## 5 Conclusions  
Based on existing published documentation and regular literature, lots of research results with passable optimal performances, and very complicated control methodologies to neural networks, feedback linearization, sliding mode control, and backstepping methodologies for the trajectory tracking subject of a swarm of WMRs have been investigated. For enhancing these improvable control properties, a nonlinear \(H_{2}\) control method that is analytically inferred is a global solution without any approximations. In this paper, the nonlinear trajectory tracking subject of a swarm of WMRs is completely constructed to promote the tracking capability and reduce the cost of the effort of design, implementation, and computational consumption. Especially, it is actually a simple and easy-to-implement control method. Incidentally, it is challenging to find out this control method due to a solution of one nonlinear time-varying differential equation that needs to satisfy the \(H_{2}\) performance index. Fortunately, an \(H_{2}\) closed-form solution of this nonlinear trajectory tracking subject can be inferred directly by applying to an appropriate mathematical conversion method. According to the results of the simulation and practical experiment, this nonlinear \(H_{2}\) control method not only fulfills the simulation results for tracking the predefined trajectories in a circular and S-type scenario, but also achieves perfect and engaged trajectory tracking performance in the real world. Finally, we can also assert that this nonlinear \(H_{2}\) control method gives some outstanding benefits when a swarm of WMRs performs trajectory tracking missions, such as security, transportation, patrol, and so forth.  
Conceptualization, Y.-H.C.; methodology, Y.-H.C.; software, Y.-H.C.; formal analysis, Y.-H.C.; investigation, Y.-H.C.; writing--original draft preparation, Y.-H.C.; writing--review and editing, S.-J.L.; supervision, S.-J.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the Ministry of Science and Technology of Taiwan, grant number MOST 107-2218-E-020-001. The authors declare no conflict of interest. The authors declare no conflict of interest.  
Appendix A Proof of \(H_{2}\) performance index with nonlinear time-varying Riccati-like equation.  
Figure 19: Results of S-type trajectory tracking torques concerning initial point (\(x_{c}=0\) m, \(y_{c}=-1.9\) m).  
**Proof.** First, consider the cost function \(Y_{2}(u_{i2})\). It was evident that Equation (7) can be rewritten as:  
\[\begin{split} Y_{2}(u_{i2})=\min_{u_{i2}}\sum\limits_{i=1}^{n} \Big{[}e_{i2}{}^{T}(0)J_{i2}(e_{i2}(0),0)e_{i2}(0)+\int_{0}^{t_{f}}\!\!\Big{[}e _{i}{}^{T}(t)\Big{(}j_{i2}(e,t)+J_{i2}(e,t)W_{i2}(e,t)\\ +W_{i2}^{T}(e,t)J_{i2}(e,t)+Q_{i2}\Big{)}e_{i2}(t)+u_{i2}^{T}(t)R_{i2}u_{i 2}(t)+u_{i2}^{T}(t)\Delta_{i2}^{T}(e,t)J_{i2}(e,t)e_{i2}(t)\\ +e_{i2}^{T}(t)J_{i2}(e,t)\Delta_{i2}(e,t)u_{i2}(t)\Big{]}dt\end{split} \tag{12}\]  
By some manipulations and using Riccati-like Equation (12), we obtain:  
\[\begin{split} Y_{2}(u_{i2},0)=\min_{u_{i2}}\sum\limits_{i=1}^{n} \Big{[}e_{i2}{}^{T}(0)J_{i2}(e_{i2}(0),0)e_{i2}(0)+\int_{0}^{t_{f}}\!\!\Big{[} \Big{(}u_{i2}(t)+R_{i2}^{-1}\Delta_{i2}^{T}(e,t)J_{i2}(e,t)e_{i2}(t)\Big{)}^{T }\\ R_{i2}\Big{(}u_{i2}(t)+R_{i2}^{-1}\Delta_{i2}^{T}(e,t)J_{i2}(e,t )e_{i2}(t)\Big{)}\Big{]}dt\end{split} \tag{13}\]  
Then, by choosing control law like that in Equation (8), we can conclude that:  
\[\begin{split} Y_{2}(u_{i2}^{*}(t))=\min_{u_{i2}}\sum\limits_{i=1} ^{n}J_{i2}(u_{i2})=\min_{u_{i2}}\sum\limits_{i=1}^{n}\Big{[}e_{i2}{}^{T}(0)J_ {i2}(e_{i2}(0),0)e_{i2}(0)\Big{]}\end{split} \tag{14}\]  
This is the \(H_{2}\) control performance index in Equation (7). \(\Box\)