This dissertation aims to realize the goal of developing robust control solutions that can enable legged robots to navigate complex unknown environments. The idea of creating articulated-legged machines that can mimic animal locomotion has fueled the imagination of many researchers. These legged robots are designed to assist humans in their day-to-day tasks and challenging scenarios such as monitoring remote, inhospitable environments, disaster response, and other dangerous environments. Despite several decades of research, legged robots have yet to reach the dexterity or dynamic stability needed for real-world deployments. A fundamental gap exists in the understanding and development of reliable and scalable algorithms required for the real-time planning and control of legged robots. The overarching goal of this thesis is to formally develop computationally tractable, robust controllers based on nonlinear hybrid systems theory, model predictive control, and optimization for the real-time planning and control of agile locomotion in quadrupedal robots. Toward this objective, this thesis first investigates layered control architectures. In particular, we propose a two-level hierarchical control architecture in which the higher level is based on a reduced-order model predictive control (MPC), and the lower level is based on a full-order quadratic programming (QP) based virtual constraints controller. Specifically, two MPC architectures are explored: 1) An event-based MPC scheme that generates the optimal center of mass (COM) trajectories using a reduced-order linear inverted pendulum (LIP) model, and 2) A time-based MPC scheme that computes the optimal COM and ground reaction forces (GRF) using the reduced-order single rigid body (SRB) dynamics model. The optimal COM trajectories in the event-based MPC and the optimal COM trajectories, along with the ground reaction forces in the time-based MPC, are then tracked by the low-level virtual constraints controller. The event-based MPC scheme is numerically validated on the Vision 60 platform in a physics-based simulation environment. It has significantly reduced the computational burden associated with real-time planning-based MPC schemes. However, owing to the quasi-static nature of the optimal trajectories generated by the LIP model, we explored a time-based MPC scheme using Single Rigid Body Dynamics. This time-based MPC scheme is also numerically validated using the mathematical model of the A1 quadrupedal robot. Most MPC schemes use a reduced-order model to generate optimal trajectories. However, the abstraction and unmodeled dynamics in template models significantly increase the gap between reduced- and full-order models, limiting the robot's full scope and potential. In the second part of the thesis, we aim to develop a computationally tractable robust model predictive control (RMPC) scheme based on convex QPs to bridge this gap. The RMPC framework considers the single rigid body model subject to a set of unmodeled dynamics and plans for the optimal reduced-order trajectory and GRFs. The generated optimal GRFs of the high-level RMPC are then mapped to the full-order model using a low-level nonlinear controller based on virtual constraints and QP. The key innovation of the proposed RMPC framework is that it allows the integration of the hierarchical controller with Reinforcement Learning (RL) techniques to train a neural network to compute the vertices of the uncertainty set numerically. The proposed hierarchical control algorithm is validated numerically and experimentally for robust and blind locomotion of the A1 quadrupedal robot on different indoor and outdoor terrains and at different speeds. The numerical analysis of the RMPC suggests significant improvement in the performance of the rough terrain locomotion compared to the nominal MPC. In particular, the proposed RMPC algorithm outperforms the nominal MPC by over 60% during rough terrain locomotion over 550 uneven terrains. Our experimental studies also indicate a significant reduction in the gap between the reduced full-order models by comparing the desired and actual GRFs. Finally, the last part of the thesis presents a formal approach for synthesizing robust $mathcal{H}_2$- and $mathcal{H}_infty$-optimal MPCs to stabilize the periodic locomotion of legged robots. The proposed algorithm builds on the existing optimization-based control stack. We outline the set of conditions under which the closed-loop nonlinear dynamics around a periodic orbit can be transformed into a linear time-invariant (LTI) system using Floquet theory. We then outline an approach to systematically generate parameterized $mathcal{H}_2$- and $mathcal{H}_infty$- robust controllers using linear matrix inequalities (LMIs). We subsequently established a set of conditions guaranteeing the existence of such robust optimal controllers. The proposed $mathcal{H}_2$- and $mathcal{H}_infty$-optimal MPCs are extensively validated both numerically and experimentally for the robust locomotion of the A1 quadrupedal robot subject to various external disturbances and uneven terrains. Our numerical analysis suggests a significant improvement in the performance of robust locomotion compared to the nominal MPC. / Doctor of Philosophy / Legged robots have always been envisioned to work alongside humans, assisting them in mundane day-to-day tasks to challenging scenarios such as monitoring remote locations, planetary exploration, and supporting relief programs in disaster situations. Furthermore, research into legged locomotion can aid in designing and developing powered prosthetic limbs and exoskeletons. With these advantages in mind, several researchers have created sophisticated-legged robots and even more complicated algorithms to control them. Despite this, a significant gap exists between the agility, mobility, and dynamic stability shown by the existing legged robots and their biological counterparts. To work alongside humans, legged robots have to interact with complex environments and deal with uncertainties in the form of unplanned contacts and unknown terrains. Developing robust control solutions to accommodate disturbances explicitly marks the first step towards safe and reliable real-world deployment of legged robots. Toward this objective, this thesis aims to establish a formal foundation to develop computationally tractable robust controllers for the real-time planning and control of legged robots. Initial investigations in this thesis report on the use of layered control architectures, specifically event-based and time-based Model Predictive Control(MPC) schemes. These layered control architectures consist of an MPC scheme built around a reduced-order model at the high level and a virtual constraints-based nonlinear controller at the low level. Using these layered control architectures, this thesis proposed two robust control solutions to improve the rough terrain locomotion of legged robots. The first proposed robust control solution aims to mitigate one of the issues of layered control architecture. In particular, layered control architectures rely on a reduced order model at the high level to remain computationally tractable. However, the approximation of fullorder models with reduced-order models limits the full scope and potential of the robot. The proposed algorithm aims to bridge the gap between reduced- and full-order models with the integration of model-free Reinforcement Learning (RL) techniques. The second algorithm proposes a formal approach to generate robust optimal control solutions that can explicitly accommodate the disturbances and stabilize periodic legged locomotion. Under some mild conditions, the MPC control solution is analyzed, and an auxiliary feedback control solution that can handle disturbances explicitly is proposed. The thesis also theoretically establishes the sufficient conditions for the existence of such controllers. Both the proposed control solutions are extensively validated using numerical simulations and experiments using an A1 quadrupedal robot as a representative example.

Cette thèse porte sur le développement de lois de commande pour la marche desrobots bipèdes. Le sous actionnement engendré par le basculement, volontaire ouinvolontaire, du pied en appui sur le sol représente une difficulté majeure. Nousabordons ce problème par l’étude de robots plans avec pieds ponctuels.La première partie de la thèse est une compilation des informations issuesde la littérature que nous avons jugées intéressantes. Nous traitons dans unpremier temps de la modélisation adoptée, puis effectuons une revue des différentesméthodes existantes, et présentons la mise en oeuvre expérimentale de l’une d’entre elle : la méthode HZD.Dans une deuxième partie, nous procédons à une étude de la dissipation relativede l’énergie cinétique du robot lorsque le pied impacte le sol. Nous utilisons les résultats issus de cette étude pour planifier des trajectoires de marche dissipant peu d’énergie. De telles trajectoires ont a priori le mérite de préserver la structure du robot et de générer moins de bruit. A contrario, des trajectoires dissipant la majorité de l’énergie du robot sont utilisées pour un arrêt rapide. Une étude numérique a montré que ces résultats sont robustes à des incertitudes de modèle.Enfin, dans une dernière partie, afin de compenser les difficultés liées au sousactionnement, nous proposons d’utiliser le degré de liberté supplémentaire offert par un changement de l’échelle de temps dans les équations de la dynamique (Time Scaling) pour la classe de robots considérée. En utilisant par ailleurs un changement de coordonnées et de feedback, nous dérivons de nouvelles formes normales exactes et approximatives. / This thesis addresses the general problem of the walking control of biped robots. The foot of the robot in contact with the ground may tip over and cause the robot to be undercatuated. This is a major difficulty in term of control. This problem is addressed by considering planar biped robots with point feet.In a first part, we present a standard way of modeling such systems, a litterature review of the existing methods, and then report experimental results of the walking control of a biped robot using the HZD method.In a second part, we perform an analytic and numeric study of the relativekinetic energy dissipation when the foot of the robot impacts the ground. Usingthis study, we design trajectories with low energy dissipation at impact, which a priori result in gaits preserving the hardware of the robot and causing less noise. On the contrary, trajectories dissipating almost all the kinetic energy are used to quickly stop the robot.Finally, in an attempt to alleviate the burden due to underactuation, we proposeto investigate the additional degree of freedom provided, in the control design, by a change of time scale in the dynamic equations (Time-Scaling) for the considered class of biped robots. Using feedback transformations, we derive new exact and approximative normal forms.

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Existem métodos de integração de equações algébrico diferenciais não lineares (DAEs) considerados clássicos pela literatura. Porém, neste trabalho, através uma abordagem geométrica, apresenta-se um método de integração de DAEs. Tal método é inspirado na teoria de desacoplamento de sistemas não lineares explícitos, quando se considera que as saídas são restrições algébricas. Neste caso, a DAE pode ser identificada como dinâmica zero. O resultado principal desta abordagem é que, dada uma DAE, sob certas condições, é possível a construção de um sistema explícito, de tal maneira, que as soluções desse sistema explícito convergem para as soluções da DAE. / Classical methods for numerical integration of diferential algebraic equations (DAEs) can be formal in the literature. In this work, using a diferential geometric approach, a numerical method of integration of DAEs is established. This method is inspired in the decoupling theory of nonlinear explicit systems, when one considers that the outputs are algebraic constraints. The main result is the construction of an explicit system, whose solutions converge to the solutions of the DAE.

Existem métodos de integração de equações algébrico diferenciais não lineares (DAEs) considerados clássicos pela literatura. Porém, neste trabalho, através uma abordagem geométrica, apresenta-se um método de integração de DAEs. Tal método é inspirado na teoria de desacoplamento de sistemas não lineares explícitos, quando se considera que as saídas são restrições algébricas. Neste caso, a DAE pode ser identificada como dinâmica zero. O resultado principal desta abordagem é que, dada uma DAE, sob certas condições, é possível a construção de um sistema explícito, de tal maneira, que as soluções desse sistema explícito convergem para as soluções da DAE. / Classical methods for numerical integration of diferential algebraic equations (DAEs) can be formal in the literature. In this work, using a diferential geometric approach, a numerical method of integration of DAEs is established. This method is inspired in the decoupling theory of nonlinear explicit systems, when one considers that the outputs are algebraic constraints. The main result is the construction of an explicit system, whose solutions converge to the solutions of the DAE.