A New Robust Finite-Time Control Approach for Robotic Manipulators
Dongya Zhao1,3, Shaoyuan Li1*, Quanmin Zhu4, Feng Gao2
1 Institute of Automation, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China
2 School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China
3 College of Mechanical and Electronic Engineering, China University of Petroleum, Dongying, 257061, China
4 Faculty of CEMS, University of the West of England, Coldharbour Lane, Bristol BS16 1QY, UK
*Corresponding Author’s E-mail: syli@sjtu.edu.cn
Abstract
In this study, a new robust finite-time stability control approach for robot systems is developed based on
finite-time Lyapunov stability principle and proved with backstepping method. The corresponding stability
analysis is presented to lay a foundation for theoretical understanding to the underlying design issues as well as
safe operation for real systems. A case study of a two-link robot model is presented to demonstrate the
effectiveness of the proposed approach.
Keywords: Robust control, finite-time stability, stabilization, robot control.
1
Introduction
Generally, robot dynamic behaviors can be described by a class of second order multi-dimensional nonlinear
1
models. Due to unknown or changing payload, friction, backlash and flexible joints, the system uncertainties
are frequently encountered in robot operations. It has been noticed that neglecting the effects induced by model
uncertainties can significantly decrease performance in terms of tracking accuracy and attainable velocity.
Robust control is one of the important approaches to deal with uncertain systems. Owing to its simple structure
and easy implementation, various robust control schemes have been developed for robot manipulators [1-4].
Though these robust control algorithms have achieved many remarkable successes in both theory and
applications, most of them are merely guaranteed by asymptotic stability [5-8], which require infinite time to
converge to system equilibrium states. To achieve fast convergence, the control gains of asymptotic stability
control (ASC) need to be greatly increased. The high gain request is undesirable and can not be implemented in
practice sometime. Consequently, it is important to develop a fast convergent and strong robust control
approach accommodating both theory and applications for robot control.
Recently, finite-time stability control (FTSC) has drawn an increasing attention in nonlinear control system
design. Obviously from its name, the FTSC approach can stabilize system states to equilibrium in a finite time.
Roughly speaking, there are four state feedback based approaches for nonlinear system finite-time control [9],
time optimal control, such as bang-bang control [10, 11], homogeneous system finite-time stability approach
[12, 13], finite-time Lyapunov stability approach [14, 15] and terminal sliding mode control (TSMC) method
[16, 17]. Compared with ASC, the FTSC offers some superior properties such as fast response, high tracking
precision, strong disturbance rejection and insensitivity to system uncertainty [18-22]. These properties are
particularly useful for high precision control of robot manipulators. The efforts to develop finite-time control
for robot systems have been found in TSMC [23, 24] and nonsmooth proportional-derivative (PD) based
control [25]. Regarding to the TSMC scheme, the system states can reach a nonlinear finite-time convergent
sliding mode, namely, terminal sliding mode (TSM) in a finite time, then converge to the equilibrium along the
2
TSM in a finite-time. Literature [23] presented a discontinuous non-singular TSMC for rigid robot. Literature
[24] developed a continuous TSMC with fast TSM-type reaching law, which can avoid the chattering and
makes the tracking error converging to a residual set under system uncertainty. By using the homogenous
system results on finite-time stability, literature [25] developed two FTSC approaches for robot systems:
inverse dynamics plus nonsmooth PD control and gravity compensation plus nonsmooth PD control. In its
conclusions, literature [25] claimed that two challenging issues have not been resolved completely. One is the
relation of values of the controller parameters to the setting time the other is the solution uniqueness in forward
time. Since the requirement of uniqueness of the solution in forward time is not guaranteed in some robots,
such as revolute joint robot manipulators, this condition limits the range of application of the gravity
compensation nonsmooth PD control. Especially, system uncertainties were not considered in literature [25]. As
the survey papers [26, 27] claimed, the uncertainties are frequently encountered in robotic systems, which must
be coped with by some techniques to guarantee performance in terms of tracking accuracy.
To advance the research in this field, a novel robust finite-time stability control (RFTSC) approach is
developed in this study. Based on the results of finite-time Lyapunov stability [18, 19] and differential
inequalities [28, 29], the proposed approach can stabilize the tracking errors to zero in a finite time. In addition
to solving the two challenging problems presented by [25], system uncertainties are addressed explicitly in this
study. The corresponding finite-time stability is proved with backstepping method. As a result, a novel
systematic finite-time stability controller design procedure is proposed for controlling of generic robots. It
should be mentioned that the main difference between the current study and reference [25] is on the control
algorithms and the stability analysis. The RFTSC is developed in light of the results on finite-time Lyapunov
stability and differential inequalities with backstepping method. The control approaches [25] are developed
from the homogenous system results with finite-time stability. In summary, this study takes into the following
3
two considerations. The first is, for applications, the RFTSC provides a systemic design procedure for robot
manipulators finite-time control in the presence of system uncertainty. Thus the RFTSC may offer an
alternative, but more effective, for robot control. The second is, for theory, finite-time convergence has been an
important and challenging topic in theoretical studying. Recently, finite-time stability control for nonlinear
systems has been extensively studied [20-22]. Hopefully, to establish a basis for further development, the study
can provide a new insight and application incentive in aspect of the theoretical development.
The rest of this paper is organized as follows. In Section 2, the problem formulation is given. In Section 3,
the main results of this paper are presented as two theorems with proofs. In Section 4, a case study is described
to initially validate the proposed approach. Finally, in section 5, concluding remarks are given.
2
Problem formulation
Consider the following general robot system model [23, 24, 30]
M  q  q  C  q, q  q  G  q   
where q, q, q  R
n
(1)
are the vectors of joint angular position, velocity and acceleration, respectively.
M  q   M 0  q   M  q   Rnn
is
symmetric
and
positive
definite
inertia
matrix,
C  q, q   C0  q, q   C  q, q   Rnn and C  q, q  q is the vector of centrifugal and Coriolis torques,
G  q   G0  q   G  q   Rn is the vector of gravitational torques,   R n is the vector of applied joint
torque. Here M 0  q  , C0  q, q  and G0  q  are nominal parts, whereas M  q  , C  q, q  and
G  q  represent the perturbations in the system matrices. Then the dynamical model of robot system (1) can
be rewritten as [26, 27]
M 0  q  q  C0  q, q  q  G0  q       q, q, q 
where
(2)
  q, q, q   M  q  q  C  q, q  q  G  q   Rn is the lumped system uncertainty.
4
Remark 1
It is realistic to suppose that the dynamics of robot system is known partially. In robot robust
controller design, it is often to make an assumption that the dynamics of robot system is known partially. This
technique has been successfully used in many robot robust control literatures such as [2, 23, 24, 26, 27, 34].
The robot dynamic model has the following properties [30]
(P1) The matrix M  q  satisfies
M  q   m for some constants m  0 for all q .
(P2) The vector G  q  satisfies G  q   g for some constant g  0 for all q .
In this paper,  denotes L2 norm for vector and induced norm for matrix, respectively.
In this study, we consider the set-point control of robot systems. Suppose q  R
d
the position error and velocity error are defined as x1  q  q
d
n
is the desired position,
and x2  q , respectively. Equation (2) can be
written as the following second order multi-dimensional nonlinear model
x1  x2



x  M 01  x1  q d  C0  x1  q d , x2  x2  G0  x1  q d       x1  q d , x2 , x2 

 2


(3)
Finite-time stability requires essentially that a control system is stable in the sense of Lyapunov and its
trajectories tend to equilibrium in a finite time. The definition of finite-time stability and some lemmas that will
be used in the stability analysis are attached in Appendix.
The previous main results for finite-time control of system (3) without considering system uncertainty, that is,
  x1  q d , x2 , x2   0 can be described as follows [25]
The inverse dynamics nonsmooth PD regulator


  G  x1  q d   C  x1  q d , x2  x2  M  x1  q d  l1sig  x1   l2 sig  x2  
1

2

(4)
The gravity compensation nonsmooth PD regulator
  G  x1  qd   l1sig  x1   l2 sig  x2 
1
where 0  1  1 ,
2
(5)
2  21 1  1  , l1  0 and l2  0 . The notations sig  y  , y  Rn , 0    1

5
are defined as in [31] sig  y    y1 sign  y1  ,



Remark 2
T
, yn sign  yn   .


Three issues, the explicit estimation of setting time, the solution uniqueness in forward time and
system uncertainty, have not been adequately addressed in control laws (4) and (5). The first issue relies on the
explicit construction of the Lyapunov function. Though it is possible to find a Lyapunov function for the closed
loop system constructed from controller (4), it does not seem to be easy to do so for constructing the closed
loop system from controller (5) since the closed loop system is not homogeneous. For the second issue, the
stability analysis of closed loop system by controller (4) has been satisfactorily achieved. However, for the
closed loop system from controller (5), the results rely on assumption of uniqueness of the solution in forward
time. The uniqueness condition may be verified for prismatic joint robots, but may not be guaranteed for
general robot systems. For the third issue, system uncertainty has not been considered in both controller designs.
Especially, when gravitational vector has parameter perturbation the condition of Lemma 3 of the literature [25]
is not held. Without the support of Lemma 3 of [25], system stability can not be analyzed by using the
homogeneous system results. It should be noticed that system uncertainties are frequently encountered in robot
control in practice, it must be coped with in the controller design to guarantee performance.
Suppose
  x1   Rn with   0   0 , z  x2    x1  , then model (3) can be written as

x1    x1   z


z  M 01  x1  q d  C0  x1  q d , x2  x2  G0  x1  q d       x1  q d , x2 , x2     x1 



Remark 3

(6)
  x1  is a stabilizing state feedback control law for x1    x1  , which will guarantee
finite-time stability of x1 . The terms
  x1  and   x1  in expression (6) can be designed by backstepping
method [32, 33]. The specified formulations of
  x1  and   x1  will be given in the following part of this
paper.
The objective of this paper is to derive a control approach to accommodate the issues rising in Remark 2 and
6
guarantee the finite-time stability of expression (6) under the following assumptions [23, 24, 34].

(A1) The matrix C x1  q , x2
d

C  x1  q d , x2   0  1 x1  q d   2 x2
satisfies
2
for some
constant  0 , 1 ,  2  0 .
(A2) The joint torque vector
 satisfies
  0  1 x1  q d  2 x2
2
for some constants
0 , 1 , 2  0 .
Under assumptions (A1)-(A2) and properties (P1)-(P2), literature [34] demonstrated the following
assumption for system uncertainty.

(A3) The system uncertainty  x1  q , x2 , x2
d

satisfies
  x1  q d , x2 , x2   b0  b1 x1  q d  b2 x2
2
for some constants b0 , b1 , b2  0 .
(A4) The desired joint position q
d
is bounded constant vector.
(A5) The joint angular position q and velocity q are measurable.
Remark 4
The bounds of robot system uncertainty can be measured by the polynomial
b0  b1 x1  q d  b2 x2 . Parameters b0 , b1 and b2 can be determined by  0 , 1  2 , 0 , 1 , 2 ,
2
m , g . The method to estimate  0 , 1  2 can be found in literatures [1, 23, 24, 26, 27, 34]. The values of
1 , 2 , 3 are related to parameters of robot dynamic equation and controller gains. The details of how to
obtain 1 , 2 , 3 have been studied by literatures [23, 24, 34]. The details to estimate b0 , b1 , b2 have
been proved in [34]. The successful application of this system uncertainty measure method can be found in
literature [23] and [24]. In the controller synthesis, it only requires the values of b0 , b1 , b2 . The values of
 0 , 1 ,  2 , 1 , 2 , 3 are not explicitly required in the controller synthesis. They are only important in
stability analysis. In practical application, the parameters b0 , b1 , b2 can also be obtained by using trial and
error method. Gradually increase them from zero until the performance is satisfied.
7
3
Robust finite-time control for robotic manipulators
This is the main section of the study. The proposed control approaches will be developed based on finite-time
Lyapunov stability principles with backstepping design method. The limiting case that supposes
  x1  q d , x2 , x2   0 will be studied firstly in this section. Similar to inverse dynamics nonsmooth PD
regulator (4), the accurate robot system model is known a prior and can be used in the controller design. The
result of the limiting case is given as Theorem 1. Under the inspiration of Theorem 1, the general case, namely,
  x1  q d , x2 , x2   0 is further studied. The corresponding result is summarized as Theorem 2. Though the
controller is designed under assumption that the dynamics of robot system is known partly opposed to the
gravity compensation nonsmooth PD regulator (5) that supposes only the gravity vector is known exactly, the
Theorem 2 gives a more effective approach in the viewpoint of applications.
3.1
Finite-time control without system uncertainty


When the system uncertainty  x1  q , x2 , x2  0 the control law is designed as
d


  x1    K1sig  x1 



  C0  x1  q d , x2  x2  G0  x1  q d   M 0  x1  q d    x1   x1  K 2 sig  z 



where
K1  diag k11 ,
, k1n  , K2  diag k21 ,

(7)
, k2n  are positive definite diagonal matrices,
0    1.
It is obvious that the derivative of
  x1  will be infinite as x1i  0 and x1i  0 in expression (6) and
(7). To avoid this problem, the following definition of
 x1i  1 x1i

 1

i  x1i    i x1i

0


  x1  is presented.
x1i  0
and
x1i  0
x1i  0
and
x1i  0
(8)
x1i  0
8
where x1i is the i-th element of vector x1 ,
positive constant, i  1,
Remark 5
i  x1i  is the i-th element of vector 1  x1  ,  i is a small
,n .
With the specified
i  x1i  , the singularity can be avoided in the control law. It should be noticed
that it is different from the approach to cope with singularity problem in TSMC [34]. The control law of TSMC
is set to 0 as arbitrary position error x1i  0 . A small positive constant  i is employed in this study to cope
with singularity problem without making the control law
  0 while the position error x1i  0 .
Substituting (7) into (6), yields the following closed loop system dynamic equation


 x1   K1sig  x1   z



 z   K 2 sig  z   x1
(9)


Theorem 1 Under assumptions (A4)-(A5), if system uncertainty  x1  q , x2 , x2  0 , the equilibrium
 x1, x2    0,0
Proof
d
of the closed loop system (6) subjected to control law (7) is globally finite-time stable.
The proof proceeds with backstepping method [32, 33].
Consider Lyapunov function V1 
1 T

x1 x1 for differential equation x1   K1sig  x1  . Differentiating V1
2
with respect to time, yields
n
V1   x1T K1sig  x1    k1i x1i

1
(10)
i 1
Define k1_ min  min k1i  , the expression (10) satisfies the following inequality
n
V1   k1_ min  x1i
1
(11)
i 1
Using Lemma 3 (see Appendix), leads to the following expression
 n

V1   k1_ min   x12i 
 i 1 
Define  
1
2
(12)
1  1
   1 , k1  2 k1_ min , the expression (12) can be written as
,
2
2
V1  k1V1
(13)
9
It is obvious that k1V1 is positive semidefinite. According to Lemma 1 (see Appendix), the equilibrium of
differential equation x1   K1sig  x1 

T1 
is globally finite-time stable with the setting time estimation
1
1
V1  x1  0   .
k1 1  
Take the Lyapunov function V2  V1 
1 T
z z for closed loop system dynamic equation (9), differentiating
2
V2 with respect to time, yields
V2   x1T K1sig  x1   z T K 2 sig  z 


(14)
Define k2_ min  min k2i  , k2  2 k2 _ min , the following inequality can be established


1 n

V2  k1V1  k2   zi2 
 2 i 1 

(15)
According to Lemma 2 (see Appendix), (15) yields


1 n

V2  k  V1    zi2    kV2
 2 i 1  


(16)

where k  min k1 , k2 .
Consider the Lemma 1, the origin of system (9) is a global finite-time stable equilibrium, the setting time
estimation is T2 
Remark 6
1
1
V2  x1  0  , x2  0   .
k 1  
□


Theorem 1 suppose that the system uncertainty  x1  q , x2 , x2  0 , control law (7) leads to
d
the closed loop equation (9) which satisfies the assumptions and the conditions given in Lemma 1. It should be
noticed that control law (4) [25] is developed by using homogenous system results on finite-time stability [12,
13], control law (7) of this paper is proposed in light of the finite-time Lyapunov stability approach [18, 19]. In
viewpoint of application, control law (7) can cover the control law (4).


When system uncertainty  x1  q , x2 , x2  0 , the uniqueness of solution in forward time is not
d
guaranteed in general because of the complexity of the closed loop systems. Control law (5) can not work in
10
this situation. Lemma 1 can not be employed directly to check the finite-time convergence for the robot systems
with uncertainty. In the rest of this section, the second theorem of this paper will address this problem by using
the results on differential inequalities [28, 29] (Definition 2, Lemma 4 and Lemma 5 in Appendix).
3.2 Robust finite-time control with system uncertainty


When system uncertainty  x1  q , x2 , x2  0 the control law is designed as
d
  x1    K1sig  x1 

   0  1

(17)

 0  C0  x1  q d , x2  x2  G0  x1  q d   M 0  x1  q d    x1   x1  K2 sig  z 


 z T M 1 x  q d T

0  1

b0  b1 x1  q d  b2 x2
 1   z T M 1  x  q d 
0
1

0

where
K1  diag k11 ,

, k1n  , K2  diag k21 ,

2


z 0
z 0
, k2n  are positive definite diagonal matrices,
0    1 ,   x1  is defined as expression (8), b0 , b1 , b2  0 are positive constants.
Remark 7
The control law  1 is employed to cope with system uncertainty [26, 27].
A control gain tuning strategy is proposed as follows. First, tune the control gains K1 , K 2 and
 using
trial and error method. Second, gradually increase b0 from zero. Meanwhile, b1 and b2 is also increased
from zero. Finally, the previously tuned gains may need to be changed slightly utilizing a trial and error
method.
Substituting (17) into (6), leads the following closed loop system dynamic equation


x1   K1sig  x1   z



1
d
d
 z   K 2 sig  z   x1  M 0  x1  q    x1  q , x2 , x2    1

Theorem 2 Under assumptions (A1)-(A5), the equilibrium
 x1, x2    0,0

(18)
of the closed loop system (6)
11
subjected to control law (17) is globally finite-time stable.
Proof The proof proceeds with backstepping method.
Consider Lyapunov function V3  t  
1 T

x1 x1 for differential equation x1   K1sig  x1  , we can obtain
2
the result like Theorem 1, that is
 n

V3   k1_ min   x12i 
 i 1 
where  
1
2
  k1V3  t 
(19)
1  1
   1 , k1  2 k1_ min .
,
2
2
To prove the equilibrium of differential equation x1   K1sig  x1 

is finite-time stable, the following
proof will proceed in two steps.
First, prove condition (i) of Definition 1 (see Appendix).
It is obvious k1V3  0 , then V3  0 . Because expression (19) has x1 , then x1 is bounded in terms of L2
norm [32]. From the differential equation x1   K1sig  x1  , x1 is also bounded. According to Barbalat’s

lemma [32], x1  0 as t   . It means that the equilibrium of x1   K1sig  x1 

is global
asymptotically stable. It is also implied that the equilibrium is satisfied with global Lyapunov stability.
Second, prove condition (ii) of Definition 1.
According to Lemma 5, V3 =0 as t  t0 
x1  0 as t  t0 
1
1
V3  t0  , t 0 is the initial time. It also means that
k1 1  
1

1
V3  t0  . It shows that the equilibrium of x1   K1sig  x1  is finite-time
k1 1  
convergent.
Based on Definition 1, the equilibrium of x1   K1sig  x1  is globally finite-time stable. The setting time

can be estimated by T3  t0 
1
1
V3  t0  .
k1 1   
12
Take the Lyapunov function V4  V3 
1 T
z z for closed loop system dynamic equation (18), substituting
2
 1 into (18), then, differentiating V4 with respect to time along the resulting equation, leads to the following
expression

V4   x1T K1sig  x1   zT K 2 sig  z   z T M 01  x1  q d    z T M 01  x1  q d  b0  b1 x1  q d  b2 x2



2

V4   x1T K1sig  x1   z T K 2 sig  z   z T M 01  x1  q d    z T M 01  x1  q d  b0  b1 x1  q d  b2 x2


V4   x1T K1sig  x1   zT K2 sig  z   zT M 01  x1  q d 


 b  b
0
1
x1  q d  b2 x2
2
  
(20)
2
 (21)
(22)
According to (A3), it yields
V4   x1T K1sig  x1   z T K 2 sig  z 


(23)
First, we prove that the condition (i) of Definition 1 is held.
It is obvious that x1 K1sig  x1   0 , z K 2 sig  z   0 , then V4  0 . Since x1 and z appear in (23),

T

T
they are bounded in terms of L2 norm [32]. Following the definitions of
  x1  and z , we conclude that
x2 is bounded. According to (A4), q d and q d are bounded constant vector. Thus, q , q and q are
bounded. From (18), we know z is bounded as well. Consider the definitions of
  x1  and z again, x2
is bounded. Therefore, x1 and x2 are uniformly continuous since x1 and x2 are bounded. Form
Barbalat’s lemma [32], x1  0 and x2  0 as time t   . This means that the origin of (18) is globally
asymptotic stable. It is obvious that global asymptotic stability implies global Lyapunov stability.
Second, we show that condition (ii) of Definition 1 is satisfied.
Similar to Theorem 1, we have
V4  kV4


where k  min k1 , k2 ,  
(24)
1  1
 1.
,
2
2
Using Lemma 5, we have V4  0 as t  t0 
V31  t0 
k 1  
, t 0 is initial time. It also means that x1  0 ,
13
z  0 as t  t0 
1
1
V3  t0  . According to the definition of z , x2  0 as x1  0 and z  0 .
k1 1  
This shows finite-time convergence.
According to Definition 1, the origin of system (18) is a global finite-time stable equilibrium, the setting time
can be estimated by T4  x   t0 
Remark 8
1
1
V4  t0  .
k 1  
□
It should be noted that control law (5) presented by [25] is based on the homogenous system
results on finite-time stability. Since it requires uniqueness of the solution in forward time, control law (5) can
only be used in some special robots such as prismatic joint robots. By employing the results on differential
inequalities [28, 29], Theorem 2 can guarantee the finite-time stability of robot control systems without
requiring the uniqueness of solution in forward time condition. Then control law (17) proposed by this paper
can be applied to more general robot manipulators.
Remark 9
System uncertainties frequently encountered in robot control practice must be coped with in robot
controller design to guarantee specified performance [26, 27]. This issue is not resolved in the study of [25].
Therefore its control law (5) can not be applied to deal with parameter perturbation incurred in gravitational
vector. The new study has accommodated system uncertainty in design of control law (17).
Remark 10
Theorem 2 extends the result of Theorem 1 indeed. In case of no uncertainty, that is,
  x1  q d , x2 , x2   0 , Theorem 2 is equivalent to Theorem 1.
Remark 11
Since discontinuous term  1 of control law (17) is employed to cope with system uncertainty,
chattering phenomenon will occur in practical implementation. The following robust saturation approach can be
used to remove chattering effect in a practical controller implementation [1, 26, 27, 32]
14


 z T M 1 x  q d T

0  1

b0  b1 x1  q d  b2 x2
T

1
d
 z M  x  q 
0
1
1  
 z T M 1 x  q d T

0  1

b0  b1 x1  q d  b2 x2



where



2

z 
(25)
2

z 
  0 a small positive constant denotes saturation limitation. Under robust saturation approach,
system states will converge to a small residual set containing equilibrium point. The residual set can be made
arbitrarily small by choosing
 small enough. In the limit, as   0 , it will recover the performance of
discontinuous controller. In practice, trial and error method can be employed to obtain
 . Gradually increase
it from zero to an appropriate value. Of course, from practical viewpoint, to obtain higher control precision,
should be chosen as small as feasible. However, too small
selection of
about

 will cause chattering phenomenon. Thereby, the
 should be tradeoff between chattering phenomenon and control precision. The more details
 can be found in the literatures [1, 26, 27, 32].
Remark 12
If the controller parameter
  1 , the finite-time stable control law (7) and the robust finite-time
stable control law (17) will become the asymptotic stability control law (26) and robust asymptotic stability
control law (27), respectively. The control laws (26) and (27) can be developed by directly using the
backstepping design procedure [32, 33] with the robust control technique for robotic manipulators [26, 27].

  x1    K1 x1


d
d
d

  C  x1  q , x2  x2  G  x1  q   M  x1  q    x1   x1  K 2 z


  x1    K1 x1

    0  1
(26)
(27)
 0  C0  x1  q d , x2  x2  G0  x1  q d   M 0  x1  q d    x1   x1  K 2 z 


 z T M 1 x  q d T

0  1

b0  b1 x1  q d  b2 x2
 1   z T M 1  x  q d 
0
1

0


2

z 0
z 0
15
Remark 13
The output of real-life actuator is bounded by a saturation value which they will never be able to
exceed [35, 36]. If initial values of q  0  and q  0  are chosen appropriately near the desired q
then
d
d
and q ,
M  x1  q d   m , C  x1  q d , x2   c and G  x1  q d   g . Suppose m, c, g  T , T is the
saturation bound of actuator.
Illumined by Remark 7 of literature [25] and literature [35, 36], a variation of control law (7) can be given as




  x1    s K1sig  x1 



  s C  x1  q d , x2  x2  G  x1  q d   s M  x1  q d    x1   x1  K 2 sig  z 






 
 , 0  L  T . By using this saturation controller, local finite-time stability can be
L
where s     Lsat 
achieved in the free of system uncertainty for actuator saturation case. The robot finite-time control approach is
currently being studied in the presence of system uncertainty for actuator saturation case and will be reported as
soon as it is completely.
Remark 14
If control gains K1 and K 2 are positive definite diagonal matrices, finite-time stability can be
guaranteed by using the proposed approach. Large K1 and K 2 can achieve fast convergence rate, but too
large control gains will excite high frequency dynamics of robot system. High gains are not expected in
industrial application sometime. The control gain matrices K1 and K 2 can be obtained by using trial and
error method. Gradually increase them from zero until the performance is satisfied.
Remark 15
In general, there are two reference tracking control methods for robotic manipulators. One is
point-to-point control the other is trajectory tracking control. A point-to-point robot can be taught a discrete set
of points in its joint space, but there is no control on the path of the joint in between the taught points. Set point
control method is a foundational approach for point-to-point control [30]. In this study, the proposed approach
is finite-time stability set point control method for robotic manipulators. Then, it is possible to consider
16
reference tracking by using our approach in practice.
4
Performance validation for two-link rigid robot system
In this section, we will study the performance of the RFTSC by using a general two-link rigid robot [23, 24]
11  q2  12  q2    q1      q2  q1



 22   q2  
0
12  q2 
2  q2  q1   q1    1  q1 , q2  


 g 
  q2  q2   q2   2  q1 , q2  
where
11  q2    m1  m2  r12  m2r22  2m2r1r2 cos  q2   J1
12  q2   m2 r22  m2 r1r2 cos  q2 
 22  m2 r22  J 2
  q2   m2 r1r2 sin  q2 
 1  q1 , q2    m1  m2  r1 cos  q2   m2r2 cos  q1  q2 
 2  q1 , q2   m2r2 cos  q1  q2 
The parameter values were assigned as r1  1m , r1  0.8m , J1  5kgm , J1  5kgm , m1  0.5kg
and m1  1.5kg . The desired joint position signals were set as q1  1rad , q2  1rad . The initial values of
d
d
the system were selected as q1  0   0 , q2  0  0 , q1  0   0 and q2  0  0 .
Four typical cases were studied. Case 1 was used to demonstrate the Theorem 1, the performance of the
FTSC and the ASC were obtained in the free of system uncertainty, respectively. Case 2 was used to
demonstrate Theorem 2 and the performance of applying the RFTSC in the presence of system uncertainty.
Case 3 was used for further testing the effectiveness of the RFTSC to cope with system uncertainty while the
robot manipulator picked an object ( 0.5kg ) up suddenly at t  5sec . Case 4 was used for testing the
performance of RFTSC at high frequency in the presence of system uncertainty. During 3.5sec  t  5sec ,
17
high frequency force disturbances 10sin 100t  Nm were added to both of two joints of robot manipulator.
For comparison purpose, the performances of corresponding asymptotic stability control law (26) and (27)
were also demonstrated in the following four case studies.
Robot manipulators are multivariable system. For showing coupling effects in such a multivariable system,
step inputs were commanded in each joint position at different instants. q1 was commanded at t  0 , q2
d
d
was commanded at t  0.2sec in Case 1 and Case 2. q1 was commanded at t  0 , q2 was commanded
d
d
at t  1sec in Case 3 and Case4.
Case 1
In the free of the system uncertainty
The control parameters of finite-time stability controller (7) were chosen as K1  diag 1.8,1.8 ,
K2  diag 1.8,1.8 and   3 5 . The control parameters of asymptotic stability control controller (26)
were chosen as K1  diag 1.8,1.8 , K2  diag 1.8,1.8 .
Fig. 1 shows the performance of FTSC in the free of the system uncertainty, solid line is the desired position,
dashed line is the joint position. From Fig. 1, we can see FTSC stabilizes joint 1 arrives at steady state after
t  1.7sec , joint 2 enters steady state after t  1.9sec . For comparing purpose, the performance of ASC (26)
which is a limiting case of control law (7) with
  1 is illustrated in Fig. 2. From this figure, we can see the
ASC needs more time to stabilize the robot system to arrive at steady state. It is obvious that the FTSC exhibits
the superior behavior such as finite-time convergence, fast response and high tracking precision in comparison
with ASC. Since step inputs were commanded in each joint position at different instants, the coupling effects
can be seen from Fig. 1 and Fig. 2. These two figures also show the proposed approach can cope with coupling
effects effectively.
Case 2
In the presence of the system uncertainty
ˆ 1  0.4kg , m
ˆ 2  1.2kg . The boundary
The nominal values of m1 and m2 were specified as be m
18
parameters of system uncertainty in (A3) were assumed to be b0  0.9 , b1  0.1 and b2  0.1 . The control
parameters of control law (17) were chosen as K1  diag 1.8,1.8 , K2  diag 1.8,1.8 and
  3 5.
Same nominal values and boundary parameters were chosen for asymptotic stability control law (27). The
control parameters of control law (27) were chosen as K1  diag 1.8,1.8 , K2  diag 1.8,1.8 .
Table 1: Nominal values and perturbations
m̂1
0.4kg
m1
0.1kg
 20% 
m̂2
1.2kg
m2
0.3kg
 20% 
Fig. 3 illustrates the performance of RFTSC in the presence of the system uncertainty. Fig. 4 shows the
performance of the limiting case of control law (17) with
  1 , that is, control law (27). From Fig3, we can
see both of joint 1 and joint 2 enters steady state after t  1.9sec . From Fig. 4, we can see the robust ASC
needs more time to drive the joints to their desired positions. From comparing Fig. 3 with Fig. 4, we can see the
performance of RFTSC is better than the robust ASC’s in the presence of the system uncertainty by using same
controller parameters. These two figures also show the proposed approach can cope with coupling effects
effectively.
Case 3 Picking up an object suddenly.
For further testing the robustness of the RFTSC, it was assumed that the robot manipulator suddenly picked
an object ( 0.5kg ) up at t  5sec , namely, the mass of link 2 is then increased form 1.5kg to 2.0kg after
t  5sec . The nominal values of m1 and m2 were specified as mˆ 1  0.4kg , mˆ 2  1.2kg . The boundary
parameters of system uncertainty in (A3) were assumed to be b0  0.9 , b1  0.1 and b2  0.1 . The control
parameters of control law (17) were chosen as K1  diag 1.8,1.8 , K2  diag 1.8,1.8 and
  3 5.
Same nominal values and boundary parameters were chosen for asymptotic stability control law (27). The
19
control parameters of control law (27) were chosen as K1  diag 1.8,1.8 , K2  diag 1.8,1.8 . For
showing coupling effects, q1 was commanded at t  0 , q2 was commanded at t  1sec in this Case.
d
d
Table 2: Nominal values and perturbations
m̂1
0.4kg
m1
0.1kg
 20% 
m̂2
1.2kg
m2
0.8kg
 53% 
Table 1 and Table 2 show that system uncertainty increase from 20% to 53% after t  5sec in this
case. The nominal values, boundary parameters and controller parameter of RFTSC control law (17) and ASC
control law are same, except the parameter
 , which is the key parameter to achieve finite time stability.
Fig. 5 shows the performance of RFTSC, from which we can see that the RFTSC is robust to the plant
parameter variation. Fig. 6 illustrates the performance of the limiting case of control law (17) with
 1,
namely, the control law (27). From Fig. 6, we can see that the robust ASC is robust to small system uncertainty
( 20% parameter perturbation, 0 ~ 5sec ) but lose to work well under large system uncertainty ( 53%
parameter perturbation, 5 ~10sec ). The simulation results of this case illustrate that the RFTSC has stronger
robustness than robust ACS to cope with system uncertainty by using the same control parameters. These two
figures also show the proposed approach can cope with coupling effects effectively.
Case 4 Performance at high frequency.
To show the performance of both controllers (ASC and RFTSC) at high frequency, high frequency force
disturbances 10sin 100t  were added to both of two joints of the robot manipulator in the presence of the
system uncertainty during 3.5sec  t  5sec . The nominal values of m1 and m2 were specified as
ˆ 1  0.4kg , m
ˆ 2  1.2kg . The boundary parameters of system uncertainty in (A3) were assumed to be
m
b0  0.9 , b1  0.1 and b2  0.1 . For showing coupling effects, q1d was commanded at t  0 , q2d was
commanded at t  1sec in this Case.
20
First, for a more fair comparison, the control gains of robust ASC (27) were turned to high gains,
K1  diag 4, 4 and K2  diag 4, 4 . It can be seen from Fig.7 and Fig. 8 that the step responses of
robust ASC with high control gains are similar to those of RFTSC with low control gains
K1  diag 1.8,1.8 , K2  diag 1.8,1.8 . It is obvious that both control approaches entered to steady state
after t  3.5sec . High frequency forces disturbance were added to both joints of robot manipulator after
t  3.5sec . By studying the simulation results presented by Fig.7 and Fig. 8, it can be found that both of the
controllers (ASC and RFTSC) have similar performance at low frequency and high frequency however the
maximum absolute value of control inputs of robust ASC ( 184Nm and 104Nm ) are much larger than those
of RFTSC ( 59Nm and 28Nm ). The larger control inputs of robust ASC are caused by high control gains.
Considering the saturation property of control input signals in practice robot control, high gains are undesirable.
Second, for further testing RFTSC, the performance with low control gains ( K1  diag 0.8,0.8 ,
K2  diag 0.8,0.8 ) and high control gains ( K1  diag 4, 4 , K2  diag 4, 4 ) at high frequency
( 10sin 100t  , 3.5sec  t  5sec ) were presented, respectively. The simulation results are shown in Fig. 9.
(a) and (b) are low control gain performances of joint 1 and joint 2, respectively. (c) and (d) are high control
gain performances of joint 1 and joint 2, respectively. From these simulation results (Fig 9), it can be seen that
the main difference among different control gains of RFTSC is the response rate. High gain results in fast
response.
The better performance of RFTSC comes from the finite-time stability control itself. Note that a power
  0.6 is employed in control law RFTSC however   1 in control law robust ASC. The simulation
results also conform some conclusions of literatures [19, 22], which have proved that finite-time stability
control approach has stronger robustness and faster convergence rate around the equilibrium point than
asymptotic stability control approach.
21
5
Conclusions
This paper has studied the issues associated with the finite-time stability control of robot system based on
finite-time Lyapunov stability. A systematic design procedure is proposed for robot finite-time control with
backstepping algorithm. The study offers an alternative finite-time stability control approach for robotic
manipulators and gives the explicit estimation of the setting time by using the controller parameters. This
approach also removes the constraint of the uniqueness of solution of closed loop systems in forward time.
Since the system uncertainties are accommodated in the controller design, this approach has the strong
robustness and is more applicable. The stability analysis and a numerical example are presented to illustrate the
effectiveness of the proposed approach. It should be mentioned that sound bench tests need to be conducted by
simulations and lab demonstrations before applying the approach to control of real robotic manipulators. The
robot finite-time controller is under the author’s research in the presence of system uncertainty and actuator
saturation.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant: 60825302, 60534020,
60774015 and 60604018), the High Technology Research and Development Program (2006AA04Z173), the
Specialised Research Fund for the Doctoral Program of Higher Education of China (20060248001), and partly
by Shanghai Natural Science Foundation (07JC14016)
Appendix
The following Definition 1 and Lemma 1 are given in [18, 19]
Consider the system of differential equations
22
x  f  x  , f  0  0 , x  R n , x  0   x0
Definition 1
(28)
The origin is said to be a finite-time stable equilibrium of (28) if there exists an open
neighborhood N  D of the origin , and a function T : N 0   0,   , called the settling-time function,
such that, the following statements hold:
(i) Lyapunov stability: For every open neighborhood U  of 0 there exists an open subset U  of N

containing 0 such that, for every x U \ 0 , x  t ,0, x0  U for all t   0, T  x0  .

(ii) Finite-time convergence: For every x0  N \ 0 , every solution x  t ,0, x0  is defined on 0,T  x0  ,
x  t ,0, x0   N \ 0 for all t  0, T  x0   , and lim x  t , 0, x0   0 .
t T  x 
The origin is said to be a globally finite-time stable equilibrium if it is a finite-time stable equilibrium with
D  N  Rn .
Lemma 1
Suppose there exists a continuously differentiable function V : D  R , real number k  0 and
   0 1 , and a neighborhood U  D of the origin such that V is positive definite on U and
V  kV  is negative semidefinite on U , where V  x  
V
 x  f  x  . If f : D  R n is continuous on
x
an neighborhood D of the origin x  0 and the differential equation (28) possesses unique solutions in
forward time for all initial conditions, except possibly the origin, then the origin is a finite-time stable
equilibrium of (28). Moreover, if T is the settling time, then T  x0  
1
1
V  x0  for all x in
k 1   
some open neighborhood of the origin.
The following Lemma 2 and Lemma 3 are given in [24].
Lemma 2
Assume a1  0 , a2  0 and 0  c  1, the following inequality holds
 a1  a2 
Lemma 3
Suppose a1 , a2 ,
c
 a1c  a2c
an are positive numbers and 0  p  2 , then the following inequality holds
23
a
2
1
 a12 
 an2    a1p  a2p 
p
 anp 
2
The following results on differential inequalities are given in [28, 29].
Definition 2 If g V , t  is a scalar function of scalars V  t  , t in some open connected set D  R2 , then a
function V  t  on t0 , t1  is a solution of the differential inequality
V  t   g V  t  , t 
(29)
on t0 , t1  if V  t  is continuous on t0 , t1  and its derivative on t0 , t1  satisfies (29).

Lemma 4 Let g y  t  , t

be continuous on an open connected set D  R
2
and assume that the initial value
problem for the scalar equation
y  t   g  y  t  , t  , g  t0   y0
(30)
has a unique solution. If y  t  is a solution of (30) on t0 , t1  and V  t  is a solution of (29) on t0 , t1  with
V  t0   y  t0  , then V  t   y  t  for t0  t  t1 .
Lemma 5 Assume that a continuous positive definite function V  t  satisfies the differential inequality
V  t   kV   t  , t  t0 , V  t0   0
(31)
where k  0 , 0    1 are constants. Then, for any given t 0 , V  t  satisfies the inequality
V 1  t   V 1  t0   k 1  t  t0  , t0  t  t1
(32)
V 1  t0 
and V  t   0 , t  t1 , with t1 given by t1  t0 
.
k 1  
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Fig. 1 Performance of FTSC in the free of the system uncertainty
28
Fig. 2 Performance of ASC in the free of the system uncertainty
Fig. 3 Performance of RFTSC in the presence of the system uncertainty
29
Fig. 4 Performance of robust ASC in the presence of the system uncertainty
Fig. 5 Performance of RFTSC with picking an object ( 0.5kg ) at t  5sec
30
Fig. 6 Performance of robust ASC with picking an object ( 0.5kg ) at t  5sec
Fig. 7 Performance of RFTSC at high frequency
31
Fig. 8 Performance of robust ASC at high frequency
32
Fig. 9 Performance of RFTSC with low and high control gains at high frequency
33