1. Forward Kinematics

2. 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 armsRobot 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 a i 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  i 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, d i Joint Angle,  

9. Joints Parameters Link Offset, d i 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,   The angle between the mutual perpendiculars (i.e. links) incident on joint axes i. Variable for a revolute joint

10. Assigning Frames to Links Z i coincident with joint axis i Origin i at intersection of joint axis i and mutual perpendicular with axis i+1 X i coincident with mutual perpendicular pointing in the direction of axis i+1 If a i = 0 (i.e. the axes intersect) then X i is perpendicular to axes i and i+1 Y i chosen 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: X n = X n-1 when  n = 0 Origin {n} such that d n =0 Prismatic Joint n: X n such that  n = 0 Origin {n} at intersection of joint axis n and X n when d n =0

12. Denavit Hartenberg Parameters a i = the distance from Z i to Z i+1 measured along X i  i = the angle between Z i to Z i+1 measured about X i d i = the distance from X i-1 to X i measured along Z i  i = the angle between X i-1 to X i measured about Z i

13. Link transformation

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