chapter 4. dynamic analysis and forces...

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)



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



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



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):



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)



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



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.



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:

-

-



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)



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



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



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=



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

+ −

+ − = =

+ −



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



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



16

Dynamic Equation of Motion for General Robots

1 max

2

( )

1 1 1 1

( )

1( , )

, 1 ~

( )

1 1( )

2 2

n n


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

=

= =

= = −



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

=



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 )?

•



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

êp

1

ˆˆ

i i iz −=

1ˆ

ip −

i-th link

êv

ê

ii-th joint

end-effector

각속도( )가

속도에 기여하는 양

ii-th joint

end-effector

각속도( )가

각속도에 기여하는 양



20


( ) ( ) ( )

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



21

iz

{ }joint 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



22


( ) ( )

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



23


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

= = =

= = = = =

= = =

→ = + = +

− = =


( )

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

qq

= = = =

= =

−

=

= =

= − = −

=

+ −

= =

→

+

정리하면, 1 1

n n

j k i i

j k

jk

q q G = =

+ =



24


( ) ( )( )( ) ( ) ( ) ( )

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



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



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



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 +



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


civi

1,i if −1,i iN −

, 1i if +−

, 1i iN +−

im g

iz

ix

iy

,i cir



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

− − = − − − + − =



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



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



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

chapter 4. dynamic analysis and forces...

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