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.

Control barrier functions (CBFs) are one of the many used approaches for achieving safety in robot autonomy. This thesis tackles several challenges present in control barrier functions in different aspects, including optimization, control, planning and navigation.

This thesis is composed of three parts. In Part I, we point out the optimization infeasibility between CBF constraint and input constraint, and address the feasibility problem in optimal control for quadratic programming using control barrier functions under input constraints. We also notice that the potential conflict between input constraints and safety constraints also exists in the discrete-time domain together with model predictive control. This conflict is formally identified in reachability analysis and reachability is later enhanced in proposed formulations.

In the Part II, we focus on obstacle avoidance for control and trajectory generation in a tight environment, which requires polytopic obstacle avoidance. We analyze the optimization problem for obstacle avoidance between polytopes. The novel optimizations for obstacle avoidance in continuous domain and discrete-time domain are proposed to solve this challenge.

In Part III of the thesis, we discuss several applications of motion planning and navigation using control barrier functions.We firstly address the motion planning problem within a finite state machine which unifies a mid-level planner and a low-level safety-critical controller with application to autonomous driving. Next, we propose parallelism for motion planning with control barrier functions with application to autonomous racing. Finally, we explore the possibility of safety-critical motion planning for high dimensional systems such as Cassie, a life-size bipedal robot.

Safety is a primary concern for deploying autonomous robots in the real world. Model-based control theoretic tools provide formal safety guarantees when a mathematical model of the dynamics of the system is available. However, constructing analytical models that accurately predict future states from sensory measurements and designing controllers that use such models can be costly, labor-intensive, or even unattainable for intricate real robotic systems. In contrast, learning-based control strategies have demonstrated the ability for robots to execute complex tasks directly from data, eliminating the need for analytical dynamics models. However, these learning-based control policies are typically trained intensively on vast datasets collected through simulations, which may not encompass the full range of complexities found in real-world interactions. Furthermore, the resulting policies often lack interpretability, making it hard to formally analyze their robustness properties.

This dissertation focuses on establishing core principles for developing reliable and intelligible controllers for real-world autonomous systems. Specifically, it introduces formal techniques that leverage approximate model knowledge when available, and utilize data to nimbly adapt to the intricacies of the real world. The thesis unfolds in two main sections, each examining how control-theoretic model-based strategies and data-driven methodologies can mutually enhance each other. In the first section, the emphasis is on the application of control theory principles to enhance the interpretability and trustworthiness of data-driven approaches. Conversely, the second section flips this narrative, investigating how data can boost control-theoretic approaches and transfer the assurances provided by model-based controllers on approximate models to the actual system being controlled.

Delving into the first part of the dissertation, a principled reward design methodology for model-free policy optimization problems is presented. By exploiting the underlying geometric structures of the system, the proposed reward functions guide the policy search towards safe and stabilizing control policies. It is demonstrated that these policies can be learned directly on hardware from only a few minutes of experimental data. Following this, the dissertation draws on nonlinear control methods to propose an end-to-end distributional shift prevention mechanism for learning-based policies. Preventing distributional shift is, in fact, a critical matter for assuring the safety of data-driven controllers, as operating in unexplored regions can lead to the unexpected behavior of these policies. Taking raw high-dimensional sensory observations as input, the proposed mechanism constitutes an effective safety layer for a wide variety of applications, from robotic manipulation to autonomous driving.

The second part of the dissertation introduces several approaches, ranging from reinforcement learning to Bayesian inference methods, able to safely bridge the gap between an approximate dynamics model and the real system when using model-based controllers. By quantifying the uncertainty within the learning model, it also presents a safe online learning strategy that empowers the system to assess whether its current information is adequate for ensuring safety, or if acquiring new measurements is necessary. Additionally, it puts forward a data selection method that ascertains the impact of individual data points on the overall decision-making process.

Aerial vehicles for physical interactions as a form of a freely-floating manipulator are ofgrowing interest in recent times. Aerial vehicles like quadrotors enable us to address issues such as last-mile delivery and search & rescue. Using multiple such vehicles collaboratively increases the scope of manipulation beyond what a single vehicle can achieve. This dissertation presents theoretical and experimental contributions toward using multiple quadrotors for collaborative tasks, with an emphasis on cable-suspended payloads.

The dissertation studies the problem of collaborative aerial manipulation from two overarchingviews. The first half of the dissertation has an “individualistic view” and presents methods and algorithms for a single quadrotor/quadrotor with a cable-suspended payload. An L1 adaptation scheme is implemented on a geometric attitude control for the quadrotor on SO(3) in the presence of model uncertainties and disturbances. Next, the extended Kalman filter is modified to estimate states on S2. The concept of variation on manifolds is employed to linearize the system states and compute the variations in the state. Optimal and obstacle-free trajectories for a quadrotor with a suspended payload are generated using direct collocation. The planning method exploits the differential flatness for generating the trajectories in the flat space and converts non-differentiable obstacle avoidance constraints into smooth constraints using dual variables.

The second part of the dissertation has a “collaborative view” and presents results for twotypes of aerial manipulation, multiple quadrotors carrying a payload using suspended cables (parallel-aerial-manipulator) and a series of quadrotors connected using a flexible cable (serial-aerial-manipulator). The dissertation presents experimental results for grasping and controlling payload using a cable-suspended gripper using more than one quadrotor. Finally, the dissertation models and computes coordinate-free geometric dynamics for multiple quadrotors connected in series using a single flexible cable and shows that the system is differentially-flat.