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![Page 1: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/1.jpg)
1
Dynamics
• Differential equation relating input torques and forces to the positions (angles) and their derivatives.
• Like force = mass times acceleration.
)(),()( qgqqqcqqD
anglesjoint of vector theis
uesinput torq of vector theis
q
![Page 2: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/2.jpg)
2
Euler-Lagrange Equations
• Equations of motion for unconstrained system of particles is straightforward (F = m x a).
• For a constrained system, in addition to external forces, there exist constrained forces which need to be considered for writing dynamic equations of motion.
• To obtain dynamic equations of motion using Euler-Lagrange procedure we don’t have to find the constrained forces explicitly.
![Page 3: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/3.jpg)
3
Holonomic ConstraintsF
(x,y)
m
l222 lyx
mjtqq nj ,,1,0),,,(
sconstraint holonomicfor expression General
1
![Page 4: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/4.jpg)
4
Nonholonomic Constraints
)(sin)(
)(cos)(
ttry
ttrx
mjtqqqqf nnj ,,1,0),,,,,,( 11
x
y
General expression for nonholonomic constraints is:
Nonholonomic constraints contain velocity terms which cannot be integrated out.
A rear powered front steering vehicle
![Page 5: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/5.jpg)
5
krr ,,1
Consider a system of k particles, with corresponding coordinates,
kiqqrr
nii
n
,,1),,,(
and ,,
1
1
Often due to constraints or otherwise the position of k particles can be written in terms of n generalised coordinates (n < k),
In this course we consider only holonomic constraints and for those constraints one can always find in principle n (n < k) independent generalised coordinates.
![Page 6: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/6.jpg)
6
Virtual Displacements
ir
kidtt
rdq
q
rdr i
n
jj
j
ii ,,2,1,
1
Define virtual displacements from above by setting dt=0.
kiqq
rr
n
jj
j
ii ,,2,1,
1
In our case dqj are independent and satisfy all the constraints. If additional constraints have to be added to dqj to finally arrive at a statement of virtual displacements which have only independent dqj, we can replace dqj with jq
![Page 7: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/7.jpg)
7
Virtual Work
forces ngconstraini theare
forces external theare )(a
j
j
f
f
k
jj
Ta
jj
k
jj
T
j rffrFW1
)(
1
)(
Let Fi be total force on every particle, then virtual work is defined as:
Constraining forces do no work when a virtual displacement, i.e., displacement satisfying all the constraints, takes place (as is the case with holonomic constraints), so in equilibrium
0)(1
k
jj
T
j rfW
![Page 8: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/8.jpg)
8
D’Alembert’s Principle
D’Alembert’s principle states that, if one introduces a fictitious additional force, the negative of the rate of change of the momentum of particle i, then each particle will be in equilibrium.
dt
drmprpfW j
jj
k
jj
T
jj
,0)(1
are) but t independennot are ( ij δqr
![Page 9: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/9.jpg)
9
Generalised Forces
is called the i-th generalised force. The equations of motion become:
.,,2,1,
where
1i
11 11
niq
rf
qqq
rfrf
k
j i
jT
j
i
n
ii
k
j
n
ii
i
jT
j
k
jj
T
j
.0))((11
i
k
j i
jT
jj
n
ii q
q
rrm
![Page 10: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/10.jpg)
10
F
(x,y)
m
l
222 lyx
cos
sin
coordinate dGeneralise
;sin
cos
1,1
1
1
1
F
F
F
FF
q
l
l
y
xr
nk
y
x0))((11
i
k
j i
jT
jj
n
ii q
q
rrm
Write the equation of motion.
![Page 11: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/11.jpg)
11
F
(x,y)
m
l
222 lyx
cos
sin
coordinate dGeneralise
;sin
cos
1,1
1
1
1
F
F
F
FF
q
l
l
y
xr
nk
y
x
0)(
0))((
1
11
yym
xxm
rrm i
k
j i
jT
ji
n
ii
FlFlFl
l
lF
q
rf T
j i
jT
j
22
1
2
1
sincos
sin
cos
2222
2222
))(cossin()cos(
))(cossin()sin(
llx
y
llx
x
Fml
:ismotion ofequation The
![Page 12: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/12.jpg)
12
Euler-Lagrange Equations of Motion
k
j i
jT
jj
i
jT
jj
k
j i
jT
jj
q
r
dtd
rmq
rrm
dtd
q
rrm
1
1
)()(
)(
i
j
l
n
l li
j
i
j
i
j
i
j
i
n
i
jT
jj
q
vq
r
q
r
dtd
q
r
q
v
rrv
1
2
11
and
Then
)(
![Page 13: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/13.jpg)
13
k
j i
jT
jj
i
jT
jj
k
j i
jT
jj q
vvm
q
vvm
dtd
q
rrm
11
)(
ii
k
j i
jT
jj
k
jj
T
jj qK
q
Kdtd
q
rrmvvm
11
)(21
K
be K toenergy kinetic theDefine
.,2,1,0
0))((111
niqK
q
Kdtd
qqK
q
Kdtd
rrm
i
ii
n
iii
ii
i
k
j i
jT
jj
n
ii
![Page 14: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/14.jpg)
14
K-VL
niqL
q
Ldtd
qV
i
ii
i
i
i
Where
.,2,1,
Then
such that
forces),or torques(external tau and energy) (potential V
exist theresuppose Finally,
![Page 15: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/15.jpg)
15
F
(x,y)
m
l
y
xyxmK21
.,2,1, niqL
q
Ldtd
i
ii
Write the dynamic equations of motions for this system.
![Page 16: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/16.jpg)
16
F
(x,y)
m
l
Fl
mly
xyxmK
2
2
21
21
.,2,1, niqL
q
Ldtd
i
ii
Flml
2
:ismotion of
equation dynamic The
![Page 17: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/17.jpg)
17
Expression for Kinetic Energy
B
T
B
T
B
dmzyxvzyxv
dxdydzzyxzyxvzyxvK
mdxdydzzyx
),,(),,(21
),,(),,(),,(21
),,(
BBody
0)(or 1
1,
1,
1
:asgiven is ),,( mass of Centre
B
c
B
c
B
c
B
c
B
c
ccc
dmrrrdmm
r
ydmm
zydmm
yxdmm
x
zyx
![Page 18: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/18.jpg)
18
Attach a coordinate frame to the body at its centre of mass, then velocity of a point r is given by:
mass of centre theofocity linear vel theis
mass of centre sbody' the toattached
frame theoflocity angular ve theis
where
)(
c
c
v
rSvv
4321
)()(21
KKKK
dmrSvrSvK c
T
B
c
![Page 19: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/19.jpg)
19
c
T
cc
T
B
c vvmdmvvK21
21
1
)0 (since 0)(21
)(21
2 c
B
T
c
T
B
c rdmrSvdmrSvK
0)(21
3 dmvrSK c
T
B
)()(21
)()(21
)()(21
4
T
TT
B
T
B
JSSTr
dmSrrSTr
dmrSrSK
BBB
BBB
BBB
T
B
dmzyzdmxzdm
yzdmdmyxydm
xzdmxydmdmx
dmrrJ
2
2
2
Where
![Page 20: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/20.jpg)
20
BBB
BBB
BBB
dmyxyzdmxzdm
yzdmdmzxxydm
xzdmxydmdmzy
I
)(
)(
)(
Where
22
22
22
0
0
0
)(
xy
xz
yz
S
IJSSTrK TT
21
)()(21
4
![Page 21: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/21.jpg)
21
00000
4
0
21
21
21
IRIR
IK
R
TTT
T
T
Frame for I and Omega
The expression for the kinetic energy is the same whether we write it in body reference frame or the inertial frame but it is much easier to write I in body reference frame since it doesn’t change as the body rotates but its value in the inertial frame is always changing.
So we write the angular velocity and the inertia matrix in the body reference frame.
![Page 22: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/22.jpg)
22
Jacobian and velocity
frame) referencebody in is ( )()(
,)(
i
T
ii
vci
qqJqR
qqJv
i
ci
qqJqRIqRqJqJqJmqKn
i
T
iii
T
v
T
vi
T
iicici1
)()()()()()()(21
qqDqK T )()(21
D(q) is a symmetric positive definite matrix and is known as the inertia matrix.
![Page 23: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/23.jpg)
23
Potential Energy V
B B
c
TTT mrgrdmgrdmgV
![Page 24: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/24.jpg)
24
Two Link Manipulator
0z
1
2
11, lm
22 , lm
p
1z
2z
0x
1x
2x
Links are symmetric, centre of mass at half the length.
21,,, Find21
cc vv
21
2211
T
2
T
1
22211121
ggV21
21
)(21
)(21
K
Find
cc
TT
c
T
cc
T
c
rmrm
IIvvmvvm
![Page 25: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/25.jpg)
25
0z
1
2
11, lm
22 , lm
p
1z
2z
0x
1x
2x
Links are symmetric, centre of mass at half the length.
1000
100
0
0
1
1
1
1111
111
l
slcs
clsc
A
1000
0100
0
0
122111212
122111212
21
2
0
slslcs
clclsc
AAT
1000
100
0
0
1
2
2
2222
222
l
slcs
clsc
A
![Page 26: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/26.jpg)
26
0z
1
2
11, lm
22 , lm
p
1z
2z
0x
1x
2x21
2211
T
2
T
1
22211121
ggV21
21
)(21
)(21
K
cc
TT
c
T
cc
T
c
rmrm
IIvvmvvm
Revision Questions:
1. What are m1 and m1?
2. In which frame are vc1 and vc2 specified?
3. In which frame are I1 and I2 specified?
4. In which frame are 1 and 2 specified?
5. What is g?
6. What are rc1 and rc2?
![Page 27: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/27.jpg)
27
K-VL niqL
q
Ldtd
i
ii
Where.,2,1,
qqDqK T )()(21
mrgV c
T
.,2,1
,21
)(,
,,
,,
nk
qV
qqq
d
q
dqqd k
k
jiji j
ji
i
jk
jijjk
Euler-Lagrange Equations – The general form.
and
![Page 28: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/28.jpg)
28
k
k
k
ji
j
ik
i
jk
ijk qV
q
d
q
d
q
dc
and 21
Let
,,,
)(),()( qgqqqCqqD
],,[ ,],,[)(
21
)(
where
11
1
,,,
1
n
T
n
i
n
i k
ji
j
ik
i
jkn
iiijkkj
qg
d
q
d
q
dqqcC
![Page 29: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/29.jpg)
29
01
0
1
1
0
1
0
0
11
0
0
2/
)( cc r
l
RSr
0z
1
2
11, lm
22 , lm
p
1z
2z
0x
1x
2x
0
1cr
0
2cr
1
2cr
Revision
The linear velocity of a rotating vector is the cross-product of its angular velocity and the vector itself.
Proof: Let be the vector rotating with an angular velocity then its derivative with time can be written as follows.
0
1cr1
0
2
1
11
0
01
2
0
1
1
0
0 )()( 22
22
2
cc
cc
c rzrzrr
r
1
0
0
01
0
0 )(z 111
ccc rrr
![Page 30: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/30.jpg)
30
0z
1
2
11, lm
22 , lm
p
1z
2z
0x
1x
2x
0
1cr
0
2cr
1
2cr
1000
100
0
0
1
1
1
1111
111
l
slcs
clsc
A
1000
0100
0
0
122111212
122111212
21
2
0
slslcs
clclsc
AAT
0
0 ;
0
0
2/ 1
1
0
10
2
2
0
1
222
l
Rrr
l
Rr ccc
;
0
0
2/1
1
0
0
1
l
Rrc
![Page 31: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/31.jpg)
31
r.manipulatolink - twofor the )( and ),(),( Evaluate 1
1
0
0
0
0 221 ccc rzrzrz
![Page 32: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/32.jpg)
32
qJl
l
rvcvcc 1,12
2
111
11
0
00
0cos)2/(
0sin)2/(
qJlll
lll
rvcvcc 2,22
2
121221211
21221211
00
)cos()cos()2/(cos
)sin()sin()2/(sin
2
1211
11
00
00
,
1
0
0
21
2211
T
2
T
1
22211121
ggV21
21
)(21
)(21
K
cc
TT
c
T
cc
T
c
rmrm
IIvvmvvm
![Page 33: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/33.jpg)
33
Write the expressions for K and V. From these obtain:
).( and ),,(),( qgqqCqD
![Page 34: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/34.jpg)
34
2
2
21
2
2222
221
2
2
2
122112
21
2
1
2
12
2
1111
)2/(
)cos)2/()2/((
)2/(2)2/(()2/(
zz
zz
zzzz
Ilmd
Iqllllmdd
IIllllmlmd
0;0;;
:sin)2/( ;0
222122112221
2212121111
cchchc
hqllmcc
)cos()2/(
)cos()2/(cos))2/((
))sin()2/(sin(;sin)2/(
21222
2122112111
21
21211221111
qqglm
qqglmqglmlm
VVV
qqlqlgmVqlgmV
![Page 35: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/35.jpg)
35
Newton-Euler Formulation
ii gm
if 1
1
i
i
i fR
i1
1
i
i
iR icir , icir ,1
iciiii
i
ii amgmfRf ,1
1
)()( ,11
1
,1
1
iiiiiicii
i
iciii
i
ii IIrfRrfRii
![Page 36: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/36.jpg)
36
Newtonian Mechanics
1. Every action has an equal and opposite reaction. Thus if body 1 applies a force f and torque tau to body 2, then body 2 applies a force –f and torque of –tau to body 1.
2. The rate of change of linear momentum equals the total force applied to the body.
3. The rate of change of the angular momentum equals the total torque applied to the body.
![Page 37: 1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.](https://reader036.fdocuments.in/reader036/viewer/2022070412/5697bf8a1a28abf838c8a456/html5/thumbnails/37.jpg)
37
Basic Relationships
000 )(
dtId
fdtmvd )(
ωIIωII
ωIIRS
ωRIRRISRhR
ωRIRISωRIIωRh
RIωRωRIRh
h
RR
T
TTT
T
T
)()(
)(
)(
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bygiven is momentumAngular
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40
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41
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