# An Underactuated Drift-Free Left Invariant Control System on a Specific Lie Group

Authors: Camelia Pop Arieşanu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The paper presents a geometrical overview on an optimal control problem on a special Lie group. The Hamilton-Poisson realization of the dynamics offers us the possibility to study the system from mechanical geometry point of view.


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## Body

## 1. Introduction

Recent work in nonlinear control has drawn attention to drift-free systems with fewer degrees than state variables. These arise naturally in problems of motion planning for wheeled robots subject to nonholonomic controls [1], models of kinematic drift effects in space subjects to appendage vibrations or articulations [1], the molecular dynamics [2], the autonomous underwater vehicle dynamics [3], the car's dynamics [4], and spacecraft dynamics [5]. The purpose of our paper is to study a class of left-invariant, drift-free optimal control problems on a specific Lie group G. The class of all control-affine left-invariant, drift-free optimal control problems on G can be reduced to a class of two typical controllable left-invariant control systems on G. The left-invariant, drift-free optimal control problems involve finding a trajectory-control pair on G, which minimizes a cost function and satisfies the given dynamical constrains and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*G using the optimal Hamiltonian on 𝒢*, where the maximum principle yields the optimal control. The energy-Casimir method is used to give sufficient conditions for nonlinear stability of the equilibrium states. Around this equilibrium states, we are able to find the periodical orbits using Moser's theorem. In the last paragraph, we have studied the numerical integration via three methods: Lie-Trotter algorithm, Kahan's algorithms, and Runge-Kutta 4th method. Numerical simulations and a comparison between these three methods are presented too.


## 2. The Geometrical Picture of the Problem

LetG be the Lie group given by(2.1)G={[1x2x40e-x1x3001]∈M3(R)∣x1,x2,x3,x4∈R}.Proposition 2.1.
The Lie algebra𝒢 of G is generated by
(2.2)A1=[0000-10000],A2=[010000000],A3=[000001000],A4=[001000000],
and the Lie algebra structure of 𝒢 is given by the following
(2.3)[⋅,⋅]A1A2A3A4A10A2-A30A2-A20A40A3A3-A400A40000.Proposition 2.2.
The minus-Lie-Poisson structure on𝒢*≃(R4)*≃R4 is generated by the matrix
(2.4)Π-=[0-x2x30x20-x40-x3x4000000].
An easy computation leads one to the following.Proposition 2.3.
The following two systems are drift-free-left invariant controllable systems onG, namely:
(2.5)Ẋ=X(A1u1+A2u2+A3u3),(2.6)Ẋ=X(A1u1+A2u2+A3u3+A4u4),
where X∈G,Ai are the matrix defined above, and ui∈C∞(ℝ,ℝ),i=1,4¯.Proof.
Since the span of the set of Lie brackets generated byA1,A2, and A3 coincides with 𝒢 the proposition is a consequence of a result due to Jurdjevic and Sussmann [6].


## 3. An Optimal Control Problem for the System (2.5)

LetJ be the cost function given by(3.1)J(u1,u2,u3)=12∫0tf[c1u12(t)+c2u22(t)+c3u32(t)]dtc1>0,c2>0,c3>0.Then we have the following.Proposition 3.1.
The controls that minimizeJ and steer the system (2.5) from X=X0 at t=0 to X=Xf at t=tf are given by
(3.2)u1=1c1x1,u2=1c2x2,u3=1c3x3,
where xi’s  are solutions of
(3.3)ẋ1=-1c2x22+1c3x32,ẋ2=1c1x1x2-1c3x3x4,ẋ3=-1c1x1x3+1c2x2x4,ẋ4=0.Proof.
Let us consider the optimal Hamiltonian given by(3.4)H(x1,x2,x2,x4)=12(x12c1+x22c2+x32c3).
It is in fact the controlled Hamiltonian Hopt given by
(3.5)Hopt=x1u1+x2u2+x3u3-12(c1u12+c2u22+c3u32),
which is reduced to 𝒢* via Poisson reduction. Then the optimal controls are given by
(3.6)u1=1c1x1,u2=1c2x2,u1=1c3x3,
where xi’s are solutions of the reduced Hamilton's equations given by
(3.7)[x1x2x3x4]t=Π-⋅∇H,
which is nothing else than the required (3.3).Remark 3.2.
It is easy to see from (3.3) that x4=constant, and so the dynamics (3.3) can be put in the equivalent form
(3.8)ẋ1=-1c2x22+1c3x32,ẋ2=1c1x1x2-kc3x3,ẋ3=-1c1x1x3+kc2x2.The goal of our paper is to study some geometrical and dynamical properties for the system (3.8).Proposition 3.3.
The dynamics (3.8) has the following Hamilton-Poisson realization:
(3.9)(R3,Π,H),
where
(3.10)Π=[0-x2x2x20-k-x3k0]
and the Hamiltonian
(3.11)H(x1,x2,x3)=12(x12c1+x22c2+x32c3).Proof.
Indeed, it is not hard to see that the dynamics (3.8) can be put in the equivalent form
(3.12)[ẋ1,ẋ2,ẋ3]t=Π⋅∇H,
as required. Moreover, the function C given by
(3.13)C=kx1+x2x3
is a Casimir of our configuration. Indeed,
(3.14)(∇C)tΠ=0,
as desired.Remark 3.4.
The phase curves of the dynamics (3.8) are intersections of
(3.15)x12c1+x22c2+x32c3=const.,
with
(3.16)kx1+x2x3=const.,
see Figure 1.Figure 1
The phase curves of the system (3.8) (c1=1, c2=2, c3=3, and k=10).Proposition 3.5.
The dynamics (3.8) has an infinite number of Hamilton-Poisson realizations.Proof.
An easy computation shows us that the triples(3.17)(R3,{⋅,⋅}ab,Hcd),
where
(3.18){f,g}ab=-∇Cab⋅(∇f×∇g),(∀)f,g∈C∞(R3,R),Cab=aC+bH,Hcd=cC+dH,a,b,c,d∈R,ad-bc=1,
define Hamilton-Poisson realizations of the dynamics (3.8), as required.Remark 3.6.
The above proposition tells us in fact that (3.8) is unchanged, so the trajectories of motion in ℝ3 remain the same when H and C are replaced by linear combinations of H and C with coefficients which form a real matrix with det one.


## 4. Stability and Periodical Orbits

It is not hard to see that the equilibrium states of our dynamics (3.8) are(4.1)e1M=(M,0,0),M∈R,e2M=(-kc1c2c3,-c2c3M,M),M∈R,e3M=(kc1c2c3,c2c3M,M),M∈R.LetA be the matrix of the linear part of the system (3.8), that is,(4.2)A=(0-2c2x22c3x31c1x21c1x1-kc3-1c1x3kc2-1c1x1),
then the characteristic roots of A(e1M), respectively, A(e2M), respectively, and A(e3M) are given by(4.3)λ1=0,λ2,3=±c2c3M2-c1k2c1c2c3,
respectively,(4.4)λ1=0,λ2,3=±2ic1c3M,
respectively,(4.5)λ1=0,λ2,3=±2ic1c3M,
so we can conclude with the following.Proposition 4.1.
The equilibrium statese1M,M∈ℝ, are spectrally stable if M∈(-(c1k/c2c3),(c1k/c2c3)).Proposition 4.2.
The equilibrium statese2M and e3M,M∈ℝ, are spectrally stable for any M∈R.We can now pass to discuss the nonlinear stability of the equilibrium statese1M,e2M, and e3M, M∈ℝ.Proposition 4.3.
The equilibrium statese1M,M∈ℝ*, are nonlinearly stable if M∈(-(c1k/c2c3),(c1k/c2c3)).Proof.
We will make the proof using energy-Casimir method (see [7]). Let
(4.6)Hφ=H+φ(C)=x122c1+x222c2+x322c3+φ(kx1+x2x3)
be the energy-Casimir function, where φ:R→R is a smooth real-valued function defined on R.
Now, the first variation ofHφ is given by
(4.7)δHφ=x1c1δx1+x2c2δx2+x3c3δx3+φ̇(kx1+x2x3)⋅(kδx1+x2δx3+x3δx2),
where
(4.8)φ̇=∂φ∂(kx1+x2x3).
This equals zero at the equilibrium of interest if and only if
(4.9)φ̇(kM)=-Mkc1.
The second variation of Hφ is given by
(4.10)δ2Hφ=1c1(δx1)2+1c2(δx2)2+1c3(δx3)2+φ̈⋅(kδx1+x2δx3+x3δx2)2+2φ̇⋅δx2δx3.
Since M∈(-(c1k/c2c3),(c1k/c2c3)) and if we choose the function φ such that
(4.11)φ̇(kM)=-Mkc1,φ̈(kM)>0,
we can conclude that the second variation of Hφ at the equilibrium of interest is positive definite and, thus, e1M are nonlinearly stable.Similar arguments lead us to the following result.Proposition 4.4.
The equilibrium statese2M,M∈ℝ* and e3M,M∈ℝ* are nonlinearly stable for any M∈ℝ*.In order to find the periodical orbits around the equilibrium statese1M, we make use of the property that the dynamics described by a Hamilton-Poisson system takes place on the symplectic leaves of the Poisson configuration manifold, to prove the existence of periodic orbits by looking for periodic orbits of the symplectic Hamiltonian completely integrable system obtained by the restriction of the Lorenz system to the regular coadjoint orbits of G*. This procedure will be implemented around nonlinearly stable equilibrium states. The procedure is the following: we consider the system restricted to a regular coadjoint orbit of G* that contains a nonlinearly stable equilibrium, and then we will get the existence of periodic solutions for the restricted system. These periodic solutions are periodic solutions also for the unrestricted system.Proposition 4.5.
Neare1M=(M,0,0), M∈(-(c1k/c2c3),(c1k/c2c3)), the reduced dynamics has, for each sufficiently small value of the reduced energy, at least 1 periodic solution whose period is close to
(4.12)2πc1c2c3k2c12-c2c3M2.Proof.
Indeed, we have successively the following:(i)
the restriction of our dynamics (3.8) to the coadjoint orbit
(4.13)kx1+x2x3=kM
gives rise to a classical Hamiltonian system,(ii)
the matrix of the linear part of the reduced dynamics has purely imaginary roots. More exactly(4.14)λ2,3=±ik2c12-c2c3M2c1c2c3,(iii)
consider the following:(4.15)span⁡(∇C(e1M))=V0,
where
(4.16)V0=ker⁡(A(e1M))=span⁡(100),(iv)
the reduced Hamiltonian has a local minimum at the equilibrium statee1M (see the proof of Proposition 4.3).
Then our assertion follows via the Moser-Weinstein theorem with zero eigenvalue; see [8] for details.Remark 4.6.
The existence of the periodical orbits around the equilibrium pointse2 and e3 remains an open problem, the Moser-Weinstein theorem with zero eigenvalue being inconclusive.


## 5. Lax Formulation and Numerical Integration of the Dynamics (3.8)

Proposition 5.1.
The dynamics (3.8) allows a formulation in terms of Lax pairs.Proof.
Let us take the following:(5.1)L=[0l12l13-l120l23-l13-l230],l12=2c3x2c2c1+2x3c1c2+2(c2-22-c1c2)c2c1k-8c3c2c1,l13=-2c3x1c1-2c3(2-c1c2)c2c1x2-2(2-c1c2)c1c2x3+c22-c1c2-4c2c1c2k-42c3(2-c1c2)c2c1,l23=-2c3(2-c1c2)c1c2x1+2c3c2x2+2x3+42c3c2-k,B=[0b12b13-b120b23-b13-b230],b12=c2(22x3+k)-22-c1c2k-42c2c32c2c1c3,b13=-22(c2c2c3x1+2k)+c2(2-c1c2)(-4c3(2+x2)+2c2k)4c2c1c3,b23=-2-c1c2c12x1+x2-c222c3k+2,
then, using MATHEMATICA 7.0, we can put the system (3.8) in the equivalent form
(5.2)L.=[L,B],
as desired.We will discuss now the numerical integration of the dynamics (3.8) via the Kahan integrator, Lie-Trotter integrator [9], and also via Runge-Kutta 4th steps integrator, and we will point out some properties of Kahan and Lie-Trotter integrators.It is easy to see that for the equations (3.8), Kahan's integrator can be written in the following form:
(5.3)x1n+1-x1n=-hc2x2n+1x2n+hc3x3n+1x3n,x2n+1-x2n=h2c1(x1n+1x2n+x2n+1x1n)-hk2c3(x3n+x3n+1),x3n+1-x3n=-h2c1(x1n+1x3n+x3n+1x1n)+hk2c2(x2n+x2n+1).Using MATHEMATICA 8.0, we can prove the following proposition which shows the incompatibility of the Kahan's integrator with the Poisson structure of the system (3.8).Proposition 5.2.
Kahan's integrator (5.3) has the following properties:(i)
it is not Poisson preserving;(ii)
it does not preserve the CasimirC of the Poisson configuration (ℝ3,Π);(iii)
it does not preserve the HamiltonianH of the system (3.8).We will discuss now the numerical integration of the dynamics (3.8) via the Lie-Trotter integrator.To begin with, let us observe that the Hamiltonian vector fieldXH splits as follows:(5.4)XH=XH1+XH2+XH3,
where(5.5)H1=12c1x12,H2=12c3x22,H3=12c3x32.
Their corresponding integral curves are, respectively, given by(5.6)[x1(t)x2(t)x3(t)]=Ai[x1(0)x2(0)x3(0)],i=1,3,
where(5.7)A1=[1000e(a/c1)t000e-(a/c1)t],a=x1(0),A2=[1-bc2t00100kc2t1],b=x2(0),A3=[10cc3t01-kc3t001],c=x3(0).
Then the Lie-Trotter integrator is given by(5.8)x1n+1=x1n-bc2tx2n+(cc3t+kbc2c3t2)x3n,x2n+1=e(a/c1)tx2n-kc3te(a/c1)tx3n,x3n+1=kc2te(-a/c1)tx2n-k2c2c3t2e(-a/c1)tx3n.Now, a direct computation or using MATHEMATICA leads us to the following.Proposition 5.3.
The Lie-Trotter integrator (5.8) has the following properties:(i)
it preserves the Poisson structureΠ;(ii)
it preserves the CasimirC of the Poisson configuration (ℝ3,Π);(iii)
it does not preserve the HamiltonianH of the system (3.8);(iv)
its restriction to the coadjoint orbit(𝒪k,ωk), where
(5.9)Ok={(x1,x2,x3)∈R3∣kx1+x2x3=const.}and ωk is the Kirillov-Kostant-Souriau symplectic structure on 𝒪k, gives rise to a symplectic integrator.Remark 5.4.
If we compare these methods, see Figures2, 3, and 4, with the 4th-step Runge-Kutta method, we can see that Lie-Trotter integrator and Kahan's integrator give us a weak approximation of our dynamics. However, Kahan's integrator and the Lie-Trotter integrator have the advantage of being more easily implemented.Figure 2
The 4th-step Runge-Kutta integrator of the system (3.8) (c1=1, c2=2, c3=3, k=10, and x1(0)=x2(0)=x3(0)=1).Figure 3
The Kahan's integrator of the system (3.8) (c1=1, c2=2, c3=3, k=10, and h=1, x1(0)=x2(0)=x3(0)=1).Figure 4
The Lie-Trotter integrator of the system (3.8) (c1=1, c2=2, c3=3, k=10, and x1(0)=x2(0)=x3(0)=1).


## 6. Conclusion

The paper analyses a drift-free left invariant controllable system on a special Lie group. The Hamilton-Poisson realization of the system allows us to study the system from the mechanical geometry point of view. This means that we can use specific tools as energy-Casimir method for nonlinear stability, the Moser's theorem to find the periodical orbits, and Poisson integrators to make the numerical integration of the dynamics. In addition, we use non-Poisson integrators (Kahan's integrator and Runge-Kutta 4th-step integrator) to make a comparison between the obtained results. Numerical simulations via MATHEMATICA 8.0 are presented too. Similar problems have been studied on the Lie groupsSO(3),SO(4) (see [10]), on the Heisenberg Lie groups H(3) and H(4), or SE(2,ℝ)×SO(3) (see [11]).



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