In this paper, we describe a planner capable of generating walking trajectories by using the centroidal dynamics and the full kinematics of a humanoid robot model. The interaction between the robot and the walking surface is modeled explicitly through a novel contact parametrization. The approach is complementarity-free and does not need a predefined contact sequence. By solving an optimal control problem we obtain walking trajectories. In particular, through a set of constraints and dynamic equations, we model the robot in contact with the ground. We describe the objective the robot needs to achieve with a set of tasks. The whole optimal control problem is transcribed into an optimization problem via a Direct Multiple Shooting approach and solved with an off-the-shelf solver. We show that it is possible to achieve walking motions automatically by specifying a minimal set of references, such as a constant desired Center of Mass velocity and a reference point on the ground.
We present a trajectory planning and control architecture for bipedal locomotion at a variety of speeds on a highly underactuated and compliant bipedal robot. A library of compliant walking trajectories are planned offline, and stored as compact arrays of polynomial coefficients for tracking online. The control implementation uses a floating-base inverse dynamics controller which generates dynamically consistent feedforward torques to realize walking using information obtained from the trajectory optimization. The effectiveness of the controller is demonstrated in simulation and on hardware fo…
Italian Institute of Technology（意大利技术研究院）；University of Manchester（曼彻斯特大学）；University of Genoa（热那亚大学）
The paper presents a planner to generate walking trajectories by using the centroidal dynamics and the full kinematics of a humanoid robot. The interaction between the robot and the walking surface is modeled explicitly via new conditions, the \emph{Dynamical Complementarity Constraints}. The approach does not require a predefined contact sequence and generates the footsteps automatically. We characterize the robot control objective via a set of tasks, and we address it by solving an optimal control problem. We show that it is possible to achieve walking motions automatically by specifying a m…
Walking motion planning based on Divergent Component of Motion (DCM) and Linear Inverted Pendulum Model (LIPM) is one of the alternatives that could be implemented to generate online humanoid robot gait trajectories. This algorithm requires different parameters to be adjusted. Herein, we developed a framework to attain optimal parameters to achieve a stable and energy-efficient trajectory for real robot's gait. To find the optimal trajectory, four cost functions representing energy consumption, the sum of joints velocity and applied torque at each lower limb joint of the robot, and a cost func…
A safety-critical measure of legged locomotion performance is a robot's ability to track its desired time-varying position trajectory in an environment, which is herein termed as "global-position tracking". This paper introduces a nonlinear control approach that achieves asymptotic global-position tracking for three-dimensional (3-D) bipedal robot walking. Designing a global-position tracking controller presents a challenging problem due to the complex hybrid robot model and the time-varying desired global-position trajectory. Towards tackling this problem, the first main contribution is the c…
Many methods have been developed in trajectory planning in order to achieve smooth and accurate motion with considering the constraints of kinematics constraints such as angular position, velocity, acceleration, and jerk. The problem of using the combination of n-order polynomials is that there is no ideally match between the segments of trajectory path at the via point in terms of the number of kinematic constraints. It leads to generate undesirable trajectory path at the via point that connects between two segments of the trajectory path. In this paper, we aim to investigate the effect of in…
The Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Google X（谷歌X）；The University of Southern California（南加利福尼亚大学）；New York University（纽约大学）
Cost functions have the potential to provide compact and understandable generalizations of motion. The goal of Inverse Optimal Control (IOC) is to analyze an observed behavior which is assumed to be optimal with respect to an unknown cost function, and infer this cost function. Here we develop a method for characterizing cost functions of legged locomotion, with the goal of representing complex humanoid behavior with simple models. To test this methodology we simulate walking gaits of a simple 5 link planar walking model which optimize known cost functions, and assess the ability of our IOC me…
Indirect Optimal Control for Constructing Libraries of Periodic Trajectories: Application to Bipedal Walking
Institute for Adaptive Mechanical Systems, University of Stuttgart（斯图加特大学自适应机械系统研究所）；Systems Modeling and Simulation, Saarland University（萨尔大学系统建模与仿真研究所）
Bipedal robots have received much attention because of the variety of motion maneuvers that they can produce, and the many applications they have in various areas including rehabilitation. One of these motion maneuvers is walking. In this study, we presented a framework for the trajectory optimization of a 5-link (planar) Biped Robot using hybrid optimization. The walking is modeled with two phases of single-stance (support) phase and the collision phase. The dynamic equations of the robot in each phase are extracted by the Lagrange method. It is assumed that the robot heel strike to the groun…
Discrete time model predictive control for humanoid walking with step adjustment
Robert Bosch Centre for Cyber Physical Systems, Indian Institute of Science（印度科学学院罗伯特·博世网络物理系统中心）；Computer Science and Automation, Indian Institute of Science（印度科学学院计算机科学与自动化系）
The ability to realize nonlinear controllers with formal guarantees on dynamic robotic systems has the potential to enable more complex robotic behaviors -- yet, realizing these controllers is often practically challenging. To address this challenge, this paper presents the end-to-end realization of dynamic bipedal locomotion on an underactuated bipedal robot via hybrid zero dynamics and control Lyapunov functions. A compliant model of Cassie is represented as a hybrid system to set the stage for a trajectory optimization framework. With the goal of achieving a variety of walking speeds in all…
This paper presents a novel Adaptive-frequency MPC framework for bipedal locomotion over terrain with uneven stepping stones. In detail, we intend to achieve adaptive foot placement and gait period for bipedal periodic walking gait with this MPC, in order to traverse terrain with discontinuities without slowing down. We pair this adaptive-frequency MPC with a kino-dynamics trajectory optimization for optimal gait periods, center of mass (CoM) trajectory, and foot placements. We use whole-body control (WBC) along with adaptive-frequency MPC to track the optimal trajectories from the offline opt…
This paper presents a framework that leverages both control theory and machine learning to obtain stable and robust bipedal locomotion without the need for manual parameter tuning. Traditionally, gaits are generated through trajectory optimization methods and then realized experimentally -- a process that often requires extensive tuning due to differences between the models and hardware. In this work, the process of gait realization via hybrid zero dynamics (HZD) based optimization is formally combined with preference-based learning to systematically realize dynamically stable walking. Importa…
We present a framework to generate periodic trajectory references for a 3D under-actuated bipedal robot, using a linear inverted pendulum (LIP) based controller with adaptive neural regulation. We use the LIP template model to estimate the robot's center of mass (CoM) position and velocity at the end of the current step, and formulate a discrete controller that determines the next footstep location to achieve a desired walking profile. This controller is equipped on the frontal plane with a Neural-Network-based adaptive term that reduces the model mismatch between the template and physical rob…
Natural and Robust Walking using Reinforcement Learning without Demonstrations in High-Dimensional Musculoskeletal Models
Max-Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Hertie Institute for Clinical Brain Research（赫蒂临床脑研究所）；Center for Integrative Neuroscience（综合神经科学中心）；Goatstream；Meta AI；University of Stuttgart（斯图加特大学）；University of Tübingen（图宾根大学）；Heidelberg University（海德堡大学）
Oregon State University（俄勒冈州立大学）；Collaborative Robotics and Intelligent Systems Institute（协作机器人与智能系统研究所）
In this paper, we describe an approach to achieve dynamic legged locomotion on physical robots which combines existing methods for control with reinforcement learning. Specifically, our goal is a control hierarchy in which highest-level behaviors are planned through reduced-order models, which describe the fundamental physics of legged locomotion, and lower level controllers utilize a learned policy that can bridge the gap between the idealized, simple model and the complex, full order robot. The high-level planner can use a model of the environment and be task specific, while the low-level le…
Enhancing Model-Based Step Adaptation for Push Recovery through Reinforcement Learning of Step Timing and Region
Automation and Control Institute, TU Wien（维也纳工业大学自动化与控制研究所）；Autonomous Systems Lab, TU Wien（维也纳工业大学自主系统实验室）；Institute of Robotics and Mechatronics (DLR), German Aerospace Center（德国航空航天中心机器人与机电一体化研究所（DLR））
A Three-Stage Offline SDRE-Based Control Framework for Human Motion Reproduction on a Suspended Bipedal Robot
Department of Applied Mathematics, National Yang Ming Chiao Tung University（应用数学系，阳明交通大学）；Department of Electrical Engineering, National Central University（电气工程系，国立中央大学）
Indian Institute of Technology (BHU) Varanasi（瓦拉纳西印度理工学院（BHU））；Ohio State University（俄亥俄州立大学）；Indian Institute of Science（印度科学学院）
In this work, we demonstrate robust walking in the bipedal robot Digit on uneven terrains by just learning a single linear policy. In particular, we propose a new control pipeline, wherein the high-level trajectory modulator shapes the end-foot ellipsoidal trajectories, and the low-level gait controller regulates the torso and ankle orientation. The foot-trajectory modulator uses a linear policy and the regulator uses a linear PD control law. As opposed to neural network-based policies, the proposed linear policy has only 13 learnable parameters, thereby not only guaranteeing sample efficient …
Multiple chaotic central pattern generators with learning for legged locomotion and malfunction compensation
Beijing University of Aeronautics and Astronautics（北京航空航天大学）；Bernstein Center for Computational Neuroscience（伯恩斯坦计算神经科学中心）；Georg-August-Universität Göttingen（哥廷根大学）；University of Southern Denmark（南丹麦大学）
Terrain geometry is, in general, non-smooth, non-linear, non-convex, and, if perceived through a robot-centric visual unit, appears partially occluded and noisy. This work presents the complete control pipeline capable of handling the aforementioned problems in real-time. We formulate a trajectory optimization problem that jointly optimizes over the base pose and footholds, subject to a heightmap. To avoid converging into undesirable local optima, we deploy a graduated optimization technique. We embed a compact, contact-force free stability criterion that is compatible with the non-flat ground…
Italian Institute of Technology（意大利技术研究院）；Technical University of Munich（慕尼黑工业大学）；University of Isfahan（伊斯法罕大学）；Isfahan University of Technology（伊斯法罕理工大学）
Cyclic motions are fundamental patterns in robotic applications including industrial manipulation and legged robot locomotion. This paper proposes an approach for the online modulation of cyclic motions in robotic applications. For this purpose, we present an integrated programmable Central Pattern Generator (CPG) for the online generation of the reference joint trajectory of a robotic system out of a library of desired periodic motions. The reference trajectory is then followed by the lower-level controller of the robot. The proposed CPG generates a smooth reference joint trajectory convergen…
Motor Control Insights on Walking Planner and its Stability
Rehabilitation Research Institute of Singapore（新加坡康复研究院）；Nanyang Technological University（南洋理工大学）；University of Edinburgh（爱丁堡大学）；Kyungpook National University（庆北国立大学）
