2.1. Butterfly Optimization Algorithm (BOA)

This optimization technique solves the global optimization problems by imitating the food finding as well as mating characteristics of butterflies. The location of nectar or mating partner is basically identified by the sense of smell of butterflies. The fitness of a butterfly will vary according to its fragrance intensity, and it varies from location to location. This fragrance will play an important role in creating a social knowledge network among butterflies. The global search occurs mainly when a butterfly finds and senses the fragrance of abother butterfly and moves towards it. Whereas local search is the phase when it is unable to identify and sense any fragrance from others, then it moves randomly to somewhere else; this is known as local search [Reference Arora and Singh35].

Three essential elements form the basis of the overall sensing and processing approach used as the sensory modality (c), stimulus intensity (I), and power exponent (a). A key parameter in the BOA is the fluctuation of I and f’s formulation. I is related to the fitness function. f is relative, though, so other butterflies ought to be able to detect it. c is utilized, in accordance with Steven’s power law (Stevens 1975), to differentiate smell from other modalities. Now, f increases more quickly than I when the butterfly with fewer I approaches the butterfly with more I. In order to obtain this level of absorption, the power exponent parameter a is used to vary f. Utilizing these ideas, the smell in BOA is derived as function of stimulus intensity, which is mentioned below in Eq. (1).

(1) \begin{equation}f=cI^{a}\end{equation}

The value of a and c is taken in the range between 0 and 1. The value $a=1$ represents zero absorption of smell, i.e., the smell emitted by a single butterfly is received in identical manner by other butterflies. Whereas the vale $a=0$ shows the emitted fragrance cannot be sensed by other butterflies. The convergence of the BOA model primarily depends upon the value of a and c. The BOA is performed in three phases such as Initialization, Iteration, and Final phase. Every BOA run begins with the initialization phase, followed by an iterative search, and a final phase where the algorithm is eventually stopped after the global best result is shown. The fitness function and its searching space are defined by the algorithm at the initialization stage. Additionally, values are assigned to the variables used in BOA. The method then generates an initial group of butterflies for optimization after the values have been set. Then, by evaluating the fitness value and fragrance, the locations of the butterflies are created within the searching space. The algorithm initialization phase stops, and iteration phase starts with the artificial butterflies that are created in the initialization phase. The parameters of the BOA are depicted in Table II.

Table II. Representation of butterfly optimization algorithm parameters.

In the second phase, which is the iteration phase, the model runs through a number of iterations. Every butterfly in the solution space moves to a new location after each iteration, after which their fitness values are assessed. The algorithm begins by determining each butterfly’s fitness value at each of its various locations throughout the solution space. The smell of the butterflies is created by using Eq. (1). The iteration phase is based on the two phases, such as global and local search phase. The movement of the butterfly to the best butterfly ( $G^{\ast}$ ) in the solution space is termed as global search and it is shown in Eq. (2). The local random movement is known as local search when the butterflies is not able to sense the fragrance of best butterflies it will take a random walk by using Eq. (3). The Eq. (3) represents here the local search of the BOA. The equation of universal and local search, as derived by the author [Reference Arora and Singh35], is mentioned in the Eqs. 3 and 4, respectively.

(2) \begin{equation}Z_{i}^{t+1}=Z_{i}^{t}+(R^{2}\times G^{\ast}-Z_{i}^{t})\times f_{i}\end{equation}

(3) \begin{equation}Z_{i}^{t+1}=Z_{i}^{t}+(R^{2}\times Z_{j}^{t}-Z_{k}^{t})\times f_{i}\end{equation}

These two local search and global search can be switched by a parameter $\beta$ , which represents the switch probability. The value of stimulus intensity (c) greatly influences the convergence of the algorithm. The value of c should not be too big or too small, and to avoid the premature convergence, the author [Reference Arora and Singh35] demonstrates the calculation of c as present in Eq. (4).

(4) \begin{equation}c^{i+1}=c^{i}+\left(\frac{0.025}{c^{i}\times Max\_ it}\right)\end{equation}

Where i and Max_it denote the current iteration and maximum iteration of the algorithm, respectively. The iteration phases continue until the stopping criteria are fulfilled. The location of the manipulator, such as the manipulator reaching to the desired point, is defined as the stopping criteria in this path navigational problem. The details of the three phases are mentioned below in the flowchart of BOA and pseudocode of BOA.

2.2. Proposed improved BOA (IBOA)

It has been found from the literature that the possibility of capture in the local minima is high in the conventional BOA technique. Consequently, this study introduces an improved BOA (IBOA) technique by using the Levy flight, and a nonlinear strategy of decreasing inertia weight factor is implemented in the basic BOA architecture. The Levy flight [Reference Yahya and Saka36] is one operator that is used for short range through local search and occasional movements for long-distance search. As the butterflies are searching for the food in a very unpredictable region so the Levy flight will improve the efficacy of the searching and also the diversity among the butterflies. While the algorithm is not able to detect any nearby solution, then the butterfly position will be updated by the operator by using Eq. (5).

(5) \begin{equation}Z_{i}^{t+1}=Z_{i}^{t}+(L\hat{e}vy \left(D\right) Z_{j}^{t}-Z_{k}^{t})\times f_{i}\end{equation}

Here, the current iteration is denoted by t, and position vector dimension is denoted by D.

(6) \begin{equation}L\hat{e}vy \left(z\right)=0.01\times \frac{R_{1}\times \rho }{\left| R_{2}\right| ^{\frac{1}{\alpha }}}\end{equation}

Here, R1 and R2 denote two random number in [0,1], $\alpha$ is a constant ( $\alpha=1.5$ here) and $\rho$ calculated as mentioned below in Eq. (7).

(7) \begin{equation}\rho =\left(\frac{\lceil (1+\alpha )\times \sin \frac{\pi \alpha }{2}}{\lceil \left(\frac{1+\alpha }{2}\right)\times \alpha \times 2^{\left(\frac{\alpha -1}{2}\right)}}\right)^{\frac{1}{\alpha }}\end{equation}

In BOA, the search agents are the butterflies. All the butterflies are searching for the best butterfly, which is having a large fragrance value. If the best butterfly is trapped in the local optimal, then the other butterflies will also be trapped. Therefore, to overcome this, in this paper, a nonlinear decreasing inertia weight [Reference Guimin, Jianyuan and Qi37] has been introduced during the global search phase. The weights are updated as shown below:

(8) \begin{equation}W=W_{min}\left(\frac{W_{max}}{W_{min}}\right)^{\left(\frac{1+\vartheta t}{Max\_ it}\right)}\end{equation}

The butterflies positions will be updated based on the nonlinear decreasing inertia weight, as per Eq. (9).

(9) \begin{equation}Z_{i}^{t+1}=W\times Z_{i}^{t}+(R^{2}\times G^{\ast}-Z_{i}^{t})\times f_{i}\end{equation}

Here, $W_{min}$ and $W_{max}$ represent the minimum and maximum inertia weights of 0.4 and 0.95, respectively. The acceleration factor is represented by $\vartheta$ with a value 10.

The pseudocode of the IBOA is presented below.

The flowchart of the suggested IBOA is shown below in Figure 1.

Figure 1. Illustrate the flowchart of IBOA technique.

2.3. Mathematical model of path navigation

The workspace modeling, path searching, and path smoothness are the three key components of the robot navigational problem. In a real-world setting, a robot must navigate complex and rapidly changing conditions. To standardize the experimental environment, the WEBOT simulation platform was employed for modeling the environment. The robot route finding problem is then converted into using algorithms to guide the robot from the starting position to the desired location, avoiding obstacles along the way and securing an optimal path that is free from collisions. Based on environmental modeling, the challenge of identifying an optimal path is transformed into the task of optimizing a fitness function to achieve the best path. Considering an obstacle-free path, the distance between the start and to goal is formulated as an objective function as shown in Eq. (10).

(10) \begin{equation}F_{D}=\sqrt{\left(X_{st}-X_{go}\right)^{2}-\left(Y_{st}-X_{go}\right)^{2}}\end{equation}

where $F_{D}$ represents the shortest length from start and goal point. ( $X_{st}, Y_{st})$ and ( $X_{go}, Y_{go})$ represent the start and goal point, respectively. However, in real working environments, the robot will face obstacles that hinder its direct forward movement. The workspace model of the manipulator is assigned with a Cartesian reference system with some nodes as $Z_{i}=(X_{i},Y_{i}), i=1,2,3,4\ldots n$ . The total length between the search agents formed by the model is shown in Eq. (11).

(11) \begin{equation}D=\sum _{k=1}^{m}\sqrt{\left(X_{k+1}-X_{k}\right)^{2}+\left(Y_{k+1}-Y_{k}\right)^{2}},(m\lt n)\end{equation}

The barrier of the path is designed by using Eq. (12). If any point in the solution space comes into contact with the barrier, a penalty is added to the fitness function, and it is represented in Eq. (13), where p is the penalty function [Reference Wang, Wang, Xu, Pan and Chu38].

(12) \begin{equation}O_{b}=\frac{\sqrt{\left(X_{i}-X_{o}\right)^{2}+\left(Y_{i}-Y_{o}\right)^{2}}}{R_{o}}-1\begin{cases} O_{b}\gt 0, \textit{without } \textit{barrier} \\ O_{b}\lt 0, with \textit{ barrier} \end{cases}\end{equation}

where $(X_{o},Y_{o}) \;and\; R_{o}$ represents the position of the center and radius of obstacle o, respectively, and 0 = 1, 2, 3,……

(13) \begin{equation}P=P+mean(min \left(O_{b},0\right))\end{equation}

The objective function by using the penalty function for a barrier free path represents the shortest path length is mentioned in Eq. (14).

(14) \begin{equation}PL=D\times (1+w\times P)\end{equation}

Where w represents the penalty factor with a value of 10. [Reference Wang, Wang, Xu, Pan and Chu38]

Table III. Analysis of path traveled and time taken for navigation, based on simulation and experimental analysis of a single NAO.

3.1. Experimental results and discussion

The simulation investigation for path identification is performed using Webots platform. To do the simulation task, the developed algorithm is implemented in Webots and programed in MATLAB. A simulation platform measuring 230 x 150 units with variously shaped barriers is created in Webots to evaluate the efficacy of the suggested methods. Simulated navigation between two points, i.e., start and goal point, utilizing the improved butterfly optimization technique, is carried out for single humanoid robot in intricate terrain full of obstacles. The path travelled and time taken to complete the path for a single NAO during simulation and experiments are depicted in TableIII.  The simulation results of both single and multiple NAO are displayed in Figures 2 and 4, respectively. To compare simulation findings with the IBOA’s efficiency, experimental platforms that mimic a simulation environment in a lab setting are created. The navigation testing results for one humanoid and two humanoids are portrayed in Figures 3 and 5, respectively.

The results comparison indicates that the suggested controller performed satisfactorily with few variations in the difficult environmental conditions. The results comparison indicates that the suggested controller performed satisfactorily with few variations in the difficult environmental conditions. The primary reason for the slight variation in the results is the conditions considered during the movements of the manipulator from a particular position to a desired location. The ideal condition is considered during travel of the robot in the simulation platform, but because of the several factors such as floor smoothness, irregular route, friction between leg and floor, and internet connectivity, it is not taken into account while movement in the experimental platform.

Figure 2. Illustrate the Webots simulation result of single humanoid robot.

Figure 3. Portrayed the experimental results of single humanoid NAO.

Figure 4. Illustrate the Webots simulations results of multiple NAO.

Figure 5. Illustrate the real laboratory experimental outcomes of multiple NAO.

Table IV shows the path length travelled for multiple humanoids to complete navigation during simulation and experiment.

Table V shows the time required for multiple humanoids to complete navigation during simulation and experiment.

4.1. Statistical evaluation of results of single NAO

The trial data for both simulation and real-world experiments of single humanoid NAO are presented in this section.

4.1.1. Histogram plots

The histogram plots of virtual and real laboratory results obtained by using single humanoid robot are presented in Figure 6. The plots are shown in two categories: the simulation and experimental path travelled, represented in Figure 6(a), while the simulation required time and experimental required time by a single humanoid are represented in Figure 6(b). The mean value and standard deviation are shown in the histogram plots for both the path length and required time by single humanoid to reach the goal point.

Figure 6. Histogram plot of virtual and experimental (a) path travelled (b) Time taken by single NAO.

4.1.2. Normality plots

The Anderson-Darling method has been followed to perform the normality test. The test outcomes are presented in Figure 7. The tests are performed separately for each parameter, i.e, simulation and experimental path traveled and time taken. The mean, standard deviation, P-value, etc. are shown in the graph.

Figure 7. Normality plot of (a) Simulation path travelled (b) Experimental path travelled (c) Simulation time taken (d) Experimental time taken by single NAO.

4.1.3. Probability plots

The probability plots of virtual and real laboratory path travelled are shown in Figure 8(a), whereas the probability plot of virtual and experimental time consumed by the single NAO is shown in Figure 8(b). The curve follows the normal distribution with 95% confidence intervals. The p-Value, mean, andstandard deviation of all the parameters are shown in the plots.

Figure 8. Probability plots of simulation and experimental (a) path travelled (b) time taken by using single NAO.

4.1.4. Surface plots

The surface plots among simulation and experimental path travelled and deviation in path travelled are presented in Figure 9(a). The time consumed in virtual and experimentation and the percentage variation in time taken are shown in Figure 9(b).

Figure 9. (a) Surface plot of simulation, experimental and deviation in path travelled obtained by using single NAO (b) Simulation, experimental and deviation in time taken by using single NAO.

4.2. Statistical evaluation of results of multiple NAO

The experimental data for both simulation and real-world experimentation of multiple NAOs are presented in this section. The notations used in the analysis are mentioned below.

N1_: Humanoid NAO 1

N2: Humanoid NAO 2

PT: Path Travelled

TT: Time Taken

SPT: Simulation Path Travelled

EPT: Experimental Path Travelled

STT: Simulation Time Taken

ETT: Experimental Time Taken

DPT: Deviation in Path Travelled

DTT: Deviation in Time Taken

4.2.1. Histogram plots

The histogram plots of virtual and real experimentation result obtained by using multiple humanoid NAO robots are presented in Figure 10. The plots have shown in two categories, the simulation and experimental path traveled of N₁ and N₂ exhibited in Figure 10(a) and (b) while the simulation required time and experimental required time by N₁ and N₂ represented in Figure 10(c) and (d). The mean value and standard deviation are shown in the histogram plots for both the path traveled and time taken by N₁ and N₂ to reach the goal point.

Figure 10. Histogram plots of (a, b) SPT and EPT obtained by N1 and N2 (c, d) STT and ETT obtained by N1 and N2.

4.2.2. Normality plots

The Anderson-Darling method has been followed to perform the normality test. The test results are shown in Figures 11 and 12. The tests are performed separately for each parameter, i.e, SPT, EPT, STT & ETT by N₁ and N₂. The mean, standard deviation, P-value, etc. are shown in the graph.

Figure 11. (a, b) Normality plots of SPT results of N1 and N2. (c, d) Normality plots of EPT results of N1 and N2.

Figure 12. (a, b) Normality plots of STT results of N1 and N2 (c, d) Normality plots of ETT results of N1 and N2.