Kinematics

1
Kinematics
????????

• ???? ???? ???? ??????
• ??? 1?2?3 ???? Introduction to robotics
  mechanics and control

2
Kinematics
????????

• ?????? ?????? ???????? ???? ???? ????????? ???
  ??????? ???.
• ???????? ????? ?? ??????? ?? ???? ???? ????
  ?????? ? ??????? ??? ?????? ????.
• ????????
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• ???? ?? ?? ???? ?????? ????
• ???? ?? ???? ?? ? ?? ??????? ?? ???? ???? ??????

3
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• ?? ??? ???????? ?????? ???????? ????? ????
  ????? ????? ?? ?????? ? ??? ???? ?? ? ????? ?????
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  ???? ?????? ?????.
• ???? ?????? ??????? ??????? ?? ?? ??? ?? ???????
  ????? ??? ??? ? ??? ?????? ??? ??? ?????? ????
  ?????.

4
Manipulators

• ????? ??????? ? ?? ????? ????? ?? ?????? ?????
  ??? (link) ????? ??????? ???? ????? (joints) ??
  ?? ???? ??????.
• ???? ?? ???? ??? ????? ? ?? ??? ? ??
• ???? ?????.
• ???? ???? ???? ????? ????? ?? ????? ? ?? ?????
  ???????? ??????? ?????.
• ?? ?????? ????? ?? ????? ???? ?????? ?? ?? ???
  ???? ?? ?? ?? ????? ??? ?????.

5
??? ??? ?????? ???????? ?????
Articulated
Cylindrical
SCARA

• Cartesian

Spherical
6
Manipulators

• Robot Configuration

Cartesian PPP
Cylindrical RPP
Spherical RRP
Hand coordinate n normal vector s sliding
vector a approach vector, normal to the tool
mounting plate
SCARA RRP (Selective Compliance Assembly Robot
Arm)
Articulated RRR
7
Manipulators

• ????? ????? ?? ?????? ??? ?????? ?????
• ????? ??????
• ??????? ???? 1-3 ?? ???? ????? ?????? ???? ????
  ??????
• ??????? ???? 4-6 ?? ???? ??? ???? ?? ????? ????
  ??????.
• ??????? ????? 7-n ?? ???? ????? ?? ?????? ???
  ??????? ???? ??????.
• ???? ????? Degree of Freedom (DOF)
• ???? ???? Workspace
• ????? ??? Payload (load capacity)
• ??? ? ?????????? Precision v.s. Repeatability

8
??? ???????? ???? ???????? ?????

• ???????? ?? ?? ??? ????? ?????? ????? ????? ????
  ???? ?? ?????? ????? ?? ?? ???? ?? ?? ??? ????
  ?????? ? ?? ???? ???? ?????.

• Given joint variables
• End-effector position and orientation, -Formula?

9
??? ???????? ????? ???? ???????? ?????

• ???????? ?? ????? ???? ??????? ????? (P) ? ??????
  ????? ???? ???? ?? ????? ???? ?? ??? ???? ?????
  ???(R) ?????? ????? ????? ?? ????? ?? ???? ????
  ?? ????? ?? ?????? ???? ??? ???? ????.

Position (P) Orientation (R) of the
end-effector
10
???????? ?????
K-1
(q1 qn)
(x,y,z,qx,qy,qz)
11
????
12
??? ???????? ????? ???? ???????? ?????

• ??? ???????? ????? ??? ?? ?? ???????? ?????? ???
  ????
• ??????? ????? ??? ??? ???? ? ?? ????? ????? ???
  ?? ?????????? ???? ?? ???? ????? closed form ????
  ?????.
• ??? ?? ????? ???? ????.
• ??? ?? ????? ?? ?????? ????? ????

13
???????? ?????? ? ?????
Link Space n variables (q1 qn)
Tool Space 6 variables (x,y,z,qx,qy,qz)
Forward K
Inverse K
14
???????? ???????? ?????

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  ???
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• ????? ??? ????? ?????? ????? ????? ???
• ?? ?????? ??? ??? ????? ????? ? ????? ????? ??
  ?????? ???? ?????? ??? .????? ????? ?? ?? ????
  ???? ??? ??? ?????? ?????? ???? ???? ?? ??? ????
  ???? ?????? ????.

15
????? ?????? ?? ?????

• ???? ????? ?????? ?? ????? ???? ??? ?? ????? ???
  ?? ??????
• ??? ?????????/ ???????? ?????
• ??? ????? ??? ??? ? ????
• ?????? ?? ???? ???? ????
• ????? ????- ????? ??????
• ????? ?????? ?? ??????????? ???? ?? ???????
  ?????.

16
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• ???????? ????? ???? ?? ?????? ????? ?? ????? ??
  ??? ????? ?? ????? ?? ????? ??????? ???? ????
  ???? ?????? ?????? ????? ? ???? ????? ???.
• ????? ????? ?????? ?? ????? ?????? ??????? ?????
  ?? ?????? ???? ???? ?? ?? ???? ?????? ????
  ??????.
• ?? ?????? ???? ????? ?? ????? ???? ?????? ???
  ???? ????? ???????? ? ???? ???? ?? ???? ?????
  ?????? ???? ??????.
• ?? ????? ???? ??? ?? ?????? ????? ??? ????? ????
  ?? ?? ???? ?? ?????? ????? ?????? ???? ??????
  ???? ?? ???? ?????? ???? ???? ????.
• ???? ?????? ???? ?? ????? ?????? ????? ???

17
??????? ???? ??????
Joint frame
Tool frame
T
P
W
R
World frame
18
???????? ??????? ???? ??????

• ???? ????? ???? ??? ????? ????? ???? ?????? ?? ??
  ????? ???? ????? ????? ?????? ???? ???????? ????
  ?? ???? ??????? ???? A
• ?? ??????? ???? ???? ????
• ?????? ?? ???? ?????? ??? ????
• ???? ?? ??? ??? ???
• ???? ???? AP ?????? ???? P
• ?? ???? ?? ???? A ???? ?????.

19
The Use of Multiple Coordinate Frames in Robotics
A very simple robot arm with one joint and one
gripper. The world, camera, joint, and gripper
coordinate frames are indicated
20
???????? ??????? ???? ??????

• ??? ????? ?? ???? ????? ???? ? ???? ?????? ??????
  ???? ????? ????? ??? ???
• ?? ??? ??? ???? ?? ???? ?? ??
• ??? 4 ???? ??????? ?????
• ??? ???? ? ?? ?? ?? ???????
• ?? ????
• ? ?? ????? ?? ???? ?? ?????
• ??????? ?????? ???? ?????.

• Oxyz
• Ouvw

21
????? ???? ? ?????

• ?? ???? ?? ??? ?? ?? ????? ????? ?? ????? ?????
  ?????? ?? ??????? ?? ???? ???? ?????.
• Point represented in OXYZ

Vector represented in OXYZ
22
??? ?????
??? ???? ??? ????? ????? ?????? ?? ?? ?????
?????? ???? ???.
Let and be arbitrary vectors in
and be the angle from to ,
then Where q is the angle between the vectors
and is the norm. X.Y0 if X is
perpendicular to Y.
23
??? ?????
????? ??? ???? ?????? ??????

• Mutually perpendicular

• Unit vectors

O
24
????? ???

• ???? ????? ????? ????? ?? ???? P ?? ??? ??? ????
  ??? ??? ???? ?????
• ??? ????? ?? ?????? ???? ????
• ????? ????? ???? ??????? ?? ????
• ????? ?? ?? ???? ????? ???.
• ??? ?? ?? ???? ?? ?????? ??
• ?????? ?? ?? ??? ???? ??? ???
• ???? ???? ???? ?? ??? ?? ???
• ?? ??? ?? ??? ???? ?????.

25
????? ??????

• ???? ?? ???? ?????? B ????? ?? ?? ?? ?? ???
  ??????.
• ???? ????? ???? P ?? ???
• ??? ??? ?? ?? ????
• ?????? A ???? ????
• ???? ??? ????? ?????? ?
• ???? ???? ?????? B ??
• ???? ?? A??????

Rotation of B with respect to A
Translation of the origin of B with respect to
origin of A
26
????? ??????
????? ?? ?? ??? ?????? ?? ?? ???? ?? ?? ????
????? ?? ?? ???? ???? ??? ???? ??? ???? ????
OXYZ ? ????? ???? ??????? ?? ?? ??? ???? ??? ???
Ouvw
Point represented in Ouvw
Two frames coincide gt
27
????? ??????

• ??? ??? ??? ???? ?? ???? ???? ????? ?????

????? ?????? ?????? ???? ????? ???? ?????? ?? ??
?? ??? ????
28
???? ????? ????

• Px , Py , and Pz represent the projections of
  P onto OX, OY, OZ axes, respectively
• Since

29
?????? ????? ????

• Rotation about x-axis with

????? ???? ?? ???? x ?? ?????? ?? ?????? ??
?????? ????? ??? ???? ???
30
?????? ????? ????

• Rotation about x axis with

31
?????? ??? ????? ????
?? ???? ????? ?????? ?????? ????? ???? ?? ???????
???? ?? ???? ????

• Rotation about x-axis with q
• Rotation about y-axis with q
• Rotation about z-axis with q

32
?????? ????? ???? ???? ????? ?? ???? ?????? ????

• Obtain the coordinate of from the
  coordinate of

??? ?????? ????? ?? ???? uvw ?? ???? xyz ?? ?????
????? ?????? ?? ???? ?? ????? ??? ?????? ?????
????? ?? ???? ????.
Dot products are commutative!
lt 3X3 identity matrix
33
????

• 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.

34
????

• 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.

35
?????? ????? ??????

• ??? ?????? ????? ?????? ?? ?? ???? ???? ?????? ??
  ????? ?? ?? ?? ?????? ????? ???? ???? ??? ??
  ???????? ?????? ?? ????? ?? ?????? ??? ?????
  ????? ????? ??????? ?????? ?? ???? ?????? ??
  ?????? ??? ??????? ?????
• 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

36
????

• ?????? ????? ?????? ??? ?? ??????

Pre-multiply if rotate about the OXYZ axes
Post-multiply if rotate about the OUVW axes
37
??????

• ????? ?? ??? ???? ?? ???? ?????? ?????? ?????
  ?????. ???? ???? ??? ?? ???? ?????? B ? A
  ?? ?? ????? ????? ??????????

38
????? ??????

• ???? ?? ???? ?????? B ????? ?? ?? ?? ?? ???
  ??????.
• ???? ????? ???? P ?? ???
• ??? ??? ?? ?? ????
• ?????? A ???? ????
• ???? ??? ????? ?????? ?
• ???? ???? ?????? B ??
• ???? ?? A??????

Rotation of B with respect to A
Translation of the origin of B with respect to
origin of A
39
???? ??? ???

• Two special cases
• 1. Translation only
• Axes of B and A are parallel
• 2. Rotation only
• Origins of B and A are coincident

40
????? ????

• ???? ????? ??????? ? ????? ???????? ?? ?? ??????
  ???? ????? ?? ??? ?????? ? ?? ????? ???? ?? ?? ??
  ?? ?? ?????? ???? ???? ???.
• ?????? ?? ?????? ?? ???
• ?? ?????? ??? ?????? ?
• ????? ????? ?????.

41
????? ????

• Coordinate transformation from B to A
• Homogeneous transformation matrix

?? ??? ?? ?????? ?? ????? ?????
????? ?? ?????? ??? ?? ?? ?????? ???? ???
Rotation matrix
Position vector
Scaling
42
????? ????

• ????? ???
• 1. Translation
• 2. Rotation

43
????

• Translation along z-axis with h

44
????

• Rotation about the x-axis by

45
????? ????

• ??? ??? ?????? ?????? ????? ????? ?????? ??
  ??????? ?? ????? ??? ?????? ??? ???? ?????? ?????
  ?? ?? ?????? ???? ?? ?? ?? ??? ????? ? ?????? ???
  ?? ???? ????
• 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

46
????

• Find the homogeneous transformation matrix (T)
  for the following operation

47
????

• For the figure shown below, find the 4x4
  homogeneous transformation matrices and
  for i1, 2, 3, 4, 5

Can you find the answer by observation based on
the geometric interpretation of homogeneous
transformation matrix?
48
????? ???

• ?????? ???? ???? ????? ???? ???? ?? ??? ??? ????
  ?? 9 ??????? ????.??? ??? ???? ??? ???? ??????
  ???? ?????
• Any easy way?

Euler Angles Representation
49
Orientation Representation

• Euler Angles Representation ( , , )
• Many different types
• Description of Euler angle representations

50
Euler Angle I
z
w'
w"
w'"
f
v'"
v "
?
v'
y
u'"
?
u'
u"
x
51
Orientation Representation

• Euler Angle I

52
Euler Angle I
Resultant eulerian rotation matrix
53
Euler Angle II
z
w'
w"
w"'
?
v"'
?
v'
v"
?
y
u"'
u"
Note the opposite (clockwise) sense of the third
rotation, f.
u'
x
54
Orientation Representation

• Matrix with Euler Angle II

55
Orientation Representation

• Description of Roll Pitch Yaw