the city college of new york 1 dr. jizhong xiao department of electrical engineering city college of...
Post on 19-Dec-2015
217 views
TRANSCRIPT
The City College of New York
1
Dr. Jizhong Xiao
Department of Electrical Engineering
City College of New York
Kinematics of Robot Manipulator
Introduction to ROBOTICS
The City College of New York
2
Outline• Review
• Robot Manipulators– Robot Configuration– Robot Specification
• Number of Axes, DOF• Precision, Repeatability
• Kinematics– Preliminary
• World frame, joint frame, end-effector frame• Rotation Matrix, composite rotation matrix• Homogeneous Matrix
– Direct kinematics• Denavit-Hartenberg Representation• Examples
– Inverse kinematics
The City College of New York
3
Review• What is a robot?
– By general agreement a robot is:• A programmable machine that imitates the actions or
appearance of an intelligent creature–usually a human.
– To qualify as a robot, a machine must be able to: 1) Sensing and perception: get information from its surroundings 2) Carry out different tasks: Locomotion or manipulation, do
something physical–such as move or manipulate objects3) Re-programmable: can do different things4) Function autonomously and/or interact with human beings
• Why use robots?
4A: Automation, Augmentation, Assistance, Autonomous
4D: Dangerous, Dirty, Dull, Difficult
–Perform 4A tasks in 4D environments
The City College of New York
4
Manipulators
• Robot arms, industrial robot– Rigid bodies (links) connected
by joints– Joints: revolute or prismatic– Drive: electric or hydraulic – End-effector (tool) mounted
on a flange or plate secured to the wrist joint of robot
The City College of New York
5
Manipulators• Robot Configuration:
Cartesian: PPP Cylindrical: RPP Spherical: RRP
SCARA: RRP
(Selective Compliance Assembly Robot Arm)
Articulated: RRR
Hand coordinate:
n: normal vector; s: sliding vector;
a: approach vector, normal to the
tool mounting plate
The City College of New York
6
Manipulators
• Motion Control Methods– Point to point control
• a sequence of discrete points• spot welding, pick-and-place, loading & unloading
– Continuous path control• follow a prescribed path, controlled-path motion• Spray painting, Arc welding, Gluing
The City College of New York
7
Manipulators
• Robot Specifications– Number of Axes
• Major axes, (1-3) => Position the wrist• Minor axes, (4-6) => Orient the tool• Redundant, (7-n) => reaching around
obstacles, avoiding undesirable configuration
– Degree of Freedom (DOF)– Workspace– Payload (load capacity)– Precision v.s. Repeatability
Which one is more important?
how accurately a specified point can be reached
how accurately the same position can be reached if the motion is repeated many times
The City College of New York
8
What is Kinematics
• Forward kinematics
Given joint variables
End-effector position and orientation, -Formula?
),,,,,,( 654321 nqqqqqqqq
),,,,,( TAOzyxY x
y
z
The City College of New York
9
What is Kinematics• Inverse kinematics
End effector position
and orientation
Joint variables -Formula?
),,,,,,( 654321 nqqqqqqqq
),,,,,( TAOzyx
x
y
z
The City College of New York
10
Example 1
0x
0y
1x1y
)/(cos
kinematics Inverse
sin
cos
kinematics Forward
11
1
1
lx
ly
lx
l
The City College of New York
11
Preliminary
• Robot Reference Frames– World frame– Joint frame– Tool frame
x
yz
x
z
y
W R
PT
The City College of New York
12
Preliminary• Coordinate Transformation
– Reference coordinate frame OXYZ
– Body-attached frame O’uvw
wvu kji wvuuvw pppP
zyx kji zyxxyz pppP
x
y
z
P
u
vw
O, O’
Point represented in OXYZ:
zwyvxu pppppp
Tzyxxyz pppP ],,[
Point represented in O’uvw:
Two frames coincide ==>
The City College of New York
13
Preliminary
• Mutually perpendicular • Unit vectors
Properties of orthonormal coordinate frame
0
0
0
jk
ki
ji
1||
1||
1||
k
j
i
Properties: Dot Product
Let and be arbitrary vectors in and be the angle from to , then
3R
cosyxyx
x yx y
The City College of New York
14
Preliminary
• Coordinate Transformation– Rotation only
wvu kji wvuuvw pppP
x
y
zP
zyx kji zyxxyz pppP
uvwxyz RPP u
vw
How to relate the coordinate in these two frames?
The City College of New York
15
Preliminary• Basic Rotation
– , , and represent the projections of onto OX, OY, OZ axes, respectively
– Since
xp Pyp zp
wvux pppPp wxvxuxx kijiiii
wvuy pppPp wyvyuyy kjjjijj
wvuz pppPp wzvzuzz kkjkikk
wvu kji wvu pppP
The City College of New York
16
Preliminary• Basic Rotation Matrix
– Rotation about x-axis with
w
v
u
z
y
x
p
p
p
p
p
p
wzvzuz
wyvyuy
wxvxux
kkjkik
kjjjij
kijiii
x
z
y
v
wP
u
CS
SCxRot
0
0
001
),(
The City College of New York
17
Preliminary• Is it True?
– Rotation about x axis with
cossin
sincos
cossin0
sincos0
001
wvz
wvy
ux
w
v
u
z
y
x
ppp
ppp
pp
p
p
p
p
p
p
x
z
y
v
wP
u
The City College of New York
18
Basic Rotation Matrices– Rotation about x-axis with
– Rotation about y-axis with
– Rotation about z-axis with
uvwxyz RPP
CS
SCxRot
0
0
001
),(
0
010
0
),(
CS
SC
yRot
100
0
0
),(
CS
SC
zRot
The City College of New York
19
Preliminary• Basic Rotation Matrix
– Obtain the coordinate of from the coordinate of
uvwxyz RPP
wzvzuz
wyvyuy
wxvxux
kkjkik
kjjjij
kijiii
R
xyzuvw QPP
TRRQ 1
31 IRRRRQR T
uvwP
xyzP
<== 3X3 identity matrix
z
y
x
w
v
u
p
p
p
p
p
p
zwywxw
zvyvxv
zuyuxu
kkjkik
kjjjij
kijiii
Dot products are commutative!
The City College of New York
20
Example 2
• A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.
)2,3,4(uvwa
2
964.4
598.0
2
3
4
100
05.0866.0
0866.05.0
)60,( uvwxyz azRota
The City College of New York
21
Example 3• A point is the coordinate w.r.t. the
reference coordinate system, find the corresponding point w.r.t. the rotated OU-V-W coordinate system if it has been rotated 60 degree about OZ axis.
)2,3,4(xyza
uvwa
2
964.1
598.4
2
3
4
100
05.0866.0
0866.05.0
)60,( xyzT
uvw azRota
The City College of New York
22
Composite Rotation Matrix
• A sequence of finite rotations – matrix multiplications do not commute– rules:
• if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix
• if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix
The City College of New York
23
Example 4• Find the rotation matrix for the following
operations:
Post-multiply if rotate about the OUVW axes
Pre-multiply if rotate about the OXYZ axes
...
axis OUabout Rotation
axisOW about Rotation
axis OYabout Rotation
Answer
SSSCCSCCSSCS
SCCCS
CSSSCCSCSSCC
CS
SCCS
SC
uRotwRotIyRotR
0
0
001
100
0
0
C0S-
010
S0C
),(),(),( 3
The City College of New York
24
Coordinate Transformations• position vector of P in {B} is transformed to position vector of P in {A}
• description of {B} as seen from an observer in {A}
Rotation of {B} with respect to {A}
Translation of the origin of {B} with respect to origin of {A}
The City College of New York
25
Coordinate Transformations• Two Special Cases
1. Translation only– Axes of {B} and {A} are
parallel
2. Rotation only– Origins of {B} and {A}
are coincident
1BAR
'oAPBB
APA rrRr
0' oAr
The City College of New York
26
Homogeneous Representation• Coordinate transformation from {B} to {A}
• Homogeneous transformation matrix
'oAPBB
APA rrRr
1101 31
' PBoAB
APA rrRr
10101333
31
' PRrRT
oAB
A
BA
Position vector
Rotation matrix
Scaling
The City College of New York
27
Homogeneous Transformation• Special cases
1. Translation
2. Rotation
10
0
31
13BA
BA RT
10 31
'33
oA
BA rIT
The City College of New York
28
Example 5• Translation along Z-axis with h:
1000
100
0010
0001
),(h
hzTrans
111000
100
0010
0001
1
hp
p
p
p
p
p
hz
y
x
w
v
u
w
v
u
x
y
z P
u
vw
O, O’hx
y
z
P
u
vw
O, O’
The City College of New York
29
Example 6• Rotation about the X-axis by
1000
00
00
0001
),(
CS
SCxRot
x
z
y
v
wP
u
11000
00
00
0001
1w
v
u
p
p
p
CS
SC
z
y
x
The City College of New York
30
Homogeneous Transformation
• Composite Homogeneous Transformation Matrix
• Rules:– Transformation (rotation/translation) w.r.t
(X,Y,Z) (OLD FRAME), using pre-multiplication
– Transformation (rotation/translation) w.r.t (U,V,W) (NEW FRAME), using post-multiplication
The City College of New York
31
Example 7• Find the homogeneous transformation matrix
(T) for the following operations:
:
axis OZabout ofRotation
axis OZ along d ofn Translatio
axis OX along a ofn Translatio
axis OXabout Rotation
Answer
44,,,, ITTTTT xaxdzz
1000
00
00
0001
1000
0100
0010
001
1000
100
0010
0001
1000
0100
00
00
CS
SC
a
d
CS
SC
The City College of New York
32
Homogeneous Representation• A frame in space (Geometric
Interpretation)
x
y
z),,( zyx pppP
1000zzzz
yyyy
xxxx
pasn
pasn
pasn
F
n
sa
101333 PR
F
Principal axis n w.r.t. the reference coordinate system
(X’)
(y’)(z’)
The City College of New York
33
Homogeneous Transformation
• Translation
y
z
n
sa n
sa
1000
10001000
100
010
001
zzzzz
yyyyy
xxxxx
zzzz
yyyy
xxxx
z
y
x
new
dpasn
dpasn
dpasn
pasn
pasn
pasn
d
d
d
F
oldzyxnew FdddTransF ),,(
The City College of New York
34
Homogeneous Transformation
21
10
20 AAA
Composite Homogeneous Transformation Matrix
0x
0z
0y
10 A
21A
1x
1z
1y 2x
2z2y
?i
i A1 Transformation matrix for adjacent coordinate frames
Chain product of successive coordinate transformation matrices
The City College of New York
35
Example 8• For the figure shown below, find the 4x4 homogeneous transformation
matrices and for i=1, 2, 3, 4, 5
1000zzzz
yyyy
xxxx
pasn
pasn
pasn
F
ii A1
iA0
0x 0y
0z
a
b
c
d
e
1x
1y
1z
2z2x
2y
3y3x
3z
4z
4y4x
5x5y
5z
1000
010
100
0001
10
da
ceA
1000
0100
001
010
20 ce
b
A
1000
0001
100
010
21 da
b
A
Can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?
The City College of New York
36
Orientation Representation
• Rotation matrix representation needs 9 elements to completely describe the orientation of a rotating rigid body.
• Any easy way?
101333 PR
F
Euler Angles Representation
The City College of New York
37
Orientation Representation• Euler Angles Representation ( , , )
– Many different types– Description of Euler angle representations
Euler Angle I Euler Angle II Roll-Pitch-Yaw
Sequence about OZ axis about OZ axis about OX axis
of about OU axis about OV axis about OY axis
Rotations about OW axis about OW axis about OZ axis
The City College of New York
39
Orientation Representation• Euler Angle I
100
0cossin
0sincos
,
cossin0
sincos0
001
,
100
0cossin
0sincos
''
'
w
uz
R
RR
The City College of New York
40
Euler Angle I
cossincossinsin
sincos
coscoscos
sinsin
cossincos
cossin
sinsincoscossin
sincos
cossinsin
coscos
''' wuz RRRR
Resultant eulerian rotation matrix:
The City College of New York
41
Euler Angle II, Animated
x
y
z
u'
v'
=v"
w"
w'=
u"
v"'
u"'
w"'=
Note the opposite (clockwise) sense of the third rotation, .
The City College of New York
42
Orientation Representation
• Matrix with Euler Angle II
cossinsinsincos
sinsin
coscossin
coscos
coscossin
sincos
sincoscoscossin
cossin
coscoscos
sinsin
Quiz: How to get this matrix ?
The City College of New York
43
Orientation Representation• Description of Roll Pitch Yaw
X
Y
Z
Quiz: How to get rotation matrix ?