Robot Motion Planning Based on an Adaptive Slime Mold Algorithm and Motion Constraints

Abstract

The rapid advancement of artificial intelligence technology has significantly enhanced the intelligence of mobile robots, facilitating their widespread utilization in unmanned driving, smart home systems, and various other domains. As the scope, scale, and complexity of robot deployment continue to expand, there arises a heightened demand for enhanced computational power and real-time performance, with path planning emerging as a prominent research focus. In this study, we present an adaptive Lévy flight–rotation slime mold algorithm (LRSMA) for global robot motion planning, which incorporates LRSMA with the cubic Hermite interpolation. Unlike traditional methods, the algorithm eliminates the need for a priori knowledge of appropriate interpolation points. Instead, it autonomously detects the convergence status of LRSMA, dynamically increasing interpolation points to enhance the curvature of the motion curve when it surpasses the predefined threshold. Subsequently, it compares path lengths resulting from two different objective functions to determine the optimal number of interpolation points and the best path. Compared to LRSMA, this algorithm reduced the minimum path length and average processing time by (2.52%, 3.56%) and (38.89%, 62.46%), respectively, along with minimum processing times. Our findings demonstrate that this method effectively generates collision-free, smooth, and curvature-constrained motion curves with the least processing time.

1. Introduction

In recent years, autonomous mobile robots and unmanned ground vehicles have garnered significant attention from researchers. Among the various research endeavors, path planning emerged as a crucial domain, focusing on efficiently charting a viable, secure, and uninterrupted route from the robot’s initial position to its designated destination within an unknown or known environment [1,2,3]. Fundamentally, path planning can be conceptualized as an optimization problem, wherein the objective is to attain the optimal values of target parameters while adhering to a set of constraints [4,5].

Traditional path planning methods have gained widespread application in the field of robot motion planning [6,7,8]. With the rapid development of robot path planning technology, swarm intelligence algorithms have introduced novel ideas for motion planning and have shown promising results in addressing motion planning challenges in complex environments [9,10,11]. Among these algorithms, the ant colony algorithm has emerged as one of the most widely utilized methods [12,13,14,15]. For instance, ref. [16] reduced the computational burden of the algorithm in dynamic environments and enhanced its robustness. However, this algorithm often suffers from slow convergence and a tendency to become trapped in deadlocks. Another commonly employed algorithm is particle swarm optimization [17,18,19]. While the parameters of particle swarm optimization are straightforward and easily adjustable, the convergence accuracy requires improvement. Additionally, several other swarm intelligence algorithms have been proposed for robot motion planning, such as the whale optimization algorithm [20,21], artificial bee colony optimization algorithm [22,23], chicken swarm optimization algorithm [24], and gray wolf optimization algorithm [25,26].

However, the swarm intelligence optimization algorithm faces drawbacks including limited precision, sluggish convergence, and susceptibility to local optimal solutions, particularly evident when addressing complex optimization problems [27]. To address these challenges, researchers have dedicated considerable efforts to further optimizing the performance of swarm intelligence algorithms. Typically, this involves augmenting algorithm components or integrating other algorithms [28]. As part of these endeavors, the slime mold algorithm has been proposed.

The slime mold algorithm (SMA) was initially proposed by Nakagaki et al. [29] in 2000. Since then, it has found widespread applications across a number of domains such as traffic network node selection [30], robot motion planning [31], medical image classification [32], and feature selection [33]. In 2020, building upon this theory, ref. [34] introduced an SMA for studying myxobacterial activity and kinetics. Incorporating an adaptive guided differential mutation method, ref. [35] enhanced the local search capability of individual populations and promoted population diversity to mitigate premature convergence of the algorithm. Furthermore, ref. [36] introduced a rotation perturbation method with local optimization ability to improve the convergence precision of the SMA. Building on this research, an enhanced slime mold algorithm for path planning was proposed. This algorithm features real-time monitoring capabilities and employs a convergence stagnation monitoring strategy based on tolerance. Additionally, an individual rotation disturbance mutation mechanism is employed to guide the population in escaping from local optima.

Path planning for autonomous mobile robots and unmanned ground vehicles must not only address obstacle avoidance within the environment but also consider motion characteristics [37,38]. To tackle these challenges, interpolation-based methods can be utilized to construct curves that pass through nearly any given data point. In the path planning procedure, through the establishment of a node-based coding system, the positions of interpolation points can be determined using the interpolation algorithm, thereby simulating the trajectory of mobile robots. Common interpolation algorithms include cubic spline interpolation, Bessel interpolation, and cubic Hermite interpolation, among others.

Cubic spline interpolation is a classical piecewise interpolation method that constructs a smooth curve by interpolating a series of interpolation points within a defined interval that adheres to cubic polynomial definitions. Ref. [39] initially proposed a path planning approach based on the particle swarm optimization and the cubic Ferguson curve optimization, introducing an innovative idea for path planning based on the particle swarm algorithm. However, this method suffers from high coding dimensionality. Ref. [40] introduced a path planning method based on the slime mold algorithm and cubic B-spline interpolation, effectively addressing path planning challenges in complex environments. While the resulting path curves are smooth, they may not fully comply with kinematic constraints. The shape of the Bessel curve is entirely reliant on control points, with only the starting point and endpoint guaranteed to lie on the curve. To address this, ref. [41] developed a model based on the Bessel curve and proposed a novel chaotic particle swarm optimization (CPSO) to enhance control point placement. Although this approach generates the path planning curve by determining the optimal position of control points, identifying control points for the Bessel curve remains challenging. A new path planning method was proposed in ref. [42], utilizing the Bessel curve for path planning and the Hermite curve for trajectory planning. The algorithm’s module current consumption serves as the evaluation metric for path planning, providing an accurate motion planning method for wheeled robots while adhering to position and speed constraints. Although the Hermite interpolation polynomial ensures alignment of interpolation point values with function and derivative values, monitoring the current consumption of its modules poses challenges. Ref. [43] proposed an improved Hermite interpolation method for joint motion planning. This is achieved by developing a node configuration scheme for the Hermite curve and performing motion planning based on the new scheme, thereby generating a curve that satisfies the position and velocity constraints. This approach effectively mitigates velocity fluctuations in point-to-point motion planning for industrial robots while adhering to the velocity constraint but is not directly applicable to wheeled robots.

To address the aforementioned limitations, we propose a global adaptive motion planning approach capable of automatically conducting motion planning tasks. This approach ensures adherence to the motion characteristics of the robot by considering the maximum curvature constraint. To achieve adaptive motion planning tailored to different robot characteristics, this study introduces a methodology that integrates an enhanced slime mold algorithm (SMA) with the cubic Hermite interpolation. Through this integration, the approach automatically generates interpolation points for the cubic Hermite curve and employs a well-suited encoding scheme to produce a motion curve that is free from collisions, smooth, and adherent to curvature constraints within the predefined motion range. This is achieved with minimal processing time, thereby ensuring efficient motion planning.

This paper builds upon the research presented in [41]. The innovative approach presented in this paper eliminates the need for a priori knowledge of the appropriate number of interpolation points. Initially, instances where the curvature of the motion curve exceeds the prescribed threshold are improved through adaptive augmentation of interpolation points. Subsequent to this improvement, the optimal path and the optimal number of interpolation points are determined via a comparative analysis of path lengths utilizing two distinct objective functions. Furthermore, this paper introduces a method for monitoring the population’s status based on the Hamming closeness theory. This method facilitates the identification of convergence stagnation within the slime mold population, thereby providing a foundation for interpolation point updates.

The structure of this paper is as follows: Section 2 provides an overview of the Lévy flight–rotation slime mold algorithm (LRSMA). Section 3 delves into the underlying theory of LRSMA and its improvements. Section 4 conducts simulations that demonstrate the dynamic path planning advantages of LRSMA for mobile robots. Finally, Section 5 summarizes the key conclusions and discusses the significance of the findings.

2. Overview of the Lévy Flight–Rotation Slime Mold Algorithm (LRSMA)

In the Lévy flight–rotation slime mold algorithm (LRSMA), the variable neighborhood Lévy mechanism is employed to update the optimal solution, thereby enhancing the convergence accuracy of the algorithm. Real-time monitoring is conducted using a tolerance-based convergence stagnation monitoring strategy. Additionally, an individual rotation perturbation and variation mechanism is utilized to guide the population in escaping local optima.

2.1. Elite-Based Variable Neighborhood Lévy Flight Learning Strategy

The SMA is susceptible to falling into local optima in the later stages of iteration, resulting in stagnation in population convergence. To address this challenge, this paper introduces an elite-based variable neighborhood Lévy flight learning strategy to perturb and mutate the position of the population’s optimal fitness. Following mutation, a comparison between the optimal fitness before and after the process is conducted, and individuals exhibiting superior fitness are retained. The formula for the variable neighborhood Lévy flight learning strategy is defined as follows:

$$\{\begin{array}{c}X\_best{(t)}_1=\alpha \times (X\_best(t)+stepsize\times L\acute{e}vy(\beta )),\\ \alpha =1+\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}\times r,\end{array}$$

(1)

$$\{{ }_{{ }_{{ }_{{\sigma }_{\mu }={\left\{\frac{\Gamma (1+\beta )\times \sin (\pi \beta /2)}{\Gamma \left[(1+\beta )/2\right]\times \beta \times 2^{(\beta -1)/2}}\right\}}^{1/\beta },}}^{v\sim N(0,{\sigma }_v^2),}}^{{ }_{\mu \sim N(0,{\sigma }_{\mu }^2),}^{L\acute{e}vy(\beta )=\frac{\mu }{{\left|v\right|}^{1/\beta }},}}$$

(2)

where $t$ is the iterative index, $\alpha$ is the variable neighborhood coefficient, $X\_best{(t)}_l$ is the fresh optimal position after Lévy flight, and $X\_best{(t)}_i$ is the optimal position of the current iterative myxomycete population. $stepsize$ is the step factor, which is used to adjust the random search range. $L\acute{e}vy(\beta )$ is the basic Lévy flight random value. The value range of the index is $\left\{0,1,2\right\}$. For each value of the index, if the fitness value corresponding to $X\_best{(t)}_l$ after Lévy flight is less than $X\_best{(t)}_i$ or the index reaches its maximum value, the value selection and optimal position search processes are terminated. $r_l$ is a random value within the range of $\left[0, 0.5\right]$. $\Gamma (\ast )$ is gamma function, and $\beta$ remains as a constant. $\mu$ and $v$ are random variables obeying a normal distribution and being the standard deviations of ${\sigma }_{\mu }$ and ${\sigma }_v$, respectively. In this paper, ${\sigma }_v=1$.

2.2. Tolerance-Based Individual Rotation Perturbation Mutation Mechanism

In this study, an individual rotation disturbance mutation mechanism based on tolerance is employed to enhance the accuracy of SMA local search. By utilizing a convergence stagnation monitoring strategy based on tolerance, the population’s status is continuously monitored in real time to prevent prolonged convergence stagnation. In this paper, the tolerance parameters are updated according to whether the fitness difference of the optimal individual changes. The formula for updating the tolerance parameter is defined as follows:

$$\tau (t+1)=\{\begin{array}{ll}\tau (t+1),& if T(t)<T_{max} and \left|\Delta F(t)\right|<F_{min},\\ 0,& if (T(t)=T_{max} and \left|\Delta F(t)\right|<F_{min}) or \left|\Delta F(t)\right|>F_{min},\end{array}$$

(3)

where $t$ is the current iteration index, and $\tau (t)$ is the tolerance parameter at iteration, with a value range of $\left\{0,\cdot \cdot \cdot ,{\tau }_{\mathit{max}}\right\}$. In this paper, ${\tau }_{\max }=2$. $\tau (t+1)$ is the updated tolerance parameter after the completion of the current iteration. $\left|\Delta F(t)\right|$ is the absolute value of the fitness difference between the $t$-th iteration and the previous iteration, calculated as $\left|\Delta F(t)\right|=\left|F(t)-\right|F(t-1)$. $F_{\mathit{min}}$ is the threshold of fitness difference, which remains constant. As $\tau (t)>0$, the myxomycete population does not necessarily enter a state of convergence stagnation. When $\tau (t)={\tau }_{\mathit{max}}$ and $\left|\Delta F(t)\right|<F_{\mathit{min}}$, the algorithm falls into convergence stagnation.

When the algorithm detects a state of convergence stagnation, certain individuals undergo disturbance and mutation through rotational operations. This step is essential for re-guiding the myxobacterial population and enhancing the diversity of its distribution. Specifically, the individual with the highest fitness ranking is chosen for random rotational disturbance, as defined in the following formula:

$$X{(t)}_r=X(t)+\omega \times \frac{1}{{\Vert X(t)\Vert }_2}\times (R\cdot X(t)),$$

(4)

where $X{(t)}_r$ is the individual position after rotation disturbance and $X(t)$ is the slime mold position at the $t$-th iteration. $\omega$ is the rotation factor, with a value range of $[0.1, 1]$. Assuming that the search space is D-dimensional, $R$ is a random $1\times D$ matrix whose dimensions are uniformly distributed in the interval $\left[-1, 1\right]$. $R\cdot X(t)$ is the point multiplication of the random matrix $R$ and the individual position $r$ of slime mold. ${\Vert X(t)\Vert }_2$ is the 2-norm, which is also the Euclidean norm.

Lastly, the selection of slime mold individuals before and after the disturbance and mutation was conducted according to the greedy principle. The fitness values of the corresponding individuals before and after the mutation were compared, and those individuals with superior fitness were selected.

2.3. Elite-Based Simulated Annealing Mechanism

When computing the optimal solution of the population disturbed by the individual rotation disturbance mutation mechanism based on tolerance, the Metropolis criterion of the simulated annealing algorithm was used to receive poor solutions with a certain probability. The Metropolis criterion is expressed as follows:

$$f(x)=\{\begin{array}{cc}1& ,D{F}_r<DF,\\ {\exp }^{-(D{F}_{\mathrm{r}}-DF)/\lambda }<ran{d}_s& ,D{F}_r\ge DF.\end{array}$$

(5)

where $D{F}_r$ is the optimal fitness after the rotation disturbance, $D{F}_{ }$ is the optimal fitness before the disturbance, and $ran{d}_s$ is a randomly generated number between 0 and 1. In this paper, $\lambda =1000$.

2.4. Steps for Path Planning Based on LRSMA

Figure 1 shows the flow chart of LRSMA, which consists of the following steps:

Step 1: Initialization of the slime mold population.

Step 2: Computation of the fitness and weight coefficient of slime molds.

Step 3: Perturb the optimal individual of the population using the elite-based variable neighborhood Lévy flight learning strategy.

Step 4: Monitor the state of the algorithm using the convergence stagnation method based on tolerance. If convergence stagnation is detected, apply the rotation disturbance and mutation mechanism to certain slime mold individuals to identify superior ones.

Step 5: Selection of the optimal individual after the disturbance using the simulated annealing mechanism.

3. Curvature-Constrained Path Planning Based on an Adaptive LRSMA

3.1. Problem Statement

Path planning involves determining an optimal route from the starting point to the desired destination within an unknown or known environment, aiming to minimize distance, time, or a combination of both along with other metrics. In scenarios involving path planning with multiple constraints, these constraints are typically converted into objective function evaluation indices for combinatorial optimization. To explore the optimal route complying with constraints such as feasibility, safety, and minimum distance, three fundamental indices are chosen in this study: path length, collision risks, and curvature costs, to evaluate the generated path.

When the curvature of a curve surpasses the predefined threshold, augmenting interpolation points proves to be an effective method for ameliorating curvature. Consequently, when the curvature cost function is utilized, it diverges from the collision risk function, where weights are directly assigned, and weighted indices are summed to formulate the objective function. Given the operational context of mobile robots and a specified number of interpolation points m, with only path length and collision risks selected as evaluation indices for the objective function, if the curvature of the resulting optimal solution surpasses the threshold, achieving the optimal path may necessitate either integrating curvature as an objective function or augmenting interpolation points. A comparative analysis of the paths derived from these two scenarios is imperative to determine the optimal path.

3.2. Adaptive LRSMA

When employing interpolation functions, variations in the number of interpolation points yield different resultant curves. Hence, selecting an appropriate number of interpolation points is pivotal for attaining the optimal solution. To facilitate the automatic generation of the optimal solution, it is crucial to discern when the curvature of the generated optimal solution, given the current number of interpolation points, exceeds the threshold, and seize the opportune moment to augment interpolation points, thereby attaining an optimal solution that complies with curvature constraints following the incorporation of new interpolation points.

In this paper, an adaptive LRSMA is proposed to automatically increase interpolation points when the curvature of the optimal solution exceeds the threshold, thereby producing a collision-free path that adheres to curvature constraints while minimizing path length. The aforementioned LRSMA introduces a convergence stagnation monitoring approach based on tolerance. This approach monitors convergence status by evaluating the discrepancy in fitness values among individual slime molds and perturbs the population falling into local optima. Given the inherent randomness of the slime mold algorithm’s optimization process, if the distance between any two myxomycete individuals is small in the previous iteration and large in the current iteration, it may revert to being smaller again in subsequent iterations. To address this challenge, we introduce maturity as a novel evaluation index to gauge the convergence status of the population with the present number of interpolation points. Population maturity is measured by the resemblance among individuals within the population, which is represented by Hamming closeness derived from the positions of current population individuals and historical optimal individuals.

Suppose, in a D-dimensional space, the myxomycete population comprises N individuals, and $X\_bes{t}_i(t)$ is the optimal position of the $i$-th individual from the beginning to the $t$-th iteration, defined at $X\_bes{t}_i(t)=((x{b}_i^1(t),y{b}_i^1(t)),(x{b}_i^2(t),y{b}_i^2(t)),\cdot \cdot \cdot ,(x{b}_i^m(t),y{b}_i^m(t)))$. Let $U=2\times \mathrm{m}\times D$, the collective current positions of all myxomycete individuals and the optimal position attained throughout the iterative process, formulate a matrix Q(t) with $N$ rows and $U$ columns:

$$\{\begin{array}{l}Q(t)={(Q_1(t),Q_2(t),\cdot \cdot \cdot ,Q_N(t))}^T,\\ Q_i(t)=\\ ((x_i^1(t),y_i^1(t),x_i^2(t),y_i^2(t),\cdot \cdot \cdot ,x_i^m(t),y_i^m(t),x{b}_i^1(t),y{b}_i^1(t),x{b}_i^2(t),y{b}_i^2(t),\cdot \cdot \cdot x{b}_i^m(t),y{b}_i^m(t)).\end{array}$$

(6)

Normalizing the formula above yields an $N\times U$ order matrix $Q^{\prime }(t)$ In this matrix, the element $Q_{ab}^{\prime }(t)$ in row $a$ and column $b$ is computed as follows:

$$Q_{\mathrm{ab}}^{’}(t)=\frac{Q_{ab}(t){-}_{1\le u\le N,1\le v\le U}^{\min }{Q}_{\mathrm{uv}}(t)}{{ }_{1\le c\le N,1\le e\le U}^{\max }{Q}_{ce}(t){-}_{1\le u\le N,1\le v\le U}^{\min }{Q}_{\mathrm{uv}}{(t)}^{\prime }},$$

(7)

where both $a$ and $b$ are integers, with a value range of $a\in \left[1, N\right]$,$b\in \left[1, U\right]$. ${ }_{1\le u\le N,1\le v\le U}^{ \min }{Q}_{\mathrm{uv}}(t)$ is the minimum value of $Q(t)$, and ${ }_{1\le c\le N,1\le e\le U}^{ \max }{Q}_{ce}(t)$ is the maximum value of $Q(t)$.

Assuming $Q_f^{\prime }(t)$ is any row vector within the matrix $Q_f^{ }(t)$, which can be construed as a fuzzy set, the similarity between these two fuzzy sets at random $Q_f^{\prime }(t)$ and $Q_g^{\prime }(t)$ can be represented by the Hamming closeness function of $H_{fg}(t)$ defined by

$$H_{fg(t)}=1-\frac{1}{U}{\displaystyle \sum _{v=1}^U\left|Q_{fv}^{\prime }(t)-Q_{gv}^{\prime }(t)\right|},$$

(8)

where $Q_{\mathrm{fv}}^{\prime }(t)$ is the $i$-th element of $Q_f^{\prime }(t)$ and $Q_{\mathrm{gv}}^{\prime }(t)$ is the $v$-th element of $Q_g^{\prime }(t)$.

The population maturity can be determined based on the average closeness of the population. The calculation formula is as follows:

$$\{\begin{array}{c}\delta (t)=\frac{A(t)}{\rho },\\ A(t)=\frac{2}{N(N-1)}{\displaystyle \sum _{f=1}^{N-1}{\displaystyle \sum _{g=f+1}^N{H}_{fg(t)}^{ }},}\end{array}$$

(9)

where $\delta (t)$ is the population maturity, $A(t)$ is the average closeness of the population, and $\rho$ is the maturity adjustment coefficient, with $\rho \in \left[0.8, 1\right]$.

In this study, we compute population maturity employing the convergence stagnation monitoring approach based on tolerance. When the curvature of the optimal solution in the current iteration surpasses the predetermined threshold, and if $\delta (t)>1$, it signifies a high concentration of positions within the population. This concentration indicates a high population maturity, suggesting that the population has achieved full convergence and has identified the optimal solution given the current interpolation points. Under such circumstances, increasing the interpolation points becomes necessary.

3.3. Coding Based on Cubic Hermite Interpolation

Common interpolation methods include cubic B-spline interpolation and cubic Hermite interpolation. Given that the curvature of the cubic B-spline function frequently surpasses the threshold, we choose cubic Hermite interpolation as the preferred method for path planning.

Let the function $f(x)$ possess $(n+1)$ distinct points within its domain of definition $\left[a,b\right]$. Corresponding to these points are function values $y_0,y_1,\cdot \cdot \cdot y_n$ and derivatives $d_0,d_1,\cdot \cdot \cdot d_n$. The piecewise cubic polynomial $H(x)$ is defined by

$$\begin{array}{l}H(x)=y_i,\\ H^{\prime }(x)=d_i.\end{array}$$

(10)

The optimal path of the intelligent mobile robot is determined through the integration of the enhanced slime mold algorithm with the Hermite interpolation method. Each path node corresponds to the turning points of an individual interpolation segment, where all nodes are represented as one slime mold individual. Denoting the quantity of path nodes as m, their coordinates are expressed as $(x_n^1,y_n^1),(x_n^2,y_n^2),\dots (x_n^m,y_n^m)$. The starting point coordinates are denoted as $(x_s,y_s)$, and the endpoint coordinates as $(x_e,y_e)$. Utilizing the cubic Hermite spline interpolation method, the coordinates $(x_1,y_1),(x_2,y_2)\cdot \cdot \cdot ,(x_n,y_n)$ of $n$ interpolation points can be obtained. Consequently, the starting point, $n$ interpolation points, and the endpoint collectively compose the trajectory of the mobile robot.

3.4. Objective Function

The objective function presented in this paper pertains to three indices: path length, collision risk, and curvature cost. It serves to compute the fitness value of the evaluated path. The objective function, denoted as $f_o$, is defined by

$$f_o=\{\begin{array}{l}{\eta }_1\times f_1+{\eta }_2\times f_2, Condition,\\ {\eta }_1\times f_1+{\eta }_2\times f_2+{\eta }_3\times f_3, Other,\end{array}$$

(11)

where $f_1$ is the path length, encompassing the distance from the starting point through each interpolation point to the endpoint. ${\eta }_1$ is the penalty coefficient of the path length function, set at ${\eta }_1=1$. $f_2$ is the average distance from all interpolation points to all obstacles, with ${\eta }_2$ serving as the collision penalty coefficient. A higher value of ${\eta }_2$ corresponds to a reduced likelihood of selecting collision-prone paths, with ${\eta }_2=1000$ employed in this paper. $f_3$ is the maximum curvature of the path, with ${\eta }_3$ as the curvature penalty coefficient. A greater ${\eta }_3$ value corresponds to a decreased likelihood of selecting paths with significant curvature, set at ${\eta }_3=1000$ in this paper. The term $Condition$ denotes scenarios where the shortest path is derived without imposing curvature constraints, while others represent paths conforming to curvature constraints. If a collision occurs within the path, $f_2>0$. If the obtained path failed to circumvent obstacles, $f_2=0$. As long as the maximum curvature in the path exceeds the predetermined threshold, $f_3>0$. If the obtained path has no predetermined threshold of curvature, $f_3=0$.

3.5. Path Planning Process

Figure 2 illustrates the path planning flow chart based on the adaptive LRSMA, which comprises the following sequential steps:

Step 1: Initialization of algorithm parameters and the population: This involves defining the number of the population as $N$, defining the maximum number of iterations as $T_{\max }$, specifying the upper boundary ($UB$) and lower boundary ($LB$) of the search scope, defining the coordinates of the starting point $(x_s,y_s)$ and endpoint $(x_e,y_e)$, and defining the coordinates of path nodes $(x_{n\_1},y_{n\_1}),(x_{n\_2},y_{n\_2}),\cdot \cdot \cdot ,(x_{n\_m},y_{n\_m})$, where the path nodes correspond to the positions of each slime mold individual, denoted as $X$. Additionally, the proportion of individual rotation disturbance mutation based on tolerance is defined as $p_r$.

Step 2: The cubic spline interpolation method is employed to compute the coordinates of interpolation points between the starting point, each path node, and the endpoint, denoted as $(x_1,y_1),(x_2,y_2),\cdot \cdot \cdot (x_n,y_n)$.

Step 3: The fitness value of each individual is computed and sorted to determine the optimal fitness value ($bF$) and the worst fitness value ($wF$).

Step 4: Update the weight coefficient $W(SIndex)$ of individual slime molds.

Step 5: Utilizing Equations (8)–(10), the position of each individual myxomycete is updated.

Step 6: Employing Formula (11), the elite-based variable neighborhood Lévy flight learning strategy is applied to the optimal position, identifying the optimal position.

Step 7: Examine if the population has reached convergence. If the convergence process has been completed, the $N\ast p_r$ optimal slime mold individuals with high fitness rankings undergo rotation. If the updated fitness values are lower than those of the original individuals, the positions of slime mold individuals are updated.

Step 8: Compare the fitness values of the updated optimal individuals from step 7 with those of the original optimal individuals. Based on this comparison, determine whether to execute the replacement operation using the simulated annealing criterion. Lastly, record the position and fitness value of the optimal individual throughout the iteration process.

Step 9: Examine whether the curvature of the optimal solution exceeds the predetermined threshold. If it surpasses the threshold, proceed to step 10; otherwise, proceed to step 12.

Step 10: Examine if the maximum number of iterations has been reached. If not, proceed to step 12; otherwise, terminate the algorithm.

Step 11: Examine whether the positions of the slime mold population are stagnant. If not, repeat steps 2 through 9. Otherwise, introduce a new interpolation point and repeat steps 2 through 9.

Step 12: If the maximum number of iterations has not been reached, repeat steps 2 through 9. Otherwise, terminate the algorithm and obtain the optimal position of the slime mold individual along with its corresponding fitness value.

4. Simulation Results and Analysis

4.1. Simulation Environment and Parameters

All simulations were conducted using a uniform setup on a single PC, maintaining consistency in the operating platform and environment throughout. Specifically, the operating system employed was a 64-bit version of Windows 10, with compilation facilitated by Visual Studio Code version 1.68.1. The CPU used was an Intel Xeon E3-1231 v3, operating at a frequency of 3.4 GHz, and manufactured in Vietnam, coupled with 16 GB of DDR4-2400 RAM, produced by Kingston in China.

In this study, a two-dimensional environment was selected for simulation purposes. In the simulation scenario, the center point of the robot is regarded as its representative, disregarding the size of the robot during the path planning process. Concurrently, obstacles are simplified as circles with varying radii. Two distinct scenarios were devised, with the starting point located at (−10, 10) and the endpoint at $\left(10, 10\right)$. Scenario 1 encompasses 25 circular obstacles, while scenario 2 encompasses 9 circular obstacles, as delineated in

Table 1. Distribution of obstacles.

Scenario	Obstacle Coordinates	Radius
Scenario 1	(−5.9, −2.4), (−5.9, −0.4), (−5.9, −6.3), (−5.9, −4.3), (−5.9, −8.3), (−2.7, 6.6), (−2.7, 8.6), (−2.7, 2.6), (−2.7, 4.6), (−2.7, 0.7), (0.7, −2.4), (0.7, −0.4), (0.7, −4.3), (0.7, −6.3), (0.7, −8.3), (4.5, 6.2), (4.5, 8.2), (4.5, 2.3), (4.5, 4.3), (4.5, 0.3), (7.7, −2.1), (7.7, −0.1), (7.7, −6.0), (7.7, −4.0), (7.7, −8.0)	1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0
Scenario 2	(−5.8, 5.6), (5.9, −6.4), (0.1, −0.5), (−5.8, −6.2), (6.2, 6.4), (6.8, −0.3), (−5.7, −0.1), (0.4, 5.7), (0.4, −6.6)	3.0, 3.0, 3.0, 3.0, 3.0, 2.0, 2.0, 2.0, 2.0

. To evaluate the effectiveness of the adaptive LRSMA, comparisons were conducted against the approach of combining SMA with the cubic B-spline interpolation [25] and the approach of combining LRSMA with the cubic Hermite interpolation [24], with parameter values consistent with refs. [24,25]. The population size ($N$) was set at $N=20$, while the switching probability ($z$) between search and development modes was set at $z=0.3$. The fitness difference threshold ($F_{min}$) was defined at $F_{min}=1$, and the proportion of individual rotation disturbance mutation based on tolerance ($p_r$) was defined at $p_r=0.5$. Additionally, a maturity adjustment coefficient ($\rho$) of $\rho =0.9$ was employed, alongside a curvature threshold of 1. Each algorithm underwent 100 iterations within each scenario, with termination occurring upon reaching the maximum number of iterations.

4.2. Comparison of the Algorithm Combining Cubic B-Spline Interpolation with LRSMA and the Algorithm Combining Cubic Hermite Interpolation with Adaptive LRSMA

To evaluate the effectiveness of the adaptive LRSMA, a comparative analysis was conducted between two algorithms: one combining cubic B-spline interpolation with LRSMA and the other combining cubic Hermite interpolation with adaptive LRSMA. In the simulation based on the cubic B-spline interpolation with the LRSMA approach, we initialize the number of interpolation points at three and four, iterating 30 times. For the simulation experiments based on the other algorithm, we initialize the number of interpolation points at three, iterating 40 times. Effective curvatures are selected for the second simulation. Throughout all simulation experiments, the objective function remains consistent, following the first expression in Formula (11), thereby disregarding curvature constraints. These simulation experiments aim to contrast the length and curvature of the optimal path when different numbers of interpolation points are used. Figure 3a–c illustrate the optimal paths generated by the two algorithms in scenario 1. It is evident from Figure 3 that the paths generated by the combination of cubic B-spline interpolation with LRSMA all exceed the search scope or surpass the curvature threshold when different numbers of interpolation points are used. Figure 4a–c illustrate the optimal paths generated by the two algorithms in scenario 2. In both scenarios, the path length generated by the algorithm combining cubic B-spline interpolation with LRSMA is shorter than that generated by the other algorithm. The algorithm combining cubic Hermite interpolation with LRSMA eliminates the need for manually increasing interpolation points. It can automatically detect the stagnation state of the algorithm and add new interpolation points. The value range of interpolation points is $\left[3, 4\right]$, ensuring that all generated paths remain within the search range, with their curvatures falling within the prescribed threshold.

4.3. A Comparison between the LRSMA and the Adaptive LRSMA

To assess the effectiveness of the adaptive LRSMA, a comparative analysis was conducted between the algorithm combining cubic Hermite interpolation with LRSMA and the algorithm combining cubic Hermite interpolation with adaptive LRSMA. In the simulations based on the combination of cubic Hermite interpolation with the LRSMA, the number of interpolation points was initially set at three and four, with 30 iterations. Conversely, the simulations based on the combination of cubic Hermite interpolation with adaptive LRSMA began with three interpolation points and iterated 40 times. Throughout all simulations, consistency was maintained in the objective function, following the first expression in Formula (11) and disregarding curvature constraints. These simulations aimed to contrast the length, maximum curvature, and path planning time of the optimal path when different numbers of interpolation points are employed.

Table 2. Comparison of simulation results: the algorithm combining cubic Hermite interpolation with LRSMA vs. the algorithm combining cubic Hermite interpolation with adaptive LRSMA in scenario 1.

Algorithm	Number of Interpolation Points	Minimum Path Length (Meters)	Average Path Length (Meters)	Maximum Curvature Value $\left({\mathit{m}}^{-1}\right)$	Curvature Threshold Compliance $\left({\mathit{m}}^{-1}\right)$	Minimum Path Planning Time (Seconds)	Average Path Planning Time (Seconds)
The algorithm combining cubic Hermite interpolation with LRSMA	3	31.56	32.96	1.12	No	3.76	3.75
The algorithm combining cubic Hermite interpolation with LRSMA	4	31.73	33.96	0.96	YES	4.17	4.00
The algorithm combining cubic Hermite interpolation with adaptive LRSMA	[3, 4]	31.72	35.96	0.85	YES	5.44	5.58

and

Table 3. Comparison of simulation results: the algorithm combining cubic Hermite interpolation with LRSMA vs. the algorithm combining cubic Hermite interpolation with adaptive LRSMA in scenario 2.

Algorithm	Number of Interpolation Points	Minimum Path Length (Meters)	Average Path Length (Meters)	$\mathbf{Maximum} \mathbf{Curvature} \mathbf{Value} \left({\mathit{m}}^{-1}\right)$	$\mathbf{Curvature} \mathbf{Threshold} \mathbf{Compliance} \left({\mathit{m}}^{-1}\right)$	Minimum Path Planning Time (Seconds)	Average Path Planning Time (Seconds)
The algorithm combining cubic Hermite interpolation with LRSMA	3	31.89	34.70	1.31	No	2.48	2.33
The algorithm combining cubic Hermite interpolation with LRSMA	4	32.05	35.00	0.97	YES	2.54	2.53
The algorithm combining cubic Hermite interpolation with adaptive LRSMA	[3, 4]	32.08	35.00	0.78	YES	3.09	3.68

demonstrate that in both scenarios 1 and 2, the average path length and the minimum and average path planning times of the optimal path produced by the algorithm combining cubic Hermite interpolation with LRSMA increase when the number of interpolation points is four compared to three. Additionally, augmenting the number of interpolation points in the combination of cubic Hermite interpolation with adaptive LRSMA moderately reduces the maximum curvature of the curve. However, this increase in interpolation points also results in escalated computational burden and an increase in the length of the optimal path.

Regarding path length, in both scenarios, the path generated by the algorithm combining cubic Hermite interpolation with adaptive LRSMA exhibits slight variations compared to that produced by the combination of cubic Hermite interpolation with LRSMA. In terms of maximum curvature, in both scenarios, when the number of path nodes is three, the curvature of the optimal path produced by the combining of cubic Hermite interpolation with LRSMA surpasses the prescribed threshold. However, when the number of path nodes is four, the curvature of the optimal path produced by the combination of cubic Hermite interpolation with LRSMA complies with the prescribed threshold. Conversely, the maximum curvature of the path produced by the algorithm combining cubic Hermite interpolation with adaptive LRSMA consistently remains within the prescribed threshold.

In scenario 1, compared with the algorithm combining cubic Hermite interpolation with adaptive LRSMA, the minimum and average path planning times of the algorithm combining cubic Hermite interpolation with LRSMA are reduced by 45.78% and 38.89%, respectively. In scenario 2, compared with the algorithm combining cubic Hermite interpolation with adaptive LRSMA, the minimum and average path planning times of the combination of cubic Hermite interpolation with LRSMA are reduced by 62.46% and 32.07%, respectively.

These findings suggest that initializing the number of interpolation points at three for both algorithms enables them to find a curve satisfying the curvature constraint when the number of interpolation points is increased to four.

Figure 5 illustrates the iterative curve of path planning utilizing the algorithm combining cubic Hermite interpolation with adaptive LRSMA in scenario 1 and scenario 2. Across both scenarios, the algorithm demonstrates the ability to identify instances where the population converges towards local optima after the iteration starts for a period of time. Subsequently, the algorithm initiates a process of reconvergence, eventually resulting in convergence once more after a sufficient number of iterations. Remarkably, in both scenarios, 40 iterations prove to be sufficient for achieving satisfactory path planning results.

4.4. The Optimal Path Planning Results

To obtain the optimal path using two LRSMAs in scenario 1 and scenario 2, consideration of the curvature constraint is imperative. The number of interpolation points is initialized at three, and the objective function adheres to the second expression in Formula (11). The simulation results are presented in

Table 4. The path planning results of two algorithms in scenarios 1 and 2 when considering the curvature constraint.

Scenario	Number of Interpolation Points	Minimum Path Length (Meters)	Maximum $\mathbf{Curvature} \mathbf{Value} \left({\mathit{m}}^{-1}\right)$	Minimum Path Planning Time (Seconds)
Scenario 1	3	32.52	0.95	9.98
Scenario 2	3	33.22	0.91	8.56

.

A comparison between Table 3 and Table 4 reveals that when factoring in the curvature constraint, the algorithm combining cubic Hermite interpolation with adaptive LRSMA demonstrates a reduction in path length of 2.52% in scenario 1 and 3.56% in scenario 2 compared to the path length produced by the combination of cubic Hermite interpolation with the LRSMA approach.

Table 5. Comparison of ultimate path planning results: the algorithm combining cubic Hermite interpolation with LRSMA vs. the algorithm combining cubic Hermite interpolation with adaptive LRSMA.

Scenario	Algorithm	Number of Optimal Interpolation Points	Path Length (Meters)	Maximum $\mathbf{Curvature} \mathbf{Value} \mathbf{\left(}{\mathit{m}}^{\mathbf{-}\mathbf{1}}\mathbf{\right)}$	Total Path Planning Time (Seconds)
Scenario 1	The algorithm combining cubic Hermite interpolation with LRSMA	4	31.73	0.84	17.91
	The algorithm combining cubic Hermite interpolation with adaptive LRSMA	4	31.72	0.85	15.42
Scenario 2	The algorithm combining cubic Hermite interpolation with LRSMA	4	32.05	0.97	13.58
	The algorithm combining cubic Hermite interpolation with adaptive LRSMA	4	32.08	0.78	11.65

presents the final path planning results of the algorithm combining cubic Hermite interpolation with LRSMA and the combination of cubic Hermite interpolation with adaptive LRSMA approaches in both scenarios. The simulation results underscore that the algorithm combining cubic Hermite interpolation with adaptive LRSMA can yield a smooth path that adheres to curvature constraints while minimizing path length. Furthermore, the path length generated by this algorithm closely aligns with that produced by the combination of cubic Hermite interpolation with LRSMA. Notably, the algorithm combining cubic Hermite interpolation with adaptive LRSMA can autonomously achieve the optimal path while adhering to the curvature constraint, thereby reducing the time required for identifying the optimal solution.

5. Conclusions

In this research, we present a global robot motion planning approach that integrates the improved slime mold algorithm with cubic Hermite interpolation. The algorithm eliminates the requirement for manually increasing interpolation points. Specifically, if the number of interpolation points is insufficient and the calculated path curvature exceeds the predefined threshold, the algorithm can adaptively increase the number of interpolation points to compute a path that meets the curvature threshold, thereby improving the curvature of the motion curve. Subsequent to this improvement, the best path and the optimal number of interpolation points are determined through a comparative analysis of path lengths utilizing two distinct objective functions. The simulation results demonstrate that this approach can generate a collision-free path that adheres to curvature constraints while minimizing path length.

However, the proposed algorithm has only been evaluated in simulation settings and has not yet undergone validation on real robots. Our subsequent steps will include real-world testing to further refine the algorithm in terms of computational cost and other relevant aspects. Additionally, the algorithm requires evaluation with multiple robots.

In practical applications, the algorithm is not well suited for open spaces with sparse obstacles. It is more effective in narrow passages, where fewer interpolation points often result in path curvatures that exceed the predefined threshold. Consequently, this algorithm is applicable to slow-moving robots in constrained passages, such as for transportation operations in mining areas or post-disaster material transportation in earthquake-affected regions.