Partial Derivatives Computation via Autodiff#
In this notebook, we show how to use the autodiff feature of \(\partial\textrm{SGP4}\). Due to the fact that it is written in pytorch, it automatically supports automatic differentiation via torch.autograd.
In this notebook, we show how these partial derivatives can be constructed: for more advanced examples on how to use these gradients for practical applications, see the tutorials on state_transition_matrix_computation, covariance_propagation, graident_based_optimization, orbit_determination.
import dsgp4
import torch
We create a TLE object
#as always, first, we create a TLE object:
tle=[]
tle.append('0 COSMOS 2251 DEB')
tle.append('1 34454U 93036SX 22068.91971155 .00000319 00000-0 11812-3 0 9996')
tle.append('2 34454 74.0583 280.7094 0037596 327.9100 31.9764 14.35844873683320')
tle = dsgp4.tle.TLE(tle)
print(tle)
TLE(
0 COSMOS 2251 DEB
1 34454U 93036SX 22068.91971155 .00000319 00000-0 11812-3 0 9996
2 34454 74.0583 280.7094 0037596 327.9100 31.9764 14.35844873683320
)
Now, as shown in the tle_propagation tutorial, we can propagate the TLE. However, instead of using the standard API, we require torch.autograd to record the operations w.r.t. the time.
Partials with respect to time#
Let’s compute the partials of the \(\partial \textrm{SGP4}\) output w.r.t. the propagation times
Single TLEs#
Let’s first see the case of single TLEs, propagated at various future times:
#let's take a random tensor of 10 tsince elements, where we track the gradients:
tsince=torch.rand((10,),requires_grad=True)
#the state is then:
state_teme = dsgp4.propagate(tle,
tsinces=tsince,
initialized=False)
#now, we can see that the gradient is tracked:
print(state_teme)
tensor([[[ 1.3804e+03, -6.9927e+03, 1.9941e+02],
[ 1.9805e+00, 5.8818e-01, 7.1998e+00]],
[[ 1.3693e+03, -6.9959e+03, 1.5917e+02],
[ 1.9889e+00, 5.4514e-01, 7.2009e+00]],
[[ 1.3919e+03, -6.9891e+03, 2.4137e+02],
[ 1.9716e+00, 6.3304e-01, 7.1984e+00]],
[[ 1.3802e+03, -6.9928e+03, 1.9855e+02],
[ 1.9806e+00, 5.8726e-01, 7.1998e+00]],
[[ 1.4227e+03, -6.9782e+03, 3.5449e+02],
[ 1.9472e+00, 7.5389e-01, 7.1932e+00]],
[[ 1.3498e+03, -7.0008e+03, 8.8896e+01],
[ 2.0035e+00, 4.6996e-01, 7.2023e+00]],
[[ 1.3533e+03, -7.0000e+03, 1.0125e+02],
[ 2.0010e+00, 4.8318e-01, 7.2021e+00]],
[[ 1.4273e+03, -6.9765e+03, 3.7138e+02],
[ 1.9435e+00, 7.7193e-01, 7.1923e+00]],
[[ 1.3452e+03, -7.0019e+03, 7.2326e+01],
[ 2.0069e+00, 4.5223e-01, 7.2025e+00]],
[[ 1.4423e+03, -6.9703e+03, 4.2694e+02],
[ 1.9313e+00, 8.3124e-01, 7.1889e+00]]],
grad_fn=<TransposeBackward0>)
Let’s now retrieve the partial derivatives of the SGP4 output w.r.t. time.
Since the state is position and velocity (i.e., \([x,y,z,v_x,v_y,v_z]\)), these partials will be all the elements of type:
(1)#\[\begin{equation}
\dfrac{d \pmb{x}}{d t}=[\dfrac{dx}{dt}, \dfrac{dy}{dt}, \dfrac{dz}{dt}, \dfrac{d^2x}{dt^2}, \dfrac{d^2y}{dt^2}, \dfrac{d^2z}{dt^2}]^T=[v_x, v_y, v_z, \dfrac{dv_x}{dt}, \dfrac{dv_y}{dt}, \dfrac{dv_z}{dt}]^T
\end{equation}\]
Note
One thing to be careful about is that \(\partial\textrm{SGP4}\), mirroring the original \(\textrm{SGP4}\), takes the time in minutes, and returns the state in km and km/s. Hence, the derivatives will have dimensions coherent to these, and to return to SI, conversions have to be made.
partial_derivatives = torch.zeros_like(state_teme)
for i in [0,1]:
for j in [0,1,2]:
tsince.grad=None
state_teme[:,i,j].backward(torch.ones_like(tsince),retain_graph=True)
partial_derivatives[:,i,j] = tsince.grad
#let's print to screen the partials:
print(partial_derivatives)
tensor([[[ 1.1883e+02, 3.5291e+01, 4.3199e+02],
[-9.1182e-02, 4.6190e-01, -1.3206e-02]],
[[ 1.1934e+02, 3.2709e+01, 4.3206e+02],
[-9.0451e-02, 4.6212e-01, -1.0541e-02]],
[[ 1.1830e+02, 3.7982e+01, 4.3190e+02],
[-9.1941e-02, 4.6165e-01, -1.5985e-02]],
[[ 1.1884e+02, 3.5236e+01, 4.3199e+02],
[-9.1166e-02, 4.6190e-01, -1.3150e-02]],
[[ 1.1683e+02, 4.5234e+01, 4.3159e+02],
[-9.3969e-02, 4.6090e-01, -2.3475e-02]],
[[ 1.2021e+02, 2.8198e+01, 4.3214e+02],
[-8.9167e-02, 4.6246e-01, -5.8878e-03]],
[[ 1.2006e+02, 2.8991e+01, 4.3213e+02],
[-8.9394e-02, 4.6240e-01, -6.7062e-03]],
[[ 1.1661e+02, 4.6316e+01, 4.3154e+02],
[-9.4270e-02, 4.6078e-01, -2.4593e-02]],
[[ 1.2042e+02, 2.7134e+01, 4.3215e+02],
[-8.8863e-02, 4.6253e-01, -4.7904e-03]],
[[ 1.1588e+02, 4.9875e+01, 4.3133e+02],
[-9.5256e-02, 4.6036e-01, -2.8271e-02]]])
Batch TLEs#
Let’s now see how it works for batch TLEs. The API is basically identical:
#we load 6 TLEs:
inp_file="""0 PSLV DEB
1 35350U 01049QJ 22068.76869562 .00000911 00000-0 24939-3 0 9998
2 35350 98.6033 64.7516 0074531 99.8340 261.1278 14.48029442457561
0 PSLV DEB *
1 35351U 01049QK 22066.70636923 .00002156 00000-0 63479-3 0 9999
2 35351 98.8179 29.5651 0005211 45.5944 314.5671 14.44732274457505
0 SL-18 DEB
1 35354U 93014BD 22068.76520028 .00021929 00000-0 20751-2 0 9995
2 35354 75.7302 100.7819 0059525 350.7978 9.2117 14.92216400847487
0 SL-18 DEB
1 35359U 93014BJ 22068.55187275 .00025514 00000-0 24908-2 0 9992
2 35359 75.7369 156.1582 0054843 50.5279 310.0745 14.91164684775759
0 SL-18 DEB
1 35360U 93014BK 22068.44021735 .00019061 00000-0 20292-2 0 9992
2 35360 75.7343 127.2487 0071107 32.5913 327.9635 14.86997880798827
0 METEOR 2-17 DEB
1 35364U 88005Y 22067.81503681 .00001147 00000-0 84240-3 0 9995
2 35364 82.5500 92.4124 0018834 303.2489 178.0638 13.94853833332534"""
lines=inp_file.splitlines()
#let's create the TLE objects
tles=[]
for i in range(0,len(lines),3):
data=[]
data.append(lines[i])
data.append(lines[i+1])
data.append(lines[i+2])
tles.append(dsgp4.tle.TLE(data))
#we also create 9 random times, tracking the gradients:
tsinces=torch.rand((6,),requires_grad=True)
#let's initialize the TLEs:
_,tle_batch=dsgp4.initialize_tle(tles)
#let's propagate the batch:
state_teme = dsgp4.propagate_batch(tle_batch,
tsinces=tsinces)
Now, let’s retrieve the partial of each TLE, at each propagated time, and store them into a Nx2x3 matrix:
#let's retrieve the partials w.r.t. time:
partial_derivatives = torch.zeros_like(state_teme)
for i in [0,1]:
for j in [0,1,2]:
tsinces.grad=None
state_teme[:,i,j].backward(torch.ones_like(tsinces),retain_graph=True)
partial_derivatives[:,i,j] = tsinces.grad
#let's print to screen the partials:
print(partial_derivatives)
tensor([[[ 5.7180e+01, -3.6004e+01, 4.4363e+02],
[-2.0204e-01, -4.2680e-01, -4.5892e-03]],
[[ 2.4651e+01, -6.5092e+01, 4.4366e+02],
[-4.1144e-01, -2.3154e-01, -1.0460e-02]],
[[-1.0796e+02, -3.3918e+01, 4.4222e+02],
[ 9.6697e-02, -4.8886e-01, -1.3957e-02]],
[[-3.7333e+01, -1.0605e+02, 4.4122e+02],
[ 4.5411e-01, -1.9882e-01, -6.7121e-03]],
[[-7.8860e+01, -8.1715e+01, 4.4170e+02],
[ 3.0359e-01, -3.9254e-01, -1.6069e-02]],
[[ 4.6672e+01, -3.7112e+02, -2.3730e+02],
[ 3.8606e-02, 2.4403e-01, -3.7421e-01]]])
Partials with respect to TLE parameters#
Let’s now tackle the case in which we are interested in the partials of the \(\partial\textrm{SGP4}\) output w.r.t. the TLE parameters.
Single TLEs#
We first tackle the case of single TLE, propagated at multiple times:
In this case, we want the Jacobian of the output state, w.r.t. the following TLE parameters \(\textrm{TLE}=[n,e,i,\Omega,\omega,M,B^*,\dot{n},\ddot{n}]\), where:
\(n\) is the mean motion (also known as
no_kozaiin the original implementation) [rad/minute];\(e\) is the eccentricity [-];
\(i\) is the inclination [rad];
\(\Omega\) is the right ascension of the ascending node [rad];
\(\omega\) is the argument of perigee [rad];
\(M\) is the mean anomaly [rad];
\(B^*\) is the Bstar parameter [1/earth radii]
\(\dot{n}\) mean motion first derivative [radians/\(\textrm{minute}^2\)]
\(\ddot{n}\) mean motion second derivative [radians/\(\textrm{minute}^2\)]
(2)#\[\begin{equation}
\dfrac{\partial \pmb{x}}{\partial \textrm{TLE}}=
\begin{bmatrix}
\frac{\partial x}{\partial B^*} & \frac{\partial x}{\partial \dot{n}} & \frac{\partial x}{\partial \ddot{n}} & \frac{\partial x}{\partial e} & \frac{\partial x}{\partial \omega} & \frac{\partial x}{\partial i} & \frac{\partial x}{\partial M} & \frac{\partial x}{\partial n} & \frac{\partial x}{\partial \Omega} \\
\\
\frac{\partial y}{\partial B^*} & \frac{\partial y}{\partial \dot{n}} & \frac{\partial y}{\partial \ddot{n}} & \frac{\partial y}{\partial e} & \frac{\partial y}{\partial \omega} & \frac{\partial y}{\partial i} & \frac{\partial y}{\partial M} & \frac{\partial y}{\partial n} & \frac{\partial y}{\partial \Omega} \\
\\
\frac{\partial z}{\partial B^*} & \frac{\partial z}{\partial \dot{n}} & \frac{\partial z}{\partial \ddot{n}} & \frac{\partial z}{\partial e} & \frac{\partial z}{\partial \omega} & \frac{\partial z}{\partial i} & \frac{\partial z}{\partial M} & \frac{\partial z}{\partial n} & \frac{\partial z}{\partial \Omega} \\
\\
\frac{\partial v_x}{\partial B^*} & \frac{\partial v_x}{\partial \dot{n}} & \frac{\partial v_x}{\partial \ddot{n}} & \frac{\partial v_x}{\partial e} & \frac{\partial v_x}{\partial \omega} & \frac{\partial v_x}{\partial i} & \frac{\partial v_x}{\partial M} & \frac{\partial v_x}{\partial n} & \frac{\partial v_x}{\partial \Omega} \\
\\
\frac{\partial v_y}{\partial B^*} & \frac{\partial v_y}{\partial \dot{n}} & \frac{\partial v_y}{\partial \ddot{n}} & \frac{\partial v_y}{\partial e} & \frac{\partial v_y}{\partial \omega} & \frac{\partial v_y}{\partial i} & \frac{\partial v_y}{\partial M} & \frac{\partial v_y}{\partial n} & \frac{\partial v_y}{\partial \Omega} \\
\\
\frac{\partial v_z}{\partial B^*} & \frac{\partial v_z}{\partial \dot{n}} & \frac{\partial v_z}{\partial \ddot{n}} & \frac{\partial v_z}{\partial e} & \frac{\partial v_z}{\partial \omega} & \frac{\partial v_z}{\partial i} & \frac{\partial v_z}{\partial M} & \frac{\partial v_z}{\partial n} & \frac{\partial v_z}{\partial \Omega} \\
\\
\frac{\partial \dot{n}}{\partial B^*} & \frac{\partial \dot{n}}{\partial \dot{n}} & \frac{\partial \dot{n}}{\partial \ddot{n}} & \frac{\partial \dot{n}}{\partial e} & \frac{\partial \dot{n}}{\partial \omega} & \frac{\partial \dot{n}}{\partial i} & \frac{\partial \dot{n}}{\partial M} & \frac{\partial \dot{n}}{\partial n} & \frac{\partial \dot{n}}{\partial \Omega} \\
\\
\frac{\partial \ddot{n}}{\partial B^*} & \frac{\partial \ddot{n}}{\partial \dot{n}} & \frac{\partial \ddot{n}}{\partial \ddot{n}} & \frac{\partial \ddot{n}}{\partial e} & \frac{\partial \ddot{n}}{\partial \omega} & \frac{\partial \ddot{n}}{\partial i} & \frac{\partial \ddot{n}}{\partial M} & \frac{\partial \ddot{n}}{\partial n} & \frac{\partial \ddot{n}}{\partial \Omega}
\end{bmatrix}
\end{equation}\]
#as always, first, we create a TLE object:
tle=[]
tle.append('0 COSMOS 2251 DEB')
tle.append('1 34454U 93036SX 22068.91971155 .00000319 00000-0 11812-3 0 9996')
tle.append('2 34454 74.0583 280.7094 0037596 327.9100 31.9764 14.35844873683320')
tle = dsgp4.tle.TLE(tle)
print(tle)
tle_elements=dsgp4.initialize_tle(tle,with_grad=True)
TLE(
0 COSMOS 2251 DEB
1 34454U 93036SX 22068.91971155 .00000319 00000-0 11812-3 0 9996
2 34454 74.0583 280.7094 0037596 327.9100 31.9764 14.35844873683320
)
#let's select 10 random times:
tsince=torch.rand((10,))
#and let's propagate:
state_teme=dsgp4.propagate(tle,tsince)
#now we can build the partial derivatives matrix, of shape Nx6x9 (N is the number of tsince elements, 6 is the number of elements in the state vector, and 9 is the number of elements in the TLE):
partial_derivatives = torch.zeros((len(tsince),6,9))
for k in range(len(tsince)):
for i in range(6):
tle_elements.grad=None
state_teme[k].flatten()[i].backward(retain_graph=True)
partial_derivatives[k,i,:] = tle_elements.grad
#let's print them to screen:
print(partial_derivatives)
tensor([[[-2.7184e-05, 0.0000e+00, 0.0000e+00, 9.3720e+02, 1.9122e+03,
-6.2828e+01, 1.9270e+03, -1.4037e+04, 7.0024e+03],
[-1.4554e-04, 0.0000e+00, 0.0000e+00, 6.3800e+03, 4.3616e+02,
-1.5214e+01, 4.2466e+02, 7.4512e+04, 1.3427e+03],
[-1.8805e-04, 0.0000e+00, 0.0000e+00, 7.3696e+03, 6.8569e+03,
1.4119e+01, 6.9006e+03, 1.9040e+02, 0.0000e+00],
[-2.9565e-07, 0.0000e+00, 0.0000e+00, 9.2625e-01, -1.4080e+00,
-7.0737e+00, -1.4165e+00, 1.0472e+01, -4.4273e-01],
[-1.6639e-07, 0.0000e+00, 0.0000e+00, 4.3726e+00, 7.3645e+00,
-1.3401e+00, 7.3870e+00, 3.3568e+00, 2.0088e+00],
[-1.1234e-06, 0.0000e+00, 0.0000e+00, 6.0347e+00, -5.2318e-02,
2.0517e+00, -6.7107e-02, 3.8330e+01, 0.0000e+00]],
[[-2.2675e-05, 0.0000e+00, 0.0000e+00, 9.3589e+02, 1.9141e+03,
-5.2993e+01, 1.9290e+03, -1.4051e+04, 7.0030e+03],
[-1.2244e-04, 0.0000e+00, 0.0000e+00, 6.3739e+03, 4.2592e+02,
-1.3351e+01, 4.1438e+02, 7.4508e+04, 1.3400e+03],
[-1.5754e-04, 0.0000e+00, 0.0000e+00, 7.3612e+03, 6.8570e+03,
1.1266e+01, 6.9007e+03, 1.3710e+02, 0.0000e+00],
[-2.4923e-07, 0.0000e+00, 0.0000e+00, 9.3289e-01, -1.4050e+00,
-7.0738e+00, -1.4136e+00, 1.0516e+01, -4.3201e-01],
[-1.3913e-07, 0.0000e+00, 0.0000e+00, 4.3552e+00, 7.3651e+00,
-1.3401e+00, 7.3877e+00, 3.1286e+00, 2.0108e+00],
[-9.4630e-07, 0.0000e+00, 0.0000e+00, 6.0462e+00, -4.1794e-02,
2.0517e+00, -5.6517e-02, 3.8333e+01, 0.0000e+00]],
[[-6.6819e-05, 0.0000e+00, 0.0000e+00, 9.4742e+02, 1.8961e+03,
-1.4278e+02, 1.9109e+03, -1.3921e+04, 6.9969e+03],
[-3.3522e-04, 0.0000e+00, 0.0000e+00, 6.4302e+03, 5.1938e+02,
-3.0361e+01, 5.0812e+02, 7.4561e+04, 1.3654e+03],
[-4.4754e-04, 0.0000e+00, 0.0000e+00, 7.4373e+03, 6.8558e+03,
3.7310e+01, 6.8994e+03, 6.2346e+02, 0.0000e+00],
[-6.6920e-07, 0.0000e+00, 0.0000e+00, 8.7196e-01, -1.4317e+00,
-7.0724e+00, -1.4403e+00, 1.0108e+01, -5.2984e-01],
[-4.0160e-07, 0.0000e+00, 0.0000e+00, 4.5129e+00, 7.3588e+00,
-1.3399e+00, 7.3809e+00, 5.2104e+00, 1.9919e+00],
[-2.5591e-06, 0.0000e+00, 0.0000e+00, 5.9390e+00, -1.3787e-01,
2.0514e+00, -1.5320e-01, 3.8284e+01, 0.0000e+00]],
[[-1.2760e-04, 0.0000e+00, 0.0000e+00, 9.6007e+02, 1.8744e+03,
-2.4905e+02, 1.8890e+03, -1.3772e+04, 6.9881e+03],
[-5.9282e-04, 0.0000e+00, 0.0000e+00, 6.4995e+03, 6.2990e+02,
-5.0495e+01, 6.1898e+02, 7.4657e+04, 1.3951e+03],
[-8.2377e-04, 0.0000e+00, 0.0000e+00, 7.5256e+03, 6.8529e+03,
6.8135e+01, 6.8962e+03, 1.1980e+03, 0.0000e+00],
[-1.1551e-06, 0.0000e+00, 0.0000e+00, 7.9903e-01, -1.4630e+00,
-7.0691e+00, -1.4717e+00, 9.6080e+00, -6.4554e-01],
[-7.5131e-07, 0.0000e+00, 0.0000e+00, 4.6954e+00, 7.3496e+00,
-1.3394e+00, 7.3712e+00, 7.6697e+00, 1.9691e+00],
[-4.4549e-06, 0.0000e+00, 0.0000e+00, 5.8066e+00, -2.5158e-01,
2.0505e+00, -2.6760e-01, 3.8163e+01, 0.0000e+00]],
[[-3.7251e-05, 0.0000e+00, 0.0000e+00, 9.3999e+02, 1.9079e+03,
-8.4182e+01, 1.9228e+03, -1.4005e+04, 7.0011e+03],
[-1.9588e-04, 0.0000e+00, 0.0000e+00, 6.3933e+03, 4.5839e+02,
-1.9259e+01, 4.4695e+02, 7.4523e+04, 1.3488e+03],
[-2.5537e-04, 0.0000e+00, 0.0000e+00, 7.3878e+03, 6.8567e+03,
2.0313e+01, 6.9004e+03, 3.0610e+02, 0.0000e+00],
[-3.9608e-07, 0.0000e+00, 0.0000e+00, 9.1180e-01, -1.4143e+00,
-7.0735e+00, -1.4229e+00, 1.0376e+01, -4.6600e-01],
[-2.2685e-07, 0.0000e+00, 0.0000e+00, 4.4103e+00, 7.3631e+00,
-1.3400e+00, 7.3855e+00, 3.8521e+00, 2.0043e+00],
[-1.5076e-06, 0.0000e+00, 0.0000e+00, 6.0094e+00, -7.5167e-02,
2.0516e+00, -9.0100e-02, 3.8321e+01, 0.0000e+00]],
[[-1.4198e-04, 0.0000e+00, 0.0000e+00, 9.6266e+02, 1.8696e+03,
-2.7208e+02, 1.8842e+03, -1.3741e+04, 6.9860e+03],
[-6.4955e-04, 0.0000e+00, 0.0000e+00, 6.5148e+03, 6.5385e+02,
-5.4859e+01, 6.4299e+02, 7.4683e+04, 1.4015e+03],
[-9.1003e-04, 0.0000e+00, 0.0000e+00, 7.5445e+03, 6.8520e+03,
7.4816e+01, 6.8953e+03, 1.3223e+03, 0.0000e+00],
[-1.2588e-06, 0.0000e+00, 0.0000e+00, 7.8311e-01, -1.4697e+00,
-7.0682e+00, -1.4784e+00, 9.4972e+00, -6.7060e-01],
[-8.3262e-07, 0.0000e+00, 0.0000e+00, 4.7344e+00, 7.3474e+00,
-1.3392e+00, 7.3688e+00, 8.2021e+00, 1.9641e+00],
[-4.8638e-06, 0.0000e+00, 0.0000e+00, 5.7771e+00, -2.7623e-01,
2.0502e+00, -2.9240e-01, 3.8128e+01, 0.0000e+00]],
[[-2.3639e-04, 0.0000e+00, 0.0000e+00, 9.7706e+02, 1.8406e+03,
-4.0934e+02, 1.8551e+03, -1.3563e+04, 6.9715e+03],
[-9.9493e-04, 0.0000e+00, 0.0000e+00, 6.6091e+03, 7.9646e+02,
-8.0867e+01, 7.8601e+02, 7.4873e+04, 1.4394e+03],
[-1.4587e-03, 0.0000e+00, 0.0000e+00, 7.6551e+03, 6.8452e+03,
1.1463e+02, 6.8882e+03, 2.0606e+03, 0.0000e+00],
[-1.8648e-06, 0.0000e+00, 0.0000e+00, 6.8746e-01, -1.5094e+00,
-7.0609e+00, -1.5183e+00, 8.8192e+00, -8.1986e-01],
[-1.3573e-06, 0.0000e+00, 0.0000e+00, 4.9620e+00, 7.3322e+00,
-1.3380e+00, 7.3530e+00, 1.1368e+01, 1.9337e+00],
[-7.2855e-06, 0.0000e+00, 0.0000e+00, 5.5959e+00, -4.2310e-01,
2.0482e+00, -4.4015e-01, 3.7854e+01, 0.0000e+00]],
[[-2.1347e-05, 0.0000e+00, 0.0000e+00, 9.3551e+02, 1.9147e+03,
-5.0063e+01, 1.9296e+03, -1.4056e+04, 7.0032e+03],
[-1.1556e-04, 0.0000e+00, 0.0000e+00, 6.3721e+03, 4.2287e+02,
-1.2796e+01, 4.1132e+02, 7.4506e+04, 1.3391e+03],
[-1.4850e-04, 0.0000e+00, 0.0000e+00, 7.3586e+03, 6.8570e+03,
1.0416e+01, 6.9007e+03, 1.2123e+02, 0.0000e+00],
[-2.3538e-07, 0.0000e+00, 0.0000e+00, 9.3487e-01, -1.4041e+00,
-7.0738e+00, -1.4127e+00, 1.0529e+01, -4.2882e-01],
[-1.3108e-07, 0.0000e+00, 0.0000e+00, 4.3500e+00, 7.3653e+00,
-1.3401e+00, 7.3879e+00, 3.0606e+00, 2.0114e+00],
[-8.9351e-07, 0.0000e+00, 0.0000e+00, 6.0496e+00, -3.8659e-02,
2.0517e+00, -5.3361e-02, 3.8334e+01, 0.0000e+00]],
[[-7.5956e-05, 0.0000e+00, 0.0000e+00, 9.4953e+02, 1.8926e+03,
-1.5984e+02, 1.9074e+03, -1.3896e+04, 6.9956e+03],
[-3.7613e-04, 0.0000e+00, 0.0000e+00, 6.4412e+03, 5.3712e+02,
-3.3593e+01, 5.2592e+02, 7.4574e+04, 1.3702e+03],
[-5.0552e-04, 0.0000e+00, 0.0000e+00, 7.4516e+03, 6.8555e+03,
4.2258e+01, 6.8990e+03, 7.1577e+02, 0.0000e+00],
[-7.4801e-07, 0.0000e+00, 0.0000e+00, 8.6031e-01, -1.4367e+00,
-7.0720e+00, -1.4454e+00, 1.0029e+01, -5.4842e-01],
[-4.5490e-07, 0.0000e+00, 0.0000e+00, 4.5425e+00, 7.3574e+00,
-1.3398e+00, 7.3795e+00, 5.6055e+00, 1.9883e+00],
[-2.8643e-06, 0.0000e+00, 0.0000e+00, 5.9181e+00, -1.5612e-01,
2.0513e+00, -1.7156e-01, 3.8269e+01, 0.0000e+00]],
[[-8.5479e-05, 0.0000e+00, 0.0000e+00, 9.5163e+02, 1.8891e+03,
-1.7716e+02, 1.9039e+03, -1.3872e+04, 6.9943e+03],
[-4.1784e-04, 0.0000e+00, 0.0000e+00, 6.4523e+03, 5.5514e+02,
-3.6874e+01, 5.4400e+02, 7.4588e+04, 1.3750e+03],
[-5.6535e-04, 0.0000e+00, 0.0000e+00, 7.4661e+03, 6.8551e+03,
4.7282e+01, 6.8985e+03, 8.0949e+02, 0.0000e+00],
[-8.2773e-07, 0.0000e+00, 0.0000e+00, 8.4846e-01, -1.4419e+00,
-7.0715e+00, -1.4505e+00, 9.9482e+00, -5.6728e-01],
[-5.1013e-07, 0.0000e+00, 0.0000e+00, 4.5724e+00, 7.3560e+00,
-1.3398e+00, 7.3780e+00, 6.0066e+00, 1.9846e+00],
[-3.1740e-06, 0.0000e+00, 0.0000e+00, 5.8968e+00, -1.7466e-01,
2.0511e+00, -1.9021e-01, 3.8252e+01, 0.0000e+00]]])
Batch TLEs:#
As for the time derivatives, the API stays practically identical:
#we load 6 TLEs:
inp_file="""0 PSLV DEB
1 35350U 01049QJ 22068.76869562 .00000911 00000-0 24939-3 0 9998
2 35350 98.6033 64.7516 0074531 99.8340 261.1278 14.48029442457561
0 PSLV DEB *
1 35351U 01049QK 22066.70636923 .00002156 00000-0 63479-3 0 9999
2 35351 98.8179 29.5651 0005211 45.5944 314.5671 14.44732274457505
0 SL-18 DEB
1 35354U 93014BD 22068.76520028 .00021929 00000-0 20751-2 0 9995
2 35354 75.7302 100.7819 0059525 350.7978 9.2117 14.92216400847487
0 SL-18 DEB
1 35359U 93014BJ 22068.55187275 .00025514 00000-0 24908-2 0 9992
2 35359 75.7369 156.1582 0054843 50.5279 310.0745 14.91164684775759
0 SL-18 DEB
1 35360U 93014BK 22068.44021735 .00019061 00000-0 20292-2 0 9992
2 35360 75.7343 127.2487 0071107 32.5913 327.9635 14.86997880798827
0 METEOR 2-17 DEB
1 35364U 88005Y 22067.81503681 .00001147 00000-0 84240-3 0 9995
2 35364 82.5500 92.4124 0018834 303.2489 178.0638 13.94853833332534"""
lines=inp_file.splitlines()
#let's create the TLE objects
tles=[]
for i in range(0,len(lines),3):
data=[]
data.append(lines[i])
data.append(lines[i+1])
data.append(lines[i+2])
tles.append(dsgp4.tle.TLE(data))
#we also create 6 random times, tracking the gradients:
tsinces=torch.rand((6,))
#let's now initialize the TLEs, activating the gradient tracking for the TLE parameters:
tle_elements,tle_batch=dsgp4.initialize_tle(tles,with_grad=True)
#let's now propagate the batch of TLEs:
state_teme = dsgp4.propagate_batch(tle_batch,tsinces)
Finally, we can build the matrix that contains the partial of the SGP4 output w.r.t. the TLE parameters, for each TLE:
#now we can build the partial derivatives matrix, of shape Nx6x9 (N is the number of tsince elements, 6 is the number of elements in the state vector, and 9 is the number of elements in the TLE):
partial_derivatives = torch.zeros((len(tsinces),6,9))
for k in range(len(tsinces)):
for i in range(6):
tle_elements[k].grad=None
state_teme[k].flatten()[i].backward(retain_graph=True)
partial_derivatives[k,i,:] = tle_elements[k].grad
#let's print them to screen:
print(partial_derivatives)
tensor([[[-4.5642e-05, 0.0000e+00, 0.0000e+00, -1.3664e+03, 9.4478e+02,
3.3895e+01, 9.2021e+02, -3.2021e+04, -6.4385e+03],
[-2.2545e-04, 0.0000e+00, 0.0000e+00, 2.0294e+03, -4.9629e+02,
-1.8026e+01, -5.4239e+02, -6.7916e+04, 3.0428e+03],
[ 3.6253e-04, 0.0000e+00, 0.0000e+00, -1.3881e+04, 7.0426e+03,
-3.5086e+00, 7.0257e+03, 4.5703e+01, 0.0000e+00],
[-1.5704e-07, 0.0000e+00, 0.0000e+00, 3.0031e+00, -3.2061e+00,
6.6833e+00, -3.1952e+00, 4.8505e+00, 5.7024e-01],
[-1.3699e-07, 0.0000e+00, 0.0000e+00, 6.7659e+00, -6.7649e+00,
-3.1531e+00, -6.7610e+00, -3.5072e+00, 9.6711e-01],
[-5.5091e-07, 0.0000e+00, 0.0000e+00, -1.1228e+00, -9.4534e-02,
-1.1154e+00, -4.0051e-02, 3.9013e+01, 0.0000e+00]],
[[-8.0306e-06, 0.0000e+00, 0.0000e+00, -5.0992e+03, 5.2614e+02,
2.2033e+00, 5.2422e+02, -6.5461e+04, -3.5128e+03],
[-9.1220e-06, 0.0000e+00, 0.0000e+00, -1.1010e+03, -9.5769e+02,
-7.6949e+00, -9.5969e+02, -3.7114e+04, 6.1947e+03],
[ 2.5588e-05, 0.0000e+00, 0.0000e+00, -1.0039e+04, 7.0386e+03,
1.1141e+00, 7.0437e+03, -1.1550e+02, 0.0000e+00],
[-2.1186e-08, 0.0000e+00, 0.0000e+00, 5.0342e+00, -6.5178e+00,
3.6462e+00, -6.5199e+00, 2.8621e+00, 1.0064e+00],
[ 1.0290e-08, 0.0000e+00, 0.0000e+00, 1.9299e+00, -3.6955e+00,
-6.4300e+00, -3.6972e+00, -5.3299e+00, 5.4971e-01],
[-1.2481e-07, 0.0000e+00, 0.0000e+00, 5.1949e+00, -1.0590e-02,
-1.1439e+00, -7.8453e-03, 3.9115e+01, 0.0000e+00]],
[[-1.0210e-03, 0.0000e+00, 0.0000e+00, 6.8213e+02, -1.6197e+03,
2.9051e+02, -1.6404e+03, 1.2950e+04, -6.7880e+03],
[ 5.6088e-03, 0.0000e+00, 0.0000e+00, -6.9387e+03, -6.2658e+02,
5.8482e+01, -6.2559e+02, -6.9854e+04, -1.3691e+03],
[-1.8029e-04, 0.0000e+00, 0.0000e+00, 2.4669e+03, 6.7121e+03,
7.1367e+01, 6.7913e+03, 1.3672e+03, 0.0000e+00],
[ 5.3471e-06, 0.0000e+00, 0.0000e+00, -1.4288e+00, 1.5024e+00,
7.2321e+00, 1.5134e+00, -8.0795e+00, 6.7804e-01],
[ 1.7537e-06, 0.0000e+00, 0.0000e+00, -2.2066e+00, -7.4597e+00,
1.3796e+00, -7.5028e+00, -8.4025e+00, -1.7767e+00],
[-2.1914e-05, 0.0000e+00, 0.0000e+00, 7.1464e+00, -3.1671e-01,
1.8681e+00, -3.2754e-01, 3.7514e+01, 0.0000e+00]],
[[-4.1331e-03, 0.0000e+00, 0.0000e+00, 4.8139e+03, -2.9842e+02,
1.6269e+02, -2.7500e+02, 6.5136e+04, -2.7085e+03],
[-7.1212e-04, 0.0000e+00, 0.0000e+00, 6.8869e+02, -1.7380e+03,
3.7592e+02, -1.7620e+03, -2.9324e+04, -6.3858e+03],
[ 9.1307e-03, 0.0000e+00, 0.0000e+00, -1.0126e+04, 6.7235e+03,
1.0007e+02, 6.7726e+03, 1.9440e+03, 0.0000e+00],
[ 6.6237e-06, 0.0000e+00, 0.0000e+00, -5.3289e+00, 6.9847e+00,
2.9636e+00, 7.0099e+00, 4.9130e+00, 1.9075e+00],
[ 4.3232e-06, 0.0000e+00, 0.0000e+00, 8.6657e-01, -2.9542e+00,
6.7120e+00, -2.9730e+00, -1.2479e+01, -2.9706e-01],
[-2.6048e-05, 0.0000e+00, 0.0000e+00, 5.3598e+00, -4.7766e-01,
1.8608e+00, -4.5003e-01, 3.7214e+01, 0.0000e+00]],
[[-3.2274e-03, 0.0000e+00, 0.0000e+00, 4.9141e+03, -1.2074e+03,
1.9085e+02, -1.2069e+03, 4.3002e+04, -5.4877e+03],
[ 2.4299e-03, 0.0000e+00, 0.0000e+00, -3.5001e+03, -1.2403e+03,
1.4899e+02, -1.2755e+03, -5.7011e+04, -4.2498e+03],
[ 4.3177e-03, 0.0000e+00, 0.0000e+00, -7.0483e+03, 6.7279e+03,
5.7591e+01, 6.8103e+03, 1.0943e+03, 0.0000e+00],
[ 4.4628e-06, 0.0000e+00, 0.0000e+00, -3.4989e+00, 4.6611e+00,
5.8553e+00, 4.6853e+00, -4.1652e+00, 1.3771e+00],
[ 8.3629e-07, 0.0000e+00, 0.0000e+00, 1.8756e+00, -6.0077e+00,
4.4551e+00, -6.0499e+00, -1.0302e+01, -1.3025e+00],
[-1.5942e-05, 0.0000e+00, 0.0000e+00, 6.4941e+00, -2.9147e-01,
1.8662e+00, -2.6673e-01, 3.7692e+01, 0.0000e+00]],
[[-1.7827e-06, 0.0000e+00, 0.0000e+00, -6.0292e+02, 7.6122e+02,
6.1426e+03, 7.5833e+02, 7.1336e+03, 3.8856e+03],
[ 5.4816e-06, 0.0000e+00, 0.0000e+00, -4.1835e+03, -6.1837e+03,
2.5800e+02, -6.1607e+03, 4.1623e+04, -6.4098e+02],
[ 1.0834e-05, 0.0000e+00, 0.0000e+00, 5.9495e+03, -3.8233e+03,
8.0417e+02, -3.8087e+03, -6.7945e+04, 0.0000e+00],
[-5.1654e-08, 0.0000e+00, 0.0000e+00, -7.5270e-01, 6.4857e-01,
-3.8556e+00, 6.4732e-01, 4.2900e+00, 6.2455e+00],
[ 4.1567e-07, 0.0000e+00, 0.0000e+00, 6.3441e+00, 3.9251e+00,
-1.6105e-01, 3.9181e+00, -3.3627e+01, 7.6815e-01],
[ 2.6039e-07, 0.0000e+00, 0.0000e+00, 3.7111e+00, -6.2227e+00,
-5.0294e-01, -6.2109e+00, -2.2023e+01, 0.0000e+00]]])