  We present a method of design of control systems for $n$ bodies in the real
line $\Bbb R^1$ and on the unit circle $ S^1$, to be collision-free and
controllable. The problem reduces to designing a control-affine system in $\Bbb
R^n$ 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.
