Direct methods for trajectory optimization are widely used for planning locally optimal trajectories of robotic systems. Many critical tasks, such as locomotion and manipulation, often involve impacting the ground or objects in the environment. Most state-of-the-art techniques treat the discontinuous dynamics that result from impacts as discrete modes and restrict the search for a complete path to a specified sequence through these modes. Here we present a novel method for trajectory planning of rigid-body systems that contact their environment through inelastic impacts and Coulomb friction. This method eliminates the requirement for a priori mode ordering. Motivated by the formulation of multi-contact dynamics as a Linear Complementarity Problem for forward simulation, the proposed algorithm poses the optimization problem as a Mathematical Program with Complementarity Constraints. We leverage Sequential Quadratic Programming to naturally resolve contact constraint forces while simultaneously optimizing a trajectory that satisfies the complementarity constraints. The method scales well to high-dimensional systems with large numbers of possible modes. We demonstrate the approach on four increasingly complex systems: rotating a pinned object with a finger, simple grasping and manipulation, planar walking with the Spring Flamingo robot, and high-speed bipedal running on the FastRunner platform.

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A direct method for trajectory optimization of rigid bodies through contact

This two-part paper discusses the analysis and control of legged locomotion in terms of N-step capturability: the ability of a legged system to come to a stop without falling by taking N or fewer steps. We consider this ability to be crucial to legged locomotion and a useful, yet not overly restrictive criterion for stability. In this part (Part 1), we introduce a theoretical framework for assessing N-step capturability. This framework is used to analyze three simple models of legged locomotion. All three models are based on the 3D Linear Inverted Pendulum Model. The first model relies solely on a point foot step location to maintain balance, the second model adds a finite-sized foot, and the third model enables the use of centroidal angular momentum by adding a reaction mass. We analyze how these mechanisms influence N-step capturability, for any N > 0. Part 2 will show that these results can be used to control a humanoid robot.

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Capturability-based analysis and control of legged locomotion, Part 1: Theory and application to three simple gait models

This three-part paper discusses the analysis and control of legged locomotion in terms of N-step capturability: the ability of a legged system to come to a stop without falling by taking N or fewer steps. We consider this ability to be crucial to legged locomotion and a useful, yet not overly restrictive criterion for stability. Part 1 introduces the theoretical framework for assessing N-step capturability. Formal definitions of N-step capturability and related terms are given, and general disturbance robustness metrics based on capturability are proposed. Part 2 uses the theoretical framework developed in the current part to analyze N-step capturability for three simple gait models. Part 3 describes how the results for the simple models were used to control a complex lower body humanoid robot with two six degree of freedom legs.

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Capturability-Based Analysis and Control of Legged Locomotion

In this paper, the concept of divergent component of motion (DCM, also called “Capture Point”) is extended to 3-D. We introduce the “Enhanced Centroidal Moment Pivot point” (eCMP) and the “Virtual Repellent Point” (VRP), which allow for the encoding of both direction and magnitude of the external forces and the total force (i.e., external plus gravitational forces) acting on the robot. Based on eCMP, VRP, and DCM, we present methods for real-time planning and tracking control of DCM trajectories in 3-D. The basic DCM trajectory generator is extended to produce continuous leg force profiles and to facilitate the use of toe-off motion during double support. The robustness of the proposed control framework is thoroughly examined, and its capabilities are verified both in simulations and experiments.

Three-Dimensional Bipedal Walking Control Based on Divergent Component of Motion

https://hal.archives-ouvertes.fr/hal-01676474/document

Models, feedback control, and open problems of 3D bipedal robotic walking

This two-part paper discusses the analysis and control of legged locomotion in terms of N-step capturability: the ability of a legged system to come to a stop without falling by taking N or fewer steps. We consider this ability to be crucial to legged locomotion and a useful, yet not overly restrictive criterion for stability. Part 1 introduced the N-step capturability framework and showed how to obtain capture regions and control sequences for simplified gait models. In Part 2, we describe an algorithm that uses these results as approximations to control a humanoid robot. The main contributions of this part are (1) step location adjustment using the 1-step capture region, (2) novel instantaneous capture point control strategies, and 3) an experimental evaluation of the 1-step capturability margin. The presented algorithm was tested using M2V2, a 3D force-controlled bipedal robot with 12 actuated degrees of freedom in the legs, both in simulation and in physical experiments. The physical robot was able to recover from forward and sideways pushes of up to 21 Ns while balancing on one leg and stepping to regain balance. The simulated robot was able to recover from sideways pushes of up to 15 Ns while walking, and walked across randomly placed stepping stones.

Capturability-based analysis and control of legged locomotion, Part 2: Application to M2V2, a lower-body humanoid

Bipedal robots are currently either slow, energetically inefficient and/or require a lot of control to maintain their stability. This paper introduces the FastRunner, a bipedal robot based on a new leg architecture. Simulation results of a Planar FastRunner demonstrate that legged robots can run fast, be energy efficient and inherently stable. The simulated FastRunner has a cost of transport of 1.4 and requires only a local feedback of the hip position to reach 35.4 kph from stop in simulation.

FastRunner: A fast, efficient and robust bipedal robot. Concept and planar simulation

This paper presents a new technique for estimating the center of mass of articulated rigid body systems. This estimation technique uses the statically equivalent serial chain, a serial chain representation of any multilink branched chain whose end-effector locates directly the center of mass. This technique works based on a knowledge of only the kinematic architecture of the system and does not require the total mass, or any of the individual body's mass or length properties. This constitutes an advance in center of mass estimation, providing an alternative to techniques requiring continuously a force plate. A comparison of these estimation techniques is presented. The modeling and estimation technique is then implemented on a human subject.

Estimation of the Center of Mass: From Humanoid Robots to Human Beings

The estimation of the centre of mass position in humans is usually based on biomechanical models developed from anthropometric tables. This method can potentially introduce errors in studies involving elderly people, since the ageing process is typically associated with a modification of the distribution of the body mass. In this paper, an alternative technique is proposed, and evaluated with an experimental study on 9 elderly volunteers. The technique is based on a virtual chain, identified from experimental data and locating the subject's centre of mass. Its configuration defines the location of the centre of mass, and is a function of the anatomical joint angles measured on the subject. This method is a valuable investigation tool in the field of geronto-technology, since it overcomes some of the problems encountered with other CoM estimation methods.

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Estimation of the centre of mass from motion capture and force plate recordings: A study on the elderly

This paper proposes an analysis of the manipulability of the Center of Mass (CoM) of humanoid robots. Starting from the dynamic equations of humanoid robots, the operational space formulation is used to express the dynamics of humanoid robots at their CoM and under their specific characteristics: a free-floating base, forces at contact points, and dynamic balance constraints. After a review of the kinematic manipulability of the CoM, the concept of dynamic manipulability of the CoM is introduced. The latter represents the ability of a humanoid robot to generate a spatial motion under a stability criterion. The size and shape of the dynamic manipulability of the CoM are a function of the joint torque limitations, the contact forces and the zero moment point used as a stability criteria. Two calculations of the CoM dynamic manipulability are proposed, a fast ellipsoid approximation, and the exact polyhedron computation. A case study illustrates the proposed approach on the HOAP3 humanoid robot and its use for mechanical design optimization.Copyright © 2010 by ASME

Sebastien Cotton

Papers

Capturability-based analysis and control of legged locomotion, Part 2: Application to M2V2, a lower-body humanoid

FastRunner: A fast, efficient and robust bipedal robot. Concept and planar simulation

Estimation of the Center of Mass: From Humanoid Robots to Human Beings

Estimation of the centre of mass from motion capture and force plate recordings: A study on the elderly

On the manipulability of the center of mass of humanoid robots, application to design