  Spring-driven jumping robots use an energised spring for propulsion, while
the onboard motor only serves as a spring-charging source. A common mechanism
in designing these robots is the rhomboidal linkage, which has been combined
with linear springs (spring-linkage) to create a nonlinear spring, thereby
increasing elastic energy storage and jump height for a given motor force. The
effectiveness of this spring-linkage has been examined for individual designs,
but a generalised design theory of this class of system remains absent. This
paper presents an energetics analysis of the spring-linkage and provides
insight into designing an ideal constant force spring, which stores the maximum
energy for a given motor force. A quasi-static analysis shows that the
force-displacement relationship of the spring-linkage changes with the
orientation and type of the spring, but is independent of the linkage scale.
Combining different types and orientations of springs within the linkage
enables higher elastic energy storage than using single springs. Placing two
translational springs at the diagonals of the rhomboidal linkage creates an
ideal spring that could increase the jump height of prior robots by 50-160%.
