an introduction to riemannian geometry - uni … · jim mainprice - introduction to riemannian...

An Introduction to Riemannian

Geometry

Jim Mainprice, PhD, Humans to Robots Research (HRM) Group

IMPRS - Boot Camp - October 11th 2017

http://imprs.is.mpg.deSystem Mensch

https://ipvs.informatik.uni-stuttgart.de/mlr/HRM/

http://imprs.is.mpg.de

Jim Mainprice - Introduction to Riemannian Geometry - October 11th 2017

Outline

• 1 - Why geometry matters

• Feature maps

• Dimensionality reduction

• 2 - Differential geometry

• Manifolds

• Differentiable Maps

• Diffeomorphisms

• Tangent spaces

• 3 - Riemannian geometry

• Riemannian metric

• Calculus on the sphere

• Pullback metric

• Induced metric

• Nash embedding theorem

• Geodesics

• Harmonic maps

• 4 - Example in motion

optimization

2


Why Geometry Matters

Shortest path on the globe is not the same as shortest path on the map

3

A

≠

[Slide Courtesy: Frank C. Park RSS17]


2D Maps of the Globe

4



Nonlinear Feature Maps

• Feature maps are nonlinear transformation of the data space

• The classification problem is linear in but not in

5

X

= φ

φ H

φ H X


Dimensionality Reduction

= φ

• Data lies on an embedded non-linear manifold

• Manifold learning: Find an embedding to Euclidean space that minimally

distorts the metric

• Can be used for feature extraction

• Embedding is nonlinear, Metric in feature space is linear

6

X φ H


Outline


• Feature maps



• Manifolds


• Diffeomorphisms

• Tangent spaces




• Pullback metric

• Induced metric


• Geodesics

• Harmonic maps


optimization

7


What is a Manifold

• A manifold M is a topological space

• Set of points with neighborhood for each points

• Each point of M has a neighborhood

homeomorphic to Euclidean space

• A coordinate chart is a pair

8

Example: 4 charts of the circle

Abstract manifold chart

ϕ

W U !

1.2). By

! yU is

definition

pair .U;'/, w

from to an

n M

homeomorphism

U1 2 U3

ˆ1 U2

ˆ3 U4

ϕ : U → U ⊆ Rn


What is an smooth-Manifold

• An atlas A is a collection of coordinate

charts that cover the manifold

• said to be smooth of there exists transition maps

between its charts

• A smooth-manifold (M, A) is a topological

manifold with a smooth atlas

9

ϕ

n M

homeomorphism

U1 1 U2

ψ

ψ ϕ−1

Transition map between two

coordinate charts


Differentiable Maps

• A map f is said to be differentiable if its first order

derivative exists for all points of the domain U

• f is said of class C(r) if it has continuous partial

derivative up to order r

• When r = ∞, f is said to be smooth

10

f : U ⊂ Rn→ R

m

∂rf(x)

∂xr, ∀x ∈ U


Jacobian and Hessian

• Jacobian matrix is the matrix of first order

derivative of f

• Hessian is the matrix of second order

derivatives of f

• Differential d of f at x is the map solution

of the first order linearization of f

11

Jf (x) =

0

B

B

B

B

@

∂f1

∂x1· · ·

∂f1

∂xn...

. . ....

∂fm

∂x1· · ·

∂fm

∂xn

1

C

C

C

C

A

B

@

C

A

Hf (x) =

0

B

B

B

B

B

@

∂2f1

∂x21

· · ·

∂2f1

∂x2n

.... . .

...

∂2fm

∂x21

· · ·

∂2fm

∂x2n

1

C

C

C

C

C

A

dxf : Rn→ R

m

s.t.

f(x+ δ) = f(x) + dxf(δ) + o(δ)


What is a Diffeomorphism

• A differentiable map f : M → N between two manifolds M

and N, is called a diffeomorphism if

• it is a bijection

• its inverse f^−1 : N → M is differentiable

• Inverse Function Theorem states that a differentiable map

is invertible iif it’s Jacobian determinant does not vanish on

the domain

12

Diffeomophism of the plane


smooth-Manifolds

Transition maps are diffeomorphisms,

hence their Jacobians are invertible

because they map euclidean spaces of

the same dimension

13

ϕ

n M

homeomorphism

U1 ˆ1 U2

ψ

ψ ϕ−1


Examples of Manifolds with Global Charts

• Euclidean Space

• Implicit smooth surfaces defined by

• Space of n-dimensional Gaussian distributions

• (Euclidean point, Positive matrix)

14

ϕ = I

f : U ⊂ R2→ R

m ϕ = f

ϕ = (µ,Σ) ∈ Rn× P (n)

→ Rm


Example: 1D Gaussian Distributions

• The space of 1D Gaussian distribution is

parametrized by mean and standard deviation

• Can also be parameterized by mean and variance

• Smooth reparametrization i.e., change of coordinates

15

ψ ϕ−1(x) = (µ,pσ2)

ϕ

n M

homeomorphism

U1 ˆ1 U2

ψ

ψ ϕ−1

Std Dev. Variance

Manifold of Gaussian Distributions


Example: The SphereTwo charts of the sphere

• In general, manifolds require multiple

coordinate charts

• Here it needs two charts that can be obtained

by stereographic projection

16

S2 = (x, y, z) ∈ R3 | x2 + y2 + z2 = 1

ϕN (p) =

x

1− z,

y

1− z

,with p ∈ S2

Stereographic projection


Submanifolds of Euclidean Space

• A submanifold of Euclidean space can be

defined through the kernel of a nonlinear

smooth map f

• Dimension of the manifold is m - n

• For example, the unit sphere S^n can be

defined this way

• n = 1, m = dim(E)

17

M = f−1(0)

f : U ∈ Rm

→ Rn,with n ≤ m

f(x) = 1− kxk2

Sphere as a submanifold

of Euclidean space


Product Manifolds

• If M and N are manifolds, so is M x N

• The resulting chart is the product of

the charts on both manifolds

18

x =

x =

Cylinder = S1× U ⊂ R

Torus = S1× S1


What Are Tangent Vectors

• Tangent vectors indicate a direction on the

manifold

• To specify a tangent vector we can specify

a curve on the manifold and take the

derivative of the curve at the point p on M

19

γ(t) =

n M

homeomorphism

γn : [−1, 1] 7! M


What is the Tangent Space

• Suppose two differentiable curves are given

• Equivalent at p iif the derivative of their pushfoward through a local-coordinate chart coincide at 0

• Any such curves leads to an equivalence class denoted:

• The tangent space of M at p, , is then defined as the set of all tangent vectors at p ; it does not depend on the choice of coordinate chart

20

Equivalence class of tangent vectors

n M

homeomorphism

TpMv pv p

γ1(t)

t) γ2(t)

ϕ γ1,ϕ γ2 : (−1, 1) ! Rn

γ1(0) = γ2(0) = p

TpM

v = γ(0)

ϕ


Tangent Space Basis• Consider the set of all smooth function on M:

• The tangent space can also be defined as the set of all derivations:

• A basis for the tangent space can be defined for a given

coordinate chart in the following way:

21

n M

homeomorphism

TpMv p

ϕ

U1

8i 2 1, . . . , n, 8f 2 C∞(M) :

8 2 8 2

∂

∂xi

p

(f)df= (∂i(f ϕ−1))(ϕ(p))

x1

1 x2

2 ∂

∂x1

∂

∂x2

: C∞(M)

D : C∞(M) → R


Differential of Smooth Maps

• Consider a smooth map f. A differential (or pushfoward) is a

function that maps the tangent spaces of M and N

• It corresponds to the Jacobian of f in local coordinates

22

n M

homeomorphism

TpM

v p

f : M → N

→ N

TfpNdpf


Outline


• Feature maps



• Manifolds


• Diffeomorphisms

• Tangent spaces




• Pullback metric

• Induced metric


• Geodesics

• Harmonic maps


optimization

23


Riemannian metric

• A Riemannian metric is a an inner

product g(u, v) defined over the tangent

space varying smoothly over M

• In coordinates, it corresponds to a

smooth family of positive-definite

matrices

• A smooth manifold with a Riemannian

metric is a Riemannian manifold (M, g)

24

2 h i

Ap =

0

B

@

a11 · · · a1N...

. . ....

aN1 · · · aNN

1

C

A> 0

ZZ

8u, v 2 TpM, gp(u, v) = hu, vig = uTApv 2 R

0 1


Riemannian of Euclidean Space

• In Euclidean space, the associated metric is

constant and equal to the identity

25

ϕ = I

8p 2 Rn, 8u, v 2 TpR

n, g(u, v) = uT v

Ap =

0

B

@

1 · · · 0...

. . ....

0 · · · 1

1

C

A= I


Calculus on the Sphere

A curve on the sphere

has the following arc length

26

:2 2 2 2 2 2 2 2 2 2

S2 = (x, y, z) ∈ R3 | x2 + y2 + z2 = 1

Sphere is defined by

Coordinates:

γ(t) =(

x(t), y(t), z(t))

ds2 = dx2 + dy2 + dz2 = dφ2 + sin2φ dθ2

ϕϕ(x, y, z) =

tan−1(y

x), cos−1(z)

= (θ, φ)



Calculus on the Sphere

27

Calculating lengths and area on the sphere

using spherical coordinates:

0 2 2

• Length on the sphere

C =

Z T

0

q

φ2 + θ2sin2φ dt

ϕ

A =

ZZ

A

|sin(φ)| dφ dθ

• Area on the sphere



Pullback Metric and Embeddings

f defines an embedding of M in some other

manifold N if it is an injective homeomorphism

28

n M

homeomorphism

TpM

v p

f : M → N

→ N

TfpNdpf

gN (f∗gN )(v, w) = gN (df(v), df(w))

f∗gN = JTANJ

The pullback metric of g trough f

In coordinates:

J is the Jacobian of f at p

pullback metric


2 2 2 2 2 2 2 2 2 2

Induced Metric on Submanifolds of

Euclidean space

• The induced metric is the inner product

from Euclidean space restricted to M

• We can get it by pullback of the identity

through a coordinate-chart

29

f : S27! R

3

pullback metric

ϕ

Ap = JTf Jf =

1 00 sin2 θ

2 3

6

4

7

5

Jf (θ, φ) =

0

@

cos θ cosφ − sin θ sinφcos θ sinφ sin θ cosφ− sin θ 0

1

Af(θ, φ) =

8

>

<

>

:

x = sin θ cosφ

y = sin θ sinφ

z = cos θ


Nash Embedding Theorems

• Every Riemannian manifold can be

isometrically embedded into

some Euclidean space

• Under particular conditions n ≥ m+1

• Harder conditions lead to n ≥ m(3m+11)/2

30

Ap = JTf Jf

2

n M

homeomorphism

TpM

v p

dpf

pullback metric

= I

2 Rn,

7!

TfpRn

8

f : Mm7! R

n


What are Geodesics

• Geodesics correspond to curves for which

where nabla is an affine connection

intuitively this means that the acceleration is either 0 or

orthogonal to the tangent space, it generalizes the

notion of straight line to curved spaces

Connections allow to differentiate tangent vectors from

different tangent spaces, we will not see them.

31

rγ γ = 0Z

q


Geodesic Equation• Geodesic locally minimize the length functional

• The solution to the Euler-Lagrange equation linked to the minimization of this functional leads to the geodesic equation:

32

L(γ) =

Z T

0

q

gγ(t)(γ(t), γ(t))dt

r

d2xλ

dt2+ Γ

λµν

dxµ

dt

dxν

dt= 0

Z

q

n M

homeomorphism

TpM

v p

γ(t) =

M, gp(0

In coordinates (the Christoffel symbols

characterize the metric connection)

With this definition of length every Riemannian Manifold can become a metric space

The geodesic equation has a unique solution,

given an initial position and an initial velocity


Coordinate Invariant Distortion Measure

• Consider a smooth map f : M → N between two compact Riemannian manifolds

• (M, g): local coord. x = ( x_1, ⋯ , x_m), Riemannain metric G

• (N, h): local coord. y = ( y_1, ⋯ , y_n), Riemannain metric H

• Isometry

• Map preserving length, angle, and volume - the ideal case of no distortion

• if dim(M) <= dim(N), f is an isometry when for all p in M

33

n M

homeomorphism

TpM

v p

f : M → N

→ N

TfpNdpf

JTf H(f(p))Jf (p) = G(p)

(f(p))



Coordinate Invariant Distortion Measure• Comparing the pullback metric to

• Global distortion measure:

34

n M

homeomorphism

TpM

v p

f : M → N

→ N

dpf

pullback metric

H(f(p))

)) JTHJ(p)

p) G(p)

)) JTHJ(p) p) G(p)

ZM

σ(λ1, · · · , λm)√

detG dx1 · · · dxmwith lambdas roots of det(JTHJ − λG) = 0

√



Harmonic Maps

• Define the functional

• Variational equation throughout Euler-Lagrange formula, solutions are known as the harmonic maps (ones that locally minimize E)

• Intuitively these f wraps marble (N) with rubber (M), harmonic maps correspond to elastic equilibria

35

E(f) =

ZM

tr(JTHJG−1)√

detG dx1 · · · dxm

n M

homeomorphism

→ N f :



Example of Harmonic Maps

• Lines

• Geodesics

• Laplace’s Equation

• 2R Spherical mechanism

36

→ ∫ → ∫ ⊤22 ∑= ∑= Γ ℝ ℝ → ℝ → ∗ ∗ 1/ 2

0 = 0 1 = 1

0 = 1 =

E(f) =

Z 1

0

f2dt

ZZ

E(f) =

Z 1

0

fTHfdtZ

r2f = 0

r

E(f) = 2(1 + 22)Z p[Slide Courtesy: Frank C. Park RSS17]


Things to Know about Riemannian Manifolds

• Riemannian metrics can be used to define

various properties of a manifold such as

• angles, lengths of curves, areas (or volumes)

• curvature

• gradients of functions

• divergence of vector fields.

• Every manifold admits a Riemannian metric

• Can be used to model most of modern physics

• Theory of gravitation

• Theory of electro-magnetism

37


Outline


• Feature maps



• Manifolds


• Diffeomorphisms

• Tangent spaces




• Pullback metric

• Induced metric


• Geodesics

• Harmonic maps


optimization

38

Jim Mainprice - The Geometry of Human-Robot Space Sharing, Max Planck Institute, Tübingen, Germany

The space around us is fundamentally non-Euclidean

39


How can we warp workspace geometry ?

40

Riemannian workspace metric

B(x) = BWS(x)B(x) = I

Euclidean workspace metric

Geodesics x∗ = argminx∈W (x1,x2)

1

2

Z

xTB(x)xdt

Z

[Mainprice IROS16]


Pullback Metric Definition

41

Euclidean space Riemannian Workspace

W =(

R3, BWS(x) = JT

φWSJφWS

)

4 5

) = φWS(x) =

2

6

4

α1φ1(x)...

αd+1φd+1(x)

3

7

5

[RieMO, Ratliff 15]

Workspace Geometry map

• Decomposition in local

coordinate systems

• Geodesics well defined

B(x) = I

[Mainprice IROS16]


How does the metric operate on a

manipulation problem ?

42

“Pullback” metric

A(x) = JTφ B(x)Jφ

kxk2B(x) = xTB(x)x = qT

(

JTφ B(x)Jφ

)

q

Z

xg

3

φ(q)q1

q2q3

x = Jφq

Riemannian Workspace Metric

BWS(x) = JTφWS

JφWS

[RieMO, Ratliff 15]

[Mainprice IROS16]


Local coordinates around arbitrary objects ?

43

Cylindrical coordinate systemSpherical coordinate system

Local coordinates

Mesh

Point cloud

[Mainprice IROS16]


Electrical Coordinate Systems

System has 3 coordinates

• r : value of the potential

• u : 1st coordinate of field line

• v : 2nd coordinate of field line

44

[Harmonic parameterization by electrostatics, Wang 13]

Charge simulation (i.e., find charges e_i)

Electric potential at p

P (p 2 R3) =

Z

ρ(x)

||p− x||2dΩ(x)

8

>

>

<

>

>

:

e1P (T1,p1) + . . .+ enP (Tn,p1) = V...

e1P (T1,pn) + . . .+ enP (Tn,pn) = V

[Mainprice IROS16]


Motion Optimization Objective

45

q1[Zucker CHOMP 13]

. . . qNqt Configuration space

Cartesian space

Workspace geometry

x(1)1

(1)x(2)1

(2)z(1,1)1 z

(1,2)1 z

(2,1)1 z

(2,2)1

2)x(1)N

(1)x(2)N

z(1,1)N z

(1,2)N z

(2,1)N z

(2,2)N

Results in a differentiable function network

with discretized trajectory using N way points

X

(

ξ =

q1 . . . qNT

>

>

:

c(ξ) =

Z T

0

mX

i=1

cti(

φi(qt),d

dtφi(qt), . . . ,

dk

dtkφki (qt)

)

dt

Finite time horizon cost functional

[Mainprice IROS16]


Augmented Lagrangian Optimizer

46

Configuration Space

Smoothness

Collision

Terminal Potential

Joint Limits

Joint Postural

Task Space

Smoothness

Constraints Objectives

Compute Lagrange Multipliers

Unconstrained Proxy

Local Approximation

Optimization Step

Constraints

Objectives

X

(

ξ =

q1 . . . qNT

Trajectory vector

Local Optimizer

while not StopCondition() do

v f(ξi) ;

gξ rf(ξi) ;

ξi+1 ξi − ηiB−1i gξi

;[Mainprice IROS16]


Augmented Lagrangian Optimizer

• Augmented Lagrangian Solver

• Divide between constraints and objectives

• Computes Lagrange multipliers at each iteration

• Creates an unconstrained proxy solved by local optimization

47


Fixed Goal Configurations Comparisons

• STOMP 0th order

• CHOMP 1st order

• “ Gauss-Newton” 2nd order

48

[Mainprice IROS16]


Goal Sets (Null space) Comparisons

49

• STOMP 0th order

• CHOMP 1st order

• “ Gauss-Newton” 2nd order

[Mainprice IROS16]


That’s it !!! (links are clickable)

• John M. Lee (2003). Introduction to Smooth Manifolds.

(Book) Springer New York.

• John M. Lee (1997). Riemannian manifolds: an introduction to curvature.

(Book) Springer Science & Business Media.

• Jürgen Jost (2008). Riemannian geometry and geometric analysis.

(Book) Springer Science & Business Media.

• Aasa Feragen and François Lauze (2014). A Very Brief Introduction to Differential and Riemannian Geometry.

(Presentation) University of Copenhagen. Faculty of Science.

• Frank C. Park (2017). Manifolds, Geometry, and Robotics.

(Keynote) Robotics Science and System (RSS).

• Jim Mainprice, Nathan Ratliff, and Stefan Schaal (2016). Warping the workspace geometry with electric potentials for motion

optimization of manipulation tasks.

(Paper) IROS.

50

http://www.springer.com/us/book/9781441999818



http://image.diku.dk/MLLab/SummerSchools/presriemann_aasa.pdf

http://www.roboticsconference.org/docs/invitedtalks/park-manifoldsgeometryandrobotics.pdf

http://ieeexplore.ieee.org/document/7759488

an introduction to riemannian geometry - uni … · jim mainprice - introduction to riemannian...

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