The symmetric product and geodesic invariance in geometry

The symmetric product and geodesic

invariance in geometry, mechanics, and

control theory

Andrew D. Lewis

20/02/1996

Notes for Slide 0

In this work we present a product for vector fields on a manifold with an affine connection. Thisproduct we dub the symmetric product. The product itself first arose in the work of Crouch [1981]on gradient dynamical systems on Riemannian manifolds. It was rediscovered and named thesymmetric product by Lewis [1995] in the investigation of controllability for a class of mechanicalsystems, but has implications beyond that application. We mention that elements of this product,in its control theoretic guise, have appeared previously in the literature. In particular, we mentionthe work of Bonnard [1984] where special brackets are investigated on the Lie algebra for groupinvariant control systems. These brackets turn out to give exactly the symmetric product. We arebeginning to investigate the symmetric product in geometry, Lie groups, and mechanical systemswith constraints as well as further understanding its role in control of mechanical systems.

Andrew D. Lewis California Institute of Technology

1. The symmetric product in geometry

1.1. Some geometry

• Let M be a differentiable manifold with an affine connection ∇ (in

particular the Levi-Civita connection associated with a Riemannian

metric). Denote by Zg the geodesic spray associated with this affine

connection.

• Call an integrable distribution D ⊂ TM totally geodesic if every

geodesic which starts tangent to a leaf of the associated foliation

remains on that leaf.

• Call a distribution D geodesically invariant if every geodesic with its

initial velocity in D evolves so that its subsequent velocities remain

in D.

Notes for Slide 1

For the Levi-Civita connection it is true that the orthogonal complement of a totally geodesicdistribution is totally geodesic. This leads to a decomposition of a Riemannian manifold intototally geodesic components.

Clearly a distribution which is totally geodesic will be geodesically invariant. However, thereexist distributions which are geodesically invariant but are not integrable (and hence not totallygeodesic).


1.2. The symmetric product

• Be thinking about Lie algebras, the Lie bracket for vector fields, and

involutive ( =⇒ integrable by Frobenius) distributions.

• Call a R-vector space with a bilinear product (a, b) 7→ 〈a : b〉

satisfying 〈a : b〉 = 〈b : a〉 a R-symmetric algebra.

• Make the R-vector space of vector fields on M a R-symmetric

algebra with the symmetric product

〈X : Y 〉 = ∇XY +∇YX.

1 Theorem: A distribution D is geodesically invariant if and only

if it is closed under symmetric product.

Notes for Slide 2

We present the symmetric product in a manner which, as much as possible, illustrates its similaritywith the Lie bracket. Recall that the Lie bracket makes the set of vector fields on a manifoldinto a Lie algebra. Also recall that a distribution is involutive (i.e., closed under Lie bracket) ifand only if it is integrable (i.e., the subspaces of the distribution are the tangent spaces of localsubmanifolds).


• We present some elements of the proof here.

• For a vector field X on M define a vector field on TM by

X lift(vx) =d

dt

∣

∣

∣

∣

t=0

(vx + tX(x)) = X i(x)∂

∂vi

and call this the vertical lift of X .

• A simple computation verifies the formula

[X lift, [Zg, Ylift]] = 〈X : Y 〉lift. (SYM)

• It is also easy to verify that X lift is tangent to D ⊂ TM if and only

if X takes its values in D.

• Also, by definition, D is geodesically invariant if and only if Zg is

tangent to D ⊂ TM .

Notes for Slide 3

Where does equation (SYM) come from? It originally arose in some computations for control ofmechanical systems. We will not be getting into the details of those computations in this talk.However, we do mention that the proof we present of the Theorem 1 relies on this formula to agreat extent.


• It is now easy to show that if D is geodesically invariant then it is

closed under symmetric product.

◦ Let X and Y be vector fields with values in D. Thus X lift and

Y lift are tangent to D ⊂ TM .

◦ Since D is geodesically invariant, Zg is tangent to D ⊂ TM .

Therefore [X lift, [Zg, Ylift]] = 〈X : Y 〉lift is tangent to D ⊂ TM .

◦ Thus 〈X : Y 〉 takes its values in D.

• The converse may be proved with a coordinate computation.

• The key ingredient in the above proof is the formula (SYM).

Notes for Slide 4

Both directions in the proof actually follow from the coordinate calculations. However, we presentthe intrinsic portion of the proof as it gives some intuition about how (SYM) is important.

We mention that we have not yet given a proof of Theorem 1 without the use of (SYM).However, it is probably not too difficult to do so using some facts that have not been statedhere.


1.3. Other applications in geometry

• In the case where the affine connection is the Levi-Civita connection

associated with a Riemannian metric on g, the existence of a

geodesically invariant distribution whose orthogonal complement is

also geodesically invariant implies a reduction of the structure group

in the orthonormal frame bundle O(M).

◦ Not all geodesically invariant distributions admit a geodesically

invariant orthogonal complement.

◦ If D is totally geodesic, then its orthogonal complement is also

totally geodesic (and hence geodesically invariant).

◦ Not all examples of geodesically invariant distributions admitting

geodesically invariant orthogonal complements are totally

geodesic.

Notes for Slide 5

To construct a non-integrable geodesically invariant distribution whose orthogonal complementis geodesically invariant we proceed as follows. Let G be a Lie group with a bi-invariant Rie-mannian metric. This is possible, for example, if G is compact. In this case, every left-invariantdistribution is geodesically invariant. Thus any non-integrable left-invariant distribution willprovide what we are looking for.


• When the affine connection is a left-invariant affine connection on a

Lie group, the symmetric product appears in contexts other than

determining geodesically invariant distributions.

◦ Along any geodesic, one may measure the variation of “nearby”

geodesics with Jacobi fields. A Jacobi field along a geodesic may

be pulled back to the Lie algebra. The equations describing the

evolution of the corresponding Lie algebra element involve the

symmetric product.

Notes for Slide 6

The relationship between the symmetric product and Jacobi fields for left-invariant affine con-nections on Lie groups is not well understood. It is merely a computational artifact at themoment.


2. Geodesic invariance in control of mechanical

systems

2.1. Simple mechanical control systems

• The data for a simple mechanical control system is:

1. A Riemannian metric g on an n-dimensional configuration

manifold Q (describes the kinetic energy).

2. A function V on Q (describes potential energy). Today V = 0.

3. m one-forms F 1, . . . , Fm on Q (describe the input forces). We

shall use Ya = (F a)♯.

Notes for Slide 7

The one-forms F 1, . . . , Fm are the primitive objects which describe the forces. That is to say,the objects in physics which one typically calls forces are naturally one-forms. Nevertheless, it isthe vector fields we actually use in computations.


• The governing equations are

∇q(t)q(t) = uaYa(q(t)) (LCS)

where ∇ is the Levi-Civita connection for the kinetic energy metric.

• It is possible to write the equations of motion on TQ. Recall that

Zg was the geodesic spray and that X lift is the vertical lift of the

vector field on Q.

• The governing equations may then be written as

v = Zg(v) + uaY lifta (v)

where v ∈ TQ.

• The drift vector field is Zg and the control vector fields are Y lifta ,

a = 1, . . . ,m.

Notes for Slide 8

The solutions of the governing equations in the absence of forces are simply the geodesics of theRiemannian metric.


2.2. Configuration controllability of (LCS)

• Let q0 ∈ Q, T > 0, and let U be a neighbourhood of q0. Define

RUQ(q0, T ) = {q′ ∈ Q | there exists a solution

(q, u) of (LCS) which is at q0 at rest

at t = 0 and at q′ (not necessarily at rest)

at t = T and q(t) ∈ U}

and RUQ(q0,≤ T ) =

⋃

0≤t≤T RUQ(q0, t).

• We say that (LCS) is locally configuration accessible at q if

RUQ(q,≤ T ) contains an open subset of Q for each U and each T

sufficiently small.

Notes for Slide 9

Observe that we always start at rest. For the systems we are considering, this is equivalent tostarting at an equilibrium point. It is worth emphasising that we do not restrict ourselves byasking that the final state have zero velocity.

There is also a notion of controllability in the configuration variables. To save confusion, wedo not mention these things here. However, we do have results for configuration controllability.We refer the reader to [Lewis 1995] for the details.

Observe that our controllability definitions are the natural modifications of the usual con-trollability definitions from nonlinear control theory. The modification is that we are no longerlooking at the states of the system but rather just the configurations. This is a physically naturalthing to do, and, as we shall see, also roots out the pertinent mathematical structure of thesystem.


2.3. A simple example

• Consider the “robotic leg” below

• Q = S1 × S

1 × R+ and

g = Jdθ ⊗ dθ +mr2dψ ⊗ dψ +mdr ⊗ dr

where m is the mass of the ball and J is the moment of inertia of

the body.

• For this example ask the following question:

Starting at a given configuration at rest, which configurations

are accessible by applying suitable combinations of the force

and torque?

Notes for Slide 10

This example, though simple, captures a few important issues involved in our controllabilitydefinitions and computations.

1. If we think of this as a control system on TQ, then this control system is not controllable.The reason for this is that the system, even with inputs, preserves angular momentumand so the solutions must evolve on surfaces of constant angular momentum. In spite ofthis, the system is locally configuration accessible. So in this example our controllabilitydefinitions capture something that the usual definitions fail to capture.

2. This system is more than locally configuration accessible. It is actually controllable. Thatis to say, we can steer the system from one configuration to another. We will not discussthis here, but we refer the reader to [Lewis 1995] for a discussion of related issues.


2.4. Decompositions for simple mechanical control systems

• Denote Y = {Y1, . . . , Ym}.

• Denote by Cver the smallest geodesically invariant distribution

containing Y.

• Denote by Chor the smallest totally geodesic distribution containing

Y.

2 Theorem: Let q ∈ Q.

(i) The set of velocities accessible from q is an open subset of the

vertical lift of Cver.

(ii) RUQ(q,≤ T ) is an open subset of the integral manifold through q

of Chor.

• The symmetric product is important in proving this result.

Notes for Slide 11

Note that (LCS) is locally configuration accessible at q if rank((Chor)q) = dim(Q).Theorem 2 captures explicitly how the geometry of simple mechanical control systems appears

in the controllability of these systems. It is interesting to see how the notion of geodesic invarianceappears in the result. We mention, without going into details, that the symmetric product wasthe crucial ingredient in arriving at this theorem.


• Recall that the orthogonal complement of a totally geodesic

distribution is totally geodesic.

• Denote by F the foliation associated with Chor and by F⊥ the

foliation associated with C⊥hor

.

• This defines (at least locally) the following decompositions.

Q

||②②②②②②②②

!!❈❈

❈❈❈❈

❈❈

Q/F⊥ Q/F

• The system on Q/F⊥ is locally configuration accessible.

• The dynamics on Q/F represent “configuration uncontrollable”

dynamics.

• Almost all of the above carries through for general affine

connections, not just Levi-Civita connections. Who cares??

Notes for Slide 12

By “configuration uncontrollable” we mean that, starting at zero velocity, we cannot throughany application of inputs, affect these variables.


3. Geodesic invariance in constrained mechanics

• Linear constraints in mechanics typically model “rolling without

slipping” (e.g., wheels). The constraint is a distribution D on Q.

• For example, consider the rolling penny:

At each configuration q = (x, y, θ, ψ), Dq describes the directions

the system may move without violating the constraint.

Notes for Slide 13

One may consider more general types of constraints (which are nonlinear in velocity). We arenot sure how or if our methods may be applied to those problems.


• D⊥ is the orthogonal complement of D and P : TQ→ TQ is the

projection onto D and P ′ : TQ→ TQ is the projection onto D⊥.

• For the Lagrangians we are considering (L = KE), the constrained

equations of motion are (using the Lagrange-d’Alembert principle)

∇q(t)q(t) = λ(t)

P ′(q(t)) = 0

where λ is a section of D⊥ (the constraint force).

Notes for Slide 14

What we present here is a generalisation of work of Bloch and Crouch [1995].Everything we say in this section may be generalised appropriately for arbitrary affine con-

nections.The Lagrange-d’Alembert principle is the appropriate variational principle for mechanical

systems with constraints. There is actually some debate over this, and we refer the readerto [Lewis and Murray 1995] for a discussion of these issues.

In the classical development of the equations of motion for constrained systems, λ is aLagrange multiplier.


• By covariantly differentiating the constraints we may solve for the

constraint force to arrive at the equivalent equations

∇q(t)q(t) + (∇q(t)P′)(q(t)) = 0

=⇒ ∇′q(t)q(t) = 0

where ∇′ is a new affine connection defined by

∇′XY = ∇XY + (∇XP

′)(Y ).

• This connection has the property that ∇′XY is a section of D for

every vector field X and every section Y of D. In particular,

◦ D is geodesically invariant with respect to ∇′, and

◦ the affine connection ∇′ restricts to a vector bundle connection

in D.

Notes for Slide 15

The object ∇′ is an affine connection since (X,Y ) 7→ (∇XP′)(Y ) defines a (1, 2) tensor field.

It is interesting to see that we have defined ∇′ so that D is geodesically invariant. In thiscase, it is also true that the affine connection reduces to a connection in D. This is not true,however, for arbitrary geodesically invariant distributions, even in the Levi-Civita case.

Speaking of Levi-Civita connections, it is interesting to ask when the connection ∇′ is aLevi-Civita connection. The answer is best obtained by looking at the holonomy groups of ∇′.This will be future work.


• Since our control theoretic setup for unconstrained systems extends

to general affine connections, we may directly apply those methods

to problems with constraints.

• The control problem is

∇′q(t)q(t) = uaP (Ya(t)) (CLCS)

• We have the following result:

3 Theorem: Let q ∈ Q. The set of configurations of (CLCS)

accessible from q with zero initial velocity is an open subset of the

leaf through q of the smallest totally geodesic (w.r.t. ∇′!)

distribution containing the inputs.

Notes for Slide 16

The fact that one only has to consider the projection of the inputs to D make sense since thepart of the force in D⊥ will be absorbed by the Lagrange multiplier.


• We may perform a sanity check of our constructions and results.

◦ If D is integrable then

(X,Y ) 7→ −(∇XP′)(Y )

is the second fundamental form for the integral manifolds. In this

case ∇′ reduces to the induced Levi-Civita connection on the

leaves.

◦ If D is totally geodesic then ∇XP′ = 0 for every vector field X .

In this case ∇′ = ∇.

Notes for Slide 17

Recall that the second fundamental form is derived when one decomposes the original affineconnection into its tangential and normal pieces. It is easy to show that (∇XP

′)(Y ) is orthogonalto D when X and Y are in D and so it easily follows that it must be (up to sign) the secondfundamental form. Also recall that the second fundamental form vanishes when the submanifoldin question is totally geodesic as, in this case, the geodesics of the induced affine connectionagree with those of the full system.


◦ Since

1. we can only reach configurations contained in leaves of the

smallest totally geodesic distribution containing the inputs,

and

2. the inputs are contained in D,

the set of reachable configurations is contained in the leaves of

the involutive closure of D. That is, we cannot reach more

configurations that the constraints will allow.

Notes for Slide 18

These sanity checks give us no surprising results, but at least indicates that our constructioneasily recovers the obvious results.


4. Wrap up

• We have explored the relationship of the symmetric product with

the concept of geodesic invariance for affine connections.

• These ideas have applications in geometry, mechanics (with

constraints), and control theory for mechanical systems.


• Future work includes:

◦ On the geometry side we would like to fully understand how the

presence of geodesically invariant distributions affects the

structure of the affine connection (especially its holonomy

groups).

◦ On the control theory side, we would like to see if an

understanding of the symmetric product can be useful in

designing controllers for mechanical systems.

◦ For systems with constraints, we would like to understand the

relationship of ∇′ with other recent work on constrained systems.

Notes for Slide 20

We have briefly stated a few preliminary results about how one may reduce the structure groupof O(M) in the presence of a distribution which is geodesically invariant and whose orthogonalcomplement is geodesically invariant. However, a more general understanding might be valuable.

In nonlinear control theory, understanding the Lie bracket has enabled certain researchers toimplement control laws for certain classes of control systems, especially nonholonomic systems.It would be interesting to see if a similar benefit may be gained by better understanding thesymmetric product.

Recent work for constrained systems includes that of Bloch, Krishnaprasad, Marsden, andMurray [1996]. Understanding their momentum equation in terms of our connection ∇′ may beinteresting.


References

Bloch, A. M. and Crouch, P. E. [1995] Another view of nonholonomic mechanical control systems,in Proceedings of the 34th IEEE Conference on Decision and Control, pages 1066–1071,Institute of Electrical and Electronics Engineers, New Orleans, LA.

Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E., and Murray, R. M. [1996] Nonholonomic

mechanical systems with symmetry, Archive for Rational Mechanics and Analysis, 136(1),21–99.

Bonnard, B. [1984] Controllabilite de systemes mecaniques sur les groupes de Lie, SIAM Journalon Control and Optimization, 22(5), 711–722.

Crouch, P. E. [1981] Geometric structures in systems theory, Institution of Electrical Engineers.Proceedings. D. Control Theory and Applications, 128(5), 242–252.

Lewis, A. D. [1995] Aspects of Geometric Mechanics and Control of Mechanical Systems, Ph.D.thesis, California Institute of Technology, Pasadena, California, USA.URL: http://www.cds.caltech.edu/

Lewis, A. D. and Murray, R. M. [1995] Variational principles for constrained systems: Theory

and experiment, International Journal of Non-Linear Mechanics, 30(6), 793–815.


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