1. Direct Kinematic Model
Chapter-3
Direct Kinematic Model
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2. Rotation Matrices in 3D ,lets return from homogenous representation
Rotation around the Z-Axis
Rotation around the Y-Axis
Rotation around the X-Axis
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3. Homogeneous Matrices in 3D
H is a 4x4 matrix that can describe a translation, rotation, or both in one matrix O P Y X Z N A
Translation without rotation Y O N X
Rotation part:
Could be rotation around z-axis, x-axis, y-axis or a combination of the three. Z
Rotation without translation A
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4. Y X Z J I K N O A T P
Substituting for
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5. Notice that H can also be written as:
Product of the two matrices
Notice that H can also be written as:
H = (Translation relative to the XYZ frame) * (Rotation relative to the XYZ frame)
* (Translation relative to the IJK frame) * (Rotation relative to the IJK frame)
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6. One more variation on finding H:
The Homogeneous Matrix is a concatenation of numerous translations and rotations J I K Y N T P X A O Z
One more variation on finding H:
H = (Rotate so that the X-axis is aligned with T)
* ( Translate along the new t-axis by || T || (magnitude of T))
* ( Rotate so that the t-axis is aligned with P)
* ( Translate along the p-axis by || P || )
* ( Rotate so that the p-axis is aligned with the O-axis)
This method might seem a bit confusing
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7. You have a robotic arm that starts out aligned with the xo-axis.
The Situation:
You have a robotic arm that starts out aligned with the xo-axis.
You tell the first link to move by θ1 and the second link to move by θ2.
The Question:
What is the position of the end of the robotic arm?
Solution:
1. Geometric Approach
This might be the easiest solution for the simple situation. However, notice that the angles are measured relative to the direction of the previous link. (The first link is the exception. The angle is measured relative to it’s initial position.) For robots with more links and whose arm extends into 3 dimensions the geometry gets much more complicated.
2. Algebraic Approach
Involves coordinate transformations.
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8. Link Description
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9. Other basic joints Revolute Joint 2 DOF ( Variable - θ1, θ2)
Prismatic Joint
1 DOF (linear) (Variables - d)
Spherical Joint
3 DOF ( Variables - θ1, θ2, θ3)
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10. History of Matrix Representation
Method introduced in 1955 by Denavit and Hartenburg
Matrix representation is known as the Denavit-Hartenburg (D-H) representation of linkages
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11. Denavit-Hartenburg (D-H) representation
Describe the rotation and translation relationship between adjacent link.
D-H representation results in a 4 X 4 homogenous transformation matrix representing each link’s coordinate system at the joint with respect previous link’s coordinate system
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12. Denavit-Hartenberg Notation
Z(i - 1)
Y(i -1)
Y i
Z i
X i
a i
a(i - 1 )
d i
X(i -1) θi
( i - 1)
Each joint is assigned a coordinate frame. Using the Denavit-Hartenberg notation, you need 4 parameters to describe how a frame (i) relates to a previous frame ( i -1 ).
THE PARAMETERS/VARIABLES: , a , d, θ
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13. The Parameters 1) θi ( Joint Angle)
Amount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi axis.
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i θi
You can align the two axis just using the 4 parameters
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14. Z(i - 1)
Y(i -1)
Y i
Z i
X i
a i
a(i - 1 )
d i
X(i -1) θi
( i - 1)
2) a(i-1) (Link Length)
Technical Definition: a(i-1) is the length of the perpendicular between the joint axes. The joint axes is the axes around which revolution takes place which are the Z(i-1) and Z(i) axes. These two axes can be viewed as lines in space. The common perpendicular is the shortest line between the two axis-lines and is perpendicular to both axis-lines.
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15. a(i-1) cont...
Visual Approach - “A way to visualize the link parameter a(i-1) is to imagine an expanding cylinder whose axis is the Z(i-1) axis - when the cylinder just touches the joint axis i the radius of the cylinder is equal to a(i-1).”
It’s Usually on the Diagram Approach - If the diagram already specifies the various coordinate frames, then the common perpendicular is usually the X(i-1) axis. So a(i-1) is just the displacement along the X(i-1) to move from the (i-1) frame to the i frame.
If the link is prismatic, then a(i-1)
is a variable, not a parameter.
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i θi
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16. 3) (i-1) (Link Twist Angle)
Technical Definition: Amount of rotation around the common perpendicular so that the joint axes are parallel.
i.e. How much you have to rotate around the X(i-1) axis so that the Z(i-1) is pointing in the same direction as the Zi axis. Positive rotation follows the right hand rule.
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i θi
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17. Denavit Hartenburg VDP Reference [3] Text [3]:
Use bond length, bond angles, and torsion as parameters for Denavit Hartenburg (DH_.
This representation is useful for determining the appropriate rigid-body transformation
to apply to any link in a series of attached links.
Figure 2 depicts the quantities that appear in (1). The fictitious bond is used to define 0.
If bi is not rotatable, then i is a constant; otherwise, i is a conformation parameter, included in .
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18. 4) d(i-1) (JOINT DISTANCE)
Technical Definition: The displacement along the Zi axis needed to align the a(i-1) common perpendicular to the ai common perpendicular.
In other words, displacement along the
Zi to align the X(i-1) and Xi axes.
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i
θ i
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19. DH parameters Four DH parameters: First, translate by along z axis
Then, translate by along x axis
and rotate by about z axis
and rotate by about x axis
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20. DH parameters These four DH parameters,
represent the following homogeneous matrix:
First, translate by along z axis
Then, translate by along x axis
and rotate by about z axis
(JOINT PARAMETER)
and rotate by about x axis
(LINK PARAMETER)
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21. DH parameters
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22. The Denavit-Hartenberg Matrix
Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame.
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23. DH1
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24. DH2
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25. Denavit-Hartenberg Convention
Number the joints from 1 to n starting with the base and ending with the end-effector.
Establish the base coordinate system. Establish a right-handed orthonormal coordinate system at the supporting base with axis lying along the axis of motion of joint 1.
Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1.
Establish the origin of the ith coordinate system. Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis.
Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel.
Establish Yi axis. Assign to complete the right-handed coordinate system.
Find the link and joint parameters

26. Direct Kinematics Algorithm
1) Draw sketch
2) Number links. Base=0, Last link = n
3) Identify and number robot joints
4) Draw axis Zi for joint I
5) Determine link length ai-1 between Zi-1 and Zi
6) Draw axis Xi-1
7) Determine link twist i-1 measured around Xi-1
8) Determine the joint offset di
9) Determine joint angle i around Zi
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27. Planar 2-R Manipulator
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28. Planar 2-R Manipulator
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29. VDP

30. Planar 2-R Manipulator
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31. Planar 2-R Manipulator
X=a1 c1 + a2 c12
Y=a1 s1 + a2 s12
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32. Three Link RLL Manipulator
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33. VDP

34. VDP

35. X = y = z =
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36. Example 1: DH parameters2 3
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37. VDP

38. DH parameters 1 2 3
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39. Example I 3 Revolute Joints a0 a1 d2 Joint 1 Joint 2 Joint 3 Link 1Z0 X0 Y0 Z3 X2 Y1 X1 Y2 d2 Z1 X3 Z2
Joint 1
Joint 2
Joint 3
Link 1
Link 2
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40. Link Coordinate Frames
Assign Link Coordinate Frames:
To describe the geometry of robot motion, we assign a Cartesian coordinate frame (Oi, Xi,Yi,Zi) to each link, as follows:
establish a right-handed orthonormal coordinate frame O0 at the supporting base with Z0 lying along joint 1 motion axis.
the Zi axis is directed along the axis of motion of joint (i + 1), that is, link (i + 1) rotates about or translates along Zi; a0 a1 Z0 X0 Y0 Z3 X2 Y1 X1 Y2 d2 Z1 X3 Z2
Joint 1
Joint 2
Joint 3
Link 1
Link 2

41. Link Coordinate Frames
Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis.
the Xi axis lies along the common normal from the Zi-1 axis to the Zi axis , (if Zi-1 is parallel to Zi, then Xi is specified arbitrarily, subject only to Xi being perpendicular to Zi); a0 a1 Z0 X0 Y0 Z3 X2 Y1 X1 Y2 d2 Z1 X3 Z2
Joint 1
Joint 2
Joint 3

42. Link Coordinate Frames
Assign to complete the right-handed coordinate system.
The hand coordinate frame is specified by the geometry of the end-effector. Normally, establish Zn along the direction of Zn-1 axis and pointing away from the robot; establish Xn such that it is normal to both Zn-1 and Zn axes. Assign Yn to complete the right-handed coordinate system. a0 a1 Z0 X0 Y0 Z3 X2 Y1 X1 Y2 d2 Z1 X3 Z2
Joint 1
Joint 2
Joint 3

43. Link and Joint Parameters
Joint angle : the angle of rotation from the Xi-1 axis to the Xi axis about the Zi-1 axis. It is the joint variable if joint i is rotary.
Joint distance : the distance from the origin of the (i-1) coordinate system to the intersection of the Zi-1 axis and the Xi axis along the Zi-1 axis. It is the joint variable if joint i is prismatic.
Link length : the distance from the intersection of the Zi-1 axis and the Xi axis to the origin of the ith coordinate system along the Xi axis.
Link twist angle : the angle of rotation from the Zi-1 axis to the Zi axis about the Xi axis.

44. Example I a0 a1 d2 D-H Link Parameter TableZ0 X0 Y0 Z3 X2 Y1 X1 Y2 d2 Z1 X3 Z2
Joint 1
Joint 2
Joint 3
D-H Link Parameter Table
: rotation angle from Zi-1 to Zi about Xi
: distance from intersection of Zi-1 & Xi
to origin of i coordinate along Xi
: distance from origin of (i-1) coordinate to intersection of Zi-1 & Xi along Zi-1
: rotation angle from Xi-1 to Xi about Zi-1
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45. Example

46. Three link cylindrical robotai αi di i 1 2 3
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47. VDP

48. VDP

49. Example II: PUMA 260 Number the joints Establish base frame
Establish joint axis Zi
Locate origin, (intersect. of Zi & Zi-1) OR (intersect of common normal & Zi )
Establish Xi,Yi t
PUMA 260

50. Link Parameters : angle from Xi-1 to Xi about Zi-16 90 5 8
-90 4 3 2 13 1 J -l
: angle from Xi-1 to Xi
about Zi-1
: angle from Zi-1 to Zi about Xi
: distance from intersection
of Zi-1 & Xi to Oi along Xi
Joint distance : distance from Oi-1 to intersection of Zi-1 & Xi along Zi-1
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51. Example: Puma 560
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52. VDP

53. VDP

54. VDP

55. VDP

56. VDP

57. VDP

58. VDP

59. VDP

60. VDP

61. VDP

62. VDP

63. VDP