Super-twisting sliding mode based nonlinear control for planar dual arm robots

Bulletin of Electrical Engineering and Informatics
Vol. 9, No. 5, October 2020, pp. 1844∼1853
ISSN: 2302-9285, DOI: 10.11591/eei.v9i5.2143 r 1844
Super-twisting sliding mode based nonlinear control for
planar dual arm robots
Tung Lam Nguyen, Huu Tinh Vu
School of Electrical Engineering, Hanoi University of Science and Technology, Vietnam
Article Info
Article history:
Received Jan 22, 2020
Revised Mar 10, 2020
Accepted Apr 12, 2020
Keywords:
Dual arm robot
Nonlinear control
Siding mode control
Super-twisting
ABSTRACT
In this paper, a super-twisting algorithm sliding mode controller is proposed for a
planar dual arm robot. The control strategy for the manipulator system can effec-
tively counteract chattering phenomenon happened with conventional sliding mode
approach. The modeling is implemented in order to provide the capability of maneu-
vering object in translational and rotational motions. The control is developed for a
2n-link robot and subsequently simulations is carried out for a 4-link system. Com-
parative numerical study shows that the designed controller performance with good
tracking ability and smaller chattering compared with basic sliding mode controller.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Tung Lam Nguyen,
School of Electrical Engineering,
Hanoi University of Science and Technology, Ha Noi, Vietnam.
Email: lam.nguyentung@hust.edu.vn
1. INTRODUCTION
Initially, robots are designed with an idea of developing a mechanism to replace/support human when
interacting with the environment [1-4]. Along with the speedy evolution of control engineering, computer
science, and mechanical technologies, robots have an inclination toward possessing the human form [5-7]. One
of the earliest humanoid robots but currently attracted researcher’s attentions is a dual arm manipulator [8].
In comparison to a single arm manipulator, the dual arm manipulator is capable of handling sizable objects,
objects that having a certain degree of flexibility, and assembling mechanical parts [9]. In recent years, with
the development of modern dual-arm robots (DAR), this concept becomes more and more important for the
automation of processes in the life sciences [10]. Due to the human-like structure of the robotic arms, it can
used for a wide number of application.
Various applications of in automatic measurement field in [11-13]. The other applications are also
used in industrial manipulation tasks for performing automatic assembly operations [14, 15], or using for
automation of cell production system for cellular phones [16]. Primarily designed to work in coordinated
operations, the dual arm manipulator suffers from inherent kinematic and strong dynamic couplings. The phe-
nomenon poses a challenge in dual arm manipulator control problems. The situation become even worse when
the bi-manual manipulator holding a rigid object due to the formation of closed kinematic chains. Individual
arms are dynamically engaged through the manipulated object, thus in coordinated modes, two arms operate in
constrained motions. This leads to a requirement of complex control schemes that can handle the manipulator
and object dynamics. Various control methods have been developed for the . Classical algorithms as X.Yun at
el. [17] apply nonlinear feedback techniques approach, N. Sarkar at el. [18] propose control scheme integrates
hybrid position/force control and vibration suppression. In paper [19], a fuzzy force control framework is
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Bulletin of Electr Eng & Inf ISSN: 2302-9285 r 1845
proposed for dual-industrial robot systems. Besides, there are also many researches consider adaptive control
algorithms for the . Zhi Liu at el. [20] use an adaptive neural control scheme to deal with the problem of
output hysteresis. N. Yagiz at el. [21] develop a sliding mode controller to investigate the performance of two
coordinated planar arms in transporting a load to its new location with friction-assisted handling. In [22], the
authors propose a sliding mode control combined with a parameter estimator for the robot system. The result
show a practical approach since many parameters are not convenient to measure. In this study, we develop a
generalized model for model with 2n degrees of freedom and introducing a control approach for trajectory
tracking using super-twisting algorithm. The feasibility of the control is confirmed using the 4 DoF model
to numerically investigate tracking quality and the adaptability to the disturbance of the super-twisting sliding
mode controller.
2. MATHEMATICAL MODEL
The configuration of model consisting of two planar arm robot that is shown in Figure 1. Each arm
robot is composed of a set of links connected together by revolute joints. In this study, the model with 2n
joints is assumed through that all joints have only a single Degree-of-Freedom (DoF). With the ith
joint, we
associate a joint variable, denoted by θi. Additional, mi, Ii, Li represente the mass, mass moment of inertia
and length of related links, respectively. The mass of load is denoted by m(t), and , d1, d2 denote the width of
the rectangular load and the distance between the bases of the robot arms.There is viscous frictions acting on
all of the joints denoted by bi.
Figure 1. model coordinates
The forward kinematic problem is to determine the position and orientation of the end-effector, given
the values for the joint variable of the robot. To perform the kinematic analysis of 2n links model, we represent
used Denavit–Hartenberg conventions that can be referred to [23]. By using the generalized frame oxyz which
is shown in Figure 1, the kinematic equations of right arm robot are



xn = L1 cos(θ1) + L2 cos(θ1 + θ2) + ... + Ln cos(θ1 + θ2 + ... + θn) +
d2
2
yn = L1 sin(θ1) + L2 sin(θ1 + θ2) + ... + Ln−1 sin(θ1 + θ2 + ... + θn)
(1)
Kinematic equations of left arm robot in the generalized frame oxyz are



x2n = Ln+1 cos(θn+1) + Ln+2 cos(θn+1 + θn+2) + ... + L2n cos(θn+1 + θn+2 + ... + θ2n) −
d2
2
y2n = Ln+1 sin(θn+1) + Ln+2 sin(θn+1 + θn+2) + ... + L2n sin(θn+1 + θn+2 + ... + θ2n)
(2)
Super-twisting sliding mode based nonlinear control for planar dual arm robots (Tung Lam Nguyen)

1846 r ISSN: 2302-9285
(a) When the load at fixed position (b) When the load at fixed position
Figure 2. Representation of forces, (a) When the load at fixed position and (b) When the load at fixed position
Where as the kinematic equations describe the motion of the robot, the dynamic equations explicitly
describe the relationship between force and motion. Firstly, we concerned about the term of dynamic which
handling the load. Representation of forces acting on load is illustrated in Figure 2 (a). Where F1, F2 are forces
of right arm, left arm tips to the load. The friction forces Fs1 Fs2 and their components Fs1xy, Fs2xy, Fs1z,
Fs2z between the arm tips and the load surface. The force F1, F2 can be write in frame oxy as
F1x = −F1 cos ϕ − Fs1xy sin ϕ, F1y = −F1 sin ϕ + Fs1xy cos ϕ
F2x = F2 cos ϕ − Fs2xy sin ϕ, F2y = F2 sin ϕ + Fs2xy cos ϕ (3)
Next, we used the Euler-Lagrange equations to derive the dynamical equations in the n-link robot. Following
[23], the motion equation of model in matrix form as
D(q)q̈ + C(q, q̇)q̇ − g(q) = τ (4)
when the robot perform the transportation, the motion of robot can be effect by external disturbances, viscous
friction forces and load torques. The motion equation of model after handling the load were given as
D(q)q̈ + C(q, q̇)q̇ − g(q) = τ + τ0 − ω − β (5)
where state variables q = [θ1, θ2, ..., θ2n]T
; τ0 is the load torque ; ω, β is 2n × 1 vectors represent for the
external disturbance torques and the viscous friction forces, respectively. We specified as
ω = [ω1, ω2, ..., ω2n]
T
, β =
h
b1θ̇1, b2θ̇2, ..., b2nθ̇2n
iT
(6)
3. CONTROLLER DESIGN
3.1. Sliding mode controller
The state space form of model with joint variables q = [θ1, θ2, ..., θ2n]T
can be written as
q̈ = D(q)−1
[τ0 − ω − β − C(q, q̇) − g(q)] + D(q)−1
τ (7)
The state space form of model can be rewritten as form of a non-linear dynamic system
ẋ = f(x, t) + B(x, t)u (8)
Where the state variables in the state space x ∈ R4n
, x = [q, q̇]T
= [θ1, θ2, ..., θ2n, θ̇1, θ̇2..., θ̇2n]T
.
The state equations without the control inputs f(x, t) ∈ R4n
, f(x, t) = [θ̇1, θ̇2..., θ̇2n, f1, f2, ..., f2n]T
with the
component [f1, f2, ..., f2n]T
= D(q)−1
[τ0 − ω − β − C(q, q̇) − g(q)]. Meanwhile, u ∈ R2n
is generalized
torque input vector and B = [zeros(2n), D(q)]T
∈ R4n×2n
is the elements of which are the coefficients of the
Bulletin of Electr Eng & Inf, Vol. 9, No. 5, October 2020 : 1844 – 1853

generalized control inputs in the state equations. With ∆(x) = xr − x are the difference between the reference
value and the system response. We define the sliding surface in the state space form as
s = [G]∆(x) = [G][e ė]
T
(9)
Where s ∈ R4n
, λi(i = 1, 2, ..., 2n) are the coefficients of sliding surface, and [G] ∈ R2n×4n
as
[G] =





λ1 0 0 0 1 0 0 0
0 λ2 0 0 0 1 0 0
0 0
... 0 0 0
... 0
0 0 0 λ2n 0 0 0 1





(10)
In sliding mode controlled systems, the control input is changed to drives, and maintains the system
states on a sliding surface[24], [25]. Therefore, consider the control law u= ueq (s = 0)
uN (s 6= 0)
Where the equivalent control ueq is the value the switched signal to maintain sliding, and which is
formally obtained as the solution to the algebraic equation ṡ = 0 when s = 0, must satisfy
ṡ = [G]ẋr − [G][f(x, t) + B(x, t)ueq] = 0 (11)
ueq = [GB(x, t)]−1
[Gẋr − Gf(x, t)] (12)
Let define Lyapunov candidate function has to be positive definite V (s) = 1
2 sT
s. To drive x reach the
sliding surface the uN signal have to make sure V̇ (s) = sT
ṡ < 0 for s 6= 0. Therefore, uN = ueq + ∆ we
have
ṡ = [G]ẋr − [G][f(x, t) + B(x, t)(ueq + ∆)] (13)
= −[G]B(x, t)∆ = 0 (14)
If we denoted that ṡ = −[K]sign(s) with
[K] =





k1 0 0 0
0 k2 0 0
0 0
... 0
0 0 0 k2n





(15)
∆ = [GB(x, t)]−1
[K]sign(s) (16)
Combining with (12), the total control input is found by
u = [GB(x, t)]−1
[Gẋr − Gf(x, t) + [K]sign(s)] (17)
3.2. Super-twisting sliding mode controller
The sliding set of order r defined by s = ṡ = s̈ = ... = s(r−1)
= 0. The first-order sliding mode tries
to keep s = 0, second-order sliding mode verified s = ṡ = 0. We purpose [z1, z2]T
= [s, ṡ]T
, the problem of
second-order sliding mode is reduced to the stabilization in finite time of the auxiliary system as below
(
˙
z1 = z2
˙
z2 = ϕ(x, t, u) + γ(x, t)u̇
(18)

1848 r ISSN: 2302-9285
Where,





ϕ(x, t, u) =
∂ṡ
∂t
+
∂ṡ
∂x
[f(x, t) + B(x, t)u]
γ(x, t) =
∂ṡ
∂x
B(x, t)
(19)
The super-twisting algorithm to stabilize the robot manipulator requested that for some positive constants
q, C, K1, K2, U, and,
|ϕ̇(·)| + U|γ̇(·)| ≤ C, 0 < K1 ≤ γ(·) ≤ K2, |
ϕ(·)
γ(·)
| < qU, 0 < q < 1
The super-twisting sliding mode controller is given by
usw = −β
p
|z1|sign(z1) + u1 (20)
1 = −u (|u| > U)
αsign(z1) (|u| ≤ U) The effective control signal u for super-twisting algorithm consists of two terms: u =
ueq + usw. Similar sliding mode controller, ueq is equivalent control signal that guarantees s = ṡ = 0 or
ṡ + λs = 0.
[G]ẋr − [G][f(x, t) + B(x, t)ueq] + λs = 0 (21)
ueq = [GB(x, t)]−1
([Gẋr − Gf(x, t)] + λs) (22)
The term of usw is control signal of super-twisting algorithm. Hence, combining (20) and (22) the super-
twisting algorithm have the control input
u = [GB(x, t)]−1
([Gẋr − Gf(x, t)] + λs) − β
p
|z1|sign(z1) + u1 (23)
4. RESULT AND DISCUSSION
4.1. Trajectory planning
To illustrate the effectiveness and applicability of super-twisting sliding mode controller, we design
the super-twisting sliding mode controller for model with four DoF (n = 2), and simulation results are
compared with conventional sliding mode We used point-to-point motion for the trajectory planning, which
is specified initial and final configuration of end-effector. In this case, the inverse kinematic solution must be
used to convert end-effector configuration to a sequence of joint configurations. The reference trajectories are
given by
x1(t) = xf1 + (xi1 − x(f1))e−10t2
y1(t) = yf1 + (yi1 − y(f1))e−10t2
x2(t) = xf2 + (xi2 − x(f2))e−10t2
y2(t) = yf2 + (yi2 − y(f2))e−10t2
Where x1(t), y1(t), x2(t), y2(t), are the reference trajectories of right arm and left arm robot;
xi1, yi1, xi2, yi2, are the start positions, xf1, yf1, xf2, yf2, are the destination positions. Each periods, the
robot represents the amount of time t = 2s. In last period reach to the final configuration, the robot must to
rotate ϕ = −(π/4) + (π/4)e−10(t−4)2
. Hence, the reference trajectories in this period t = 4 → 6s is
x1(t) = xm + (xi1 − xm)e−10t2
+
d1
2
cos ϕ y1(t) = ym + (yi1 − ym)e−10t2
+
d1
2
sin ϕ
x2(t) = xm + (xi2 − xm)e−10t2
−
d1
2
cos ϕ y2(t) = ym + (yi2 − ym)e−10t2
−
d1
2
sin ϕ

4.2. Numerical simulation
From Figure 3(a), we can see that, both the basic sliding mode controller and super-twisting sliding
mode controller solve successfully the problem of trajectory tracking. The actual value of the rotation angle at
the joints are very closely track to the reference value. Besides, we also find that the quality of super-twisting
controller is better when the maximum deviation angle at joints is only about 0.4 × 10−3
rad, while in sliding
mode controller is about 0.9 × 10−3
rad. The value of rotation of the object respect to the ox axis in both
controllers is very close to the set value. In the period from 0 to 4 seconds this deviation angle is equal to 0
degrees, and starts from the 4th second onward. The object begins to rotate with the trajectory as described
in above, until the end of the cycle at the 6th second the object’s rotation angle respect to ox is 45 degrees.
The deviation made by the basic sliding controller is clearly larger the deviation made by the super-twisting
sliding mode controller.
(a) The reference and the actual joint angles
(b) Error tracking of the joint variables
Figure 3. The reference and the actual joint angles, (a) The reference and the actual joint angles,
(b) Error tracking of the joint variables
The Figures 4 (a) and (b) depict the torque of joints and the interaction forces between the robot
arm and the object, respectively. The control torque signals as well as the interaction forces from the basic
sliding mode controller are greatly vibrated (chattering), while in the super-twisting sliding mode controller,

1850 r ISSN: 2302-9285
the vibration range and vibration amplitude are very small. Also from the simulation results we see that the
force acting on the object is match with the requirement to move the object according to the trajectory. From
second to 4th second, force F2 is greater than F1 to move objects from left to right; the forces Fs1xy and Fs2xy
are equally assured so that the object does not rotate in the direction of oz because at this time the object is
moving in parallel with the ox axis. From the 4th to the 6th seconds, during this period the object is rotating
so that the forces Fs1xy and Fs2xy are no longer equal. Summary, in the absence of noise, the quality of the
super-twisting sliding mode controller is better than the basic sliding mode controller. We assume the noise
signal is a function of time ω = [sin 10πt sin 20πt − sin 10πt − sin 20πt]T
. It can be seen that both
controllers are affected by noise, especially in the basic sliding mode controller, when the noise appears, the
vibration amplitude and frequency of the control torque signal and the interaction force also increase. For super-
twisting controller, the effect of noise is much smaller than the basic sliding mode controller. In general, both
two controllers response well to the the requirements of trajectory control. However, in terms of chattering
reduction and feasibility in practical, the super-twisting sliding mode controller will be more dominant than
the basic sliding mode controller as shown in Figure 5 and The numerical simulation values can be found in
Table 1.
(a) Torque control of the joints
(b) Interaction forces
Figure 4. Control torques and interaction forces, (a) Torque control of the joints,

(a) Error tracking of the joint variables
Figure 5. Result simulation with uncertain disturbance torque, (a) Error tracking of the joint variables,
Table 1. Simulation parameters of model
m1 = m3 = 5(kg) L1 = L3 = 0.5(m) d2 = 0.4(m) b1 = b2 = b3 = b4 = 100(Nms)
m2 = m4 = 4(kg) L2 = L4 = 0.4(m) d1 = 0.2(m) mload = 0.5(kg)
I1 = I3 = 0.1(kgm2) k1 = k3 = 0.25(m) µ = 0.3 g = 9.8(m2)
I2 = I4 = 0.08(kgm2) K2 = K4 = 0.2(m)
5. CONCLUSION
The paper successfully built a generalized model for model 2n degrees of freedom. According to
an analysis of the numerical results, the super-twisting sliding mode controller shows adaptability external
disturbances, trajectory tracking while the robot handling and transportation the object. Furthermore, the
super-twisting sliding mode controller has restricted significantly chattering phenomenon which is the diffi-
cult problem of the basic sliding mode controller.
ACKNOWLEDGEMENT
This research is supported by Hanoi University of Science and Technology research grant
T2018-PC-053.

1852 r ISSN: 2302-9285
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BIOGRAPHIES OF AUTHORS
Tung Lam Nguyen received the B.S degree in Control and Automation Engineering from Hanoi
University of Science and Technology, Hanoi, Vietnam, 2005, the M.S degree from Asian Institute
of Technology, 2007, and the Ph.D from The University of Western Australia, 2014. He is current
working as a lecturer at Department of Industrial Automation, School of Electrical Engineering,
Hanoi University of Science and Technology. His research interests include motion control, control
system and its applications.
Huu Tinh Vu received the B.S degree in Control and Automation Engineering from Thuyloi Uni-
versity, Hanoi, Vietnam, 2018. He is current working as a student for M.S degree at Department
of Industrial Automation, School of Electrical Engineering, Hanoi University of Science and Tech-
nology. His researches are in fields of electronics, digital systems, signal processing, manipulator
systems.

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