The CRS A255 Robot

NMT EE 589 & UNM ME 482/582

ROBOT ENGINEERING

Dr. Stephen BruderNMT EE 589 & UNM ME 482/582

Dr. Stephen Bruder

1. The CRS A255 Robot An In Class Example

Thursday 4th Oct 2012ME 482/582: Robotics Engineering

Consider the CRS A255 Robot (brochure)○ A 5 DOF “RRRRR” Anthropomorphic Robotic Arm

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http://www.youtube.com/watch?v=0PKQYOt8XWU

Dr. Stephen Bruder

1. The CRS A255 Robot Forward Kinematics


Step 1: Identify the Joint Axes

Axis

#1

Axis #2

Axis #3

Axis #4Axis #5

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Dr. Stephen Bruder



Step 2: Assign link origns

Axis

#1

Axis #2

Axis #3

Axis #4Axis #5

origin of frame{1}

origin of frame{2}

origin of frame{3}

origin of frames {4 & 5}

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Dr. Stephen Bruder



Step 3: Assign z-axes

Axis

#1

Axis #2

Axis #3

Axis #4Axis #5z1

z2

z3

z4z5

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Dr. Stephen Bruder



Step 4: Assign x-axes

Axis

#1

Axis #2

Axis #3

Axis #4Axis #5z1

z2

z3

z4z5

x1

x2

x3 x4,5

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Dr. Stephen Bruder



Step 6: Add a base/Tool Frame

Axis

#1

Axis #2

Axis #3

Axis #4Axis #5z1

z2

z3

z4 z5

x1

x2

x3 x4,5

z0

x0

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Dr. Stephen Bruder



Build DH Table

D-H

params. i

1

2

3

4

5

i - 1 i-1a id i

z1

z2

z3

z4 z5

x1

x2

x3 x4,5

z0

x0

1d

0 0 1d ( )t190 0 0 ( ) 90t 2

0 2a 03( ) 90t

0 3a 0 ( ) 90t 4

90 0 0 ( )t5

2a

3a

From measurements:• d1=10 in• a2=10 in• a3=10 in

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Dr. Stephen Bruder



A quick visual check of the D-H table via Robotics TB

z1

z2

z3

z4 z5

x1

x2

x3 x4,5

z0

x0

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Dr. Stephen Bruder



Real World vs Model

1 1RW 2 2 90RW

3 2 3RW 4 2 3 4RW

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Dr. Stephen Bruder



Real World vs Model○ Config#2

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Dr. Stephen Bruder



Real World vs Model○ Config#3

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Dr. Stephen Bruder



Build the T-matrices1 1

1 11

1

0

0 0

0 0

0 0 1

0 0 0 1

c s

s cT

d

2

12

2

2 2

0 0

0 0 1 0

0 0

0 0 0 1

s c

c sT

3 3

3

2

3 32

0

0 0

0 0 1 0

0 0 0 1

cT

s c a

s

4 4 3

4 434

0

0 0

0 0 1 0

0 0 0 1

c

sT

s a

c

5 5

5 5

45

0 0

0 0 1 0

0 0

0 0 0 1

c s

s cT

1 2 1 2 1

1 2 2 1 102

2 2 1

0

0

0

0 0 0 1

c s c c s

s s c s cT

c s d

1 23 1 23 1 2 1 2

23 1 1 23 1 2 1 203

23 23 2 2 10

0 0 0 1

c c c s s a c s

c s s s c a s sT

s c a c d

1 234 1 234 1 1 3 23 2 2

0 1 234 234 1 1 1 3 23 2 24

234 234 2 2 1 3 230

0 0 0 1

c s c c s c a c a s

s s c s c s a c a sT

c s a c d a s

1 5 1 5 234 5 1 1 5 234 1 234 1 3 23 2 2

0 1 5 5 1 234 1 5 1 5 234 234 1 1 3 23 2 25

5 234 234 5 234 2 2 1 3 23

0 0 0 1

s s c c s c s c s s c c c a c a s

c s c s s c c s s s c s s a c a sT

c c c s s a c d a s

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Dr. Stephen Bruder

1. The CRS A255 Robot Inverse Kinematics


A 6 DOF World and a 5 DOF Robot?○ Looking at the A255 robot it seems reasonable to solve for

the 3D position, inclination of z5 to the x/y plane, and rotation about axis#5 (screwdriver type applications).

○ Considering the positional equations

11 12 131 5 1 5 234 5 1 1 5 234 1 234 1 3 23 2 2

21 22 230 1 5 5 1 234 1 5 1 5 234 234 1 1 3 23 2 25

31 32 335 234 234 5 234 2 2 1 3 23

=

0 0 0 10 0 0 1

x

yd

z

r r r ps s c c s c s c s s c c c a c a s

r r r pc s c s s c c s s s c s s a c a sT T

r r r pc c c s s a c d a s

1 3 23 2 2 (1.1)xc a c a s p

1 3 23 2 2 (1.2)ys a c a s p

2 2 1 3 23 (1.3)za c d a s p

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Dr. Stephen Bruder



From (1.1) and (1.2) we get

Hence,

3 23 2 211

3 23 2 2

y

x

p a c a sTan

p a c a s

11 1 (1.4)180y

x

pTan and

p

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Dr. Stephen Bruder

1. The CRS A255 Robot

{0}

5 234 5 234 234 3 23 2 2 1 11 21 1 1 12 22 1 1 13 23 1 1 1

5 5 1 21 11 1 1 22 12 1 1 23 13 1 1 1

5 234 234 5 234 2 2 3 23 31 32 33 1

0 0

0 0 0 1 0 0 0 1

x y

y x

z

c s s s c a c a s c r r s c r r s c r r s c p p s

s c c r r s c r r s c r r s c p p s

c c c s s a c a s r r r d p

Inverse Kinematics


Looking at the length of {1} to {5}○ This should only be a fn of 3

1 15 0 = dT TT

2 2 2 2 2 22 3 2 3 3 1 12 2x y z za s a a a d p p d p p

2 2 2 2 2 21 1 2 3

32 3

2

2x y z zd p p d p p a a

sa a

{1}{2}

{3}{4}

{5}15ORGd

13 32

(1.5181

0 )aT dan n

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Dr. Stephen Bruder



Adding Eqn. (1.2) Cos(2) to Eqn. (1.3) Sin(2)

This is case 7, hence

2 3 23 2 2 2 2 2 3 23 2 1 2 1x zc a c a s s a c a s c p c s p d

3 3 2 1 2 1x za c c p c s p d

2 2 1 1 3 3, ,x zb c where a bac s p c p d c a cand

2 2 21 1

2 (1.6)b a b c

Tan Tana c

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Dr. Stephen Bruder



Finally, inclination of z5 to the x/y plane

And, 5 is given

i i

The X0/Y0 plane

5z

5 0

0

234

5 3,3

ˆ ˆiSin z z

s

R

4 2 3 (1.7)i

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Dr. Stephen Bruder

1. The CRS A255 Robot Motion Kinematics


Build the Jacobian Matrix○ Let’s use the indirect method to build Jv

1 3 23 2 20

5 1 3 23 2 2

2 2 1 3 23

ORG

c a c a s

d s a c a s

a c d a s

1

1

1 3 23 2 205

1 3 23 2 2

0v

s a c a sd

c a c a sJ

2

2

1 2 2 3 2305

1 2 2 3 23

3 23 2 2

v

c a c a sd

s a c a s

a c a s

J

3

3 1 2305

3 1 23

3 23

3

v

a c sd

a

a

J s s

c

4

0

4

5

0

0

0v

dJ

5

0

5

5

0

0

0v

dJ

1 3 23 2 2 1 2 2 3 23 3 1 23

1 3 23 2 2 1 2 2 3 23 3 1 23

3 23 2 2 3 23

0 0

0 0

0 0 0v

s a c a s c a c a s a c s

c a c a s s a c a s a s s

a c a s c

J

a

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Dr. Stephen Bruder

1. The CRS A255 Robot Motion Kinematics


Build the Jacobian Matrix○ Let’s use the direct method to build J

Hence,

1

01

0ˆ 0

1

J z

2

10

2 1ˆ

0

s

z cJ

3

10

3 1ˆ

0

s

z cJ

4

10

4 1ˆ

0

s

z cJ

5

1 2340

5 234 1

234

ˆc c

z c s

s

J

1 1 1 1 234

1 1 1 234 1

234

0

0

1 0 0 0

s s s c c

c c c c sJ

s

1 3 23 2 2 1 2 2 3 23 3 1 23

1 3 23 2 2 1 2 2 3 23 3 1 23

3 23 2 2 3 23

1 1 1 1 234

1 1 1 234 1

234

0 0

0 0

0 0 0

0

0

1 0 0 0

v

s a c a s c a c a s a c s

c a c a s s a c a s a s s

a c a s a c

s s s c c

c c c c

J

s

JJ

s

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The CRS A255 Robot