In order to characterize the controllability properties in the attitude control and the orbital transfer we present a general theorem which can be applied to every system where the free motion is Poisson stable.
3.3.1 Poisson Stability
Definition 3.2.Let q˙=X(q) be a (smooth) differential equation onM. We noteq(t, q0)the solution with q(0) =q0. We assume it is defined on IR. The point q0 is Poisson stable if for each V neighborhood of q0, for each T > 0 there exist t1, t2≥T such that q(t1, q0), q(−t2, q0)∈V. The vector field X is said to be Poisson stable if almost every point is Poisson stable.
Theorem 3.1 (Poincar´e theorem). If X is a conservative vector field and if every trajectory is bounded, then X is Poisson stable.
Corollary 3.1. Let −→H be a Hamiltonian vector field, if every trajectory is bounded, then −→H is Poisson stable.
Example 3.1. Consider the free motion in the orientation of a rigid body. Let H(R, Ω) = 12(I1Ω12+I2Ω22+I3Ω32) be the kinetic energy. The free motion corresponds to Euler-Lagrange equations and induces a Hamiltonian system.
The trajectories are bounded because the attitudeR∈SO(3) and the angular velocity is bounded,H being is a first integral.
Example 3.2.Consider Kepler equation: ăq =−à q
|q|3, H = 1 2q˙2− à
|q| be the energy. If we restrict our equation to the elliptic domainΣewhere the trajec- tories are ellipses, the vector field is Poisson stable. It corresponds to bounded trajectories.
3.3.2 General Results About Controllability Notations
• V(M): set of (smooth) vector fields onM.
• Iff :M →IR smooth,LXf =df(X) = ∂f
∂q.X(q): Lie derivative.
• Lie bracket:X, Y ∈V(M),[X, Y] =LY ◦LX−LX◦LY = ∂X
∂q (q)Y(q)−
∂Y
∂q(q)X(q) in local coordinates
• If ˙q=X(q), we noteq(t, q0) the maximal solution withq(0) =q0. We set:
(exptX)(q0) =q(t, q0), exptXis a local diffeomorphism onM,Xis complete if exptX is defined∀t∈IR.
Definition 3.3.A polysystem D is a family {Xi, i ∈ I} of vector fields.
We note D : q → SpanD(q) the associated distribution; D is involu- tive if [Xi, Xj] ∈ D,∀Xi, Xj ∈ D. We note DA.L. the Lie algebra gen- erated by D computed by: D1 = SpanD, D2 = Span{D1 + [D1, D1]},..., Dk =Span{Dk−1+ [D1, Dk−1]},DA.L.= k≥1Dk.
Definition 3.4.Let q˙ = f(q, u) be a system on M. Let U be the control domain and we can restrict our controls to the set U of piecewise mappings with values in U. Letq(t, q0, u)be the solution withq(0) =q0associated to u.
We note:
• A+(q0, t) =
u(.)
q(t, q0, u): accessibility set in timet,A+(q0) =
t≥0
A+(q0, t)
the accessibility set
• A−(q0, t): set of points which can be steered to q0 in time t, A−(q0) =
t≥0A−(q0, t).
The system is controllable in timet ifA+(q0, t) =M,∀q0, and controllable if A+(q0) =M,∀q0. The system is locally controllable atq0 if ∀t≥0, A+(q0, t) andA−(q0, t)are neighborhood ofq0.
We introduce the polysystem D={f(q, u);u∈U} and the setST(D) = {expt1X1◦...◦exptkXk;Xi ∈ D, k ≥ 0, ti >0, t1+...+tk =T} and the pseudo semi-groupS(D) =
T≥0
ST(D);G(D) is the pseudo-group{expt1X1◦ ...◦exptkXk;Xi ∈D, k∈IN, ti∈IR}.
Lemma 3.1.We have:A+(q0, T) =ST(D)(q0),A+(q0) =S(D)(q0).
Definition 3.5.LetDbe a polysystem,Dis said to be controllable if S(D)(q0)
= M, ∀q0 and weakly controllable if the orbit O(q0) = G(D)(q0) = M,∀q0. The polysystem is said to satisfy the rank condition ifDA.L.(q0) =Tq0M,∀q0.
The following two results are standard ones.
Theorem 3.2 (Chow). Let D be a polysystem on a connected manifold M. If it satisfies the rank condition, thenD is weakly controllable.
Proposition 3.3.AssumeD satisfies the rank condition. Then for eachq0∈ M, for each V neighborhood of q0, there exist U+, U− open sets respectively contained inV A+(q0),V A−(q0).
Theorem 3.3 (Lobry). LetD be a polysystem on a connected manifoldM. Assume:
1. DA.L.(q0) =Tq0M,∀q0, 2. ∀Xi∈D, Xi is Poisson stable.
ThenD is controllable.
Proof . Let q0, q1 ∈ M, one must construct a trajectory joining q0 to q1. Using the previous proposition, we takeU− andU+contained respectively in A−(q0) and A+(q1). Let q0 ∈U− andq1∈U+, the problem is equivalent to steerq0toq1. From Chow’s theorem, we can joinq0toq1using concatenation of integrals curves or vector fields in D, with positive or negative time t.
Each curve with t <0 is replaced by a curve of the same vector field, with positive time using Poisson stability (replacing a trajectory if necessary by a neighboring trajectory). This proves the assertion.
This result can be improved to get controllability conditions with only one Poisson stable vector field.
3.3.3 Controllability and Enlargement Technique (Jurdjevi´c-Kupka)
We restrict our analysis to polysystems satisfying the rank condition. Then D is controllable if and only if ∀q0, S(D)(q0) =M.
Definition 3.6.Let D,D be two polysystems. They are called equivalent if
∀q0, S(D)(q0) = S(D)(q0). The union of all polysystems equivalent to D is called the saturated, with notation satD.
Proposition 3.4.Let D be a polysystem.
1. If X ∈D andX is Poisson stable, then−X∈D.
2. The convex hullconvD⊂satD.
Proof . For condition 1, use the proof of theorem (3.3).
For 2,ifX∈D, thenλX∈satD,∀λ >0. IfX, Y ∈D, from Baker-Campbell- Hausdorff formula we have:
n times
(exp t
nX)◦(exp t
nY) = expt(X+Y) +◦(1 n).
Taking the limit whenngoes to +∞, we haveX+Y ∈satD.
Theorem 3.4.Let q˙=F0(q) +
p
i=1
uiFi(q)be a system on a connected man- ifold M, withui∈ {−1,+1}. Assume:
1. F0 is Poisson stable
2. {F0, ..., Fp}A.L.(q0) =Tq0M,∀q0.
Then the polysystem is controllable. Moreover, condition 2. is also necessary in the analytic case.
Proof .LetD={F0(q) + pi=1uiFi(q),ui∈ {−1,+1}}.We check:
• DA.L. ={F0,ã ã ã, Fp}A.L.
• F0= 1
2[(F0+F1) + (F0−F1)]∈satD
• −F0∈satD
• ±Fi∈satD, fori= 1,ã ã ã, p.
Hence the symmetric polysystem{±Fi;i= 0, ..., p} ∈satD and the control- lability result follows from Chow’s theorem. The necessary condition in the analytic case follows from Nagano-Sussmann theorem.
In the next section, we apply this result to characterize controllability computing Lie brackets.
3.3.4 Application to the Attitude Problem
Theorem 3.5.Consider the system (3.3) describing the attitude control prob- lem, with m = 1, i.e. a single pair of opposite gas jets. Then the system is controllable except when twobi1 are zero or√a3b11=±√a1b31.
Sketch of proof. The free motion is Poisson stable and the system is control- lable if and only if the rank condition is satisfied.
First of all, consider the subsystem describing the evolution of the angular velocity: ˙Ω = Q(Ω) +u1b1. We must have{Q, b1}A.L. of rank 3 at each point.
Since Q is homogeneous, this condition is satisfied if and only if the Lie algebra of constant vector fields in {Q, b1}A.L. is of rank 3. A computation shows that this Lie subalgebra is generated by: f1 = b1, f2 = [[Q, f1], f1], f3 = [[Q, f1], f2], f4 = [[Q, f1], f3] and f5 = [[Q, f2], f2].
The conditions follow.
Geometric Interpretation
Consider Euler equation with no applied torque, describing the evolution of the angular velocity. The system can be integrated using the conservation of the energy and of the momentum. If we fix the energy to H = 1, we can represent the trajectories. The system has three pairs of singular points corresponding to the axis Ei, the rigid body describing stationary rotations.
Singular points corresponding to the major and the minor axis are centers and for the intermediate axis we have saddles. Each trajectory is periodic except separatrices connecting opposite saddle points.
If the applied torque is oriented along one axis, we control only the cor- responding stationary rotation. This corresponds to the first condition. The condition √a3b11 = ±√a1b31 means that the torque is oriented in one of the plane filled by separatrices.
3.3.5 Application to the Orbital Transfer
We restrict our analysis to the case when the mass is constant, the system being given by equation (3.4).
First we make Lie bracket computations using cartesian coordinates and the thrust being decomposed in the tangential/normal frame. Computations give us:
Proposition 3.5.Let x= (q,q)˙ with q∧q˙= 0, then:
1. {F0, Ft, Fc, Fn}A.L.(x) = IR6 and coincides with
Span{F0(x), Ft(x), Fc(x), Fn(x),[F0, Fc](x),[F0, Fn](x)}.
2. The dimension of {F0, Ft}A.L.(x) is four.
3. The dimension of {F0, Fn}A.L.(x) is three.
4. The dimension of {F0, Fc}A.L.(x) is four if L(0) = 0 and three if L(0) = 0.
Also the orbit corresponding to each direction can be characterized using the orbital coordinates and system (3.7).
Proposition 3.6. Consider the single input system: ˙x=F0(x)+uG(x), where GisFt,Fc orFn andxis restricted to the elliptic domain, then:
1. If G=Ft, then the orbit is the 2D–elliptic domain.
2. IfG=Fn, then the orbit is of dimension three and is the intersec- tion of the 2D–elliptic domain witha=a(0)constant.
3. IfG=Fc, the orbit is of dimension four ifL(0) = 0, and of dimen- sion three ifL(0) = 0 and is given by a=a(0),|e|=|e(0)|.
Theorem 3.6.Consider the single input system: x˙ =F0(x) +uG(x), where G isFt,Fc orFn andx is restricted to the elliptic domain, then each point in the orbit is accessible.