The collision avoidance and the controllability for $n$ bodies in dimension one

We present a method of design of control systems for $n$ bodies in the real line ${\mathbb{R}}^1$ and on the unit circle $S^1$, to be collision-free and controllable. The problem reduces to designing a control-affine system in ${\mathbb{R}}^n$ and in $n$-torus $T^n,$ respectively, that avoids certain obstacles. We prove the controllability of the system by showing that the vector fields that define the control-affine system, together with their brackets of first order, span the whole tangent space of the state space, and then by applying the Rashevsky-Chow theorem.