Legged robots have the potential to provide extreme mobility in highly rugged terrain. Developing such locomotion capabilities within these robots is challenging for several reasons, including their inherent hybrid and nonlinear dynamics. Moreover, uncertainties in the robot's model arising due to external payloads, joint friction, wear and tear, and uncertainties introduced by environmental factors such as changing contact conditions or force perturbations can significantly hinder the performance and reliability of these robots. Developing safe, reliable, and robust feedback controllers is crucial as we begin to deploy such robots into environments with humans. Moreover, the underlying feedback controllers must be able to quickly adapt to rapidly changing environmental factors that can significantly affect their performance.

Recent developments in Reinforcement Learning have shown tremendous success and impressive results on several legged robot platforms to navigate challenging terrain. While these methods can generate very complex behaviors, they are highly sample inefficient as they do not take into account any knowledge of the dynamical structure of the robot. Model-based methods such as Control Lyapunov Functions (CLFs), Control Barrier Functions (CBFs), and Model Predictive Control (MPC), on the other hand, provide us with a set of tools to achieve desired control objectives while remaining within specified constraints for the closed-loop system. The performance of both RL-based control and model-based methods can significantly degrade if an accurate model of the underlying system is not known. In such cases, achieving the best performance will require learning from real-world data. This thesis develops planning algorithms and feedback controls through the lens of model-based techniques to achieve safe, stable, and robust legged locomotion on challenging terrain. In particular, we present a trajectory generation method to achieve aperiodic running with precise foot placement for a 2D bipedal robot model. We then develop a novel coordinate-free geometric MPC for the Cassie biped and validate our approach through several hardware experiments. We apply the developed trajectory generation method and geometric MPC on a quadruped robot to navigate challenging terrain with visual feedback. Using tools from model-based methods such as CLFs and CBFs, we then develop novel reward function shaping methods to achieve safe and robust locomotion policies for bipedal and quadrupedal robots. We show that our method can significantly reduce the sample complexity to learn a stabilizing controller, which allows us to finetune policies directly on hardware using only a few seconds to a few minutes of data.