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Differential Geometry

A Short Summary

Basilio Bona

DAUIN Politecnico di Torino

July 2009

Geometric and Probabilistic Fundamentals in Robotics

B. Bona (DAUIN) Differential geometry July 2009 1 / 24

Introduction

Differential geometry is a mathematical discipline that uses the methods ofdifferential and integral calculus to study problems in geometry.

The theory of plane and space curves and of surfaces in thethree-dimensional Euclidean space formed the basis for its initialdevelopment in the eighteenth and nineteenth century.

Since the late nineteenth century, differential geometry has grown into afield concerned more generally with geometric structures on differentiablemanifolds.

Differential geometry is closely related with differential topology and withthe geometric aspects of the theory of differential equations.

Basic concepts of differential geometry are required to study the behaviourof mechanical systems subject to nonholonomic constraints, such aswheeled mobile robots and free flying bodies subject to angular momentconservation.

B. Bona (DAUIN) Differential geometry July 2009 2 / 24

Differentiable manifolds

A differentiable manifold is a space that is locally diffeomorphic to Rn.

Figure: A differential manifold.

B. Bona (DAUIN) Differential geometry July 2009 3 / 24

Tangent space

Consider a point x in a vector space Rn. The tangent space TxRn is the

space of all vectors that are the derivatives of trajectories on the manifoldthat pass through the point x.

If the manifold is not Rn, but a generic differential manifold M, the notionof tangent space is still valid.

Figure: A tangent space.

B. Bona (DAUIN) Differential geometry July 2009 4 / 24

Vector fields

A vector field f : Rn 7 TxRn is a smooth mapping that assigns to each

point x a tangent vector f(x) in the tangent space

x f(x)

Smooth means that the mapping is differentiable up to what is necessary.

Figure: A vector field.

B. Bona (DAUIN) Differential geometry July 2009 5 / 24

In other words, a vector field assigns a tangent vector to each point x inthe manifold, in an appropriately smooth way (notice the differencebetween point and vector).

Vector fields can be studied as differential operators and can be used todifferentiate functions and other vector fields.

There is one-to-one correspondence between ordinary differential equationsand vector fields.

Vector fields are to be thought as right-hand sides of differential equations

x = f(x)

Of great importance is the Lie bracket operator between vector fields: it isdefined as an appropriate composition of differential operators, and itfrequently arises in the composition of flows of differential equations.

B. Bona (DAUIN) Differential geometry July 2009 6 / 24

Flows

A flow of f is defined when a vector field f(x) is used to specify adifferential equation

x(t) = f(x(t))

The flow ft(x) is then the mapping that associates to each point x thevalue at time t of the solution of the above differential equation, evolvingfrom x(t) at t = 0, or

d

dtft(x) = f(

ft(x))

For time-invariant linear systems, the differential equation takes the form

x(t) = Ax(t)

and the flow becomes the well know linear operator

ft(x) = eAtx hence ft = e

At

B. Bona (DAUIN) Differential geometry July 2009 7 / 24

Lie bracket

Given two vector fields g and h the composition of their flow is in generalnon commutative

gt

ht 6=

ht

gt

A Lie bracket [g,h] can be defined using the vector fields as

[g,h] =h

xg

g

xh = Jhg Jgh

Properties:

[f,g] = [g, f]

[f, [g,h]] + [g, [h, f]] + [h, [f,g]] = 0

if [f,g] = 0, the two vector fields commute.

f always commute with itself: [f, f] = 0

B. Bona (DAUIN) Differential geometry July 2009 8 / 24

Lie algebra

A vector space V over R is a Lie algebra, if there exists a binary operationV V V, denoted [, ], that a) is skew-symmetric, and b) satisfy theJacobi identity.

The vector space of smooth vector fields on Rn, equipped with the Liebracket as the operator between space elements, is a Lie algebra, denotedby X(Rn).

B. Bona (DAUIN) Differential geometry July 2009 9 / 24

Lie derivative

A Lie derivative Lfg evaluates the change of one vector field g along theflow of another vector field f.

The Lie derivatives are represented by vector fields, as infinitesimalgenerators of flows (active diffeomorphisms) on a manifold M. Looking atit the other way around, the diffeomorphism group of M has theassociated Lie algebra structure, of Lie derivatives, in a way directlyanalogous to the Lie group theory.

In our case the manifold M is the (flat) Euclidean space Rn.

B. Bona (DAUIN) Differential geometry July 2009 10 / 24

Given two vector fields f,g : D 7 Rn the Lie derivative is

Lfg(x) =g

xf

hence[f,g] = Lfg(x) Lgf(x)

When the vector field f is a real valued function f (x) : Rn 7 R, the Liederivative is

Lgf (x) =f

xg(x)

A Lie product is a nested set of Lie brackets, as, for example, in

[[f,g], [f , [f,g]]]

B. Bona (DAUIN) Differential geometry July 2009 11 / 24

Given two real valued functions f1(x) and f2(x), the following holds

Lg(f1f2) = (Lgf1)f2 + (Lgf2)f1

Given two real valued functions f1(x) and f2(x) and two vector fields g,h,the following holds

[f1g, f2h] = f1f2[g,h] + f1(Lgf2)h f2(Lhf1)g

Another characterization of Lie brackets

L[g,h]f = LgLhf LhLgf

B. Bona (DAUIN) Differential geometry July 2009 12 / 24

Geometrical interpretation

Consider the dynamic system associated to the vector fields g1 and g2:

x(t) = g1(x)u1(t) + g2(x)u2(t)

if the two inputs u1, u2 are not simultaneously active, the solution to theabove differential equation can be obtained composing the two flows; inparticular

u(t) =

u1(t) = +1 u2(t) = 0 t [0, )u1(t) = 0 u2(t) = +1 t [, 2)u1(t) = 1 u2(t) = 0 t [2, 3)u1(t) = 0 u2(t) = 1 t [3, 4)

where is an infinitesimal time duration. The solution x(t) at the finaltime t = 4 of the composition of flows is

x(t) = (g2

g1

g2

g1 )x(0)

= x(0) + 2(

g2x

g1x(0) g1x

g2x(0)

)

+ O(3)

B. Bona (DAUIN) Differential geometry July 2009 13 / 24

Hence, neglecting the higher order infinitesimal,

x(t) x(0) = [g1,g2]x(0)

Figure: A Lie bracket motion.

B. Bona (DAUIN) Differential geometry July 2009 14 / 24

An important conclusion derived from the previous analysis is that thesystem can achieve infinitesimal motions not only along the two field f,g,but also along the infinitesimal direction specified by the Lie bracket [f,g].

More complicated motions are possible along higher order Lie brackets,such as [f, [f,g]].

Usually the following notations are used

ad0f g = g

adfg = [f,g]

adkf g = [f, adk1f g], k 1

B. Bona (DAUIN) Differential geometry July 2009 15 / 24

Example

Let

f(q) =

[q2

sin q1 q2

]

; g(q) =

[0q1

]

Then (with si = sin qi , ci = cos qi )

[f, g] = Jgf Jfg =

[0 01 0

] [q2

s1 q2

]

[0 1

c1 1

] [0q1

]

=

[q1

q1 + q2

]

[f, [f, g]] =

[1 01 1

] [q2

s1 q2

]

[0 1

c1 1

] [q1

q1 + q2

]

=

[q1 q2

q1 + q2 s1 c1q1

]

B. Bona (DAUIN) Differential geometry July 2009 16 / 24

Dynamic systems

The dynamic system

x(t) = g1(x)u1(t) + g2(x)u2(t)

is said to be a driftless system, since, as long as the control inputs remainzero, the state does not vary.

A system described by a drift vector field is, for example, the following

x(t) = f(x) + g1(x)u1(t) + g2(x)u2(t)

B. Bona (DAUIN) Differential geometry July 2009 17 / 24

Example

Assuming the SISO linear time-invariant dynamical system, given by thefollowing matrix differential equation

x(t) = Ax(t) + bu(t)

where f(x) = Ax is the drift vector field, and g(x) = b is the input vectorfield, we can compute the various Lie brackets

[f,g] =g

xf

f

xg =

b

xAx

0

Ax

xb

Ab

= Ab = [g, f]

[f, [f,g]] = [Ax,Ab] =(Ab)

xAx

0

+Ax

xAb

A2b

= A2b

[f, [f, [f,g]]] = A3b etc.

that corresponds to the columns of the controllability matrix, i.e., thedirections along which it is possible to move the system.

B. Bona (DAUIN) Differential geometry July 2009 18 / 24

Distributions

It is convenient to think about a vector field f as a vector in Rn

f(x) =[f1(x) fn(x)

]T

If (x) is a smooth function Rn 7 R, then (x)f(x) is also a vector fieldand the sum of two vectors field is also a vector field. Thus the vector fieldis a module over the ring of smooth functions defined on U Rn, denotedby C(U).

A module is a linear space for which the scalars are elements of a ring.Also, the space of vector fields is a vector space over the field of reals.

Now we have this definition: given a set of smooth vector fieldsf1(x), . . . , fm(x), we define the distribution (x) to be

(x) = span {f1(x), , fm(x)}

i.e., the elements of at the point x are of the form

1(x)f1(x) + 2(x)f2(x) + + m(x)fm(x)

B. Bona (DAUIN) Differential geometry July 2009 19 / 24