The Lagrangian Method for a Basic Bicycle

The ground plan in order to disentangle the hard problem of modelling the motion of a bicycle is to start from a very simple model and to outline the proper mathematical scheme: for this reason the first step we perform lies in a planar rigid body (simulating the bicylcle frame) pivoting on a horizontal segment whose extremities, subjected to nonslip conditions, oversimplify the wheels. Even in this former case, which is the topic of lots of papers in literature, we find it worthwhile to pay close attention to the formulation of the mathematical model and to focus on writing the proper equations of motion and on the possible existence of conserved quantities. In addition to the first case, being essentially an inverted pendulum on a skate, we discuss a second model, where rude handlebars are added and two rigid bodies are joined. The geometrical method of Appell is used to formulate the dynamics and to deal with the nonholonomic constraints in a correct way. At the same time the equations are explained in the context of the cardinal equations, whose use is habitual for this kind of problems. The paper aims to a threefold purpose: to formulate the mathematical scheme in the most suitable way (by means of the pseudovelocities), to achieve results about stability, to examine the legitimacy of certain assumptions and the compatibility of some conserved quantities claimed in part of the literature.


Introduction 1.The Equations of the Model
A very simple scheme can be formulated by assuming that the body is a planar rigid system  sketched by three points A, B and 0 P ; A and B, performing the two contact points of the wheels, belong to a horizontal plane and 0 P is the centre of mass of  .The rigid body can lean with respect to the vertical direction and bend with respect to a fixed horizontal direction.Let O be the projection of 0 P on B A − and take a fixed reference frame { } where I are the moments of inertia with respect to the body reference frame { } 0 , , , P i j k , which is supposed to be principal.
The only kinetic constraint we are going to consider is B ∧ = i  0 , whose expression in the Lagrangian coordinates is 0 0 sin cos 0.
bψ ξ ψ η ψ The first kind Lagrangian equations of motion are the lagrangian coordinates and λ is the unknown multiplier, will be suitably handled if one defines the pseudovelocities (see [1] for the concepts and the method we are pursuing) We point out the following relationships involving U, V and the real velocities: where a a b = and 1 3 1 2 sin cos cos sin .
Together with the kinetic constraint, Equations (4) give the set of possible velocities in terms of ( ) It is known (see [1]) that linear kinetic constraints allow to refine the equations of motion (3) in a way similar to the holonomic case: as a matter of fact, the constraints identify a subspace in the tangent space of the lagrangian coordinates, giving the virtual displacement of the system.The geometrical method consists of projecting the equations according to T d dt where Γ is defined in (6).Joining to the kinetic constraint (2) and the definition (4) we get, dividing by suitable constants, , , are dimensionless constants.Since the rigid body is practically plane and contained in the plane orthogonal to j, it is reasonable to assume > ) From here on we adopt (9).
The seven ODEs (7) contain the seven unknown quantities ( ) 0 0 , , , , , , U V W ξ η ψ θ .With respect to the first kind system (3) they have the advantage of no exhibiting multipliers and of reducing the kinetic variables of one unity.
It is not at all worthless to explain (7) in the context of the the cardinal equations of dynamics, seeing that many models in literature (some of them are [2]- [5]) rest on such equations: the first three equations in (7) are respectively where L is the angular momentum and ( ) ( ) Hence no force exists along i (first equation) and (second equation), along 3 e (and not k , as stated in [6]).Finally, third equation is simply due to the fact that the only external force with a non zero momentum along i is the weight force.
Notice that any of the three equations do not give rise to a conserved quantity: the only evident constant of motion is the total energy Actually, even if the system is nonholonomic, the first integral 0 ≡   can be achieved starting from and performing the usual calculations as in the holonomic case, achieving at last 0 =  .In the same matter of integral invariants for the system, we find it not correct to claim, as in [7], that the absence of 0 ξ , 0 η , ψ from  entails three constant of motion, in order that four conserved quantities (includ- ing the total energy) can be obtained: as a matter of fact, the equations of the model are not and cyclic variable does not mean conserved quantity.Besides that, even if  is written in terms of U and V , ψ is not a cyclical coordinate, being implicitily in such variables.For this reason we question the validity of d 0 14) in [7]), which would imply an integral invariant.
With regards to the same subject, we emphasize that equation 16) in [7]), giving rise to the conservation of sin U hV θ + , is not correct in our advice, since U does not refer to a lagrangian coordinate.

The Mathematical Problem
We perform now a brief analysis of (7).It is evident that the first four equations in (7) form the sub-system The last three equations in (7) give simply 0 ξ , 0 η , ψ once (12) has been solved.
Statement 1.1 System (7) admits locally one solution, for any set of data ( ) Proof.The assigned data provide ( ) , by means of (6).Furthermore where we defined (see also (8) and ( 9) Therefore ( 12) can be solved in an appropriate time interval.Finally one obtains 0 ξ , 0 η e ψ from the last three equations in (7) and the rest of the given data.□ Remark 1.1 If in (7) we let 0 h = , we are dealing with the simpler case of a bar on a horizontal plane with one point moving at each time along the direction of the bar: since gives the orbits on the phase plane ( ) , U V , namely each point of the U -axis and the semi-ellipses ( ) Again for 0 h = , the special case 0 B P ≡ , that concerns with one typical instance in nonholonomic con- straints (see for instance [1]), cannot be recovered from the system of Remark 1.2, but it requires the definition of the pseudovelocity 0 cos U ξ ψ =  .We are going now to investigate the stability of the system at 0 θ = .For what concerns with the initial data, we can certainly assume with no loss in generality ( ) Let us first check whenever ( ) 0 t θ ≡ is a solution of (7).
Statement 1.2 ( ) 0 t θ = is solution of (7) if and only if U is constant and (7), first three equations: ( ) We incidentally notice that if On the other hand, U constant and 0 V ≡ make us write (12), second and third equation, as (it is physically correct to assume ( ) The following statement also follows from the previous analysis: if the angle ψ is constant (that is B A − never changes direction), then U has to be constant and ( ) t θ has to be zero.
Corollary 1.1 Assume ( ) If 0 θ = is solution, then V must be zero at any time; on the other hand, the set of data Our aim points now to discuss the stability of ( ) 0 t θ = .Thre physical problem requires 0 > 0 U (see (5)).Incidentally we notice that if is the solution of (7) starting from ( ) is the solution of the same equations corresponding to ( ) It may be helpful by the way to set 0 t = in (7) in order to figure out the behaviour of the solution for short times: that is, assuming for istance 0 > 0 U , 0 > 0 V , U and W are initially increasing, V decreasing.Let us now show the following Proposition 1.1 The equilibrium point ( ) of (12) is unstable.Proof.We set (12) in normal form: calling where det is computed in (13), it is easily found  can be calculated (see (12) for  ).We now compute the Jacobian matrix of ( ) where ( ) Since 0 > 0 U , the polynomial ( ) ψ σ in brackets is such that ( ) = +∞ , so that there exists one real and positive eigenvalue σ and standard results in this sense (see e.g.[8]) can be implemented.□ The linear approximation ( ) which give the equation for θ : whose solution contains 1 e t σ , 1 > 0 σ . As to ( ) V t , the linear approximation gives which diverges the same.An analytical investigation can be performed directly for system (7): choosing for istance, as it is natural, ( ) shows that ϑ initially increases, so that P 0 enters the quarter ( ) ) , F , 2 F ; on the other hand, using also which is (11) with the appropriate G, one should acquire information about the maintenance of > 0 U ( A B − does not change verse), of < 0 Z (the transverse velocity of O is opposite to the position of 0 P with respect to the vertical direction) and of > 0 W (absence of inversion with respect to ϑ ).Such as analysis will be not ex- panded, in order not to overload the Section.
Remark 1. 3 We have not imposed any constraint on the velocity of A yet: the velocity A  can be calculated a posteriori by means of (5) and the angle β between i and A  is such that ( ) where a a b = .Remark 1. 4 It is sometimes assumed in literature (more or less expressly) to know U e V: in that case system (7) is obviously simpler, but such an assumption corresponds to impose the constraints 0 , with given U and V. Hence ( ) cos sin , sin cos , 0, 0 must appear on the right-hand side of (3), with u λ , v λ unknown multipliers.As a whole, we get seven equations in the sev- en unknown quantities q , λ , u λ and v λ .

Adding a Stabilizing Device
Following the approach in [6], we add an external force in order to modify the dynamics of the system and to infer the stability of the stationary solution.
We impose a force , , , , the same for the other coefficients.Notice that a force along 1 j in B would have no effects.Computing the Lagrangian components θ F of the vector of forces in the tangent space and taking the pro- jection T θ Γ F (see (6)), one can check that the term to add to the right-hand side of (7), first three equations, is ( ) The conclusion of Statement 1.1 about existence and uniqueness is not altered, since the matrix A of ( 12) is still the same.
Let us investigate about the effect of stabilization by the external device in the case of the force in A only: ( ) (actually the overlap of e F does not change the substance).Moreover, A F has to vanish at the equilibrium: It can be easily seen that the characteristic polynomial (17) changes into ( ) ( ) ( ) where the partial derivatives are calculated in . The following Proposition sets a selection of choices for f.Proposition 1.2 (o) If 0 > 0 a then the system is unstable.1) for 0 < 0 a : a) if 1 0 a ≥ or 2 0 a ≥ then the system is unstable, b) if 1 < 0 a and 2 < 0 a and ( ) ( ) ], then the system is unstable [resp.stable].
2) For 0 0 a = : c) if 1 > 0 a or 2 > 0 a , then the system is unstable, d) if 1 0 a ≤ and 2 0 a ≤ , then the (real or complex) eigenvalues different from zero have negative real part.Proof.
, then ( ) = +∞ , a real posi- tive eigenvalue certainly exists.1) a) If 0 < 0 a then at least one real negative eigenvalue 1 σ exists and p can be written as .Likewise, if 1 0 a ≥ , then it must be 1 < 0 b and we conclude in the same way.
b) It has to be checked the sign of 1 b : from (22) we see that 1 b must solve ( ) ( ) The latter equation has a unique positive [resp.negative] solution 1 b if and only if ( ) ( ) ].For 1 < 0 b we conclude as before; on the other hand, if 1 > 0 b the real part of the (real or complex) solutions of 2 The linear approximation ( ) (7) with the "new" F encompassing the external force a f is which generalizes (18).The partial derivatives are calculated at the equilibrium ( ) 0 ,0,0,0 U .The equation for θ replacing ( 19) is ( ) (see (21) for the definition of 2 a , 1 a , 0 a ).Hence the stable case 1), b) in Proposition 1.2 is asymptotic stabil- ity for θ .Case 2), d) concerns with ( ) (real or complex) and (assume where 0 W  comes from (23), second equation.Obviously each specific case (coincident eigenvalues, 1 a or 2 a equal to zero, ...) can be examined deeper.
Remark 1.5 A simple guess for f is Inversely, the achieved conditions can be also read in terms of finding 0 U , for a given external force a f as in (20), in order to get stability.In particular, the case 1 1 α = studied in [6] concerns with a counterbalance effect, so that the term UV in (7), second equation, vanishes.
Remark 1. 6 The simplyfing assumption 0 U   sometimes used in models makes sense only around the equilibrium position: far from 0 X the linear approximation U constant would force the system to non reasonable predictions.Besides that, the same assumption is not a consequence of the equations, as we pointed out in Remark 1.1.

The Equations of Motion
We now consider a rigid device simulating the front wheel, adding to the body  a rigid part  (say the front wheel together with handlebars) hung in A and forming the angle β (front steering) between the direc- tion i and a direction fixed in the body frame  : For the sake of simplicity, we may imagine  as a rigid bar laying on 0 z = , with no active force operating on it.We now consider the five lagrangian coordinates ( ) . The angular velocity of  is hence The Lagrangian function of the whole system is 1 T = +    , where  is the same as (1) and where β is a new lagrangian coordinate and m  and I  are respectively the mass of  and the central inertial momentum of  with respect to the direction 3 e .The equation of motion with respect to β is simply Equation ( 24) gives is the angle between 1 e and  .Remark 2.1 If no further constraints are enjoined, the system is unstable the same: actually, changements in (7) are not essential: still keeping ( ) is again an equilibrium point for the system and (13), (17) are replaced respectively by ( ) which has in the same way one real positive solution.
We now add the kinetic constraint of no skidding of  at A : A possible way to face the problem is to neglect the mass m  of the anterior part, so that the Lagrangian function is the same as (1).However, a complication is, in our point of view, the role of β , which does not ap- pear in  , but only in the constraint (25).This is a nontrivial point for the theory-building of the correct equations of motion: the way we are going to follow is not to neglect the mass m  and consider 1  as the Lagrangian function.Even more, if we think of the problem as a "bicycle'" model, the front mass is not at all unsignificant for the overall frame.
We now consider the set of Lagrangian coordinates where 0 ξ , 0 η , ψ , θ , β , 1 λ and 2 λ are the seven unknown quantities.Concening with the initial condi- tions for (26) we can choose, with no loss in generality: are imposed.In order to reduce (26) and eliminate the multipliers, we define, similarly to (4), the pseudovelocities ( ) Joining ( 29) with ( 2) and (25), the lagrangian velocities q  are written in terms of the parameters ) , , ( Y W U : The equations of motion replacing ( 7) are now T , , , , together with (2), ( 25) and ( 29): straightforward computations lead to where and, as before, Even in this case the equations of motion can be led back to the cardinal equations: indeed, calling  the rigid part containing A, B and 0 P (30), the second cardinal equation of the whole system    using B for calculating the momenta and projecting along 3 e writes where is the momentum of the external forces of the whole system.Since the constraints are smooth, the force in A realizing the kinetic constraint (25) can be modelled as so that (see also (10)) Finally, the fourth equation in (30), namely (24), is simply the second cardinal equation written only for  and with respect to the point A , where all the momenta of the external forces vanish.
As we already remarked in Section 1, the overview of the system in the frame of the cardinal equations does not determine any conserved quantity: the only evident one is the energy conservation ≡ , 0 W ≡ can be a solution.Substituting in the first integrals we see that 0 0 θ = , 0 0 y = , that is the stationary solution.
We are going to investigate the stability of the stationary solution.Proposition 2.1 The equilibrium point ( ) 0 0 , 0, 0, 0 U = X , 0 Y = of (36) is unstable.Proof.The Jacobian matrix at the equilibrium is In any case, if (30) is accepted, the assumption U constant allows the immediate calculation of β , irrespec- tive of θ and ψ .

3 e0, , , i j k such that ψ is the angle  1 e i , θ is the angle  3 ke
being the upward vertical, and a body reference frame { } .The mutual disposal of the two frames is given by

M
the resultant momentum of the external forces.Actually, since no friction is present, the constraint A Φ in A is along 3 e , while the constraints in B can be modelled as a force B Φ along 3 e plus a horizontal force ( ) v B Φ perpendicular to B A = (see Figure 2), acting the constraint (2):

Figure 2 .
Figure 2. The system of forces.
either two real positive solu- tions or two complex solutions with positive real part 1 2 b −

=
. The rest of the eigenvalues are the roots of ( ) positive root exists, while if 2 > 0 a then either a real positive eigenvalue exists or the real part of the complex roots is positive.d) In that case the eigenvalues are 0 (twice) and the two roots of which are nonposi- tive if real or with nonpositive real parts if complex.□

= q : system ( 3 )
of the first kind Lagrangian equations is now replaced by the seven equations velocities q  are not independent, because of (2), (25): if on the one hand recall, a a b = .


Carring out all the computations, here omitted, one gets exactly (33).The second equation in (30) is again ascribable to the momentum balance of the system along i , similarly to what discussed in Remark 1.1:

(
It is evident that the unique solution of (30) corresponding to the initial data