With the continuous development of robotics, parallel robots are increasingly used in military and medical applications due to their compact structure, high stiffness, high load-bearing capacity, better isotropy, and small working space [
1]. FRR-assisted fracture repositioning with a parallel mechanism is a feasible and accurate method of repositioning that is more minimally invasive in operation, reduces the operator’s physical effort in manual pulling and repositioning, and minimizes X-ray radiation exposure to patients and medical personnel [
2,
3]. Accuracy and precision are important performance indicators in the FRR resetting process, which directly affects the success of fracture resetting surgery [
4,
5]. Due to the complex parallel structure of the resetting robot, there are structural errors, such as hinge mounting and assembly [
6,
7], and there are dynamic errors arising from non-linear factors in the resetting process, which can lead to reduced precision in the resetting process of the resetting robot and have a significant impact. This can lead to reduced accuracy during repositioning, which can have a significant impact on the repositioning procedure. Therefore, there is an urgent need to compensate for the errors of the resetting robot, to eliminate or reduce the end position errors of the fracture resetting robot, and to improve the accuracy of the resetting process [
8,
9]. This leads to more accurate fracture repositioning [
10,
11].
Fortunately, a number of methods for compensating for positional errors at the end of a parallel mechanism have attracted the interest of researchers. Firstly, the fracture repositioning robot posture error modelling of the parallel mechanism is mainly performed using the D–H transformation matrix and closed-loop vector method. Lee Sungcheul [
12] used a combination of linear uniform transformation matrix and D–H method to build the error model, but the D–H matrix method is not easy to obtain obvious expressions, so this paper uses the closed-loop vector to complete the error modeling. This paper uses a closed-loop vector to complete the error model. The structural error parameter compensation values of the mechanism are obtained by an optimization algorithm. To date, new intelligent optimization algorithms have been developed to search for optimal solutions, including genetic algorithms, neural networks, particle swarm algorithms [
13], ant colony algorithms [
14], etc. G. Gungor [
15] and others have performed online estimations of errors and compensation of errors based on least squares for parallel robots. However, the least squares algorithm is extremely sensitive to the effects of noise and tends to cause unstable iterative results. Angelidis A and Vosniakos G C. [
16] and Nguyen [
17] used artificial neural nets to measure and compensate for the end-of-execution errors of industrial robots, respectively. According to Wang Ruizhou [
18] the use of the multi-objective particle swarm optimization algorithm was proposed to optimize the parameters and, thus, improve the accuracy of the parallel mechanism motion. In [
19], different pseudo-random number assignment strategies are introduced to study the effect of controlled randomness on the search scheme of the particle swarm optimization algorithm; however, the particle swarm algorithm is prone to premature convergence and was shown not to be globally convergent. Based on this, this paper introduces the whale optimization algorithm [
20]. The whale optimization algorithm has the advantages of a simple process and fast convergence, but the whale optimization algorithm, as a new population intelligence optimization algorithm, still has some shortcomings. In [
21], a chaotic feedback adaptive whale optimization algorithm was proposed for the disadvantage of low accuracy in finding the best complex function optimization problems. In [
22], a chaotic search strategy-based whale optimization algorithm (CWOA) is proposed, which addresses the problem that exploration and exploitation capabilities are difficult to coordinate and easily fall into a local optimum. In [
23], the Lévy flying whale optimization algorithm is used to improve the convergence of the algorithm. An adaptive decision operator-based whale optimization algorithm (IWOA) was proposed in [
24] to improve the convergence speed of the algorithm. Scholars have made many improvements and have now achieved better experimental results, making WOA optimization relatively mature. However, the whale optimization algorithm still does not fully solve the problem of global search capability and convergence speed optimization, and is prone to fall into the problem of local optimality. Based on this [
25], a chaotic whale optimization algorithm incorporating differential evolution was proposed to improve the performance of the algorithm, and, in [
26], a differential evolution (M-WODE) algorithm based on the multi-objective whale optimization algorithm was proposed to ensure the diversity of solutions and enhance the local search ability of M-WODE. In short, the optimized whale optimization algorithm is widely used, but it has not been applied for the end position error compensation of parallel mechanisms, especially for fracture resetting robots with parallel mechanisms.
This paper analyzes the research of recent years and combines different improvement methods. A differential evolution-based Cauchy opposition-based learning whale optimizations algorithm is proposed to compensate for the end-position error of the FRR. This paper analyzes the mechanism error existing in the FRR of the parallel mechanism and establishes its error model, analyzes the influence of the drive rod parameters on the end-position error, and uses the improved whale optimization algorithm (CRLWOA-DE) to find the optimal drive rod parameters to compensate the dynamic platform end-position error of the FRR. Finally, simulation experiments are used to verify the superiority of the algorithm for error compensation in the FRR reset process.