Optimal Routing of a Sailboat in Steady Winds

1
Optimal Routing of a Sailboat in Steady Winds

• Presentation by Nate Greiner
• Advisor Mike Hennessey
• Center for Applied Mathematics
• Tuesday, August 30

2
Sail Boat Kinematics
3
Boat Polar
4
The Dual Simulation

• Wind Variables Speed and Direction
• Vary Direction w/ Constant Wind Speed
• Vary Wind Speed w/ Constant Direction
• Dr. Hennessy has previously researched the first,
  and I am interested in the latter
• Variables with respect to position and heading

5
The Problem

• e 0.05
• w 5.0 Knots
• a p/4
• L 1 nm
• e w 0.25 Knots
• 1

6
VMG and Outer Boundary
7
Two Legged Route
O
C
A
B
When time is Minimized
8
Background Theory Calculus of Variations

• Used to find smooth optimal curves, shapes,
  control efforts, trajectories, etc.
• Variation operator point wise incremental
  changes with t fixed

Neighboring curve
f
f
f
t
9
Typical Variational Calculus Mechanics Formulation

• Find u(t) to minimize J
• Initial conditions given
• Example problem- minimum time

10
Consider augmented J, J
11
dJ (after simplification)
12
Minimum Time Formulation

• given (so )
• and since H is not an explicit function of time
  (i.e. )
• It can be shown that H 0

13
Zermelo Problem Formulationfor 1D Y Field

• Kinematics Equations
• Hamiltonian

14
Euler-Lagrange Equations with Vanishing H

• H 0
• Hamiltonian not explicit function of time

15
Solution
16
1D X Field With Isotropic Polar
17
Shooting Method

• Know initial x and y
• Know final x and y
• Dont know initial ?
• So must iterate on initial ? to make the path go
  through final x and y

Goes through (1,-1)
18
Matlab Code

• M-File
• function zdot zdot(t,z)
• x z(1) y z(2) theta z(3) eps 0.05
  wbar 5.0 L 1.0
• v -wbar (1.0- (1.0 - eps) x / L) vel -v
• wbar and L are given in knots
• zdot vel cos(theta) vel sin(theta) -((1
  - eps) wbar / L) sin(theta)
• Command
• gtgt t,z ode23(_at_zdot, 0.00.0010.7650', 0.0
  0.0 -1.452pi/3')

19
Simulation Results
0.892 Hrs
Y
0.785 Hrs
0.831 Hrs
0.765 Hrs
X
20
Future Research

• Using a more realistic polar
• Make the variation of wind non-linear
• Adding in the effects of current
• Look at the sensitivity of Epsilon
• Both wind speed and direction variable

21
Any Questions?