A method for capturing accuracy and pose repeatability of articulated industrial robots

Journal of Science & Technology 119 (2017) 032-036
32
A Method for Capturing Accuracy and Pose Repeatability of Articulated
Industrial Robots
Duong Minh Tuan*, Le Duc Do
Hanoi University of Science and Technology – No. 1, Dai Co Viet Str., Hai Ba Trung, Ha Noi, Viet Nam
Received: December 21, 2016; accepted: June 9, 2017
Abstract
This paper presents a low cost method for measuring accuracy and pose repeatability of industrial robots,
which can be used for robot home calibration, accuracy improvements, robot control compensation or
evaluating robot features. The method is proposed based on the kinematical relations of the robot joints and
the geometrical data from the robot controller with respect to several defined coordinate systems. The
experimental setup consists of three dial gauges, capturing position measurements of a KUKA KR6/2 robot
end-effector. The positions of the end-effector are employed to estimate robot pose accuracy and
repeatability. Moreover, experimental results demonstrate the efficiency of the method and can be used for
improving the robot accuracy as well as robot pose repeatability in industrial applications.
Keywords: Pose Repeatability, Accuracy, Industrial robot, KUKA KR 6/2
1. Introduction
Recently, industrial*robots (IRs) have been
more and more used to realize continuous operations
such as prototyping, premachining of cast parts as
well as end-machining of middle tolerance parts [1].
In addition, for repetitive tasks such as placing,
welding, and assembling, the repeatability of IRs
typically ranging from 0.03 to 0.1 mm and their
several millimeters accuracy are commonly used.
Thus, measurement approaches for accuracy and
repeatability have been used for enhancing actual
robot positioning accuracy. Lombard and Perrot [2]
have implemented an automatic method to measure
robot accuracy. The most widely applied approach is
to use theodolite systems for calibration
measurements [3]. Another high precision devices
like Coordinate Measuring Machines (CMMs) have
been widely used for industrial dimension
measurement [4]. Laser Tracking Systems (LTS) [5]
can combine the advantages of a large working space,
high accuracy and dynamic pose measurements.
However, they are high cost, and hence it is
prohibitive to calibrate and measure robot poses with
low costs. A simple approach to measure an error
along a direction by a dial gauge has been discussed
in [6]. However, this could only give accuracy for
one direction. Using three dial gauge fixture for
industrial robots has been investigated in [7,8].
Herein, we employ a general low cost method with a
simple measurement fixture using three dial gauges to
* Corresponding author: Tel: 0947 036 686
Email: tuan.duongminh@hust.edu.vn
capture the robot poses for robot accuracy evaluation
and calibrations. The measurement results show the
efficiency of the method.
2. Method
The method and the experimental setup
including a KUKA KR6/2 (6kg of payload) robot at
the laboratory of Machine-Tools and Tribology are
presented in this section. To derive the accuracy and
the repeatability from the experimental data, the
coordinate systems must be defined for the proposed
measurement method. All coordinate systems are
depicted in Fig. 1. The world frame so-called w is
fixed. The User frame can be defined at an arbitrary
location in the robot cell (Fig. 2) with a designed
point of a working table as the origin of the frame.
The Tool frame has the origin at the Tool Center
Point (TCP), see Fig.1.
2.1 Principle for Robot Pose Measurements
KUKA Robot language runing on VxWork - a
real time operating system (parallel with Windows),
controls the robot. This can provide the position and
the orientation of the end-effector (Tool frame) via 6
articulated joint angles. Thus, the robot pose can be
displayed on the screen of the KUKA Control Panel
(KCP). For example, when the robot moves to a
specified location described by pose k for which the
pose variables are defined. That means that the pose
variables are determined by three position
components ( , y and z )k k kx and three orientation
terms, Yaw ( ),kA Pitch ( )kB and Roll ( ).kC As
illustrated in Fig. 1, the robot changes pose 1 into
Journal of Science & Technology 119 (2017) 032-036
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pose 2, it needs a transformation matrix called 1 2T
and these poses are expressed in the World
coordinates defined by 1
wT and 2 ,
wT respectively.
Similarly, with respect to the User frame, one can
derive two transformation matrices for pose 1 and
pose 2 written as 1
u F and 2
u F , respectively. The
controlled transformation matrix from the World
coordinate system to the Tool one describing pose k
is evaluated as
1 1 2 2 3 3 4 4 5 5 6
0 1
w c w c
w c w k k
k k k k k k k
R p
T T T T T T T
, (1)
where w ckT is the transformation matrix from the
w frame to Tool frame of pose k; 1i ikT
 presents the
transformation matrix from joint frame i to joint
frame i-1, (i=1,2,,6); and w ckR is the controlled
rotational matrix from the w frame to Tool frame of
pose k calculated as
cos sin 0 cos 0 sin
sin cos 0 0 1 0
0 0 1 sin 0 cos
1 0 0
 0 cos sin
0 sin cos
k k k k
w c
k k k
k k
k k
k k
C C B B
R C C
B B
A A
A A
(2)
with w ckp is the controlled position vector
expressed in the w system reads as  Tw ck k k kp x y z .
Moreover, we obtain the relation for controlled
transformation matrices of poses 1 and 2 with respect
to different frames as seen in Fig. 1
1
2 1 2
w c w c cT T T  11 12 1 2c w c cT T T  . (3)
Consequently, the controlled transformation
matrix from pose 1 to pose 2 can be determined as
follows
1 1
1 2 2
2 0 1
c c
c R pT
, (4)
where 1 2
cR and 1 2
cp are the rotation matrix and
the position vector from pose 1 to pose 2 which are
used to evaluate the pose measurement data,
respectively. On one hand, the transformation matrix
from pose 1 to pose 2 is also calculated by the KRL
software using joint sensor measurements (in each
robot joint). On the other hand, the matrix is specified
by the commands in the robot program on the KCP.
The real or measured transformation matrix from
pose 1 to pose 2 can be computed as
1
2 1 2
u m u m mF F T  and 11 2 1 2m u m u mT F F  , (5)
1 1
1 2 2
2 0 1
m m
m R pT
, (6)
with 1 2
mR and 1 2
mp are the rotation matrix and
the position vector from pose 1 to pose 2,
respectively.
The robot absolute pose accuracy or the
positioning and orientation accuracy are defined as a
difference between the measured transformation
matrix 1 2
mT and the controlled counterpart 1 2 .
cT
Obviously, these matrices are independent on the
World and the User frames.
2.2 Experimental Setup
Fig. 1. Coordinate system definition
Fig. 2. Experimental setup for robot’s pose
measurements of KUKA KR6/2.
Fig. 2 shows the experimental setup for robot
pose measurements with a fixture supporting three
dial gauges with a specified space in which a tooling
cube (A=50mm, made of aluminum) attached to the
robot end-effector can be easily placed. The tooling
Journal of Science & Technology 119 (2017) 032-036
34
P
cube is fixed on the end-effector at the origin of the
Tool frame and three orthonormal faces with respect
to x , y and z axes are well calibrated. The three
dial gauges have resolution of 0.05 mm and their
stroke is of 18 mm. When the three orthonormal faces
of the tooling cube touch the tips of the gauges, the
slide rods of the gauges are depressed in the x , y
and z directions. If the movement range of the cube
is within the stroke, the dial gauges can directly
indicate the change of the tooling cube position. The
setup is fixed on the table and referred to the
reference plate (User frame).
2.3 Positioning Accuracy
The Tool Center Point can be measured. We call
a reference point on the gauge fixture is P with which
the TCP at pose k can be calculated by
1 1
2 2
3 3
(0.5 )
(0.5 ) ,
(0.5 )
k x
k y
k z
x P A M D
y P A M D
z P A M D
(7)
where ,kx ky and kz are the coordinates of the
TCP k with respect to the User frame. ,xP yP and zP
are the coordinates of the point P on the gauge fixture
which can be determined from measurement
dimensions as shown in Fig. 3. 1 ,D 2D and 3D are
the maximum travel limits of gauges 1, 2 and 3,
respectively.
Fig. 3. Dimensions of the experimental setup.
The measurement procedure of the TCP can be
addressed as follows:
At first, the robot moves to a designed pose k.
The tooling cube is installed on designed positions
1 ,P position. As the fixture moves so that the cube
touches the tips of the dial gauges generating 1 ,P 2P
and 3P points, then fixes the fixture in that place. The
indications of the dial gauges are recorded. The next
step is to measure and record the coordinates of point
P in Fig. 3. At last, the measures
xP , yP and zP are
substituted into Eq. (7). Therefore, the actual position
coordinates; u mkp ( , ,k k kx y z ) of the robot end-effector
with respect to the User is achieved.
2.4 Orientation Accuracy
To determine the orientation of the robot end-
effector, the dial gauges can be placed so that they are
not orthogonal together i.e. their axes do not intersect
into a point. Consequently, three measuring points
1 1 1 1 ( , , )P x y z , 2 2 2 2 ( , , )P x y z and 3 3 3 3 ( , , )P x y z create
a plane on which the orientation of the robot end-
effector is defined. By investing this plane for each
sample one can obtain the orientation of the robot
tool frame.
Mathematically, the plane passing the three
points is constructed and the 1 1 1 1 (x , y , z )P lies in the
plane yields the plane equation as
1 1 1( ) ( ) ( ) 0a x x b y y c z z , (8)
where a, b and c are the direction numbers of the
plane, respectively. On the other hand, the 2P and 3P
lying in the plane yields
2 1 2 1 2 1
3 1 3 1 3 1
( ) ( ) ( ) 0
( ) ( ) ( ) 0
a x x b y y c z z
a x x b y y c z z
. (9)
With three variables and three equations (8) and
(9), the system of equations is solved to obtain the
solution as
 
 
3 1 2 1 2 1 3 1
2 1 3 1 3 1 2 1
( )( ) ( )( ) ,
 ( )( ) ( )( )
a y y z z y y z z
x x y y x x y y
 (10)
 
 
3 1 2 1 2 1 3 1
3 1 2 1 2 1 3 1
( )( ) ( )( ) ,
 ( )( ) ( )( )
b x x y y x x y y
x x z z x x z z
 (11)
 
 
3 1 2 1 2 1 3 1
2 1 3 1 3 1 2 1
( )( ) ( )( ) .
 ( )( ) ( )( )
c x x y y x x y y
x x y y x x y y
 (12)
Thereby, the normal vector of the measured
plane can be evaluated by its cosines
 cos / ; cos / ; cos /m m mz z za q b q c q   , (13)
where 2 2 2q a b c . The direction vector
projected on z axis is
cos cos cosm m mz z zz l m h   , (14)
where , , andl m h are unit direction vectors of
the User frame. Similarly, the direction vector of the
O
X
P2
P3
P1
Z Y
D2
M2
M3D3
M1
D1 P
Journal of Science & Technology 119 (2017) 032-036
35
x axis is cos cos cosm m mx x xx l m h   when the
x axis passing 1P and 2P
2 1
2 1
2 1
cos ( ) /
cos ( ) /
cos ( ) /
m
x
m
x
m
x
x x r
y y r
z z r


, (15)
where 2 2 22 1 2 1 2 1( ) ( ) ( )r x x y y z z .
Therewith, the direction of y axis is computed as
cos cos cosm m my y yy z x l m h   .
The rotation matrix of the Tool coordinate
system with respect to the User frame reads as
cos cos cos
cos cos cos
cos cos cos
m m m
x y z
u m m m m
k x y z
m m m
x x z
R
  
  
. (16)
Consequently, we combine the position
measurement results and derive the actual or
measured transformation matrix from the User frame
k given by
0 1
u m u m
u m k k
k
R p
F
. (17)
Substituting u mkF into Eq. (5) the measured
transformation matrix 1 2
mF from pose 1 to pose 2 is
explicitly obtained in which both the robot position
accuracy and orientation accuracy are evaluated.
Thus, we can clearly define the accuracy of the robot
formulated as
1 1 1
2 2 2
1 1 1
2 2 2
c m
c m
R R R
p p p
. (18)
If we reset all measurements of dial gauges of
the pose 1 to zero then the robot pose repeatability
can be derived properly. The pose 2 of the robot is
repeatedly measured with respect to the pose 1 as the
reference for repeatability study. Thus the 1 2
mp is the
positioning repeatability and the 1 2
mR describes the
orientation repeatability of the robot.
3. Results and Discussion
This section describes the measurements using
the proposed method on the KUKA robot.
Experimental data obtained for 1 2
mp shows the
efficiency of the method. The KR 6/2 robot has the
repeatability of 0.1 .mm 1 2mp was calculated by
invoking the Eq. (18) and shown in Table 1.
Table 1. Positioning repeatability (mm)
Direction x Axis y Axis z Axis
Max errors 0.125 0.165 0.115
Min errors -0.145 -0.115 -0.170
Mean -0.025 -0.036 0.045
STD 0.045 0.036 0.069
The mean and the standard of deviation (STD)
of experimental data were performed for 48 samples
or 48 times. Obviously, Fig. 4 describes that the
current robot position repeatability is mostly inside a
sphere (radius of 0.1 mm as the specified
repeatability of the robot by KUKA Company). Of
course there are some experimental measurements
outside the sphere, larger than 0.1 mm. However, the
measurements are inside the sphere with 95% of
confidence (as the Gauss distribution).
Fig. 4. Measurements of repeatability.
The orientation accuracy represented by the
rotational matrix in (16) can be derived from the
corresponding triangle. As shown in Fig. 5, the
triangles connecting three measurement points
1, 2 3( , , and )P P P describe the orientation errors from
which orientation repeatability 1 2
mR can be derived.
Fig. 5. Orientation repeatability described as triangles
( 1, 2 3, , andP P P ).
With 48 samples for the robot pose we
computed the orientating repeatability as
Journal of Science & Technology 119 (2017) 032-036
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0.031 0.002 0.257
1 0.002 0.031 0.022
2
0.00001 0.00001 0.031
0.646 0.0008 0.258
0.0008 0.646 0.455
0.00001 0.00001 0.645
mR
. (19)
These above values in (19) are Means and STDs
of each entry of the rotational matrix from pose 1 to
pose 2. It is obvious that the changes of the
orientation of the robot end-effector and positioning
repeatability are acceptable if the robot can be used
for material manipulating. In terms of operations with
continuous paths such as part machining the changes
in the orientation might be large and can cause
significant errors on machined parts.
The experimental results easily obtained from
the measurement approach with very low cost dial
gauge fixture. This setup can be successfully used for
large working space of IRs in robot cells or in
industrial applications. The equipment is very flexible
and be placed at anywhere the robot can reach for
experiments. The calibration of the gauges and fixture
can be done with ease. Therewith the method is
helpful for robot calibration at work. In addition to
that, the robot accuracy can be correctly estimated
with low cost.
Moreover, the pose repeatability can be easily
obtained from the method. The method helps evaluate
the robot accuracy and repeatability after long time
use. More importantly, based on the evaluation the
robot accuracy can be enhanced in order to be used in
high accuracy process such as robot milling.
Therewith the robot errors must be compensated by
improving the robot control model [9]. In addition,
the stiffness of the robot should be taken into account
since it strongly influences on the positioning
accuracy, in particular, for different poses and
positions in the IRs volumes. The larger distance of
the end-effector is, the larger deflection by the robot
end-effector becomes (lower accuracy) [10].
However, compared with the accuracy the pose
repeatability is less influenced by target location.
Thus, the repeatability is an important robot
characteristic. Herein, the load was neglected due to
the light tooling cube. Factors affecting the
repeatability will be discussed in the future work.
4. Conclusion
The study presented the low cost measurement
method in detail, which can be used for any type of
industrial robot topologies. Both pose accuracy and
pose repeatability can be obtained from the method.
Interestingly, with very low cost experimental setup
we were able to demonstrate that this approach is
successfully realized on this equipment with the
KUKA KR6/2 robot. The experimental data were
processed to compute the position repeatability and
orientation repeatability comprehensively. Therefore,
studying robot accuracy by this setup is convenient
and flexible, particularly in large workshops.
Furthermore, with the measured accuracy and
repeatability solutions to improve the robots can be
found, e.g. improving robot control laws with error
compensation. Thus the robot accuracy and pose
repeatability can be enhanced for continuous
operations.
Acknowledgments
This research is funded by Hanoi University of
Science and Technology (HUST) under project
number T2016-PC-058.
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