Page 103 - Introduction to Autonomous Mobile Robots

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hence it is locally exponentially stable if the eigenvalues of the matrix Chapter 3

k – ρ 0 0
A = 0 – ( k – ) k– β (3.62)
k
α
ρ
0 k – ρ 0
A
all have a negative real part. The characteristic polynomial of the matrix is

( λ + k ) λ +( 2 λ k – k ) – k k ) (3.63)
(
ρ
ρ β
ρ
α
and all roots have negative real part if
k > 0 ; k– β > 0 ; k – k > 0 (3.64)
ρ
α
ρ
which proves the claim.
For robust position control, it might be advisable to apply the strong stability condition,
which ensures that the robot does not change direction during its approach to the goal:

---k >
k > 0 ; k < 0 ; k + 5 β 2 0 (3.65)
---k –
β
α
ρ
3 π ρ
This implies that α ∈ I 1 for all t, whenever α 0() ∈ I 1 and α ∈ I 2 for all , whenever
t
α 0() ∈ I respectively. This strong stability condition has also been verified in applica-
2
tions.