IEEE Robotics & Automation Magazine - December 2022 - 109

variables ()z 01
kz kaa
and ()z 02
11 1
- 11 , and :,zR kz k
n
are contained in X | != "
X =-| "
1
z1
z ,2
defined as follows:
X
X
z | " zR zk ei ,,f, m,
" zR zk ei
z
1
2
:: (),, ,,
:
=- =
=- =
1
( 'zR z i
n
22 2
22
ia
!< <#
!< <#
!< <#
11
ia
2m:( ),
n
1122
112
p
-
p
-
p
f n,
m ()min
M
Proof of System Stability
Before proceeding with the proof of stability, the following
Mq is symmetric and
useful technical lemmas and properties must be clarified:
* Property 1: The inertia matrix ()
Mq Cq q
o
positive definite, and () (, ) is a skew-symmetric
o
- 2
matrix with the property [S1]
() (, ).
Mq Cq q20o -=o
* Property 2: There is a parameter vector that depends on
the manipulator parameters (),Mq (, ),Cq qo
Gq ()
(), Fqo
and satisfies the linear relationship
() (, )( )( )( ,, ,) ,
Mq Cq qGqFqqqP
vt
U
o
++ U vt+=
oo o
where (, ,, )qq aa ! Rnl
o
joint variables, and PRn
# is the regression matrix of known
! is an unknown constant parameter
vector that describes the mass of the manipulator.
* Lemma 1: For bounded initial conditions, if there
exists a C1
that () () , where ,:RRn
# Vx c-+t
xt is uniformly bounded.
VV log
21
=+
=
=12 f ,, ,
2
1
kk zz
kk
a
T
a
T
2 2
a
2 2a - T
where [, ,, ], kk B22 0
in and a
kk kkaa aa i
f
VV zC qq qG qM q
zM qz
21 2
2
2
1
2
1
1
=+ -- -
++
T [( ,)
T o () 2
=+ -- -
+- +
Vz CqqGqM q
zM qC qq z
T[( ,)
o
2
T[( )( ,)]. (S2)
2 2
o
2
2
2Cq qz2
o
Vz Cq qG qM qk z
kk zz
o 2
=- -- -
+
is zero, and we can obtain
[( ,)
zJ z2
T
1 s
a
T
1 1a - T
1 1
+
zz
.
T
2 2
kk zz
a
T
2 2a - T
Txa a
o
o
2
2 2
According to property 2, there is a parameter vector based on
the robot manipulator that satisfies () (, )( )
Mq Cq qG qaa
o++ =
o
() () ]
.
o
2
kk zz
zz
a
T
2 2a - T
2 2
According to property 1, the value of (/ )[ ()zM q -
(, )]
12 T
2
11
<<
2
(S3)
o
11 1
kk zz
zz
T .
2 2
a
T
2 2a - T
2 2
o xa a
T .
() () ]
o
221222 ai=- as ;;1
<<1B .0 The derivative of V2
T
ci
is given by
oo xao
oo
() () ]
2 2
12
12
<< ## <<
ll " are class l
continuous and positive definite Lyapunov
xVxx
function V(x) satisfying () () (),ll such
Vxo
functions, and t and c are two positive constants, then the
solution ()
To prove the stability of the whole system, we choose the
Lyapunov candidate function as
+ zM qz2
2
1
2
T () ,
(S1)
xk ,ic i22
[, ,, ], in are positive constants. Thus, we
can obtain
where [, ,, ],
kk k i
21 22
rr= rr T
2
12
T
f
f
.
21 1
Vk zz sgnz
1
=- +- -
=- -+ -
kz kz zz P
11
<<
<< <<
2
=
Then, we obtain
Vk zk z
21 1
- 11 demand according to Theorem 1, and the
signal z2
We can deduce that
kz kaa
References
[S1] H. L. Tong and C. J. Harris, Adaptive Neural Network Control of
Robotic Manipulators, vol. 19. Singapore: World Scientific, 1998.
[S2] J.-J. E. Slotine et al., Applied Nonlinear Control, vol. 199.
Englewood Cliffs, NJ, USA: Prentice-Hall, 1991.
also contains kz .kaa
- 11
22 2
o #< << <-- 2
2
22 10 .
the error signal z1
(S8)
satisfies the
2 ///Pj
2
r r
22 |. (S7)
i 1
; UUij j
===i 1
iiji
j
22
1
r
2iKi
()i
i== =1 j 1
l
22
1
i
n
n
kz zP
i =
i
2
2
i
n
K KK Kn KP , in1 f= ,, , and k2
=12 f ,, ,
r =RU=
r r
i
j
l
1 ij j
// G
/ /
n
n
l
2iUij j
(, ,, ),qq
o
U aa P
o
Ur
where (, ,, )qq
o
ij
U aa ! Rnl#
o
unknown constant parameter vector that describes the mass
of the manipulator, and the upper bound Pj
jl1 f= ,, .
that satisfies UUr ij $; ; , in1 f= ,, , jl1 f= PRn
r
,, .
We can obtain
[( ,, ,) ]
Vz qq Pk z
kk zz
o =-U
T
2 xa <<2
o
T
2
+
zJ z2
1 s
a
T
1 1a - T
1 1
ao
+
kk zz
a
T
2 2a - T
zz
22 = )
TT +
-
11
zz
.
T
2 2
.
2 2
Consider the Moore-Penrose inverse as follows:
()
0 z
1
,
,
When z [, ,, ],00 0 T
2 = f
.
2 = 00 0
otherwise
f
we can deduce that
Vk .
21 1
2
=- #z 0
z [, ,, ],00 0 T
2 = f
Y
define zq2
.
r
x=- -22 2
Ksgn zk z
()
2
a
T
2 2
a
-
-
kk zz
x
ao
T
2 2
- z2
()
T +
kk zz
zJ z
T
1 s
a
T
2
1 1a - T
,
1 1
=
(S6)
[, ,, ]T
.
(S4)
satisfies P Pjr $; j ; ,
is the regression
! is an
matrix of known joint variables, and there exists an upper bound
ij
,, ,, ,in1
=12 f
z20 22 12
z10 zR :m
aa
2
! 11 ,
respectively. The control variables of the visual servoing system
will remain in the compact set
X and X which is
where p =202V (). The robust FoV constraint control strategy
can hold the following properties:
1) The output control variables never violate the preset constraints,
i.e.,
;;1 im 6t 02 .
xk ,ic i11
3) The tracking errors z1
=12 2f ,, ,,
2) All the closed-loop signals are constrained within preset
boundaries.
and z2
asymptotic convergence to zero, i.e., () ()
System Stability. "
xt xtd
of the closed-loop system achieve
1 " as t " 3 .
Please see the detailed stability analysis in " Proof of
(S5)
Thus, the system is asymptotically stable according to the
Barbalat lemma [S2]. If
=-a
o
as
the sliding mode surface and design the control input x as
follows:
DECEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
109
IEEE Robotics & Automation Magazine - December 2022

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2022
