Forward Kinematics Where is my hand ?

Examples

Denavit-Hartenberg Specialized description of articulated figures (joints) Each joint has only one degree of freedom rotate around its z-axis translate along its z-axis What’s so interesting about 6 DOF ?

Denavit-Hartenberg 1. Compute the link vector a i and the link length 2. Attach coordinate frames to the joint axes 3. Compute the link twist α i 4. Compute the link offset d i 5. Compute the joint angle φ i 6. Compute the transformation (i-1) T i which transforms entities from link i to link i-1

Denavit-Hartenberg This transformation is done in several steps : Rotate the link twist angle α i around the axis x i Translate the link length a i along the axis x i Translate the link offset d i along the axis z i Rotate the joint angle φ i around the axis z i 5

Denavit-Hartenberg 6

Multiplying the matrices : In DH only φ and d are allowed to change. 7

Denavit-Hartenberg Video

Example 1 D-H Link Parameter Table : rotation angle from X i-1 to X i about Z i-1 : distance from origin of (i-1) coordinate to intersection of Z i-1 & X i along Z i-1 : distance from intersection of Z i-1 & X i to origin of i coordinate along X i : rotation angle from Z i-1 to Z i about X i a0a0 a1a1 Z0Z0 X0X0 Y0Y0 Z3Z3 X2X2 Y1Y1 X1X1 Y2Y2 d2d2 Z1Z1 X3X3 Z2Z2 Joint 1 Joint 2 Joint 3

Example 2 : rotation angle from X i-1 to X i about Z i-1 : distance from origin of (i-1) coordinate to intersection of Z i-1 & X i along Z i-1 : distance from intersection of Z i-1 & X i to origin of i coordinate along X i : rotation angle from Z i-1 to Z i about X i

Example 3