## 1 Introduction  
Over the past few decades, scientific and technological progression and innovation have led to the universal application of wheeled mobile robots (WMRs) in daily life. These WMRs with more precise motion capability and powerful controllers are applied to inspection, security, and transportation, etc. According to the above, the accurate motion control of WMRs is more and more critical in the robotics industry, and many researchers have also stepped into this attractive topic in recent decades [1; 2; 3; 4; 5; 6]. The real-time tracking control design of WMRs is always an essential subject for the controller with a straightforward implementation configuration. From the existing studies, it is still an open challenge question to find out how to effectively develop a control design of WMRs to precisely track the desired trajectories in robotics. The trajectory tracking design of the WMRs system has the capability of handling the tracking error between the actual path of the controlled WMRs and the desired trajectory to converge to approach zero under the effects of slippage, disturbances, and measurement noises. Therefore, the WMRs manipulating precisely is more and more critical in the trajectory tracking topic under external disturbance effects. By the survey of existing literature, many research results have worked on tracking control of WMRs, which focused on four categorizations: (1) sliding mode control [7; 8; 9; 10; 11; 12]; (2) feedback linearization [13; 14]; (3) backstepping [15; 16; 17],; and (4) neural networks and Fuzzy approaches [18; 19; 20; 21; 22; 23]. Other designs have also stated the combination of the fuzzy theory orsliding mode and neural network control, as presented in [19; 21; 23]. According to efficient experiences, it is challenging to implement low-cost microchips with computational consumption based on this kind of control algorithm methodology and very complicated theory structure.  
Base on the above reasons, a progressive nonlinear control method, which provides an easy hardware implementation and a high-performance trajectory tracking for a swarm of WMRs, will be proposed in this recommended research. For achieving the objectives of easy hardware implementation and the lowest consumption in the computational calculation, a revolutionary nonlinear control method was inferred for the trajectory tracking problem of a swarm of WMRs. We try to solve a nonlinear and complex time-varying Riccati-like equation directly [24; 25; 26], and this is equal to solving an \(H_{2}\) closed-form solution for the trajectory tracking problem of a swarm of WMRs based on an optimal performance index. This \(H_{2}\) closed-form solution is also a key and primary contribution to this research. Considering the above statement, the subject of a nonlinear optimal control design must solve one nonlinear time-varying Riccati-like equation, which is very difficult to explain, and it is difficult to find out the solution. Fortunately, this closed-form solution can be derived from the appropriately selected state variable transformation and tracking error dynamics analysis in this research. According to this closed-form solution, one nonlinear optimal control method contains a straightforward implementation structure for the trajectory tracking problem of a swarm of WMRs that can be constructed. This paper presents the following sections: The introduction and literature review will be stated in Section 1; the mathematical model and tracking error dynamics of a swarm of WMRs will be described in Section 2; problem formulation and the optimal controller design objective with a closed-form solution for a swarm of WMRs trajectory tracking will be introduced in Section 3; and the simulation performance verifications of the trajectory tracking ability of a swarm of WMRs by the proposed method are demonstrated in Section 4. Finally, the conclusions are summarized in Section 5.