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PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachersand educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it withoutpermission.

Lecture 2 Robot Kinematics

byS.K. Saha

Aug. 03’16 (W)@JRL301 (Robotics Tech.)

@ McGraw-Hill Education 2

Outline• Links and Joints

• Kinematic chains

• Degrees-of-freedom (DOF)

• Pose (≡ Configuration)

• Homogeneous transformation

• Denavit-Hartenberg Parameters

• Conclusions

@ McGraw-Hill Education 3

Link

Link

Joint

Joint

TAL Robot

@ McGraw-Hill Education 4

Transformations

• Robot Architecture– Links: A rigid body with 6-DOF– Joints: Couples 2 bodies. Provide

restrictions• Relationship between joint motion (input)

and end-effector motion (output) – Transformations between different

coordinate frames are required

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Joints or Kinematic Pairs• Lower Pair

– Surface contact: Hinge joint of a door

• Higher pair– Line or point contact: Roller or ball rolling

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Lower Pair: Revolute Joint

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Lower Pair: Prismatic Joint

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Lower Pair: Helical Joint

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Lower Pair: Cylindrical Joint

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Lower Pair: Spherical Joint

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Closed Kinematic Chain

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Open Kinematic Chain

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Degrees-of-Freedom (DOF)• Number of independent (or minimum)

coordinates required to fully describe pose or configuration (position + rotation)– A rigid body in 3D space has 6-DOF

• DOF = Coordinates - Constraints– Grubler formula (1917) for planar

mechanisms, DOF = 3 (r-1) – 2p– Kutzbach formula (1929) for spatial

systems, DOF = 6 (r-1) – 5p

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A 4-bar Mechanism

n = 3 (r − 1) − 2p= 3(4-1) − 2×4= 1

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A Robot Manipulator

n = 6 (r − 1) − 5p= 6(7-1) − 5×6 = 6

@ McGraw-Hill Education 16

n = 3 (r − 1) − 2p= 3(5-1) − 2×6 = 0

@ McGraw-Hill Education 17

F

p΄

o

p

U

MOM

V

P

W

O

X

Z

Y

Pose ≡ Position + Orientation

Translation: 3Rotation: 3

Total: 6

A moving body Pose or Configuration

Position (noun) or Translation (verb):

Easy (unique)

Orientation (noun) or Rotation

(verb): Difficult (non-unique)

@ McGraw-Hill Education 18

[ ,

[ ] ,

[ ]

0

0001

u]

v

w

F

F

F

CαSα

SαCα

⎡ ⎤⎢ ⎥

≡ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥

≡ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥

≡ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−

. . . (5.20)

Example 5.6 Elementary Rotations @ Z [5.13(a)]

Fig. 5.13

ClueCoordinate

transformation of Class XII

@ McGraw-Hill Education 19

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ααα−α

≡10000

CSSC

ZQ . . . (5.21)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−≡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−≡

γγγγ

ββ

ββ

CSSC

CS

SC

XY

00

001;

0010

0QQ

. . . (5.22)

@ McGraw-Hill Education 20

Z Y X

C C C S S S C C S C S SS C S S S C C S S C C S

S C S C C

α β α β γ α γ α β γ α γα β α β γ α γ α β γ α γ

β β γ β γ

≡ =

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

− ++ −

−

Q Q Q Q

Rotations about Z Y (new) X (new) axes

ZYX-Euler angles: 12 sets

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Non-commutative Property: Geometrically

Fig. 5.20

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Non-commutative Property …

Fig. 5.21

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W.R.T. fixed frame: QZY = QYQZ =⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

010001100

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−≡

001010100

9009001090090

Yoo

oo

CS

SCQ

But, QYZ = QZQY = ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

001100010

Non-commutative Property

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −≡

100001010

1000909009090

Zoo

oo

CSSC

Q

Hence, QZY ≠ QYZ

@ McGraw-Hill Education 24

Orientation Description

• Euler angles representation

• Direction cosine representation

• Euler parameters representation, etc.

@ McGraw-Hill Education 25

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

@ McGraw-Hill Education 26

p = o + p′ . . . (5.45)[p]F = [o]F + Q[p’]M . . . (5.46)

⎥⎦

⎤⎢⎣

⎡ ′⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡1][

1][

1][

TF MF poQp

0. . . (5.47)

MF ][][ pTp ′= . . . (5.48)

Homogenous Transformation

@ McGraw-Hill Education 27

TTT ≠ 1 or T−1 ≠ TT . . . (5.49)

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=−

1][

T

TT1

0oQQT F . . . (5.50)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

≡

1000110020100001

T

. . . (5.51)

Example 5.12 Pure Translation

T: Homogenous transformation matrix (4 × 4)

Fig. 5.24 (a)

@ McGraw-Hill Education 28

rt TTT ≡ . . . (5.53)30 30 0 230 30 0 10 0 1 00 0 0 1

3 1 0 22 21 3 0 12 20 0 1 00 0 0 1

T

o o

o o

C SS C

⎡ ⎤−⎢ ⎥⎢ ⎥≡⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤

−⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

. . . (5.54)

Example 5.14 General Motion

Fig. 5.24 (c)

@ McGraw-Hill Education 29

PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachersand educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it withoutpermission.

Lecture 3 Robot Kinematics

byS.K. Saha

Aug. 17’16 (W)@JRL301 (Robotics Tech.)

@ McGraw-Hill Education 30

Summary of Last Class

• Kinematic chain: Links and joints• DOF: Parameters-constraints• Position: Simple (like good friend in the

hostel)• Orientation: Confusing and SERIOUS

attention to be paid

@ McGraw-Hill Education 31

Denavit and Hartenberg (DH) Parameters—Frame Allotment

• Serial chain - Two links connected

by revolute joint, or- Two links connected

by prismatic joint

Fig. 5.27

@ McGraw-Hill Education 32

• Joint axis i: Link i-1 + link i• Link i: Fixed to frame i+1 (Saha) / frame i (Craig)

DH Variablesbi and θi

[Screw@Z]

Constantsai and αi

[Screw@X]Saha XiXi+1@Zi ZiZi+1@Xi+1

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

Z’’’i

Zi+1

@ McGraw-Hill Education 33

Revolute Joint

Fig. 5.28

• DH@Z (Variable)– Joint offset (b)– Joint angle (θ)

• DH@X (Const.)– Link length (a)– Twist angle (α)

Z’’’i

Zi+1

@ McGraw-Hill Education 34

Tb =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1000100

00100001

ib. . . (5.49a)

Tθ =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

100001000000

ii

ii

CθSθθSCθ

. . . (5.49b)

Mathematically• Translation along Zi

• Rotation about Zi

@ McGraw-Hill Education 35

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

100000000001

ii

ii

CαSααSCαTα = . . . (5.49d)

Ta =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

100001000010

001 ia

. . . (5.49c)

• Translation along Xi+1

• Rotation about Xi+1

@ McGraw-Hill Education 36

Ti = TbTθTaTα . . . (5.50a)

Ti =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

−

10000 iii

iiiiiii

iiiiiii

bCαSαSθaSαCθCαCθSθCaSαSθCαSθCθ θ

. . . (5.50b)

• Total transformation from Frame i to Frame i+1

Rotation Matrix

Position

@ McGraw-Hill Education 37

Three-link Planar Arm

Ti =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ θ−

10000100

00

iiii

iiii

SθaCθSθCaSθCθ

• DH-parameters

, for i=1,2,3

Link bi θi ai αi

1 0 θ1 (JV) a1 02 0 θ2 (JV) a2 03 0 θ3 (JV) a3 0

• Frame transformations(Homogeneous)

Fill-up the DH parameters

Fill-up with the elements

@ McGraw-Hill Education 38

Conclusions

• DOF and Constraints• Rotation representations• DH Parameters• Configuration and Homogeneous

Transformation• RoboAnalyzer software• Examples

@ McGraw-Hill Education 39

THANK YOUsaha@mech.iitd.ac.in

http://sksaha.com

@ McGraw-Hill Education 40

PROPRIETARY MATERIAL. © 2014, 2008 The McGraw-Hill Companies, Inc. All rights reserved. No part of this PowerPoint slide may be displayed,reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachersand educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this PowerPoint slide, you are using it withoutpermission.

Lecture 4 (SIT Sem. Rm.)Forward and Inverse Kinematics

(Ch. 6)by

S.K. SahaAug. ??, 2016 (?)@JRL301(Robotics Tech.)

@ McGraw-Hill Education 41

Recap• Orientation representations

– Non-commutative

• Direction cosines: Has disadv. of 9 param.

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

@ McGraw-Hill Education 42

Forward and Inverse Kinematics

Inverse: 1st soln..Inverse: nth soln.

Forward: One soln.S

olve N

on-lin. eqns.M

ultiply + A

dd

@ McGraw-Hill Education 43

Three-link Planar Arm

Ti =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ θ−

10000100

00

iiii

iiii

SθaCθSθCaSθCθ

• DH-parameters

, for i=1,2,3

Link bi θi ai αi

1 0 θ1 (JV) a1 02 0 θ2 (JV) a2 03 0 θ3 (JV) a3 0

• Frame transformations(Homogeneous)

Fill-up the DH parameters

Fill-up with the elements

Fig. 5.29 A three-link planar arm

@ McGraw-Hill Education 44

DH Parameters of Articulated ArmLink bi θi ai αi

1 0 θ1 (JV) 0 − π/2

2 0 θ2 (JV) a2 0

3 0 θ3 (JV) a3 0

@ McGraw-Hill Education 45

Matrices for Articulated Arm1 1

1 11 0 1 0 0

0 0 0 1

c 0 s 0s 0 c 0

−⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥−⎢ ⎥⎣ ⎦

T

2 2 2 2

2 2 2 22

c s 0 a cs c 0 a s0 0 1 00 0 0 1

−⎡ ⎤⎢ ⎥⎢ ⎥≡⎢ ⎥⎢ ⎥⎣ ⎦

T

3 3 3 3

3 3 3 33

c s 0 a cs c 0 a s0 0 1 00 0 0 1

−⎡ ⎤⎢ ⎥⎢ ⎥≡⎢ ⎥⎢ ⎥⎣ ⎦

T

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−−−+−+−

≡

1000sasa0cs

)cac(ascsscs)cac(acssc-cc

233222323

2332211231231

2332211231231

)(T … (6.11)

@ McGraw-Hill Education 46

Inverse Kinematics

• Unlike Forward Kinematics, general solutions

are not possible.

• Several architectures are to be solved

differently.

@ McGraw-Hill Education 47

Two-link Arm

X2θ2a2

a1

X1

X3

Y2

Y1 Y3

θ1px

py

12211

12211

sasapcacap

y

x

+=+=

21

22

21

22

2 2 aaaapp

c yx −−+=

222 1 cs −±=

θ2 = atan2 (s2, c2)

Δpsap)ca(a

s xy 222211

−+= 22

22122

21 2 yx ppcaaaaΔ +=++≡

Δpsap)ca(a

c yx 222211

++= θ1 = atan2 (s1, c1)

θ1

θ2

RoboA

nalyzer

@ McGraw-Hill Education 48

Inverse Kinematics of 3-DOF RRR Arm

321 θθθφ ++=123312211 cacacapx ++=

123312211 sasasap y ++=

122113 cacac φ apw xx +=−=122113 sasas φ apw yy +=−=

… (6.18a)

… (6.18b)

… (6.18c)

… (6.19a)… (6.19b)

@ McGraw-Hill Education 49

w2x + w2

y = a12+ a2

2 + 2 a1a2c2

21

22

21

22

2 2 aaaawwc 21 −−+

= 222 1 cs −±=

θ2 = atan2 (s2, c2) . . . (6.21)

2121221 ssa)ccaa(wx −+=

2121221y sca)sca(aw ++=

Δwsaw)ca(a

s xy 222211

−+=

Δwsaw)ca(a

c yx 222211

++=

22221

22

21 2 yx wwcaaaaΔ +=++≡

θ1 = atan2 (s1, c1) . . . (6.23c)

θ3 = ϕ - θ1 − θ2 . . . (6.24)

… (6.22a)… (6.22b)

… (6.20a)

… (6.20b,c)

… (6.23a,b)

@ McGraw-Hill Education 50

Numerical Example

3 52 2

3 3 12 2

T

⎡ ⎤− +⎢ ⎥

⎢ ⎥⎢ ⎥

≡ +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1 0 32

1 02

0 0 1 00 0 0 1

• An RRR planar arm (Example 6.15). Input

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

Rotation Matrix

Origin of end-effectorframe

4.23

1.86

0

Do it yourself Verify using RoboAnalyzer

@ McGraw-Hill Education 51

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 θ2 = 30o.

The use of negative value would result in :

…(6.30c)

θ2 = 30o

θ1 = 0o.

θ3 = 30o.

θ1 = 0o θ2 = 30o, and θ3 = 30

θ1 = 30o θ2 = -30o, and θ3 = 60o

@ McGraw-Hill Education 52

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

@ McGraw-Hill Education 53

Velocity Analysis

[ ]1 2where andJ j j j n=

ii

i ie

, if Joint is revolutei ⎡ ⎤

≡ ⎢ ⎥×⎣ ⎦

ej

e aprismatic isointJif, i

ieii ⎥

⎦

⎤⎢⎣

⎡×

≡ae

0j

et Jθ=

1

twistof end - effector : ; Joint rates : ee

en

θ

θ

⎡ ⎤⎡ ⎤ ⎢ ⎥≡ =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦

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)

@ McGraw-Hill Education 54

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 e T≡ ≡

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

≡

≡

@ McGraw-Hill Education 55

Example: Singularity of 2-link RR Arm

⎥⎦

⎤⎢⎣

⎡+

−−−≡

12212211

12212211

cacacasasasa

J θ2 = 0 or π

@ McGraw-Hill Education 56

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

@ McGraw-Hill Education 57

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 = 2π x 10/60 = 1.047 rad/sec• Power = Torque x Speed = 0.513 kW• Simple but sufficient for approximation

@ McGraw-Hill Education 58

Practical Application

Subscript l for load; m for motor;G = ωl/ωm (< 1); η: Motor + Gear box efficiency

Trapezoidal Trajectory

@ McGraw-Hill Education 59

Accelerations & Torques

Ang. accn. during t1:

Ang. accn. during t3:

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

Torque during t1: T1 =

Torque during t2: T2 =

Torque during t3: T3 =

@ McGraw-Hill Education 60

RMS Value

@ McGraw-Hill Education 61

Motor Performance

@ McGraw-Hill Education 62

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.

@ McGraw-Hill Education 63

Dynamics and Control Measures

12n rω ω≤

• Rule of Thumb

nω

rω

: closed-loop natural frequency

: lowest structural resonant frequency

… (7.51)

@ McGraw-Hill Education 64

Manipulator Stiffness

21 2

1 1 1

ek k kη= + ek ≡

η ≡

equivalent stiffness

gear ratio … (7.48)

@ McGraw-Hill Education 65

Link Material Selection

• Mild (low carbon) steel: Sy = 350 Mpa; Su = 420 Mpa

• High alloyed steelSy = 1750-1900 Mpa; Su = 2000-2300 Mpa

• Aluminum• Sy = 150-500 Mpa; Su = 165-580 Mpa

@ McGraw-Hill Education 66

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

@ McGraw-Hill Education 67

Summary

• Forward Kinematics• Inverse kinematics

– A spatial 6-DOF wrist-portioned has 8 solutions

• Velocity and Jacobian• Mechanical Design

@ McGraw-Hill Education 68

THANK YOUsaha@mech.iitd.ac.in

http://sksaha.com