Recursive Robot Dynamics

Lecture 1

Prof. S.K. Saha ME Dept., IIT Delhi

What is recursive?

1

#n

#1

#i

i

n

#1: vel., accn.

1: rate, accn

+ #0

#i: vel., accn.

#n: vel., accn.

#n: inertia force

n: torque

i: torque

#i inertia force

#1:inertia force

#0: vel., accn. = 0

1: torque

Why recursive?

• Efficient, i.e., less computations and CPU time

Compute jt. torque Compute jt. accn. (Inverse) (Forward)

0 5 10 15 20 25 30

0

2000

4000

6000

8000

10000

12000

14000

Total number of joints

Co

mp

uta

tio

na

l C

ou

nts

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

Proposed

Balafoutis

Featherstone

Angeles

0 5 10 15 20 25 30

0

0.5

1

1.5

2

2.5x 10

4

Total number of joints

Co

mp

uta

tio

na

l C

ou

nts

1-, 2- and 3-DOF joints

Proposed

Mohan and Saha

Lilly and Orin

Fetherstone

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.

Unrealistic: Constraint Violation

#0

#1

1

#2

#3

2

3 θ3

θ1 θ2

Review of Dynamic Formulations

• Euler-Lagrange (EL)

• Newton-Euler (NE) i i i

i i i i i i

m

v f

I ω ω I ω n

i

i i

d L L

dt q q

i i i i i i M t WM t w

, ,i i ii

i i i i

i i im

I O ω nω 1 ΟM W t ,w

O 1 v fΟ Ο

Starting point for the Recursive Dynamics (using the DeNOC matrices)

DeNOC matrices

Newton-Euler to Euler-Lagrange

• Analytical expressions of vector and matrices,

Decomposition of inertia Matrix, Recursive algorithms,

Dynamics model simplifications, etc.

,

Decoupled form of the velocity transformation matrix

Minimal order Equations of motion

(Euler-Lagrange)

Newton-Euler Equations of

motion ×

Eliminate constraint forces and moments from the NE equations.

Rate of Independent coordinate θ.

Link velocities t (ω and v) = ×

DeN

OC

m

atri

ces

=

Example: A Moving Mass

: Vertical component Reaction

: Horizontal component Motion

Mass, m

Force,

f

fvf

vf

f

x

Equation of Motion

Newton’s 2nd law: Velocity constraint: NOC: Euler-Lagrange:

cff mce

x ][ic

][i

ef

[ ] [ ( ) ] [ ]T Tv cf f f mx f mx i i j i i

External force,

Mass, m

i

j

c

cfReaction,

Note that

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

Uncoupled NE Equations

• Newton-Euler (NE) equations for the ith rigid body

• NE equations in compact form

where

i

i i i i i i

i im

I ω ω I ω n

v f

i i i i i i M t WM t w

, , ,i i ii

i i i i

i i im

I O ω nω 1 ΟM t W w

O 1 v fΟ Ο

Uncoupled NE Equations

#0

1

#n

#1

#i

i

n

E C

i i i i i i i M t WM t w w

E C Mt WMt w w

1 2

1 2

diag [ ]

diag [ ]

n

n

M M M M

W W W W

• Separate the bodies n bodies

• 6n NE equations

Kinematic (Velocity) Constraints

Bij: the 6n 6n twist-propagation matrix

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

( )

ij

ij j i

c

1 O

B r d 1 1i

i

i i

ep

e d

DeNOC Matrices

• NNlNd: the 6n n Decoupled Natural Orthogonal Complement

Coupled Equations of Motion

• Pre-multiplication by NT

• Equations in compact form

I : nn Generalized inertia matrix (GIM)

C : nn Matrix of convective inertia (MCI) terms

: n-dimensional vector of generalized forces due to driving torques/forces, and those resulting from the gravity, environment and dissipation.

Iθ Cθ τ

τ

)()( CETTwwNWMttMN

n coupled EL equations

- no partial differentiation T C N w 0

Generalized Inertia Matrix (GIM)

• Generalized inertia matrix (GIM)

where

• Each element of the GIM

• Mass matrix of composite body

15

T

ij i i ij ji p M B p

1, 1 1,

T

i i i i i i i M M B M B

Td dI N MN

l

T

l MNNM ~

Vector of Convective Inertia (VCI)

• Vector of Convective Inertia

where and

• Each element of h

'

1, 1

where , and

and

T

i i i

T

i i i i i n n

i i i i i i

h p w

w w B w w w

w M t W M t

'~wNθChT

d

)('~ WMtMt'Nw T

d θWNNt' )( ll

Generalized Force (Joint Torque)

• Generalized Force

where

• Each element is obtained recursively

1, 1

,

where , and

T E

i i i

E E T E E E

i i i i i n n

p w

w w B w w w

T Edτ N w

ET

l

EwNw ~

Example: One-link 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 ]02

1

2

1[][ ;]100[][ 11 de

20 0 0

[ ] 0 1 012

0 0 1

ma

2

I

2

22

0

[ ] [ ] 012

0 0 1

s s c

maT s c c1 2

I Q I Q

100

0

0

cs

sc

Q

Ref: “Introduction to Robotics” by Saha

Moment of inertia about O

where

( )

[ )] 0

T

T

h

p MW WM p

e I(e e) (e Ie

f

ndeewN ])([1

TTET

l

TT mg ]00[][;]00[][ 11 fn

mgas2

11

Equation of motion: 21 1

3 2ma mgas

1

#n

#1

#i

i

n

#0

1 1 1

2 2 2 1

1n n n n

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

β p Ω p

β p Ω p

B β B α

β p Ω p

B β B α

Recursive Inverse Dynamics

1 1 1 1 1 1 , 1

1 1 1 1 1 1 21 2

n n n n n n

T

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 W M α B γ

1 1 1

1 1 1

T

n n n

T

n n n

T

p γ

p γ

p γ

Recursive Robot Dynamics

Lecture 2

Prof. S.K. Saha ME Dept., IIT Delhi

Summary of Previous Lecture

• Why recursive?

• Define DeNOC (Decoupled Natural Orthogonal Complement) matrices from velocity constraints

• Derived NE EL equations (Constrained minimum set)

• Analytical expressions for the GIM, and other matrices

Equation of motion (Dynamic Model)

21 1

3 2ma mgas

One-link Arm

Chinese (PRC)

Spanish (Mexico)

RoboAnalyzer (Free: www.roboanalyzer.com)

where

( ) [ )] 0T Th p MW WM p e I(e e) (e Ie

f

ndeewN ])([1

TTET

l

TT mg ]00[][;]00[][ 11 fn

mgas2

11

Equation of motion: 21 1

3 2ma mgas

2

11

1( )

3

T T TI i m ma p Mp e Ie (e d) (e d)

Verify the results using MATLAB’s symbolic tool (MuPAD)

Example: Two-link Manipulator ;

11 12 21 1 1

21 22 2 2

( ); ;

i i i h

i i h

I h Cθ τ

22 :ScalarTi 2 2 22 2p M B p

2222222222222 ][][][][][ ddeIeTT mi

22 : 6 6 identity matrix

1 OB

O 1

2

2

2 2

: 6-dim. vector

ep

e d

2

2 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 s e d

2 2

2 2 2

0

0

0 0 1

c s

s c

Q

22 2 2

22

2 2 2 2 2 2 2

0

[ ] [ ] 02 3 120 0 1

T

s s c

mas c c

I Q I Q

2222222222222 ][][][][][ ddeIeTT mi

2 2 222 2 2 2 2 2 2

1 1 1

12 4 3i 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

e d Q d

e d r

r1 1 1

1 1 1

0

0

0 0 1

c s

s c

Q

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

1 1 1 1 1

12 2 4 3 2i m a m a a c m a m a m a a c

1

1

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

1

1

1

: 6 6 sym. matrixm

I OM

O 1

X3

Y3

11δ

1δIBMBMM 2

11

11212111 ~

~~~

m

T

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 c

e I e d d

Vector of Convective Inertia

Link Joint ai

(m)

bi

(m)

i

(rad)

i

(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

1

2

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

an

d I

nert

ia p

ara

me

ters

Inverse Dynamics Results

Joint Torques

No gravity (horizontal)

With gravity (vertical

Equation of motion for one-link arm

21 1

3 2ma mgas

Forward Dynamics & Simulation

2

3 1( )

2mgas

ma Forward Dynamics:

Integration (numerical): 1 2

1 2

2 12

;

3 1( sin )

2

y y

y y

y mga yma

1st order form:

( , )ty f y

Integrate (say, numerically) to obtain y(t) using, e.g., Runge-Kutta method (ode45 of MATLAB)

Simulation

Simulation using RA

Recursive Forward Dynamics

θ

34

Step 2: UDUT Decomposition

, where andT T

Analytically

UDU θ I UDU τ Cθ

Equation of the motion

The joint accelerations are then solved as

1 1θ U D U

T

Hence forward dynamics requires three steps

Step 1: Computation of φ

Step 3: Recursive computation of

Iθ Cθ τ

Observations

• Derivations appear to be complex for the simpler manipulators

• It was for demonstration only

• The computations will be done algorithmically

• Due to recursive natures, calculations are fast

• The algorithm should be used for complex robotic systems like 6- or more-DOF robots

Recursive Robot Dynamics

Lecture 3

Prof. S.K. Saha ME Dept., IIT Delhi

Summary of Previous Lecture

• Use of RoboAnalyzer (RA) for inverse dynamics of one-link arm

• GIM for 2-link manipulator

• Inverse dynamics of KUKA robot

• Forward dynamics and simulation

• Use of RA for simulation

• Some observations for using recursive dynamics

Stew

art

Pla

tfo

rm

Parallel Manipulators

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 y

f f

I d f d f r f r f

m c f f m c f f

=

Equations of motion: 3 per link

Base, #0

#1

#3

C1

#2

f12

f21

f01

f32

f23

f03

d1

r1

#1

#2

a3

2

3

1

3 2

1 4

a2

a0

a1

Base, #0

#3

4

1

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

1

1 1 0 '

1 1

1 1

1 1

0 ' 1 1

1 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 I

fs m c

f m c

fd d r r I

f m c

f m c

fd d r r I

fs

f

3

3 3

3 3

: 9 9 matrix x : 9 1 : 9 1

x

y

m c

m c

A b

• Ghosh and Mallik, Theory of Machines and Mechanisms

1 (Forward); (Backward)y

x=A b LUx=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

#3

C1

#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 Mt WMt 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

3 2

1

a2

a1

#3 1

f03 a0

• The DeNOC matrices for 3-link serial manipulator

Proof of C = JTf03

1 1 1

2 21 2 2

3 31 32 3 3

18 1 :18 18 :18 3 3 1

0' 0 '

0 '

l d

s s

s

N N

t 1 p

t B 1 p

t 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 w

1 B B w

N w 1 B w w B w

1 w

w

03 23 3 23 3 03

3

03 23

C

n n d f r fw

f f2 3

32

6 6

( )

0 '

T

s

1 r d 1B

1

f03

f23

r3

d3 n03

n23

C3

23

23

32 12 2 12 2 32

2 2 32 3

32 126 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 f

w

f f

03 01 1 03 1 01

1

03 01

C

n n p f d fw

f f

C2

r3 2

3

1

3 2

1

r2

1

f03

p2

p1

d3

Constraint Torque 1 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)i

ii i

i

ep

e d

03 01 1 03 1 01

1 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

Scalar

T

T

T T

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

p w e ρ f e ρ f

p w e a f e a f

C2

a3 2

3

1

3 2

1

r2

1

f03

p1

1

d2

Equations of Motion

*1 03[ ,0,0]:

T T

E C

known

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

1

0

0

x

y

a s a s a s a c a c a c

a s a s a c a c f

a s a c f

A xb

Adv.: Reduced size of 33 (instead of 99) for inverse dynamics

Subsystem Recursive Method • Cut-open in a suitable location • Make serial systems • Apply serial-chain methods

( )I I I I I C I I θ C θ τ τ

( )II II II II II C II I θ C θ τ τ

Subsystem equations

: 2 eqs., 2 (x, y)? [2R: done]

: 1 eq., 3 (, x, y)? [1R: done]

1 0 2 3 a a a a

#1 #3

4

3

1

1

Base, #0

#2

#1

#2

a3

2

3

1

3 2

1 4

a2

a0

a1

Base, #0

#3

4

1

4

f12 (-)

f21 ()

123

2-3

I I II II J J θ

3 123 2 12 2 121 1

2 1 2 2 3 123 2 12 2 121 1

;

I I I II II IIen ln d en ln d

I II a s a s a sa s

a c a c a ca c

A N N A N N

J J

1 2 3 a a a

[0,0] ( )

( )T

II T

II II II II II C II

J λ

I θ C θ τ τ

1( )

( )T

I I I I I C II C

J λ

Subsystem II:

Subsystem I:

System + Motion

Solve

System + Motion Driving torque, 1

Adv.: Subsystem recursion.; Maximum size: 22 (not 99 or 33); Can use existing serial-chain dyn. algo.

123

3

1

1

2

;

II

I II

II

θ

Sub-system

(link) Mass (Kg)

Length (m)

I(#1) 1.5

0.038

II(#2) 5

0.2304

II(#3) 3

0.1152

0 0.5 1 1.50

50

100

150

200

250

300

350

400

Time (s)

Jo

int

an

gle

s (

de

g)

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)

Dri

vin

g t

orq

ue

(N

m)

3

1

2

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

Recursive Robot Dynamics

Lecture 4

Prof. S.K. Saha ME Dept., IIT Delhi

International 2013

Int.: 2009 Indian: 2013

Summary of Previous Lecture

• Dynamics (classical way) of 4-bar mechanism using FBD

• Cut-open 3+0 constrained dynamics of 4-bar

• Cut-open 2+1 constrained dynamics of 4-bar

• Inverse dynamics using ReDySim

Forward Dynamics of 4-bar Mechanism

[0,0] ( )

( )T

II T

II II II II II C II

J λ

I θ C θ τ τ

11 ( )

( )T

I I I I I C II C

J λ

Constrained equations of motion:

1

:5 1:5 5 :5 1

)

0 ) -

I I T I I I

II II T II II II

I II I I II II

I C

xA b

O (J

I -(J θ C θ

J -J O λ -J J θ

0II II

II

J -J

0θVelocity constraints:

1x=A b

+ (jt. accn.) jt.

vel. & pos.

DAE [Diff. Algeb. Eqn.) formulation: System appr.]

System-level Lagrange Multiplier Method

[0,0] ( )

( )T

II T

II II II II II C II

J λ

I θ C θ τ τ

Constrained equations of motion:

1

2

T

I J θ

J O λ

2

II II I I II II

II

J

θ

J -J -J J θθ

jt. vel. & pos. (jt. accn.)

1:3 1:3 3 :3 2

( )

0 ( )

T

I I TI I I I

II II TII II II II

I C

θI J

O J-λ

I Jθ τ -C θ

11 ( )

( )T

I I I I I C II C

J λ

Velocity constraints:

1 1 1

1 2

:2 2

( ) )T

I

λ JI J (JI

1

1( )T θ I J λ

Adv.: Inversions of smaller (33 and 22) matrices

Five-bar 2-DOF Manipulator

Base, #0

#1

2

1

2

1

a2

a0

a1 1

Cut

here

#1

2

1

2

1

a2

a0

a1 1

-

Base, #0

( )I θ C θ τ τI I I I I I

( )I θ C θ τ τII II II II II II

Subsystem equations

: 2 eqs., 1 (2 )? [2R done]

: 2 eqs., 3 (1, x, y)? [2R done]

Symmetric; Need to combine: 4 eqs., 4 (1, 2, x, y)?

2 2

Subsystem Recursion for 5-bar

Base, #0

#1

2

1

2

1

a2

a0

a1 1

Cut

here

#1

2

1

2

1

a2

a0

a1 1

-

Base, #0

( )I θ C θ τ τI I I I I I

( )I θ C θ τ τII II II II II II

Subsystem equations

: 1 eq., 1 (2 )? [1R done]

: 3 eqs., 3 (1, x, y)? [3R done]

No need to combine: Solved

2 2

Inverse Dynamics (w/o Gravity) Link lengths: 1 m; Mass of links: 6 kg

1 11 1

2[ sin( )]

2

f ii

T tt

T T

Inverse Dynamics (w/Gravity)

Two-DOF Parallel Manipulator

#0 O5

O3

#3 #2

#1

O4

F2

C2

C1

r2

c12

d1

O2

#4

F3

C3

C4

r3

c43

d4

O1

Cut here

P

x2

l1 l2

θ1

ao

1 2 0 l l a 0

Kinematic constraints

Link bi

(m)

θi

(m)

ai

(m)

αi

(m)

1 0 1 0 /2

2 b2 [JV] 0 0 0

Sliding velocity (constant): 0.1 m/sec.

Link mi

(kg)

ri,x, ri,y

(m)

ri,z Ii,xx, Ii,yy

(kg-m2)

1,2 655 0 .625 399

Two RP serial manipulators; Combined: 4 eqs., 4 (f1, f2, x, y)?

Three-DOF RRR Parallel Robot

Subsystem I: 5 eqs., 5 (D1, 2x, 2y, 7x, 7y )? Subsystem II: 1 eq.. 1 (D2)? Subsystem III: 1 eq.. 1 (D3)?

Inverse Dynamics of 3-DOF RRR Robot

Driving Torques (w/o gravity)

Driving Torques (w/gravity)

Six-DOF Parallel Robot: Stewart Platform

#0

O1

O2

Subsystem I (3-DOF)

3 eqs.; 4 (x, y, x, f1)?

O1

O4

#3

#2

#1

O3

f λ

One particular limb

f1

fI

Moving platform

O2

#4

#5 O5

#3

#2

#1

O3

O4

O6 Cut joint

Dimensionless

links C3

C2

r3

c23

d2

C1

d23

d3 14

14

2I

T

1

T * g T λ

I I I I 2 I

TI 3

1I

*

0

( ) 0 +

1

f

e a

N w w = e a f

e

A A

f

FBD of VII Subsystem

-fIλ -fII

λ

-fIIIλ

-fIVλ -fV

λ

-fVIλ

pp1 pp2

pp3

pp4 pp5

pp6

VII VII VII

c c c+ × =•

I ω ω I ω n

VII VII

cm =•

c f

VII

λ

I

c

p1 p6

λ

VI

f

n p 1 p 1

f

=

VII

c λ λ

I VI= f f f

Subsystem VII (6-DOF) 6 eqs.; Zero?

Inverse Dynamics Algorithm ;

VII

VII

6c

1 61 p1 p6

6c 1 6

VI:6 61

:6 1

:6 1

Ii

i

i

i

f

f

J

τ

h

n xs p s p

s sf y

1τ J h

;

Ref: Sadana, M., 2009, Dynamic Analysis of 6-DOF Motion Platform, M. Tech Project Report, IIT Delhi

Trajectory and actuating force

Using GUI

More Robots using ReDySim

Planar biped

Waking leg (also used as carpet scrapping

mechanism)

Legged Robots

Spatial biped

Spatial quadruped

Flexible Rope using ReDySim

0

Ms

M1

Base

Mi Torsion

spring

Detail of the

module Mi

Conclusions • Purpose of Recursive Robot Dynamics

– Efficiency – Numerical stability

• DeNOC matrices for serial-chain • NE to EL derivations • Constrained equations of motion for serial-chain systems • Schemes for Inverse and Forward Dynamics (Simulation) • RoboAnalyzer software for robot dynamics • Parallel robot application • Four-bar mechanism from FBD • Cut-open system and subsystem recursion • ReDySim software • Five-bar, 3-DOF RRR, Stewart platform dynamics • Custom-made GUI for Stewart platform • Simulation of walking robots and rope using ReDySim

Acknowledgements

• Mr. Majid Koul

• Mr. Rajivlochana C.G.

• Dr. Suril V. Shah

• Mr. Mukesh Sadana

• Dr. Himanshu Chaudhary

• Students of IIT Delhi and other institutes who used the materials mentioned in the lectures and given feedbacks.

Thank you

For comments/questions:

saha@mech.iitd.ac.in

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