# if there are any advantages of time regularization such as Sundman transformation in gravity-assisted trajectory optimization?

I recently came across an article titled "A Time Regularization Scheme for Spacecraft Trajectories Subject to Multi‑Body Gravity,"(https://doi.org/10.1007/s40295-023-00364-0) which introduces a novel time regularization scheme. I'm intrigued by this concept, but I'm confused about some of the details and concepts discussed.

I want to know if there is any chance to apply time regularization to trajectory optimization. if there are more messages about time regularization? What are the real advantages of time regularization?

What are the real advantages of time regularization?

The key advantage of time regularization applies to eccentric orbits and using a fixed step size integrator. The issue is that there's always a tradeoff between errors inherent to the integration technique itself, which makes smaller step sizes better, and errors due to using finite precision arithmetic which makes larger step sizes better. This in turn means that there's a sweet spot in the time step size that balances out these two competing error sources.

Taking equal time steps means either too large of a step size at periapsis or too small of a step size at apoapsis. Equal time steps oftentimes is not optimal with regard to computation time and with regard to accuracy. Adaptive step size integrators are one way to address this key challenge of finding the optimal step size, but they have their own problems, and they also benefit from time regularization. Yet another approach is to use extended precision arithmetic so that the sweet spot occurs at a much shorter time step. This approach unfortunately drastically increases computation time.

The Sundman transform regularizes time via $$dt = c r ds$$ where $$t$$ is time, $$c$$ is a constant scale factor, $$r$$ is the distance to the central body, and $$s$$ is the regularized equivalent of time. This has been generalized to $$dt = c r^n ds$$ where $$n$$ is a constant exponential power. It turns out that

• $$n=0$$ results in equal time steps, or equivalently equal mean anomaly steps.
• $$n=1$$ (the original form of the Sundman transform) results in equal steps in eccentric anomaly.
• $$n=2$$ results in equal steps in true anomaly.
• A number of papers suggest that using $$n=3/2$$ is a good choice that comes close to hitting that sweet spot. Many call this the "intermediate anomaly."

I want to know if there is any chance to apply time regularization to trajectory optimization.

The above works great in a patched conic sense, but maybe not so great with perturbations from non-gravitational forces and from third bodies. This is why so many papers suggest different time regularization schemes; the paper found by the author the question is far from the only one. The goal of time regularization is to drastically decrease computation time without sacrificing accuracy.

With regard to trajectory optimization, one typically starts with a solution found using a patched conic approach (fast but not particularly accurate) and then fine tunes that solution using numerical integration. This numerical integration will be performed multiple times with small tweaks each run to burn timing, magnitude, and direction. Since it is performed many times over, time regularization can make the difference between starting the optimization before lunch and having the results shortly after returning from lunch versus starting the optimization before leaving work and hopefully having the results back the next day (or maybe the next week).

• Here's a nice article by John Baez on some of the mathematical properties of the $dt=r ds$ parametrization. johncarlosbaez.wordpress.com/2015/03/17/… Commented Mar 23 at 16:41
• Thank for your answer. you have mentioned adaptive step size integrators, so what is the problem with this method? and if there are any other useful time regularization schemes being used?
– Liu
Commented Mar 25 at 7:10