2009 Bai - A globally stable SM-PD-INP-D tracking control of robot manipulators.pdf

A Globally Stable SM-PD-I NP-D Tracking Control of Robot Manipulators
Bai-shun Liu Fan-Cai Lin
Department of Battle & Command
Academe of Naval Submarine
QingDao, China
E-mail: baishunliu@163.com E-mail: lfcai_777@yahoo.com.cn.
Abstract — This paper deals with the tracking control of robot
manipulators. Synthesizing the strong robustness of the sliding
mode control and the good flexibility of PD-I NP-D control,
Proposed is a simple class of robot tracking controller
consisting of a linear sliding mode term plus a linear PD
feedback plus an integral action driven by an NP-D controller.
By using Lyapunov’s direct method and LaSalle’s invariance
principle, the simple explicit conditions on the controller gains
to ensure global asymptotic stability are provided. The
theoretical analysis and simulation results show that: i) the
SM-PD-I NP-D controller has the faster convergence, better
flexibility and stronger robustness with respect to initial errors;
ii) the proposed control laws can not only achieve the
asymptotically stable trajectory tracking control but also the
tracking errors quickly tend to almost zero without oscillation
as time increases; iii) after the sign function is replaced by
saturation function in the SM-PD-I NP-D control law, the highfrequency
oscillation of the control input vanishes.
Keywords — Manipulators; Robot control; PID control;
global stability; Tracking control.
I. INTRODUCTION
For the tracking control of robot manipulators, many
control strategies have been proposed in the literature [1-4] .For
example, feedback linearization method, PD feedback with
feedforward compensation, PID control, sliding mode
control, adaptive control, sliding mode linear PID control,
sliding mode nonlinear PID control, and so on. Among them,
PID-like controllers with little prior knowledge of system are
one of the most popular control strategies because they can
not only effectively deal with nonlinearity and uncertainties
of dynamics but also the globally or semiglobally
asymptotically stable tracking control can be achieved
accordingly. Hence, most robots employed in industrial
operation are controlled through PID-like controllers.
A major drawback of PID-like tracking controller is that
it often produces the larger oscillation of the controlled
system, even may lead to instability due to unlimited integral
action. Hence, in this paper, synthesizing the strong
robustness of the sliding mode control and the good
flexibility of PD-I NP-D control [7] , we develop a new class of
global position tracking controller for robot manipulators
leading to a linear sliding mode term plus a linear PD
feedback plus an integral action driven by NP-D controller.
The novel controllers not only can effectively avoid the
drawback due to unlimited integral action but also do not
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978-1-4244-4520-2/09/$25.00 ©2009 IEEE
require the exact knowledge of robot dynamics. The simple
explicit conditions on the controller gains to ensure globally
asymptotically stable position tracking control are provided.
Throughout this paper, we use the notation λm (A)
and
λ (A) M
to indicate the smallest and largest eigenvalues,
respectively, of a symmetric positive define bounded matrix
n
A (x) , for any x ∈ R . The norm of vector x is defined as
T
x = x x , and that of matrix A is defined as the
T
corresponding induced norm A ( A A)
= .
The remainder of the paper is organized as follows.
Section II summarizes the robot model and its main
properties. Our main results are presented in Section III and
IV, where we propose SM-PD-I NP-D control law and provide
the conditions on the controller gains to ensure global
asymptotic stability, respectively. Simulation examples are
given in Section V. Conclusions are presented in Section VI.
II. PROBLEM FORMULATION
The dynamic system of an n-link rigid robot manipulator
system [1] can be written as,
M ( q)
q̇ + C(
q,
q̇
) q̇
+ Dq̇
+ g(
q)
= u
(1)
where q is the n × 1 vector of joint positions, u is the n × 1
vector of applied joint torques, M (q)
is the n × n symmetric
positive define inertial matrix, C( q,
q̇)
q̇
is the n × 1 vector
of the Coriolis and centrifugal torques, D is the n × n
positive define diagonal friction matrix, and g(q)
is the
n ×1 vector of gravitational torques obtained as the gradient
of the robot potential energy U (q)
due to gravity.
A list of properties [1] of the model (1) is recalled as,
0 < λ ( M ) ≤ M ( q)
≤ λ ( M )
(2)
ς
T
m
M
( Ṁ
( q)
− 2C(
q,
q̇
)) ς = 0
C ( q,
ς ) v = C(
q,
v)ς
λ M
n
∀ς ∈ R
(3)
∀ q ,
n
,ς v ∈ R
(4)

2
0 < C q̇
≤ C(
q,
q̇
) q̇
≤ C q̇
m
g(
q)
≤ κ
g
M
2
∀q
̇
n
, q ∈ R (5)
n
∀ q ∈ R
(6)
where C
m
, CM
and κ
g
are all positive constants.
For convenience, we define hyperbolic tangent vector
n
function tanh( •) ∈ R and hyperbolic secant diagonal matrix
n×
n
function sec h(•) ∈ R , as follows,
T
tanh( ξ ) = [tanh( ξ1),
⋯,tanh(
ξn )] ,
sech(
ξ ) = diag[sech(
ξ1),
⋯,sech(
ξn
)],
and then one can obtain,
T
T
T
tanh ( ξ )tanh( ξ ) ≤ tanh ( ξ ) ξ ≤ ξ ξ
(7)
where K
P
, K
D
, K
η
, K
IP
and K
ID
are the suitable positive
define diagonal n × n matrices;
Discussion 1 Notice that the integral action in (11) and
(10) can be rewritten as: σ̇
= −K
IP
tanh( e)
− K
IDė
and
σ ̇ = −K I
(tanh( e)
+ ė
) , respectively. Although they are all
asymptotically stable, the former adds a gain matrix K
ID
,
which make us more easily inject the suitable damping to
adjust the integral action. This means that the controller (11)
should have the faster convergence, better flexibility than the
one (10), and then the controller (11) could yield higher
control performance.
Remark 1 It is well known that sliding mode control
easily excites high-frequency oscillation of control action in
practice, which degrades the performance of the system and
may even lead to instability. To avoid this drawback, the sign
function can be replaced by saturation function, and then a
continuous control law can be obtained, as follows.
|| tanh( ξ ) ||≤
n
(8)
u = −K
e − K
ė
− K
t
( K tanh( e(
τ )) K ė
( τ ))
−
∫
+
0
IP
P
D
η
tanh( η)
ID
dτ
(12)
λ (sec 2 M
h ( ξ )) = 1
(9)
n
where ξ = [ ξ 1
, ⋯ , ξ ]∈ R .
n
III. SM-PD- I NP-D CONTROL LAW
The position tracking control of most of the industrial
robots are controlled through PID-like controllers. The
textbook version of the SM-PD-NI tracking controller [1] can
be described as,
u = −K
P
e − K
D
t
∫0
ė − K sgn( η)
− K η(
τ ) dτ
(10)
η
n
where e( t)
= q(
t)
− qd
( t)
∈ R is the tracking errors, q d
(t)
is the desired trajectories, and η = tanh( e)
+ ė
; K
P
, K
D
and
K
I
are suitable positive define diagonal n × n matrices;
sgn(• ) is the normal sign function.
Although the controller (10) has been shown in practice
to be effective for position tracking control of robot
manipulators, unfortunately, it often produces the larger
oscillation of the controlled system due to 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 suitable damping is injected into the
integrator. Following this idea, a novel class of tracking
controller is proposed, as follows,
u = −K
e − K
ė
− K
t
( K tanh( e(
τ )) K ė
( τ ))
−
∫
+
0
IP
P
D
η
sgn( η)
ID
dτ
I
(11)
IV.
CLOSED-LOOP ANALYSIS
For analyzing the stability of the closed-loop system, it is
convenient to introduce the following notation.
t
1
Defining z(
t)
=
−
∫
tanh( e(
τ )) dτ
− K
IP
K
IDe(0)
, and then
0
the control law (11) can be rewritten as,
u = −( K + K ) e − K ė − K z − sgn( η)
(13)
P
ID
D
IP
The closed-loop system dynamics is obtained by
substituting the control action u from (13) into the robot
dynamic model (1),
M ( q)
̇̇ e + C(
q,
q̇
) ė
+ Dė
+ ( K
P
+ K
+ K ė
D
+ K
IPz
+ Kη sgn( η)
= ρ
K η
ID
) e
(14)
where ρ = −M
( q)
q̇̇
d
− C(
q,
q̇
) q̇
d
− Dq̇
d
− g(
q)
Using (2), (5) and (6), the upper bound of || ρ || can be
rewritten as [1] ,
|| ρ || ≤ λM
( M ) AM
+ CMV
+ λM
( D)
VM
+ κ
g
≡ γ + γ || ė
||
0
1
2
M
+ C
M
V
M
|| ė
||
(15)
where γ
0
and γ 1
are positive constants; AM
and V M
are the
upper bound of q̇ ̇
d
and q̇ d
, respectively.
Now, the objective is to provide conditions on the
controller gains K
P
, K
η
, K
D
, K
IP
and K
ID
guaranteeing

global asymptotic stability of the closed-loop system (14).
This is established in the following.
Theorem Consider the robot dynamics (1) together with
the control law (11). There exist gain matrices K
P
, K
η
, K
D
,
K
IP
and K
ID
such that
λ ( K + K − K ) > 4λ
( M )
(16)
m
P
ID
IP
M
3 1
λ
m
( K
D
+ D)
≥ γ
1
+ CMVM
+ nCM
+ λM
( M ) (17)
2 2
1
λ
m
( K
P
+ K
ID
− K
IP
) ≥ ( γ
1
+ CMVM
)
(18)
2
λ
η
≥
m
( K ) γ
0
(19)
hold, and then the closed-loop system (14) is globally
asymptotically stable, i.e., lim e(
t)
= 0 .
t→∞
Proof: To carry out the stability analysis, we consider the
following Lyapunov function candidate:
1 T
T
V = ė
M ( q)
ė
+ tanh ( e)
M ( q)
ė
2
1 T
1 T
+ e ( K
P
+ K
ID
− K
IP
) e + ( e + z)
K
2
2
+
n
∑
i=
1
( K
Di
+ D )ln(cosh( e ))
i
i
IP
( e + z)
(20)
where K
Di
and Di
are the diagonal element of the
matrices K
D
and D , respectively; cosh(• ) and ln(• ) are the
hyperbolic cosine function and natural logarithm function,
respectively.
I. Positive definition of Lyapunov function (20).
Now, using (2) and (7), the following inequality holds,
1 T
T
1 T
ė
M ( q)
ė
+ tanh ( e)
M ( q)
ė
+ e ( K
P
+ K
ID
− K
4
4
1
T
= ( ė
+ 2tanh( e))
M ( q)(
ė
+ 2tanh( e))
4
T
1 T
− tanh ( e)
M ( q)tanh(
e)
+ e ( K
P
+ K
ID
− K
IP
) e
4
1 T
T
≥ e ( K
P
+ K
ID
− K
IP
) e − tanh ( e)
M ( q)tanh(
e)
4
1
≥
4
n
∑
i=
1
( K
Pi
+ K
IDi
− K
IPi
− 4λ
M
( M ))tanh
2
( e )
i
IP
) e
(21)
where K
Pi
, K
IPi
and K
IDi
are the diagonal element of the
matrices K
P
, K
IP
and K
ID
, respectively.
Substituting (21) into (20), and using (16), and the fact
T T T T
ln(cosh( e )) ≥ 0 , for any ( e , ė , z ) ≠ 0 ,weobtain,
1
V ≥ e
4
+
n
∑
i=
1
( K
1 T
+ ( e + z)
K
2
i
1
M ( q)
e + e
4
̇T
̇
Di
( K
+ D )ln(cosh( e ))
i
IP
T
P
( e + z)
> 0
i
+ K
ID
− K
IP
) e
(22)
This shows that the Lyapunov function (20) is positive
define.
II. Time derivative of Lyapunov function (20).
The time derivative of Lyapunov function (20) along the
trajectories of the closed-loop system (14) is,
1
V̇
T
T
= ė
M ( q)
̇̇ e + ė
Ṁ
( q)
ė
2
T
T
+ ė
( K
P
+ K
ID
− K
IP
) e + ( ė
+ ż
) K
IP
( e + z)
T
T
+ tanh ( e)
Ṁ
( q)
ė
+ tanh ( e)
M ( q)
ė̇
2 T
+ (sech
( e)
ė
) M ( q)
ė
+ ė
( K + D)tanh(
e)
D
(23)
Substituting z ̇( t)
= tanh( e)
and M ( q)̇
ė
from (14) into
(23), and using (3), we have,
V̇
T T
= ( ė
+ tanh( e))
ρ −η
K sgn( η)
T
− ė
( K
D
2 T
+ (sech
( e)
ė
)
(sech
≤ (
+ D)
ė
− tanh
T
η
( e)(
K
M ( q)
ė
+ tanh
T
P
+ K
ID
− K
IP
T
( e)
C ( q,
q̇
) ė
) e
Now, using (2), (4), (5), (8) and (9), we get,
2
= (sech
+ (
nC
T
T
( e)
ė
) M ( q)
ė
+ tanh ( e)
C(
q,
q̇
) ė
2
T
T
( e)
ė
) M ( q)
ė
+ tanh ( e)
C(
q,
ė
)( ė
+ q̇
)
M
+ λ
M
( M )) || ė
||
2
+ C
M
V
M
|| tanh( e)
|| || ė
||
Substituting (15) and (25) into (24), we obtain,
V̇
≤||
ė
+ tanh( e)
|| ( γ + γ || ė
||)
T
T
−η
K sgn( η)
− ė
( K
T
− tanh ( e)(
K
+ C
+ K
2
nCM
+ λ ( M )) || ė
M
||
V || tanh( e)
|| || ė
||
M
η
M
P
0
ID
D
1
− K
+ D)
ė
IP
) e
d
(24)
(25)
(26)
By || η || = || tanh( e)
+ ė
|| and || η || ≤||
tanh( e)
|| + || ė
|| , the
inequality (26) can be rewritten as,

V̇
T
≤ γ || η || −η
K
+ ( γ +
1
1
0
T
ė
( K
D
+ ( γ + C V
sgn( η)
−
T
+ D)
ė
− tanh ( e)(
K
nC
M
M
M
η
P
+ K
2
+ λ ( M )) || ė
M
||
) || tanh( e)
|| || ė
||
ID
− K
IP
) e
(27)
The parameter values of the system are selected as:
2
m = m 1kg
, l = l 1m
, g = 10m/
s .
1 2
=
1 2
=
2 2
Inserting 2 || tanh( e)
|| || ė || ≤||
tanh( e)
|| + || ė
|| into (27)
and using (7), we get,
V̇
T
≤ γ || η || −η
K
0
T
− ė
( K
D
sgn( η)
T
+ D)
ė
− tanh ( e)(
K
3 1
+ ( γ
1
+ CMVM
+ nCM
+ λ
2 2
1
2
+ ( γ
1
+ CMVM
) || tanh( e)
||
2
Noticing
η
n
∑
i=
1
P
+ K
M
ID
− K
IP
( M )) || ė
||
)tanh( e)
T
η sgn( η)
= | ηi
| and || η || ≤ ∑|
ηi
|
inequality (28) can be rewritten as,
2
n
i=
1
(28)
, the
Figure 1. The two-link robot manipulators
V̇
≤ −(
λ ( K
m
η
) − γ ) || η ||
⎡λm
( K
D
+ D)
−
T
− ė
⎢
⎢
3 1
( γ
1
+ CMVM
+ nC
⎢⎣
2 2
⎡λm
( K
P
+ K
ID
− K
T
− tanh ( e)
⎢
⎢
1
− ( γ
1
+ CMVM
)
⎢⎣
2
0
M
IP
⎤
⎥ė
+ λM
( M )) ⎥
⎥⎦
) ⎤
⎥ tanh( e)
⎥
⎥⎦
(29)
From (17), (18), (19), (22) and (29), we can conclude
V̇ ≤ 0 . Using the fact that the Lyapunov function (20) is a
positive define function and its time derivative is a negative
semidefine function, we conclude that the closed-loop
system (14) is stable. In fact, V̇ = 0 means e = 0 and ė = 0 .
By invoking the LaSalle’s invariance principle [5] ,itiseasyto
know that the closed-loop system (14) is globally
asymptotically stable, i.e., lim e(
t)
= 0 .
t→∞
V. SIMULATIONS
To illustrate the effect of the controller given in this
paper, a two-link manipulators shown in Figure 1 is
considered. The dynamics (1) is of the following form [6]
2
⎧M
11q̇̇
1
+ M
12q̇̇
2
+ F11q̇
2
+ 2F12q̇
1q̇
2
+ G1
= u1
⎨
2
⎩M
12q̇̇
1
+ M
22q̇̇
2
+ F11q̇
1
+ 2F12q̇
1q̇
2
+ G2
= u2
2 2
where: M = m + m ) l + m l + 2m l l cos( ) ,
11
(
1 2 1 2 2 2 1 2
q2
2
2
M
22
= m2l2
, M
12
= m2l2
+ m 2
l 1
l 2
cos( q 2
) ,
F 11
= F 12
= −m 2
l 1
l 2
sin( q 2
) , G
2
= m2gl2
sin( q1
+ q2)
,
and G = m + m ) gl sin( q ) + m gl sin( q + ) .
1
(
1 2 1 1 2 2 1
q2
Figure 2. The desired (dashed) and actual (solid) trajectories with the
control law (11).
Figure 3. The desired (dashed) and actual (solid) trajectories with the
control law (12).
The desired trajectories are taken as: q d 1
= 2sin( t)
and
q d 2
= 2cos( t)
for Link1 and Link2, respectively. The
simulation is implemented by using the control laws (11) and
(12), respectively. The controller gains matrices are given as:
K P
= diag(500,500) , K D
= diag(150,150)
,
K IP
= diag(200,200) , K ID
= diag(50,50)

and K
η
= diag(10,10)
.
ii) SM-PD-I NP-D control law has the faster convergence,
better flexibility and stronger robustness with respect to
initial position errors, and then the tracking errors quickly
tend to almost zero without oscillation as time increases;
iii) After the sign function is replaced by saturation
function, the high-frequency oscillation of the control input
vanishes.
Figure 4. Position errors with the control law (11).
Figure 7. Control input with the control law (12).
Figure 5. Position errors with the control law (12).
Figure 6. Control input with the control law (11).
The simulations with sampling period of 2ms are
implemented. Figure 2 ~ Figure 5 present the response of the
robot manipulators with the control laws (11) and (12),
respectively. Figure 6 and Figure 7 are the control inputs
produced by the control laws (11) and (12), respectively.
From simulation results, it is easy to see that:
i) Using the proposed control laws (11) and (12), the
asymptotically stable tracking control can all be achieved;
VI. CONCLUSIONS
In this paper, we have presented a solution to the tracking
problem control of robot manipulators without exact
knowledge of the robot dynamics. The explicit conditions on
the controller gains to ensure global asymptotic stability of
the overall closed-loop system are given in terms of some
information exacted from the robot dynamics. An attractive
feature of our scheme is that SM-PD-I NP-D tracking controller
has the faster convergence, better flexibility and stronger
robustness with respect to initial errors, and then the tracking
errors quickly tend to almost zero without oscillation as time
increases. Our findings have been corroborated numerically
on a two DOF vertical robot manipulator.
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