pyqlaw

Q-law feedback control for low-thrust orbital transfer in Python

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Capabilities:

• Q-law formulated in Keplerian & SMA-MEE (MEE with semilatus rectum replaced by semimajor axis)
• Coasting capabilities with efficiency parameters [3]
• Thrust duty cycles
• Battery level tracking

Installation

To install, run

pip install pyqlaw

and to uninstall

pip uninstall pyqlaw

The package depends on the following basic python modules:

• sympy, numpy, tqdm, matplotilb, numba, scipy, spiceypy

Overview of Q-law

Q-law is a Lyapunov controller defined in terms of orbital elements, and can be used as a feedback controller to construct suboptimal low-thrust, many-revolution transfers. Q-law is very sensitive to the problem (initial & final orbital elements, choice of orbital elements, thruster specs = control authority) as well as its various hyperparamters, which must be chosen carefully. In general, the following should be kept in mind:

• For numerical stability, always work with canonical scales.
• Be very careful with initial/final orbits not to contain singular elements (e.g. inclination ~ 0 deg in Keplerian elements representation).
• Q-law is not suitable for high control authority applications (e.g. interplanetary transfer with 0~very few revolutions).
• Taking larger integration time steps t_step (or angle steps, if use_sundman = True) makes the algorithm ``faster'' (less time until reaching the targeted elements), but may also lead to instability/high jitter once the spacecraft is close to the target; an appropriate value must be found on a problem-to-problem basis.

For more discussions, see for example:

• Petropoulos, A. E. (2004). Low-thrust orbit transfers using candidate Lyapunov functions with a mechanism for coasting. Collection of Technical Papers - AIAA/AAS Astrodynamics Specialist Conference, 2(August), 748–762. https://doi.org/10.2514/6.2004-5089
• Petropoulos, A. E. (2005). Refinements to the Q-law for low-thrust orbit transfers. AAS/AIAA Space Flight Mechanics Meeting.
• Hatten, N. (2012). A Critical Evaluation of Modern Low-Thrust, Feedback-Driven Spacecraft Control Laws.

Basic usage

Test files are included in ./tests/. Here, we present a basic example. Before starting, please note a couple of things:

• Dynamics & spacecraft parameters are given in non-dimensional quantities, scaling GM = 1.0 (which may be modified, but it is numerically desirable to use this scaling).
• All angles are defined in radians.
• Due to instability of Gauss's equation and the Lyapunov feedback control law, some of the elements should not be smaller in magnitude than a certain safe-guarding threshold value. This is why some of the starting elements in the following example are not set to 0, but a mild value (e.g. 1e-2, 1e-3).

We start by importing the module

import pyqlaw

Construct initial and final Keplrian elements to target, along with weighting

# initial and final elements (always in order: [SMA, ECC, INC, RAAN, AOP, TA])
oe0 = np.array([1.0, 1e-2, 1e-2, 1e-3, 1e-3, 1e-3])
oeT = np.array([1.1, 5e-3, 0.2, 0.0, 0.0])
woe = [1.0, 1.0, 1.0, 1.0, 0.0]

Provide spacecraft parameters (max thrust and mass-flow rate), max transfer time, and time-step (note that integration is done using fixed-steps):

# spacecraft parameters
mass0 = 1.0
tmax = 1e-3
mdot = 1e-4
tf_max = 300.0
t_step = 0.1

Construct the problem object, then set the problem parameters

prob = pyqlaw.QLaw()
prob.set_problem(oe0, oeT, mass0, tmax, mdot, tf_max, t_step, woe=woe)
prob.pretty()  # print info

Transfer:
  sma  : 1.0000e+00 -> 1.1000e+00 (weight: 1.00)
  ecc  : 1.0000e-02 -> 5.0000e-03 (weight: 1.00)
  inc  : 1.0000e-02 -> 2.0000e-01 (weight: 1.00)
  raan : 1.0000e-03 -> 0.0000e+00 (weight: 1.00)
  aop  : 1.0000e-03 -> 0.0000e+00 (weight: 0.00)

solve the problem

prob.solve()
prob.pretty_results()   # print info

qlaw:  99%|█████████████████████████████████████████████████████████████████▏| 2970/3000 [03:03<00:01, 16.20it/s]
Target elements successfully reached!
Final state:
  sma  : 1.1000e+00 (error: 1.8297e-05)
  ecc  : 5.0141e-03 (error: 1.4138e-05)
  inc  : 1.9994e-01 (error: 5.7253e-05)
  raan : 4.5828e-04 (error: 4.5828e-04)
  aop  : 1.2206e-01 (error: 1.2206e-01)

Some conveninence methods for plotting:

fig1, ax1 = prob.plot_elements_history()
fig2, ax2 = prob.plot_trajectory_3d()

Reporting issues

If you find a bug in pyqlaw, please report it by creating a new issue on the repository. Please make sure to include an example the demonstrates the issue.

Contributing

If you have an idea for improving the module, please feel free to submit an issue with your proposal for enahncement!

References

[1] Petropoulos, A. E. (2003). Simple Control Laws for Low-Thrust Orbit Transfers. AAS Astrodynamics Specialists Conference.

[2] Petropoulos, A. E. (2004). Low-thrust orbit transfers using candidate Lyapunov functions with a mechanism for coasting. AIAA/AAS Astrodynamics Specialist Conference, August. https://doi.org/10.2514/6.2004-5089

[3] Petropoulos, A. E. (2005). Refinements to the Q-law for low-thrust orbit transfers. Advances in the Astronautical Sciences, 120(I), 963–982.

[4] Shannon, J. L., Ozimek, M. T., Atchison, J. A., & Hartzell, C. M. (2020). Q-law aided direct trajectory optimization of many-revolution low-thrust transfers. Journal of Spacecraft and Rockets, 57(4), 672–682. https://doi.org/10.2514/1.A34586

[5] Leomanni, M., Bianchini, G., Garulli, A., Quartullo, R., & Scortecci, F. (2021). Optimal Low-Thrust Orbit Transfers Made Easy: A Direct Approach. Journal of Spacecraft and Rockets, 1–11. https://doi.org/10.2514/1.a34949

[6] Modified Equinoctial Elements (careful with typos in this document!)

[7] Hatten, N. (2012). A Critical Evaluation of Modern Low-Thrust, Feedback-Driven Spacecraft Control Laws.