M.LeGallic,J.Tillet,F.LeBars,L.Jaulin TightSlalomControlforSailboatRobots

Tight Slalom Control for Sailboat Robots

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin

IRSC 2018, Southampton September 1, 2018

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Tight slalom

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

wind

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

We consider a mobile robot [1]

_ x = f ( x , u ) p = g ( x )

with an input vector u = (u ₁ , . . . , u _m ) and a pose vector p = (p ₁ , . . . ,p _{m+ 1} ) .

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Control m + 1 state variables and not only m of them.

Perform a path following instead of a trajectory tracking.

The controller makes p collinear (instead of equal) to the required ˙ eld.

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Method

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Dubins car: 





˙

x ₁ = cos x ₃

˙

x ₂ = sin x ₃

˙

x ₃ = u

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Output

e = x ₃ + atanx ₂ . Thus

˙

e = ˙ x ₃ + x ˙ ₂

1 + x ₂ ² = u + sin x ₃ 1 + x ₂ ² . We want

˙

e + e = 0.

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

We get:

u = −x ₃ − atan x ₂ − ^{sin x}

1 +x

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Generalization

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

We want to follow the eld ψ( p ) .

This can be translated into ϕ (ψ (p), p) = ˙ 0, where ϕ ( r , s ) = 0 ⇔ ∃λ > 0 , λ r = s .

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

We dene

e = ϕ (ψ ( p ), _ p ) = ϕ

ψ( g ( x )), ∂ g

∂ x ( x ) · f ( x , u )

. Since dime = dimu = m , we can make e → 0.

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Van der Pol cycle

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

The car has to follow the limit cycle of the Van der Pol equation:

ψ ( p ) =

p ₂

− 0 . 01 p ₁ ² − 1 p ₂ − p ₁

.

We take g ( x ) = (x ₁ ,x ₂ ) ^T to build paths in the (x ₁ ,x ₂ ) -space.

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

We get that nal controller is u = − sawtooth

x ₃ − atan2

− _x

100

− 1

x ₂ − x ₁ ,x ₂

+

1001

− 1

x

+x

· sin x

+x

·

x1x2 cosx3

50

+

1001

− 1

sin x

+ cos x

x

+

1001

− 1

x

+x

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Controller

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

We consider sailboat [2]:



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

˙

x ₁ = v cos θ

˙

x ₂ = v sin θ θ ˙ = −ρ ₂ v sin2 u ₁

˙

v = ρ ₃ k w _ap k sin (δ _s − ψ _ap ) sin δ _s − ρ ₁ v ² σ = cosψ _ap + cos u ₂

δ s =

π + ψ _ap if σ ≤ 0

− sign ( sin ψ _ap ) ·u ₂ otherwise w _ap =

−a sin (θ) − v

−a cos (θ)

ψ _ap = angle w _ap

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

The path to follow is

e ( p ) = 10sin p ₁ 10

− p ₂ = 0 where e (p) is the error.

We want e ˙ = − 0 . 1 e .

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

Thus

cos p ₁ 10

˙ p ₁ − p ˙ ₂

| {z }

˙ e( p )

= − 1 10

10sin p ₁ 10

− p ₂

| {z }

e( p )

We take p ˙ ₁ = 1, to go to the right. Thus:

ψ ( p ) = p ˙ ₁

˙ p ₂

=

1

cos ^p ₁₀

+ ₁₀ ¹ 10sin ^p ₁₀

− p ₂

(1) which is attracted by the curve p ₂ = 10sin ^p ₁₀

.

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

The controller is

u ₁ = − ¹ ₂ arcsin tanh

ω ˆ ρ

v

ω ˆ = −( sawtooth( θ − atan2 (b, 1 )) + _{1 +b} ^{b ˙}

b ˙ = ₁₀ ¹ cos x ₃ · cos ^x ₁₀

− sin ₁₀ ^x

− ₁₀ ¹ sin x ₃ b = cos ^x ₁₀

+ sin ^x ₁₀

− ₁₀ ¹ x ₂

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

wind

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots

L. Jaulin.

Automation for Robotics.

ISTE editions, 2015.

L. Jaulin and F. Le Bars.

An Interval Approach for Stability Analysis; Application to Sailboat Robotics.

IEEE Transaction on Robotics, 27(5), 2012.

M. Le Gallic, J. Tillet, F. Le Bars, L. Jaulin Tight Slalom Control for Sailboat Robots