Beyond Basins of Attraction: Quantifying Robustness of Natural Dynamics

Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the basins of attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond basins of attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.Comment: 15 pages. This work has been accepted to IEEE Transactions on Robotics (2019

Orbit Characterization, Stabilization and Composition on 3D Underactuated Bipedal Walking via Hybrid Passive Linear Inverted Pendulum Model

A Hybrid passive Linear Inverted Pendulum (H-LIP) model is proposed for characterizing, stabilizing and composing periodic orbits for 3D underactuated bipedal walking. Specifically, Period-l (P1) and Period -2 (P2) orbits are geometrically characterized in the state space of the H-LIP. Stepping controllers are designed for global stabilization of the orbits. Valid ranges of the gains and their optimality are derived. The optimal stepping controller is used to create and stabilize the walking of bipedal robots. An actuated Spring-loaded Inverted Pendulum (aSLIP) model and the underactuated robot Cassie are used for illustration. Both the aSLIP walking with PI or P2 orbits and the Cassie walking with all 3D compositions of the PI and P2 orbits can be smoothly generated and stabilized from a stepping-in-place motion. This approach provides a perspective and a methodology towards continuous gait generation and stabilization for 3D underactuated walking robots

Motion Planning and Control of Dynamic Humanoid Locomotion

Inspired by human, humanoid robots has the potential to become a general-purpose platform that lives along with human. Due to the technological advances in many field, such as actuation, sensing, control and intelligence, it finally enables humanoid robots to possess human comparable capabilities. However, humanoid locomotion is still a challenging research field. The large number of degree of freedom structure makes the system difficult to coordinate online. The presence of various contact constraints and the hybrid nature of locomotion tasks make the planning a harder problem to solve. Template model anchoring approach has been adopted to bridge the gap between simple model behavior and the whole-body motion of humanoid robot. Control policies are first developed for simple template models like Linear Inverted Pendulum Model (LIPM) or Spring Loaded Inverted Pendulum(SLIP), the result controlled behaviors are then been mapped to the whole-body motion of humanoid robot through optimization-based task-space control strategies. Whole-body humanoid control framework has been verified on various contact situations such as unknown uneven terrain, multi-contact scenarios and moving platform and shows its generality and versatility. For walking motion, existing Model Predictive Control approach based on LIPM has been extended to enable the robot to walk without any reference foot placement anchoring. It is kind of discrete version of \u201cwalking without thinking\u201d. As a result, the robot could achieve versatile locomotion modes such as automatic foot placement with single reference velocity command, reactive stepping under large external disturbances, guided walking with small constant external pushing forces, robust walking on unknown uneven terrain, reactive stepping in place when blocked by external barrier. As an extension of this proposed framework, also to increase the push recovery capability of the humanoid robot, two new configurations have been proposed to enable the robot to perform cross-step motions. For more dynamic hopping and running motion, SLIP model has been chosen as the template model. Different from traditional model-based analytical approach, a data-driven approach has been proposed to encode the dynamics of the this model. A deep neural network is trained offline with a large amount of simulation data based on the SLIP model to learn its dynamics. The trained network is applied online to generate reference foot placements for the humanoid robot. Simulations have been performed to evaluate the effectiveness of the proposed approach in generating bio-inspired and robust running motions. The method proposed based on 2D SLIP model can be generalized to 3D SLIP model and the extension has been briefly mentioned at the end

3LP: a linear 3D-walking model including torso and swing dynamics

In this paper, we present a new model of biped locomotion which is composed of three linear pendulums (one per leg and one for the whole upper body) to describe stance, swing and torso dynamics. In addition to double support, this model has different actuation possibilities in the swing hip and stance ankle which could be widely used to produce different walking gaits. Without the need for numerical time-integration, closed-form solutions help finding periodic gaits which could be simply scaled in certain dimensions to modulate the motion online. Thanks to linearity properties, the proposed model can provide a computationally fast platform for model predictive controllers to predict the future and consider meaningful inequality constraints to ensure feasibility of the motion. Such property is coming from describing dynamics with joint torques directly and therefore, reflecting hardware limitations more precisely, even in the very abstract high level template space. The proposed model produces human-like torque and ground reaction force profiles and thus, compared to point-mass models, it is more promising for precise control of humanoid robots. Despite being linear and lacking many other features of human walking like CoM excursion, knee flexion and ground clearance, we show that the proposed model can predict one of the main optimality trends in human walking, i.e. nonlinear speed-frequency relationship. In this paper, we mainly focus on describing the model and its capabilities, comparing it with human data and calculating optimal human gait variables. Setting up control problems and advanced biomechanical analysis still remain for future works.Comment: Journal paper under revie

Sequential Motion Planning for Bipedal Somersault via Flywheel SLIP and Momentum Transmission with Task Space Control

In this paper, we present a sequential motion planning and control method for generating somersaults on bipedal robots. The somersault (backflip or frontflip) is considered as a coupling between an axile hopping motion and a rotational motion about the center of mass of the robot; these are encoded by a hopping Spring-loaded Inverted Pendulum (SLIP) model and the rotation of a Flywheel, respectively. We thus present the Flywheel SLIP model for generating the desired motion on the ground phase. In the flight phase, we present a momentum transmission method to adjust the orientation of the lower body based on the conservation of the centroidal momentum. The generated motion plans are realized on the full-dimensional robot via momentum-included task space control. Finally, the proposed method is implemented on a modified version of the bipedal robot Cassie in simulation wherein multiple somersault motions are generated

사람 보행 분석 연구와 그 결과를 활용한 휴머노이드 로봇 보행 패턴 생성

학위논문 (박사) -- 서울대학교 대학원 : 융합과학기술대학원 융합과학부(지능형융합시스템전공), 2020. 8. 박재흥.발의 미끄러짐은 보행의 안정성을 떨어트리는 요인 중 하나이다. 보행 중 발에 발생하는 수평 전단력이 발과 지면 사이의 마찰력보다 커지면, 발은 접촉을 상실하고 미끄러지게 된다. 여기서, 발과 지면 사이의 마찰력은 발에 작용하는 수직력에 의해 결정되게 된다. 즉, 휴머노이드 로봇 보행 패턴 생성의 측면에서 보자면, 로봇 발에 발생하는 수평력과 수직력을 어떻게 설계하는지에 따라 보행 중 미끄러짐의 가능성이 바뀐다는 것이다. 선형 역진자 모델은 휴머노이드 로봇의 무게 중심 궤적 생성을 위해 자주 사용되어왔다. 선형 역진자 모델은 로봇의 무게 중심 높이를 일정하게 유지하도록 제한한다. 무게 중심의 높이 제한 때문에 로봇의 수직 방향의 가속도는 보행 속도와 관련 없이 항상 중력 가속도가 된다. 그러나 수평 방향의 가속도는 보행 속도가 증가하면 비례하여 증가한다. 따라서 빠른 보행 속도에서는 수직력에 비례하는 마찰력에 비해 수평 전단력이 커지면서 발의 미끄러짐이 발생할 수 있다. 선형 역진자 모델에 의한 일정한 수직 높이 구속 조건이 로봇 발의 미끄러짐을 유발할 수 있다는 것을 시사한다. 무게 중심의 적절한 수직 움직임을 생성함으로써 휴머노이드 로봇 보행 중 발의 미끄러짐을 줄일 수 있다. 인간공학 분야에서는 Available Coefficient of Friction(aCOF)과 Utilized Coefficient of Friction(uCOF)을 이용하여 사람 보행 중 발의 미끄러짐 가능성을 예측하는 연구들이 수행됐다. 여기서, aCOF는 두 물체의 재질이나 상태에 의해 결정되는 마찰 계수이다. 반면, uCOF는 보행 중 지지하는 발에 가해지는 수평 전단력과 수직력의 비이다. 인간공학 연구들에 따르면, uCOF가 aCOF를 초과할 때 발은 접촉을 상실하고 미끄러지게 된다. 로봇 발의 미끄러짐 감소를 위해서는 로봇 보행 중 발에 발생하는 uCOF가 로봇 발과 지면 사이의 aCOF 보다 작아지도록 적절한 수직 방향의 무게 중심 궤적을 생성하는 것이 필요하다. 다양한 형태의 수직 방향의 무게 중심 궤적 생성이 가능한데, 간단하면서도 효율적인 방법은 무게 중심의 에너지가 보존되도록 수직 방향의 무게 중심 궤적을 생성하는 것이다. 기존 선형 역진자 모델을 이용해 수평 방향의 무게 중심 궤적을 생성하고, 운동 에너지와 위치 에너지가 교환되면서 전체 에너지가 보존되는 수직 방향의 무게 중심 궤적을 추가하는 것이다. 무게 중심의 에너지 보존 원리를 이용하여 무게 중심의 양의 일(Mechanical Work) 생성을 최소화함으로써 관절의 양의 일 생성을 감소시키고, 이를 통해 보행 중 에너지 효율을 높이는 것이 가능하다. 이 논문은 발과 지면 사이의 aCOF 보다 작도록 보행 중 uCOF를 유지하면서 무게 중심의 양의 일을 최소화하는 적절한 수직 방향의 무게 중심 궤적을 생성하는 것을 목표로 한다. 발의 미끄러짐이 감소하면서 에너지 효율이 높은 휴머노이드 로봇 보행 패턴 생성을 위해, 먼저 사람 보행 중 uCOF에 관한 연구와 사람 보행 중 관절의 일에 관한 연구를 선행한다. 사람 보행에 관한 분석 연구와 사람 보행의 원리 이해를 통해 최적화 알고리즘 기반 수직 방향의 무게 중심 궤적 생성 방법이 제시된다. 제시된 알고리즘을 이용하여 구해진 수직 방향의 무게 중심 궤적을 휴머노이드 로봇 보행 실험에 적용한다. 궁극적으로 이 논문은, 수직 방향의 무게 중심 궤적을 추가함으로써 기존 선형 역진자 모델의 한계를 극복하여, 미끄러짐의 가능성이 감소하고 에너지 효율이 높은 휴머노이드 로봇 보행 패턴을 생성한다.Foot slippage is one of the factors responsible for the increasing instability during human walking. A slip occurs when the horizontal shear force acting on the foot becomes greater than the frictional force between the foot and the ground, which is proportional to the vertical force. For humanoid robot walking, the possibility of a slip depends upon how the horizontal shear force and vertical force both acting on the foot are designed. In the linear inverted pendulum model (LIPM), which is commonly used to generate the center of mass (COM) trajectory of humanoid robots, the vertical height of the COM is kept constant. The constant height of the COM restricts that the vertical force is always equal to the gravitational force at any walking speed. However, upon increasing the walking speed, the horizontal ground reaction force increases in proportion with the forward and lateral accelerations of the COM. This increase in the horizontal ground reaction force, while the vertical ground force is being constant, suggests that the robot-foot slippage can occur because of the restriction of the vertical motion by the LIPM constraint. By generating the appropriate vertical motion, the robot-foot slippage can be reduced during humanoid robot walking. Researchers in the field of ergonomics have been conducted studies on the relationship between the available coefficient of friction (aCOF) and the utilized coefficient of friction (uCOF) to predict the potential for a slip during human walking. The aCOF is both the static and dynamic coefficient of friction between two objects in contact, and it depends on the properties of the objects. The uCOF is the ratio of the horizontal shear force to the vertical force applied by the supporting foot. Foot slippage occurs when the uCOF exceeds the aCOF. Various types of vertical motion can set the maximum value of the uCOF to be less than the aCOF between the foot and floor for humanoid robot walking. One of the simple and energy-efficient methods is to minimize the mechanical work of the COM by introducing added vertical motion. Therefore, the COM pattern would become more energy efficient by exchanging kinetic energy and potential energy. This thesis aims to generate the appropriate vertical motion of the COM to maintain the utilized coefficient of friction (uCOF) less than the available coefficient of friction between the foot and the ground, and to minimize the mechanical work during humanoid robot walking. Before generating a slip-safe and energy-efficient COM trajectory for humanoid robot walking, studies on analyzing the COM patterns, mechanical work, and uCOF during human walking are conducted to understand the principle of walking. Vertical motions at various speeds are generated using an optimization method. Subsequently, the generated COM motion patterns are used as reference trajectories of the COM for humanoid robot walking. This thesis suggests a way to generate slip-safe and energy-efficient COM patterns, which, in turn, overcome the limitations of the LIPM by adding vertical COM motion.Chapter 1 Introduction 1 1.1 Research Background 1 1.2 Contributions of Thesis 3 1.3 Overviews of Thesis 4 Chapter 2 Dynamics of Walking 5 2.1 Walking Model 5 2.1.1 Linear Inverted Pendulum Model 5 2.1.2 Spring-Loaded Inverted Pendulum Model 6 2.1.3 Extrapolated Center of Mass Dynamics 9 2.2 Walking Theory 11 2.2.1 Step-to-Step Transition 11 Chapter 3 HumanWalking Analysis 13 3.1 Motion Capture for Walking 13 3.1.1 Motion Capture Technology 13 3.1.2 Joint Kinematics and Kinetics 15 3.2 Joint and COM During Human Walking 17 3.2.1 Introduction 17 3.2.2 Methods 19 3.2.3 Change of Joint Angle and the COM 20 3.2.4 Discussion 26 3.3 Slipping During Human Walking 27 3.3.1 Introduction 27 3.3.2 Methods 31 3.3.3 Change of uCOF and GRF 34 3.3.4 Interaction Effect Between Heel Area and Speed 36 3.3.5 Discussion 39 3.4 Mechanical Work During Human Walking 44 3.4.1 Introduction 44 3.4.2 Methods 46 3.4.3 Calculation for Joint Mechanical Work 48 3.4.4 Change of Joint Mechanical Work 51 3.4.5 Change of Stride Parameters 53 3.4.6 Discussion 54 Chapter 4 Robot Walking Pattern Generation 59 4.1 Introduction 59 4.2 Forward and Lateral COM 61 4.2.1 XcoM Method 61 4.2.2 Preview Control Method 63 4.3 Vertical COM 64 4.3.1 Calculation for uCOF 64 4.3.2 Calculation for ZMP 65 4.3.3 Calculation for COM Mechanical Work 66 4.3.4 Optimization for Vertical COM Generation 68 4.3.5 Results of Optimization for Vertical COM 73 4.4 Slipping During Robot Walking 75 4.4.1 Robot Simulation 75 4.4.2 Robot Experiments 77 4.5 Mechanical Work During Robot Walking 81 4.5.1 Robot Simulation 81 4.5.2 Robot Experiments 82 4.6 Discussion 87 4.6.1 Tracking Errors in Robot Experiments 87 4.6.2 Effect of Vertical Motions on Real Net Power 91 4.6.3 Trade-Off Between Efficiency and Stability 92 4.6.4 Difference Between Human and Robot 93 Chapter 5 Conclusions 95 Bibliography 97 Abstract (Korean) 111Docto

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