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I keep reading that the Powered Explicit Guidance equations (PEG) (detailed in the Technical note linked below) is (an approximation of) the most efficient way to get to space. That being said, I know of ascent trajectory optimisation tools that 1/ offer trajectory optimisation based on a number of phases with prescribed flight behaviours such as constant pitch/rate, Gravity Turn etc. and 2/ can't model the tangent steering profile that the PEG is based on.

If PEG is the most optimal, and given its' ability to adapt to dispersions if computed onboard, why do mission designers consider these other flight phases- and why do some trajectory optimisation softwares not even provide the capability of modelling the linear tangent law?

Is this a limitation of on-board compute? I wouldn't expect so since these algorithms ran on the Shuttle missions.

Are these flight phases used only in the initial part of the trajectory, where assumptions in PEG aren't valid (e.g. presence of an atmosphere)? If so, when does the transition to PEG occur? At some threshold altitude?

These are questions I'm considering as I put together my own simulated project, with the goal of working in Flight Physics/GNC domains when I graduate.

Thanks for your help.

TN: https://ntrs.nasa.gov/citations/19660006073

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Shuttle used open-loop guidance (pre-programmed table of pitch / yaw / roll) for first stage, switching to PEG at SRB sep. This was to avoid exceeding first stage dynamic pressure/structural limits based on winds of the day - the table was calculated day-of-launch based on measured winds.

At least in the early 2000s, Atlas vehicles used a similar scheme.

See for more info:

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    $\begingroup$ The SLS (Space Launch System) also follows a similar scheme. It's much more important to avoid potentially nasty problems at max-q (really nasty, as in potentially tearing a launch vehicle apart level of nastiness) than it is to be slightly fuel inefficient. $\endgroup$ Nov 29, 2022 at 13:07
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The Powered Explicit Guidance is a successful algorithm with many practical applications. However, there are areas of improvement that can be made. As can be seen, there are many approximations made which reduce the optimality of PEG. There is also no way to completely constrain the final po- sition with the original implementation; however, Fill’s improvements should allow for this. In addition, while PEG is able to combine multiple thrust arcs, such as a constant thrust arc followed by a constant acceleration arc, there is no computation of optimal bang-bang switching times for the throttle command between maximum and minimum thrust levels. In fact, PEG was derived assuming that the thrust profile is a known function of time.

-- An Investigation of Fuel Optimal Descent, Jeremy Ryan Rea, 2009 [1]

Having worked on guidance algorithms for on-orbit maneuvers and soft landings on the Moon, I'll add that PEG is useful in the boost stage of rockets (high thrust long burn times) but not as useful when it comes to precise maneuvering (maybe high thrust, possibly very low burn times (10-30 seconds)). Coworkers have disagreed with me on that by correctly pointing out that Orion uses PEG for all of its maneuvering.

In my experience, one of the main issues with PEG is that it targets a Cartesian state with an open-loop time to go: matching the orbital energy is important for transfers (e.g. trans-lunar injections, probably also for lunar orbit injections) but may increase the cost of cleanup maneuvers. In fact, if the Cartesian state is achieved at the wrong epoch, then the orbit may be completely different. Hence, one approach is to use an orbital element targeting scheme (Ruggiero, Q-Law, etc.) to meet the orbital parameters with high accuracy and have the true anomaly as a free parameter. This ensures correct orbital insertion and allows mission designers to then perform clean-up maneuvers to fix specific orbital elements.

The other issue with PEG is that it requires very good knowledge of the acceleration of the vehicle throughout the burn. In turn, this means low-noise inertial measurement units (and low noise aggregation (random walk is small)), and those can be prohibitively expensive (several million dollars each). That's why a common scheme used on non-governmental missions is to have a cheaper IMU and two cut-off thresholds: (noisy) acceleration and max burn time. Then, mission designers simply upload a $\Delta v$ (velocity change) to achieve and an orientation the vehicle should have throughout the burn.

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    $\begingroup$ It could be that Orion uses PEG simply because that algorithm is well known and is easily adaptable to on-orbit operations. That doesn't necessarily mean it's the best algorithm. Orion ran into multiple cost hiccoughs, including multiple software cost hiccoughs. Why develop a new, untested guidance algorithm when they already had one that more or less worked? $\endgroup$ Nov 30, 2022 at 11:16
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    $\begingroup$ A link to Rea's PhD thesis would be useful. Here's one: repositories.lib.utexas.edu/handle/2152/18393 . $\endgroup$ Nov 30, 2022 at 11:23

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