Further Development of the Mathematical Model of a Snakeboard

Transcription

1 ISSN , Regular and Chaotic Dynamics, 007, Vol 1, No 3, pp c Pleiades Publishing, Ltd, 007 RESEARCH ARTICLES Further Development of the Mathematical Model of a Snakeboard A S Kuleshov * Department of Mechanics and Mathematics, Lomonosov Moscow State University Main Building of MSU, Leninskie Gory, Moscow, Russia Received March 9, 007; accepted May, 007 Abstract This paper gives the further development for the mathematical model of a derivative of a skateboard known as the snakeboard As against to the model, proposed by Lewis et al [1] and investigated by various methods in [1 13], our model takes into account an opportunity that platforms of a snakeboard can rotate independently from each other This assumption has been made earlier only by Golubev [13] Equations of motion of the model are derived in the Gibbs Appell form Analytical and numerical investigations of these equations are fulfilled assuming harmonic excitations of the rotor and platforms angles The basic snakeboard gaits are analyzed and shown to result from certain resonances in the rotor and platforms angle frequencies All the obtained theoretical results are confirmed by numerical experiments MSC000 numbers: 70F5, 70E55, 70E60, 70E18 DOI: 10113/S Key words: Snakeboard, Gibbs Appell equations, dynamics, analysis of motion 1 INTRODUCTION The Snakeboard is one of the modifications of a well-known skateboard It allows the rider to propel himself forward without having to make contact with the ground even if a motion occurs uphill The motion of the snakeboard becomes possible due to a specific features of its construction and due to the special coordinated motions of legs and a torso of the rider The first snakeboard has appeared in 1989 and from this moment till now it has found a lot of fans among the amateurs of extreme sports Soon after the invention of the snakeboard the first attempts were made to give a mathematical description of the basic principles of human snakeboarding The basic mathematical model for the snakeboard investigated by various methods in many papers [1 13] was proposed by Lewis et al [1] In our paper we give the further development of the model proposed in [1] Fig 1 The Snakeboard The Snakeboard see Figs 1 consists of two wheel-based platforms upon which the rider is to place each of his feet These platforms are connected by a rigid crossbar with hinges at each platform to allow rotation about the vertical axis To propel the snakeboard the rider first turns both of his feet * kuleshov@mechmathmsusu 31

2 3 KULESHOV Fig Scheme of the Snakeboard Fig 3 Instruction on the Snakeboarding Fig Mathematical model for the Snakeboard in see Fig 3 By moving his torso through an angle, the snakeboard moves through an arc defined by the wheel angles The rider then turns both feet so that they point out, and moves his torso in the opposite direction By continuing this process the snakeboard may be propelled in the forward direction without the rider having to touch the ground The mathematical model of the snakeboard considered in this paper is represented in Fig We assume that the snakeboard moves on the xy plane, and let Oxy be a fixed coordinate system with origin at any point of this plane Let x and y be the coordinates of the system center of mass point G and θ the angle between the central line of the snakeboard and the Ox-axis In the basic model treated in [1] platforms could rotate through the same angle in opposite directions with respect to a central line of the snakeboard by other words, for this model ϕ f = ϕ b = ϕ, see Figs 3 We suppose that platforms can rotate independently and their positions are defined by two independent variables ϕ f and ϕ b Fig The motion of the rider is modeled by a rotor, represented in the form of a dumb-bell in Fig Its angle of rotation with respect to the crossbar is denoted by δ REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

3 FURTHER DEVELOPMENT OF THE MATHEMATICAL MODEL OF A SNAKEBOARD 33 Let l be the distance from the system center of mass G to the location of the wheels points A and B We assume that GA = GB = l, see Fig The platforms of the snakeboard are assumed to move without lateral sliding This condition is modeled by constraints which may be shown to be nonholonomic For the front platform corresponding constraint has a form and for the rear platform it has a form ẋ sin ϕ f θ ẏ cos ϕ f θ l θ cos ϕ f = 0 1 ẋ sin ϕ b θ ẏ cos ϕ b θ l θ cos ϕ b = 0 We can solve equations 1 and with respect to ẋ and ẏ Then ẋ = ẏ = l θ sin ϕ f ϕ b cos ϕ b cos ϕ f θ cos ϕ f cos ϕ b θ, l θ sin ϕ f ϕ b cos ϕ b sin ϕ f θ cos ϕ f sin ϕ b θ 3 Further we describe a motion of platforms using new variables ψ 1 and ψ, connected with variables ϕ f and ϕ b by relations ψ 1 = ϕ f ϕ b, ψ = ϕ f ϕ b Control of the snakeboard is realized by rotations of the platforms through ϕ f and ϕ b and by rotation of the rotor through δ We assume that the variables δ, ψ 1 and ψ are known functions of time t, ie δ = δ t, ψ 1 = ψ 1 t, ψ = ψ t These variables are the controlled variables in this problem EQUATIONS OF MOTION We derive now equations of motion of the given model of a snakeboard in the form of the Gibbs Appell equations For this purpose we use the method, which was applied earlier by Ispolov and Smolnikov [1] to study the skateboard dynamics We introduce pseudovelocity V by the formula From this formula we have V = l θ sin ψ 1 cos ψ 1 cos ψ θ = V l sin ψ 1 cos ψ 1 cos ψ 5 Using we can rewrite expressions 3 for ẋ and ẏ as follows: ẋ = V cos θ V sin ψ sin θ cos ψ 1 cos ψ, ẏ = V sin θ V sin ψ cos θ cos ψ 1 cos ψ 6 In a case, when ϕ f = ϕ b = ϕ and therefore ψ = 0 equations 6 have a very simple form ẋ = V cos θ, ẏ = V sin θ So, in this case pseudovelocity V can be interpreted as velocity of the system s center of mass In general case, this pseudovelocity is velocity of a point which can be obtained if we take a projection of the system s instantaneous center onto the line passing through the central line of a snakeboard ie passing through the crossbar REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

4 3 KULESHOV In order to derive differential equations of snakeboard motion in the Gibbs Appell form, let us obtain the Gibbs function [15] This function is known also as the energy of acceleration, see [1, 16] It is well known that the Gibbs function of a rigid body can be calculated using the formula see [16] S = m W C 1 ω Θ C ω [ω Θ C ω] ω, 7 where m is the mass of the body, W C is acceleration of the body s center of mass, ω and ω are the angular velocity and angular acceleration of the body, Θ C is a tensor of inertia of the body The Gibbs function of a snakeboard is the sum of the Gibbs functions of a crossbar, of a rotor and of a platforms Thus, we find, according to formula 7, the Gibbs functions for each of these parts and then we take their sum We denote by m b the mass of the crossbar and let J b be its moment of inertia with respect to the vertical axis, passing through its center of mass Then the Gibbs function of the crossbar can be calculated in the form S b = m b ẍ ÿ J b θ Let m r be the mass of the rotor and J r is its moment of inertia with respect to the vertical axis, passing through its center of mass The angular velocity of the rotor is the sum of the angular velocity of the crossbar and the angular velocity of the rotor with respect to the crossbar: ω = θ δ e z Therefore, the Gibbs function of the rotor have a form S r = m r ẍ ÿ J r θ δ Now we calculate the Gibbs function of the platforms Suppose that platforms have equal masses and equal moments of inertia with respect to their vertical axes Suppose also that the center of mass of the front platform is located at point A and the center of mass of the rear platform is located at point B Then the center of mass of the front platform has the following coordinates with respect to the coordinated system Oxy x A = x l cos θ, y A = y l sin θ The center of mass of the rear platform has the following coordinates x B = x l cos θ, y B = y l sin θ Corresponding accelerations can be written in the form: ẍ A = ẍ l θ sin θ l θ cos θ, ÿ A = ÿ l θ cos θ l θ sin θ, ẍ B = ẍ l θ sin θ l θ cos θ, ÿ B = ÿ l θ cos θ l θ sin θ Let m p be the mass of the platforms and J p is their moment of inertia Then the Gibbs function of the platforms may be written as follows: θ ϕf θ ϕb S p = m p ẍ A ÿ A ẍ B ÿ B J p Omitting the terms which don t contain the generalized accelerations we obtain S p = m p ẍ ÿ m p l θ J p θ J p θ ψ Finally the Gibbs function of the system has a form S =S b S r S p = m b m r m p ẍ ÿ Jb m p l θ J r J 8 p θ J r δ Jp ψ θ Further we denote the total mass of the system by m: m = m b m r m p REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

5 FURTHER DEVELOPMENT OF THE MATHEMATICAL MODEL OF A SNAKEBOARD 35 Denote also expression J b m p l by J Then expression 8 may be rewritten as follows: S = m ẍ ÿ J J r J p θ J r δ Jp ψ θ 9 From 5 6 we find θ = V sin ψ 1 V ψ 1 cos ψ ψ 1 1 sin ψ 1 ψ sin ψ V sin ψ 1 l cos ψ 1 cos ψ l cos ψ 1 cos ψ, ẍ = V cos θ sin ψ sin θ sin θ sin ψ cos θ V sin ψ 1 cos ψ 1 cos ψ cos ψ 1 cos ψ l cos ψ 1 cos ψ ψ cos ψ ψ 1 sin ψ 1 ψ sin ψ sin ψ cos ψ 1 cos ψ cos ψ 1 cos ψ V sin θ, ÿ = V sin θ sin ψ cos θ cos θ sin ψ sin θ cos ψ 1 cos ψ cos ψ 1 cos ψ ψ 1 sin ψ 1 ψ sin ψ sin ψ cos ψ 1 cos ψ V cos θ ψ cos ψ cos ψ 1 cos ψ V sin ψ 1 l cos ψ 1 cos ψ Substituting expressions for ẍ, ÿ and θ into expression 9 we finally obtain S = m [1 V sin ψ cos ψ 1 cos ψ J J r J p sin ] ψ 1 ml cos ψ 1 cos ψ mv V [ ψ cos ψ 1 cos ψ sin ψ cos ψ J J ] r J p ψ ml 1 sin ψ 1 cos ψ 1 mv V ψ 1 sin ψ 1 ψ sin ψ [ cos ψ 1 cos ψ 3 sin ψ J J ] r J p V J r δ Jp ψ sin ψ 1 ml sin ψ 1 l cos ψ 1 cos ψ This expression for S contains only the terms which depend on V The other terms are omitted here The corresponding Gibbs Appell equation, describing the rate of change of the pseudovelocity V, has a form: S/ V = 0 Thus, we have the complete system of the equations of snakeboard motion: where ẋ = V cos θ V sin ψ sin θ, cos ψ 1 cos ψ θ = V sin ψ 1 l cos ψ 1 cos ψ, ẏ = V sin θ V sin ψ cos θ cos ψ 1 cos ψ, P 1 t V P t V = Q t, sin ψ P 1 t = 1 cos ψ 1 cos ψ k sin ψ 1 cos ψ 1 cos ψ, ψ sin ψ cos ψ k ψ 1 sin ψ 1 cos ψ 1 ψ 1 sin ψ 1 ψ sin ψ 10 P t = cos ψ 1 cos ψ sin cos ψ 1 cos ψ 3 ψ k sin ψ 1, d 1 δ d ψ sin ψ 1 Q t =, cos ψ 1 cos ψ k = J J r J p ml, d 1 = J r ml, d = J p ml REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

6 36 KULESHOV The latter equation in system 10 determines the dependence of the pseudovelocity V on the controlled variables δ = δ t, ψ 1 = ψ 1 t and ψ = ψ t Suppose that V 0 = V 0 = 0 Then the solution of the latter equation of system 10 with zero initial condition may be written as follows: V = cos ψ 1 t cos ψ t k sin ψ 1 t cos ψ 1 t cos ψ t sin ψ t t d 1 δt d ψ t sin ψ 1 t k sin ψ 1 t cos ψ 1 t cos ψ t sin ψ t 0 After finding the expression for V t, we obtain from the third equation of system 10 θ t = θ 0 t 0 V t 1 l 11 sin ψ 1 t 1 cos ψ 1 t 1 cos ψ t 1 dt 1 1 Further, using formula 1, we can obtain from the first two equations of system 10 t x t = x 0 y t = y 0 0 t 0 V t 1 cos θ t 1 sin ψ t 1 sin θ t 1 dt 1, cos ψ 1 t 1 cos ψ t 1 V t 1 sin θ t 1 sin ψ t 1 cos θ t 1 dt 1 cos ψ 1 t 1 cos ψ t 1 Thus, the problem of snakeboard dynamics at arbitrary controlled variables δ t, ψ 1 t and ψ t is completely solved in terms of quadratures However, the calculation of integrals in for given controlled variables and the analysis of the exact solution is a rather complicated problem Below we assume that the controlled variables vary sinusoidally 13 3 BASIC ASSUMPTIONS FOR FURTHER INVESTIGATION Observations of actual snakeboard riders suggest that sinusoidal inputs provide a good starting point for our investigations δ = a r sin ω r t, ψ 1 = a 1 sin ω 1 t, ψ = a sin ω t Suppose also that the amplitudes a 1 and a in the expressions for ψ 1 and ψ satisfy the inequalities a 1 05 rad, a 05 rad This assumption is completely justified by the features of snakeboard s construction According to this assumption, since 1 sin ω 1 t 1, 1 sin ω t 1 then 05 ψ 1 05 and 05 ψ 05 In this intervals for the angles ψ 1 and ψ the following approximative formulas sin ψ 1 ψ 1, sin ψ ψ, cos ψ 1 1 ψ 1, cos ψ 1 ψ are valid Thus we consider parameters a 1 and a as small parameters in this problem We will neglect terms of order higher than second on the parameters a 1 and a The snakeboard is assumed to have its initial condition at the origin in the space state, ie x 0 = y 0 = 0, θ 0 = 0 REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

7 FURTHER DEVELOPMENT OF THE MATHEMATICAL MODEL OF A SNAKEBOARD 37 Taking into account all these assumptions we have the following simplified formula for the pseudovelocity V t: V t d 1a r a 1 ω r t 0 sin ω r t 1 sin ω 1 t 1 dt 1 d a 1 a ω t 0 sin ω 1 t 1 sin ω t 1 dt 1 1 Thus we can conclude that the character of the snakeboard motion depends on the relations between the frequencies ω r, ω 1 and ω 1, ω NONRESONANT CASE In this case, when ω r ω 1 ω, the expressions for basic variables in the problem have rather complicated form For their simplicity we denote by A αi and B αi the following values: A αi = ω r 1 i αω 1, B αi = ω 1 1 i αω Then, integrating formula 1, we have for V t V t = d 1a r a 1 ωr 1 i1 sin A 1i t d a 1 a ω A 1i 1 i1 sin B 1i t B 1i Substituting the obtained expression for V t in the integral 1 and omitting the terms, which contain a 1, a order more than second, we find after integration: θ t = d 1a r a 1 ω r sin A i t sin ω rt 16l A 1i A i A 11 A 1 Thus, the expression for θ t contains the parameter a 1 to the second order On the other hand, if A 1i 0 and A i 0 i = 1,, then the function sin A i t A 1i A i sin ω rt A 11 A 1 is limited for arbitrary t Consequently, in the general case the function θ t has second order dependence on a 1 and we can assume approximately: ẋ V t, ẏ V t ψ t Integrating, we obtain the following formulas for x t and y t: x t = d 1a r a 1 ωr ω r ω 1 A 11 A 1 1 i cos A 1i t A 1i d a 1 a ω ω 1 ω B 11 B 1 1 i cos B 1i t B 1i, y t = d 1a r a 1 a ωr 1 j sin ω r 1 i B 1j t 8 A 1i ω r 1 i B 1j i,j=1 One can see that at some ratios of frequencies ω r, ω 1 and ω the denominators of the functions in the obtained expressions for V t, θ t, x t and y t can vanish Below we enumerate all existing cases: 1 A 1i = 0, i = 1,, ie ω r = ±ω 1 ; A i = 0, i = 1,, ie ω r = ±ω 1 ; 3 ω r 1 i B 1j = 0, i, j = 1,, ie ω r = ω 1 ± ω or ω r = ω 1 ± ω REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

8 38 KULESHOV Moreover, in all cases listed above we can separate two different subcases: when B 1i 0 ie ω 1 ±ω and when B 1i = 0 ie ω 1 = ±ω If one of the listed ratios is carried out during the motion of the snakeboard, then there is a certain resonance in the system To each resonance there corresponds a certain type of motion of the snakeboard The basic principle of motion of the rider on the snakeboard consists in skilful attainment of this or that resonant ratio appropriate to a desirable type of maneuver The motion of the snakeboard in resonant cases will be investigated below System 10 was investigated numerically at the following parameters: m = 75 kg, J = 0 kg m, J r = 1 kg m, J p = 0013 kg m, l = 085 m 15 These values of parameters as a whole correspond to an actual physical situation The parameters included in the law of changing of the controlled variables have been chosen corresponding to nonresonant case: a r = 07 rad, a 1 = 03 rad, a = 0 rad, 16 ω r = 1/ rad/sec, ω 1 = 1/ 3 rad/sec, ω = 1/ rad/sec Numerical analysis of system 10 in the nonresonant case has shown that for all t the exact and approximate solutions are sufficiently close to each other Figure 5 shows the position of the center of mass of the snakeboard along the trajectory A similar trajectory is characteristic for beginners who make their first steps in snakeboarding Now we consider the subcase of nonresonant case, when ω r ω 1 and ω 1 = ω In this particular case the formulas for θ t and y t don t change while the formulas for V t and x t have a form: V t = d 1a r a 1 ωr x t = d 1a r a 1 ω r 1 i1 sin A 1i t d a 1 a ω 1 A 1i ω r ω 1 A 11 A 1 1 i cos A 1i t A 1i ω 1 t sin ω 1 t cos ω 1 t, d a 1 a 8 ω 1 t sin ω 1 t For this particular case we also make numerical analysis of the system 10 For the numerical analysis we use the values of parameters and the following values of frequencies ω r = 1/ rad/sec, ω 1 = ω = 1/ 3 rad/sec Fig 5 Trajectory of the system s center of mass in the nonresonant case subcase ω 1 ω REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

9 FURTHER DEVELOPMENT OF THE MATHEMATICAL MODEL OF A SNAKEBOARD 39 Fig 6 Trajectory of the system s center of mass in the nonresonant case subcase ω 1 = ω Figure 6 shows the position of the center of mass of a snakeboard along the trajectory in this subcase 5 1:1 RESONANCE FORWARD MOTION We begin study of resonant cases with the case when ω r = ω 1 = ω = ω Let us introduce the dimensionless time τ using the formula τ = ωt Formula 1 for the pseudovelocity V gives the following expression: V τ = d 1a r d a a 1 ω τ sin τ cos τ Thus, when the frequency of rotation of the rotor is equal to the frequencies of rotations of the platforms, the pseudovelocity V grows linearly in time The same fact is valid for the velocity of the system s center of mass, see formulas 6 Hence, if the rider, moving on the snakeboard, will achieve, that the frequency of rotation by a torso has coincided with frequency of rotation by stops of legs, he can propel himself forward with increasing velocity This is the main principle of snakeboard dynamics An approximate formula for θ τ has the form: θ τ = d 1a r a 1 8l sin τ τ cos τ sin3 τ 3 Evidently, the function θ τ as well as V τ is a linearly growing function of time τ Therefore, we can consider the angle θ as a small angle up to some moment of time If we consider that the angles θ and ψ 1, ψ are small at θ 05 rad, then we can give the following estimation for this moment: J r a r a 1 8ml τ 1 ie τ ml J r a r a 1 Substituting in this formula the values of parameters we have the following estimation for the interval of dimensionless time during which we can consider the angle θ as a small angle: τ 768 For small values of θ we have for x τ and y τ: τ sin τ, x τ = d 1a r d a a 1 8 y τ = d 1a r a 1 a sin τ τ cos τ sin3 τ 3 For numerical analysis of system 10 in the case of 1:1 resonance we used the values of parameters and the following values of frequencies ω r = ω 1 = ω = ω = 1 rad/sec REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

10 330 KULESHOV Numerical analysis of system 10 has shown, that for all τ 768 the exact and approximate solutions are sufficiently close to each other Figure 7 shows the position of the center of mass of the snakeboard along the trajectory for the case of 1:1 resonance Analyzing the formulas for V τ and x τ we can conclude that even for a r = 0 the rotor doesn t rotate the snakeboard may be propelled forward By other words, the rider can propel snakeboard using only legs This fact emphasize the difference between two type, of boards: the snakeboard and the skateboard Fig 7 Trajectory of the system s center of mass at 1:1 resonance 6 :1 RESONANCE ROTATIONAL MOTION We consider now the case when the frequencies ω r and ω 1 are connected by a the ratio ω r = ω 1 Suppose that ω 1 ω Then, using formula 1, we obtain V t = d 1a r a 1 ω 1 3 sin 3 ω 1 t d a 1 a ω 1 i1 sin B 1i t B 1i The corresponding expression for the function θ t has the form: θ t = d 1a r a 1 ω 1 t sin ω 1 t cos ω 1 t l 3 sin3 ω 1 t cos ω 1 t Thus, as in the case of 1:1 resonance, the function θ t grows linearly in time We can consider this angle as small for the interval ω 1 t l d 1 a r a 1 Substituting to this inequality the values of parameters we find, that ω 1 t 1381 For this interval of t we can consider the angle θ t as a small angle and obtain for x t and y t the following expressions: d 1a r a 1 sin ω 1 t cos ω 1 t 9 d a 1 a ω ω 1 ω B 11 B 1 x t = 8d 1a r a 1 9 y t = d 1a r a 1 a ω i cos B 1i t B 1i, [ sin ω 3ω 1 t sin ω 3ω 1 t 3 ω 3ω 1 ω 3ω 1 ] 1 i sin B 1i t B 1i REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

11 FURTHER DEVELOPMENT OF THE MATHEMATICAL MODEL OF A SNAKEBOARD 331 For numerical investigation of system 10 we choose the values of parameters and the following values of frequencies ω r = rad/sec, ω 1 = 1 rad/sec, ω = 1/ rad/sec Figure 8 shows the position of the center of mass of the snakeboard along the trajectory for the case ω r = ω 1, ω 1 ω, resulting in an almost pure rotation of the snakeboard s central line axis of θ τ = 003 rad Fig 8 Trajectory of the system s center of mass at ω r = ω 1, ω 1 ω Now we consider the first subcase: ω r = ω 1 = ω, ω 1 = ω = ω As in the previous case of 1:1 resonance we will use the dimensionless time τ Then expression for θ τ may be written as follows: θ τ = d 1a r a 1 τ sin τ cos τ l 3 sin3 τ cos τ 17 Expressions for other functions have a form: x τ = 8d 1a r a 1 9 sin 3 τ d a 1 a ω τ sin τ cos τ, d 1a r a 1 sin τ cos τ d a 1 a τ sin τ, 9 8 τ sin τ cos τ 3 sin3 τ cos τ V τ = d 1a r a 1 ω 3 y τ = d 1a r a 1 a In this particular case we choose the following values of frequencies ω r = ω = rad/sec, ω 1 = ω = ω = 1 rad/sec Figure 9 shows the position of the the center of mass of the snakeboard along the trajectory for this subcase The second particular case takes place when ω r = ω 1 = ω, ω = 3ω In this case function θ τ is determined by formula 17, while the formulas for V τ, x τ and y τ have a form: x τ = 9d a 1 a 8 V τ = a 1ω sin 3 τ 6 y τ = d 1a r a 1 a 6 8d 1 a r 7d a cos τ, sin τ d 1a r a 1 sin τ cos τ 8d 1a r a 1, sin τ 3 sin 6τ sin τ τ 6 REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

12 33 KULESHOV Fig 9 Trajectory of the system s center of mass at ω r = ω 1, ω 1 = ω In this particular case we choose the following values of frequencies ω r = ω = rad/sec, ω 1 = ω = 1 rad/sec, ω = 3ω = 3 rad/sec Fig 10 Trajectory of the system s center of mass at ω r = ω 1, ω = 3ω 1 Figure 10 shows the position of the the center of mass of the snakeboard along the trajectory for this subcase 7 RESONANCE OF THE THIRD TYPE In the end of our investigation we consider one of the cases of the third type see Section These cases are determined by relation between ω r, ω 1 and ω in the form ω r 1 i B 1j = 0, i, j = 1, All these cases are investigated similarly and here we study only the case when ω r = ω 1 ω REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

13 FURTHER DEVELOPMENT OF THE MATHEMATICAL MODEL OF A SNAKEBOARD 333 If ω 1 = ω = ω, then we obtain the first subcase of :1 resonance We already investigate this case above Therefore, suppose ω 1 ω Then we have the following expressions for the main functions: V t = ω 1 ω d 1 a r a 1 sin ω t sin ω 1 ω t ω ω 1 ω d a 1 a ω sin ω1 ω t sin ω 1 ω t, ω 1 ω ω 1 ω θ t = ω 1 ω d 1 a r a [ 1 sin ω ω 1 t sin ω 1 ω t sin 3ω ] 1 ω t, 16l ω ω ω 1 ω ω 1 ω ω 1 ω 3ω 1 ω [ ] x t = ω 1 ω d 1 a r a 1 cos ω 1 ω t ω 1 ω cos ω t ω ω 1 ω 1 ω ω ω 1 ω d a 1 a ω y t = ω 1 ω d 1 a r a 1 a 8ω ω 1 ω d 1 a r a 1 a 16 ω 1 ω [ cos ω 1 ω t ω 1 ω cos ω 1 ω t ω t sin ω t cos ω t [ sin ω1 ω t ω 1 ω ω 1 ω ω 1ω ω 1 ω ], sin ω ] 1t ω 1 For numerical investigation of the system 10 in this case we choose the values of parameters and the following values of frequencies ω r = rad/sec, ω 1 = 3 rad/sec, ω = 1 rad/sec Figure 11 shows the position of the center of mass of the snakeboard along the trajectory in this case Fig 11 Trajectory of the system s center of mass in the case ω r = ω 1 ω Thus, the basic types of human snakeboarding are investigated The obtained results will be coordinated in many respects with the results of observations of actual snakeboard riders and with authors snakeboard experience 8 CONCUSION In this paper we give a further development of the mathematical model of a snakeboard, proposed by Lewis et al [1] We assume that platforms of a snakeboard can rotate independently Therefore, REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007

14 33 KULESHOV the position of platforms is described by two independent coordinates ϕ f and ϕ b From these coordinates we pass to the coordinates ψ 1 = ϕ f ϕ b, ψ = ϕ f ϕ b Then in the case ψ = 0 we have the Lewis model of a snakeboard The rider s motion is modeled by a rotor Its angle of rotation we denote by δ As in the paper [1] we suppose that the variables δ, ψ 1 and ψ vary sinusoidally: δ = a r sin ω r t, ψ 1 = a 1 sin ω 1 t, ψ = a sin ω t For the sinusoidal inputs we give a complete analysis of a possible snakeboard gaits We conclude that each of these gaits is defined by certain resonance condition between frequencies of rotation of a rotor and platforms For 1 : 1 resonance forward motion we note, that even if the rider doesn t rotate his torso, he can propel snakeboard forward using only his legs All analytical conclusions are confirmed by numerical experiments This research was supported financially by the Russian Foundation for Basic Research Grant REFERENCES 1 Lewis, A D, Ostrowski, J P, Murray, R M, and Burdick, J W, Nonholonomic Mechanics and Locomotion: the Snakeboard Example, Proc of the IEEE ICRA, San Diego, May 199, pp Ostrowski, J P, Burdick, J W, Lewis, A D, and Murray, R M, The Mechanics of Undulatory Locomotion: The Mixed Kinematic and Dynamic Case, Proc of the IEEE ICRA, Nagoya, Japan, May 1995, pp Ostrowski, J P, The Mechanics and Control of Undulatory Robotic Locomotion, PhD thesis, California Institute of Technology, Pasadena, California, USA, Sept 1995 Bloch, A M, Krishnaprasad, P S, Marsden, J E, and Murray, R M, Nonholonomic Mechanical Systems with Symmetry, Arch Rational Mech Anal, 1996, vol 136, pp Koon, W S and Marsden, J E, Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction, SIAM J Control Optim, 1997, vol 35, pp Robinson, D, Newtonian Exercise on a Snakeboard, Physical Education, 1999, vol 3, pp Ostrowski, J P, Desai, J P and Kumar, V, Optimal Gait Selection for Nonholonomic Locomotion Systems, International Journal of Robotics Research, 000, vol 19, pp Bullo, F and Lewis, A D, Kinematic Controllability and Motion Planning for the Snakeboard, IEEE Transactions on Robotics and Automation, 003, vol 19, pp Blankenstein, G, Symmetries and Locomotion of a D Mechanical Network: the Snakeboard, Lecture Notes for the Euron/GeoPlex Summer School, Bertinoro, Italy, July Duindam, V and Stramigioli, S, Energy-Based Model-Reduction and Control of Nonholonomic Mechanical Systems, Proc of the IEEE ICRA, 00, pp Iannity, S and Lynch K, Minimum Control-Switch Motions for the Snakeboard: A Case Study in Kinematically Controllable Underactuated Systems, IEEE Transactions on Robotics and Automation, 00, vol 0, pp Shammas, E, Choset, H, and Rizzi, A, Toward Automated Gait Generation for Dynamic System with Non- Holonomic Constraints, Proceedings of the IEEE ICRA, Orlando, Florida, 006, pp Golubev, Y F, A Method for Controlling the Motion of a Robot Snakeboarded, Prikl Mat Mekh 006, vol 70, no 3, pp [J Appl Math Mech Engl Transl, vol 70, no 3, pp ] 1 Ispolov, Y G and Smolnikov, B A, Skateboard Dynamics, Computer Methods in Applied Mechanics and Engineering, 1996, vol 131, pp Ardema, M D, Analytical Dynamics: Theory and Applications, New-York: Kluwer Academic Publishers, Lurie, A I, Analytical Mechanics, Berlin: Springer-Verlag, 00 REGULAR AND CHAOTIC DYNAMICS Vol 1 No 3 007