IEEE Robotics & Automation Magazine - December 2022 - 70

6xx x @
ff
12
,,..., W
q
f T
q in (7) can be seen as the combined
effect of all activated muscles. Finally, but most importantly,
the number of enhancement nodes increases with tracking
demands. According to the BLS, accuracy is improved by
inserting additional feature nodes and enhancement nodes,
which is similar to the contractile mechanism in which
impedance is increased by activating more muscle fibers.
●
Learning process: In previous studies, e.g., in [12] and
[13], stiffness and damping are updated by a decayed
factor and tracking errors, while in (5),
D ()tj
K ()tj
6xx fxxR= jj
lent
ff WWq
r tt T
12
,, ,, and the equivaq
q
f
T
@
j
q
st iffness () =R 1= ^hj()KK W and dampq
=1
` j
j
()
j
q
KK
ij
tt DD
!
ij
tt to
!
j
q
ing () =R 1= ^h
j
{d
DD Wj
T
r tt . Due to i and
{k
j ,
{k
{d
i and
can be different functions for ij ,! and the impedance
factors satisfy () () and () ()
imitate the difference in muscle fiber activations. The BLSbased
learning structure takes advantage of sparse autoencoder
characteristics to obtain better features, and the stiffness
and damping adaptation is realized by adding new enhancement
nodes and updating weights by the linear inverse functions
in [23], which causes a nonlinear learning process.
Moreover, referring to the control structure in [12], the proposed
framework offers some advantages: no force sensor,
control in the joint and Cartesian space, and adaptive control
with unknown parameters of the robot dynamics model.
Simulations and Experiment
There are three groups of comparative simulations. The first
two simulations are based on the data set provided in [27]
Table 1. Definitions of different force fields.
Force Field Expressions
DF
FDF =*
CF
VF
P-DF
MF
;;EE x ! ^-02 02h
100
FCF =; E
10
F ==-;
o
VF vv
Kx,K
Z
]
]
]
]
]
]
]
]
]
]
]
]
]
]
]]
;
;
FMF =[
100
-;
10
E,
13
18
E;
13
18
F =-PDF
x
x
2
1
E,
-18
13
012h
E
100^ DX ., / EE
-
D
D D
X
X X
x ! ^00 2,. h
y ! ^0450 6., .
x ! ^00 2,. h
y ! ^02 045@
--02 0@
0450 6h
18
13
E= G
o
o
x
y
., .
,
x
y
\XxyX XXc
1000 12h
6 ,, D /
=@
!
!
^
^
^
DX - .,
T
D
D
.,
., .
X
X x
y
!
!
^
^
-02 0@
02 045@
.,
., .
h
;;
x
y
x
y
c
x
y
otherwise
,. ,.
,
j
q
and
are calculated by using different transformation
f
functions for each muscle. Then we can get x =
f T
and on https://www.imperial.ac.uk/human-robotics/
software/, which are carried out on a planar arm using the
two-joint model of a human arm/robot, which is detailed in
[8]. In [8], there are two kinds of force field: a position-dependent
DF field and a velocity dependent external force (VF)
field. In [5], the authors defined two other force fields: a constant
interaction force (CF) field and a position-dependent
DF (P-DF). In the third simulation, we introduce a new
mixed force (MF) field that combines the aforementioned
four force fields to compare the effect of position tracking and
force matching in a complex environment. The expressions of
these force fields are listed in Table 1.
Line-Tracking Task in the VF field
The first simulation takes place in the VF field, and the comparison
method is from [8]. The reference trajectory starts at
x [, .]00 31
s =
T 13=
.
and ends at x 00 55
ping function is zi xx ,=^h
impedance factors are xxw2
where w 05 1
s . The parameters are l 5 ,=
{ =^h
j
k
.,
and transformation functions for
x and { =^h
j xx ,w10
d
x
x !^h is a randomly selected factor. The iteration
times are 50. As the robust term is specific to robot control,
it is not considered in the simulation as in [12]. Each
iteration is completed within a periodic time and refines the
parameters and calculations based on the results of the previous
iteration.
The simulation results are presented in Figure 4. Figure
4(a) shows the evolution of the trajectories in the first and
the last three trials. The final trajectory is almost a direct line
between the start and the end as planned before adaptation.
Figure 4(b) and (c) show that joint torques change with time
and the feedforward torque eventually approaches that which
is exerted by external forces. The adjustment process of the
feedforward torque and impedance of the shoulder joint with
the iterations is displayed in Figure 4(d), (i), and (j). These
variables can approach the final states rapidly but don't converge
to the final state within 50 periods, which is inconsistent
with the experimental results that converge in exponential
form within 10 trials [21].
Figure 4(e) shows trajectories under the control of the proposed
method. We can see that the trajectories converge within
the first three iterations and the final trajectories are
straighter, benefiting from the faster and more efficient convergence
rate of the feedforward torque and impedance factors,
as illustrated in Figure 4(h), (k), and (l) (fewer than 10
iterations). Similar to the results in [8], impedance factors
increase initially and then decrease to the final value within
the next trials.
Tracking Tasks in Different Force Fields
The second simulation group is taken within four kinds of
force fields in Table 1: DF, CF, VF, and P-DF to complete the
same task that the actuator moves along the following four
line segments first:
1) from [. ,. ]
2) from [. ,. ]
xs =-0120 43 to x = [. ,. ]
xs =-0085 0345 to x = [. ,. ]
d 0120 43
d 0085 0515
70 * IEEE ROBOTICS & AUTOMATION MAGAZINE * DECEMBER 2022
d = [, .] with duration
c 2 ,= the feature map
https://www.imperial.ac.uk/human-robotics/software/ https://www.imperial.ac.uk/human-robotics/software/
IEEE Robotics & Automation Magazine - December 2022

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