Forward Kinematics

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Forward Kinematics

• Professor Nicola Ferrier
• ME Room 2246, 265-8793
• ferrier_at_engr.wisc.edu

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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

Today
Next Lecture
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Industrial Robot
sequence of rigid bodies (links) connected by
means of articulations (joints)
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Robot Basics Modeling
?

• Kinematics
• Relationship between the joint angles, velocities
  accelerations and the end-effector position,
  velocity, acceleration

?
?
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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 gt degrees of
  freedom
• Workspace
• Portion of environment where the end-effector can
  access

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Modeling Robot Manipulators

• Open kinematic chain
• sequence of links with one end constrained to
  the base, the other to the end-effector

End-effector
Base
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Modeling Robot Manipulators

• Motion is a composition of elementary motions

End-effector
Joint 2
Joint 1
Joint 3
Base
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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,q2,¼,qn l1¼,ln,?1¼,?n)

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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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P
Z
BP (x,y,z)
Z
AP (x,y,z)
Y
B
Y
A
X
X
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Pose of a Rigid Body

• Pose Position Orientation

How do we do this?
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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

YA
XA
ZA
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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

YA
XA
ZA
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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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P
Z
AP (x,y,z)
W
BP (u,v,w)
V
Y
O
X
U
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P
Z
AP (x,y,z)
W
BP (u,v,w)
V
Y
O
X
U
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P
Z
AP (x,y,z)
W
BP (u,v,w)
V
Y
O
X
U
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P
Z
AP (x,y,z)
W
BP (u,v,w)
V
Y
O
X
U
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Rotation Matrices
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Rotation Matrices
1
X axis expressed wrt Ouvw
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Rotation Matrices
1
Y axis expressed wrt Ouvw
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Rotation Matrices
1
Z axis expressed wrt Ouvw
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Rotation Matrices
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Rotation Matrices
X axis expressed wrt Ouvw
Y axis expressed wrt Ouvw
Z axis expressed wrt Ouvw
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Rotation Matrices
1
U axis expressed wrt Oxyz
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Rotation Matrices
U axis expressed wrt Oxyz
V axis expressed wrt Oxyz
W axis expressed wrt Oxyz
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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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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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Elementary Rotations X axis
Z
Y
X
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Elementary Rotations X axis
Z
Y
X
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Elementary Rotations Y axis
Z
Y
X
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Elementary Rotations Z-axis
Z
Y
X
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Composition of Rotation Matrices

• Express P in 3 coincident rotated frames

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Composition of Rotation Matrices

• Recall for matrices
• AB ¹ BA
• (matrix multiplication is not commutative)

RotZ,90
RotY,-90
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Composition of Rotation Matrices

• Recall for matrices
• AB ¹ BA
• (matrix multiplication is not commutative)

RotZ,90
RotY,-90
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RotZ,90
RotY,-90
RotZ,90
RotY,-90
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Rotz,90Roty,-90 ¹ Roty,-90 Rotz,90
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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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Decomposition of Rotation Matrices

• Euler Angles
• Roll, Pitch, and Yaw

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Decomposition of Rotation Matrices

• Angle Axis

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Decomposition of Rotation Matrices

• Angle Axis
• Elementary Rotations

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Pose of a Rigid Body

• Pose Position Orientation

Ok. Now we know what to do about
orientationlets get back to pose
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Spatial Description of Body

• position of the origin with an orientation

Z
B
Y
A
X
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Homogeneous Coordinates

• Notational convenience

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Composition of Homogeneous Transformations

• Before
• After

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Homogeneous Coordinates

• Inverse Transformation

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Homogeneous Coordinates

• Inverse Transformation

Orthogonal no matrix inversion!