Elsevier

Mechatronics

Volume 68, June 2020, 102364
Mechatronics

Centroidal-momentum-based trajectory generation for legged locomotion

https://doi.org/10.1016/j.mechatronics.2020.102364Get rights and content

Abstract

This paper presents a trajectory optimization framework for planning dynamic legged locomotion based on a robot’s centroidal momentum (CM), which is the aggregation of all the links’ momenta at the robot’s Center of Mass (CoM). This new framework is built around CM dynamic model driven by Ground Reaction Forces (GRFs) parameterized with Bézier polynomials. Due to the simple form of CM dynamics, the closed-form solution of the robot’s CM can be obtained by directly integrating the Bézier polynomials of GRFs. The CM can be also calculated from the robot’s generalized coordinates and velocities using Centroidal Momentum Matrices (CMM). For dynamically feasible motions, these CM values should match, thereby providing equality constraints for the proposed trajectory optimization framework. Direct collocation methods are utilized to obtain feasible GRFs and joint trajectories simultaneously under kinematic and dynamic constraint. With the closed-form solutions of CM due to the parameterization of GRFs in the formulation, numerical error induced by collocation methods in the solution of trajectory optimization can be reduced, which is crucial for reliable tracking control when applied to real robotic systems. Using the proposed framework, jumping trajectories of legged robots are obtained in the simulation. Experimental validation of the algorithm is performed on a planar robot testbed, proving the effectiveness of the proposed method in generating dynamic motions of the legged robots.

Introduction

Agile quadrupeds such as cats and squirrels are capable of performing highly dynamic maneuvers over a variety of terrains. When navigating in challenging environments, they can plan trajectories that fully utilize their physical capabilities and inherent dynamics. Such abilities, if successfully implemented on robots, possess profound potential in scenarios such as hazardous environment reconnaissance, disaster response and rescue.

Recent advances in quadruped robots has shown promising results in matching the dynamic capabilities of their natural counterparts: robust walking gaits have been performed by the BigDog [1], Spot and SpotMini from Boston Dynamics, though details of their control algorithms are yet to be published. ANYmal [2] is able to perform stable locomotion such as walking and trotting. Park et al. [3] displayed robust bounding motion on the MIT Cheetah 2 for a wide range of speeds in untethered 3D tests. Similar maneuverability has also been achieved on HyQ2Max [4] in creating trotting and self-righting while using hydraulic actuation. As the performance of the quadrupedal robots keeps advancing, so is the need for accurate algorithms capable of planning trajectories that fully utilize a robot’s dynamical capabilities to create agile motions, or to navigate challenging environments while respecting the robot’s physical restrictions.

The impediment to fast trajectory planning is partly rooted in the high degrees of freedom (DoF) present in robots. Methods based on a robot’s full-body dynamics have proven effectiveness in producing expressive motions, as shown in [5], [6], [7], [8], but would often suffer from long computational time. As a work-around, simplified models including the point-mass model [9] and other ones stemming from the linear inverted pendulum (LIP) model introduced by [10] have been used, such as the Reaction Mass Pendulum (RMP) [11] which augment the LIP models with a reaction-mass ellipsoid to capture the change in centroidal momenta for tasks such as balancing. A modified Spring Loaded Inverted Pendulum (SLIP) model is used by [12] to achieve running long jumps. Though less restrictive, these models still require task-specific modifications to its dynamic models to complement the full-body motions, or part of the dynamics will be left undecided or uncontrolled.

In recent years, the centroidal momentum (CM) of a robot, which is the aggregation of all links’ momenta at the robot’s Center of Mass (CoM) [13], has gained increasing attention as a reduced-order dynamic model in planning trajectories for robots with complex dynamics [14], [15], [16], [17]. A robot’s CM enjoys simple dynamics driven by the ground reaction forces (GRF) in legged locomotion. By including the GRF as optimization variables, a robot’s CM can be obtained through numerical integration, from which the corresponding joint angles can be recovered from kinematic equations without solving the complete second-order equation of motions. However, the use of CM dynamics in trajectory optimization for robots are often limited to a subsection of the optimization: either to validate transition feasibility between consecutive contact states [15], or to generate contact wrenches, which tends to rely on pre-defined contact locations and also requires additional process for joint trajectories [14]. Two notable exceptions come from the work of Dai et al. [17] and Fernbach et al. [15], where the contact states and forces are solved simultaneously as a hard linear complementarity problem. This method is shown to produce expressive motions and allows automatic gait scheduling, but often at the cost of long computation time when incorporating the full joint-space rigid body dynamics, or loss of generality

In this paper, we propose to augment the CM-dynamic-based trajectory optimization scheme by parameterizing the GRF and swing leg trajectories with Bézier polynomials. The optimizer simultaneously selects the Bézier polynomial coefficients and the contact leg’s joint trajectories, under an equality constraint which unifies the CM directly integrated from GRF through equations of motion, to the CM calculated from the robot’s generalized states and velocities using Centroidal Momentum Matrix [13], as shown in Fig. 1. This ensures that the obtained CM always respect the robot’s full-body dynamics, even between optimization nodes, unlike other collocation-based methods which approximate the solution of the model using polynomials. A similar approach can be found in [14], but is restricted to using a basic power basis polynomial for the CoM and GRF only, and relies on additional processing for the joint trajectories which requires prior knowledge of the exact contact locations.

The remainder of this paper is organized as follows: Section 2.1 describes the planar robot model used in formulating the proposed trajectory optimization algorithm. Section 2.2 presents two ways of obtaining a robot’s centroidal momentum with ground contact. Detailed formulation of the proposed multi-phase optimization framework, including transition map between different motion phases is presented in Section 3. Experimental validations are presented in Section 4 with reference trajectories obtained from the proposed method.

Section snippets

Simplified planar model

In this paper, a planar model shown in Fig. 2 is adopted to study a quadruped’s behavior in its sagittal plane, where the most of dynamic motions happen. The model consists of five massive links, with the torso link in the middle connecting two identical two-link legs through ideal revolute joints. The physical parameters of the planar model specified in Table 1.

The configuration of this five-link body can be described with its body angles:qb=(qTor,qF1,qF2,qH1,qH2).where subscripts F and H

Trajectory optimization

The values of a robot’s centroidal momentum derived in the section above should agree for any dynamically feasible trajectories. An equality constraint that unifies the two calculated values forms the basis of the proposed motion planning framework. This section presents the setup of this multi-phase trajectory optimization framework, followed by a test example to compare it with two methods

Experimental validation

This section introduces the planar testbed used in validating the proposed trajectory optimization framework, together with the experiment result of a 3-phase forward jumping motion. The optimization was based on the planar floating-base model described in Section 2.1 and formulated with the same structure proposed in Section 3.

Conclusion and future work

In this paper we present a trajectory optimization framework for planning legged locomotion based on a robot’s centroidal momentum, which enjoys simple dynamics dominated by the GRFs and the gravity. By parameterizing the GRFs as Bézier polynomials, the centroidal momentum of the robot can be solved analytically instead of through numerical integration. This avoids interpolating or curve-fitting the output trajectories between collocation points, which may result in sub-optimal trajectories or

CRediT authorship contribution statement

Chuanzheng Li: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft. Yanran Ding: Investigation, Resources, Validation, Software. Hae-Won Park: Conceptualization, Methodology, Supervision, Project administration, Funding acquisition, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they do not have any financial or nonfinancial conflict of interests

Acknowledgment

This project is supported by NAVER LABS Corp. under grant 087387, Air Force Office of Scientific Research under grant FA2386-17-1-4665, and National Science Foundation under grant 1752262.

Chuanzheng Li received his B.S. degree in Mechatronics from Zhejiang University, Hangzhou, China in 2014, and the M.S. degree from the Mechanical Science and Engineering Department, University of Illinois at Urbana-Champaign, Champaign, IL, USA in 2017. He is currently in the Ph.D. program at University of Illinois at Urbana-Champaign supervised by Dr. Hae-Won Park, working primarily on the design of mechatronic systems and the real-time control of legged robots.

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    Chuanzheng Li received his B.S. degree in Mechatronics from Zhejiang University, Hangzhou, China in 2014, and the M.S. degree from the Mechanical Science and Engineering Department, University of Illinois at Urbana-Champaign, Champaign, IL, USA in 2017. He is currently in the Ph.D. program at University of Illinois at Urbana-Champaign supervised by Dr. Hae-Won Park, working primarily on the design of mechatronic systems and the real-time control of legged robots.

    Yanran Ding received B.S. degree from UM-SJTU joint institute, Shanghai Jiao Tong University, Shanghai, China in 2015. He has received the M.Sc. degree from the Mechanical Science and Engineering Department, University of Illinois at Urbana-Champaign, Champaign, IL, USA in 2017, and he is currently working towards his Ph.D. degree at the Dynamic Robotics Laboratory. His research interests include the design and control agile robotics systems. His research include mechanism design, actuation systems and concentrates on using applied optimization and control theory to enable legged robots to achieve dynamic motions. He is one of the best student paper finalists in IROS 2017.

    Hae-Won Park is an Assistant Professor of Mechanical Engineering at the Korean Advanced Institute of Science and Technology (KAIST). He received B.S. and M.S. degrees from Yonsei University, Seoul, Korea, in 2005 and 2007, respectively, and the Ph.D. degree from the University of Michigan, in 2012, all in mechanical engineering. His research interests lie at the intersection of control, dynamics, and mechanical design of robotic systems, with special emphasis on legged locomotion robots. He is the recipient of the 2018 National Science Foundation (NSF) CAREER Award, NSF’s most prestigious awards in support of early-career faculty.

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