  The robotic locomotion community is interested in optimal gaits for control.
Based on the optimization criterion, however, there could be a number of
possible optimal gaits. For example, the optimal gait for maximizing
displacement with respect to cost is quite different from the maximum
displacement optimal gait. Beyond these two general optimal gaits, we believe
that the optimal gait should deal with various situations for high-resolution
of motion planning, e.g., steering the robot or moving in "baby steps." As the
step size or steering ratio increases or decreases, the optimal gaits will
slightly vary by the geometric relationship and they will form the families of
gaits. In this paper, we explored the geometrical framework across these
optimal gaits having different step sizes in the family via the Lagrange
multiplier method. Based on the structure, we suggest an optimal locus
generator that solves all related optimal gaits in the family instead of
optimizing each gait respectively. By applying the optimal locus generator to
two simplified swimmers in drag-dominated environments, we verify the behavior
of the optimal locus generator.
