Forward Kinematics

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Forward Kinematics
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Forward Kinematics
Focus on links chains May be combined in a tree
structure
Degrees of Freedom Number of independent
position variables (i.e. joints )
Joints 1 Degree of Freedom Rotation or
Translation
Pose Specification of parameters for all joint
DoFs
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Robotics
Robot arms - focus on end effector Forward
Kinematics Where am I Joint space
Cartesian space
Inverse Kinematics Where should I
be Cartesian space Joint space

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Resources

• Kinematics Notes
• cs236927 Technion
• Robotteknik

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Links

• A link is a rigid body that defines the
  relationship between two adjacent joint axes of a
  manipulator
• Joint axis i is defined by a line in space about
  which link i rotates (or translates) relative to
  link i-1
• A link is specified by two parameters that
  define the relative position and orientation of
  the joint axes incident on the link

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Link Length ai

• Link length is the distance along a line mutually
  perpendicular to axes i and i1.
• The mutual perpendicular always exists and is
  unique except with axis i and i1 are parallel.
• Parallel axes give infinitely many perdendiculars

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Link Twist ai

• Link twist is the angle between the projections
  of axes i and i1 onto a plane whose normal is
  the mutually perpendicular to axes i and i1.
• The angle is measure from axis i to axis i1 in
  the right hand sense about the mutual
  perpendicular.

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Joints (connecting links)

• Joints define the relationship between adjacent
  links
• A joint can be described with two parameters The
  angle is measure from axis i to axis i1 in the
  right hand sense about the mutual perpendicular.
• Link Offset, di
• Joint Angle, qi

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Joints Parameters

• Link Offset, di
• The distance along the axis of joint i between
  the intersections of the mutual perpendiculars
  with axes i-1 and i
• Variable for a prismatic joint
• Joint Angle, qi
• The angle between the mutual perpendiculars (i.e.
  links) incident on joint axes i.
• Variable for a revolute joint

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Assigning Frames to Links

• Zi coincident with joint axis i
• Origini at intersection of joint axis i and
  mutual perpendicular with axis i1
• Xi coincident with mutual perpendicular pointing
  in the direction of axis i1
• If ai 0 (i.e. the axes intersect) then Xi is
  perpendicular to axes i and i1
• Yi chosen by the right-hand rule

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First and Last Links

• Base frame (0) is arbitrary
• Make life easy
• Coincides with frame 1 with joint parameter is
  0
• Frame n (last link)
• Revolute Joint n
• Xn Xn-1 when qn 0
• Origin n such that dn0
• Prismatic Joint n
• Xn such that qn 0
• Origin n at intersection of joint axis n and Xn
  when dn0

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Denavit Hartenberg Parameters
ai the distance from Zi to Zi1 measured along
Xi ?i the angle between Zi to Zi1 measured
about Xi di the distance from Xi-1 to Xi
measured along Zi ?i the angle between Xi-1 to
Xi measured about Zi
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Link transformation
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Demonstrations

• Jack
• Norm Badler, U Penn
• Danse Interactif, SIGGRAPH 94
• Ken Perlin, NYU