IEEE Robotics & Automation Magazine - December 2022 - 108

W=
sgnBSSESW
arccos
^
e
BS SW SE EW
BS SW SE EW
#$ $
^^
<<<<
##
##
hh
h
o
(2)
where BS is the vector from the base to the shoulder of the
robot, SE is the vector from the shoulder to the elbow of
the robot, SW is the vector from the shoulder to the wrist
of the robot, and EW is the vector from the elbow to the
wrist of the robot. To implement humanoid control, the
swivel angle W is projected to a null space by utilizing
the robot's redundancy. The joint velocity in the null space
qN can be calculated by
o
o
qJ .NE
= W,W
+
JRE
!
W
(3)
33 is a mapping relationship between the swivel
#
angle W and the null-space joint velocities, which are computed
from the robot's elbow to the base. Wo
is the differential
of W . , ! R31
#
is defined as the direction vector of the swivel-angle
motion, expressed as follows:
,
W =
<<
SE EW
SE EW
#
#
.
(4)
To impel the of the IBVS task constraint's procedure, the
BLF is applied to design the IBVS control strategy and choose
the Lyapunov function candidate as follows:
V1 =
=
,, ,
2
1
log
f
kk zz
kk
a
T
kk kkaa aa i
11 11
11
a
T
a
a - T
ci
im xA ,
,
where [, ,, ], 11 0
=12 2f and <<1 0 and ka1
o
111121 ai=- as ;;1
d
(5)
T kk A xk ,ic i11
determines the
boundary value's constraint. For the visual servoing task, the
IBVS position constraint controller qx
the desired feature point coordinates and restrict them in the
camera's FoV, which is presented as
qJ xk kz zk z
.
x=- -
+
sd a
T
[(o
111121
=
f
constants, and qx
o
a
where [, ,, ], in
kk kk i
T
,, ,
11 11 11
T )],
(6)
=12 f are positive
that drive the robot to achieve the specified location. In
Images FeatureExtraction
Algorithm
x1
xd
+
z1
α
IBVS
Controller
ψd, ψd
.
Null-Space
Projection
q
.
τ
+
+
q
.
N
α
+
-
q
x2
Robot ψ
Kinematics
Figure 4. The control strategy structure of IBVS with humanoid
control.
108 * IEEE ROBOTICS & AUTOMATION MAGAZINE * DECEMBER 2022
-
q
.
IBVS-SMC
z2
τ
Torque
Controller
kk kkaa aa i
in a
=
f
,, ,
221222 ai=- as x i2;;1
=12 f and <<1B .0 Parameter ka2
T kk ,B
ci
determines
where [, ,, ], 22 0
k ,ci2
the boundary constraint of the joint velocity. The control
block diagram is presented in Figure 4.
Stability Analysis
Theorem 1: Consider the overall dynamic model of the visual
servoing control system and the IBVS controller constraint
(7) with the torque controller constraint (8), combined with
an SMC. The initial values of the output error signals z1
z2 are finite, and the initial values of the control input
represents the required joint velocities kk kk i
= [, ,, ], i ,, , n12 f=
rr= rr T
f
12
f
T
VV log
21
=+
2
1
kk zz
kk
a
T
22 22
22
a
T
a
a - T
where [, ,, n],KK K
221222
K
KP ,ij
l
r =RU=
r
1
is proposed to track
the system dynamics modeling, the image feature coordinates
are related to the joint velocities through the task
Jacobian matrix.
Differentiating V1
zJ () .zx
with respect to time, we can obtain
12 a=+ -
oo To stabilize the visual servoing system
sd
and constrain the visual feature points in the camera's FoV,
we design the visual servoing position constraint controller
a as follows:
a
=+ -
=- -+ -
qI JJ q
Jx kk zz kz IJ JJ
.
x ()oqq N
[(o
+
sd a
T
++ +
11 11 11
a
T )] () .
qq E Wo ,W
(7)
In this work, we assume that the anthropomorphic robotic
arm is far from singularity and that pseudoinverse of the Jacobian
matrix exists.
Sliding Mode Torque Control
For robot manipulator control, a torque-level controller is
investigated to actuate the robot to the desired position.
The term () (, )( )
++
oo
oo
Mq Cq qGqqqPaa aa
#
++ U=
oo
ij
oo
,, ,
Mq Cq qG qaa of the torque controller
is unknown, which causes poor control performance. A
parameter vector based on the robot manipulator exists to
satisfy () (, )( )( , ,,) ,
U (, ,,)qq aa ! Rnl
where
joint variables, and there exists an upper bound ijUr
satisfies UUr ij $; ; , in1 f= ,, , jl1 f= PRn
r
=-a
o
is the regression matrix of known
that
! is an
unknown constant parameter vector that describes the
mass of the manipulator, and the upper bound Pj
fies P Pjr $; j ; , jl1 f= ,, . To overcome this problem and
improve control accuracy, zq2
satisis
defined as the
sliding surface, and the sliding mode torque controller is
designed as follows:
.
r
x=- -22 2
Ksgn zk z
()
a
T
a
kk zz
x
22 22
2
-
-
ao
T
- z2
()
T +
kk zz
zJ z
T
a
T
11 11
12
s
a - T
,
(8)
ij jr in1 f= ,, , and
are positive constants.
Considering the stability of the visual servoing system, we
choose the Lyapunov function candidate V2
as
+ zM qz
2
1
T
22 (9)
() ,
and
IEEE Robotics & Automation Magazine - December 2022

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