geometric aspects of robotics and controlbanavar/notes/sc624_ls_geometry2.pdf · lecture slides -...

GEOMETRIC ASPECTS OF ROBOTICS AND CONTROL

(Slides for SC 624)

(Jan 2006)

R. N. BanavarProfessor, Systems and Control Engineering,

I. I. T. Bombay

([email protected])

Lecture Slides - SC 624

SC 624

Jan 2005

Outline

• Week 1 - Topology, differentiable manifolds, smooth functions

• Week 2 - Group actions, orbits, left/right invariant vector fields, fibre

bundles

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SC 624

Jan 2005

IRn, open and closed sets

• IRn - open ball of radius r about p

Ur(p) = x ∈ IRn : ‖x − p‖ < r

• A subset U of IRn is open if for each p ∈ U there exists an r such that

Ur(p) ⊂ U

• Open subsets of IRn have the following properties

– The empty set and IRn are both open

– The union of any arbitrary collection of open sets is open

– The intersection of a finite collection of open sets is open

• A subset C of IRn is said to be closed if its complement IRn − C is

open.

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• An open rectangle in IRn is a subset of the form

(a1, b1) × ... × (an, bn)

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An arbitrary set and a topology

• For an arbitrary, nonempty set X , a topology for X is a collection τ

of subsets of X that has the following properties

– The null set and X belong to τ

– The union of any arbitrary collection of τ belongs to τ

– The intersection of any finite collection of τ belongs to τ

• (X, τ ) is called a topological space

• Elements of τ are called the open sets of the topology

• A subset C of X is said to be closed if its complement X −C is open.

• A map f : X → Y of X into Y is said to be continuous if f−1(U)

(set-theoretic) is open in X whenever U is open in Y .

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• A continuous, one-to-one map h of X onto Y for which h−1 : Y → X

is also continuous is called a homeomorphism

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Examples of topologies

• Topology on Rn as open balls (seen before) - standard topology

• Discrete topology - τ is all subsets of X

• Trivial topology - τ = X, φ

• Subspace topology -Y is a subset of a toplogical space (X, τ and

the collection

τY = Y⋂

U |U ∈ τWith this topology, Y is called a subspace of X .

• Limit points - If A is a subset of the topological space X , and x is a

point of X such that every neighbourhood of x intersects A in some

point other than x itself.

• A topological space X is called a Hausdorff space if for each pair

x1, x2 of distinct points of X , there exist neighbourhoods U1 and U2 of

x1 and x2, respectively, that are disjoint.

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Jan 2005Features of a set arising from topology

• Compactness - A topological space X is compact if, whenever

Uαα∈A is a family of open sets whose union is X , then there are

finitely many of the Uαs whose union is X .

• Connectedness - A space X is connected if it cannot be expressed

as the union of two disjoint open subsets of X.

1. All open or closed rectangles in IRn are connected.

2. The torus T and the Mobius strip M , being continuous images of

closed rectangles in IR2, are connected.

• Fixed point If X is an arbitrary space and f : X → X is a map,

then a point x0 ∈ X for which f (x0) = x0 is called a fixed point of f .

If a space X is such that every continuous map of X into itself has a

fixed point, then X is said to have the fixed point property.

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Jan 2005

Manifolds

• The study of differential geometry in our context is motivated by the

need to study dynamical systems that evolve on spaces other than the

usual Euclidean space.

• Single pendulum. double pendulum

• Charts and Atlases

• Definition 1.1 A manifold is a topological space M with the

following property. For any x ∈ M , there exists a neighbourhood B

of x which is homeomorphic to Rn (for some fixed n ≥ 0)

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Charts and Atlases

• We start with a topological space X and a positive integer n

• An n-dimensional chart on X is a pair (U, φ) where U is open and

φ is a homeomorphism

• X is said to be locally Euclidean if there exists a positive integer n

such that for each x ∈ X , there is an n-dimensional chart (U, φ) on X

with x ∈ U

• Two charts (U1, φ1) and (U2, φ2) and U1

⋂U2 = 0 then φ1 φ−1

2 and

φ1 φ−12 are homeomorphisms

• If both are C∞ then the two charts are C∞ -related

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• An atlas of dimension n on X is a collection (Uα, φα)α∈A of

n-dimensional charts on X , any two of which are C∞-related and such

that⋃

α∈A Uα = X

• A maximal atlas contains every chart on X that is admissible to it - or

whenever (U, φ) is a chart on X that is C∞-related to every

(Uα, φα), α ∈ A, then there exists an α0 ∈ A such that U = Uα0 and

φ = φα0

• Example: The charts (US, φS) and (UN, φN) together form an atlas

on S1.

• Consider the standard atlas on R. Now consider the chart

((−π/2, π/2), tan). Is this admissible ? How about (R, φ) where

φ(x) = x3 ?

• A maximal n-dimensional atlas for a topological manifold X is called a

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differentiable structure on X and a topological manifold together

with some differntiable structure is called a differentiable or

smooth or C∞ manifold.

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On C∞ functions

Lemma 1.1 Any closed set in IRn is the set of zeros of some

non-negative C∞ function on IRn i. e. if A ⊂ IRn is closed, then there

exists a C∞ function f : IRn → IR with f (x) ≥ 0 for all x ∈ IRn and

A = f−1(0)

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Stereographic projections

• A circle S1 - subset of R2

• Consider US = S1 − N and UN = S1 − S

• Map φs : US → R (homeomorphism ?)

φs(x1, x2) =

x1

1 − x2φ−1

s (y) = (2y

y2 + 1,y2 − 1

y2 + 1)

• φN : UN → R

φN(x1, x2) =x1

1 + x2φ−1

N (y) = (2y

y2 + 1,−y2 + 1

y2 + 1)

• UN

⋂US = S1 − N, S

• φS φ−1N (y) = y−1 = φN φ−1

S (y)

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Jan 2005

A Sphere

•

• A sphere S2 - subset of R3

• Consider US = S2 − N and UN = S2 − S

• Map φs : US → R (homeomorphism ?)

φs(x1, x2, x3) = (

x1

1 − x3,

x2

1 − x3) φN(x1, x2, x3) = (

x1

1 + x3,

x2

1 + x3)

• φ−1s : R2 → US

φ−1s (y) =

1

(1 + ‖y‖2)(2y1, 2y2, ‖y‖2 − 1)

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Jan 2005

Coordinate systems and smooth maps

• Charts assign local coordinates to the manifold

• Projection onto the ith coordinate P i : Rn → R

• Coordinate function P i φ : U → R

• F : X → R is C∞ on X , if, for every chart (U, φ) in the differentiable

structure for X , the coordinate expression F φ−1 is a C∞ real- valued

function on the open subset φ(U) of Rn.

• A bijection F : X → Y for which both F and F−1 : Y → X are C∞ is

called a diffeomorphism

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Tangent vectors and spaces

• A tangent vector v to a point p on a surface will assign to every

smooth real-valued function f on the surface a “directional derivative”

v(f ) = f (p) · v.

• Draw a curve α(.) : M = R → R on the plane. Let α ∈ C∞. Considerdα(t)

dt |t=p. What is this expression ? It takes a curve α(∈ C∞) to the

reals at a point p.

• A tangent vector (derivation) to a differentiable manifold X at a

point p is a real-valued function v: C∞ → R that satisfies

– (Linearity) v(af + bg) = a v(f) + b v(g)

– (Leibnitz Product Rule) v(fg) = f (p)v(g) + v(f )g(p) for all

f, g ∈ C∞(X) and all a, b ∈ R

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• The set of all tangent vectors to X at p is called the tangent space

to X at p and denoted Tp(X).

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More on tangent vectors

Lemma 1.2 Let v be a tangent vector to X at p and suppose that f

and g are elements of C∞(X) that agree on some neighbourhood of p in

X. Then v(f ) = v(g)

Lemma 1.3 If X is an n-dimensional C∞ manifold and p ∈ X, then

the dimension of the vector space Tp(X) is n.

Lemma 1.4 Let X be a C∞ manifold, p ∈ X and v ∈ Tp(X). Then

there exists a smooth curve α ∈ X, defined on some interval about 0 in

IR, such that α′(0) = v.

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Jan 2005

Maps between manifolds and ”the linear approximation”

Consider a smooth map f : X → Y . At each p ∈ X we define a linear

transformation

f∗p : Tp(X) → Tf(p)(Y )

called the derivative of f at p, which is intended to serve as a ”linear

approximation to f near p,” defined as

• For each v ∈ Tp(X) we define f∗p(v) to be the operator on C∞(f (p))

defined by (f∗p(v))(g) = v(g f ) for all g ∈ C∞(f (p)).

Lemma 1.5 (Chain Rule) Let f : X → Y and g : Y → Z be smooth

maps between differentiable manifolds. Then g f : X → Z is smooth

and for every p ∈ X

(g f )∗p = g∗f(p) f∗p

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The cotangent space

Definition 1.2 For any f ∈ C∞(p) we define an operator

df(p) = dfp : Tp(X) → IR called the differential of f at p by

df(p)(v) = dfp(v) = v(f )

for every v ∈ Tp(X). Since dfp is linear, it is an element of the dual

space of Tp(X) called the cotangent space of X at p and denoted by

T ∗p (X)

• basis - ∂∂xi and dual basis dxi

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Submanifolds

Definition 1.3 A subset X ′ of an n-dimensional smooth manifold X

will be called a k-dimensional submanifold of X if for for each

p ∈ X ′ there exists a chart (U, φ) of X at p such that φ|U⋂

X ′ onto an

open set in some copy of IRk in IRn.

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Jan 2005

Vector fields

• A vector field on a smooth manifold X is a map V that assigns to

each p ∈ X a tangent vector V(p) in Tp(X).

• If this assignment is smooth, the vector field is called smooth or C∞.

• The collection of all C∞ vector fields on a manifold X denoted by

X (X) is endowed with an algebraic structure as follows: Let

V, W ∈ X (X), a ∈ R and f ∈ C∞(X)

– V + W (p) = V(p) + W(p) (∈ X (X))

– a(V)(p) = aV(p) (∈ X (X))

– (fV)(p) = f (p)V(p) (∈ X (X))

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Tensors and tensor products

• Vector space V and the dual space V∗

• T 2(V) - space of bilinear forms on V - covariant tensors of rank 2

• Symmetric - A(v, w) = A(w, v) or skew-symmetric -

A(v, w) = −A(w, v)

• Inner product on V - symmetric, covariant tensor g of rank 2 that is

– positive definite

• Tensor product of α and β (∈ V∗) - α ⊗ β

(α ⊗ β)(v, w) = α(v)β(w)

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Jan 2005

Wedge Product

• Symmetric product - 12(α ⊗ β + β ⊗ α)

• Skew-symmetric product - 12(α ⊗ β − β ⊗ α)

• Wedge product - α ∧ β = (α ⊗ β − β ⊗ α)

• Λ2(V) - skew-symmetric forms - is a linear subspace of T 2(V)

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Jan 2005

Covariant tensor fields and Riemannian metric

• A covariant tensor field of rank 2 on a smooth manifold X is a map A

that assigns to each p ∈ X an element A(p) ∈ T 2(Tp(X))

• An element g of T 2(X) which at each p ∈ X is symmetric,

nondegenerate and positive definite is called a Riemannian metric on

X .

• An element Ω of T 2(X) which at each p ∈ X is skew-symmetric is

called a 2-form on X .

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Jan 2005

Equivalent notions

• One-form

Smooth assignment of covectors -C∞(X)-module homomorphism

from X (X) to C∞(X).

–• Covariant tensor field of rank 2

– C∞(X) bilinear map from X (X) ×X (X) to C∞(X).

• Manufacturing 2-forms from 1-forms. Leads to definition of a d

operator and exterior calculus.

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Jan 2005

Exterior calculus

dΘ(V,W) = V(ΘW) − W(ΘV) − Θ([V,W])

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Jan 2005

Group Actions and Orbits

• Let M be a manifold and let G be a Lie group. A left action of a Lie

group G on M is a smooth mapping Φ : G × M → M such that

– Φ(e, x) = x for all x ∈ M

– Φ(g, φ(h, x)) = Φ(gh, x) for all g, h ∈ G and x ∈ M

– Φ(g, ·) is a diffeomorphism for each g ∈ G

• Given a group action G on M ,for a given point x ∈ M , the set

Orbx = gx|g ∈ G

is called the group orbit through x.

Example: Action of SO(3) on IR3

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Group Actions and Orbits

• Consider the projection of S3 onto S2 in the following manner

• Characterize any element of S3 as a complex 2-tuple (z1, z2) ∈ S3

where z1, z2 ∈ C and |z1|2 + |z2|2 = 1

• Parametrize as - z1 = cos(φ/2)eiθ1, z2 = sin(φ/2)eiθ2

• Define a left-action of U(1) = G on S3 as

g(z1, z2) = (gz1, gz2)

• The group orbit for (cos(φ/2)eiθ1, sin(φ/2)eiθ2) is

(cos(φ/2)ei(θ1+α), sin(φ/2)ei(θ2+α)) : α ∈ IR

• Now we project this group orbit to S2. We associate a single element of

S2 with this orbit

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Orbits and Projections

• Notice that any element (z1, z2) in the orbit satisfies

z1

z2= tan(φ/2)ei(θ1−θ2)

• Question: Can we associate an element of S2 with this orbit - the

characteristic feature of the orbit is the ratio (a complex number C) ?

• Recall the stereographic projection of S2 on the plane IR2.

• φ−1s : IR2 → US

φ−1s (z) =

1

(1 + ‖z‖2)(2x, 2y, ‖z‖2 − 1)

where z = (x, y) ∈ IR2.

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Orbits and Fibers

• P(z1, z2) = φ−1s (z1

z2)

• P(φ, (θ1 − θ2)) = (sin φ cos(θ1 − θ2), sin φ sin(θ1 − θ2), cos φ)

• The fiber P−1(x) = g.(z1, z2) : g ∈ U(1),P(z1, z2) = x

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Principal Bundle

Definition: Let X be a manifold and G a Lie group. A C∞ (smooth)

principal bundle over X with structure group G consists of a manifold P ,

a smooth map P : P → X and a smooth right action (p, g) → p.g of G

on P which satisfies

• The action of G on P preserves the fibers of P

P(p.g) = P(p) ∀p ∈ P and ∀g ∈ G

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• (Local triviality): For each x0 ∈ X there exists an open set V

containing x0 and a diffeomorphism Ψ : P−1(V ) → V × G of the form

Ψ(p) = (P(p), ξ(p))

where ξ : P−1(V ) → G satisfies

ξ(p.g) = ξ(p)g ∀p ∈ P−1(V ) and ∀g ∈ G

P is called the bundle space, X the base space and P the projection of

the bundle

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Infinitesimal Generators

• Suppose Φ : G × M → M is an action. For ξ ∈ G, φξ : IR × M → M

defined by Φξ(t, x) = Φ(exp(tξ), x) is an IR-action on M . The vector

field on M defined by

ξM(x) =d

dt|t=0Φ(exp(tξ), x)

is called the infinitesimal generator of the action corresponding to ξ.

• Consider the Lie algebra G of n × n real matrices. The IR-action on M

(the group IRn) is defined as

Φξ(t, x) = etξx

and the infinitesimal generator is the vector field on IRn

ξIR(x) =d

dt|t=0e

tξx = ξx Recall x = Ax of linear systems ?

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Jan 2005

Left (Right) Invariant Vector Fields

• A Lie group G acts on its tangent bundle (TG considered as the

manifold) by the tangent map.

Lh : TG → TG (g, vg) → (hg, TgLh(vg))

• Given ξ ∈ G, the left (right) action of G on ξ is TeLgξ (or TeRgξ)

• For a left (right) invariant vector field X

(ThLg)X(h) = X(gh) (ThRg)X(h) = X(hg)

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