Post on 11-Nov-2020
DeNOC-based Dynamic
Modelling of Multibody
Systems
(5th in a series)
Prof. S.K. SAHA Naren Gupta Chair Professor
Dept. of Mech. Eng.
IIT Delhi, INDIA
Mini-symposium, Fukuoka Univ., Japan Nov. 16, 2013
Fukuoka Univ., Japan Nov. 16, 2013
IIT Delhi, India
Total: 4
2011
USA
2012
P.R. China
Japan
S. Korea 2
Plan of Presentation
Introduction
Serial systems
RoboAnalyzer
Tree-type systems
Modeling & Simulation
Closed-loop systems
Multibody Dynamics for Rural Applications (MuDRA)
Conclusions
3
Introduction to IIT Delhi
Indian Institute of Technology (IIT)
7 + 9 new IITs = 16
Centers of Excellence
IIT Delhi (1961): Celebrating 50 Years
320 acres
13 Departments
11 Centers
3 Schools
8000 UG/PG/PH. D
Kanpur •
• Kharagpur
Bombay •
• Madras
Guwahati
•
• Roorkee
Pakistan
Australia
Delhi •
*2008
*2008
*2008
2008*
*2008 (Hyderabad w/Japan-JICA)
2008*
*2009
*2009
*2012
4
5
6
7
PAR Lab.@IIT Delhi ( 10 Faculty + 20 Students)
Pro
gra
mm
e for
Auto
nom
ous R
obotics
(Sponsore
d b
y B
ARC/B
RN
S)
Vis
it us a
t http
://robotic
s.iitd
.ac.in
8
Camera
F/T Sensor
KRC2 Controller
KUKA KR-5
Immersive
Environment
Exo-Skeleton
Project 4
Project 5
Project 3
Project 1
Layout of Four Interlinked Projects
Video
Mechatronics Laboratory, IIT Delhi
Estd. 2001
2011
Bro
chure
Annual Report
Robocon 10
Space robots (Toshiba, Japan)
1995
2013
2003
2006
2012
2000
Evolution of DeNOC
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)
2009 RoboAnalyzer Software (IITD)
11
Serial Chain: DeNOC-Based
1997 IEEE Trans. on Rob. & Aut.
V. 13, N. 2, Apr., pp. 301-304
PUMA Robot 12
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
13
Simple System
Newton’s 2nd law: Modelling
Given ; Find Inverse dynamics
Given ; Find Forward dynamics
Integrations
Actually
xmf xm , ffm,
CRO
Force, Mass, m
xf
dtxxdtxx ;
m
fx
14
Methods
τθθ
LL
dt
d)(
Newton-Euler (NE) Euler’s: Newton’s:
Euler-Lagrange (EL)
Kane’s, Hamilton’s …
Orthogonal Complement based, e.g., Decoupled Natural Orthogonal Complement (DeNOC)
15
DeNOC for Simple System
Newton’s 2nd law:
Velocity constraint:
Natural Orthogonal Complement:
Constrained motion (Euler-Lagrange):
cff mce x ][ic ][i
Force,
Mass, m
ef
i
j
xmfmT
ce
T ciffi ][)(][
Free-body diagram
c
cfReaction, Note that
0][][
][
ji
fi
c
T
c
T
f
16
Uncoupled NE Equations
•The 6n uncoupled equations of motion 17
Velocity Constraints: DeNOC Matrices
Bij: the 6n 6n twist-propagation matrix
pi: the 6n-dimensional joint-rate propagation vector or twist generator 18
Definition: DeNOC Matrices
• NNlNd: the 6n n Decoupled Natural Orthogonal Complement 19
Coupled Equations
• n coupled Euler-Lagrange equations
- no partial differentiation 20
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
21
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
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 Inverse Dynamics Algorithm
Saha (1999): ASME
22
Recursive Forward Dynamics
& Simulation
Articulated body matrix
1999 IJRR, V. 18, N. 1, pp. 20-36
23
Chinese (PRC)
Spanish (Mexico)
Books + RoboAnalyzer (Free: www.roboanalyzer.com)
24
Tree-type and Closed Systems
Base
A link or
body
0
#k
#β(k)
25
1i
ηi
1i
Mi
ki
ki
ηi
Intra-modular Constraints (Inside the module)
, 1 1t A t pk k k k k k
Module Mi
thTwist of the k link
ωt
o
k
k
k
-1 t tk kterms of
Inter-modular Constraints (Between the modules)
,i i i i t A t N θ
Module Mi and Mβ
11
and
t
t t θ
t
i k i k
ii
t ti Module twist in terms of
Module twist and module joint rate
1i
ηi
1β
1i
Mi
ηβ
ki
1β
ηβ
ki
ηi
Mβ
26
DeNOC Matrices
,
00
1 1
,
θt
t θ
t qt θ
tθ
i i
ss
and
l dN N and are the desired DeNOC matrices written in terms of module imformation
, l dt N N q
, 1, ...,t i i sSubstituing for
t qFor (s+1) modules generalized twist and rate
00
1,0 11
2,0 2,1 2
,0 ,1 , 1
,
's's
;
's
, if
l d
i
ss s s s s
j i j iM
1N O
A 1 ON
A A 1N NN
O NA A A 1
A O and
M0
i
Modules
Detail inside
the module
j
27
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 C
d 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
Module
s
NE equations for entire system
, ,E F C Mt ΩMEt w w = w w wwhere
28
DeNOC matrices
Dynamics using the DeNOC Matrices
Analytical expressions of vector and matrices, Decomposition inertia
Matrix, Recursive algorithms, Dynamics model simplifications, etc.
,
Decoupled form of the velocity transformation matrix
Minimal-order Equations of
motion
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
matr
ices
=
29
Mi
M0
Modules
30
Inter- and intra- modular recursive
( ) ( ) , ( )
T
i i i
T
i i i i i
N w
w w A w
,
, ,
i
i i
i i i i
i i i i i i i
i i i i i i i
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
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
T
j j j
j j
T
j k k j
r
r
p w
w w
w A w
k = 1:ηi , r=1:εk
i = 1:s
i = s:1
k = 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 iA N , and wrench due to foot-ground interaction F
iw F
orw
ard
recu
rsio
n
Back
ward
recu
rsio
n
1
1
,
( ) ( ) ,, (( )) , , ( )
( ) ( ) , ( )
ˆ ˆ ˆˆ ˆ ˆ; ;
ˆ
ˆ ˆ
ˆˆ ˆ ˆ;
ˆ ˆ ˆ
i i i T
i i i i i i
T
i i i i
i i i
T
i i i i i i i i i
T
i i ii ii i i i i
T
i i i i i
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
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
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
T
j j j j j j j j j
T
j j j j
j j j
T
j j j j j j j j j
T
j j k j j k
j j
T
j j k j
j
m m
m
r
r
r
r
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 i
i i i i
T
i i i i
μ A N q μ
q φ Ψ μ
i = 1:s
, ( ) ( ) ( )
( ) ( ) ( )
( ), 1
, 1
j k j j j
j j j
T
j j j j
r
r
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 )F
i(w and initial conditions and i iq q
Ste
p 1
Ste
p 2
Ste
p 3
Inverse dynamics Forward dynamics
31
Inverse Dynamics Forward Dynamics
0 5 10 15 20 25 30
-1000
0
1000
2000
3000
4000
5000
6000
7000
Total number of joints
Co
mp
uta
tio
na
l C
ou
nts
Al l 1-DOF joints
Proposed
Balafoutis
Featherstone
Angeles
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
2000
4000
6000
8000
10000
12000
Total number of joints
Co
mp
uta
tio
na
l C
ou
nts
Al l 1-DOF joints
Proposed
Mohan and Saha
Lilly and Orin
Fetherstone
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
1-D
OF
Mu
ltip
le-D
OF
This book addresses dynamic modelling methodology and simulation of tree-type robotic systems.
Such analyses are required to visualize the motion of a system without really building it.
Novel treatment of tree-type robots using
1) Kinematic modules;
2) Decoupled Natural Orthogonal Complements (DeNOC)
Unified representation of the multiple-degrees-of freedom-joints
Efficient Recursive Dynamics Algorithms
Examples: Biped, Quadruped. Hexapod, etc.
2013 ¥10,000 32
ReDySim?
Recursive Dynamic Simulator is Free
http://www.roboanalyzer.com
or
http://www.redysim.co.nr/download
It has three modules
1. Open- and closed-loop multi-body systems
2. Free-floating system
3. Legged robot with ground interactions
33
A Robotic Gripper (Inverse Dynamics)
#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
#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
#2
#1
#4
#3
O0, O1
O2
O3
O4 0.1
0.05
0.05
0.05
0.05
34
Planar biped
#5
#3
#2
#6
Y0
X0
φ1 θ5
θ2
θ6
θ3
O1
#7 #4
θ7 θ4 X6
X5
X2
X3
X4 X7
X1
#1
Y1 O0
O4 O7
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
#5
#3
#2
#6
Y0
X0
φ1 θ5
θ2
θ6
θ3
O1
#7 #4
θ7 θ4 X6
X5
X2
X3
X4 X7
X1
#1
Y1 O0
O4 O7
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
35
Legged robots
36
0
Ms
M1
Base
Mi Torsion
spring
Detail of the
module Mi
Agrawal (2013): MUB
New Application: Chains and Ropes
37
Four-bar mechanism (Forward Dynamics)
0.0895
0.038
0.1152
0.2304
#0
#1
1
#2
#3
2
3 θ3
θ1 θ2 38
Closed-loop systems
39
Recursive Dynamics for Closed-loops
(Without Cut-method) Courtesy: Prof. Shimizu
Saha (2001): MSM
Khan (2005): ASME
MUB
40
Solve for Unactuated Joint-rates
DeNOC:
41
Use in MuDRA
Multibody Dynamics for Rural Applications (MuDRA) Pose rural mechanisms as research
problems
Use of modern tools, e.g., Multibody Dyn.
Use of modern software
Able to publish
Benefits of MuDRA Establishing a culture
Rural problems may get solved
42
Carpet Cleaning: Traditional
43
Carpet Scrapping Machine
Purpose: To reduce human effort
Straight line generating machine
8
b5
7
4
Y
X
Carpet
Path of point T
Path of coupler point C
C
3
2
8
1
6
5
Not on scale
44
Tree-types: Double Recursion
#20
#10 #11
#30 #1
#2
#1
#0
Subsystem III Subsystem II Subsystem I
10
30
20
11
1
2
1
#0
#0
C
X
Y
C
C
-
-
B
B
-
D
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) 45
Results
0 0.2 0.4 0.6 0.8 1 1.2 1.4-3000
-2000
-1000
0
1000
2000
3000
Time (sec)
Fo
rce (
N)
1x
1y
0 0.2 0.4 0.6 0.8 1 1.2 1.4-100
-50
0
50
100
Time (sec)
To
rqu
e (
N-m
)
D
:Proposed
D
:ADAMS
ADAMS
Animation 46
Preface
This book has evolved from the passionate desire of the authors in using the modern concepts of multibody dynamics for the design improvement of the machineries used in the rural sectors of India and The World. In this connection, the first author took up his doctoral research in 2003 whose findings have resulted in this book. …. (contd.)
2009 ¥10,000 47
Conclusions
DeNOC for serial-chain systems
RoboAnalyzer
Tree-type and Closed-loop systems
Use in MuDRA
48
Conclusions
Components of a software
How much should we know? RecurDyn
DeNOC-based modeling and simulation
RoboAnalyzer and ReDySim
MuDRA
Acknowledgements
49
1st Int. & 16th Nat. Conf. on Machines and Mechanisms
Dec. 18-20, 2013
IIT Roorkee
Advances in Robotics: 2nd Int. Conf. of Robotics Society of India
July 1-4, 2015
Goa
iNaCoMM 2013
AiR-2015
• Goa (3h)
• Roorkee
(187km: 4h)
50
THANK YOU
ありがとうございました
http://web.iitd.ac.in/~saha
saha@mech.iitd.ac.in
51
Par
alle
l R
ob
ots
(SDD’08-1
0)
52