## 3 Mathematical Models and Control Design
### Mathematical Model  
Problem formulation: The problem of multi-agent collision avoidance is presented, developed, and evaluated in a centralized environment. All remaining members of the group have constant access to each agent's positions. A group of \(N\) WMRs traveling on the  
Figure 3: Overview of the collision avoidance and object detection algorithm. ground (the Euclidean plane) is taken into consideration; they may be distinguished by the existence of (static) impediments. Specifically, a model resembling a unicycle with a passive wheel describes each WMR [1; 21]. \[\left\{ \begin{matrix}\dot{X}_{i}=\cos(\theta_{i})v_{i}\\ Y_{i}=sin(\theta_{i})v_{i}\\ \dot{\theta}_{i}=\omega_{i}\end{matrix} \right. \tag{1}\]  
where \((X_{i},\ Y_{i})\in R^{2}\) is the location of the middle point between the two activated wheels on the Euclidean plane, along the axle that connects them, \(i=1,\ldots,N\); \(\theta_{i}\) is the orientation of the ith robot; \(v_{i}\) is longitudinal velocity, and \(\omega_{i}\) is angular velocity. \(\left(X_{i}^{*},\ Y_{i}^{*}\right)\in R^{2}\). For every agent, find the control inputs for its center of mass, \(v_{i}\), and \(\omega_{i}\) in such a way that the \(i^{th}\) agent is guided to the intended location while avoiding collisions with both static barriers and other team members (Figure 4). Take into consideration the scenario where single-integrator dynamics describes each virtual [1] agent's motion. \[\dot{x}_{i}=u_{i} \tag{2}\]  
The boundary of the region in the multi-agent collision avoidance problem for WMRs is a fundamental concept that defines the safe operational space for the robots. It ensures that the WMRs maintain a minimum distance from obstacles and other agents, thereby avoiding collisions. The boundary is enforced using control barrier functions (CBFs), which guarantee forward invariance of the safe set. This approach is particularly important in dynamic and unpredictable environments, such as agricultural fields, where collision  
Figure 4: Diagram of the WMR. avoidance is critical for the safe and efficient operation of WMRs. Therefore, the boundary of the region would be  
\[\partial P_{j}=\left\{x\in R^{2}:\left\|x-p_{j}\right\|^{2}E_{j}-p_{j}^{2}=0\right\} \tag{3}\]  
where \(p_{j}>0\), and \(E_{j}=E_{j}^{T}>0\). Initial deployment without obstacle collisions: the agents' starting locations satisfy  
\[\left\|x_{i}(0)-x_{f}(0)\right\|>r_{i}+r_{j} \tag{4}\]  
Initial deployment of agents without collisions: the agents' starting locations satisfy  
\[\left\|x_{i}^{*}-p_{j}^{c}\right\|^{2}>\left(r_{i}+\overline{p}_{j}(\varnothing ^{*})\right) \tag{5}\]  
Desired deployment without obstacles colliding: the agents' intended positions fulfill  
\[\left\|x_{i}^{*}-x_{j}^{*}\right\|>r_{i}+r_{j} \tag{6}\]  
Desired deployment of agents without collisions: each agent's target placements meet  
\[l_{i}\cap\left(\cup_{j=1,...,m}\partial p_{j}\right)=\varnothing \tag{7}\]  
A multi-agent system with \(N>1\) WMRs presents significant challenges, including scalability, coordination, and adaptability to dynamic environments. By leveraging control barrier functions (CBFs) and control Lyapunov functions (CLFs), the system can achieve collision-free navigation while ensuring stability and goal convergence. Consequently, take into consideration a multi-agent system with dynamics that consists of \(N>1\).