1. 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.
joint θ d a α
nnn1 θ1 d1 0 -90
2 θ2 d2 0 90
3 0 d3 0 0
4
5
6
     
C1 0 -S1 0 C2 0 S2 0 1 0 0 0
 S1 0 C1 0  1  S2 0 -C2 0  2  0 1 0 0 
T10 =  T2 =   T3 =  
 0 -1 0 d1   0 1 0 d2   0 0 1 d3 
0 0 0 1 0 0 0 1 0 0 0 1
   
C1 C2 -S1 C1 S2 -S1 d2 C1 C2 -S1 C1 S2 C1 S2 d3 - S1 d2
 S 1 C2 C1 S1 S2 C1 d 2 0  1 C2
S C1 S1 S2 S1 S 2 d3 + C 1 d 2 
T20 =  T3 =  
 -S2 0 C2 d1   -S2 0 C2 C2 d3 + d1 
0 0 0 1 0 0 0 1
 
1 0 0 0
 0 1 0 d2 
0
if θ1 = θ2 = 0, d3 = 0: T3 =   (Zero Position)
 0 0 1 d1 
0 0 0 1
1

2. 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.
joint θ d a α
1 θ1 0 0 0
2 0 d2 0 -90
3 θ3 0 0 90
4 θ4 d4 0 0
   
C1 -S1 0 0 1 0 0 0
 S1 C1 0 0  1  0 0 1 0 
A0 =  A =  
1  0 0 1 0  2  0 -1 0 d2 
0 0 0 1 0 0 0 1
   
C3 0 S3 0 C4 -S4 0 0
 S3
 0 -C3 0  3  S4 C4 0 0 
2
A3 = A =  
 0 1 0 0  4  0 0 1 d4 
0 0 0 1 0 0 0 1
2

3.     
C1 -S1 0 0 1 0 0 0 C1 0 -S1 0

 S1 C1 0 0  0 0 1 0   S1 0 C1 0 
A0 = 
2
 = 
 0 0 1 0 
 0 -1 0 d2   0 -1 0 d2 
0 0 0 1  0 0 0 1 0 0 0 1

    
C3 0 S3 0 C4 -S4 0 0 C3 C4 -C3 S4 S3 S3 d4

 S3 0 -C3 0  S4 C4 0 0   S3 C4 -S3 S4 -C3 -C3 d4 
A2 = 
4
 = 
 0 1 0 0 
 0 0 1 d4   S4 C4 0 0 
0 0 0 1  0 0 0 1 0 0 0 1

  
C1 0 -S1 0 C3 C 4 -C3 S4 S3 S3 d4

 S1 0 C1 0  S3 C4 -S3 S4 -C3 -C3 d4 

A0 = A02A24 = 
4
 0 -1 0 d2 
 S4 C4 0 0 
0 0 0 1  0 0 0 1

 
C1 C3 C4 - S1 S4 -C1 C3 S4 - C4 S1 C1 S3 C1 S3 d4
 C3 C4 S1 + C1 S4 -C3 S1 S4 + C1 C4 S1 S3 S1 S3 d4 
= 
 -S3 C4 S3 S4 C3 C 3 d4 + d 2 
0 0 0 1
 
1 0 0 0
 0 1 0 0 
if θ1 = θ3 = θ4 = 0, d2 = 0: A04= 
 0
 (Zero Position)
0 1 d4 
0 0 0 1
 
-1 0 0 0
 0 0 1 d4 
if θ1 = θ3 = θ4 = 90, d2 = D: A 4= 
0 
 0 1 0 D 
0 0 0 1
3