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Forward Kinematics

Forward Kinematics. 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

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Forward Kinematics

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  1. Forward Kinematics

  2. 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

  3. 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

  4. Resources • Kinematics Notes • cs236927 Technion • Robotteknik

  5. 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

  6. Link Length ai • Link length is the distance along a line mutually perpendicular to axes i and i+1. • The mutual perpendicular always exists and is unique except with axis i and i+1 are parallel. • Parallel axes give infinitely many perdendiculars

  7. Link Twist ai • Link twist is the angle between the projections of axes i and i+1 onto a plane whose normal is the mutually perpendicular to axes i and i+1. • The angle is measure from axis i to axis i+1 in the right hand sense about the mutual perpendicular.

  8. 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 i+1 in the right hand sense about the mutual perpendicular. • Link Offset,di • Joint Angle, qi

  9. 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

  10. Assigning Frames to Links • Zi coincident with joint axis i • Origini at intersection of joint axis i and mutual perpendicular with axis i+1 • Xicoincident with mutual perpendicular pointing in the direction of axis i+1 • If ai = 0 (i.e. the axes intersect) then Xiis perpendicular to axes i and i+1 • Yichosen by the right-hand rule

  11. 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 dn=0 • Prismatic Joint n: • Xn such that qn = 0 • Origin {n} at intersection of joint axis n and Xnwhen dn=0

  12. Denavit Hartenberg Parameters ai = the distance from Zi to Zi+1 measured along Xi i = the angle between Zi to Zi+1 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

  13. Link transformation

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

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