  Gait plan is a procedure that is typically applied on the ground robots,
e.g., quadrupedal robots; the tilt-rotor, a novel type of quadrotor with eight
inputs, is not one of them. While controlling the tilt-rotor relying on
feedback linearization, the tilting angles (inputs) are expected to change
over-intensively, which may not be expected in the application. To help
suppress the intensive change in the tilting angles, a gait plan procedure is
introduced to the tilt-rotor before feedback linearization. The tilting angles
are specified with time in advance by users rather than given by the control
rule. However, based on this scenario, the decoupling matrix in feedback
linearization can be singular for some attitudes, combinations of roll angle
and pitch angle. It hinders the further application of the feedback
linearization. With this concern, Two Color Map Theorem is established to
maximize the acceptable attitude region, where the combinations of roll and
pitch will give an invertible decoupling matrix. That theorem, however,
over-restricts the choice of the tilting angles, which can rule out some
feasible robust gaits. This paper gives the generalized Two Color Map Theorem;
all the robust gaits can be found based on this generalized theorem. The
robustness of three gaits that satisfy this generalized Two Color Map Theorem
(while violating Two Color Map Theorem) are analyzed. The results show that
Generalized Two Color Map Theorem completes the search for the robust gaits for
a tilt-rotor.
