  We rely on Nagumo's invariance theorem to develop a new approach for
navigation in unknown environments of arbitrary dimension. The idea consists in
projecting the nominal velocities (that would drive the robot to the target in
the absence of obstacles) onto Bouligand's tangent cones (referred to as the
safety velocity cones) when the robot is close to the boundary of the free
space. The proposed projection-based controller is explicitly constructed to
guarantee safety and convergence to a set of Lebesgue measure zero that
contains the target. For specific free spaces such as Euclidean sphere worlds,
the convergence to the target is guaranteed from almost all initial conditions
in the free space. We provide a version of the controller (generating a
continuous and piecewise smooth closed-loop vector field) relying on the
robot's current position and local range measurements (e.g., from LiDAR or
stereo vision) without global prior knowledge about the environment.
