1. Introduction
1.1. 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
; A and B, performing the two contact points of the wheels, belong to a horizontal plane and 
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 
on 
and take a fixed reference frame
, 
being the upward vertical, and a body reference frame 
such that 
is the angle
, 
is the angle
. The mutual disposal of the two frames is given by
and sketched in Figure 1.
Set
: the geometrical restrictions
, 
, 
lead us to consider the four lagrangian coordinates
. Therefore
. The angular velocity of the rigid body is
.
The Lagrangian function of the system writes
(1)
where 
and
, 
and 
are the moments of inertia with respect to the body reference frame
, which is supposed to be principal.
The only kinetic constraint we are going to consider is
, whose expression in the Lagrangian coordinates is
(2)
The first kind Lagrangian equations of motion
(3)
where 
are the lagrangian coordinates and 
is the unknown multiplier, will be suitably handled if one defines the pseudovelocities (see [1] for the concepts and the method we are pursuing)
(4)
We point out the following relationships involving U, V and the real velocities:
(5)
where 
and 
Together with the kinetic constraint, Equations (4) give the set of possible velocities in terms of
:
(6)
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 
where 
is defined in (6). Joining to the kinetic constraint (2) and the definition (4) we get, dividing by suitable constants,
(7)
where
(8)
are dimensionless constants. Since the rigid body is practically plane and contained in the plane orthogonal to j, it is reasonable to assume
, 
, so that
(9)
From here on we adopt (9).
The seven ODEs (7) contain the seven unknown quantities
. 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 
the resultant momentum of the external forces. Actually, since no friction is present, the constraint 
in A is along
, while the constraints in B can be modelled as a force 
along 
plus a horizontal force 
perpendicular to 
(see Figure 2), acting the constraint (2):
(10)
Hence no force exists along 
(first equation) and 
(second equation), along 
(and not
, as stated in [6] ). Finally, third equation is simply due to the fact that the only external force with a non zero momentum along 
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
:
(11)
Actually, even if the system is nonholonomic, the first integral 
can be achieved starting from
and performing the usual calculations as in the holonomic case, achieving at last
.
In the same matter of integral invariants for the system, we find it not correct to claim, as in [7] , that the absence of
, 
, 
from 
entails three constant of motion, in order that four conserved quantities (including 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 
and
, 
is not a cyclical coordinate, being implicitily in such variables. For this reason we question the validity of 
(Equation (14) in [7] ), which would imply an integral invariant.
With regards to the same subject, we emphasize that equation 
(Equation (16) in [7] ), giving rise to the conservation of
, is not correct in our advice, since U does not refer to a lagrangian coordinate.
1.2. 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
(12)
The last three equations in (7) give simply
, 
, 
once (12) has been solved.
Statement 1.1 System (7) admits locally one solution, for any set of data
, 
, 
, 
, 
, 
,
.
Proof. The assigned data provide 
and
, by means of (6). Furthermore
(13)
where we defined (see also (8) and (9))
(14)
Therefore (12) can be solved in an appropriate time interval. Finally one obtains
, 
e 
from the last three equations in (7) and the rest of the given data. □
Remark 1.1 If in (7) we let
, 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 
equations reduce to
The energy conservation 
gives the orbits on the phase plane
, namely each point of the 
-axis and the semi-ellipses
. Moreover
, 
, that is
,
.
Again for
, the special case
, that concerns with one typical instance in nonholonomic constraints (see for instance [1] ), cannot be recovered from the system of Remark 1.2, but it requires the definition of the pseudovelocity
.
We are going now to investigate the stability of the system at
.
For what concerns with the initial data, we can certainly assume with no loss in generality
, 
, 
, hence
.
Let us first check whenever 
is a solution of (7).
Statement 1.2 
is solution of (7) if and only if U is constant and
.
Proof. Set 
in (7), first three equations:
which entails
,
. We incidentally notice that if 
then U, V consistent with 
are those we discussed in Remark 1.3.
On the other hand, U constant and 
make us write (12), second and third equation, as
(15)
(it is physically correct to assume
), which implies
. □
Remark 1.2 The following statement also follows from the previous analysis: if the angle 
is constant (that is 
never changes direction), then U has to be constant and 
has to be zero.
Corollary 1.1 Assume
,
; then 
is solution of (7) if and only if
.
Proof. If 
is solution, then 
must be zero at any time; on the other hand, the set of data
, 
, 
, 
give univocally the solution
, 
,
. □
Our aim points now to discuss the stability of
. Thre physical problem requires 
(see (5)). Incidentally we notice that if 
is the solution of (7) starting from
, then 
is the solution of the same equations corresponding to
.
It may be helpful by the way to set 
in (7) in order to figure out the behaviour of the solution for short times:
(16)
that is, assuming for istance
, 
, 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 
is computed in (13), it is easily found
so that 
can be calculated (see (12) for
).
We now compute the Jacobian matrix of 
at the equilibrium
: calculations lead to
where
. The eigenvalues 
are found by solving
(17)
Since
, the polynomial 
in brackets is such that 
and
, so that there exists one real and positive eigenvalue 
and standard results in this sense (see e.g. [8] ) can be implemented. □
The linear approximation 
entails 
and
(18)
which give the equation for
:
(19)
whose solution contains
,
. As to
, 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, 
, 
and
, (16) shows that 
initially increases, so that P0 enters the quarter
. Furthermore, setting
, 
one writes (7) as
with appropriate
,
; on the other hand, using also
which is (11) with the appropriate G, one should acquire information about the maintenance of 
(
does not change verse), of 
(the transverse velocity of O is opposite to the position of 
with respect to the vertical direction) and of 
(absence of inversion with respect to
). Such as analysis will be not expanded, in order not to overload the Section.
Remark 1.3 We have not imposed any constraint on the velocity of A yet: the velocity 
can be calculated a posteriori by means of (5) and the angle 
between 
and 
is such that
where
.
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
, 
, with given U and V. Hence 
must appear on the right-hand side of (3), with
, 
unknown multipliers. As a whole, we get seven equations in the seven unknown quantities
, 
, 
and
.
1.3. 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 
in
, where 
(horizontal versor perpendicular to
) and 
in
; we expect
, the same for the other coefficients. Notice that a force along 
in 
would have no effects.
Computing the Lagrangian components 
of the vector of forces in the tangent space and taking the projection 
(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 
does not change the substance). Moreover, 
has to vanish at the equilibrium:
(20)
It can be easily seen that the characteristic polynomial (17) changes into
(21)
where the partial derivatives are calculated in
. The following Proposition sets a selection of choices for f.
Proposition 1.2 (o) If 
then the system is unstable.
1) for
:
a) if 
or 
then the system is unstableb) if 
and 
and 
[resp.
], then the system is unstable [resp. stable].
2) For
:
c) if 
or
, then the system is unstabled) if 
and
, then the (real or complex) eigenvalues different from zero have negative real part.
Proof.
(o) Call
. If
, then
: since
, a real positive eigenvalue certainly exists.
1) a) If 
then at least one real negative eigenvalue 
exists and p can be written as
(22)
If
, then
: since
, the equation 
has either two real positive solutions or two complex solutions with positive real part
. Likewise, if
, then it must be 
and we conclude in the same way.
b) It has to be checked the sign of
: from (22) we see that 
must solve
The latter equation has a unique positive [resp. negative] solution 
if and only if 
[resp.
]. For 
we conclude as before; on the other hand, if 
the real part of the (real or complex) solutions of 
is negative.
2) c) If 
then
. The rest of the eigenvalues are the roots of
. If 
then a real positive root exists, while if 
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 nonpositive if real or with nonpositive real parts if complex. □
The linear approximation 
of (7) with the “new” 
encompassing the external force 
is 
and
(23)
which generalizes (18). The partial derivatives are calculated at the equilibrium
. The equation for 
replacing (19) is
(see (21) for the definition of
, 
,
). Hence the stable case 1), b) in Proposition 1.2 is asymptotic stability for
. Case 2), d) concerns with 
so that 
(real or complex) and (assume
)
where 
comes from (23), second equation. Obviously each specific case (coincident eigenvalues, 
or 
equal to zero, ...) can be examined deeper.
Remark 1.5 A simple guess for f is
, with 
and 
constant: in that case 
or 
gives instability. The stability region located by case 1), b) in the quarter 
is not empty:
actually, 
and 
corresponds to the set
, possibly cut on the upper part by the line
, if the latter value is lower than
. Such a set has a nonempty intersection with
: indeed, it is sufficient to take, for each fixed
, 
large enough in module.
Furthermore, the case 2), d) is simply
.
Inversely, the achieved conditions can be also read in terms of finding
, for a given external force 
as in (20), in order to get stability. In particular, the case 
studied in [6] concerns with a counterbalance effect, so that the term UV in (7), second equation, vanishes.
Remark 1.6 The simplyfing assumption 
sometimes used in models makes sense only around the equilibrium position: far from 
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.
2. A Two-Body Model
2.1. 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 
(front steering) between the direction 
and a direction fixed in the body frame
:
For the sake of simplicity, we may imagine 
as a rigid bar laying on
, 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
, where 
is the same as (1) and
where 
is a new lagrangian coordinate and 
and 
are respectively the mass of 
and the central inertial momentum of 
with respect to the direction
. The equation of motion with respect to 
is simply
(24)
Equation (24) gives
where 
is the angle between 
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
, where
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
: 
which gives
(25)
A possible way to face the problem is to neglect the mass 
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
, which does not appear 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 
and consider 
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
: system (3) of the first kind Lagrangian equations is now replaced by the seven equations
(26)
where
, 
, 
, 
, 
, 
and 
are the seven unknown quantities. Concening with the initial conditions for (26) we can choose, with no loss in generality:
Let us change for the sake of convenience 
into the variable
(27)
so that
. The velocities 
are not independent, because of (2), (25): if on the one hand 
and 
are arbitrary, on the other hand once 
has been fixed the initial velocities
(28)
are imposed. In order to reduce (26) and eliminate the multipliers, we define, similarly to (4), the pseudovelocities
(29)
Joining (29) with (2) and (25), the lagrangian velocities 
are written in terms of the parameters
:
where 
and, we recall,
.
The equations of motion replacing (7) are now
, 
together with (2), (25) and (29): straightforward computations lead to
(30)
(31)
where
(32)
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 
(30), the second cardinal equation of the whole system 
using B for calculating the momenta and projecting along 
writes
(33)
where 
is the momentum of the external forces of the whole system. Since the constraints are smooth, the force in 
realizing the kinetic constraint (25) can be modelled as
so that (see also (10))
. On the other hand, the first cardinal equation for the whole system along 
is 
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
, similarly to what discussed in Remark 1.1:
Finally, the fourth equation in (30), namely (24), is simply the second cardinal equation written only for 
and with respect to the point
, 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
2.2. The Mathematical Problem
System (30), (31) consists of eight ODEs for the eight variables
, 
, 
, 
, 
, 
, 
,
. The five equations (30) form a sub-system for 
together with the initial conditions
(34)
while the constant value Y is deduced from (24), (27) and (28)):
(35)
Once (30) has been solved, (31) and (28) allow to solve
, 
,
.
As in the case of Section 1, we incidentally remark that, if 
and Y as in (35) is solution of (30), then
, 
is solution of the same system, as we expect.
Statement 2.1 For any set of data (34), (35) system (30) admits one solution.
Proof. System (30) can be concisely written as
(36)
with 
and
the normal form being 
with
and
It is evident that the unique solution of (30) corresponding to the initial data
, 
, 
, 
is 
and W, 
identically zero. We notice that no other solutions such that the plane 
is vertical are possible:
Statement 2.2 If
, then
, 
,
. Conversely, if
, 
and
, 
, 
, then
.
Proof. Set 
in (30) and call
:
The first two equations give
; on the other hand, eliminating 
from second and third equations yields
, hence 
and
. But 
is consistent with the second equation only if 
and the third equation leads to
.
Conversely, if
, 
, 
are replaced in (30), one gets 
that, together with the initial conditions, gives
. □
Still concerning with the initial data assignment, we notice that if 
(which is a reasonable condition for
) we get from (35)
: we wonder whether solutions such that 
(meaning 
and 
constant) are possible. From (30), fourth equation, one gets
,
. If 
we clearly have
, 
and 
for
. If on the other hand
, by replacing 
and 
in the first two equations of (30) we achieve the first integrals
so that only
, 
can be a solution. Substituting in the first integrals we see that
, 
, that is the stationary solution.
We are going to investigate the stability of the stationary solution.
Proposition 2.1 The equilibrium point
, 
of (36) is unstable.
Proof. The Jacobian matrix at the equilibrium is
whose eigenvalues are
, 
,
. The linearized system
gives
with
, 
,
. □
We remark that, as bigger is 
as longer is the time when the planar body 
falls to the ground
.
3. Discussing Some Specific Assumptions
It is evident that the stability of the system can be achieved by introducing an external force as in Paragraph 1.3: instead of replay such a theme, we prefer to discuss some assumptions recurring in literature which indeed semplify the mathematical problem.
First of all, let us see what happens if we let
, known constant. If 
then 
and the solution is 
(that is
) and
. The point 
draws the circle
,
.
On the other hand, if
, the angle 
can be calculated by (30), fourth and fifth equations, regardless of the rest of the system:
(37)
By integrating one gets in terms of
:
In any case, if (30) is accepted, the assumption U constant allows the immediate calculation of
, irrespective of 
and
.
The same system (30) is worth considering together with the assumption, not uncommon in literature [2] [9] -[11] ,
. Actually, assuming
, 
constant, would lead to the invariant quantity 
and, by integration,
. Hence also 
is proportional to 
and (30), second equation, can be written in terms of 
only:
On the other hand, also (30), first equation, turns out to be written in terms of
, since 
and
: it should be checked that two obtained equations show compatibility.
Hence, in our mind the angle 
also must be governed by the equations of motion which do not make room for an assumption such as
.
Nevertheless, assume that the mass of 
is negligible, so that 
and
, as it is found in some models: in that case fourth and fifth equations in (30) have to be disregarded and 
cannot longer be computed by means of (37). Following this point of view, the system is not closed, since 
enters only the constraint condition (25) and not in the Lagrangian function
. This is the reason why an additional condition (say a constitutive law) is needed, as for instance
.