Weather routing
The weather routing problem has the following parts:
- a predicted surface wind direction and speed, at increments of longitude, latitude, and time
- an expected surface current direction and speed, at increments of longitude, latitude, and time
- 'polar data' describing maximum speed of a sailboat at points of sail for a given speed of wind over water
- regions for sailing (the open ocean) and not (the land, shallows, restricted areas, etc.)
- a starting location and time, and a destination
Given the above information and a specific path, progress and arrival time are determined. The weather routing problem, conversely, is to determine the path which results in the earliest arrival time.
Go
This runs in only 37 seconds which is surprisingly quick compared to Julia. However, I've just noticed that I'm using an out of date version of Julia (1.0.4) so hopefully the latest version will be able to close the gap. <lang go>package main
import (
"fmt" "io/ioutil" "log" "math" "strconv" "strings"
)
type matrixF = [][]float64 type pred = func(float64) bool
/*
Structure that represents a polar CSV file's data. Note 0 degrees is directly into the wind, 180 degrees is directly downwind.
- /
type SailingPolar struct {
winds []float64 // vector of windspeeds degrees []float64 // vector of angles in degrees of direction relative to the wind speeds matrixF // matrix of sailing speeds indexed by wind velocity and angle of boat to wind
}
/*
Structure that represents wind and surface current direction and velocity for a given position. Angles in degrees, velocities in knots.
- /
type SurfaceParameters struct{ windDeg, windKts, currentDeg, currentKts float64 }
// Checks for fatal errors. func check(err error) {
if err != nil { log.Fatal(err) }
}
// Reads a sailing polar CSV file and returns a SailingPolar struct containing the file data. // A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma. // The first line of file contains labels for the wind velocities that make up columns, and // the first entry of each row makes up a column of angle of sailing direction from wind in degrees. func getPolarData(fileName string) *SailingPolar {
content, err := ioutil.ReadFile(fileName) check(err) lines := strings.Split(string(content), "\n") header := strings.Split(lines[0], ";") var winds, degrees []float64 var speeds matrixF for _, col := range header[1:] { t, err := strconv.ParseFloat(col, 64) check(err) winds = append(winds, t) } for _, line := range lines[1:] { if line == "" { break // ignore final blank line if there is one } cols := strings.Split(line, ";") f, err := strconv.ParseFloat(cols[0], 64) check(err) degrees = append(degrees, f) var temp []float64 for _, col := range cols[1:] { t, err := strconv.ParseFloat(col, 64) check(err) temp = append(temp, t) } speeds = append(speeds, temp) } return &SailingPolar{winds, degrees, speeds}
}
const R = 6372800.0 // Earth's approximate radius in meters
/* various helper methods which work with degrees rather than radians. */
// Converts degrees to radians. func deg2Rad(deg float64) float64 { return math.Mod(deg*math.Pi/180+2*math.Pi, 2*math.Pi) }
// Converts radians to degrees. func rad2Deg(rad float64) float64 { return math.Mod(rad*180/math.Pi+360, 360) }
// Trig functions. func sind(d float64) float64 { return math.Sin(deg2Rad(d)) } func cosd(d float64) float64 { return math.Cos(deg2Rad(d)) } func asind(d float64) float64 { return rad2Deg(math.Asin(d)) } func atand(x, y float64) float64 { return rad2Deg(math.Atan2(x, y)) }
// Calculates the haversine function for two points on the Earth's surface. // Given two latitude, longitude pairs in degrees for a point on the Earth, // get distance in meters and the initial direction of travel in degrees for // movement from point 1 to point 2. func haversine(lat1, lon1, lat2, lon2 float64) (float64, float64) {
dlat := lat2 - lat1 dlon := lon2 - lon1 a := math.Pow(sind(dlat/2), 2) + cosd(lat1)*cosd(lat2)*math.Pow(sind(dlon/2), 2) c := 2 * asind(math.Sqrt(a)) theta := atand(sind(dlon)*cosd(lat2), cosd(lat1)*sind(lat2)-sind(lat1)*cosd(lat2)*cosd(dlon)) theta = math.Mod(theta+360, 360) return R * c * 0.5399565, theta
}
// Returns the index of the first element of 'a' for which 'p' returns true or -1 otherwise. func findFirst(a []float64, p pred) int {
for i, e := range a { if p(e) { return i } } return -1
}
// Returns the index of the last element of 'a' for which 'p' returns true or -1 otherwise. func findLast(a []float64, p pred) int {
for i := len(a) - 1; i >= 0; i-- { if p(a[i]) { return i } } return -1
}
// Calculate the expected sailing speed in a specified direction in knots, // given sailing polar data, a desired point of sail in degrees, and wind speed in knots. func boatSpeed(sp *SailingPolar, pointOfSail, windSpeed float64) float64 {
winds := sp.winds degrees := sp.degrees speeds := sp.speeds udeg := findLast(degrees, func(t float64) bool { return t <= pointOfSail }) odeg := findFirst(degrees, func(t float64) bool { return t >= pointOfSail }) uvel := findLast(winds, func(t float64) bool { return t <= windSpeed }) ovel := findFirst(winds, func(t float64) bool { return t >= windSpeed }) if udeg == -1 || odeg == -1 || uvel == -1 || ovel == -1 { return -1 } var frac float64 switch { case odeg == udeg && uvel == ovel: frac = 1 case odeg == udeg: frac = (windSpeed - winds[uvel]) / (winds[ovel] - winds[uvel]) case uvel == ovel: frac = (pointOfSail - degrees[udeg]) / (degrees[odeg] - degrees[udeg]) default: frac = ((pointOfSail-degrees[udeg])/(degrees[odeg]-degrees[udeg]) + (windSpeed-winds[uvel])/(winds[ovel]-winds[uvel])) / 2 } return speeds[udeg][uvel] + frac*(speeds[odeg][ovel]-speeds[udeg][uvel])
}
// Calculates the expected net boat speed in a desired direction versus the wind ('azimuth'). // This is generally different from the actual boat speed in its actual direction. // Directions are in degrees ('pointos' is point of sail the ship direction from the wind), // and velocity of wind ('ws') is in knots. func sailingSpeed(sp *SailingPolar, azimuth, pointos, ws float64) float64 {
return boatSpeed(sp, pointos, ws) * cosd(math.Abs(pointos-azimuth))
}
// Calculates the net direction and velocity of a sailing ship. // Arguments are sailing polar data, direction of travel in degrees from north, wind direction in // degrees from north, wind velocity in knots, surface current direction in degrees, and // current velocity in knots. func bestVectorSpeed(sp *SailingPolar, dirTravel, dirWind, windSpeed, dirCur, velCur float64) (float64, float64) {
azimuth := math.Mod(dirTravel-dirWind, 360) if azimuth < 0 { azimuth += 360 } if azimuth > 180 { azimuth = 360 - azimuth } vmg := boatSpeed(sp, azimuth, windSpeed) other := -1.0 idx := -1 for i, d := range sp.degrees { ss := sailingSpeed(sp, azimuth, d, windSpeed) if ss > other { other = ss idx = i } } if other > vmg { azimuth = sp.degrees[idx] vmg = other } dirChosen := deg2Rad(dirWind + azimuth) wx := vmg * math.Sin(dirChosen) wy := vmg * math.Cos(dirChosen) curX := velCur * math.Sin(deg2Rad(dirCur)) curY := velCur * math.Cos(deg2Rad(dirCur)) return rad2Deg(math.Atan2(wy+curY, wx+curX)), math.Sqrt(math.Pow(wx+curX, 2) + math.Pow(wy+curY, 2))
}
// Calculates the trip time in minutes from (lat1, lon1) to the destination (lat2, lon2). // Uses the data in SurfaceParameters for wind and current velocity and direction. func sailSegmentTime(sp *SailingPolar, p SurfaceParameters, lat1, lon1, lat2, lon2 float64) float64 {
distance, dir := haversine(lat1, lon1, lat2, lon2) _, vel := bestVectorSpeed(sp, dir, p.windDeg, p.windKts, p.currentDeg, p.currentKts) // minutes/s * m / (knots * (m/s / knot)) = minutes return (1.0 / 60.0) * distance / (vel * 1.94384)
}
/* Structure that represents a point in 2-D space. */ type Point2 struct{ x, y int }
func (p Point2) add(p2 Point2) Point2 { return Point2{p.x + p2.x, p.y + p2.y} } func (p Point2) equals(p2 Point2) bool { return p.x == p2.x && p.y == p2.y } func (p Point2) String() string { return fmt.Sprintf("[%d, %d]", p.x, p.y) }
/*
Structure that consists of a tuple of latitude and longitude in degrees. NB: This uses latitude (often considered to be y) first then longitude (often considered to be x). This latitude, then longitude ordering is as per ISO 6709 (en.wikipedia.org/wiki/ISO_6709).
- /
type Position struct{ lat, lon float64 }
/* Structure that represents a Position with the SurfaceParameters of wind and current at the Position. */ type GridPoint struct {
pt Position sp SurfaceParameters
} type MatrixG = [][]*GridPoint
/*
Type alias for a matrix of GridPoints, each Position point with their SurfaceParameters. A Vector of TimeSlice can give the surface characteristics for an ocean region over time.
- /
type TimeSlice = MatrixG
/* Structure that represents a routing problem. */ type RoutingProblem struct {
timeInterval float64 // the minutes duration for each TimeSlice timeFrame []TimeSlice // a vector of sequential timeslices for the ocean region obstacleIndices []Point2 // the Cartesian indices in each TimeSlice for // obstacles, such as land or shoals, where the ship may not go startIndex int // the TimeSlice position for time of starting start Point2 // starting location on grid of GridPoints finish Point2 // destination / finish location on grid of GridPoints allowRepeatVisits bool // whether the vessel may overlap its prior path, usually false
}
/* Structure that represents a timed path. */ type TimedPath struct {
duration float64 // minutes total to travel the path path []Point2 // vector of Cartesian indices of points in grid for path to travel
}
func (t TimedPath) String() string { return fmt.Sprintf("%g %v", t.duration, t.path) } func (t TimedPath) equals(t2 TimedPath) bool { return t.String() == t2.String() }
func findMin(a []float64) (float64, int) {
min := a[0] idx := 0 for i, e := range a[1:] { if e < min { min = e idx = i + 1 } } return min, idx
}
var ntuples = [][2]int{{-1, -1}, {-1, 0}, {-1, 1}, {0, -1}, {0, 1}, {1, -1}, {1, 0}, {1, 1}} var neighbors = make([]Point2, len(ntuples))
func init() {
for i := 0; i < len(ntuples); i++ { neighbors[i] = Point2{ntuples[i][0], ntuples[i][1]} }
}
func contains(points []Point2, point Point2) bool {
for _, p := range points { if p.equals(point) { return true } } return false
}
// Returns a list of points surrounding 'p' which are not otherwise excluded. func surround(p Point2, mat TimeSlice, excluded []Point2) []Point2 {
xmax := len(mat) ymax := len(mat[0]) var res []Point2 for _, x := range neighbors { q := x.add(p) if (0 <= q.x && q.x < xmax) && (0 <= q.y && q.y < ymax) && !contains(excluded, q) { res = append(res, q) } } return res
}
// Get the route (as a TimedPath) that minimizes time from start to finish for a given // RoutingProblem (sea parameters) given sailing polar data (ship parameters). func minimumTimeRoute(rp *RoutingProblem, sp *SailingPolar, verbose bool) *TimedPath {
timedPaths := []*TimedPath{&TimedPath{0, []Point2{rp.start}}} completed := false minPath := &TimedPath{1000, []Point2{}} for i := 0; i < 1000; i++ { var newPaths []*TimedPath if verbose { fmt.Printf("Checking %d paths of length %d\n", len(timedPaths), len(timedPaths[0].path)) } for _, tpath := range timedPaths { le := len(tpath.path) if tpath.path[le-1] == rp.finish { completed = true newPaths = append(newPaths, tpath) } else { p1 := tpath.path[le-1] num := int(math.Round(tpath.duration)) den := int(math.Round(rp.timeInterval)) slice := rp.timeFrame[(num/den)%len(rp.timeFrame)] for _, p2 := range surround(p1, slice, rp.obstacleIndices) { if !rp.allowRepeatVisits && contains(tpath.path, p2) { continue } gp1 := slice[p1.x][p1.y] gp2 := slice[p2.x][p2.y] lat1 := gp1.pt.lat lon1 := gp1.pt.lon lat2 := gp2.pt.lat lon2 := gp2.pt.lon t := sailSegmentTime(sp, gp1.sp, lat1, lon1, lat2, lon2) path := make([]Point2, len(tpath.path)) copy(path, tpath.path) path = append(path, p2) newPaths = append(newPaths, &TimedPath{tpath.duration + t, path}) } } } set := make(map[string]*TimedPath) for _, np := range newPaths { set[np.String()] = np } timedPaths = timedPaths[:0] for k := range set { timedPaths = append(timedPaths, set[k]) } if completed { var durations []float64 for _, x := range timedPaths { durations = append(durations, x.duration) } minDur, _ := findMin(durations) var finished []*TimedPath for _, x := range timedPaths { le := len(x.path) if x.path[le-1] == rp.finish { finished = append(finished, x) } } durations = durations[:0] for _, x := range finished { durations = append(durations, x.duration) } minFinDur, idx := findMin(durations) if verbose { fmt.Printf("Current finished minimum: %v, others %v\n", finished[idx], minDur) } if minDur == minFinDur { minPath = finished[idx] break } } } return minPath
}
/*
The data is selected so the best time path is slightly longer than the shortest length path. The forbidden regions are x, representing land or reef. The allowed sailing points are . and start and finish are S and F.
x . . F . . x . x . . . . . . . x x x . . x x x . . . . . x x x x . x x x . . . x x . x . x . . . x x . x . . . . . x . . x . x . . . . . . x . . . . S . . . . .
- /
// These need to be changed to 0-based for Go. var ftuples = [][2]int{
{1, 8}, {2, 1}, {2, 8}, {3, 5}, {3, 8}, {4, 1}, {4, 5}, {4, 6}, {4, 8}, {5, 1}, {5, 5}, {5, 6}, {5, 8}, {6, 3}, {6, 4}, {6, 5}, {6, 6}, {6, 8}, {6, 9}, {7, 1}, {7, 4}, {7, 5}, {7, 6}, {8, 8}, {8, 9}, {9, 1}, {9, 7}, {9, 9},
}
var forbidden = make([]Point2, len(ftuples))
func init() {
for i := 0; i < len(ftuples); i++ { forbidden[i] = Point2{ftuples[i][0] - 1, ftuples[i][1] - 1} }
}
// Create regional wind patterns on the map. func surfaceByLongitude(lon float64) SurfaceParameters {
switch { case lon < -155.03: return SurfaceParameters{-5, 8, 150, 0.5} case lon < -155.99: return SurfaceParameters{-90, 20, 150, 0.4} default: return SurfaceParameters{180, 25, 150, 0.3} }
}
// Vary wind speeds over time. func mutateTimeSlices(slices []TimeSlice) {
i := 1 for _, slice := range slices { for j := 0; j < len(slice); j++ { for k := 0; k < len(slice[j]); k++ { x := slice[j][k] x.sp = SurfaceParameters{x.sp.windDeg, x.sp.windKts * (1 + 0.002*float64(i)), x.sp.currentDeg, x.sp.currentKts} } } i++ }
}
func main() {
startPos := Point2{0, 3} // 0-based endPos := Point2{8, 3} // ditto slices := make([]MatrixG, 200) for s := 0; s < 200; s++ { gpoints := make([][]*GridPoint, 9) for i := 0; i < 9; i++ { gpoints[i] = make([]*GridPoint, 9) for j := 0; j < 9; j++ { pt := Position{19.78 - 1.0/60.0 + float64(i)/60, -155.0 - 5.0/60.0 + float64(j)/60} gpoints[i][j] = &GridPoint{pt, surfaceByLongitude(pt.lon)} } } slices[s] = gpoints } mutateTimeSlices(slices) routeProb := &RoutingProblem{10, slices, forbidden, 0, startPos, endPos, false} fileName := "polar.csv" sp := getPolarData(fileName) tp := minimumTimeRoute(routeProb, sp, false) fmt.Println("The route taking the least time found was:\n", tp.path, "\nwhich has duration", int(tp.duration/60), "hours,", int(math.Round(math.Mod(tp.duration, 60))), "minutes.")
}</lang>
- Output:
The route taking the least time found was: [[0, 3] [0, 4] [1, 5] [2, 6] [3, 6] [4, 6] [5, 6] [6, 6] [7, 5] [7, 4] [8, 3]] which has duration 4 hours, 44 minutes.
Julia
Brute force optimization search, practical for shorter path lengths, but would require a better algorithm for paths over twice this size. <lang julia>module SailingPolars
using DelimitedFiles
export SailingPolar, SurfaceParameters, getpolardata, deg2rad, rad2deg, cartesian2polar export polar2cartesian, haversine, inverse_haversine, boatspeed, bestvectorspeed export sailingspeed, sailsegmenttime
"""
Structure to represent a polar CSV file's data.
Contains a matrix, speeds, of sailing speeds indexed by wind velocity and angle of boat to wind winds is a list of wind speeds degrees is a list of angles in degrees of direction relative to the wind Note 0.0 degrees is directly into the wind, 180 degrees is directly downwind. """ struct SailingPolar
winds::Vector{Float32} degrees::Vector{Float32} speeds::Matrix{Float32} # speeds[wind direction degrees, windspeed knots]
end
"""
struct SurfaceParameters
Structure that represents wind and surface current direction and velocity for a given position Angles in degrees, velocities in knots """ struct SurfaceParameters
winddeg::Float32 windkts::Float32 currentdeg::Float32 currentkts::Float32
end
""" function getpolardata(filename)
Read a sailing polar CSV file and return a SailingPolar containing the file data.
A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma. The first line of file contains labels for the wind velocities that make up columns, and the first entry of each row makes up a column of angle of sailing direction from wind in degrees """ function getpolardata(filename)
datacells, headercells = readdlm(filename, ';', header=true) winds = map(x -> parse(Float32, x), headercells[2:end]) degrees = datacells[:, 1] speeds = datacells[:, 2:end] return SailingPolar(winds, degrees, speeds)
end
const R = 6372800 # Earth's approximate radius in meters
"""
deg2rad(deg)
Convert degrees to radians """ deg2rad(deg) = (deg * π / 180.0 + 2π) % 2π
"""
rad2deg(rad)
Convert radians to degrees """ rad2deg(rad) = (rad * (180.0 / π) + 360.0) % 360.0
"""
cartesian2polard(x, y)
Convert x, y coordinates to polar coordinates with angle in degrees """ cartesian2polard(x, y) = sqrt(x * x + y * y), atand(x, y)
"""
polard2cartesian(r, deg)
Convert polar coordinates in degrees to cartesian x, y coordinates """ polard2cartesian(r, deg) = r .* sincosd(deg)
"""
function haversine(lat1, lon1, lat2, lon2)
Calculate the haversine function for two points on the Earth's surface.
Given two latitude, longitude pairs in degrees for a point on the Earth, get distance in meters and the initial direction of travel in degrees for movement from point 1 to point 2. """ function haversine(lat1, lon1, lat2, lon2)
dlat = lat2 - lat1 dlon = lon2 - lon1 a = sind(dlat / 2)^2 + cosd(lat1) * cosd(lat2) * sind(dlon / 2)^2 c = 2.0 * asind(sqrt(a)) theta = atand(sind(dlon) * cosd(lat2), cosd(lat1) * sind(lat2) - sind(lat1) * cosd(lat2) * cosd(dlon)) theta = (theta + 360) % 360 return R * c * 0.5399565, theta
end
"""
function inverse_haversine(lat1, lon1, distance, direction)
Calculate an inverse haversine function.
Takes the point of origin in degrees (latitude, longitude), distance in meters, and initial direction in degrees, and returns the latitude and longitude of the endpoint in degrees after traveling the specified distance. """ function inverse_haversine(lat1, lon1, distance, direction)
lat2 = asind(sind(lat1) * cos(distance / R) + cosd(lat1) * sin(distance / R) * cosd(direction)) lon2 = lon1 + atand(sind(direction) * sin(distance / R) * cosd(lat1), cos(distance / R) - sind(lat1) * sind(lat2)) return lat2, lon2
end
"""
function boatspeed(sp::SailingPolar, pointofsail, windspeed)
Calculate the expected sailing speed in a specified direction in knots, given sailing polar data, a desired point of sail in degrees, and wind speed in knots """ function boatspeed(sp::SailingPolar, pointofsail, windspeed)
winds, degrees, speeds = sp.winds, sp.degrees, sp.speeds udeg = findlast(t -> t <= pointofsail, degrees) odeg = findfirst(t -> t >= pointofsail, degrees) uvel = findlast(t -> t <= windspeed, winds) ovel = findfirst(t -> t >= windspeed, winds) if any(t -> t == nothing, [udeg, odeg, uvel, ovel]) return -1.0 end frac = (odeg == udeg && uvel == ovel) ? 1.0 : (odeg == udeg) ? (windspeed - winds[uvel]) / (winds[ovel] - winds[uvel]) : (uvel == ovel) ? (pointofsail - degrees[udeg]) / (degrees[odeg] - degrees[udeg]) : ((pointofsail - degrees[udeg]) / (degrees[odeg] - degrees[udeg]) + (windspeed - winds[uvel]) / (winds[ovel] - winds[uvel])) / 2 return speeds[udeg, uvel] + frac * (speeds[odeg, ovel] - speeds[udeg, uvel])
end
"""
sailingspeed(sp::SailingPolar, azimuth, pointos, ws)
Calculate the expected net boat speed in a desired direction versus the wind (azimuth). This is generally different from the actual boat speed in its actual direction. Directions are in degrees (pointos is point of sail, the ship direction from wind), and velocity of wind (ws) is in knots. """ sailingspeed(sp, azimuth, pointos, ws) = boatspeed(sp, pointos, ws) * cosd(abs(pointos - azimuth))
"""
function bestvectorspeed(sp::SailingPolar, dirtravel, dirwind, windspeed, dircur, velcur)
Calculate the net direction and velocity of a sailing ship.
Arguments are sailing polar data, direction of travel in degrees from north, wind direction in degrees from north, wind velocity in knots, surface current direction in degrees, and current velocity in knots. """ function bestvectorspeed(sp::SailingPolar, dirtravel, dirwind, windspeed, dircur, velcur)
azimuth = (dirtravel - dirwind) % 360.0 azimuth = azimuth < 0 ? azimuth + 360.0 : azimuth azimuth = azimuth > 180.0 ? 360.0 - azimuth : azimuth VMG = boatspeed(sp, azimuth, windspeed) other, idx = findmax([sailingspeed(sp, azimuth, x, windspeed) for x in sp.degrees]) if other > VMG azimuth = sp.degrees[idx] VMG = other end dirchosen = deg2rad(dirwind + azimuth) wx, wy = VMG * sin(dirchosen), VMG * cos(dirchosen) curx, cury = velcur * sin(deg2rad(dircur)), velcur * cos(deg2rad(dircur)) return rad2deg(atan(wy + cury, wx + curx)), sqrt((wx + curx)^2 + (wy + cury)^2)
end
"""
function sailsegmenttime(sp::SailingPolar, p::SurfaceParameters, lat1, lon1, lat2, lon2)
Calculate the trip time in minutes from (lat1, lon1) to the destination (lat2, lon2). Uses the data in SurfaceParameters for wind and current velocity and direction. """ function sailsegmenttime(sp::SailingPolar, p::SurfaceParameters, lat1, lon1, lat2, lon2)
distance, dir = haversine(lat1, lon1, lat2, lon2) dir2, vel = bestvectorspeed(sp, dir, p.winddeg, p.windkts, p.currentdeg, p.currentkts) # minutes/s * m / (knots * (m/s / knot)) = minutes return (1 / 60) * distance / (vel * 1.94384)
end
end # module
module SailingNavigation
export Position, lat, lon, GridPoint, TimeSlice, TimedPath, closestpoint, surround export RoutingProblem, minimumtimeroute
using GeometryTypes using ..SailingPolars
- NB: This uses latitude (often considered to be y) first then longitude (often considered to be x).
- This latitude, then longitude ordering is as per ISO 6709 (en.wikipedia.org/wiki/ISO_6709)
- Position is a Float32 2-tuple of latitude and longitude in degrees
Position = Point2f0
- latitude from Position
lat(p::Position) = p[1]
- longitude from Position
lon(p::Position) = p[2]
- A GridPoint is a Position with the SurfaceParameters of wind and current at the Position
mutable struct GridPoint
pt::Position sp::SurfaceParameters
end
"""
TimeSlice
A TimeSlice is a matrix of GridPoints, each Position point with their SurfaceParameters A Vector of TimeSlice can give the surface characteristics for an ocean region over time. """ TimeSlice = Matrix{GridPoint}
"""
mutable struct RoutingProblem
timeinterval: the minutes duration for each TimeSlice timeframe: a vector of sequential timeslices for the ocean region obstacleindices: the Cartesian indices in each timeslice for
obstacles, such as land or shoals, where the ship may not go
startindex: the timeslice position for time of starting start: starting location on grid of GridPoints finish: destination / finish location on grid of GridPoints allowrepeatvisits: whether the vessel may overlap its prior path, usually false """ mutable struct RoutingProblem
timeinterval::Float64 # minutes between timeframe slices timeframe::Vector{TimeSlice} obstacleindices::Vector{Point2{Int}} startindex::Int start::Point2{Int} finish::Point2{Int} allowrepeatvisits::Bool
end
"""
mutable struct TimedPath
duration: minutes total to travel the path path: vector of Cartesian indices of points in grid for path to travel """ mutable struct TimedPath
duration::Float64 path::Vector{Point2{Int}}
end
"""
closestpoint(p, mat)
Get the closest GridPoint in matrix mat to a given position p. p: Cartesian indices of a Position (latitude, longitude in degrees) in grid of GridPoints mat: matrix of Gridpoints """ closestpoint(p, mat) = findmin(gp -> haversine(p[1], p[2], gp.pt[1], gp.pt[2])[1], mat)[2]
function surround(p, mat, excluded)
neighbors = Point2{Int}[(-1, -1), (-1, 0), (-1, 1), (0, -1), (0, 1), (1, -1), (1, 0), (1, 1)] (xmax, ymax) = size(mat) return filter(q -> 1 <= q[1] <= xmax && 1 <= q[2] <= ymax && !(q in excluded), [x + p for x in neighbors])
end
"""
function minimumtimeroute(rp::RoutingProblem, sp::SailingPolar, verbose=false)
Get the route (as a TimedPath) that minimizes time from start to finish for a given RoutingProblem (sea parameters) given sailing polar data (ship parameters). """ function minimumtimeroute(rp::RoutingProblem, sp::SailingPolar, verbose=false)
timedpaths = [TimedPath(0.0, [rp.start])] completed, mintime, minpath = false, 1000.0, TimedPath(1000.0, []) for i in 1:1000 newpaths = TimedPath[] verbose && println("Checking ", length(timedpaths), " paths of length ", length(timedpaths[1].path) - 1) for tpath in timedpaths if tpath.path[end] == rp.finish completed = true push!(newpaths, tpath) else p1 = tpath.path[end] slice = rp.timeframe[div(Int(round(tpath.duration)), Int(round(rp.timeinterval))) % length(rp.timeframe) + 1] for p2 in surround([p1[1], p1[2]], slice, rp.obstacleindices) !rp.allowrepeatvisits && p2 in tpath.path && continue gp1, gp2 = slice[p1[1], p1[2]], slice[p2[1], p2[2]] lat1, lon1, lat2, lon2 = gp1.pt[1], gp1.pt[2], gp2.pt[1], gp2.pt[2] t = sailsegmenttime(sp, gp1.sp, lat1, lon1, lat2, lon2) path = deepcopy(tpath.path) push!(path, p2) push!(newpaths, TimedPath(tpath.duration + t, path)) end end end timedpaths = unique(newpaths) if completed mindur = minimum(map(x -> x.duration, timedpaths)) finished = filter(x -> x.path[end] == rp.finish, timedpaths) minfindur, idx = findmin(map(x -> x.duration, finished)) verbose && println("Current finished minimum: ", finished[idx], ", others $mindur") if mindur == minfindur minpath = finished[idx] break end end end return minpath
end
end # module
using GeometryTypes using .SailingNavigation, .SailingPolars
- =
The data is selected so the best time path is slightly longer than the shortest length path. The forbidden regions are x, representing land or reef. The allowed sailing points are . and start and finish are S and F.
x . . F . . x . x . . . . . . . x x x . . x x x . . . . . x x x x . x x x . . . x x . x . x . . . x x . x . . . . . x . . x . x . . . . . . x . . . . S . . . . . =# const forbidden = Point2{Int}.([
[1, 8], [2, 1], [2, 8], [3, 5], [3, 8], [4, 1], [4, 5], [4, 6], [4, 8], [5, 1], [5, 5], [5, 6], [5, 8], [6, 3], [6, 4], [6, 5], [6, 6], [6, 8], [6, 9], [7, 1], [7, 4], [7, 5], [7, 6], [8, 8], [8, 9], [9, 1], [9, 7], [9, 9],
])
- Create regional wind patterns on the map.
function surfacebylongitude(lon)
return lon < -155.03 ? SurfaceParameters(-5.0, 8, 150, 0.5) : lon < -155.99 ? SurfaceParameters(-90.0, 20, 150, 0.4) : SurfaceParameters(180.0, 25, 150, 0.3)
end
- Vary wind speeds over time.
function mutatetimeslices!(slices)
for (i, slice) in enumerate(slices), x in slice x.sp = SurfaceParameters(x.sp.winddeg, x.sp.windkts * (1 + 0.002 * i), x.sp.currentdeg, x.sp.currentkts) end
end
const startpos = Point2{Int}(1, 4)
const endpos = Point2{Int}(9, 4)
const pmat = [Position(19.78 - 1/60 + i/60, -155.0 - 5/60 + j/60) for i in 0:8, j in 0:8]
const gpoints = map(pt -> GridPoint(pt, surfacebylongitude(lon(pt))), pmat)
const slices = [deepcopy(gpoints) for _ in 1:200]
mutatetimeslices!(slices)
const routeprob = RoutingProblem(10.0, slices, forbidden, 1, startpos, endpos, false) const filename = "polar.csv" const sp = getpolardata(filename) const tp = minimumtimeroute(routeprob, sp)
println("The route taking the least time found was:\n ", tp.path,
"\nwhich has duration $(div(tp.duration, 60)) hours, $(rem(tp.duration, 60)) minutes.")
</lang> The polar CSV file used for this solution, named polar.csv, is as follows. Note that this is a very detailed polar, chosen to stress the testing of the code. Most polar files are far smaller, with fewer choices for angle and wind speed.
TWA\TWS;0;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18;19;20;21;22;23;24;25;26;27;28;29;30;35;40;60;70 40;0;0.53;0.54;0.49;0.4;0.31;0.21;0.16;0.11;0.08;0.05;0.03;0.02;0.01;0;0;0;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.05;-0.1;-0.11 41;0;0.61;0.62;0.56;0.47;0.36;0.25;0.19;0.14;0.1;0.07;0.04;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;0;0;-0.04;-0.09;-0.1 44;0;0.89;0.91;0.82;0.69;0.56;0.42;0.33;0.24;0.18;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;-0.02;-0.06;-0.06 45;0;0.99;1.02;0.92;0.78;0.64;0.49;0.39;0.29;0.22;0.15;0.1;0.07;0.04;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.05;-0.05 46;0;1.11;1.14;1.02;0.87;0.73;0.57;0.45;0.35;0.26;0.18;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;0;-0.01;-0.04;-0.05 47;0;1.23;1.25;1.14;0.97;0.82;0.66;0.53;0.41;0.31;0.22;0.15;0.1;0.07;0.04;0.02;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.01;-0.03;-0.04 48;0;1.37;1.37;1.26;1.08;0.93;0.76;0.61;0.48;0.36;0.26;0.19;0.13;0.08;0.05;0.03;0.02;0.01;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.02;-0.03 49;0;1.5;1.5;1.39;1.2;1.05;0.87;0.71;0.56;0.42;0.31;0.22;0.15;0.1;0.07;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;0;-0.02;-0.02 50;0;1.65;1.64;1.52;1.33;1.18;1;0.81;0.65;0.49;0.37;0.26;0.19;0.13;0.08;0.05;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;0;0;-0.01;-0.02 51;0;1.79;1.77;1.67;1.46;1.32;1.13;0.92;0.74;0.57;0.43;0.31;0.22;0.15;0.1;0.07;0.05;0.03;0.02;0.02;0.01;0.01;0;0;0;0;0;0;0;0;-0.01;-0.02 53;0;2.1;2.07;1.99;1.76;1.62;1.4;1.14;0.95;0.74;0.57;0.43;0.31;0.22;0.16;0.1;0.08;0.06;0.04;0.03;0.02;0.01;0.01;0.01;0;0;0;0;0;0;-0.01;-0.01 54;0;2.26;2.22;2.16;1.92;1.78;1.55;1.28;1.06;0.84;0.65;0.5;0.37;0.27;0.19;0.13;0.1;0.07;0.06;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;0;-0.01 55;0;2.43;2.39;2.34;2.09;1.95;1.7;1.42;1.18;0.95;0.74;0.57;0.43;0.32;0.23;0.16;0.12;0.09;0.07;0.05;0.04;0.03;0.02;0.01;0.01;0;0;0;0;0;0;-0.01 60;0;3.29;3.33;3.33;3.08;2.93;2.64;2.29;1.98;1.66;1.36;1.1;0.88;0.68;0.53;0.39;0.32;0.26;0.21;0.17;0.13;0.1;0.07;0.05;0.04;0.03;0.02;0.01;0;0;0;0 70;0;5.2;5.53;5.74;5.59;5.5;5.22;4.84;4.46;3.94;3.51;3.08;2.65;2.26;1.9;1.55;1.38;1.22;1.06;0.92;0.78;0.66;0.55;0.46;0.37;0.3;0.24;0.18;0.03;0;0;0 80;0;6.8;7.43;7.97;8.02;8.23;8.34;8.2;7.9;7.37;6.91;6.43;5.9;5.32;4.72;4.12;3.83;3.55;3.25;2.96;2.67;2.4;2.13;1.88;1.65;1.43;1.22;1.04;0.37;0.09;0.01;0 90;0;7.59;8.5;9.4;9.73;10.4;11.16;11.53;11.56;11.3;11.05;10.77;10.44;9.83;9.07;8.34;8;7.65;7.27;6.88;6.46;6.04;5.61;5.15;4.74;4.33;3.88;3.51;1.72;0.67;0.12;0.03 100;0;7.34;8.25;9.16;9.86;10.5;11.95;12.79;13.5;14.02;14.4;14.37;14.5;14.4;13.92;13.52;13.19;12.79;12.51;12.1;11.66;11.22;10.77;10.26;9.72;9.2;8.58;8.01;4.87;2.51;0.7;0.23 110;0;7.09;7.97;8.84;9.74;10.09;11.85;12.75;13.84;14.99;16.02;16.33;17.1;17.83;17.99;18.32;18.14;17.81;17.84;17.6;17.3;17.05;16.83;16.53;16.03;15.59;15.03;14.37;10.26;6.41;2.32;0.86 120;0;6.59;7.42;8.3;9.1;9.56;10.83;11.6;13.1;13.87;14.66;15.75;16.67;17.63;18.43;19.62;20.17;20.6;21.12;21.55;21.75;21.91;22.07;21.9;21.58;21.29;20.92;20.29;16.47;12.03;5.49;2.26 129;0;6.14;6.93;7.83;8.52;9.09;9.89;10.57;12.42;12.87;13.43;15.23;16.16;17.08;18.07;19.48;20.35;21.22;21.93;22.85;23.44;23.98;24.55;24.59;24.55;24.51;24.46;24;21.56;17.75;9.64;4.25 130;0;6.07;6.87;7.76;8.44;9.02;9.8;10.48;12.29;12.73;13.27;15.08;16.03;16.97;17.96;19.36;20.25;21.15;21.88;22.82;23.44;24.03;24.6;24.66;24.68;24.67;24.64;24.24;22;18.33;10.11;4.5 135;0;5.72;6.57;7.36;8.02;8.65;9.38;10.11;11.52;11.97;12.55;13.85;15.31;16.31;17.33;18.54;19.48;20.35;21.28;22.3;23.08;24.09;24.63;24.69;24.78;24.79;24.91;24.82;23.74;20.98;12.39;5.78 136;0;5.66;6.5;7.28;7.93;8.57;9.3;10.04;11.34;11.82;12.42;13.62;15.06;16.17;17.2;18.35;19.29;20.15;21.12;22.15;22.96;24.07;24.6;24.67;24.76;24.75;24.85;24.81;23.98;21.45;12.8;6.03 139;0;5.42;6.31;6.92;7.67;8.34;9.08;9.86;10.86;11.32;12.03;12.99;14.3;15.73;16.76;17.76;18.71;19.53;20.6;21.66;22.54;23.92;24.44;24.53;24.64;24.58;24.65;24.67;24.47;22.68;13.79;6.73 140;0;5.35;6.22;6.79;7.59;8.26;9;9.8;10.72;11.16;11.89;12.79;14.06;15.5;16.62;17.57;18.51;19.32;20.43;21.49;22.4;23.84;24.36;24.46;24.58;24.51;24.57;24.61;24.56;23.02;14.08;6.96 141;0;5.29;6.12;6.67;7.48;8.18;8.93;9.74;10.57;11.02;11.77;12.62;13.82;15.26;16.47;17.38;18.32;19.04;20.28;21.31;22.07;23.53;24;24.21;24.29;24.43;24.48;24.55;24.61;23.33;14.31;7.18 142;0;5.23;6.02;6.57;7.39;8.1;8.86;9.67;10.43;10.88;11.64;12.45;13.59;15.03;16.24;17.14;18.06;18.77;19.98;21.01;21.75;23.18;23.65;23.86;23.95;24.34;24.39;24.48;24.61;23.61;14.54;7.4 143;0;5.16;5.93;6.45;7.3;8;8.78;9.54;10.27;10.75;11.5;12.28;13.36;14.8;16.01;16.9;17.81;18.5;19.69;20.72;21.43;22.84;23.31;23.52;23.61;24.05;24.27;24.41;24.57;23.85;14.8;7.6 144;0;5.09;5.83;6.33;7.23;7.92;8.66;9.41;10.13;10.62;11.39;12.13;13.13;14.57;15.78;16.65;17.56;18.24;19.41;20.43;21.12;22.5;22.97;23.19;23.28;23.73;24.08;24.33;24.49;24.04;15;7.8 145;0;5.02;5.73;6.23;7.15;7.85;8.55;9.28;9.98;10.51;11.27;11.98;12.92;14.35;15.56;16.42;17.31;17.97;19.13;20.14;20.82;22.17;22.64;22.87;22.96;23.42;23.81;24.23;24.41;24.19;15.14;7.98 146;0;4.96;5.64;6.12;7.07;7.77;8.43;9.15;9.84;10.38;11.16;11.83;12.71;14.12;15.35;16.19;17.07;17.72;18.86;19.86;20.51;21.84;22.31;22.56;22.65;23.12;23.48;23.94;24.33;24.3;15.3;8.16 148;0;4.82;5.45;5.91;6.9;7.59;8.21;8.89;9.55;10.14;10.89;11.55;12.29;13.7;14.92;15.74;16.6;17.23;18.32;19.3;19.91;21.2;21.67;21.95;22.05;22.53;22.87;23.38;24.13;24.39;15.52;8.46 149;0;4.76;5.35;5.81;6.78;7.49;8.09;8.78;9.42;10.01;10.76;11.41;12.1;13.48;14.71;15.52;16.36;16.98;18.06;19.03;19.63;20.89;21.37;21.67;21.77;22.26;22.61;23.12;23.98;24.37;15.57;8.58 150;0;4.69;5.26;5.7;6.67;7.37;7.96;8.64;9.26;9.86;10.6;11.24;11.89;13.26;14.48;15.29;16.11;16.73;17.79;18.74;19.33;20.55;21.04;21.37;21.48;21.98;22.34;22.83;23.69;24.11;15.6;8.67 155;0;4.33;4.74;5.16;6.16;6.79;7.33;7.96;8.51;9.15;9.81;10.4;10.85;12.14;13.37;14.1;14.87;15.48;16.42;17.3;17.8;18.88;19.39;19.86;20;20.54;20.97;21.45;22.25;22.77;15.38;8.89 160;0;4.09;4.41;4.83;5.77;6.39;6.94;7.55;8.04;8.67;9.28;9.83;10.24;11.46;12.69;13.39;14.11;14.73;15.6;16.41;16.87;17.85;18.4;18.97;19.15;19.72;20.2;20.65;21.35;21.84;14.95;8.74 162;0;4;4.29;4.69;5.62;6.23;6.77;7.38;7.86;8.48;9.07;9.6;10;11.18;12.42;13.1;13.81;14.43;15.27;16.06;16.5;17.43;18;18.62;18.81;19.39;19.89;20.33;20.99;21.48;14.76;8.61 168;0;3.74;3.93;4.35;5.15;5.75;6.31;6.93;7.34;7.92;8.45;8.95;9.35;10.44;11.68;12.32;12.99;13.63;14.39;15.11;15.5;16.31;16.94;17.7;17.92;18.53;19.08;19.49;19.99;20.46;14.34;8.3 170;0;3.69;3.85;4.27;5.04;5.65;6.22;6.82;7.23;7.8;8.31;8.8;9.22;10.27;11.51;12.15;12.81;13.45;14.19;14.9;15.28;16.06;16.7;17.51;17.73;18.34;18.91;19.31;19.77;20.22;14.24;8.24 174;0;3.57;3.69;4.11;4.83;5.43;6.01;6.62;7;7.55;8.03;8.5;8.93;9.95;11.19;11.81;12.45;13.11;13.81;14.48;14.84;15.57;16.24;17.11;17.35;17.98;18.57;18.95;19.33;19.77;14.03;8.13 180;0;3.51;3.6;4.03;4.71;5.31;5.91;6.51;6.88;7.41;7.88;8.33;8.79;9.78;11.02;11.63;12.26;12.93;13.61;14.26;14.61;15.31;15.99;16.9;17.15;17.79;18.39;18.77;19.09;19.52;13.87;8.07
- Output:
The route taking the least time found was: Point{2,Int64}[[1, 4], [1, 5], [2, 6], [3, 7], [4, 7], [5, 7], [6, 7], [7, 7], [8, 6], [8, 5], [9, 4]] which has duration 4.0 hours, 43.697879668707344 minutes.
Wren
A reasonably faithful translation though I haven't bothered to split the code up into modules (which would mean separate files in Wren) and have dispensed altogether with four functions which aren't actually called.
As Wren uses 0-based indexing the points in the minimum path have coordinates one less than those in the Julia results.
As you'd expect, this takes many times longer than Julia to run (about 24.5 minutes versus 3 minutes 20 seconds) but gets there in the end :) <lang ecmascript>import "io" for File
/*
Class that represents a polar CSV file's data. Contains a matrix, 'speeds', of sailing speeds indexed by wind velocity and angle of boat to wind. 'winds' is a list of wind speeds. 'degrees' is a list of angles in degrees of direction relative to the wind. Note 0 degrees is directly into the wind, 180 degrees is directly downwind.
- /
class SailingPolar {
construct new(winds, degrees, speeds) { _winds = winds _degrees = degrees _speeds = speeds // speeds[wind direction degrees, windspeed knots] } winds { _winds } degrees { _degrees } speeds {_speeds }
}
/*
Class that represents wind and surface current direction and velocity for a given position. Angles in degrees, velocities in knots.
- /
class SurfaceParameters {
construct new(windDeg, windKts, currentDeg, currentKts) { _windDeg = windDeg _windKts = windKts _currentDeg = currentDeg _currentKts = currentKts } windDeg { _windDeg } windKts { _windKts } currentDeg { _currentDeg } currentKts { _currentKts }
}
// Reads a sailing polar CSV file and returns a SailingPolar object containing the file data. // A sailing polar file is a CSV file, with ';' used as the comma separator instead of a comma. // The first line of file contains labels for the wind velocities that make up columns, and // the first entry of each row makes up a column of angle of sailing direction from wind in degrees. var getPolarData = Fn.new { |fileName|
var lines = File.read(fileName).split("\n") var header = lines[0].split(";") var winds = header[1..-1].map { |x| Num.fromString(x) }.toList var degrees = [] var speeds = [] for (line in lines[1..-1]) { if (line == "") break // ignore final blank line if there is one var cols = line.split(";") degrees.add(Num.fromString(cols[0])) speeds.add(cols[1..-1].map{ |x| Num.fromString(x) }.toList) } return SailingPolar.new(winds, degrees, speeds)
}
var R = 6372800 // Earth's approximate radius in meters
/* Class containing various helper methods which work with degrees rather than radians. */ class D {
// Converts degrees to radians. static deg2Rad(deg) { (deg*Num.pi/180 + 2*Num.pi) % (2*Num.pi) }
// Converts radians to degrees. static rad2Deg(rad) { (rad*180/Num.pi + 360) % 360 }
// Trig functions. static sin(d) { deg2Rad(d).sin } static cos(d) { deg2Rad(d).cos } static asin(d) { rad2Deg(d.asin) } static atan(x, y) { rad2Deg(x.atan(y)) }
}
// Calculates the haversine function for two points on the Earth's surface. // Given two latitude, longitude pairs in degrees for a point on the Earth, // get distance in meters and the initial direction of travel in degrees for // movement from point 1 to point 2. var haversine = Fn.new { |lat1, lon1, lat2, lon2|
var dlat = lat2 - lat1 var dlon = lon2 - lon1 var a = D.sin(dlat/2).pow(2) + D.cos(lat1) * D.cos(lat2) * (D.sin(dlon/2).pow(2)) var c = 2 * D.asin(a.sqrt) var theta = D.atan(D.sin(dlon) * D.cos(lat2), D.cos(lat1)*D.sin(lat2) - D.sin(lat1) * D.cos(lat2) * D.cos(dlon)) theta = (theta + 360) % 360 return [R * c * 0.5399565, theta]
}
// Returns the index of the first element of 'a' for which 'pred' returns true or -1 otherwise. var findFirst = Fn.new { |a, pred|
for (i in 0...a.count) if (pred.call(a[i])) return i return -1
}
// Returns the index of the last element of 'a' for which 'pred' returns true or -1 otherwise. var findLast = Fn.new { |a, pred|
for (i in a.count-1..0) if (pred.call(a[i])) return i return -1
}
// Calculate the expected sailing speed in a specified direction in knots, // given sailing polar data, a desired point of sail in degrees, and wind speed in knots. var boatSpeed = Fn.new { |sp, pointOfSail, windSpeed|
var winds = sp.winds var degrees = sp.degrees var speeds = sp.speeds var udeg = findLast.call(degrees) { |t| t <= pointOfSail } var odeg = findFirst.call(degrees) { |t| t >= pointOfSail } var uvel = findLast.call(winds) { |t| t <= windSpeed } var ovel = findFirst.call(winds) { |t| t >= windSpeed } if ([udeg, odeg, uvel, ovel].any { |t| t == -1 }) return -1 var frac = (odeg == udeg && uvel == ovel) ? 1 : (odeg == udeg) ? (windSpeed - winds[uvel])/(winds[ovel] - winds[uvel]) : (uvel == ovel) ? (pointOfSail - degrees[udeg])/(degrees[odeg] - degrees[udeg]) : ((pointOfSail - degrees[udeg])/(degrees[odeg] - degrees[udeg]) + (windSpeed - winds[uvel])/(winds[ovel] - winds[uvel]))/2 return speeds[udeg][uvel] + frac * (speeds[odeg][ovel] - speeds[udeg][uvel])
}
// Calculates the expected net boat speed in a desired direction versus the wind ('azimuth'). // This is generally different from the actual boat speed in its actual direction. // Directions are in degrees ('pointos' is point of sail the ship direction from the wind), // and velocity of wind ('ws') is in knots. var sailingSpeed = Fn.new { |sp, azimuth, pointos, ws|
return boatSpeed.call(sp, pointos, ws) * D.cos((pointos - azimuth).abs)
}
// Calculates the net direction and velocity of a sailing ship. // Arguments are sailing polar data, direction of travel in degrees from north, wind direction in // degrees from north, wind velocity in knots, surface current direction in degrees, and // current velocity in knots. var bestVectorSpeed = Fn.new { |sp, dirTravel, dirWind, windSpeed, dirCur, velCur|
var azimuth = (dirTravel - dirWind) % 360 azimuth = (azimuth < 0) ? azimuth + 360 : azimuth azimuth = (azimuth > 180) ? 360 - azimuth : azimuth var VMG = boatSpeed.call(sp, azimuth, windSpeed) var other = -1 var idx = -1 for (i in 0...sp.degrees.count) { var ss = sailingSpeed.call(sp, azimuth, sp.degrees[i], windSpeed) if (ss > other) { other = ss idx = i } } if (other > VMG) { azimuth = sp.degrees[idx] VMG = other } var dirChosen = D.deg2Rad(dirWind + azimuth) var wx = VMG * (dirChosen.sin) var wy = VMG * (dirChosen.cos) var curX = velCur * (D.deg2Rad(dirCur).sin) var curY = velCur * (D.deg2Rad(dirCur).cos) return [D.rad2Deg((wy + curY).atan(wx + curX)), ((wx + curX).pow(2) + (wy + curY).pow(2)).sqrt]
}
// Calculates the trip time in minutes from (lat1, lon1) to the destination (lat2, lon2). // Uses the data in SurfaceParameters for wind and current velocity and direction. var sailSegmentTime = Fn.new { |sp, p, lat1, lon1, lat2, lon2|
var h = haversine.call(lat1, lon1, lat2, lon2) var distance = h[0] var dir = h[1] var vel = bestVectorSpeed.call(sp, dir, p.windDeg, p.windKts, p.currentDeg, p.currentKts)[1] // minutes/s * m / (knots * (m/s / knot)) = minutes return (1 / 60) * distance / (vel * 1.94384)
}
/* Class that represents a point in 2-D space. Need value type semantics for comparisons etc. */ class Point2 {
construct new(x, y) { _x = x _y = y } x { _x } y { _y }
+ (other) { Point2.new(x + other.x, y + other.y) }
== (other) { x == other.x && y == other.y } != (other) { !(this == other) }
toString { "[%(_x), %(_y)]" }
}
/*
Class that consists of a tuple of latitude and longitude in degrees. NB: This uses latitude (often considered to be y) first then longitude (often considered to be x). This latitude, then longitude ordering is as per ISO 6709 (en.wikipedia.org/wiki/ISO_6709).
- /
class Position {
construct new(lat, lon) { _lat = lat _lon = lon } lat { _lat } lon { _lon }
}
/* Class that represents a Position with the SurfaceParameters of wind and current at the Position. */ class GridPoint {
construct new(pt, sp) { _pt = pt _sp = sp } pt { _pt } pt=(value) { _pt = value } sp { _sp } sp=(value) { _sp = value }
}
/*
Class that consists of a matrix of GridPoints, each Position point with their SurfaceParameters. A Vector of TimeSlice can give the surface characteristics for an ocean region over time.
- /
class TimeSlice {
construct new(gridPoints) { _gridPoints = gridpoints } gridPoints { _gridPoints }
}
/*
Class that represents a routing problem and requiring the following parameters: * timeinterval: the minutes duration for each TimeSlice * timeframe: a vector of sequential timeslices for the ocean region * obstacleindices: the Cartesian indices in each timeslice for obstacles, such as land or shoals, where the ship may not go * startindex: the timeslice position for time of starting * start: starting location on grid of GridPoints * finish: destination / finish location on grid of GridPoints * allowrepeatvisits: whether the vessel may overlap its prior path, usually false.
- /
class RoutingProblem {
construct new(timeInterval, timeFrame, obstacleIndices, startIndex, start, finish, allowRepeatVisits) { _timeInterval = timeInterval // minutes between timeFrame slices _timeFrame = timeFrame _obstacleIndices = obstacleIndices _startIndex = startIndex _start = start _finish = finish _allowRepeatVisits = allowRepeatVisits }
timeInterval { _timeInterval } timeFrame { _timeFrame } obstacleIndices { _obstacleIndices } startIndex { _startIndex } start { _start } finish { _finish } allowRepeatVisits { _allowRepeatVisits }
}
/*
Class that represents a timed path and requires the following parameters: * duration: minutes total to travel the path * path: vector of Cartesian indices of points in grid for path to travel.
- /
class TimedPath {
construct new(duration, path) { _duration = duration _path = path } duration { _duration } path { _path }
toString { "%(_duration) %(_path)" }
== (other) { this.toString == other.toString } != (other) { this.toString != other.toString }
}
var findMin = Fn.new { |a|
var min = a[0] var idx = 0 for (i in 1...a.count) { if (a[i] < min) { min = a[i] idx = i } } return [min, idx]
}
var ntuples = [ [-1, -1], [-1, 0], [-1, 1], [0, -1], [0, 1], [1, -1], [1, 0], [1, 1] ] var neighbors = List.filled(ntuples.count, null) (0...ntuples.count).each { |i| neighbors[i] = Point2.new(ntuples[i][0], ntuples[i][1]) }
// Returns a list of points surrounding 'p' which are not otherwise excluded. var surround = Fn.new { |p, mat, excluded|
var xmax = mat.count var ymax = mat[0].count return neighbors.map { |x| x + p }.where { |q| return (0 <= q.x && q.x < xmax) && (0 <= q.y && q.y < ymax) && !excluded.contains(q) }.toList
}
// Get the route (as a TimedPath) that minimizes time from start to finish for a given // RoutingProblem (sea parameters) given sailing polar data (ship parameters). var minimumTimeRoute = Fn.new { |rp, sp, verbose|
var timedPaths = [TimedPath.new(0, [rp.start])] var completed = false var minPath = TimedPath.new(1000, []) for (i in 0...1000) { var newPaths = [] verbose && System.print("Checking %(timedPaths.count) paths of length %(timedPaths[0].path.count)") for (tpath in timedPaths) { if (tpath.path[-1] == rp.finish) { completed = true newPaths.add(tpath) } else { var p1 = tpath.path[-1] var num = tpath.duration.round var den = rp.timeInterval.round var slice = rp.timeFrame[(num/den).truncate % rp.timeFrame.count] for (p2 in surround.call(p1, slice, rp.obstacleIndices)) { if (rp.allowRepeatVisits || !tpath.path.contains(p2)) { var gp1 = slice[p1.x][p1.y] var gp2 = slice[p2.x][p2.y] var lat1 = gp1.pt.lat var lon1 = gp1.pt.lon var lat2 = gp2.pt.lat var lon2 = gp2.pt.lon var t = sailSegmentTime.call(sp, gp1.sp, lat1, lon1, lat2, lon2) var path = tpath.path.toList path.add(p2) newPaths.add(TimedPath.new(tpath.duration + t, path)) } } } } var set = {} for (np in newPaths) set[np.toString] = np timedPaths = set.values.toList if (completed) { var minDur = findMin.call(timedPaths.map { |x| x.duration }.toList)[0] var finished = timedPaths.where { |x| x.path[-1] == rp.finish }.toList var mi = findMin.call(finished.map { |x| x.duration }.toList) var minFinDur = mi[0] var idx = mi[1] if (verbose) { System.print("Current finished minimum: %(finished[idx]), others %(minDur)") } if (minDur == minFinDur) { minPath = finished[idx] break } } } return minPath
}
/*
The data is selected so the best time path is slightly longer than the shortest length path. The forbidden regions are x, representing land or reef. The allowed sailing points are . and start and finish are S and F.
x . . F . . x . x . . . . . . . x x x . . x x x . . . . . x x x x . x x x . . . x x . x . x . . . x x . x . . . . . x . . x . x . . . . . . x . . . . S . . . . .
- /
// These need to be changed to 0-based for Wren. var ftuples = [
[1, 8], [2, 1], [2, 8], [3, 5], [3, 8], [4, 1], [4, 5], [4, 6], [4, 8], [5, 1], [5, 5], [5, 6], [5, 8], [6, 3], [6, 4], [6, 5], [6, 6], [6, 8], [6, 9], [7, 1], [7, 4], [7, 5], [7, 6], [8, 8], [8, 9], [9, 1], [9, 7], [9, 9]
]
var forbidden = List.filled(ftuples.count, null) (0...ftuples.count).each { |i| forbidden[i] = Point2.new(ftuples[i][0]-1, ftuples[i][1]-1) }
// Create regional wind patterns on the map. var surfaceByLongitude = Fn.new { |lon|
return (lon < -155.03) ? SurfaceParameters.new(-5, 8, 150, 0.5) : (lon < -155.99) ? SurfaceParameters.new(-90, 20, 150, 0.4) : SurfaceParameters.new(180, 25, 150, 0.3)
}
// Vary wind speeds over time. var mutateTimeSlices = Fn.new { |slices|
var i = 1 for (slice in slices) { for (j in 0...slice.count) { for (k in 0...slice[j].count) { var x = slice[j][k] x.sp = SurfaceParameters.new(x.sp.windDeg, x.sp.windKts * (1 + 0.002 * i), x.sp.currentDeg, x.sp.currentKts) } } i = i + 1 }
}
var startPos = Point2.new(0, 3) // 0-based var endPos = Point2.new(8, 3) // ditto var slices = List.filled(200, null) for (s in 0...200) {
var gpoints = List.filled(9, null) for (i in 0..8) { gpoints[i] = List.filled(9, null) for (j in 0..8) { var pt = Position.new(19.78 - 1/60 + i/60, -155.0 - 5/60 + j/60) gpoints[i][j] = GridPoint.new(pt, surfaceByLongitude.call(pt.lon)) } } slices[s] = gpoints
} mutateTimeSlices.call(slices) var routeProb = RoutingProblem.new(10, slices, forbidden, 0, startPos, endPos, false) var fileName = "polar.csv" var sp = getPolarData.call(fileName) var tp = minimumTimeRoute.call(routeProb, sp, false) System.print("The route taking the least time found was:\n %(tp.path) \nwhich has duration " +
"%((tp.duration/60).truncate) hours, %((tp.duration%60).round) minutes.")</lang>
- Output:
The route taking the least time found was: [[0, 3], [0, 4], [1, 5], [2, 6], [3, 6], [4, 6], [5, 6], [6, 6], [7, 5], [7, 4], [8, 3]] which has duration 4 hours, 44 minutes.