  In this article, we give a condition for the global controllability of affine
nonlinear control systems with drifts on Euclidean spaces. Under regularity
assumptions, the condition is necessary and sufficient in the codimension-1 and
codimension-2 cases, and holds for systems of higher codimensions under mild
restrictions. We then investigate motion planning problems for codimension-1
affine systems, and give proof of the global existence of the lift to control
curves for certain drifted systems using the homotopy continuation method.
