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Lagrangian Dynamics for Simple Pendulum
q = q Q = T
xG = a cos x^ ^ asin
yG = a sin
y acos
^ ^
2 2 2 G
(^21) 2 G
(^2222221) 2
(^21) 2 G
(^221) 2
1 K mx y J ma sin a cos J J ma
P = m g y = m g a sin
L KP J ma mgasin
2 2 2 G
T
L L
dt
d Q q
L
q
L
dt
d
^
mgacos
L
J ma
L
dt
d J ma
L 2
G
2 G
J ma mgacos T
2 G ^
a
Y
A
B X
T
m, J (^) G G
gravity
Lagrangian Dynamics for Spring-Mass
q = x q x Q = FEXT
2 2 K 1 mx^2 2 P 1 kx L = K - P = 2 2
(^21) 2
(^1) mx kx
Q
q
L
q
L
dt
d
FEXT
x
L
x
L
dt
d
m x x
L
mx x
L
dt
d
kx x
L
mx kxF EXT
k
m FEXT
x
x
2 33
2 m 2 a mr J J
L
^ ^ ^
2 33
2 m 2 a m r J J 2 mrr
L
dt
d
gma mr cos
L
2 33
T m a m r J 2 J 3 2 m 3 r 3 r 3 gm 2 am 3 r 3 cos
2 33
2 2
33 3
mr r
L
33 3
m r r
L
dt
d
mr m gsin r
L
3
2 33 3
F m r m r m 3 gsin
2 3 3 33
F mr mg sin
T 2 mrr gm a m r cos
0 m r
m a m r J J 0
3
2 33
333 2 33
(^33)
2 3
2 33
2 2
Lagrangian Dynamics for Two Link Anthropomorphic Manipulator (Double
Pendulum)
Two solid rigid bars with revolute joints A and B
Lengths d 2 and d 3 - mass centers at a 2 and a 3 from proximal ends
Masses m 2 and m 3 - centroidal mass moments of inertia J 2 and J (^3)
2 CCW from positive x axis T 2 is torque of ground on bar 2 about pin A, CCW positive
3 CCW from centerline of bar 2 T 3 is torque of bar 2 on bar 3 about pin B, CCW positive
Gravity g acts along negative y axis
q 2 2 q 2 2 Q 2 T 2
q 3 3 q 3 3 Q 3 T 3
x 2 a 2 cos 2 x 2 a 2 2 sin 2
y 2 a 2 sin 2 y 2 a 2 2 cos 2
x 3 d 2 cos 2 a 3 cos 2 3 x 3 d 2 2 sin 2 a 3 2 3 sin 2 3
y 3 d 2 sin 2 a 3 sin 2 3 y 3 d 2 2 cos 2 a 3 2 3 cos 2 3
2 2 3 2 3
(^21) 2 2 2
(^21) 3
2 2 3 3
(^21) 2
2 2 2 2 K 1 m x y m x y J J
2 2 3 2 3
(^21) 2 2 2
1
3 2 3 2 2 3 3
2 2 3
2 2 3 3
(^21) 2
2 2 3 2
(^21) 2
2 2 2 2
1
J J
K m a md ma mda cos
P m 2 y 2 gm 3 y 3 g
P m 2 a 2 m 3 d 2 g sin 2 m 3 a 3 gsin 2 3
a (^3)
d 3
d 2
a (^2)
Y
A
B
X
C
T 2
T 3
m 2 , J (^2)
m 3 , J (^3)
md a sin ma cos g
L
3 2 3 2 2 3 3 3 3 2 3 3
md a sin m a cos g
T m a J mda cos md a sin
3 2 3 2 2 3 3 3 3 2 3
3 2 3 3 2 3 2 3 3 2 3 2 3 3
2 3 3 3
ma cos g
mda sin
ma J
T m a J md a cos
3 3 2 3
2 3 2 3 3 2
3 3
2 3 3
3 3 2 3 3 2
2 3 3 3
3
2 J (^) B m 3 a 3 J
2 3 2 3 3
2 3 2
2 J (^) A JBm 2 a 2 m d J 2 md a cos
C JB m 3 d 2 a 3 cos 3
D m 3 d 2 a 3 sin 3
G 2 m 2 a 2 m 3 d 2 gcos 2
G 3 m 3 a 3 gcos( 2 3 )
2 3
2 T 2 JA 2 C 3 2 D 2 3 D 3 G G
3
2 T 3 C 2 JB 3 D 2 G
3
2 3 2 2
2 3
2 3
3
2
B
A
3
2
G
G G
D
D 2 D
C J
J C
T
T
3
2 3 2 2
2 3
2 3
3
2
1
B
A
3
2
G
G G
D
D 2 D
T
T
C J
J C
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
x 10
-15 (^) Planar two link manipulator
Time [sec]
Validate torque equations [N.m]
T
T
4 2 2 2 3 ^23 ^ ^23 ^4 ^234 ^ ^234
4 2 2 3 2 3 4 2 3 4
x d sin d sin a sin
x d cos d cos a cos
^
4 2 2 2 3 ^23 ^ ^23 ^4 ^234 ^ ^234
4 2 2 3 2 3 4 2 3 4
y d cos d cos a cos
y d sin d sin a sin
^
2 2 4 2 3 4
(^21) 2 3 2 3
(^21) 2 2 2
1
2 4
2 2 4 4
(^21) 3
2 2 3 3
(^21) 2
2 2 2 2
1
J J J
K m x y m x y m x y
2 2 4 2 3 4
(^21) 2 3 2 3
(^21) 2 2 2
1
4 3 4 2 3 2 3 4 4
4 2 3 2 2 3 3 4 2 4 2 2 3 4 3 4
2 2 3 4
2 2 4 4
(^21) 2 3
2 2 4 3
(^21) 2
2 2 4 2
1
3 2 3 2 2 3 3
2 2 3
2 2 3 3
(^21) 2
2 2 3 2
(^21) 2
2 2 2 2
1
J J J
m da cos
m dd cos m da cos
m d m d ma
K m a md ma mda cos
P m 2 y 2 gm 3 y 3 gm 4 y 4 g
P m 2 a 2 m 3 d 2 m 4 d 2 g sin 2 m 3 a 3 m 4 d 3 gsin 2 3 m 4 a 4 gsin 2 3 4
i i i
Q
q
L
q
L
dt
d L K P
2 2 2
T
L L
dt
d
4 3 4 ^234 ^4
3 2 3 4 2 3 2 3 3 4 2 4 2 3 4 3 4
4 2 3 4
2 3 2 3 4 4
2 4 3
2 2 2 3 3
2 4 2
2 3 2
2 2 2 2
m da 2 2 cos
mda m d d 2 cos md a 2 cos
m a md m d J m a m d J m a J
L
4 3 4 ^234 ^44344 ^234 ^4
4 2 4 2 3 4 3 4 4 2 4 3 4 2 3 4 3 4
3 2 3 4 2 3 2 3 3 3 2 3 4 2 3 3 2 3 3
4 2 3 4
2 3 2 3 4 4
2 4 3
2 2 2 3 3
2 4 2
2 3 2
2 2 2 2
m da 2 2 cos m da 2 2 sin
m da 2 cos m da 2 sin
mda m d d 2 cos md a md d 2 sin
m a md m d J ma md J ma J
L
dt
d
2
m a md m d gcos m a m d gcos m a gcos
L
4 3 4 2 3 4 4 4 3 4 4 2 3 4 4
4 2 4 2 3 4 3 4 4 2 4 3 4 2 3 4 3 4
3 2 3 4 2 3 2 3 3 3 2 3 4 2 3 3 2 3 3
4 2 3 4
2 3 2 3 4 4
2 4 3
2 2 2 3 3
2 4 2
2 3 2
2 2 2 2
m a md md gcos ma md gcos ma gcos
mda 2 2 cos m da 2 2 sin
mda 2 cos mda 2 sin
mda mdd 2 cos mda mdd 2 sin
T ma md md J ma md J ma J
4 2 4 3 4 4 3 4 4 3 4
4 2 4 3 4 4 3 4 4 2 4
3 2 3 4 2 3 3 4 2 4 3 4 2 3
2 4 2 4 3 4 4 3 4 4 4
2 3 2 3 4 2 3 3 4 2 4 3 4 3
4 4 2 4 3 4 4 3 4 4 4
2 4 4
3 3 2 3 4 2 3 3 4 2 4 3 4 4 3 4 4
4 3
2 4 4
2 4 3
2 3 3
2 3 2 3 4 2 3 3 4 2 4 3 4 4 3 4 4
2 3 4
2 4 4
2 4 3
2 4 2
2 3 3
2 3 2
2 2 2 2
m a md md gcos ma md gcos ma gcos
2 md a sin m da sin
2 md a sin m da sin
2 md a mdd sin mda sin
md a sin m da sin
md a mdd sin mda sin
ma J md a cos mda cos
mda md d cos mda cos 2 m da cos
ma md ma J J
2 mda md d cos 2 mda cos 2 m da cos
m a md ma md md ma J J J T
2 4
2 3
2 A 2 3 4 2 3 2 4 3 4
D E E F G G G
T J A B 2 D E 2 E F 2 E F
3 3 3
T
L L
dt
d
4 2 3 4
2 3 2 3 4 4
2 4 3
2 3 3 3
mda m d d cos m da cos m da 2 2 cos
ma m d J m a J
L
4 3 4 ^23 ^44344 ^23 ^4
4 2 4 2 3 4 4 2 4 2 3 4 3 4
4 2 3 4
2 4 4 4
m da cos m da sin
m da cos md a sin
m a J
L
dt
d
4 4 ^234
4 2 4 2 2 3 4 3 4 4 3 4 2 3 2 3 4 4 4
ma gcos
m d a sin m da sin
L
4 4 ^234
4 2 4 2 2 3 4 3 4 4 3 4 2 3 2 3 4 4
4 3 4 2 3 4 4 3 4 4 2 3 4
4 2 4 2 3 4 4 2 4 2 3 4 3 4
4 2 3 4
2 4 4 4
m a gcos
m da sin m da sin
m da cos m da sin
m da cos md a sin
T ma J
4 4 ^234
2 4 3 4 4 3
2 4 3 4 4 4 2 4 3 4 2
4 3 4 4 2 3
4 4
2 4 4
4 4 3 4 4 3
2 4 4
4 4 2 4 3 4 4 3 4 4 2
2 4 4 4
m a gcos
m da sin
mda sin m da sin
2 m da sin
m a J
m a J m da cos
T ma J m d a cos m da cos
2 3
2 T 4 B 2 C 3 JC 4 2 F 2 3 EF 2 F G
4
2 J (^) C m 4 a 4 J
3 4 3 4 4
2 4 3
2 J (^) B JCm 3 a 3 m d J 2 m da cos
2 ^323423 ^3424 ^34
2 4 2
2 3 2
2 J (^) A JBm 2 a 2 md m d J 2 m da m dd cos 2 m da cos
A JB m 3 d 2 a 3 m 4 d 2 d 3 cos 3 m 4 d 2 a 4 cos 3 4
B JC m 4 d 3 a 4 cos 4 m 4 d 2 a 4 cos 3 4
C JC m 4 d 3 a 4 cos 4
D m 3 d 2 a 3 m 4 d 2 d 3 sin 3
E m 4 d 2 a 4 sin 3 4
F m 4 d 3 a 4 sin 4
G 4 m 4 a 4 gcos 2 3 4
G 3 m 3 a 3 m 4 d 3 g cos 2 3
G 2 m 2 a 2 m 3 d 2 m 4 d 2 g cos 2
(^234)
2 4
2 3
2 A 2 3 4 2 3 2 4 3 4
D E E F G G G
T J A B 2 D E 2 E F 2 E F
(^34)
2 4
2 T 3 A 2 JB 3 C 4 2 F 2 4 2 F 3 4 DE 2 F G G
(^4)
2 3
2 T 4 B 2 C 3 JC 4 2 F 2 3 EF 2 F G
4
3 4
2 3 4
2 4
2 3
2 2
3 4
2 4
2 3
4
3
2
C
B
A
4
3
2
G
G G
G G G
E F F 0
D E 0 F
0 D E E F
2 F 0 0
0 2 F 2 F
2 D E 2 E F 2 E F
B C J
A J C
J A B
T
T
T
4
3 4
2 3 4
2 4
2 3
2 2
3 4
2 4
2 3
4
3
2
1
C
B
A
4
3
2
G
G G
G G G
E F F 0
D E 0 F
0 D E E F
2 F 0 0
0 2 F 2 F
2 D E 2 E F 2 E F
T
T
T
B C J
A J C
J A B
Linear State Space Model for Two Link Manipulator
{y} f({y}) {y} [A ]{y }
y
y
y
y
{y}
y
y
y
y
{y } LINEAR
3
2
3
2
4
3
2
1
3
2
3
2
4
3
2
1
linearize about nominal values of y
3
2 3 2 2
2 3
2 3
3
2
1
B
A
3
2
G
G G
D
D 2 D
T
T
C J
J C
2 3 2 3 3
2 3 2
2 J (^) A JBm 2 a 2 m d J 2 md a cos
3
2 J (^) B m 3 a 3 J
C JB m 3 d 2 a 3 cos 3
D m 3 d 2 a 3 sin 3
G 2 m 2 a 2 m 3 d 2 gcos 2
G 3 m 3 a 3 gcos( 2 3 )
H
G
C J
J C
J J C
A
B 2 3 A B
2
T md a sin magcos( )
T mda sin 2 md a sin ma md gcos magcos( )
T D G
T D 2 D G G
H
G
3 3 2 3
2 3 3 2 3 3 2
3 2 3 3 2 3 2 2 3 2 2 3 3 2 3
2 2 3 2 3 3 3
3
2 3 2
2 3 2 3
2 2 3
3
2
3
2
2 3 2
2 3 2 3
A
B 2 A B
3 3 A 3
3 2 A B
3 A
B 2 2 A B
3 B A 3
3
2
H/ H/ H/ 0
G/ G/ G/ G/
C J
J C
J J C
H
G
C/ J /
0 C/
J J C
H
G
C J
J C
J J C
2 C C/ J J /
3 ^23
2 3
3 2 3 3 2 3 3 3 2 3
2 3 3 2 3 3 3
2 2 2 3 2 2 3 3 2 3
G/ 2 D
G/ 2 D
G/ mda cos 2 mda cos magsin( )
G/ ma md gsin ma gsin( )
H/ 0
H/ 2 D
H/ mda cos magsin( )
H/ magsin( )
3
2 2
3 3 2 3
2 3 3 2 3 3 2
2 3 3 2 3
C/ D
J / 2 D
3
A 3
3
2
3
2
2 3 2
2 3 2 3
A
B 2 A B
2 3 A B A
2 A B B
2 B B A B B 2 2 (^3) A B
2
H/ H/ H/ 0
G/ G/ G/ G/
C J
J C
J J C
H
G
J J 2 J C C 22 J J J C C
2 J J C J J 2 J C C
J J C
D
(^)
H
G
J J 2 J C C 22 J J J C C
2 J J C J J 2 J C C
J J C
D
P
P
2 A B A
2 A B B
2 B B A B B 2 2 (^33) A B
23
3
2
3
2
33
23
2 3 2
2 3 2 3
A
B 2 3 A B
2
3
2
0 P 0 0
0 P 0 0
H/ H/ H/ 0
G/ G/ G/ G/
C J
J C
J J C
0 P 0 0
0 P 0 0
H/ H/ H/ 0
G/ G/ G/ G/
C J
J C
J J C
A
33
23
2 3 2
2 3 2 3
A
B 2 A B
LINEAR
