot Cartesian workspace does not

depend on L, although its value depends on L.

Because the second and the third revolute joints are the most inﬂuential joints on the translation motions

of the robot end-eﬀector, and because the ﬁrst revolute joint does not aﬀect manipulator dexterity, θ1is null,

and the wrist angles θ4,θ5and θ6are set to 45◦so that the corresponding wrist conﬁguration is far from

singularities.

0.53

0.53

0.53

0.53

0.51

0.51

0.46

0.46

0.46

0.46

0.46

0.46

0.46

0.41

0.41

0.41

0.41

0.41

0.35

0.35

0.35

0.35

0.35

0.30

0.30

0.30

0.25

0.25

0.21

0.21

0.05

0.05

0.21

x[m]

z[m]

-2 -1.5 -1 -0.5 00.5 11.5 2 2.5 3

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

(a)

0.1

0.250.3

0.35

0.4

0.4

0.4

0.1

0.1

0.15

0.15

0.2

0.2

0.25

0.45

0.45

0.35

0.4

0.4

0.45

0.45

0.5

0.5

0.5

θ2[deg]

θ3[deg]

0 50 100

-200

-150

-100

-50

0

(b)

Figure 3: Contours of the inverse condition number of JN: (a) in the robot Cartesian workspace and (b) in the

robot joint space θ2,θ3

Figure 3(a) depicts the isocontours of the inverse condition number of JNbased on the Frobenius norm,

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C. Lecerf, S. Caro, S. Garnier and B. Furet 6

κF(JN)−1, throughout the robot’s Cartesian workspace. The higher κF(JN)−1, the better the dexterity. On

the contrary, the lower κF(JN)−1is, the closer the robot to singularities.

Likewise, Fig. 3(b) shows the isocontours of κF(JN)−1throughout the robot joint space. The blacker the

color, the closer the manipulator to singularities. The oblique black line characterizes the conﬁgurations in

which the wrist center is located on the ﬁrst joint axis. The horizontal black line in Fig. 3(b) characterizes the

singularities when the arm is folded.

Figures 3(a)-(b) are useful in choosing the optimal robot conﬁgurations for joint stiﬀness identiﬁcation as

explained in Section 4.

3 Cartesian Stiﬀness Matrix Formulation

The Cartesian stiﬀness matrix of a robot depends on its conﬁguration, link stiﬀness, control loop stiﬀness and

actuators mechanical stiﬀness. In this paper, the last two sources of stiﬀness are considered. The links of the

robot are assumed to be rigid, the damping is neglected and the stiﬀness of the joints is represented with linear

torsional springs.

Conservative congruence transformation was proposed by Chen and Kao [23] to deﬁne the spatial CaSM of

a serial robot. We obtain the relation:

w=KX∆X (5)

with

KX=J−T(Kθ−KC)J−1(6)

wis the 6-dimensional wrench vector composed of the forces and torques applied on the end-eﬀector and

expressed in F0.KXis the 6 ×6 CaSM of the robot expressed in F0.∆X is the 6-dimensional vector composed

of the translational and rotational displacements of the end-eﬀector expressed in F0.Jis the kinematic Jacobian

matrix of the robot deﬁned in Eq. (1). Kθis the diagonal joint stiﬀness matrix deﬁned as follows:

Kθ=

kθ100000

0kθ20 0 0 0

0 0 kθ30 0 0

0 0 0 kθ40 0

0 0 0 0 kθ50

0 0 0 0 0 kθ6

(7)

Where kθi,i= 1,...,6, is the ith joint stiﬀness value. KCis the complementary stiﬀness matrix (CoSM)

deﬁned in [15] that takes the form:

KC=∂JT

∂θ1

w∂JT

∂θ2

w∂JT

∂θ3

w∂JT

∂θ4

w∂JT

∂θ5

w∂JT

∂θ6

w(8)

It is noteworthy that this expression is equivalent to Salisbury’s when there is no force or torque applied on

the robot end-eﬀector. Matrix KCis not null and aﬀects matrix KX. This expression respects the principle

of virtual work and has been tested on several robots [15, 24]. The identiﬁcation of the joint stiﬀness values

kθi,i= 1,...,6, is the subject of the next section because they are required to evaluate KXfor any robot

conﬁguration.

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C. Lecerf, S. Caro, S. Garnier and B. Furet 7

4 Joint Stiﬀness Matrix Evaluation

Figure 4 shows a procedure to evaluate the joint stiﬀness matrix Kθ, which was expressed in Eq. (7). First, the

zones of the robot workspace and joint space where the robot has good dexterity are identiﬁed in Figs. 3(a)-

(b). Then, for a given wrench applied on the robot end-eﬀector, the areas of the previous zones where the

complementary stiﬀness matrix KCis negligible with respect to Kθare determined. Certain robot conﬁgurations

are chosen from those areas for the tests. For each test, a given wrench is applied on the robot end-eﬀector,

and its displacements (translations and rotations) are measured. Finally, the joint stiﬀness values are obtained

by a given number of tests.

Identify the zones of the joint space where the robot has a good dexterity

Zones where the robot has a good dexterity are known

Identify the areas of the previous zones where KCis negligible with respect to Kθ

Zones where KX≈J−T(Kθ)J−1are known

Select a robot conﬁguration

An optimum robot conﬁguration is selected for the test

Apply a wrench (force and moment) on the robot end-eﬀector

The robot end-eﬀector is loaded with a given wrench

Measure the displacements (translations and rotations) of the end-eﬀector

The displacements of the robot end-eﬀector are found

Evaluate the joint stiﬀness values with Eq.(21)

i=n, where nis the number of tests required and determined in Section 4.2

Kθis found

i < n

i=i+ 1

Figure 4: Procedure for the evaluation of the joint stiﬀness values

4.1 Inﬂuence of KCon KX

From Eq. (6), KXdepends on both Kθand KC. It makes sense that joint stiﬀness identiﬁcation is easier

when KCis negligible with respect to Kθ. Consequently, this section analyzes the inﬂuence of KCon KX.

From Eq. (8), the higher the wrench applied on the robot end-eﬀector, the higher the inﬂuence of KCon KX.

Consequently, the worst-case scenario appears when the force and the moment applied on the robot end-eﬀector

are at the maximum. Let the components of the force vector along x0,y0and z0axes be equal to 2200 N, and

the components of the moment vector about x0,y0and z0axes be equal to 400 Nm. For the sake of clarity,

the ﬁrst three joint stiﬀness values are assumed to be equal to those found in [24], and the other three joint

stiﬀness values are chosen arbitrarily as shown in Table 2.

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C. Lecerf, S. Caro, S. Garnier and B. Furet 8

Table 2: Joint stiﬀness values given in [24] and expressed in [Nm/rad]

kθ1kθ2kθ3kθ4kθ5kθ6

1409800 400760 935280 360000 370000 380000

The norm δp of the robot end-eﬀector small displacement screw is expressed as:

δp =qδpx

2+δpy

2+δpz

2(9)

where δpx,δpyand δpzare its displacements along the x0,y0and z0axes, respectively. Let δrx,δryand δrz

be the small rotations of the robot end-eﬀector about x0,y0and z0axes, respectively.

Let δpKCand δpKCbe the point-displacement of the robot end-eﬀector obtained with Eqs. (5) and (6)

assuming that matrix KCis not null and null, respectively. Likewise, let δrxKC,δryKC,δrzKCand δrxKC,

δryKC,δrzKCbe the small rotations of the robot end-eﬀector about the x0,y0and z0axes obtained with qs.(5)

and (6), respectively, assuming that matrix KCis not null and null, respectively.

To analyze the inﬂuence of KCon KXthroughout the robot workspace, we introduce indices νpand νr,

which characterize the inﬂuence of KCon the evaluation of the robot translational and rotational displacements,

respectively:

νp=δpKC−δpKC

max δpKC, δpKC(10)

and

νr= max nδrxKC−δrxKC,δryKC−δryKC,δrzKC−δrzKCo(11)

Figures 5(a)-(b) illustrate the isocontours of νpand νrthroughout the robot joint space (θ2, θ3). The blacker

the color, the higher the inﬂuence of KCon the evaluation of the end-eﬀector displacements. The shapes of

Fig. 5(a) and Fig. 5(b) are similar. This ﬁnding means that the robot conﬁgurations for which the inﬂuence

of KCon the end-eﬀector translational displacements is at its maximum are the same as those for which the

inﬂuence of KCon the end-eﬀector rotational displacements is at its maximum. As shown in Fig. 3(b) and

Figs. 5(a)-(b), the robot conﬁgurations for which the inﬂuence of KCon KXare at their maximum are also

those for which κF(JN)−1is at its minimum, i.e., close to singularity.

Nevertheless, the νpand νrvalues are very small, νp≤0.016 and νr≤0.025 deg throughout the robot joint

space. Accordingly, KCis negligible with respect to Kθ, and Eq.(6) can be reduced to:

KX≈J−TKθJ−1(12)

The above-mentioned tests were conducted with diﬀerent joint stiﬀness values than those given in Table 2.

The shapes of the νpand νrisocontours remained the same, and KCremained negligible with respect to Kθ.

Equation (12) and the robot conﬁgurations for which νpand νrare at their minimum re used in the next section

to identify the joint stiﬀness values of the robot. Then, Eq. (6) is be used to evaluate KXthroughout the robot

Cartesian workspace for the sake of accuracy.

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C. Lecerf, S. Caro, S. Garnier and B. Furet 9

0.002

0.002

0.002

0.002

0.002

0.004

0.004

0.004

0.004

0.006

0.006

0.006

0.006

0.008

0.008

0.008

0.01

0.01

0.01

0.012

0.012

0.012

0.002

0.002

0.002

0.002

0.002

0.014

0.014

0.004

0.004

0.004

0.006

0.016

θ2[deg]

θ3[deg]

0 50 100

-200

-150

-100

-50

0

(a)

0.005

0.005

0.005

0.005

0.005

0.005

0.01

0.01

0.01

0.01

0.01

0.01

0.015

0.015

0.015

0.015

0.015

0.005

0.005

0.005

0.02

0.02

0.02

0.02

0.01

0.01

0.025

0.025

θ2[deg]

θ3[deg]

0 50 100

-200

-150

-100

-50

0

(b)

Figure 5: Isocontours of (a) νpand (b) νrin the joint space (θ2, θ3)

4.2 Identiﬁcation of the Joint Stiﬀness Values

For the robot conﬁgurations in which KCis negligible with respect to Kθ, Eq. (5) takes the form:

w=J−TKθJ−1δd(13)

As a consequence, the 6-dimensional robot end-eﬀector small displacement screw δdcan be expressed as:

δd=JKθ

−1JTw(14)

Let the joint compliances1be the components of the 6-dimensional vector x, namely:

x=h1/kθ11/kθ21/kθ31/kθ41/kθ51/kθ6iT

(15)

1The compliance stands for the inverse of the stiﬀness

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From Eq. (14), it turns out that

δd=

P6

j=1 xjJ1jPi=6

i=1 Jij wi

...

...

...

...

P6

j=1 xjJ6jPi=6

i=1 Jij wi

(16)

where xjis the jth component of vector x, i.e., xj= 1/kθj,j= 1,...,6.

By isolating the components of vector xin Eq. (16), the joint compliances can be expressed with respect to

the displacements of the robot end-eﬀector as follows:

A x =δd(17)

where Ais a 6 ×6 as follows:

A=

J11P6

i=1 Ji1wiJ16P6

i=1 Ji6wi

... ... ...

... ... ...

... ... ...

... ... ...

J61P6

i=1 Ji1wiJ66P6

i=1 Ji6wi

(18)

Therefore, a 6-dimensional wrench vector, a 6-dimensional EE displacement vector and a 6 ×6Amatrix

are associated with each test. When only one test is considered, Ais a 6-dimensional square matrix. If it is

nonsingular, then Eq. (17) has a unique solution, namely:

x=A−1δd(19)

When several tests are considered, the equation system (17) becomes overdetermined. Assuming that ntests

are considered, n > 1, matrix Abecomes 6n×6. Because matrix Ais no longer square, the joint compliance

vector xcannot be calculated using Eq. (19). Because the number of equations is higher than the number of

unknows, it is usually not possible to ﬁnd a vector xthat veriﬁes all 6nequations. Accordingly, we look for the

vector xthat minimizes the following error:

minimize E(x)≡1

2kAx −δdk2

2(20)

over x

The value of xthat minimizes the Euclidean norm of the approximation error of the system is

x0=ATA−1ATδd=AIδd(21)

where AIis the generalized inverse of A, which is also known as the left Moore-Penrose generalized inverse of

A. The Matlab function ”pinv” has been used to compute this matrix.

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5 Robustness of the Joint Stiﬀness Identiﬁcation Method with Re-

gard to Measurement Noise

Alici and Shirinzadeh [24] introduced a method to identify the stiﬀness values of the ﬁrst three joints of a

six-revolute robot by measuring only translational displacements of its end-eﬀector. Therefore, they did not

considered the coupling between the rotational and translational Cartesian motions. On the contrary, our

method considers both rotational and translational displacements of the end-eﬀector, and the stiﬀness values

of the six actuated joints can be identiﬁed as explained in the previous section. Finally, a graphical user

interface (GUI), which is shown in Fig. 6, was developed to test our method and to analyze the sensitivity of

the results to measurement noise.

Figure 6: Graphical user interface — robustness of the joint stiﬀness identiﬁcation method

5.1 Measurement Noise

This graphical user interface requires the knowledge of the geometric and stiﬀness models of the robot under

study. The displacements (translations and rotations) of the robot end-eﬀector are assessed for a given wrench

and given joint stiﬀness values. Then, assuming that the joint stiﬀness values are no longer known, the GUI

aims at evaluating them from the end-eﬀector displacements.

The GUI returns the exact joint stiﬀness values when a signiﬁcant wrench is applied on the robot end-

eﬀector and when the robot does not reach any singularity. To be more realistic, the errors are considered at

the identiﬁcation stage. The diﬀerent types of errors, which are given in Table 3, are considered and assumed

to follow a normal distribution.

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Table 3: Sources of errors

Source of errors Magnitude (±)

Position measuring system (tracker resolution) 0.03 mm

Orientation measuring system (tracker resolution) 0.0003 rad

Geometric parameters errors (geometric calibration) 0.03 mm

Force measurement (force sensor) 0.25 N

Torque measurement (torque sensor) 0.0125 Nm

Joint angles (encoder resolution) 0.0001 rad

5.2 Optimal Number of Experiments

From the equation system (17), it is apparent that the higher the number of tests, the higher the degree of

constraint of the equation system, the more accurate the solution.

Let us assume that the actual joint stiﬀness values of the robot are those given in Table 2, and the wrench

applied on the robot end-eﬀector is:

wef f =

0 N

2200 N

2200 N

200 Nm

200 Nm

0 Nm

(22)

Figure 7 illustrates the mean and standard deviation of the joint stiﬀness values obtained by using the proposed

identiﬁcation method and for diﬀerent numbers of tests. From Eq. (17), matrix Ahas the size of 6n×6 for n

tests.

We can see that the higher n, the lower the standard deviation of the joint stiﬀness values, i.e., the more

accurate the evaluation of the joint stiﬀness values. Obviously, the higher the number of tests, the more expensive

the identiﬁcation of the joint stiﬀness values. Therefore, the user must compromise between identiﬁcation

accuracy and identiﬁcation cost. Figure 7 shows that ﬁve tests are a good compromise.

5.3 Optimal Robot Conﬁgurations

The joint stiﬀness identiﬁcation method is based on Eq. (21), which requires matrix Ato be invertible. To this

end, the robot conﬁguration must be chosen such that κF(JN)−1, where JNis deﬁned in Eq.(4), is as high as

possible.

Figure 8 depicts three zones in the joint space that were obtained from Figs. 3(a) and 3(b). The sensitivity

of the solution of Eq. (21) to errors is at its minimum in the light grey zones. On the contrary, this sensitivity

reaches its maximum in the black zones. Therefore, Table 4 gives θ2and θ3ranges associated with good robot

conﬁgurations for joint stiﬀness identiﬁcation.

Furthermore, matrix KCshould be negligible for all robot conﬁgurations used for the joint stiﬀness identi-

ﬁcation. Thus, the light grey zones shown in Fig. 8 are correct according to Figs. 5(a) and 5(b).

The GUI shown in Fig. 6 allows the user to analyze the sensitivity of the results to the errors described in

Table 3. The broken lines correspond to the joint stiﬀness values obtained with diﬀerent sets of errors. We

can notice that the sixth joint stiﬀness value, i.e., kθ6, is the most sensitive to errors because the distance

between the broken lines is a maximum for this joint stiﬀness value. It means that the identiﬁcation of kθ6is

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C. Lecerf, S. Caro, S. Garnier and B. Furet 13

1 test

1 test

2 tests

2 tests

3 tests

3 tests

5 tests

5 tests

10 tests

10 tests

50 tests

50 tests

kθ1

kθ2

kθ3

kθ4

kθ5

kθ6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

2

4

6

8

1

12

14

16

Asize: 6×612 ×618 ×630 ×660 ×6300 ×6

×105[Nm/rad]

×105[Nm/rad] Standard deviation

Mean

Figure 7: Mean and standard deviation of the joint stiﬀness values as a function of the number of tests

more diﬃcult than the identiﬁcation of its counterparts. This result makes sense because the initial value of

kθ6given in Table 2 is higher than those of kθ4and kθ5. The ﬁrst three joint stiﬀness values are determined

correctly because the ﬁrst three components of weff given in Eq. (22), i.e., the forces applied on the robot

end-eﬀector along the x0,y0and z0axes, are high enough to cause a signiﬁcant translational displacement of

the end-eﬀector.

The user can easily check the sensitivity of the results to variations in each parameter by using the scrollbars.

The lower the torque applied on the robot end-eﬀector, the lower the standard deviations of kθi,i= 1,...,6.

Finally, the user must pay attention to the measurement system used to assess the end-eﬀector orientation

because the corresponding results are very sensitive to measurement noise.

6 Conclusions

The subject of this paper was to develop a new methodology for the joint stiﬀness identiﬁcation of six-revolute

industrial serial robots. A robust procedure for joint stiﬀness identiﬁcation was proposed, and the Kuka KR240-

2 robot was used as an illustrative example throughout the paper. First, the robot kinematic model was

obtained to determine the optimal robot conﬁgurations according to the condition number of its kinematic

Jacobian matrix. Then, its stiﬀness model was developed through its Cartesian stiﬀness matrix KXand its

complementary stiﬀness matrix KC. The links of the robot were assumed to be much stiﬀer than the joints.

Because the stiﬀness model and, as a consequence, joint stiﬀness identiﬁcation become simple when KCis

negligible with respect to Kθ, we determined the robot conﬁgurations that minimized the inﬂuence of KCon

KX. The robustness of the method was also studied with a sensitivity analysis of the results to measurement

hal-00632989, version 1 - 17 Oct 2011

C. Lecerf, S. Caro, S. Garnier and B. Furet 14

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

θ2[deg]

θ3[deg]

050 100

-200

-150

-100

-50

0

Figure 8: Selection of robot conﬁg-

urations in the joint space

Table 4: Optimal robot conﬁgurations:

ranges of θ2and θ3

Zone θ2θ3

1 0◦to 110◦−245◦to −170◦

2 0◦to 25◦0◦to 29◦

3 100◦to 146◦0◦to 29◦

errors and to the number of experimental tests. Moreover, the proposed methodology will be improved in a

future work that will identify the link stiﬀness in addition to the joint stiﬀness of industrial robots. Finally,

experimental tests will be performed to validate the proposed method.

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