  Many robotic systems locomote using gaits - periodic changes of internal
shape, whose mechanical interaction with the robot's environment generate
characteristic net displacements. Prominent examples with two shape variables
are the low Reynolds number 3-link "Purcell swimmer" with inputs of 2 joint
angles and the "ideal fluid" swimmer. Gait analysis of these systems allows for
intelligent decisions to be made about the swimmer's locomotive properties,
increasing the potential for robotic autonomy. In this work, we present
comparative analysis of gait optimization using two different methods. The
first method is variational approach of "Pontryagin's maximum principle" (PMP)
from optimal control theory. We apply PMP for several variants of 3-link
swimmers, with and without incorporation of bounds on joint angles. The second
method is differential-geometric analysis of the gaits based on curvature
(total Lie bracket) of the local connection for 3-link swimmers. Using
optimized body-motion coordinates, contour plots of the curvature in shape
space give visualization that enables identifying distance-optimal gaits as
zero level sets. Combining and comparing results of the two methods enables
better understanding of changes in existence, shape and topology of
distance-optimal gait trajectories, depending on the swimmers' parameters.
