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

VOLUME 8, 2020 108019

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),

108020 VOLUME 8, 2020

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,

VOLUME 8, 2020 108021

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)

108022 VOLUME 8, 2020

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

VOLUME 8, 2020 108023

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

108024 VOLUME 8, 2020

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

VOLUME 8, 2020 108025

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

108026 VOLUME 8, 2020

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.

VOLUME 8, 2020 108027

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

108028 VOLUME 8, 2020

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.

REFERENCES

[1] W. Khalil and E. Dombre, Modeling, Identiﬁcation and Control of Robots.

London, U.K.: Butterworth-Heinemann, 2004, pp. 291–311.

[2] T. Boaventura, J. Buchli, C. Semini, and D. G. Caldwell, ‘‘Model-based

hydraulic impedance control for dynamic robots,’’ IEEE Trans. Robot.,

vol. 31, no. 6, pp. 1324–1336, Dec. 2015.

[3] M. Rauscher, M. Kimmel, and S. Hirche, ‘‘Constrained robot control using

control barrier functions,’’ in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst.

(IROS), Daejeon, South Korea, Oct. 2016, pp. 279–285.

[4] K. J. Gucwa and H. H. Cheng, ‘‘RoboSim: A simulation environment for

programming virtual robots,’’ Eng. Comput., vol. 34, no. 3, pp. 475–485,

Jul. 2018.

[5] S. Haddadin, A. De Luca, and A. Albu-Schäffer, ‘‘Robot collisions:

A survey on detection, isolation, and identiﬁcation,’’ IEEE Trans. Robot.,

vol. 33, no. 6, pp. 1292–1312, Dec. 2017.

[6] M. Schumacher, J. Wojtusch, P. Beckerle, and O. von Stryk, ‘‘An intro-

ductory review of active compliant control,’’ Robot. Auton. Syst., vol. 119,

pp. 185–200, Sep. 2019.

[7] J. Wu, J. Wang, and Z. You, ‘‘An overview of dynamic parameter identiﬁca-

tion of robots,’’ Robot. Comput.-Integr. Manuf.,vol. 26, no. 5, pp. 414–419,

Oct. 2010.

[8] M. Da Lio, A. Doria, and R. Lot, ‘‘A spatial mechanism for the measure-

ment of the inertia tensor: Theory and experimental results,’’ J. Dyn. Syst.,

Meas., Control, vol. 121, no. 1, pp. 111–116, Mar. 1999.

[9] C. Urrea and J. Pascal, ‘‘Design, simulation, comparison and evaluation of

parameter identiﬁcation methods for an industrial robot,’’ Comput. Electr.

Eng., vol. 67, pp. 791–806, Apr. 2018.

[10] J. Swevers, W. Verdonck, and J. De Schutter, ‘‘Dynamic model identiﬁca-

tion for industrial robots,’’ IEEE Control Syst., vol. 27, no. 5, pp. 58–71,

Oct. 2007.

[11] C. G. Atkeson, C. H. An, and J. M. Hollerbach, ‘‘Estimation of inertial

parameters of manipulator loads and links,’’ Int. J. Robot. Res., vol. 5, no. 3,

pp. 101–119, Sep. 1986.

[12] M. Gautier and W. Khalil, ‘‘Direct calculation of minimum set of inertial

parameters of serial robots,’’ IEEE Trans. Robot. Autom., vol. 6, no. 3,

pp. 368–373, Jun. 1990.

[13] W. Khalil and F. Bennis, ‘‘Comments on ‘direct calculation of minimum

set of inertial parameters of serial robots,’’’ IEEE Trans. Robot. Autom.,

vol. 10, no. 1, pp. 78–79, Feb. 1994.

[14] G. Zak, B. Benhabib, R. G. Fenton, and I. Saban, ‘‘Application of the

weighted least squares parameter estimation method to the robot calibra-

tion,’’ J. Mech. Des., vol. 116, no. 3, pp. 890–893, Sep. 1994.

[15] J. Swevers, C. Ganseman, D. B. Tukel, J. de Schutter, and H. Van Brussel,

‘‘Optimal robot excitation and identiﬁcation,’’ IEEE Trans. Robot. Autom.,

vol. 13, no. 5, pp. 730–740, Oct. 1997.

[16] J. Swevers, W. Verdonck, B. Naumer, S. Pieters, and E. Biber, ‘‘An exper-

imental robot load identiﬁcation method for industrial application,’’ Int. J.

Robot. Res., vol. 21, no. 8, pp. 701–712, Aug. 2002.

[17] C. Gaz, F. Flacco, and A. De Luca, ‘‘Identifying the dynamic model

used by the KUKA LWR: A reverse engineering approach,’’ in Proc.

IEEE Int. Conf. Robot. Automat. (ICRA), Hong Kong, May/Jun. 2014,

pp. 1386–1392.

VOLUME 8, 2020 108029

T. Xu et al.: Dynamic Identification of the KUKA LBR iiwa Robot

[18] 1.0, KUKA System Technology (KST), Version 2, KUKA FastResearchIn-

terface, Augsburg, Germany, 2011.

[19] T. Lee, P. M. Wensing, and F. C. Park, ‘‘Geometric robot dynamic identi-

ﬁcation: A convex programming approach,’’ IEEE Trans. Robot., vol. 36,

no. 2, pp. 348–365, Apr. 2020.

[20] H. Jin, Z. Liu, H. Zhang, Y. Liu, and J. Zhao, ‘‘A dynamic parameter identi-

ﬁcation method for ﬂexible joints based on adaptive control,’’ IEEE/ASME

Trans. Mechatronics, vol. 23, no. 6, pp. 2896–2908, Dec. 2018.

[21] H. Jamshidifar, H. Askari, and B. Fidan, ‘‘Parameter identiﬁcation and

adaptive control of carbon nanotube resonators,’’ Asian J. Control, vol. 20,

no. 4, pp. 1329–1338, Jul. 2018.

[22] C. Yang, Y. Jiang, W. He, J. Na, Z. Li, and B. Xu, ‘‘Adaptive parameter

estimation and control design for robot manipulators with ﬁnite-time

convergence,’’ IEEE Trans. Ind. Electron., vol. 65, no. 10, pp. 8112–8123,

Oct. 2018.

[23] Z. Dong, X. Yang, M. Zheng, L. Song, and Y. Mao, ‘‘Parameter identiﬁca-

tion of unmanned marine vehicle manoeuvring model based on extended

Kalman ﬁlter and support vector machine,’’ Int. J. Adv. Robot. Syst., vol. 16,

no. 1, pp. 1–10, 2019.

[24] P. Agand, M. A. Shoorehdeli, and A. Khaki-Sedigh, ‘‘Adaptive recurrent

neural network with Lyapunov stability learning rules for robot dynamic

terms identiﬁcation,’’ Eng. Appl. Artif. Intell., vol. 65, pp. 1–11, Oct. 2017.

[25] H. Su, W. Qi, Y. Hu, J. Sandoval, L. Zhang, Y. Schmirander, G. Chen,

A. Aliverti, A. Knoll, G. Ferrigno, and E. De Momi, ‘‘Towards model-

free tool dynamic identiﬁcation and calibration using multi-layer neural

network,’’ Sensors, vol. 19, no. 17, p. 3636, Aug. 2019.

[26] H. Su, W. Qi, C. Yang, J. Sandoval, G. Ferrigno, and E. D. Momi, ‘‘Deep

neural network approach in robot tool dynamics identiﬁcation for bilateral

teleoperation,’’ IEEE Robot. Autom. Lett., vol. 5, no. 2, pp. 2943–2949,

Apr. 2020.

[27] A. Jubien, M. Gautier, and A. Janot, ‘‘Dynamic identiﬁcation of the KUKA

LightWeight robot: Comparison between actual and conﬁdential KUKA’s

parameters,’’ in Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatronics,

Besacon, France, Jul. 2014, pp. 483–488.

[28] S. A. Kolyubin, A. S. Shiriaev, and A. Jubien, ‘‘Reﬁning dynamics identiﬁ-

cation for co-bots: Case study on KUKA LWR4+,’’ IFAC-PapersOnLine,

vol. 50, no. 1, pp. 14626–14631, 2017.

[29] C. Gaz, F. Flacco, and A. De Luca, ‘‘Extracting feasible robot parameters

from dynamic coefﬁcients using nonlinear optimization methods,’’ in Proc.

IEEE Int. Conf. Robot. Automat. (ICRA), Stockholm, Sweden, May 2016,

pp. 2075–2081.

[30] C. Gaz, M. Cognetti, A. Oliva, P. R. Giordano, and A. De Luca, ‘‘Dynamic

identiﬁcation of the franka emika panda robot with retrieval of feasible

parameters using penalty-based optimization,’’ IEEE Robot. Autom. Lett.,

vol. 4, no. 4, pp. 4147–4154, Oct. 2019.

[31] V. Mata, F. Benimeli, N. Farhat, and A. Valera, ‘‘Dynamic parameter

identiﬁcation in industrial robots considering physical feasibility,’’ Adv.

Robot., vol. 19, no. 1, pp. 101–119, Jan. 2005.

[32] N. D. Vuong and M. H. Ang, Jr., ‘‘Dynamic model identiﬁcation for

industrial robots,’’ Acta Polytech. Hung., vol. 6, no. 5, pp. 51–68, 2009.

[33] C. D. Sousa and R. Cortesão, ‘‘Physical feasibility of robot base inertial

parameter identiﬁcation: A linear matrix inequality approach,’’ Int. J.

Robot. Res., vol. 33, no. 6, pp. 931–944, May 2014.

[34] P. M. Wensing, S. Kim, and J.-J.-E. Slotine, ‘‘Linear matrix inequalities

for physically consistent inertial parameter identiﬁcation: A statistical

perspective on the mass distribution,’’ IEEE Robot. Autom. Lett., vol. 3,

no. 1, pp. 60–67, Jan. 2018.

[35] C. D. Sousa and R. Cortesao, ‘‘Inertia tensor properties in robot dynamics

identiﬁcation: A linear matrix inequality approach,’’ IEEE/ASME Trans.

Mechatronics, vol. 24, no. 1, pp. 406–411, Feb. 2019.

[36] S. Traversaro, S. Brossette, A. Escande, and F. Nori, ‘‘Identiﬁcation of

fully physical consistent inertial parameters using optimization on man-

ifolds,’’ in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), Daejeon,

South Korea, Oct. 2016, pp. 5446–5451.

[37] Y. R. Stürz, L. M. Affolter, and R. S. Smith, ‘‘Parameter identiﬁcation of

the KUKA LBR iiwa robot including constraints on physical feasibility,’’

IFAC-PapersOnLine, vol. 50, no. 1, pp. 6863–6868, Jul. 2017.

[38] C. Gaz and A. De Luca, ‘‘Payload estimation based on identiﬁed coef-

ﬁcients of robot dynamics—With an application to collision detection,’’

in Proc. IEEE/RSJ Int. Conf. Intell. Robot. Syst. (IROS), Vancouver, BC,

Canada, Sep. 2017, pp. 3033–3040.

[39] B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo, Robotics: Modelling,

Planning and Control. London, U.K.:Springer, 2010, pp. 247–301.

[40] M. Safeea and P. Neto, ‘‘KUKA sunrise toolbox: Interfacing collabora-

tive robots with MATLAB,’’ IEEE Robot. Autom. Mag., vol. 26, no. 1,

pp. 91–96, Mar. 2019.

[41] C. Hennersperger, B. Fuerst, S. Virga, O. Zettinig, B. Frisch, T. Neff,

and N. Navab, ‘‘Towards MRI-based autonomous robotic US acquisi-

tions: A ﬁrst feasibility study,’’ IEEE Trans. Med. Imag., vol. 36, no. 2,

pp. 538–548, Feb. 2017.

[42] W. Khalil, M. Gautier, and P. Lemoine, ‘‘Identiﬁcation of the payload

inertial parameters of industrial manipulators,’’ in Proc. IEEE Int. Conf.

Robot. Automat. (ICRA), Roma, Italy, Apr. 2007, pp. 4943–4948.

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.

VOLUME 8, 2020 108031