The thesis shows how to achieve bipedal robotic walking on flat-ground, up-slope and rough terrain by using Human-Inspired control. We begin by considering human walking data and find outputs (or virtual constraints) that, when calculated from the human data, are described by simple functions of time (termed canonical walking functions). Formally, we construct a torque controller, through model inversion, that drives the outputs of the robot to the outputs of the human as represented by the canonical walking function; while these functions fit the human data well, they do not apriori guarantee robotic walking (due to do the physical differences between humans and robots). An optimization problem is presented that determines the best fit of the canonical walking function to the human data, while guaranteeing walking for a specific bipedal robot; in addition, constraints can be added that guarantee physically realizable walking. We consider a physical bipedal robot, AMBER, and considering the special property of the motors used in the robot, i.e., low leakage inductance, we approximate the motor model and use the formal controllers that satisfy the constraints and translate into an efficient voltage-based controller that can be directly implemented on AMBER. The end result is walking on flat-ground and up-slope which is not just human-like, but also amazingly robust. Having obtained walking on specific well defined terrains separately, rough terrain walking is achieved by dynamically changing the extended canonical walking functions (ECWF) that the robot outputs should track at every step. The state of the robot, after every non-stance foot strike, is actively sensed and the new CWF is constructed to ensure Hybrid Zero Dynamics is respected in the next step. Finally, the technique developed is tried on different terrains in simulation and in AMBER showing how the walking gait morphs depending on the terrain.

Impact at each leg transition is one of the main causes of energy dissipation in most of the current bipedal walking robots. Minimizing impact can reduce the energy loss. Instead of controlling the joint angle profiles to reduce the impact which requires significant amount of energy, installing elastic mechanisms on the robots structure is proposed in this research, enabling the robot to reduce the impact, and to store part of the energy in the elastic form which returns the energy to the robot. Practically, this motivates the development of the bipedal walking robots with adjustable stiffness elasticity which itself creates new challenging problems. This thesis addresses some of the challenges through five consecutive stages. Firstly, an adjustable compliant series elastic actuator (named ACSEA in this thesis) is developed. The velocity control mode of the electric motor is used to accurately control the output force of the ACSEA. Secondly, three different conceptual designs of the adjustable stiffness artificial tendons (ASAT) are proposed each of which is added at the ankle joint of a bipedal walking robot model. Simulation results of the collision phase (part of the gait between the heel-strike and the foot-touch-down in bipedal walking) demonstrate significant improvements in the energetics of the bipedal walking robot by proper stiffness adjustment of ASAT. In the third stage, in order to study the effects of ASATs on reducing the energy loss during the stance phase, a simplified model of bipedal walking is introduced consisting of a foot, a leg and an ASAT which is installed parallel to the ankle joint. A linear spring, with adjustable stiffness, is included in the model to simulate the generated force by the trailing leg during the double support phase. The concept of impulsive constraints is used to establish the mathematical model of impacts in the collision phase which includes the heel-strike and the foot-touch-down. For the fourth stage, an energy-feedback-based controller is designed to automatically adjust the stiffness of the ASAT which reduces the energy loss during the foot-touch-down. In the final stage, a speed tracking (ST) controller is developed to regulate the velocity of the biped at the midstance. The ST controller is an event-based time-independent controller, based on geometric progression with exponential decay in the kinetic energy error, which adjusts the stiffness of the trailing-leg spring to control the injected energy to the biped in tracking a desired speed at the midstance. Another controller is also integrated with the ST controller to tune the stiffness of the ASAT when reduction in the speed is desired. Then, the local stability of the system (biped and the combination of the above three controllers) is analyzed by calculating the eigenvalues of the linear approximation of the return map. Simulation results show that the combination of the three controllers is successful in tracking a desired speed of the bipedal walking even in the presence of the uncertainties in the leg’s initial angles. The outcomes of this research show the significant effects of adjustable stiffness artificial tendons on reducing the energy loss during bipedal walking. It also demonstrates the advantages of adding elastic elements in the bipedal walking model which benefits the efficiency and simplicity in regulating the speed. This research paves the way toward developing the dynamic walking robots with adjustable stiffness ability which minimize the shortcomings of the two major types of bipedal walking robots, i.e., passive dynamic walking robots (which are energy efficient but need extensive parameters tuning for gait stability) and actively controlled walking robots (which are significantly energy inefficient). / May 2008

Impact at each leg transition is one of the main causes of energy dissipation in most of the current bipedal walking robots. Minimizing impact can reduce the energy loss. Instead of controlling the joint angle profiles to reduce the impact which requires significant amount of energy, installing elastic mechanisms on the robots structure is proposed in this research, enabling the robot to reduce the impact, and to store part of the energy in the elastic form which returns the energy to the robot. Practically, this motivates the development of the bipedal walking robots with adjustable stiffness elasticity which itself creates new challenging problems. This thesis addresses some of the challenges through five consecutive stages. Firstly, an adjustable compliant series elastic actuator (named ACSEA in this thesis) is developed. The velocity control mode of the electric motor is used to accurately control the output force of the ACSEA. Secondly, three different conceptual designs of the adjustable stiffness artificial tendons (ASAT) are proposed each of which is added at the ankle joint of a bipedal walking robot model. Simulation results of the collision phase (part of the gait between the heel-strike and the foot-touch-down in bipedal walking) demonstrate significant improvements in the energetics of the bipedal walking robot by proper stiffness adjustment of ASAT. In the third stage, in order to study the effects of ASATs on reducing the energy loss during the stance phase, a simplified model of bipedal walking is introduced consisting of a foot, a leg and an ASAT which is installed parallel to the ankle joint. A linear spring, with adjustable stiffness, is included in the model to simulate the generated force by the trailing leg during the double support phase. The concept of impulsive constraints is used to establish the mathematical model of impacts in the collision phase which includes the heel-strike and the foot-touch-down. For the fourth stage, an energy-feedback-based controller is designed to automatically adjust the stiffness of the ASAT which reduces the energy loss during the foot-touch-down. In the final stage, a speed tracking (ST) controller is developed to regulate the velocity of the biped at the midstance. The ST controller is an event-based time-independent controller, based on geometric progression with exponential decay in the kinetic energy error, which adjusts the stiffness of the trailing-leg spring to control the injected energy to the biped in tracking a desired speed at the midstance. Another controller is also integrated with the ST controller to tune the stiffness of the ASAT when reduction in the speed is desired. Then, the local stability of the system (biped and the combination of the above three controllers) is analyzed by calculating the eigenvalues of the linear approximation of the return map. Simulation results show that the combination of the three controllers is successful in tracking a desired speed of the bipedal walking even in the presence of the uncertainties in the leg’s initial angles. The outcomes of this research show the significant effects of adjustable stiffness artificial tendons on reducing the energy loss during bipedal walking. It also demonstrates the advantages of adding elastic elements in the bipedal walking model which benefits the efficiency and simplicity in regulating the speed. This research paves the way toward developing the dynamic walking robots with adjustable stiffness ability which minimize the shortcomings of the two major types of bipedal walking robots, i.e., passive dynamic walking robots (which are energy efficient but need extensive parameters tuning for gait stability) and actively controlled walking robots (which are significantly energy inefficient).

Impact at each leg transition is one of the main causes of energy dissipation in most of the current bipedal walking robots. Minimizing impact can reduce the energy loss. Instead of controlling the joint angle profiles to reduce the impact which requires significant amount of energy, installing elastic mechanisms on the robots structure is proposed in this research, enabling the robot to reduce the impact, and to store part of the energy in the elastic form which returns the energy to the robot. Practically, this motivates the development of the bipedal walking robots with adjustable stiffness elasticity which itself creates new challenging problems. This thesis addresses some of the challenges through five consecutive stages. Firstly, an adjustable compliant series elastic actuator (named ACSEA in this thesis) is developed. The velocity control mode of the electric motor is used to accurately control the output force of the ACSEA. Secondly, three different conceptual designs of the adjustable stiffness artificial tendons (ASAT) are proposed each of which is added at the ankle joint of a bipedal walking robot model. Simulation results of the collision phase (part of the gait between the heel-strike and the foot-touch-down in bipedal walking) demonstrate significant improvements in the energetics of the bipedal walking robot by proper stiffness adjustment of ASAT. In the third stage, in order to study the effects of ASATs on reducing the energy loss during the stance phase, a simplified model of bipedal walking is introduced consisting of a foot, a leg and an ASAT which is installed parallel to the ankle joint. A linear spring, with adjustable stiffness, is included in the model to simulate the generated force by the trailing leg during the double support phase. The concept of impulsive constraints is used to establish the mathematical model of impacts in the collision phase which includes the heel-strike and the foot-touch-down. For the fourth stage, an energy-feedback-based controller is designed to automatically adjust the stiffness of the ASAT which reduces the energy loss during the foot-touch-down. In the final stage, a speed tracking (ST) controller is developed to regulate the velocity of the biped at the midstance. The ST controller is an event-based time-independent controller, based on geometric progression with exponential decay in the kinetic energy error, which adjusts the stiffness of the trailing-leg spring to control the injected energy to the biped in tracking a desired speed at the midstance. Another controller is also integrated with the ST controller to tune the stiffness of the ASAT when reduction in the speed is desired. Then, the local stability of the system (biped and the combination of the above three controllers) is analyzed by calculating the eigenvalues of the linear approximation of the return map. Simulation results show that the combination of the three controllers is successful in tracking a desired speed of the bipedal walking even in the presence of the uncertainties in the leg’s initial angles. The outcomes of this research show the significant effects of adjustable stiffness artificial tendons on reducing the energy loss during bipedal walking. It also demonstrates the advantages of adding elastic elements in the bipedal walking model which benefits the efficiency and simplicity in regulating the speed. This research paves the way toward developing the dynamic walking robots with adjustable stiffness ability which minimize the shortcomings of the two major types of bipedal walking robots, i.e., passive dynamic walking robots (which are energy efficient but need extensive parameters tuning for gait stability) and actively controlled walking robots (which are significantly energy inefficient).

Bipedal robots offer a uniquely flexible platform capable of navigating complex, human-centric environments. This makes them ideally suited for a variety of missions, including disaster response and relief, emergency scenarios, or exoskeleton systems for individuals with disabilities. This, however, requires significant advances in humanoid locomotion and control, as they are still slow, unnatural, inefficient, and relatively unstable. The work of this dissertation the state of the art with the aim was of increasing the robustness and efficiency of these bipedal walking platforms. We present a series of control improvements to enable reliable, robust, natural bipedal locomotion that was validated on a variety of bipedal robots using both hardware and simulation experiments. A huge part of reliable walking involves maximizing the robot's control authority. We first present the development of a model predictive controller to both control the ground reaction forces and perform step adjustment for walking stabilization using a mixed-integer quadratic program. This represents the first model predictive controller to include step rotation in the optimization and leverage the capabilities of the time-varying divergent component of motion for navigating rough terrain. We also analyze the potential capabilities of model predictive controllers for the control of bipedal walking. As an alternative to standard trajectory optimization-based model predictive controls, we present several optimization-based control schemes that leverage more traditional bipedal walking control approaches by embedding a proportional feedback controller into a quadratic program. This controller is capable of combining multiple feedback mechanisms: ground reaction feedback (the "ankle strategy"), angular momentum (the "hip strategy"), swing foot speed up, and step adjustment. This allows the robot to effectively shift its weight, pitch its torso, and adjust its feet to retain balance, while considering environmental constraints, when available. To enable the robot to walk with straightened legs, we present a strategy that insures that the dynamic plans are kinematically and dynamically feasible to execute using straight legs. The effects of timing on dynamic plans are typically ignored, resulting in them potentially requiring significantly bending the legs during execution. This algorithm modifies the step timings to insure the plan can be executed without bending the legs beyond certain angle, while leaving the desired footsteps unmodified. To then achieve walking with straight legs we then presented a novel approach for indirectly controlling the center of mass height through the leg angles. This avoids complicated height planning techniques that are both computationally expensive and often not general enough to consider variable terrain by effectively biasing the solution of the whole-body controller towards using straighter legs. To incorporate the toe-off motion that is essential to both natural and straight leg walking, we also present a strategy for toe-off control that allows it to be an emergent behavior of the whole-body controller. The proposed approach was demonstrated through a series of simulation and experimental results on a variety of platforms. Model predictive control for step adjustment and rough terrain is illustrated in simulation, while the other step adjustment strategies and straight leg walking approaches are presented recovering from external disturbances and walking over a variety of terrains in hardware experiments. We discuss many of the practical considerations and limitations required when porting simulation-based controller development to hardware platforms. Using the presented approaches, we also demonstrated a important concept: using whole-body control frameworks, not every desired motion need be directly commanded. Many of these motions, such as toe-off, may simply be emergent behaviors that result by attempting to satisfy other objectives, such as desired reaction forces. We also showed that optimization is a very powerful tool for walking control, able to determine both stabilizing inputs and joint torques. / Ph. D.

Human walking is dynamic, stable, and energy efficient. To achieve such remarkable legged locomotion in robots, engineers have explored bipedal robots developed based on two paradigms: trajectory-controlled and passive-based walking. Trajectory-controlled bipeds often deliver energy-inefficient gaits. The reason is that these bipeds are controlled via high-impedance geared electrical motors to accurately follow predesigned trajectories. Such trajectories are designed to keep a biped locally balanced continually while walking. On the other hand, passive-based bipeds provide energy-efficient gaits. The reason is that these bipeds adapt to their natural dynamics. Such gaits are stable limit-cycles through entire walking motion, and do not require being locally balanced at every instant during walking. However, passive-based bipeds are often of round/point foot bipeds that are not capable of achieving and experiencing standing, stopping, and some important bipedal gait phases and events, such as the double support phase. Therefore, the goals of this thesis are established such that the aforementioned limitations on trajectory-controlled and passive-based bipeds are resolved. Toward the above goal, comprehensive simulation and experimental explorations into bipedal walking have been carried out. Firstly, a novel systematic trajectory-controlled gait-planning framework has been developed to provide mechanical insights into bipedal walking in terms of gait characteristics and energy efficiency. For the same purpose, a novel mathematical model of passive-based bipedal walking with compliant hip-actuation and compliant-ankle flat-foot has been developed. Finally, based on mechanical insights that have been achieved by the aforementioned passive-based model, a physical prototype of a passive-based bipedal robot has been designed and fabricated. The prototype experimentally validates the importance of compliant hip-actuation in achieving a highly dynamic and energy efficient gait.

This thesis presents an approach towards experimental realization of underactuated bipedal robotic walking using human data. Human-inspired control theory serves as the foundation for this work. As the name, "human-inspired control," suggests, by using human walking data, certain outputs (termed human outputs) are found which can be represented by simple functions of time (termed canonical walking functions). Then, an optimization problem is used to determine the best fit of the canonical walking function to the human data, which guarantees a physically realizable walking for a specific bipedal robot. The main focus of this work is to construct a control scheme which takes the optimization results as input and delivers human-like walking on the real-world robotic platform - AMBER. To implement the human-inspired control techniques experimentally on a physical bipedal robot AMBER, a simple voltage based control law is presented which utilizes only the human outputs and canonical walking function with parameters obtained from the optimization. Since this controller does not require model inversion, it can be implemented efficiently in software. Moreover, applying this methodology to AMBER, experimentally results in robust and efficient "human-like" robotic walking.

Military, Medical, Exploratory, and Commercial robots have much to gain from exchanging wheels for legs. However, the equations of motion of dynamic bipedal walker models are highly coupled and non-linear, making the selection of an appropriate control scheme difficult. A temporal difference reinforcement learning method known as Q-learning develops complex control policies through environmental exploration and exploitation. As a proof of concept, Q-learning was applied through simulation to a benchmark single pendulum swing-up/balance task; the value function was first approximated with a look-up table, and then an artificial neural network. We then applied Evolutionary Function Approximation for Reinforcement Learning to effectively control the swing-leg and torso of a 3 degree of freedom active dynamic bipedal walker in simulation. The model began each episode in a stationary vertical configuration. At each time-step the learning agent was rewarded for horizontal hip displacement scaled by torso altitude--which promoted faster walking while maintaining an upright posture--and one of six coupled torque activations were applied through two first-order filters. Over the course of 23 generations, an approximation of the value function was evolved which enabled walking at an average speed of 0.36 m/s. The agent oscillated the torso forward then backward at each step, driving the walker forward for forty-two steps in thirty seconds without falling over. This work represents the foundation for improvements in anthropomorphic bipedal robots, exoskeleton mechanisms to assist in walking, and smart prosthetics. / Master of Science

This thesis poses a feedback control method for obtaining humanlike bipedal walking on a human-inspired hybrid biped model. The end goal was to understand better the fundamental mechanisms that underlie bipedal walking in the hopes that this newfound understanding will facilitate better mechanical and control design for bipedal robots. Bipedal walking is hybrid in nature, characterized by periodic contact between a robot and the environment, i.e., the ground. Dynamic models derived from Lagrangians modeling mechanical systems govern the continuous dynamics while discrete dynamics were handed by an impact model using impulse-like forces and balancing angular momentum. This combination of continuous and discrete dynamics motivated the use of hybrid systems for modeling purposes. The framework of hybrid systems was used to model three-dimensional bipedal walking in a general setup for a robotic model with a hip, knees, and feet with the goal of obtaining stable walking. To achieve three-dimensional walking, functional Routhian reduction was used to decouple the sagittal and coronal dynamics. By doing so, it was possible to achieve walking in the two-dimensional sagittal plane on the three-dimensional model, restricted to operate in the sagittal plane. Imposing this restriction resulted in a reduced-order model, referred to as the sagittally-restricted model. Sagittal control in the form of controlled symmetries and additional control strategies was used to achieve stable walking on the sagittally-restricted model. Functional Routhian reduction was then applied to the full-order system. The sagittal control developed on the reduced-order model was used with reduction to achieve walking in three dimensions in simulation. The control schemes described resulted in walking which was remarkably anthropomorphic in nature. This observation is surprising given the simplistic nature of the controllers used. Moreover, the two-dimensional and three-dimensional dynamics were completely decoupled inasmuch as the dynamic models governing the sagittal motion were equivalent. Additionally, the reduction resulted in swaying in the lateral plane. This motion, which is generally present in human walking, was unplanned and was a side-effect of the decoupling process. Despite the approximate nature of the reduction, the motion was still almost completely decoupled with respect to the sagittal and coronal planes.

This thesis covers the study of an actively assisted passive walker with discontinuous and impulsive actuation. The dynamics of the passive and active portions are derived, and a comprehensive mathematical model is proposed. An actuation method is also proposed to study the use of multiple discrete actuation events in a walking gait. Two key cases are considered: actuation at the stance point and at the EA point of a non-kneed walker. An experimental walker was designed that is capable of passive walking and has an experimental implementation of the proposed actuation system. A thorough characterization of the model is then performed, with experimental validation to show that: at high ramp angles, energy injection results in an increase in BOA of ~38% on a stable walking gait at a Ct of 0.086, and at low ramp angles, injection results in a stride length increase of ~29% at a Ct of 0.06.