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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.

<p>Successful navigation requires an accurate mental spatial representation of the environment that can be updated during movement. Experiments with animals and humans have demonstrated the existence of two forms of spatial representation: egocentric (observer-centered) and allocentric (environment-centered). Unfortunately, specifically how humans use these two systems is not well understood. The current dissertation was focused on providing evidence differentiating human use of egocentric and allocentric spatial reference frames, specifically examining the characteristics and contributions from body-based sources. Two empirical chapters are presented that include experiments involving two common spatial tasks. In Chapter 2, updating of feature relations within a room-sized environment was examined by having observers provide directional judgments to learned features with respect to an imagined orientation that was either congruent or incongruent with their physical orientation. The information available for updating the physical orientation was manipulated across experiments. Performance differences between congruent and incongruent conditions demonstrated the reliance on egocentric representations for updating, and differentiated body- and knowledge-based components of the egocentric updating process. The specificity of the body-based component was examined in Chapter 3 by having observers detect changes made to a tabletop spatial scene following a viewpoint shift resulting from their movement, scene rotation or both. The relation between the extent of observer movement and the magnitude of the experienced viewpoint shift was manipulated. Change detection performance was best when the extent of observer movement most closely matched the viewpoint shift, and declined as the match declined. Thus, body-based cues contributed specific information for updating self-to-feature relations that facilitated scene recognition. Throughout the course of the research program it has become clear that humans rely on egocentric representations to complete these tasks, and sensory and motor modalities involved in self-motion are integrated for updating spatial relations of novel environments.</p> / Doctor of Philosophy (PhD)

No-vision walking is supported in the central nervous system (CNS) by a spatial updating process. This process involves the iterative updating of a mental representation of the environment using estimates of distance traveled gleaned from locomotive kinematic activity. An effective means of examining the online regulation of this process is a continuous pointing task, which requires performers to walk along a straight-line forward trajectory while keeping their right arm straight and index finger fixated on a stationary ground-level target beside the walking path. In the current thesis, no-vision continuous pointing was examined in typically calibrated and recalibrated perceptual-motor states. Shoulder and trunk joint angles provided the basis for perceptual measures that reflected spatial updating performance and kinematic measures that reflected its underlying CNS online regulation. In the typically calibrated conditions, no-vision walking demonstrated a slight perceptual underestimation of distance traveled (Study 1). In the recalibrated conditions, no-vision walking demonstrated: a) perceptual underestimation and overestimation following adaptation periods involving walking with low and high visual gains, respectively (Study 2); and b) partial recalibration following exposures to vision and arm gains (Study 3). The latter was suggested as being impacted by task specific changes in CNS multisensory integration resulting from the development of a robust task prior and/or the altering of sensory cue weights. Importantly, this thesis used a novel trajectory parsing procedure to quantify discrete CNS perceptual updating units in the shoulder plane of elevation trajectory. The starts and ends of these updating units were consistently timed to the late left-to-early right foot swing phase of the step-cycle, regardless of perceptual-motor state. This was suggested to reflect perceptual units that were purposely timed, but indirectly mapped, to this kinematic event. The perceptual differences in Studies 1 and 2 were at least partially reflected in these units. / Thesis / Doctor of Philosophy (PhD) / It is well understood that humans can effectively walk without vision to environmental locations up to 15 metres away. However, less is known about how these walking movements are controlled during the course of forward progression. This thesis fills this knowledge gap using a task that requires participants to walk forward along a straight path while keeping their right index finger pointed toward a ground-level target beside the walking path. The patterns of arm movements performed during this task are indicative of the control strategies used by the performer to mentally update their positions in space. One of the key contributions of this work is showing that humans perform this mental updating in a repetitive manner, and that these repetitions are consistently linked to early forward movements of the right leg. This pattern is maintained when walking without vision is performed in a variety of different contexts.

No / There is an increasing interest in modelling of agents interacting as crowd and a simulation of such scenarios that map to real-life situations. This paper presents a generic state-based abstract model for crowd behaviour that can be mapped onto different agent-based systems. In particular, the abstract model is mapped into the simulation framework NetLogo. We have used the model to simulate a real-life case study of high density diverse crowd such as the Hajj ritual at the mosque in Mecca (Makkah). The computational model is based on real data extracted from videos of the ritual. We also present a methodology for extracting significant data, parameters, and patterns of behaviour from real-world videos that has been used as an early stage validation to demonstrate that the obtained simulations are realistic.

This thesis synthesizes general and scalable distributed nonlinear control algorithms with application to legged robots. It explores both naturally decentralized problems in legged locomotion, such as the collaborative control of human-lower extremity prosthesis and the decomposition of high-dimensional controllers of a naturally centralized problem into a net- work of low-dimensional controllers while preserving equivalent performance. In doing so, strong nonlinear interaction forces arise, which this thesis considers and sufficiently addresses. It generalizes to both symmetric and asymmetric combinations of subsystems. Specifically, this thesis results in two distinct distributed control algorithms based on the decomposition approach. Towards synthesizing the first algorithm, this thesis presents a formal foundation based on de- composition, Hybrid Zero Dynamics (HZD), and scalable optimization to develop distributed controllers for hybrid models of collaborative human-robot locomotion. This approach con- siders a centralized controller and then decomposes the dynamics and parameterizes the feedback laws to synthesize local controllers. The Jacobian matrix of the Poincaré map with local controllers is studied and compared with the centralized ones. An optimization problem is then set up to tune the parameters of the local controllers for asymptotic stability. It is shown that the proposed approach can significantly reduce the number of controller parameters to be optimized for the synthesis of distributed controllers, deeming the method computationally tractable. To evaluate the analytical results, we consider a human amputee with the point of separation just above the knee and assume the average physical parameters of a human male. For the lower-extremity prosthesis, we consider the PRleg, a powered knee-ankle prosthetic leg, and together, they form a 19 Degrees of Freedom (DoF) model. A multi-domain hybrid locomotion model is then employed to rigorously assess the performance of the afore-stated control algorithm via numerical simulations. Various simulations involving the application of unknown external forces and altering the physical parameters of the human model unbeknownst to the local controllers still result in stable amputee loco- motion, demonstrating the inherent robustness of the proposed control algorithm. In the later part of this thesis, we are interested in developing distributed algorithms for the real-time control of legged robots. Inspired by the increasing popularity of Quadratic programming (QP)-based nonlinear controllers in the legged locomotion community due to their ability to encode control objectives subject to physical constraints, this thesis exploits the idea of distributed QPs. In particular, this thesis presents a formal foundation to systematically decompose QP-based centralized nonlinear controllers into a network of lower-dimensional local QPs. The proposed approach formulates a feedback structure be- tween the local QPs and leverages a one-step communication delay protocol. The properties of local QPs are analyzed, wherein it is established that their steady-state solutions on periodic orbits (representing gaits) coincide with that of the centralized QP. The asymptotic convergence of local QPs' solutions to the steady-state solution is studied via Floquet theory. Subsequently, to evaluate the effectiveness of the analytical results, we consider an 18 DoF quadrupedal robot, A1, as a representative example. The network of distributed QPs mentioned earlier is condensed to two local QPs by considering a front-hind decomposition scheme. The robustness of the distributed QP-based controller is then established through rigorous numerical simulations that involve exerting unmodelled external forces and intro- ducing unknown ground height variations. It is further shown that the proposed distributed QPs have reduced sensitivity to noise propagation when compared with the centralized QP. Finally, to demonstrate that the resultant distributed QP-based nonlinear control algorithm translates equivalently well to hardware, an extensive set of blind locomotion experiments on the A1 robot are undertaken. Similar to numerical simulations, unknown external forces in the form of aggressive pulls and pushes were applied, and terrain uncertainties were introduced with the help of arbitrarily displaced wooden blocks and compliant surfaces. Additionally, outdoor experiments involving a wide range of terrains such as gravel, mulch, and grass at various speeds up to 1.0 (m/s) reiterate the robust locomotion observed in numerical simulations. These experiments also show that the computation time is significantly dropped when the distributed QPs are considered over the centralized QP. / Doctor of Philosophy / Inspiration from animals and human beings has long driven the research of legged loco- motion and the subsequent design of the robotic counterparts: bipedal and quadrupedal robots. Legged robots have also been extended to assist human amputees with the help of powered prostheses and aiding people with paraplegia through the development of exoskeleton suits. However, in an effort to capture the same robustness and agility demonstrated by nature, our design abstractions have become increasingly complicated. As a result, the en- suing control algorithms that drive and stabilize the robot are equivalently complicated and subjected to the curse of dimensionality. This complication is undesirable as failing to compute and prescribe a control action quickly destabilizes and renders the robot uncontrollable. This thesis addresses this issue by seeking nature for inspiration through a different perspective. Specifically, through some earlier biological studies on cats, it was observed that some form of locality is implemented in the control of animals. This thesis extends this observation to the control of legged robots by advocating an unconventional solution. It proposes that a high-dimensional, single-legged agent be viewed as a virtual composition of multiple, low-dimensional subsystems. While this outlook is not new and forms precedent to the vast literature of distributed control, the focus has always been on large-scale systems such as power networks or urban traffic networks that preserve sparsity, mathematically speaking. On the contrary, legged robots are underactuated systems with strong interaction forces acting amongst each subsystem and dense mathematical structures. This thesis considers this problem in great detail and proposes developments that provide theoretical stability guarantees for the distributed control of interconnected legged robots. As a result, two distinctly different distributed control algorithms are formulated. We consider a naturally decentralized structure appearing in the form of a human-lower extremity prosthesis to synthesize distributed controllers using the first control algorithm. Subsequently, the resultant local controllers are rigorously validated through extensive full- order simulations. In order to validate the second algorithm, this thesis considers the problem of quadrupedal locomotion as a representative example. It assumes for the purposes of control synthesis that the quadruped is comprised of two subsystems separated at the geometric center, resulting in a front and hind subsystem. In addition to rigorous validation via numerical simulations, in the latter part of this thesis, to demonstrate that distributed controllers preserve practicality, rigorous and extensive experiments are undertaken in indoor and outdoor settings on a readily available quadrupedal robot A1.

This study focuses on developing a better comprehension of the mobility of lightweight autonomous vehicles with varying locomotion platforms on sand. This research involves four segments. The first segment is a review of military criteria for the development of lightweight unmanned ground vehicles, followed by a review a review of current methodologies for evaluating the terramechanic (vehicle-ground interaction) mobility measures of heavyweight wheeled and tracked vehicles, and ending with a review of the defining properties of deformable terrain with specific emphasis on sand. These present a basis for understanding what currently defines mobility and how mobility is quantified for traditional heavyweight wheeled and tracked vehicles, as well as an understanding of the environment of operation (sandy terrain) for the lightweight vehicles in this study. The second segment involves the identification of key properties associated with the mobility and operation of lightweight vehicles on sand as related to given mission criteria, so as to form a quantitative assessment system to compare lightweight vehicles of varying locomotion platforms. A table based on the House of Quality shows the relationships&#8212;high, low, or adverse&#8212;between mission profile requirements and general performance measures and geometries of vehicles under consideration for use. This table, when combined with known values for vehicle metrics, provides information for an index formula used to quantitatively compare the mobility of a user-chosen set of vehicles, regardless of their methods of locomotion. This table identifies several important or fundamental terramechanics properties that necessitate model development for robots with novel locomotion platforms and testing for lightweight wheeled and tracked vehicles so as to consider the adaptation of counterpart heavyweight terramechanics models for use. The third segment is a study of robots utilizing novel forms of locomotion, emphasizing the kinematics of locomotion (gait and foot placement) and proposed starting points for the development of terramechanics models so as to compare their mobility and performance with more traditional wheeled and tracked vehicles. In this study several new autonomous vehicles&#8212;bipedal, self-excited dynamic tripedal, active spoke-wheel&#8212;that are currently under development are explored. The final segment involves experimentation of several lightweight vehicles and robots on sand. A preliminary experimentation was performed evaluating a lightweight autonomous tracked vehicle for its performance and operation on sand. A bipedal robot was then tested to study the foot-ground interaction with and sinkage into a medium-grade sand, utilizing a one of the first-developed walking gaits. Finally, a comprehensive set of experiments was performed on a lightweight wheeled vehicle. While the terramechanics properties of wheeled and tracked vehicles, such as the contact patch pressure distribution, have been understood and models have been developed for heavy vehicles, the feasibility of extrapolating them to the analysis of light vehicles is still under analysis. A wheeled all-terrain vehicle was tested for effects of sand gradation, vehicle speed, and vehicle payload on measures of pressure and sinkage in the contact patch, and preliminary analysis is presented on the sinkage of the wheeled all-terrain vehicle. These four segments&#8212;review of properties of sandy terrain and measures of and criteria for the mobility of lightweight vehicles operating on sandy terrain, the development of the comparison matrix and indexing function, modeling and development of novel forms of locomotion, and physical experimentation of lightweight tracked and wheeled vehicles as well as a bipedal robot&#8212;combine to give an overall picture of mobility that spans across different forms of locomotion. / Master of Science

Biological systems offer novel and efficient solutions to many engineering applications, including marine propulsion. It is of interest to determine how fish interact with the water around them, and how best to utilize the potential their methods offer. A stingray-like fin was chosen for analysis due to the maneuverability and versatility of stingrays. The stingray fin was modeled in 2D as a sinusoidal wave with an amplitude increasing from zero at the leading edge to a maximum at the trailing edge. Using this model, a parametric study was performed to examine the effects of the fin on surrounding water in computational fluid dynamics (CFD) simulations. The results were analyzed both qualitatively, in terms of the pressure contours on the fin and vorticity in the trailing wake, and quantitatively, in terms of the resultant forces and the mechanical power requirements to actuate the desired fin motion. The average thrust was shown to depend primarily on the relationship between the swimming speed and the frequency and wavelength (which both are directly proportional to the wavespeed of the fin), although amplitude can be used to augment thrust production as well. However, acceleration was shown to significantly correlate with a large variation in lift and moment, as well as with greater power losses. Using results from the parametric study, the potential for power regeneration was also examined. Relationships between frequency, velocity, drag, and power input were determined using nonlinear regression that explained more than 99.8% of the data. The actuator for a fin was modeled as a single DC motor-shaft system, allowing the combination of the energetic effects of the motor with the fin-fluid system. When combined, even a non-ideal fin model was able to regenerate more power at a given flow speed than was required to swim at the same speed. Even in a more realistic setting, this high proportion of regenerative power suggests that regeneration and energy harvesting could be both feasible and useful in a mission setting. / Master of Science / Animals interact with the world much differently than engineered systems, and can offer new and efficient ways to solve engineering problems, including underwater vehicles. To learn how to move an underwater vehicle in an environmentally conscious way, it is useful to study how a fish’s movements affect the manner in which it moves through the water. Through careful study, the principles involved can be implemented for an efficient, low-disturbance underwater vehicle. The particular fish chosen for in-depth study was the stingray, due to its maneuverability and ability to travel close to the seafloor without disturbing the sediment and creatures around it. In this work, computational analysis was performed on a model of a single stingray fin to determine how the motion of the fin affects the water around it, and how the water affects the fin in turn. The results were analyzed both in terms of the wake behind the fin and in terms of how much power was required to make the fin move in a particular way. The speed of the fin motion was found to have the strongest effect in controlling swimming speed, although the lateral motion of the fin also helped with accelerating faster. Additionally, the potential for a robotic stingray fin to harness power from the water around it was examined. Based on results from simulations of the fin, a mathematical model was formulated to relate energy harvesting with the flow speed past the fin. This model was used to determine how worthwhile it was to use energy harvesting. Analysis of the model showed that harvesting energy from the water was quite efficient, and would likely be a worthwhile investment for an exploration mission.