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GEOMETRIC ASPECTS OF ROBOTICS AND CONTROL
(Slides for SC 624)
(Jan 2006)
R. N. BanavarProfessor, Systems and Control Engineering,
I. I. T. Bombay
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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Lecture Slides - SC 624
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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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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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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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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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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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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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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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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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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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Exterior calculus
dΘ(V,W) = V(ΘW) − W(ΘV) − Θ([V,W])
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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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