IEEE Robotics & Automation Magazine - December 2022 - 107

converted to each other through the formula VW ,Vccee
where W ec
ce
=
8033#
ce
tion matrix and translation vector, respectively. []tce #
33# skew-symmetric matrix. In this article, Wce
R []# ceB where RS ()O 3 and tR3
ce !
R ce
tR
[]
,
is the screw-transformation matrix Wce
ce !
=
are the rotais
the
is a constant
matrix because the camera is rigidly attached to the end
effector. Hence, we can obtain
sJ (, ,) ,zs qqsd
oo=
where JLWJ is the Jacobian task.
+
sceq
_
Define [] !
sqo
TT RTm n
2
as a new state variable of the
visual servoing system; combined with the dynamics model
of the robotic manipulators, the visual servoing system
dynamics can be expressed as
dt
d s
; E =
=
qo
Mq Cq qq Gq
sd
(, ,) o
Jz sq q
-1()[( ,)
For the convenience of expression, define xs ,
the system dynamics can be described as
()[( ,)
o =
where [, ,, ], vm2= represents the
vector of m image feature points, and x1
xx xx R
111121vf !
=
set ;;# im Similarly, the joint velocity
T
xk ,ic i11
=12 f ,, ,
where kc1
=12 2f ,, ,.
vector of the robotic manipulator [, ,, ],
in is required to satisfy xk ,ic i22
are the set of positive constants.
xx xx i
221222
=
f
;;# in
and kc2
Preliminary
BLF
Definition 1 [20]: BLF is a scalar function
V(x) defined on an open region
p including the origin relative to the
system
xf (),x=o
and V(x) is a continuous,
positive-definite function,
which has continuous first-order partial
derivatives at each point of
p .
When x approaches the boundary
of
p , ()
and some positive constant
b, (( ))Vx tb# , 6t 0$ .
Methodology
Problem Formulation
The visual servoing system consists of
an anthropomorphic 7-DoF redundant
robotic arm with a depth camera
mounted at the end effector, as depicted
in Figure 3. Oq
origin of tool frame Fq
is the coordinate
at the geometric
Base
Feature Point
AprilTag
Visual Target Plane
Figure 3. A schematic diagram of the coordinate transformation relationship and swivel
angle defined in the Sawyer robot. S: shoulder; E: elbow; W: wrist.
DECEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
107
Vx . " 3 For the solution of
xf ()x=o
Oc
S
xq
yq
Oq
zq
zc
xc
yc
ψ
W
=12 f ,, , ,
oo o=- - ()],
Tv
12s
2
-1
x
must stay in the
xJ x
xM qC qq qG q
x-- ()]
oo
G.
1 = xq ,= o and
2
●
(1)
center point of the end effector. Oc
frame ,Fc
is the origin of camera
which is situated at the projection center of the
depth camera. The four points marked in the AprilTag represent
four visual feature points. The swivel angle W is
defined in the Sawyer robot, which is described in detail in
this section. The purpose of this research is to propose a
visual controller constraint based on IBVS, which is combined
with the humanoid control of an anthropomorphic
robotic manipulator to make it more intelligent in human-
robot collaboration while ensuring the following:
●
The anthropomorphic robotic manipulator can follow a
time-varying swivel angle given by the operator and rotate
the corresponding angle around the axis formed by the
shoulder and the wrist.
Image feature points are continuously restricted to the
camera's FoV without violating preset safety boundaries.
Humanoid-IBVS Control
IBVS control utilizes image features to define the control
law, and we select points as features. Before proceeding
with controller design, we predefine the image feature
points tracking error as
zx ,xd11
zz zz R
111121v
=
= [, ,, ],f !
zz zz R
221222nf !
=- zx ,22 a=- where
Tv vm2= are the error signals
Tn correspond to the error
between the current and the desired feature points, respectively;
[, ,, ]
signals between the current and the reference joint velocities
of the manipulator; and α is the reference velocity, which are
designed later.
To simultaneously accomplish the constrained FoV task
and humanoid control motion in the redundant robot manipulator,
the swivel angle is employed to accomplish human-like
control. In Figure 3, the swivel angle W consists of the angle
formed by the actual plane (SEW) and the reference plane
(BSW) around the axis
SW . W can be calculated using geometric
relationships.
E
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