ACS6121 - Unit 02 (Network Robotics)

Question 1

What is a graph in the context of networked robotics?

Answer 1

A mathematical structure representing robots as nodes and their communication links as edges.

Question 2

Define a spanning tree.

Answer 2

A sub-graph that connects all nodes without forming cycles.

Question 3

What does the Laplacian matrix 𝐿 represent?

Answer 3

L = D_out − A; it captures network structure and is used in consensus analysis.

Question 4

What is consensus in multi-agent systems?

Answer 4

The process where agents adjust their states to agree on a common value over time.

Question 5

How is consensus achieved in a network?

Answer 5

Using the control law
𝑢_𝑖 = ∑(𝑥_𝑗 − 𝑥_𝑖), which leads to 𝑥˙ = − 𝐿𝑥.

Question 6

What is the significance of the second smallest eigenvalue 𝜆_2 of the Laplacian?

Answer 6

If 𝜆_2>0, the graph is connected, ensuring convergence to consensus.

Question 7

What is a strongly connected digraph?

Answer 7

A directed graph where there is a path from any node to every other node.

Question 8

What is the adjacency matrix used for?

Answer 8

To represent which nodes are connected and how (binary or weighted connections).

Question 9

Give one real-world application of consensus protocols.

Answer 9

Autonomous vehicle platooning – keeping equal distances and same speed.

Question 10

What kind of graph ensures each node is reachable by one unique directed path from a root?

Answer 10

A directed tree.

Question 11

1. Multiple Choice:
   What does a Laplacian matrix help analyze in networked robotics?
   A) Sensor accuracy
   B) Communication speed
   C) Graph connectivity and consensus
   D) CPU usage

Answer 11

→ Correct Answer: C) Graph connectivity and consensus

Question 12

2. True/False:
   A graph is connected if 𝜆_2 = 0.

Answer 12

→ False (connected if 𝜆_2>0)

Question 13

3. Fill in the Blank:
   A spanning tree is a connected, acyclic subgraph that includes __________.

Answer 13

→ All nodes of the original graph

Question 14

4. Short Answer:
   Why is strong connectivity important in a directed graph?

Answer 14

→ It ensures that each node can reach every other node, which is necessary for global coordination.

Question 15

5. Multiple Choice:
   Which of the following matrices is diagonal?
   A) Adjacency matrix
   B) Laplacian matrix
   C) Out-degree matrix
   D) Weighted matrix

Answer 15

→ Correct Answer: C) Out-degree matrix