Exercise DET-05.04 – Position of a robot arm

Prof. Dr. Thorsten Jungmann (last updated 2026-11-02)

Relates to DET-05 Codes, code converters and standard components

Task

The rotary encoder at a joint of a robot arm delivers the position of the joint in the Gray code, while the controller calculates in the binary code. The code disc has 16 sectors and four tracks. A code converter turns the Gray code word with the bits $A$ , $B$ , $C$ and $D$ into the binary code word with the bits $Y_{3}$ , $Y_{2}$ , $Y_{1}$ and $Y_{0}$ , $Y_{3}$ with the weight 8. This is the code table:

Sector	$A$	$B$	$C$	$D$	$Y_{3}$	$Y_{2}$	$Y_{1}$	$Y_{0}$
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	0	1
2	0	0	1	1	0	0	1	0
3	0	0	1	0	0	0	1	1
4	0	1	1	0	0	1	0	0
5	0	1	1	1	0	1	0	1
6	0	1	0	1	0	1	1	0
7	0	1	0	0	0	1	1	1
8	1	1	0	0	1	0	0	0
9	1	1	0	1	1	0	0	1
10	1	1	1	1	1	0	1	0
11	1	1	1	0	1	0	1	1
12	1	0	1	0	1	1	0	0
13	1	0	1	1	1	1	0	1
14	1	0	0	1	1	1	1	0
15	1	0	0	0	1	1	1	1

a) Minimize $Y_{3}$ with a Karnaugh map.

b) Minimize $Y_{2}$ with a Karnaugh map.

c) Minimize $Y_{1}$ with a Karnaugh map.

d) Draw the Karnaugh map of $Y_{0}$ . You do not need to write down its expression. What happens to the loops from a) to d)?

e) Express the four outputs with XOR gates with two inputs, as a chain from the highest bit down. How many XOR gates does the code converter need?

f) Check your chain for sector 11.

A colleague has built this code converter with a fault and reports: $Y_{3}$ and $Y_{2}$ agree with the code table in all 16 rows, but $Y_{1}$ and $Y_{0}$ do not:

Row	$A$	$B$	$C$	$D$	$Y_{1}$ from the code table	$Y_{1}$ in the simulator	$Y_{0}$ from the code table	$Y_{0}$ in the simulator
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	1	1
2	0	0	1	0	1	1	1	1
3	0	0	1	1	1	1	0	0
4	0	1	0	0	1	0	1	0
5	0	1	0	1	1	0	0	1
6	0	1	1	0	0	1	0	1
7	0	1	1	1	0	1	1	0
8	1	0	0	0	1	1	1	1
9	1	0	0	1	1	1	0	0
10	1	0	1	0	0	0	0	0
11	1	0	1	1	0	0	1	1
12	1	1	0	0	0	1	0	1
13	1	1	0	1	0	1	1	0
14	1	1	1	0	1	0	1	0
15	1	1	1	1	1	0	0	1

Simulation of the faulty code converter from Gray to binary, with four logic inputs and three XOR gates, in which all inputs and all outputs are low

Open in the simulator

Circuit as text (File → Import From Text…)

<cir f="1" ts="0.000005" ic="1.500424758475255" cb="50" pb="50" vr="5" mts="5e-11">
  <L x="96 64 96 16" f="2"/>
  <L x="160 64 160 16" f="2"/>
  <L x="224 64 224 16" f="2"/>
  <L x="288 64 288 16" f="2"/>
  <M x="688 128 736 128" f="2"/>
  <Xor x="448 256 592 256" f="0"/>
  <M x="688 256 736 256" f="2"/>
  <Xor x="448 400 592 400" f="0"/>
  <M x="688 400 736 400" f="2"/>
  <Xor x="448 544 592 544" f="0"/>
  <M x="688 544 736 544" f="2"/>
  <w x="96 128 688 128" f="0"/>
  <w x="96 240 448 240" f="0"/>
  <w x="96 384 448 384" f="0"/>
  <w x="96 64 96 128" f="0"/>
  <w x="96 128 96 240" f="0"/>
  <w x="96 240 96 384" f="0"/>
  <w x="160 272 448 272" f="0"/>
  <w x="160 64 160 272" f="0"/>
  <w x="592 256 688 256" f="0"/>
  <w x="224 416 448 416" f="0"/>
  <w x="224 64 224 416" f="0"/>
  <w x="592 400 640 400" f="0"/>
  <w x="640 400 688 400" f="0"/>
  <w x="640 400 640 496" f="0"/>
  <w x="400 496 640 496" f="0"/>
  <w x="400 496 400 528" f="0"/>
  <w x="400 528 448 528" f="0"/>
  <w x="288 560 448 560" f="0"/>
  <w x="288 64 288 560" f="0"/>
  <w x="592 544 688 544" f="0"/>
  <x x="66 22 76 25" f="0" si="16" te="A"/>
  <x x="130 22 140 25" f="0" si="16" te="B"/>
  <x x="194 22 205 25" f="0" si="16" te="C"/>
  <x x="258 22 269 25" f="0" si="16" te="D"/>
  <x x="756 134 775 137" f="0" si="16" te="Y3"/>
  <x x="756 262 775 265" f="0" si="16" te="Y2"/>
  <x x="756 406 775 409" f="0" si="16" te="Y1"/>
  <x x="756 550 775 553" f="0" si="16" te="Y0"/>
</cir>

g) In which rows do $Y_{1}$ and $Y_{0}$ deviate, and why do both outputs deviate in the same rows? Which input of which gate is connected wrongly? Confirm it in the simulation.

Solution

a) $Y_{3} = A$
b) $Y_{2} = \overline{A} \cdot B + A \cdot \overline{B} = A \oplus B$
c) $Y_{1} = \overline{A} \cdot \overline{B} \cdot C + \overline{A} \cdot B \cdot \overline{C} + A \cdot \overline{B} \cdot \overline{C} + A \cdot B \cdot C = A \oplus B \oplus C$
d) a chequerboard of eight single ones: the loops shrink with every further input, from eight cells to single cells, and no loop can minimize $Y_{0}$
e) a chain, in which each output is the XOR of the output above it and one more input: $Y_{3} = A$ , $Y_{2} = A \oplus B$ , $Y_{1} = A \oplus B \oplus C$ , $Y_{0} = A \oplus B \oplus C \oplus D$ ; three XOR gates
f) sector 11 is the Gray code word 1110; the chain gives 1011, the number 11
g) rows 4, 5, 6, 7, 12, 13, 14 and 15, exactly those with $B = 1$ ; the XOR gate of $Y_{1}$ receives $A$ instead of $Y_{2}$ , and $Y_{0}$ is formed from the wrong $Y_{1}$

▶️ Full worked solution

Wattsinside

Exercise DET-05.04 – Position of a robot arm

Task

Backlinks

Sector	$A$	$B$	$C$	$D$	$Y_{3}$	$Y_{2}$	$Y_{1}$	$Y_{0}$
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	0	1
2	0	0	1	1	0	0	1	0
3	0	0	1	0	0	0	1	1
4	0	1	1	0	0	1	0	0
5	0	1	1	1	0	1	0	1
6	0	1	0	1	0	1	1	0
7	0	1	0	0	0	1	1	1
8	1	1	0	0	1	0	0	0
9	1	1	0	1	1	0	0	1
10	1	1	1	1	1	0	1	0
11	1	1	1	0	1	0	1	1
12	1	0	1	0	1	1	0	0
13	1	0	1	1	1	1	0	1
14	1	0	0	1	1	1	1	0
15	1	0	0	0	1	1	1	1

Row	$A$	$B$	$C$	$D$	$Y_{1}$ from the code table	$Y_{1}$ in the simulator	$Y_{0}$ from the code table	$Y_{0}$ in the simulator
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	1	1
2	0	0	1	0	1	1	1	1
3	0	0	1	1	1	1	0	0
4	0	1	0	0	1	0	1	0
5	0	1	0	1	1	0	0	1
6	0	1	1	0	0	1	0	1
7	0	1	1	1	0	1	1	0
8	1	0	0	0	1	1	1	1
9	1	0	0	1	1	1	0	0
10	1	0	1	0	0	0	0	0
11	1	0	1	1	0	0	1	1
12	1	1	0	0	0	1	0	1
13	1	1	0	1	0	1	1	0
14	1	1	1	0	1	0	1	0
15	1	1	1	1	1	0	0	1

Sector	$A$	$B$	$C$	$D$	$Y_{3}$	$Y_{2}$	$Y_{1}$	$Y_{0}$
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	0	1
2	0	0	1	1	0	0	1	0
3	0	0	1	0	0	0	1	1
4	0	1	1	0	0	1	0	0
5	0	1	1	1	0	1	0	1
6	0	1	0	1	0	1	1	0
7	0	1	0	0	0	1	1	1
8	1	1	0	0	1	0	0	0
9	1	1	0	1	1	0	0	1
10	1	1	1	1	1	0	1	0
11	1	1	1	0	1	0	1	1
12	1	0	1	0	1	1	0	0
13	1	0	1	1	1	1	0	1
14	1	0	0	1	1	1	1	0
15	1	0	0	0	1	1	1	1

Row	$A$	$B$	$C$	$D$	$Y_{1}$ from the code table	$Y_{1}$ in the simulator	$Y_{0}$ from the code table	$Y_{0}$ in the simulator
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	1	1
2	0	0	1	0	1	1	1	1
3	0	0	1	1	1	1	0	0
4	0	1	0	0	1	0	1	0
5	0	1	0	1	1	0	0	1
6	0	1	1	0	0	1	0	1
7	0	1	1	1	0	1	1	0
8	1	0	0	0	1	1	1	1
9	1	0	0	1	1	1	0	0
10	1	0	1	0	0	0	0	0
11	1	0	1	1	0	0	1	1
12	1	1	0	0	0	1	0	1
13	1	1	0	1	0	1	1	0
14	1	1	1	0	1	0	1	0
15	1	1	1	1	1	0	0	1

Sector	$A$	$B$	$C$	$D$	$Y_{3}$	$Y_{2}$	$Y_{1}$	$Y_{0}$
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	0	1
2	0	0	1	1	0	0	1	0
3	0	0	1	0	0	0	1	1
4	0	1	1	0	0	1	0	0
5	0	1	1	1	0	1	0	1
6	0	1	0	1	0	1	1	0
7	0	1	0	0	0	1	1	1
8	1	1	0	0	1	0	0	0
9	1	1	0	1	1	0	0	1
10	1	1	1	1	1	0	1	0
11	1	1	1	0	1	0	1	1
12	1	0	1	0	1	1	0	0
13	1	0	1	1	1	1	0	1
14	1	0	0	1	1	1	1	0
15	1	0	0	0	1	1	1	1

Row	$A$	$B$	$C$	$D$	$Y_{1}$ from the code table	$Y_{1}$ in the simulator	$Y_{0}$ from the code table	$Y_{0}$ in the simulator
0	0	0	0	0	0	0	0	0
1	0	0	0	1	0	0	1	1
2	0	0	1	0	1	1	1	1
3	0	0	1	1	1	1	0	0
4	0	1	0	0	1	0	1	0
5	0	1	0	1	1	0	0	1
6	0	1	1	0	0	1	0	1
7	0	1	1	1	0	1	1	0
8	1	0	0	0	1	1	1	1
9	1	0	0	1	1	1	0	0
10	1	0	1	0	0	0	0	0
11	1	0	1	1	0	0	1	1
12	1	1	0	0	0	1	0	1
13	1	1	0	1	0	1	1	0
14	1	1	1	0	1	0	1	0
15	1	1	1	1	1	0	0	1