SC 618: An introduction to manifoldsbanavar/resources/SC618/sc618_manifold… · SC 618: An...

Motivation Mathematical preliminaries Submanifolds Optional

SC 618: An introduction to manifolds

Ravi N [email protected] 1

1Systems and Control Engineering,IIT Bombay, India

Geometric MechanicsMonsoon 2014

September 23, 2014

Smooth manifolds SC 618


Outline

1 Motivation

2 Mathematical preliminaries

3 Submanifolds of Rn

4 Optional material



Engineering applications

• Articulated mechanisms, rolling mechanisms, robotic manipulators andunderwater vehicles - configurations spaces are not necessarily the realline OR copies of the real line OR Euclidean space.

• Issues: Develop kinematic level and dynamic level modellingframework in non-Euclidean settings. Backdrop for path planning,feedback control and stabilization.

• Other reasons: Interconnected mechanisms and more complexmechanisms- satellite with tethers, mobile robots with internal drivingmechanisms - need for a better theoretical framework.

• All these issues motivate better mathematical tools. We explore thegeometric framework OR differential geometry



A few problems

• The acrobot

Figure: Two link manipulator



Wheeled mobile robots

• Wheeled mobile robots, fingers handling an object (rolling contact)

• Objective - move from one configuration to another

• Constraints - pure rolling (no sliding or slipping)

x

y

θ

φ

Figure: Vertical coin on a plane



Underwater vehicles

• Kinematics

x = vx cos θ − vy sin θ

y = vx sin θ + vy cos θ

θ = ωz

• Dynamics

m11vx −m22vyωz + d11vx = Fxm22vy +m11vxωz + d22vy = 0

m33ωz + (m22 −m11)vxvy + d33ωz = τz



Classification of constraints in mechanical systems

Holonomic constraintsRestrict the allowable configurations of the system

Nonintegrable constraints

Do not restrict the allowable configurations of the system but restrictinstantaneous velocities/accelerations

Velocity level constraints - parking of a car, wheeled mobile robots, rollingcontacts in robotic applications.Acceleration level - fuel slosh in spacecrafts/launch vehicles, underwatervehicles, underactuated mechanisms (on purpose or loss of actuator)systems - serial link manipulators.






Do not restrict the allowable configurations of the system but restrictinstantaneous velocities/accelerationsVelocity level constraints - parking of a car, wheeled mobile robots, rollingcontacts in robotic applications.

Acceleration level - fuel slosh in spacecrafts/launch vehicles, underwatervehicles, underactuated mechanisms (on purpose or loss of actuator)systems - serial link manipulators.






Do not restrict the allowable configurations of the system but restrictinstantaneous velocities/accelerationsVelocity level constraints - parking of a car, wheeled mobile robots, rollingcontacts in robotic applications.Acceleration level - fuel slosh in spacecrafts/launch vehicles, underwatervehicles, underactuated mechanisms (on purpose or loss of actuator)systems - serial link manipulators.



Velocity level constraints

• Rolling coin : Configuration variables

qqq = (x, y, θ, φ) ∈M = R1 × R1 × S1 × S1

x

y

θ

φ

Figure: Rolling coin

• Constraints - No slip and no sliding




• Constraints of motion are expressed as

x sin θ − y cos θ = 0 No lateral motion

x cos θ + y sin θ = rφ Pure rolling

• Constraints in a matrix form

[sin(θ) − cos(θ) 0 0cos θ sin θ 0 −r

]xy

θ

φ

= 0



Bead in a slot

Figure: A bead in a slot

• Configuration variables of the bead are (x, y)

• Restriction to stay in the slot gives rise to an algebraic constraint

g(x, y) = 0

• g is the equation of the curve describing the slot

• Slot restricts the possible configurations that the bead can assume



Acceleration level constraints

Figure: A two-link manipulator

• Two link manipulator moving in a vertical plane with one actuator atits second joint (acrobot)



Acrobot

•

d11(q)q1 + d12(q)q2 + h1(q, q) + ψ1(q) = 0

d21(q)q1 + d22(q)q2 + h2(q, q) + ψ2(q) = τ2 (1)

• First equation (with the right hand side being zero) denotes the lack ofactuation at the first joint

• The acrobot can assume any configuration but cannot assumearbitrary accelerations.



Fuel slosh in a launch vehicle

• A launch vehicle with liquid fuel in its tank. The motion of the fluidand the outer rigid body are coupled.

• Unactuated pivoted pendulum model. The motion of the pendulum issolely affected by the motion of the outer rigid body

!

"

fuel tank

Figure: Fuel slosh phenomenon



• The equations of motion are of the form

qu + f1(q, q, q) = 0

qa + f2(q, q, q) = F (2)

where q4= (qu, qa), qu corresponds to the configuration variable of the

pendulum, qa corresponds to the configuration variable of the outerrigid body and F is the external force.

• While the acceleration level constraint in the acrobot arises due topurpose of design or loss of actuation, in the case of the launch vehicleit is the inability to directly actuate the fluid dynamics.



Mechanics, geometry and control

The thumb experiment

Start

Finish

Figure: Motion on a sphere (Courtesy: Marsden and Ostrowski)



Mechanics, geometry and control

• Phenomenon similar to the thumb experiment is also common to alarge class of systems which are termed blue.

• Variables describing the motion of these systems can be classified intotwo sets called the blueshape variables and the group variables.

• Cyclic motion in the shape variables produces motion in the group (fiber) variables. Biological systems (fishes, snakes, paramecia) androbotic mechanisms, the falling cat, the steering of a car, the motion ofunderactuated systems of linkages.

• The net changes in position due to changes in the shape variables isexplicable either due to an interaction with the environment or someconservation law.



Geometry

• Provides better insight, better framework for solution (control law) andan elegant setting for these problems

• Essential tools - Lie groups and differentiable manifolds



Dynamical systems and geometry

θ

S1

θ1

θ2

S1 × S1

1

Figure: Single pendulum (circle) and double pendulum (torus)



A glimpse of differential geometry

• Familiar with calculus (differentiation, integration) on the real line (R)or Rn. Extend this calculus to surfaces - say on a circle, a sphere, atorus and more complicated surfaces

• But why ? From a dynamical systems and control theory point ofview, systems need not always evolve on the real line (R) or multiplereal lines (Rn). They may evolve on non-Euclidean surfaces. On suchsurfaces, we wish to talk of ”rate of change (velocity) ⇒ differentiation,to talk of ”rate of rate of change (acceleration), to talk of accumulation⇒ integration

• Can these surfaces which are not Euclidean be locally represented asEuclidean ? This would allow us to employ the calculus that we arefamiliar with for these surfaces



Outline

1 Motivation



4 Optional material



Level Sets in Rn+1

• Consider a smooth function f : Rn+1 → R and consider the set

S = {x ∈ Rn+1 : x = f−1(c)} for some fixed c ∈ R

• Examples: a circle - x21 + x22 = c(c 6= 0)is a 1-surface ∈ R2, a spherex21 + x22 + x23 = c(c 6= 0) is a 2-surface ∈ R3

• The gradient of f is defined as

5f(·) : Rn+1 → Rn+1

• We restrict our study to those level sets such that

5f(x) 6= 0 ∀x ∈ S

We shall also call these level sets as n-surfaces.



Vector field

• A vector field X on U ⊂ Rn+1 is an assignment of a vector at eachpoint of U

• Our interest centres on smooth vector fields - here the assignment is interms of a smooth function

• Gradient vector field - It is the smooth vector field associated with eachsmooth function f : U → Rn+1, and is given by

(∇f)(p) = (p,∂

∂x1(p), . . . ,

∂

∂xn+1(p))



A parametrized curve in Rn+1 and an integral curve

• A parametrized curve in Rn+1 is smooth mapping from an openinterval of the real-line (R) to Rn+1

• α(·) : (a, b)→ Rn+1 is a parametrized curve implies that each of theαi(·)s (i = 1, . . . , n+ 1) is a smooth function on (a, b)

• Associated with a parametrized curve is the notion of a velocity vector.At any point p on the parametrized curve, the associated velocityvector is

vα(p)4= α(t)|t=p

• A parametrized curve α(·) on an open interval (α, β) is said to be theintegral curve of a smooth vector field X if

vα(p) = X(p) ∀p ∈ (a, b)



Integral curves and solutions of differential equations

• In engineering we are often interested in solving a system of equationsof the following form

x1 = f1(x1, . . . , xn+1)

...

xn+1 = fn+1(x1, . . . , xn+1)

where the fis are smooth functions on an open interval (a, b) and withan initial condition as x(p) = (x1(p), . . . , xn+1(p)). Let us now try tounderstand what does it mean in a geometric sense to seek a solutionof this equation.

• A solution to this set of equations describes an integral curve of thevector field f(·).



Parametrized curves on a level set

• Consider a smooth curve α(·) : I → S where 0 ∈ I ⊂ R, S is a level setof a smooth function f and let α(0) = p. Then

vp4=d(f ◦ α)(t)

dt|t=0

is a tangent vector to the level set S at p.

• The set of all tangent vectors at a point p to a level set S forms avector space and is denoted by Sp

• It can be shown that the Sp is orthogonal to 5f(p)



Tangent vector field

• A tangent vector field on a surface S is the assignment of a tangentvector at each point p ∈ S. If this assignment is smooth, we have asmooth tangent vector field on the surface

• Associated with a surface S are two smooth unit normal vector fieldsdefined as

5f(p)

‖5f(p)‖ and − 5f(p)

‖5f(p)‖



Smooth functions: f : X ⊂ Rm → Y ⊂ Rn

• Differentiable (or smooth functions) help us to characterize thederivative maps.

• From linear algebra, recall that the rank of an n×m matrix A isdefined in 3 equivalent ways:

1 The dimension of the subspace V (⊂ Rm) spanned by the n rows of A.2 The dimension of the subspace W (⊂ Rn) spanned by the m columns of

A.3 The maximum order of any non vanishing minor determinant.

• The rank of Df(x) (the the n×m Jacobian) is referred to the rank off at x. And since f is smooth, and the value of the determinant is acontinuous function of its entries, the rank remains constant in someneighborhood of x.

• Diffeomorphisms have non-singular Jacobians.

• So rank of f is the same as the rank of h ◦ f , where h is adiffeomorphism.



Submersion

Let f : X ⊂ Rm → Y ⊂ Rn be smooth. Then

• If the Jacobian Df(p) is onto for all p ∈ X, then f is called asubmersion. (this definition makes sense only when n ≤ m.)

• Note that Jacobian maps vectors from Rm to Rn, and in coordinates isexpressed as

[Df(p)] =

∂f1∂x1

. . . ∂f1∂xm

......

...∂fn∂x1

. . . ∂fn∂xm

x=p



Immersion

Let f : X ⊂ Rm → Y ⊂ Rn be smooth, Then

• If the Jacobian Df(p) is one-to-one (injective) for all p ∈ X, then f iscalled a immersion. (this definition makes sense only in the case whenn ≥ m.)

• Once again the Jacobian maps vectors from Rm to Rn, and incoordinates is expressed as

[Df(p)] =

∂f1∂x1

. . . ∂f1∂xp

......

...∂fn∂x1

. . . ∂fn∂xm

x=p



Outline

1 Motivation



4 Optional material



A submanifold of Rn

Three equivalent definitions

Each of these definitions is equivalent to the other and makes M(⊂ Rn) anm-dimensional submanifold of Rn.

• For every x ∈M ⊂ Rn, there exists a neighbourhoood U containing xand a smooth submersion f : U→Rn−m such that U ∩M is a level setof f .

• For every x ∈M ⊂ Rn, there exists a neighbourhoood U containing xand a smooth function g : V ⊂ Rm→Rn−m such that U ∩M is agraph of g.

• For every x ∈M ⊂ Rn, there exists a neighbourhoood U containing xand an smooth embedding ψ : V ⊂ Rm→Rn such that ψ(V ) = M ∩ U .

Examples seen : S1, O(2), SO(3), O(n), Sn



S1

The unit circle - S1

• Consider the function f(x, y) = x2 + y2 : U→R1. Note that f is asmooth submersion on U and U ∩ S1 is a level set of f .

• Consider the function g(x) =√

1− x2 and note that U ∩ S1 = (x, g(x))- the graph of g.

• Consider the function ψ(x) = (x,√

1− x2) in a neighbourhoodV (⊂ R1) which satisfies U ∩ S1 = ψ(V ), is an immersion andψ−1(a, b) = a is continuous in U ∩ S1.



O(2)

2× 2 real, orthogonal matrices - O(2)

O(2) = {A ∈ R2×2|ATA = I}

• Parameterize A as A =

[a bc d

]with

a2 + c2 = 1, ab+ cd,= 0, b2 + d2 = 1.

• Let f4= (f1, f2, f3) where

f1(·) = a2 + c2, f2(·) = ab+ cd, f3(·) = b2 + d2, U→R1

This is a smooth submersion and U ∩O(2) is a level set of f .

U ∩O(2) = f−1(1, 0, 1)



O(2) (contd.)

• The function f1 allows us to characterize the 4-tuple as (a, b, c, g1(·)).Similarly, the next two functions further reduce this characterization to

the form (a, g3(·), g2(·), g1(·)). Consider the function g(x)4= (g1, g2, g3)

and note that the set U ∩O(2) = (x, g(x)) - the graph of g.

• Consider the function ψ(x) = (x, g(x)) in a neighbourhood V (⊂ R1)which satisfies U ∩O(2) = ψ(V ), is an immersion andψ−1(a, b, c, d) = a is continuous in U ∩O(2).



Immersed submanifold

Let f : X ⊂ Rm → Y ⊂ Rn be smooth, Then

If the Jacobian Df(p) is one-to-one (injective) for all p ∈ X, and f isone-to-one on X then f(X) is called an immersed submanifold. (onceagain, this definition makes sense only in the case when n ≥ m.)



Examples

• Helix f : R→R3, f(t) = (cos(2πt), sin(2πt), t).

• Circle f : R→R2, f(t) = (cos(2πt), sin(2πt)).

• Spiralling curve f : (1,∞)→R2, f(t) = ( 1t

cos(2πt), 1t

sin(2πt)).

• Spirals to a circle f : (1,∞)→R2, f(t) = ( t+12t

cos(2πt), t+12t

sin(2πt)).

• A sleeping figure eightf : (1,∞)→R2, f(t) = (2 cos(t− 0.5π), sin 2(t− 0.5π)).



Coordinates for a submanifold

• Choosing a basis for Rm, one could assign coordinates to these n-tuples

• So we introduce the notion of coordinate functions x4= (x1, . . . , xm).

• The ith coordinate of a point p on the manifold is xi(p) : M → R

• Coordinate function xi : U → R

• We could choose another set of coordinates as well - say y



Change of coordinates

• How do we move from one coordinate system to another ?

• Consider a smooth function f(·) : M → R and consider two coordinatesystems x and y.

f ◦ x−1 = f ◦ y−1 ◦ y ◦ x−1

• Then the partial derivative of f(·) with respect to xj (the jth partial inthe x coordinate system) is given by

Dj(f ◦ x−1) = Dj(f ◦ y−1 ◦ y ◦ x−1) = Dk(f ◦ y−1) ·Dkj (y ◦ x−1)

• Note

f ◦ y−1 : Rn → R y ◦ x−1 : Rn → Rn



Example 1: R2

• Think of the Cartesian plane and two coordinate systems (x, y) and(r, θ).

• Now x = r cos θ and y = r sin θ

• Compute ∂f∂x

and ∂f∂y

with the chain rule mentioned in the last slide.



Example 2: Sphere -S2

Sphere

Choose spherical coordinates as (θ, φ) ∈ ((−π, π), (0, 2π)) or

Cartesian as(x, y, z = ±

√1− (x2 + y2) = (cos(θ) sin(φ), cos(θ) cos(φ), sin(θ)).



Example 3: Cylinder - S1 × R1

Cylinder

Choose cylindrical coordinates as (θ, z) ∈ ((−π, π),R) or

Cartesian as (x,±√

1− x2, z) = (cos(θ), sin(θ), z).



Tangent (velocity)vectors

• The two tuple (p, vp) constitutes a tangent vector at p ∈M , wherec : R ⊃ (−a, a)→M is any curve that satisfies

c(0) = p vp = c′(0)

• The set of all tangent vectors at p is called the tangent space at p.This is a vector space and is denoted by TpM .

• For an m-dimensional manifold M , the tangent space at each point hasdimension m.



Coordinates for the tangent space

Coordinates for M ∩ ULet x1, . . . , xm be coordinates for M ∩ U .

Then draw curves of the form

αi : I→M ∩ U αi(t) = (p1, . . . , pi + t, . . . , pm) i = 1, . . . ,m

Coordinates for TpM

Now define∂

∂xi|p4=dαidt|t=0 = (0, . . . , 1, . . . , 0)

∂∂xi|p defines a tangent vector at αi(0) = p.

The set of tangent vectors

{ ∂

∂x1|p, . . . ,

∂

∂xm|p}

forms a basis for the tangent space TpM at p.



Coordinates for the tangent space

Coordinates for M ∩ ULet x1, . . . , xm be coordinates for M ∩ U .Then draw curves of the form

αi : I→M ∩ U αi(t) = (p1, . . . , pi + t, . . . , pm) i = 1, . . . ,m

Coordinates for TpM

Now define∂

∂xi|p4=dαidt|t=0 = (0, . . . , 1, . . . , 0)

∂∂xi|p defines a tangent vector at αi(0) = p.

The set of tangent vectors

{ ∂

∂x1|p, . . . ,

∂

∂xm|p}

forms a basis for the tangent space TpM at p.



Tangent bundle

• The disjoint union of all tangent spaces to a manifold forms thetangent bundle.

TM =⋃p∈X

TpM

• TM has a smooth manifold structure (an atlas for M induces an atlasfor TM .)

Optional

• Call π : TM →M the canonical projection defined by

vp → p ∀vp ∈ TpX

• If {(Uα, φα)} is an atlas for X, then {(π−1(Uα), Dφα)}is an atlas forTX.



Vector fields

• A vector field on a smooth manifold M is a map X that assigns to eachp ∈M a tangent vector X(p) in Tp(M).

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

Optional

• The collection of all C∞ vector fields on a manifold M denoted byX (M) is endowed with an algebraic structure as follows: LetV,W ∈ X (X), a ∈ R and f ∈ C∞(M)

• V +W(p) = V(p) +W(p) (∈ X (M))• a(V)(p) = aV(p) (∈ X (M))• (fV)(p) = f(p)V(p) (∈ X (M))



The cotangent space

DefinitionThe dual space of Tp(M) is called the cotangent space of X at p anddenoted by T ∗p (M)

DifferentialsFor any f ∈ C∞(p) we define an operator df(p) = dfp : Tp(M)→ R calledthe differential of f at p by

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

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

Basis and dual basisBased on coordinates x1, . . . , xm, the basis for the tangent space atp (TpM) is { ∂

∂xi} and the dual basis at p (T ∗pM) is {dxi}



Cotangent bundle

• The disjoint union of all cotangent spaces to a manifold forms thecotangent bundle.

T ∗M =⋃p∈X

T ∗pM

• T ∗M has a smooth manifold structure (an atlas for X induces an atlasfor T ∗M .)

Optional

• Call π : T ∗M →M the canonical projection defined by

vp → p ∀vp ∈ T ∗pM

• If {(Uα, φα)} is an atlas for M , then {(π−1(Uα), (Dφ)−1)}is an atlasfor T ∗M .



Casting a mechanical system in the manifold setting

• Rolling wheel or coin : Configuration variables evolve on a smoothmanifold

qqq = (x, y, θ, φ) ∈M = R1 × R1 × S1 × S1

x

y

θ

φ

Figure: Rolling coin

• Constraints - No slip and no sliding




• Constraints of motion are expressed as

x sin θ − y cos θ = 0 No lateral motion

x cos θ + y sin θ = rφ Pure rolling

• Constraints in a matrix form

[sin(θ) − cos(θ) 0 0cos θ sin θ 0 −r

]xy

θ

φ

= 0



Manifold language

• p = (x0, y0, θ0, φ0).[sin(θ0) − cos(θ0) 0 0

]∈ T ∗pM[

cos θ0 sin θ0 0 −r]∈ T ∗pM

vxvyvθvφ

∈ TpM• This is just kinematics.

• The vector field describing the motion of the coin sits in TM .



Outline

1 Motivation



4 Optional material



Two important results

The inverse function theorem

TheoremGiven a function f : Rn → Rn that is continuously differentiable in a regionU around a point a ∈ Rn, and whose derivative is non-singular at a, thereexists an open set (or a neighbourhood) V containing f(a) in which thefunction f has an inverse f−1 which is differentiable and further thisinverse is given by

(Df−1)(y) = [Df(f−1(y))]−1 ∀y ∈ V



The implicit function theorem

TheoremGiven a functionf : Rn × Rm → Rm such that f(a, b) = 0 and f iscontinuously differentiable in a region U × V around (a, b), and further

Dij+nf(a, b) 1 ≤ i, j ≤ m is non-singular.

Then there exists a neighbourhood W ⊂ U containing a and a neighbourhoodP ⊂ V containing b, and a differentiable function g(·) : W → P such that

f(x, g(x)) = 0 ∀x ∈W



Derivations: an alternate viewpoint to tangent vectors

• A tangent vector v to a point p on a surface will assign to everysmooth real-valued function f on the surface a “directional derivative”v(f) = 5f(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 M at a pointp 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

• A vector field X on a differentiable manifold M is a linear operatorthat maps smooth functions to smooth functions. Mathematically

X : C∞(X)→ C∞(X)



Maps between manifolds and ”the linear approximation”

Consider a smooth map f : M → N . At each p ∈M we define a lineartransformation

f∗p : Tp(M)→ Tf(p)(N)

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

• For each v ∈ Tp(M) 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(Chain Rule) Let f : M → N and g : N → P be smooth maps betweendifferentiable manifolds. Then g ◦ f : M → P is smooth and for every p ∈ X

(g ◦ f)∗p = g∗f(p) ◦ f∗p



An alternate viewpoint of maps between manifolds

The derivative of f at p could be constructed as follows. Once againf(·) : M → N . (The notation f∗p is now changed to Tpf .)

• Choose a parametrized curve c(·) : (−ε, ε)→ X with c(0) = p andc(0) = vp.

• Construct the curve f ◦ c. Then define

Tpf · vp =d

dt|t=0(f ◦ c)(t)

• With coordinates, we have

Tx(p)(y ◦ f ◦ x−1) =∂

∂xj(y ◦ f ◦ x−1)|x(p) The Jacobian at x(p)

• The rank of f at p is the rank of the Jacobian matrix at x(p) and thisis independent of the choice of the charts



Critical points and regular points

Consider a smooth map f : M → N

• A point p ∈M is called a critical point of f if Tpf is not onto.

• A point p ∈M is called a regular point of f if Tpf is onto.

• A point y ∈ N is called a critical value if f−1(y) contains a criticalpoint. Otherwise, y is called a regular value of f .



Submersion theorem

TheoremIf f : Xn → Y k is a smooth map, and y ∈ f(X) ⊂ Y is a regular value of f ,then f−1(y) is a regular submanifold of X of dimension n− k.

• Example: f(x, y) = x2 + y2 − 1. Take f−1(0).

• Show that O(n) = {A ∈Mn(R)|ATA = I} is a submanifold of Mn(R).What is its dimension ?Hint: Consider a map f : Rn×n → Sn×n (symmetric matrices). Letf(A) = ATA and examine the value I.


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