IEEE Robotics & Automation Magazine - December 2022 - 30

Remark 1
We note that the adaptation law dio
in () ()dhor includes two parts: the signal hoito
ii ax=-o
to
tt
of the desired trajectory
is
obtained by visual sensing (introduced in the section " Visual
Sensing of Human Motion " ), motion prediction (introduced
in the section " Human Motion Prediction " ), and
filtering (introduced in the section " Signal Filtering " ), and
-ax r
is the force feedback for correcting the motion error
this means it only relies on
xr 0 ,=
ii ax=-o
to
dhor
tt , i.e.,
iho 0 ,=
to
delays in
where Le
between the human and robot. We note that if there exists
no external torque, i.e.,
vision prediction, and this method is always subject to
model overfitting, as explained before. If vision prediction
is not involved in () ()
cooperation are inevitable based on only force feedback.
Based on (5), with
xi ii ii i
rr dr dr d
=- +- +pp
-=
-xi i
where Ko
M ,r D ,r Kr and measured rx by force
oo
denotes the human arm stiffsensor,
ri can be calculated. If i tracks ri precisely, (5) can
be rewritten as () () ().MD K
For later stability analysis, we consider that the human
motor control can be described by an impedance model as
roK (),h
ness gain.
Control Framework
Hybrid Visual-Haptic Control
In this section, we design a hybrid control framework
including visual servoing and impedance control. This
framework combines the complementary advantages of
these two methods, as will be illustrated by experimental
results.
We design the robot control torque as follows:
xi ii i
=+ +
xx xx
xici ic
ffb r
f
f
f
f
bP rD r
MC G
KK
o
=+ +
=- -- -
ir . In fx b, KP and KD
,
() (, )( ),
() (),
o
i
oo
(6)
where ffx denotes the feedforward torque for compensating
for the robot's dynamics, and fbx denotes the feedback torque
for tracking
differential gains.
We define the virtual vector () ,
ciK rr
=- ii
Kc denotes a positive gain. Considering that there exist
uncertainties in the robot's dynamics, we utilize a radial basis
function NN (RBFNN) to solve this issue, so that ffx can be
redesigned as
~ici ic
to
)T
SZ MC G
SZ
x~ ~wZ
o
ff
()
~i v~
== -
=+ +
=t
SZ SZ
TT
C6
where the input vector
o
"
() (, )( ),
()() ,
-+
)
i
t@
Z ,, ,,
iiT TTT
= 6
basis function, and ~TSZt
~ TSZ)
(7)
o cco @ S(Z) denotes the
() is the approximate value of
(). ~to denotes the weight adaptation law in RBFNNs,
C denotes a gain matrix, and the positive constant v is
designed to improve the robustness.
30 * IEEE ROBOTICS & AUTOMATION MAGAZINE * DECEMBER 2022
() () (),
o
denote proportional and
c -+ o where
Stability Analysis
In this section, we analyze the stability of our proposed method.
In particular, we consider a Lyapunov function candidate
L as follows:
LL LL
LMeK e
LNP
LK
er r
ph
ho
2
1
+ u
i
u
i ,
2
2
1
=+ +
=+
=
=
2
1
o
ep h
,
(),
,
22
2
(8)
is used to verify the stability of the tracking error,
Lp is used to verify the stability of the prediction error, and
Lh is used to verify the stability of the error of synchronization
between the human and robot. () ()hho
() represents the prediction error, e iid
resents the angle tracking error, and ii hi=-u
ii=-u
t
tt
ih tpT+
ax bp
QX and
+ and ()=+iho taxbp
()
t
method, xp
t
iho tQ .X=
t
Qa b= [, ] and Xx 1p
t
= [, ], so ()ih tpT+=
T
=- reprepresents
the
motion error between the human and robot.
We define
t can be rewritten as ()ih tpT+=
According to the gradient descent
will eventually converge to a positive constant
when p is given; therefore, X is a positive definite matrix. Also,
the parameter Qt adaptation law can be expressed as
t =- ix+
o
u
QN ,Phr
where N, P R12
#
!
are positive matrices, and P+
generalized inverse matrix of P.
Remark 2
Q can be estimated according to the LS method in (2) or the
gradient descent method in (9). For facilitating stability analysis,
the gradient descent method is utilized in this section.
According to (5), the desired impedance model can be
. Considering Q 0=o
rewritten as
Me De Kerr rrx++ =
po
and further taking the derivative of L and substituting (9),
we obtain
oo oo
op o
oo
2
ep h
rr
oo
2
=- -- ++ -
+- +
=- -- ++
+- -
rr
=- -- ++
+- -
uh
()
()
()
()
rr
oo
oo
De eNPNXNXe
NP X
rd rhoh
rr
oo
2
+ uuhr r
2
h
ooxi ix x o
2
De eNPNXNXe
PX
xi x
2
axr
ooxi ix x o
2
De eNPNXNXe
PX
2
=- -- - PXia x # 0
NP NX De
+
axr
2
NXuhr
d o
2 () .
2
r
+ uuhr r
ix
2
h
(10)
rh r
rr
NXuhr
+ uuhr r
ix
2
h
u
o
to
LL LL
eM eKeNPK
eDeNP
De eNPN PX
De eNPNXNX
+- -+ - o
()
() () ()
() ()
h +
rr hhoh rh
rr hh rr ho
uuhr
+
+
uu
2
o
to
2
r
xi ii xi i
xi ix xi ii
xi ix
+
+
ii iiuuohh uuo
o
T
u to oo o
uo h
xi iaxi i
xi ix xax
=+ +
=+ ++
=- -+ -+ -
=- -+ -+ +- +
=- -(9)
denotes
the
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