The idea of rigid formation control based on distance constraints is very simple but very

important!

Notation:

: nodes

: edges

: distance constraint, the length of .

Goal:

‖ ‖

Procedure:

Rigidity function:

( ) [‖ ‖

‖ ‖

]

Define the error vector as

( )

The goal is to make . Define the Lyapunov function as

‖ ( ) ‖

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𝑉 (𝑟(𝑒) 𝑑)𝑇𝑟 (𝑒)

(𝑟(𝑒) 𝑑)𝑇𝑟 (𝑒)

𝜓𝑇 𝜕𝑟(𝑒)

𝜕𝑒 𝑒

Choose

𝑒 𝜕𝑟(𝑒)

𝜕𝑒

𝑇

𝜓

Then

𝑉 𝜓𝑇 𝜕𝑟(𝑒)

𝜕𝑒

𝜕𝑟(𝑒)

𝜕𝑒

𝑇

𝜓 ≤

Rigidity matrix:

( )

( )

( )

Rigidity matrix is important, especially its rank. For a rigid graph, ( ) . For a

minimally rigid graph (edge number=2n-3), ( ) is by . Then ( ) is of full rank. (see

2012IJRNC,2009ECC).

So when approaching to a rigid formation, 𝜕𝑟(𝑒)

𝜕𝑒

𝜕𝑟(𝑒)

𝜕𝑒 𝑇

is a square matrix and of full rank.

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In the above, I made a classic mistake.

( ( ) ) ( )

So choose

(

)

If we choose ( )

, usually it is difficult to get . After all, we need . Then

(

)

And

is the rigid matrix, instead of ( )

. So when approaching to a rigid formation,

is a

square matrix and of full rank.

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Then

≤

so it is locally exponentially stable!!