The ability to control compliance of robotic joints is desirable because the resulting robotic mechanisms can adapt to varying task requirements and can take advantage of natural limb and joint dynamics. The implementation of controllable compliance in robots, however, is often constrained by the inherent instability of active compliance methods and by the limited availability of the custom, nonlinear springs needed by passive compliance methods. This work overcomes a major limitation of passive compliance by producing designs for two novel mechanisms capable of generating a wide variety of specifiable, nonlinear elastic relationships. One of these designs is physically implemented as a quadratic “spring” and is used to create a passively compliant robot joint with series-elastic actuation. A simple feed-forward algorithm is then experimentally shown to be sufficient to control independently and simultaneously both joint angle and joint compliance, regardless of the presence of external forces on the joint. We believe that this is the first physically constructed system to use antagonistic quadratic springs to successfully demonstrate open-loop, independent, and simultaneous control of both joint angle and joint stiffness. Because this approach better emulates the underlying joint mechanics used by animals, it may improve both the quality and variety of robotic movements.