Time-optimal trajectory planning for a six-degree-of-freedom manipulator: a method integrating RRT and chaotic PSO

3.3.1. Initialization of chaotic sequence

The choice of initialization strategy determines the distribution of the initial population in the solution space. In the traditional particle swarm algorithm, the initial population is generated randomly. While this method is simple and easy to implement, it does not ensure a uniform distribution of individuals in the search space, which can negatively affect the algorithm's search speed and optimization performance. Therefore, chaotic sequences are introduced during the initialization phase to ensure a uniform distribution of the initial solutions.

Key parameters in chaotic mappings play a crucial role in determining the system's dynamic behavior, and their selection directly influences the distribution of the initial population. Specifically:

Control Parameter ($$ r $$): For the proposed composite chaotic system, the value of $$ r $$ governs the chaotic behavior of the system. Lower $$ r (r<0.5) $$ values generally lead to periodic behavior in the chaotic mapping. When a smaller $$ r $$ is chosen, the generated sequences cluster within a limited search space region, thereby diminishing the algorithm's exploratory potential. In contrast, higher $$ r (r>0.8) $$ values promote chaotic behavior, facilitating the generation of more widely dispersed initial solutions. This dispersion enhances the diversity of the initial population, thereby improving the algorithm's global search capability.

Initial Value $$ (x0) $$: To achieve a well-distributed initial population, an appropriate initial value $$ (x0) $$ must be carefully selected. It is imperative to ensure that $$ x0 $$ is uniformly distributed across the entire search space to maximize the coverage and effectiveness of the algorithm.

Chaotic systems exhibit ergodicity, meaning they can cover nearly all states within a certain range. By introducing chaotic sequences during the population initialization phase of the improved PSO [sine-tent-cosine particle swarm optimization (SPSO)] algorithm, more uniformly distributed initial particles can be generated. This uniformity and diversity improve the algorithm's ability to cover the search space, addressing the problem of unexplored regions. In high-dimensional optimization problems, chaotic sequences help prevent the clustering or insufficient coverage of local regions often caused by traditional random methods. A more evenly distributed population lays a solid foundation for subsequent global searches. The pseudo-randomness and sensitivity to initial conditions inherent in chaotic systems enhance the ability of the particles to "jump" during the global search phase. In multi-modal problems, chaotic sequences assist particles in escaping local optima, thereby strengthening global search capabilities.

The fitness function value of the random numbers generated using chaotic mapping shows significant improvement compared to those generated by conventional uniformly distributed random number generators. Replacing the conventional method with chaotic mapping yields better results, especially when the search space contains many local optima, making it easier to find the global optimal solution. Chaotic mapping enhances the randomness and diversity of particles.

Chaos mapping is used to generate chaotic sequences, which are sequences of randomness produced by simple deterministic systems. These sequences describe complex chaotic phenomena that are sensitive to initial conditions, non-periodic, and internally stochastic, generated by deterministic nonlinear systems. In optimization, chaotic mapping can serve as an alternative to pseudo-random number generators to produce chaotic numbers between 0 and 1. Experimental results have demonstrated that using chaotic sequences for population initialization, selection, crossover, and mutation influences the algorithm process and often achieves better results than pseudo-random numbers.

Chaotic systems ^[16] can be classified into low- and high-dimensional categories. High-dimensional systems are more complex, with numerous control parameters and greater computational demands. In contrast, low-dimensional systems are simpler, require fewer control parameters and are easy to implement. Nevertheless, certain limitations are inherent to low-dimensional systems, such as constrained chaotic behavior, discontinuous chaotic intervals, and non-uniform data distribution within the generated chaotic sequences.

To develop chaotic systems with better performance, Sine, Tent, and Cosine chaotic mappings are combined to form a new composite chaotic system. This system effectively overcomes the shortcomings of low-dimensional chaos and is less complex and easier to implement than high-dimensional chaos, which is calculated by :

where $$ r=0.7 $$, $$ x(i) $$ and $$ x(i+1) $$ represent the state variables before and after the chaotic mapping, respectively.

To verify whether the chaotic mapping used in this study generates sequences with sufficient uniformity, we evaluated its ability to initialize particle positions reasonably within the search space.

The most commonly used chaotic initialization methods include logistic, circle, and singer mappings. However, the hybrid combination of Sine, Tent, and Cosine chaotic mappings enhances randomness, which introduces diversity during the population initialization phase. This helps the algorithm explore a broader solution space and reduces the likelihood of being trapped in local optima.

Based on the characteristics of the hybrid chaotic mapping, chaotic sequences are generated in the range of [0, 1]. These sequences are then scaled to match the upper and lower bounds of the search space, completing the population initialization.

The logistic chaotic mapping is expressed as follows:

Where $$ a $$ is the control parameter, taking values in the range (0, 4]. The chaotic behavior becomes more pronounced as $$ a $$ increases, reaching a state of full chaos when $$ a $$ = 4.

The Circle chaotic mapping is expressed as follows:

Where $$ a $$ and $$ b $$ are control parameters, with commonly used values of $$ a=0.5 $$ and $$ b=0.2 $$. The mod function represents the modulus operation, ensuring that the output remains within the interval [0, 1].

The Singer chaotic mapping is expressed as follows:

Where $$ a $$ is the control parameter, taking values in the range [0.9, 1.08]. This range ensures the mapping exhibits chaotic behavior suitable for complex optimization tasks.

To verify that the hybrid chaotic mapping initialization method produces results with better randomness and diversity, as well as more uniform distribution compared to logistic chaotic mapping, circle chaotic mapping, singer chaotic mapping, and random initialization, a comparison was conducted using frequency distribution histograms and scatter plots.

The settings for the experiment were: $$ lb=0, ub=1, $$ and $$ NP $$ = 1,000. The interval [0, 1] was divided into ten equal-length subintervals, and the population distribution frequency for each interval was calculated. The results are shown in Figure 5.

Figure 5. Comparison of various chaotic mappings.

The frequency histogram demonstrates that the sequences generated by the hybrid chaotic mapping are more uniformly distributed within the interval [0, 1] compared to other mappings, such as Logistic, Circle, and Singer. The frequency distribution closely aligns with the theoretical values. The scatter plot further validates that the points generated by the hybrid chaotic mapping exhibit no significant clustering and uniformly cover the entire search space, effectively enhancing population diversity during the initialization phase. In contrast, Logistic mapping shows data clustering in certain intervals, which increases the likelihood of searching only local regions. Circle and Singer mappings produce sequences with uneven distribution near the boundaries, potentially limiting global exploration capabilities. Hybrid chaotic mapping combines the advantages of multiple mappings, achieving the best uniformity in distribution and avoiding the limitations of individual chaotic mappings.

By introducing hybrid chaotic mapping as the initialization method, population diversity is significantly improved, enhancing the algorithm's global search capability and reducing the risk of being trapped in local optima.

3.3.2. Improved inertia weights and learning factors

The values of inertia weights ^[17]$$ www $$ and learning factors $$ ccc $$ in the PSO algorithm directly affect its convergence performance. Addressing the shortcomings of the basic particle swarm algorithm - specifically its slow convergence speed for complex functions and low optimization accuracy - larger weights ^[18] enhance the algorithm's global search capability, while smaller weights improve local search ability.

To improve these aspects, this paper introduces adaptive inertia weights in the global search stage, which decrease nonlinearly with the number of evolutionary generations. This approach enhances global searching ability in the early stages and ensures particle diversity in the latter stages by obtaining smaller random values, thus improving the solution's accuracy. The proposed inertia weight function is expressed as follows:

where $$ \alpha $$ = 3, $$ \beta $$ = 5, $$ w_{max} $$ is the maximum inertia weight, $$ w_{min} $$ is the minimum inertia weight, $$ t $$ is the current number of iterations of the particle, $$ T_{max} $$ is the maximum number of iterations, and $$ rand $$ is a random number between (0, 1).

The learning factors $$ c_1 $$ and $$ c_2 $$ of the traditional PSO algorithm are fixed values, which can affect its optimization speed and solving accuracy. The factor $$ c_1 $$ represents the particle's self-learning and summarization ability, reflecting the individual's self-perception, while $$ c_2 $$ represents the particle's ability to learn from better-performing particles, reflecting the particle's social perception.

Since a larger $$ c_1 $$ improves global search capability and a larger $$ c_2 $$ enhances local search ability, it is advantageous to set $$ c_1 $$ large and $$ c_2 $$ small in the pre-search stage, and $$ c_1 $$ small and $$ c_2 $$ large in the post-search stage. Therefore, $$ c_1 $$ is set to decrease linearly with the number of iterations, while $$ c_2 $$ increases linearly. This introduces asynchronous learning factors, defined by:

Where $$ c_{1, start}=2.5, c_{1, end}=0.5, c_{2, start}=0.5, c_{2, end}=2.5, $$ and $$ t $$ and $$ T_{max} $$ represent the number of current iterations and total iterations, respectively.

3.5.1. Compare objects and test functions

To evaluate the performance of the proposed SPSO algorithm, it is compared with the elementary PSO, swallow tern algorithm (STOA) ^[19], and Sine Cosine algorithm (SCA) ^[20]. All algorithms were implemented in MATLAB 2023a and executed on a Windows 11 system with 16 GB of memory. For each algorithm, the population size was set to 30, and the number of iterations was 1,000. The minimum (min), mean, maximum (max), standard deviation (std), optimal value, and running time of each algorithm were recorded to assess their ability to find the optimal value.

In this paper, six benchmark test functions are selected to evaluate the performance of the proposed SPSO algorithm. Each algorithm is run independently 30 times on these functions, as listed in Table 2. The experiments tracked the minimum (min), mean, maximum (max), standard deviation (std), optimal value (value), and running time (time) of each algorithm. These metrics are used to evaluate the optimization capabilities of the algorithms. The results are summarized in Table 3.

As can be seen from Table 3, functions F1-F3 are multidimensional test functions with multiple local optimal solutions. These functions simulate the complexity of many real-world problems and test the algorithm's balance between global exploration and local exploitation. Specifically, they assess how well the algorithm avoids falling into local optima and finds global optima during the search process. Functions F4-F6 are hybrid functions, designed to test the algorithm's ability to handle complex hybrid optimization problems. These functions may contain locally optimal solutions of varying shapes, sizes, and distributions, thereby increasing the challenge for the algorithm. The design of these hybrid functions ensures a balance between global and local optima, comprehensively assessing the algorithm's global and local search capabilities.

From Table 3, it is evident that among the multidimensional test functions, F1 is characterized by multiple peaks, making it difficult to find the optimal solution. This causes the SPSO algorithm to converge with less than ideal accuracy during certain iterations, although it still ranks second relative to other algorithms. F1 is a multi-modal function with multiple local optima, simulating the complexity of real-world optimization problems. In the case of the F1 function, SPSO demonstrates the best performance, with the smallest value and optimal solution closest to the global optimum, while also requiring less runtime. In contrast, PSO performs poorly, yielding significantly higher minimum values. Such problems challenge the algorithm's ability to balance global exploration and local exploitation, particularly in avoiding entrapment in local optima. As shown in Table 3 and Figure 7, when optimizing the F1 function, SPSO achieves the optimal fitness value of 23.475 in only 400 iterations, while STOA and SCA require approximately 800 iterations, and PSO takes 1,000 iterations. The optimal fitness value achieved by SPSO is significantly better than those of the other algorithms, indicating its superior global search capability. The F1 function is characterized by a multi-peak distribution, which often leads traditional algorithms to become trapped in local optima. However, SPSO effectively avoids this issue through the use of chaotic sequences and dynamic inertia weights.

Figure 7. Comparison of various chaotic mappings.

In the F2 test function, the SPSO algorithm finds the optimal value faster than other algorithms, with the smallest standard deviation and highest stability. F2 is a nonlinear optimization function characterized by complex nonlinear relationships and constraints. It evaluates the algorithm's capability to optimize high-dimensional nonlinear problems, focusing on efficiency and stability in large-scale complex scenarios. For the F2 function, SPSO not only finds the smallest minimum value but also exhibits the lowest standard deviation, indicating the highest stability. Additionally, SPSO achieves the shortest runtime, reflecting its high efficiency. For the F2 function, SPSO finds the optimal solution (fitness value of 0.000054567) in only 300 iterations, outperforming other algorithms in both the number of iterations and the quality of the optimal solution. The F2 function requires high search efficiency, and SPSO achieves rapid convergence by dynamically adjusting the weights of global and local search through adaptive learning factors.

F3 is a function with multiple local optima, designed to test an algorithm's performance in multi-local optima scenarios. This function requires optimization algorithms to avoid local optima and successfully locate the global optimum. On the F3 function, SPSO achieves a balanced performance, finding a smaller optimal value with a lower standard deviation and shorter runtime. This highlights its high stability and efficiency in multi-local optima problems. When optimizing the F3 function, SPSO reaches a fitness value of 2.1259 after 400 iterations, whereas PSO achieves a suboptimal fitness value only after 600 iterations. SPSO also outperforms SCA and STOA in this scenario.

F4 is a hybrid function that combines a global optimum with multiple local optima. Its objective is to evaluate how well an algorithm balances local search and global search in complex optimization spaces. SPSO performs exceptionally well on the F4 function, achieving a minimum value close to the global optimum with the lowest standard deviation, indicating its strong capability in solving hybrid optimization problems. For the F4 function, SPSO identifies the optimal solution (fitness value of 0.013754) in approximately 500 iterations. Although the optimal values obtained by STOA and SCA are similar, SPSO requires significantly fewer iterations. For the F4 hybrid function, the global search capability of SPSO proves particularly advantageous, as its dynamic parameter adjustment effectively prevents premature convergence to local optima.

F5 is a relatively simple optimization problem, typically with a single global optimum. This function tests the algorithm's performance in simpler optimization scenarios. SPSO demonstrates exceptional stability on the F5 function, achieving a minimum value close to the global optimum with shorter runtime and higher stability. For the F5 function, SPSO, SCA, and STOA all eventually converge to a fitness value of 0.998. However, SPSO reaches this value in fewer than 100 iterations, while SCA requires approximately 150 iterations, and STOA takes about 700 iterations.

F6 is a near-optimal solution problem, containing solutions close to the global optimum. It evaluates how quickly an algorithm can converge to near-optimal solutions and handle minor errors. SPSO outperforms other algorithms on the F6 function, achieving the smallest minimum value with shorter runtime, indicating its strong capability in solving near-optimal solution problems. For the F6 chaotic characteristic function, SPSO finds the optimal solution in approximately 300 iterations, achieving a fitness value of 0.000307, which outperforms the other algorithms. This function places high demands on randomness and distribution, and SPSO's chaotic sequence initialization strategy significantly enhances the randomness and coverage of the search process.

In testing hybrid functions, SPSO ranks in the top two overall when solving complex functions. In summary, whether facing multi-peak or hybrid functions, SPSO ranks first overall, demonstrating the shortest time, greater stability, and smallest standard deviation.

In this paper, the optimization curve of the test function is plotted based on the number of iterations and the adaptation size, as shown in Figure 7. From the convergence curves of functions F1 to F6, it is evident that SPSO exhibits fast convergence speed and strong optimization capabilities. Specifically, for the F1 function, SPSO reaches the optimal value in approximately 400 iterations, whereas STOA reaches the optimal value in about 800 iterations. For the F5 function, all the algorithms eventually stabilize, but the SPSO algorithm converges faster and achieves better results compared to the other algorithms.