Joint Stiffness Identification of Industrial Serial Robots

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.

Full text

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 nδrxKC−δrxKC,δryKC−δryKC,δrzKC−δrzKCo(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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C. Lecerf, S. Caro, S. Garnier and B. Furet 10
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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C. Lecerf, S. Caro, S. Garnier and B. Furet 11
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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C. Lecerf, S. Caro, S. Garnier and B. Furet 12
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
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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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