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A 'Sidewinding' Locomotion Gait for Hyper-Redundant Robots

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A 'Sidewinding' Locomotion Gait for Hyper-Redundant Robots A “Sidewinding” Locomotion Gait for Hyper-Redundant Robots 3.W. Burdick, J. Radford Dept. of Mechanical Engineering, Mail Code 10444, CALTECH, Pasadena, CA 91125. G.S. Chirikjian Dept. of Mechanical Engineering, 124 Latrobe Hall, Johns Hopkins University, Baltimore, MD 21218. Abstract While some mobile hyper-redundant robots are a hybrid between a snakelike vehicle and a tracked This paper considers the kinematics of a novel or wheeled vehicle [HiMoSO], the focus of this paper form of hyper-redundant mobile robot locomotion which is analogous to the ‘sidewinding’ locomotion and its companions [ChBSlb, ChB93a, ChB93bl of desert s akes. TI$= form o locomotio can be s 1 Gomotion. wLich gises from internal robo generated Ey a repetitive travAing wave of mecha- Len2mg or twisting. more extensive review o! nism bending. Using a continuous backbone curve hyper-redundant robotic locomotion research can model, we develop algorithms which enable travel be found in [ChB93a]. While the robotics liter- in a uniform direction as well as changes in direc- ature is devoid of sidewinding gait analysis, biolo- tion. gists have empirically examined sidewinding [Gray, Secor, SJB]. Here we give the first quantitative de- 1. Introduction scription of a sidewinding motion. Hyper-redundant robotic systems have a very large 2. Qualitative Description of Sidewinding or infinite degree of kinematic redundancy. They are analogous to snakes, worms, or elephant trunks. In previous work [ChBSla, Ch921, two of the authors have developed novel methods for kine- matic analysis and modeling of such systems. Further, these basic methods were subsequently used to develop efficient algorithms for hyper- redundant robot obstacle avoidance [ChBgOa], lo- comotion [ChBSlb, ChB93a, ChB93b1, and grasp ing [ChBSlb, Ch92, ChB93bl. We define hyper-redundant robot “locomotion” as the net displacement of a hyper-redundant mobile robot which arises from internally indvced knd- ing and twisting of the mechanism. Actuatable Figure 1: Sidewinding snake wheels, tracks, or legs are not necessary. A gait is a distinct repetitive sequence of mechanism de- If the snake is moving uniformly across a sandy formations which results in net displacement. In surface, a set of parallel tracks, which are neither [ChBSlb, ChB93a, ChB93bl we developed two parallel nor perpendicular to the direction of me classes of hyper-redundant robot locomotion gaits tion, will be left in the sand after the snake passes which are based on standing or traveling waves of (Fig. 1). Let these tracks be called ground con- mechanism distortion. These gaits are idealizations tact tracks, or GCTs. The portions of the snake in of gaits used by inchworms, earthworms, and slugs. contact with the ground will henceforth be termed These locomotion schemes have been implemented, ground contact segments, or GCSs. Net snake dis- and their viability demonstrated in an actual 30 placement, or locomotion, is produced by moving degree-of-freedom hyper-redundant robot mecha- the snake to sequential GCTs. nism [ChB92b, Ch92, ChB93al. However, other possible forms of hyper-redundant locomotion do The cycle begins with the head of the snake lift- not fit into these gait classes. Snakes also em- ing from a current GCT and moving toward the ploy sidewinding, concertina, and undulating gaits next GCT. The lifted portion is termed an arch [Gray]. This paper develops a novel locomotion segment, or AS. The body is peeled away from the scheme which is qualitatively identical to sidewind- most forward GCT until almost 1/4 of the snake’s ing. This gait is a useful addition to our previously body is cantilevered. A point just behind the head developed gait repetoire. Empirical evidence indi- touches the ground, establishing a point on the cates that snakes generate larger accelerations and next GCT. Successive body segments are “layed” travel faster when employing a sidewinding gait. down on along this newly established GCT, while Thus, sidewinding would be most useful for imple- segments are simultaneously “peeled” away from menting fast gross displacement, while the algo- the prior GCT. In this way, a GCS effectively trav- rithms in [ChB93a, ChB93bI would be most useful els the length of a GCT, even though the GCSs are for precise locomotory movements. at rest with respect to the ground. LO1 1050-4729/93 $3.00 0 1993 IEEE Aftgr cptainEeyt of the sn&e is in c nt ct wit t e orwar the Drocess is reDeate8. f’or most snakes, the body stiaddles three GCTs, or two GCTs during transitory phases. The combina- tion of an adjacent GCS and an AS will be termed a basic gait segment. The net direction of travel is the sum of two components: one parallel to the GCT and one along a vector from the beginning of one AS to the beginning of the next. Experiments indicate that sidewinding generates Figure 2: Definitions of K(s,t),T(s,t) the greatest acceleration and speed of all gaits. This is logical, as: (1) the GCSs afford a large sur- face area of contact between the ground and the [ChBSla], which measures how the robot twists snake; (2) the contact friction is static, and not about the backbone, is required to completely spec- sliding, friction; and (3) the contact reaction forces ify a hyper-redundant robot configuration. As- are distributed over two or more GCSs, adding sta- suming an axially symmetric robot, roll can be bility. Additionally, desert snakes prefer sidewind- neglected. The backbone curve is an abstraction ing because only small portions of the snake are of a real hyper-redundant robot geometry. To ap- in contact with hot desert sand, thus preventing ply this framework to discretely segmented hyper- overheating. redundant morphologies, a “fitting” procedure is required. Fitting procedures determine the actua- tor displacements so that the discrete morphology 3. Kinematics of Backbone Curves robot exactly or closely follows the backbone curve In the rest of this paper, we abstract the impor- model [ChBSla, Ch921. The spatial Stewart plat- tant macroscopic kinematic phenomena in terms form fitting algorithm of [ChBSla] is used in the of a continuous backbone czLme. In this paradigm, ensuing example. which is the basis of our previous work, motion planning is reduced to determining the proper time 4. Uniform Direction Sidewinding varying behavior of the backbone curve. We re- vie here the essential kinematics of nonextensible e e we Consider vnifo m m tion o er 3 terrain. baclbone curves. !&ewinding can viewecf ils a Yorm ortrav%ng wave locomotion, in which the AS shape propagates from The Cartesian position of points on a nonextensible the head to the tail. For uniform motion, the AS backbone curve can be intrinsicly parametrized by: wave shape is constant. For turning motions, the AS shape is time varying. This section synthesizes qs,t) = 1’ ;(U, t)du (3.1) the backbone curve shape functions which imple- ment a uniform direction sidewinding gait over flat where s is the arclength parameter at time t: i.e., terrain. This can be done as follows: (1) find the IdZ/’lds) = 1. Z(s, t) is a vector from the base of the static AS and GCS shapes; (2) ensure that the backbone curve, located at s = 0, to the point on AS and GCS shapes blend smoothly at their in- the backbone curve denoted by s. Distance is nor- tersection; and (3) convert the resulting shape to a malized so that s E [0,1]. i;(s,t) is the unit length traveling wave form (i.e., for shape function S(s,t): backbone curve tangent at s. The parametriza- S(s,t) +- S (6-wt,t)). The backbone curve shape tion of (3.1) has the following interpretation. The can in turn be used to control a discrete morphol- backbone curve is “grown” from the base by propa- ogy system through a fitting procedure. gating the curve forward along the tangent vector, which is varying its direction according to G(s, t). We assume tyji s = 0 is t4“bead; oitp g””1- Any spherical kinematic representation can be used bone curve, w 1 e s = 1 is t e rear o t e ac - bone urye su e t e rob t cont to parametrize G(s) in (3.1) [ChB92c]. Here we use grounh via‘ nP&g. rstmilly, tgere wi~%et% the following parametrization for Z(s, t): basic gait segments (some of these basic gaits seg- ments may be not be fully formed at any instant). I- -, Index these segments by j, with j = 1 indexing the segment closed to the head. Let the arc-length of the jth fully formed GCS be denoted by L,. Sim- ilarly, let Laj be the arc-length of a fully formed AS. Let sjl(t) and sjz(t) denote the most forward (3.2) and rearward points of the jth GCS (see Fig. 3). K(s) and T(s)are angles which determine the di- Similarly, let sj3(t) denote the most rearward point rection of G(s) (see Fig. 2). By convention, the of the fh AS. sjl(t),sjz(t), and sjg(t) move along initial conditions K(O;b= T(0)= 0 are assumed. In summary, the back one curve is a function of a the backbone curve with wave speed w. reduced set of intrinsic “shape functions,” K(s,t) The total backbone shape can then be constructed and T(s,t). Additionally, the roll distribution as the piecewise sum of shape functions which sep- 102 where &(s) i; the vkctor tangent to the AS in the interval s E [sj,~,sj,~]. U', is parameterized by Ka and T,, as in Equation (3.2). In this local coordi- nate system, the arch segment must satisfy: Figure 3: Schematic of an arch segment arately control the shapes of the arch segments and pJ (0) = xj,2 = tangent to GCT at sj,2 the ground contact segments: fll)= Z(Sj,3) - ?(Sj,2) = [dll, dn, 0lT (4.7) N $(I) = xj,3= tangent to GCT at sj,3 = Kg(s - atj(t))W(s - "tj(t), Sj,l,Sj,2) j=1 where a ' indicates differentiation with respect to Ka(s - atj(t))W(S - atj(t),sj,2, sj,3) + s.
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