Restricted Discrete Invariance and Self-Synchronization For Stable Walking of Bipedal Robots

Models of bipedal locomotion are hybrid, with a continuous component often generated by a Lagrangian plus actuators, and a discrete component where leg transfer takes place. The discrete component typically consists of a locally embedded co-dimension one submanifold in the continuous state space of the robot, called the switching surface, and a reset map that provides a new initial condition when a solution of the continuous component intersects the switching surface. The aim of this paper is to identify a low-dimensional submanifold of the switching surface, which, when it can be rendered invariant by the closed-loop dynamics, leads to asymptotically stable periodic gaits. The paper begins this process by studying the well-known 3D Linear Inverted Pendulum (LIP) model, where analytical results are much easier to obtain. A key contribution here is the notion of \textit{self-synchronization}, which refers to the periods of the pendular motions in the sagittal and frontal planes tending to a common period. The notion of invariance resulting from the study of the 3D LIP model is then extended to a 9-DOF 3D biped. A numerical study is performed to illustrate that asymptotically stable walking may be obtained.Comment: Conferenc

Virtual Constraints and Hybrid Zero Dynamics for Realizing Underactuated Bipedal Locomotion

Underactuation is ubiquitous in human locomotion and should be ubiquitous in bipedal robotic locomotion as well. This chapter presents a coherent theory for the design of feedback controllers that achieve stable walking gaits in underactuated bipedal robots. Two fundamental tools are introduced, virtual constraints and hybrid zero dynamics. Virtual constraints are relations on the state variables of a mechanical model that are imposed through a time-invariant feedback controller. One of their roles is to synchronize the robot's joints to an internal gait phasing variable. A second role is to induce a low dimensional system, the zero dynamics, that captures the underactuated aspects of a robot's model, without any approximations. To enhance intuition, the relation between physical constraints and virtual constraints is first established. From here, the hybrid zero dynamics of an underactuated bipedal model is developed, and its fundamental role in the design of asymptotically stable walking motions is established. The chapter includes numerous references to robots on which the highlighted techniques have been implemented.Comment: 17 pages, 4 figures, bookchapte

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

Omnidirectional Walking Pattern Generator Combining Virtual Constraints and Preview Control for Humanoid Robots

This paper presents a novel omnidirectional walking pattern generator for bipedal locomotion combining two structurally different approaches based on the virtual constraints and the preview control theories to generate a flexible gait that can be modified on-line. The proposed strategy synchronizes the displacement of the robot along the two planes of walking: the zero moment point based preview control is responsible for the lateral component of the gait, while the sagittal motion is generated by a more dynamical approach based on virtual constraints. The resulting algorithm is characterized by a low computational complexity and high flexibility, requisite for a successful deployment to humanoid robots operating in real world scenarios. This solution is motivated by observations in biomechanics showing how during a nominal gait the dynamic motion of the human walk is mainly generated along the sagittal plane. We describe the implementation of the algorithm and we detail the strategy chosen to enable omnidirectionality and on-line gait tuning. Finally, we validate our strategy through simulation experiments using the COMAN + platform, an adult size humanoid robot developed at Istituto Italiano di Tecnologia. Finally, the hybrid walking pattern generator is implemented on real hardware, demonstrating promising results: the WPG trajectories results in open-loop stable walking in the absence of external disturbances

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

From Bipedal to Quadrupedal Locomotion, Experimental Realization of Lyapunov Approaches

Possibly one of the most significant innovations of the past decade is the hybrid zero dynamics (HZD) framework, which formally and rigorously designs a control algorithm for robotic walking. In this methodology, Lyapunov stability, which is often used to certificate a dynamical system's stability, was introduced to the control law design for a hybrid control system. However, the prerequisites of precise modeling to apply the HZD methodology can often be too restrictive to design controllers for uncertain and complex real-world hardware experiments. This thesis addresses the problem raised by noisy measurements and the intricate hybrid structure of locomotion dynamics. First, the HZD methodology's construction is based on the full-order, hybrid dynamics of legged locomotion, which can be intractable for control synthesis for high-dimensional systems. This thesis studies the general structure of hybrid control systems for walking systems, ranging from 1D hopping, 2D walking, 2D running, and 3D quadrupedal locomotion on rough terrains. Further, we characterize a walking behavior--gait--as a solution (execution) to a hybrid control system. To find these solutions, which represent a "gait," we employed advanced numerical methods such as collocation methods to parse the solution-finding problem into the open- and closed-loop trajectory optimization problems. The result is that we can find versatile gaits for ten different robotic platforms efficiently. This includes bipedal running, bipedal walking on slippery surfaces, and quadrupedal robots walking on sloped terrains. The numerous solution-finding examples expand the applicability of the HZD framework towards more complex dynamical systems. Further, for the uncertain and noisy real-world implementation, the exponential stability of the continuous dynamics is an ideal but restrictive condition for hybrid stability. This condition is especially challenging to satisfy for highly dynamical behaviors such as bipedal running, which loses ground support for a short period. This thesis observes the destabilizing effect of the noisy measurements of the phasing variable. By reformulating the traditional input-to-state stability (ISS) concept into phase-uncertainty to state stability, we are able to synthesize a robust controller for bipedal running on DURUS-2D. This time+state-based controller formally guarantees stability under noisy measurements and stabilizes the 1.75 m/s running experiments. Lastly, robotic dynamics have long been characterized as the interconnection of rigid-body dynamics. We take this perspective one step further and incorporate controller design into the formulation of coupled control systems (CCS). We first view a quadrupedal robot as two bipedal robots connected via some holonomic constraints. In a dimensional reduction manner, we develop a novel optimization framework, and the computational performance is reduced to a few seconds for gait generation. Furthermore, we can design local controllers for each bipedal subsystem and still guarantee the overall system's stability. This is done by combining the HZD framework and the ISS properties to contain the disturbance induced by the other subsystems' inputs. Utilizing the proposed CCS methods, we will experimentally realize quadrupedal walking on various outdoor rough terrains.</p

Evolutionary robotics and neuroscience

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Locomoção bípede adaptativa a partir de uma única demonstração usando primitivas de movimento

Doutoramento em Engenharia EletrotécnicaEste trabalho aborda o problema de capacidade de imitação da locomoção humana através da utilização de trajetórias de baixo nível codificadas com primitivas de movimento e utilizá-las para depois generalizar para novas situações, partindo apenas de uma demonstração única. Assim, nesta linha de pensamento, os principais objetivos deste trabalho são dois: o primeiro é analisar, extrair e codificar demonstrações efetuadas por um humano, obtidas por um sistema de captura de movimento de forma a modelar tarefas de locomoção bípede. Contudo, esta transferência não está limitada à simples reprodução desses movimentos, requerendo uma evolução das capacidades para adaptação a novas situações, assim como lidar com perturbações inesperadas. Assim, o segundo objetivo é o desenvolvimento e avaliação de uma estrutura de controlo com capacidade de modelação das ações, de tal forma que a demonstração única apreendida possa ser modificada para o robô se adaptar a diversas situações, tendo em conta a sua dinâmica e o ambiente onde está inserido. A ideia por detrás desta abordagem é resolver o problema da generalização a partir de uma demonstração única, combinando para isso duas estruturas básicas. A primeira consiste num sistema gerador de padrões baseado em primitivas de movimento utilizando sistemas dinâmicos (DS). Esta abordagem de codificação de movimentos possui propriedades desejáveis que a torna ideal para geração de trajetórias, tais como a possibilidade de modificar determinados parâmetros em tempo real, tais como a amplitude ou a frequência do ciclo do movimento e robustez a pequenas perturbações. A segunda estrutura, que está embebida na anterior, é composta por um conjunto de osciladores acoplados em fase que organizam as ações de unidades funcionais de forma coordenada. Mudanças em determinadas condições, como o instante de contacto ou impactos com o solo, levam a modelos com múltiplas fases. Assim, em vez de forçar o movimento do robô a situações pré-determinadas de forma temporal, o gerador de padrões de movimento proposto explora a transição entre diferentes fases que surgem da interação do robô com o ambiente, despoletadas por eventos sensoriais. A abordagem proposta é testada numa estrutura de simulação dinâmica, sendo que várias experiências são efetuadas para avaliar os métodos e o desempenho dos mesmos.This work addresses the problem of learning to imitate human locomotion actions through low-level trajectories encoded with motion primitives and generalizing them to new situations from a single demonstration. In this line of thought, the main objectives of this work are twofold: The first is to analyze, extract and encode human demonstrations taken from motion capture data in order to model biped locomotion tasks. However, transferring motion skills from humans to robots is not limited to the simple reproduction, but requires the evaluation of their ability to adapt to new situations, as well as to deal with unexpected disturbances. Therefore, the second objective is to develop and evaluate a control framework for action shaping such that the single-demonstration can be modulated to varying situations, taking into account the dynamics of the robot and its environment. The idea behind the approach is to address the problem of generalization from a single-demonstration by combining two basic structures. The first structure is a pattern generator system consisting of movement primitives learned and modelled by dynamical systems (DS). This encoding approach possesses desirable properties that make them well-suited for trajectory generation, namely the possibility to change parameters online such as the amplitude and the frequency of the limit cycle and the intrinsic robustness against small perturbations. The second structure, which is embedded in the previous one, consists of coupled phase oscillators that organize actions into functional coordinated units. The changing contact conditions plus the associated impacts with the ground lead to models with multiple phases. Instead of forcing the robot’s motion into a predefined fixed timing, the proposed pattern generator explores transition between phases that emerge from the interaction of the robot system with the environment, triggered by sensor-driven events. The proposed approach is tested in a dynamics simulation framework and several experiments are conducted to validate the methods and to assess the performance of a humanoid robot