1 @ McGraw-Hill Education Chapters 8-9 - IIT Kanpur · related to Robot Kinematics & Dynamics ... – Links: A rigid body has 6-DOF – Joints: ... @ McGraw-Hill Education 16 ZY X

@ McGraw-Hill Education 1

Robot Dynamics

Prof. S.K. SahaDept. of Mech. Eng.

IIT Delhi

Mar. 16, 2016@Advanced Robotics (TEQUIP), IIT Kanpur

Chapters 8-9


Learning Objectives

• Get acquainted with the terminologies related to Robot Kinematics & Dynamics

• Formulations using classical EL and NE formulations

• Get exposed to the philosophy of the DeNOC-based dynamics (SK Saha’s)


Kinematic Chain

• Series of links connected by joints– Links: A rigid body has 6-DOF– Joints: Couples 2 bodies. – Provide restrictions (Constraints)

• Example of joints (next few slides)


Revolute Joint: Five constraints

Fig. 5.1 A revolute joint


Prismatic Joint: Five constraints

Fig. 5.2


Cylindrical Joint: Four constraints

Fig. 5.4


Helical Joint: ??? constraints

Fig. 5.3


Spherical Joint: ??? constraints

Fig. 5.5 A spherical jointFig. 5.6


Closed and Open Chain

• Simple Kinematic Chain: When each and every link is coupled to at most two other links– Closed: If each and every link coupled to two

other links Mechanism– Open: If it contains only two links (end ones)

that are connected to only one link Manipulator


Closed-chain

Fig. 5.8 Fig. 5.10


Open-chain

Fig. 5.9


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 (n-1) – 2j– Kutzbach formula (1929) for spatial

systems, DOF = 6 (n-1) – 5j


Motions

• Positions (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 (NCP): 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

NCP: Mathematically

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −≡

100001010

1000909009090

Zoo

oo

CSSC

Q

Hence, QZY ≠ QYZ


Dynamics

Newton’s 2nd law: ModellingGiven ; Find Inverse dynamicsGiven ; Find Forward dynamics

Integrations

Actually

xmf =xm , ffm ,

CRO

Force, Mass, m

xf

∫∫ == dtxxdtxx ;mfx =


Inverse vs. Forward Dynamics

Inverse Dynamics

Find joint torques/forces for given joint motions and end-effector moment/force

Forward Dynamics

Find end-effector motion for known joint torques/forces


Euler-LagrangeEuler-Lagrange

• Generalized CoordinatesCoordinates that specify the configuration, i.e., the position and orientation, of all the bodies or links of a mechanical system

• They can have several representations


g gEuler-Lagrange Formulation

d L Lidt q qi i

ϕ⎛ ⎞∂ ∂⎜ ⎟− =⎜ ⎟∂ ∂⎝ ⎠

L (Lagrangian) = T – U;T: Kinetic energy; U: Potential energy;qi: Generalized coordinate;φi : Generalized force.


Kinetic and Potential Energies

• Kinetic Energy

• Potential Energy

∑=

−=n

iT

iimU1

gc

∑=

⎟⎠⎞⎜

⎝⎛ +=∑

==

n

i iiTii

Tiim

n

i iTT1 2

1

1ωIωcc


A Moving Mass: Euler-Lagrange

m2

21 xmT =

xY

X

• Generalized Coordinate: x

• Kinetic Energy:

• Potential Energy: U = 0

• Lagrangian: L=T - U

xmxL

dtd

=∂∂ )( 0; =

∂∂

=∂∂

xLxm

xL

• Generalized Force: f

f

fxL

xL

dtd

=∂∂

−∂∂ )(

(Dynamic) Equation of Motion

or Dynamic Model

fxm =


Example: One-DOF Arm2

2 21 1( ) ; 2 2 12

( )2 2

a maT m2a aU mg c

θ θ

θ

≡ +

= −

θ−=θ∂∂

θ=θ∂∂ mgasLmaL

dtd

21;

31)( 2

τθθ =+ mgasma31 2

21

)1(2

2 θθ camg6

maU-TL2

−−≡=


Example: Two-DOF Arm

τγhθI =++••

⎥⎦

⎤⎢⎣

⎡≡

2221

1211

i i i i

I

22 21 2

11 1 1, 2 1 1 2 2 2,ZZ ZZm ai a I m (a a a c ) I4 4

= + + + + +

22 1 2 2

12 21 2 2,ZZa a a ci i m ( ) I4 2

= = + +

22

22 2 2,ZZai m I4

= +


21 21 2 1 2 2 1 2 2 2 2

a ah m a a s m s2

= − θ θ − θ

222 1 2 2 12

mh a a s θ=

1 21 1 1 2 1 1 12 a am g c m g(a c c )

2 2γ = + +

22 1 12 am g c

2γ =

Other Terms


Newton-Euler Equations• Newton’s 2nd law(Linear equations of motion)

ccf mdtdm ==

• Euler’s rotational equations of motion

)c][c]([c][c][c][c][ ωIωωIn ×+=

)F

][F]([F][F

][F][F][ ωIωωIn ×+=

TC][F][ QIQI ≡


DeNOC-based Dynamics

Decoupled Natural Orthogonal Complement matrices


Space robots (Toshiba, Japan)

1995

2013

2003

2006

2012

2000

Long Chain (With IIT Madras)

Industrial Robots (IITD)1999

2007

Parallel robots (Univ. of Stuttgart, Germany)

Closed-loop (Ph. D)

3-DOF parallel (McGill, Canada)

Flexible serial (Ph.D)

Tree-type (Ph. D)Book by Springer & ReDySim

Book by Springer

2011

2005

Machine Tool (Ph. D)

Engine Cam (M.S.-IITM)

RIDIM (IITD)

2009RoboAnalyzer Software (IITD)

2015Reduced-order closed-loop (Ph. D)DeNOC

In the syllabus of 2 courses at John Hopkins Univ., USA


Simple System

: Vertical component Reaction: Horizontal component Motion

Mass, m

Force,

f

efvf

vff

x


Using DeNOC

Newton’s 2nd law:

Velocity constraint:

NOC:

Euler-Lagrange:

cff mce =+

x][ ic =

][ i

ef

[ ] [ ( ) ] [ ]T Tv cf f f m x f m x+ + = ⇒ =i i j i i

External force,

Mass, m

i

j

ccfReaction,

Note that

[ ] ( ) [ ] 0Tv cf f+ =i j


Complex Systems• Newton-Euler (NE)

Euler’s:

Newton’s:

34

3 scalar eqs.3 scalar eqs.


Uncoupled NE Equations

•The 6n uncoupled equations of motion35


Velocity Constraints: DeNOC Matrices

Bij: the 6n × 6n twist-propagation matrix

pi: the 6n-dimensional joint-rate propagation vector or twist generator36


Definition: DeNOC Matrices

• N≡NlNd: the 6n × n Decoupled Natural Orthogonal Complement37


Coupled Equations

• n coupled Euler-Lagrange equations

- no partial differentiation38

[ ] ( ) [ ] 0Tv cf f+ =i j


Recursive Expressions• For the n × n GIM, each element

• For the n × n MCI, each element

• For the n × n generalized forces

Composite body mass matrix

39


Example: One-DOF Arm

11

2

( )

1 3

T

T T

I i

m

ma

≡ =

= + × ×

=

p Mp

e Ie (e d) (e d)

where and m

⎡ ⎤ ⎡ ⎤≡ ≡ =⎢ ⎥ ⎢ ⎥×⎣ ⎦ ⎣ ⎦

e I Op M M

e d O 1

;

TT asac ]021

21[][ ;]100[][ 11 θθ≡≡ de

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100000001

12][

2ma2I

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

θθθθθθ

==10000

12][][ 2

22

scscsc

maT21 QIQI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡θθθ−θ

=10000

cssc

Q


where

( )

[ )] 0

T

T

h

θ

= +

= × + × =

p MW WM p

e I(e e) (e Ie

⎥⎦

⎤⎢⎣

⎡×==

fn

deewN ])([1TTET

lτ

TT mg ]00[][;]00[][ 11 =≡ fn τ

θττ mgas21

1 −=

Equation of motion:

21 13 2

ma mgasθ τ θ= −


Example: Two-link Manipulator;

11 12 21 1 1

21 22 2 2

( ); ;

i i i hi i h

ττ

=⎡ ⎤ ⎡ ⎤ ⎡ ⎤≡ ≡ ≡ ≡⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

I h Cθ τ

22 : ScalarTi ≡ 2 2 22 2p M B p

2222222222222 ][][][][][ ddeIe TT mi +=

22 : 6 6 identity matrix⎡ ⎤

≡ ×⎢ ⎥⎣ ⎦

1 OB

O 1

22

2 2

: 6-dim. vector⎡ ⎤

≡ ⎢ ⎥×⎣ ⎦

ep

e d

22 2

2

: 6 6 sym. matrixm

⎡ ⎤≡ = ×⎢ ⎥

⎣ ⎦

I OM M

O 1

X3

Y3


2 2 2 2 2 2 2 21 1[ ] [0 0 1] ; [ ] [ 0]2 2

T Ta c a sθ θ≡ ≡e d

2 2

2 2 2

00

0 0 1

c ss cθ θθ θ

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Q

22 2 22

22 2 2 2 2 2 2

0

[ ] [ ] 02 3 120 0 1

T

s s cma s c c

θ θ θ

θ θ θ

⎡ ⎤−⎢ ⎥⎢ ⎥= = −⎢ ⎥⎢ ⎥⎣ ⎦

I Q I Q

2222222222222 ][][][][][ ddeIe TT mi +=

2 2 222 2 2 2 2 2 2

1 1 112 4 3

i m a m a m a= + =

X3

Y3


;

21 12 1 1( ) : ScalarTi i= ≡ 2 2 2p M B p

21 2 1 2 1 1 1 2 2 1 1 1 1 2 1[ ] [ ] [ ] [ ] ([ ] [ ] )T Ti m= + + +e I e d d r d

211 2

: 6 6 matrix( )

⎡ ⎤≡ ×⎢ ⎥− + ×⎣ ⎦

1 OB

r d 1 1

[ ]2 1 2 1 1 2 2 2 12 2 12

1 1 1 1 1 1 1 1 2 1

1 1[ ] 0 0 1 ; [ ] [ ] [ 0]2 2

1 1[ ] [0 0 1] ; [ ] [ ] [ 0]2 2

T T

T T

a c a s

a c a s

θ θ

θ θ

≡ = =

≡ = =

e d Q d

e d r

r11 1

1 1 1

00

0 0 1

c ss cθ θθ θ

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Q

2 2 221 2 2 2 1 2 2 2 2 2 2 2 1 2 2

1 1 1 1 112 2 4 3 2

i m a m a a c m a m a m a a cθ θ= + + = +

11

1 1

: 6-dim. vector⎡ ⎤

≡ ⎢ ⎥×⎣ ⎦

ep

e d


11 1 1 11 1 : ScalarTi ≡ p M B p

11 : 6 6 identity matrix⎡ ⎤

≡ ×⎢ ⎥⎣ ⎦

1 OB

O 1

11

1

: 6 6 sym. matrixm

⎡ ⎤≡ ×⎢ ⎥⎣ ⎦

I OM

O 1

X3

Y3

⎥⎦

⎤⎢⎣

⎡

×−×

=+=11δ1δI

BMBMM 211

11212111 ~

~~~

mT

( )12 21

1 1 2 2 1 2 1( )

( )m=−

= + + + × ×c c

I I I r d δ 1

2121~ cδ m=

211~ mmm +=

11 1 1 1 1 1 1 1 1 1 1

2 2 21 1 2 2 2 1 2 1 2 2

[ ] [ ] [ ] [ ] [ ]1 ( )3

T Ti m

m a m a m a m a a cθ

= +

= + + +

e I e d d


Vector of Convective Inertia

Link Joint ai(m)

bi(m)

αi(rad)

θi(rad)

1 r 0.3 0 0 JV [0]

2 r 0.25 0 0 JV [0]

22 2 2 2 1 2 2 1

12

Th m a a sθ θ′= =p w 1 1 2 1 2 2 2 2 11( )2

Th m a a sθ θ θ θ′= = − +1 p w

Link mi ri,x ri,y ri,z Ii,xx Ii,xy Ii,xz Ii,yy Ii,yz Ii,zz

(kg) (m) (kg-m2)

1 0.5 0.15 0 0 0 0 0 0.00375 0 0.00375

2 0.4 0.125 0 0 0 0 0 0.00208 0 0.00208

DH

and

Iner

tia p

aram

eter

sInverse Dynamics Results


Joint Torques

No gravity (horizontal

)

With gravity (vertical


1

#n

#1

#i

i

n

#0

1 1 1

2 2 2 1

1n n n n

θ

θ

θ −

=

= +

= +

α p

α p α

α p α

1 1 1 1 1 1

2 2 2 2 2 2

21 1 21 1

, 1 1 , 1 1

n n n n n n

n n n n n n

θ θ

θ θ

θ θ

− − − −

= +

= +

+ +

= +

+ +

β p Ω p

β p Ω p

B β B α

β p Ω p

B β B α

DeNOC-based Recursive Inverse Dynamics

1 1 1 1 1 1 , 1

1 1 1 1 1 1 21 2

n n n n n nT

n n n n n n n n n

T

− − − − − − −

= +

= + +

= + +

γ M β W M α

γ M β W M α B γ

γ M β WM α B γ

1 1 1

1 1 1

Tn n n

Tn n n

T

τ

τ

τ

− − −

=

=

=

p γ

p γ

p γ


Computational complexity

Algorithm Multiplications/Divisions (M)

Additions/Subtractions (A)

n=6

Hollerbach (1980) 412n-277 320n-201 2195M 1719A

Luh et al. (1980) 150n-48 131n+48 852M 834A

Walker and Orin (1982) 137n-22 101n-11 800M 595A

RIDIM (Saha, 1999) 120n-44 97n-55 676M 527A

Khalil et al. (1986) 105n-92 94n-86 538M 478A

Angeles et al. (1989) 105n-109 90n-105 521M 435A

ReDySim (Shah et al., 2013) 94n-81 82n-75 483M 417A

Balafoutis et al. (1988) 93n-69 81n-65 489M 421M

Table 9.1 Computational complexities for inverse dynamics


UDUT & Recursive Forward Dynamics

θ

Step 2: UDUT Decomposition

, where andUDU θ I UDU τ CθT T≡ = −ϕ ϕ =

Equation of the motion

The joint accelerations are then solved as 1 1θ U D UT− − −= ϕ

Hence forward dynamics requires three steps

Step 1: Computation of φ

Step 3: Recursive computation of

φ is obtained from inverse dynamics algorithm by substituting

Step 1: Computation of φ

θ 0=

… (9.38)


UDUT decomposition

, ,

ˆˆ

T Ti j i j i i

Ti i i

u

m

=

=

p B ψ

p ψ

Step 2: UDUT Decomposition

112 1

22

ˆ 0 01ˆ00 1

, where , and0ˆ0 00 0 1

I UDU U D

n

nT

n

mu umu

m

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ≡ ≡⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦Element of matrices U and D are obtained as

In above equation is obtained recursively for i = n… 1 asψ i

where

ˆˆ

ii

im≡ψ

ψ ˆˆ i i i=ψ M p

ˆ ;M M B M BTi i i 1,i i 1 i 1,i+ + +≡ + ˆ ˆ ;M M ψ ψT

i i i i≡ −

nn MM =ˆ

,and … (9.39c)

… (9.39d)


Recursive Forward Dynamics

T ≡UDU θ φ

θ

θStep 3: Recursive computation of

The solution of require three recursive steps

1ˆ U−=ϕ ϕ 1 ˆD−=ϕ ϕ θ U T−= ϕStep 1 Step 2 Step 3

ˆ ˆ ,i i imϕ = ϕ

, 1 , 1 1

1 1 1 1, 2

where ,

and

μ B μ

μ p μi i i i i

i i i i iθ− − −

− − − − −

≡

≡ +

, 1ˆ Ti i i i i+ϕ = ϕ −p η

, 1 1, 1

1 1 1 1, 2

whereˆand,

Ti i i i i

i i i i i

+ + +

+ + + + +

=

= ϕ +

η B η η ψ η

1ˆi ψ μθ ϕ= − Ti i i,i-… (9.39a)

… (9.39b)

… (9.40) … (9.41a)

… (9.41b)


Computational complexity

Algorithm Multiplications/Divisions (M)

Additions/Subtractions (A)

n=6 N=7 n=10

ReDySim(Shah et al., 2013)

135n-116 131n-123 694M663A

829M794A

1234M1187A

Saha (2003) 191n-284 187n-325 862M 797A

1053M984A

1626M1545A

Featherstone (1983) 199n-198 174n-173 996M 871A

1195M1045A

1792M1527A

Valasek (1996) 226n − 343 206n − 345 1013M 891A

1239M1097A

1917M 1715A

Brandl et al. (1986) 250n − 222 220n − 198 1278M 1122A

1528M1342A

2278M2002A

Walker and Orin (1982) n3/6+23 n2/2+ 115n/3-47

n3/6+7n2+ 233n/3-46 633M 480A

842M898A

1653M 1209A

Table 9.6 Computational complexities for forward dynamics


Tree-type and Closed Systems

Base

A link or body

0

#k

#β(k)

54


1i

ηi

1i

Miki

ki

ηi

Intra-modular Constraints (Inside the module)

, 1 1t A t pk k k k k kθ− −= +Module Mi

thTwist of the k link ω

to

kk

k

⎡ ⎤= ⎢ ⎥⎣ ⎦

-1 t tk kterms of

Inter-modular Constraints (Between the modules)

,i i i iβ β= +t A t N θModule Mi and Mβ

11

and

t

t t θ

t

i k i k

ii

η η

θ

θ

θ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥≡ ≡ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

t ti βModule twist in terms of

Module twist and module joint rate

1i

ηi

1β

1i

Mi

ηβ

ki

1β

ηβ

ki

ηi

Mβ

55


Upon pre-multiplication

Dynamics using DeNOC matrices

Generalized inertia matrix (GIM)

Matrix of Convective inertia (MCI)

Equation can be written as,

and

( )

T

T

T E

F T F

l d

=

= +

=

=

=

I N MN

C N MN WMEN

τ N w

τ N w

N N N

( ) , = 0 T T T T E F T T Cd l d l d l+ = +N N M t ΩME t N N (w w ) N N wwhere

Iq Cq τ τF+ = +

Generalized external force

Generalized force other than driving(e.g. Link ground interaction)

Mi

Modules

NE equations for entire system

, ,E F C+ = + +Mt ΩMEt w w = w w wwhere


DeNOC matrices

Summary: Using DeNOC Matrices

,

Decoupled form of the velocity transformation matrix

Minimal-order Equations of motion

(Euler-Lagrange)

Newton-Euler Equations of

motion×

Rate of Independent coordinate θ.

Link velocitiest (ω and v) = ×

DeN

OC

m

atric

es

=


Mi

M0

Modules

58

Inter- and intra- modular recursive

( ) ( ) , ( )

Ti i i

Ti i i i iβ β β

=

= +

N w

w w A w

τ

,

, ,

i

i i

i i i i

i i i i i i i

i i i i i i i

β β

β β β β

= +

= +

+

t A t N θ

t A t +A t +N θ N θ

w =M t Ω M E t

, ( )

, ( ) , ( )

, 1

, 1

, 1

, 1

0,

,

j k j j j

j j

j k j k j j j j j j

j j j j j

j k

k j j k k j k

r

r

r

r

r

r

β β

β β β β

= + θ =

= θ >

= + + θ + θ =

= θ + θ >

= < ε

= + = ε

t A t p

p

t A t A t Ω p p

Ω p p

w

M t Ω M E t

Inter-modular

( ) ( )

( ) , ( )

, 1

, 1

Tj j j

j j

Tj k k j

r

rβ β

β β

τ =

= >

= + =

p w

w w

w A w

k = 1:ηi , r=1:εk

i = 1:s

i = s:1k = ηi :1, r =εk:1

Intra-modular

Joint torques ( )i τ

Joint motions , , )i i i(q q q , inertia parameters ( )iM , twist and motion propagations

,( and )i iβA N , and wrench due to foot-ground interaction Fiw

Forw

ard

recu

rsio

n B

ackw

ard

recu

rsio

n

1

1

,

( ) ( ) ,, (( )) , , ( )

( ) ( ) , ( )

ˆ ˆ ˆˆ ˆ ˆ; ;ˆ

ˆ ˆ

ˆˆ ˆ ˆ;ˆ ˆ ˆ

i i i Ti i i i i i

Ti i i i

i i i

Ti i i i i i i i i

Ti i ii ii i i i i

Ti i i i i

β β ββ β

β β β

−

−

= = =

= −

=

= − = +

= +

= +

Ψ M N I N Ψ Ψ Ψ I

φ φ N η

φ I φ

M M Ψ Ψ η Ψ φ η

M M A M A

η η A η

,

, ,

*

i

i i

i i i i

i i i i i

i i i i i i i

β β

β β β β

= +

=

+

t A t N θ

t A t +A t +N θ

w =M t Ω M E t

, ( )

, ( ) , ( )

, 1

, 1

, 1

, 1

0,

,

j j

j k j j j

k k

j k j k j j j j

j j j

j k

k j j k k j k

r

r

r

r

r

r

β β

β β β β

= + θ =

= θ >

= + + θ =

= θ >

= < ε

= + = ε

t A t p

p

t A t A t Ω p

Ω p

w

M t Ω M E t

Inter-modular

,

( ) ( ) , , ,

,

( ) ( ) ,

ˆˆ ˆ ˆˆ ˆ; ; /

ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ;ˆ ˆ ˆ , 1

ˆ , 1

, 1

, 1

Tj j j j j j j j j

Tj j j j

j j j

Tj j j j j j j j j

Tj j k j j k

j j

Tj j k j

j

m m

m

r

r

r

r

β β β β

β β β

= = =

ϕ = ϕ −

ϕ = ϕ

= − = ϕ +

= + >

= =

= + >

= =

ψ M p p ψ ψ ψ

p η

M M ψ ψ η ψ η

M M A M A

M

η η A η

η

k = 1:ηi, r=1:εk

i = 1:s

i = s:1 k = ηi :1, r =εk:1

, ( )( ) ( ) ( )( )

,

i ii i i i

Ti i i i

ββ β β= +

= −

μ A N q μ

q φ Ψ μ

i = 1:s, ( ) ( ) ( )

( ) ( ) ( )

( ), 1

, 1j k j j j

j j j

Tj j j j

r

rβ β β β

β β β

θ

θ

θ

= >

= =

= ϕ −

μ A p μ

p μ

ψ μ

+

+

k = 1:ηi, r=1:εk

Intra-modular

Independent accelerations iq

Joint torques ( )iτ , inertia parameters ( )iM , twist and motion propagations ,( and )i j iA N , wrench

due to foot-ground interaction )Fi(w and initial conditions and i iq q

Step

1

Step

2

Step

3

Inverse dynamicsForward dynamics


Four-bar Mechanism• Separate all links• Draw Free-body diagrams (FBD)

12 12

1 1 1 1 01 1 01 1 21 1 21

1 1 01 21 1 1 01 21;x y

x y y x x y y x

x x x y y yf f

I d f d f r f r f

m c f f m c f f

θ τ

− −

− + + −

= + = +

=

Equations of motion: 3 per linkBase, #0

#1#3C1

#2

f12

f21

f01

f32

f23

f03

d1

r1

#1

#2

a3

2

3

1

θ3

θ2

θ14

a2

a0

a1

Base, #0

#3

θ4

τ1

9 equations 13 – 4 (3rd

law) = 9 unknowns


11 1 1 1 1 1

01 1 1

01 1 1

122 2 2 2 2 2

12 2 2

23 2 2

233 3 3 3 3

03

03

11 1 0'

1 1

1 11 1

0' 1 11 1

y x y x

x x

y x

xy x y x

x x

x y

yy x y x

x

y

d d r r Ifs m cf m cfd d r r If m cf m cfd d r r Ifsf

τ θ

θ

⎡ ⎤− −⎡ ⎤⎢ ⎥⎢ ⎥− ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥

− − ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ =−⎢ ⎥⎢ ⎥

− ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥− − ⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦

3

3 3

3 3

: 9 9 matrix x : 9 1 : 9 1

x

y

m cm c

θ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

× × × A b• Ghosh and Mallik, Theory of Machines and Mechanisms

1 (Forward); (Backward)y

− ⇔x=A b L Ux =b; Ly=b Ux=y

Disadv.: Need to calculate even the reactions for inv. dyn.


Three-link Serial with f03 as External• Join first three links form

Equations of motion

Base, #0

#1#3C1

#2

f12

f21

f01

f32

f23

f03

d1

r1

Base, #0

033 3

( ) ( )

: 3 eqs.T

T T T T E Cd l d l

E C

×

+ = +

+ = +J f

N N M t W M t N N w w

I θ Cθ τ τ0 1 2 3

2 3×

+ + + = → =0

a a a a 0 J θ 0

#1

#2

a3

2

3

1

θ3

θ2

θ1

a2

a1

#3 τ1

f03 a0


• The DeNOC matrices for 3-link serial manipulator

Proof of τC = JTf03

1 1 1

2 21 2 2

3 31 32 3 318 1 :18 18 :18 3 3 1

0 ' 0 '

0 '

l d

s s

s

θθθ

× × × ×

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

N N

t 1 pt B 1 pt B B 1 p

32 21 31 21 32 31

2

1 21 2 32 3

21 31 1

32 2 2 32 3

3

3

( )

0 '

T T T

C

C T C T C

T T C

T C T C C T Cl

C

C

s

= ⇒ =

⎡ ⎤+ +⎢ ⎥

⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= = +⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎢ ⎥⎣ ⎦

B B B B B B

w

w B w B w1 B B w

N w 1 B w w B w1 w

w

∵

03 23 3 23 3 033

03 23

C + − × + ×⎡ ⎤= ⎢ ⎥+⎣ ⎦

n n d f r fw

f f2 3

326 6

( )0'

T

s×

+ ×⎡ ⎤= ⎢ ⎥⎣ ⎦

1 r d 1B

1

f03

f23

r3

d3n03

n23

C3


23

23

32 12 2 12 2 32

2 2 32 332 126 1

03 23 3 23 3 03 2 3 03 23

03 23

( ) ( )

C C T C −

×−

+ − × + ×⎡ ⎤⎢ ⎥

= + = ⎢ ⎥+⎢ ⎥

⎢ ⎥⎣ ⎦+ − × + × + + × +⎡ ⎤

⎢ ⎥+⎣ ⎦

n

f

n n d f r f

w w B wf f

n n d f r f r d f f +

f f

2

03 12 2 3 3 03 2 12

2

03 12

( )C

+ + + + × − ×⎡ ⎤⎢ ⎥= ⎢ ⎥

+⎢ ⎥⎣ ⎦

p

n n r d r f d fw

f f

03 01 1 03 1 011

03 01

C + + × − ×⎡ ⎤= ⎢ ⎥+⎣ ⎦

n n p f d fw

f f

C2

r3

2

3

1

θ3

θ2

θ1

r2

τ1

f03

p2

p1

d3


Constraint Torque1 1

2 2

3 3

0 '

0 '

T C

T C T Cd

T C

s

s

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

p w

N w p w

p w

( 1, 2, 3) ii

i ii

⎡ ⎤= = ⎢ ⎥×⎣ ⎦

ep

e d

03 01 1 03 1 011 1 1 1 1 1

03 01

1 03 01 1 03 1 01

1 1 03 01

1 1 03 1 1 03

( )

( )

( ) ( )

( ) ( )

C T C T T

ScalarT

T

T T

τ+ + × − ×⎡ ⎤

⎡ ⎤= = × ⎢ ⎥⎣ ⎦ +⎣ ⎦

+ + × − ×

+ × +

× = ×

n n p f d fp w e e d

f f

= e n n p f d f

e d f f

= e ρ f e ρ f

2 2 2 2 2 03 2 2 03

3 3 3 3 3 03 2 3 03

( ) ( )

( ) ( )

C T C T T

C T C T T

τ

τ

= = × = ×

= = × = ×

p w e ρ f e ρ f

p w e a f e a f

C2

a3

2

3

1

θ3

θ2

θ1

r2

τ1

f03

p1

ρ1

d2


Equations of Motion

*1 03[ ,0,0]: T T

E C

known τ+ = +

J fτ

Iθ Cθ τ τ

3

1 1 2 2 3 3

⎡ ⎤⎢ ⎥= × × ×⎢ ⎥⎣ ⎦a

J e ρ e ρ e ρ

*1 1 1 2 12 3 123 1 1 2 12 3 123 1*2 2 12 3 123 2 12 3 123 03* 3 123 3 123 033

:3 3 :3 1:3 1

100

x

y

a s a s a s a c a c a ca s a s a c a c f

a s a c f

τ ττ

τ× ××

⎡ ⎤ ⎡ ⎤− − − + +⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ = − − + ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥−⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦A xb

Adv.: Reduced size of 3×3 (instead of 9×9) for inverse dynamics


Sub-system(link) Mass (Kg) Length (m)

I(#1) 1.5 0.038II(#2) 5 0.2304II(#3) 3 0.1152

0 0.5 1 1.50

50

100

150

200

250

300

350

400

Time (s)

Join

t ang

les

(deg

)

0 0.5 1 1.5-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (s)

Driv

ing

torq

ue (N

m)

θ3

θ1

θ2

Free: http://www.redysim.co.nr/download


Why recursive?• Efficient, i.e., less computations and CPU

timeCompute jt. torque Compute jt.

accn. (Inverse)

(Forward)

0 5 10 15 20 25 300

2000

4000

6000

8000

10000

12000

14000

Total number of joints

Com

puta

tiona

l Cou

nts

Equal number of 1-, 2- and 3-DOF joints

ProposedBalafoutisFeatherstoneAngeles

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5x 10

4

Total number of joints

Com

puta

tiona

l Cou

nts

1-, 2- and 3-DOF joints

ProposedMohan and SahaLilly and OrinFetherstone

Ref. : Shah, S.,V. Saha, S.K., Dutt, J.K, Dynamics of Tree-type Robotic Systems, Springer 2013


Why recursive? (contd.)• Numerically stable Simulation is realistic

Recursive Non-recursive(Forward) (Forward)

Ref. : Mohan, A., and Saha, S.K., A recursive, numerically stable, and efficient simulation algorithm for serial robots with flexible links, Multibody System Dyn., V. 21, N. 1,pp. 1—35.


• Dynamic modelling• Simulation: Tee-type systems• To visualize the motion• Symbolic equationsDeNOC for Tree-type• Multiple-degrees-of freedom-joints• Efficient Recursive Algorithms• Examples:

– Biped– Quadruped– Hexapod– Long-chain (like ropes)

2013 ¥10,000 69


ReDySim?

• Recursive Dynamic Simulator is Freehttp://www.roboanalyzer.com

orhttp://www.redysim.co.nr/download

Demo• Modules

1. Open- and closed-loop multi-body systems2. Free-floating system3. Legged robot with ground interactions4. Symbolic computations

70


A Robotic Gripper (Inverse Dynamics)

#2

#1

#4

#3

O0, O1

O2

O3

O4 0.1

0.05

0.05

0.05

0.05

#2

#1#4

#3 3

2

4

θ1

θ3

θ2

θ4

#0

X0

Y0

O0, O1

O2

O3

O4

X4

X1

X2

Y1

X3

1

71


Planar biped

#5

#3

#2

#6

Y0

X0

φ1θ5

θ2

θ6

θ3

O1

#7 #4

θ7 θ4X6

X5

X2

X3

X4X7

X1

#1

Y1O0

O4O7

O6

O3

O2 , O5

#5 #2

#6

O1

#4

#1

O4 O7

O6 O3

O2 , O5

0.5

0.5

0.5

0.25

0.15

72


More robots

73


Agrawal (2013): MUBNew Application: Chains and Ropes

74

0

Ms

M1

Base

Mi Torsionspring

Detail of the module Mi


MuDRA: Carpet Cleaning

75

Multibody Dynamics for Rural Applications (MuDRA)Pose rural mechanisms as research problems


Tree-types: System Recursion

#20

#10

#11

#30#1

#2

#1

#0

Subsystem IIISubsystem IISubsystem I

10

30

20

11

1

2

1

#0

#0

C

X

YC

C

-

-

B

B

-

τD

E

ASME J. of Mech. Des., Dec. 2007

• Unknowns: 6+3 & Eqs. 2+1

• Unsolvable independently (Indeterminate subsystems)

• Unknowns: 4 & Eqs.: 4

• Solvable independently (determinate subsystem)

76


Results

0 0.2 0.4 0.6 0.8 1 1.2 1.4-3000

-2000

-1000

0

1000

2000

3000

Time (sec)

Forc

e (N

)

λ1xλ1y

0 0.2 0.4 0.6 0.8 1 1.2 1.4-100

-50

0

50

100

Time (sec)

Torq

ue (N

-m)

τD :ProposedτD :ADAMS

ADAMS Animation

77


Int.: 2009Indian: 2013


Carpet Srapper vs. Robot Leg

8

b5

7

4

Y

X

Carpet

Path of point T

Path of coupler point C

C

3

2

8

1

6

5

Leg

mec

hani

sm b

y C

ecca

relli

79


Using ReDySim (As a Robotic Leg)

80


Summary

• Kinematics• Dynamics• DeNOC-based modeling• In-house RoboAnalyzer & ReDySim

software• Recursive dynamics

– Open, Tree and Closed-loop systems• Several practical examples


Acknowledgements

• Dr. Suril V. Shah• Mr. Rajeevlochana• Mr. Amit Jain• Ms. Joyti Bahuguna• Dr. Sandipan Bandyopadhyay• …..


Thank you

Email: [email protected]://sksaha.com

Questions / Comments / Suggestions?

1 @ McGraw-Hill Education Chapters 8-9 - IIT Kanpur · related to Robot Kinematics & Dynamics ... – Links: A rigid body has 6-DOF – Joints: ... @ McGraw-Hill Education 16 ZY X

Documents