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Digital Object Identifier 10.1109/ACCESS.2023.0322000
Analysis and Compensation of Kinematic and
Hysteresis Errors in Industrial Robots
WEI HE1, KAI GUO1,(Member, IEEE), Jie Sun1
1Key Laboratory of High Efficiency and Clean Mechanical Manufacture of Ministry of Education, Department of Mechanical Engineering, Shandong University,
Jinan 250061, China
Corresponding author: Kai Guo (e-mail: kaiguo@sdu.edu.cn).
This work was supported by the National Key R&D Program of China under Grant 2022YFB3206700, National Natural Science
Foundation of China under Grant 52375452 and Grant 52175419, National Key R&D Program of China under Grant 2023YFB4703900,
Aeronautical Science Foundation of China under Grant 2023M0440Q3001, Key R&D Program of Shandong Province under Grant
2022CXGC020202, Open Fund of Laboratory of Aerospace Servo Actuation and Transmission under Grant LASAT-2022-A01-03,
Shandong Province Enterprise Innovation Enhancement Project under Grant 2022TSGC2414.
ABSTRACT Industrial robots are extensively utilized in handling, assembly, and welding tasks owing to
their expansive workspace, scalability, flexibility, and cost-effectiveness. However, their inadequate absolute
positioning accuracy significantly impedes their application in precise operational scenarios. To enhance
robot positioning accuracy, the hysteresis error induced by gear meshing backlash is considered. Firstly,
the impact of joint hysteresis on robot positioning errors is analyzed, the notion of modified joint space
is introduced, and the similarity theory of error in modified joint space is analyzed. Secondly, for the
problem of parameter overfitting of the universal Kriging model, a method of dynamically determining
the basis function set by using the genetic algorithm is proposed. Finally, the target trajectory is corrected
by a feed-forward iterative compensation algorithm. An experiment on a tandem industrial robot SMART5
NJ 220-2.7 is conducted to demonstrate the effectiveness of the compensation. The experimental results
show that the error caused by joint hysteresis is significant, with joint 1 notably affecting yaxis positioning
accuracy, while joints 2 and 3 predominantly influence xaxis positioning accuracy. Furthermore, cross-
validation tests verified the good anti-overfitting effect of optimized Kriging for models with multiple input
parameters and the good fitting accuracy of the modified space model for hysteresis errors. Moreover, after
employing MJS&GPS+GA error modeling and feed-forward iteration compensation, the average absolute
positioning error of the trajectory decreased by 81% to 0.09252 mm, and the maximum absolute positioning
error decreased by 59% to 0.27713 mm.
INDEX TERMS Industrial robot, optimized Kriging, coordinate identification, error compensation.
I. INTRODUCTION
COMPARED to conventional specialized equipment, in-
dustrial robots offer a larger workspace, a greater degree
of freedom, excellent scalability, and a low cost [1], [2].
They are widely used for tasks such as handling, assembly,
and welding. Nevertheless, industrial robots with the tandem
open chain structure exhibit amplification properties for com-
ponent manufacturing and assembly errors, which typically
yield an absolute positioning accuracy of ±1∼2mm [3].
The poor absolute positioning accuracy seriously hinders the
promotion and utilization of offline programming techniques
for robots in fine manipulation scenarios [4], [5].
Currently, the primary techniques for enhancing the po-
sitioning accuracy of robots encompass controlling error
sources, implementing online compensation, and performing
kinematic calibration [6]. Controlling error sources, which re-
lies on reasonable design, high-precision manufacturing, and
environmental controls, is a fundamental means of ensuring
robot accuracy. However, heightened accuracy requirements
invariably escalate manufacturing costs significantly [7]. On-
line compensation relies on additional sensors that monitor
real-time tool center point (TCP) positions and adjust control
commands to correct errors based on feedback signals [8].
However, due to its expensive and intricate nature, the promo-
tion of online compensation is limited. Kinematic calibration
analyzes the sources of error, constructs corresponding error
model, and improves accuracy through error prediction and
feed-forward correction. Compared to the former methods,
kinematic calibration offers superior cost-effectiveness and
broader applicability [9].
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W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
Robot calibration methods can be categorized into two
types: kinematic model-dependent and kinematic model-
independent calibration. Kinematic model-dependent calibra-
tion [10] begins with establishing the kinematic model of the
robot. Common models [11] include the Denavit-Hartenberg
(D-H), modified Denavit-Hartenberg (MD-H) models. Sub-
sequently, kinematic parameters are identified by measur-
ing datasets and correcting these parameters to mitigate er-
rors. However, this method necessitates constructing a model
specific to the kinematic parameters of the robot, limiting
its applicability across robots with varying configurations.
Moreover, correcting kinematic parameters demands elevated
controller privileges, significantly restraining the widespread
application of this technique [12].
To overcome traditional kinematic model-dependent cal-
ibration limitations, researchers have adopted an innovative
"black box" approach for robot calibration. This approach
avoids depending on certain kinematic models and instead
utilizes nonlinear models such as neural networks [13], [14]
and Kriging models [15] to establish mappings between the-
oretical postures or joint angles and observed errors. Then,
an error modeling and compensation method based on the
positioning error similarity principle is developed. In con-
trast to kinematic model-dependent calibration, this kine-
matic model-independent approach demonstrates broad ap-
plicability across robots with diverse configurations, as it
does not depend on specific structural details. However, in
order to obtain accurate results, the robot must demonstrate
exceptional levels of multi-directional repeatability [16].
Practically, the robot performs well in unidirectional re-
peatability [17], about ±0.07 mm, but encounters challenges
in multi-directional repeatability, primarily due to joint hys-
teresis errors [18], [19]. Traditional calibration methods typ-
ically rely on the kinematic model, theoretical positions, or
joint spaces. However, these methods often fail to effectively
mitigate joint hysteresis errors caused by joint backlash.
Consequently, despite compensation efforts, the positioning
errors of the robots remain 3-6 times less precise than their
unidirectional repeatability, which is difficult to further im-
prove.
According to Literature [20], joint hysteresis can be char-
acterized as periodic functions of joint angles (for rotational
joints) or displacements (for translational joints), influenced
by the joint direction. Therefore, for the errors caused by
joint backlash, this paper introduces joint direction coeffi-
cients within the joint space and proposes the concept of
a modified joint space. To tackle the overfitting problem
for the universal Kriging in this modified joint space with
multiple input parameters, a optimized Kriging using genetic
algorithm (GA) for basis function optimization is proposed.
Furthermore, a surrogate model between the modified joint
space and the positioning error is developed. Finally, com-
pensation for positioning error is achieved by a feed-forward
iterative compensation algorithm.
The remainder of this paper is structured as follows: In Sec-
tion II, the similarity of position errors in the modified joint
space is investigated through analysis of the robot kinematic
model. Section III introduces an error modeling approach
based on the optimized Kriging algorithm, and a feed-forward
iterative compensation algorithm is proposed. In Section IV,
the results of the experimental verification are presented and
analyzed. Finally, conclusions are summarized in Section V.
II. IMPACT OF JOINT HYSTERESIS ON TCP ERROR
A. MODELING TCP ERROR
Based on the MD-H model proposed by Denavit and Harten-
berg, the homogeneous transformation matrix of adjacent
linkages is formulated as follows:
Ti−1
i=Rot (zi−1, θi) Tr (zi−1,di) Tr (xi,ai)
Rot (xi, αi) Rot (yi, βi)(1)
where θi,di,ai,αiand βiare the joint angle, the joint distance,
the link length, the link twist, and the extra rotation parameter,
respectively. Rot () and Tr () are the rotation transformation
matrix and the translation transformation matrix, respectively.
Defining ˜
θi,˜
di,˜
ai,˜αiand ˜
βias the errors corresponding to
the parameters of the MD-H model, the transformation matrix
of the TCP relative to the base coordinates for a tandem robot
with njoints, can be represented as:
ˆ
T=
n
Y
i=1
ˆ
Ti−1
i=
n
Y
i=1 Ti−1
i+˜
Ti−1
i(2)
where ˆ
Ti−1
iand ˜
Ti−1
iare the real transformation matrix and
the differential transformation matrix of adjacent linkages,
respectively. Each kinematic parameter follows the assump-
tion of small displacements for differential transformations.
Therefore, ˜
Ti−1
ican be estimated as:
˜
Ti−1
i=∂Ti−1
i
∂θi
˜
θi+∂Ti−1
i
∂di
˜
di+∂Ti−1
i
∂ai
˜
ai
+∂Ti−1
i
∂αi
˜αi+∂Ti−1
i
∂βi
˜
βi
(3)
According to (2) and (3), the error is determined by ne-
glecting terms of second order and higher:
∆P=
n
X
i=1 ∂T
∂θi
˜
θi+∂T
∂di
˜
di+∂T
∂ai
˜
ai+∂T
∂αi
˜αi+∂T
∂βi
˜
βi(4)
where ∂T
∂•=T1
2T2
3. . . ∂Ti−1
i
∂•. . . Tn−1
n,•is the robot kine-
matic model parameters.
B. ANALYSIS OF ERROR SPATIAL SIMILARITY
According to (1), for the i-th joint of a tandem robot, the kine-
matic parameters ai,αi,βiand di(for the rotational joint) or θi
(for the translational joint) of the MD-H model are constants,
and only the joint distance di(for the translational joint) or
joint angle θi(for the rotational joint) are variables. Hence, the
robot positioning error depends on joint distance and angle,
highlighting its excellent unidirectional repeatability [21].
Due to gear meshing backlash, when the direction of joint
movement (rotation) changes, the actual joint distance (the
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W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
joint angle) deviates from the theoretical value, leading to
joint reversal error [22]. Research has demonstrated that the
joint distance (the joint angle) is determined by the direction
of motion and the theoretical value. Typically, hysteresis is as-
sumed constant, exhibiting equal magnitude in both forward
and reverse joint motion [21]. Thus, the actual joint distance
(the actual joint angle) can be described as:
˜
•i=•i+ξibi(•i)(5)
with
ξi=1,Forward
−1,Reverse
where ξiand bi(•i)are the joint motion direction coefficients
and the joint backlash, respectively. Based on the preceding
analysis, it is evident that joint backlash primarily affects
the TCP position by influencing the actual joint distance in
translational joints or the actual joint angle in rotational joints
of the robot. When considering only static error sources, if the
theoretical values and motion directions of each robot joint
are similar, the TCP error will also be similar. Therefore,
this paper introduces motion direction coefficients for each
joint into the n-dimensional joint space of an n-degree-of-
freedom tandem robots, thereby establishing a modified joint
space [•1,· · · ,•n, ξ1,· · · , ξn](where •is the theoretical joint
distance or the theoretical joint angle). This modification
accounts for the influence of joint motion directions on the
absolute positioning accuracy of the robot.
III. OPTIMIZED KRIGING MODELING AND ERROR
COMPENSATION
A. KRIGING METHOD
Given the continuity and spatial similarity of the TCP posi-
tioning errors in the modified joint space, this paper employs
the Kriging interpolation method. It establishes a surrogate
model linking theoretical joint distances (or angles), motion
direction coefficients, and positioning errors in the modified
joint space.
According to Kriging interpolation principle, for m
sets of modified joint space data X= [x1,· · · ,xm]T,
(xi= [•i1,· · · ,•in, ξi1,· · · , ξin ]) and their corresponding er-
ror outputs ∆= [δ1,· · · , δm]Tobtained through experimen-
tation, the prediction error for any given modified joint space
point xcan be defined as:
ˆ
δ(x) = E(β,x) + z(x)(6)
where E(β,x)is the trend error, formulated as a linear
combination of regression functions fi(x), that depicts the
change in the error mean. z(x)is the stochastic process error
with zero mean. Xand ∆satisfy normalization conditions.
The covariance between sample points xiand xjcan then be
defined as:
Cov (z(xi),z(xj)) = σ2˜
Q(λ,xi,xj)(7)
where σ2is the process variance of the stochastic error,
˜
Q(λ,xi,xj)is the correlation model with parameters λ. Usu-
ally, Gaussian model is used in engineering:
˜
Q(λ,xi,xj) = exp −
2n
X
k=1
λk|xi,k−xj,k|2!(8)
Based on similarity theory, the prediction error for any
given set of modified joint space point xis expressed as a
linearly weighted interpolation of the output ∆of the sample
point set X:
ˆ
δ(x) = wT(x)∆(9)
where w(x)is the vector of weight coefficients. Then the
estimate of δis:
ˆ
δ(x) = fT(x)ˆ
β+qT(x)Q−1(∆−Fˆ
β)(10)
with
q(x) = ˜
Q(λ,x,x1),˜
Q(λ,x,x2),· · · ,˜
Q(λ,x,xm)T
Q=
˜
Q(λ,x1,x1)· · · ˜
Q(λ,x1,xm)
.
.
.....
.
.
˜
Q(λ,xm,x1)· · · ˜
Q(λ,xm,xm)
f(x)=[f1(x),f2(x),· · · ,fp(x)]T
F= [f1(x),f2(x),· · · ,fm(x)]T
In the above equation, the least squares estimation of β
and the prediction variance with respect to the point xare
respectively:
ˆ
β=FTQ−1F−1FTQ−1∆(11)
φ(x) =ˆσ2h1−qT(x)Q−1q(x) + FTQ−1q(x)−f(x)T
FTQ−1F−1FTQ−1q(x)−f(x)i(12)
where ˆσ2is an estimated value of the process variance σ2,
which can be expressed as:
ˆσ2=1
m(∆−Fˆ
β)TQ−1(∆−Fˆ
β)(13)
From the above analysis, it is evident that both ˆ
βand ˆσ2
depend on the matrix Q. Once the regression model E(β,x)
and the correlation model ˜
Q(λ,xi,xj)are established, matrix
Qdepends on λ. Thus, the Kriging modeling process can
be reformulated as optimizing λ. The maximum likelihood
estimation of λcan be denoted as:
argminψ(λ) = |Q(λ)|1
mσ(λ)2,(λ(i)≥0) (14)
where argmin is the value of variable λthat minimizes the
function ψ(λ).
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W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
B. OPTIMIZED KRIGING
The candidate basis function set ffor universal Kriging con-
sists of polynomials, specifically in the form of products of
the modified joint space parameter [•1,· · · ,•n, ξ1,· · · , ξn]:
fc=1,•1,•2, . . . , ξn,•2
1, . . . , ξ2
n,..., •η
1, ξη
n]1×Cη
2n+η
(15)
where ηis the highest order of the basis function. For the
universal Kriging, the basis function set is fixed in a regres-
sion model with a fixed order. It is usually believed that the
high-order terms in the set can predict the nonlinear error.
However, universal Kriging are susceptible to overfitting due
to the increased variable dimension, and higher-order basis
functions may not always enhance the precision of the surro-
gate model [23], [24]. To address this, this paper proposes a
optimized Kriging algorithm that optimizes the basis function
set using the GA algorithm. Additionally, the generalized
pattern search (GPS) method is employed instead of Hooke-
Jeeves (HJ) that emphasizes solution efficiency, aiming to
achieve an accurate proxy model in a modified joint space
with multiple input parameters and high nonlinearity. The
process is described below (as Figure 1):
(1) Initialization
From (15), it is evident that the selection of optimal basis
functions involves a discrete optimization problem. This
selection process can be represented by a vector , where
"1" denotes a function is selected and "0" denotes it is not.
For the selected basis function set, the GPS algorithm is
used to solve the relevant parameter λand construct the
Kriging surrogate model.
(2) Calculating fitness
Through K-fold cross validation, the spatial interpola-
tion precision of the Kriging model fitted by each set
of basis function set is quantitatively compared. In de-
tail, the dataset {X,∆}is partitioned into Ksubsets
{X1,∆1},{X2,∆2},...,{XK,∆K}. During each cy-
cle, one subset is selected as the validation dataset, and
a surrogate model is constructed from the remaining
subsets. The process is repeated Ktimes, ensuring each
subset is used once as a validation dataset. Consequently,
a total of Kresults can be obtained as:
∆cv(i) = ∆i−ˆ
∆(Xi),(i= 1,...,K)(16)
During the K-fold cross-validation process, a total of
KKriging models need to be sequentially constructed,
with each model generated using (K−1) m/Ksamples.
According to the estimation in (13), the highest order η
must satisfy Cη
2n+η≤(K−1)m/K[25]. Additionally,
since ∆cv is in the form of vectors, the root-mean-square
error (RMSE) is used as the fitness for the GA algorithm:
Rcv =r1
n∆T
cv∆cv (17)
(3) Update population
The GA is used to perform operations such as selection,
crossover, and mutation to update the optimal basis func-
tion set and obtain an updated population.
Start
Population
Calculate the fitness
End
Satisfy termination
condition?
Optimum solution
Yes
No Selection
Elite Anti-elite
Crossover
Mutation
GA
operation
FIGURE 1. Optimization process for the basis function set of the
optimized Kriging algorithm.
(4) Iterative optimization
The best basis function set is derived by iterating steps
(2)-(4) until the termination condition is met.
C. FEED-FORWARD ITERATIVE COMPENSATION
The trajectory planning process based on the feedforward
iterative compensation algorithm is illustrated in Figure 2.
Initially, the theoretical interpolation point set corresponding
to the target trajectory and the theoretical joint angles cor-
responding to each interpolation point are calculated. Subse-
quently, according to the theoretical joint angles of adjacent
interpolation points, the modified joint space for the inter-
polation point is established. The Kriging model, which was
created in Section III, is used to determine the positioning
error of each interpolation point. If the error meets the thresh-
old requirement, the set of interpolated points is outputted.
If not, the iterative optimization process will continue until
it reaches the preset maximum number of iterations or the
RMSE satisfies the threshold requirement.
The specific iterative compensation process is depicted in
Figure 3, and its main steps are as follows:
(1) Calculate the joint angle corresponding to the i-th inter-
polation point by inverse kinematics:
θi,k=IK (Pi,k,qn)(18)
where IK () is the robot inverse kinematics calculation
and qnis the robot theoretical kinematics parameter vec-
tor.
(2) The modified joint space of the i-th interpolation point is
established based on the joint angles of adjacent interpo-
lation points:
˜
θi,k=θi,ksign(θi,k−θi−1,k)(19)
(3) Calculate the TCP error corresponding to the modified
joint space based on the Kriging error model established
in Section III:
ˆ
δi,k=Kr(˜
θi,k)(20)
(4) Calculate the RMSE of the trajectory:
Rk=v
u
u
t
n
X
i=1
εk2,n(21)
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W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
Target
trajectory
Interpolation
point set
Joint angle
sequence
(1)
Modified
joint angle
sequence (2)
Calculate
TCP error
(3)
Satisfy termination
condition?
Corrected
interpolation
point set (5)
Calculate the
root mean
square error (4)
Modified
target
trajectory
No
Yes
Iterative compensation
FIGURE 2. Trajectory planning process based on feed-forward iterative compensation algorithm.
with
εk=ˆ
δi,k−1−ˆ
δi,k
where nis the total number of interpolation points.
(5) If the RMSE of the trajectory does not meet the prede-
fined threshold and the number of iterations has not ex-
ceeded the preset maximum, the interpolation points are
adjusted using error feedforward compensation theory:
Pm
i,k=Pm
i,k−1+εk−1(22)
IV. EXPERIMENTAL VALIDATION AND RESULTS
In this study, the SMART5 NJ 220-2.7 industrial robot of
COMAU was selected as the research object, and the Radian
Core laser tracker of API was used to measure the position
of the spherical mounted retroreflector (SMR) (as Figure
4). Firstly, the position of the TCP in the base coordinate
system and its corresponding position error are determined
using coordinate transformation techniques. After that, the
optimized Kriging approach is applied to create the surrogate
model between the modified joint space and the TCP error.
Finally, a feed-forward iterative compensation algorithm is
utilized to compensate for the positioning error.
A. COORDINATE SYSTEM IDENTIFICATION
1) SMR and TCP Offset Identification
This paper focuses on analyzing the TCP position. As the
TCP and the SMR do not coincide, accurately determining
the TCP positioning error requires precise identification of
offsets between the SMR and both the TCP and robot flange.
The SMR offset relative to the robot flange is determined
using the circle-point-analysis (CPA) method [8], detailed in
Figure 5. The process is outlined as follows:
(1) Record the initial coordinates oSMR of the SMR when the
robot is at zero position. Rotate axis 6 to measure the
real-time coordinate values of the SMR, and fit Circle 6
based on these measurements. Orient zflange axis along the
normal line A6 of Circle 6. Repeat these steps for axis 5
to fit Circle 5, using the common perpendicular between
the zflange axis and the normal line A5 of Circle 5 as the
xflange axis. Determine the yflange axis of flange coordinate
system.
(2) Establish the rotation matrix from the robot flange to the
laser tracker:
Rflange
laser =[ xflange yflange zflange ](23)
Subsequently, the coordinates of the origin oflange of the
flange coordinate system in the laser tracker coordinate
system can be expressed as:
oflange
laser =o6−(d−d6)·zflange (24)
where d=−−→
o5o6·zflange,d6is the theoretical length of the
linkage 6, o5and o6are the location coordinates of the
centers of circles 5 and 6 in the laser tracker coordinate
system, respectively.
(3) Using the flange origin oflange and the rotation matrix
Rflange
laser , the position of oSMR in the robot end-flange co-
ordinate system, namely, the offset of the SMR relative
to the robot flange, can be calculated as follows:
PSMR
flange = (Rflange
laser )−1(oSMR −oTCP)(25)
For TCP calibration, the commonly used 4-point calibra-
tion suffers from insufficient accuracy, while dedicated equip-
ment tends to be expensive and complex to operate. This study
presents a concise device that utilizes the CPA to determine
the offset of the TCP relative to the SMR (as Figure 6). The
procedure can be delineated below:
(1) The TCP calibration device is mounted at the end of the
tool, with the SMR fixed on the target sphere seat. When
the robot is in the zero position, the real-time coordinate
values of the SMR are measured while the tool and the
calibration device are rotating. A circular TCP is then
fitted based on these measurement points. The direction
normal to the circular TCP is defined as the zTCP axis,
with the positive direction of the ySMR axis as the yTCP
axis, and the xTCP axis is established according to the
right-hand rule.
(2) Construct the rotation matrix from the TCP coordinate
system to the laser tracker:
RTCP
laser =[ xTCP yTCP zTCP ](26)
Subsequently, the coordinates of the origin oTCP of the
TCP coordinate system in the laser tracker coordinate
system can be expressed as follows:
oTCP =oSMR,TCP −(l2−l0)zTCP (27)
where oSMR,TCP is the coordinates of the center of the
circle TCP in the laser tracker coordinate system, l0is
the theoretical length of ooSMR,TCP, and l2is the length
of ooTCP. The length l2can be calculated based on the
VOLUME 11, 2023 5
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2024.3486716
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W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
Initial
Iteration 1
Iteration 2
Iteration k
…
FIGURE 3. Process of error iterative compensation.
SMR
Axis 5
Laser
Tracker
Axis 6
Robot
Spindle
holder
yworkpiece
zworkpiece
xworkpiece
FIGURE 4. Experiment platform.
measurements from the displacement sensor. First, a cali-
bration rod of length l1is placed into the TCP calibration
device, yielding a sensor reading of u1. Next, the TCP
calibration device is mounted on the spindle tool, result-
ing in a sensor reading of u2. The distance from the oTCP
(Tool center point) to the end face oof the fixing ring is
then given by l2=l1+Kd(u2−u1), where Kdis the
amplification coefficient of the displacement sensor.
(3) Based on the TCP origin oTCP and the rotation matrix
RTCP
laser , the position of the TCP coordinate system origin
in the SMR coordinate system, i.e., the TCP offset with
respect to the SMR, can be calculated:
PTCP
SMR =RSMR
laser −1(oTCP −oSMR)(28)
Circle 5
o5
×
o6
A5
Circle 6
A6
xflange
zflange yflange
×
zSMR ySMR
xSMR
SMR Ⅰ
FIGURE 5. Schematic diagram of the SMR offset relative to the flange.
××
l0
l2
(u2)
l1
(u1)
TCP calibration device
Displacement
sensor
SMR Ⅱ
Fixing ring
SMR Ⅰ
oSMR,TCP
oTCP
yTCP
zTCP
xTCP
×zSMR ySMR
xSMR
Circle TCP
o
SMR Ⅱ
FIGURE 6. Schematic diagram of the TCP offset relative to the SMR.
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2024.3486716
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
2) Base Coordinate System Identification
Given that the laser tracker coordinate system and the robot
base coordinate system are both coordinate systems of the
same scale, they can be aligned through rotation and trans-
lation. Thus, this paper utilizes the singular value decomposi-
tion (SVD) algorithm for coordinate system transformation,
defined by the following mathematical model:
F(R,t) = argmin
m
X
j=1
∥(Rpj+t)−qj∥2(29)
where argmin is the variable minimizing the objective func-
tion, Rand tis the rotation and translation matrices, pjand
qjis the coordinate of the TCP in the laser tracker coordinate
system and the robot base coordinate system respectively,
and mis the number of sampling points. Further, a partial
derivation of (29) yields:
∂F
∂t= 2mt+ 2R
m
X
j=1
pj−2
m
X
j=1
qj= 0 (30)
Then:
¯q −R¯p =t(31)
where ¯q =1
m
m
P
j=1
qjand ¯p =1
m
m
P
j=1
pjare the center coor-
dinates of the point set in the laser tracker and robot base
coordinate system, respectively. Given that the rotation matrix
Ris an orthogonal matrix, substituting (31) into (29) yields:
F= argmin
m
X
j=1
∥(R·aj)−bj∥2
= argmin
m
X
j=1 aT
jaj−2bT
jRaj+bT
jbj
(32)
where aj=pj−¯p,bj=qj−¯q. Since aT
jajand bT
jbjare
constants, minimizing (32) is equivalent to maximizing the
function m
P
j=1
bT
jRaj:
Q= argmax
m
X
j=1
bT
jRaj
= argmax tr BTRA= argmax tr RABT
(33)
where A= [a1,a2,· · · ,am]T,B= [b1,b2,· · · ,bm]T.
Based on the principle of singular value decomposition of a
matrix, ABT=UΣVT, the above equation can be expressed
as: Q= argmax tr RUΣVT
= argmax tr ΣVTRU (34)
where U,Vand Σare the left singular, right singular, and
diagonal matrices respectively. Here, U,Vand Σare or-
thogonal matrices, and Qachieves its maximum value when
VTRU =I[26], i.e.:
R=VUT
t=¯q −R¯p (35)
After the base coordinate system, the TCP offset and the
SMR offset are identified, the measurements of the SMR are
converted to TCP coordinates referenced to the base coordi-
nate system.
B. EXPERIMENTAL VALIDATION
1) Analysis of Joint Hysteresis Error
To quantitatively analyze the influence of joint hysteresis
loops on positioning error, the robot postures shown in Table
1 were chosen for the study. During testing, the studied joint
moves unidirectionally from its initial angle. Once the robot
reaches the specified angle, it pauses, and a laser tracker
measures the TCP position. After reaching the final angle, the
joint reverses direction, repeating this process four times. The
errors in the x,yand zaxes for the first three joints, measured
during forward and reverse rotation, are depicted in Figure 7.
As shown in Figure 7, the robot exhibits poor multi-
directional repeatability but relatively good unidirectional
repeatability. Specifically, the robot demonstrates unidirec-
tional repeatability within 0.1 mm, which forms the ba-
sis for compensating joint hysteresis. In contrast, its multi-
directional repeatability is poorer, with a maximum of about
1mm. Particularly, joint 1 notably affects the yaxis hysteresis
error, while joints 2 and 3 have a larger influence on the
xaxis hysteresis error. When comparing positioning errors
during joint forward and reverse rotations, it appears that
these hysteresis errors exhibit a certain symmetry around the
robot zero position.
2) Selection of Basis Function Set
In the robot workspace, 300 sequences of target points were
produced randomly using the Latin hypercube sampling ap-
proach, constrained by the joint angle range (see Table 2). The
positions of the SMR were measured using a laser tracker, and
the errors in the x,y, and zaxes of the TCP were calculated.
Four Kriging methods were employed to model the x,y,
and zaxis positioning errors of the TCP. The first approach,
JS&HJ, utilizes universal Kriging to construct a surrogate
model in the joint space. The second approach, MJS&HJ,
differs by constructing a surrogate model in a modified joint
space. The third approach, MJS&GPS, employs generalized
pattern search to determine the best correlation coefficient
in the modified joint space and builds a surrogate model.
The fourth approach, MJS&GA+GPS, introduces optimized
Kriging with a GA optimizing the basis function set based on
MJS&GPS. Using 10-fold cross-validation, box plots depict-
ing estimation errors and variances for the aforementioned
four strategies are presented in Figure 8. The results indicate
that the optimized Kriging algorithm (MJS&GA+GPS) in the
modified joint space achieves the smallest estimation error,
yielding models with higher accuracy.
The optimization results for the second-order Kriging ba-
sis functions are shown in Figure 9. After optimization, the
number of basis functions in the x,y, and zdirections was de-
creased from 91 to 34, 28, and 44, respectively. This reduction
helps to alleviate the risk of overfitting in the surrogate model.
VOLUME 11, 2023 7
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2024.3486716
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
TABLE 1. Joint angles set in the experiment.
Joint Initial angle (◦) Final angle (◦) Step (◦) Target angle {Code} (◦)
1 {-30, 0, -90, 0, 0, 0} {30, 0, -90, 0, 0, 0} 10 -20{1}, -10{2}, 0{3}, 10{4}, 20{5}
2 {0, -30, -90, 0, 0, 0} {0, 30, -90, 0, 0, 0} 10 -20{1}, -10{2}, 0{3}, 10{4}, 20{5}
3 {0, 0, -70, 0, 0, 0} {0, 0, -130, 0, 0, 0} 10 -80{1}, -90{2}, -100{3}, -110{4}, -120{5}
1
2
3
4
5
0 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
- 0 . 3
- 0 . 2
- 0 . 1
0 . 0
0 . 1
0 . 2
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
- 0 . 8
- 0 . 4
0 . 0
0 . 4
0 . 8
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
0 . 0 0
0 . 1 5
0 . 3 0
0 . 4 5
0 . 6 0
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
- 0 . 3
- 0 . 2
- 0 . 1
0 . 0
0 . 1
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
- 0 . 8
- 0 . 4
0 . 0
0 . 4
0 . 8
1 . 2
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
- 1 . 0 0
- 0 . 7 5
- 0 . 5 0
- 0 . 2 5
0 . 0 0
0 . 2 5
0 . 5 0
0 . 7 5
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
1
2
3
4
5
0 . 0
0 . 1
0 . 2
0 . 3
T a r g e t a n g l e c o d e
E r r o r ( m m )
F o r w a r d
R e v e r s a l
( a ) ( c )( b )
( d ) ( e ) ( f )
( g ) ( i )( h )
FIGURE 7. Impact of joint hysteresis on the TCP error for joints 1, 2, and 3. (a) xaxial error during rotation of joint 1. (b) yaxial error during rotation
of joint 1.(c) zaxial error during rotation of joint 1.(d) xaxial error during rotation of joint 2. (e) yaxial error during rotation of joint 2. (f) zaxial error
during rotation of joint 2. (g) xaxial error during rotation of joint 3. (h) yaxial error during rotation of joint 3. (i) zaxial error during rotation of joint 3.
TABLE 2. Setting of robot joint rotation range.
Joint Range limitation (°) Joint Range limitation (°)
A1 (-20,20) A4 (-15,15)
A2 (0,-40) A5 (-55,15)
A3 (-150,-100) A6 (-15,15)
Moreover, a significant amount of basis functions linked to
the coefficients of joint motion direction ξwere remained,
indicating a robust association between joint motion direction
and TCP error. Specifically, for the xaxis, 16 out of 25 basis
functions related to the joint motion direction coefficients
ξ2and ξ3were retained, while for the yaxis, 10 out of 13
basis functions associated with the joint motion direction
coefficient ξ1were retained. The results support the analysis
in Section IV-B1, confirming that the motion direction of joint
1 has a significant impact on the yaxis error, whereas the
motion directions of joints 2 and 3 significantly affect the x
axis error.
Figure 10 illustrates the trend of the fitness value of the
optimal individual across iterations during the optimization
of the optimized Kriging algorithm. It shows a consistent
8VOLUME 11, 2023
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2024.3486716
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
2 5 % ~ 7 5 %
1 . 5 I Q R
M e d i a n l i n e
M e a n
O u t l i e r s
x
y
z
x
y
z
x
y
z
x
y
z
J S & H J
M J S & H J
M J S & G P S
M J S & G A
0.000
0.005
0.010
0.015
0.020
0.025
P r o c e s s v a r i a n c e ( m m )
2 5 % ~ 7 5 % 1 . 5 I Q R
M e d i a n l i n e M e a n
O u t l i e r s
x
y
z
x
y
z
x
y
z
x
y
z
J S & H J
M J S & H J
M J S & G P S
M J S & G A
- 0 . 5 0
- 0 . 2 5
0 . 0 0
0 . 2 5
0 . 5 0
E r r o r ( m m )
( a ) ( b )
FIGURE 8. Estimation of positioning error. (a) Box plot of estimated errors. (b) Box plot of estimated mean square errors.
1111000000101- -
000100010100- - - -
10111100011- - - - - -
1000010101- - - - - - - -
111011111- - - - - - - - - -
01111111- - - - - - - - - - - -
0101011- - - - - - - - - - - - - -
100100- - - - - - - - - - - - - - - -
01000- - - - - - - - - - - - - - - - - -
1000- - - - - - - - - - - - - - - - - - - -
010- - - - - - - - - - - - - - - - - - - - - -
0 0 - - - - - - - - - - - - - - - - - - - - - - - -
0- - - - - - - - - - - - - - - - - - - - - - - - 1
ξ
6
ξ
5
ξ
4
ξ
3
ξ
2
ξ
1
θ
6
θ
5
θ
4
θ
3
θ
2
θ
11
1
θ
1
θ
2
θ
3
θ
4
θ
5
θ
6
ξ
1
ξ
2
ξ
3
ξ
4
ξ
5
ξ
6
( c )
1010011010010- -
010110000011- - - -
00001101000- - - - - -
1100110000- - - - - - - -
000000010- - - - - - - - - -
00000100- - - - - - - - - - - -
0000110- - - - - - - - - - - - - -
110111- - - - - - - - - - - - - - - -
00000- - - - - - - - - - - - - - - - - -
0000- - - - - - - - - - - - - - - - - - - -
010- - - - - - - - - - - - - - - - - - - - - -
0 0 - - - - - - - - - - - - - - - - - - - - - - - -
0- - - - - - - - - - - - - - - - - - - - - - - - 1
ξ
6
ξ
5
ξ
4
ξ
3
ξ
2
ξ
1
θ
6
θ
5
θ
4
θ
3
θ
2
θ
11
1
θ
1
θ
2
θ
3
θ
4
θ
5
θ
6
ξ
1
ξ
2
ξ
3
ξ
4
ξ
5
ξ
6
( b )
0101110110010- -
000110011010- - - -
01011000100- - - - - -
0010000000- - - - - - - -
110010000- - - - - - - - - -
01010001- - - - - - - - - - - -
0011100- - - - - - - - - - - - - -
100000- - - - - - - - - - - - - - - -
10111- - - - - - - - - - - - - - - - - -
1001- - - - - - - - - - - - - - - - - - - -
100- - - - - - - - - - - - - - - - - - - - - -
0 0 - - - - - - - - - - - - - - - - - - - - - - - -
0- - - - - - - - - - - - - - - - - - - - - - - - 1
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
ξ
6
ξ
5
ξ
4
ξ
3
ξ
2
ξ
1
θ
6
θ
5
θ
4
θ
3
θ
2
θ
11
1
θ
1
θ
2
θ
3
θ
4
θ
5
θ
6
ξ
1
ξ
2
ξ
3
ξ
4
ξ
5
ξ
6
( a )
FIGURE 9. Optimized Results of Second-Order Kriging Basis Functions (1 = Retained, 0 = Not Retained). (a) xaxis. (b) yaxis. (c) zaxis.
decrease in RMSE for the x,y, and zaxes as the number
of iterations increases. This improvement indicates that op-
timizing the basis functions enhances the fitting accuracy of
the surrogate model. Specifically, RMSE stabilizes at 0.077,
0.078, and 0.079 after 36, 38, and 42 iterations, respectively.
Concurrently, the analysis reveals anisotropic errors across
the x,y, and zaxes, attributed to varying effects of robot joint
angles on error in each direction.
3) Error Compensation
Once the surrogate model for modified joint space and po-
sitioning errors was acquired, the error compensation pro-
cess described in Section III was employed. The reference
trajectory to be compensated, selected within the constraints
outlined in Table 2, is a circular path with a diameter of 100
mm. The TCP errors of the trajectory before and after com-
pensation are shown in Figure 11, and the specific statistical
results are displayed in Table 3. The results of the experi-
ment indicate that after compensation, the average absolute
positioning error of the robot decreased from 0.48798 mm
to 0.09252 mm, and the maximum absolute positioning error
reduced from 0.67515 mm to 0.27713 mm. This indicates
0 1 0 2 0 3 0 4 0 5 0
0 . 0 7
0 . 0 8
0 . 0 9
0 . 1 0
0 . 1 1
0 . 1 2
F i t n e s s
x
y
z
I t e r a t i o n
FIGURE 10. Fitness curves for optimized Kriging.
improvements of 81% and 59% in average and maximum
absolute position accuracy, respectively.
VOLUME 11, 2023 9
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content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2024.3486716
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
W. He et al.: Analysis and Compensation of Kinematic and Hysteresis Errors in Industrial Robots
FIGURE 11. Comparison of positioning error before and after compensation. (a) Trajectory curve. (b) Box plot of the TCP positioning error.
TABLE 3. Statistical results of the positioning errors.
Compensation Axis Mean (mm) Standard deviation (mm) Minimum value (mm) Maximum value (mm)
Before x 0.12116 0.14273 -0.19255 0.42845
y 0.04521 0.15876 -0.29607 0.39625
z -0.41837 0.07204 -0.65461 -0.21336
Absolute 0.48798 0.06632 0.28676 0.67515
After x 0.00842 0.05887 -0.18544 0.27516
y 8.54477E-4 0.06139 -0.15276 0.20987
z -0.00196 0.053 -0.20833 0.16586
Absolute 0.09252 0.03948 0.00254 0.27713
V. CONCLUSION
This paper focuses on addressing errors caused by hysteresis
in robot joints. Starting with the kinematic model of the robot,
the mechanism of the joint hysteresis on the TCP position is
analyzed. The joint motion direction coefficients are further
introduced to put forward the concept of modified joint space.
Within the modified joint space, the spatial similarity of the
positioning error of the robot is analyzed. In addition, a GA-
based optimized Kriging method is introduced to dynamically
determine the basis function set. Building upon this, a trajec-
tory error compensation strategy is proposed based on a feed-
forward iterative compensation algorithm.
The experimental results show that the error caused by
joint hysteresis is significant, with joint 1 notably affecting y
axis positioning accuracy, while joints 2 and 3 predominantly
influence xaxis positioning accuracy. The TCP errors exhibit
a certain symmetry around the robot zero position. Further-
more, cross-validation experiments confirmed the effective
prevention of overfitting by optimized Kriging in models
with numerous input parameters, and the redesigned modi-
fied joint space had acceptable fitting accuracy for hysteresis
errors. Finally, after employing MJS&GPS+GA error mod-
eling and feed-forward iteration compensation, the average
absolute positioning error of the trajectory decreased by 81%
to 0.09252 mm, and the maximum absolute positioning error
decreased by 59% to 0.27713 mm.
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WEI HE received the M.S. degree in mechanical
engineering at Taiyuan University of Technology,
Taiyuan, China, in 2019. He is currently pursu-
ing the Ph.D. degree in mechanical engineering at
Shandong University, Jinan, China.
His currently research interests include machin-
ing with industrial robot, chatter suppression, and
manufacturing process monitoring.
KAI GUO (Member, IEEE) received the Ph.D. de-
gree in fluid power transmission and control from
Zhejiang University, Hangzhou, China, in 2015.
He is currently an Professor with the Key Lab-
oratory of High Efficiency and Clean Mechanical
Manufacture of Ministry of Education, Depart-
ment of Mechanical Engineering, Shandong Uni-
versity, Jinan, China. His current research inter-
ests include the modeling and control of industrial
robots, intelligent manufacturing.
JIE SUN received the Ph.D. degree in Mechanical
Engineering from Zhejiang University, Hangzhou,
China, in 2004.
He is currently a Professor with the School of
Mechanical Engineering, Shandong University, Ji-
nan, China. His current research interests include
the high-speed cutting mechanism of difficult-to-
machine materials, deformation control and cor-
rection of NC machining of large structure com-
ponents, and remanufacturing.
VOLUME 11, 2023 11
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2024.3486716
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