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In this paper, we form a constrained optimization problem for spherical four-bar motion generation. Instead of using local optimization methods, all critical points are found using homotopy continuation solvers. The complete solution set provides a full view of the optimization landscape and gives the designer more freedom in selecting a mechanism. The motion generation problem admits 61 critical points, of which two must be selected for each four-bar mechanism. We sort solutions by objective value and perform a second order analysis to determine if the solution is a minimum, maximum, or saddle point. We apply our approximate synthesis technique to two applications: a hummingbird wing mechanism and a sea turtle flipper gait. Suitable mechanisms were selected from the respective solution sets and used to build physical prototypes. 

Articulation between body segments of small insects and animals is a three degree-of-freedom (DOF) motion. Implementing this kind of motion in a compact robot is usually not tractable due to limitations in small actuator technologies. In this work, we concede full 3-DOF control and instead select a one degree-of-freedom curve in SO(3) to articulate segments of a caterpillar robot. The curve is approximated with a spherical four-bar, which is synthesized through optimal rigid body guidance. We specify the desired SO(3) motion using discrete task positions, then solve for candidate mechanisms by computing all roots of the stationary conditions using numerical homotopy continuation. A caterpillar robot prototype demonstrates the utility of this approach. This synthesis procedure is also used to design prolegs for the caterpillar robot. Each segment contains two DC motors and a shape memory alloy, which is used for latching and unlatching between segments. The caterpillar robot is capable of walking, steering, object manipulation, body articulation, and climbing.