Using the finite - time disturbance observer (FTO) for robotic manipulator almega 16
KHOA HỌC
CÔNG NGHỆ
P-ISSN 1859-3585 E-ISSN 2615-9619
USING THE FINITE - TIME DISTURBANCE OBSERVER (FTO)
FOR ROBOTIC MANIPULATOR ALMEGA 16
DÙNG BỘ QUAN SÁT NHIỄU VỚI THỜI GIAN HỮU HẠN CHO TAY MÁY ROBOT ALMEGA 16
Vo Thu Ha
machine not to stick exactly to the given trajectory.To
know the exact external noise components, it is necessary
to incorporate a noise observation device (DOB) to
estimate these disturbances. When applying the DOB noise
monitor in the mechanical hand movement system, control
can be based on the noise monitor [1-3], estimate and
compensate the friction component [4,5], control force or
tissue. non-sensor torque [6-8], error diagnosis and
isolation (FDI) [9-11]. The DOB turbulence monitor has been
widely used in hand machine motion control for a variety of
purposes. The basic idea of DOB is to use the motion state
variables of the robot and the torque of the joints as input
values and then estimate all the unknown internal and
external torque. In [5], the Nonlinear Noise Observer
(NDOB) was established to estimate the friction component
so that accurate real friction component values can be
known with fast time. The NDOB is done by choosing a
certain nonlinear function. But the downside of the NDOB
is that choosing such a nonlinear function is not
straightforward. In [9], the use of the generalized
momentum observer (GMO) has the advantage of not only
avoiding acceleration calculations to reduce the effect of
noise in site measurements, but also creating disturbance
observations at superlative form. GMOs are able to realize
FDI such as predicting random effects as well as saturation
actuator error. The GMO Observer is easy to implement and
has reliable results and the GMO has become a popular and
widely used method in many hand-operated applications.
However, the downside of GMOs is that the failure to return
diagnostic results and slow response isolation (FDI) results
in reduced sensitivity and response speed when the GMO is
used in the case of collision detection. In [10] there was a
solution for the GMO set, by treating the collision detection
case as an extrinsic perturbation. Although many DOB
observers have been developed and used for mechanical
hand movement systems [5,9,10,13,14]. However, this DOB
observer shows that the asymptotic convergence rate and
the estimated bias of the perturbations will not converge
quickly to zero. So for the conventional DOB convergence
rate is is best exponentially while the FTO can achieve a
faster convergence rate with convergence in finite time.
Given their finite time characteristics, a number of FTOs
have been designed and applied to different systems with
ABSTRACT
This paper presents build a finite time observator (FTO) and applies it to the
Almega16 robot motion system. The main content of the article is to design a
FTO so that the observation of the external noise of the Almega16 robot motion
system will converge to the desired true value over a period of time. finite, is
done by estimating the external noise quantities and then feeding them into the
available Robot controller. The advantage when applying the FTO disturbance
monitor is that it is possible to eliminate the inverse inertia matrix component in
the dynamic equation. The results achieved showed that the Almega16 robot
movement system ensures that the errors of the rotating joints quickly reach
zero with a small transition time, making the closed system stable according to
Lyapunov standards.
Keywords: Robot Almega 16, Finite - time observer, Lyapunov standards.
TÓM TẮT
Bài báo trình bày xây dựng bộ quan sát nhiễu với thời gian hữu hạn (FTO) và
ứng dụng cho hệ chuyển động Robot Almega16. Nội dung chính bài báo là thiết
kế bộ quan sát nhiễu với thời gian hữu hạn (FTO) sao cho việc quan sát các nhiễu
ngoại của hệ thống chuyển động Robot Almega16 sẽ hội tụ về giá trị thực mong
muốn với một khoảng thời gian hữu hạn, được thực hiện bằng cách là ước lượng
các đại lượng nhiễu ngoại sau đó đưa vào bộ điều khiển Robot có sẵn. Ưu điểm
khi ứng dụng bộ quan sát nhiễu FTO là có thể loại bỏ thành phần ma trận quán
tính nghịch đảo trong phương trình động lực học. Kết quả đạt được, cho thấy hệ
chuyển động Robot Almega16 đảm bảo sai số của các khớp quay nhanh chóng
đạt tới không với thời gian quá độ nhỏ, làm cho hệ thống kín ổn định theo tiêu
chuẩn Lyapunov.
Từ khóa: Robot Almega 16, bộ quan sát nhiễu với thời gian hữu hạn, tiêu
chuẩn Lyapunov.
University of Economics - Technology for Industries
Email: vtha@uneti.edu.vn
Received: 16/4/2021
Revised: 20/5/2021
Accepted: 25/6/2021
1. INTRODUCTION
In the kinetic equation of industrial manipulator [26],
there are always external noise components and internal
noise. Especially the external noise components inside are
unknown or not exactly known and these are the
components that cause the movement of the hand
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P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY
versatile applications [15-20]. In this paper, a new DOB is planetary gear and Small air gap. It is influenced by friction
shown. form is based on the requirement of fast and such as static friction, friction, viscous friction and so on.
accurate estimation of robot disturbances. Based on the Therefore, the first three joints are the basic chain that
robot's dynamic model, the finite time control concept is ensures movement in 3D (X, Y, Z) space. The basis for the
used in the observer design. The resulting FTO can allow for study of the next steps in robot manipulator motion
estimation of the disturbance for a finite time, this ensures systems. The problem with the controller is that: should
that the estimated error can disappear after a certain time. design the quality control ensures precise orbit grip that
The proposed FTO also eliminates the need for acceleration does not depend on the parameters of the model
calculation. Its finite-time convergence feature makes uncertainty and the impact on channel mix between
observing perturbations more accurate and fast. The match-axis error between joint angles and the angle joints
structure of the paper is presented as follows: Part 1, actually put a small (< 0.1%).
problematic, Part 2 is the introduction of the Almega robot
control object 16. Part 3 is detailed about robot dynamics
and stability in time. finite time. Part 4 is an introduction to
the FTO Observer. Part 5 is the application of the FTO
controller to the Almega 16. Robot motion system. Part 6 is
the conclusion.
3. PRELIMINARIES
3.1. Robot Dynamic Model
The dynamic of an n-link rigid manipular, [1-4, 6] can be
written as
(1)
τ + τd = M(q)q+C(q,q)q+ G(q)+ τf
2. OBJECT CONTROL
Where q is the n x 1 joint variable vector, τ is a n x 1
generalized torque vector M(q) is the nxn inertia matrix,
The Almega 16 robot is shown in Figure 1, as follows
[25]. This is a vertical welding robot with fast, rhythmic and
precise movement characteristics, including six–link axes,
each one link axes is equipped with a permanent magnet
synchronous servo motor and closed loop control. In the
article using only three-link axes as the research object,
specifically the main specifications of the three joints as
follows.
H(q,q) is the n x 1 Coriolis/centripetal vector, G(q) is the n
x1 gravity vector.
n denotes the lumped friction effect
τd R
from both the motor and link sides and are always
described with the following Coulomb-viscous model,
namely
(2)
τf = F sgn(q)+F q
c
v
with
F diag F ,.....,F ,F diag F ,.....,F ,F ,F (1 i n)
cn vn
c
c1
v
v1
ci vi
are the Coulomb and viscous friction coefficients for the ith
joint. Such a friction model could capture most dynamic
property of the friction in a rigid joint. The equivalent
motor torque at the link side through a reduced
τ Rn ,τd Rn
amplification is denoted as
is the
internal/external disturbances which could be an external
force, unmodeled or uncertain robot dynamics. The exact
meaning of τd decides on the specific application. The
observed disturbance for a manipulator can be further
utilized in FDI and disturbance rejection control. For
example, τd is deemed as the physical impact with the
environment for collision detection scenario and, thus, τd
can indicate the occurrence of the collision. The robot
dynamics model in Equation (1) has the following property.
Figure 1. Six-link Almega 16 arm
First joint: Rotation angle: 1350. Center tops from top
to bottom: 28cm. Center line of axis I to the center of the
cylinder: 35cm. Second joint: Rotation angle: 1350. The
length between the center of the axis I and II is 65cm. Third
joint: Angle of rotation: 900 and -450. The length between
the two centers of axis I and II is 47cm. The total volume of
the Almega16 Robot: V = 0.12035m3. Total weight of the
robot: 250kg. The mass of joints is as follows: m0 = 100kg,
m1 = 67kg, m2 = 52kg, m3 = 16kg, m4 = 10kg, m5 = 4kg, m6 = 1kg.
In which 1: The matrix
is skew-symmetry
M(q)-2C(q,q)
[21], and it follows that
T
(3)
M(q) = C(q,q)+C (q,q)
3.2. Disturbance Observer GMO
In order to estimate the external beeb-type noise
components for hand-operated systems, there are many
different monitors. One of the most commonly used
observers is the observed (GMO) observed in Reference [9].
Combined with the generalized momentum p, in Equation
The motion system Almega16 Robot is a nonlinear
system that has constant model parameters and is
interfering with the channel between the component
motion axes. According to the literature as follows [26], the
first three joints have fully integrated the dynamics of the
freedom arm. The motor connected to the joint is usually a
(1) could be rewritten to
T
(4)
p τ +C (q,q)q- G(q)+ τd
The GMO component p is estimated as follows:
45
Website: https://tapchikhcn.haui.edu.vn Vol. 57 - No. 3 (June 2021) ● Journal of SCIENCE & TECHNOLOGY
KHOA HỌC
CÔNG NGHỆ
P-ISSN 1859-3585 E-ISSN 2615-9619
2 b α
T
zb2 = zb3 +M-1(q)τa +K
e
(14)
2
ˆ
ˆ
(5)
p = τ +C (q,q)q- G(q)+ τd
ˆ
3 b α
ˆ
(6)
τd = K0 (p -p)
3
zb3 = K
e
(15)
where ( )denotes the estimated value and
.
zb1 = q, zb2 = q, zb3 = M-1(q)τd
eb = q- q, K1, K2 , K3 Rnxn are diagonal gain matrices.
and
ˆ
ˆ
Where
. So the estimate of the external
K diag k 0
0i
0
ˆ
disturbance is given is:
T
Moreover, the corresponding powers are selected as
ˆ
ˆ
(7)
τd = K0 (τ+C (q,q)q- G(q)+ τd )dt
2
α1 = α,α2 = 2α -1,α3 = 3α -1 and < α <1. The operator
From equations (5) and (6), are determined:
3
ˆ
α
ˆ
(8)
τd = K0 (τd - τd )
.
is denoted as
or convert to a Laplace image which will be written in
the following format:
α
x
= x α sgn(x), x Rn and α > 0 (16)
K0
Consequently, the disturbance observation ˆ is
τd
(9)
τd
ˆ
τd =
s+K0
According to reference [9] shows that the component
ˆ is a first order inertial function τ . So the external
computed as
ˆ
τd = M(q)zb3 (17)
τd
d
From (17) shows, the proposed FTO is a ternary system
that can simultaneously estimate the joint velocity and the
external perturbation component. It shows that the joint
velocity can be obtained instantly from the robot control
system. From the formula (13) - (15), it is possible to
downgrade the observational equation for external
disturbance state variables leading to a reduction in the
computational heavy process. Therefore, the downgrade
FTO is determined as follows:
perturbation estimation ˆ component of the GMO will
τd
converge exponentially and depend on the observation
matrix Ko. Therefore, the GMO observer always has an
estimated bias in the outer perturbations.
3.3. Consider steady state in finite time
Consider the following nonlinear system
n
(10)
x = f(x), x R ,f(0) = 0
1 r α
(18)
-1
1
where f satisfies the locally Lipschitz continuous
condition. Some basic knowledge about finite time
homogeneity and stability (FTS) in the document [22,23].
zr1 = zr2 +M (q)τa +K
e
2 r α
2
zr2 = K
e
(19)
and
zb1 = q,zb2 = M-1(q)τd
4. FINITE-TIME OBSERVER OF ROBOTIC DISTURBANCE
ˆ
Where
The main content of this paper is to design a finite time
observer so that the observation of the noise td can
converge to its true value in a finite time. In this section 4
will be presented on the content of constructing the FTO
observer to estimate the external perturbations. After the
estimation is complete, the state variables estimate the
external perturbations to the existing control system such
as the PID controller,.... When the FTO Observer is
connected, the calculation and elimination will be reduced.
remove the inverse inertial matrix in the kinetic equation.
1
2
ˆ
eb = q- q,α1 = α,α2 = 2α -1, < α < 1. From formula (19),
we determine the formula to calculate the estimate of the
external disturbance is determined as follows:
ˆ
τd M(q)zr2 (20)
The decremented FOT observer will estimate the
perturbed state variables faster than the original
unremarked design. And from formula (12) shows that still
exists the inverse matrix component of M(q). To remove
the inverse inertial matrix component of M(q), it is
necessary to rearrange the original system from equation
(12) into a transformed equation with different state
variables. Multiplying both sides of Equation (12) by M(q)
yields the following:
4.1. Finite-Time Observer Design
From Equation (1), the acceleration qcan be written as
1
1
q = M (q)τd M (q)(τ C(q,q)q- G(q)- τf ) (11)
Put: τa = τ - C(q,q)q- G(q)- τf , then (11) is rewritten as
follows as:
M(q)q = τa + τd (21)
-1
-1
Additionally, the left side of Equation (21) could be
altered using the generalized momentum p, namely
q = M (q)τd +M (q)τa (12)
where M-1(q)τd is treated as the system disturbances
with M-1(q)τa the system input. According to reference
[20], the FTO monitor for manipulators is specifically
designed as follows:
p-M(q)q = τa + τd (22)
Reorganizing Equation (22) and employing Property 1,
the derivative of the generalized momentum p is rewritten
as
1 b α
p = τ + τ
1
zb1 = zb2 +K
e
(13)
(23)
p
d
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P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY
T
V
Where τp = M(q)q+ τa = τ+φ(q,q) = C (q,q)q- G(q)- τf .
V =
e
eT
n
The system should observe that the externally
implemented perturbation variables have been altered and
have different state variables. A set of FTOs reduced from
tertiary to quadratic is replaced as follows:
α
2
e -K epi
1
α
α
1
2
=
k
epi
- eTdK2 epi
2i
di
i=1
(29)
n
α
α
α
+α
2
1
2
2
= eTdK2 epi
-
k k epi
- edTK2 epi
1 m α
1
1i 2i
zm1 = zm2 + τp +K
e
(24)
i=1
n
2 m α
+α
2
α
2
1
zm2 = K
e
(25)
= - k k epi
0
1i 2i
i=1
ˆ
ˆ
ˆ
where zm1 = q,zm2 = τd and em = q- q .
Where k1i is a positive defined diagonal element of K1
and such that ep = ed = 0. Apply to LaShalle's theorem, to
ensure that the asymptotic convergence of the deviation e
to 0 is guaranteed. The next content of the paper will
present about demonstrating the observer's finite
convergence of time. According to Definition 2 [27],
Equations (28) and (29) are orderly homogeneous with
respect to weight. Hence, consider equations (28) and (29)
that have a negative identity. From Theorem 1, [27], the
error system is the global FTO. In other words, the
estimated deviations of the turbulent state variables will
disappear for a finite time. From that it can be concluded,
the proposed downgrade FTO is stable and with
convergent efficiency in finite time...
The control structure diagram with FTO is shown in
figure 2.
5. SIMULATION RESULTS
Afer building up the algorithms and control programs,
we will proceed to run the simulation program to test
computer program. The FTO was Simulink with Table 1.
Figure 2. The control structure diagram with FTO of robotic disturbance
finite-time observer
Table 1. The Parameter of FTO
The Parameter value of the
Symbol
The parameter
joint axis
The obtained FTO given in Equations (24) and (25) is
structurally similar to the GMO defined in Equations (5) and
(6) as both observer shares the same system states and
input. The obtained FTO observer is represented in
Equations (26) and (27) and has an estimation structure of
external perturbation variables similar to that identified in
Equations (5) and (6), since both of these observers use
system state variables and as input signals.
qd
Desired joint position
qd1 = qd2 = qd3 = sint
K1
K2
α
Scalar
K1 = [200, 200, 200]
K2 = [10000, 10000, 10000]
α = 1
Constant
Power coefficient
Disturbance
τd
sint
After simulation we have results position and position
tracking error is depicted Figure 3÷ 7.
4.2. Consider the stability and convergence of the FTO
The observation errors are given as:
* Desired joint position is sin(t)
1
α
ep = ed -K1 ep (26)
α
2
ed = -K2 ep
(27)
ˆ
ˆ
Where ep = p-p,ed = τd - τd ,e = [ep ,ed ] .
Using Lyapunov standard to prove the stability of the
FTO is proposed as follows;
Given a positive defined function, (28):
edTed
2
k2i
edTed
2
n
n
e
pi
α
α
+1
V =
k
τ
2 dτ +
=
epi
+
2
(28)
α +1
2i
0
i=1
i=1
2
where k2i is the ith diagonal element of K2. Then its
derivative is
Joint 1
47
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CÔNG NGHỆ
P-ISSN 1859-3585 E-ISSN 2615-9619
0.004
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
e3 = qd3 q3
-0.004
10
5
0
1
2
3
4
8
9
6
7
Joint 2
Time(s)
Figure 4. Express the response between the set angles q and real q
3000
2500
2000
1500
1000
500
0
-500
20
40
60
80
100
120
140
160
180
200
Joint 3
qd1
Figure 3. Performing deviation between the angles q set and q real
q
Figure 5. Express the control moments of the controller joints
1.2
1
0.8
0.6
0.4
0.2
0
0.004
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
e1 = qd1 q1
-0.2
0
2
4
6
8
10
Time (s)
12
14
16
18
20
Figure 6. Performing the response of the actual joint position and the actual
joint position
10
5
0
1
2
3
4
6
7
8
9
Time(s)
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0.004
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
0
0
5
10
Time(s)
15
20
10
5
1
2
3
4
6
7
8
9
Time(s)
Figure 7. The deviation response controls the actual joint position and
estimated joint position
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P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY
[11]. Haddadin S., De Luca A., Albu-Schäffer A., 2017. Robot collisions: A
survey on detection, isolation, and identification. IEEE Trans. Robot, 33, 1292–
1312.
Comment: The simulation results of the real state and
the above estimate prove that the synthesized FTO
observer is correct, showing that the joints of the Almega
16 robot are good with small setting time, less oscillation,
over-adjustment.
[12]. Cho C.N., Kim J.H., Kim Y.L., Song J.B., Kyung J.H., 2012. Collision
detection algorithm to distinguish between intended contact and unexpected
collision. Adv. Robot, 26, 1825–1840.
6. CONCLUSION
[13]. Mohammadi A., Tavakoli M., Marquez H.J., Hashemzadeh F., 2013.
Nonlinear disturbance observer design for robotic manipulators. Control Eng.
Pract., 21, 253–267.
The order reduction FTO has omitted acceleration and
inverse matrix matrix, resulting in the FTO making the
estimation deviations converge to 0 in a finite time.
Although FTO has a more complex formula and thus the
calculation time increases slightly compared to other
observers. The theory and simulation results show that
using the FTO observatory to estimate the perturbation
components outside the Almega 16 robot motion system
has ensured the real joint position closely follows the
estimated joint position and ensures the actual matching
position is closely related to the matching position with
small error.
[14]. Nikoobin A., Haghighi R., 2009. Lyapunov-based nonlinear disturbance
observer for serial n-link robot manipulators. J. Intell. Robot. Syst., 55, 135–153.
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Trường Đại học Kinh tế - Kỹ thuật Công nghiệp
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