Dynamic humanoid locomotion: Hybrid zero dynamics based gait optimization via direct collocation methods

Hybrid zero dynamics (HZD) has emerged as a popular framework for dynamic and underactuated bipedal walking, but has significant implementation difficulties when applied to the high degrees of freedom present in humanoid robots. The primary impediment is the process of gait design–it is difficult for optimizers to converge on a viable set of virtual constraints defining a gait. This dissertation presents a methodology that allows for the fast and reliable generation of efficient multi-domain robotic walking gaits through the framework of HZD, even in the presence of underactuation. To achieve this goal, we unify methods from trajectory optimization with the control framework of multi-domain hybrid zero dynamics. We present a novel optimization formulation in the context of direct collocation methods and HZD where we rigorously generate analytic Jacobians for the constraints. Two collocation methods, local collocation and pseudospectral (global) collocation, are developed within an unified framework, and their performance in different circumstances is comparatively studied. As a result, solving the resulting nonlinear program becomes tractable for large-scale NLP solvers, even for systems as high-dimensional as humanoid robots. We experimentally validate our methodology on the spring-legged prototype humanoid, DURUS, showing that the optimization approach yields dynamic and stable walking gaits for different walking configurations, including unrestricted 3D dynamic walking.Ph.D

A Legendre-Gauss pseudospectral collocation method for trajectory optimization in second order systems

© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksPseudospectral collocation methods have proven to be powerful tools to solve optimal control problems. While these methods generally assume the dynamics is given in the first order form xdot = f(x, u, t), where x is the state and u is the control vector, robotic systems are typically governed by second order ODEs of the form qddot = g(q, qdot, u, t), where q is the configuration. To convert the second order ODE into a first order one, the usual approach is to introduce a velocity variable v and impose its coincidence with the time derivative of q. Lobatto methods grant this constraint by construction, as their polynomials describing the trajectory for v are the time derivatives of those for q, but the same cannot be said for the Gauss and Radau methods. This is problematic for such methods, as then they cannot guarantee that qddot = g(q, qdot, u, t) at the collocation points. On their negative side, Lobatto methods cannot be used to solve initial value problems, as given the values of u at the collocation points they generate an overconstrained system of equations for the states. In this paper, we propose a Legendre-Gauss collocation method that retains the advantages of the usual Lobatto, Gauss, and Radau methods, while avoiding their shortcomings. The collocation scheme we propose is applicable to solve initial value problems, preserves the consistency between the polynomials for v and q, and ensures that qddot = g(q, qdot, u, t) at the collocation points.Peer ReviewedPostprint (author's final draft

From Bipedal Walking to Quadrupedal Locomotion: Full-Body Dynamics Decomposition for Rapid Gait Generation

This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot — the result is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics. This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one diagonally symmetric ambling gait at 0.35 m/s on a quadrupedal robot — the Vision 60, with 36 state variables and 12 control inputs — both in simulation and through outdoor experiments. This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds

From Bipedal Walking to Quadrupedal Locomotion: Full-Body Dynamics Decomposition for Rapid Gait Generation

This paper systematically decomposes a quadrupedal robot into bipeds to rapidly generate walking gaits and then recomposes these gaits to obtain quadrupedal locomotion. We begin by decomposing the full-order, nonlinear and hybrid dynamics of a three-dimensional quadrupedal robot, including its continuous and discrete dynamics, into two bipedal systems that are subject to external forces. Using the hybrid zero dynamics (HZD) framework, gaits for these bipedal robots can be rapidly generated (on the order of seconds) along with corresponding controllers. The decomposition is achieved in such a way that the bipedal walking gaits and controllers can be composed to yield dynamic walking gaits for the original quadrupedal robot — the result is the rapid generation of dynamic quadruped gaits utilizing the full-order dynamics. This methodology is demonstrated through the rapid generation (3.96 seconds on average) of four stepping-in-place gaits and one diagonally symmetric ambling gait at 0.35 m/s on a quadrupedal robot — the Vision 60, with 36 state variables and 12 control inputs — both in simulation and through outdoor experiments. This suggested a new approach for fast quadrupedal trajectory planning using full-body dynamics, without the need for empirical model simplification, wherein methods from dynamic bipedal walking can be directly applied to quadrupeds

Online Trajectory Planning Through Combined Trajectory Optimization and Function Approximation: Application to the Exoskeleton Atalante

Autonomous robots require online trajectory planning capability to operate in the real world. Efficient offline trajectory planning methods already exist, but are computationally demanding, preventing their use online. In this paper, we present a novel algorithm called Guided Trajectory Learning that learns a function approximation of solutions computed through trajectory optimization while ensuring accurate and reliable predictions. This function approximation is then used online to generate trajectories. This algorithm is designed to be easy to implement, and practical since it does not require massive computing power. It is readily applicable to any robotics systems and effortless to set up on real hardware since robust control strategies are usually already available. We demonstrate the computational performance of our algorithm on flat-foot walking with the self-balanced exoskeleton Atalante

Dynamic Walking: Toward Agile and Efficient Bipedal Robots

Dynamic walking on bipedal robots has evolved from an idea in science fiction to a practical reality. This is due to continued progress in three key areas: a mathematical understanding of locomotion, the computational ability to encode this mathematics through optimization, and the hardware capable of realizing this understanding in practice. In this context, this review article outlines the end-to-end process of methods which have proven effective in the literature for achieving dynamic walking on bipedal robots. We begin by introducing mathematical models of locomotion, from reduced order models that capture essential walking behaviors to hybrid dynamical systems that encode the full order continuous dynamics along with discrete footstrike dynamics. These models form the basis for gait generation via (nonlinear) optimization problems. Finally, models and their generated gaits merge in the context of real-time control, wherein walking behaviors are translated to hardware. The concepts presented are illustrated throughout in simulation, and experimental instantiation on multiple walking platforms are highlighted to demonstrate the ability to realize dynamic walking on bipedal robots that is agile and efficient