IEEE Robotics & Automation Magazine - December 2018 - 89

including the smoothing model-reference SMC (SMRSMC),
smoothed quasicontinuous second-order SMC (SQC2S),
and quasicontinuous third-order SMC (QC3S). Simulation
results have shown that SQC2S can deliver the shortest settling times as well as relatively smooth control torques compared with SMRSMC and QC3S. Thus, SQC2S is chosen for
application to the dual-arm space robot and comparison
with the controller proposed in the "Adaptive Variable
Structure Controller" section.
The sliding surfaces are selected as
s = eo 1 + Ke 1 = e 2 + Ke 1,

(4)

where s ! R 7 and K ! R 7 # 7 is a constant positive diagonal
matrix. The selection of this sliding surface ensures that, once
the sliding manifold is maintained, the tracking error can
s = 0 , then tlim
e 1 = 0. The conconverge to zero, i.e., if tlim
"3
"3
*
x
trol torque is given by
*

xi

= - ki

so i + | s i | 1/2 sgn ^s ih
, i = 1, 2, ..., 7,
| so i | + | s i | 1/2 + o

(5)

with o as a small positive scalar. Let M denote a matrix or a
vector and M = (m ij), i = 0 for a vector.
Define M = ^ m ij h . Then, k i satisfies the following
condition:
k i $ " b 2 c 2 At - 1 Bt Xo + Xp d + K eo 1
+ c 2 r At - 1 At At - 1 x max ,i
= W max i .
(6)
Here, b 2, c 2, and r are positive scalars introduced to represent the bound of the matrices. Specifically, A - At =
DA # r At , b 1 Bt # B # b 2 Bt , c 1 At -1 # A -1 #
c 2 At - 1 ; (5) shows that x * depends on s and so . Because so
will induce a serious chattering effect if directly derived from
s, a first-order differentiator [16] is used to obtain an estimation of s and so . The first-order differentiator takes the form
1/2
zo 0i = - b 0i z 0i - s i sgn (z 0i - s i) + z 1i,
zo 1i = - b 1i sgn (z 0i - s i),

(7)

where b 0i and b 1i (i = 1, 2, f, 7) are positive constants. By
replacing s i and so i with z 0i and z 1i, respectively, in (5), the
chattering effect exhibited in control outputs can be reduced.
The preceding control laws ensure that the system is asymptotically stable, proof of which has been provided in [15].
AVSC Controller
In contrast to SMC, which restricts the error dynamics to a
predetermined sliding manifold, the AVSC approach is able
to update the control gain in real time and enforce the error
states to the origin along a parabolic trajectory. Such a path is
closer to the natural behavior of the system. As a result, the
chattering effect is eliminated, and the overall settling time is
reduced. In addition, the AVSC method ensures robustness
against system uncertainties and disturbances through gain
adaptation [12].

The AVSC algorithm consists of three stages. In stage one,
referred to as the reaching phase, the control law is designed to
enforce the system's reaching of the condition v (e) = 0.
Here, v (e) = 0 represents a virtual parabolic trajectory determined by the error states. Once the error dynamics arrive at
the vicinity of v (e) = 0, they enter what is called the convergence phase. During the convergence phase, the error states
are driven to the origin of the error phase plane. The adaptive
control gain is tuned online during this phase. The tuning is
done by comparing the expected error dynamic behavior of
the nominal system with the measured behavior of the error
dynamics between two sampling instances. When the error
states arrive at the vicinity of the origin of error space, the
phase known as the constrained phase starts. In this final
stage, an appropriate robust control approach, e.g., SMC, can
be used to constrain the error at the origin in the error phase
plane. The error dynamics in the error phase plane for AVSC
are shown in Figure 3. It is worth mentioning that the main
advantage of the AVSC controller lies with individual setpoint tracking. In the case of trajectory tracking, because the
change of two adjacent desired states is quite small, the proposed controller would continue to operate in phase 3. Thus,
the controller would still be robust in trajectory tracking; the
system performance will be similar to that of SMC.
The control laws for phases 1 to 3 take the following form.
Phase 1 (reaching phase):
*

xi

= - fi - M maxi sgn (v i) .

(8)

The term M maxi is selected such that M maxi 2 h i max . As
addressed in (3), fi and h i represent the ith element of
f (e, t) and h (e, t), respectively.
Phase 2 (convergence phase):
*

xi

= - fi + M i (t),

(9)

with M i (t) = l ci (t) - l mi (t) + M i (t - T), where
l ci (t)

=

eo 12i (t)
,
2e 1i (t)

l mi (t)

=

.
e1

eo 12i (t) - eo 21i (t - T)
,
2 [e 1i (t) - e 1i (t - T)]

Initial Error States
← Phase 3

e1

Phase 1

Phase
e2
t=T
t=2
2T
T

t=0

σ (e ) = 0

Figure 3. An example of how error dynamics converge to zero
under AVSC [12].

december 2018

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