The basic structural goal in soft robotics is to create robots which are controllably stiff. Fluidic elastomer actuator (FEA) soft robots are currently the most successful type of controllably stiff robot because they are fast and inexpensive. It is not possible to control their stiffness separately from their motion, however, and in untethered applications either speed or battery life is severely sacrificed. To improve soft robots, it helps to better understand the mechanics of the natural muscular hydrostats which inspired their development. Muscle is about 80% water. A muscular hydrostat and a balloon have structure for basically the same reason: the cell membranes and muscle fibers (analogous to membrane of a balloon) bear tension, and the fluid in the muscle cells (analogous to the air in a balloon) bears compression. Muscular hydrostats are approximately tensegrity structures. By definition, tensegrity structures contain only single force elements, meaning that the materials can be optimized to the type of load they bear. The stiffness and shape of the structures can be changed separately via the loading of the elements. I present fluid-membrane tensegrity structures made with the following design principles: (1) actuation is caused by tendons driven by electric motors; (2) both shape change and motion rely primarily on buckling rather than stretching; (3) the total mass of fluid in the system stays constant. This “tensegristat” design will reduce unnecessary material deformation and allow better untethered speed and efficiency, stiffness modulation separate from motion, and large-scale shape changes. While there have been a few robots that use a combination of tendon and pneumatic drive, almost all rely on external air compressors, and I have not found any research on tendon-fluid combinations in which untethered, mobile applications were explored. My work addresses this gap. Here I first present an intuitive taxonomy of ways pressure affects stiffness of inflated structures. Then I show the possibility of using tensegristat designs for large scale shape change, illustrated with a morphing amphibian robot. Thirdly I show the possibility of using energy storing fluids in tensegristat robots, and finally I present a way to 3D print durable tendons into soft robots.