Robots that are designed for NASA's missions to foreign planets, as with disaster relief efforts on earth, face tough challenges with harsh environments and locomotion over extreme terrain. Soft-bodied robots could address many of these challenges by conforming and adapting to their environments. This dissertation presents the modeling, design, and control of a set of soft and flexible robotic spines that assist quadruped robots in their locomotion over extreme terrain. These spines are tensegrity systems, consisting of rigid bodies held together in a network of flexible cables, used as a practical method of producing soft behavior.

First, the geometry and possible movements of these spines are discussed. Next, the inverse statics problem is solved for these spines, in order to calculate the tensions of the cables which control the spine's vertebrae. The resulting inverse statics optimization algorithm is tested in a hardware experiment, demonstrating pseudo-static open-loop positioning of the spine. Using this model, a design of a quadruped robot with a tensegrity spine is proposed and prototyped. Simulations show that this quadruped robot's tensegrity spine can lift and position its feet, as a way to assist with locomotion and balance. Hardware experiments validate the simulation's motions of the robot's spine and feet.

Then, control systems are investigated for these tensegrity spines. A set of closed-loop controllers, which use model-predictive control (MPC) in combination with the inverse statics algorithm, are proposed and simulated against dynamics models of the spine. Two different MPC formulations are used, both of which show low-error tracking in simulation.

Finally, given the ongoing challenges with MPC, an energy-based stability criterion is derived for a class of high-dimensional, nonlinear, possibly hybrid robotic systems.

These systems, termed `statically conservative', include networks of cables in tension, similar to tensegrity spines. The stability criterion is applied to these cable networks, giving conditions for stabilizing controllers. An example controller is proposed for a cable-driven robot with slack cables. Simulations of this system and its controller to validate the stability proof. These control approaches show promise for future hardware implementation of walking locomotion in quadruped robots with tensegrity spines.