Implementación del kitsch en el diseño editorial
4.1 PREÁMBULO DE METODOLOGÍA
This section examines the simplifications that occur when symmetries are added to nonholonomic systems with constraints. The symmetry involved is group invariance with the Lie group G, where Gacts on the configuration manifold Q. The action ofG will be denoted byΦg, and mapsg 7→ gq. The action gives
the manifoldQa principal bundle structure, π : Q → Q/G. The base space isM =Q/G. The reader is referred to Marsden and Ratiu [99] for more background on Lie groups and geometric mechanics.
For the Lie group, G, there is a corresponding Lie algebra g, defined using the tangent space of the group identity, TeG. The Lie algebra elements act as vector fields on the configuration space Qvia the
infinitesimal generator,
ξ·q≡ξQ(q)≡
d
dtΦexpξt.
Assume that the entire system is invariant under the group action. In particular, the constraints are G- invariant, i.e.,
TΦ∗gA(q,q˙)≡A(TΦg(q,q˙)) =A(q,q˙) = 0.
Group invariance means that the system may be reduced. The Ehresmann connection form reduces to a
principal connection form.
Definition 26 [15] A principal connection on the principal bundleπ : Q → Q/Gis a mapA : T Q7→ g
that is linear on each tangent space (e.g.,Ais a Lie-algebra valued one-form) and is such that 1. A((Qq)) =ξ,∀ξ∈gandq∈Q.
2. AisAd-equivariant:
A(TqΦg(vq)) = AdgA(vq),
for allvq ∈TqQandg∈G, whereAddenotes the adjoint action ofGong.
Notions of horizontality and verticality are inherited from the Ehresmann connection, except that now the G-invariant vertical space can be, pointwise, isomorphically identified with the Lie algebra,g. The equations of motion reduce also to the Lie algebra, and in a trivialization the oscillatory control equation (3.49) is now given by
˙
r =Ya(r)ua
ξ =−Aloc(r)Ya(r)ua,
(3.62)
where the Lie group evolution is obtained from the reconstruction equation, ˙
g=−gξ , g(0) =g0. (3.63)
The local form of the principal connection is a Lie-algebra valued 1-form,Aloc:T M →g.
Lie algebra rank condition. Due to the identification of the vertical vector space with the Lie algebra, Theorem 20 may be reinterpreted for systems evolving on a principal bundle with a principal connection. Theorem 22 [137] Under the assumption that the control vector fields, Ya ∈ X(M), span the tangent
spaceTπ(q)M forq ∈Q, the Lie algebra rank condition atqis equivalent to
spannAhYihk. . . ,hYih2,hYih1, Yih0iii o=g. (3.64)
The principal connection can be used to define the notion of parallel transport. Parallel transport is given by flow along horizontal vector fields,
Pt0X,t = Φ0X,th, X ∈ X(M). (3.65) Using the parallel transport, it is possible to define a covariant derivative for elements inC∞M,
∇Xf =LXhf (3.66)
Corollary 9 [137] Under the assumption that the control vector fields,Ya∈ X(R), span the tangent space
Tπ(q)Rforq∈Q, the Lie algebra rank condition atq is equivalent to
spannA∇Yik· · · ∇Yi2B(Y h i1, Y h i0) o =g, (3.67) fork ∈Z+\ {0}.
Because the principal connection isAd-equivariant and its higher order covariant derivative are all vertical, the curvature form may be identified with an element of the associated adjoint bundle, eg, of the principal bundle,
e
g≡Q×Gg≡(Q×g)/G. (3.68)
Sections of the associated adjoint bundle are in1−1correspondence withAd-equivariant Lie algebra valued functions. The local form of the principal connection,Aloc, is such a section.
The local curvature form may be obtained from the principal connection, via covariant differantiation on the associated adjoint bundle,
Bloc(X, Y)≡∇eXAloc(Y), ∀X, Y ∈ X(R). (3.69)
Corollary 10 [137] Under the assumption that the control vector fields, Ya ∈ X(M), span the tangent
spaceTπ(q)M forq ∈Q, the Lie algebra rank condition atqis equivalent to the condition that
spann∇eYik· · ·∇eYi2Bloc(Yi1, Yi0)(r)
o
=g, (3.70)
fork ∈Z+\ {0}.
The Lie algebra rank condition has been simplified to the condition that the local form of the curvature and its covariant derivatives span the tangent to the fiber.
Controllability of Systems with Constraints. It was shown in the analysis for systems with an Ehres- mann connection that weak controllability does not place the same restrictions on the Lie algebra as strong controllability. Analogous results hold for the principal connection.
Theorem 23 For the system (3.62), the following are equivalent:
1. the system is small-time weakly controllable,
2. span Yh ik. . . , Yh i3, Yh i2, Y h i1 (q) =VqQ, 3. spann n ∇eYik· · ·∇eYi2Bloc(Yi1, Yi0)(q) o S { Aloc(Yi0))} o =g, fork ∈Z+\ {0}anda= 1. . . m.
The Averaged Expansions. Suppose that the system (3.62) were to be weakly fiber controllable, and there was no need to stabilize the base space. Then the averaged expansions from Section 3.1 may be simplified. Suppose furthermore that controllability was obtained through the curvature and higher-order covariant derivatives.
Third-order averaging using the averaged expansion of Equation (3.19) is
˙ r ˙ g = 0 −g 1 2V (α,β) (1,0)(t)Bαβ(r, s) + 13V (α,β,γ) (1,1,0)(t)Bαβγ(r, s) (3.71)
whereBαβ is the local form of the curvature form, defined as follows,
Bαβ ≡ Bloc(Yα, Yβ),
andBαβγis the covariant derivative of the local curvature form,
Bαβγ ≡∇eYαBloc(Yβ, Yγ).
Due to the structure of the vector fields, there will never be a contribution to the base space unless the first order average is non-vanishing. This means that all expansions will only affect the fiber space. In the average, the evolution of the base space is trivial ince, in the average, the base variables are constant. Replacing them by their averages in the equations of motion reduces Equation (3.71) to
˙ g=−g 1 2V (α,β) (1,0)(t)Bαβ(r, s) + 1 3V (α,β,γ) (1,1,0)(t)Bαβγ(r, s) (3.72)
The equations have been simplified to evolution of the fiber only. Higher-order averages have the same simplifying property.
3.3
Examples
Several examples will be examined to demonstrate the applicability of this approach. In some cases, the con- trollers constructed using the generalized averaging theory are compared to known stabilizing controllers. The first example in Section 3.3.1 is a three-state nonholonomic integrator. Being the simplest type of un- deractuated nonlinear driftless control system, all of the controller strategies are worked out to demonstrate how different controllers may be derived from different choices of the α-parametrization. Additionally, the difference between averaged coefficients with and without definite integrals will be studied using the nonholonomic integrator.
In Section 3.3.2, the Hilare robot is examined. Because the Hilare robot evolves on the Lie group SE(2), the Lie group exponential is used instead. It is possible to provide a stabilizing controller without the use of coordinate transformations. The essential information needed for stability is found in the averaged coefficients and Jacobi-Lie brackets. Lastly, a discretized control strategy is given for the kinematic car in Section 3.3.3. The controller is quickly derived using the procedure of this chapter. The discretized strategy can be improved via modification to a continuous feedback strategy.