Forward kinematics robotics m tech.

3
CS 4733, Class Notes: Forward Kinematics II 1 Stanford Manipulator - First Three Joints Figure 1: Stanford Robotic Arm. The frame diagram shows the first three joints, which are in a R-R-P configuration (Revolute-Revolute-Prismatic. 0 S 2 0 1 0 0 0 0 -C 2 0 0 1 0 0 1 0 d 2 0 0 1 d 3 0 0 1 0 0 0 1 C 1 C 2 -S 1 C 1 S 2 -S 1 d 2 C 1 C 2 -S 1 C 1 S 2 C 1 S 2 d 3 - S 1 d 2 S 1 C 2 C 1 S 1 S 2 C 1 d 2 0 S 1 C 2 C 1 S 1 S 2 S 1 S 2 d 3 + C 1 d 2 -S 2 0 C 2 d 1 -S 2 0 C 2 C 2 d 3 + d 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 d 2 0 0 1 d 1 0 0 0 1 2 T 3 = 1 d 1 0 1 T 2 = 0 C 2 0 S 2 C 1 0 -S 1 S 1 0 C 1 0 -1 0 T 1 T 3 = T 2 0 if θ 1 = θ 2 = 0, d 3 = 0 (Zero Position) joint θ d a α nnn1 θ 1 d 1 0 -90 2 θ 2 d 2 0 90 3 0 d 3 0 0 4 5 6

Transcript of Forward kinematics robotics m tech.

Page 1: Forward kinematics robotics m tech.

CS 4733, Class Notes: Forward Kinematics II

1 Stanford Manipulator - First Three Joints

Figure 1: Stanford Robotic Arm. The frame diagram shows the first three joints, which are in a R-R-P configuration (Revolute-Revolute-Prismatic.

0 S2 0 1 0 0 00 -C2 0 0 1 0 01 0 d2 0 0 1 d3

0 0 1 0 0 0 1

C1 C2 -S1 C1 S2 -S1 d2 C1 C2 -S1 C1S2 C1 S2 d3 - S1 d2

S1C2 C1 S1 S2 C1 d2 0 S1 C2 C1 S1S2 S1 S2 d3 + C1 d2

-S2 0 C2 d1 -S2 0 C2 C2 d3 + d1

0 0 0 1 0 0 0 1

1 0 0 00 1 0 d2

0 0 1 d1

0 0 0 1

2

T3 =

1

d1 01 0

T2 =

0 C2

0 S2 C1 0 -S1

S1 0 C1

0 -1 00 0 0

0T1 =

T3 =

T2 =0

if θ1 = θ2 = 0, d3 = 0: T3 = 0

(Zero Position)

1

joint θ d a α

nnn1 θ1 d1 0 -902 θ2 d2 0 903 0 d3 0 0456

Page 2: Forward kinematics robotics m tech.

2 4-DOF Gantry Robot

Figure 2: Gantry Robot Arm. This arm is in a R-P-R-R configuration. θ1 , θ3 , θ4 are the revolute joint angle variables and d2 is the prismatic joint variable. d4 is a constant.

0 00 1 -1 00 0

-S4 0C4 00 10 0

40 S4

1 0

A1 =

0 0

2

0 10 00 01 0

0 C4

A3 =

2

00 d2

1

0 0

-S1 0C1 00 10 0

S3 -C3

0 0

C1

S1

0 0

A0 =1

C3 0S3 00 10 0

3A2 =d4

1

joint θ d a α

1 θ1 0 0 02 0 d2 0 -903 θ3 0 0 904 θ4 d4 0 0

Page 3: Forward kinematics robotics m tech.

C1 -S1 0 0 1 0 0 0 S1 C1 0 0 0 0 1 00 0 1 0 0 -1 0 d2

0 0 0 1 0 0 0 1

S3 0 C4 -S4 0 0 -C3 0 S4 C4 0 0

0 0 0 0 1 d4

0 1 0 0 0 1

C1 0 -S1 0 C3 C4

S1 0 C1 0 S3 C4

0 -1 0 d2 S4

0 0 0 1 0

C1 C3 C4 - S1S4 -C1 C3 S4 - C4S1 C1 S3

C3 C4 S1 + C1 S4 -C3S1S4 + C1 C4 S1 S3

-S3C4 S3S4 C3

0 0 0

if θ1 = θ3 = θ4 = 90, d2 = D: A0 =

=

S3 d4 -C3 d4

0 1

C3 0 S3 00 10 0

2A0 =

C1 0 -S1 0S1 0 C1 00 -1 0 d2

0 0 0 1

=

A2 =

-C3S4

-S3S4

C4

0

S3 -C3

0 0

C1 S3 d4

S1S3 d4

C3 C4

S3 C4

S4

0

-C3S4

-S3S4

C4

0

4

S3

-C3

0 0

S3 d4 -C3 d4

0 1

4 2 4 A0 = A0 A2 =

=C3 d4 + d2

1

1 0 00 1 00 0 10 0 0

if θ1 = θ3 = θ4 = 0, d2 = 0: A0 =4

0 d4

D1

3

00 d4

1

0 1 0 0

(Zero Position)

-1 00 00 10 0

4