IEEE Robotics & Automation Magazine - March 2023 - 77

R
Kt =
G()
T
S
S
S
S
-
-
cos
sin
sini
i
i
cosi
cos
sin
cos
sin
i
i
i
i
L V
-
-
L
L
L
X
W
W
W
W
increment in the predicted time domain can be obtained
as follows:
.
T
HD0 Vd .#r
It should be noted that the discretized model in the kine- dd XVkd XVk11max2 11 ], =-CCro
T
matic model can be written as follows given the system state
X(t) by expressing the output as () ().
Yt Xt=
dV (),( ).
20 max-= -Using
the nonlinear
feedforward difference method and rewriting the velocity
input V(t) in incremental form,
*
where () is the velocity input of the system in the previous
moment and DVk is the velocity increment in the current
moment. The matrix AI ,3= and I3 is the identity matrix.
() (()( ))-1
Vk 1()
BT
() is the input matrix,
= · KtN
KtG
T
Kt Kt KtNG G
T
=
() is the pseudo-inverse of KtG (), and T is the sampling
period.
Here N is the prediction time horizon, and NC is the control
time horizon. To constrain the actual velocity of the RRDS,
the following constraints were established for the motion and
input velocities of each wheel:
)rr r
ro
ro
XX X
VV V
minmax
minmax
##
##
ro
(5)
where Vr and Xor denote the predicted velocity input at future
time steps from k to kN 1C+- and the predicted actual
motion velocity at future time steps from k to
kN ,1+respectively.
The upper and lower limits of the system input
are Vmaxr
and
V ,min
r
upper and lower limits of motion velocity are Xmax
ro
HDV .r
max =- min
and
X ,min
ro
motion velocity of the RRDS is given by ()XV k 1 +
VV kk Vk kV1TT T
(| )] ,, ,[ ,, ,] ,
kN kFZFZZ II I
+- CH
R
Z1 =
T
S
S
S
S
R
Fq =
T
S
S
S
S
S
I
I
4
4
hh
()
00
h
I4
I4 II
Fk
44 N
g
g
j
g
Fk 1
+
hh
X
W
V
W
W
W
()
,
g
g
j
g
h
Fk N 1
+()
V
X
W
W
W
W
W
where
Z1 and Fq are the decomposition matrices of the
coefficients C and H in the predictive velocity model,
respectively, and (),,Fk
+= -
~~ 01 N 1f
is the input
coefficient matrix corresponding to the RRDS system for
future time steps from k to
kN .1+- By substituting the
motion velocity of the RRDS into (5), the constraint for the
C 1 == =01 04 44g
Her e [( |),( |),,
T
f
DD DDr=+
qq
N
T
respectively, where rr TheVV .
ro maxmin=- or
Here () () (),()()( ), () is the
limited motion velocity, Xtd () is the specified trajectory,
and ()tpt
et Xt Xt et Vt Xt Vtdd=- =- o
12
is the estimated human-robot uncertain environrespectively,
where XX . According to (4), the
ro =-C
ment, respectively.
The stability of the system can be achieved by establishing
v () (/ )( )( )( /) () () (/ )
Lyapunov Vt=+ 2 + 12
is the estimated error of the weight b .
12 et et
1
bbTuu where bb b=-u
t
,
In the passive training stage, the main purpose of our design is
to make the RRDS track the training trajectory specified by the
doctor at a limited velocity and suppress uncertain motion environments
to ensure the safety of the human-robot system.
After the passive training stage, the rehabilitee achieved
a certain leg support strength. The RRDS design movement
pattern must conform to the walking characteristics
of the rehabilitee. The rehabilitee should express the
desire to participate in the training process. Next, the
RRDS function should be switched directly to the active
training stage.
ACTIVE TRAINING
As the system enters the active training stage, the RRDS needs to
identify the velocity of the rehabilitee to achieve the coordination
of the human-robot velocity and stable tracking in a limited learning
time to ensure the safety of the rehabilitee.
MARCH 2023 IEEE ROBOTICS & AUTOMATION MAGAZINE
77
TT
1
2
12 et et
ut BM Xt et et
ad 12
() ip (8)
=- -t
p
()
(()()()(t)) .
t
Xk 1
Vk Vk Vk
Yk Xk BV k
+= +
=- +
==o
() () ()
() () ()
() () ()
AX kBVk
1 D
(4)
mi 11nmin ZVkd VZ Vk
To minimize the actual velocity tracking error of the
Here [],[ZZ dd ddmin max
minmin -= --ro
=- 20
TT TT T
11
T
1
T
1
max
max
rr
max
robot and the velocity input increment of each wheel, the
objective function can be established as ()min XXJ
^h
=- T
ro
ro d
rr
XX VVd
PD TKD-+
oo rr Combining the objective function
.
and (6), the optimization problem for the motion velocity constraint
can be obtained as follows:
) =+rr
DD D
JVPV bV
d
HD
0 #
min
V
where ,( ),Pb Vk22 X1TT^HPHK HP C
=+ =- - r h and
h
^
d
P and K are positive definite symmetric matrices with
appropriate dimensions.
Using quadratic programming to solve (7), the velocity
increment DVkr () can be obtained. The prediction model in
Vk and obtain the velocity
(4) can then be used to calculate ()
constraint of each wheel.
COMPENSATION TRACKING CONTROL
In the passive training stage, the controller can be designed
using model (2) as follows:
2
1 TT
(7)
(6)
HH H=- -=-- T
(),( ),
2min
IEEE Robotics & Automation Magazine - March 2023

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