The Gateway Manifold states

[rslippert]

Apolune states [X, Y, Z, vX, vY, vZ] near Gateway orbit for -.176 <= Z <= -.200
Manifold = [
[1.019254, 0.0 , -0.176 , 0.000 , -0.098198 , 0.0 ],
[1.019126, 0.0 , -0.177 , 0.000 , -0.097808 , 0.0 ],
[1.019626, 0.0 , -0.178 , 0.000 , -0.098762 , 0.0 ],
[1.018148, 0.0 , -0.179 , 0.000 , -0.09475 , 0.0 ],
[1.021154, 0.0, -0.180 , 0.000 , -0.101888 , 0.0 ],
[1.021244, 0.0 , -0.181 , 0.000 , -0.101752 , 0.0 ],
[1.021325, 0.0, -0.181619, 0.0 , -0.101736, 0.0 ], # 9:2 Gateway orbit
[1.020866, 0.0, -0.182 , 0.000 , -0.100492 , 0.0 ],
[1.021428, 0.0 , -0.183 , 0.0 , -0.1015100, 0.0 ],
[1.022326, 0.0 , -0.184 , 0.000 , -0.10332 , 0.0 ],
[1.023558 ,0.0 , -0.185 , 0.000 , -0.10590 , 0.0 ],
[1.024452, 0.0 , -0.186 , 0.0 , -0.10764, 0.0 ],
[ 1.025165, 0.0 , -0.187 , 0.0 , -0.10893 , 0.0 ],
[ 1.02574 , 0.0 , -0.188 , 0.0 , -0.109882 , 0.0 ],
[ 1.026449, 0.0 , -0.189 , 0.00 , -0.111121 , 0.0 ],
[ 1.027056, 0.0 , -0.190 , 0.0 , -0.112112 , 0.0 ],
[ 1.02765 , 0.0 , -0.191 , 0.000 , -0.113056 , 0.0 ],
[ 1.028288, 0.0 , -0.192 , 0.000 , -0.114078 , 0.0 ],
[ 1.028908 ,0.0 , -0.193 , 0.000 , -0.115046 , 0.0 ],
[ 1.02934 , 0.0 , -0.194 , 0.000 , -0.115582 , 0.0 ],
[ 1.0297 , 0.00 , -0.195 , 0.000 , -0.115946 , 0.0 ],
[ 1.030154 ,0.0 , -0.196 , 0.000 , -0.116512 , 0.0 ],
[ 1.030654 , 0.0, -0.197 , 0.000 , -0.11716 , 0.0 ],
[ 1.03126 , 0.0 , -0.198 , 0.000 , -0.11802 , 0.0 ],
[ 1.032562 ,0.0 , -0.199 , 0.000 , -0.120324 , 0.0 ],
[ 1.032916 ,0.0 , -0.200 , 0.000 , -0.120612 , 0.0 ]]

[rslippert]

The following code predicts stable NRHO apolune states near the Gateway 9:2 manifold for -.1822 < Z < -.1805
This can be used for station keeping at Gateway apolune , Gateway orbit is at Z=-0.181619
The resulting orbit remains stable for 11 orbits (given no perturbations)

def VY( X): #predict VY given X
return(-2.346408659*X + 2.2947005114)
#----------------------------------------------------------------------
def predict_NRHO( Z ): # predict NRHO apolune states for -.1822 < Z < -.1805
X = -561067.73159 *Z**3+ -303871.6211 ZZ+ -54860.72303 *Z+ -3300.61965
return( [X, 0.0, Z, 0.00 , VY(X), 0.0 ] )