These are the exact notes and presentation of my Seminar in November 2000 to present my Mars research. This was the weekly meeting of the ASE department’s “Research Seminar” class. There were several grad students there and also Cesar Ocampo (my official adviser at the time, in addition to Roger Broucke; Dr. Ocampo set up the seminar, you will see his name on the flyer), Wallace Fowler, PAM Abusali, and David G. Hull. I saw Dr. Abusali in the hallway at Woolrich Labs the following semester. He was surprised I was still on campus. He said he thought that presentation was my dissertation defense. Dr. Broucke was not there.
Optimizing the Earth-to-Mars Trajectory By William H. Clark II, P.E. [Diagram: Orbit illustration showing the Sun at the center, Earth’s orbit, Mars’ orbit, a spacecraft trajectory from Earth to Mars, and a Conjunction point aligned with Earth and Mars.] © 2000 by William H. Clark II, P.E. all rights reserved
Optimizing the Earth to Mars Trajectory I. Introduction II. Theory III. Numerical Optimization Methods IV. Conclusions & Recommendations V. Future: Modeling the Semiconductor Junction Free download of program and code at: http://whcii.home.mindspring.com/astro/marvin.html
Optimizing the Earth to Mars Trajectory Bill Clark, P.E.
[Handwritten URL at top: http://whcii.home.mindspring.com/astro/marvin.html] INTRODUCTION <<slide 1>> Summary I’d like to talk about some research I have been doing on the Earth to Mars trajectory. Specifically, the trajectory of the NASA mission to be launched to Mars in April of next year. I have created a computer simulation using the exact ephemeredes of the planets. All motion is assumed coplanar. Mars’ orbit is at a 3° inclination to the ecliptic, so coplanar motion is not a bad approximation. Other assumptions will have negligible affect on this analysis, and they will be noted in context as appropriate. The title of the report is “Optimizing” the trajectory. Or, as NASA puts it, “Getting there faster and cheaper.” That is important of course, but I have placed more emphasis on getting there “safer.” Just last week NASA announced that the $1 billion rock collecting mission to Mars was postponed for ten years because of the two mission failures. One more failure and they’ll probably postpone doing anything substantial on Mars for fifty years, or indefinitely. Personally, I think the last two Mars missions failed because of a high-risk trajectory. Granted, they claim mechanical or computer code flaws were the cause of the problems, but no one really knows for sure. The Russians tried to get to Mars what, six times; and never did make it. They have hardware just as good as ours – maybe better, since everything doesn’t go to the lowest bidder. [Handwritten note in margin: How does NASA get to Mars now?] It’s not hard to make a case for faulting the trajectory in all these failures. It becomes even more of an issue when you look at a typical scenario. <<slide 2, 3>> NASA’s plot of Mars Pathfinder vs. Nominal missions NASA’s standard flight path is a five-thrust trajectory for a Hohmann type transfer. The s/c reaches Mars at the apoapse of the transfer ellipse. At this point Mars is going about 2 km/sec faster than the s/c along nearly parallel paths. Basically, the s/c has to get to the intersection point before Mars, then just wait there and make a quick reactive maneuver into orbit. A more proactive approach is shown in the Pathfinder trajectory, used for the successful mission that landed on July 4, 1997. The angle of intersection is less acute, and the relative velocity is lower too – both of which increase the margin for error in making the final approach. The only problem is that NASA had to use a lot more fuel on this trajectory.
So this will be the approach taken to the optimization problem: [Handwritten note: slide]
THEORY <<slide 6>> Hohmann Transfer Leading into the theory for the trajectory analysis, consider the geometry of the flight path from Earth to Mars as a Hohmann Transfer. The two planets are in circular orbits here, and the s/c is on a two-body ellipse. <<slide 7>> Syzygygy Now look at the phase angle of the three bodies in heliocentric coordinates. Earth and Mars have a constant slope because their speed is constant on a circular path. Earth moves slower, so it has a lower slope. The s/c starts at Earth, and ends up at Mars and there are three conjunctions along the way: [Handwritten note: label each one]
[Handwritten note at top right: on board] So, my efforts have been focused on trying to find a more efficient “Pathfinder” type of trajectory. The hope was to make an approach that could be adjusted to imitate the so-called free-return Apollo mission to the moon type of flight path. [Handwritten note: enlarge 1. rotating, 2. non-rotating] <<slide 13>> Dr. Schutz’ class notes [Handwritten note: (~3 Body Problem)] These figure-8’s, from Dr. Schutz’ orbital mechanics class notes this past spring, are for the Earth-moon system. If anything happens, the s/c will always return to Earth. Theoretically, there should be the same kind of flight path for the Sun-Mars system. Making this kind of an approach would take the urgency out of the split-second maneuver typically done for the Mars capture sequence, and result in a low anxiety approach that the s/c can do with more margin for error because the s/c is following a natural periodic orbit. <<slide 14>> free return from Mars, showing relative velocity vectors The general idea is to find an approach geometry that puts the s/c on a path similar to the free-return trajectory. The most important thing is the relative velocity of the s/c with respect to Mars, which should be aligned with the intended trajectory as indicated. Thus, the flight path should be configured to intersect Mars’ orbit at some point before apoapse. [Handwritten text crossed out: three topical areas of project] -> [Handwritten text added: OK] [Handwritten note pointing to paragraph: 1) inspiration, 2) perspiration setup, 3) numerical analysis] So, that was the 1% of inspiration for this project: look for a special approach to Mars. The other 99% of perspiration was in setting up the model and trying to configure it to find this mission profile. I will first discuss a fairly simple theory that was the basis for setting up the problem on the computer. Then I will talk about the method by which the trajectory was optimized. The mission was divided into several segments that were studied individually and optimized in sequence, making the problem easier to study and faster to analyze on the computer. Keep in mind that all of the results I will be presenting were generated on a 300 Mhz PC. The entire simulation takes less than 60 seconds on this machine. Everything is integrated using a Runge-Kutta 7/8 variable step integrator set to maximum tolerance, which gives a consistent 13 or 14 significant digits from beginning to end. So limiting the analysis to a PC has not affected accuracy, except at one point which will be noted in context. The idea is that this kind of program can run on board a s/c, to make real time flight path analyses of the mission profile.
[Handwritten note at top right: on board] SIMULATION <<slide 11>> trajectory.gif with s/c illustration Now, to create the model for the Earth-to-Mars trajectory. First, fixed endpoints are needed so it is an official two point boundary value problem: assume a 200 km Earth parking orbit and a 100 km Mars parking orbit. Since the integration is already interrupted at conjunction to apply a thrust there, it is convenient to divide the flight path into two optimization problems starting at conjunction: [Handwritten note in margin: slide of Two point boundary value problem]
INITIAL CONDITIONS <<slide 13>> algorithm for initial conditions Since an actual mission is being simulated, the J2000 ephemeredes must be extrapolated to get the position of Earth and Mars at conjunction. The position of the s/c at this time is also needed, which is not so easy to find. In setting up the problem, it was stated that the Earth-to-conjunction part of the trajectory is assumed ideal already. So, based on two-body dynamics the position of Earth can be found at the beginning of the Hohmann Transfer. This is identified as the positive x-axis. The time for Earth to go from this point to conjunction is known too. So the s/c is situated on the Hohmann ellipse at that time from leaving Earth. This, of course, is an artificial convention, fixing the endpoints of the first half of the trajectory. It will still be optimized for the minimum thrust, though, as will be shown in a moment. The idea here is that our fuel allowance will be able to save more time on the approach to Mars because the s/c is near apoapse there and going much slower than in the first half of the flight path. Shortening the last half of the trajectory will be easier to do, and a more effective use of limited fuel. The point at which the mid-course correction should be applied is fixed also, at a true anomaly of 90° and as a tangential deltaV, for reasons stated earlier. Given these reasonable criteria, then, you can say unequivocally that the Earth-to-conjunction flight path has the optimal end points already. We only need to find the best path between them. Likewise, for the conjunction-to-Mars flight path, the initial point is optimal, and we will only need to first designate a final end point and then optimize the trajectory. [Handwritten note pointing to paragraph below: bold] More formally, the Earth trajectory has a free final time, fixed position; and the Mars trajectory has a fixed final time and a free position. Both have a fixed initial position and time.
EARTH <<slide 14>> Earth deltaV well First, consider the trajectory from conjunction to Earth. The objective is to minimize the deltaV to reach the 200-km parking orbit, starting from conjunction. Since every iteration of the loop must return a value, the loop is set up to target Earth’s SOI, then once that is achieved, continue the trajectory to Earth itself. Every iteration must not only return a value, but also overall have a gradient driving the flight path toward the SOI; then toward the minimal flight path to the final parking orbit. [Handwritten note: start @ zero, go beyond earthly build up to global min.] In particular, a prograde orbit is sought, so the code is configured to return values as indicated in the figure. The objective is to simply find the minimum deltaV to reach Earth, and with this kind of mapping the loop finds the minimum total deltaV for a given deltaV applied at conjunction. Then the global minimum total deltaV should logically be easy to find by simply varying the deltaV at conjunction. <<slide 15>> Earth algorithm Consider on iteration of this search routine. Once SOI is reached, the coordinates are transferred to geocentric and the trajectory is integrated until the range to Earth starts increasing. At each step of the integrator, a deltaV is calculated to the 200-km parking orbit by first circularizing the orbit and then doing a Hohmann Transfer to the final parking orbit. The optimal trajectory for this iteration is that with the lowest final deltaV, which is added to the deltaV applied at conjunction to get the total deltaV for that iteration. Initially the deltaV at conjunction is incremented so that the flight path of the s/c is beyond Earth, from which point there is a constant negative gradient. Then the thrust is decreased by half steps in a one dimensional search routine until a minimum with a specific tolerance (nominally 1 m/sec) is reached. Will this method work? Will it be fast and effective? <<slide 16>> one dimensional search to Earth First consider the effectiveness of the mapping. Here the search
Does the resulting trajectory work? Does it go through Earth, etc?? <<slide 17>> Earth approach Here are the x- and y-coordinates as the s/c approaches Earth, in geocentric coordinates. The s/c never gets closer than 200 km from the surface of Earth. In fact, the s/c goes down and then back up again, making the escape thrust at a higher altitude. <<slide 18>> Earth escape maneuvers The geometry of the trajectory found by this method is just what was proposed to be the optimal Earth escape scenario: starting from the 200 km parking orbit, the s/c makes a Hohmann Transfer to a targeting orbit above 200 km, then executes the escape thrust from this orbit. This thrust puts the s/c on a hyperbolic escape trajectory with periapse at 200 km. It therefore has a little help from the gravity of Earth and makes a sling shot trajectory away from Earth. The flight path thus initiated takes the s/c directly to the designated point at conjunction, with no further maneuvers or thrusts necessary. It is essentially on a minimum energy Hohmann transfer ellipse, as hoped for.
MARS <<slide 19>> relative velocity vector at Mars approach The trajectory from conjunction to Mars is a little more complex. The basic idea is to apply a large thrust at conjunction, to arrive at Mars a lot faster but also altering the approach geometry so that a lower thrust will get you into the “targeting orbit.” The position of Mars at the end of the trajectory is varied to optimize the relative velocity vector, as indicated. <<slide 20>> Mars algorithm The code is set up to accept a thrust at conjunction, then to optimize the trajectory to Mars. First there is a loop that adjusts the phase angle of Mars until the s/c reaches SOI. This is a very non-linear problem, so the code does not do a complete integration of the trajectory from conjunction, as was possible with Earth. Instead, the velocity vector at SOI is re-positioned on the circumference of SOI in the neighborhood of the approach trajectory, and the best path to the parking orbit is selected from among these sub-trajectories. (This assumes the problem can be made exact with a more powerful computer and better optimization methods; although there is not much of a margin for error to make up.) Now, we are at the moment of truth. All this work just to see if it is possible to find a “free return” approach to Mars. . . . The algorithm is set up to try approach trajectories to Mars, at about 1.5 degree increments, looking for a trajectory that slides right into Mars orbit with a very small deltaV. <<slide 21>> deltaV to phasing orbit Bingo. The computer found a low deltaV trajectory – almost 5 km/sec lower than the other Pathfinder type mission profiles. Is it a viable trajectory, though? Yes, it is 3 degrees wide, which is a sizeable target to aim for on the SOI. Also, … <<slide 22>> Mars fly by The final range is 3415 km, and the radius of mars is 3400 km; so the s/c gets close to Mars, then makes a Hohmann Transfer up to the 100 km parking orbit that is the endpoint of the simulation. Is this a safe trajectory? <<slide 23>> deltaV to reach Mars parking orbit There is a constant drop in deltaV needed to reach the final parking orbit (a pattern which is mirrored if this final end point is not reached by the s/c). This is confirmed by the actual data point values. <<slide 24>> mars approach data
THE TOTAL MISSION <<slide 25>> catapult We have thus far managed to optimize the Earth-to-Mars flight path in two separate parts. The entire trajectory can now be investigated, putting these parts together. Typically, the Mars trajectory is first optimized for a given deltaV, then the Earth trajectory. The best way to view the results is just to look at some total values for various scenarios. <<slide 26>> Hohmann Notice the bold values, which are the approximate time and total deltaV for a minimum energy Hohmann Transfer. As noted, this flight profile can beat both the time and the deltaV considerations.
CONCLUSION Despite the clumsy, unconventional methods used, a whole set of trajectories have been found which seem to lie in the “impossible” region of NASA’s mission planning “pork chop” plots. I guess you could call them the T-bone trajectories. Admittedly, there are some refinements to be made. You might even apply some actual optimization ideas like integrating a transition matrix to get the next step size, or search methods like Fibonacci or the Golden Rule. These may generate even better times and thrusts. Actually, though, the importance of this analysis is that the ideas developed can be incorporated into any existing trajectory model. [Handwritten note: slide]
[Page 20 is mostly illegible / faded text]
[Handwritten note: fig 5-8.gif] subroutine CATAPULT
[Handwritten note: fig 5-2.gif] subroutine INITIAL
[Handwritten note: fig 5-4.gif] subroutine HAWKEYE
[Handwritten note: fig 5-6.gif] subroutine HAWKEYM
[Handwritten note: fig 5-3.gif] [Diagram showing a graph with “terminal thrust” on the y-axis. A spacecraft is shown with a dashed trajectory curving downwards across zones A, B, C, and D. An “Earth” is shown at the bottom surrounded by a dashed “200 km parking orbit” and an outer dashed “target quadrant” / “SOI” (Sphere of Influence). An arrow points from the spacecraft’s trajectory towards the Earth’s orbit.]
[Handwritten note: fig 5-5.gif] [Diagram labeled “Earth Escape Geometry”. A solid central circle represents Earth. A dashed circular orbit is labeled “parking orbit”, an elliptical dashed orbit is labeled “Hohmann Transfer”, and a second dashed elliptical orbit is labeled “phasing orbit”. Tangential thrust vectors “dv1” and “dv2” are shown. A solid line showing the spacecraft’s path loops around Earth and escapes outward, labeled “hyperbolic escape”.]
[Handwritten note: fig 5-1.gif] Rotating: restricted 3 body [Graph showing a horizontal figure-8 orbital path along the x-axis, centered around 0.] Fig. 5. A near “free-return” flyby of the Moon illustrated in the rotating coordinate system. The initial conditions given above were used and the point of closes approach to the center of the Moon was about 1700 km, or an altitude of about zero (radius of the Moon is 1738 km). Nonrotating: restricted 3 body [Graph showing an elliptical orbital path originating from the origin, looping out to a dashed point labeled “moon @ t=0”, and returning.] Fig. 6. The near free-return trajectory illustrated in nonrotating coordinates, where the X axis coincides with the x-axis at t=0.
[Handwritten note: fig 5-9.gif] [Graph titled “deltaV to phasing orbit at Mars”. The y-axis shows “deltaV, km/sec” ranging from 0.00E+00 to 9.00E+00. The x-axis shows “increments along SOI, pi/220” from 1 to 37. A line graph remains relatively high near 8.00E+00 on both ends but features a steep drop/spike downward to around 3.00E+00 near the x-axis increment of 27.]
[Handwritten note: fig 5-7.gif] [Diagram labeled “Mars Capture Geometry”. A solid central circle represents Mars. It is surrounded by a “parking orbit” and a “phasing orbit”. Thrust vectors “dv1” and “dv2” are indicated. A “Hohmann Transfer” ellipse is shown. The outer dashed circle represents the “sphere of influence”. A dashed line indicates the “mars orbit”. A solid line enters the sphere of influence from the bottom right (“relative velocity vector”), loops around Mars, and exits the frame. A parallel dotted line on the bottom left is labeled “relative velocity at x-axis”.]
Mars centered coordinates – True Rotating System Figure 5 [Graph showing a single curved trajectory starting from a high negative Y-rotating value, swooping upwards across the X-axis into positive Y territory, then dipping slightly back down and curving sharply towards the origin on the right side of the plot.]
[Clean version of the “Mars Capture Geometry” diagram from page 29] [Diagram labeled “Mars Capture Geometry”. A solid central circle represents Mars. It is surrounded by a “parking orbit” and a “phasing orbit”. Thrust vectors “dv1” and “dv2” are indicated. A “Hohmann Transfer” ellipse is shown. The outer dashed circle represents the “sphere of influence”. A dashed line indicates the “mars orbit”. A solid line enters the sphere of influence from the bottom right (“relative velocity vector”), loops around Mars, and exits the frame. A parallel dotted line on the bottom left is labeled “relative velocity at x-axis”.]
[Clean version of the “Earth Escape Geometry” diagram from page 26] [Diagram labeled “Earth Escape Geometry”. A solid central circle represents Earth. A dashed circular orbit is labeled “parking orbit”, an elliptical dashed orbit is labeled “Hohmann Transfer”, and a second dashed elliptical orbit is labeled “phasing orbit”. Tangential thrust vectors “dv1” and “dv2” are shown. A solid line showing the spacecraft’s path loops around Earth and escapes outward, labeled “hyperbolic escape”.]
Optimizing the Earth-to-Mars Trajectory By William H. Clark II, P.E. [Diagram: Orbit illustration showing the Sun at the center, Earth’s orbit, Mars’ orbit, a spacecraft trajectory from Earth to Mars, and a Conjunction point aligned with Earth and Mars.] © 2000 by William H. Clark II, P.E. all rights reserved
Optimizing the Earth to Mars Trajectory I. Introduction II. Theory III. Numerical Optimization Methods IV. Conclusions & Recommendations V. Future: Modeling the Semiconductor Junction Free download of program and code at: http://whcii.home.mindspring.com/astro/marvin.html
Optimizing the Earth to Mars Trajectory Bill Clark, P.E.
- April 2001 ephemeredes
- Coplanar motion *
- Earth & Mars in elliptical 2-body motion
- Three Body Problem (3BP) inside Earth SOI, in geocentric coordinates
- 5BP heliocentric path from Earth to Mars (perturbed by Earth, Mars, and Jupiter)
- 3BP inside Mars SOI, in Mars-centered coordinates
[Handwritten URL at top: http://whcii.home.mindspring.com/astro/marvin.html] INTRODUCTION <<slide 1>> Summary I’d like to talk about some research I have been doing on the Earth to Mars trajectory. Specifically, the trajectory of the NASA mission to be launched to Mars in April of next year. I have created a computer simulation using the exact ephemeredes of the planets. All motion is assumed coplanar. Mars’ orbit is at a 3° inclination to the ecliptic, so coplanar motion is not a bad approximation. Other assumptions will have negligible affect on this analysis, and they will be noted in context as appropriate. The title of the report is “Optimizing” the trajectory. Or, as NASA puts it, “Getting there faster and cheaper.” That is important of course, but I have placed more emphasis on getting there “safer.” Just last week NASA announced that the $1 billion rock collecting mission to Mars was postponed for ten years because of the two mission failures. One more failure and they’ll probably postpone doing anything substantial on Mars for fifty years, or indefinitely. Personally, I think the last two Mars missions failed because of a high-risk trajectory. Granted, they claim mechanical or computer code flaws were the cause of the problems, but no one really knows for sure. The Russians tried to get to Mars what, six times; and never did make it. They have hardware just as good as ours – maybe better, since everything doesn’t go to the lowest bidder. [Handwritten note in margin: How does NASA get to Mars now?] It’s not hard to make a case for faulting the trajectory in all these failures. It becomes even more of an issue when you look at a typical scenario. <<slide 2, 3>> NASA’s plot of Mars Pathfinder vs. Nominal missions NASA’s standard flight path is a five-thrust trajectory for a Hohmann type transfer. The s/c reaches Mars at the apoapse of the transfer ellipse. At this point Mars is going about 2 km/sec faster than the s/c along nearly parallel paths. Basically, the s/c has to get to the intersection point before Mars, then just wait there and make a quick reactive maneuver into orbit. A more proactive approach is shown in the Pathfinder trajectory, used for the successful mission that landed on July 4, 1997. The angle of intersection is less acute, and the relative velocity is lower too – both of which increase the margin for error in making the final approach. The only problem is that NASA had to use a lot more fuel on this trajectory.
So this will be the approach taken to the optimization problem: [Handwritten note: slide]
- Assume the s/c trajectory from Earth to conjunction is already optimal
- Concentrate efforts to save time and/or energy on the conjunction-to-Mars half of the trajectory
THEORY <<slide 6>> Hohmann Transfer Leading into the theory for the trajectory analysis, consider the geometry of the flight path from Earth to Mars as a Hohmann Transfer. The two planets are in circular orbits here, and the s/c is on a two-body ellipse. <<slide 7>> Syzygygy Now look at the phase angle of the three bodies in heliocentric coordinates. Earth and Mars have a constant slope because their speed is constant on a circular path. Earth moves slower, so it has a lower slope. The s/c starts at Earth, and ends up at Mars and there are three conjunctions along the way: [Handwritten note: label each one]
- Earth-s/c
- Earth-Mars
- Mars-s/c
- Earth is on a parabolic path
- The x-axis is the Earth-Mars conjunction because the orbit of Earth is mirrored about it. <<slide 11>>
- The s/c is on a straight line path that is an asymptote to Earth’s trajectory
[Handwritten note at top right: on board] So, my efforts have been focused on trying to find a more efficient “Pathfinder” type of trajectory. The hope was to make an approach that could be adjusted to imitate the so-called free-return Apollo mission to the moon type of flight path. [Handwritten note: enlarge 1. rotating, 2. non-rotating] <<slide 13>> Dr. Schutz’ class notes [Handwritten note: (~3 Body Problem)] These figure-8’s, from Dr. Schutz’ orbital mechanics class notes this past spring, are for the Earth-moon system. If anything happens, the s/c will always return to Earth. Theoretically, there should be the same kind of flight path for the Sun-Mars system. Making this kind of an approach would take the urgency out of the split-second maneuver typically done for the Mars capture sequence, and result in a low anxiety approach that the s/c can do with more margin for error because the s/c is following a natural periodic orbit. <<slide 14>> free return from Mars, showing relative velocity vectors The general idea is to find an approach geometry that puts the s/c on a path similar to the free-return trajectory. The most important thing is the relative velocity of the s/c with respect to Mars, which should be aligned with the intended trajectory as indicated. Thus, the flight path should be configured to intersect Mars’ orbit at some point before apoapse. [Handwritten text crossed out: three topical areas of project] -> [Handwritten text added: OK] [Handwritten note pointing to paragraph: 1) inspiration, 2) perspiration setup, 3) numerical analysis] So, that was the 1% of inspiration for this project: look for a special approach to Mars. The other 99% of perspiration was in setting up the model and trying to configure it to find this mission profile. I will first discuss a fairly simple theory that was the basis for setting up the problem on the computer. Then I will talk about the method by which the trajectory was optimized. The mission was divided into several segments that were studied individually and optimized in sequence, making the problem easier to study and faster to analyze on the computer. Keep in mind that all of the results I will be presenting were generated on a 300 Mhz PC. The entire simulation takes less than 60 seconds on this machine. Everything is integrated using a Runge-Kutta 7/8 variable step integrator set to maximum tolerance, which gives a consistent 13 or 14 significant digits from beginning to end. So limiting the analysis to a PC has not affected accuracy, except at one point which will be noted in context. The idea is that this kind of program can run on board a s/c, to make real time flight path analyses of the mission profile.
[Handwritten note at top right: on board] SIMULATION <<slide 11>> trajectory.gif with s/c illustration Now, to create the model for the Earth-to-Mars trajectory. First, fixed endpoints are needed so it is an official two point boundary value problem: assume a 200 km Earth parking orbit and a 100 km Mars parking orbit. Since the integration is already interrupted at conjunction to apply a thrust there, it is convenient to divide the flight path into two optimization problems starting at conjunction: [Handwritten note in margin: slide of Two point boundary value problem]
- Conjunction-to-Earth, integrating back in time
- Conjunction-to-Mars, integrating forward
- “free return” approach geometry at Mars
- gravity assist fly-by at Earth
INITIAL CONDITIONS <<slide 13>> algorithm for initial conditions Since an actual mission is being simulated, the J2000 ephemeredes must be extrapolated to get the position of Earth and Mars at conjunction. The position of the s/c at this time is also needed, which is not so easy to find. In setting up the problem, it was stated that the Earth-to-conjunction part of the trajectory is assumed ideal already. So, based on two-body dynamics the position of Earth can be found at the beginning of the Hohmann Transfer. This is identified as the positive x-axis. The time for Earth to go from this point to conjunction is known too. So the s/c is situated on the Hohmann ellipse at that time from leaving Earth. This, of course, is an artificial convention, fixing the endpoints of the first half of the trajectory. It will still be optimized for the minimum thrust, though, as will be shown in a moment. The idea here is that our fuel allowance will be able to save more time on the approach to Mars because the s/c is near apoapse there and going much slower than in the first half of the flight path. Shortening the last half of the trajectory will be easier to do, and a more effective use of limited fuel. The point at which the mid-course correction should be applied is fixed also, at a true anomaly of 90° and as a tangential deltaV, for reasons stated earlier. Given these reasonable criteria, then, you can say unequivocally that the Earth-to-conjunction flight path has the optimal end points already. We only need to find the best path between them. Likewise, for the conjunction-to-Mars flight path, the initial point is optimal, and we will only need to first designate a final end point and then optimize the trajectory. [Handwritten note pointing to paragraph below: bold] More formally, the Earth trajectory has a free final time, fixed position; and the Mars trajectory has a fixed final time and a free position. Both have a fixed initial position and time.
EARTH <<slide 14>> Earth deltaV well First, consider the trajectory from conjunction to Earth. The objective is to minimize the deltaV to reach the 200-km parking orbit, starting from conjunction. Since every iteration of the loop must return a value, the loop is set up to target Earth’s SOI, then once that is achieved, continue the trajectory to Earth itself. Every iteration must not only return a value, but also overall have a gradient driving the flight path toward the SOI; then toward the minimal flight path to the final parking orbit. [Handwritten note: start @ zero, go beyond earthly build up to global min.] In particular, a prograde orbit is sought, so the code is configured to return values as indicated in the figure. The objective is to simply find the minimum deltaV to reach Earth, and with this kind of mapping the loop finds the minimum total deltaV for a given deltaV applied at conjunction. Then the global minimum total deltaV should logically be easy to find by simply varying the deltaV at conjunction. <<slide 15>> Earth algorithm Consider on iteration of this search routine. Once SOI is reached, the coordinates are transferred to geocentric and the trajectory is integrated until the range to Earth starts increasing. At each step of the integrator, a deltaV is calculated to the 200-km parking orbit by first circularizing the orbit and then doing a Hohmann Transfer to the final parking orbit. The optimal trajectory for this iteration is that with the lowest final deltaV, which is added to the deltaV applied at conjunction to get the total deltaV for that iteration. Initially the deltaV at conjunction is incremented so that the flight path of the s/c is beyond Earth, from which point there is a constant negative gradient. Then the thrust is decreased by half steps in a one dimensional search routine until a minimum with a specific tolerance (nominally 1 m/sec) is reached. Will this method work? Will it be fast and effective? <<slide 16>> one dimensional search to Earth First consider the effectiveness of the mapping. Here the search
- starts at zero
- increments to 1.0 E-4
- increments to 2.0 E-4
- steps back to the global minimum
- takes a total of 8 iterations
Does the resulting trajectory work? Does it go through Earth, etc?? <<slide 17>> Earth approach Here are the x- and y-coordinates as the s/c approaches Earth, in geocentric coordinates. The s/c never gets closer than 200 km from the surface of Earth. In fact, the s/c goes down and then back up again, making the escape thrust at a higher altitude. <<slide 18>> Earth escape maneuvers The geometry of the trajectory found by this method is just what was proposed to be the optimal Earth escape scenario: starting from the 200 km parking orbit, the s/c makes a Hohmann Transfer to a targeting orbit above 200 km, then executes the escape thrust from this orbit. This thrust puts the s/c on a hyperbolic escape trajectory with periapse at 200 km. It therefore has a little help from the gravity of Earth and makes a sling shot trajectory away from Earth. The flight path thus initiated takes the s/c directly to the designated point at conjunction, with no further maneuvers or thrusts necessary. It is essentially on a minimum energy Hohmann transfer ellipse, as hoped for.
MARS <<slide 19>> relative velocity vector at Mars approach The trajectory from conjunction to Mars is a little more complex. The basic idea is to apply a large thrust at conjunction, to arrive at Mars a lot faster but also altering the approach geometry so that a lower thrust will get you into the “targeting orbit.” The position of Mars at the end of the trajectory is varied to optimize the relative velocity vector, as indicated. <<slide 20>> Mars algorithm The code is set up to accept a thrust at conjunction, then to optimize the trajectory to Mars. First there is a loop that adjusts the phase angle of Mars until the s/c reaches SOI. This is a very non-linear problem, so the code does not do a complete integration of the trajectory from conjunction, as was possible with Earth. Instead, the velocity vector at SOI is re-positioned on the circumference of SOI in the neighborhood of the approach trajectory, and the best path to the parking orbit is selected from among these sub-trajectories. (This assumes the problem can be made exact with a more powerful computer and better optimization methods; although there is not much of a margin for error to make up.) Now, we are at the moment of truth. All this work just to see if it is possible to find a “free return” approach to Mars. . . . The algorithm is set up to try approach trajectories to Mars, at about 1.5 degree increments, looking for a trajectory that slides right into Mars orbit with a very small deltaV. <<slide 21>> deltaV to phasing orbit Bingo. The computer found a low deltaV trajectory – almost 5 km/sec lower than the other Pathfinder type mission profiles. Is it a viable trajectory, though? Yes, it is 3 degrees wide, which is a sizeable target to aim for on the SOI. Also, … <<slide 22>> Mars fly by The final range is 3415 km, and the radius of mars is 3400 km; so the s/c gets close to Mars, then makes a Hohmann Transfer up to the 100 km parking orbit that is the endpoint of the simulation. Is this a safe trajectory? <<slide 23>> deltaV to reach Mars parking orbit There is a constant drop in deltaV needed to reach the final parking orbit (a pattern which is mirrored if this final end point is not reached by the s/c). This is confirmed by the actual data point values. <<slide 24>> mars approach data
THE TOTAL MISSION <<slide 25>> catapult We have thus far managed to optimize the Earth-to-Mars flight path in two separate parts. The entire trajectory can now be investigated, putting these parts together. Typically, the Mars trajectory is first optimized for a given deltaV, then the Earth trajectory. The best way to view the results is just to look at some total values for various scenarios. <<slide 26>> Hohmann Notice the bold values, which are the approximate time and total deltaV for a minimum energy Hohmann Transfer. As noted, this flight profile can beat both the time and the deltaV considerations.
CONCLUSION Despite the clumsy, unconventional methods used, a whole set of trajectories have been found which seem to lie in the “impossible” region of NASA’s mission planning “pork chop” plots. I guess you could call them the T-bone trajectories. Admittedly, there are some refinements to be made. You might even apply some actual optimization ideas like integrating a transition matrix to get the next step size, or search methods like Fibonacci or the Golden Rule. These may generate even better times and thrusts. Actually, though, the importance of this analysis is that the ideas developed can be incorporated into any existing trajectory model. [Handwritten note: slide]
- Applying a large thrust (up to 20% of the total thrust budget) at conjunction
- Escape thrust from a phasing orbit above the parking orbit at Earth to use a gravity-assisted fly by.
- Capture at Mars via a phasing orbit below the final parking orbit for a “free return” type of approach.
Mars Pathfinder
by
Bar X Software - Austin, Texas
Est. 1993
------------------------------------------------------------
Your mission is to reach Mars by adjusting values
of the trajectory parameters to (1) minimize
fuel usage, in km/sec of thrust, and (2) to
reach Mars as quickly as possible. . .
You can change the trajectory in one place,
about half way to Mars. You select MAGNITUDE and
DIRECTION for this thrust. The program will apply
the thrust at the best time and place, and
optimize all the other four maneuvers of the
flight path based upon your criteria.
Now enter the THRUST MAGNITUDE in km/sec.
Values should be between zero
(a standard Hohmann Transfer) and 2.5.
0.9
Now the THRUST DIRECTION, in degrees above/below
the current velocity direction. Be careful! A value
below zero, with a low thrust and you will never reach
Mars because you will pass inside its orbit!
0
The program will now compute the minimum energy
trajectory to get you to Mars in the least time,
using the values you have specified.
------------------------------------------------------------
Mission Profile
------------------------------------------------------------
The program uses the J2000 Ephemerides. The first
Earth-Mars conjunction occurs, in days after
1/1/2000 (midnight is at 0.5 days) at
530.5542993317358000
The position of Earth and Mars at this time is
(units, degrees)
264.0131915092446000
Where zero degrees is the Vernal Equinox by
convention. All subsequent coordinates will be in
this coordinate system, with the z-axis perpendicular
to the orbit plane. The true anomalies of Earth and
Mars at mid-course correction (about conjunction) are
161.0758434292446000
-112.2199327613671000
The motion is assumed to be co-planar.
------------------------------------------------------------
The mission begins with the spacecraft (s/c) in a
200 km parking orbit around Earth. The s/c maneuvers
to slightly higher circular orbit using a minimum
energy Hohmann transfer with a thrust of
(units, km/sec)
0.2081942111215296
The interplanetary trajectory begins with a thrust of
3.5271197931170930
at the following geocentric (earth-centered)
coordinates, initiatiating a hyperbolic escape
trajectory (perturbed by the sun and mars):
-924.9960563278269000
7273.5824133303440000
The initial velocity coordinates are:
10.5589685087009200
-1.6540587935915460
The time (relative to the mid-course thrust) is
-95.6722078139787000
The s/c leaves Earths sphere of influence (soi)
at the following geocentric coordinates and time:
35174.8110972940900000
-892091.6278970391000000
0.0171518339681476
-2.5227454825702580
-92.0593889987111000
Its heliocentric (sun-centered) coordinates
at this time are
-1.4803812019254480D+008
1.3472550464435410D+007
-3.3442660548611180
-32.2838240415630200
------------------------------------------------------------
The s/c is now on an elliptical heliocentric course
(perturbed by Earth, Mars, and Jupiter). The mid-
course correction occurs at time zero, and at the
following heliocentric coordinates
-2.4205138214124150D+007
-1.7324114651203530D+008
26.5277319107005300
-8.3936763444886430
(This velocity is after the thrust is applied.)
The s/c reaches Mars soi at the heliocentric
coordinates and time of,
1.9761816025430460D+008
-8.2036726393335330D+007
11.0958881978198400
20.3924950190578500
117.3631915916114000
(Time, measured after the mid-course corrrection.)
The mars-centered (geocentric) coordinates at this
point are
-501623.2796767056000000
204908.3278136402000000
2.8181954966896100
-3.9286208224952470
------------------------------------------------------------
The s/c follows a hyperbolic capture orbit to Mars
(perturbed by the sun), and applies a thrust of
1.5040643848000960
to circularize the orbit at coordinates, and time,
-3314.8306409049450000
-1225.0545500430200000
1.3545883474034440
6.7643054615615870
1.3621115389107490
A final thrust, used to target a specific latitude
and longitude on the surface of Mars, of
0.0142136164151653
brings the s/c via Hohmann Transfer into a 100 km
parking orbit around Mars. The total time and
thrust required for this scenario are
220.0629520382686000
6.1536920054538850
------------------------------------------------------------
Your total thrust requirement in km/sec was
6.1536920054538850
The total time required for this mission in days was,
220.0629520382686000
Please refer to the above data printout for a more
complete mission profile.
------------------------------------------------------------
Copyright 2000, William H. Clark II
All Rights Reserved.
------------------------------------------------------------
------------------------------------------------------------
Please enter 1 if you want to try another mission.
.
[Page 20 is mostly illegible / faded text]
[Handwritten note: fig 5-8.gif] subroutine CATAPULT
- Initialize RK78 constants and global constants
- (down arrow to) subroutine HAWKEYM
- given dV at conjunction, find time to reach Mars and total dV
- (down arrow, 1st iteration only, to) subroutine HAWKEYE
- find minimum dV, time to reach Earth
- (down arrow to check) Total time / Total dV -> are these optimal?
- if no: increment dV to Mars -> (up arrow to) subroutine HAWKEYM
- (down arrow to) subroutine HAWKEYM
[Handwritten note: fig 5-2.gif] subroutine INITIAL
- Find 2001 Earth-Mars conjunction
- (down arrow)
- Transform coordinates to a 2-body Hohmann transfer orbit
- (down arrow)
- Earth at t=0 on Hohmann ellipse is the positive x-axis
- (down arrow)
- Find the phase angle of Earth at conjunction in J2000 epoch TC = time Earth takes to get to conjunction from t=0
- (down arrow)
- Find coordinates of s/c on Hohmann ellipse in 2-body system for s/c to go from Earth at t=0 to t= TC
- (down arrow)
- Initial conditions for s/c with Earth and Mars at first conjunction in J2000 epoch
[Handwritten note: fig 5-4.gif] subroutine HAWKEYE
- initial positions of s/c and Earth are constant
- objective is to minimize total thrust to 200 km parking orbit
- final position of Earth is constant at x-axis, per Hohmann Transfer
- subroutine INITIAL -> dV = 0
- dv1 = dv1 + dv
- (down arrow to) Conjunction (heliocentric)
- numerical integration (variable time step)
- subroutine TAILHOOKE
- (down arrow to) 2*SOI
- fixed time step
- (down arrow to) Earth SOI (geocentric)
- variable time step
- subroutine EARTH
- (down arrow to) 200 km parking orbit (2-body Hohmann Transfer)
- MIN?
- if no: (up arrow to) subroutine TRAPE -> increment dV -> back to dv1 = dv1 + dv
- MIN?
[Handwritten note: fig 5-6.gif] subroutine HAWKEYM
- deltaV at conjunction is constant, dV1
- vary phase angle at conjunction until s/c crosses Mars’ SOI
- subroutine INITIAL -> d∅ = 0
- d∅1 = d∅1 + d∅
- (down arrow to) Conjunction (heliocentric)
- numerical integration (variable time step)
- subroutine TAILHOOKM
- (down arrow to) 2*SOI
- fixed time step
- reach SOI?
- if no: (up arrow to) increment d∅ -> back to d∅1 = d∅1 + d∅
- (down arrow to) Mars SOI (geocentric)
- shift state vector @ SOI around circumference of SOI
- variable time step
- subroutine PHOENIX
- (down arrow to) 100 km parking orbit (2-body Hohmann Transfer)
- save min. of all iterations -> MIN?
[Handwritten note: fig 5-3.gif] [Diagram showing a graph with “terminal thrust” on the y-axis. A spacecraft is shown with a dashed trajectory curving downwards across zones A, B, C, and D. An “Earth” is shown at the bottom surrounded by a dashed “200 km parking orbit” and an outer dashed “target quadrant” / “SOI” (Sphere of Influence). An arrow points from the spacecraft’s trajectory towards the Earth’s orbit.]
[Handwritten note: fig 5-5.gif] [Diagram labeled “Earth Escape Geometry”. A solid central circle represents Earth. A dashed circular orbit is labeled “parking orbit”, an elliptical dashed orbit is labeled “Hohmann Transfer”, and a second dashed elliptical orbit is labeled “phasing orbit”. Tangential thrust vectors “dv1” and “dv2” are shown. A solid line showing the spacecraft’s path loops around Earth and escapes outward, labeled “hyperbolic escape”.]
[Handwritten note: fig 5-1.gif] Rotating: restricted 3 body [Graph showing a horizontal figure-8 orbital path along the x-axis, centered around 0.] Fig. 5. A near “free-return” flyby of the Moon illustrated in the rotating coordinate system. The initial conditions given above were used and the point of closes approach to the center of the Moon was about 1700 km, or an altitude of about zero (radius of the Moon is 1738 km). Nonrotating: restricted 3 body [Graph showing an elliptical orbital path originating from the origin, looping out to a dashed point labeled “moon @ t=0”, and returning.] Fig. 6. The near free-return trajectory illustrated in nonrotating coordinates, where the X axis coincides with the x-axis at t=0.
[Handwritten note: fig 5-9.gif] [Graph titled “deltaV to phasing orbit at Mars”. The y-axis shows “deltaV, km/sec” ranging from 0.00E+00 to 9.00E+00. The x-axis shows “increments along SOI, pi/220” from 1 to 37. A line graph remains relatively high near 8.00E+00 on both ends but features a steep drop/spike downward to around 3.00E+00 near the x-axis increment of 27.]
[Handwritten note: fig 5-7.gif] [Diagram labeled “Mars Capture Geometry”. A solid central circle represents Mars. It is surrounded by a “parking orbit” and a “phasing orbit”. Thrust vectors “dv1” and “dv2” are indicated. A “Hohmann Transfer” ellipse is shown. The outer dashed circle represents the “sphere of influence”. A dashed line indicates the “mars orbit”. A solid line enters the sphere of influence from the bottom right (“relative velocity vector”), loops around Mars, and exits the frame. A parallel dotted line on the bottom left is labeled “relative velocity at x-axis”.]
Mars centered coordinates – True Rotating System Figure 5 [Graph showing a single curved trajectory starting from a high negative Y-rotating value, swooping upwards across the X-axis into positive Y territory, then dipping slightly back down and curving sharply towards the origin on the right side of the plot.]
[Clean version of the “Mars Capture Geometry” diagram from page 29] [Diagram labeled “Mars Capture Geometry”. A solid central circle represents Mars. It is surrounded by a “parking orbit” and a “phasing orbit”. Thrust vectors “dv1” and “dv2” are indicated. A “Hohmann Transfer” ellipse is shown. The outer dashed circle represents the “sphere of influence”. A dashed line indicates the “mars orbit”. A solid line enters the sphere of influence from the bottom right (“relative velocity vector”), loops around Mars, and exits the frame. A parallel dotted line on the bottom left is labeled “relative velocity at x-axis”.]
[Clean version of the “Earth Escape Geometry” diagram from page 26] [Diagram labeled “Earth Escape Geometry”. A solid central circle represents Earth. A dashed circular orbit is labeled “parking orbit”, an elliptical dashed orbit is labeled “Hohmann Transfer”, and a second dashed elliptical orbit is labeled “phasing orbit”. Tangential thrust vectors “dv1” and “dv2” are shown. A solid line showing the spacecraft’s path loops around Earth and escapes outward, labeled “hyperbolic escape”.]