Geometric Representations and Transformations

J. Irving Vasquez

Centro de Innovacion y Desarrollo Tecnologico en Computo

20 de febrero de 2020

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Introduccion

Image from[Kostas]

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Body Frame

Figura: A sensor (red point) is not always in the center of the body frame.

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Definitions

Espacio euclidiano Un espacio euclidiado de dimension n, es elconjunto de tuplas de dimension n,(x1, x2 . . . , xn),

que satisface la metrica(∑n

i=1 x2i

) 12

Point Una tupla de n-elementos, (x1, x2 . . . , xn)

Linea Dados dos puntos, una linea esta formada lacombinacion lineal de ellos.

αp1 + (1− α)p2 α ∈ R

Segmento de linea Un segmento de linea es la combinacionconvexa de dos puntos.

αp1 + (1− α)p2 α ∈ [0, 1]

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Definitions

Convex set A set X ⊂ Rn is called convex if and only if, for anypair of points in X , all the points along the linesegment that connects them are contained in X .

λx1 + (1− λ)x2 ∈ X

Convex Hull The convex hull of a set of points, S , in En is theboundary of the smallest convex set in En containingS .

Polygon A polygon is a set of vertices and edges, Q = (V ,E ),where V is a set of points on the plane and E is a setof 2-d vertices tuples.

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Geometric Modeling

The world W (R2 or R3) contains closed sets:

RobotsA ⊆ WObstaclesO ⊆ W

Representation approaches:

BoundarySolid

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Polygonal Models

We can construct bodies by primitives and set theory operations.

Boundary representation

2D, convex polygons, G = {V ,E}Solid

Intersection of half planes

y < mx + b

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Nonconvex polygons

Nonconvex polygon

A collection of convex polygons

O = O1 ∪ O2 ∪ · · · ∪ On (1)

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Polyhedral models

A 3D world

Boundary:A set of vertices, edges and faces P = {V ,E ,F}Solid: half spaces representation

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3D triangles

A triangle = {(x1, y1, z1), (x2, y2, z2), (x3, y3, z3)}A set of triangles represents a body.Triangular meshes

Figura: Triangular Mesh of the Bebop drone

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Semi algebraic models

for algebraic models, the functions are linear

for semi-algebraic models, the functions can be anypolynomial with real-valued coefficients.

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Assignment

Implement in Python:

Notebook: “representacion poligono”Represent a robot with a polygonPlot the Polygon

Figura: A triangular representing a robot.

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Assignment

Display the following geometric models in RViz:A Polygon

A Mesh (binary .stl or Ogre .mesh in 1.0, COLLADA (.dae) in1.1)

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Rigid Motions

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We will use an analytic approach

The superscript is used to denote the reference frame:

p0 = [5, 6] , p1 = [−2.8, 4.2] (2)

p is a geometric entity and it is independent from itscoordinate representation.

a vector has direction and magnitude.

Thus, v1 and v2 are invariant with respect to the choice ofcoordinate systems,

The representation by coordinates of these vectors dependsdirectly on the choice of reference coordinate frame.

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Rotations

Rotations in the plane

We can represent the coordinates of the point p1 in the frameO0 by getting the proyections of the verctors over the frame.

We obtain:R0

1 =[x0

1 |y01

]where x0

1 and y01 are the coordinates in frame O0 of unit

vectors x1 and y1.

It is called Rotation Matrix.

R01 =

[cos θ − sin θsin θ cos θ

]

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Special orthogonal group

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Assigment

Mestre que la inversa de una rotacion por el angulo θ es iguala la transpuesta.

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Rotations in three dimensions

In three dimensions, each axis of the frame x1, y1 y z1 isprojected onto coordinate frame x0, y0 y z0. The resultingrotation matrix is given by

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Rotation in z

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Rotation in z

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The basic rotations

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Rigid Body

A rigid body is a geometric representation of a physical bodyso that its shape does not change over time.

Figura: Two rigid bodies in the environment (A Kinect Sensor and a box).

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The rotational transformation

Given the coordinates p1 of the point p (i.e., given thecoordinates of p with respect to the frame x1 y1 z1 ), we wishto determine the coordinates of p relative to a fixed referenceframe x0 y0 z0.

p = ux1 + vy1 + wz1

Recalling that the dot product of two unit vectors gives theprojection of one onto the other

p0 =

p · x0

p · y0

p · z0

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The rotational transformation

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The rotational transformation

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Rigid motions

We can use rotation matrices to represent rigid motions thatcorrespond to pure rotations (In the same frame).

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Rigid motions

We can rotate a point in the same reference frame by doing:

p0b = Rz,πp

0a

In general,p0b = Rp0

a

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Note!

A rotation matrix can be interpreted in three distinct ways:

It represents a coordinate transformation relating thecoordinates of a point p in two different frames.

It gives the orientation of a transformed coordinate frame withrespect to a fixed coordinate frame.

It is an operator taking a vector and rotating it to a newvector in the same coordinate system.

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Composition of Rotations

Consider two sequential rotations:

p0 = R01p

1

p1 = R12p

2

p0 = R02p

2

Then:

p0 = R01R

12p2

Composition law (post-multiplication)

R02 = R0

1R12

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Composition of Rotations

Exercise. Suppose a rotation matrix R represents a rotation ofangle φ about the current y -axis followed by a rotation of angle θabout the current z-axis. What is the rotation matrix?

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Rotations with respect of a fixed frame

Some times it is desired to perform a sequence of rotations,each about a given fixed coordinate frame, rather than aboutsuccessive current frames.

The matrices are multiplied in the reverse order

R02 = R0

1

[(R0

1

)−1RR0

1

]= RR0

1

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Note!

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Assignment

Lets suppose a mobile robot rotated by θ = 35o, that observes afeature point at probot = [10, 32, 52].

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Parameterized representations

An arbitrary rotation ca be represented using only threeindependent variables.

Some conventions are:

Euler anglesYaw, pitch, rollAxis-angle

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Euler Rotations

Given three sucesive angles (φ, θ, ψ):

First rotate about z by φThen rotate about y by θFinally rotate about the current z by ψ

RZYZ = Rz,φRy ,θRz,ψ

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Roll, pitch, yaw angles

Given three sucesive angles (φ, θ, ψ):

First rotate about z by φThen rotate about y by θFinally rotate about the current x by ψ

RZYX = Rz,φRy ,θRx ,ψ

Figura:

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Assigment

https://github.com/irvingvasquez/ra_programas

Resolver el ejercicio Rotaciones de Euler

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Gimbal Lock

When all three gimbals are lined up, the system can only move intwo dimensions.

Figura:

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Quaternions

A generalization of complex numbers

William Hamilton (1843)

i2 = j2 = k2 = ijk = −1

q = a + bi + cj + dk

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Performing rotations

We need the quaternion

q = a + bi + cj + dk

and its inverse

q = a− bi − cj − dk

then

p = [xyz ]T

pnew = qpq

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Euler to Quaternion

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Quaternion to Euler

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Quaternion to Rotation

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