Approach to Decomposing Joint Transformation Matrix
Extracting Joint to Joint transformation matrices from the robot joint vectors which
include a position vector and a compound transformation matrix.
YA
A
θ3
{C}
{D}
P=(x,y,z)
θ4
1
θ2
{B}
A
θ1
PDORG
XA
{A}
Step 1: Set a point vector #P to the home position via #P=#PPOINT(0,0, …,0).
Step 2: Execute the ROTJOINT V+ program which outputs a compound transformation matrix for the
point #P. Set the rotation angle to 30° just for the joint in question, with all others to zero.
Step 3: Set up transformation equations and successively evaluate the LHS with the RHS from the
program output and the prior calculations.
T  ABT 1 CAT
B
C
T  CAT 1 DAT  CAT 1 ABT CBT
C
D
Step 4: Replace the non-zero or one numeric values in the rotation matrix portion of the transformation
with either sinθ = 0.50 or cosθ=0.866.
Alternative: Manually generate all joint frame work in trigonometric equation form and compare the
results by entering θ = 0°, 30°, 60°, 90°, etc. for selected joints.
Adept 550 SCARA robot joint configuration
The Adept 550 follows the convention in frame structuring, namely, the joint rotation or translation is
about or along Z axis and joint-to-joint frame rotation is done about X axis. The following is a complete
set of transformation matrices which has been numerically verified with ROTJOINT.
Joint 1
z
Joint 2
z
c1
R  R R R R   s1
 0
0
1
1
2
2
3
3
4
c12
R   s12
 0
s12
 c12
0
x
 s1 0 c2
c1 0  s2
0 1  0
 c12c4  s12s4
  c12s4  s12c4

0
0
4
Joint 4
x
y
x
0
4
Joint 3
 s2
c2
0
y
x
z
z
0 1 0
0  c4


0 0  1 0   s4
1 0 0  1  0
 s4
 c12s4  s12c4
 c12c4  s12s4
0
c4
0
0  c124  2s12s4
0    s124  2c12s4
 1 
0
0
 s12

0
0  when θ4=0. 4 R   c12
 0
 1
 c12
 s12
0
0
0
1
s124  2c12s4
 c12c4  2s12s4
0
0
0 
 1
0
0  when θ4=90°.
 1
The Adept transformation vectors carry the compound values of 04T . The 4th column vector is
0
P4org  L1c1  L2c12
L1s1  L2 s12
D  d3 
T
where L1=300 mm, L2=250 mm, and D is the maximum reach in Z direction. The model number “550”
comes from L1+L2.
Adept Six 300 robot joint configuration
Assuming that the Adept six axis robot follows the convention in frame structuring, namely, the joint
rotation or translation is about or along Z axis and joint-to-joint frame rotation is done about X axis. The
following is a set of the transformation matrices which has been partially verified with ROTJOINT.
Joint 1
Joint 2
Joint 3
z
x
x
x
y
 s1 0
c1 0,
0 1
1 0 0 c2  s2

1
0 1  s2 c2
2 R  0
0  1 0  0
0
c3  s3 0

2
c3 0
3 R   s3
 0
0 1
1 0 0  c4  s4
 0  1  s
3
4 R  0
  4 c4
0 1 0   0
0
1 0 0 c5  s5

4
0 1  s5 c5
5 R  0
0  1 0  0
0
1 0 0  c6  s6
 0  1  s
5
6 R  0
  6 c6
0 1 0   0
0
0
1
Joint 5
Joint 6
z
z
y
y
x
y
y
x
x
c1
R   s1
 0
0  c2
0   0
1  s2
0 c4
0   0
1  s4
0  c5
0   0
1  s5
0 c6
0   0
1  s6
 c1c23  c1s23  s1 

0
0 1 2
 s1s23 c1 
3 R 1 R 2 R 3 R   s1c23
 s23  c23
0 
0
6
Joint 4
z
z
y
z
R 30R 63R
 s2
0
c2
 s4
0
c4
0
1,
0
0
 1,
0 
 s5
0
0 1,
c5 0
 s6 0 
0  1,
c6
0 
c4c5c6  s 4 s6

3
3 4 5
s5c6
6 R4 R5 R6 R  
 s4c5c6 c 4 s6
 c4c5 s6  s 4 c6
 s5 s6
 s4c5 s6 c 4 c6
c4 s5 
 c5 
s4 s5 