It is sometimes necessary to change your orbit semi-major axis, ascending node and inclination with a low-thrust engine. It is easy to do, as long as you can point your engine along orbit normal and tangential to the orbit.
It is easiest to see how this is done by looking at the Gauss’ Variational equations, simplified for small eccentricity.
I is inclination, is semi-major axis, is the gravitational parameter, is argument of perigee, is true anomaly, is ascending node. and the orbit tangential acceleration and is the orbit normal acceleration.
The resulting simulation is shown below. Mode 0 is semi-major axis change, Mode 1 is ascending node change, Mode 2 is inclination change and Mode 3 is off. It is best to change inclination and ascending node at the highest semi-major axis. You should change ascending node at the lowest inclination. The burns are done where the rate of changes are higher. Some change in inclination and ascending. node will happen when the other is being corrected.
The script for this simulation with the controller is part of the Spacecraft Control Toolbox Release 2020.1 coming in May.
A popular way of launching a small satellite is to bring it up on an International Space Station resupply mission. The Spacecraft Control Toolbox has functions to help you animate the orbit of your spacecraft near the ISS. A function, ISSOrbit, generates the orbital elements for the ISS. ISSOrbit generates Keplerian Elements from the latest 2-line elements. We use the function CoplanarOrbit to create an orbit 50 m below the ISS. There are no disturbances and the gravity model is for a point mass Earth.
DrawSpacecraft.m is a function that will draw any number of spacecraft in the viewer. This is the ISS and our, very small, NanoSatellite. The MATLAB camera controls allow you to zoom in or rotate the view. The view is with respect to the first satellite entered in the argument list, which in this case is the nano satellite.
DrawSpacecraft also does animation and will create an avi file. You can see the animation on our YouTube Channel or by clicking the video below. We converted the avi file to an mp4 file using a movie converter.
The script is an m-file that you can download, just to view, here.
We developed a round trip mission to Mars using Direct Fusion Drive. Key parameters are:
Specific power of 1.2 kW/kg
Exhaust velocity of 110 km/s
40 MW engine
Payload of 55 MT out, 40 MT return
Outward leg 128 days
Inward leg 110 days
Stay on Mars 650 days
The following plot shows the trajectory. Some additional time would be needed to enter and exit the Mars and Earth’s orbit. That could be shortened by using a nuclear thermal engine tug.
The payload is based on the NASA Deep Space Habitat. It could be replaced by a less massive habitat and a lander. Two spacecraft, one including a lander, is another possibility. A dual spacecraft mission would enhance safety.
This mission plan has the DFD decelerating the spacecraft and going into Mars orbit. Some time could be saved using aerodynamic braking. The mass of the aeroshell would need to be included as part of the “engine” mass.
It may from time to time necessary to damp nutation with thrusters on a momentum bias spacecraft. For example, nutation can happen when you transition from stationkeeping mode, which uses thrusters, to normal mode, in which you use low control authority actuators, such as magnetic torquers, for control.
We’ve written a script that simulation a momentum bias spacecraft. Let’s show the results.
The spacecraft body rate is 0.01 rad/s. This is much greater than you will ever see on your own spacecraft. The peak roll angle is about 18 degrees! The last plot shows the thruster control. In this simulation we don’t apply any control so the line is flat.
All you need is one thruster pulse, properly timed, to drive the x angular rate to zero. You must time it properly so as not to not leave a roll error. What we do is measure the peak angular rate, compute the pulsewidth needed to drive the rate to zero, and turn on the thruster when the roll angle is zero and the nutation rate is at its peak with the appropriate sign. The following plot shows the results.
The results aren’t perfect, as they would not be operationally. We did both simulations with the same Spacecraft Control Toolbox http://www.psatellite.com/products/sct/ script. This is a link to the m-file, saved as a zip file.
You won’t be able to run this without our toolboxes but you can see how we implemented “manual” nutation control. This script, and the new function RHSGyrostat.m will be available in SCT 2020.1 coming soon!
Over 100 new functions were added or had major updates in Version 2019.1. Improvements were made to dozens of existing functions to improve their performance and expand their applications. Built-in demos were added to many functions to make them easier to use in your applications
In the Aircraft Control Toolbox, we added new tools for aircraft simulation. This includes a model builder allowing you to create mass and aerodynamic models from a CAD model that you load from in a Wavefront OBJ format. The model builder GUI is shown below. This tool is matched with a new 6 degree of freedom aircraft simulation with an easier-to-use model viewer, shown under the GUI. The simulation lets you plug in your own aerodynamics and engine models, use the built-in defaults or other models from the toolbox.
The Spacecraft Control Toolbox has many new features. For example, you can now create cross-scale constellations and control them, or any other constellation, using control laws recently developed at PSS and presented at IWSCFF in 2019. A cross-scale constellation is shown below under active control.
We added the Fusion Toolbox for the development of nuclear fusion reactors. This function includes physics models, reactor models, thermal models and many other tools. Both core reactor and balance of plant functions are included.
The below shows a plot from one of the nuclear fusion tools which finds the fusion reaction rate for the aneutronic Deuterium-Helium3 fuel cycle as a function of ion temperature.
One concept Gateway may be in a polar orbit with an apolune of 70,000 km and perilune of 3,000 km. One concept is for the lander and Orion to meet at Gateway. Our alternative is for Artemis to stay in a low lunar orbit and be met there by Orion, the cargo transfer vehicle and the tanker. There are many orbit maneuver sequences that will get us from Gateway to our 15 km altitude orbit. A simple one is shown below. We first lower apogee to 3,000 km we then do a Hohmann transfer from the 3,000 km orbit to the 1753 km orbit (that is 15 km altitude). The maneuver to lower apogee is shown below.
The delta-v for the first maneuver is 0.49 km/s and for the Hohmann transfer is 0.39 km/s.
While in low lunar orbit in between landings the lunar lander will do high resolution photo surveys of the surface. These will be used to train the neural network for landing navigation.
NASA would like a crew to land on the moon by 2024.
We didn’t have time to write a proposal, but here is our design. We propose a single stage vehicle, that can land from and return to a 15 km circular orbit. It uses 2 Blue Origins BE-3U engines that use cryogenic hydrogen and oxygen. An Orion capsule houses the astronauts. The Orion would take astronauts to and from Gateway and to and from the Earth. Lockheed Martin is building the Orion spacecraft. The European Space Agency is building the service module. A separate transport would bring fuel and payload to the lander. In the future, the lander could be refueled from lunar water.
The dimensions are in meters. The Orion is shown below. We purchased the model from https://hum3d.com.
The landing gear were scaled from the Apollo Lunar module.
It is interesting to compare its size with the Apollo Lunar Module. The Artemis is designed to fit into the 10 m SLS fairing. This a fully reusable lunar vehicle that can be refueled. It is designed for a long-term, sustainable, lunar base.
We use two toroidal hydrogen tanks and two spherical oxygen tanks. The cylinder on the outside is the solar array producing 34 kW of power. Of course, numerous details are omitted. We developed this model using our Spacecraft Control Toolbox. The design script will be available in the Spacecraft Control Toolbox Version 2019.1 due in mid-November.
Other elements of the lander were designed for different purposes. The GN&C system is based on our Army Precision Attitude Control System.
Our control system is based on a robotic lander we designed some time ago. We have full C++ code for the control and guidance system.
The architecture for Earth/Moon transportation system is shown below. Eventually, a Direct Fusion Drive freighter would be the main way of moving cargo between Earth orbit, lunar orbit and Gateway. The lander would remain in lunar orbit. Humans would go to the moon using fast orbital transfer, much like during Apollo.
Our next blog post will show how we get from Gateway to and from our 15 km starting orbit. A subsequent post will demonstrate our lunar landing guidance that uses a neural network for navigation based on images of the surface. Using it for landing would require higher resolution images than we have today, but short of building a lunar GPS system, it might be more cost-effective to have a satellite assembling images from low lunar orbit.
We will also update this blog post from time to time. Stay tuned!
As an intern during Summer 2019, one of my tasks was to size and plan a flight for a remote-controlled aircraft testing a rotational detonation engine (RDE). The aircraft needed to reach a speed of Mach 3, remain as close to the airport as possible, and conduct maneuvers in the same fashion as a real aircraft would. To accomplish this, I used a new trajectory model developed by PSS.
After creating the RDE analytical model outputting specific fuel consumption and sizing the aircraft to carry 25 kilograms of hydrogen, I was ready to map the trajectory of the flight. I inputted dry mass, initial fuel mass, wing aspect ratio, and wing area, then plugged in the RDE specific fuel consumption function. The next step was to build flight segments for take-off, climb, turns, cruise, etc. Segments can be simulated separately allowing the user to fine tune parameters like velocity, heading, and pitch. Initial aircraft flight conditions can be set and tracked through segments using the MATLAB debugger. Here is a sample segment of a take-off followed by a turn and climb:
The trajectory I built encompasses a climb to 10 km, an acceleration to Mach 3, and several heading changes to remain in transmitting range of the airport and to set up an approach. A photo of the climb portion and deceleration from Mach 3 is included. The climb shows take-off and initial climb to 200 m, and a turn before continuing the climb. The deceleration phase shows slowing from Mach 3 and a turn to set up an approach to the airport. The total flight time is about 15 minutes. In working at PSS, I have not only learned about aircraft design and cycle analysis, but also CubeSats, space environments and disturbances and improved my coding skills. This summer has been a lot of fun and overall incredible!