ME 192 Exam 1 Solutions
19/24/14
Name _________________________
Score ___________________
1. SCARA stands for Selectively Compliant Assembly Robot Arm. Name the non-compliant direction(s).
(4) Non-compliant in the (angular) directions of Yaw and Pitch at the wrist joint. (The robot can reach an
object in a vertical direction only without additional degrees of freedom added to the end effector.)
The two key rules in setting up the joint frames that the robots in the lab also follow:
a) (3) Between joints (fixed rotation): Rotate Frame i about its X axis to orient Frame i+1.
b) (3) Within joints (variable rotation): Rotate about or translate along Z axis following frame rotation.
2. In the below, frame {1} transforms into {2} by rotating 90° about its X1 and translating to the {2} position.
Then frame {2} rotates about its Z2 by -90° to set its X2 orientation. Complete the labeling of the axes below
and set up a three-step transformation (rotate-translate-rotate) for frame 2. Also show the inverse of the
resulting matrix and the D-H table entries for steps 1 and 2 prior to the Z2 rotation.
1
0
1

T

2
0

0
0 1
0  1 0 0
1 0 0  0

0 0 1  0
0
0
(3)
a  0
1 0  d   1
0 1 0  0

0 0 1  0
0 0
1 0 0  0
0 0 0  0

0 1 0   1
 
0 0 1  0
(3)
(2)
0
y
(2)
0  a
 1  90
a  a 
1 0 
1

(3) D  H  



d

d
0 d
2

y

0 0 
 2  0 
1 0
0 0
(3) 21T  
0  1

0 0
a 
0  1 0 
0 0  d

0 0
1 
1
(3)
d
2
x a
1
3. In the below transformation matrix, evaluate or describe what the vectors represent:
r21 r22 r23  __________
__________
= The projection
of ŷ of __________
frame {i} onto_______
the three axes of {i+1}. (2)
 r11 r12 r13 p1 
r r
T   21 22
r31 r32

0 0
A
B
r23
r33
0
p2 
p3 

1
r13
 p1
r23
p2
r33  __________
__________
__________
= The projection
of ẑ of frame
{i} onto______
the three axes of {i-1}. (2)
T
p3  __________
__________
= The origin__________
of frame {i} relative
to that______
of {i-1}.
T
(r1  r3 )  r2  __________
__________________________
-1 (3)
The (2, 4) element of BAT when P= 1 1 1 = -r12 - r22 - r32 (4)
T
4. The below V+ commands are written for a SCARA robot. Given the position P = [x, y, z, 0, 180, r], and #Q
= [α, β, d, γ] with all parameters predefined, find the new values of P and #Q.
DECOMPSE V[1]= P ; Extract the Cartesian coordinates and the roll angle
SET P=P:TRANS(100, -V[2], -V[3]+300, 0, 0, -V[6]) ; Redefine P via a vector addition.
DECOMPSE W[1]= #Q ; Extract the joint roll angles and displacements.
SET #Q=#PPOINT(-W[1], -W[2]+30, -W[3]+60, -W[4]+90)
(5) P = [x+100, 0, 300, 0, 180, 0]
(5) #Q = [- α, - β +30, -d+60, -γ +90]
; Redefine # Q
(2)
5. In a SCARA robot, the roll angle on the Z axis “r” is given in terms of the base frame. If r needs to be kept
constant when the arm moves (e.g. the axes rotate.) how should the wrist angle γ be set?
(5) Since roll angle r = α + β + (180 – γ) , keep γ equal to α + β + (180) to counter-rotate by α + β.
What is the values of the 4th column of the transformation matrix of a SCARA robot when the joint values
are #Q = [α, β, d, γ] and the link lengths are L1 and L2, and the maximum Z height is D.
(5) P = [L1cos α + L2cos (α + β), L1sin α + L2sin (α + β), D-d, 1]
6. What rotation about one of the principal axes will result in the below
0
1 0

 Axis: X Axis
(4) 0  1 0


0 0  1
Angle: 180°
In a matrix form, show that rot ( yˆ ,180)  rot ( xˆ ,180)rot ( zˆ,180)  rot ( zˆ,180)rot ( xˆ,180)
 1 0 0  1 0 0   1 0 0  1 0 0 1 0 0 
 0 1 0   0  1 0   0  1 0   0  1 0 0  1 0 

 

 


 0 0  1 0 0  1  0 0 1  0 0 1 0 0  1
(2.5)
(2.5)
7. Show the transpose of a rotation matrix RT = R-1. Choose an algebraic proof based on the dot product
properties of orthonormal matrices. Follow the argument that because RR-1=I, if RRT = I, then, RT = R-1.
a
R  b
 c
g
h 
i 
d
e
f
a
R   d
 g
T
a 2  d 2  g 2

R  R T   ba  ed  hg
 ca  fd  ig

b
e
h
c
f 
i 
ab  de  gh ac  df  gi  1 0 0

b 2  e 2  h 2 bc  ef  hi   0 1 0  I (7)
cb  fe  hi c 2  f 2  i 2  0 0 1

(3) (The above is due to that all 3x3 elements of RRT are the vector dot products of two rows or two
columns of orthonormal matrices which are either 0 or 1. Since RR-1=I and RRT = I, it follows that RT = R-1.
8. Find the transformation matrix
B
Going either clockwise or counterclockwise
(6) T  T
C
D
B
C
1 B
A
E
A
T T
C
1 E
D
T
D
A
E
9. The linked rotation matrices of a SCARA robot is expressed by 40 R  10 R 21 R32 Q 43 R , where
i 1
i
R is a variable
2
3
rotation matrix and Q is a fixed rotational relationship between Z2 and Z3. If a subscript “0” denotes a
zero rotation and θ denotes a non-zero rotation, then find the single joint rotations from a composite
rotation. Start by explaining why
(3) 40 R0  40R0,0,,0  32Q :
Because 10 R0  21R0  43R0  I
1
(3) 10 R  _______ 40R1 ( 40 R0 )  Express in terms of 40 R and 40 R0 and/or their inverses.
1
(3) 21 R  _______ 40R 2 ( 40 R0 )
Only one side of 40 R  is applicable.
1
(3) 43 R ( 40 R0 ) 40 R 4 ______
0
0
Note that Program ROTJOINT outputs both 4 R0 and 4 R , and their inverses. The six axis version of the above can be
i




derived from 6 R 1 R  Q 2 R3 R Q 4 R Q 5 R Q 6 R and 6 R0   Q Q Q Q
0
0
1

2
3
4
5
0
1

3
4
5