Post on 26-Mar-2018
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
‖ ( ) ‖
******************************************************************************
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.
******************************************************************************
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.
******************************************************************************
Then
≤
so it is locally exponentially stable!!