Recent work in human-robot interaction has revealed the need for compliant, human-friendly devices. One such device, known as the MARIONET, is a cable-driven single joint actuator with the intended applications of physical rehabilitation and assistive devices. In this work, the stability of the nonlinear system is determined in regards to its equilibria in a wide variety of configurations. In certain configurations, the canonical version of this mechanism experiences an interesting mathematical behavior known as "catastrophes". This behavior may be disadvantageous toward control or even safety. Several cases are thoroughly investigated, two cases where each of two degrees of freedom loses control, and the final case explores the use of a mechanical advantage such as a block and tackle. The study concludes that for a range of design options, the MARIONET does not suffer from any catastrophes. However, the unique behaviors such as a unidirectional bifurcation produced by certain configurations may have use outside of our objectives, perhaps as a type of switch or valve.