## About Michael Paluszek

Michael Paluszek is President of Princeton Satellite Systems. He graduated from MIT with a degree in electrical engineering in 1976 and followed that with an Engineer's degree in Aeronautics and Astronautics from MIT in 1979. He worked at MIT for a year as a research engineer then worked at Draper Laboratory for 6 years on GN&C for human space missions. He worked at GE Astro Space from 1986 to 1992 on a variety of satellite projects including GPS IIR, Inmarsat 3 and Mars Observer. In 1992 he founded Princeton Satellite Systems.

# Maximum Achievable Velocity Change

The rocket equation gives the ratio of the initial mass to the final mass given a velocity change and an exhaust velocity.

$$\frac{m_i}{m_f} = e^\frac{\Delta V}{V_e}$$

This seems to say that given enough fuel we could get an infinite velocity change! To see what the maximum possible velocity change could be we need to account for the structural fraction. The structural fraction multiplied by the mass of fuel gives the mass of the structure needed to support and contain the fuel. The rocket equation now is as follows

$$\frac{m_h + (1+f)m_p}{m_h + fm_p} = e^\frac{\Delta V}{V_e}$$

where $m_p$ is the mass of propellant, $f$ is the structural fraction, and $m_h$ is the mass of all other hardware. If we let the mass of propellant go to infinity, and solve for the velocity change, we get:

$$\frac{\Delta V}{V_e} = \log{\frac{1+f}{f}}$$

The following plot shows the ratio of velocity change to exhaust velocity for a range of structural fractions.

# Reaction Wheel Friction Models

Reaction wheels are used in many spacecraft for attitude control. A reaction wheel is a momentum exchange device because it controls the spacecraft by exchanging momentum with the rest of the spacecraft. Momentum is exchanged via a motor that is fixed to the spacecraft. As with all rotating parts it is subject to friction. Friction needs to be modeled as part of the design process.

The standard way to model friction is with three terms. One is damping which is proportional to wheel speed. The faster the wheel spins the more friction torque is produced. Ultimately, this limits the net control torque. At some speed the motor is just balancing the friction torque. The second component is Coulomb friction that is constant but flips signs when the wheel speed changes sign. the third is static friction. It is like Coulomb friction but only exists at zero speed.

An alternative friction model is known as the bristle friction model. This models friction as bristles that bend. It also has the same friction components described above but they are applied though the bristle state.

Both models can be made to produce similar results as shown in the following figure.

The static friction is clearly seen. The wheel speeds are nearly the same. The middle plot is of the bristle state. The problem with these models is when the torque is low and the wheel speed passes through zero. We then get limit cycling as shown below.

This is due to numerical error.

We can eliminate the limit cycling by using a very small integration time step with the bristle friction model. An alternative approach is to use the first model and multiply the sum of the static and Coulomb friction with a sigmoid, or s function.

The coefficient of the sigmoid function is beta. Very small betas remove the static friction, and all Coulomb friction, near zero speed. The large betas retain the form of the friction and eliminate the limit cycling!

These models can be found in the Spacecraft Control Toolbox 2015.1 . This particular script will be available in 2015.2 which will be released in July.

# Two Stage to Orbit with the Launch Vehicle Toolbox

The Launch Vehicle Toolbox (LVT) combines the Spacecraft Control Toolbox, the Aircraft Control Toolbox and additional libraries of launch vehicle functions and scripts. We’ve used it internally to support a number of contracts.

We have studied two stage to orbit vehicles for a number of years. Our design, known as Space Rapid Transit, uses an aircraft first stage (the Ferry) with a turbo-ramjet engine to take the launch vehicle to 40 km and Mach 6.5. The turbo-ramjet engine is dual fuel using jet fuel for the turbofan and hydrogen for the ramjet. The turbofan core would be based on an existing modern jet engine. A hydrogen fueled turbo-ramjet was tested by MBB for their Sanger launch vehicle. Hydrogen fueled ramjets have been tested by NASA. The SR-71 engine was an early operational turbo-ramjet.

The Orbiter uses a cryogenic hydrogen/oxygen engine to enter the transfer ellipse and then circularize the orbit. The Ferry engine can operate in pure turbofan mode for efficient low-speed operations such as moving the Orbiter between airfields.

TSTODemo.m is a LVT script that models the trajectory from takeoff through circular orbit insertion. The TSTO stack starts on the runway in takeoff mode. When it is moving at the takeoff speed it pulls up and climbs. It transitions from turbofan to ramjet and climbs to the separation altitude and velocity. The simulation works with flight path and heading angles. You can try flying the vehicle in a variety of trajectories. The following figure shows the trajectory up to Ferry/Orbiter separation.

The Space Rapid Transit vehicle is documented in this paper:

Paluszek, M. and J. Mueller, Space Rapid Transit – A Two Stage to Orbit
Fully Reusable Launch Vehicle, IAC-14,C4,6.2, International Astronautical Congress, Toronto, Ontario Canada, October 2014.

The Orbiter starts at the termination condition. The script computes a transfer orbit and the necessary velocity changes to get the Orbiter into an ISS altitude orbit. Part of the delta-V is the drag loss. The Orbiter trajectory is not simulated. The architecture of LVT makes it easy to build these kind of analysis and simulation scripts. Your aren’t locked into a specific design path as can happen with GUI based tools.

For more information go to Launch Vehicle Toolbox for MATLAB.

# SolidWorks Interface in SCT 2015.1

Version 2015.1 will have a new DXF file format exporter to export CAD models built in the Spacecraft Control Toolbox into SolidWorks. The following figure shows the Lunar Lander model in the Spacecraft Control Toolbox CAD window.

Exporting requires just two lines of code:
 g = BuildCADModel( 'get model' ); ExportDXF(g,'LunarLander'); 
Rodger Stephens of Prism Engineering provided SolidWorks models from the DXF file. The file opened in SolidWorks with 7 parts creating an assembly called LunarLander-1.

Each part contains planes, sketches, and surfaces.

The Spacecraft Control Toolbox has always had DXF import capability but now it can export in a format that is supported by most CAD packages. This will speed the process of going from conceptual designs in the Spacecraft Control Toolbox to detailed designs in SolidWorks and other CAD packages.

# Patched Conics

Patched conics are a useful approximation when dealing with orbits that are under the influence of multiple planets or moons. The idea is that only one planet’s or moon’s gravitational field is active at any one time. For example, at the start of a mission from Earth orbit to the Moon, we assume that only the Earth’s gravity acts on the spacecraft. For each planet or moon we define a sphere of influence where that body’s gravity is greater than all other sources. In the Earth/Moon system the Moon’s sphere of influence extends to about 66,000 km from the moon.

A new function in the Spacecraft Control Toolbox Release 2015.1 is PatchedConicPlanner.m. It allows you to explore trajectories in a two-body system. The following figure shows the trajectory of the spacecraft and the orbit of the Moon in the Earth-centered frame. The trajectories assume that the spacecraft is only under the influence of the Earth. The spacecraft is in an elliptical orbit designed to have its apogee just behind the Moon.

The next figure shows the spacecraft in the Moon centered frame. The blue line is the trajectory of the spacecraft assuming that the Moon was not there. The green line is the hyperbolic trajectory of the spacecraft starting from the patch point computed assuming the Earth’s gravity had no influence on the trajectory. Notice the sharp turn due to the Moon’s gravity. The function returns the Moon-centered orbital elements along with other useful quantities.

The following shows a closeup of the trajectory. The miss distance, as expected, is less for the hyperbolic trajectory. The plot clearly shows a good place for a delta-v maneuver to put the spacecraft into lunar orbit.

This function allows you quickly explore the effect of different patch points and to try different spacecraft transfer orbits. While a “high-fidelity” analysis requires numerical orbit propagation that includes the Moon, Sun and Earth’s gravitational fields, PatchedConicPlanner.m, let’s you generate good starting trajectories for mission planning.

# Heading to the Moon

We have transitioned our lunar lander work from the Spacecraft Control Toolbox to VisualCommander. Here is a simulation of the lander heading to the moon on the elliptical transfer orbit designed in our Landing on the Moon blog post.

The model was discussed in our Moon Lander Design blog post. We exported it from the Spacecraft Control Toolbox as a Wavefront obj file. The textures were applied by Amazing3D Graphics. Amazing 3D Graphics builds very high quality models with low polygon counts that are ideal for simulation and games.

The attitude control system is our Precision ACS system. In the next few weeks we’ll be adding software to perform mid-course corrections, lunar orbit insertion and lunar landing. Say tuned!

# Moon Lander Design

Our last post showed the mission planning script for our lunar lander. The next step was to layout the lander. We did this using the BuildCADModel function in the Spacecraft Control Toolbox. The propulsion system is designed to meet the requirements of the mission plan. We use six 1 N HPGP thrusters for attitude control and one 220 N thruster for orbit maneuvers and landing. We have two HPGP tanks for the fuel. There are two cameras. One is used as a star camera for attitude determination and navigation and the second, which is articulated, is used for optical navigation, descent navigation and science. The IMU and C&DH box can bee seen in the drawing.

The solar array has two degrees-of-freedom articulation. The high gain antenna is also articulated. We adapted the landing legs from the Apollo Lunar Module. The thruster layout is shown in the following figure and is done using the ThrusterLayout function in the toolbox.

We get full 6 degree-of-freedom attitude control and z-axis velocity change control. We use the 220 N engine as the primary engine for landing but can also use four of the 1 N thrusters for fine terminal control.

We are working on the science payload for the mission. One experiment will be to mine helium-3 from the surface. Helium-3 would be a fuel for advanced nuclear fusion power plants and nuclear fusion propulsion systems.

# Landing on the Moon

There is a lot of interest in lunar landing missions for both scientific exploration and commercial purposes. Commercial applications might include mining helium-3 for future nuclear fusion power plants on earth and mining water for rocket fuel.

The Spacecraft Control Toolbox makes it easy to do preliminary planning for lunar missions. In this blog we present a single MATLAB script that takes a spacecraft from a low Earth parking orbit to the lunar surface! Here is the final segment, the descent to the moon.

We ended up with a 30 kg dry mass for a spacecraft that can use an ECAPS 220 N HPGP thruster for delta-v.

The published script can be found here:

Lunar Mission Planning as a published MATLAB script

You can also send us an email to find out more about our Lunar Mission Design Tools.

# 2014 International Astronautical Congress

From the movie “2001: A Space Odyssey”, 1968. Dr. Heywood Floyd is talking with Elena, a colleague from Russia:

Elena, “Well, I hope that you and your wife can come to the I.A.C conference in June.”
Floyd, “We’re trying to get there. I hope we can.”

I was able to attend the IAC conference in 2014 in Toronto, Ontario, Canada. I presented two papers:

“Direct Fusion Drive for a Human Mars Orbital Mission”

“Space Rapid Transit – A Two Stage to Orbit Fully Reusable Launch Vehicle”.

2001: A Space Odyssey was the “theme” for my two papers. I had a photo of the Discovery II spacecraft in my DFD talk and an image from an online simulator of the full Orion III launch vehicle in my SRT talk. Both papers were well received. I got a good question from an engineer from Reaction Engines Limited about separation. We have done some separation simulations but have not testing our separation control mode in depth. He noted that the D-21 program, and example of high speed separation I gave, was not really successful.

When I wasn’t in my sessions, I spent my time in the exhibits hall talking with the representatives at the booths, handing out business cards and flyers about Princeton Satellite Systems. Some of our customers, including KARI from Korea and the Canadian Space Agency, had exhibits.

I spoke at length with Astrobotics, a company that plans to land a rover on the moon. They were founded by a professor from Carnegie Mellon. I suggested that our flight control experience could be of value to them. Their work shows the feasibility of helium-3 mining on the moon. We would need helium-3 mining if we were ever to use DFD for base load terrestrial power generation.

I chatted with the Aerospace Corporation. I worked with them on GPS IIR while at GE Astro Space. I explained that they might be interested in working with us on DFD particularly in applying it to Air Force applications like space based radar.

SpaceX had the crew chairs and displays from their Dragon Capsule in their exhibit. It was the coolest exhibit in the hall! I had a nice chat with their marketing person on DFD. SpaceX and Boeing recently were awarded contracts to develop the Commercial Crew vehicle.

Lockheed Martin had a huge exhibit. They had a 3D printer running

I talked with them about A2100, a comsat under design at GE Astro Space when I left. I also talked about ControlPlan applications for MUOS, a satellite Lockheed Martin is building for the Navy. We developed antenna beam optimization for MUOS using our ControlPlan multi-objective optimization package.

I spoke with Surrey about their new comsat program and suggested that we could help as we have extensive comsat experience. Surrey now has a U.S. branch. I spoke with the Canadian Armed Forces about their satellite programs. They were interested in our Kestrel Eye work.

The CN Tower is in the middle of the convention center:

On the way home I ate at the Apropos restaurant in the Air Canada terminal. It is really cool! You order through an iPad and pay using a credit card terminal next to the iPad. Besides being high tech, the food was really good! The restaurant can be seen in the following picture.

# Attitude Maneuvers with the CubeSat Control System

The CubeSat control system is designed to work with either thrusters or reaction wheels. It has a number of handy built in maneuver modes such as pointing at the sun, nadir pointing or pointing at a specific latitude and longitude on the ground. Here is the spacecraft shown in the VisualCommander interface.

The movie in the link below shows attitude maneuvers in the VisualCommander interface. The interface has pages for the various subsystems and attitude control system functions. We start by seeing the spacecraft in a polar orbit on the Summary page. The solar arrays are reorienting themselves so that their cell faces are pointed at the sun. We switch the 3D display to look along the boresight of the telescope. We then go to the ACS page and select a sun pointing maneuver. We go back to the Summary page and see that the sun appears in the display. We then return to the ACS page and command nadir pointing. The remainder of the movie shows the reorientation maneuver to nadir pointing.

For more information on our simulation frameworks including our real-time control system framework, ControlDeck, go to Simulation Framework page.

For more information on VisualCommander go to VisualCommander page.

You can also send us an email to find out more about our CubeSatControl System. All of these products are available now.