로봇공학 (KAU AME)
로봇공학, Chapter 4
1
Chapter 4. Dynamic Analysis and Forces
로봇 동역학
1. Newtonian mechanics vs. Lagrangian mechanics
2. Robot dynamics:
- Forward dynamics & Inverse dynamics
- 2-DOF manipulator의 예
3. Static force relationship between joint forces/torques
and end-effector ones
로봇공학 (Robotics)
로봇공학 (KAU AME)
로봇공학, Chapter 4
2
Robot dynamics
1
2
3
1 1 1q q q→ →
2 2 2q q q→ →
3 3 3q q q→ →
1
2
n
Robot armdynamics
1
2
n
q
q
q
1) Applied joint torques ➔ 각 관절에 발생하는 가속도 계산
Inverse dynamics
Forward dynamics
2) Required Joint torque 계산 Desired Joint trajectory
Dynamic Relationship(Dynamic equation of
Motion)
torques applied at
joints
Joint accelerations
ˆev
ˆe
로봇공학 (KAU AME)
로봇공학, Chapter 4
1
2
3
3
1
2
n
11 1
22 2
nn n
→ →
Joint torques
Inverse dynamics
Forward dynamics
조인트 가속도, 속도, 위치
( , , )
( , , )
( , , )
( , , )
x y z
x y z
p p p
v v v
말단부 위치 및 자세
말단부 속도
Position
Orientaion
Velocity
RPY rate
로봇공학 (KAU AME)
로봇공학, Chapter 4
4
Robot Dynamics
▪ Robot dynamics 해석의 필요성
• 로봇의 동적 거동(Dynamic behavior) 예측
• 로봇이 원하는 힘과 속도를 낼 수 있는 구동기(motor) 선정
• Model-based Control: 궤적 추종제어 알고리즘에서 로봇 동역학 식을 이용하여
필요한 토크 계산 → Computed Torque Method (CTM)
• 로봇이 핸들링 할 수 있는 가반 하중(Payload) 계산
( )
( )
d mvF ma
dt
d IM I
dt
= =
= =
1) Translational equation of motion:
: (3-DOF)
2) Rotational equation of motion:
: (3-DOF)
FC.G.
CGM
ma
C.G.
I
=
▪ Single rigid-body dynamics (6-DOF):
로봇공학 (KAU AME)
로봇공학, Chapter 4
5
Robot Dynamics
▪ Robot manipulator의 특징
• 다자유도, 다물체(Multi-body) 시스템
• 3차원 운동 (3-dimensional motion)
• 각 관절의 운동을 직전 링크에 대한 상대 운동으로 기술
→ 운동방정식을 관절좌표계(일반좌표계)에 대하여 표현
▪ Robot dynamics 방정식 유도 방법
1) Newtonian mechanics approach (Newton-Euler formulation)
2) Lagrangian mechanics approach (Lagrangian formulation)
로봇공학 (KAU AME)
로봇공학, Chapter 4
6
4.2 Lagrangian Mechanics: An Overview
2 2
( )
( )
1 1
2 2
d mvF ma
dtd I
M Idt
K mv I
P mg
•
= =
= =
•
= +
=
Newtonian Mechanics
- Translation:
- Rotation:
Lagrangian Mechanics
- Kinectic energy (KE): ( ) ( )
- Potential energy(PE):
병진운동 회전운동
21( (
2
( 1 ~ )
( 1 ~ )
( 1 ~ )
i
i i
i
i
h kx
L K P
d L LQ i n
dt q q
q i n
Q i n
+
−
− = =
= →
= →
) )
- Lagrnagian:
-
: generlaized coordinates
Lagrnagian equation of
( )
m
ot
: genera
ion:
중력 스프링 힘
일반좌표계=관절좌표계
( / )
lized force
: KE( ) PE( )
일반좌표에 해당하는 운동을 일으키는 외부 힘 토크
속도와 각속도 와 위치 를 일반좌표계의 함수로 표현
로봇공학 (KAU AME)
로봇공학, Chapter 4
7
Lagrangian Mechanics
1
1
( 1 ~ )
( ~ )
( ~ )
i
i i
n
n
d L LQ i n n
dt q q
q q q
Q
•
− = =
•
•
Lagrnagian eq. of motion:
: joint variables
: joint torques/Forces
Revolute J
원 연립미방
다자유도 로봇 매니퓰레이터의 의 경우
- 일반 좌표계
- 일반 힘
i
i i
i
ii
Rotational motion
d L L
dt
Translational motion
d L LF
dt dd
→
− =
• →
− =
oint
Prismatic Joint
◆ Lagrangian mechanics is based on the differentiation of energy terms only,
with respect to the system’s variables and time.
로봇공학 (KAU AME)
로봇공학, Chapter 4
8
Examples
▪ 예제 4.1~ 4.4
• 각각 Newton 역학과 Lagrange 역학을 이용하여 운동방정식을 유도하고
결과 비교
1 1 1 1
1 1 1 1
2 2 2 2
2 2
ˆ ( )
ˆ ˆ,
ˆ ( ) ( sin ) (1 cos )
ˆ ( cos )
m p x i y j xi
m v xi m a xi
m p x i y j x l i l j
m v x l
•
= + =
→ = =
= + = + + −
→ = +
Ex. 4.2 (Cart-pendulum system)
position:
position:
-
속도: 가속도:
-
속도:
2 2
sin
ˆ ( sin ) (1 cos )
i l j
m a x l i l j
+
→ = + + − 가속도:
1
1 1 1 1 1 1 1
2
2 2 2 1 2 12 1 2 12
ˆ ( ) ( sin ) ( cos )
ˆ ( ) ( sin ) ( cos )
m
p x i y j l i l j
m
p x i y j x l i y l j
•
= + = + −
= + = + + −
Ex. 4.3 (double pendulum system)
position:
position:
-
-
로봇공학 (KAU AME)
로봇공학, Chapter 4
9
2 2
1 1 1 1 1 1 1 1
2
1
1 1 1
(
1 1
2 21
2A
c
K m v I v
I
P m gl
= +
=
=
link 1 A rotation)
* link (C.G. translation + C.G. rotation)
link 1: ( : link1 c.g. , = )
은 점에 대하여
각 의 운동 = 의 에 대한
- 의 속도
11 1 1
2 2
2 2 2 2 2 2 2 1 2
2 2 1 1 2 12
1 2 1 2
sin sin2
1 1
2 2( sin sin )
( ) ( )
c
lm g
K m v I v
P m g l l
L K K P P
=
= + = +
= +
•
= + − +
link 2: ( : link2 c.g. , )
Lagrange's eq. of motion:
- Lagrangian:
- 의 속도
1
1 1
2
2 2
d L L
dt
d L L
dt
− =
− =
- For joint 1:
- For joint 2:
• Ex. 4.4 (Two link robot manipulator)
로봇공학 (KAU AME)
로봇공학, Chapter 4
10
4.3 Effective Moment of Inertia
211 12 1 11 1 1 2
212 22 2 22 2 2 1
D D G
D D G
•
+ + + =
2-DOF manipulator
Inertial force effect
Centrifugal force effect
Gravitational effect
Joint torques
Coriolis force effectInertial force interaction
between two links
( ) ( , ) ( ) ( )
( ) :
:
( ) :
( , ) :
( ) :
H q q C q q G q t
t
q
H q
C q q
G q
+
•
+ =
Input torque
Output state vector
Inertial
General form of Euler-Lagrange system dynam
matrix
Coriolis and centrifual vecto
ics
r
Gravitational vector
로봇공학 (KAU AME)
로봇공학, Chapter 4
11
4.4 Multi-DOF Manipulators
V
• Equations for a multiple-degree-of-freedom robot are very long and
complicated,
• But can be found by calculating the kinetic and potential
energies of the links and the joints,
• By defining the Lagrangian and by differentiating the Lagrangian equation
with respect to the joint variables.
• The kinetic energy of a rigid body with motion in three dimension :
__1 1
2 2GK V mV h= +
• The kinetic energy of a rigid body in planar motion
22
2
1
2
1IVmK +=
A rigid body in three-dimensional motionand in plane motion.
V
로봇공학 (KAU AME)
로봇공학, Chapter 4
12
General Formulation
0 0 0
00
1 1
( )
( )
( ) ( ) ( )
( , , ,1)T
i
ij
i
i i i i i i
i ij ji i
i i i i i
j jj
r
dm
p T r R r p
dq dqdp Td
x
v T r r rdt dt q dt dt
y z
U= =
•
= = +
→ =
=
=
=
Position of element mass in base frame
iz
{ }joint i
i
1 iz −
ix
iy
x y
z
idm
ir
0
i i ip T r=
( )idm link i의미소질량
0
ci i ip T r=
( )
2 2 2
1 1
1
( )
1 1( ) ( )
2 2
1( ) ( )
2
1[
2i
i
T
i i i i i i i i
Ti i
p ri i i
p r
i
i i i
ip ir
T
p i i i
J
r
dm
dK x y z dm Trace v v dm
dq dqTrace r r dm
dt dt
K dK Trace
U U
rrU dm
= =
=
•
= + + =
=
→ = =
Kinetic energy of element mass
1
]i
T
ir r p
p
U q q=
로봇공학 (KAU AME)
로봇공학, Chapter 4
13
Pseudo-inertia matrix
2 2 2 2
2 2 2 2
2 2 2 2
( ) ( ) / 2
( ) ( ) / 2
( ) ( ) / 2
z i i i y z x
y i i i z x y
x i i i x y z
I r dm x y dm x dm I I I
I r dm z x dm y dm I I I
I r dm y z dm x dm I I I
•
= = + = + − = = + = + − = = + = + −
Mass moment of inertia of link i w.r.t. {i}-frame
<Note>
2
2
20
,
i i i i
i i i iT
i i i i
i i i i
i i i i
i i i
i i i
x dm xydm xzdm xdm
yxdm y dm yxydm yzdm zxdm
x
zdm ydmJ rr dm
zxdm zydm z dm zdm
xdm x m
dm ydm zdm dmy
•
= =
= = =
=
Pseudo-inertia matrix
Symmetric link
의 경우
,i i i i i idm y m zdm z m
= =
2
2
2
y z x
xy xz i i
z x y
T yx yz i ii i i i
x y z
zx zy i i
i i i i i i i
I I II I m x
I I II I m y
J rr dm
I I II I m z
m x m y m z m
+ −
+ − = =
+ −
로봇공학 (KAU AME)
로봇공학, Chapter 4
14
Kinetic Energy
0
00
1 1
0
1 2 1 2
( , , ,1)
( ) ( ) ( )
( )0
0 0 0 1
j
j
j
T
i
ij
i
i i i
i ij ji i
i i i i i
j jj
j j j j j j
j j j j j j jij i i j
j jj j j
j
a
a
d
p T r
dq dqdp Tdv T r r r
dt dt q dt dt
c s c s s c
A s c c c s sTA A A A A A A A
s cq q
r x y z
U
Uq
= =
• =
→ = = =
− −
• = = =
=
=
)
0 1 0 0
1 0 0 0(
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
)
Revolute joint
revolute)
Prismatic joint
j
j
j j j j j j
j j j j j j j j
j j j
j j
j
a
a
i
s c c c s s
A A c s c s s cA Q A
q
ii
A
−
− − − − → = = =
→
0
1 2 1 2
1 2
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0(
0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0
( )
prismatic)
,
j
j
i
j j
j j
jii j j i
j j
j j k k i
j
ij
k
k
ij
AA Q A
q d
ATA A A A A Q A A
q q
A A Q A Q A A j kq
U
UU
= = =
=
=
=
마찬가지로
로봇공학 (KAU AME)
로봇공학, Chapter 4
15
Kinetic Energy & Potential Energy
0
0
1 1
( , ,( )
( )
,0)T
x y z
T
i i i i
T
i i i i
i i
g g gP m g T r
P P m g
L P
g
T r
K
= =
•
= −
•
= = −
= −
=
n n
Tota
Potential energy of l
l potential
ink i
Lagrangia
ene
n:
rg
y
1 1 1 1
2
( )
1
1( )
2
1(
2
i iT
i ip ir r p
i i p r
i act i
i
iK K Trace U U q
I
J q
q
= = = =
=
•
= =
+
n n
n
Total kin
actuator inerti
etic
a
energy
)에 의한 K.E
izi
1 iz −
ix
iy
x y
z
im
0
ci i ip T r=
ir
로봇공학 (KAU AME)
로봇공학, Chapter 4
16
Dynamic Equation of Motion for General Robots
1 max
2
( )
1 1 1 1
( )
1( , )
, 1 ~
( )
1 1( )
2 2
n n
Lagrange's eq. of motion:
i
i i
i iT
ip ir r p i act i
i p r ii i
n
i ac
i
n nT
pj p t i ij j ipi j
j p i j j
d L LQ i n
dt q q
L KK Trac J
Trace U J
e U U q q I qq q
I q DU q q I
= = =
= =
=
=
− = =
= = +
= + +
( )
( ) ( )
1 1 1 1 1
( )
1 1 1
(
act i
n n n n nij ij
ij j i act i j ij j i act i k j
j j j j ki k
i i i
n n n
ij j i act i ijk k j i i
j j k
ij p
q
dD Dd LD q I q q D q I q q q
dt q dt q
L K P
q q q
D q I q D q q D
D Trace U
= = = = =
= = =
→ = + + = + +
= − =
+ + + =
=
최종적으로max( , )
max( , , ) 1
)
( ), )n
nT
j p pi
p i j
nT T
ijk pjk p pi i p ppi
p i j k p
J U
D Trace UU J U D m g r
=
= =
= = −
로봇공학 (KAU AME)
로봇공학, Chapter 4
17
(Ref.) Lagrangian Formulation (Asada & Slotine’s Book)
0.
[1]
1 1( )
2 21 1
( )2 2
[2]
ˆ ( , , )
G i i
i i ci ci i G
T T
i ci ci
T
x y
i i i
T
i i i zc
h I
K m v v h
m v v I
g g gm gP gr
=
= +
=
=
+
= −
* angular momentum)
Kinetic energy of i-th link
Potential energy of link i
각운동량(
ci
i
i
v
I
base frame i-th link
base frame i-th link
base frame i-th link MOI)
는 에 대한 무게중심의 속도
는 에 대한 의 각속도
는 에 대한 의 질량관성모멘트(
iz
{ }joint i
( , , )link i i ii l m I
i
1 iz −
ix
iy
civi
x y
z0,ˆ
cir
ir
xx xy xz
yx yy yz
zx zy zz
i
I I I
I I I
I I I
I
=
로봇공학 (KAU AME)
로봇공학, Chapter 4
18
Lagrangian Formulation
1
2
1
(3 )
(3 )
Jacobian relationship between joint velocity and end-effector velocity
x
y
e z
e x
ny
z n
qv
qv
v v
q
n
nq
−
•
= =
− − − −
− −
P
O
J
J
1 2
1 2
1 1, 1
1 0
P P P Pn
O O O On
Pi i i e i
Oi i
z p z
z
− − −
−
=
• •
= =
For revolute joints For prismatic joints
Pi
Oi
J J J JJ =
J J J J
J J
J J
ci i
i
How to determine the velocity(v ) and angular velocity(ω )
of the centroid and the moment of inetia(I )?
•
로봇공학 (KAU AME)
로봇공학, Chapter 4
3-19
1ˆ ˆ
e ip p −− 1
ˆ ˆ ˆ( )i e ip p −= −
1ˆ
i iz −=
i-th Revolute joint
i-th Prismatic joint
ˆep
1
ˆˆ
i i iz −=
1ˆ
ip −
i-th link
ˆev
ˆe
ii-th joint
end-effector
각속도( )가
속도에 기여하는 양
ii-th joint
end-effector
각속도( )가
각속도에 기여하는 양
로봇공학 (KAU AME)
로봇공학, Chapter 4
20
Lagrangian Formulation
( ) ( ) ( )
1 1 2 2
( ) ( ) ( )
1 1 2 2
ˆ
ˆ
ˆ
ˆ
i i ici p p pi i
i i ii o o oi i
x
y
ci z
i x
y
z
v J q J q J q
J q J q J q
v
v
v v
•
+ + + =
+ + +
→ =
Similary, the velocity of the centroid of the i-th link is given by
1 1( ) ( ) ( )
1
( ) ( ) ( )
1
( )
(
( )
1
( )
)
1,
0
0
i i i
p pi p
i i i
o oi o
n n
i
j
i
p
i
o
j c
i
ipj
ioj
q qJ J J qJ
J J J qq q
z
J
z
rJ
J
− −
= =
=
For revolute joints For prismatic joints
where ( )
1
( )
1 0
(1 )
i
pj i
i
oj
J z
J
j i
−
−
=
<주의!> {j-1} 좌표계 원점에서i-th link 중심까지의 거리
로봇공학 (KAU AME)
로봇공학, Chapter 4
21
iz
{ }joint i
( , , )link i i ii l m I
1ˆ
ii z −
ix
iy
civi
x y
z
0,ˆ
cir
ir
1
1,
( )
1
( )
ˆ( )
ˆ
ˆ
( ) (1 )
i
pj j j j
i
oj j j
j ci
j
J q z q
J q z q j i
r−
−
−=
=
1ˆ
jj z −
1,j cir −
( ) ( ) ( ) ( )
1 1 2 2
( ) ( ) ( ) ( )
1 1 2 2
ˆ
ˆ
i i i i
ci p p pi i p
i i i i
i o o oi i o
v J q J q J q J q
J q J q J q J q
= + + + =
= + + + =
im g
로봇공학 (KAU AME)
로봇공학, Chapter 4
22
Lagrangian Formulation
( ) ( )
1 1
( ) ( ) ( )
1
)
( )
)
(
(
1 1( )
2 2
1
2
1
2
n nT T
i ci
ici p
ici i i i
i i
nT T
T i i T i i
i p p o i o
i
T
i o
K m v v I
m q J J q q J I J
v J q
q
Hq
q
q
J
= =
=
•
= +
=
= +
=
Kinetic energy of manipulator links
Manipula
tor i
( ) ( )( )2
( ) ( ) ( ) ( )
1
1 1
0,
1
( )
1
1
2
0
1
2
( , , )
nT T
i i i i
i p p o i o
i
n n
ij i j
i j
nT
i ci
i act i
i
T
i
x y z
H m J J J I J
H q q I q
g gP m g g gr
=
= =
=
=
+
= +
=
•
= − =
n
n
Potential energy of of manipulator links
ertia tenso
r
1( ~ )nq qconfiguration
dependent!
( )i actI : actuator inertia at joint
로봇공학 (KAU AME)
로봇공학, Chapter 4
23
Lagrangian Formulation
1 1 1
1 1 1 1 1
,
1
1
2
~
n n n
ij i j ij j
i j ji i i
n n n n nij ij
ij j j ij j k j
j j j j
i
i
k
i
ki
L KH q q H q
q q q
dH Hd L
d L LQ i n
dt q q
H q q H q q qdt q dt q
= = =
= = = = =
= = =
→ = + = +
− = =
Lagrange's eq. of motion:
( )
0, 1, (
1 1 1 1
0, ( ))
1 1
1
1 1
2 2
1
2
cj i c
n n n njk
jk j k j k
j k j ki i i
n ncjT T j
j i
j j
pi
i
T
e
i j pi
j ji i
i i
nij jk
ij j
j i
i
k
xt
ih
r r
HKH q q q q
q q q
rPm g G G m g J
q q
Q
H H
J
H
q
q
F
Jq
= = = =
= =
−
=
= =
= − = −
=
+ −
= =
→
+
정리하면, 1 1
n n
j k i i
j k
jk
q q G = =
+ =
로봇공학 (KAU AME)
로봇공학, Chapter 4
24
Lagrangian Formulation
( ) ( )( )( ) ( ) ( ) ( )
1
( )
1
1 1 1
1
2
nT T
i i i i
i p p o i o
i
ij jk
k in
T j
i j pi
j
n n n
i
ij
j j j
j
k
jk
j k
i
H m J J J I J
H H
q q
G m g J
H q q q
h
h
=
=
= = =
•
= +
−
+
=
= −
General dynamic equation of motion for robot manipulators
, ( ( , ) ( ) )1 ~k i i Hq C q q qn GG i + + =+ = =
각가속도에 의한 관성력 모멘트
Centrifugal force 및 Coriolis force에 의한 moment
Gravity force에 의한 moment
Joint torques
로봇공학 (KAU AME)
로봇공학, Chapter 4
25
(Example) 2-DOF Manipulator
1 1 1, ,l m I
1
1x
0x
0y
1y2
2 2 2, ,l m I
( )
1
1 1 (1) (1)21 1
1 1 (1)
1 0 0, 1
(1) (1) (1)
1 1 1 0
1 2 2
2
1 2 2
1 1 00
1 1 0
1 0 0 (0,0,1)
1 12 12
1 12 12
c c
c p p
c c
p c
T
o o o
c c
c
c c
qql s l s
qv q q J q J ql c l c
J z r
q J q J q J z
l s l s l sv
l c l c l c
=− −
= = = =
=
= = = = =
− − −
=
+
( )
(2) (2)
1 2
(2) (2) (2) (1) (1)
2 1 2 1 2 0 11 1 (0,0,1)
p p
T
o o o o o
q J J q
q J J q J q J J z z
=
= = = = = = =
( ) ( )( )( ) ( ) ( ) ( )(1) (1) (1) (1) (2) (2) (2) (2)
1 1 2 2
2 2 222 21 2 1 2 1 2 2 2 1 21 1
2 222 2 1
( ) ( ) ( ) ( )
1
2 2 2
0 ( 2 2) ( 2)0
0 0 ( 2)0 0
T T T T
p p o o p p o o
z zz c c c cc
nT T
i i i i
i p p o i
zc
o
i
c c
m J J J I J m J J J I J
I II m l l l l c m l l l cm
H m J J J
l
I Im l l l c m l
I J=
= + + +
+ + + = + + +
= +
+
2
11 12
12 22
z
H H
H H
로봇공학 (KAU AME)
로봇공학, Chapter 4
26
2-DOF Manipulator
1 111 122 112 121
2
111 121 122 2 1 2 2 112 2 1 2 1 2
2 211 212 221 222
2
222 212 221 211 2 1
( )
1
2 1
( ,
1
2
)
* 0, ( 2) (2 2)
( , )
* 0, 0, ( 2)
,c c
c
ij jk
ijk
k i
nT j
i j pi
j
h q q h h h h
h h h m l l s q h m l l s q q
h q q
H Hh
q q
G m
h h h h
h h h h m l
J
l
g
s q
g=
= −
= −
= + + +
= = = − = −
= + + +
= + = =
(1) (2)
1 1 1 2 1 1 1 2 2 1
(1) (2)
2 1 2 2 2 2 2
11 12 1 1 1 1
12 22 2 2 2
1 ( 12 1)
12
( , ) ( ) (
( , ) ( )
(0, ,0)
T T
p p c c
T T
p p c
T
G m g J m g J m gl c m g l c l c
G m g J m g J m gl c
H H q h q q G q
H G
g
H q h q q q
= − − = + +
= − − =
•
+ + =
−
=
2-DOF manipulator dynamics
2
)
( )
t
t
로봇공학 (KAU AME)
로봇공학, Chapter 4
27
2-DOF Manipulator
( )
2 2 2 2 2
1 1 2 2 2 1 2 2 2 2
2 2 2
2 2 2 2 2 1 2 1 1 2 12 2 1 1( ) 1
2 2 2 2
2 2 2 2 1 2 2 2 2 2
1 4 1 1
3 3 3 2
1 1 1
2 2 2
1 1 1 1 1
3 2 3 2 2
act
m l m l m l C m l m l C
m l S m l S m glC m glC m glC I
m l m l C m l m l S m glC
= + + + +
+ + + + + +
= + + + +
12 2( ) 1actI +
로봇공학 (KAU AME)
로봇공학, Chapter 4
28
Newton-Euler formulation (Ref.) Asada & Slotine’s Book
1, , 1
1, , 1 , , 1 1, 1,( ) ( ) ( )
Translational eq. of motion
Rotational eq. of motion
i i i
cii i i i i i i ci
ici i i i i i ci i i i ci i i
ii i i i i i i i
dI dII I I
dt
dvF f f m g m m v
dt
dHM N N r f r f
dt
dH
dt dt
− +
− + + − −
= = + =
•
= − + = =
•
+
= − + − − + − =
1,i i iN −
=
{ }joint i
( , , )link i i ii l m I
civi
1,i if −1,i iN −
, 1i if +−
, 1i iN +−
im g
iz
ix
iy
,i cir
로봇공학 (KAU AME)
로봇공학, Chapter 4
29
(Example) 2-DOF Manipulator
0,1 1,2 1 1 1
0,1 1,2 1, 1 1,2 0, 1 0,1 1 1
1,2 2 2 2
1,2 1, 2 1,2 2 2
0,1 1 1,2 2
0,1 1,2
1
2
,
(
(
for link 1
for lin
scalar)
scalar)
k 2
c
c c
c
c
f f m g m v
N N r f r f I
f m g m v
N r f I
N N
f f
I
I
•
− + =
− + − =
•
+ =
− =
= =
→
과 을 소거하면
2 1, 2 2 2 1, 2 2 2 2
1 2 0, 1 1 1 0,1 2 2 0, 1 1 0,1 2 1 1
c c c
c c c c
r m v r m g I
r m v r m v r m g r m g I
− − = − − − + − =
로봇공학 (KAU AME)
로봇공학, Chapter 4
30
4.5 Static Force Analysis
▪ Position control → end-effector(hand)가 주어진 궤적을 추종
▪ Force control→ end-effector와 접촉면 사이의 force/torque를 일정하게 유지 또
는 주어진 force/torque 궤적을 추종
[ ]
[ ]
H T
x y z x y z
H T
x y z x y z
F f f f m m m
D d d pd
−
=
−
=
−
•
=
•
External Force/torque at hand frame
Differen
Cartesian space (Task space, Operational space)
Joint spac
tial motion of the hand
Joint torques (revolu
e
te) an
1 2 6
1 2 6
[ ]
[ ]
T
T
T T T T
D d qd d
=
−
=
==
d Forces (prismatic)
Differential motion of joints
1
2
3
H F
로봇공학 (KAU AME)
로봇공학, Chapter 4
31
Static Force Analysis
H H TT TTW FF W Dp qD T
•
= = = =
•
=
Total virtual work Total
Principle of virt
virtual work
at task
ual work
Differential relati
o
sp
ns
ace at joint space
hip between joint m
otio
)
(
)
(
)
(
H
H T H H T T
T H
T T
T
p J qD JD
W F D F JD T D
FF
F
T
p
J
q
J
=
=
→ ==
== =
=
→
n and end-effector motion
Required joint torques/forces(or trajectory)
Desired force/torque (or trajectory)at hand frame(task space)
ManipulatorJacobian
로봇공학 (KAU AME)
로봇공학, Chapter 4
32
H.W. #3
▪ 예제 Example 4.1 ~4.5, 4.8~4.10 (8 probs.) = 40점
▪ 연습문제 Problem 1, 2, 3, 7, 8 (5 probs.)= 50점
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