1. Robot Modeling and the Forward Kinematic Solution
ME 4135
Lecture Series 4 – PART 2
6 DOF Articulating Arm

2. Another? 6dof Articulating Arm – (The Figure Contains Frame Skelton)

3. LP Table
Frames
Link
Var  d l 
S 
C 
S 
C 
0 → 1 1 R 1
90 -1 S1 C1
1 → 2 2 2 a2 0 S2 C2
2 → 3 3 3 a3 S3 C3
3 → 4 4 4 a4
-90 S4 C4
4 → 5 5 5 S5 C5
5 → 6 6
6* d6 S6 C6
* With End Frame in Better Kinematic Home, otherwise is (6 - 90), which is a problem!

4. Using MathCad:

5. A Matrices, in Robot shorthand

6. A Matrices, cont.

7. Leads To:
A FKS of:

8. Solving for FKS Pre-process {A2*A3*A4} to collect angular terms
They are the planer arm issue as in the previous robot model

9. Then Continuing:
Then Form: A1* {A2*A3*A4}*A5*A6
Simplify for FKS!

10. Simplifies to:
nx = R11 = C1·(C5·C6·C234 - S6·S234) - S1·S5·C6 ny = R21 = C1·S5·C6 + S1·(C5·C6·C234 - S6·S234) nz = R31 = S6·C234 + C5·C6·S234 ox = R12 = S1·S5·S6 - C1·(C5·S6·C234 + C6·S234) oy = R22 = - C1·S5·S6 - S1·(C5·S6·C234 + C6·S234) oz = R32 = C6·C234 - C5·S6·S234 ax = R13 = C1·S5·C234 + S1·C5 ay = R23 = S1·S5·C234 - C1·C5 az = R33 = S5·S234 dx = C1·(C234·(d6·S5 + l4) + l3·C23 + l2·C2) + d6·S1·C5 dy = S1·(C234·(d6·S5 + l4) + l3·C23 + l2·C2) - d6·C1·C5 dz = S234·(d6·S5 + l4) + l3·S23 + l2·S2

11. And Again Physical Verification:

12. And Finally of the FKS:
Remember – these “Physical Verifications” must be checked against the robot’s Frame skeleton – not just prepared!

13. You should Develop Frame Skeleton for each of the Various Arm Types
SCARA
Cylindrical
Prismatic
Gantry
Cantilevered

14. And Proceeding from the text
It is often possible to find that robots are assembled from Arms and various Wrist
Thus Arms ‘control’ the Positional issues of POSE
And Wrist ‘adjust’ the Orientation Issues of POSE
Hence these POSE issues can be treated separately
See text for Wrist Details
Spherical
RPY of various arrangements