2-Lecture_kin1 [Compatibility Mode]

8/6/2019 2-Lecture_kin1 [Compatibility Mode]

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Robotics

A robot consists of:

Links

Joints Number of joints specify the degrees of freedom

Problems of interest to control a robot are:


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1-Forward Kinematics

Given angles of both joints

Find the coordinates of Point A & B

In the base frame (common frame)


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1-Forward Kinematics

Position and orientation matrix (POSE)

Establish a systematic approach to findpose by using homogeneous

transformation


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2- Inverse Kinematics

Given 1 and 2 we know the coordinates

of A (Forward kinematics) Now given the coordinates of B find 1 and

2 (Inverse Kinematics)


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2- Inverse Kinematics

Solution is difficult

Not unique

May not exist


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3- Velocity Kinematics

Find the velocity of each joint given the

velocity of the tool Systematic approach to find the Jacobian

matrix is called cross-product form


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4- Path Planning & Trajectory

Generation Path planning trajectory generation

trajectory tracking

Define path then define time position-timetrajectory then follow it.


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5-Vision

Use camera to give feedback on robot

position

Use camera to give position of externalobjects


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6- Dynamics

Robots are mechanical objects on motion

Need to find differential equations to

define the model Equations are usually non-linear

Use Lagrangian dynamics


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7- Position Control

Tracking problem

Disturbance Rejection problem

Using Feed forward control and thesecond method of Lypanove.


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8-Force Control

May be used insteadof position control

Avoids uncertainty in the position

Ensures constant normal forcs along thetrajectory

Use Hybrid control and impedance control


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Kinematics: Pose (position and

orientation) of a Rigid Body


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You want to put your hand on the cup

Suppose your eyes tell you where the

mug is and its orientation in the robot

base frame(big assumption)

In order to put your hand on the object,

you want to align the coordinate frame

of your hand w/ that of the object

This kind of problem makes

representation of pose important...

Why are we studying pose?


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Representing Position:

Vectors


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Representing Position: vectors

x

y

p

5

2

2

5p (column vector)

25p (row vector)

x

y

z

25

2

p

p


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Vectors are a way to transformbetween two different reference

frames w/ the same orientation

The prefix superscript denotes the

reference frame in which the vector

should be understood

2

5pb

0

0pp

Same point, two different

reference frames

p

5

2

b

px

bx

by

py


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Note that I am denoting the axes as

orthogonalunit basis vectors

bxA vector of length one pointing

in the direction of the base

frame xaxis

yaxisby

py pframe y axis

This means perpendicular

p

5

2

b

px

bx

by

py


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What is this unit vector you speak of?

22

yx aaa Vectorlength/magnitude:

a

1 aDefinition of unit vector:

y

x

a

a

aThese are the elements ofa:

5

2

xb

yb

22

yx aa

aa

You can turn a into a unit vector of

the same direction this way:


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yyxx bababa

)cos(ba

And what does orthogonal mean?

Unit vectors are orthogonal iff(if and

only if) the dot product is zero:

xp is orthogonal to iff

First, define the dot product:

0ba when: 0a0b

0cos

or,

or, a

b

xp

yp

yp 0 yx pp


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A couple of other random things

bbb yxp 25

5

2

b xb

yb

x

y

zx

y

z

right-handed

coordinate frame

left-handed

coordinate frame

Vectors are elements of nR


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The importance of differencing two vectors

err

b

eff

b

object

b xxx

The effneeds to make a Cartesian

displacement of this much to reach the

object

object

bx

err

bxeff

bx


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The importance of differencing two vectors

errb

effb

objectb xxx

bbx

by

eff

effx

effy

objectobjectx

objecty

err

The effneeds to make a Cartesian displacement

of this much to reach the object


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Representing Orientation: Rotation

Matrices

The reference frame of the hand and

the object have different orientations

We want to represent and difference

orientations just like we did for

positions


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333231

232221

131211

aaa

aaa

aaa

A

332313

322212

312111

aaa

aaa

aaaT

A

333231

232221

131211

aaa

aaa

aaa

2

5p 25Tp

Before we go there review of matrix

transpose

TTT BABA Important property:


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2221

1211

aa

aaA

2221

1211

bb

bbB

and matrix multiplication

2222122121221121

2212121121121111

2221

1211

2221

1211

babababa

babababa

bb

bb

aa

aaAB

bab

baabababa T

y

x

yxyyxx

Can represent dot product as a matrix multiply:


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Same point - different reference frames

xa

p

5

2

ya yb

xb

8.3

8.3

2

5pa

8.3

8.3pb

for the moment, assumethat there is no difference

in position


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a

b

)cos()cos( bbabal

yyxx bababa

)cos(ba

Another important use of the dot product:

projection

l


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p

5

2

8.3

8.3

B-frames xaxis written

in A frame

B-frames yaxis writtenin A frame


xa

ya

b

ax

b

a

y


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xba

ba pppx cos

xa

pa

5

2

ya

8.3

8.3

b

ax


in A frame

b

a

y B-frames yaxis writtenin A frame



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pxp aba

x

b

5

2

8.3

8.3


in A frame

B-frames yaxis writtenin A frame

pyp aba

y

b


xa

pa

ya

b

ax

b

a

y


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pxp ABA

x

B

pyp

A

B

A

y

B

xA

p

5

2

yA

8.3

8.3

B

Ax

B

Ay

py

x

py

px

py

px

pA

T

B

A

T

B

A

AT

B

A

AT

B

A

A

B

A

A

B

A

B

Rotation matrix

pRp AT

B

AB

py

xp A

T

B

A

T

B

AB


2*1 2*1

2*2


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pRp AT

B

AB py

xp A

T

B

A

T

B

AB

The rotation matrix

By the same reasoning:

From last page:

pRp BT

A

BA py

xp B

T

A

B

T

A

BA


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The rotation matrix

p

ya

b

ax

b

ay

p

a

by

xb

yb

BABABA yxR

T

A

B

T

A

B

B

A

y

xR

a

bx


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Example 1: rotation matrix

xa

ya

xb

yb

cossin

sincosb

a

b

a

b

a yxR

sin

cosb

ax

cos

sinb

ay

cossin

sincosa

bR


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yA

zA

xA

45

zB

xB

yB

B

A

B

A

B

A

B

A zyxR

2

1

21

21

21

001

0

0BA

R

21

21

2

1

2

1

0

0100

B

AR


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ax

ay

az

bzbx

by

cs

ssccs

scscc

ss

csccs

ccscc

Rca

002

2

2


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Rotations about x, y, z

100

0cossin0sincos

zR

cos0sin010

sin0cos

yR

cossin0

sincos0

001

xR

These rotation matrices encode the basis vectors of the

after-rotation reference frame in terms of the before-

rotation reference frame


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Remember those double-angle formulas

sincoscossinsin

sinsincoscoscos


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Example 1: composition of rotation matrices

C

B

B

A

C

A RRR

p

ya

xb

yb

xa

yc

xc

12

21212121

21212121

22

22

11

11

cossin

sincos

cossin

sincos

ssccsccs

csscssccRca

1212

1212

cs

sc


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Example 2: composition of rotation matrices

ax

ay

az

bx

cx

csssccs

scscc

cs

sc

cs

sc

RRR cb

b

a

c

a

00010

0

1000

0


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Recap of rotation matrices

T

a

b

Ta

b

B

A

B

A

B

A

y

xyxR

baT

ab

ab RRR

1

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