Robotic Arm Trajectory Planning Based on Improved Slime Mould Algorithm

Abstract

The application of robotic arms in the industrial field is continuously becoming greater and greater. The impact force generated by a robotic arm in a gripping operation leads to vibration and wear. To address this problem, this paper proposes a trajectory planning method based on the improved Slime Mould Algorithm. An interpolation curve under the joint coordinate system is constructed by using seven non-uniform B-spline functions, with time and impact force as the optimization objectives and angular velocity, angular acceleration, and angular acceleration as the constraints. The original algorithm introduces Bernoulli chaotic mapping to increase the diversity of the population, adaptively adjusts the feedback factor, improves the crossover operator to accelerate the global convergence, and combines the original algorithm with an improved artificial bee colony search strategy guided by the global optimal solution, adding a quadratic interpolation method to increase the diversity of the population and to accelerate the global convergence speed. Combined with the improved artificial swarm search strategy guided by the global optimal solution, the quadratic interpolation method is added to enhance the local utilization ability. The simulation and real-machine experimental results show that the improved algorithm shortens the movement time of the robotic arm, reduces the joint impacts, minimizes the vibration and wear, and prolongs the service life of the robotic arm.

1. Introduction

With the rapid development of science and technology, robotics has become a hot spot in today’s technological field, and its technology and applications have been gradually improved and popularized. Robotic arms, due to their high efficiency, good safety, good adaptability to harsh conditions, and other characteristics, have transcended into industry, agriculture, medical care, and many other fields, enhancing productivity for economic development and social construction [1,2,3]. In modern industrial automation, robotic arms, as important actuators, are widely used in various production lines and machining scenarios, and their efficient and accurate movement capabilities bring great convenience and benefits to the production process [4,5]. However, robotic arms must follow a specific pathway when performing a task to ensure that the workpiece can be accurately grasped, placed, or processed. Therefore, trajectory planning algorithms are one of the key technologies for robotic arm control, and are of great significance for improving the efficiency and accuracy of robotic arm motion [6,7].

The study of robotic arm trajectory planning algorithms aims to solve the problem of how to make a robotic arm efficiently plan the motion trajectory in complex environments [8,9,10]. Robotic arm trajectory planning, at present, mainly adopts B-spline curve and polynomial interpolation as the optimization objective function of the robotic arm trajectory; polynomial interpolation is mainly based on cubic polynomials, quintic polynomials, and 3-5-3 polynomials, and B-spline interpolation is mainly based on five-times B-spline and seven-times B-spline [11], Particle Swarm Optimization [12], Cuckoo Search [13], and Tuna swarm optimization [14] for trajectory optimization.

With the deepening of the research on robot trajectory planning, the variety of algorithms has become more and more diverse in order to adapt to different needs. Zhao et al. [15] examined the jitter problem generated by a robotic arm during the working process, and proposed a time-seeking optimization algorithm for improved sparrow search optimization, which effectively improves the convergence efficiency of the algorithm and the solution accuracy. On the other hand, Youmin Wang et al. [16] found that the B-spline function has the characteristics of high computational efficiency and low memory consumption when using the B-spline curve for trajectory planning of the robotic arm. Xuhua Yuan et al. [17] optimized the B-spline curve by incorporating multiple control points and optimized the curve using a swarm search strategy algorithm as a way of enhancing the accuracy of the curve fitting and reducing the time required for planning. By incorporating dynamic interference coefficients into the traditional bat algorithm for the local search process, Sun and Diem et al. [18] optimized the conversion strategy between the local and global search of the bat algorithm, and then optimized the trajectory profile of the robotic arm by optimizing the bat algorithm. Liu et al. [19] used 3-5-3 spline polynomials to generate smooth curves with continuous velocity, acceleration, and additive acceleration, and used an improved multivariate universe algorithm to optimize the performance for time and energy consumption. Additionally, they constructed an objective function using a weighting method to select the desired optimum from the set of Pareto solutions to improve the efficiency of the robotic arm motion and reduce energy consumption. The Slime Mould Algorithm (SMA) was proposed by Li et al. [20] in 2020, and was inspired by the diffusion and foraging behaviors of Sticky Mushroom and belongs to the meta-inspired algorithms. It is characterized by its fast convergence speed and strong optimization-seeking ability. Although this algorithm is widely used, there are still many areas that need to be improved. Xiong et al. [21] examined the problem of the search process of the SMA, which has high randomness and easily converges to the local optimum with decreasing accuracy. They proposed an effective global optimization method via an improved Slime Mould Algorithm (EISMA) combining multiple strategies.

This paper proposes a trajectory optimization method for robotic arms based on an improved Slime Mould Algorithm. For the position and time series of each joint of the robotic arm in the joint coordinate system, a seventh-order non-uniform B-spline function is used to construct interpolation curves in the joint coordinate system. The time and impact are considered as optimization objectives, and the angular velocity, angular acceleration, and angular acceleration of the robotic arm joints are used as constraints. The improved Slime Mould Algorithm is used to optimize the objectives. Firstly, Bernoulli chaotic mapping is added to improve the diversity of the population. Then, adaptive adjustable feedback factors, crossover operators, and improved search strategies for artificial bee colonies are added to the global search process to enhance the algorithm’s convergence accuracy and speed, as well as its ability to escape from local optima. Finally, the effectiveness and practicality of the proposed method are verified through simulation and real-machine experiments, providing new ideas and methods for the trajectory planning of robot arms in complex environments.

2. Relevant Knowledge

2.1. Optimal Objective Function Model for Trajectory Planning

When the robotic arm moves in the operation space, the end-effector of the robotic arm will move to the middle trajectory point one by one in the Cartesian coordinate system of the operation space; $\left\{O_{\mathrm{i}}\right\}$ is the collection of trajectory points during the movement of the end-effector of the robotic arm, $i=0,1,\dots ,m$ in the collection, $O_0$ in the collection indicates the initial trajectory point of the end-effector of the robotic arm, while $O_{\mathrm{m}}$ indicates the trajectory point of the target point of the end-effector of the robotic arm, and then the inverse is applied. Then, by using inverse kinematics to solve for each joint of the robotic arm, it can be concluded that each trajectory point corresponds to each joint of the robotic arm under the joint coordinate system; the position coordinate set is $\left\{{\theta }_{\mathrm{j}}\right\}$, the set of $j=0,1,\dots ,n$, and n represents the number of joints of the robotic arm. The parameters of the optimization objective of the robotic arm are determined as follows:

$$F_1=\sum _{i=1}^m{t}_i$$

(1)

$$F_2=\sum _{j=1}^N\sqrt{{\displaystyle \frac{1}{T}}{\int }_0^T{\left({\stackrel{⃛}{\theta }}_j\right)}^{2}dt}$$

(2)

where $F_1$ represents the total time for the movement of the robot arm from the starting point to the target point, $t_{\mathrm{i}}$ represents the time taken by the end-effector of the robot arm to move from the trajectory point $O_{\mathrm{i}}$ to the trajectory point $O_{\mathrm{i}+1}$, $i=1,2,L,\dots ,m-1$, $F_2$ represents the impacts suffered by the joints of the robot arm, and ${\stackrel{⃛}{\theta }}_j$ denotes the additive acceleration of the rotation of the joint j. The optimization objective is normalized and the optimization objective function obtained is as follows:

$$F_f=K_t{F}_1+K_j{F}_2+K_p{F}_p$$

(3)

In the formula, $K_{\mathrm{t}}$ denotes the time coefficient, $K_{\mathrm{j}}$ denotes the impact coefficient, and $K_{\mathrm{p}}$ denotes the penalty coefficient, and then each coefficient is assigned a value, such as $K_{\mathrm{t}}=0.4$, $K_{\mathrm{j}}=0.6$, and $K_{\mathrm{p}}=0.5$, respectively, where $F_{\mathrm{p}}$ denotes the penalty term, which is set to an initial value of 0, and when the joints in the robotic arm exceed a certain constraint in the process of movement, it will cause the value of $F_{\mathrm{p}}$ to increase by one.

The following three constraints are to be satisfied by the robotic arm joints when the robotic arm performs the movement:

$$\left\{\begin{array}{c}{g}_1=\left|{\dot{\theta }}_j\right|-{\dot{\theta }}_{jmax}\le 0\hfill \\ g_2=\left|{\ddot{\theta }}_j\right|-{\ddot{\theta }}_{jmax}\le 0\hfill \\ g_3=\left|{\stackrel{⃛}{\theta }}_j\right|-{\stackrel{⃛}{\theta }}_{jmax}\le 0\hfill \end{array}\right.$$

(4)

where $g_1$ denotes the angular velocity constraint of the robotic arm joint, $g_2$ denotes the angular acceleration constraint of the robotic arm joint, and $g_3$ denotes the angular additive acceleration constraint of the robotic arm joint; ${\dot{\theta }}_j$ denotes the angular velocity of the rotation of the robotic arm joint j, whereas ${\dot{\theta }}_{jmax}$ denotes the maximum angular velocity of the rotation of the robotic arm joint j; ${\ddot{\theta }}_j$ denotes the angular acceleration of the rotation of the robotic arm joint j, whereas ${\ddot{\theta }}_{jmax}$ denotes the maximum angular acceleration of the rotation of the robotic arm joint j; and ${\stackrel{⃛}{\theta }}_j$ denotes the impact of the rotation of the robotic arm joint j, whereas ${\stackrel{⃛}{\theta }}_{jmax}$ denotes the maximum additive acceleration of the rotation of the joint j.

2.2. Seven-Times B-Spline Interpolation Curve Model

The basic B-spline curve has many good characteristics; its high flexibility is the most important feature, and the local parameters inside the B-spline curve can be modified. Under the constraints of the joint angle, joint angular velocity, joint angular acceleration, and joint impact of each joint of the robotic arm, the trajectories of each rotating joint of the robotic arm can be interpolated and fitted in the joint coordinate system through the use of the seventh B-spline curve, so that the resulting joint angle, joint angular velocity, joint angular acceleration, and joint impact of the robotic arm are all continuous and smooth function curves. The following is the formula for calculating the P-sub B-spline curve:

$$A_{\left(u\right)}=\sum _{i=0}^n{N}_i{p}_{\left(u\right)}{d}_i,a\le u\le b$$

(5)

where $d_{\mathrm{i}}$ is the distance to the control vertex in the sense of a scalar; $\{N_{\mathrm{i}},p_{\left(\mathrm{u}\right)}\}$ is a p-times B-spline curve basis function; u is a parametric variable; and $A_{\left(\mathrm{u}\right)}$ denotes a scalar.

The equation defined by the recursive method of the basis functions of the B-spline curve is as follows:

$$\left\{\begin{array}{c}{N}_{i,0}\left(u\right)=\left\{\begin{array}{c}1,u_i\le u\le u_{i+1}\hfill \\ 0,other\hfill \end{array}\right.\hfill \\ N_{i,p}\left(u\right)={\displaystyle \frac{(u-u_i)N_{i,p-1}\left(u\right)}{u_{i+1}-u_i}}+{\displaystyle \frac{(u_{i+k+1}-u)N_{i+1,p-1}\left(u\right)}{(u_{i+k+1}-u_{i+1})}}\hfill \end{array}\right.$$

(6)

where p denotes the number of B-spline curves; i denotes the ordinal number of the B-spline curve. The interval of $N_{i,p}\left(u\right)$ is $\left(u_i,u_{i+k+1}\right)$. The node vector of the B-spline curve basis function $\left\{N_{i,p}\left(u\right)\right\}$ is $U=\left[u_0,u_1,u,\dots ,u_{2k}\right]$. A point $u_i$ in the interval can have at most $k+1$ non-zero k-times B-spline curves $B_{r,k}\left(u\right)$, where $r\in \left[i-1,i\right]$ and all the other k-times B-spline basis functions are 0. Therefore, the B-spline curve is transformed to be represented by the following equation:

$$p(u)=\sum _{r=i-k}^i{d}_r{B}_{r,k}\left(u\right),u\in [u_i,u_{i+1}]$$

(7)

The domain of definition of the B-spline interpolation curve is then transformed to the canonical domain of definition, resulting in a set of nodal vector equations as follows:

$$\left\{\begin{array}{c}u\in [u_k,u_{n+k}]=[0,1]\hfill \\ u_0=u_1=\dots u_k=0\hfill \\ u_{n+k}=u_{n+k+1}=\dots u_{n+2k}=1\hfill \end{array}\right.$$

(8)

After that, the normalization of the time nodes starts, resulting in the formula for its calculation as follows:

$$u_i=u_{i-1}+{\displaystyle \frac{|\mathrm{\Delta }{t}_{i-k-1}|}{{\displaystyle \sum _{r=0}^{n-1}}|\mathrm{\Delta }{t}_r|}}$$

(9)

This results in an equation that satisfies the constraints of the B-spline interpolation curve:

$$p(u_{i+k})=\sum _{r=i}^{i+k}{d}_r{B}_{r,k}\left(u_{i+k}\right)=P_i$$

(10)

Here, $u_{i+k}\in \left[u_k,u_{n+k}\right]$ and $i=0,1,\dots ,n$, where p denotes the number of B-spline curves; i denotes the ordinal number of the B-spline curve.

The $k-1$ additional constraints can be obtained by means of tangential vector boundary constraints. Thus, they can be obtained through seven B-spline curves:

$$\left\{\begin{array}{c}\dot{p}\left(u_s\right)=v_s,\ddot{p}\left(u_s\right)=a_s\hfill \\ \dot{p}\left(u_e\right)=v_e,\ddot{p}\left(u_e\right)=a_e\hfill \end{array}\right.$$

(11)

where $v_s$ denotes the angular velocity of the robotic arm joint start, while $v_e$ denotes the angular velocity of the robotic arm joint stop; $a_s$ denotes the angular acceleration of the robotic arm joint start, while $a_e$ denotes the angular acceleration of the robotic arm joint stop; and $\dot{p}$ and $\ddot{p}$ represent the first derivative and second derivative of the spline curve, respectively.

The vectorial derivatives of each order can be found based on the De Boor recursion formula:

$$p^l\left(u\right)=\sum _{r=i-k+1}^i{d}_r^l{B}_{r,k-l}\left(u\right)$$

(12)

$$d_r^l=\left\{\begin{array}{c}{d}_r,l=0\hfill \\ {\displaystyle \frac{(k+1+l)(d_r^{l-1}-d_{r-1}^{l-1})}{(u_{r+k+1-l}-u_r)}},l=1,2,3,\dots ,r;j=i-k+1,\dots ,i\hfill \end{array}\right.$$

(13)

In accordance with the above obtained formulae, n + 7 equations can be derived and the equations are represented according to the following matrix:

$$A_j{d}_j=p_j$$

(14)

$$\left\{\begin{array}{c}{A}_j\in R^{(n+7)\times (n+7)}\hfill \\ d_j={[d_{j,0}{d}_{j,1}···d_{j,(n+6)}]}^T\hfill \\ p_j=[p_{j,0}{p}_{j,1}···p_{j,(n+6)}···v_{j,s}{v}_{j,e}{a}_{j,s}{a}_{j,e}]\hfill \end{array}\right.$$

(15)

The control points can be solved inversely from Equation (14):

$$d_j=A_j^{-1}{p}_j$$

(16)

The time node vectors can be obtained by Equation (9); the control vertex vectors can be obtained by Equation (16); the basis functions of the B-spline curves can be obtained by Equation (6); the trajectory curves of the joints of the robotic arm in the intervals $t\in \left[t_0,t_n\right]$ can be obtained by Equation (7); and the trajectory curve of the robotic arm in the intervals can be obtained by Equation (12), which can obtain the curves of any order derivative, and thus can be constructed to construct the constraints when the optimal trajectory curves of the robotic arm are being solved.

3. Research Content

3.1. Fundamentals of the Standard Slime Mould Algorithm

3.1.1. Proximity to Food

The process of realizing the individual convergence behavior of slime moulds can be represented through the use of mathematical modeling, and the mathematical model developed with Equation (17) is also proposed to simulate the contraction of slime moulds when they are performing convergence behavior:

$$X_{(\mathrm{t}+1)}=\left\{\begin{array}{c}{X}_{b(t+1)}+vb\ast (w\ast X_{A\left(t\right)}-X_{B\left(t\right)}),r<p\hfill \\ vc\ast X_{\left(t\right)},r\ge p\hfill \end{array}\right.$$

(17)

In the formula, the value of $v_b$ is a randomly selected number in the interval $\left[-a,a\right]$, the value of $v_c$ is a parameter that is continuously oscillated in the interval $\left[-1,1\right]$ and finally converges to zero infinitely, and t denotes the number of iterations of the Slime Mould Algorithm in the current search space, where $X_{b\left(t\right)}$ is the position of the individual with the optimal fitness value generated by the iteration in the current search space, and $X_{\left(t\right)}$ is the position of the new slime mould individual generated by the iteration in the current search space, while $X_{A\left(t\right)}$ and $X_{B\left(t\right)}$, respectively, denote the position of two random slime mould individuals generated by iteration in the space; W is the weight of the random slime mould individual in the new slime mould individual.

The update formula for the control parameter P during the iteration process is as follows:

$$p=tan\left|S_{\left(i\right)}-DF\right|$$

(18)

where $i=1,2,\dots ,n$,$S_{\left(i\right)}$ denotes the fitness value of the ith slime mould individual, and $DF$ denotes the fitness value of the current optimal individual; the parameters $vb$ and a and the weight coefficients W in the iterative process position updating formulae are computed as follows, respectively:

$$vb=\left[-a,a\right]$$

(19)

$$a=arctanh(-\left({\displaystyle \frac{t}{T}}\right)+1)$$

(20)

$$W\left(SmellInde{x}_{\left(i\right)}\right)=\left\{\begin{array}{c}1+r\ast log({\displaystyle \frac{bF-S_{\left(i\right)}}{bF-wF}}+1),\begin{array}{cc}& i<{\displaystyle \frac{N}{2}}\end{array}\\ 1-r\ast log({\displaystyle \frac{bF-S_{\left(i\right)}}{bF-wF}}+1),\begin{array}{cc}& i\ge {\displaystyle \frac{N}{2}}\end{array}\end{array}\right.$$

(21)

$$SmellInder=sort\left(N\right)$$

(22)

where N is the initialized population size of the slime moulds, $S_{\left(t\right)}$ denotes the individual fitness of the ith slime mould during the iteration process, r denotes a random number within the interval [0,1], T denotes the maximum number of iterations, $i<\left(N/2\right)$ denotes the slime mould individual in the first half of $S\left(i\right)$, $bF$ denotes the optimal fitness value obtained when iterating within the current search space, and $wF$ denotes the worst fitness value obtained when iterating within the current search space; meanwhile, SmellInder denotes the initialization of the slime moulds.

3.1.2. Proximity to Food

In the search space, each slime mould individual performs a position update with the position update equation:

$$X_{(t+1)}=\left\{\begin{array}{c}rand\ast \left(UB-LB\right)=LB,rand<z\hfill \\ X_{b\left(t\right)}+vb\ast (W\ast X_{A\left(t\right)}-X_{B\left(t\right)}),r<p\hfill \\ vc\ast X_{\left(t\right)},r\ge p\hfill \end{array}\right.$$

(23)

where $UB$ and $LB$ are the upper and lower bounds of the search space, the value of $rand$ is a randomly selected number between the interval [0,1], and z is a custom parameter and is optimal with a value of z of 0.04 after several experiments.

3.1.3. Access to Food

Here, the value of $vb$ is oscillating randomly between the interval $\left[-a,a\right]$, and the value of $vb$ is also gradually converging to zero as the number of iterations of the viscous bacteria optimization algorithm increases. Meanwhile, the value of $vc$ is randomly oscillating between the interval [−1,1], and finally it constantly converges to zero, and the formula of $vb$ is as follows:

$$vb=\left[-a,a\right]$$

(24)

$$vb=1-{\displaystyle \frac{t}{T}}$$

(25)

3.2. Population Initialization Based on Bernoulli Chaotic Mapping

The Slime Mould Algorithm has the problem of over-dependence on the initial population position when iteratively updating, and the use of the original random position updating formula for population initialization produces an uneven distribution of the initial population, which affects the solution accuracy and convergence speed of the Slime Mould Algorithm. Therefore, chaos mapping is added to the population initialization of the Slime Mould Algorithm to improve the diversity and traversal of the initial population, and most of the intelligent optimization algorithms currently use Tent mapping, circle mapping, and logistic mapping for the initial population mapping, and Tent mapping and Circle mapping when taking values within [0, 2]. There is a phenomenon of small periods, and it is easy to fall into the fixed point, and logistic mapping has more stringent requirements when initializing the population, which can easily produce an uneven distribution and less diversity of the slime mould population during initialization, so the improved Slime Mould Algorithm adopts the Bernoulli chaotic map to increase the diversity of the initial population of slime mould and the uniformity of the location distribution, and the formula of the Bernoulli chaotic map is as follows:

$$X_{(t+1)}=\left\{\begin{array}{c}{\displaystyle \frac{X_t}{1-\mu }},0<X_{\left(t\right)}\le (1-\mu )\hfill \\ {\displaystyle \frac{X_{\left(t\right)}-1+\mu }{\mu }},(1-\mu )<X_{\left(t\right)}<1\hfill \end{array}\right.$$

(26)

where u is the chaos parameter, and it is experimentally concluded that Bernoulli chaotic mapping has the best performance when the value of u is 0.4.

Let the number of iterations t of the slime mould population be 5000, and initialize the population using Logistic Chaos Mapping and Bernoulli Chaos Mapping, respectively, as shown in Figure 1 and Figure 2.

3.3. Adaptive Adjustable Feedback Factors

In the Slime Mould Algorithm, the feedback factor $vc$ mainly reflects the feedback relationship between the food concentration and the quality of the slime mould individuals, because the value of $vc$ will oscillate randomly in the interval [−1, 1], and finally converge to zero, and the feedback factor can not accurately reflect the feedback relationship between the quality of the individual slime moulds and the food concentration, so the slime mould algorithm is slower and less precise when converging. Therefore, this paper adds an adaptive adjustable feedback factor into the original algorithm: when the algorithm is in the early stage of the iteration phase, because the slime mould individual is in the global search space to complete the exploration, and at this time the slime mould individual food concentration will be relatively small, the value of the feedback factor should be rapidly reduced to reduce the feedback relationship between the two, so as to enhance the global exploration ability of the slime mould optimization algorithm; when the algorithm is in the iteration phase, the feedback factor cannot accurately reflect the feedback relationship between slime mould individual quality and food concentration. When the algorithm is in the late iteration stage, because the food concentration in the search space has increased, the feedback factor should be maintained in a relatively stable state, so as to improve the speed of the mucilaginous individuals in finding the highest concentration of food in the local search. The adaptive adjustable feedback factor is calculated as follows:

$$v_c=(e^{{\displaystyle \frac{(t_{max}-t)}{(t_{max}-1)}}}\ast {\displaystyle \frac{1}{(e-1)}}{)}^k$$

(27)

In the formula, t denotes the number of iterations in the current iteration process, while $t_{max}$ denotes the maximum number of iterations in the iteration process; k in the formula denotes the adjustable factor, and the values of the adjustment parameter k were tested and compared, as shown in Figure 3.

As can be seen from Figure 3, the speed of change in the optimized feedback factor change curve will be accelerated with the increase in the adjustment factor k. And it can be concluded from the analysis of the experimental data that, when the value of k is too large, it will result in the convergence of the slime mould optimization algorithm in the pre-iterative period to be too fast, which leads to the algorithm easily falling into the local optimal solution, and the global search capability will be reduced; when the value of k is too small, it will lose the role of the feedback factor, and cannot play the role of balancing the search ability of the viscous bacteria optimization algorithm, and the convergence speed of the global search will be reduced. After a large number of experiments and analyses, it is concluded that when k = 4, the feedback factor plays a relatively optimal feedback role.

3.4. Intersection Operators

In order to improve the convergence speed of the Slime Mould Algorithm, an improved crossover operator is added to update the position of the slime mould individual; that is, the current slime mould individual and the optimal individual in the population are cross-operated with a certain probability $P_{\mathrm{t}}$, and the mathematical model formula of the crossover operator is as follows:

$$\left\{\begin{array}{c}{X}_{A1(t+1)}=L\ast X_{A\left(t\right)}+(1-L)\ast X_{best\left(t\right)}\\ X_{A2(t+1)}=L\ast X_{best\left(t\right)}+(1-L)\ast X_{A\left(t\right)}\end{array}\right.$$

(28)

where t denotes the number of iterations in the iterative process, while $X_{A1}$ and $X_{A2}$ denote the location information of the two newborn individuals generated through the crossover operation, respectively, where $X_A$ denotes the location information of the individual in the current iterative process, and $X_{best}$ denotes the location information of the optimal slime mould individual in the population of the current iterative process, and L is a random number in the interval (0, 1).

It can be concluded from Equation (28) that the location information of the generated offspring is mainly determined by the parents and the parameter L, where the parameter L is used to regulate the proportion of information that the newborn individuals can obtain from the two parents, so that there are relatively more genes of the excellent parents in the newborn individuals, and also so that the mucus individuals in the newborn individuals’ populations can maintain the diversity that the original populations have. Therefore, in this paper, the original stochastic parameters were optimized, mainly by incorporating the Laplace coefficients into the stochastic parameters, and the optimized parameter L is given by the following equation:

$$L=\left\{\begin{array}{c}\mu -\lambda \ast {log}_e(r),\begin{array}{cc}& r\le {\displaystyle \frac{1}{2}}\end{array}\\ \mu +\lambda \ast {log}_e(r),\begin{array}{cc}& r>{\displaystyle \frac{1}{2}}\end{array}\end{array}\right.$$

(29)

where $\mu$ and $\lambda$ are Laplace coefficients, $\lambda$ is a number greater than 0 that controls the scale, the value of $\mu$ is a natural number, and r is a random number within the interval [0, 1]. It can be concluded through Equation (29) that the modified coefficient L uses the newly quoted gravitational coefficient $\lambda$ to regulate the relative distance between the parent and the offspring, and a smaller value of $\lambda$ indicates that the relative distance between the offspring and the parent is shorter and the population is richer, and under the regulation of the Laplace coefficient $\mu$ and the Laplace coefficient $\lambda$ both together, the newly generated offspring can obtain a better parent in the process of updating the location information.

3.5. Improving Search Strategies for Artificial Swarms

In this paper, an artificial bee colony search strategy is introduced into the Slime Mould Algorithm in the global search process, and some improvements are obtained; the original mathematical model of the artificial bee colony search strategy formula is as follows:

$$Z_{i,j}=x_{i,j}+{\varphi }_{i,j}(x_{i,j}-x_{k,j})$$

(30)

where $Z_{i,j}$ denotes the candidate solution, $x_{i,j}$ denotes the current slime mould individual, $x_{k,j}$ denotes a random slime mould individual, k and j are random parameters, $k\in \left\{0,1,\dots ,M\right\}$ and $j\in \left\{0,1,\dots ,d\right\}$, and are fixed values, where d denotes the dimension and the value of k is unequal to i, and the value of ${\varphi }_{i,j}$ is a random number in the interval of [−1, 1]; it can be concluded from Equation (30) that the candidate solution is obtained by randomly generating the two individuals in the search space and performing a difference operation on the two individuals in the iteration process.

In this paper, the original artificial bee colony search strategy is optimized, mainly introducing the guidance of the global optimal position, and its optimized search strategy is formulated as follows:

$$Z_{i,j}=x_{i,j}+{\varphi }_{i,j}(x_{i,j}-x_{k,j})+{\mathrm{\Omega }}_{i,j}(p_{g,j}-x_{i,j})$$

(31)

$\mathrm{\Omega }$ in the equation is a random number within the interval [0, 1.5] and $p_{g,j}$ is the optimal position in the search space.

3.6. Secondary Interpolation

The principle of the algorithm is as follows: Suppose X and Y are two random positions of the population in a D-dimensional problem, where $X=\left(x_1,x_2,K,x_D\right)$, $Y=\left(y_1,y_2,K,y_D\right)$ and $Z=\left(z_1,z_2,K,z_D\right)$ are the optimal positions iterated to the current one in the process of performing the local search, and the fitness values corresponding to X, Y, and Z are, respectively, $F\left(X\right)$, $F\left(y\right)$, and $F\left(Z\right)$. Then, the position updating formula of the viscous bacterial optimization algorithm with the addition of quadratic interpolation is shown as follows (32), which results in the resulting position updating to produce a new individual $X=(x_1,x_2,K,x_D)$

$$x_q={\displaystyle \frac{(z_q^2-y_q^2)\ast F\left(X\right)+(x_q^2-z_q^2)\ast F\left(Y\right)+(y_q^2-x_q^2)\ast F\left(Z\right)}{2\ast [(z_q-y_q)\ast F(X)+(x_q-z_q)\ast F(Y)+(y_q-x_q)\ast F(Z\left)\right]}}$$

(32)

where q is an integer within the intervals [1, D].

4. Analysis of Experimental Results

4.1. Comparison Algorithm Initialization Parameter Settings

In order to verify the performance of the improved slime mould optimization algorithm, this paper selects the Butterfly Optimization Algorithm (BOA), the Slime Mould Algorithm (SMA), the Gray Wolf Algorithm (GWO), and the Sparrow Search Algorithm (SSA) for comparison with the improved Slime Mould Algorithm (ISMA). Six different test functions were selected to test the above five optimization algorithms to analyze the performance and effectiveness of the improved slime mould optimization algorithm. $F_1$ and $F_2$ are single-modal benchmark functions, $F_3$ and $F_4$ are multimodal benchmark functions, and $F_5$ and $F_6$ are composite benchmark functions. In order to ensure the fairness of the experiment, all five algorithms used the same experimental conditions, and all algorithms were run 30 times under each test function. The specific information of the benchmark test functions is shown in

Table 1. Benchmark functions.

Function	Dim	Range	${\mathit{F}}_{min}$
$F_{1\left(X\right)}={\sum }_{i=1}^{n-1}\left[100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2\right]$	30	[−30, 30]	0
$F_{2\left(X\right)}={\sum }_{i=1}^n{\left[(x_i+0.5)\right]}^2$	30	[−100, 100]	0
$F_{3\left(X\right)}=-20exp(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{x}_i^2})-exp({\displaystyle \frac{1}{n}}{\sum }_{i=1}^{n}cos(2\pi x_i))+20+e$	30	[−32, 32]	0
$F_{4\left(X\right)}={\sum }_{i=1}^n-x_{i}sin\left(\sqrt{\left|x_i\right|}\right)$	30	[−500, 500]	−12,569.5
$\begin{array}{c}{F}_{5\left(X\right)}=0.1\left\{{sin}^2\left(3\pi x_1\right)+{\sum }_{i=1}^n{(x_i-1)}^2\left[1+{sin}^2(3\pi x_i+1)\right]\right.\hfill \\ \left.\begin{array}{cc}\begin{array}{cc}& \end{array}& +{(x_n-1)}^2[1+{sin}^2(2\pi x_n\left)\right]\end{array}\right\}+{\sum }_{i=1}^{n}u(x_i,5,100,4)\hfill \end{array}$	30	[−50, 50]	0
$F_{6\left(X\right)}={\displaystyle \frac{1}{4000}}{\sum }_{i=1}^n{x}_i^2-{\prod }_{i=1}^{n}cos\left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)+1$	30	[−600, 600]	0

. The initialization parameters were also set uniformly, with the maximum number of iterations all set to $t_{max}=5000$, the size of the population set to $N=50$, and the dimension $D=30$.

4.2. Comparative Simulation Analysis

The improved slime mould optimization algorithm was tested with benchmark functions and compared to the other four algorithms.

Table 2. Comparison of the average test results of five algorithms.

Function	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{F}}_3$	${\mathit{F}}_4$	${\mathit{F}}_5$	${\mathit{F}}_6$
BOA	28.9127	5.4496	0.0138	−3382.5132	2.8340	0.0033
GWO	27.9115	1.2608	1.505 × 10${}^{-5}$	−6046.7061	1.0984	0.0110
SSA	28.5236	2.8898	1.007 × 10${}^{-15}$	−9807.0834	2.2639	0
SMA	29.5465	4.5133	1.007 × 10${}^{-15}$	−3977.0257	2.4101	5.07 × 10${}^{-15}$
ISMA	0.0584	0.0008	8.882 × 10${}^{-16}$	−7496.7903	0.0075	0

shows the comparison results of the mean values, and

Table 3. Comparison of standard deviations of test results for five algorithms.

Function	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{F}}_3$	${\mathit{F}}_4$	${\mathit{F}}_5$	${\mathit{F}}_6$
BOA	0.0230	0.5947	0.0027	211.0252	0.2431	0.0016
GWO	0.8900	0.4473	6.225 × 10${}^{-6}$	575.4946	0.26468	0.0199
SSA	0.5105	0.7589	6.4863 × 10${}^{-16}$	1447.6617	0.4481	0
SMA	2.6847	0.3899	6.4863 × 10${}^{-16}$	340.8653	0.3008	2.7749 × 10${}^{-14}$
ISMA	0.0885	0.0006	0	1832.0243	0.0116	0

shows the comparison results of the standard deviations. In order to visualize the optimization speed and performance of the ISMA, the convergence curves of some basic test functions are given in this paper, as shown in Figure 4, Figure 5, Figure 6. Through the convergence curve, it can be more intuitively concluded that the convergence accuracy of the ISMA is higher than that of the SMA, BOA, GWO, and SSA, and the convergence speed is also faster than that of the other four algorithms. As a test function to test the development performance of the algorithm, the convergence curves of the $F_1$ and $F_2$ test functions show that the ISMA has indeed improved the convergence accuracy; the convergence curves of the $F_3$ and $F_4$ test functions show that the convergence speed of the ISMA is faster; and the convergence curves of $F_5$ and $F_6$ are the convergence curves of the composite benchmark test functions, which illustrate the fact that the improved viscous optimization algorithm not only improves the optimality search of the original algorithm, but also increases the convergence accuracy of the original algorithm. The improved algorithm not only improves the optimization accuracy of the original algorithm, but also improves the convergence speed of the algorithm.

In MATLAB R2023b, using the Robot toolbox, the improved Slime Mould Algorithm is used to plan the trajectory of the robotic arm, and the coordinates of the starting point of the robotic arm are set to be 450, −100, 821, and the coordinates of the target point are set to be −508, −601, 244. A part of the node positions are selected on the path from the start point to the target point to find out the corresponding angles of each joint of the robotic arm, as shown in

Table 4. Position coordinates and joint angles of path nodes.

Node	(x, y, z)/mm	${\mathit{\theta }}_1$/rad	${\mathit{\theta }}_2$/rad	${\mathit{\theta }}_3$/rad	${\mathit{\theta }}_4$/rad	${\mathit{\theta }}_5$/rad	${\mathit{\theta }}_6$/rad
1	(450, −100, 821)	1.9698	−95.3302	−52.5182	−109.9252	−85.6002	1.9698
2	(458, −280, 762)	−12.6766	−100.2120	−53.7348	−104.2469	−65.9316	−1.5196
3	(396, −413, 724)	−40.5661	−109.2507	−56.2990	−92.4981	−38.4071	−8.6572
4	(274, −580, 640)	−29.4233	−103.5337	−55.7650	−96.7164	−78.2400	−5.8801
5	(89, −700, 517)	−51.5939	−112.3181	−57.7720	−88.0456	−52.5424	−10.9649
6	(−129, −726, 422)	−73.2598	−118.2951	−59.9670	−78.4775	13.9345	−17.0033
7	(−354, −687, 330)	−101.4153	−128.6296	−63.0296	−66.9342	14.9857	−22.9221
8	(−508, −601, 243)	−117.7092	−133.2772	−64.8753	−59.3292	41.8628	−27.2202

. The motion diagram of some nodes during the movement of the robot arm from the starting point to the target point in three-dimensional space, as shown in Figure 7; The time-varying curves of the angle, angular velocity, angular acceleration, and angular acceleration of each joint are shown in Figure 8.

As shown in Figure 8. This paper adopts a seven-times B-spline curve function to interpolate the motion curves of each joint of the robotic arm, so as to obtain the higher-order smooth continuity, velocity, and acceleration with no obvious sudden change, and the motion trajectory of each joint of the robotic arm is relatively smooth, which demonstrates the improvement of the viscous bacterial optimization algorithm for the optimization of the objective function.

4.3. Real-Machine Experimental Verification

In order to verify the validity of the experiment, the trajectory optimization experiment is verified by using an ER3A-C60 model industrial robotic arm from China and Anhui Erfurt Intelligent Equipment Co., Ltd. (Wuhu City, China), as shown in Figure 9 below in the real environment. Six obstacles are set up in the working area of the robotic arm, which are cylinders and rectangles, and the starting point is set to be Position 1, and the termination point to be Position 6, and the robotic arm moves from Position 1 to Position 6 in order to achieve a reasonable trajectory optimization. After the completion of trajectory optimization in Matlab, the host computer sends the coordinate information of the key points in the optimization process to the robotic arm controller, and it can be concluded from Figure 9 that the algorithm of this paper is relatively smooth for the movement of each link in the process of the real-machine experiments and there is no phenomenon of sudden change in speed and acceleration, which proves the effectiveness of the algorithm of this paper.

5. Conclusions

In this paper, a robotic arm trajectory optimization method based on the improved viscous mode optimization algorithm is proposed.

1.

Firstly, the population is initialized by Bernoulli chaotic mapping, and then an adaptive adjustable feedback factor and an improved artificial bee colony search strategy are added to the global search process to improve the convergence accuracy and convergence speed of the algorithm, as well as the ability of jumping out of the locally optimal solution.

2.

The improved Slime Mould Algorithm is integrated into the trajectory curve of each joint of the robotic arm to optimize its trajectory. And test function experiments are carried out to compare it with the four algorithms of the sparrow algorithm, butterfly algorithm, standard viscous mode algorithm, and Gray Wolf Algorithm. By comparing the results produced by the test function, it can be concluded that the improved viscous mode optimization algorithm not only has a high convergence speed, but also has a relatively high convergence accuracy.

3.

Using MATLAB R2023b software to carry out simulation experiments, the improved Slime Mould Algorithm is integrated into the movement of the robotic arm, and it is concluded through simulation experiments that the improved Slime Mould Algorithm reduces the maximum impact force of the joints of the robotic arm, and the trajectory curves of the joints of the robotic arm also become smoother.

4.

In order to further verify the effectiveness of the algorithm, the algorithm is experimentally verified in a real environment, based on an industrial robotic arm, and the experimental results show that the algorithm of this paper has a relatively smooth motion of each link in the process of the real experiment, and there is no phenomenon of sudden changes in speed and acceleration. The effectiveness and feasibility of the algorithm are proved.