1. Introduction
In modern society, mobile robots have been widely introduced into the harsh manufacturing environment and some other industries to reduce labor intensity for people. Many topics associated with mobile robots have been studied, including navigation, path planning, trajectory tracking, and simultaneous localization and mapping (SLAM) [
1,
2,
3,
4]. In this regard, MRPP plays a significant role in the field of mobile robots. It is mainly responsible for finding a collision-free and optimal (or approximately optimal) traveling path [
5]. This work is motivated by such a context and aims to explore a fast and efficient methodology for MRPP problems.
At present, MRPP-related algorithms mainly include the artificial potential field method, grid method, ant colony algorithm, genetic algorithm and neural network method. Due to its simple mathematical model, small computation time, and strong, real-time nature, the artificial potential field method is widely used [
6]. Nevertheless, it is a local path-planning approach in nature that, considering its weaknesses of target unreachability and local minimum points, may lead to the failure of path planning. The grid method is used to model the mobile environment and then to convert the robot path into connections between grids, paving the way for algorithm deployment [
7]. It is a global path-planning approach in nature. However, the number of grids increases dramatically when the mobile environment becomes larger. Thus, it is difficult to achieve the desired routes in terms of large and complex environments. In addition, some bionic intelligent optimization algorithms have been proposed and successfully applied to MRPP-related problems. Representative approaches include the ant colony algorithm (ACO), particle swarm optimization (PSO), genetic algorithm (GA), and the neural network (NN). However, it should be noted that some weaknesses of these approaches restrict their applications in MRPP, including the local optimum, low search efficiency, and large burden of computation [
8,
9,
10]. Therefore, this work focuses on designing a more efficient procedure and methodology to generate solutions for MRPP problems.
In real-world applications, major efforts have concentrated on strengthening by hybridizing one candidate bionic intelligent optimization algorithm with other sophisticated skills. The ACO is one example: some of the most recent ACO-based path-planning algorithms are reviewed as follows. To reduce search blindness, Zhao et al. [
11] embedded the directional, heuristic information and unevenly distributed initial pheromone into the ACO architecture. The rollback strategy was introduced by Wu et al. [
12] to make more ants reach the goal, and the death strategy was also utilized to reduce the influence of an invalid pheromone on the algorithm evolution. Wang et al. [
13] designed an improved, pseudo-random proportional rule for the ant state transition which greatly improved the convergence speed. A novel penalty strategy to volatilize the poor path pheromone was embedded into the ACO by Yue et al. [
14], which efficiently improved the utilization rate of the ant exploration. Chen et al. [
15] tried to improve the visibility of heuristic information by virtue of the principle of infinite step length. In addition, the pheromone update rules, as well as the evaporation rate, were dynamically adjusted to accelerate the convergence speed. The evaluation function of the A-star algorithm and the curvature suppression operator were introduced into the ACO by Dai et al. [
16]; these additions were in favor of accelerating the convergence and improving the smoothness of the final path. Luo et al. [
17] proposed an attenuation-factor-based heuristic information mechanism, and adjusted the pseudo-random ratio of state selection by means of an adaptive function. The adaptive adjustment factor and pheromone volatilization factor were embedded into the pheromone update rule by Miao et al. [
18]. Such modifications efficiently balanced the global search ability and convergence. As can be noted from the above literature, different skills have been embedded into the basic algorithm with respect to the aspects of path formulation, path adjustment, solution modification, individual update, etc. The proposed ETLBO is motivated and designed by these observations and is developed by virtue of TLBO.
Many practical applications have proven TLBO to be superior to some conventional metaheuristics [
19,
20,
21]. TLBO simulates the behaviors of the teacher and students in the classroom, where the so-called teacher and learner phases are utilized to update the population in the algorithm evolution. Some modified, TLBO-related algorithms have already been proposed by many researchers; please refer to the book by Rao [
22] for the most recent studies. However, some challenges still exist when applying the TLBO approach to practical optimization problems [
23]. First, the adaptation of TLBO to a specified application requires an appropriate problem conversion and solution representation. Second, TLBO tends to get trapped in a local optimum in real-world applications, which is a common weakness of swarm-based metaheuristics. In this context, this paper develops an efficient, TLBO-based algorithm for MRPP.
This work is motivated by the great demand for a fast and efficient path-planning approach for a mobile robot in a complex environment. In the current research, an effort is made to address such an issue with a TLBO-version approach. Four main contributions associated with the methodology design are stated as follows:
For the algorithm deployment, a divide-and-conquer design, coupled with the Dijkstra method, is developed to realize the problem transformation;
The interpolation method is embedded into the proposed algorithm to smooth the traveling route as well as to reduce the problem dimensionality;
An opposition-based learning strategy is utilized to modify the algorithm initialization process;
To balance between exploitation and exploration, a novel, individual update method is established by hybridizing TLBO with DE.
The rest of this work is organized as follows.
Section 2 formulates the MRPP, and
Section 3 presents a brief introduction of TLBO and DE. The proposed methodology is formally described in
Section 4. Computational studies are designed and carried out in
Section 5. Finally,
Section 6 displays conclusions and future work.
4. Proposed ETLBO for MRPP
This section presents a detailed introduction of the proposed methodology for the MRPP problem.
4.1. Divide-and-Conquer Design for MRPP
As is illustrated in
Figure 3, a divide-and-conquer design is embedded into the algorithm deployment. In this regard, the investigated MRPP problem is broken down into multiple simple sub-problems, for which an appropriate divide-and-conquer grid in each column between the origin and destination must be optimized to form a simple route [
25]. Then, local optimal routes between any two divide-and-conquer grids are generated by fast and sophisticated methods to form a solution to the original MRPP problem.
Two decisions must be made when dealing with the converted problem. First, an appropriate divide-and-conquer grid in each column between the origin and destination must be determined in order to construct the simple route. Second, local optimal routes between any two divide-and-conquer grids must be obtained to form a solution to the original MRPP problem. The prior decision is to be tackled by the proposed ETLBO, while the latter decision is optimized by the Dijkstra approach. More details are presented here to better explain the application of the Dijkstra approach. Let us denote the grid in a (row, column) format. For example, (1, 3) represents the grid located at row 1 and column 3. On this basis, all workable grids in column 3 and column 4 can be expressed by sets A = {(1, 3), (2, 3), (3, 3), (6, 3), (7, 3), (8, 3), (9, 3), (10, 3)} and B = {(1, 4), (3, 4), (6, 4), (7, 4), (8, 4), (9, 4), (10, 4)}, respectively. The Dijkstra approach is then utilized to obtain optimal routes between any two positions from sets A and B, respectively. Since these positions are located in columns adjacent to each other, the decision process of the Dijkstra approach is time-saving and always ensures optimality.
4.2. Interpolation Application in ETLBO
As stated in the previous subsection, ETLBO is used to determine the locations of every divide-and-conquer grid at each column between the origin and destination. To reduce the problem dimensionality and to smooth the traveling routes, the interpolation method is embedded into the application of the ETLBO. The corresponding illustrative schematic diagram is depicted in
Figure 4.
It can be observed from
Figure 4 that some divide-and-conquer grids are to be optimized by the solution array of the ETLBO, such as grids ④ and ⑦, while the other divide-and-conquer grids are obtained by the interpolation method. In the case in which an interpolated grid is located at an obstacle grid, the nearest workable grid in that column is selected as the divide-and-conquer grid. In this regard, the B-Spline interpolation method is selected to pave the way for ELTBO deployment according to some existing MRPP-related studies [
26].
4.3. Pyramid of ETLBO
ETLBO is established by virtue of the outstanding performance of TLBO. Meanwhile, the opposition-based learning mechanism and the DE-related offspring method are utilized to enhance the algorithm performance.
4.3.1. Opposition-Based Learning Initialization Method
The opposition-based learning mechanism is a simple but efficient method to strengthen the algorithm’s performance in swarm computation [
27]. In ETLBO, the opposition-based learning mechanism is used to create intimal solutions with high qualities. In this regard,
solutions are initialized in a random way. Then,
opposition-based solutions are generated based on the opposition-based learning concept. Finally,
individuals with better evaluations are selected to form the initial population.
As stated previously, TLBO is run by virtue of
individuals, and each candidate can be marked as an array
in which
. In this regard, the search space is defined by
. In the initialization, let us denote the
random solution and its associated, opposition-based solution by
and
, respectively. According to “Definition 2” stated by Mahdavi et al. [
27], the following formula holds true:
Based on the above concepts, the initialization process in ETLBO is implemented as follows:
Step 1. Set and . Then, go to step 2.
Step 2. If condition holds, go to step 3; otherwise, go to step 7;
Step 3. If condition holds, go to step 4; otherwise, go to step 5;
Step 4. Set and . Then, go to step 5;
Step 5. Set , if condition holds, go to step 6; otherwise, go to step 3;
Step 6. Set ; then, go to step 2;
Step 7. Select best individuals from to form the initial population.
4.3.2. Hybrid Offspring Generation Method
An efficient swarm-based metaheuristic should make full use of both the local and global information on the solutions found. Local information on solutions can be helpful for exploitation, while global information tends to guide the search for exploring promising areas. The evolution of TLBO mainly depends on the global information, and thus TLBO has an obvious advantage in that learner individuals will converge at a single attractor/teacher individual. It can be observed that DE evolves mainly based on the distance and direction information, which are types of local information in nature. In other words, DE has the advantage of not being biased toward any prior defined guide, and such an attribute makes DE able to maintain population diversity and explore local searches.
The key reason for employing the hybridization of different metaheuristics is that the hybrid approach can make full use of the strengths of each algorithm while simultaneously overcoming the associated limitations. In this regard, instead of employing a single updated learner individual in the TLBO, the proposed ETLBO makes use of the integration of two offspring individuals. The first offspring individual is computed by the TLBO, and the second one is obtained by the DE operators. Detailed mathematical expressions are described as follows.
- (1)
Teacher phase
Given a candidate solution,
, the corresponding offspring individual,
, is calculated by the hybridization of the teacher phase in TLBO and differential learning in DE:
where
represents the offspring individual obtained by Equation (4) in TLBO, while
represents the offspring individual computed by Equations (6)–(11) in DE. The hybridization factor
is a random number that lies in the interval [0, 1].
- (2)
Learner phase
Analogous to the teacher phase, the offspring individual
of a candidate solution
is generated by taking the following equation:
where
represents the offspring individual obtained by Equation (5) in TLBO, while
is the offspring individual computed by Equations (6)–(11) in DE. The hybridization factor
is a random number that lies on interval [0, 1].
4.4. Framework of Proposed Methodology for MRPP
By virtue of the previous statements, the flowchart of the proposed ETLBO for the MRPP problem is illustrated in
Figure 5. Simulations are to be conducted in the next section to validate the algorithm performance.
6. Conclusions
This research proposes a so-called ETLBO algorithm to realize an efficient path planning for a mobile robot. Four strategies are introduced to accelerate the basic TLBO algorithm and optimize the final path. Firstly, a divide-and-conquer design coupled with the Dijkstra method is developed to realize the problem transformation so as to pave the way for the algorithm deployment. Then, the interpolation method is utilized to smooth the traveling route as well as to reduce the problem dimensionality. Thirdly, the algorithm initialization process is modified by an opposition-based learning strategy. Finally, a hybrid individual update method is prosed to strike a balance between exploitation and exploration.
Simulation experiments are designed to validate the proposed algorithm. In terms of benchmark function problems, the proposed ETLBO performs better than the compared algorithms in a statistical manner and obtains consistent and more accurate solutions for most optimization function problems. With respect to MRPP problems, ETLBO outperforms some state-of-are algorithms in both optimality and efficiency, and thus can be implemented on the real-world robot for path planning.
Since the local optimal path planning in current algorithm is solved by the Dijkstra method, it is important to modify the existing divide-and-conquer design in future research work to reduce the computation burden when faced with more complicated practical problems. In addition, the proposed methodology can be adjusted and adapted to the dynamic environment in which certain real-time factors are seriously considered.