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Page 108 Zhou et al. Intell Robot 2023;3(1):95-112 I http://dx.doi.org/10.20517/ir.2023.05

0.5 0.2
Reference Reference
0.4 AFOFTSMC 0 AFOFTSMC
CSMC CSMC
0.3 FTSMC -0.2 FTSMC
Hip joint (rad) 0.1 Knee joint (rad) -0.4
0.2
-0.6
-0.24
-0.1 0 0.364 -0.8 -0.23
0.362
-0.25
-0.2 0.36 -1
-0.26
0.333 0.334 0.335 0.336 0.337 0.228 0.23 0.232 0.234 0.236 0.238 0.24
-0.3 -1.2
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Time(s) Time(s)
(a) Tracking profiles of hip joint (b) Tracking profiles of knee joint
Figure 3. Trajectory tracking performance of the robotic exoskeleton by using CSMC, FTSMC, and AFOFTSMC control strategy. (a) tra-
jectory tracking of the hip joint. (b) trajectory tracking of the knee joint. CSMC: conventional sliding mode control; FTSMC: fast terminal
sliding mode control; AFOFTSMC: adaptive fractional order fast terminal sliding mode controller.

15 0 1 0
( ) isthetrackingerrorunderthecontrolofCSMC, = 15 10 , = 0.01 0.01 , = .
0 10 0 1
2) FTSMC

An FTSMC control law [27] is also designed for comparison, and the control input is shown below:

−1
=[ ( )] [ ( ( + 1) − ( )) − (1 − ) ( )
(51)

+ ( ( )) − 2 | ( )| [ ( )]].
In Formula (51), is a sliding mode variable, defined as follows:

( ) = 1 ( ) + ¤ ( ) + 2 ( ) ( ) (52)

15 0 1 0
( ) isthetrackingerrorunderFTSMcontrol, 1 = 15 10 , 2 = 0.05 0.05 , = .
0 10 0 1
To test the robustness of the three algorithms to external disturbances, the same external disturbances are set
for the three controllers respectively, and the external disturbances are set as:

5 ( )
( ) = ( ). (53)
−5 ( )

The simulation results are shown in Figure 3-6, which are position tracking of the hip joint and knee joint,
control input signals, position tracking errors, and sliding mode surface function respectively. In Figure 3,
compared with CSMC and FTSMC, the control algorithm proposed in this paper can track the gait trajectory
more accurately, and the tracking trajectory of AFOFTSMC is closer to the reference trajectory.

Moreover,itcanbeseenfromFigure4thatthecontrolinputsofAFOFTSMCdonotneedtogivegreatercontrol
efforts to maintain higher track tracking accuracy and effectively eliminate the impact of external interference.
According to the comparison of tracking errors in Figure 5, the AFOFTSMC algorithm has the smallest steady-
state tracking error, which can ensure that the system error converges in a finite time. In Figure 6, the sliding
mode variables of the three controllers can move rapidly into the quasi-sliding mode band, and the control
algorithm proposed in this paper can effectively reduce the width of the quasi-sliding mode band.