Let us review the definitions and main results related with accessibility for ACCS [Bullo and Lewis 2005a;b, Cort´es and Mart´ınez 2003, Lewis and Murray 1997, Ostrowski and Burdick 1997].
Definition 6.2.1. Let Σ = (Q, ∇,Y , U) be an ACCS and vx ∈ TxQ.
1. Thereachable set from vxat time T in Q is
122 6.2. Accessibility and controllability for mechanical systems
2. Thereachable set from vxat time T in T Q is
RΣ,T Q(vx, T ) = {Υ(T ) | (Υ, u) : I ⊂ R → T Q × U satisfies (6.1.2)
and Υ(0) = vx}.
3. Thereachable set RΣ,Q(vx, ≤ T ) from vxup to T in Q is
RΣ,Q(vx, ≤ T ) =
[
0≤t≤T
RΣ,Q(vx, t).
4. Thereachable set RΣ,T Q(vx, ≤ T ) from vxup to T in T Q is
RΣ,T Q(vx, ≤ T ) =
[
0≤t≤T
RΣ,T Q(vx, t).
Once the reachable sets are defined, we can introduce the notion of accessibility and con- trollability.
Definition 6.2.2. Let Σ = (Q, ∇,Y , U) be an ACCS and vx∈ TxQ.
1. The systemΣ is accessible from vxif there existsT > 0 such that
int RΣ,T Q(vx, ≤ t) 6= ∅
for everyt ∈ (0, T ].
2. The system Σ is configuration accessible from vx if there exists T > 0 such that
int RΣ,Q(vx, ≤ t) 6= ∅ for every t ∈ (0, T ].
3. The systemΣ is small–time locally controllable from vx if there existsT > 0 such that
vx∈ int RΣ,T Q(vx, ≤ t) for every t ∈ (0, T ].
4. The system Σ is small–time locally configuration controllable from vx if there exists
T > 0 such that x ∈ int RΣ,Q(vx, ≤ t) for every t ∈ (0, T ].
Remark 6.2.3. Observe that if a system is accessible, then it is configuration accessible. Anal- ogously, if the system is small–time locally controllable, then it is small–time locally configu- ration controllable. The converses are not necessarily true.
Remark 6.2.4. According to Ostrowski and Burdick [1997], the notion of small–time locally controllable only has sense if the initial velocity is assumed to be zero, otherwise we cannot guarantee that the trajectory stays in a neighbourhood of the initial condition. There exist results related with controllability for mechanical systems in the literature on control theory as long as the initial velocity is zero; see for instance [Lewis and Murray 1997].
Before proceeding, we need an assumption about the control set; for more details see [Bullo and Lewis 2005a;b;c]. In the following statement, conv (U ) is the convex hull of the open control set (that is, the smallest convex set containing U ), and aff (U ) is the affine hull (that is, the smallest affine subspace of Rkcontaining U ). See Appendix B for definitions of convexity and affinity.
Definition 6.2.5. The control set U ⊂ Rkisalmost proper if 1. 0 ∈ conv(U ), and
2. aff(U ) = Rk.
Using the notation in Chapter 3, this assumption on the control set guarantees that the span of the vector fields {f0, f1, . . . , fk} defining a control–affine system is equal to the span of the
vector fields {f0+Pks=1usfs | u ∈ U }. This is useful to analyze the structure of the reachable
set.
If the mechanical system is studied as a control–affine system on T Q; see Equation (6.1.2), then the notions of accessibility reviewed in §3.2 and described in [Nijmeijer and van der Schaft 1990, Sussmann and Jurdjevic 1972] can be used. Thus, according to Definition 3.2.4, the ac- cessibility distribution of the system (6.1.2) is Lie∞ Z, Y1V, . . . , YkV. The system is accessi- ble if
Lie∞(Z, Y1V, . . . , YkV)vx = TvxT Q
because of Proposition 3.2.6.
At first, the mechanical system is defined on Q. Thus if the initial velocity is taken to be zero, there is a characterization of the accessibility in terms of constructions on Q without considering vector fields on the tangent bundle. In order to obtain results, it is necessary to study the symmetric product of two vector fields X, Y ∈ X(Q) denoted by hX : Y i and defined as follows
hX : Y i = ∇XY + ∇YX. (6.2.3)
The geometric meaning of the symmetric product is the characterization of the geodesic invari- ance of a distribution; see [Bullo and Lewis 2005b] for more details.
Theorem 6.2.6. ([Bullo and Lewis 2005a, Cort´es and Mart´ınez 2003, Lewis and Murray 1997]) Let Σ = (Q, ∇,Y , U) be an ACCS with the control set U being almost proper and Sym∞(Y ) be the smallest distribution such that Y ⊂ Sym∞(Y ), hX : Y i ∈ Γ (Sym∞(Y )) for each X, Y ∈ Γ (Sym∞(Y )). If Lie∞(Sym∞(Y ))x = TxQ, then Σ is configuration
accessible from0x. IfSym∞(Y )x = TxQ, then Σ is accessible from 0x.
This result is related with the fact that V0x(τQ) = Sym
∞(Y )
xand H0x(T Q) = Lie
∞(Sym∞(Y ) x) .
To try to generalize the results of accessibility for non–zero initial velocities, we compute some Lie brackets of the control vector fields and the drift vector field. In natural local coordinates (x, v) for T Q, Z = vi ∂ ∂xi − Γ i jlvjvl ∂ ∂vi, Y V = Yi ∂ ∂vi, (6.2.4)
where Γijl, Yi∈ C∞(Q), and then we have
[Z, YV] = −Yi ∂ ∂xi + vi∂Y l ∂xi + Y ivj(Γl ij + Γlji) ∂ ∂vl. (6.2.5)