03-05-2011, 10:54 AM
Abstract
A nonsmooth (hybrid) 3-D mathematical model ofa snake robot (without wheels) is developed and experimentallyvalidated in this paper. The model is based on the framework ofnonsmooth dynamics and convex analysis that allows us to easilyand systematically incorporate unilateral contact forces (i.e., betweenthe snake robot and the ground surface) and friction forcesbased on Coulomb’s law of dry friction. Conventional numericalsolvers cannot be employed directly due to set-valued force lawsand possible instantaneous velocity changes. Therefore, we showhow to implement the model for numerical treatment with a numericalintegrator called the time-stepping method. This methodhelps to avoid explicit changes between equations during simulationeven though the system is hybrid. Simulation results for theserpentine motion pattern lateral undulation and sidewinding arepresented. In addition, experiments are performed with the snakerobot “Aiko” for locomotion by lateral undulation and sidewinding,both with isotropic friction. For these cases, back-to-back comparisonsbetween numerical results and experimental results are given.Index Terms—3-D snake robot, kinematics, nonsmooth dynamics,time-stepping method.
I. INTRODUCTION
WHEELED mechanisms constitute the backbone of mostground-based means of transportation. Unfortunately,rough terrain makes it hard, if not impossible, for such mechanismsto move. To be able to move in various terrains, such asgoing through narrow passages and climb on rough ground, thehigh-mobility property of snakes is recreated in robots that lookand move like snakes.Snake robots most often have a high number of DOFs, andthey are able to move forward without using active wheels orlegs. Due to the high number of DOF, it can be quite expensiveand time-consuming to build and maintain a snake robot. Thismotivates the development of accurate mathematical models ofsnake robots. Such models can be used for synthesis and testingof various serpentine motion patterns intended for serpentinelocomotion.Manuscript received July 19, 2007; revised November 29, 2007. This paperwas recommended for publication by Associate Editor S.Ma and Editor F. Parkupon evaluation of the reviewers’ comments.A. A. Transeth is with theDepartment ofApplied Cybernetics, SINTEF Informationand Communication Technology (ICT), NO-7465 Trondheim, Norway(e-mail: aksel.transeth[at]sintef.no).R. I. Leine and Ch. Glocker are with the Institut f¨ur Mechanische Systeme(IMES) Centre of Mechanics, Eidgen¨ossische Technische Hochschule (ETH)Z¨urich, CH-8092 Zurich, Switzerland (e-mail: remco.leine[at]imes.mavt.ethz.ch;christoph.glocker[at]imes.mavt.ethz.ch).K. Y. Pettersen is with the Department of Engineering Cybernetics,Norwegian University of Science and Technology, NO-7491 Trondheim,Norway (e-mail: kristin.y.pettersen[at]itk.ntnu.no).Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TRO.2008.917003The first working snake robot was built in 1972 [1]. Thisrobot was limited to planar motion, but snake robots capableof 3-D motion have appeared more recently [2]–[6]. Togetherwith the robots, mathematical models of both the kinematicsand the dynamics of snake robots have also been developed.Purely kinematic 3-D models have been presented in [6]–[8],where frictional contact between the snake robot and the groundis not included in the model. Hence, contact between the snakerobot and the ground surface is either modeled with frictionlesspassive wheels, or the parts of the snake robot that touches theground are defined as anchored to the ground [9]. A modelof the dynamics of motion is needed to describe the frictionforces a snake robot without wheels is subjected to when movingover a surface. Most mathematical models that describe thedynamics of snake robot motion have been limited to planar(2-D) motion [10]–[13], and 3-D mathematical models of snakerobots have only recently been developed [3], [7]. 3-D modelsfacilitate testing and development of 3-D serpentine motionpatterns such as sinus lifting and sidewinding. A description ofthese motion patterns is found in, e.g., [4]. A physics enginecalled the open dynamics engine (ODE) has been employedto simulate a 15-link snake robot instead of deriving explicitexpressions for its dynamics in [14]. Such software makes iteasy to change the geometry of a snake robot if needed, and thetime needed to prepare a working model is relatively short [15].On a flat surface, the ability of a snake robot to move forwardis dependent on the friction between the ground surface and thebody of the snake robot. Hence, unilateral contact forces andfriction forces are important parts of the mathematical model ofa snake robot. The friction forces have usually been based on aCoulomb or viscous-like frictionmodel [11], [12], and Coulombfriction has most often been modeled using a sign function [12],[16]. The contact between a snake robot and the ground surfacecan sometimes be approximated by a no-sideways-slip constraintfor snake robot with wheels [17], [18]. However, suchan approximation is not valid for wheel-less snake robots. Theunilateral contact forces have been modeled as a mass-springdampersystem in [3] (i.e., compliant contact), and each link hasonly a single and fixed contact point with the ground surface.When running simulations, direct implementations or approximationsof the sign function can lead to an erroneous descriptionof sticking contacts or very stiff differential equations. Also, amass-spring-damper model introduces a very stiff spring thatleads to stiff differential equations. In addition, it is not clearhow to determine the dissipation parameters of the contact unambiguouslywhen using a compliant model [19]. The ODEimplements a form of rigid body contact (i.e., not compliantcontact). However, the implementation of this engine trades off simulation accuracy in order to increase simulation speed andstability [15], [20].In this paper, we develop a nonsmooth (hybrid) 3-D mathematicalmodel of a snake robot with cylindrical links withoutwheels. Set-valued force laws for the constitutive description ofunilateral contact forces and friction forces in a 3-D setting aredescribed in the framework of nonsmooth dynamics and convexanalysis [19], [21], [22].Moreover, themodel has amoving contactpoint on the surface of each link for contact with the groundsurface instead of just a fixed point for each link. The latter is anapproach employed in prior publications on mathematical modelsof 3-D snake robot motion. Stick-slip transitions (based on aset-valued Coulomb friction law) and impacts with the groundaremodeled as instantaneous transitions. This results in an accuratemodel of spatial Coulomb friction where both the directionof the friction force and a true stick-phase are taken properly intoaccount. For wheel-less snake robots, it is important to describethe frictional contact between the wheel-less snake robot andthe ground in an accurate manner, both with respect to stick-sliptransitions and the direction of the friction force while slidingalong the ground surface. This latter property also distinguisheswheel-less snake robots from, e.g., legged mechanisms that mostoften try to “stick” to the ground rather than sliding along it.The dynamics of the snake robot is described by an equalityof measures, which includes the Newton–Euler equations forthe nonimpulsive part of the motion as well as impact equations.A particular choice of coordinates results in an effectiveway of writing the system equations. The set-valuedness of theforce laws allows us to write each constitutive law with a singleequation and avoids explicit switching between equations(for example, when a collision between the snake robot and theground surface occurs) even though this is a hybrid system. Thisis advantageous since the snake robot links repeatedly collideswith the ground surface during, e.g., locomotion by sidewinding.A discretization of the equality of measures gives the so-calledtime-stepping method (see [22] and references therein) that weuse for numerical simulation. The description of the model andthe method for numerical integration are presented in this paperin such a way that people who are new to the field of nonsmoothdynamics can use this paper as an introduction to nonsmoothmodeling of robot manipulators with impacts and friction. Inaddition, we present experimental results that validate the mathematicalmodel. To the best of our knowledge, this is the firsttime such a back-to-back comparison between simulation andexperimental results is presented for 3-D snake robot motion.The experiments are performed with the snake robot “Aiko” inFig. 1 built at the Norwegian University of Science and Technology(NTNU)/SINTEF Advanced Robotics Laboratory.The paper is organized as follows. A short introduction to themodeling procedure is given in Section II. The kinematics of thesnake robot with the ground surface as a unilateral constraintis described in Section III. Then, the groundwork for findingthe friction and ground contact forces is laid in Section IV. Thenonsmooth dynamics is presented in Section V, while the serpentinemotion patterns employed in this paper are described inSection VI. The numerical treatment of the mathematical modelis given in Section VII. Simulations and experimental validationsare given in Section VIII. Conclusions and suggestions forfurther research are presented in Section IX.
II. SUMMARY OF THE MATHEMATICAL MODEL
This section contains a brief outline of howto derive the nonsmoothmathematical model of the snake robot. This preliminarysection is meant to motivate and ease the understanding of theforthcoming deduction of the system equations.The snake robot model consists of n links connected by n − 1two-DOF cardan joints (i.e., rotational joints). Let u ∈ R6n bea vector containing the translational and rotational velocities ofall the links of the snake robot (the structure of the snake robottogether with the coordinates and reference frames are describedfurther in Section III). Let the differential measures du and dtbe loosely described for now as a “possible differential change”in u and time t, respectively, while a more precise definitionis given in Section V. The use of differential measures allowsfor instantaneous changes in velocities that occur for impactsbetween the snake robot and the ground surface. The systemequations for the snake robot can now be written asMdu − hdt − dR = τ C dt (1)which is called the equality of measures [23], where M ∈R6n×6n is the mass matrix, h ∈ R6n consists of the smoothforces, τ C ∈ R6n contains all the joint actuator torques, anddR ∈ R6n accounts for the normal contact forces/impulsesfrom the ground, the Coulomb friction forces/impulses, andthe bilateral constraint forces/impulses in the joints. Note: Weallow in this paper for instantaneous changes in velocitiesusually associated with collisions. Hence, the (normal contact/friction/constraint) forces are not always defined due to theinfinite accelerations. In these cases, we have impulses insteadof forces. The nonsmooth equality of measures (1) allows us toformulate in a uniform manner both the smooth and nonsmoothphases of motion. This is achieved partly by representing thecontact forces/impulses as contact impulse measures.
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