1 @ McGraw-Hill Education Lecture 2 (III-LT1) Robot ...sksaha.com/sites/default/files/upload_data/documents/02-04kin.pdf · @ McGraw-Hill Education 4 Orientation Description 1. Direction

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1

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permission.

Lecture 2 (III-LT1)

Robot Kinematics (Ch. 5)by

S.K. SahaAug. 07, 2015 (F)@JRL301 (Robotics Tech.)


2

F

p΄

o

p

U

MOM

V

P

W

O

X

Z

Y

Pose Position + Rotation

Translation: 3

Rotation: 3

Total: 6

A moving body Pose or Configuration


3

xp

zp

ypF][p . . . (5.8)

[ ] , [ ] , and [ ]

1 0 00 1 0

00 1F F F

x y z

. . . (5.10)

Position Description

p = px x + py y + pz z . . . (5.9)


4

Orientation Description

1. Direction cosine representation

2. Fixed-axes rotations

3. Euler angles representation

4. Single- and double-axes rotations

5. Euler parameters representation

I will illustrate the first TWO only


5

u = ux x + uy y + uz z

. . . (5.11a)

v = vx x + vy y + vz z

. . . (5.11b)

w = wx x + wy y + wz z

. . . (5.11c)

Direction Cosine Representation

Refer to Fig. 5.12

p = puu + pvv + pww

. . . (5.12)


6

p = (puux + pvvx + pwwx)x + (puuy + pvvy + pwwy)y

+ (puuz + pvvz + pwwz)z . . . (5.13)

px = uxpu + vxpv + wxpw . . . (5.14a)

py = uypu + vypv + wypw . . . (5.14b)

pz = uzpu + vzpv + wzpw . . . (5.14c)

Substitute eqs. (5.11a-c) into eq. (5.12)

[p]F = Q [p]M . . . (5.15)


7

[p]F = Q [p]M . . . (5.15)

xwxvxu

zwzvzu

ywyvyuQpp

TTT

TTT

TTT,][,][x

wx

vx

u

zw

zv

zu

yw

yv

yu

F

up

wp

vp

xp

zp

yp

M

.. . (5.16)

uTu = vTv = wTw = 1, and

uTv(vTu) = uTw(wTu) = vTw(wTv) = 0 … (5.17)

Q is called Orthogonal

Orientation description 1


8

u v = w, v w = u, and w u = v . . . (5.18)

QTQ = QQT = 1 ; det (Q) = 1; Q1 = QT . . . (5.19)

Due to orthogonality


9

[ ,

[ ] ,

[ ]

0

0

001

u]

v

w

F

F

F

CαSα

SαCα

. . . (5.20)

Example 5.6 Rotations [Elementary] (Fig. 5.13a)In

terp

reta

tion 1


10

100

0

0

CS

SC

ZQ . . . (5.21)

CS

SC

CS

SC

XY

0

0

001

;

0

010

0

QQ

. . . (5.22)


11

Non-commutative Property: An Illustration

Fig. 5.20 Successive rotation of a box about Z and Y-axes


12

Non-commutative Property (contd.)

Fig. 5.21 Successive rotation of a box about Y and Z-axes


13




permission.

Lecture 3

Robot Kinematics (Ch. 5)by

S.K. SahaAug. 10, 2015 (M)@JRL301 (Rob. Tech.)


14

Recap

• Orientation representations

– Non-commutative

• Direction cosines: Has disadv. of 9 param.

• Fixed-axes (RPY) rotations (12 sets)


15

Homogeneous Transformation

F

p΄

o

p

U

MOM

V

P

W

O

X

Z

Y

Task: Point P is known in moving frame M. Find P in fixed frame F.

Fig. 5.23 Two coordinate frames


16

p = o + p . . . (5.45)

[p]F = [o]F + Q[p’]M . . . (5.46)

1

][

1

][

1

][T

F MF poQp

0. . . (5.47)

MF ][][ pTp . . . (5.48)

Homogenous Transformation


17

TTT 1 or T1 TT . . . (5.49)

1

][T

TT1

0

oQQT F . . . (5.50)

1000

1100

2010

0001

T

. . . (5.51)

Example 5.10 Pure Translation

T: Homogenous transformation matrix (4 4)

Fig. 5.24 (a)


18

. . . (5.52)

Example 5.11 Pure Rotation

30 30 0 0

30 30 0 0

0 0 1 0

0 0 0 1

3 10 0

2 2

1 30 0

2 2

0 0 1 0

0 0 0 1

T

o o

o o

C S

S C

Fig. 5.24 (b)


19

Like rotation matrices homogeneous transformation

matrices are non-commutative, i. e.,

Non-commutative Property

TATB TBTA


20

Denavit and Hartenberg (DH) Parameters

• Serial chain

- Two links connected

by revolute or

prismatic joint

• Four parameters– Joint offset (b)

– Joint angle ()

– Link length (a)

– Twist angle ()

Fig. 5.27 Serial manipulator


21

• Joint axis i: Link i-1 + link i

• Link i: Fixed to frame i+1 (Saha) / frame i (Craig)

DH Variables

bi and i

[Screw@Z]

Constants

ai and i

[Screw@X]

Saha XiXi+1@Zi ZiZi+1@Xi+1

Craig Xi-1Xi@Zi ZiZi+1@Xi

Z’’’i

Zi+1


22

• bi (Joint offset): Length of the intersections of the

common normals on the joint axis Zi, i.e., Oi and

Oi. It is the relative position of links i 1 and i.

This is measured as the distance between Xi

and Xi + 1 along Zi.

Z’’’i

Zi+1


23

• i (Joint angle): Angle between the orthogonal projections of

the common normals, Xi and Xi + 1, to a plane normal to the

joint axes Zi. Rotation is positive when it is made counter

clockwise. It is the relative angle between links i 1 and i.

This is measured as the angle between Xi and Xi + 1 about Zi.

Z’’’i

Zi+1


24

• ai (Link length): Length between the O’i and Oi

+1. This is measured as the distance between

the common normals to axes Zi and Zi + 1 along

Xi + 1.

Z’’’i

Zi+1


25

• i (Twist angle): Angle between the orthogonal

projections of joint axes, Zi and Zi+1 onto a plane

normal to the common normal. This is measured as

the angle between the axes, Zi and Zi + 1, about axis Xi

+ 1 to be taken positive when rotation is made counter

clockwise.

Z’’’i

Zi+1


26

Revolute Joint

Fig. 5.28

• DH@Z (Variable)– Joint offset (b)

– Joint angle ()

• DH@X (Const.)– Link length (a)

– Twist angle ()

Z’’’i

Zi+1


27

Tb =

1000

100

0010

0001

ib. . . (5.60a)

T =

1000

0100

00

00

ii

ii

CθSθ

θSCθ. . . (5.60b)

Mathematically• Translation along Zi

• Rotation about Zi


28

1000

00

00

0001

ii

ii

CαSα

αSCαT = . . . (5.60d)

Ta =

1000

0100

0010

001 ia

. . . (5.60c)

• Translation along Xi+1

• Rotation about Xi+1


29

Ti = TbTTaT . . . (5.61a)

Ti =

1000

0 iii

iiiiiii

iiiiiii

bCαSα

SθaSαCθCαCθSθ

CaSαSθCαSθCθ

. . . (5.61b)

• Total transformation from Frame i to Frame i+1

Rotation

Matrix

Po

sitio

n

Do it yourself!


30

Spherical-type Arm

• DH-parameters

Link bi i ai i

1 0 1 (JV) 0 /2

2 b2 2 (JV) 0 /2

3 b3

(JV)

0 0 0

Fill-up the DH parameters

Fig. 5.32 A spherical arm

C:/Users/SAHA/Desktop/RoboAnalyzer.lnk


31

PUMA 560

i Variable

DH

Constant

DH

bi i ai i

1 0 1 0 -/2

2 0 2 a2 0

3 B3 3 a3 -/2

4 b4 4 0 /2

5 0 5 0 -/2

6 0 6 0 0

Fig. 5.35 PUMA 560 and its frames


32




permission.

Lecture 4 (SIT Sem. Rm.)

Forward and Inverse Kinematics

(Ch. 6)by

S.K. SahaAug. 12, 2015 (W)@JRL301(Robotics Tech.)


33


Inverse: 1st soln.

.

Inverse: nth soln.

Forward: One soln.S

olv

e

Non

-lin. e

qns.

Multip

ly

+ A

dd


34

Three-link Planar Arm

Ti =

1000

0100

0

0

iiii

iiii

SθaCθSθ

CaSθCθ

• DH-parameters

, for i=1,2,3

Link bi i ai i

1 0 1 (JV) a1 0

2 0 2 (JV) a2 0

3 0 3 (JV) a3 0

• Frame transformations

(Homogeneous)

Fill-up the DH

parameters

Fill-up with the elements

Fig. 5.29 A three-link planar arm


35

DH Parameters of Articulated Arm

Link bi i ai i

1 0 1 (JV) 0 π/2

2 0 2 (JV) a2 0

3 0 3 (JV) a3 0


36

Matrices for Articulated Arm1 1

1 1

10 1 0 0

0 0 0 1

c 0 s 0

s 0 c 0

T

2 2 2 2

2 2 2 2

2

c s 0 a c

s c 0 a s

0 0 1 0

0 0 0 1

T

3 3 3 3

3 3 3 3

3

c s 0 a c

s c 0 a s

0 0 1 0

0 0 0 1

T

1000

sasa0cs

)cac(ascsscs

)cac(acssc-cc

233222323

2332211231231

2332211231231

)(T … (6.11)

../../roboanalyzer/RoboAnalyzer6/RoboAnalyzer6.exe


37

Inverse Kinematics

• Unlike Forward Kinematics, general solutions

are not possible.

• Several architectures are to be solved

differently.


38

Two-link Arm

X22a2

a1

X1

X3

Y2

Y1 Y3

1px

py

12211

12211

sasap

cacap

y

x

21

2

2

2

1

22

22 aa

aappc

yx

2

22 1 cs

2 = atan2 (s2, c2)

Δ

psap)ca(as

xy 22221

1

22

221

2

2

2

1 2 yx ppcaaaaΔ

Δ

psap)ca(ac

yx 22221

1

1 = atan2 (s1, c1)

1

2

RoboA

naly

zer


../../../../saha-software/roboanalyzer/RoboAnalyzer 7.1


39

Inverse Kinematics of 3-DOF RRR Arm

321 θθθφ

123312211 cacacapx

123312211 sasasapy

122113 cacac φ apw xx

122113 sasas φ apw yy

… (6.18a)

… (6.18b)

… (6.18c)

… (6.19a)

… (6.19b)


40

w2x + w2

y = a12+ a2

2 + 2 a1a2c2

21

2

2

2

1

22

22 aa

aawwc 21

2

22 1 cs

2 = atan2 (s2, c2) . . . (6.21)

2121221 ssa)ccaa(wx

2121221y sca)sca(aw

Δ

wsaw)ca(as

xy 22221

1

Δ

wsaw)ca(ac

yx 22221

1

22

221

2

2

2

1 2 yx wwcaaaaΔ

1 = atan2 (s1, c1) . . . (6.23c)

3 = - 1 2 . . . (6.24)

… (6.22a)

… (6.22b)

… (6.20a)

… (6.20b,c)

… (6.23a,b)


41

Numerical Example

3 5

2 2

3 31

2 2T

10 3

2

10

2

0 0 1 0

0 0 0 1

• An RRR planar arm (Example 6.15). Input

where = 60o, and a1 = a2 = 2 units, and a3 = 1 unit.

Rotation

Matrix

Origin

of end-

effector

frame

4.23

1.86

0

Do it yourself Verify using RoboAnalyzer


C:/Users/SAHA/Desktop/RoboAnalyzer.lnk


42

Using eqs. (6.13b-c), c2 = 0.866, and s2 = 0.5,

Next, from eqs. (6.16a-b), s1 = 0, and c1= 0.866.

Finally, from eq. (6.17) ,

Therefore …(6.30b)

The positive values of s2 was used in evaluating 2 = 30o.

The use of negative value would result in :

…(6.30c)

2 = 30o

1 = 0o.

3 = 30o.

1 = 0o 2 = 30o, and 3 = 30

1 = 30o 2 = -30o, and 3 = 60o


../../../book/robotics/revision07-08/ch6ikin3.m


43

Watch

• Forward and Inverse Kinematics: Watch 3/3 of

IGNOU Lectures [29min]

https://www.youtube.com/watch?v=duKD8cvtBTI

• For more clarity: Watch 12 of Addis Ababa

Lectures [77 min]

[https://www.youtube.com/watch?v=NXWzk1toze4

• Robotics (13 of Addis Ababa Lectures): Inverse

Kinematics [82 min]

https://www.youtube.com/watch?v=ulP3YiJLiEM

https://www.youtube.com/watch?v=duKD8cvtBTI

https://www.youtube.com/watch?v=NXWzk1toze4

https://www.youtube.com/watch?v=ulP3YiJLiEM


44

Velocity Analysis

1 2where andJ j j j n

i

i

i ie

, if Joint is revolutei

ej

e aprismatic isointJif, i

iei

i

ae

0j

et Jθ

1

twistof end - effector : ; Joint rates : e

e

e

n

ωt θ

v

Jacobian maps joint rates into end-effector’s velocities. It

depends on the manipulator configuration.

nee1e aeaeae

eeeJ

n2

n221

1

. . (6.86)


45

Jacobian of a 2-link Planar Arm

ee 2211 aeaeJ

1 1 2 12 2 12

1 1 2 12 2 12

Hence, Ja s a s a s

a c a c a c

1 2where [0 0 1]e eT

1 1 2

1 1 2 12 1 1 2 12[ 0]

a a a

e

Ta c a c a s a s

2 2

2 12 2 12[ 0]

a a

e

Ta c a s


46

Example: Singularity of 2-link RR Arm

12212211

12212211

cacaca

sasasaJ 2 = 0 or


47




permission.

Lecture 4 (SIT Sem. Rm.)


(Ch. 6)by

S.K. SahaAug. 12, 2015 (W)@JRL301(Robotics Tech.)


48




permission.

Statics and Manipulator

Design (Ch. 7)


49

Principle of Virtual Work

• Relation between two virtual displacements

(Can be derived from velocity expression)

θτxw TTe

θJx

θτθJw TTe

TT

e τJw

e

TwJτ

… (7.28)

… (7.29)

… (7.32)

… (7.31)


50

Example: 2-link RR Planar Arm

1 1 1 01 1

1 2 2 1 2

[ ] [ ] e n

T

x y

τ

a f sθ (a a cθ )f

y

T faτ 2212222 ][][ ne

fJτT

2

1

τ

ττ

0

y

x

f

f

f

00

0

2

22121

a

acθasθaT

J


51

Two Jacobian Matrices

• From

Statics

00

0

2221

21

a acθa

sθa

J

• From

Kinematics

12212211

12212211

cacaca

sasasaJ


52

Jacobian from Statics in Frame 1

00

00

0

100

0

0

100

0

0

][

12212211

12212211

2221

21

22

22

11

11

1

cθacθ acθa

sθ asθ asθa

a acθa

sθa

cs

sc

cs

sc

J

… (7.34)

• Without the last row, it is the same as

the one from kinematics Should be!


53

Manipulator Design

• High investment in robot usage low

technological level of mechanical structure

• Functional Requirements

• Kinetostatic Measures

• Structural Design and Dynamics

• Economics


54

Functional Requirements of a

Robot

• Payload

• Mobility

• Configuration

• Speed, Accuracy and Repeatability

• Actuators and Sensors


55

bmin b bmax, for 0 360o o


56

• Dexterity

• Manipulability

• Nonredundant manipulator square

Jacobian

Dexterity and Manipulability

det( )Jdw

det( )T

mw JJ

det( )mw J d mw w

… (7.44)


57

Motor Selection (Thumb Rule)

• Rapid movement with high torques (>

3.5 kW): Hydraulic actuator

• < 1.5 kW (no fire hazard): Electric

motors

• 1-5 kW: Availability or cost will

determine the choice


58

Simple Calculation

2 m robot arm to lift 25 kg mass at 10

rpm

• Force = 25 x 9.81 = 245.25 N

• Torque = 245.25 x 2 = 490.5 Nm

• Speed = 2 x 10/60 = 1.047 rad/sec

• Power = Torque x Speed = 0.513 kW

• Simple but sufficient for approximation


59

Practical Application

Subscript l for load; m for motor;

G = l/m (< 1); : Motor + Gear box efficiency

Trapezoidal Trajectory


60

Accelerations & Torques

Ang. accn. during t1:

Ang. accn. during t3:

Ang. accn. during t2: Zero (Const. Vel.)

Torque during t1: T1 =




61

RMS Value


62

Motor Performance


63

Final Selection

• Peak speed and peak torque

requirements , where TPeak is max of

(magnitudes) T1, T2, and T3

• Use individual torque and RMS values

+ Performance curves provided by the

manufacturer.

• Check heat generation + natural

frequency of the drive.


64

Dynamics and Control

Measures

1

2n r

• Rule of Thumb

n

r

: closed-loop natural frequency

: lowest structural resonant frequency

… (7.51)


65

Manipulator Stiffness

2

1 2

1 1 1

ek k k ek

equivalent stiffness

gear ratio … (7.48)


66

Link Material Selection

• Mild (low carbon) steel:

Sy = 350 Mpa; Su = 420 Mpa

• High alloyed steel

Sy = 1750-1900 Mpa; Su = 2000-2300

Mpa

• Aluminum

• Sy = 150-500 Mpa; Su = 165-580 Mpa


67

Driver Selection

• Driver of a DC motor: A hardware unit

which generates the necessary current

to energize the windings of the motor

• Commercial motors come with

matching drive systems


68

Summary

• Forward Kinematics

• Inverse kinematics

– A spatial 6-DOF wrist-portioned has 8

solutions

• Velocity and Jacobian

• Mechanical Design


69

Thank [email protected]

http://sksaha.com

http://web.iitd.ac.in/~saha

Documents

1 @ McGraw-Hill Education Lecture 2 (III-LT1) Robot ...sksaha.com/sites/default/files/upload_data/documents/02-04kin.pdf · @ McGraw-Hill Education 4 Orientation Description 1. Direction