Robot Modeling and the Forward Kinematic Solution

ME 4135Lecture Series 4 – PART 26 DOF Articulating Arm

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

l2

l3l4

LP TableFrames Link Var d l S C S C

0 → 1 1 R 1 0 0 90 -1 0 S1 C1

1 → 2 2 R 2 0 a2 0 0 1 S2 C2

2 → 3 3 R 3 0 a3 0 0 1 S3 C3

3 → 4 4 R 4 0 a4 -90 -1 0 S4 C4

4 → 5 5 R 5 0 0 90 1 0 S5 C5

5 → 6 6 R 6* d6 0 0 0 1 S6 C6

* With End Frame in Better Kinematic Home, otherwise is (6 - 90), which is a problem!

Using MathCad:

A Matrices, in Robot shorthand

1

1 0 1 01 0 1 00 1 0 00 0 0 1

C SS C

A

2

22

2 2 0 22 2 0 20 0 1 00 0 0 1

C S l CS C l S

A

3

33

3 3 0 33 3 0 30 0 1 00 0 0 1

C S l CS C l S

A

4

44

4 0 4 44 0 4 40 1 0 00 0 0 1

C S l CS C l S

A

A Matrices, cont.

5

5 0 5 05 0 5 00 1 0 00 0 0 1

C SS C

A

66

6 6 0 06 6 0 00 0 10 0 0 1

C SS C

Ad

Leads To:A FKS of:

01 2 3 4 5 6nT A A A A A A

Solving for FKSPre-process {A2*A3*A4} to collect

angular termsThey are the planer arm issue as

in the previous robot model

Then Continuing:Then Form: A1* {A2*A3*A4}*A5*A6Simplify for FKS!

Simplifies to:nx = R11 = C1·(C5·C6·C234 - S6·S234) - S1·S5·C6ny = 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·C5ay = R23 = S1·S5·C234 - C1·C5az = 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

And Again Physical Verification:

And Finally of the FKS:

Remember – these “Physical Verifications” must be checked against the robot’s Frame skeleton –

not just prepared!

You should Develop Frame Skeleton for each of the Various Arm TypesSCARACylindricalPrismatic

◦Gantry◦Cantilevered

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