Received May 31, 2020, accepted June 4, 2020, date of publication June 9, 2020, date of current version June 19, 2020.
Digital Object Identifier 10.1109/ACCESS.2020.3000997
Dynamic Identification of the KUKA LBR iiwa
Robot With Retrieval of Physical Parameters
Using Global Optimization
TIAN XU 1,2, JIZHUANG FAN1, (Member, IEEE), YIWEN CHEN2, XIANYAO NG2,
MARCELO H. ANG, JR.2, (Senior Member, IEEE), QIANQIAN FANG1,2,
YANHE ZHU 1, (Member, IEEE), AND JIE ZHAO 1, (Member, IEEE)
1State Key Laboratory of Robotics and System, School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150080, China
2Advanced Robotics Centre, Department of Mechanical Engineering, National University of Singapore, Singapore 117608
Corresponding authors: Jizhuang Fan (fanjizhuang@hit.edu.cn), Marcelo H. Ang, Jr. (mpeangh@nus.edu.sg), Yanhe Zhu
(yhzhu@hit.edu.cn), and Jie Zhao (jzhao@hit.edu.cn)
This work was supported in part by the National Key Research and Development Plan under Grant 2018YFB1308501, in part by the Major
Research Plan of the National Natural Science Foundation of China under Grant 91648201, in part by the China Scholarship Council under
Grant 201906120149, and in part by the Agency for Science, Technology and Research (A*STAR), Singapore, under Grant A18A2b0046
[Human Robot Collaborative Artificial Intelligence (AI) for Advanced Manufacturing and Engineering (AME)].
ABSTRACT This paper focuses on the problem of extracting the physical dynamic parameters which are
fundamental for computing the positive-definite link mass matrix. To solve this problem, a minimal set
of dynamic parameters were firstly identified by the standard least squares method. In order to simplify
the dynamics model, a new set of essential dynamic parameters were calculated by eliminating the poorly
identified parameters with an iterative approach. Based on these dynamic parameters with better identifica-
tion quality, a universally global optimization framework was proposed here to retrieve the set of physical
dynamic parameters of a serial robot, in which parameter bounds, linear and nonlinear constraints with
physical consistency can be easily considered, such as the triangle inequality of the link inertia tensors,
the total link mass limitations, other user-defined constraints and so on. Finally, validation experiments
were conducted on the KUKA LBR iiwa 14 R820 robot. The results show that the proposed optimization
framework is effective, and the identified dynamic parameters can predict the robot joint torques accurately
for the validation trajectories.
INDEX TERMS Dynamic parameter identification, physical parameters, nonlinear global optimization,
KUKA LBR iiwa robot.
I. INTRODUCTION
Accurate dynamic parameters [1] of a robotic manipula-
tor is important in many robotic applications [2]. Dynamic
parameters are critical in the design of control laws based on
dynamics model [3], in simulating the robot motion in some
software [4] or in implementing some human-robot inter-
action algorithms, such as collision detection and reaction
control [5], impedance control [6] and so on.
There are three main methods for obtaining the dynamic
parameters of a robotic manipulator: physical experiments,
computer aided design (CAD) techniques and dynamic
The associate editor coordinating the review of this manuscript and
approving it for publication was Christopher Kitts .
parameter identification [7]. In physical experiment method,
each link of the robot needs to be isolated and the dynamic
parameters are obtained by using special measurement
devices. For example, the mass of links can be weighted
directly, the position vector of the center of mass can be
estimated by determining counterbalanced points of the link
and the diagonal elements of the inertia tensor can be obtained
by pendular motions [1], [7]. However, the accuracy of this
method depends on the measurement devices and the pro-
cedure is very tedious; hence, it should be conducted by
the manufacturer before assembling the robot [8]. In the
CAD method, the dynamic parameters of a robot can be
obtained by analysing the 3-dimensional (3D) model in the
CAD software. However, 3D models are usually not provided
108018 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ VOLUME 8, 2020
T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
FIGURE 1. The general procedure of an identification method for
obtaining dynamic parameters of the robot.
by the manufacturer and the parameters estimated with this
method are inaccurate [7]. In addition, the friction parameters
cannot be estimated by the first two methods. In the dynamic
parameter identification method, the dynamic parameters are
estimated by minimizing the errors between a function of
the robot variables and an identification model. This method
has been extensively used [9]. Compared to the above two
methods, dynamic parameter identification can obtain more
accurate dynamic parameter estimates. In general, an identi-
fication procedure (Figure 1) usually consists of modeling,
experiment design, data acquisition and signal processing,
parameters estimation and model validation [10].
This paper presents a global nonlinear optimization frame-
work for identifying the physical parameters of a KUKA LBR
iiwa 14 R820, which can be generalized to any serial manipu-
lator. In the nonlinear problem, the parameters bounds and the
physical constraints can be easily added. To reduce the error
accumulation, the objective function was calculated by the
errors between the measured joint torque and the estimated
torque. In addition, instead of using the base parameters to
calculate the objective function, the essential parameters with
better identified quality were used.
Atkeson et al. [11] first derived the equations of lin-
earizing the inverse dynamics model of a robotic arm and
identified its minimal set of inertia parameters, the so-called
base parameters which are a set of independent identifiable
parameters, by the standard least squares method. However,
the universal derivation for analytical form of the base param-
eters set was not given in that work. To solve this problem,
Gautier and Khalil [12] and Khalil and Bennis [13] proposed
a direct method for determining and identifying the base
parameters of serial manipulator, namely, regrouping param-
eters by means of closed-form functions of the geometric
parameters of the robot. In these two works above, the applied
identification methods are all based on standard least squares
method, which is sensitive to the singularity of observation
matrix. To avoid the accumulation of estimation errors and
allow the calculation of the confidence intervals, the weighted
least squares method, adding a weighted matrix multiplier,
was proposed in [14]. However, a disadvantage of both the
standard least squares method and the weighted least squares
method is sensitivity to measurement noise. To overcome this
problem, Swevers et al. [15] proposed an optimal excitation
trajectories method of using finite Fourier series. In addi-
tion, the maximum-likelihood estimation method [16] was
also formulated because of its asymptotically unbiased and
efficient property.
Based on these previously excellent works, robotics
researchers nowadays continue to explore and expand the
methods of dynamic identification of the robot. A reverse
engineering approach for identifying the dynamics model of
KUKA LBR iiwa robot was presented in [17]. It can calcu-
late the dynamic parameters inversely by using the known
mass matrix and gravity vector information in KUKA Fast
Research Interface (FRI) [18]. However, this method is obvi-
ously not a universal approach. For instance, not all robot
companies can provide the mass matrix and gravity vector
information to users in advance. Recently, a lot of parameter
identification methods for the robot dynamics have also been
proposed, such as the convex programming approach [19],
adaptive control algorithm [20]–[22], extended Kalman filter
method [23], neural networks method [24]–[26] and so on.
However, the common problem for these methods are that
the identification accuracy can not be guaranteed. To fur-
ther simplify the dynamics model of the robot and improve
the quality of identified parameters, a new set of dynamic
parameters, namely essential parameters, was identified by
eliminating the poorly identified base parameters in an iter-
ative way [27], [28]. The identification of the base parame-
ters and essential parameters are usually sufficient for many
robotic applications, however, the retrieval of a set of feasi-
ble dynamic parameters, physical parameters [29], is also
needed. This is the case, for example, in conducting the
feedback linearization control laws under hard real-time con-
straints by the Newton-Euler method [30] or calculating the
mass matrix of links used for robot collision detection. In this
case, the symmetrical and positive-definite property of the
mass matrix must be satisfied. However, the mass matrix
calculated by the identified base parameters and essential
parameters can only satisfy the symmetrical property but the
positive-definite property can not be guaranteed. Thanks to
the physical parameters, the positive-definite property of the
mass matrix can be ensured by using them.
To obtain the physical parameters, the physical consis-
tency of the identified parameters was considered in [31].
Then, the work [32] considered a nonlinear optimization
problem to guarantee the physical feasibility of the identi-
fied parameters. However, the constraints considered in that
work, such as only considering the positive link mass and
the positive diagonal elements of the inertia tensor, were
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
too simple. To solve this problem, some researchers pro-
posed the framework of linear matrix inequalities to obtain
the physically consistent set of parameters by semi-definite
programming algorithm [33]. Afterwards, this method was
enhanced by considering the triangle inequality of the inertia
tensors [34], [35], which was originally presented in [36].
However, these methods require expressing constraints as
linear matrix inequalities [30]. Recently, a local optimiza-
tion identification framework has been proposed to identify
the physical parameters of the KUKA iiwa robot dynam-
ics in [37] by considering nonlinear constraints. The base
parameters and the physical parameters of the robot were
eventually identified and given. But the drawback of this
method is that the optimal solution was based on a local opti-
mization. In addition, the objective function of the nonlinear
optimization was computed based on the base parameters,
not the essential parameters which are with better identifi-
cation quality, in that work. Gaz et al. [30] proposed a global
nonlinear optimization framework, based on the inverse engi-
neering method presented in their previous work [38], to
identify the physical parameters of the iiwa robot dynamics.
However, the objective function described in that paper was
the errors between the identified based parameters and the
one computed by the physical parameters. Compared with
the objective function calculated by the errors between the
measured joint torque and the torque estimated by the base
parameters, the method in [30] will introduce the accumu-
lation errors because the variables for calculating the errors
were all obtained by estimation. In addition, no previous
work can provide the identified results of all the three kinds
of dynamic parameters simultaneously for a robot, namely
the base parameters, essential parameters, and the physical
parameters.
The main contributions of this work are described as
follows. (1) A universally global optimization framework for
identifying the physical parameters of serial manipulators
is presented, that account for the parameters bounds, linear
and nonlinear constraints. (2) The essential parameters with
better identification quality were first used to calculate the
objective errors instead of the base parameters. (3) The base
parameters, essential parameters and the physical parameters
of KUKA LBR iiwa 14 R820 robot were simultaneously
obtained by the method presented in this paper.
The paper is organized as follows. The identifica-
tion of essential parameters was shown in Section II.
Section III describes the calculation of the physical parame-
ters. In Section IV, the experiment results were given. Finally,
discussion and conclusion were given in Section V.
II. ESSENTIAL PARAMETERS IDENTIFICATION
A. INVERSE DYNAMICS IDENTIFICATION MODEL
By considering the rigid link dynamics with Lagrange
method or Newton-Euler algorithm [39], the iiwa robot
motion equations can be described as
τJ=M(q)¨q+C(q,˙q)˙q+g(q)+Fv˙q+Fcsign(˙q),(1)
where q,˙q,¨q∈ <n×1, with nis the degrees of freedom of
the robot, are the vectors of joint position, velocities and
acceleration, respectively. M(q)∈ <n×nis the symmetric and
positive-definite inertia matrix of links, C(q,˙q)∈ <n×nis the
centrifugal and Coriolis matrix of links, and g(q)∈ <n×1is
the gravity vector of links. Fvand Fc∈ <n×nare the diagonal
matrices of the viscous and Coulomb friction parameters.
τJ∈ <n×1is the joint torque readings of the torque sensors
built in the robot.
To derive the linear model below of dynamics equation
for dynamic parameters identification, the modified Newton-
Euler method [1] is used in our case.
τJ=Y(q,˙q,¨q)π, (2)
where Y(q,˙q,¨q)∈ <n×12nis the observation matrix which
depends only on the motion data, π=[πT
1, . . . , π T
n]T∈
<12n×1is the dynamic parameters of links, where for each
link i,i=1,...,n, the parameter vector πiis defined as
πi=[Mi,MXi,MYi,MZi,XXi,XYi,
XZi,YYi,YZi,ZZi,FVi,FCi]T.(3)
Because some inertial parameters are completely uniden-
tifiable and some others can only be identified in linear
combinations, the set of parameters πto be identified can be
reduced to a minimal set of parameters, which are referred
to as base parameters πb∈ <b×1,bis the base parameters
number and it is 57 in this paper. The regrouping relationships
between πand πbcan be found in [12]. Therefore, (2) can be
modified to
τJ=Yb(q,˙q,¨q)πb,(4)
where Yb(q,˙q,¨q)∈ <n×bis the observation matrix with
full rank after deleting the linearly correlated columns in
Y(q,˙q,¨q).
In order to identify the dynamic model, the measured data
of joint positions qand joint torques τJare needed. The
joint velocities ˙qand accelerations ¨qare calculated from the
measured joint positions q. After Kdata samplings of the
robot during the experiments, one can apply the standard
least-squares method to solve the base parameters by
τJ(1)
.
.
.
τJ(K)
| {z }
τJ
=
Yb(q(1),˙q(1),¨q(1))
.
.
.
Yb(q(K),˙q(K),¨q(K))
| {z }
Yb
πb,(5)
πb=(YT
bYb)−1YT
bτJ,(6)
where Yb∈ <Kn×band τJ∈ <Kn×1are the stacked matrix of
Yband the stacked vector of measured joint torque vectors τJ,
respectively.
In order to reduce high frequency noise, the measured
data of joint position and joint torque is filtered. Details
about the data acquisition and signal processing can be found
in Section II-Cand Section IV-B. In order to understand
the identified quality of the individual parameter in (6),
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
an indicator, which is called the relative standard deviations
(RSD) [37], is adopted. The calculation process of the RSD
for the base parameters follows the equations below.
σπj,r%=100(σπj/πj),
σπj=√Cπ(j,j),
Cπ=σ2
ρ(YT
bYb)−1,
σ2
ρ= kτJ−Ybπbk2/(Kn −b),
(7)
where σπj,r%is the RSD of the j-th identified parameter in
πb,σπjis the standard deviation of the estimation error of
the parameter j,j=1,...,b.Cπis the variance-covariance
matrix of the estimation error, and σ2
ρis the variance of the
parameter estimation.
If the RSD of an identified parameter is very large and
its value is close to zero, then this parameter can be con-
sidered as a poorly identifiable parameter because its con-
tribution to the dynamics model is negligible. Therefore, this
parameter can be removed in order to simplify the dynamics
model and a new set of parameters can be defined, which is
called as essential parameters denoted by πe. The essential
parameters are calculated using an iterative procedure starting
from the base parameters estimation. At each iteration the
base parameter which has the maximum RSD is cancelled.
Then, a new set of parameters with new RSD is repetitively
identified. The procedure ends when all parameters with
absolute values less than 0.01 and corresponding RSD values
larger than 40% are removed. It should be noted that the
values 0.01 and 40% are chosen that work best for this
problem. The readers should consider the trade-off between
torque prediction accuracy by essential parameters and bet-
ter identification quality of essential parameters for their
own case to determine the values. The identified essential
parameters will be used to calculate the objective function
of the proposed nonlinear optimization problem of extract-
ing the physical parameters in this paper. More details and
results about the identified essential parameters are presented
in Section IV.
B. OPTIMAL EXCITATION TRAJECTORIES
In order to improve the accuracy of the least-squares solu-
tion of (6), an optimal excitation trajectory is needed to
obtain a well-conditioned observation matrix. In this paper,
the classical parameterization Fourier series [15] are selected
as optimal excitation trajectories and the optimal criterion
to be minimized is the condition number of the observation
matrix. The optimal excitation trajectory equation and the
criterion rule with its constraints are shown as,
qi(t)=
L
X
l=1ai,l
ωflsin(ωflt)−bi,l
ωflcos(ωflt),i=1,...,n,
(8)
min
ai,l,bi,l
cond(Yb),∀i= {1,...,n},∀l= {1,...,L},(9)
with constraints
XL
l=1
ai,l
l,XL
l=1
bi,l
l,XL
l=1l·ai,lT
=0,
XL
l=1
1
lqa2
i,l+b2
i,l≤ωfqi,max ,
XL
l=1qa2
i,l+b2
i,l≤ ˙qi,max ,
(10)
where Lis the number of sine and cosine terms, ai,land
bi,lare optimized trajectory coefficients, the fundamental
frequency is ωf
2π. The function cond(.) is used to solve the
condition number of observation matrix Yb. The first con-
straint in (10) is to ensure the zero initial joint positions,
velocities and accelerations of optimal excitation trajecto-
ries. The second one and the third one are to make the
optimal trajectories satisfy the joint positions limits and
joint velocities limits of KUKA LBR iiwa robot. More
details and results about the optimal trajectories are shown
in Section IV.
C. DATA ACQUISITION AND SIGNAL PROCESSING
The joint position and joint torque data are collected from the
sensors in iiwa robot when the optimal excitation trajectories
are run. Unfortunately, the joint torque data contains mea-
surement noise. In addition, it is necessary to select an appro-
priate method to process the joint velocity and acceleration
estimates used for dynamic parameters identification based
on the measured joint position. Therefore, it is necessary to
conduct signal processing to clean up the measured data,
improve the signal-to-noise ratio of joint torque and joint
position measurements and have an exact calculation of joint
velocity and acceleration estimates.
Since the measured data are periodic, the signal-to-noise
ratio can be improved by data averaging without using a
lowpass noise filter. Averaging filter improves the quality
of the data with the square root of the number of measured
periods. The averaging of a signal xconsisting of Mperiods
of Ksamples is given by
x(k)=1
M
M
X
m=1
xm(k),(11)
where xm(k) indicates the kth sample within the mth period
and x(k) denotes the average of x. For robot dynamic identifi-
cation, the signal xcorresponds to the measured joint torque
signal and joint position signal.
Calculation of the observation matrix requires the esti-
mates of the joint velocity ˙qand joint acceleration ¨q.
Although the common numerical differentiation method and
a low-pass filter can help to obtain the joint velocity and
joint acceleration with less noise in off-line case, however,
to obtain good filtering results, the filter coefficients of
the low-pass filter must be tuned very carefully. The tun-
ing process is time-consuming and the accuracy of esti-
mated joint velocity and acceleration would be reduced if
the filter coefficients were not selected well. To avoid this,
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
a frequency-domain differentiation method originally pro-
posed in [10] is introduced here to estimate the joint velocity
and acceleration. Firstly, the averaged joint position data q(t)
are transformed to the frequency domain data Q(k) by using
the discrete Fourier transform (DFT). Next, a rectangular
window multiplier is applied, where the spectrum is set to
zero at all but the selected frequencies, changes the data
to Qf(k). Afterwards, the resulting spectrum is then multi-
plied by the continuous-time frequency-domain representa-
tion of differentiators to velocity and acceleration by jω(k)
and −ω(k)2, respectively, at the selected frequency, where
ω(k)=2πkfs/K,Kis the number of samples of the signal
in one sampled period, kis the number of the kth sample
and fsis the sample frequency. Finally, a transformation
back into the time domain using the inverse discrete Fourier
transform (IDFT) yields the estimates of the first and second
time derivative of the joint position, that are the velocity
and acceleration. The velocity signal and acceleration signal
are almost free of noise, that is, noise is removed from all
frequencies except the selected ones. The measured averaged
joint torques τJare filtered by the low-pass Butterworth
filter.
III. RETRIEVAL OF PHYSICAL PARAMETERS
The physical parameters are suitable combinations of geo-
metric and inertial data of the robot bodies, which specify the
mass, the position of the center of mass, and the symmetric
inertia matrix for each robot link. For link i, the physical
parameters are defined as
µi=[mi,rCix,rCiy,rCiz,Iixx ,Iixy,
Iixz,Iiyy ,Iiyz,Iizz ,fvi,fci]T,(12)
where miis the mass of link i, and rCix,rCiy,rCizTis the vec-
tor from the center of the i-th link frame to the center of mass
of link i.fvi and fci are the viscous and Coulomb friction coef-
ficients of link i, respectively. Iixx ,Iixy ,Iixz,Iiyy ,Iiyz,IizzT
form the link iinertia tensor matrix ICiwith respect to the
center of mass of link i
ICi=
Iixx Iixy Iixz
Iixy Iiyy Iiyz
Iixz Iiyz Iizz
.(13)
The relationship between the dynamic parameters πiin (3)
and the physical parameters µiin (12) are shown below. The
relationship of mass of link ibetween (3) and (12) is Mi=mi,
the relationship of product of the mass and centroid vector
is MSi=mirCi, where MSi=[MXi,MYi,MZi]T,rCi=
[rCix,rCiy,rCiz]T, and the relationship between inertia tensor
Iirelated to the origin of the i-th link frame and ICirelated
to the center of mass of link iis Ii=ICi+mi(rT
CirCiE−
rCirT
Ci) according to the parallel axis theorem, where Ii
is formed by [XXi,XYi,XZi,YYi,YZi,ZZi] in (3), Eis the
3×3 identity matrix.
Mi=mi,XXi=Iixx +mi·r2
Ciy+r2
Ciz,
MXi=mi·rCix,XYi=Iixy −mi·rCix·rCiy,
MYi=mi·rCiy,XZi=Iixz −mi·rCix·rCiz,
MZi=mi·rCiz,YYi=Iiyy +mir2
Cix+r2
Ciz,
FVi=fvi,YZi=Iiyz −mi·rCiy·rCiz,
FCi=fci,ZZi=Iizz +mi·r2
Cix+r2
Ciy.
(14)
In fact, only the estimated dynamic parameters ˆπare not
sufficient and an estimation of the physical parameters ˆµ
is needed at some time. For example, the inertia matrix
M(q) is needed for calculating the residual torque errors
during robot collision detection. The physical parameters are
needed because the inertia matrix computed by the dynamic
parameters by Newton-Euler method can not satisfy the prop-
erty of positive-definite matrix. To obtain consistent physical
parameters, a global optimization method and framework
with constraints guaranteeing physical feasibility, i.e., posi-
tive masses and positive definite inertia tensors, is proposed
in this paper. Instead of using the identified base parameters,
the objective function f(µ) for the constrained nonlinear opti-
mization problem, namely the modulus of predicted torque
errors vector between the averaged measurement torque and
the estimated torque by the identified essential parameters,
is defined as
min
µf(µ)=w
wτJ−Yeπe(µ)w
w,(15)
where µ=µT
1, . . . , µT
nTis physical parameters set of
robot. It should be noted that all the measured data in the
right hand side of (15) have been averaged by (11). Yeis
the corresponding stacked observation matrix for the essential
parameters πe, which can be obtained by eliminating the
column of Ybwith respect to the ignored poorly identifi-
cation parameters in πb.πe(µ) is the nonlinear relationship
between πeand µ. Firstly, the relationship π(µ) can be
established by (14), then the regrouping relationship πb(π)
can be established by the method described in [12]. Finally,
the relationship πe(πb) can be established by eliminating the
poorly identified parameters in πb.
To obtain a feasible solution, the lower bounds LB and
upper bounds UB should be considered, which are referred
to bounds described in [29].
LB ≤µ≤UB (16)
In order to guarantee a physically meaningful result, con-
straints on the physical parameters µare introduced. Physical
constraints regard the mass of each link to be positive, namely
mi>0, which can be implemented by the lower bounds LB.
In addition, the total sum of the link masses must be in a given
range, which is described as
mrob,min ≤
n
X
i=1
mi≤mrob,max.(17)
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
Besides, the inertia tensor of each link should be positive
definite, which impose the triangular inequality of the diag-
onal elements of the inertia tensor matrix [37] and prevent
big differences between them. The constraints of the inertia
tensor are described as
Iixx ≤Iiyy +Iizz,
Iiyy ≤Iixx +Iizz,
Iizz ≤Iixx +Iiyy,
max Iixx ,Iiyy,Iizz ≤100 ·min Iixx ,Iiyy,Iizz ,
3Ijzz,Ikyy T≤min Ijxx ,Ijyy,min (Ikxx ,Ikzz )T,
max |Iixy|,|Iixz |,|Iiyz|≤0.1·min Iixx ,Iiyy ,Iizz,
∀i= {1,...,7},∀j= {1,3,5},∀k= {2,4}.
(18)
In (18), the biggest diagonal term of the inertia can not
exceed to the smallest one too much. Moreover, the smallest
diagonal term of the inertia of links one to five is forced to
correspond to the axis parallel to the link length. It should
be noted that the fifth constraint in (18) is only used to make
the identified physical parameters compatible with the iiwa
robot model in Gazebo in our case. In fact, it is not necessary
in the proposed framework. If the reader’s robot is not an
iiwa robot, this constraint can be ignored. Finally, the non-
diagonal elements of the inertia tensor matrix should be small
compared to the diagonal ones.
TABLE 1. The proposed physical parameters retrieval algorithm.
In order to describe the retrieval algorithm of physical
parameters more clearly, the pseudo-code of the global opti-
mization framework proposed in this paper for the opti-
mization problem is shown in Table 1. The first step of the
algorithm is to load the averaged and stacked vector τJand
Ye, which are collected and calculated by the sensors reading
of iiwa robot. Then, the constrained nonlinear optimization
problem is solved by κtimes, at a given step k=1, . . . , κ,
in a loop. At every step in the loop, the initial value for
the optimization iteration is randomly selected and updated
between the lower and the upper bounds using a uniform
distribution. Afterwards, the nonlinear optimization problem
is solved by a global optimization algorithm with considering
the lower and the upper bounds and the constraints mentioned
above. In this paper, the applied global optimization method
is the multi-start algorithm because it is a global optimiza-
tion method and it contributes to obtaining better optimal
solution in our case compared with other global optimization
algorithms in MATLAB, such as GlobalSearch.Multi-start
approach can find a global solution or multiple minima solu-
tions, which starts a local solver (such as fmincon) from
multiple start points. The iteration process will end when the
termination conditions are satisfied, such as the maximum
iteration numbers. The proposed parameter retrieval frame-
work makes contribution to finding a global optimization
solution compared with those local optimization methods.
In addition, it is easy to introduce some constraints, such
as the triangle inequality of the inertia tensors, eventually
ensuring the algorithm to obtain a meaningful solution. More
experiment results can be found in next section.
FIGURE 2. The MDH link frames of the KUKA iiwa robot.
TABLE 2. The kinematic parameters of the KUKA iiwa robot.
IV. EXPERIMENT RESULTS AND VALIDATION
A. KUKA LBR IIWA 14 R820 ROBOT
The KUKA LBR iiwa 14 R820 robot is equipped with n=7
revolute joints and a torque sensor is mounted in each joint
after each of the gearboxes. Moreover, the robot is equipped
with joint position encoders. Figure 2 and Table 2 show the
iiwa robot and its kinematic parameters according to the mod-
ified Denavit-Hartenberg (MDH) convention, respectively.
It is possible to control the iiwa robot by KUKA Sunrise
Toolbox [40], iiwa stack [41] or other software packages.
In this paper, the authors control the robot through the iiwa
stack, which can support Robot Operating System (ROS)
programming and help the users to control the robot easily.
This software package is able to provide functions to read
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
the sensors data of joint position qand joint torque τJ. These
data will be of fundamental importance for the identification
of the dynamic model of the robot. Moreover, it supports lots
of control modes, such as joint position control mode,joint
velocity control mode, and impedance control mode in both
joint space and Cartesian space, which can be selected by the
users based on their requirements.
B. ROBOT EXCITATION, DATA ACQUISITION AND
SIGNAL PROCESSING
In the parametrized Fourier series (8), the number of sine and
cosine terms is set to L=5 and the fundamental frequency
to ff=0.05 Hz here. Therefore, the duration of the trajectory
is 20 s. The joint position and joint velocity limits of (10)
used in this paper are listed in Table 3. By considering the
optimization problem described by (8)∼(10), the optimal
excitation trajectories are designed and shown in Figure 3,
which gives a condition number of cond(Yb)=156.3.
In this paper, our iiwa robot is fixed on a wooden desk,
therefore, large motion velocity of the robot will make the
desk vibrate and deteriorate the collected data. So, for a
trade-off between good data measurement and low condition
number, the velocity limits are reduced compared with the
ones in iiwa manual, which makes the condition number more
than 100.
TABLE 3. Joint position and velocity limits for trajectory optimization.
FIGURE 3. Optimal excitation trajectories for KUKA iiwa robot.
The optimal joint position commands shown in Figure 3 are
sent for the seven joints of robot in the joint position control
mode by iiwa stack package. The position and torque of each
joint are measured with a sampling frequency of fs=500 Hz,
M=20 periods of K=10000 samples in one period to
construct the observation matrix Yb. Then the averaging filter
FIGURE 4. The computed joint acceleration signals comparison between
the acceleration filtered by the central differentation method (green
lines) and the acceleration filtered by the frequency filtering method (red
dashed lines). The blue lines denote the desired acceleration command of
the excitation trajectories.
described in (11) is used for the measured position and torque.
From the averaged position data, the joint velocity and joint
acceleration are calculated by the frequency filtering method
described in the end of Section II. In addition, the computed
joint acceleration of the robot for the excitation trajectories is
shown in Figure 4 by using the frequency filtering method and
the central differentation method, respectively. The results
show that the obtained acceleration signal by the frequency
filtering method is more clear than the one by the classical
central differentation method. The averaged joint torque is
filtered by a low-pass Butterworth filter with an order of 4
and a cutoff frequency of 6 Hz.
C. ESTIMATION OF ESSENTIAL PARAMETERS
By using the averaged and filtered joint position and torque
and the estimated joint velocity and acceleration calculated
above, the observation matrix Ybcan be computed and
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
TABLE 4. Identified base parameters ˆ
πband RSD σ.
TABLE 5. Identified essential parameters ˆ
πeand RSD σ.
the least-squares optimization problem (6) is solved. The
results of the identified 57 base parameters ˆπbare given
in Table 4 together with their RSD, which are indicators of
the quality of the individual parameter estimates. However,
from the Table 4, it can be found that the RSD of the four
parameters, XY5,XY6,MX6,FV5, are large and the values
of the four parameters are close to zero. Therefore, their
contribution to the dynamics model is negligible and the four
parameters can be removed when calculating the essential
parameters. The calculation procedure has been described at
the end of Section II-A. Finally, the identified 53 essential
parameters ˆπeare given in Table 5 together with their RSD,
which are all no more than 40%. Figure 5 compares the
averaged measured torque for the excitation trajectories with
the predicted torque based on the identified base parameters
and essential parameters. The curves indicate that the joint
torque can be predicted accurately for all joints of iiwa robot
FIGURE 5. Model validation based on comparison between mesaured
torque filtered by a low-pass filter (black lines) and the predicted torque
by the identified essential parameters ˆ
πe(red lines), predicted torque by
the identified base parameters ˆ
πb(blue dashed lines) and predicted
torque by the identified physical parameters ˆ
µ(green dashed lines) for
the excitation trajectories.
except for the seventh joint. One reason for this is that the
mass and absolute range of torque of joint seven are much
smaller compared to the other joints, which makes it hard to
predict the torque of seventh joint accurately.
D. ESTIMATION OF PHYSICAL PARAMETERS
The physical parameters µin (12) of the iiwa robot have
been retrieved by solving the nonlinear optimization problem
presented in Table 1. In order to solve the nonlinear opti-
mization problem, the algorithm multi-start and its related
functions in MATLAB are used. The optimization algorithm
in Table 1 is launched for κ=5 times. In every iteration
loop, the maximum iteration times of the objective function
are set to 100000. The total optimization time is 9854.5 sin
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
TABLE 6. Lower bounds (LB) and upper bounds (UB) of the link mass,
position of center of mass and the friction coefficients of iiwa robot and
the optimal solution ˆ
µobtained in this paper and the one ˆ
µ[37]
reported in [37].
MATLAB on a laptop computer with i5-8250U-1.60Ghz and
16 GB memory. For the bounds of link mass, a non-negative
mass for each link is considered and a common upper limit,
such as 6 kg, is assumed for simplicity. By considering the
light seventh joint with a payload in our case, the upper mass
limit for it is increased to 5 kg. The total sum of the link
masses mrob,min and mrob,max are set to 16 kg and 26 kg,
respectively. In addition, for each link, the center of mass is
simply assumed to be located inside the smallest parallel box
which includes the link geometry [29]. For the bounds of the
elements in the inertia matrices, the mass distribution and the
position of the center of mass for each link is evaluated with
the aid of CAD tools. By assuming the friction coefficients as
non-negative, the upper limits are set to 1 and the lower limits
of viscous friction coefficients are set to 0.1. Trading off
between the desire of strict bounds, so as to limit the feasible
set when searching for a candidate solution, and the need of
larger bounds, not to cut off any potential good candidate,
the used LB and UB are eventually selected in Table 6 and
Table 7. In order to cover the largest surface of f(µ) within
the feasible set, the starting point µ0at each search iteration
is randomly selected by µ0=LB +(UB −LB)u,u∈(0,1).
The final optimization solution ˆµobtained with the pro-
posed optimization framework is reported in Table 6 and
Table 7. The objective function value f(ˆµ) at the optimal
solution is 75.06 N.m. From the table, it can be seen the
solution is in the range of the LB and UB. In addition, the mass
and the elements of inertia matrix in the optimal solution
satisfy the constraints (17) and (18) completely. The predicted
torque estimated by the identified physical parameters ˆµfor
TABLE 7. Lower bounds (LB) and upper bounds (UB) of the elements of
the link inertia matrix Iof iiwa robot and the optimal solution ˆ
µobtained
in this paper and the one ˆ
µ[37] reported in [37].
the excitation trajectories are plotted in Figure 5. Compared
with the measured torque, the physical parameters can also
predict the joint torque accurately like the identified base
and essential parameters. In general, the reported solution
ˆµcannot be qualified as the ‘true’ physical parameters.
Nonetheless, it satisfies the dynamics model equations, and
can be safely adopted, e.g., to compute the inverse dynamics
by means of a Newton-Euler algorithm.
E. IDENTIFIED RESULTS VALIDATION
In order to validate the joint torque prediction accuracy by the
identified base parameters, essential parameters and physical
parameters, three reference trajectories qref ,j(t), j=1,2,3,
are designed and the comparison is done between the mea-
sured joint torque and the predicted torque calculated by the
estimated parameters based on robot tracking the reference
trajectories. The excitation trajectories in Figure 3 is selected
as the first reference trajectory, qref ,1(t), for a direct valida-
tion and the comparison between the measured torque and
predicted torque is shown in Figure 5. The results show the
estimated joint torque by the identified parameters, which
includes base parameters, essential parameters and physi-
cal parameters, can be predicted accurately compared with
the actual measurement joint torque. The second reference
trajectory is designed as a cosine trajectory qref ,2(t)=A
(cos ωt−B), in which A=[π
12 ,π
6,π
12 ,π
6,π
12 ,π
6,π
12 ],
B=0.1 and ω=0.03. The third reference trajectory is
designed with very slow joint velocity. In order to quantify
the errors between the measured torque and predicted torque,
the root mean square (RMS) errors are computed and the
resulting RMS values for the three reference trajectories are
given in Table 8. From the table, it can be found that the
total RMS values of predicted torque errors are approximate
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
TABLE 8. The RMS of predicted torque errors between the measured
torque and estimated torque by the identified parameters for three
reference trajectories.
calculated by the identified parameters for each reference
trajectory, which indicates high precision of the joint torque
prediction by the identified parameters. In addition, the RMS
values in the third reference trajectory are largest and the
ones in the second reference trajectory are smallest. One
reason is that the viscous and Coulomb friction model used
in this paper is not accurate enough. The final reference
trajectory with lower motion velocity and acceleration leads
to more complex joint friction torque, such as inclusion
of the Stribeck effect after motion just starting. However,
the simple viscous and Coulomb friction model used in this
paper cannot model it completely. Therefore, the joint torque
prediction accuracy for the final reference trajectory is worse
than that for the first two reference trajectories. The reason
for the second reference trajectories with smallest RMS is
that the measured torques of all joints except for the seventh
joint in the second reference trajectories are the smallest ones
compared with other two trajectories.
F. COMPARISON WITH OTHER METHODS AND AN
APPLICATION FOR THE IDENTIFIED
PHYSICAL PARAMETERS
In this subsection, the results of comparison of estimated
torque is given in Figure 6 and Table 9 between the proposed
method and the methods in [30] and [37]. The predicted
torques in Figure 6 are calculated by the identified physical
parameters obtained by the three different methods based on
the excitation trajectories data and the same friction model in
our case. From the figure and table, it can be seen that the
torque estimates from the proposed method is better than that
of previous methods in [30] and [37]. For example, the total
RMS value and the objective function value of the nonlinear
optimization in the proposed method are all the smallest one
according to Table 9.
In addition, an application for calculating the momentum
observer used for robot collision detection by the identified
physical parameters is given in Figure 7 and Table 10 based
on the three different methods. For more details about the
momentum observer and collision detection refer to [5]. The
data for calculating the residual signal is based on the second
reference trajectories. It should be noted that there is no actual
FIGURE 6. Estimated torque comparison between measured torque
filtered by a low-pass filter (black lines) and the predicted torque by the
identified physical parameters from the proposed method (red lines),
the method in [30] (blue dashed lines) and the method in [37] (green
dashed lines) for the excitation trajectories, respectively.
TABLE 9. The RMS of predicted torque errors between the measured
torque and estimated torque by the identified physical parameters from
three different methods for the excitation trajectories.
collision occurring during the robot motion in the second
reference trajectories. In ideal condition, the residual signal
in Figure 7 should be zero when there is no collision for robot.
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
FIGURE 7. Comparison of momentum observer output calculated by the
identified physical parameters from the proposed method (red lines),
the method in [30] (blue dashed lines) and the method in [37] (green
dashed lines) for the second reference trajectories, respectively.
TABLE 10. The maximum residual values for each joint estimated by the
identified physical parameters from three different methods for
the second reference trajectories.
However, in fact, the residual signal is not zero even with-
out collision because of the inaccurate dynamics. To detect
a possible collision, a constant detection threshold for each
joint should be set which should be larger than the corre-
sponding maximum residual value in Table 10. Otherwise,
false positives would happen. For example, the thresholds
of joint 4 can be set to 0.5 (>0.4140), 0.7 (>0.6224) and
0.9 (>0.7976) according to the values in Table 10 calculated
by the proposed method and the methods in [30] and [37],
respectively. It should be known that larger thresholds will
reduce detection sensitivity. Therefore, the residual signal
estimated by the parameters obtained from our method can
contribute to higher detection sensitivity than the ones by
other two methods. In general, collisions usually occur at the
last four joints, therefore, more attention should be focused
on them.
Compared with the method in [30], the main difference
is that the objective function, predicted torque errors, is cal-
culated based on the identified essential parameters and
the corresponding observation matrix in our case. However,
the objective function was calculated based on the identi-
fied base parameters in the supplementary material of [30].
Compared with using the base parameters to calculate objec-
tive function, one direct advantage of using essential param-
eters with high identification quality is that the predicted
accuracy of joint torque is significantly improved, such as
the RMS value and objective function value in Table 9. The
reason that the predicted results shown in [30] are accurate
may be that it benefits from using the reverse engineering
method which can contribute to obtaining base parameters
with lower RSD values, and using the more complete friction
model. Thanking to the method in [30], it gave us inspiration
to propose our framework in this paper.
Compared with the method in [37], one difference of the
proposed method is using a global optimization instead of a
local optimization. In addition, the objective function of the
nonlinear optimization is computed based on the essential
parameters and the corresponding observation matrix, not
the base parameters in [37]. One advantage of the proposed
method is that the predicted accuracy of joint torque is
improved compared to that of the method in [37]. In addition,
another advantage is that the identified physical parameters
by the proposed method can be used to obtain higher colli-
sion detection sensitivity compared with the ones calculated
by other two methods in [30] and [37]. Both the identified
physical parameters in this paper and the ones reported in [37]
are shown in Table 6 and 7. Although the values of two sets of
parameters are different, both of them can be used to predict
the joint torques. It should be noted that, however, compared
with the iiwa robot used in [37], there is a difference that our
iiwa robot is fixed with a payload. Therefore, it can be seen
that m7in our case is larger than the one in [37] in Table 6.
Of course, payload contribution can be also removed by the
method in [42] to obtain comparable parameters for the mass
of link 7. In addition, although the used LB and UB are not
shown in [37], one point can be certain that our LB and
UB are different from theirs. For example, m7is equal to
0.3543 in [37], which is obviously not in the range (1 ∼5) of
LB and UB in our case. By exploring the relationship between
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T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
the identified elements of the link inertia matrix in Table 7 and
the constraints in (18), it can be found that both the optimal
ones in this paper and the ones of [37] are satisfied with the
triangular inequality constraints.
V. DISCUSSION AND CONCLUSION
In this paper, the problem of extracting the physical param-
eters, which described the dynamics of the robot, was
addressed by the proposed global optimization framework.
The base dynamic parameters with RSD values were firstly
identified by the standard least squares method. Based on the
results of identified base parameters, the essential parameters
were calculated and obtained by ignoring the parameters
close to zero with high RSD values. Afterwards, based on
the identified essential parameters with better identification
quality, a set of physical parameters were retrieved by solving
a nonlinear optimization problem, taking into account several
constraints including the physical bounds on the total mass of
the robot and the triangle inequalities of the link inertia ten-
sors. Finally, the proposed framework for physical parameters
extraction and all the identified parameters were validated by
comparison experiments. In addition, a comparison between
the proposed method and the methods in [30] and [37] is
given. Then, an application of the identified physical param-
eters for calculating momentum observer is shown. The
experiment results show that the identified physical param-
eters by the proposed method can obtain better prediction
accuracy of joint torque and contribute to obtaining higher
collision detection sensitivity compared with the methods
in [30] and [37].
In our proposed framework, the physical constraints can
be easily added and avoids using complex penalty functions
like [30]. A major feature of the proposed framework is
that it is a universal algorithm of the physical parameters
retrieval for serial manipulators. In addition, it can be eas-
ily modified to include further nonlinear and physical con-
straints. In the nonlinear problem, the objective function was
computed by using the identified essential parameters with
better identification quality, instead of the base parameters.
The total RMS value and the objective function value at the
optimal solution by the proposed method were all the smallest
when compared with the methods in [30] and [37]. This
means that the proposed framework obtains more accurate
torque prediction and better optimization solution by using
the essential parameters. The identified physical parameters
in Table 6 and Table 7 can satisfy both the set bounds and
physical constraints. Therefore, the optimal solution is a set of
physically meaningful solution. The solution can be regarded
as the ‘optimal’ one, but it is not the ‘true’ one. It is possible
to get the true solution only if the strictly accurate bounds and
constraints can be added. Nonetheless, the optimal solution in
this paper satisfies all dynamics model equations and it can
be safely adopted, such as to compute the inverse dynamics,
collision detection and so on by means of Newton-Euler
method or Lagrange method. The base parameters, essential
parameters and physical parameters of iiwa robot are all
identified and given in this paper. To the best of the authors’
knowledge, this is the first paper that show all the three kinds
of dynamic parameters of an iiwa robot, which can benefit to
the researchers in robotics to select the preferred one for their
research. The total RMS values in Table 8 are approximate for
each reference trajectory, which indicates the high precision
of the joint torque prediction of iiwa robot by all the three sets
of identified parameters.
The bounds used in this paper are not strictly accurate,
which limits the possibility of accessing to the true physical
parameters solution. In addition, more meaningfully physical
constraints need to be further explored. Therefore, in the
future work, the authors would like to explore more accurate
bounds and consider other possibly physical constraints, and
use the computed dynamics by the identified parameters for
robot collision detection and reaction control.
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TIAN XU received the M.S. degree in mecha-
tronics engineering from the Harbin Institute of
Technology (HIT), Harbin, China, in 2017, where
he is currently pursuing the Ph.D. degree with the
School of Mechatronics Engineering. He is also
a joint-training Ph.D. Student with the Advanced
Robotics Centre, National University of Singa-
pore, Singapore. His research interest includes safe
human–robot interaction control.
JIZHUANG FAN (Member, IEEE) received the
Ph.D. degree in mechanical engineering from the
Harbin Institute of Technology (HIT), Harbin,
in 2007. He is currently an Associate Professor and
a Ph.D. Supervisor with the School of Mechatron-
ics Engineering, HIT. His main research interests
include bionic robots and mechatronic devices.
YIWEN CHEN received the B.S. degree in
mechanical engineering from Chongqing Uni-
versity, Chongqing, China, in 2019. He is cur-
rently pursuing the M.S. degree with the School
of Mechanical Engineering, National University
of Singapore, Singapore. His research interests
include neural networks and robotics.
XIANYAO NG received the B.S. degree in mecha-
tronics engineering from the University of New
South Wales, Sydney, Australia, in 2016. He is
currently pursuing the Ph.D. degree with the
School of Mechanical Engineering, National Uni-
versity of Singapore, Singapore. His research
interests include robotic perception and computa-
tional fabrication.
108030 VOLUME 8, 2020
T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot
MARCELO H. ANG, JR. (Senior Member, IEEE)
received the Ph.D. degree in electrical engineer-
ing from the University of Rochester, Rochester,
NY, USA, in 1988. His work experience includes
heading the Technical Training Division of Intel’s
Assembly and Test Facility in the Philippines,
research positions at the East West Center, Hawaii,
and the Massachusetts Institute of Technology,
and a Faculty position as an Assistant Profes-
sor of electrical engineering at the University of
Rochester. In 1989, he joined the Department of Mechanical Engineering,
National University of Singapore, where he is currently an Associate Pro-
fessor and the Acting Director of the Advanced Robotics Centre. He teaches
both the graduate and undergraduate levels in the following areas: robotics;
creativity and innovation, applied electronics, and instrumentation; advanced
computing; and product design and realization. He is also active in consulting
work in these areas. His research interests include robotics, mechatron-
ics, and applications of intelligent systems methodologies. In addition to
academic and research activities, he is actively involved in the Singapore
Robotic Games as its Founding Chairman and the World Robot Olympiad as
a member of the Advisory Council.
QIANQIAN FANG received the M.S. degree in
mechanical engineering from the Harbin Institute
of Technology (HIT), Harbin, China, in 2017,
where she is currently pursuing the Ph.D. degree
with the School of Mechatronics Engineering.
She is also a joint-training Ph.D. Student with
the Advanced Robotics Centre, National Univer-
sity of Singapore, Singapore. Her research inter-
est includes lower limb rehabilitation exoskeleton
robot.
YANHE ZHU (Member, IEEE) received the Ph.D.
degree in mechatronics engineering from the
Harbin Institute of Technology (HIT), Harbin,
China, in 2004. He is currently a Professor and a
Ph.D. Supervisor with the School of Mechatronics
Engineering, HIT. His research interests include
modular robots and exoskeleton robots.
JIE ZHAO (Member, IEEE) received the Ph.D.
degree in mechatronics engineering from the
Harbin Institute of Technology (HIT), Harbin,
China, in 1996. He is currently a Professor and a
Ph.D. Supervisor with the School of Mechatronics
Engineering, HIT. He is also the Leader of the
Subject Matter Expert Group of Intelligent Robots,
National Key Research and Development Pro-
gram, supervised by the Ministry of Science and
Technology, China. His research interests include
the design, modeling, and control of bionic and industrial robots.
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