PD with sliding mode control

Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

Contents lists available at ScienceDirect

Robotics and Computer-Integrated Manufacturing
journal homepage: www.elsevier.com/locate/rcim

PD with sliding mode control for trajectory tracking of robotic system
P.R. Ouyang n, J. Acob, V. Pano
Department of Aerospace Engineering, Ryerson University, Toronto, Canada

art ic l e i nf o

a b s t r a c t

Article history:
Received 21 August 2012
Received in revised form
7 August 2013
Accepted 14 September 2013
Available online 22 October 2013

Good tracking performance is very important for trajectory tracking control of robotic systems. In this
paper, a new model-free control law, called PD with sliding mode control law or PD–SMC in short, is
proposed for trajectory tracking control of multi-degree-of-freedom linear translational robotic systems.
The new control law takes the advantages of the simplicity and easy design of PD control and the
robustness of SMC to model uncertainty and parameter fluctuation, and avoid the requirements for
known knowledge of the system dynamics associated with SMC. The proposed control has the features
of linear control provided by PD control and nonlinear control contributed by SMC. In the proposed
PD–SMC, PD control is used to stabilize the controlled system, while SMC is used to compensate the
disturbance and uncertainty and reduce tracking errors dramatically. The stability analysis is conducted

for the proposed PD–SMC law, and some guidelines for the selection of control parameters for PD–SMC
are provided. Simulation results prove the effectiveness and robustness of the proposed PD–SMC. It is
also shown that PD–SMC can achieve very good tracking performances compared to PD control under the
uncertainties and varying load conditions.
& 2013 Elsevier Ltd. All rights reserved.

Keywords:
PD control
Sliding mode control
Robotic system
Stability
Trajectory tracking

1. Introduction
Because of its simple form and popularity in engineers, PD/PID
control has been widely used in many industrial applications such
as robotic control, process control, and automatic control [1–12].
PD/PID control is a model-free linear control, and the control gains
can be adjusted easily and separately. Indeed, a simple linear and
decoupled PD/PID controller with appropriate control gains may
lead to acceptable tracking performances for many applications.
It is well known that PD control with desired gravity compensation can guarantee global and asymptotic stability for a point-set
tracking problem [1,4]. However, such a design relies on prior
knowledge of the gravitational loading vector. The uncertainty of
parameters for a controlled system will affect the final tracking
performance of the controlled system. Researches on the global
stability of trajectory tracking with robotic manipulators under PD
control were given in [5–7]. Nunes and Hsu [7] proposed a causal
PD controller with feedforward terms for the global tracking control
of a robot manipulator where the derivative of tracking error was

estimated through a lead filter. PID control is also applied for
tracking control of robotic manipulators [8–11]. It was demonstrated [10,11] that a PID tracking control law with a feedforward
term can guarantee the semiglobal stability of robotic systems.
To improve the trajectory tracking performance of robot manipulators, significant efforts have been made for seeking advanced

n

Corresponding author. Tel.: þ 1 416 979 5000.
E-mail address: (P.R. Ouyang).

0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
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control strategies. Achievements were obtained in developing
adaptive control and robust control approaches that ensure globally
asymptotical convergence of tracking errors [12–14]. Sliding mode
control (SMC) [15–18] is one of the advanced controllers that have
been developed considerably in robotic areas.
SMC or variable structure control evolved from the pioneering
working in Russia in the early 1960s and has been studied extensively
to control nonlinear dynamic systems with modeling uncertainties,
time varying parameter fluctuation, and external disturbances [15,16].
SMC has been utilized in many different applications such as the
design of robust regulators, model-reference systems, adaptive schemes, tracking systems, and state observers. SMC was successfully
applied to problems such as automatic flight control, control of electric
motors, chemical processes, space systems, and robotics. The developments and applications of SMC are detailed in some literature
reviews [17,18].
SMC is characterized as high robustness. The sliding mode
behavior is insensitive to model uncertainties and disturbances.
Different types of SMC were proposed [19] to deal with tracking
problems. One problem associated with SMC is the so-called

chattering phenomenon that is high frequency oscillations of the
controller output making the trajectories rapidly oscillating about
the sliding manifold; another problem is the difficulty in the
calculation of what is known as the equivalent control where
certain knowledge of the system dynamics is required [15–20].
To avoid the calculation of equivalent control in standard SMC is a
motivation for the presented research.
A translational motion system, such as a CNC machine system,
is a simple robotic system driven by multiple axes [21–24]. In such


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P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

a robotic system, each axis is driven and controlled separately and
follows the command signal produced by the interpolator for the
purpose of coordination of the axes. One of the main requirements
for robotic systems is the good tracking performance of the
system, and many control algorithms, such as PD control and
SMC, are developed [21–24]. Another motivation of this research is
to develop a simple and effective control method for translational
motion systems.
A hybrid control scheme that switched between PD control and
SMC was proposed in [20] for tracking control of robot manipulators, where PD control is used in the reaching phase and the
semicontinuous sliding mode control is applied in the sliding
mode phase. That study is a start point for our current research. A
general goal of this current research is to find a simple and easy
control law for trajectory tracking performance improvement of
robotic systems. Considering the popularity and simplicity of PD

control in industrial applications, we focus on the combination of
PD control and SMC for the application of translational robotic
systems, and propose a new control law to deal with trajectory
tracking control problems.
This paper presents a new controller called PD–SMC that
combines PD control and SMC for trajectory tracking. In the
proposed approach, a PD control law is designed to stabilize the
nominal model and the SMC is used to provide the robustness and
to compensate the uncertainty and disturbance of the controlled
system. Model-free is a unique feature of the proposed PD–SMC
that is distinct from a standard SMC. This paper is organized as
follows. First, the dynamic model and the proposed PD–SMC
controller are discussed in Section 2. Then the stability analysis is
conducted in Section 3, followed by some simulation verifications
for complex shape tracking problems under different conditions in
Section 4. Finally, some conclusions are presented in Section 5.

denoted as M À N g 0.
For positive definite matrices, the following properties [25] will
be used in this paper:






If
If
If
If

If

M g 0 then M À 1 g 0.
M Z N g 0 then N À 1 Z M À 1 g 0.
M g 0 and λ 40 is a real number, then λM g 0.
M g 0 and N g 0, then M þ N g 0, MNM g 0, and NMN g 0.
MN ¼ NM, then MN g 0.

A translational robotic system can be described as a second
order system [21–23] as follows:
M X€ þC X_ þKX þ D ¼ F

where M, C, and K represent the mass, damping, and stiffness
matrices of the robotic system, respectively. X is the axis position
vector, F represents the control input force, and D is the combination of friction, disturbance, and model uncertainty that are
bounded.
We define the tracking error vector and its derivatives as
follows:
8
>
< E ¼ Xd À X
E_ ¼ X_ d À X_
ð2Þ
>
: E€ ¼ X€ À X€
d
where X d ; X_ d ; and X€ d are the desired position, velocity, and
acceleration vectors, respectively. We assume all these vectors are
bounded.
Substituting Eq. (2) into Eq. (1), the dynamic model can be

rewritten in the form of tracking errors as
M E€ þ C E_ þ KE ¼ P ÀF

Before starting the detailed discussion of the proposed PD–SMC
law, the following notations are introduced.
λm ðM Þ and λM ðMÞ represent the smallest and the largest
eigenvalues of a positive define matrix M, respectively. The norm
pffiffiffiffiffiffiffi
of a vector x is defined as ‖x‖ ¼ xT x, and the norm of a matrix M is
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


‖M‖ ¼ λM M T M .
If a square matrix M is positive definite, then it is denoted as
M g 0; if a square matrix M ÀN is positive definite, then it is

For a translational robotic system, it is well known that the
matrices M, C, and K are constant [21–23]. Assume that the desired
trajectories and the first and the second derivatives are bounded,
for the desired control force P, we have
P r ‖M X€ d þ C X_ d þKX d þ D‖ r ‖M X€ d þ C X_ d þKX d ‖ þ ‖D‖ ¼ P b
ð4Þ
where P b is the boundary of the control force P.
From Eq. (3), one can see that the system is stable and the
tracking error will converge to zero if we have F¼P. That is the

The desired eight curve shape
0.15

0.1

0.1

0.05

0.05
Y axis (m)

Y axis (m)

The desired ellipse
0.15

0

0

-0.05

-0.05

-0.1

-0.1

0.1

0.2

0.3

0.4

ð3Þ

where P ¼ M X€ d þ C X_ d þ KX d þ D represents the desired control
force vector.

2. Dynamic model and proposed PD–SMC law

0

ð1Þ

0.5

0.6

0

X axis(m)

0.1

0.2

0.3
X axis(m)

Fig. 1. The desired trajectory shapes.

0.4

0.5

0.6


P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

5

x 10-3

Tracking errors in x-axis

40
20

Forces(N)

Position errors(m)

3
2
1

-20

-2

3

0

x 10-3

0.5

1
Time(sec.)

1.5

-60

2

1.5

2

10

Forces(N)

5

-1

0
-5

-2

-10

-3

-15

x 10-3

1
Time(sec.)

Controlled Forces in y-axis for ellipse

0

2

0.5

15
PD
SMC
PD-SMC

0

0

Tracking errors in y-axis for ellipse

1

-4

PD
SMC
PD-SMC

-40

2

Position errors(m)

0

0
-1

0.5

1
Time(sec.)

1.5

-20

2

PD
SMC
PD-SMC
0

0.5

1
Time(sec.)

1.5

2

Controlled Forces in y-axis for eight curve

Tracking errors in y-axis for eight curve
60
40

1

20

0

Forces(N)

Position errors(m)

Controlled Forces in x-axis

60

PD
SMC
PD-SMC

4

191

-1

0
-20

-2

-3

PD
SMC
PD-SMC
0

0.5

1
Time(sec.)

PD
SMC
PD-SMC

-40

1.5

2

-60

0

0.5

1
Time(sec.)

1.5

2

Fig. 2. Comparison of tracking performances with uncertainties. (a) Control results in the x-axis for both shapes, (b) control results in the y-axis for ellipse shape and

(c) control results in the y-axis for eight curve shape.


192

P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

idea of computing torque control [26]. To find the controlled input
force F, the dynamic model should be known accurately, that is
only possible theoretically but not practically.
To solve this problem, we propose the following robust PD–SMC
law for trajectory tracking control of a translational robotic system.
F ¼ K P E þ K D E_ þ Η signðE_ þ λEÞ

ð5Þ

where K P and K D are the proportional and derivative control gains of
PD control, Η is the SMC gain, λ is the slide surface slope constant,
and sign() is the sign function.
Remark 1. From Eq. (5), one can see that the proposed PD–SMC
control law is a combination of PD control and SMC. Therefore,
it has the features of linear PD control and nonlinear SMC.
Comparing the proposed PD–SMC control law with a standard
SMC law [15–20], one can see that the PD control part in Eq. (5) is
used to replace the equivalent control part of standard SMC.
Remark 2. The proposed PD–SMC control law in Eq. (5) only
involves the tracking errors and the derivative of the tracking
errors and the dynamic model is not included in the control law.
One feature of the proposed PD–SMC is that it is a model-free
control law that is superior to a standard SMC where a normalized

2

model is needed in order to calculate the equivalent control part of
the standard SMC [16]. Therefore, it is easy to implement the PD–
SMC control law for real applications.
Applying Eq. (5) to Eq. (3), the controlled system can be written as


M E€ þ ðC þ K D ÞE_ þ ðK þK P ÞE ¼ P À Η sign E_ þ λE
ð6Þ

Theorem. Consider the translational robotic system (1) with the
proposed PD–SMC control law (5), the controlled system will be
globally stable and the final tracking error and its derivative are
convergent to zeros, provided that the control gains and parameters
are chosen as follows:
8
λ 40
>
>
>
>
< H Z Pb 4 0
ð7Þ
λm ðC þ K D Þ 4 λ U λM ðMÞ
>
>
>
>

:
λm ðK þ K P Þ 4 λ2 U λM ðMÞ
Remark 3. The conditions for choosing control parameters in
Eq. (7) are conservative. Such a conclusion will be demonstrated
through the following example verifications, see Section 4.4.

Tracking errors in x-axis

x 10-3

2

Tracking errors in y-axis for eight curve

0
Position errors(m)

0
Position errors(m)

x 10-3

-2
-4
-6

-2
-4
-6
PD

PD
-8
-10

-8

SMC
PD-SMC
0

0.5

1

1.5

-10

2

SMC
PD-SMC
0

0.5

Time(sec.)

10

Tracking errors in x-axis

x 10-3

1

10

x 10-3

PD

Position errors(m)

Position errors(m)

8

SMC
PD-SMC

6
4
2

SMC
PD-SMC

6

4
2
0

0
-2

2

Tracking errors in y-axis for eight curve

PD
8

1.5

Time(sec.)

0

0.5

1
Time(sec.)

1.5

2

-2

0

0.5

1

1.5

Time(sec.)

Fig. 3. Tracking errors for the eight curve shape under different initial errors. (a) Negative initial error and (b) Positive initial error.

2


P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

16

x 10-3

12

Controlled Forces in x-axis

120

PD
SMC

PD-SMC

14

PD
SMC
PD-SMC

100
80
60

10
Forces(N)

Position errors(m)

Tracking errors in x-axis

8
6

40
20

4

0

2

-20

0

-40

-2

0

0.5

1
Time(sec.)

1.5

-60

2

0

0.5

PD
SMC
PD-SMC

20
Forces(N)

Position errors(m)

2

30

0.015
0.01

10

0.005

0

0

-10

20

1.5

40

0.02

-0.005

1
Time(sec.)

Controlled Forces in y-axis for ellipse

Tracking errors in y-axis for ellipse

0.025

0

x 10-3

0.5

1
Time(sec.)

1.5

-20

2

PD
SMC
PD-SMC
0

0.5

1
Time(sec.)

1.5

2

Controlled Forces in y-axis for eight curve

Tracking errors in y-axis for eight curve

120

PD
SMC
PD-SMC

15

PD
SMC
PD-SMC

100
80
60

10

Forces(N)

Position errors(m)

193

5

40
20
0
-20

0

-40

-5

0

0.5

1
Time(sec.)

1.5

2

-60

0

0.5

1
Time(sec.)

1.5

2

Fig. 4. Comparison of tracking performances with varying friction and loading. (a) Control results in the x-axis for both shapes, (b) Control results in the y-axis for ellipse
shape and (c) Control results in the y‐axis for eight curve shape.


194

P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

If we choose the control gains according to (7), we can make
sure that
(
ðC þ K D Þ À λ M g 0
ð19Þ
K þK p g 0

3. Stability analysis

Preposition. Let matrix Q be a symmetric matrix expressed as
A

Q¼

B

B

T

!
ð8Þ

C

Let S be the Schur complement [25] of matrix A in Q, that is
S ¼ C À BT A À 1 B

ð9Þ

Then the matrix Q is positive definite if and only if A and S are
both positive definite [25]. It means
If A g 0 and S g 0, then Q g 0.
To prove the stability of the proposed PD–SMC control law,
first, we prove that the following matrix Q is positive definite:
"
#

K þK p λM
L¼
ð10Þ
λM
M
Proof. Choosing PD control gains to make sure that the matrix
K þ K p g 0; we know that matrix M is symmetric positive definite,
i.e., M ¼ M T ; M g 0. From conditions (7), we have
À
Á
λm K þ K p 4 λ2 λM ðM Þ
ð11Þ
From (11) we conclude
K þ Kp Àλ M g 0
2

ð12Þ

As K þ K p g 0 and M g 0, according to the property of positive
definite matrix, we have
M

À1

À λ ðK þ K P Þ
2

À1

g0

ð13Þ

Furthermore, based on (13) and M g 0, from the property
MNM g 0, we have
M À λ M ðK þ K P Þ À 1 M g 0
2

ð14Þ

It means that the first two items in Eq. (18) are negative
definite.
Based on condition (7), we have
ðE_ þ λE ÞΗ signðE_ þ λEÞ ¼ jE_ þ λE jΗ 4 jE_ þ λE jP b 4 ðE_ þ λE ÞP
T

T

T

T

T

T

T

T

ð20Þ
From Eq. (20), we have
ðE_ þ λET ÞðP À Η signðE_ þ λEÞÞ o0
T

ð21Þ

According to Eqs. (19) and (21), we can demonstrate
V_ r 0

ð22Þ

_
Since function V is a positive definite function and Vis
a
negative definite function, the robotic system in Eq. (1) controlled
by the proposed PD–SMC in Eq. (5) is globally asymptotically
stable based on the Lyapunov method, and the tracking error and
derivative are zeros.
It should be mentioned that the standard SMC law will cause
the controlled system chattering due to the switching action of the
control law of the sign function, and the same chattering problem
exists in the PD–SMC in Eq. (5). To avoid the chattering problem, a
saturation function can be chosen and the proposed control law in
Eq. (5) can be modified as follows:
F ¼ K P E þ K D E_ þ Η satððE_ þ λEÞ; ΦÞ

ð23Þ

where Φ is a constant diagonal matrix that determine the

boundary layer of the sliding surface.
8



 < signðE_ þ λEÞ if E_ þ λE 4 Φ
_


sat ðE þ λEÞ; Φ ¼
ð24Þ
: ðE_ þ λEÞ=Φ
if E_ þ λE r Φ

T

Considering M ¼ M , we have
À
ÁT
À
Á
S ¼ M À λM ðK þ K P Þ À 1 λM g 0

ð15Þ

According to the Preposition and Eq. (9), we prove that the
matrix L in Eq. (10) is positive definite.
Define the following Lyapunov function:
 
E

1
λ
VðEðt Þ; E_ ðt ÞÞ ¼ ð ET E_ T ÞL _ þ ET ðC þK D ÞE
ð16Þ
2
2
E
As the matrix L is positive definite and C þ K D is also positive
definite from Eq. (7), therefore we conclude that V is a positive
definite function. Applying Eq. (10) into Eq. (16) and differentiating
V with respect to time, we obtain
"
V_ ¼ ð E_ T

E€ T Þ

K þ KP
λM

# 
λM E
þ λE_ ðC þ K D ÞE
E_
M

T
T
T
¼ E_ ðK þK P ÞE þ ðE_ þ λET ÞM E€ þ λE_ M E_ þ λE_ ðC þ K D ÞE

ð17Þ

T
T
T
V_ ¼ E_ ðK þ K P ÞE þ λE_ M E_ þ ðE_ þ λET ÞðP À Η sgnðE_ þ λEÞÞ

À ðE_ þ λET ÞððC þK D ÞE_ þ ðK þ K P ÞEÞ þ λET ðC þ K D ÞE_
Á
TÀ
¼ À λET ðK þ K P ÞE À E_ C þ K D À λM E_
T

T

ε¼

Φ
λ

ð18Þ

ð25Þ

From Eq. (25), one can see that the maximum final tracking
error can be controlled by properly choosing the boundary layer Φ
and the slope constant λ.

4. Simulation verifications
In this section, a 2 DOF translational robotic system is used as

an example to demonstrate the effectiveness and robustness of the
proposed PD–SMC. The parameters of the motion system are
assumed as follows:
M¼

Substituting Eq. (6) into (17), we have

þ ðE_ þ λET ÞðP À Η sgnðE_ þ λEÞÞ

When the control law in Eq. (23) is used, the final tracking error
E is maintained within a guaranteed precision ε that is called the
boundary layer width [16], which can be obtained as follows:

10

0

0

5

!
C¼

10

0

0

10

!
K¼

20

0

0

20

!

An ellipse shape and an eight curve shape in XY plane are
tracked using different control methods. The desired trajectories
are defined as follows:
(
xd ¼ 0:3ð1 À cos ðωt ÞÞ
An ellipse shape :
yd ¼ 0:15 sin ðωt Þ


P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

(
An eight curve shape :

xd ¼ 0:3ð1 À cos ðωt ÞÞ

yd ¼ 0:15 sin ð2ωt Þ

In the simulations, the control gains are selected according to
(7) and shown as follows:

where ω ¼ π and t A ½0; 2Š s. The desired trajectories are shown in
Fig. 1.
In all the simulations, a normal uncertainty and a nonlinear
friction are assumed as
"
D¼

3x_ þ 0:2sgnðx_ Þ þ d1
2y_ þ 0:2sgnðy_ Þ þd2

#

-5

0

for ellipse shape tracking



 3000 0

 1800
 KD ¼ 
K P ¼ 


0
0
3000

0

40
20

SMC
PD-SMC
SMC
PD-SMC

0
-20
-40
-60

100

Error ranges for different PD gains

3
2
1
0
-1
-2

10
5

-5
-10
-15
-20

100

100

Factor of PD gains
Error ranges for different PD gains

SMC
PD-SMC
SMC
PD-SMC

0

-5

100
Factor of PD gains

Controlled force for different PD gains

60
40
20

SMC
PD-SMC
SMC
PD-SMC

0
-20
-40
-60

boundary

x 10-4

Factor of PD gains

Controller force boundary

Error boundary

5

SMC
PD-SMC
SMC
PD-SMC

0
boundary

4

Controlled force for different PD gains

15
Controller force boundary

5

Error boundary

Factor of PD gains

SMC
PD-SMC
SMC
PD-SMC

-3




1800 

Controlled force for different PD gains

60

100

x 10-4




600 

λ ¼ 100

Factor of PD gains

6




1000 

0

SMC control

Error ranges for different PD gains

SMC

PD-SMC
SMC
PD-SMC

0


 1800
K D ¼ 
0

Controller force boundary

Error boundary

x 10-4

PD control

 3000
K P ¼ 
0

for eight curve shape tracking

where d1 and d2 are random noise signals to simulate the
uncertainty and disturbance.

5

195

100
Factor of PD gains

Fig. 5. Tracking performance under different PD gain factors, (a) Tracking performance in the x-axis, (b) Tracking performance in the y-axis for the ellipse shape and
(c) Tracking performance in the y-axis for the eight curve shape.


196

P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200


0 

50 


 100
H ¼ 
0


0 

100 


 0:08

Φ ¼ 

0




0:06 
0


 0:08
Φ ¼ 
0

for ellipse shape tracking




0:08 
0

for eight curve shape tracking
It should be mentioned that the condition in Eq. (7) is conservative, which lets to more flexible selections of control gains in the PD–
SMC control law. In the following parts, different cases are examined
to deal with different situations. All the parameters and control gains
listed above are used in PD control, SMC, and PD–SMC.

x 10-4

10

5

0

6

300

x 10-4

100

2
1
0
-1

SMC
PD-SMC
SMC
PD-SMC

-100
-200

100

3

Controlled force for different SMC gain

0

Error ranges for different SMC gain
SMC
PD-SMC
SMC
PD-SMC

4

200

-300

100
Factor of SMC gain

5
Error boundary in y-axis

Error ranges for different SMC gain
SMC
PD-SMC
SMC
PD-SMC

-5

First of all, the proposed PD–SMC is applied for the tracking control
of an ellipse shape and an eight curve shape, and the comparisons
with PD control and SMC are presented in Fig. 2. Fig. 2(a) shows the
tracking errors and the required control forces in the x-axis for both
shapes, and Fig. 2(b) and (c) shows the tracking errors and the
required control forces in the y-axis for the ellipse shape and the
eight curve shape, respectively. From this figure, we can see that the
proposed PD–SMC obtained much better tracking performance than
PD control, and a slight better performance than SMC. It should be
noticed that the SMC control method needs prior knowledge of the
dynamic model of the controlled system, but PD–SMC is a model-free
control method. On the other hand, the required control forces for

Controller force boundary

Error boundary in x-axis

15

4.1. Tracking control with uncertainty and noise

Controller force boundary


 100
H ¼ 
0

50

100
Factor of SMC gain
Controlled force for different SMC gain
SMC
PD-SMC
SMC
PD-SMC

0

-50

-2
-3

8

-100

100
Factor of SMC gain
x 10-4

Error ranges for different SMC gain

300

4
2

SMC
PD-SMC
SMC
PD-SMC

0
-2
-4

Controller force boundary

Error boundary in y-axis

6

200
100

100
Factor of SMC gain
Controlled force for different SMC gain
SMC
PD-SMC
SMC
PD-SMC

0

-100
-200

-6
-8

-300
100
Factor of SMC gain

100
Factor of SMC gain

Fig. 6. Tracking performance under different SMC gain factors. (a) Tracking performance in the x-axis, (b) Tracking performance in the y-axis for the ellipse shape and (c)
Tracking performance in the y-axis for the eight curve shape.


P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

corresponded tracking errors in Fig. 2, one can see that, after passing a
very short period of time (around 0.2 s), the tracking performances are
almost the same for PD–SMC and SMC methods. It demonstrated the
fast response speed of the PD–SMC with initial error conditions.
It should be mentioned that a similar conclusion is obtained for the
tracking control of an ellipse shape under initial error conditions. For
this scenario, the standard SMC is better than the PD–SMC in terms of
the response speed for correcting the initial errors.

both SMC and PD–SMC are larger than those controlled by PD control.
Such a result is anticipated, as much control effort should be paid in

order to achieve more accurate tracking performance.
4.2. Tracking control with initial errors and disturbances
To check the response speed of the proposed PD–SMC with respect
to initial errors, simulations are conducted and Fig. 3 shows two
different initial errors conditions for the eight curve shape tracking
where Fig. 3(a) is a case of negative initial errors of 0.01 m for both
axes while Fig. 3(b) is the case of positive initial errors of 0.1 m for both
axes. From this figure, one can see that both SMC and PD–SMC have
very fast responses to overcome the initial errors, while PD control is
much slower to correct the initial errors. Comparing Fig. 3 with the
x 10-4

4.3. Tracking control with varying conditions
To verify the robustness and effectiveness of the proposed
PD–SMC, a case with varying friction and mass (loading) at different
time is simulated and compared with other control laws. It is assumed
Controlled force for different boundary layer

Error for different boundary layer
SMC
PD-SMC
SMC
PD-SMC

5

0

-5

150
Controller force boundary

Error boundary in x-axis

10

100
50
0
-50

-150

100

x 10-4

100
Factor of boundary layer

4
3

Controlled force for different boundary layer
60
Controller force boundary

Error boundary in y-axis

Error for different boundary layer
SMC
PD-SMC
SMC
PD-SMC

5

2
1
0
-1
-2
-3

8

x 10

0
-20

SMC
PD-SMC
SMC
PD-SMC

-40

100

Factor of boundary layer

Factor of boundary layer
Controlled force for different boundary layer
150

4
2

SMC
PD-SMC
SMC
PD-SMC

0
-2
-4
-6
100
Factor of boundary layer

Controller force boundary

Error boundary in y-axis

20

Error for different boundary layer

6

-8

40

-60

100

-4

SMC
PD-SMC
SMC
PD-SMC

-100

Factor of boundary layer
6

197

SMC
PD-SMC
SMC
PD-SMC

100

50
0
-50
-100
-150

100
Factor of boundary layer

Fig. 7. Tracking performance under different boundary layers, (a) Tracking performance in the x-axis, (b) Tracking performance in the y-axis for the ellipse shape and
(c) Tracking performance in the y-axis for the eight curve shape.


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P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

that an additional friction of 50jvj (v is the velocity of the axis)
is added at t¼2/3 s, and an additional loading of 10 kg is added at
t¼ 4/3 s, respectively. Fig. 4 shows the tracking errors and the required
control forces for the ellipse shape and the eight curve shape under
three different control methods. From this figure, one can observe
that there are no significant changes at the changing points for both
SMC and PD–SMC when the varied conditions are added. For the
varying conditions, it is clearly shown that the PD–SMC can obtain
very good tracking performances, while PD control had much
degraded tracking performance. This simulation demonstrated the
robustness and effectiveness of the proposed PD–SMC to compensate the varying frication and the loadings.

8

x 10-4

4.4. Control parameters effect on tracking performance
In previous subsections, comparisons of tracking performances are
conducted under different conditions based on the same control
parameters for all three control methods, and it is demonstrated that
the PD–SMC can ensure very small tracking errors in all the situations
compared with PD control. In this subsection, the effect of control
parameters on tracking performances between PD–SMC and standard
SMC are examined. In the simulations, five different factors to the
normal control parameters, which are set as 0.2, 0.5, 1, 2, and 5, are
considered in the tests to explore the effects of control parameters on
the tracking errors and control forces.

Error for different slope constant

150

4
SMC
PD-SMC
SMC
PD-SMC

2
0
-2
-4
-6

x 10-4

Error boundary in y-axis

4

50
0
-50
-100

Error for different slope constant

50

SMC
PD-SMC
SMC
PD-SMC

5

100

-150

100
Factor of slope constant

Controller force boundary

6

Controller force boundary

Error boundary in x-axis

6

3
2
1
0
-1

-50

x 10-4

Error for different boundary layer

Error boundary

4
2

SMC
PD-SMC
SMC

PD-SMC

0
-2
-4
-6

100
Factor

150
Controller force boundary

6

-100

100
Factor of slope constant

SMC
PD-SMC
SMC
PD-SMC
100
Factor of slope constant
Controlled force for different slope constant

0

-2
-3

Controlled force for different slope constant

100
50

SMC
PD-SMC
SMC
PD-SMC
100
Factor of slope constant
Controlled force for different boundary layer
SMC
PD-SMC
SMC
PD-SMC

0
-50
-100
-150

100
Factor

Fig. 8. Tracking performance under different slope of slide surfaces. (a) Tracking performance in the x-axis, (b) Tracking performance in the y-axis for the ellipse shape, and
(c) Tracking performance in the y-axis for the eight curve shape.

P.R. Ouyang et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 189–200

4.4.1. The effect of PD gains
First, we examine the effect of PD control gains on tracking
performance for trajectory tracking control of the two considered
shapes. Different factors of PD control gains, from 0.2 to 5 times of
the normal values, are used to control the trajectories. Fig. 5 shows
the comparison results for tracking errors and control forces
between PD–SMC and standard SMC. From Fig. 5, one can see that
PD–SMC is slightly superior to the standard SMC for all the selected
PD control gains in terms of reducing tracking errors. But there is no
significant difference for the tracking errors and the control forces
under different PD control gain situations. It also demonstrates that
the conservative condition of Eq. (7) for the choice of the PD control
gains. Even for small control gains where Eq. (7) is invalid, the
tracking errors are still in a small range. Such a result can be
explained as follows: the PD control part of the proposed PD–SMC
mainly contributed to the normal stabilization of the controlled
system, forcing the tracking errors in the boundary layer of the slide
surface. After entering the boundary layer, the tracking performance
is dominantly controlled by the SMC control part.
4.4.2. The effect of SMC gain
In this simulation, different factors for the SMC gain H are used
to check the effects on tracking performances. Fig. 6 shows the
results under five different factor levels. From this figure, one can
see that the tracking performances did not improve by increasing
the SMC gain, rather the required control forces increase dramatically. Therefore, a reasonable SMC gain is good for the reduction
of the required control forces, and a very high SMC gain is not

necessary in terms of the small trajectory tracking errors and
limited control forces. It also shows that for relatively large SMC
gain, the proposed PD–SMC is better than the standard SMC in
terms of reducing tracking errors.
4.4.3. The effect of boundary layer
Boundary layer has some effects on the required control forces.
Fig. 7 shows the tracking performances for different boundary
layer situations. Generally speaking, the larger the boundary layer,
the larger the tracking errors for the PD–SMC, and the smaller the
control force for PD–SMC and standard SMC. If a large boundary
layer is selected, a relatively large tracking error is allowed in the
control process, and a smaller and smoother control force is
required. To balance the tracking error and the control force, a
proper choice for the boundary layer is helpful for good tracking
performance.
4.4.4. The effect of the slope of slide surface
In a large range of the slope constant λ of the slide surface, the
tracking performances controlled by the PD–SMC is better than
the standard SMC, see Fig. 8. It is observed that larger control
forces are required if a big value of the slope λ is chosen. Under the
same boundary layer condition, a large value of λ implies that a
small value of the velocity error is required; therefore, a large
control force is required to push the tracking errors approach to
the slide surface.

5. Conclusions
In this paper, we studied the trajectory tracking control
problem of robotic systems and proposed a new PD–SMC control
method. PD–SMC is a combination of PD control and SMC, having
the advantages of linear control and nonlinear control in the sense

of simplification and robustness of the control law. The proposed
PD–SMC control law is an feedback control law that only involves

199

the tracking errors and the derivative of the tracking errors. One
advantage of the proposed PD–SMC is that it is a model-free
control law that is distinct from a standard SMC. The simplicity
and easy design of the PD–SMC is another advantage compared
with a standard SMC. Simulation results demonstrated that PD–
SMC is superior to PD and as good as a standard SMC in terms of
good tracking performance under the uncertainties, disturbances,
and varying load conditions.
Different levels of the control parameters are used to examine
the effects on tracking performances. From the simulation results,
it demonstrated that the proposed PD–SMC is slightly better than
a standard SMC in terms of reducing tracking errors in most levels
of the factors. It also showed that the variations of the PD control
gains do not have significant effect on the tracking errors and the
control forces. Such a conclusion came from the nature of the
PD control in the proposed PD–SMC, which is used to bring the
tracking errors to the boundary layer of the slide surface. On the
other hand, the SMC parameters have large effects on the tracking
errors and especially the required control forces. It concluded that
a relatively low SMC gain, a smaller slope of the slide surface, and
large boundary layer can make the required control forces in small
ranges for the proposed PD–SMC.
From the simulation results, one can see that the control
parameters of the PD–SMC have significant and complicated
effects on tracking performance and the required control forces.

Therefore, it is a challenge for the proper selection of control gains
when applying the proposed PD–SMC. In future work, we will
conduct the optimization of control parameters based on the
genetic algorithm or particle swarm optimization.
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