DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 347–353

GEOMETRICAL ASPECTS IN THE DYNAMICS OF AN AUTOMOBIL WITH TWO TRAILERS

MIRCEA CRAIOVEANU, CAMELIA POP AND MIRCEA PUTA

Abstract. We study some properties related to the dynamics, control, stability and integrability of an automobil with two trailers.

1. Introduction

Let us consider a simple kinematic model of an automobil with two trailers. Its dynamics is described by the following chained form system:

(1.1)  \{ \begin{matrix}ẋ_{1} & = & u_{1} , \\ ẋ_{2} & = & u_{2} , \\ ẋ_{3} & = & x_{2}u_{1} , \\ ẋ_{4} & = & x_{3}u_{1} , \\ ẋ_{5} & = & x_{4}u_{1} . \end{matrix}

The goal of our paper is to prove that we can express this chained form system as a left invariant control system on a matrix Lie group and then using this realisation to study some of its geometrical and dynamical properties.

2. Matrix realisation of the system (1.1)

Let G_{5} be the Lie group of all 5 × 5 matrices of the form:

X = [ \begin{matrix}1 & x_{2} & x_{3} & x_{4} & x_{5} \\ 0 & 1 & x_{1} & x_{1}^{2}/2 & x_{1}^{3}/6 \\ 0 & 0 & 1 & x_{1} & x_{1}^{2}/2 \\ 0 & 0 & 0 & 1 & x_{1} \\ 0 & 0 & 0 & 0 & 1 \end{matrix} ] ,

1991 Mathematics Subject Classification. 58F05.

Key words and phrases. Automobil with two trailers, left invariant control system, Hamilton-Poisson system, linear control, Lie-Trotter integrator.

This paper is in final form and no version of it will be submitted for publication elsewhere.

where x_{i} ∈ ℝ, for each i = 1, 2, 3, 4, 5. A basis for the nilpotent Lie algebra 𝔤_{5} of this Lie group is given by {A_{1}, A_{2}, A_{3}, A_{4}, A_{5}} = {A_{1}, A_{2}, [A_{2}, A_{1}], [[A_{2}, A_{1}], A_{1}], [[[A_{2}, A_{1}], A_{1}], A_{1}]}, where

A_{1} = [ \begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ] , A_{2} = [ \begin{matrix}0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ] .

A long, but straightforward computation shows us that:

Theorem 2.1 The chained form system (1.1) can be put in the equivalent form:

(2.1)  Ẋ = X(u_{1}A_{1} + u_{2}A_{2})

or in other words, the system (1.1) can be viewed as a left invariant control system on G_{5}.

Theorem 2.2 The system (1.1) is controllable and it is a triple Lie bracket one.

3. An optimal control problem

Let J be the cost function given as usually by

(3.1)  J(u_{1}, u_{2}) = \frac{1}{2} ∫_{0}^{t_{f}} (c_{1}u_{1}^{2} + c_{2}u_{2}^{2})dt , c_{1} > 0, c_{2} > 0 .

Then we can prove:

Theorem 3.1 The controls which minimize J and steer the system (2.1) from X = X_{0} at t = 0 to X = X_{f} at t = t_{f} are given by:

(3.2)  u_{1} = \frac{1}{c_{1}} P_{1} , u_{2} = \frac{1}{c_{2}} P_{2} ,

where P_{i}’s are the solutions of:

(3.3)  \{ \begin{matrix}Ṗ_{1} & = & \frac{1}{c_{2}} P_{2}P_{3} , \\ Ṗ_{2} & = & −\frac{1}{c_{1}} P_{1}P_{3} , \\ Ṗ_{3} & = & −\frac{1}{c_{1}} P_{1}P_{4} , \\ Ṗ_{4} & = & −\frac{1}{c_{1}} P_{1}P_{5} , \\ Ṗ_{5} & = & 0 . \end{matrix}

Proof. Simply apply Krishnaprasad’s theorem [?] to this special case. Then the extremal controls are given by (3.2), where P_{i}’s are solutions of the reduced Hamilton’s equations on 𝔤_{5}^{∗} ≅ ℝ^{5}, i.e.

[ Ṗ_{1} ; Ṗ_{2} ; Ṗ_{3} ; Ṗ_{4} ; Ṗ_{5} ] = [ \begin{matrix}0 & P_{3} & P_{4} & P_{5} & 0 \\ −P_{3} & 0 & 0 & 0 & 0 \\ −P_{4} & 0 & 0 & 0 & 0 \\ −P_{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ] · ∇H ,

GEOMETRICAL ASPECTS IN THE DYNAMICS 349

where

(3.4)  H(P_{1}, P_{2}, P_{3}, P_{4}, P_{5}) = \frac{1}{2c_{1}} P_{1}^{2} + \frac{1}{2c_{2}} P_{2}^{2} ,

or equivalently

[ Ṗ_{1} ; Ṗ_{2} ; Ṗ_{3} ; Ṗ_{4} ; Ṗ_{5} ] = [ \begin{matrix}\frac{1}{c_{2}} P_{2}P_{3} , \\ −\frac{1}{c_{1}} P_{1}P_{3} , \\ −\frac{1}{c_{1}} P_{1}P_{4} , \\ −\frac{1}{c_{1}} P_{1}P_{5} , \\ 0 \end{matrix} ] · ∇H ,

as required. □ Remark 3.1 It is not hard to see that the function C given by

C(P_{1}, P_{2}, P_{3}, P_{4}, P_{5}) = −P_{3}P_{5} + \frac{1}{2}P_{4}^{2}

is a Casimir of our configuration (𝔤_{5}^{∗})_{−}, i.e.

(∇C)^{t} · [ \begin{matrix}0 & P_{3} & P_{4} & P_{5} & 0 \\ −P_{3} & 0 & 0 & 0 & 0 \\ −P_{4} & 0 & 0 & 0 & 0 \\ −P_{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ] · ∇f = 0 ,

for each f ∈ C^{∞}(ℝ^{5}, ℝ).

4. The Hamilton-Poisson structure of the system (3.3) and some of its properties

It is not hard to see that the system (3.3) can be put in the equivalent form:

(4.1)  \{ \begin{matrix}Ṗ_{1} & = & \frac{1}{c_{2}} P_{2}P_{3} , \\ Ṗ_{2} & = & −\frac{1}{c_{1}} P_{1}P_{3} , \\ Ṗ_{3} & = & −\frac{1}{c_{1}} P_{1}P_{4} , \\ Ṗ_{4} & = & −\frac{k}{c_{1}} P_{1} , \end{matrix}

where

P_{5} = k .

In all that follows we shall assume that

(4.2)  c_{1} = c_{2} = k = 1 .

Then we can prove: Theorem 4.1 The equations (4.1) – (4.2) may be explicitely integrated by elliptic functions. Proof. It is known that

P_{1}^{2} + P_{2}^{2} = l

and

\frac{1}{2}P_{4}^{2} − P_{3} = m

are constants of motion. Then an easy computation shows us that

P_{2} = \frac{1}{6}P_{4}^{3} − cP_{4} + d ,

where c, d ∈ ℝ. Therefore

P_{1} = ±\sqrt{l − P_{2}^{2}}
= ±\sqrt{l − (\frac{1}{6}P_{4}^{3} − cP_{4} + d)} .

So,

Ṗ_{4} = ∓\sqrt{l − d − \frac{1}{6}P_{4}^{3} + cP_{4}} ,

or equivalently

t = ∓ ∫_{P_{4}(0)}^{P_{4}} \frac{dP_{4}}{\sqrt{l − d − \frac{1}{6}P_{4}^{3} + cP_{4}}}

which shows that P_{4}, and hence P_{3}, P_{1} and P_{2} are elliptic functions of time as required. □

Theorem 4.2 The system (4.1) – (4.2) has a Hamilton-Poisson realisation with the phase space ℝ^{4}, with the Poisson bracket given by the matrix

(4.3)  Π = [ \begin{matrix}0 & P_{3} & P_{4} & 1 \\ −P_{3} & 0 & 0 & 0 \\ −P_{4} & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ]

and the Hamiltonian H given by:

H(P_{1}, P_{2}, P_{3}, P_{4}) = \frac{1}{2}(P_{1}^{2} + P_{2}^{2}) .

Proof. Indeed, one can easily check that

[ Ṗ_{1} ; Ṗ_{2} ; Ṗ_{3} ; Ṗ_{4} ] = [ \begin{matrix}0 & P_{3} & P_{4} & 1 \\ −P_{3} & 0 & 0 & 0 \\ −P_{4} & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ] [ \begin{matrix}P_{1} \\ P_{2} \\ 0 \\ 0 \end{matrix} ] = [ \begin{matrix}P_{2}P_{3} \\ −P_{1}P_{3} \\ −P_{1}P_{4} \\ −P_{1} \end{matrix} ] ,

as required. □ Remark 4.1 It is not hard to see that the function C given by:

C(P_{1}, P_{2}, P_{3}, P_{4}) = −P_{3} + \frac{1}{2}P_{4}^{2}

is a Casimir of our configuration (ℝ^{4}, Π).

Let us observe now that the equilibrium states of our system (4.1) – (4.2) are

(4.4)  (0, 0, α, β), (0, α, 0, β)

where α, β ∈ ℝ.

Theorem 4.3 The equilibrium states (4.4) are all spectrally stable. Proof. The proof can be obtained immediately using the matrix of the linearized system at the equilibrium of interest and so we shall omit any other details. □

Remark 4.2 It is an open problem to decide if the equilibrium states (4.4) are nonlinear stable or they are unstable.

Let us finish this section with some remarks concerning the nonlinear stability of the equilibrium state (0, 0, α, 0), α ∈ ℝ, of our system (4.1) – (4.2) via a linear control. For beginning let us consider the controlled system:

(4.5)  \{ \begin{matrix}Ṗ_{1} & = & P_{2}P_{3} + P_{4} , \\ Ṗ_{2} & = & −P_{1}P_{3} , \\ Ṗ_{3} & = & −P_{1}P_{4} , \\ Ṗ_{4} & = & −P_{1} . \end{matrix}

Theorem 4.4 The controlled system (4.5) is an Hamilton-Poisson system with the phase space ℝ^{4}, the Poisson bracket given by the matrix (4.3) and the Hamiltonian given by

H(P_{1}, P_{2}, P_{3}, P_{4}) = \frac{1}{2}(P_{1}^{2} + P_{2}^{2} + P_{4}^{2}) .

Proof. Indeed, one easily check that

[ Ṗ_{1} ; Ṗ_{2} ; Ṗ_{3} ; Ṗ_{4} ] = [ \begin{matrix}0 & P_{3} & P_{4} & 1 \\ −P_{3} & 0 & 0 & 0 \\ −P_{4} & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ] [ \begin{matrix}P_{1} \\ P_{2} \\ 0 \\ P_{4} \end{matrix} ] = [ \begin{matrix}P_{2}P_{3} + P_{4} \\ −P_{1}P_{3} \\ −P_{1}P_{4} \\ −P_{1} \end{matrix} ] ,

as required. □

Now, we shall prove via the energy-Casimir method, [?], [?], that the equilibrium state (0, 0, α, 0) of our system (4.5) is nonlinear stable. More exactly we have:

Theorem 4.5 The equilibrium state (0, 0, α, 0), α ∈ ℝ, of the system (4.5) is nonlinear stable.

Proof. Consider the energy-Casimir function:

H_{φ}(P_{1}, P_{2}, P_{3}, P_{4}) = \frac{1}{2}(P_{1}^{2} + P_{2}^{2} + P_{4}^{2}) + φ(−P_{3} + \frac{1}{2}P_{4}^{2}) ,

where φ : ℝ → ℝ is an arbitrary smooth real valued function defined on ℝ. Let φ′, φ′′ denote its first and second derivative. Now, the first variation of H_{φ} is given by:

δH_{φ} = P_{1}δP_{1} + P_{2}δP_{2} + P_{4}δP_{4} + φ′(·)(−δP_{3} + P_{4}δP_{4}) .

This equals zero at the equilibrium of interest if and only if

(4.6)  φ′(−α) = 0 .

The second variation of H_{φ} at the equilibrium of interest is given via (4.6) by:

δ^{2}H_{φ}(0, 0, α, 0) = (δP_{1})^{2} + (δP_{2})^{2} + (δP_{4})^{2} + φ′′(−α)(δP_{3})^{2} ,

which is positive definite if we choose φ such that:

φ′′(−α) > 0 .

So we can conclude that the equilibrium state (0, 0, α, 0), α ∈ ℝ, is nonlinear stable as required. □

5. Numerical integration of the system (3.3)–(4.2)

via the Lie-Trotter integrator

In this section we shall discuss the numerical integration of the system (3.3) – (4.2) and we shall point out some of its geometrical properties from the Poisson geometry point of view.

To begin with, let us observe that the Hamiltonian vector field X_{H} splits as follows:

X_{H} = X_{H_{1}} + X_{H_{2}} ,

where

H_{1} = \frac{1}{2}P_{1}^{2}, H_{2} = \frac{1}{2}P_{2}^{2} .

The integral curves of X_{H_{1}} and X_{H_{2}} are given by:

P (t) = exp(tX_{H_{1}}) · P (0) = ϕ_{1}(t, P (0))

and respectively

P (t) = exp(tX_{H_{2}}) · P (0) = ϕ_{2}(t, P (0)) .

Now, following [?], [?], [?], [?], [?], we obtain an explicit integrator for the system (3.3) – (4.2) namely:

P^{k+1} = ϕ_{1}(t, ϕ_{2}(t, P^{k})) ,

or explicitely:

(5.1)  \{ \begin{matrix}P_{1}^{k+1} & = & P_{1}^{k} + tP_{3}(0)P_{2}^{k} , \\ P_{2}^{k+1} & = & P_{2}^{k} − tP_{1}^{2}(0)P_{3}^{k} + t^{2} \frac{P_{1}(0)}{2}P_{4}^{k} − t^{3} \frac{P_{1}^{3}(0)}{6}P_{5}^{k} , \\ P_{3}^{k+1} & = & P_{3}^{k} − tP_{1}(0)P_{4}^{k} + t^{2} \frac{P_{1}^{2}(0)}{2}P_{5}^{k} , \\ P_{4}^{k+1} & = & P_{4}^{k} − tP_{1}(0)P_{5}^{k} , \\ P_{5}^{k+1} & = & P_{5}^{k} . \end{matrix}

Some of its properties are sketched in the following theorem:

Theorem 5.1 (i) The numerical integrator (5.1) preserves the Poisson structure given by the matrix

Π = [ \begin{matrix}0 & P_{3} & P_{4} & P_{5} & 0 \\ −P_{3} & 0 & 0 & 0 & 0 \\ −P_{4} & 0 & 0 & 0 & 0 \\ −P_{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ] ,

which is in fact the minus-Lie-Poisson structure of 𝔤_{5}^{∗}.

(ii) The numerical integrator (5.1) preserves the Casimir C of our configuration (ℝ^{5}, Π), given by:

C(P_{1}, P_{2}, P_{3}, P_{4}, P_{5}) = P_{2}P_{4} − \frac{1}{2}P_{3}^{2} .

(iii) Its restriction to each coadjoint orbit C(P_{1}, P_{2}, P_{3}, P_{4}, P_{5}) = constant

gives rise to an symplectic integrator with respect to the Kostant-Kirilov-Souriau symplectic form.

(iv) The numerical integrator (5.1) does not preserve the Hamiltonian H given by

H(P_{1}, P_{2}, P_{3}, P_{4}, P_{5}) = \frac{1}{2}(P_{1}^{2} + P_{2}^{2}) .

Proof. The items (i) – (iii) hold because ϕ_{1} and ϕ_{2} are flows of some Hamiltonian vector fields, hence they are Poisson maps.

Item (iv) is essentially due to the fact that

{H_{1}, H_{2}} ≠ 0 . □

References

[1] Craioveanu, M., Dupac, M., Puta, M., Some remarks on the Lie-Trotter integrator, Analele şt. ale Universităţii ”Al.I. Cuza” Iaşi, T. XLIII, s. I-a, Matematică, f.2 (1997), 259–268.

[2] Holm, D., Marsden, J.,Raţiu, T. and Weinstein, A., Nonlinear stability of fluid and plasma equilibria, Phys. Reports 123 (1985), 1–116.

[3] Krishnaprasad, P. S., Optimal control and Poisson reduction, Technical Report 93-87, Institute for System Research, University of Maryland, 1993.

[4] McLachlan, R., On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comp. 16 (1995), 151–168.

[5] Puta, M., Hamiltonian mechanical systems and geometric quantization, Math. and Appl. 260, Kluwer, 1993.

[6] Puta, M., Lie-Trotter formula and Poisson dynamics (to appear in Int. Journal Biffurcation and Chaos).

[7] Puta, M., An overview of some Poisson integrators (to appear in Proceedings of the ENU-MATH 97).

[8] Trotter, H. F., On the product of semigroups of operators, Proc. Amer. Math. Soc. 10 (1959), 545-551.

Department of mathematics, West University of Timişoara bvd. V.Pârvan 4, Timişoara 1900, ROMANIA E-mail: craiov@geometry.uvt.ro

Department of Mathematics, ”Politehnica” University of Timişoara Piaţa Horaţiu 1, Timişoara 1900, ROMANIA

Department of mathematics, West University of Timişoara bvd. V.Pârvan 4, Timişoara 1900, ROMANIA E-mail: puta@geometry.uvt.ro