c01 – 2009.01.20 advanced robotics for autonomous manipulation

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C01 – 2009.01.20Advanced Robotics for Autonomous

ManipulationGiacomo MaraniAutonomous Systems Laboratory, University of Hawaii

Department of Mechanical Engineering ME 696 – Advanced Topics in Mechanical Engineering

http://www2.hawaii.edu/~marani

C01: Geometry of robotics structures

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Summary

1. Vectors (definitions and operations)2. Coordinate systems3. Rotation matrices4. Representation of rotation matrices5. Transformation matrices6. Geometry of robotics structures

Contents1. Vectors2. Coordinate sys.3. Rotation Matrices4. Representation5. Transformation M.6. Geometry7. Example

ME696 - Advanced Robotics

Vectors: definitions

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Point

Point in a Cartesian system:Spherical reference system:Line segment: a part of a line that is bounded by two distinct

end points.Oriented line segment:

Given two oriented line segments [P-Q] and [P`-Q`], they are equipollent if they have the same direction, length and orientation.

Given an oriented segment [P-Q], a correspondent free vector is the whole class of line segments equipollent to [P-Q].

Note the difference with the bound vector.

ME696 - Advanced Robotics – C01

321 ,, cccP

,,rP

i

j

k

p1

p3

P

Q

P`

Q`


Vectors: operations

4

Dot (scalar) product

Properties:

Cross (vector) product

Properties:


)cos(2 vvvv 121

v1

v2

v v v v21 2 1

v a b v a v b ( )

)sin(2 vvvv 121

v1

v1

v1 v2

v v v v1 2 2 1

v a b v a v b ( )


Coordinate systems

5

Coordinate systems

In geometry and kinematics, coordinate systems are used not only to describe the (linear) position of points, but also to describe the angular position of axes, planes, and rigid bodies. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system).


Oa

< a >< 0 >

k

i

ji

i

j

j

k

k



Ob

Coordinate systems

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Coordinate systems

< a > { Oa, ia, ja, ka } { Ob, ib, jb, kb }

Any vector can be uniquely expressed w.r.t either <a> and :

v = c1 ia + c2 ja + c3 ka

v = h1 ib + h2 jb + h3 kb


Oa

< a >< 0 >

k

i

ji

i

j

j

k

k



Ob

Coordinate systems

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Orthogonal systems

The reference frame < a > { Oa, ia, ja, ka } is orthogonal with right-handed orientation if:

orthogonal with left-handed orientation:

If i , j , k are also of unit length the frame are orthonormal.


k i ji j

a a a

a a

,.0

k i ji j

a a a

a a

,.0

Oa

< a >

i

j

k

Algebraic vector

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Algebraic vector

Given 2 free vectors v and w in an orthonormal-right-handed reference system we have:

Dot product:

Cross product: where:


a

ccc

v

1

2

3

3

b

hhh

v

1

2

3

3

Oa

< a >

i

i

j

j

k

k



Ob

P

Q

v w v w a T a

a a a( )v w v w a

c cc cc c

v

00

0

3 2

3 1

2 1

Rotation matrices

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Position of reference systems

Given two orthonormal reference systems <a> and , the Rotation Matrix of with respect to <a> is defined by:

Properties:


ba a

ba

ba

bR i j k 3 3

ab

T abi j 0 a

bT a

bi k 0 ab

T abj k 0

ab

T abi i 1 a

bT a

bj j 1 ab

T abk k 1

ba

ba TR R 1

det( )baR 1

Oa

< a >

i

i

j

j

k

k



Ob

Rba

Rotation matrices

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Change of reference system

Problem statement:Given a free vector v with known projection w.r.t. the frame , we want to compute the projection w.r.t. the frame <a>:

Hence:

similarly:


bbbbbbb zyx kjiv

ab

ab b

ab b

abx y zv i j k

aba bRv v b

ab aRv v

Oa

< a >

i

i

j

j

k

k



Ob

Rba

Rotation matrices

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Change of reference system for the cross-product operator


a

c cc cc c

v

00

0

3 2

3 1

2 1

a a a( )v w v w

b b b( )v w v w

a bba b

ba b b

ba b

ba T aR R R Rv w v w v w v w

aba b

ba TR Rv v


Rotation matrices

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Concatenation of rotation matrices

Problem statement:

Given n reference systems, we want to compute the rotation matrix between 2 frames <h> and <k> such as:

Algorithm:1) Identify an oriented path from <k> to <h>2) Pre-multiply the vector kv with all the rotation matrices

encountered (if the arrow is not opposite the rotation matrix is transposed)


?

< 0 >

< 1 >



< j >

< k >

< h >

hkh kRv v

hhj T

ji T

ki kR R Rv v k

hhj T

ji T

kiR R R R

Representation of rotation matrices

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Exponential representation

Problem statement:The frame , initially coincident with <a>, is rotated of an angle theta around the axis specified by v.

Expanding the exponential we have:


ka

ia

ja

ib

jb

kb

ba OO

b

av

ba R R e ( , )v v

ev v p c v v p v s v v p v s

v v p v s v v p c v v p v sv v p v s v v p v s v v p c

x x x y z x z y

x y z y y y z x

x z y y z x z z

v

cs sinp

cos( ), ( ), cos( ).

1

R R( , ) ( , )v v

R R k( , ) ( , )v v 2

R k I( , )v v2

Representation of rotation matrices

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Exponential representation

Special cases:


ka

ia

ja

ib

jb

kb

ba OO

b

av

R sinsin

( , ) cos( ) ( )( ) cos( )

i

1 0 000

Rsin

sin( , )

cos( ) ( )

( ) cos( )j

00 1 0

0

Rsin

sin( , )cos( ) ( )( ) cos( )k

00

0 0 1


Rotation matrices

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Roll-Pitch –Yaw (Euler)


j0

i0

z0k

a (yaw)

b (pitch)

g (roll)

k1 k0

j1

i1

i2

j2 j1

k2

j3k3

i3 i2

g

a

b

a

b1

b2

b

ba R R R R e e e k j i k j i, , ,a b g a b g


Rotation matrices

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Roll-Pitch –Yaw (Euler)


ba R R R R e e e k j i k j i, , ,a b g a b g

ba R


Transformation matrices

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Change of reference system for points in space

Problem statement:Given a generic point P in the space, compute its coordinates w.r.t. the frames <a> and 

We have:


k

i

ji

i

j

j

k

k

P

0

Oa a

a Pb P b

Ob

a aaP P O ( )

b bbP P O ( )

a b aP P( )

a ab b

a bP O R P

Transformation matrices

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Change of reference system for points in space

Point P in : Representation of P in homogeneous coordinates:

Transformation matrix:


k

i

ji

i

j

j

k

k

P

0

Oa a

a Pb P b

Ob

bb

b

b

Pxyz

b

b

b

b

P

xyz

1

ba b

a abT

R O

0 0 0 1

a ab b

a bP O R P

aba bP T P

Geometry of robotics structures

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Multibody systems


Link Giunto

i 1

i

i 1



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Denavit-Hartenberg




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Multibody systems


1 0,

11,

1 2,

3 2,

31, 1 3,

2 1,

2 2, 2 3,

2 4,

2 5,

1 4,

1 5,

1 6,


Simulation EnvironmentRobotics Developer Studio

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RDS: Simple application example

5 Degrees of freedom linear chain.

Link 1

Link 2

Link 3

Joint 1

Joint 2

Joint 3

Link 5

Joint 5

r2oj


End of presentation

c01 – 2009.01.20 advanced robotics for autonomous manipulation

Documents

rotation matrices4

rotation matrix

advanced topics

world coordinate system

local coordinate system

kijiijjkk b ob contents1

ijk contents1

n reference systems