Lambert method for in-plane transfer

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])

[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!

[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.