CFPSO   PSO    CPSO   WUI− 45   WUII  PPSO− IM LPSO

CFPSO   PSO    CPSO   WUI− 45   WUII  PPSO− IM LPSO

CFPSO    PSO     CPSO   WUI− 45   WUII  PPSO− IM  LPSO

Fig。 15。 The consumed energy for case 7, case 8, and case 9 based on 3 robots with 50 runs。

located at (10 m, 0 m)。 From Fig。 13, one can see that the average prediction error for the position of the odor source ap- proaches to 2 m。 Among 5 robots, the best prediction error is lesser than 1 m with a higher proportion for 50 runs。 The afore- mentioned simulation results mainly illustrate that the stable prediction results can be achieved under a given finite-time cooperative controller (17), which reflects the importance of finite-time convergence for motion control。

5。3。Comparison  results

In this subsection, we will compare the search efficiency of the multi-robot system coordinated by the proposed CFPSO algorithm with several selected algorithms, which include the PSO algorithm [30], the PPSO-IM algorithm [25], the LPSO algorithm [27], the CPSO algorithm [19], the WUI-45 algorithm [19], and the WUII algorithm [19]。 It is worth mentioning that the PPSO-IM algorithm, the PSO algorithm, and the CPSO algorithm only utilize concentration magnitude information while the WUI-45 algorithm, the WUII algorithm, and the LPSO algorithm make use of both concentration magnitude infor- mation and wind information。 The parameters of six algorithms can be found in [25,19,30,27], respectively。 Furthermore, we use three evaluation indexes: success rate, search time, and consumed energy to estimate the search performance of multi- robot systems coordinated by these algorithms。 Finally, we will test all the algorithms in the nine scenarios listed in Table 9, but use the new termination conditions, that is, the maximum search time is set as 1000s for shortening the experimental time and the radius of the circle is still set as 1 m。

Remark 13。 The six algorithms are chosen because they have been used to deal with the problem of odor source localization。 Hence, it is unfair to choose other kinds of PSO algorithms that are not designed for the problem of odor source localization。 It should be indicated that the six algorithms, which include the PSO algorithm [30], the CPSO algorithm [19], the WUI-45 algorithm [19], the WUII algorithm [19], the PPSO-IM algorithm [25], and the LPSO algorithm [27], use the same initial search process as the proposed CFPSO algorithm and run at the same simulation environment。 Moreover, the simulation results not only reflect the importance on the position prediction of the odor source, but also illustrate the effectiveness of the finite-time motion control in  the dynamical   environment。

CFPSO    PSO     CPSO   WUI− 45   WUII   PPSO− IM LPSO

CFPSO    PSO     CPSO   WUI− 45   WUII   PPSO− IM LPSO CFPSO    PSO     CPSO   WUI− 45   WUII   PPSO− IM LPSO

Fig。 16。 The consumed energy for case 1, case 2, case 3, case 4, case 5, and case 6 based on 5 robots with 50 runs。

Remark 15。 In the simulations, we choose 3 robots and 5 robots to locate the odor source, respectively。 The reasons are sta- ted in the following。 From the engineering point of view, if we can use a few robots to successfully deal with this problem, we will do not use more robots due to costs。 In addition, according to the third characteristic of the odor source localization problem described in Introduction, a few robots can also sample sufficient odor clues through the appropriate design of the the robot behavior。