IEEE Robotics & Automation Magazine - December 2018 - 94

practice, before applying
the controller to an actual
spacecraft, simulation
studies are always needed
expected to reach the
to assess the limits of the
matrices according to the
corresponding contact
desired and available
motion, as stated in [15].
points along two
The attitude-tracking
errors of the system
straight lines.
under AVSC and SQC2S
are shown in Figure 9(a)
and (b), respectively, using a boxplot. With the increasing
value of d, the tracking errors present a different variation trend for the two controllers. For AVSC, the space
robot can reach its best accuracy when the model is

× 10-4

5
0
-5

4
3
2
1
0
-1

× 10-3

-0
.
-0 3
-0.2
.1
0
0.
1
0.
2
0.
3

eγ -SQC2S (°)

-0
.
-0 3
-0.2
.1
0
0.
1
0.
2
0.
3

eγ - AVSC (°)

The end effectors are

δA and δB
(a)

δA and δB
(b)

× 10-3

eθ1e (°)

eγ (°)

4
2
0
0.08
0.06
0.04
0.02
0

er1ex (mm)

accurately known, whereas the tracking accuracy of
SQC2S becomes worse when d increases. In the presence
of system uncertainties, both controllers can restrict the
attitude-tracking error to a small value and thus achieve
high-accuracy attitude control. However, although the
spacecraft attitude can be precisely controlled, the comparison depicted in Figure 9(c) demonstrates that the
pose tracking error of arm 1 under SQC2S will yield a
large value when following the specified course, especially for position tracking (as large as 6 mm for the x direction). Such errors may cause serious issues if a space
robot operates in a complicated environment, such as in
a cabined space with obstacles.
In addition, the consumed energy can be reduced based
on AVSC compared with that based on SQC2S, as shown in
Figure 10. The energy consumption of AVSC keeps increasing as d increases; in contrast, the consumed energy of
SQC2S presents a relatively stable state. A larger value of δ
indicates larger system inertia, causing more energy to be
required based on AVSC to enforce a heavier object to track
the same desired path. As for SQC2S, because the upper
bound of A and B is used in control law (6), excessive control
torques are generated for the system, even with a smaller d.
But the SQC2S method can achieve high control accuracy if
enough energy has been available.
It should also be noted that there are some disadvantages
of the AVSC method. First, similar to SMC, AVSC requires
the upper limit of the uncertainties and disturbances present
in the system. Second, because the gain of AVSC adapts continuously at each sampling interval, the gain is susceptible to
noise. A noisy state measurement can cause the gain value to
change rapidly in consecutive sampling intervals. Hence, it
must be ensured that the measured states are as noise free
as possible. Filters and observers are recommended when
unmeasured state values are estimated. Finally, AVSC is
designed such that its gain value is recalculated at each sampling instance, by looking at how the error states evolve over
time. Therefore, the sampling rate should be set such that the
controller can capture the true progress of the error states

4
2
0

120

∑ 1 f ||ti ||dt

-0.3

-0.2

-0.1
0
0.1
δA and δB

0.2

0.3

(c)
Figure 9. The tracking errors of the system under AVSC and
SQC2S with the presence of system uncertainties: (a) the
attitude-tracking error under AVSC, (b) the attitude-tracking
error under SQC2S, and (c) a comparison of the tracking error
between AVSC (left) and SQC2S (right).

IEEE ROBOTICS & AUTOMATION MAGAZINE

december 2018

i = 1 t0

er1ey (mm)

100
1.5
1
0.5
0
-0.5

80
60
40
20
0

-0.3

-0.2

-0.1

0
0.1
δA and δB

AVSC

0.2

0.3

SQC2S

Figure 10. The energy consumption of the system under AVSC
and SQC2S with the presence of system uncertainties.

IEEE Robotics & Automation Magazine - December 2018

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