Some New Remarks about the Dynamics of an Automobile with Two Trailers

The dynamics of a kinematic model of an automobile with two trailers were first described by Leonard (see [1]) as a chained form system. Later, the system was studied as a left invariant control system on a matrix Lie group in [2]. In the same paper, the Hamilton-Poisson structure of the system was presented togetherwith some geometrical and dynamical properties.The goal of our paper is to complete some of these results. The paper is structured as follows: the first part presents the Hamilton-Poisson structure of the systems from [2]. The Casimir functions corresponding to this structure are found. Beginning with these Casimirs, two new Hamilton-Poisson structures of the system are proposed. The second paragraph analyzes the nonlinear stability of the equilibrium states of the dynamics. Due to the existence of two Casimirs, we do not need to employ a function control to obtain the stability results, like in [2]. The Lax formulation of the system is the subject of the third paragraph. In the last part of the paper we discuss numerical integration of the dynamics via two methods. A numerical simulation for both results is also presented.


Introduction
The dynamics of a kinematic model of an automobile with two trailers were first described by Leonard (see [1]) as a chained form system. Later, the system was studied as a left invariant control system on a matrix Lie group in [2].In the same paper, the Hamilton-Poisson structure of the system was presented together with some geometrical and dynamical properties.The goal of our paper is to complete some of these results.
The paper is structured as follows: the first part presents the Hamilton-Poisson structure of the systems from [2].The Casimir functions corresponding to this structure are found.Beginning with these Casimirs, two new Hamilton-Poisson structures of the system are proposed.The second paragraph analyzes the nonlinear stability of the equilibrium states of the dynamics.Due to the existence of two Casimirs, we do not need to employ a function control to obtain the stability results, like in [2].The Lax formulation of the system is the subject of the third paragraph.In the last part of the paper we discuss numerical integration of the dynamics via two methods.A numerical simulation for both results is also presented.

The Geometrical Overview of the Problem
The dynamics of the car with two trailers have been studied as a mechanical problem on a matrix Lie group.Following [2], the system that describes the dynamics is given by and may be realized as a Hamilton-Poisson system with the phase space  4 ; the Poisson structure is given by the matrix and the Hamiltonian ( 2 ), and  ∈ .
We are concerned now with finding the Casimir functions of this structure.The defining equation for the Casimir functions, denoted by , is The determination of a Casimir in a finite dimensional Hamilton-Poisson system could be done via the algebraic method of Hernández-Bermejo and Fairén (see [3]).Let us observe that the rank of Π is constant and equal to 2. Then, there exist two functionally independent Casimirs associated with our structure.The Casimir function is the solution of the following equations: Journal of Applied Mathematics Proposition 1.The functions  1 and  2 given by are the Casimir functions of the configuration Π.
Remark 2. Remark 4.1 from [2] presents only one Casimir function of the configuration described above.As we can see, the rank of the Poisson configuration matrix equals 2, so there exist two Casimir functions functionally independent.The consequence of this result lies in the analyses of the nonlinear stability of the equilibrium states of the dynamics (1).In addition, we can find two new Hamilton-Poisson structures of the system (1).
The functions  and  2 are the Casimir functions for the structure Π 1 ; for the structure Π 2 , the Casimir functions are  and  1 .
Proof.Indeed, we can check that the system (1) can be put into the equivalent form as required.
Remark 4. The phase curves of the dynamics (1) are the intersection of the surfaces

Stability Problems
The equilibrium states of the dynamics (1) are Using the matrix of the linearized system at the equilibrium of interest, we obtain the following results.
Proposition 5 (see [2]).The equilibrium states  1 are spectrally stable for any real values of  and .Proposition 6.The equilibrium states  2 are spectrally stable if  > 0.
Proof.The proof can be obtained immediately using the matrix of the linearized system at the equilibrium of interest and so we will omit any other details.Now, we are able to study the nonlinear stability of the equilibrium states  1 and  2 .Proposition 7. The equilibrium states  1 are nonlinearly stable for any real values of  and .
Proof.We will make the proof using energy-Casimir method [see [4]].Let be the energy-Casimir function, where ,  :  →  are two smooth real valued functions defined on .Now, the first variation of  , at the equilibrium  1 equals zero if If we choose now the functions  and  such that the second variation of  , at the equilibrium  1 is positively defined, so we can conclude that the equilibrium points  1 are nonlinearly stable.
Proof.To obtain the nonlinear stability result, for this time, we consider the function Using the same energy-Casimir method, we can conclude that the equilibrium state  2 is nonlinearly stable if ,  < 0.
A numerical comparison of the two results is presented in Figures 1 and 2. Now, splitting the Hamiltonian vector field   as where  1 = (1/2) 2 1 and  2 = (1/2) 2 2 ; the integral curves of the vector fields   1 and   2 are given by where Now, the Lie-Trotter integrator can be written as or, explicitly, as Some of its properties are sketched in the following proposition.Proposition 9.The numerical integrator (24) preserves the Poisson structures Π, Π 1 , and Π 2 ; it preserves the Casimirs  1 and  2 , but it does not preserve the Hamiltonian  of the configuration ( 3 , Π).In addition, its restriction to the coadjoint orbits gives rise to a symplectic integrator.
Proof.The numerical integrator (24) preserves the Poisson structures and the Casimirs of the above configuration because  1 and  2 are flows of some Hamiltonian vector fields; hence they are Poisson ones.For the same reasons, the restriction to the coadjoint orbits gives rise to a symplectic integrator.The numerical integrator (24) does not preserve the Hamiltonian because The numerical simulation of the Lie-Trotter integrator, using Mathematica 8.0, is presented in Figure 2.
Let us observe now that Kahan's integrator, associated with the dynamics (1), has the following form: After a log but straightforward computation, we are leading to the following properties of Kahan's integrator.Remark 11.Using Mathematica 8 we obtained numerical simulations for a Poisson integrator (the Lie-Trotter one) and a non-Poisson integrator (Kahan one).As we can see, the Lie-Trotter integrator gives us a better approximation of the trajectory movement, but both of them are quite different from Figure 1, which is the exact solution of the system (1).

Conclusion
A lot of mechanical problems, like the cinematic model of an automobile with (n-3) trailers [7], the underwater vehicle dynamics [1], the spacecraft dynamics [8], the molecular motion in the context of coherent control of quantum dynamics [9], and the ball-plate problem [10] or the control tower problem from air traffic [11] or the Lagrange system [12] have been modeled as a set of differential equations with the configuration space on a matrix Lie group.Finding a Hamilton-Poisson formulation for these systems is an important first step.The Hamilton-Poisson formulation allows us to study the systems form mechanical geometry points of view by giving the specific tools to study the nonlinear (Lyapunov) stability of the equilibrium states (by using energy-Casimir method), the existence of the periodic orbits (by using the Lyapunov center theorem), the bifurcation phenomena, the numerical integration (by using Poisson integrators, like Lie-Trotter one), integrability, and so forth.For such a formulation, finding the Casimir functions is an important step: they could provide other Poisson structures and they play a major role in the study of the nonlinear stability with all the known energy methods.
In addition, the Hamilton-Poisson realization offers us the possibility to find the exact solution of the system as the intersection of some surfaces, the surfaces equation being given by the Hamiltonian and the Casimirs of our configuration.
Studying the system from the Poisson geometry point of view may offer us the opportunity to find the connection between the dynamical properties of the system and the geometry of the image of a vector valued constant of motion (the energy-Casimir mapping, in our case) and can help us to detect as many as possible dynamical elements (e.g., equilibria, periodic orbits, and homoclinic and heteroclinic connections) and dynamical behavior (e.g., stability, bifurcation phenomena for equilibria, periodic orbits, and homoclinic and heteroclinic connections) by just looking at the image of this mapping (see [13]).

Figure 3
Figure 3 represents the numerical simulation of the Kahan integrator.

Proposition 10 .
Kahan's integrator (27) has the following properties: it does not preserve the Poisson structures Π, Π 1 , Π 2 , it does not preserve the Casimirs  1 ,  2 , and it does not preserve Hamiltonian .