1 @ McGraw-Hill Education Lecture 2 Robot Kinematics S.K. Saha Dynamics of Multibody... · 2016. 10. 2. · @ McGraw-Hill Education 4 Transformations • Robot Architecture – Links:

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Lecture 2 Robot Kinematics

byS.K. Saha

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


Outline• Links and Joints

• Kinematic chains

• Degrees-of-freedom (DOF)

• Pose (≡ Configuration)

• Homogeneous transformation

• Denavit-Hartenberg Parameters

• Conclusions


Link

Link

Joint

Joint

TAL Robot


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


Joints or Kinematic Pairs• Lower Pair

– Surface contact: Hinge joint of a door

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


Lower Pair: Revolute Joint


Lower Pair: Prismatic Joint


Lower Pair: Helical Joint


Lower Pair: Cylindrical Joint


Lower Pair: Spherical Joint


Closed Kinematic Chain


Open Kinematic Chain


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


A 4-bar Mechanism

n = 3 (r − 1) − 2p= 3(4-1) − 2×4= 1


A Robot Manipulator

n = 6 (r − 1) − 5p= 6(7-1) − 5×6 = 6


n = 3 (r − 1) − 2p= 3(5-1) − 2×6 = 0


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)


[ ,

[ ] ,

[ ]

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


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ααα−α

≡10000

CSSC

ZQ . . . (5.21)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−≡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−≡

γγγγ

ββ

ββ

CSSC

CS

SC

XY

00

001;

0010

0QQ

. . . (5.22)


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


Non-commutative Property: Geometrically

Fig. 5.20


Non-commutative Property …

Fig. 5.21


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


Orientation Description

• Euler angles representation

• Direction cosine representation

• Euler parameters representation, etc.


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


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


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)


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)



Lecture 3 Robot Kinematics

byS.K. Saha

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


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


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


• 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


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


Tb =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

1000100

00100001

ib. . . (5.49a)

Tθ =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

100001000000

ii

ii

CθSθθSCθ

. . . (5.49b)

Mathematically• Translation along Zi

• Rotation about Zi


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

100000000001

ii

ii

CαSααSCαTα = . . . (5.49d)

Ta =⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

100001000010

001 ia

. . . (5.49c)

• Translation along Xi+1

• Rotation about Xi+1


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


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


Conclusions

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

Transformation• RoboAnalyzer software• Examples


THANK [email protected]

http://sksaha.com



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

(Ch. 6)by

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


Recap• Orientation representations

– Non-commutative

• Direction cosines: Has disadv. of 9 param.

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


Forward and Inverse Kinematics

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

Forward: One soln.S

olve N

on-lin. eqns.M

ultiply + A

dd


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


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


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)


Inverse Kinematics

• Unlike Forward Kinematics, general solutions

are not possible.

• Several architectures are to be solved

differently.


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


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)


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)


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


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


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


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)


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

≡

≡


Example: Singularity of 2-link RR Arm

⎥⎦

⎤⎢⎣

⎡+

−−−≡

12212211

12212211

cacacasasasa

J θ2 = 0 or π


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


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


Practical Application

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

Trapezoidal Trajectory


Accelerations & Torques

Ang. accn. during t1:

Ang. accn. during t3:

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

Torque during t1: T1 =




RMS Value


Motor Performance


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.


Dynamics and Control Measures

12n rω ω≤

• Rule of Thumb

nω

rω

: closed-loop natural frequency

: lowest structural resonant frequency

… (7.51)


Manipulator Stiffness

21 2

1 1 1

ek k kη= + ek ≡

η ≡

equivalent stiffness

gear ratio … (7.48)


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


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


Summary

• Forward Kinematics• Inverse kinematics

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

• Velocity and Jacobian• Mechanical Design


THANK [email protected]

http://sksaha.com

1 @ McGraw-Hill Education Lecture 2 Robot Kinematics S.K. Saha Dynamics of Multibody... · 2016. 10. 2. · @ McGraw-Hill Education 4 Transformations • Robot Architecture – Links:

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