A simple controller for the line following of sailboats

A simple controller for the line following of sailboats

L. Jaulin and F. Le Bars

ENSTA-Bretagne, IFREMER, Brest.

LabSTICC, IHSEV, OSM.

Controller of a sailboat robot

1 Line following

Heading controller













δ_r =





δ^max_r .sin θ − θ¯ if cos θ − ¯θ ≥ 0 δ^max_r .sign sin θ − ¯θ otherwise

δ^max_s = ^π₂. ^cos(^ψ−¯θ)⁺¹

2

q

.

Rudder δ_r =





δ^max_r .sin θ − θ¯ if cos θ − ¯θ ≥ 0 δ^max_r .sign sin θ − ¯θ otherwise

Sail

δ^max_s = π 2 ·



cos ψ − ¯θ + 1 2



 q

with q = log _2β^π log (2)

2 Vector field

Nominal vector field: θ^∗ = ϕ − 2.γ_∞

π .atan e r .

A course θ^∗ may be unfeasible

θ^∗ = −^2.γ_π^∞.atan ^e_r

Keep close hauled strategy.

http://youtu.be/pHteidmZpnY

3 Controller

Controller ¯θ (m_,a_,b_{, ψ, γ∞, r, ζ)} 1 e = det ^b_b⁻₋^a_a ,m ₋ a

2 ϕ =atan2(b ₋ a₎

3 θ^∗ = ϕ − ^2.γ_π^∞.atan ^e_r

4 if cos (ψ − θ^∗) + cosζ < 0

5 or (|e| < r and (cos(ψ − ϕ) + cos(ζ) < 0)) 6 then ¯θ = π + ψ − ζ.sign(e);

7 else ¯θ = θ^∗; 8 end

Without hysteresis

Controller in: m_{, θ, ψ,}a_,b_{; out: δr, δ}^max

s ; inout: q 1 e = det ^b_b⁻₋^a_a ,m ₋ a

2 if |e| > ₂^r then q = sign(e) 3 ϕ = atan2(b ₋ a₎

4 θ^∗ = ϕ − ^2.γ_π^∞.atan _r^e

5 if cos (ψ − θ^∗) + cosζ < 0

6 or (|e| < r and (cos(ψ − ϕ) + cos ζ < 0)) 7 then ¯θ = π + ψ − q.ζ.

8 else ¯θ = θ^∗

9 end

10 if cos θ − ¯θ ≥ 0 then δ_r = δ^max_r .sin θ − ¯θ 11 else δ_r = δ^max_r .sign sin θ − ¯θ

12 δ^max_s = ^π₂. ^cos(^ψ−¯θ)⁺¹

2

q

.

With hysteresis

4 Validation by simulation

5 Theoretical validation

When the wind is known, the sailboat with the heading controller is described by

˙

x ₌ f ₍x_{) .}

The system

˙

x ₌ f ₍x₎

is Lyapunov-stable (1892) is there exists V (x_{) ≥ 0 such} that

V˙ (x_{) < 0 if} x ₌ 0_. V (x_{) = 0 iff} x ₌ 0

Definition. Consider a differentiable function V (x_{) :} R^{n →} R. The system is V -stable^∗ if

V (x_{) ≥ 0 ⇒} V˙ (x_{) < 0 .}

∗Jaulin, Le Bars (2012). An interval approach for stability analysis; Application to

sailboat robotics. IEEE TRO.

Theorem. If the system is V -stable then

(i) ∀x _{(0),∃t ≥ 0 such that V (}x _{(t)) < 0}

(ii) if V (x _{(t)) < 0 then ∀τ > 0, V (}x _{(t + τ)) < 0.}

When some parameters are unknown but bounded, the state equation

˙

x ₌ f ₍x₎ becomes a differential inclusion

˙

x _∈ F₍x_). where F is a set-membership function.

.

6 Experimental validation

7 Conclusion

It is possible for a sailboat robot to move inside a corridor.

Essential, to create circulation rules when robot swarms are considered.

Essential to determine the responsible robot in case of ac- cident.