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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 eightinputs, is not one of them. While controlling the tilt-rotor relying onfeedback linearization, the tilting angles (inputs) are expected to changeover-intensively, which may not be expected in the application. To helpsuppress the intensive change in the tilting angles, a gait plan procedure isintroduced to the tilt-rotor before feedback linearization. The tilting anglesare specified with time in advance by users rather than given by the controlrule. However, based on this scenario, the decoupling matrix in feedbacklinearization can be singular for some attitudes, combinations of roll angleand pitch angle. It hinders the further application of the feedbacklinearization. With this concern, Two Color Map Theorem is established tomaximize the acceptable attitude region, where the combinations of roll andpitch will give an invertible decoupling matrix. That theorem, however,over-restricts the choice of the tilting angles, which can rule out somefeasible robust gaits. This paper gives the generalized Two Color Map Theorem;all the robust gaits can be found based on this generalized theorem. Therobustness of three gaits that satisfy this generalized Two Color Map Theorem(while violating Two Color Map Theorem) are analyzed. The results show thatGeneralized Two Color Map Theorem completes the search for the robust gaits fora tilt-rotor.

Generalized Two Color Map Theorem -- Complete Theorem of Robust Gait Plan for a Tilt-rotor