Manufacturing Automation for Assembly and Inspection MM453
Manufacturing Automation for Assembly and
Inspection
MM453
Dr. Nigel Kent
Rm S366
nigel.kent@dcu.ie
Dublin City University
Sem 1 2021
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 1/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Lecture 2
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 2/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Truth Table Reduction
Taking the switch (with lock) from the previous lecture.
• The output is on when P ′ = 1
• Each line is an expression including the S L P P′
inputs directly if they are on (= 1) or 0 0 0 0
negated (barred) if they are off (= 0). 0 0 1 1
• The output is on if any of the lines of the 0 1 0 0
truth table with the output column are on. 0 1 1 0
• Therefore these lines are combined with 1 0 0 0
OR (+) statements and the variables for 1 0 1 1
each line are combined with AND (.) 1 1 0 1
statements. 1 1 1 1
• The expression for P ′ may be simplified
using a set of rules. This is the focus of
the coming slides.
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 3/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Truth Table Reduction
Taking the switch (with lock) from the previous lecture.
• Since we are typically only interested in
the case where the output is on we look S L P P′
only at the logic condition for those lines.
0 0 0 0
Therefore 0 0 1 1
0 1 0 0
0 1 1 0
P ′ = (S.L.P ) + (S.L.P ) + (S.L.P ) + (S.L.P ) 1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 3/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Boolean Logic Theorems and Laws
To reduce this expression to something more meaningful we can
use a number of theorems and laws to minimise the expression.
Characteristic Theorem
• X.0 = 0
• X.1 = X
• X +0=X
• X +1=1
Negation Theorem
• X=X
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 4/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Boolean Logic Theorems and Laws
Inclusion Theorem
• X.X = 0
• X +X =1
Commutative Law
• X +Y =Y +X
• X.Y = Y.X
Associative Law
• X + Y + Z = X + (Y + Z) = (X + Y ) + Z
• X.Y.Z = X.(Y.Z) = (X.Y ).Z
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 5/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Boolean Logic Theorems and Laws
Distributive Law
• X.Y + X.Z = X.(Y + Z)
• (X + Y ).(W + Z) = X.W + X.Z + Y.W + Y.Z
Idempotent Law
• X.X = X
• X +X =X
Absorptive Laws
• X + X.Y = X
• X.(X + Y ) = X
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 6/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Boolean Logic Theorems and Laws
Reflective Theorems
• X + X.Y = X + Y
• X.(X + Y ) = X.Y
• X.Y + X.Y.Z = X.Y + Y.Z
Consistency Theorems
• X.Y + X.Y = X
• (X + Y ).(X + Y ) = X
DeMorgan’s Laws
• X.Y = X + Y
• X + Y = X.Y
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 7/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Solving the Logic Expression
Taking the logic expression from the switch/lock example
P ′ = (S.L.P ) + (S.L.P ) + (S.L.P ) + (S.L.P )
Using Distributive Law
P ′ = (S + S).L.P + S.L.(P + P )
Using Inclusion Theorem
P ′ = (1).L.P + S.L.(1)
Using Characteristic Theorem
P ′ = L.P + S.L
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 8/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps
There is a graphical method for simplifying logic expressions. It
relies on two laws:
• Inclusion Theorem
• A+A=1
• Characteristic Theorem
• X.1 = X
Karnaugh Maps group expressions which have complementary
variables i.e.
A.B.C + A.B.C = A.B.(C + C) = A.B.1 = A.B
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 9/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps
2 variable Karnaugh map
Take the expression A.B + A.B. To
minimise:
A
• Draw a grid representing all
possible variable states. 0 1
0
B
1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 10/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps
2 variable Karnaugh map
Take the expression A.B + A.B. To
minimise:
A
• Draw a grid representing all
possible variable states. 0 1
• Input expression states
• A.B → 10 0 1
• A.B → 11 B
1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 10/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps
2 variable Karnaugh map
Take the expression A.B + A.B. To
minimise:
A
• Draw a grid representing all
possible variable states. 0 1
• Input expression states
• A.B → 10 0 1
• A.B → 11 B
• Group and read the variables that 1 1
don’t change from the map
• Expression can be reduced to A
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 10/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps
A A
0 1 0 1
0 1 1 0 1 1
B B
1 1 1
=B =B+A
Any group of 2n adjacent terms can be grouped together,
where n is a positive integer. i.e. 21 = two, 22 = four, 23 =
eight terms and so forth.
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 11/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps
3 Variable 4 Variable
AB AB
00 01 11 10 00 01 11 10
0 00
C
1 01
CD
11
10
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 12/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Exercise
Simplify:
Y = A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D
AB
00 01 11 10
00
01
CD
11
10
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 13/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Exercise
AB
00 01 11 10
00 1 1 1
01 1 1 1 1
CD
11 1 1 1 1
10 1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 14/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Exercise
AB • Group 1 = A
00 01 11 10
00 1 1 1
01 1 1 1 1
CD
11 1 1 1 1
10 1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 14/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Exercise
AB • Group 1 = A
00 01 11 10 • Group 2 = D
00 1 1 1
• Group 3 = C.B
01 1 1 1 1
CD
11 1 1 1 1
10 1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 14/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Exercise
AB • Group 1 = A
00 01 11 10 • Group 2 = D
00 1 1 1
• Group 3 = C.B
01 1 1 1 1
CD
11 1 1 1 1
10 1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 14/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Exercise
AB • Group 1 = A
00 01 11 10 • Group 2 = D
00 1 1 1
• Group 3 = C.B
01 1 1 1 1
CD
11 1 1 1 1
10 1 1
Solution:
Y = A + D + C.B
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 14/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps - Points to note
The map is a 2-D graph but the edges can be considered to fold
around so that left and right edges meet and top and bottom
edges meet.
AB • Group 1 = B.D
00 01 11 10 • Group 2 = A.B.D
00 1
01 1 1
CD
11 1 1
10 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 15/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps - Points to note
Also diagonal corners touch, i.e.
AB • Group 1 = B.D
00 01 11 10
00 1 1
01
CD
11
10 1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 16/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps - Points to note
Certain conditions of the input variables will be irrelevant, or can’t
physically happen. These are known as “Don’t Care” states and
are indicated with an X. They may be treated as 1’s or 0’s in order
to simplify the logic expression, i.e.:
AC
00 01 11 10
00
01 X 1 1 1
BD
11 1 1 1 X
10 X
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 17/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps - Points to note
Certain conditions of the input variables will be irrelevant, or can’t
physically happen. These are known as “Don’t Care” states and
are indicated with an X. They may be treated as 1’s or 0’s in order
to simplify the logic expression, i.e.:
AC Using 1’s only
00 01 11 10 • Y = A.B.D +
00
A.B.D + C.D
01 X 1 1 1
BD
11 1 1 1 X
10 X
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 17/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Karnaugh Maps - Points to note
Certain conditions of the input variables will be irrelevant, or can’t
physically happen. These are known as “Don’t Care” states and
are indicated with an X. They may be treated as 1’s or 0’s in order
to simplify the logic expression, i.e.:
AC Using 1’s and X terms
00 01 11 10 • Y =D
00
01 X 1 1 1
BD
11 1 1 1 X
10 X
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 17/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Example
Example
• An automatic work station cycles (Y = 1) when the power P
is on and the “cycle ready” indicator C is on.
• The machine is switched on with a spring return go switch G
and is switched off with a spring return stop switch S.
• The stop switch takes precedence over the go switch.
Inputs Outputs
Solenoid Cycle -Y
G Power P
Ready
S C
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 18/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Example
Inputs
List Inputs
Outputs
List Outputs
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 19/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Example
Inputs
• G = 0 when the go button is not pressed
• G = 1 when the go button is pressed
• S = 0 when the stop button is not pressed
• S = 1 when the stop button is pressed
• P = 0 when the power is off
• P = 1 when the power is on
• C = 0 when the cycle ready indicator is off
• C = 1 when the cycle ready indicator is on
Outputs
• P ′ = 0 when the power is off
• P ′ = 1 when the power is on
• Y = 0 station idles
• Y = 1 station cycles
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 19/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Example
G S P C P′ Y
0 0 0 0 0 0
0 0 0 1 X X State can’t exist
0 0 1 0 1 0
0 0 1 1 1 1
0 1 0 0 0 0
0 1 0 1 0 0 Would be X but S is on
0 1 1 0 0 0
0 1 1 1 0 0
1 0 0 0 1 0
1 0 0 1 X X State can’t exist
1 0 1 0 1 0
1 0 1 1 1 1
1 1 0 0 0 0
1 1 0 1 0 0 Would be X but S is on
1 1 1 0 0 0
1 1 1 1 0 0
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 20/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Example
Karnaugh map for P ′
PC
00 01 11 10
00 X 1 1
01 P ′ = G.S + P.S
GS
11
10 1 X 1 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 21/22
Manufacturing Automation for Assembly and Inspection MM453
Lecture 2
Example
Karnaugh map for Y
PC
00 01 11 10
00 X 1
01 Y = C.S
GS
11
10 X 1
Dr. Nigel Kent Manufacturing Automation for Assembly and Inspection, MM453 22/22
