Post on 22-Feb-2016
description
Kinematics of Robot
Manipulator
Examples of Kinematics Calculations
• Forward kinematics
Given joint variables
End-effector position and orientation,
-Formula?
),,,,,,( 654321 nqqqqqqqq
),,,,,( TAOzyxY x
y
z
Examples of Inverse Kinematics• Inverse kinematicsEnd effector position and orientation
Joint variables -Formula?
),,,,,,( 654321 nqqqqqqqq
),,,,,( TAOzyx
x
y
z
Example 1: one rotational link
0x
0y
1x1y
)/(cos
kinematics Inverse
sincos
kinematics Forward
01
0
0
lx
lylx
l
Robot Reference Frames– World frame– Joint frame– Tool frame
x
yz
x
z
y
W R
PT
Coordinate Transformation– Reference coordinate frame OXYZ– Body-attached frame O’uvw
wvu kji wvuuvw pppP
zyx kji zyxxyz pppP
x
y
zP
u
vw
O, O’
Point represented in OXYZ:
zwyvxu pppppp
Tzyxxyz pppP ],,[
Point represented in O’uvw:
Two frames coincide ==>
Properties: Dot Product
• Mutually perpendicular • Unit vectors
Properties of orthonormal coordinate frame
0
0
0
jk
ki
ji
1||
1||1||
k
ji
Let and be arbitrary vectors in and be the angle from to , then
3R
cosyxyx
x yx y
Coordinate Transformation
• 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?
Basic Rotation• 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
Basic Rotation Matrix• Basic Rotation Matrix
– Rotation about x-axis with
w
v
u
z
y
x
ppp
ppp
wzvzuz
wyvyuy
wxvxux
kkjkikkjjjijkijiii
x
z
y
v
wP
u
CSSCxRot
00
001),(
Is it True? Can we check?
• – Rotation about x axis with
cossin
sincos
cossin0sincos0001
wvz
wvy
ux
w
v
u
z
y
x
ppp
ppppp
ppp
ppp
x
z
y
v
wP
u
Basic Rotation Matrices– Rotation about x-axis with
– Rotation about y-axis with
– Rotation about z-axis withuvwxyz RPP
CSSCxRot
00
001),(
0
0100
),(
CS
SCyRot
10000
),(
CSSC
zRot
Matrix notation for rotations
• Basic Rotation Matrix
– Obtain the coordinate of from the coordinate of
z
y
x
w
v
u
ppp
ppp
zwywxw
zvyvxv
zuyuxu
kkjkikkjjjijkijiii
uvwxyz RPP
wzvzuz
wyvyuy
wxvxux
kkjkikkjjjijkijiii
R
xyzuvw QPP
TRRQ 1
31 IRRRRQR T
uvwP
xyzP
<== 3X3 identity matrix
Dot products are commutative!
Example 2: rotation of a point in a rotating frame
• 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
2964.4598.0
234
10005.0866.00866.05.0
)60,( uvwxyz azRota
Example 3: find a point of rotated system for point in coordinate system
• 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
2964.1
598.4
234
10005.0866.00866.05.0
)60,( xyzT
uvw azRota
• Reference coordinate frame OXYZ• Body-attached frame O’uvw
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
Example 4: finding a rotation matrix• 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
SSSCCSCCSSCSSCCCS
CSSSCCSCSSCC
CSSCCS
SC
uRotwRotIyRotR
00
001
10000
C0S-010
S0C
),(),(),( 3
Reference coordinate frame OXYZBody-attached frame O’uvw
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}
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
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
Homogeneous Transformation• Special cases
1. Translation
2. Rotation
100
31
13BA
BA RT
10 31
'33
oA
BA rIT
Example 5: translation• Translation along Z-axis with h:
1000100
00100001
),(h
hzTrans
111000100
00100001
1hp
pp
ppp
hzyx
w
v
u
w
v
u
x
y
z P
u
vw
O, O’hx
y
zP
u
vw
O, O’
Example 6: rotation• Rotation about the X-axis by
1100000000001
1w
v
u
ppp
CSSC
zyx
100000000001
),(
CSSC
xRot
x
z
y
v
wP
u
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
Example 7: homogeneous transformation
• Find the homogeneous transformation matrix (T) for the following operations:
44,,,, ITTTTT xaxdzz
:
axis OZabout ofRotation axis OZ along d ofn Translatioaxis OX along a ofn Translatio
axis OXabout Rotation
Answer
100000000001
100001000010
001
1000100
00100001
100001000000
CSSC
a
dCSSC
Homogeneous Representation• A frame in space (Geometric
Interpretation)
x
y
z),,( zyx pppP
1000zzzz
yyyy
xxxx
pasnpasnpasn
F
n
sa
101333 PR
F
Principal axis n w.r.t. the reference coordinate system
(X’)
(y’)(z’)
Homogeneous Transformation
• Translation
y
z
n
sa n
sa
1000
10001000100010001
zzzzz
yyyyy
xxxxx
zzzz
yyyy
xxxx
z
y
x
new
dpasndpasndpasn
pasnpasnpasn
ddd
F
oldzyxnew FdddTransF ),,(
Homogeneous Transformation
21
10
20 AAA
ii A1
Composite Homogeneous Transformation Matrix
0x
0z
0y1
0 A2
1A
1x
1z
1y 2x
2z2y
?Transformation matrix for adjacent coordinate frames
Chain product of successive coordinate transformation matrices
Example 8: homogeneous transformation based on geometry
• For the figure shown below, find the 4x4 homogeneous transformation matrices and for i=1, 2, 3, 4, 5
1000zzzz
yyyy
xxxx
pasnpasnpasn
F
1000010100
0001
10
dace
A
ii A1
iA0
0x 0y
0z
a
b
c
d
e
1x
1y
1z
2z2x
2y
3y3x
3z
4z
4y4x
5x5y
5z
10000100
001010
20 ce
b
A
10000001
100010
21 da
b
A
Can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?
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
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
Euler Angle I, Animated
x
y
z
u'
v'
v "
w"
w'=
=u"
v'"
u'"
w'"=
Orientation Representation• Euler Angle I
1000cossin0sincos
,cossin0sincos0001
,1000cossin0sincos
''
'
w
uz
R
RR
Euler Angle I
cossincossinsin
sincoscoscoscos
sinsincossincos
cossin
sinsincoscossin
sincoscossinsin
coscos''' wuz RRRR
Resultant eulerian rotation matrix:
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, .
Orientation Representation
• Matrix with Euler Angle II
cossinsinsincos
sinsincoscossin
coscoscoscossin
sincos
sincoscoscossin
cossincoscoscos
sinsin
Quiz: How to get this matrix ?
Orientation Representation• Description of Roll Pitch Yaw
X
Y
Z
Quiz: How to get rotation matrix ?
The City College of New York
38
The City College of New York
39
Kinematics of Robot Manipulator
Jizhong XiaoDepartment of Electrical Engineering
City College of New Yorkjxiao@ccny.cuny.edu
Introduction to ROBOTICS