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Nonholonomic Mechanics and Locomotion: The Nonholonomic Mechanics and Locomot ion: The Snakeboard Example Jim Ostrowski Andrew Lewis Richard Murray Joel Burdick Department of Mechanical Engineering, California Institute of Technology Mail Code 104-44, Pasadena, CA 91125 Abstract to most methods of locomotion, including walking, running, parallel parking, undulating, and sidewinding. Analysis and simulations are performed for a simplified model Superficially, the snakeboard appears to be closely re- of a commercially available variant of the skateboard, known lated to other robotic systems with nonholonomic constraints, as the Snakeboard’. Although the model exhibits basic gait where cyclic motions in the control space of the vehicle can patterns seen an a large number of locomotion problems, the cause net motion in the constrained directions (see [ll]for an analysis tools currently available do not apply to this prob- introduction and references). However, the dynamics of our lem. The dificulty lies primarily in the way in which the non- model of the Snakeboard do not fit into the principal fibre holonomic constraints enter into the system. As a first step bundle structure which has been used to study some non- towards understanding systems represented by OUT model we holonomic systems [2]. The snakeboard seems to represent a present the equations of motion and perform some controlla- class of systems for which current analysis tools do not pro- bility analysis for the snakeboard. We also perform numerical vide any assistance. Thus, the snakeboard model: (1) is an simulations of possible gait patterns which are characteristic interesting problem in nonholonomic mechanics; (2) repre- of snakeboard locomotion. sents an unexplored class of systems which may be used for locomotion; and (3) serves as a motivating example for the development of new frameworks for exploring the relationship Introduction between nonholonomic mechanics and locomotion. In Section 1 we give a detailed description of the Snake- This paper investigates a simplified model of a commercially board and how it is used. We also present our simplified available derivative of a skateboard known as the Snake- model which is intended to capture the essential features of board. The Snakeboard (Figure 1) allows the rider to propel the Snakeboard. In Section 2 we present Lagrange’s equations him/herself forward without having to make contact with the for the snakeboard and describe a control law which allows ground. This motion is roughly accomplished by coupling a us to follow specified inputs exactly. Since the snakeboard is conservation of angular momentum effect with the nonholo- modeled as a constrained control system, it is possible to ex- nomic constraints defined by the condition that the wheels amine controllability and thereby determine whether we may roll without slipping. Snakeboard propulsion is discussed in reach all points in our state space. This analysis is presented more detail in Section 1. in Section 3. In Section 4 the above mentioned gaits are pre- We study this model for several reasons. First, the snake- sented and analysed. The failure of the snakeboard to fit into board’s means of locomotion has not appeared in prior stud- a principal bundle formulation is discussed in Section 5. In- ies of robotic locomotion. Numerous investigators have stud- cluded in this discussion is the introduction of possible tools ied and successfully demonstrated quasi-static multi-legged suggested by recent research which may make the problem locomotion devices (161. Others have considered and imple- more tractable. mented various forms of undulatory, or “snake-like,’’locomo- tion schemes [6], [4]. Beginning with Raibert [13], hopping robots have received considerable attention as well [8, 10, 11. Bipedal walking and running has also been an active area of research [9, 71. In all of these cases except [l], the robotic locomotion devices are largely anthropomorphic or zoomor- phic. The method of locomotion used for the snakeboard is significantly different from all of these approaches and does not appear to have a direct biological counterpart. There is, rear front however, some similarity to the undulatory motion of snakes, wheel Footpads wheel and it is hoped that this research will provide insight into other areas of locomotion which make use of constraints aris- ing through ground contact. Figure 1: The Snakeboard Despite its unique features, the mechanics of the snake- board’s movement has several properties which we believe to be common to many forms of locomotion. In Section 4, the simplified snakeboard model is shown to exhibit a number of 1. The Snakeboard and a simplified model gaits, each of which generates a net motion in a certain di- rection by performing loops in the controlled variables. This The Snakeboard consists of two wheel-based platforms upon general method of locomotion (i.e., generating net motions which the rider is to place each of his feet. These platforms by cycling certain control variables) appears to be generic are connected by a rigid coupler with hinges at each platform to allow rotation about the vertical axis. See Figure 1. To ‘The name Snakeboard has been trademarked. propel the snakeboard, the rider first turns both of his feet 2391 1050-4729/94 $03.00 0 1994 IEEE lie in ker{w' ,w2}, where w1 = -sin(+b t e)& + COS(4b + 0)dy - 1COs(db)d6 (1.3) w2 = - sin(4j + B)~X+ COS(^^ + e)dy + I cos(4f)de (1.4) 2. Dynamics and control of the snakeboard To investigate the dynamics of the snakeboard we use La- grange's equations which, for constrained and forced systems, are given by L back wheels Figure 2: The simplified model of the Snakeboard Here XI,. ,A, are the Lagrange multipliers, wl,, . ,wm are the constraint one-forms, and 71,. ,r,, are the external forces. The first term on the right hand side of Lagrange's in. By moving his torso through an angle, the Snakeboard equations may be regarded as an external force applied to the moves through an arc defined by the wheel angles. The rider system to ensure that the constraints are satisfied. As such, then turns both feet so that they point out, and moves his the Lagrange multipliers are a part of the solution to the torso in the opposite direction. By continuing this process problem. See [12] for a discussion of Lagrangian mechanics the Snakeboard may be propelled in the forward direction in this vein. We will only consider torques on the controlled without the rider having to touch the ground. variables 4,4b1 and 4f.The Lagrangian for the snakeboard Our simplified model of the Snakeboard is shown in Fig- is ure 2. We will use the term snakeboard to refer to this model, 1 1 1 L = -m(k2 + iz)+ -Jdz + -J7(d + but will distinguish the model from the commercially avail- 2 2 2 + able Snakeboard by using italics and capitals to describe the latter. As a mechanical system the snakeboard has a con- figuration space given by Q = SE(2) x s' x s' x s'. Here SE(2) is the group of rigid motions in the plane, and we are thinkin of this as describing the position of the board itself. By & we mean the group of rotations on R2. The three copies of s1 in Q describe the positions of the rotor and the two wheels, respectively. As coordinates for Q we shall use (2, y, e,$, &,, 4f) where (z,y, e) describes the position of the board with respect to a reference frame (and so are to be thought of as an element of SE(2)),$ is the angle of the rotor with respect to the board, and &,, and are, respectively, the angles of the back and front wheels with respect to the board. we will frequently refer to the variables ($,db,ff) as the controlled variables since they are the variables which are rider inputs in the actual Snakeboard. Parameters for the problem are: where (u1 ,u2, u3) are the input torques in the ($, (bb, 4f)di- m : the mass of the board, rections, respectively. J : the inertia of the board, Since the rider of the Snakeboard typically propels him- J, : the inertia of the rotor, self by performing cyclic motions with his feet and torso, it Jw : the inertia of the wheels (we assume was deemed desirable to devise a control law which would them to be the same), and allow one to follow any curve, t I+($(t), db(t), df(t)), in the I : the length from the board's centre of controlled variables. It turns out that such a control law is mass to the location of the wheels. derivable with some manipulation of Lagrange's equations. The wheels of the snakeboard are assumed to roll with- We outline some of this manipulation in the proof of the fol- out lateral sliding. This condition is modeled by constraints lowing proposition. which may be shown to be nonholonomic. At the back wheels Proposition 2.1 Let t I+ ($(t),db(t),df(t))be a piece- the constraint assumes the form wise smooth curve. Then there exists a control law t * - sin(4b + 0); + cos(+a + e)$ - I COS(db)d = 0. (ul(t), uz(t),u3(t)) 30 that the ($,bb,df) components of (1.1) the sohtron to Lagrange's equations are given by t I+ (@(t)ih(t), dt(t))* Similarly at the front wheels the constraint appears as Proof In Lagrange's equations, (2.1)-(2.6), and in the con- sin(4j COS(^^ 0)~lcos(4f)8 = 0. - + e); + + straints, (1.1) and (12), regard ($, $b, 4f)as known functions (1.2) oft. Substituting (2.4)-(2.6) into (2.3) qves Alternatively one can write the constraints as the kernel of two differential oneforms.
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