@ 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
@ McGraw-Hill Education 2
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)
@ McGraw-Hill Education 3
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)
@ McGraw-Hill Education 4
Revolute Joint: Five constraints
Fig. 5.1 A revolute joint
@ McGraw-Hill Education 5
Prismatic Joint: Five constraints
Fig. 5.2
@ McGraw-Hill Education 6
Cylindrical Joint: Four constraints
Fig. 5.4
@ McGraw-Hill Education 7
Helical Joint: ??? constraints
Fig. 5.3
@ McGraw-Hill Education 8
Spherical Joint: ??? constraints
Fig. 5.5 A spherical jointFig. 5.6
@ McGraw-Hill Education 9
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
@ McGraw-Hill Education 10
Closed-chain
Fig. 5.8 Fig. 5.10
@ McGraw-Hill Education 11
Open-chain
Fig. 5.9
@ McGraw-Hill Education 12
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
@ McGraw-Hill Education 13
Motions
• Positions (noun) or Translation (verb): Easy (unique)
• Orientation (noun) or Rotation (verb): Difficult (non-unique)
@ McGraw-Hill Education 14
[ ,
[ ] ,
[ ]
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 15
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ααα−α
≡10000
CSSC
ZQ . . . (5.21)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−≡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−≡
γγγγ
ββ
ββ
CSSC
CS
SC
XY
00
001;
0010
0QQ
. . . (5.22)
@ McGraw-Hill Education 16
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
@ McGraw-Hill Education 17
Non-commutative Property (NCP): Geometrically
Fig. 5.20
@ McGraw-Hill Education 18
Non-commutative Property …
Fig. 5.21
@ McGraw-Hill Education 19
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
@ McGraw-Hill Education 20
Dynamics
Newton’s 2nd law: ModellingGiven ; Find Inverse dynamicsGiven ; Find Forward dynamics
Integrations
Actually
xmf =xm , ffm ,
CRO
Force, Mass, m
xf
∫∫ == dtxxdtxx ;mfx =
@ McGraw-Hill Education 21
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
@ McGraw-Hill Education 22
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
@ McGraw-Hill Education 23
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.
@ McGraw-Hill Education 24
Kinetic and Potential Energies
• Kinetic Energy
• Potential Energy
∑=
−=n
iT
iimU1
gc
∑=
⎟⎠⎞⎜
⎝⎛ +=∑
==
n
i iiTii
Tiim
n
i iTT1 2
1
1ωIωcc
@ McGraw-Hill Education 25
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 =
@ McGraw-Hill Education 26
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
−−≡=
@ McGraw-Hill Education 27
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
= +
@ McGraw-Hill Education 28
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
@ McGraw-Hill Education 29
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 ≡
@ McGraw-Hill Education 30
DeNOC-based Dynamics
Decoupled Natural Orthogonal Complement matrices
@ McGraw-Hill Education 31
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
@ McGraw-Hill Education 32
Simple System
: Vertical component Reaction: Horizontal component Motion
Mass, m
Force,
f
efvf
vff
x
@ McGraw-Hill Education 33
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
@ McGraw-Hill Education 34
Complex Systems• Newton-Euler (NE)
Euler’s:
Newton’s:
34
3 scalar eqs.3 scalar eqs.
@ McGraw-Hill Education 35
Uncoupled NE Equations
•The 6n uncoupled equations of motion35
@ McGraw-Hill Education 36
Velocity Constraints: DeNOC Matrices
Bij: the 6n × 6n twist-propagation matrix
pi: the 6n-dimensional joint-rate propagation vector or twist generator36
@ McGraw-Hill Education 37
Definition: DeNOC Matrices
• N≡NlNd: the 6n × n Decoupled Natural Orthogonal Complement37
@ McGraw-Hill Education 38
Coupled Equations
• n coupled Euler-Lagrange equations
- no partial differentiation38
[ ] ( ) [ ] 0Tv cf f+ =i j
@ McGraw-Hill Education 39
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
@ McGraw-Hill Education 40
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
@ McGraw-Hill Education 41
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= −
@ McGraw-Hill Education 42
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
@ McGraw-Hill Education 43
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
@ McGraw-Hill Education 44
;
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
@ McGraw-Hill Education 45
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
@ McGraw-Hill Education 46
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
@ McGraw-Hill Education 47
Joint Torques
No gravity (horizontal
)
With gravity (vertical
@ McGraw-Hill Education 48
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
@ McGraw-Hill Education 49
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
@ McGraw-Hill Education 50
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)
@ McGraw-Hill Education 51
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)
@ McGraw-Hill Education 52
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)
@ McGraw-Hill Education 53
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
@ McGraw-Hill Education 54
Tree-type and Closed Systems
Base
A link or body
0
#k
#β(k)
54
@ McGraw-Hill Education 55
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
@ McGraw-Hill Education 56
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
@ McGraw-Hill Education 57
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
=
@ McGraw-Hill Education 58
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
@ McGraw-Hill Education 59
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
@ McGraw-Hill Education 60
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.
@ McGraw-Hill Education 61
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
@ McGraw-Hill Education 62
• 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
@ McGraw-Hill Education 63
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
@ McGraw-Hill Education 64
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
@ McGraw-Hill Education 65
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
@ McGraw-Hill Education 66
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
@ McGraw-Hill Education 67
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
@ McGraw-Hill Education 68
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.
@ McGraw-Hill Education 69
• 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
@ McGraw-Hill Education 70
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
@ McGraw-Hill Education 71
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
@ McGraw-Hill Education 72
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
@ McGraw-Hill Education 73
More robots
73
@ McGraw-Hill Education 74
Agrawal (2013): MUBNew Application: Chains and Ropes
74
0
Ms
M1
Base
Mi Torsionspring
Detail of the module Mi
@ McGraw-Hill Education 75
MuDRA: Carpet Cleaning
75
Multibody Dynamics for Rural Applications (MuDRA)Pose rural mechanisms as research problems
@ McGraw-Hill Education 76
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
@ McGraw-Hill Education 77
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
@ McGraw-Hill Education 78
Int.: 2009Indian: 2013
@ McGraw-Hill Education 79
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
@ McGraw-Hill Education 80
Using ReDySim (As a Robotic Leg)
80
@ McGraw-Hill Education 81
Summary
• Kinematics• Dynamics• DeNOC-based modeling• In-house RoboAnalyzer & ReDySim
software• Recursive dynamics
– Open, Tree and Closed-loop systems• Several practical examples
@ McGraw-Hill Education 82
Acknowledgements
• Dr. Suril V. Shah• Mr. Rajeevlochana• Mr. Amit Jain• Ms. Joyti Bahuguna• Dr. Sandipan Bandyopadhyay• …..
@ McGraw-Hill Education 83
Thank you
Email: [email protected]://sksaha.com
Questions / Comments / Suggestions?