chapter 2. robot kinematics: position...
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2-12008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Chapter 2. Robot Kinematics: Position Analysis로봇 기구학: 위치 해석
Forward kinematics
Inverse kinematics
Denavit-Hartenberg representation
4 x 4 Homogeneous transformation matrix
로봇공학로봇공학 (Robotics)(Robotics)
2-22008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Robot Kinematics
Forward kinematics:
• To determine where the end-effector (gripper or hand) of robot is?
• when all joint variables (ex, 로봇관절 회전각) and the linkage parameters are known.
• 4 x 4 transformation matrix
Inverse kinematics:
• To determine what each joint variable is?
• When we desire that the end-effector(hand) be located at a particular point.
Denavit-Hartenberg (D-H) representation:
• A simple and standard way of modeling robot links and joints.
2-32008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Robot Kinematics
1 2
( , , , , , )
( , , , )n
x y z φ θ ψθ θ θ→
•Given desired end-effector position
determine the required joint ang
Inverse kinemati
:
le
c
s
s
1 2( , , , )
( , , , , , )
n
x y z
θ θ θ
φ θ ψ→
•
차원 위치 및 자세각
Given joint angles
determine the end-effector posture(posit
Forward kinematics:
ion & orientation) : 3 roll-pitch-yaw
PUMA(6dof n=6)
2-42008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
( )( )
1 1 2 1 2
1 1 2 1 2
1 2
cos cos
sin sin
x l l
y l l
θ θ θ
θ θ θθ θ
•
= + +
= + +
→ = =
2-link manipulator
?, ?
의 예
2-52008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.2 Robot Mechanism
Manipulator-type robot의 특징
• 다물체(multi-body), 다자유도 시스템(multi degrees of freedom system)
• 3차원 open-loop chain 메커니즘
각 joint 및 link에서의 error가 end-point 까지 누적: Fig. 2.2(b)
내부센서(조인트 각 측정) 만으로는 실제 말단위치 측정 부정확
극복방법: 로봇 말단 위치를 외부센서(Camera, laser sensor 등)로 측정
0x0y
0z
1x 1y
1z
2x 2y
2z
1θ ( )nx n
( )ny o
( )nz a
2θ3θ
nθ
0nT
2-62008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Robot Mechanism
Closed-loop mechanism VS. Open-loop mechnism
(a) Closed-loop mechanism versus (b) open-loop mechanism
One active joint
Three active joints3DOF manipulator
Four-bar mechanism(1 DOF, closed-loop)
passive joints
2-72008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.3 Matrix Representation (Rotational/Translational Matrix)
공간상의 점 위치 벡터로 표현
♦ A point P in space :3 coordinates relative to a reference frame
^^^
kcjbiaP zyx ++=
Fig. 2.3 Representation of a point in space Fig. 2.4 Representation of a vector in space
♦ A vector P in space :3 coordinates of its tail and of its head
^^^__
kcjbiaP zyx ++=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
w
z
y
x
P__
w: scaling factor
2-82008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Matrix Representation
기준좌표계(고정)에 대한 회전좌표계(이동)의 자세 표현 (Fig. 2.5)
( , , )
( , , )
x x x
y y y
z z z
n o a
x y z
n o a
F n o a
n o a
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
(direction cosines) (Rotation matrix):
회전좌표계 각 축이
고정좌표계 각 축과 이루는
방향코사인으로 구성
unit vectors
Fig. 2.5 Representation of a frame at the origin of the reference frame
♦ Each Unit Vector is mutually perpendicular: normal, orientation, approach vector
2-92008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Matrix Representation
기준좌표계(고정)에 대한 이동좌표계(회전+위치이동)의 표현 (Fig. 2.6)
0 0 0 1
x x x x
y y y y
z z z z
n o a p
n o a pF
n o a p
⎡ ⎤⎢ ⎥⎢ ⎥= →⎢ ⎥⎢ ⎥⎣ ⎦
Rotation matrix + translation vector
Robot hand (Gripper)=end-effector
( )nn x
( )no y
( )na z
Fig. 2.6 Representation of a frame in a frame
• Example 2.3:
2-102008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
강체(Rigid Body)의 표현
3차원 공간 상의 강체 (로봇의 각 link) (Fig. 2.8)
Fig. 2.8 Representation of an object in space
♦An object can be represented in space by attaching a frameto it and representing the frame in space.
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
1000
zzzz
yyyy
xxxx
objectPaon
Paon
Paon
F
• 강체에 이동좌표계(moving frame) (n,o,a) 부착
기준좌표계에 대한 이동좌표계(강체)의
위치(position) 및 자세(orientation)의 변화
4 x 4 transformation matrix로 표현
2-112008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
강체(Rigid Body)의 표현
3차원 공간상의 강체 (로봇의 각 link) 기준좌표계(Reference frame)에 대하여 6자유도 운동 (6 DOF motion)
(x,y,z) 각 축 방향의 translation (3DOF)
(x,y,z) 각 축에 대한 Rotation (3 DOF)
( , , )
0 0 0 1
x x x x
y y y yobject x y z
z z z z
n o a p
n o a pF p p p
n o a p
⎡ ⎤⎢ ⎥⎢ ⎥= →⎢ ⎥⎢ ⎥⎣ ⎦
Rotation + translation
강체의 Rotation(3 DOF 운동)에대한 9개의 정보
6개의 constraints(구속조건)포함 (2.10)식
강체의 translation(3 DOF 운동)에대한 3개의 정보
2-122008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Rotation Matrix의 성질
0 0 0 1
x x x x
y y y yobject
z z z z
n o a p
n o a pF
n o a p
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
0n o o a a n
•↔
→ ⋅ = ⋅ = ⋅ =•
Rotation matrix : (2.10), (2.11), Three unit vectors are mutually orthogonal dot product = 0 : Each unit vector's length = 1
특성 식
1n o a→ = = =
Example 2.3 연습
2-132008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.4 Homogeneous Transformation Matrix
기준좌표계에 대한 강체의 위치(position) 및 자세(orientation)의 변화
4 x 4 homogeneous transformation matrix로 표현
Transformation matrices must be in square form.
• It is much easier to calculate the inverse of square matrices.
• To multiply two matrices, their dimensions must match.
3 3 3 1
1 3
0 0
1
0 1
0
x x x x
y y y y
z z z z
n o a p
n o a pF
n o a p
R p× ×
×
•
⎡⎡ ⎤
⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣
⎢
⎦
⎥⎣ ⎦
Homogeneous transformation matrix
1 1 2 2 2 22 1 1 1 11 2 2 1
2 2,0 1 0 1 0 1 1 0 10
R p R p R p R pF F F F
R R R p p
•
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = → = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣
+⎡
⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
⎤⎢ ⎥⎣ ⎦
Homogeneous transformation matrix
(Ex.)
의 연산
2-142008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.5 Transformations: Pure translation
( , , )
0 0
0 0( , , )
1
1
1
1
1
0 0
0 0 0
1
1
0 0
0 0
0 0
0 0 0 1
x y z
x
yx y z
z
x x x x x
y y y y ynew
z
Trans d d d
d
dT Trans d d d
d
d n o a p
d n o a pF
d
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣
•
⎦
(Fig. 2.9)
Pure translation location
에 의한 의 변화
Pure translation1)
0 0 0 1 0 0 0 1
( , , )
old
x x x x x
y y y y y
z z z z z z z z z
new y old
F
x z
n o a p d
n o a p d
n o a p n o a p d
F Trans d d d F
+⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
= ×
♦ Pure translation♦ Pure rotation about an axis♦ Combination of translations or rotations.
2-152008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Transformations: Pure rotation about an axis
1 0 0 0
0 c 0
0 c 0
0 0 0 1
1 0 0
( , ) 0 c
0 c
c 0
( , ) 0 1 0
0 c
s
sRot x s
s
s
Rot y
s
θ θ
θ θθ θ θ θ
θ θ
θ θθ
θ θ
−
•
⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥= − =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦
⎣ ⎦⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦
:
(Fig. 2.10, 2.11)
Rotation matrices
x-axis
: y-axis
기준좌표계 에 대하여 만큼 회전
기준좌표계 에 대하여
Pure Rotatio2 n)
c 0
( , ) c 0
0 0 1
s
Rot z s
θ
θ θθ θ θ θ
−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
: z-axis
만큼 회전
기준좌표계 에 대하여 만큼 회전
2-162008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
1 0 0
( , ) 0 c
0 c
x n
xyz noa y o
z a
U U RR
P P
P Rot x P P s P
P s P
P T P
θ θ θθ θ
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= × ⇔ = −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
→ = ×simply
RnoaP P=
Rotation about the x-axis.
UxyzP P=
Fig. 2.11 (2.16) (2.17)
2-172008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Transformations: Combined transformations
( , ) ( , ) ( , ) ( , )Rot x Rot y Rot y Rot xθ φ φ θ
•
× ≠ ×
(Translation + Rotation)
Successive transformation - Rotation
- Translation (Pure translation ):
의 경우 순서가
의 경
중요:
은 순서에 무 우
관
Combined Transformation 3)
( , , ) ( , , ) ( , , ) ( , , )
( , ) ( , , ) ( , ) ( , , )
x y z x y z x y z x y z
x y z x y z
Trans p p p Trans d d d Trans d d d Trans p p p
Rot x Trans p p p Rot y Trans d d d
Tran
θ φ →•
× = ×
→ → (Reference frame) successive transformation:
=
기준좌
표
계 에 대한
( , , ) ( , ) ( , , ) ( , )x y z x y zs d d d Rot y Trans p p p Rot xφ θ× ×
→
×
Premultip lying!
2-182008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
(Ex.) Transformation Matrix의 계산
0 0 0
0 0 0 ( , ) ( ,
( , , )
( , , ) ( , , )
( , ,
, ) ( , ) ( , ,
)
)
( , , ) ( , ) ( , , ) ( , )
x y z x y z
x y z x y z
R x Tr b b b R y
p x
Tr d d d
Tr d d d R y Tr b b b
y z
p x y z x
z R
y z
x y xp
θ φ
φ θ
→ → →
× × ×
• →
• ×
→임의 의 고정 기준좌표계 에 대한
초기위치 최종위치
최종위치 초기위치
point, (reference frame) transformation:
=
0
0
0
0 0 0
0 0 0
0 0 0
0 0 0
1 0 0 c 0 0 1 0 0 1 0 0 0
0 1 0 0 1 0 0 0 1 0 0 c 0
0 0 1 0 c 0 0 0 1 0 c 0
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
c 0
0 1 0
( , , )
x x
y y
z z
x
x x x
y y y
z z z
d s b n o a x
d b s n o a y
d s b s n o a z
s d
d
x y z
φ φ
θ θ
φ φ θ θ
φ φ
−
−
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
=
0
0
0
0 0 0
0 0 0
0 0 0
1 0 0
0 c
0 c 0 c
0 0 0 1 0 0 0 1 0 0 0 1
0 0 0 1
, ,
x
y y
z z
x x x
y y y
z z z
x x x x
y y y yx y z
z z z z
b n o a x
s b n o a y
s d s b n o a z
n o a p
n o a p
n o a px p y p z p
θ θ
φ φ θ θ
−
−
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥= = = =⎢ ⎥⎢ ⎥⎣ ⎦
. Then, we have
강체(Point p)에 부착된 회전좌표계의reference frame에 대한 초기자세
강체(Point p)에 부착된 회전좌표계의reference frame에 대한 최종자세
강체(Point p)의초기위치
강체(Point p)의 최종위치
2-192008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.5.4 회전좌표계에 대한 상대 변환 (Relative Transformation)
( , ) ( , , ) ( , ) ( , , )x y z x y zRot x Trans p p p Rot y Trans d d d
R
θ φ→ → →
•→
•
Robot link
link
successive transformation :
Current frame = (
( )
=
Rotating frame
)회전
의 운동
각 에 부착된 회전좌표계에 대한
좌표계
상대운동으로 기술
에 대한
예
0 1 21 2
( , ) ( , , ) ( , ) ( , , )x y z x y z
ET T T
ot x Trans p p p Rot y Trans d d dθ φ× × ×
→
⎧ ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ ⎫• ⎯⎯→ ⎯⎯→ ⎯⎯⎯→⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭⇒
reference rotating rotating end-effector
frame 0 frame 1 fra
Postmulti
me 2 frame E
Refer
ply
en
ing!
ce f
0 0 1 21 2
0 0 0 1
x x x x
y y y yE
z z zE
z
n o a p
n o a p
n o a pT T T T
⎡
=
⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
rame{0} end-effector{E} 에 대한 의 위치 및 자세
자세(orientation)
위치(position)
2-202008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Fig. 2.15 Transformations relative to the current(=rotating) frames.
♦Example 2.80 0( ,90 ) (4, 3,7) ( ,90 )xyz noaP Rot a Trans Rot o P= −
0( ,90 )Rot a(4, 3,7)Trans −
0( ,90 )Rot o
7
3
2noaP
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
0
6
0xyzP
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
2-212008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
로봇 핸드의 위치 및 자세
0 0 1 2 11 1 2 2 3 3( ) ( ) ( ) ( )n
n n nT T T T Tθ θ θ θ−
•
→ =
Robot hand(end-effector) reference frame (position/orientation)
의
에 대한 좌표
0
0 0 0 1
x x x x
y y y yn
z z z z
n o a p
n o a pT
n o a p
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
1 0 0 0
1
~ ( , , )
( , , ) , , )
( , , ) , , )
~
reference frame hand
position: orient forward kation: ( hand : (
joint ang
inematics.
le( )
n
x y z
x y z
n
x y z
p p p p n o a
p p p p n o a
θ θ
θ θ
→
•
=• =
주어진 에 대하여 에 대한 의
과 을 구하려면
원하는 위치 와 자세 에 대하여
을 구하려면 Inverse kinema tics.→
0x0y
0z
1x1y
1z2x 2y
2z
1θ ( )nx n
( )ny o
( )nz a
2θ3θ
nθ
0nT
3x 3y
3z
2-222008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.6 Inverse of Transformation Matrices
Fig. 2.16>
: Reference frame, : Robot Base frame,
: Hand frame, : end-effector frame,
: Part fra
Unknown : Robot base(R) h (
m
an
e
d H
RH
R
U U H U PE R E P E
U R
H E
H
P
T T T T TT
T
< = =
• 에 대한 ) transformation
Forward kinematics →의
로 결정됨
Fig. 2.16 The Universe, robot, hand, part, and end effecter frames.
( ) ( )( )( ) ( )( )
( ) ( )( )( )
( ) ( )
1 1
1 1
1 1
1 1,
U U H HR R E E
U U P HR P E E
U U P HR P E E
R U P E RU P E H H
U
RH
R H ER U E
RH
H
T T T T
T T T T
T T T T
T
T
T
T T T T
T T T T
− −
− −
− −
− −
=
→ =
=
= =
=
2-232008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
1
1
0 0 0 1
0
( , ) ( , )
1
x x x x
y y y y
z z z z
T
T
n o a p
n o a p
n o a p
p
R R
RT
RT
x xθ θ−
−⇒
• →
→ =
=
⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
Rotation matrix Inverse Rotation matrix orthogonal matrix!
: unitary matrix
Transformation matrix Inverse
의
의
0 0 0 1
0 1
,
x y z
x y z
x y z
T
x x x x
y y y y
z z z z
n n n p n
o o o p o
a a a p a
p
p n o a
R
p n o a
p n o a
p n o a
− ⋅
− ⋅
− ⋅
−
= = = =
⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
, where ,
2-242008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.7 Forward/Inverse Kinematics of Robots
공간상에서 기준좌표계(reference frame)에 대한 강체(rigid body)의 위치
(position)와 자세(orientation)을 구하기 위해서는
• 강체에 강체와 함께 움직이는 이동좌표계를 부착하고
• 이동좌표계 원점의 위치[(x,y,z):3 DOF]와
• 이동좌표계 각 축의 기준좌표계에 대한 자세[방향코사인 또는 Euler angle: 3 DOF]를 구함.
Robot의 경우 (Fig. 2.17)
• Robot hand(= 강체)에 hand frame(n,o,a)를 부착
• 로봇 각 링크의 parameter 값들과 주어진 joint angle에 의해 이동좌표계(n,o,a) 원점의 위치와 기준좌표계에 대한 hand 좌표계의 자세가 결정됨.
Robot hand(Gripper)=end-effector
( )nn x
( )no y
( )na z
2-252008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.7.1 Kinematic Equations for Position
( , , )
1 0 0
0 1 0
0 0 1
0 0 0 1
x
yRP
z
x y z
p
pT
p
•
⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
Cartesian coordinates (Fig. 2.18):
( , , )
( , , ) (0,0, ) ( , ) ( ,0,0)
1 0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 1 0 0
0 0 1 0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1
RP cyl
r l
T T r l Trans l Rot z Trans r
c s r
s c
l
α
α α
α αα α
•
= =
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Cylindrical coordinates (Fig. 2.19):
=
0
0
0 0 1
0 0 0 1
c s rc
s c rs
l
α α αα α α
−⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
=
A gantry robot is a Cartesian robot.
2 Linear translations and 1 rotation
2-262008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Kinematic Equations for Position
( , , )
( , , ) ( , ) ( , ) (0,0, )
0 0 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1
0 0 0 1 0 0 0 1 0 0 0 1
RP sph
r
T T r Rot z Rot y Trans r
c s c s
s c
s c r
β γ
β γ γ β
γ γ β βγ γ
β β
•
= =
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Spherical coordinates (Fig. 2.20):
=
0
0 0 0 1
C C S S C rS C
C S C S S rS S
S C rC
β γ γ β γ β γ
β γ γ β γ β γ
β β β
⋅ − ⋅ ⋅
⋅ ⋅ ⋅
−
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
=
1 2( , , , )nθ θ θ•
→
Articulated coordinates (Fig. 2.21):
Robot D-H transformation관절형 의 경우:
방법에 따른
1 Linear translation and 2 rotations
2-272008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.7.2 Kinematic Equations for Orientation
2.7.2(a) Roll-Pitch-Yaw (RPY) angles
(
( , , )
( , ) ( , ) ( ,
, , )
)
a o n
a o n
a o n
Rot a Rot o Rot n
φ φ φ
φ φ
φ θ
φ
ψ• = = =
→ → → →about curren
Roll-Pitch-Yaw (RPY) angles (Fig. 2.22):
* RPY sequence of rotation : (z y x)
:Current frame(
t axes:
현재의 회전좌 표계)에 대한 상대회전
♦Yaw: Rotation of about -axis (x-axis of the moving frame)nφ n
♦Roll: Rotation of about -axis (z-axis of the moving frame)aaφ
♦Pitch: Rotation of about -axis (y-axis of the moving frame)oφ o
( , , ) ( , , ) ( , ) ( , ) ( , )
0 0 0 0 1 0 0 0
0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 1 0 0 0 1
a o nRPY RPY Rot a Rot o Rot n
c s c s
s c c s
s c s c
φφ φ φφ φ θ θφ φ ψ ψ
θ θ
θ ψ φ θ ψ
ψ ψ
= =
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
=
2-282008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
( , , )
( , , )
( , ) ( , ) ( , )
0
0
0
0 0 0 1
a o nRPY
RPY
Rot a Rot o Rot n
c c c s s s c c s c s c
s c s s s c c s s c c c
s c s c c
φ φ φ
φ θ φ θ ψ φ ψ φ θ ψ φ ψφ θ φ θ ψ
φ θ ψφ
φ ψ φ θ ψ φ ψθ θ ψ
θ
θ
ψ
ψ
==
− +⎡ ⎤⎢ ⎥+ −⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦
=
(2-36)식 ( , )nRot n φ
( , )aRot a φ ( , )oRot o φ
2.7.2(a) Roll-Pitch-Yaw (RPY) angles
2-292008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
( , , ) ( , ) ( , ) ( , )
0
0
0
0
0
0
0
0 0 10 1 0 0
x x x
y y y
z z z
RPY Rot a Rot o Rot n
c c c s s s c c s c s c
s c s s s c c s s c
n o a
n o a
n
c c
s oc c as c
φ θ φ θ ψ φ ψ φ θ ψ φ ψφ θ φ θ ψ φ
φ θ ψ φ θ
ψ φ θ ψ φ ψθ θ ψ θ
ψ
ψ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
=
− +⎡ ⎤⎢ ⎥+ −⎢ ⎥⎢ ⎥−⎢⎣ ⎣
⎥⎦⎦
= =
( )nn x
( )no y
( )na z
θ
φψ
2.7.2(a) Roll-Pitch-Yaw (RPY) angles
• End-effector desired orientation (robot base) end-effector RPY angle
의 에 대하여 기준좌표계 에
대한 좌표계의 을 구하는 방법
2 2
2 2
tan tan tan
atan2( , )
atan2( , )
atan2( , )
y z z
x zz z
y x
z z z
z z
n n o
n ao a
n n
n o a
o a
φ θ ψ
φ
θψ
−= = =
+→
=
= − +
=
, ,
Roll:
Pitch:
Ya :
w
RPY angles Direction cosines
2-302008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.7.2(b) Euler Angles
( ,
(
) ( , ) ( , )
, , )
Euler angle (Fig. 2.24): RPY angle Euler angle
* Eule about current axer angle : (z y )
s:
:C
a
Rot a Rot o R
a
ot a z
o
φ θ ψ
•
→ → → →
책마다 정의가 약간씩 다름.
도 넓은 의미에서 의 한 종류임.
회전순서
( , , ) ( , ) ( , ) ( , )
0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1
urrent frame(
=
= (2-43)
Euler Rot a Rot o Rot a
c s c s c s
s c s c
s c
φ φ θ θ ψ ψφ φ ψ ψ
θ θ
φ θ ψ φ θ ψ=
− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
현재의 회전좌표계)에 대한 상대회전
식
( ) RPY angle .주 의 정의만 잘 익힐 것
2-312008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
, ,( ) ( , , )
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1 0 0 0 1
x y z
x
y
RH car
z
t
x
T T RPY
c c c s s s c c s c s c
s c s s s c c s s c c c
s c s c c
c c c s s s c c s
p p p
p
p
p
pc s c
s s s s s c c
φ θ ψ
φ θ φ θ ψ φ ψ φ θ ψ φ ψφ θ φ θ ψ φ ψ φ θ ψ φ ψθ θ ψ θ ψ
φ θ φ θ ψ φ ψ φ θ ψ φ ψφ θ φ θ ψ φ ψ
= ×
− +⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+ −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
− ++
=
=
1 1 2 2 1 2 1 1 23 3 3 1 21 3 3 3 1 ( , 0
0 0 0 1 0 0 0 1
0 1 1 0 1)
0
x x x x
y y y y
z z
y
z z z
n o a p
s s c c c n o a p
s c s c c n o a p
R P R P R R P R P
p
p
R I P
φ θ ψ φ ψθ θ ψ θ ψ
× × × ×
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
⎡ ⎤ ⎡ ⎤ ⎡ ⎤+=⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦= =
( ) 주
2.7.3 Forward Kinematic Equations
Robot base에 부착된 Reference coord.(기준좌표계 R)가 Cartesian coord.(직교좌표
계)이고, Robot hand에 부착된 이동좌표계의 Orientation을 기준좌표계에 대한 RPY angle로 표현한다면, Robot hand의 (즉, 이동좌표계의) Base에 대한 위치(location) 및 자세(Orientation)는?
2-322008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.8 Denavit-Hartenberg (D-H) Representation
D-H representation 방법
Open kinematic chain 형태 robot manipulator의
forward kinematics를 기술하기에 편리함.
Manipulators Succession of joints and links
• Robot joint : Revolute type (rotation) or Prismatic type (translation).
nz
joint n link n
link link .
nn의끝단에부착되어과 같이움직임
nθ1
n
nzθ
−
: (n-1) link
에 부착된 축에 대하여 회전
{ , , }
n n nx y z좌표계
1 nz −
( 1)link n− 에 부착
nx
ny
관절 좌표계(joint coordinates)의 정의
2-332008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
D-H Parameters
1nz −
nz2nz − nθ
1nθ +
1nθ −
joint n1joint n −
1link n−link n
( )1attached to link n−
( )attached to link n
( )2attached to link n−
1nx −
nxna
nd
nθ
nα
oo
Transformation between two successive frames
21 1
1
1
1
n
n n n
n
n
n
n
d
a
θ
− −−
−
−
−
•
•
•
→
→
z
x common normal between z z
y
-axis: n-th joint (revolute joint )
or (prismatic joint
variable
)
-axis: along the -axis
variab
& -axis
-axi
le
s: f
이
이
의 회전방향
직선운동 방향
ollows right-hand rule
1na −
2-342008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
두 관절 좌표계 사이의 변환(Transformation)
1 1 1
11( (0,0,
{ , , } { , , }
, ) ) ( ,0,0) ( , )n
n
n
n
n n n n n n
nn n n n n
n n n nd a
Rot z Trans d
x y z x y z
T Trans a Rot x
θ α
θ
θ
α
− − −
−−
→ → →
•
⎡ ⎤ ⎡ ⎤⎯⎯⎯⎯⎯⎯⎯→⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
= × × ×
: x
transformation:
(n-1) link n-link
두 좌표계 사이의
좌표계 좌표계
1 1
1 1
n
n
n n n
n n n nn
d
a α− −
− −
x : z z
: x x : z z
축과 축 사이 각도 축과 축 사이의 거리
축과 축 사이의 거리 축과 축 사이 각도
1
1
1
1
1(
(0, 0,
(
( ,
, )
)
,0,0)
)
n n n
n n n
n n n
n n n n
nRot z
Trans d
Trans a
Rot x
θ
α
−
−
−
−
−•
•
•
•
→→→
→
x x
x x colinear
x x .
z z
축이 축과 평행하게 됨.
축과 축이 하게 됨.
축의 원점이 축의 원점과 일치하게 됨
축이 축과 일치하게 됨.
2-352008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
두 이동좌표계 사이의 Transformation
11( (0,0,, ) ) ( ,0,0) ( , )
0 0 1 0 0 0 1 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0 0 0
0 0 1 0 0 0 1 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
n n
n
n
nn n n n n n
n n
n n n n
n n
Rot z Trans d
a
d
T A Trans a Rot x
c s
s c c s
s c
θ α
θ θθ θ α α
α α
−−= = × × ×
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
0 0 1 0 0
0 0 0 0
0 0 1 0 0
0 0 0 1 0 0 0 1
0
0 0 0 1
n
n
n
n
n
n n
n n n n
n n
n n n n n n
n n n n n n
n n
a
d
a
a
d
c s
s c c s
s c
c s c s s c
s c c c s s
s c
θ θθ θ α α
α α
θ θ α θ α θθ θ α θ α θ
α α
−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
{n}좌표계의 {n-1}좌표계에 대한 자세
{n-1}좌표계 원점에 대한{n}좌표계 원점의 거리
2-362008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
㈜ D-H 변환에서 Y축으로의 이동이 필요한 경우 (PUMA560)
1nz −
nz2nz − nθ
1nθ +
1nθ −
joint n1joint n −
1link n−link n
( )1attached to link n−
( )attached to link n
( )2attached to link n−
1nx −
nxna
nd
nθ
nα
o
onb
1
1
1
1
1(
(0, 0,
(
,
(
(
, )
)
,0,0
0
)
,0)
)
,
n n n
n n n
n n n
n n
n
n
n
nRot z
Trans d
Trans a
Rot
Tran b
x
s
θ
α
−
−
−
−
−•
•
•
•
•
→→→→→
x x
x x parallel
x x colinear )
x x
z
축이 축과 평행하게 됨.
축과 축이 하게 됨.
축과 축이 하게 됨 (동일직선상 .
축의 원점이 축의 원점과 일치하게 됨
1n n− z 축이 축과 일치하게 됨.
2-372008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
11( (0, 0,, ) ) ( ,0,0) ( , )
0 0 1 0 0 0 1 0 0
0 0 0 1 0 0 0 1 0 0
0 0 1 0 0 0 1 0 0 1 0
0 0 0 1 0
(0, ,0)
1 0 0 0
0 1 0
0 0 1 0
00 0 01 0 00 0 1 1
n n
n
n
n
nn n n n n n
n n
n n
nRot z Trans d
d
b
a
T A Trans a Rot x
c s
Tra
s c
ns bθ α
θ θθ θ
−−= = × × ×
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
×
⎡ ⎤⎢ ⎥⎢⎢⎢⎣⎣ ⎦ ⎦
1 0 0 0
0 0
0 0
0 0 0 1
0
0 0 0 1
n n
n
nn
n
n n
n n
n n n n n n
n n n n n
n
nn
n
a
a
d
c s
s c
c s c s s c
s c c
d s
dc
s
cs s
c
α αα α
θ θ α θ α θθ θ α θ α θ
α α
θθ
⎥
⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦
⎥⎥
−+
㈜ D-H 변환에서 Y축으로의 변환이 필요한 경우 (PUMA560)
1 1 1
11( (0,0,
{ , , } { , , }
, ) ) ( ,0,0) (0, ,0) ( ,n n
n n n n n n
nn n n nn
n n n nnd ba
Rot z Trans d
x y z x y z
T Trans a Trans R tb o x
θ α
θ
− − −
−−
→→ → →
•
⎡ ⎤ ⎡ ⎤⎯⎯⎯⎯⎯⎯⎯⎯⎯→⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
= × × × ×
transformation:
(n-1) link n-link
두 좌표계 사이의
좌표계 좌표계
1 1
111
)
n n
n
n n n n
n n nn nnn n
n
b
d
a
θ
α
α
− −
− −−
: x x : z z
: x x : z : y y z
축과 축 사이 각도, 축과 축 사이의 거리,
축과 축 사이의 거리, 축과축과 축 사이의 축 사거리, 이 각도
2-382008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
PUMA560
b[i]
3
2
1
6
5
4
alpha[i]a[i]d[i]theta[i]Joint #
2-392008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
(Robot Base Hand)의 Transformation
Robot base(Reference frame)에 대한 Robot hand의 위치 및 자세
1 2 11 1 2 2 3 3 1 2 3( ) ( ) ( ) ( )
: revolute joint variable
: prismatic joint
i i
i i
R R nH n n n
q
q d
T T q T q T q T q A A A A
θ
−
=•
=
= =
0x0y
0z
1x1y
1z2x 2y
2z
1θ ( )nx n
( )ny o
( )nz a
2θ3θ
nθ
0nT
3x 3y
3z
x y
z
2-402008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Example 2.19 (PUMA type)
D-H parameters
3
2
1
6
5
4
alpha[i]a[i]d[i]theta[i]Joint #
0x 0y
0z
6x6y
6z
2-412008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Example 2.20 (Stanford Arm)
D-H parameters
3
2
1
6
5
4
alpha[i]a[i]d[i]theta[i]Joint #
0x 0y
0z
6x
6y6z
2-422008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.9 Inverse Kinematic Solutions
Position level (chap. 2)에서의 Inverse kinematics
• 특별한 규칙이 있는 것은 아니고 주로 기하적 방법(geometric method)이나 transformation matrix의 element를 비교해서
solution을 찾음.
• 6DOF robot의 역기구학 해가 존재하기 위한 충분조건:
The last three revolute joint axes (wrist) intersects at a common point.
0 0 0 1
x x x x
y y y y
z z z z
i
i
n o a p
n o a p
n o a p
dθ
•
⎡ ⎤⎢ ⎥
⎛ ⎞ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦
⎛⇒
Given the desired location:
& orientation of the robot
(revolute joint) or
Inv
Solve f
erse kinem
or the joint variables: (prismatic joi
atic
nt)
s
⎞⎜ ⎟⎝ ⎠
2-432008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Ex. 2.19의 역기구학 해
1 1 2 2 3 3 4 4 5 5 6 6
1
6 6 6
1
0 0 0 1
( ) ( ) ( ) ( )
, , }
, , }
( ) ( ) [ ]
(1)
Desired orientation/location : {
{ frame
of the hand
x x x x
y y y y
z z z z
x x x x
y y
RH
n o a p
n o a p
n o a p
n o a p
n o
x y
T A A A A A A RH
z
n a
S
A
o
θ θ θ θ θ θ
−
• =
⎡ ⎤⎛ ⎞⎢ ⎥ ⎜ ⎟⎢ ⎥= ⎜ ⎟⎢ ⎥ ⎜ ⎟
⎢ ⎥ ⎝ ⎠⎣ ⎦
=
11 2 2 3 3 4 4 1 3
5 2
5 5 6 6
1 1 1 1 1 1 1 14 3 2 1 4 3 2 1 5 5 6 6
1
34 2
5
4
0 0 0 1
0 0 0 1
[ ] ( ) ( ) ( ) ( ,
,
) ( )
(2) [ ] ( ) ( )[ ]
(3)
, ,
y y
z z z z
x x x x
y y y y
z z z z
a p
n o a p
n o a p
n o a p
n o a p
A RHS A A A A A
A A A A A A A A RHS A A RHS
A A
θ θ θ θ θ
θ
θ
θ
θ
θ θ θ θ
−
− − − − − − − −
−
⎡ ⎤⎢ ⎥⎢ ⎥ = =⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥ = =⎢ ⎥⎢ ⎥⎣ ⎦
→
→
1 1 1 1 1 1 1 1 14 3 2 1 4 3 1 65 2 6 6
0 0 0 1
[ ] ( )
x x x x
y y y y
z z z z
n o a p
n o a p
n o a pA A A A A A A A RHS A θ θ− − − − − − − − −
⎡ ⎤⎢ ⎥⎢ ⎥ = =⎢ ⎥⎢ ⎥⎣ ⎦
→
2-442008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Ex. 2.20 (Stanford Arm)의 역기구학 해
5R-1P manipulator
1 1 2 2 3 3 4 4 5 5
6 6
6 6
6
0 0 0 1
( ) ( ) ( ) ( ) ( ) ( ) [ ]
(1
, , }
, , }
)
1st three joints
Desired orientation/location : of the {
{
ha
frame
nd
x x x x
y y y y
z z z z
RH
n o a p
n o a p
n o a px y
T A A A
z
A A A RHS
n o a
θ θ θ θ θ θ•
=
=
⎡ ⎤⎛ ⎞⎢ ⎥ ⎜ ⎟⎢ ⎥= ⎜ ⎟⎢ ⎥ ⎜ ⎟
⎢ ⎥ ⎝ ⎠⎣ ⎦
3
3
2
21 11 6 2 2 3 3 4 4 5 5
2
1 1 13
2 3
1
1
2
0 0 0 1 0 0 0 1
0 0 0 1
( ) ( ) ( ) , ,( )
(2)
[
last three joints
known matr
x x x x
y y y y
z z z z
x x x x
y y y y
z z z z
n o a p d
n o a p d
n o a p d
n o a p
n o a p
n o a p
s
cA A A A A A
A A A
θ θ θθ θ θθ− −
− − −
−
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥ =⎢ ⎥
⎥
→
⎢⎣ ⎦
4 4 5 5 6 6
14 5 5 6 6 4 5 6
( ) ( ) ( )
[ ( ) ( , ,)
ix]
known matrix]
A A A
A A A
θ θ
θ θ θ
θ
θ θ− →
=
=
2-452008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.11 Degeneracy and Dexterity
Degeneracy • Robot이 운동 중에 singular configuration에 들어가
degree of freedom 을 잃는 경우가 발생
1) Robot의 각 joint가 물리적 한계에 도달하는 경우 (즉, robot의운동 영역인 workspace의 경계에 있는 경우)
2) 두개 joint의 z-axis가 colinear한 경우: 어느 축을 움직여야 하
는가 결정할 수 없음.
Dexterity • Robot hand를 workspace내의 임의 point에 원하는
orientation으로 위치시킬 수 있을 때 robot이dexterous하다라고 말함.
• Robot workspace의 경계(boundary) 부분에는 hand를
원하는 자세로 보낼 수 없음. (dexterity를 잃는 경우)
2-462008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
2.12 Fundamental Problem with D-H Representation
Defect of D-H presentation : • D-H cannot represent any motion about the y-axis, because all motions
are about the x- and z-axis.
2-472008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
Methods for representation of points, vectors, frames, and 4 by 4
homogeneous transformations by matrices.
Forward and inverse kinematics of robots
RPY angles and Euler angles
Denavit-Hartenberg method to derive the forward and inverse
kinematic equations
Summary of Chapter 2
2-482008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
[Homework #1] (200점)
예제 Example 2.1 ~ 2.20 (5점 x 20 = 100)
Solve the following problems in chapter 2: (10점 x 7=70)
• 2.3, 2.7, 2.9, 2.12, 2.20, 2.21, 2.22
Solve the forward/inverse kinematics of the stanford arm. (30점)
• Derive the Eq. (2.58): forward kinematics
• Derive the Eq. (2.78)~(2.83): inverse kinematics
(Ref.) Asada and Slotine “Robot analysis and control”
Due date:
2-492008 Fall Semester, Robotics (School of AME, KAU) by sjkwon
로봇공학, Chapter 2
H.W#1 2.22 문제
1) Ref. frame을 1번 joint에 둘 경우
9000theta33
-900l202
9000theta11
00l30H
alphaadthetaJoint #
2) Ref. frame을 Text와 같이 둘에 둘 경우
9000theta33
00l30H
l100U
-900l202
9000theta11
alphaadthetaJoint #
0 0 1 2 31 2 3H HT A A A A=
0 0 1 20 0 1 2 3
U U UH HT A T A A A A= =