Vehicle model

Figure 2: a) An agent's vehicle. b) The agent's windshield, or field-of-view

Consider a set of $n$ agents, in which agent $i$ is a point vehicle located at $p_i = (p_{i1}, p_{i2})$ in the plane with orientation $\psi_i$ (see Fig. 2(a)). Each vehicle moves as a Dubins car:

$$\dot p_{i1} = v_i\cos\psi_i, \quad \dot p_{i2} = v_i\sin\psi_i, \quad \dot \psi_i = u_i,$$ (1)

in which $v_i$ is the forward speed, and the control is $u_i \in \{-\omega_i, 0, \omega_i\}$ for some fixed $\omega_i > 0$ . For fixed $v_i$ , such a vehicle either moves along a straight line ($u_i = 0$ ) or turns clockwise/counterclockwise ( $u_i = \pm \omega_i$ ) along a circle with fixed radius. We use this fact later without further elaboration. Let $X = SE(2)^n$ denote the state space, in which $x \in X$ yields the position and orientation of all agents. Some of our results require that the agents are identical, which means $v_i = v_j$ and $\omega_i = \omega_j$ for all pairs, $i,j$ , of agents.