  This article presents the four-dimensional surfaces which instruct the gait
plan for a tilt-rotor. The previous gaits analyzed in the tilt-rotor research
are inspired by animals; no theoretical base backs the robustness of these
gaits. This research deduces the gaits by diminishing the effect of the
attitude of the tilt-rotor for the first time. Four-dimensional gait surfaces
are subsequently found, on which the gaits are expected to be robust to the
attitude. These surfaces provide the region where the gait is suggested to be
planned. However, a discontinuous region may hinder the gait plan process while
utilizing the proposal gait surfaces. A Two Color Map Theorem is then
established to guarantee the continuity of each gait designed. The robustness
of the typical gaits obeying the Two Color Map Theorem and on the gait surface
is demonstrated by comparing the singular curve in attitude with the gaits not
on the gait surface. The result shows that the acceptable attitudes enlarge for
the gaits on the gait surface.
