How to use OpenGoddard 4-python library for nonlinear optimal control and orbit generation
https://github.com/istellartech/OpenGoddard https://istellartech.github.io/OpenGoddard/
How to use OpenGoddard 1 How to use OpenGoddard 2 How to use OpenGoddard 3 How to use OpenGoddard 4 ← Now here
This time, I will talk about the formulation and results using one of the standard techniques for optimization.
Low thrust orbit transition problem
With the development of electric propulsion such as the ion engine, which became famous for being installed in Hayabusa, asteroid explorers are often going from the earth to other planets by electric propulsion instead of chemical propulsion like rocket engines. I did. It is well known that the orbit transition in chemical propulsion can take the orbit called Hohmann orbit with the minimum energy required. However, this is only an engine with sufficient thrust, and at low thrust it will be a trajectory called a spiral trajectory. Most of the orbital transition problems between two points are solved as boundary value problems (Lambert's problem, also known as Lambert's problem), but here we will treat them as optimal control problems.
The original papers are as follows.
Ross, I. Michael, Qi Gong, and Pooya Sekhavat. "Low-thrust, high-accuracy trajectory optimization." Journal of Guidance, Control, and Dynamics 30.4 (2007): 921-933. https://doi.org/10.2514/1.23181
Equation of motion
\begin{align}
\dot{r} &= v_r \\
\dot{v_r} &= v_t^2 / r - 1 / r^2 + u_r \\
\dot{v_t} &= - v_r v_t / r + u_t
\end{align}
Restraint condition
Enter the initial value and the end value to limit the magnitude of thrust.
|u_r| \leq 0.01 \\
|u_t| \leq 0.01 \\
0 < t_f < 55
Evaluation function
The aim is to minimize the total thrust time required.
J = \int_{t_0}^{t_f} |u_r| + |u_t| dt
Absolute value replacement
Since the evaluation function containing absolute values is not smooth, we perform substitution using free variables (also auxiliary variables) as a standard for numerical analysis. http://www.msi.co.jp/nuopt/download/introduction/module/technic.pdf
u = u_1 - u_2 \\
|u| = u_1 + u_2
In consideration of this, describe as 3 state variables ($ r, v_r, v_t
If the evaluation function has an integral term, specify the content of the integral term in running_cost.
def running_cost(prob, obj):
ur1 = prob.controls_all_section(0)
ur2 = prob.controls_all_section(1)
ut1 = prob.controls_all_section(2)
ut2 = prob.controls_all_section(3)
return (ur1 + ur2) + (ut1 + ut2)
prob.running_cost = running_cost
result
Looking at the graph of u, which is a control variable, we can see that the history of thrust is difficult for humans to understand. Even such a difficult problem can be easily solved (although it requires some trial and error and know-how).
Summary
As mentioned above, we have explained the points that can be noted from a simple example when using OpenGoddard. The following is a brief summary.
- Scaling-Align the size of variables
- Interval update-Increase or decrease knot
- knotting method-Definition of non-contiguous parts of state variables
- Replacement of non-smooth things such as absolute values
Although the area remains as an intuitional part with know-how, it is still possible to easily find the solution (orbit) of nonlinear optimal control. It is important to note that there is no guarantee that it is the optimum value because it is a numerical solution, and that it is a locally optimal solution, but I think that there is no problem in many practices. Please try using Open Goddard!
Serpentine 1
Scipy's SLSQP is currently used in v1.0.0 as a solver for sequential quadratic planning, but the performance of this solver is not suitable for large-scale problems. Problems with a large number of variables and nodes will take a considerable amount of calculation time. It seems that Coco Rahen should use a high-class solver that uses the characteristics of sparse matrices, but it has not been verified. SNOPT and IPOPT are often mentioned in papers.
Serpentine 2
In addition to the Open Goddard explained this time, we have released a program called Open Tsiolkovsky.
- Open Goddard (Origin of the name) --- Mr. Robert Goddard of the United States called the father of modern rockets
- OpenGoddard (Contents)-Orbit design by solving optimization problems
- OpenTsiolkovsky (Origin of the name) --Konstantin Zolkovsky, the father of space travel
- OpenTsiolkovsky (contents) --Solving differential equations from rocket equations of motion to calculate rocket orbit
These two are a pair of software, the inverse problem and the forward problem.
The original reason for making it is rocket orbit generation, but as a result Open Goddard is a fairly versatile tool. It may be better to use another tool for large-scale optimal control problems, but I think that it is a good tool because simple calculations have few dependencies and are easy to install (hot self-praise).
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