IEEE Robotics & Automation Magazine - December 2018 - 91

Next, the system performance under the two types of controllers, i.e., SQC2S and AVSC, is analyzed in detail. The control parameters for the SQC2S method were chosen to realize
optimum performance. Specifically, the sliding manifold
parameter m i is tuned such that the space robot achieves its
fastest possible settling time: m i = 3. Here, b 2, c 2, and r
are determined by the maximum percentage of system
uncertainties: b 2 = c 2 = 1.3, r = 0.3. For the AVSC method,
M maxi, which represents the maximum values of the adaptive
gains, is tuned so that the control torques are kept in a similar
range compared to those provided by the SQC2S method:
M maxi = 0.3.
As described in the "Smoothed Quasicontinuous Second-Order Sliding Mode Controller" section, the value of
o should be carefully determined for SQC2S. It should not
be set too low (otherwise, the smooth properties of the
control torques may be lost) nor too large (as the system
will become unstable without sufficient control effort).
Therefore, the system has been tested with different o values, i.e., o = 0.01, 0.03, or 0.1. Tracking errors of spacecraft attitude, the orientation of the first arm end effector,
and the position of the first arm end effector are shown in
Figure 5(a)-(c). The control torque of the reaction wheel
and joint torques of the first arm are given in Figure 5(d)
and (e). Other system states and control inputs present the
same trend.
Because energy consumption plays a criticalt f role for a
spacecraft, a performance criterion E = R 7i = 1 # x i dt is
t0
calculated to evaluate the system's total energy consumption,

Initial
Posture

re(2) (t0)

Figure 4. The course of the space robot in path tracking for AVSC.

as depicted in Figure 5(f). Figure 5 shows that a smaller o
can potentially reduce the end-effector pose tracking error
but will result in a chattering effect in the control torques and
lead to more energy consumption; conversely, a larger o can
provide smooth control torques and consume less energy but
at a sacrifice of increasing tracking errors. Therefore, to
achieve a high-accuracy tracking operation, o should be
determined as a tradeoff between energy availability and
chattering-effect exclusion.
The AVSC method has been applied to the robot under
the same initial conditions to track the same hand trajectories. Figure 6 illustrates that AVSC achieves much higher

2
0

v = 0.01
v = 0.03
v = 0.1

5

-2

0

0.4

0

50

100
Time (s)

0

150

50

100
Time (s)

v = 0.01
v = 0.03
v = 0.1

-0.4
0

50

100
Time (s)
(d)

150

t1 (Nm)

-0.1

0
-0.05
v = 0.01
v = 0.03
v = 0.1

-0.1
-0.15
0

50

100
Time (s)

150

(c)

0.05

0

-0.3

50

(b)

0.1

-0.2

0

150

100
Time (s)

Energy Consumption

0

v = 0.01
v = 0.03
v = 0.1

0.2

-5

(a)

tω (Nm)

re(2) (tf)

1

v = 0.01
v = 0.03
v = 0.1

1

eγ (°)

re(1) (tf)

× 10-4

4

-4

re(1) (t0)

ere (mm)

× 10-5

eθ (°)

6

Final
Posture

60
40

v = 0.01
v = 0.03
v = 0.1

20

150

0

0

50

100
Time (s)

150

(f)

(e)

Figure 5. The system performance under SQC2S with different values for o: blue: o = 0.01, red: o = 0.03, magenta: o = 0.1. The
(a) attitude-tracking error, (b) end-effector orientation-tracking error of arm 1, (c) end-effector position-tracking error of arm 1, (d)
reaction wheel control torque, (e) joint control torques of arm 1, and (f) energy consumption.

december 2018

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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IEEE Robotics & Automation Magazine - December 2018

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