Single Camera Structure and Motion Estimation - Visual Servoing via Advanced Numerical Methods - page 217

12.4.2.4 Stability Analysis

The stability of the proposed observer is analyzed in this section using a Lyapunov-
based approach. Since the proposed observer uses a projection law, the Lyapunov
analysis is examined for three possible cases of projection.

Theorem 12.2. The observer in (12.50)-(12.54) is asymptotically stable provided
Assumptions 12.1-12.8 are satisfied, and ρ 3 ( t ) , ρ 4 ( t ) and ρ 5 ( t ) are selected ac-

cording to (12.56)-(12.58).

6

Proof. Consider a domain D ⊂ R containing e (0) and a continuously differen-

+ ,definedas

tiable Lyapunov function, V ( e , t ) : D× [0 , ∞ ) → R

1
2 e T e

V ( e ) . (12.76)

0:
After utilizing error dynamics in (12.64), (12.65), (12.69)-(12.72), the time deriva-
tive of (12.76) can be expressed as

Case 1: y 3 ≤ y 3 ( t ) ≤ y 3 or if y 3 ( t ) > y 3 and φ ( t ) ≤ 0orif y 3 ( t ) < y 3 and φ ( t ) ≥

V = e 1 ( e 4 −

y 1 b 3 e 3 − ρ 1 e 1 + Ψ 1 )+ e 2 ( e 5 − y 2 b 3 e 3 − ρ 2 e 2 + Ψ 2 )

+ e 3 (( y 3 + y 3 ) b 3 e 3 +( y 2 ω 1 − y 1 ω 2 ) e 3 + y 2 y 3 ˜ ω 1 − y 1 y 3 ˜ ω 2

− ρ 3 e 3 + y 1 b 3 e 1 + y 2 b 3 e 2 − c 1 b 3 e 6 )

+ e 4 ( b 3 u 1 e 3 + y 3 b 3 e 4 +( y 2 ω 1 −

y 1 ω 2 ) e 4 + y 2 u 1 ˜ y 1 u 1 ˜

ω 1 − ω 2

+ u 1 J 4 e 4 + q ( u 1 ) e 4 − ρ 4 ( e 4 + Ψ 1 ) − e 1 )

y 1 ω 2 ) e 5 + y 2 u 2 ˜ y 1 u 2 ˜

+ e 5 ( b 3 u 2 e 3 + y 3 b 3 e 5 +( y 2 ω 1 − ω 1 − ω 2

+ u 2 J 5 e 5 + q ( u 2 ) e 5 − ρ 5 ( e 5 + Ψ 2 ) − e 2 )

+ e 6 ( c 1 b 3 e 3 − ρ 6 e 6 ) .

Using the facts that

1

2 e 3 + 1

1

2 e 3 + 1 2 e 4 , 2 e 5

| e 3 || e 4 |≤ | e 3 || e 5 |≤

along with the bounds on the terms u 1 ( t ), u 2 ( t ), J 4 , J 5 , and re-arranging the terms,
the following inequality for V ( t ) can be developed

V

≤− ρ 1 e 1 − ρ 2 e 2 − ρ 6 e 6
−

3 ) e 3

− − −| | −| |

( ( y 3 + y 3 ) b 3 y 2 ω 1 + y 1 ω b 3 u 1 b 3 u 2 + ρ 2

J 4

4 ) e 4

− − − − − −| |

( y 3 b 3 y 2 ω 1 + y 1 ω u 1 p ( u 1 ) b 3 u 1 + ρ 2

J 5

5 ) e 5

− ( − y 3 b 3 − y 2 ω

1 + y 1

ω − u 2 − p ( u 2 ) −| b 3 | u 2 + ρ

2

+( e 1 − ρ

4 e 4 )

Ψ

1 +( e 2

− ρ

5 e 5 )

Ψ

2

+( y 2 y 3 e 3 + y 2 u 1 e 4 + y 2 u 2 e 5 ) ˜ ω 1 − ( y 1 y 3 e 3 + y 1 u 1 e 4 + y 1 u 2 e 5 ) ˜ ω 2 .