Robustness multi-objective optimization for parallel robot based on subregional meta-heuristic iteration

Parallel robots work dependably in intelligent manufacturing, logistics industry, medical engineering and numerous other fields rely on their high precision and reliability. It also has the advantage of small size which makes them well-suited for space constrained applications. Nevertheless, these are related to the structural parameters of the mechanism, and the optimized structural design is a permanent challenge.

Several existing researches focus on designing innovative structures. Riabtsev et al. (2022) present a 2-DOF active lockable joint. Ye et al. (2020) developed a 1R1T parallel mechanism as a remote center of motion mechanism. Others focus on the optimization of classical structures. Dastjerdi et al. (2020) explored how to obtain the smallest robot structural design by using dimensionally integrated analysis for parallel robots, ensuring the specified workspace was met. Quintero-Riaza et al. (2019) studied the optimal size design method for planar parallel robots, which enables the best dexterity index, force transfer efficiency and stiffness of the robots.

Comprehensive mathematical modeling is fundamental for structural optimization. Altuzarra et al. (2023) analyzed the kinematics of a three-degree-of-freedom (3-DOF) planar parallel continuum mechanism and developed a procedure for solving the fully inverse and forward kinematics of a planar 3-DOF system. The application of enhanced optimisation algorithms can enhance the efficiency of optimisation processes. Laribi et al. (2007) developed an optimal dimensional synthesis method of the Delta parallel mechanism for a prescribed workspace using a genetic algorithm-based method.

Most of the structural optimization design methods are empirical and obsolete. These methods rely heavily on multiple experiments and trials, which can lead to inaccuracies or inefficiencies. Based on the previous work (Tao et al., 2024; Xu et al., 2023), a complete mathematical description of the classical structure is developed and the specific work situation is taken into account, which further completes the professional robust multi-objective optimization via an improved meta-heuristic intelligent optimization algorithm.

Figure 1(a) demonstrates a logistics sorting parallel robot working on a conveyor line. It is constructed as a delta-type parallel mechanism, which has 3 degrees of freedom (3-DOF) for movement in the XYZ direction, and the schematic is shown in Figure 1(b), where $R$ is the radius of the fixed platform with center $O$⁠, $r$ is the radius of the actuator with center $O′$⁠, $L$ is the length of the drive arm, $l$ is the length of the slave arm, and $q$ is the drive angle.

There are already numerous contributions from scholars for mathematical modeling of the delta parallel mechanism (Altuzarra et al., 2023), and some of our previous works have advanced the theoretical study of it (Tao et al., 2024; Xu et al., 2023). To summarize, we will employ kinematic models, performance parameter models and error models.

The kinematic model includes forward, inverse and velocity models. The forward model $ffwd$ calculates the actuator position $O′$ of the mechanism based on the drive angle $q$⁠, and the inverse model $fivs$ solves for the drive angle $q$ based on the end position $X$⁠.

The velocity model calculates the mechanism actuator velocity $X˙$ based on the drive angular velocity $q˙$ with a transfer matrix called the Jacobian matrix $J$⁠.

Error models may be classified as absolute error models or probability error models.

where $ein$ is the input absolute error, $eout$ is the output absolute error, $σin$ is the standard deviation of the input error, $σout$ is the standard deviation of the output error, $Je$ is the absolute error transfer matrix, and $Jσ$ is the probability error transfer matrix.

The performance parameters models encompass the response speed, bearing capacity and stiffness of the mechanism, which are all related to the Jacobi matrix.

where $σmax()$ represents computing the maximum singular value, $σmin()$ represents computing the minimum singular value, and $T$ is the matrix transpose symbol.

Various optimization algorithms have been proposed with the objective of facilitating the rapid and optimal design of prototypes. This is essential for the diffusion and application of novel mechanical structures, by which the design parameters of mechanical structures are iterated in an accelerated manner. It is also possible to reduce the cost of testing and production, the former being achieved by numerical simulation techniques, and the latter using cost as one of the considerations for the optimization objective.

Currently, meta-heuristic optimization algorithms are the subject of extensive study due to their problem-free dependency and their ability to solve non-convex problems effectively. The whole optimization search process can be generally divided into three steps: population initialization, meta-heuristic optimization strategy and population update strategy.

Population initialization provides a set of initial random solutions within the specified boundaries. In contrast to the use of simple random number generators, chaotic mapping techniques are capable of generating initial solutions that exhibit a more extensive distribution. We have employed the Cubic chaotic mapping with good chaotic properties with the expression:

where $xn$ is a random value ranging from (0, 1), $xn+1$ is a chaotic value, and $α$ is a control parameter.

After completing the initialization, it is necessary to optimize these solutions by executing the selected meta-heuristic optimization strategy on the generated populations. Ultimately, the fitness of the original and updated solutions must be evaluated in order to determine which solutions should be retained. When faced with multi-objective optimization problems, we focus on finding the Pareto frontier. The non-dominated ordering update strategy is capable of computing the non-dominated level of hierarchical relationships among the individuals of the population, thereby obtaining the Pareto-optimal solution set under the multi-objective condition. When there are an excessive number of population individuals in the highest dominance level, the crowding distance is further computed in order to remove some of the overly concentrated solutions, thus broadening the Pareto front.

where $Dcrowd$ is the crowding distance, $Xi+1$ and $Xi−1$ are the two solutions adjacent to $Xi$⁠, $Obji$ denotes the $i$-th optimization objective, and $m$ is the total number of optimization objectives.

For the parallel robot used in logistics sorting, its optimization problem can be described as:

where $X$ represents the design variable to be optimized. $R4$ indicates that the viable domain of the design variable is a 4-dimensional solution space. $e(X)$ means to optimize the error. $k(X),−Fmax(X)$ and $Dmax(X)$ mean to optimize the performance parameters. $Xlb$ and $Xub$ are the lower and upper bounds of the design variable, respectively.

In particular, depending on our application, it is preferable to limit the optimized objectives in accordance with the respective trajectory segments. We restrict the optimization of $e(X)$ and $k(X)$ to smoothly moving trajectory segments, and the optimization of $−Fmax(X)$ and $Dmax(X)$ to trajectory segments with large acceleration, as shown in Figure 2.

Settings $Xlb=[80,60,110,140]$⁠, $Xub=[100,70,130,160]$⁠. Table 1 illustrates the candidate design parameters of the mechanism after SMI optimization and their corresponding fitness. After verifying with simulation analysis and considering the structural rationality, the third set of candidate parameters was selected as the final optimization outcome. Figure 3(a) displays the error performance of the mechanism after SMI optimization over the full route of operation. It is evident that the error of the actuator after optimization is suppressed over the full path. The reduction is more significant in the acceleration route segment, which is one of the optimization objectives. The maximum reduction ratio in the path achieves 6.81% and the average error over the full path is reduced by 1.46%. Figure 3(b) illustrates the average change rate of performance parameters after SMI optimization. It can be observed that $k$ of the mechanism becomes smaller, $Fmax$ increases and $Dmax$ decreases. This is reflected in practice resulting in a more flexible mechanism with greater load and stiffness. The optimized mechanism is more robust which broadens the range of applications.

• (1)

  A rapid and robust design method for the optimization of parallel mechanisms

A rapid and robust optimal design method for parallel mechanisms based on subregional meta-heuristic iteration (SMI) is intended to address the issues associated with the highly intricate structural design of parallel robots, which is heavily empirically dependent and inefficient due to a large number of physical experiments.

• (2)

  Improved meta-heuristics iterative strategy to accelerate the search for the global optimum

An improved meta-heuristic iterative strategy is proposed for the optimization of design parameters of parallel mechanisms. The initialization strategy is enhanced through the incorporation of chaotic mapping, thereby augmenting the diversity of initial solutions, which facilitates the rapid arrival at the global optimum and avoids local optimums. Meta-heuristics is employed to generate a novel generation of solutions, and through the implementation of a tailored policy, the algorithm is endowed with the capacity to evade local optimum. Non-dominated sorting is utilized as the update strategy, enabling the algorithm to adeptly address multi-objective optimization problems.

• (3)

  Optimization by constructing a subregional multiple optimization objective function

For the case of a logistics sorting parallel robot, a subregional multiple optimization objective function is constructed by analyzing its motion trajectory, which makes the optimization objective more directional and targeted. The actuator error is reduced by 6.81% at most and 1.46% on average across the entire path after proposed SMI optimization. Furthermore, the flexibility, bearing capacity, and stiffness performance of the mechanism are improved by 63.83%, 43.98%, and 97.51%, respectively, in comparison with the pre-optimisation stage.

In the future, more work will be carried out, including the development of more efficient meta-heuristics, the adaptive segmentation of the region for the optimization and the incorporation of additional mathematical models, such as the energy consumption and the dynamic vibration, among others.

The work presented in this article is funded by the China National Key Research and Development Project (2022YFB3303303) and the China State Key Laboratory of Mechanical Transmission Key Open Fund (SKLMT-ZDKFKT-202202).