Engineering, 2010, 2, 118-123
doi:10.4236/eng.2010.22017 Published Online February 2010 (http://www.scirp.org/journal/eng).
A Set of Globally Stable N-PID Regulators for Robotic
Manipulators
Baishun Liu, Fancai Lin, Bingli Tian
Department of Battle & Command, Academy of Naval Submarine, Qingdao, China
Received August 28, 2009; revised September 28, 2009; accepted October 4, 2009
Abstract
This paper deals with the position control of robot manipulators with uncertain and varying-time payload.
Proposed is a set of novel N-PID regulators consisting of a linear combination of the proportional control
mode, derivative control mode, nonlinear control mode shaped by a nonlinear function of position errors,
linear integral control mode driven by differential feedback, and nonlinear integral control mode driven by a
nonlinear function of position errors. By using Lyapunov’s direct method and LaSalle’s invariance principle,
the simple explicit conditions on the regulator gains to ensure global asymptotic stability are provided. The
theoretical analysis and simulation results show that: an attractive feature of our scheme is that N-PID regu-
lators with asymptotic stable integral actions have the faster convergence, better flexibility and stronger ro-
bustness with respect to uncertain and varying-time payload, and then the optimum response can be achieved
by a set of control parameters in the whole control domain, even under the case that the payload is changed
abruptly.
Keywords: Manipulators, Robot Control, PID Control, N-PID Control, Global Stability
1. Introduction
It is well known that PID controllers can effectively deal
with nonlinearity and uncertainties of dynamics, and
asymptotic stability is achieved accordingly [1–3]. Hence,
most robots employed in industrial operations are con-
trolled through PID controllers that introduce integral
action by integrating the error. It is well known that inte-
gral-action controllers with this class of integrator often
suffer a serious loss of performance due to integrator
windup, which occurs when the actuators in the control
loop saturate. Actuator saturation not only deteriorates
the control performance, causing large overshoot and
long settling time, but also can lead to instability, since
the feedback loop is broken for such saturation. To avoid
this drawback, various PID-like controllers have been
proposed to improve the transient performance. For ex-
ample, PID controllers consisting of a saturated-P, and
differential feedback plus a PI controller driven by a
bounded nonlinear function of position errors [4], a lin-
ear PD feedback plus an integral action of a nonlinear
function of position errors [5], a linear derivative feed-
back plus a PI term driven by a nonlinear function of
position errors [1], a linear PD feedback plus double in-
tegral action driven by the positions error and the filtered
position [6], a linear PD feedback plus an integral action
driven by PD controller [7], and a linear PD feedback
plus an integral action driven by NP-D controller [8], are
presented recently.
In this paper, we propose a set of new global position
controllers for robots which do not include their dynam-
ics in the control laws. Motivated by the idea that is in-
troducing the nonlinear action of the position errors into
PD-NI controller [5] and modifying the nonlinear inte-
gral action via injection of the required damping so that
the transient performance of the closed-loop system may
be improved, we develop a set of new N-PID-like regu-
lators consisting of a linear combination of the propor-
tional control mode, derivative control mode, nonlinear
control mode shaped by a nonlinear function of position
errors, linear integral control mode driven by differential
feedback, and nonlinear integral control mode driven by
a nonlinear function of position errors. The simple ex-
plicit conditions on the regulator gains to ensure global
asymptotic stability are provided.
Throughout this paper, we use the notation )(A
m

and )(A
M

to indicate the smallest and largest eigen-
values, respectively, of a symmetric positive define
bounded matrix)(xA , for any. The norm of vec-
tor
n
Rx 
x is defined asx
T
xx , and that of matrix A

B. S. LIU ET AL.119
is defined as the corresponding induced norm
 AAA T
M
 .
The remainder of the paper is organized as follows.
Section 2 summarizes the robot model and its main
properties. Our main results are presented in Sections 3
and 4, where we briefly review some known PID-like
control laws and present a set of new N-PID-like control
laws, and then provide the conditions on the controller
gains to ensure global asymptotic stability, respectively.
Simulation examples are given in Section 5. Conclusions
are presented in Section 6.
2. Problem Formulation
The dynamic system of an n-link rigid robot manipulator
system [1] can be written as:
uqgqDqqqCqqM  )(),()(  (1)
where is the vector of joint positions, is the
vector of applied joint torques, is the
symmetric positive define inertial matrix,
is the vector of the Coriolis and centrifugal
torques, is the positive define diagonal fric-
tion matrix, and is the vector of gravita-
tional torques obtained as the gradient of the robot po-
tential energy due to gravity.
q
n
1n
n
(qg
)(q
u
)
C(
1n
nn 
(qM
qqq ),
1 
Dn
) 1n
U
A list of properties [1] of the robot dynamic model (1)
is recalled as follows:
)()()(0 MqMM Mm   (2)
 0),(2)(   qqCqM
T
 (3)
n
R 
22),(0 qCqqqCqC Mm   (4)
n
Rqq   ,
where and are all positive constants.
m
CM C
For the purpose of this paper, it is convenient to in-
troduce the following definition and properties [5].
Definition 1: ),,(
x
F


with01 

, 0

, and
denotes the set of all continuous differential in-
creasing functions,
n
Rx
T
n
xfxfxf )](,),([)( 1such that
|||)(||| xxfx



|:| xRx


|)(| xf



|:| xRx
0)()/(1 xfdxd (5)
where stands for the absolute value. || 
Figure 1 depicts the region allowed for functions be-
longing to set),,( xF


. For instance, the tangent hyper-
bolic function belongs to set and the
Arimoto sine function, whose entries are given as follows:
),1),1(tanh( xF









2/1
2/||)sin(
2/1
)(



xif
xifx
xif
xf (6)
which belongs to set .
),1),1(sin( xF
The important properties of function belonging
to set
)(xf
),,( xF


are now established.
The function satisfies for all, )(xfx Tn Rx 
0)( xfxT (7)
The Euclidean norm of satisfies for all )(xf n Rx

,
22 ||||||)(|| xxf  (8)
 nxf ||)(|| (9)
Throughout this paper, we use the notation d
qqq



P
KD K
,
to indicate the position errors, is the desired joint
position, which is assumed to be constant; , ,
, , , and are all positive define
diagonal
d
q
S
KPf K
n
I
K
n
IP
KID K

matrices.
3. N-PID-Like Control Laws
3.1. PID-Like Control Law Review
To put our contribution in perspective, we will briefly
review some known PID-like control laws, as follows:
1) Semiglobally stable PID control law [1],
 t
IDP dqKqKqKu
0
)( 
.
2) Globally stable PD-NI control law [1],
 t
IDP dqKqKqKu
0
))(tanh( 
.
where )tanh(

is the hyperbolic tangent vector function.
3) Globally stable PD-NPI control law [1],
 t
ISDPdqsKqsKqKqKu
0
))(()( 
.
Figure 1. ),,( xF


functions.

B. S. LIU ET AL. 120
where is the differential function of a class of ap-
proximate potential energy function.
)(s
4) Semiglobally stable PI2D control law [6],
 t
IDP dqqKqKqKu
0
)]()([ 
 .
5) Semiglobally stable PD-IPD control law [7],
 t
IDIPDPdqKqKqKqKu
0
)]()([ 
 .
6) Globally stable PD-INP-D control law [8],


t
IDIP
DP
dqKqK
qKqKu
0
)]())(tanh([ 


.
7) Globally stable SP-D-NPI control law [4],


t
bI
bPfDP
dqfK
qfKqKqsatKu
0
))((
)()(


.
where is the normal saturated function and
is a bounded nonlinear function.
)(sat
)(
b
f
8) Globally stable PD-NI control law [5],
 t
IDP dqfKqKqKu
0
))(( 
 (10)
where is a continuous differential increasing
bounded functions shown in Figure 1.
)(f
Most of the present industrial robots are controlled
through PID-like controllers above. Although these con-
trollers have been shown in practice to be effective for
position control of robot manipulators, unfortunately
most of them often suffer a serious loss of performance,
that is, causes large overshoot and long settling time due
to unlimited integral action.
Based on the above fact, we get intuitively an idea that
the transient performance of the closed-loop system may be
improved if the nonlinear action of position errors is intro-
duced into the control law (10) and the damping is injected
into its integrator. Following this idea, a set of novel
N-PID-like regulators is developed in the next subsection.
3.2. Our N-PID Control Laws
The new nonlinear PID control laws are proposed as fol-
lows,


t
IDIP
PfDP
dqKqfK
qfKqKqKu
0
)]())(([
)(



(11)
It is worthy to note that the control law above is con-
sisting of a linear combination of the proportional control
mode, derivative control mode, nonlinear control mode,
linear integral control mode and nonlinear integral control
mode. Hence, based on the five control modes above, five
differential N-PID-like control laws with the same stability
as the control law (11) can be derived, as follows:

t
IDIPDP dqKqfKqKqKu
0
)]())(([ 

(12)






t
IDIP
PfD
dqKqfK
qfKqKu
0
)]())(([
)(



(13)
 t
IDIPD dqKqfKqKu
0
)]())(([ 
 (14)
 t
IPDPdqfKqKqKu
0
))(( 
 (15)
 t
IPPfDP dqfKqKqKqKu
0
))(()( 
 (16)
Discussion 1: It is obvious that the control law (11)
can be simplified to other PID-like control laws such as
P-NI control, NPI control, PD-INP-D control [8], and so on.
Moreover, the control law (15) is the same as the one (10)
reported by [5]. This shows that the control law (11) is a
natural extension of them and implies that its application
is not limited in the robots, too.
Discussion 2: Notice that the integral actions in con-
trol laws (10) and (11) can be rewritten as
)(qfKI




 andqKqfK IDIP  


)(

, respec-
tively. From this, it is easy to see that the latter has the
same stability as the one presented by [7,8], that is, they
are all asymptotically stable but the former has the same
stability as the classical integral action, qKI





IP
,
that is, they are all only stable. This means that the con-
troller (11)-(14) should have faster convergence, better
flexibility than the one (10), and then the controller
(11)-(14) can yield higher performance of control, too.
Moreover, integrator windup can be avoided by choosing
suitable parameters
K
andID
K
.
Discussion 3: Compared to the classical PID con-
trol and PD-NI control [5], the following observations
can be made during the control process: when
0)(




qKqfK IDIP, the integral action remains con-
stant; if the integral action is large, increases, and
then the integral action instantly decreases, vice versa.
However, the integral action produced by the classical or
nonlinear integrator [5] always increases as long as the
error does not cross over zero, only when the error
crosses over zero, the integral action will start to de-
crease. This shows that the control laws with the asymp-
totically stable integrator should have the faster conver-
gence and better flexibility, and then can yield higher
transient performance, once again.
q


4. Stability Analysis
For analyzing the stability of the closed-loop system, it is
convenient to introduce the following notation.
Defining ,
)]0()([))(()(1
0
qKqgKdqftz IDdIP
t 



B. S. LIU ET AL.121
and then the control law (11) can be rewritten as,
)()(
)(
dIPPf
DIDP
qgzKqfK
qKqKKu



 
(17)
The closed-loop system dynamics is obtained by sub-
stituting the control action from (17) into the robot
dynamic model (1),
u
0)()(
)()()(),()(


qfKzKqKK
qgqgqDKqqqCqqM
PfIPIDP
dD 
(18)
From the closed-loop system dynamics above, it is
easy to see that the origin nTTTTR
z
qq 3 0),,(  is
the unique equilibrium point.
Now, the objective is to provide conditions on the con-
troller gains, ,
P
KD KPf
K
,
IP
K
and ID
K
ensuring
global asymptotic stability of the unique equilibrium
point. This is established in the following.
Theorem: Consider the robot dynamics (1) together
with control law (11). There exists positive constant
small enough, and choose the gain matrices, ,
, and such that
a
DP
K K
Pf
KIP
KID K
IMKKK MIPIDP)(4

 (19)
2
||||)(
4
1
)()()(
qaqKKKq
qgqqUqU
IPIDP
T
d
T
d


(20)
2
||)(||
))((
)()()]()()[(
qfa
qKKKqf
qfKqfqgqgqf
IPIDP
T
Pf
T
d
T



(21)


InCMDK MMD   )( (22)
hold, and then the closed-loop system (18) is globally
asymptotically stable, i.e. 0li
m


 q t.
Proof: To carry out the stability analysis, we consider
the following Lyapunov function candidate:






n
i
q
iiDiPfii
T
IPIDP
T
T
d
T
d
IP
TT
idxDKKxf
qKKKq
qqMqfqgqqUqU
qzKqzqqMqV
10
))((
)(
2
1
)()()()()(
)()(
2
1
)(
2
1


(23)
where ; , and are the diagonal
element of the matrices , and , respec-
tively.
ii qx Pfi
KDi K
K
i
D
D
K
Pf D
1) Positive definition of Lyapunov function candidate.
Now, considering the following inequality, and using
(8), we have,
)())(4)((
4
1
)(
4
1
)()()(
)](2)[()](2[
4
1
)()(
)(
4
1
)(
4
1
qfIMKKKqf
qKKKq
qfqMqf
qfqqMqfq
qqMqf
qKKKqqqMq
MIPIDP
T
IPIDP
T
T
T
T
IPIDP
TT










(24)
Substituting (20) and (24) into (23), and using (7) and
(19), for any, we obtain, 0),,( TTTT zqq 
0)()(
2
1
))((
||||)(
4
1
10
2






qzKqz
dxDKKxf
qaqqMqV
IP
T
n
i
q
iiDiPfii
T
T
i

(25)
This shows that the Lyapunov function candidate (23)
is positive define.
2) Time derivative of Lyapunov function candidate.
The time derivative of Lyapunov function candidate (23)
along the trajectories of the closed-loop system (18) is,
)()())((
)()()(
)()()()(
)())(()(
2
1
)(
qzKqzqDKKqf
qKKKqqgqqgq
qqMqfqqMqf
qqMqqfqqMqqqMqV
IP
T
DPf
T
IPIDP
T
d
TT
TT
TTT




  









(26)
Substituting )( qfz


, and from (18) into
(26), and using (3), we have,
qqM )(
)()(
))((
)]()()[(),()(
)())(()(
qfKqf
qKKKqf
qgqgqfqqqCqf
qqMqqfqDKqV
Pf
T
IPIDP
T
d
TT
T
D
T









(27)
Now, using (2) and (5), we get,
2
||||)()())(( qMqqMqqfM
T

  (28)
By using (4) and (9), we obtain,
2
||||),()( qnCqqqCqfM
TT 
  (29)
Incorporating (21), (28) and (29) into (27), we obtain,
2
||)(||
))((
qfa
qInCIMDKqV MMD
T

 


(30)
From (22), (30) and, we can conclude.
0a0V


B. S. LIU ET AL. 122
rge
and
, in th
he n
Using the fact that the Lyapunov function candidate
(23) is a positive define function and its time derivative
is a negative semi define function, we conclude that the
equilibrium point of the closed-loop system (18) is stable.
In fact, means and. By invoking
the LaSalle’s invariance principle, it is easy to know that
the equilibrium point is globally asymp-
totically stable, i.e., .
0V
0q
),(TT qq 
0
q
0q

0
T
lim
t
Remark 1: For the control laws (12)-(16), the globally
asymptotically stable results can be derived under some
similar sufficient conditions presented by (19)-(22). The
proof can follow the similar argument and procedure. It
is omitted because of the limited space.
Discussion 4: From the proof procedure above, it is
easy to see that: 1) if we chooses gain matrix, ID K la
enough, the linear term, qKPthe nonlinear term,
fKPf e control law (11) is not necessary for
guaranteeing global asymptotic stability of the closed-
loop system and this means that the global asymptotic
stability can still be ensured by the simplified form of the
control law (11); 2) on the other hand, with the linear
term, qKP, and tlinear term, )( qfKPf
,
on
)( q

, it is
at the condition (19)-(21) is more easily satis-
fied and this results in that the control engineers have
more freedom to choose the controller parameters, and
then make them more easily tune a high performance
controller.
obvious th
5. Simulations
To illustrate the effect of the controller given in this pa-
per, two-link manipulators shown in Figure 2 are consid-
ered. The dynamics (1) is of the following form [9]





222112
2
111222112
112112
2
211212111
2
2
uGqqFqFqMqM
uGqqFqFqMqM


where: ,
)cos(2)(2212
2
22
2
12111 qllmlmlmmM 
, ,
2
2222 lmM )cos( 2212
2
2212qllmlmM 
)sin(22121211 qllmFF  ,
)sin()sin()( 212211211 qqglmqglmmG 
)sin( 21222qqglmG  .
The normal parameter values of the system are se-
lected as:, , 2. kgmm 1
21  mll1 21  /10 smg 
The desired (set point) positions are chosen as:
when , ,
st 101
1 d
q1
2

d
q;
when , , ; sts 2010 3
1 d
q2
2 d
q
when , . st200
21 dd qq
The simulation is implemented by using the control
laws (11) and (14), respectively. In simulation, Arimoto
Figure 2. The two-link robot manipulators.
sine function (6) is used as the nonlinear function of the
control laws.
The gain matrices of the control law (11) are selected as:
)310,310(diagKP

, , )150,150(diagKD
)500,500( diagKIP

, , )200,200(diagKID 
and )100,100(diagKPf

.
The gain matrices of the control law (14) are given as:
)30,60(diagKD

, , )500,1000(diagKIP
and )600,1250(diagKID

.
The simulations with sampling period of 2ms are im-
plemented. Figures 3 and 5 present the response of the
robot manipulators under the normal parameters. Figures
4 and 6 are the simulation results under the case that the
mass is substituted for when kgm1
2
st 20
kgm 3
2
s10


, corresponding to moving payload of 2kg.
From the simulation results, it is easy to see that:
1) The optimum response can all be achieved, respec-
tively, by the control laws (11) and (14) with a set of
control parameters in the whole domain of interest, even
under the case that the payload is changed abruptly;
2) These tow controllers used in simulation all have
the faster convergence, better flexibility and stronger
robustness with respect to uncertain payload, which
means that the controllers (12) and (13) should have the
same high performance of control as the controllers (11)
and (14) because they all employ the same integrators;
3) Comparing the Figures 3 and 5, the controller (11)
is easier to achieve the control of high speed and high
performance than the controller (14) because the former
has two freedom parameters.
Figure 3. Under normal parameters, the simulation results
with controller (11).

B. S. LIU ET AL. 123
exact knowledge of the payload. An attractive feature of
our scheme is that the control laws with the asymptoti-
cally stable integrators have the faster convergence, bet-
ter flexibility and stronger robustness with respect to
uncertain and varying-time payload, and then the opti-
mum response can be achieved. The explicit conditions
on the regulator gains to ensure global asymptotic stabil-
ity of the overall closed-loop system are given in terms
of some information exacted from the robot dynamics.
Our findings have been corroborated numerically on a
two DOF vertical robot manipulators.
Figure 4. Under perturbed parameters, corresponding to
moving payload, the simulation results with controller (11). 7
. References
[1] Y. X. Su, Nonlinear Control Theory for Robot Manipula-
tors, Science Publishing, Beijing, 2008.
[2] J. Alvarez-Ramirez, I. Gervantes, and R. Kelly, “PID
regulation of robot manipulators: Stability and perform-
ance,” System & Control Letters, Vol. 41, No. 2, pp.
73–83, April 2000.
[3] A. A. Pervozvanski and L. B. Freidovich, “Robust stabil-
ity of robotic Manipulators by PID controllers,” Dynamics
and Control, Vol. 9, No. 3, pp. 203–222, September 1999.
[4] S. Arimoto, “A class of quasi-natural potentials and Hy-
per-stable PID servo-loops for nonlinear robotic system,”
Transactions of the Society of Instrument and Control En-
gineers, Vol. 30, No. 9, pp. 1005–1012, September 1994.
Figure 5. Under normal parameters, the simulation results
with controller (14).
[5] R. Kelly, “Global positioning of robotic manipulators via
PD control plus a class of nonlinear integral actions,”
IEEE Transactions on Automatic Control, Vol. 43, No. 7,
pp. 934–938, July 1998.
[6] R. Ortega, A. Loria, and R. Kelly, “A semiglobally stable
output feedback PI2D regulator for robot manipulators,”
IEEE Transactions on Automatic Control, Vol. 40, No. 8,
pp. 1432–1436, August 1995.
[7] B. S. Liu and F. C. Lin, “A semiglobally stable PD-IPD
regulator for robot manipulators,” to be published in 2009
International Conference on Measuring Technology and
Mechatronics Automation proceeding.
Figure 6. Under perturbed parameters, corresponding to
moving payload, the simulation results with controller (14).
[8] B. S. Liu and B. L. Tian, “A globally stable PD-INP-D
regulator for robot manipulators,” to be published in 2009
International Conference on Information Technology and
Computer Science proceedings.
6. Conclusions
[9] S. J. Yu, X. D. Qi, and J. H. Wu, Iterative Learning Con-
trol Theory & Application, Machine Publishing, Beijing,
2005.
In this paper, we have presented a set of solution to the
problem of set point regulation for rigid robots without