Well, let's just try the random movement thing and see how it goes. The nice thing about random movement is since enemy acceleration and movement is random with an expected value of 0, it is all the same whether you are leading the target or not, making things simpler.
Spoiler
x <- 0
y <- 1000
alpha <- 0
xour <- 0
your <- 0
beta <- pi/2
enemytopspeed <- 300
ourtopspeed <- 50
turretrotationrate <- 30*pi/180
mindistance <- 50
locationmatrix <- data.frame()
for (t in 1:100) {
locationmatrix <- rbind(locationmatrix, c(t,x,y,xour,your,gamma))
#the enemy is not allowed to come within mindistance pixels of our ship in either direction
deltax <- rnorm(1,0,enemytopspeed)
deltay <- rnorm(1,0,ourtopspeed)
while(abs(x+deltax-xour) < mindistance) deltax <- deltax <- rnorm(1,0,enemytopspeed)
while(abs(y+deltay-xour) < mindistance) deltay <- rnorm(1,0,ourtopspeed)
#alpha is the angle of directional vector from our ship to enemy
x <- x+deltax
y <- y+deltay
alpha <- atan2(y-your,x-xour)
xour=xour+cos(alpha)*ourtopspeed
your=your+sin(alpha)*ourtopspeed
#beta is the angle of our turret
#rotate beta towards alpha by a maximum of rate turretrotationrate
beta <- beta + sign(alpha-beta)*min(turretrotationrate,abs(alpha-beta))
#gamma is the angle from our turret facing to the enemy ship
gamma <- alpha-beta
print(gamma)
}
names(locationmatrix) <- c("t","x","y","xour","your","gamma")
library(ggplot2)
ggplot(data=locationmatrix)+
geom_path(aes(x=x,y=y,col="ENEMY"))+
geom_point(aes(x=x,y=y,col="ENEMY"))+
geom_path(aes(x=xour,y=your))+
geom_point(aes(x=xour,y=your))
ggplot(data=locationmatrix)+
geom_path(aes(x=gamma*180/pi,y=t))+
geom_point(aes(x=gamma*180/pi,y=t))
Here are some random paths (enemy speed 300, our speed 50)
I have yet to see it escape once. Seems like the randomness is simply too much.
Here is a plot with also the enemy's apparent position in degrees from the point of view of our turret which is trying to rotate in pursuit (max turret rotation rate 30 deg / sec, assume infinite turret arc, assume ship does not rotate)
Now it seems like it's actually a pretty bad idea for our ship to close in to point blank range because then the apparent angle changes faster than the turret can follow. So let's make it keep a minimum distance of 1000.
if(sqrt((x-xour)^2+(y-your)^2)>1000){
xour=xour+cos(alpha)*ourtopspeed
your=your+sin(alpha)*ourtopspeed
}
Most of the time we are going to be hitting this speed 300 ship with no problem. What about with a turret rotation speed of 3 (Gauss cannon) rather than 30?
This certainly seems like it would affect damage output a little.
(Edit: change starting turret angle to pi/2 from -pi/2, was latter in graphs)
Con't
Now that we have a model, we can also integrate probability to hit into it. Let's say that the enemy ship is a 100 px wide sphere. Then previously in this thread we found probability to hit a coordinate less than Z for a weapon to be
G <- function(y) return(y*pnorm(y) + dnorm(y))
fEz <- function(z, a, b) return((1/2/b)*(pnorm(z/a+b/a)-pnorm(z/a-b/a)))
PrEltZ <- function(z, a, b) return(a/2/b*(G(z/a+b/a)-G(z/a-b/a)))
(note: if b=0, use normal distribution instead)
where b is the weapon spread and a is the SD of the normal distribution (presumably 50 px). Now if coordinate 0 is right in front of us then probability to hit enemy ship is PrEltZ(enemyshipleftbound)-PrEltZ(enemyshiprightbound). The enemy ship's left bound is approximately given by arc length to enemy ship middle -50 px, so (gamma*r). So we have the probability to hit is approx PrEltZ(range*gamma+50, SD, spread)-PrEltZ(range*gamma-50, SD, spread).
Putting it all together we get this code
Spoiler
x <- 0
y <- 1000
alpha <- 0
xour <- 0
your <- 0
beta <- pi/2
enemytopspeed <- 300
ourtopspeed <- 50
turretrotationrate <- 3*pi/180
mindistance <- 50
SD <- 50
spread <- 0
G <- function(y) return(y*pnorm(y) + dnorm(y))
fEz <- function(z, a, b) return((1/2/b)*(pnorm(z/a+b/a)-pnorm(z/a-b/a)))
PrEltZ <- function(z, a, b){
if(b==0) return(pnorm(z,0,a)) else return(a/2/b*(G(z/a+b/a)-G(z/a-b/a)))
}
locationmatrix <- data.frame()
for (t in 1:100) {
locationmatrix <- rbind(locationmatrix, c(t,x,y,xour,your,prob))
#the enemy is not allowed to come within mindistance pixels of our ship in either direction
deltax <- rnorm(1,0,enemytopspeed)
deltay <- rnorm(1,0,ourtopspeed)
while(abs(x+deltax-xour) < mindistance) deltax <- deltax <- rnorm(1,0,enemytopspeed)
while(abs(y+deltay-xour) < mindistance) deltay <- rnorm(1,0,ourtopspeed)
#alpha is the angle of directional vector from our ship to enemy
x <- x+deltax
y <- y+deltay
alpha <- atan2(y-your,x-xour)
range <-sqrt((x-xour)^2+(y-your)^2)
if(range>1000){
xour=xour+cos(alpha)*ourtopspeed
your=your+sin(alpha)*ourtopspeed
}
#beta is the angle of our turret
#rotate beta towards alpha by a maximum of rate turretrotationrate
beta <- beta + sign(alpha-beta)*min(turretrotationrate,abs(alpha-beta))
#gamma is the angle from our turret facing to the enemy ship
gamma <- alpha-beta
prob <- PrEltZ(gamma*range+50, SD, spread)-PrEltZ(gamma*range-50,SD,spread)
print(prob)
}
names(locationmatrix) <- c("t","x","y","xour","your","hitprobability")
library(ggplot2)
ggplot(data=locationmatrix)+
geom_path(aes(x=x,y=y,col="ENEMY"))+
geom_point(aes(x=x,y=y,col="ENEMY"))+
geom_path(aes(x=xour,y=your))+
geom_point(aes(x=xour,y=your))
#ggplot(data=locationmatrix)+
# geom_path(aes(x=gamma*180/pi,y=t))+
# geom_point(aes(x=gamma*180/pi,y=t))
ggplot(data=locationmatrix)+
geom_path(aes(y=hitprobability,x=t))
And these somewhat unflattering graphs for the Gauss (spread 0, turn rate 3)
Using these assumptions, what kinds of hit rates will we get for various turret rotation speeds? Let's stick with the Gauss (spread 0) for now, but try turn rate 0, 3, 5, 10, 15, 20 and 30. Run 1000 combats and take the average.
Turn rate Hit rate
0 0.4%
3 23.4%
5 32.4%
10 42.5%
15 49.1%
20 52.9%
30 57.5%
That is versus a small speed 300 ship so it's no wonder turn rate is important. But it also suggests that this is important enough that it should be included in any comprehensive model of space combat (and also suggests that you should put advanced turret gyros on your Gauss Conquests, incidentally). There also appears to be such a thing as "enough turn rate" with diminishing returns as the turret is mostly pointed in the right direction.
Author
Topic: Optimizing the Conquest: a Mathematical Model of Space Combat (Read 25143 times)