# Lecture Kinematic Foundations

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```					                                  Lecture 2.
Kinematic
Foundations

Preview

Deﬁnitions and
basic concepts
Lecture 2.              System, conﬁguration, rigid
body, displacement
Rotation, translation

Kinematic Foundations        Conﬁguration space,
degrees of freedom
Metrics

Group theory
Groups, commutative
Matthew T. Mason         groups
Displacements with
composition as a group
Noncommutativity of
displacements

Projective
Mechanics of Manipulation   geometry
Motivation
Spring 2011            The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Outline                                                 Kinematic
Foundations

Preview
Preview
Deﬁnitions and basic concepts                      Deﬁnitions and
System, conﬁguration, rigid body, displacement   basic concepts
System, conﬁguration, rigid

Rotation, translation                            body, displacement
Rotation, translation

Conﬁguration space, degrees of freedom           Conﬁguration space,
degrees of freedom
Metrics
Metrics
Group theory
Groups, commutative
Group theory                                       groups
Displacements with

Groups, commutative groups                      composition as a group
Noncommutativity of
displacements
Displacements with composition as a group
Projective
Noncommutativity of displacements               geometry
Motivation
The projective plane
Projective geometry                                Homogeneous coordinates
Projections
Motivation
The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Preview                                                         Kinematic
Foundations

The agenda                                                 Preview

Deﬁnitions and
Today: general concepts.                                basic concepts
System, conﬁguration, rigid

Next: rigid motions in the Euclidean plane (E2 ).
body, displacement
Rotation, translation
Conﬁguration space,

Then: rigid motions in Euclidean three space   (E3 ),   degrees of freedom
Metrics

and the sphere (S2 ).                                   Group theory
Groups, commutative
groups
Displacements with
composition as a group

Why?                                                       Noncommutativity of
displacements

Projective
geometry
Why spend so much time on such fundamental              Motivation

concepts? Because robotics is so challenging.           The projective plane
Homogeneous coordinates
Projections
“Give me six hours to chop down a tree and
I will spend the ﬁrst four sharpening the
axe.” (Abraham Lincoln)
Lecture 2.
Preview                                                         Kinematic
Foundations

The agenda                                                 Preview

Deﬁnitions and
Today: general concepts.                                basic concepts
System, conﬁguration, rigid

Next: rigid motions in the Euclidean plane (E2 ).
body, displacement
Rotation, translation
Conﬁguration space,

Then: rigid motions in Euclidean three space   (E3 ),   degrees of freedom
Metrics

and the sphere (S2 ).                                   Group theory
Groups, commutative
groups
Displacements with
composition as a group

Why?                                                       Noncommutativity of
displacements

Projective
geometry
Why spend so much time on such fundamental              Motivation

concepts? Because robotics is so challenging.           The projective plane
Homogeneous coordinates
Projections
“Give me six hours to chop down a tree and
I will spend the ﬁrst four sharpening the
axe.” (Abraham Lincoln)
Lecture 2.
Ambient space, system, conﬁguration                             Kinematic
Foundations
Deﬁnitions

Preview
Deﬁnition (Ambient                                         Deﬁnitions and
basic concepts
space)                                                     System, conﬁguration, rigid
body, displacement
Rotation, translation
Let X be the ambient space,                                Conﬁguration space,
degrees of freedom

either E2 , E3 , or S2 .                                   Metrics

Group theory
A system in the Euclidean   Groups, commutative
groups
Deﬁnition (System)                       plane             Displacements with
composition as a group
Noncommutativity of
A set of points in the space                               displacements

Projective
X.                                                         geometry
Motivation
The projective plane

Deﬁnition (Conﬁguration)                                   Homogeneous coordinates
Projections

The conﬁguration of a
system gives the location of
Same system, different
every point in the system.
conﬁguration
Lecture 2.
Rigid bodies, displacements                                           Kinematic
Foundations
Deﬁnitions

Preview

Deﬁnition (Displacement)                                        Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement
A displacement is a change of conﬁguration that                 Rotation, translation
Conﬁguration space,
preserves pairwise distance and orientation                     degrees of freedom
Metrics
(handedness) of a system.                                       Group theory
Groups, commutative
groups

Deﬁnition (Rigid body)                                          Displacements with
composition as a group
Noncommutativity of
displacements

A rigid body is a system that is capable of displacements       Projective
geometry
only.                                                           Motivation
The projective plane
Homogeneous coordinates
Projections

Rotation (rigid), dilation (not rigid), reﬂection (not rigid)
Lecture 2.
Question                                                       Kinematic
Foundations

Why focus on rigid bodies?                                Preview

Deﬁnitions and
basic concepts
Nothing is rigid.                                     System, conﬁguration, rigid
body, displacement

Lots of stuff is articulated or way soft: tissue      Rotation, translation
Conﬁguration space,

(surgery), ﬂuids, food, paper, books, . . ..          degrees of freedom
Metrics

Group theory
Groups, commutative
groups
Lots of reasons                                           Displacements with
composition as a group
Noncommutativity of
displacements

A reasonable approximation for some objects.          Projective
geometry
Even non-rigid transformations can be factored into   Motivation
The projective plane
displacement + shape change.                          Homogeneous coordinates
Projections
Some things are invariant with respect to
displacements.
Other?
Lecture 2.
Moving and ﬁxed spaces                                            Kinematic
Foundations
Basic convention

Preview

Convention                                                  Deﬁnitions and
basic concepts
System, conﬁguration, rigid
We will consider displacements to apply to every point in   body, displacement
Rotation, translation

the ambient space.                                          Conﬁguration space,
degrees of freedom
Metrics

Group theory
Example                                                     Groups, commutative
groups
Displacements with

For example, planar displacements are described as          composition as a group
Noncommutativity of
displacements
motion of moving plane relative to ﬁxed plane.
Fixed plane
Projective
geometry
Mo                   Motivation
vin
gp
lan          The projective plane
e
Homogeneous coordinates
Projections

Moving and ﬁxed planes.
Lecture 2.
Rotations and translations                                              Kinematic
Foundations
Deﬁnitions

Preview

Deﬁnition (Rotation)                                              Deﬁnitions and
basic concepts
System, conﬁguration, rigid

A rotation is a displacement with at least one ﬁxed point.        body, displacement
Rotation, translation
Conﬁguration space,
degrees of freedom
Metrics

Deﬁnition (Translation)                                           Group theory
Groups, commutative

A translation is a displacement for which all points move         groups
Displacements with
composition as a group
equal distances along parallel lines.                             Noncommutativity of
displacements

Projective
geometry
Motivation
The projective plane
Homogeneous coordinates

O                                                                Projections

about O            point on the body      point not on the body
Lecture 2.
Conﬁguration space, degrees of freedom                           Kinematic
Foundations
Deﬁnitions

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement
Deﬁnition (Conﬁguration space)                             Rotation, translation
Conﬁguration space,
degrees of freedom

Conﬁguration space is the space comprising all             Metrics

conﬁgurations of a given system.                           Group theory
Groups, commutative
groups
Displacements with
composition as a group

Deﬁnition (Degrees of freedom (DoFs))                      Noncommutativity of
displacements

Projective
The Degrees of Freedom (DOFs) of a system is the           geometry
dimension of the conﬁguration space. (Less precisely:      Motivation
The projective plane

the number of reals required to specify a conﬁguration.)   Homogeneous coordinates
Projections
Lecture 2.
Systems, conﬁguration spaces, DOFs                                      Kinematic
Foundations
Examples

Preview
System                  Conﬁguration         DOFs
Deﬁnitions and
point in plane                 x, y            2             basic concepts
point in space               x, y , z          3             System, conﬁguration, rigid
body, displacement
Rotation, translation
rigid body in plane          x, y , θ          3             Conﬁguration space,
degrees of freedom

rigid body in space     x, y , z, φ, θ, ψ      6             Metrics

rigid body in 4-space          ???            ???            Group theory
Groups, commutative
groups
Displacements with
composition as a group
Noncommutativity of
displacements
r
Projective
geometry
Motivation
The projective plane

(qu             Homogeneous coordinates

")        Projections

u
/J*.,^btonl   6Pu<t'                 a^L|*J+,          5p*<*'
Solution: DOFs of a rigid body in E4 .                          Lecture 2.
Kinematic
Foundations
Most common answer: 4 translations and 4 rotations.
Why assume four rotational freedoms? Generalizing      Preview

from E3 ? Consider:                                    Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement

Space         Translational DOFs Rotational DOFs       Rotation, translation
Conﬁguration space,

E0 (point)              0                     0        degrees of freedom
Metrics

E 1 (line)              1                     0        Group theory

E2 (plane)
Groups, commutative
2                     1        groups
Displacements with
E3                      3                     3        composition as a group
Noncommutativity of

E4                      4                     6        displacements

Projective
The correct generalization for E n is n choose 2.      geometry
Motivation
Identify rotational freedoms not with a single axis,   The projective plane
Homogeneous coordinates
which works only in E3 , but with a pair of axes.      Projections

The proof is simple, after we have covered rotation
matrices.
Rotations in E4 are useful. See the lecture on
quaternions.
Lecture 2.
Tricky Cspaces                                                    Kinematic
Foundations
Examples

Deﬁnition of degrees of freedom: Dimension of the        Preview

Deﬁnitions and
conﬁguration space. But dimension is sometimes           basic concepts
tricky.                                                  System, conﬁguration, rigid
body, displacement
Rotation, translation

Simple case: smooth manifold Cspace, e.g. point in       Conﬁguration space,
degrees of freedom

plane.                                                   Metrics

Group theory
Difﬁcult case: square piece of paper, with diagonal      Groups, commutative
groups

creases, that bends only at the creases.                 Displacements with
composition as a group
Noncommutativity of
displacements
The Cspace is a network of
Projective
line segments. Not a                                     geometry
Motivation
manifold. What is the                                    The projective plane
Homogeneous coordinates
dimension?                                               Projections

Since the Cspace is the ﬁnite
union of a bunch of line
segments, the dimension is
Balkcom and Mason 2008
one.
Lecture 2.
Conﬁguration spaces                                                 Kinematic
Foundations
Metrics
We can deﬁne a metric, or a distance function, for       Preview

any conﬁguration space. What is a metric?                Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement

Deﬁnition                                                     Rotation, translation
Conﬁguration space,
degrees of freedom

A metric d on a space X is a function d : X × X → R           Metrics

satisfying:                                                   Group theory
Groups, commutative
groups
d(x, y ) ≥ 0 (non-negativity);                           Displacements with
composition as a group

d(x, y ) = 0 if and only if x = y ;                      Noncommutativity of
displacements

Projective
d(x, y ) = d(y , x) (symmetry);                          geometry
Motivation
d(x, z) ≤ d(x, y ) + d(y , z) (triangle inequality).     The projective plane
Homogeneous coordinates
Projections

Every space has a metric! But what would make a
suitable metric for conﬁguration space? How would
you devise a suitable metric for conﬁgurations of S2 ?
E2 ? E3 ?
Lecture 2.
Solution: metrics for Cspaces                                         Kinematic
Foundations

Possibilities:                                               Preview
1. Let every pair of conﬁgurations have distance 1.         Deﬁnitions and
2. For S2 , the minimum angle to rotate one                 basic concepts
System, conﬁguration, rigid
conﬁguration to the other.                               body, displacement
Rotation, translation
3. For En decompose displacement into rotation and          Conﬁguration space,
degrees of freedom
translation, and add the distance to the angle.          Metrics

4. Same idea, but scale angle by a characteristic length.   Group theory
Groups, commutative
5. In general: pick some ﬁnite set of points, and take      groups
Displacements with
the maximum distance travelled by the points.            composition as a group
Noncommutativity of
displacements
(1): useless. (2–5): if two conﬁgurations are close, in
Projective
the sense that corresponding points are close in the         geometry
Motivation
ambient space, then distance is small. (3):                  The projective plane
Homogeneous coordinates
dimensionally inconsistent.                                  Projections

A desirable property: invariance with respect to
displacements. (2) achieves this for S2 . Unattainable
for E2 and E3 .
Lecture 2.
Digressing for group theory                                     Kinematic
Foundations
Motivation

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement
Rotation, translation
Conﬁguration space,
degrees of freedom

Why? If you can show that your mathematical          Metrics

Group theory
construct is a group, then you can use algebra—you   Groups, commutative
groups
can write and solve equations.                       Displacements with
composition as a group

Displacements are a group, and we need to use        Noncommutativity of
displacements

algebra on them.                                     Projective
geometry
Motivation
The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Group                                                                Kinematic
Foundations
Deﬁnition

Preview
A group is a set of elements X and a binary operator ◦         Deﬁnitions and
basic concepts
satisfying the following properties:                           System, conﬁguration, rigid
body, displacement

Closure:                                                  Rotation, translation
Conﬁguration space,

for all x and y in X , x ◦ y is in X .                    degrees of freedom
Metrics

Associativity:                                            Group theory
Groups, commutative

for all x, y , and z in X , (x ◦ y ) ◦ z is equal to      groups
Displacements with
composition as a group
x ◦ (y ◦ z).                                              Noncommutativity of
displacements

Identity:                                                 Projective
geometry
there is some element, called 1, such that for all x in   Motivation

X x ◦ 1 = 1 ◦ x = x.                                      The projective plane
Homogeneous coordinates
Projections
Inverses:
for all x in X , there is some element called x −1 such
that x ◦ x −1 = x −1 ◦ x = 1.
Lecture 2.
Commutativity of groups                                         Kinematic
Foundations

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement
Rotation, translation
Conﬁguration space,
degrees of freedom

Note that commutativity is not required for a group.   Metrics

Some are, some are not.                                Group theory
Groups, commutative
groups
A commutative group is also known as an Abelian        Displacements with
composition as a group

group.                                                 Noncommutativity of
displacements

Projective
geometry
Motivation
The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Examples of groups                                               Kinematic
Foundations
Example
Preview

Let the elements be the integers Z ;                     Deﬁnitions and
basic concepts
let the operator be ordinary addition +.                 System, conﬁguration, rigid
body, displacement
Rotation, translation
Is it a group? Verify the properties.                    Conﬁguration space,
degrees of freedom
Metrics
Is it commutative?
Group theory
Groups, commutative
groups

Example                                                     Displacements with
composition as a group
Noncommutativity of
displacements

Let the elements be the reals R;                         Projective
geometry
Let the operator be multiplication ×.                    Motivation
The projective plane
Homogeneous coordinates
Is it a group?                                           Projections

Other examples: positive rationals with multiplication
(commutative); nonsingular k by k real matrices with
matrix multiplication (noncommutative).
Lecture 2.
Displacements with composition are a group                      Kinematic
Foundations

Preview
Every displacement D is an operator on the ambient
Deﬁnitions and
space X, mapping every point x to some new point       basic concepts
D(x) = x .                                             System, conﬁguration, rigid
body, displacement
Rotation, translation

The product of two displacements is the composition    Conﬁguration space,
degrees of freedom

of the corresponding operators, i.e.                   Metrics

Group theory
(D2 ◦ D1 )(·) = D2 (D1 (·)).                           Groups, commutative
groups

The inverse of a displacement is just the operator     Displacements with
composition as a group
Noncommutativity of
that maps every point back to its original position.   displacements

Projective
The identity is the null displacement, which maps      geometry
every point to itself.                                 Motivation
The projective plane
Homogeneous coordinates

In other words:                                            Projections

The displacements, with functional composition,
form a group.
Lecture 2.
SE(2), SE(3), and SO(3)                                        Kinematic
Foundations
Special Euclidean and Special Orthogonal groups

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid

These groups of displacements have names:                body, displacement
Rotation, translation
Conﬁguration space,
SE(2): The Special Euclidean group on the plane.    degrees of freedom
Metrics

SE(3): The Special Euclidean group on E3 .          Group theory
Groups, commutative

SO(3): The Special Orthogonal group.                groups
Displacements with
composition as a group

Whence the names?                                        Noncommutativity of
displacements

Special: they preserve orientation / handedness.    Projective
geometry
Orthogonal: referring to the connection with        Motivation
The projective plane

orthogonal matrices, which will be covered later.   Homogeneous coordinates
Projections
Lecture 2.
Noncommutativity of rotations.                                             Kinematic
Foundations

Preview
Does SO(3) commute? NO! No, no, no.
Deﬁnitions and
z                                                     y           basic concepts
System, conﬁguration, rigid
body, displacement
Rotation, translation
Conﬁguration space,
degrees of freedom
y                                 y                       x   Metrics
x                     z                       z
Group theory
x                                   Groups, commutative
groups
Displacements with
composition as a group
Noncommutativity of
displacements

Projective
z                                 y                               geometry
Motivation
The projective plane
Homogeneous coordinates
z                       z                   Projections

y                                             y
x                             x
x
Lecture 2.
Do displacements commute?        Kinematic
Foundations

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement
Rotation, translation
Conﬁguration space,
degrees of freedom
Metrics
Does SE(3) commute?
Group theory
Does SE(2) commute?         Groups, commutative
groups

Does SO(2) commute?         Displacements with
composition as a group
Noncommutativity of
displacements

Projective
geometry
Motivation
The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Projective geometry                                               Kinematic
Foundations
Motivation

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid
Why projective geometry?                                    body, displacement
Rotation, translation
Conﬁguration space,

It’s cool.                                             degrees of freedom
Metrics

It’s useful.                                           Group theory
Groups, commutative
groups
Speciﬁcally, it gives us points at inﬁnity which are   Displacements with
composition as a group
very useful in kinematics.                             Noncommutativity of
displacements

It also gives us a useful dual mapping between         Projective
geometry
points and lines.                                      Motivation
The projective plane

It’s really cool.                                      Homogeneous coordinates
Projections
Lecture 2.
The projective plane                                              Kinematic
Foundations
Overview

Preview

Deﬁnitions and
basic concepts
The basic idea:                                             System, conﬁguration, rigid
body, displacement

Rotation, translation
Conﬁguration space,
degrees of freedom

In the Euclidean plane, when two lines intersect you    Metrics

Group theory
get a point, but some pairs of lines don’t intersect:   Groups, commutative
groups
parallel lines. Euclid’s ﬁfth postulate.                Displacements with
composition as a group

Add some points, the ideal points or the points at      Noncommutativity of
displacements

inﬁnity. One point for each set of parallel lines.      Projective
geometry
Call the new structure the projective   plane—P2 .      Motivation
The projective plane
Homogeneous coordinates
We will do it concretely using homogeneous                  Projections

coordinates.
Lecture 2.
Homogeneous coordinates                                        Kinematic
Foundations
Deﬁnition
Let the Cartesian coordinates of some point in E2 be      Preview

Deﬁnitions and
basic concepts
2
(η, ν) ∈ R                        System, conﬁguration, rigid
body, displacement
Rotation, translation
Conﬁguration space,

Then we will say that                                     degrees of freedom
Metrics

Group theory
(x, y , w)   (wη, wν, w)                 Groups, commutative
groups
Displacements with
composition as a group

are the homogeneous coordinates of the point,             Noncommutativity of
displacements

provided                                                  Projective
geometry
w =0                                Motivation
The projective plane
Homogeneous coordinates

To go from homogeneous coordinates in R3 to Cartesian
Projections

coordinates in R2 :
     
x
 y  → x/w , w = 0                  (1)
y /w
w
Lecture 2.
Homogeneous coordinates                                            Kinematic
Foundations
Invariance to scaling

Preview

Scaling the homogeneous coordinates does not change          Deﬁnitions and
basic concepts
the point!                                                   System, conﬁguration, rigid
body, displacement
Rotation, translation
                                                      Conﬁguration space,
ax                                                  degrees of freedom

 ay  → ax/aw             x/w                          Metrics

=           , a, w = 0
ay /aw        y /w                         Group theory
aw                                                   Groups, commutative
groups
Displacements with
composition as a group

So, homogeneous coordinates represent a point in   R2   by   Noncommutativity of
displacements

a line through the origin of R3 .                            Projective
geometry
Motivation
           
 wx                                The projective plane

x                                           Homogeneous coordinates

↔     wy  w = 0                      Projections

y
w
             
Lecture 2.
Homogeneous coordinates                                                Kinematic
Foundations
Central projection

Preview
In homogeneous coordinate space, embed the
Deﬁnitions and
Euclidean plane as w = 1.                                  basic concepts
System, conﬁguration, rigid

Also embed the sphere x 2 + y 2 + w 2 = 1.                 body, displacement
Rotation, translation

A line through the origin of R3 probably (!)
Conﬁguration space,
degrees of freedom
Metrics
intersects the sphere in antipodal points             Group theory
intersects the w = 1 plane at the appropriate point   Groups, commutative
groups
(x/w, y /w).                                          Displacements with
composition as a group
Noncommutativity of
displacements

Projective
geometry
Motivation
The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Homogeneous coordinates                                               Kinematic
Foundations
Ideal points

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement

The original idea: extend   E2   by adding some ideal     Rotation, translation
Conﬁguration space,
degrees of freedom
points.                                                   Metrics

Group theory
Euclidean point:                                          Groups, commutative

line through origin of R3 intersecting w = 1 plane.       groups
Displacements with
composition as a group

Ideal point:                                              Noncommutativity of
displacements

line through origin of R3 not intersecting w = 1 plane.   Projective
geometry
With Cartesian coords, no place to put ideal points. With       Motivation
The projective plane

homogeneous coordinates, there’s a perfect spot!                Homogeneous coordinates
Projections
Lecture 2.
Projective plane                                                    Kinematic
Foundations
Deﬁnition

Deﬁne the projective plane P2 to be the set of lines     Preview

through the origin of E3 .                               Deﬁnitions and
basic concepts

A line in E2 is represented by plane through origin of
System, conﬁguration, rigid
body, displacement
Rotation, translation

E3 .                                                     Conﬁguration space,
degrees of freedom
Metrics
The ideal points form a line! The line at inﬁnity. The
Group theory
equator of the embedded sphere.                          Groups, commutative
groups
Displacements with
“Parallel lines” intersect at inﬁnity.                   composition as a group
Noncommutativity of
displacements

Projective
geometry
Motivation
The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Projective plane and duality                                     Kinematic
Foundations

Preview

Deﬁnitions and
basic concepts
System, conﬁguration, rigid
body, displacement

Duality. Two points determine a line. Two lines         Rotation, translation
Conﬁguration space,
degrees of freedom
determine a point. Every axiom of the projective        Metrics

plane has a dual axiom by switching “line” and          Group theory
Groups, commutative
“point”.                                                groups
Displacements with
composition as a group
Every theorem likewise has a dual theorem, and          Noncommutativity of
displacements
every proof a dual proof. In projective geometry, you   Projective
get to prove two theorems for the price of one proof.   geometry
Motivation
The projective plane
Homogeneous coordinates
Projections
Lecture 2.
Projective plane and duality                                    Kinematic
Foundations
Dual map

Preview
Using the homogeneous coordinate construction, the
Deﬁnitions and
dual mapping between points and lines is concrete.    basic concepts

A point in P2 is represented by a line through the
System, conﬁguration, rigid
body, displacement
Rotation, translation
origin of R3 , perpendicular to a plane through the   Conﬁguration space,
degrees of freedom

origin of R3 , representing a line in R2 .            Metrics

Group theory
Groups, commutative
groups
Displacements with
composition as a group
Noncommutativity of
displacements

Projective
geometry
Motivation
The projective plane
Homogeneous coordinates
Projections

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