Algorithm 1 Guarding-Circle Collision Avoidance.

Require:  Set of robots $\mathcal{R}=\{1,\dots ,N\}$;   1:Latest pose vector ${\mathbf{p}}_i=(x_i,y_i,{\theta }_i,v_i)$ for each $i\in \mathcal{R}$;   2:Guarding circle radius $r_g$; Ensure:  Control command set $\mathcal{U}=\{u_1,\dots ,u_N\}$;      Stage 0 - Offline Configuration   3:Choose $r_g$ based on worst-case stopping distance, robot dimensions, and safety regulations.      Stage 1 - Digital Twin Update   4:Receive feature packet from the edge observer and update the state of the digital twins ${\widehat{\mathbf{p}}}_i$ for all $i\in \mathcal{R}$.      Stage 2 - Overlap Detection   5:Initialize $\mathcal{A}\leftarrow \mathsf{\varnothing }$                                                  ▹ Set of potentially colliding pairs   6:for all $i\in \mathcal{R}$ do   7:    for all $j\in \mathcal{R}$ with $j>i$ do   8:        Calculate distance $d_{ij}\leftarrow \sqrt{{(x_i-x_j)}^2+{(y_i-y_j)}^2}$   9:        if $d_{ij}\le 2r_g$ then 10:           Add pair to conflict list: $\mathcal{A}\leftarrow \mathcal{A}\cup \left\{\right(i,j\left)\right\}$ 11:        end if 12:    end for 13:end for      Stage 3 - Conflict Resolution 14:Initialize all commands: $u_k\leftarrow \mathtt{Proceed}$ for all $k\in \mathcal{R}$. 15:for all $(i,j)\in \mathcal{A}$ do 16:    if $v_i>v_j$ then                                   ▹ Example priority rule: faster robot yields 17:        $u_i\leftarrow$ DecelerateOrStop 18:    else 19:        $u_j\leftarrow$ DecelerateOrStop 20:    end if 21:end for      Stage 4 - Actuation 22:Transmit the final control command set $\mathcal{U}$ to the physical robots.