FORWAORD KINEMATICS

1. ROBOTICS
Forward Kinematics
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2. Forward Kinematics
• Modeling assumptions
• Review:
– Spatial Coordinates
• Pose = Position + Orientation
– Rotation Matrices
– Homogeneous Coordinates
• Frame Assignment
– Denavit Hartenberg Parameters
• Robot Kinematics
– End-effector Position,
– Velocity, &
– Acceleration
1st part
2nd part
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3. Industrial Robot
sequence of rigid
bodies (links)
connected by
means of
articulations
(joints)
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4. Robot Basics: Modeling
• Kinematics:
– Relationship between
the joint angles,
velocities &
accelerations and the
end-effector position,
velocity, & acceleration



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5. Modeling Robot Manipulators
• Open kinematic chain (in this course)
• One sequence of links connecting the two ends of the
chain (Closed kinematic chains form a loop)
• Prismatic or revolute joints, each with a
single degree of mobility
• Prismatic: translational motion between links
• Revolute: rotational motion between links
• Degrees of mobility (joints) vs. degrees of
freedom (task)
• Positioning and orienting requires 6 DOF
• Redundant: degrees of mobility > degrees of freedom
• Workspace
• Portion of environment where the end-effector can
access
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6. Modeling Robot Manipulators
• Open kinematic chain
– sequence of links with one end constrained to
the base, the other to the end-effector
Base
End-effector
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7. Modeling Robot Manipulators
• Motion is a composition of elementary
motions
Base
End-effector
Joint 1
Joint 2
Joint 3
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8. Kinematic Modeling of Manipulators
• Composition of elementary motion of each
link
• Use linear algebra + systematic approach
• Obtain an expression for the pose of the
end-effector as a function of joint variables
qi (angles/displacements) and link
geometry (link lengths and relative
orientations)
Pe = f(q1,q2,,qn ;l1,ln,1,n)
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9. Pose of a Rigid Body
• Pose = Position + Orientation
• Physical space, E3, has no natural
coordinates.
• In mathematical terms, a coordinate
map is a homeomorphism (1-1, onto
differentiable mapping with a
differentiable inverse) of a subset of
space to an open subset of R3.
– A point, P, is assigned a 3-vector:
AP = (x,y,z)
where A denotes the frame of reference
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10. A B
X
X
Y
Y
Z
Z
AP = (x,y,z)
BP = (x,y,z)
P
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11. Pose of a Rigid Body
• Pose = Position + Orientation
How do
we do
this?
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12. Pose of a Rigid Body
• Pose = Position + Orientation
• Orientation of the rigid body
– Attach a orthonormal FRAME to the body
– Express the unit vectors of this frame with
respect to the reference frame
XA
YA
ZA
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13. Pose of a Rigid Body
• Pose = Position + Orientation
• Orientation of the rigid body
– Attach a orthonormal FRAME to the body
– Express the unit vectors of this frame with
respect to the reference frame
XA
YA
ZA
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14. Rotation Matrices
• OXYZ & OUVW have coincident origins at O
– OUVW is fixed to the object
– OXYZ has unit vectors in the directions of the three
axes ix, jy,and kz
– OUVW has unit vectors in the directions of the three
axes iu, jv,and kw
• Point P can be expressed in either frame:
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15. O
X
U
V
Y
W
Z
AP = (x,y,z)
BP = (u,v,w)
P
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16. O
X
U
V
Y
W
Z
AP = (x,y,z)
P
BP = (u,v,w)
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17. O
X
U
V
Y
W
Z
AP = (x,y,z)
P
BP = (u,v,w)
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18. O
X
U
V
Y
W
Z
AP = (x,y,z)
BP = (u,v,w)
P
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19. Rotation Matrices
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20. Rotation Matrices
1
X axis
expressed
wrt Ouvw
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21. Rotation Matrices
1
Y axis
expressed
wrt Ouvw
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22. Rotation Matrices
1
Z axis
expressed
wrt Ouvw
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23. Rotation Matrices
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24. Rotation Matrices
Z axis
expressed
wrt Ouvw
X axis
expressed
wrt Ouvw
Y axis
expressed
wrt Ouvw
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25. Rotation Matrices
1
U axis
expressed
wrt Oxyz
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26. Rotation Matrices
U axis
expressed
wrt Oxyz
V axis
expressed
wrt Oxyz
W axis
expressed
wrt Oxyz
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27. Properties of Rotation Matrices
• Column vectors are the unit vectors of the
orthonormal frame
– They are mutually orthogonal
– They have unit length
• The inverse relationship is:
– Row vectors are also orthogonal unit vectors
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28. Properties of Rotation Matrices
• Rotation matrices are orthogonal
• The transpose is the inverse:
• For right-handed systems
– Determinant = -1(Left handed)
• Eigenvectors of the matrix form the axis
of rotation
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29. Elementary Rotations: X axis
X
Y
Z
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30. Elementary Rotations: X axis
X
Y
Z
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31. Elementary Rotations: Y axis
X
Y
Z
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32. Elementary Rotations: Z-axis
X
Y
Z
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33. Composition of Rotation Matrices
• Express P in 3 coincident rotated frames
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34. Composition of Rotation Matrices
• Recall for matrices
AB  BA
(matrix multiplication is not commutative)
Rot[Z,90] Rot[Y,-90]
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35. Composition of Rotation Matrices
• Recall for matrices
AB  BA
(matrix multiplication is not commutative)
Rot[Z,90]
Rot[Y,-90]
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36. Rot[Z,90]
Rot[Y,-90]
Rot[Z,90] Rot[Y,-90]
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37. Rot[z,90]Rot[y,-90]  Rot[y,-90] Rot[z,90]
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38. Decomposition of Rotation Matrices
• Rotation Matrices contain 9 elements
• Rotation matrices are orthogonal
– (6 non-linear constraints)
3 parameters describe rotation
• Decomposition is not unique
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39. Decomposition of Rotation Matrices
• Euler Angles
• Roll, Pitch, and Yaw
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40. Decomposition of Rotation Matrices
• Angle Axis
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41. Decomposition of Rotation Matrices
• Angle Axis
• Elementary Rotations
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42. Pose of a Rigid Body
• Pose = Position + Orientation
Ok. Now we know
what to do about
orientation…let’s get
back to pose
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43. Spatial Description of Body
• position of the origin with an orientation
A
X
Y
Z
B
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44. Homogeneous Coordinates
• Notational convenience
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45. Composition of Homogeneous Transformations
• Before:
• After
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46. Homogeneous Coordinates
• Inverse Transformation
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47. Homogeneous Coordinates
• Inverse Transformation
Orthogonal: no
matrix inversion!
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48. Literature:
Richard M. Murray, Zexiang Li, S. Shankar Sastry:
A mathematical Introduction to Robotic Manipulation,
University of California, Berkeley, 1994, CRC Press, pp.93-95.
An electronic edition of the book is available from:
http://www.cds.caltech.edu/~murray/mlswiki
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