Robotics, Geometry and Control - Differential Geometry · Robotics, Geometry and Control Geometry Introduction I The study of differential geometry in our context is motivated by

Robotics, Geometry and Control

Robotics, Geometry and Control - DifferentialGeometry

Ravi Banavar1

1Systems and Control EngineeringIIT Bombay

HYCON-EECI Graduate School - Spring 2008


Differentiable geometry

The material for these slides is largely taken from the texts

I Calculus on manifolds - Vol. 1, by M. Spivak, published by W. A.Benjamin, Inc., New York, 1965.

I Topology, geometry and gauge fields - Vol. 1 - by G. L. Naber,Springer, New York, 1998.

I Ordinary Differential Equations - V. I. Arnold, Springer

I Mechanics and Symmetry - J. E. Marsden and T. Ratiu, Springer,1994


Geometry

Introduction

I The study of differential geometry in our context is motivated bythe need to study dynamical systems that evolve on spaces otherthan the usual Euclidean space.

I Single pendulum. double pendulum

I DefinitionA manifold is a topological space M with the following property. Forany x ∈ M, there exists a neighbourhood B of x which ishomeomorphic to Rn (for some fixed n ≥ 0). (We shall need more -”smooth” manifolds)


Geometry

Charts and atlases

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

I An n-dimensional chart on X is a pair (U, φ) where U is openand φ is a homeomorphism

I X is said to be locally Euclidean if there exists a positive integern such that for each x ∈ X , there is an n-dimensional chart (U, φ)

on X with x ∈ U

I Two charts (U1, φ1) and (U2, φ2) and U1⋂

U2 6= 0 then φ1 ◦ φ−12

and φ1 ◦ φ−12 are homeomorphisms

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

I An atlas of dimension n on X is a collection {(Uα, φα)}α∈A ofn-dimensional charts on X , any two of which are C∞-related andsuch that

⋃α∈A Uα = X


Geometry

I 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

I Example: The charts (US, φS) and (UN , φN) together form anatlas on S1.

I Consider the standard atlas on R. Now consider the chart((−π/2, π/2), tan). Is this admissible ? How about (R, φ) whereφ(x) = x3 ?

I A maximal n-dimensional atlas for a topological manifold X iscalled a differentiable structure on X and a topologicalmanifold together with some differentiable structure is called adifferentiable or smooth or C∞ manifold.


Geometry

A circle

I A circle S1 - subset of R2

I Consider US = S1 − {N} and UN = S1 − {S}I Map φs : US → R (homeomorphism ?)

φs(x1, x2) =x1

1− x2 φ−1s (y) = (

2yy2 + 1

,y2 − 1y2 + 1

)

I φN : UN → R

φN(x1, x2) =x1

1 + x2 φ−1N (y) = (

2yy2 + 1

,−y2 + 1y2 + 1

)

I UN⋂

US = S1 − {N,S}I φS ◦ φ−1

N (y) = y−1 = φN ◦ φ−1S (y)


Geometry

A sphere

I

I A sphere S2 - subset of R3

I Consider US = S2 − {N} and UN = S2 − {S}I 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 )

I φ−1s : R2 → US

φ−1s (y) =

1

(1 + ‖y‖2)(2y1,2y2, ‖y‖2 − 1)


Geometry

R22

!2

!1 R12

S

N

x "

!

!

"

Figure: An atlas for a sphere and a torus


Geometry

Other examples of smooth manifolds

I Any nonempty open subset of IRn

I M(n) = {n × n real matrices} = Rn2

I GL(n) = {n × n real, invertible matrices}

I Sn - a sphere in n + 1 dimensional Euclidean space


Geometry

Coordinates

I Charts assign an n-tuple of reals to an element on the manifold

I Choosing a basis for IRn, one could assign coordinates to thesen-tuples

I So we introduce the notion of coordinate functions x 4= (x1, . . . , xn).

I The i th coordinate of a point p on the manifold is x i (φ(p)) : X → R

I Coordinate function x i ◦ φ : U → R

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

I Sometimes we skip φ and talk directly of x (the coordinate function)as a map from U to IRn.


Geometry

Change of coordinates - chain rule

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

I Consider a smooth function f (·) : X → IR and consider two coordinatesystems x and y .

f ◦ x−1 = f ◦ y−1 ◦ y ◦ x−1

I Then the partial derivative of f (·) with respect to x j (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)

I Note

f ◦ y−1 : IRn → IR y ◦ x−1 : IRn → IRn


Geometry

Change of coordinates - example

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

I Now x = r cos θ and y = r sin θ

I Compute ∂f∂x and ∂f

∂y with the chain rule mentioned in the last slide.


Geometry

A submanifold

DefinitionA subset X ′ of an n-dimensional smooth manifold X will be called ak-dimensional submanifold of X if for for each p ∈ X ′ there exists achart (U, φ) of X at p such that φ|U

⋂X ′ onto an open set is some

copy of IRk in IRn.

I Is the unit circle a submanifold of IR2 ?

I Is the set of real-symmetric n × n matrices a submanifold of IRn×n ?


Geometry

Differentiable functions

I F : X → R is C∞ on X , if, for every chart (U, φ) in the differentiablestructure for X , the coordinate expression F ◦ φ−1 is a C∞ real-valued function on the open subset φ(U) of Rn.

I A bijection F : X → Y for which both F and F−1 : Y → X are C∞ iscalled a diffeomorphism


Geometry

Tangent vectors

I Two curves c1 : IR ⊃ (−a,a)→ X and c2 : (−a,a) ⊃ IR → X in X arecalled equivalent at p if

c1(0) = c2(0) = p (φ ◦ c1)′(0) = (φ ◦ c2)′(0)

in some chart (U, φ).

I An equivalence class [c(·)] where c(0) = p is called a tangent vectorto X at p.

I The set of all tangent vectors at p is called the tangent space at p andis denoted by TpX .

I Identify TpX with Tφ(p)IRn.


Geometry

Tangent bundle

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

TX =⋃

p∈X

TpX

I TX has a smooth manifold structure (an atlas for X induces an atlasfor TX .)

I Call π : TX → X the canonical projection defined by

vp → p ∀vp ∈ TpX

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


Geometry

Vector fields

I A vector field on a smooth manifold X is a map V that assigns toeach p ∈ X a tangent vector V(p) in Tp(X ).

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

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

I V + W(p) = V(p) + W(p) (∈ X (X ))

I a(V)(p) = aV(p) (∈ X (X ))

I (fV)(p) = f (p)V(p) (∈ X (X ))


Geometry

An alternate viewpoint of tangent vectorsI 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 ) = 5f (p) · v.

I 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 thereals at a point p.

I A tangent vector (derivation) to a differentiable manifold X at apoint p is a real-valued function v: C∞ → R that satisfies

I (Linearity) v(af + bg) = a v(f) + b v(g)I (Leibnitz Product Rule) v(fg) = f (p)v(g) + v(f )g(p) for all

f , g ∈ C∞(X ) and all a, b ∈ R

I The set of all tangent vectors to X at p is called the tangent space toX at p and denoted Tp(X ).


Geometry

An alternate viewpoint of vector fields

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

X : C∞(X )→ C∞(X )


Geometry

Integral curve

DefinitionLet V be a smooth vector field on the manifold X . A smooth curveα : I → X in X is an integral curve for V if its velocity vector at eachpoint coincides with the vector assigned to that point

dαdt

= V(α(t))


Geometry

Maximal integral curve

TheoremLet V be a smooth vector field on the manifold X and p a point in X.Then there exists an interval (a(p),b(p)) ∈ IR and a smooth curveαp : (a(p),b(p))→ X such that

I 0 ∈ (a(p),b(p)) and αp(0) = p.

I αp is an integral curve for V.

I If (c,d) is an interval containing 0 and β : (c,d)→ X is an integralcurve for V with β(0) = p, then (c,d) ⊂ (a(p),b(p)) andβ = αp|(c,d). Thus, αp is called the maximal integral curve of Vthrough p at t = 0.


Geometry

Maps between manifolds and ”the linearapproximation”

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

f∗p : Tp(X )→ Tf (p)(Y )

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

I 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(Chain Rule) Let f : X → Y and g : Y → Z be smooth maps betweendifferentiable manifolds. Then g ◦ f : X → Z is smooth and for everyp ∈ X

(g ◦ f )∗p = g∗f (p) ◦ f∗p


Geometry

An alternate viewpoint of maps between manifoldsThe derivative of f at p could be constructed as follows. Once againf (·) : X → Y . (The notation f∗p is now changed to Tpf .)

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

I Construct the curve f ◦ c. Then define

Tpf · vp =ddt|t=0(f ◦ c)(t)

I With coordinates, we have

Tx(p)(y ◦ f ◦ x−1) =∂

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

I The rank of f at p is the rank of the Jacobian matrix at x(p) and this isindependent of the choice of the charts


Geometry

Immersions, submersions

I If f : X → Y is smooth, then if

1. If Tpf is onto for all p ∈ X , then f is called a submersion

2. If Tpf is one-to-one for all p ∈ X , then f is called a immersion

3. If f is an immersion and one-to-one, then f (X ) is an immersedsubmanifold

I Examples


Geometry

Critical points and regular points

Consider a smooth map f : X → Y

I A point p ∈ X is called a critical point of f if Tpf is not onto.

I A point p ∈ X is called a regular point of f if Tpf is onto.

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


Geometry

Submersion theorem

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

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

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


Geometry

The cotangent space

DefinitionFor any f ∈ C∞(p) we define an operator df (p) = dfp : Tp(X )→ IRcalled 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 dualspace of Tp(X ) called the cotangent space of X at p and denoted byT ∗p (X )

I basis - { ∂∂x i } and dual basis {dx i}


Geometry

Cotangent bundle

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

T ∗X =⋃

p∈X

T ∗p X

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

I Call π : T ∗X → X the canonical projection defined by

vp → p ∀vp ∈ T ∗p X

I If {(Uα, φα)} is an atlas for X , then {(π−1(Uα), )}is an atlas for T ∗X .


Geometry

One-forms

A one -form on a smooth manifold X is a smooth map α(·) thatassigns to each p ∈ X a cotangent vector α(p) in T ∗p X


Geometry

Pull-back and push-forward of vector fields

I Suppose f : X → Y is a diffeomorphism, then f∗ : X (X )→ X (Y ) thepush-forward of a vector field on X is given by

(f∗V )(q) = Tf−1(q)f · V (f−1(q)) ∀q ∈ Y ,V ∈ X (X )

I Suppose f : X → Y is a diffeomorphism, then f ∗ : X (Y )→ X (X ) thepull-back of a vector field on Y is given by

(f ∗V )(p) = Tf (p)f−1 · V (f (p)) ∀p ∈ X ,V ∈ X (Y )


Geometry

The Lie bracketDefinitionThe Lie bracket of two vector fields on a smooth manifold X is definedas

[·, ·] : X × Y → Z [P,Q]4= PQ −QP

where for f ∈ C∞(X )

[P,Q](f ) = P(Q(f ))−Q(P(f ))

Properties of the Lie bracket

I [X ,Y ] = −[Y ,X ]

I [X + Y ,Z ] = [X ,Z ] + [Y ,Z ]

I [fX ,gY ] = f · (X (g))Y − g(Y (f )) + f .g.[X ,Y ]

I [X , [Y ,Z ]] + [Y , [Z ,X ]] + [Z , [X ,Y ]] = 0


Geometry

A Lie groupDefinitionA smooth manifold G together with a group structure is called a Liegroup if the group operation is smooth.

(g,h)→ g · h is smooth

I The identity element of the Lie group is usually denoted by e.

I Left translation of a group

Lg : G→ G h→ gh

I Right translation of a group

Rg : G→ G h→ hg


Geometry

Examples of Lie groups

I IR or multiple copies of IR (as IRn.).

I The unit circle S1 or multiple copies of S1 (as S1 × . . .× S1).

I The set of n × n matrices with real entries.

I The set of n × n real-orthogonal matrices O(n).

I The set of n × n real-rotation matrices SO(n).


Geometry

The Lie algebra

I The Lie bracket on TeG is given by

[ξ, η] = [Xξ,Xη] bracket of vector fields

I The vector space TeG with the [·, ·] is called the Lie algebra of G andis denoted by g.


Geometry

Invariant vector fields

DefinitionA vector field V on a Lie group G is said to be left invariant if ∀g ∈ G

ThLg · X (h) = X (gh) OR Lg∗V = V

Some properties

I For X ,Y ∈ XL(G)⇒ [X ,Y ] ∈ XL(G).

I Identify XL(G) with TeG.


Geometry

Exponential maps

I The ”exponential map” maps a Lie algebra to the Lie group

exp : g→ G η → exp(η) = γη(1)

where γη(t) is the integral curve of Xη ∈ XL(G) with γη(0) = e.

I The exponential map is a diffeomorphism from a neighbourhood of 0in g onto a nbhd of e in G.

I Recall linear systems and the solution of x = Ax . We haveγA(t) = eAt . Can you make a connection ?


Geometry

Group actions

A (left) action of a Lie group G on a manifold M is a smooth map

Φ : G ×M → M

such that Φ(e, x) = x∀x ∈ MΦ(g, Φ(h, x)) = Φ(gh, x)∀g, h ∈ G, x ∈ M

III Examples: 1. Action of SO(3) on IR3. 2. Action of a group on itself


Geometry

Adjoint and co-adjoint actions

I Defineig : G→ G ig = Rg−1 ◦ Lg

I Adjoint action of G on g

Adg : g→ g Adg(η) = Teig(η)

I Co-adjoint action

Ad∗g : g∗ → g∗ [Ad∗g (α), η] = [α,Adgη]


Geometry

Orbits

I Say G acts on M. The orbit of x ∈ M is defined by

G · x = {g · x ∈ M|g ∈ G}

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Robotics, Geometry and Control - Differential Geometry · Robotics, Geometry and Control Geometry Introduction I The study of differential geometry in our context is motivated by