Nomenclature

ABSTRACT — Bouncing, balancing and swinging the leg forward can be considered as three basic control tasks for bipedal locomotion. Defining the trunk by an unstable inverted pendulum, balancing as being translated to trunk stabilization is the main focus of this paper. The control strategy is to generate a hip torque to have upright trunk to achieve robust hopping and running. It relies on the Virtual Pendulum (VP) concept which is recently proposed for trunk stabilization, based on human/animal locomotion analysis. Based on this concept, a control approach, named Virtual Pendulum Posture control (VPPC) is presented, in which the trunk is stabilized by redirecting the ground reaction force to a virtual support point. The required torques patterns generated by the controller, could partially be exerted by elastic structures like hip springs. Hybrid Zero Dynamics (HZD) control approach is also applied as an exact method of keeping the trunk upright. Stability of the motion which is investigated by Poincare ´ map analysis could be achieved by hip springs, VPPC and HZD. The results show that hip springs, revealing muscle properties, could facilitate trunk stabilization. Compliance in hip produces acceptable performance and robustness compared with VPPC and HZD, while it is a passive structure

Bioinspired template-based control of legged locomotion

cient and robust locomotion is a crucial condition for the more extensive use of legged robots in real world applications. In that respect, robots can learn from animals, if the principles underlying locomotion in biological legged systems can be transferred to their artificial counterparts. However, legged locomotion in biological systems is a complex and not fully understood problem. A great progress to simplify understanding locomotion dynamics and control was made by introducing simple models, coined ``templates'', able to represent the overall dynamics of animal (including human) gaits. One of the most recognized models is the spring-loaded inverted pendulum (SLIP) which consists of a point mass atop a massless spring. This model provides a good description of human gaits, such as walking, hopping and running. Despite its high level of abstraction, it supported and inspired the development of successful legged robots and was used as explicit targets for control, over the years. Inspired from template models explaining biological locomotory systems and Raibert's pioneering legged robots, locomotion can be realized by basic subfunctions: (i) stance leg function, (ii) leg swinging and (iii) balancing. Combinations of these three subfunctions can generate different gaits with diverse properties. Using the template models, we investigate how locomotor subfunctions contribute to stabilize different gaits (hopping, running and walking) in different conditions (e.g., speeds). We show that such basic analysis on human locomotion using conceptual models can result in developing new methods in design and control of legged systems like humanoid robots and assistive devices (exoskeletons, orthoses and prostheses). This thesis comprises research in different disciplines: biomechanics, robotics and control. These disciplines are required to do human experiments and data analysis, modeling of locomotory systems, and implementation on robots and an exoskeleton. We benefited from facilities and experiments performed in the Lauflabor locomotion laboratory. Modeling includes two categories: conceptual (template-based, e.g. SLIP) models and detailed models (with segmented legs, masses/inertias). Using the BioBiped series of robots (and the detailed BioBiped MBS models; MBS stands for Multi-Body-System), we have implemented newly-developed design and control methods related to the concept of locomotor subfunctions on either MBS models or on the robot directly. In addition, with involvement in BALANCE project (\url{http://balance-fp7.eu/}), we implemented balance-related control approaches on an exoskeleton to demonstrate their performance in human walking. The outcomes of this research includes developing new conceptual models of legged locomotion, analysis of human locomotion based on the newly developed models following the locomotor subfunction trilogy, developing methods to benefit from the models in design and control of robots and exoskeletons. The main contribution of this work is providing a novel approach for modular control of legged locomotion. With this approach we can identify the relation between different locomotor subfunctions e.g., between balance and stance (using stance force for tuning balance control) or balance and swing (two joint hip muscles can support the swing leg control relating it to the upper body posture) and implement the concept of modular control based on locomotor subfunctions with a limited exchange of sensory information on several hardware platforms (legged robots, exoskeleton)

Trajectory Optimization and Machine Learning to Design Feedback Controllers for Bipedal Robots with Provable Stability

This thesis combines recent advances in trajectory optimization of hybrid dynamical systems with machine learning and geometric control theory to achieve unprecedented performance in bipedal robot locomotion. The work greatly expands the class of robot models for which feedback controllers can be designed with provable stability. The methods are widely applicable beyond bipedal robots, including exoskeletons, and prostheses, and eventually, drones, ADAS, and other highly automated machines. One main idea of this thesis is to greatly expand the use of multiple trajectories in the design of a stabilizing controller. The computation of many trajectories is now feasible due to new optimization tools. The computations are not fast enough to apply in the real-time, however, so they are not feasible for model predictive control (MPC). The offline “library” approach will encounter the curse of dimensionality for the high-dimensional models common in bipedal robots. To overcome these obstructions, we embed a stable walking motion in an attractive low-dimensional surface of the system's state space. The periodic orbit is now an attractor of the low-dimensional state-variable model but is not attractive in the full-order system. We then use the special structure of mechanical models associated with bipedal robots to embed the low-dimensional model in the original model in such a manner that the desired walking motions are locally exponentially stable. The ultimate solution in this thesis will generate model-based feedback controllers for bipedal robots, in such a way that the closed-loop system has a large stability basin, exhibits highly agile, dynamic behavior, and can deal with significant perturbations coming from the environment. In the case of bipeds: “model-based” means that the controller will be designed on the basis of the full floating-base dynamic model of the robot, and not a simplified model, such as the LIP (Linear Inverted Pendulum). By “agile and dynamic” is meant that the robot moves at the speed of a normal human or faster while walking off a curb. By “significant perturbation” is meant a human tripping, and while falling, throwing his/her full weight into the back of the robot.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145992/1/xda_1.pd

LeggedWalking on Inclined Surfaces

The main contribution of this MS Thesis is centered around taking steps towards successful multi-modal demonstrations using Northeastern's legged-aerial robot, Husky Carbon. This work discusses the challenges involved in achieving multi-modal locomotion such as trotting-hovering and thruster-assisted incline walking and reports progress made towards overcoming these challenges. Animals like birds use a combination of legged and aerial mobility, as seen in Chukars' wing-assisted incline running (WAIR), to achieve multi-modal locomotion. Chukars use forces generated by their flapping wings to manipulate ground contact forces and traverse steep slopes and overhangs. Husky's design takes inspiration from birds such as Chukars. This MS thesis presentation outlines the mechanical and electrical details of Husky's legged and aerial units. The thesis presents simulated incline walking using a high-fidelity model of the Husky Carbon over steep slopes of up to 45 degrees.Comment: Masters thesi

Energy Shaping of Underactuated Systems via Interconnection and Damping Assignment Passivity-Based Control with Applications to Planar Biped Robots

The sought goal of this thesis is to show that total energy shaping is an effective and versatile tool to control underactuated mechanical systems. The performance of several approaches, rooted in the port-Hamiltonian formalism, are analyzed while tackling distinct control problems: i) equilibrium stabilization; ii) gait generation; iii) gait robustication. Firstly, a constructive solution to deal with interconnection and damping assignment passivity-based control (IDA-PBC) for underactuated two-degree-of-freedom mechanical systems is proposed. This strategy does not involve the resolution of any partial differential equation, since explicit solutions are given, while no singularities depending on generalized momenta are introduced by the controller. The methodology is applied to the stabilization of a translational oscillator with a rotational actuator system, as well as, to the gait generation for an underactuated compass-like biped robot (CBR). Then, the problem of gait generation is addressed using dissipative forces in the controller. In this sense, three distinct controllers are presented, namely simultaneous interconnection and damping assignment passivity-based control with dissipative forces, energy pumping-and-damping passivity-based control (EPD-PBC), and energy pumping-or-damping control. Finally, EPD-PBC is used to increase the robustness of the gait exhibited by the CBR over uncertainties on the initial conditions. The passivity of the system is exploited, as well as, its hybrid nature (using the hybrid zero dynamics method) to carry out the stability analysis. Besides, such an approach is applied to new gaits that are generated using IDA-PBC. Numerical case studies, comparisons, and critical discussions evaluate the performance of the proposed approaches

Stabilizing Highly Dynamic Locomotion in Planar Bipedal Robots with Dimension Reducing Control.

In the field of robotic locomotion, the method of hybrid zero dynamics (HZD) proposed by Westervelt, Grizzle, and Koditschek provided a new solution to the canonical problem of stabilizing walking in planar bipeds. Original walking experiments on the French biped RABBIT were very successful, with gaits that were robust to external disturbances and to parameter mismatch. Initial running experiments on RABBIT were cut short before a stable gait could be achieved, but helped to identify performance limiting aspects of both the physical hardware of RABBIT and the method of hybrid zero dynamics. To improve upon RABBIT, a new robot called MABEL was designed and constructed in collaboration between the University of Michigan and Carnegie Mellon University. In light of experiments on RABBIT and in preparation for experiments on MABEL, this thesis provides a theoretical foundation that extends the method of hybrid zero dynamics to address walking in a class of robots with series compliance. Extensive new design tools address two main performance limiting aspects of previous HZD controllers: the dependence on non-Lipschitz finite time convergence and the lack of a constructive procedure for achieving impact invariance when outputs have relative degree greater than two. An analytically rigorous set of solutions - an arbitrarily smooth stabilizing controller and a constructive parameter update scheme - is derived using the method of Poincare sections. Additional contributions of this thesis include the development of sample-based virtual constraints, analysis of walking on a slope, and identification of dynamic singularities that can arise from poorly chosen virtual constraints.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58477/1/morrisbj_1.pd

Upslope Walking with Transfemoral Prosthesis Using Optimization Based Spline Generation

Powered prosthetic devices are robotic systems that are aimed to restore the mobility of subjects with amputations above the knee by imitating the behavior of a normal human leg. Powered prostheses have diverse advantages compared to passive devices, including the possibility of reducing the metabolic cost of the user, providing net power into the walking gait and walking on diverse terrain. In particular, this thesis is focused on the capacity of powered transfemoral prostheses to adapt to diverse terrains. Since most terrains consist of flat and inclined surfaces, it is important that a transfemoral prosthesis can walk on these surfaces and have the capacity to seamlessly transition from one surface to another. However, currently available controllers require either intention recognition procedures that delay the terrain transition or a collection of parameters that require a large tuning process for each possible surface profile. In this thesis, we propose a framework that can generate automatically stable and human-like gaits for both surfaces with immediate transition between them. The new framework is based on human-inspired control and a spline-based trajectory generation. Specifically, the proposed method i) inserts a set of cubic splines that smoothly blend the flat ground joint trajectories into arbitrary upslope surface joint trajectories for the ankle and knee joints and ii) employs a low gain PD control for terrain adaptation for various unknown surfaces. This framework is implemented on the powered transfemoral prosthetic device, AMPRO II, for both flat ground and upslope walking to test its use as a nominal controller. The experimental results confirm that the proposed framework provides walking gaits for flat ground and upslope with seamlessly smooth transitioning gaits between them

Symmetry Method for Limit Cycle Walking of Legged Robots.

Dynamic steady-state walking or running gaits for legged robots correspond to periodic orbits in the dynamic model. The common method for obtaining such periodic orbits is conducting a numerical search for fixed points of a Poincare map. However, as the number of degrees of freedom of the robot grows, such numerical search becomes computationally expensive because in each search trial the dynamic equations need to be integrated. Moreover, the numerical search for periodic orbits is in general sensitive to model errors, and it remains to be seen if the periodic orbit which is the outcome of the search in the domain of the dynamic model corresponds to a periodic gait in the actual robot. To overcome these issues, we have presented the Symmetry Method for Limit Cycle Walking, which relaxes the need to search for periodic orbits, and at the same time, the limit cycles obtained with this method are robust to model errors. Mathematically, we describe the symmetry method in the context of so-called Symmetric Hybrid Systems, whose properties are discussed. In particular, it is shown that a symmetric hybrid system can have an infinite number of periodic orbits that can be identified easily. In addition, it is shown how control strategies need to be selected so that the resulting reduced order system still possesses the properties of a symmetric hybrid system. The method of symmetry for limit cycle walking is successfully tested on a 12-DOF 3D model of the humanoid robot Romeo.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133356/1/razavi_1.pd

Control Implementation of Dynamic Locomotion on Compliant, Underactuated, Force-Controlled Legged Robots with Non-Anthropomorphic Design

The control of locomotion on legged robots traditionally involves a robot that takes a standard legged form, such as the anthropomorphic humanoid, the dog-like quadruped, or the bird-like biped. Additionally, these systems will often be actuated with position-controlled servos or series-elastic actuators that are connected through rigid links. This work investigates the control implementation of dynamic, force-controlled locomotion on a family of legged systems that significantly deviate from these classic paradigms by incorporating modern, state-of-the-art proprioceptive actuators on uniquely configured compliant legs that do not closely resemble those found in nature. The results of this work can be used to better inform how to implement controllers on legged systems without stiff, position-controlled actuators, and also provide insight on how intelligently designed mechanical features can potentially simplify the control of complex, nonlinear dynamical systems like legged robots. To this end, this work presents the approach to control for a family of non-anthropomorphic bipedal robotic systems which are developed both in simulation and with physical hardware. The first is the Non-Anthropomorphic Biped, Version 1 (NABi-1) that features position-controlled joints along with a compliant foot element on a minimally actuated leg, and is controlled using simple open-loop trajectories based on the Zero Moment Point. The second system is the second version of the non-anthropomorphic biped (NABi-2) which utilizes the proprioceptive Back-drivable Electromagnetic Actuator for Robotics (BEAR) modules for actuation and fully realizes feedback-based force controlled locomotion. These systems are used to highlight both the strengths and weaknesses of utilizing proprioceptive actuation in systems, and suggest the tradeoffs that are made when using force control for dynamic locomotion. These systems also present case studies for different approaches to system design when it comes to bipedal legged robots