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Stable walking gaits for a three-link planar biped robot with one actuator
Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics. SLU, Dept Forestry Technol, Umeå, Sweden and Swedish Cluster Forest Technol, Vindeln, Sweden. (Robotics and Control Lab)
Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics. Norwegian Univ Sci & Technol, Dept Engn Cybernet, Trondheim, Norway.
Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics.
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2013 (English)In: IEEE Transactions on robotics, ISSN 1552-3098, E-ISSN 1941-0468, IEEE transactions on robotics, Vol. 29, no 3, 589-601 p.Article in journal (Refereed) Published
Abstract [en]

We consider a benchmark example of a three-link planar biped walker with torso, which is actuated in between the legs. The torso is thought to be kept upright by two identical torsional springs. The mathematical model reflects a three-degree-of-freedom mechanical system with impulse effects, which describe the impacts of the swing leg with the ground, and the aim is to induce stable limit-cycle walking on level ground. The main contribution is a novel systematic trajectory planning procedure for solving the problem of gait synthesis. The key idea is to find a system of ordinary differential equations for the functions describing a synchronization pattern for the time evolution of the generalized coordinates along a periodic motion. These functions, which are known as virtual holonomic constraints, are also used to compute an impulsive linear system that approximates the time evolution of the subset of coordinates that are transverse to the orbit of the continuous part of the periodic solution. This auxiliary system, which is known as transverse linearization, is used to design a nonlinear exponentially orbitally stabilizing feedback controller. The performance of the closed-loop system and its robustness with respect to various perturbations and uncertainties are illustrated via numerical simulations.

Place, publisher, year, edition, pages
2013. Vol. 29, no 3, 589-601 p.
Keyword [en]
Biped robots, holonomic servoconstraints, limit-cycle walking, orbital stabilization, trajectory planning with dynamic constraints, transverse linearization, underactuated mechanical systems, virtual holonomic constraints
National Category
Robotics Veterinary Science
URN: urn:nbn:se:umu:diva-39717DOI: 10.1109/TRO.2013.2239551ISI: 000320137200001OAI: diva2:395229
Swedish Research Council, 2008-4369

Funding Agency, Grant Number:

Russian Federal Agency for Science and Innovation, 02.740.11.505

Russian Federal Target Program "Research & Development in Priority Areas", 11.519.11.4007

Norwegian Research Council under KMB grant Next Generation Robotics for Norwegian Industry

Norwegian Research Council under FRIPRO Grant, 214525/F20

Russian Foundation for Basic Research, 12-01-00808

Available from: 2011-02-04 Created: 2011-02-04 Last updated: 2016-02-15Bibliographically approved
In thesis
1. Underactuated mechanical systems: Contributions to trajectory planning, analysis, and control
Open this publication in new window or tab >>Underactuated mechanical systems: Contributions to trajectory planning, analysis, and control
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Nature and its variety of motion forms have inspired new robot designs with inherentunderactuated dynamics. The fundamental characteristic of these controlled mechanicalsystems, called underactuated, is to have the number of actuators less than the number ofdegrees of freedom. The absence of full actuation brings challenges to planning feasibletrajectories and designing controllers. This is in contrast to classical fully-actuated robots.A particular problem that arises upon study of such systems is that of generating periodicmotions, which can be seen in various natural actions such as walking, running,hopping, dribbling a ball, etc. It is assumed that dynamics can be modeled by a classicalset of second-order nonlinear differential equations with impulse effects describing possibleinstantaneous impacts, such as the collision of the foot with the ground at heel strikein a walking gait. Hence, we arrive at creating periodic solutions in underactuated Euler-Lagrange systems with or without impulse effects. However, in the qualitative theory ofnonlinear dynamical systems, the problem of verifying existence of periodic trajectoriesis a rather nontrivial subject.The aim of this work is to propose systematic procedures to plan such motions and ananalytical technique to design orbitally stabilizing feedback controllers. We analyze andexemplify both cases, when the robotmodel is described just by continuous dynamics, andwhen continuous dynamics is interrupted from time to time by state-dependent updates.For trajectory planning, systems with one or two passive links are considered, forwhich conditions are derived to achieve periodicmotions by encoding synchronizedmovementsof all the degrees of freedom. For controller design we use an explicit form tolinearize dynamics transverse to the motion. This computation is valid for an arbitrarydegree of under-actuation. The linear system obtained, called transverse linearization, isused to analyze local properties in a vicinity of the motion, and also to design feedbackcontrollers. The theoretical background of these methods is presented, and developedin detail for some particular examples. They include the generation of oscillations forinverted pendulums, the analysis of human movements by captured motion data, and asystematic gait synthesis approach for a three-link biped walker with one actuator.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, Institutionen för tillämpad fysik och elektronik, 2011. 64 p.
Robotics and control lab, ISSN 1654-5419 ; 5
Underactuated mechanical systems, mechanical systems with impacts, trajectory planning, periodic trajectories, orbital stabilization, walking robots, virtual holonomic constraints, transverse linearization
National Category
Control Engineering
Research subject
Automatic Control
urn:nbn:se:umu:diva-39719 (URN)978-91-7459-149-1 (ISBN)
Public defence
2011-03-01, KBC-huset , KB3A9, Umeå Universitet, Umeå, 09:00 (English)
Available from: 2011-02-08 Created: 2011-02-04 Last updated: 2011-03-01Bibliographically approved

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La Hera, PedroShiriaev, Anton S.Freidovich, Leonid BMettin, Uwe
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