SC 618: Flows, derivatives and brackets · SC 618: Flows, derivatives and brackets Ravi N Banavar [email protected] 1 1Systems and Control Engineering, IIT Bombay, India Geometric

Introduction Flow Lie groups Push-forward and pull-back The Lie derivative and bracket Lie algebras

SC 618: Flows, derivatives and brackets

Ravi N [email protected] 1

1Systems and Control Engineering,IIT Bombay, India

Geometric MechanicsMonsoon 2014

September 15, 2014

Lectures 3, 4 and 5 SC 618


Outline

1 Linear and nonlinear systems

2 The flow of a vector field

3 Lie groups

4 Push-forward and pull-back

5 The Lie derivative and the Jacobi-Lie bracket

6 Lie algebras



Outline



3 Lie groups



6 Lie algebras



Linear systems - preliminaries

A linear system with an input

x = Ax+Bu x(t) ∈ Rn

x(t) lives in Rn, a vector space.

A linear autonomous system

x = Ax, x(t) ∈ Rn


The right-hand side of the differential equation is termed a vector field. Forthe linear system, it is a linear vector field.Linearity

A(α1x1 + α2x2) = α1Ax1 + α2Ax2



Linear systems - preliminaries

A linear system with an input

x = Ax+Bu x(t) ∈ Rn


A linear autonomous system

x = Ax, x(t) ∈ Rn

x(t) lives in Rn, a vector space.The right-hand side of the differential equation is termed a vector field. Forthe linear system, it is a linear vector field.Linearity

A(α1x1 + α2x2) = α1Ax1 + α2Ax2



Solution and flow

Solution to the set of differential equations

x(t) = eAtx0, x(0) = x0, eAt4= I +A+A2/2! + . . .

The term eAtx0 is termed the flow associated with the linear vector field Ax.



Nonlinear systems - preliminaries

A nonlinear system with an input

x = f(x) + g(x)u x(t) ∈M

f(·), g(·) are smooth functions, x(t) lives in M , a smooth manifold.

A nonlinear autonomous system

x = f(x), x(t) ∈ Rn

x(t) lives in M , a smooth manifold.

The right-hand side of the differential equation is a nonlinear vector field.Linearity does not hold.

f(α1x1 + α2x2) 6= α1f(x1) + α2f(x2)





x = f(x) + g(x)u x(t) ∈M



x = f(x), x(t) ∈ Rn

x(t) lives in M , a smooth manifold.The right-hand side of the differential equation is a nonlinear vector field.

Linearity does not hold.

f(α1x1 + α2x2) 6= α1f(x1) + α2f(x2)





x = f(x) + g(x)u x(t) ∈M



x = f(x), x(t) ∈ Rn

x(t) lives in M , a smooth manifold.The right-hand side of the differential equation is a nonlinear vector field.Linearity does not hold.

f(α1x1 + α2x2) 6= α1f(x1) + α2f(x2)



Solution and flow

Solution to the set of differential equations

x(t) = Φ(t, x0) x(0) = x0

The term Φ(t, x0) is termed the flow associated with the nonlinear vectorfield f(x).



Outline



3 Lie groups



6 Lie algebras



Flow of a vector field

Flow of X(x)

The flow of the vector field X(x), denoted by Φ(t, x0), is a mapping from(−a, a)× U → Rn (where a(> 0) ∈ R and U is an open region in thestate-space ) and satisfies the differential equation

dΦ(t, x0)

dt= X(Φ(t, x0)) ∀t ∈ (−a, a), x(0) = x0 ∈ U.

over the interval (−a, a) and with initial conditions starting in the region U .



x0

Φtx0M

Figure: Flow of a vector field



Properties of flows

The group structure

DenoteΦt(x0)

4= Φ(t, x0)

The set of transformations {Φt} : U → Rn satisfies the following properties.

• Φt+sx0 = Φt ◦ Φsx0 ∀t, s, t+ s ∈ (−a, a) (the group binary operation.)

• Φ0x0 = x0 (the group identity.)

• For a fixed t ∈ (−a, a) we have ΦtΦ−tx0 = x0 ⇒ [Φt]−1 = Φ−t

(existence of an inverse.)



The flow of a linear system

The group property

RemarkThe three properties mentioned above impart a group structure to the set{Φt}. This set is called a one-parameter (time) group of diffeomorphisms(Φt and its inverse are smooth mappings).

Linear flow

RemarkFor a linear system described by

x = Ax A ∈ Rn×n

the flow Φtx0 = eAtx0 where {eAt : t ∈ (−∞,∞)} constitutes theone-parameter group of diffeomorphisms.



The flow of a linear system

The group property

RemarkThe three properties mentioned above impart a group structure to the set{Φt}. This set is called a one-parameter (time) group of diffeomorphisms(Φt and its inverse are smooth mappings).

Linear flow

RemarkFor a linear system described by

x = Ax A ∈ Rn×n

the flow Φtx0 = eAtx0 where {eAt : t ∈ (−∞,∞)} constitutes theone-parameter group of diffeomorphisms.



Outline



3 Lie groups



6 Lie algebras



A group

DefinitionA group is a set G with a binary operation + such that

• For any x, y ∈ G, x+ y ∈ G (Closure) and (x+ y) + z = x+ (y + z)(Associativity)

• There exists a unique i ∈ G such that x+ i = i+ x = x for every x ∈ G(Existence of the identity element)

• For every x ∈ G there exists a unique y ∈ G such that x+ y = i.(Existence of inverse)



A Lie group

DefinitionA smooth manifold M together with a group structure is called a Lie groupG if the group operation + is smooth.

(g, h)→ g + h (∀g, h ∈ G) is smooth

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

• Left translation of a group

Lg : G→ G h→ g + h

• Right translation of a group

Rg : G→ G h→ h+ g



Examples of Lie groups

• R or multiple copies of R (as Rn) with the bianry operation being theusual component-wise addition +.

• The unit circle S1 with elements denoted as θ(∈ [0, 2π) and the binaryoperation being the usual addition. Similarly, multiple copies of S1 (asS1 × . . .× S1).

• The set of n× n invertible matrices with real entries with the binaryoperation being matrix multiplication. This group is called GL(n,R).

• The set of n× n real-orthogonal matricesO(n), asubsetofGL(n,R).Thesetofn ×n rotation matrices SO(n), asubset of O(n,R).



Rigid body motion

DefinitionRigid body motion is characterized by two properties

•• The distance between any two points remains invariant

• The orientation of the body is preserved. (A right-handed coordinatesystem remains right-handed)



SO(3) and SE(3)

• Two groups which are of particular interest to us in mechanics andcontrol are SO(3) - the special orthogonal group that representsrotations - and SE(3) - the special Euclidean group that representsrigid body motions. These are Lie groups.

• Elements of SO(3) are represented as 3× 3 real matrices and satisfy

RTR = I

with det(R) = 1.

• An element of SE(3) is of the form (p,R) where p ∈ R3 and R ∈ SO(3).



Frames of reference or coordinate frames

• In describing rigid body motions we always fix two frames of reference.One is called the body frame that remains fixed to the body and theother is the inertial frame that remains fixed in inertial space.

y

x

z

A

B

gab

z

x

y

pab

qb

qa

a

a

a

b

b

b

Figure: Rigid body motion



Rigid body motions and groups

• Suppose qa and qb are coordinates of a point q relative to frames A andB, respectively.

qa = pab +Rabqb

Here pab represents the position of the origin of the frame B withrespect to frame A in frame A coordinates and Rab is the orientation offrame B with respect to frame A .

• Appending a ”1” to the coordinates of a point ( to render the groupoperation as the usual matrix multiplication)

qa =

(qa1

)=

(Rab pab0 1

)(qb1

)= gabqb



Two results on rotations

Rotation in a plane

ClaimThe rotation group SO(2) can be identified with S1 (the unit circle).

Proof:S1 = {x ∈ R2 : ‖x‖ = 1}

Parametrize the elements of S1 in terms of θ ∈ [0, 2π]. For each θ ∈ [0, 2π],the counter-clockwise rotation of the vectors {(1, 0), (0, 1)} in R2 (theseform a basis) by the angle θ

(1, 0)→(cos θ sin θ) (0, 1)→(− sin θ cos θ)

is given by the matrix

Rθ =

[cos θ − sin θsin θ cos θ

]which is an element of SO(2).



Proof (contd.)

Conversely, take an element of SO(2) of the form

R =

[a1 a2

a3 a4

]Then from the properties of an element of SO(2), we have

a1a4 − a2a3 = 1; a21 + a2

3 = 1; a22 + a2

4 = 1; a1a2 + a3a4 = 0

It is possible to find a θ ∈ [0, 2π] such that that R can be represented in theform Rθ.



Euler’s theorem

Theorem(Euler’s theorem)Every A ∈ SO(3) is a rotation through an angle θ ∈ S1 about an axisω ∈ R3.

Proof: Since 1 is an eigen value of A, we have Aw = w where w ∈ R3 is aneigen vector. Choose two vectors e1 and e2 that are orthogonal to eachother as well as w. So

< w, e1 >= 0, < w, e2 >= 0, < e1, e2 >= 0

The two vectors {e1, e2} lie in the plane perpendicular to w and it followsthat {w, e1, e2} form a basis for R3. Since A is orthogonal, the matrix of Ain this basis is of the form 1 0 0

0 a1 a3

0 a2 a4

.



Proof (contd.)

(why ?) Now [a1 a3

a2 a4

]is an element of SO(2) and hence there exists a θ ∈ [0, 2π] such that[

a1 a3

a2 a4

]=

[cos θ − sin θsin θ cos θ

]It follows that A is a rotation about w through the angle θ .



Outline



3 Lie groups



6 Lie algebras



The ”star” map (f∗) associated with a smooth function f

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

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,”. Visualize this as follows.

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

• Construct the curve f ◦ c. Then define

f∗p(vp)4= Tpf · vp =

d

dt|t=0(f ◦ c)(t)

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



Push-forward and pull-back of a function

Figure: Pull-back and push-forward of functions



Push-forward and pull-back of a function

• Suppose X is a vector field on M and f : M → R is a smooth function.Then the push-forward of the function f on M by the flow of X is thefunction Φt∗f defined by

(Φt∗f)(x)4= f ◦ Φ−1

t (x) ∀x ∈M

• Suppose X is a vector field on M and f : M → R is a smooth function.Then the pull-back of the function f on M by the flow of X is thefunction Φ∗t f defined by

(Φ∗t f)(x)4= f ◦ Φt(x) ∀x ∈M



Push-forward and pull-back of a vector field

Figure: Pull-back and push-forward of vector fields



Push-forward of a vector field by another vector field

Push-forward

• Suppose Φt : M →M is the flow associated with a vector field X, thenthe push-forward of a vector field Y on M by f is the vector field(Φt∗Y ) on M defined by

(Φt∗Y )(x) = T(Φ−1

t x)[Y (Φ−1

t x)] ∀x ∈M

• In coordinates(Φt∗Y )(x) = (DΦt)(Y (Φ−1

t x))



Push forward of vector fields under a diffeomorphism f

M N

f

Vector Field X

TM

Vector Field f∗X

TN

f∗

Figure: Push-forward



Push-forward of vector fields under a diffeomorphism f

Push-forward

• Suppose f : M → N is a diffeomorphism, then the push-forward of avector field X on M by f is the vector field f∗X on N defined by

(f∗X)(f(x)) = Txf(X(x)) ∀x ∈M

• In coordinates

y = f(x) (f∗X)(y) = Df(x).X(x) =dy

dx·X(x)



Pull back of a vector field under a diffeomorphism f

M N

f

Vector Field f ∗Y

TM

Vector Field Y

TN

f ∗

Figure: Pull-back



Pull-back of vector fields under a diffeomorphism f

The pull-back

• Suppose f : M → N is a diffeomorphism, then the pull-back of a vectorfield Y on N by f is the vector field f∗Y on M defined by

f∗Y = (f−1)∗Y = Tf−1 ◦ Y ◦ f

• In coordinates

y = f(x) (f∗X)(y) = Df(x).X(x) =dy

dx·X(x)



Outline



3 Lie groups



6 Lie algebras



Operations on vector fields

The GradientConsider a smooth function g(·) : U → R. The gradient of such a function,denoted by ∇g, is defined as

∇g(x) =[

∂g∂x1

· · · ∂g∂xn

]alternate notation: grad(g).



The Lie derivative of a function

The Lie derivativeThe Lie derivative of a function f along X is

(LXf)(x) =d

dt|t=0(Φ∗t f)(x) =

d

dt|t=0f ◦ Φt(x)

In coordinates we have the familiar

(LXf)(x) =[

∂f∂x1

· · · ∂f∂xn

]X(x)

Alternate notation

(Xf)(x) =d

dt|t=0f ◦ Φt(x) = lim

t→0

f(Φt(x))− f(x)

t



The Lie derivative of a function

The Lie derivativeThe Lie derivative of a function f along X is

(LXf)(x) =d

dt|t=0(Φ∗t f)(x) =

d

dt|t=0f ◦ Φt(x)

In coordinates we have the familiar

(LXf)(x) =[

∂f∂x1

· · · ∂f∂xn

]X(x)

Alternate notation

(Xf)(x) =d

dt|t=0f ◦ Φt(x) = lim

t→0

f(Φt(x))− f(x)

t



High school physics

The cross product

• Vector space R3 and the cross-product operation ×.• (α1a1 + α2a2) × b = α1(a1 × b1) + α2(a2 × b2) - linearity. (holds in the

second argument as well.)• a× b = −b× a - skew-commutative.• a× (b× c) + c× (a× b) + b× (c× a) = 0 - the Jacobi-Lie identity.

Comment: the cross-product of two linearly independent vectors in R3

yields a vector in a new direction.

An alternate notation

a× b↔ ab a =

0 −a3 a2

a3 0 −a1

−a2 a1 0



High school physics

The cross product

• Vector space R3 and the cross-product operation ×.• (α1a1 + α2a2) × b = α1(a1 × b1) + α2(a2 × b2) - linearity. (holds in the

second argument as well.)• a× b = −b× a - skew-commutative.• a× (b× c) + c× (a× b) + b× (c× a) = 0 - the Jacobi-Lie identity.

Comment: the cross-product of two linearly independent vectors in R3

yields a vector in a new direction.

An alternate notation

a× b↔ ab a =

0 −a3 a2

a3 0 −a1

−a2 a1 0



The Lie derivative of a vector field

The pull back of a vector field

The Lie derivative of Y along X is

LXY4=

d

dt|t=0Φ∗tY

where Φ is the flow of X.

Explicitly

(LXY )(x) =d

dt|t=0(DΦt(x))−1 · Y (Φt(x))

The Lie bracketIn coordinates we have the familiar expression

d

dt|t=0(DΦt(x))−1 · Y (Φt(x)) =

∂Y

∂xX(x)− ∂X

∂xY (x) = [X,Y ](x)



The Lie derivative of a vector field

The pull back of a vector field

The Lie derivative of Y along X is

LXY4=

d

dt|t=0Φ∗tY

where Φ is the flow of X.Explicitly

(LXY )(x) =d

dt|t=0(DΦt(x))−1 · Y (Φt(x))

The Lie bracketIn coordinates we have the familiar expression

d

dt|t=0(DΦt(x))−1 · Y (Φt(x)) =

∂Y

∂xX(x)− ∂X

∂xY (x) = [X,Y ](x)



Operation on vector fields

The Jacobi-Lie bracketThe Jacobi-Lie bracket of two vector fields is an operation between twovector fields that yields another vector field. For two vector fields X and Y ,both defined from U to Rn, it is defined as

[X,Y ] = (LXY ) = (DY ) ·X − (DX) · Y

and satisfies the following properties (for any three vector fields X,Y, Z)

• [αX + βY, Z] = α[X,Z] + β[Y,Z] - linearity in the first argument (alsohold for the second argument)

• [X,Y ] = −[Y,X] - skew-commutative.

• [X, [Y,Z]] + [Z, [X,Y ]] + [Y, [X,Z]] = 0 - the Jacobi-Lie identity



More properties

The Jacobi-Lie bracketLet X generate the flow {Φt} and Y generate the flow {Ψt}. Then[X,Y ] = 0 if and only if Φt ◦Ψs = Ψs ◦ Φt for all s, t ∈ R.



Outline



3 Lie groups



6 Lie algebras



The Lie algebra - so(3)

3 × 3 skew-symmetric matrices

Recall

ω × x↔ ωx ω =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

The eigen values of ω are 0,±‖ω‖ i (Hint: The trace of a matrix is the sumof its eigen values.)

ClaimExponential of a skew-symmetric matrix is a rotation matrix

To show eω ∈ SO(3)

(eω)(eω)T = (e(ω−ω)) = I⇒ det(eω) = ±1

Now from ω = 0, eω = I and det(I) = 1. The determinant is a continuousfunction of the elements of the matrix

⇒ det(eω) = 1



Properties of the Lie algebra - so(3)

It is a vector space of dimension 3.The tangent space of the identity of SO(3) i. e. Te SO(3) = so(3).The bracket operation [·, ·] : so(3)× so(3)→ so(3) satisfies

• [αx+ βy, z] = α[x, z] + β[y, z] - linearity in the first argument (alsohold for the second argument)

• [x, z] = −[z, x] - skew-commutative.

• [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 - the Jacobi-Lie identity

Notice that the cross product relates as

[x, z] = x× z


Documents

SC 618: Flows, derivatives and brackets · SC 618: Flows, derivatives and brackets Ravi N Banavar [email protected] 1 1Systems and Control Engineering, IIT Bombay, India Geometric