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Page 10 of 14 Ma et al. Complex Eng Syst 2023;3:10 I http://dx.doi.org/10.20517/ces.2023.14

¤ 5
| | < 9 + 9 1. According to Theorem 3, the state variable , will converge into the origin within a fixed
˜ + 1 can be obtained. Then,
time 1 , then one has ≤ , | | ≤ , | 2 | < . Further | ¤ | < | ¤ | ≤ 1 max
(33) can be simplified as
¤ 2 1 4
1
6 ≤ (2 1 + 2 3 + + )( 1 max + 1 ) + ( 1 max + 1 ) 4 1
4
2 1
+ √ 3 1 ( 1 max + 1 ) +
2
2 2 1 4 4 2 3 1 (34)
1
≤ 2 2 1 max + 3 + 2 3 + + ( 1 max + ) + √ ( 1 max + )
2 4 2

+
+2 3 1 max

≤ 6 + 1
where and 1 satisfy the following constraints:

2 1 4 4 2 3 1
+ 3 + 2 3 + + + ) + √ + ) (35)
= 2 2 1 max ( 1 max ( 1 max
4 2

+ (36)
1 = 2 3 1 max

¤ ¤
On the contrary, if 1 > 1, there exists a positive constant 2, which satisfies 6 ≤ 2. One has 6 ≤ 6 + 3
for the state variable , , . Further, before the angular error converges to zero, one can obtain
3 3
6 ≤ 6 (0) + − (37)

Remark 1 The auxiliary variable in (24) can reduce the order of the third-order subsystem, which simplifies
the process of the controller design. In addition, the controller developed in this paper can guarantee that the
system state variables converge in a fixed time and the chattering problem is solved by using the error function
erf(·). Furthermore, utilizing the variable exponent coefficient in (24) avoids the common singularity problem.

4. EXPERIMENT RESULTS
To verify the effectiveness of the proposed control scheme, the trajectory tracking experiment is implemented
on a Quanser QBot 2e mobile robot platform composed of a QBot 2e mobile robot, an OptiTrack system with
12 infrared cameras, and a computer. The experimental platform is presented in Figure 2. The whole closed-
loop experiment structure is as follows: The simulation diagram is compiled on the host computer equipped
with MATLAB/Simulink to transform the simulation into an executable file. And the controlscheme is written
to the Gumstix computer embedded in the QBot 2e through wireless communication protocol. The real-time
position information of the QBot 2e is obtained by the OptiTrack positioning system. Then the host computer
calculates the information and transmits them to the embedded computer of a WMR for the input of real-time
calculation of executable files. So as to complete the trajectory experiment of the mobile robot.

In the experiment, the physical parameters of the QBot 2e are chosen as follows: = 4 kg, = 2.5 kg · m .
2
The desired reference trajectory is set as = cos(0.2 ) m, = sin(0.2 ) m. The initial values of the reference
T T T T
and practical trajectories are [ (0), (0), (0)] = [1, 0, /2] , [ (0), (0), (0)] = [0.7, −0.02, /6] ,
respectively. The main relevant parameters of the proposed control scheme are as follows: 1 = 0.001,
11 = 12 = 0.9, 21 = 0.05, 22 = 0.06, 2 = 0.00001. Choose the parameters 1 = 2, 2 = 0.5, 1 = 2 = 1
for the sliding mode surface 1 in (17) and 2 in (25), respectively.