## 5 Optimal Collision-Avoidance Trajectory Planning Based on dsRNA Splicing Evolutionary Algorithm  
### Fixed-Time Parameter Optimization of dsRNA Splicing Evolutionary Algorithm  
In recent years, many researchers have paid attention to biological evolutionary algorithms because of their conformity to the laws of nature and physics [1; 3; 4; 11; 19]. Parameter optimization has always been an important direction of its application, but the existing evolutionary algorithms often find it difficult to get the global optimal value, which is unfavorable to its practical application. In our opinion, the parameter optimization of trajectory planning for robot arm can be solved in two aspects: (1) the evolutionary algorithm, (2) the fitness objective function. By observing the evolution of dsRNA viruses, an evolutionary algorithm simulating dsRNA replication variation based on gene splicing strategy is designed to obtain semi-globally optimal parameter \(m\) of fixed-time stability system. The steps of the dsRNA splicing evolutionary algorithm are as follows:  
(a). \(N\) initial (\(-\))RNA groups are established, and the single-stranded RNA(ssRNA) includes genetic information. (b). According to genetic central dogma, (\(+\))RNA is synthesized using (\(-\))RNA as a template, then, initial dsRNA virus groups are synthesized. Here, we assume that the fitness function is expressed by the traits of (\(+\))RNA and (\(-\))RNA, respectively. The parameter \(m\) make up the gene fragment, and then \(k\) parameters make up the effective genetic fragment of the (\(-\))RNA. Notably, due to the high variability of the virus, the synthesis process of (\(+\))RNA may produce errors and the mutation probability is set to \(P_{m}\). (c). Complementary (\(-\))RNA and (\(+\))RNA are taken as a group, and the optimal fragment is observed by fitness function. When fitness function is effective, the special part effective genetic fragment of (\(-\))RNA is removed and the rest are connected by ligase. Note that this removed fragment needs to be determined according to the coding rules of optimal parameters, so that the spliced effective sequence of (\(-\))RNA reflects the expression character of the optimal fragment. This process imitates the technology of RNA splicing, the probability of success is \(P_{s}\), the (\(-\))RNA that failed to splice is assumed to be alive with an unknown gene sequence. Again, there are mutations involved in (\(-\))RNA synthesis process if the fitness function is invalid. (d). Repeating steps (c) \(\rightarrow\) (b) M times to evolve the optimal ssRNA of the current generation, and then we find out the optimal parameter set among the ssRNA in that generation. Furthermore, the virus is assumed to die after one generation of reproduction, with a 100 percent chance of surviving each generation. (e). The individual fitness function is used to find the optimal fragment of the optimal ssRNA among the end generation. For aspect (1), the above-mentioned optimization algorithm based on dsRNA virus gene splicing evolution strategy can search for the complementary regions of parameter \(m\), so its optimization range is semi-global. Meanwhile, the high variation and adaptability of the virus also provide the natural law basis for the proposed evolution algorithm. For aspect (2), the traditional optimization of the system using evolutionary algorithms alone is limited, so an essential improvement of the objection function is considered. Firstly, according to Theorem 1, an objection function can be designed without considering pose error of end-effector and base as the following:  
\[B_{1}=w_{1}\min_{i=1}^{k}(\max_{t=0}^{t=t_{f}}(\hat{x}_{i}))+w_{2}\min_{i=1}^{ k}(sum(sat_{1}(\hat{x}_{i}))), \tag{15}\]  
where \(w_{1,2}\) is the weight parameter and \(sat(\hat{x}_{i})\) is with (16)\[sat(\dot{x}_{i})=\left\{ \begin{array}{ll}0,&|\dot{x}_{i}|>\chi_{1}\\ \dot{x}_{i},&|\dot{x}_{i}|\leq\chi_{1}\end{array} \right., \tag{16}\]  
in which \(\chi_{1}\) is a parameter related the chattering of input \(\hat{x}_{i}\).