Lambert method for in-plane transfer

[tmamedzadeh]

I'm trying to solve the transfer between a circular orbit with 200km radius and an orbit with 200km perigee and 8000km apogee.

The optimal transfer is hoffman, however, izzo.lambert gives different result.

My code:

from astropy import units as u
from poliastro.bodies import Earth
from poliastro.iod import izzo
from poliastro.core.elements import coe2rv
from poliastro.util import norm
import math
import time

Earth_k = Earth.k
Req = Earth.R.to(u.km).value

def keplerian2cartesian(kepler):

    a=(kepler[0]+kepler[1]+Req*2)*1000/2 * u.m
    e=(kepler[0]-kepler[1])/(kepler[0]+kepler[1]+Req*2)

    (R,V)=coe2rv(Earth.k,a,e,kepler[2],kepler[3],kepler[4],kepler[5])

    return [R * u.m] + [V * u.m/u.s]

# Checking different transfer times
def transfer_time(DV_min,vector0,vector,period):

    init_t=10
    t_value=init_t
    (r0,r,v0,v)=(vector0[0],vector[0],vector0[1],vector[1])

    while init_t<period:

        tof = init_t * u.min

        try:
            (f_v0, f_v), = izzo.lambert(Earth_k, r0, r, tof)
        except:
            init_t+=1
            continue

        DV0=norm(f_v0-v0).value*1000
        DV=norm(f_v-v).value*1000

        if(DV0+DV<DV_min):
            t_value=init_t
            DV0_value=DV0
            DV_value=DV
            DV_min=DV0+DV

        init_t+=1

    return (t_value,DV_value,DV0_value)


# Checking different initial and final true anomalies
def transfer_tetta(kepler0,kepler):
    DV_min=100000
    final_tetta=0

    a=(kepler[0]+kepler[1]+Req*2)*1000/2
    period=math.ceil(math.sqrt((a**3/Earth.k.value)*4*(math.pi)**2)/60)

    while final_tetta<360:
        init_tetta=0
        while init_tetta<360:
            vector0=keplerian2cartesian(kepler0 + [init_tetta*math.pi/180])
            vector =keplerian2cartesian(kepler  + [final_tetta*math.pi/180])
            try:
                (init_t,DV,DV0)=transfer_time(DV_min,vector0,vector,period)
            except:
                init_tetta+=5
                continue

            if DV+DV0<DV_min:
                DV_min=DV+DV0
                (t_value,DV0_value,DV_value,init_tetta_value,final_tetta_value)=(init_t,DV0,DV,init_tetta,final_tetta)

            init_tetta+=5
        final_tetta+=5

    print("t: ", t_value, "DV_init: ", DV0_value,"DV_final: ", DV_value)
    print("DV: ", DV_min)
    print("Init tetta: ", init_tetta_value, "Final tetta: ", final_tetta_value)

# 1->2
transfer_tetta([200,  200, 64.3*math.pi/180, 0, 300*math.pi/180],[8000, 200, 64.3*math.pi/180, 0, 300*math.pi/180])

[astrojuanlu]

Hello @Tarlan0001! No shame in opening an issue and then discovering the answer yourself :) Would you be so kind to restore the original text and point out how did you fix the problem?

If you ever have other questions, feel welcome to join our chat. Cheers!

[tmamedzadeh]

@astrojuanlu Hi. I didn't solve it yet. I deleted it because it's not a matter of a ticket right now, I have to find the issue.

However, if you could help me with izzo.lambert for my problem, I would appreciate.

[astrojuanlu]

Sure thing! Could you please share the code you're using? If it's long, you can upload it to https://gist.github.com/

[astrojuanlu]

By the way, https://doi.org/10.2514/3.20941 contains a simple proof of the optimality of the Hohmann transfer. The problem with applying the Lambert problem in this case is that it's ill conditioned if the transfer angle is 180 degrees.

[tmamedzadeh]

@astrojuanlu Sure, added my code. I guess, the result should be a maneuver at 0 true anomaly (perigee) and zero maneuver at apogee.

Thank you for the paper!

[astrojuanlu]

I recommend you to simplify by code by using from poliastro.core.elements import coe2rv and from poliastro.util import norm.

Also, I am not confident at all that this should work, because of the singularity I mentioned.

[astrojuanlu]

xref: https://space.stackexchange.com/q/43386/10716

[astrojuanlu]

I gave a detailed answer with my perspective on the theoretical aspects of this at Space.SE. It's difficult for me to make sense of the code if I don't understand the research question.

[tmamedzadeh]

@astrojuanlu Thank you for the detailed answer!
Indeed, I want to get the same result with Lambert solver, as it would be with the Hoffman transfer.
Now it gives an optimal transfer starting from 0 deg true anomaly and finishing at 185 deg (which is close to Hoffman). I checked an interval of transfer times with 1 minute period.

I accepted your answer, however, if it's possible, I would appreciate your time to check the code above to ensure, that the problem is in singularity.

[tmamedzadeh]

Well, I've got the result 0 deg initial and 181 deg final with the Lambert solver. However, the DV is a little higher than with the Hoffman.
Closing this ticket, thank you!