Messenger (full version)
This trajectory optimisation problem represents a rendezvous mission to Mercury modelled as an MGA-1DSM problem. The selected fly-by sequence and other parameters are compatible with the currently flying Messenger mission. With respect to the problem Messenger (reduced), the fly-by sequence is more complex and allows for resonant fly-bys at Mercury to lower the arrival DV. As far as chemical propelled interplanetary trajectories go, this particular one is particularly complex and difficult to design. The amount of specialistic knowledge that needs to be used to obtain a successfull design is significant and, before the remarkable results from G. Stracquadanio, A. La Ferla and G. Nicosia (see below) were found, it was hardly believable that a computer, given the fly-by sequence and an ample launch window, could design a good trajectory in complete autonomy without making use of additional problem knowledge.
The code
- MATLAB: use the function messengerfull.m and pass to it the MGAproblem variable contained in messengerfull.mat
- C++: call the function "double messengerfull(const std::vector & x)" provided in the GTOPtoolbox.
- C++ (PaGMO): use the class pagmo::problem::messeger_full
- Python 2.7 (PyGMO): use PyGMO.problem.messeger_full().objfun(x)
Problem Description
The box bounds on each of the decision vector variable are given below.
State | Variable | Lower Bounds | Upper Bounds | Units |
|---|---|---|---|---|
| x(1) | t0 | 1900 | 2300 | MJD2000 |
| x(2) | Vinf | 2.5 | 4.05 | km/sec |
| x(3) | u | 0 | 1 | n/a |
| x(4) | v | 0 | 1 | n/a |
| x(5) | T1 | 100 | 500 | days |
| x(6) | T2 | 100 | 500 | days |
| x(7) | T3 | 100 | 500 | days |
| x(8) | T4 | 100 | 500 | days |
| x(9) | T5 | 100 | 500 | days |
| x(10) | T6 | 100 | 600 | days |
| x(11) | eta1 | 0.01 | 0.99 | days |
| x(12) | eta2 | 0.01 | 0.99 | n/a |
| x(13) | eta3 | 0.01 | 0.99 | n/a |
| x(14) | eta4 | 0.01 | 0.99 | n/a |
| x(15) | eta5 | 0.01 | 0.99 | n/a |
| x(16) | eta6 | 0.01 | 0.99 | n/a |
| x(17) | r_p1 | 1.1 | 6 | n/a |
| x(18) | r_p2 | 1.1 | 6 | n/a |
| x(19) | r_p3 | 1.05 | 6 | n/a |
| x(20) | r_p4 | 1.05 | 6 | n/a |
| x(21) | r_p5 | 1.05 | 6 | n/a |
| x(22) | b_incl1 | -pi | pi | n/a |
| x(23) | b_incl2 | -pi | pi | n/a |
| x(24) | b_incl3 | -pi | pi | n/a |
| x(25) | b_incl4 | -pi | pi | n/a |
| x(26) | b_incl5 | -pi | pi | n/a |
No other constraints are considered for this problem. The objective function is considered to the precision of meters per second.
Solutions:
The best solutions submitted to this problem are listed below in chronological order.
Objective Function (km/s) | Solution Vector | Credits: | Date: |
|---|---|---|---|
| 6.943 | N/A | M. Schlueter, J. Fiala, M. Gerdts, University of Birmingham (found by MIDACO solver) | 19/06/2009 |
| 6.404 | N/A | G. Stracquadanio, A. La Ferla, G. Nicosia, University of Catania (Found by SAGES Self-Adaptive- Gaussian Evolutionary Strategy) | 17/11/2009 |
| 6.047 | N/A | M. Schlueter, University of Birmingham, M. Gerdts, University of Wuerzburg, M. Munetomo and K. Akama, Hokkaido University, S. Erb and G. Ortega, ESTEC/TEC-ECM (found by MIDACO solver) | 30/11/2009 |
| 4.254 | N/A | F. Biscani and D. Izzo, ESTEC Advanced Concepts Team. Found using PaGMO | 01/12/2009 |
| 2.970 | x = [2037.98811327576208896062, 4.04999923293175267958, 0.55668295364006559200, 0.63442241846342029010, 451.61549554889830915272, 224.69404452813012085244, 221.37565155671779848490, 269.57368260182812491621, 267.38440535525694485841, 529.63112171564182517614, 0.64326985928466706710, 0.71762145795073051247, 0.69585552227505076406, 0.72983941248758110731, 0.72845456800654084795, 0.87005691197281054272, 1.69359753796273460047, 1.10000000087138438687, 1.05000002563125405253, 1.05000001880069060434, 1.05000275284308064450, 2.77281392591838837802, 1.56807099547678863161, 2.65646585499325071922, -1.57357003056070321456, -1.57633751284459244779] | G. Stracquadanio, Dept of Biomedical Engineering, Johns Hopkins University, A. La Ferla, G. Nicosia, University of Catania (Found by SAGES Self-Adaptive- Gaussian Evolutionary Strategy) | 28/02/2011 |
| 2.113 | x=[2060.69440202574969589477, 4.04269112527382556976, 0.44059489075180063855, 0.65331122144567743693, 428.89625699706778050313, 224.69403898951520659466, 221.30624564658421604690, 266.24324448899460549001, 357.59063180909026868903, 444.61732604543010438647, 0.56290791173634824318, 0.38624128635969090517, 0.69949004446927753875, 0.66884189137768290667, 0.82933310225879219857, 0.86923487511685348927, 1.57624377056926956442, 1.10012566413730894510, 1.05219311415095639894, 1.05036443853409666715, 1.72992837141351518682, 2.78624948818170814491, 1.60302706895153312949, 2.62081666961903714252, 1.56982977393428124735, 1.61939808401837570528] | G. Stracquadanio, Dept of Biomedical Engineering, Johns Hopkins University, A. La Ferla, G. Nicosia, University of Catania (Found by SAGES Self-Adaptive- Gaussian Evolutionary Strategy) | 10/04/2012 |
| 2.104 | x=[2060.69440202574969589477, 4.04269112527382556976, 0.44059489075180063855, 0.65331122144567743693, 428.89625699706778050313, 224.69403898951520659466, 221.30624564658421604690, 266.24324448899460549001, 357.59063180909026868903, 444.61732604543010438647, 0.56290791173634824318, 0.38624128635969090517, 0.69949004446927753875, 0.66884189137768290667, 0.82933310225879219857, 0.86923487511685348927, 1.57624377056926956442, 1.10012566413730894510, 1.05219311415095639894, 1.05036443853409666715, 1.72992837141351518682, 2.78624948818170814491, 1.60302706895153312949, 2.62081666961903714252, 1.56982977393428124735, 1.61939808401837570528] | M. Schlueter, M. Munetomo (found by MIDACO solver) | 17/10/2013 |
| 1.983 | x=[ 2037.793650139994270, 4.035829738138824, 0.555436051620218, 0.636393238132614, 451.447750355605706, 224.694208867341700, 221.880177091452026, 265.151670648704567, 358.288400601717910, 534.212841688461253, 0.538362042300023, 0.753339114855577, 0.719294670714628, 0.750352217362636, 0.830544140272688, 0.902346174479331, 1.424520437366302, 1.100327418086428, 1.050586612441532, 1.164323431714812, 1.072024806423244, 2.820081320974608, 1.515793529485625, 2.588292685117210, 1.756804428126312, 1.530086523658156] | M. Schlueter, (found by MIDACO solver) | 11/02/2014 |
| 1.972 | x=[2037.735811696081782, 4.036904766842642, 0.555556999357183, 0.636428972195638, 451.484382165790407, 224.694263238629617, 221.856631717467451, 265.933341444053781, 357.976995644824399, 534.106576178371711, 0.532515156187294, 0.706442511540274, 0.718411658825252, 0.759622141750054, 0.828618506885608, 0.902851485640470, 1.424762657242012, 1.100000006850630, 1.050585900931688, 1.050000000000000, 1.050000000000000, 2.820002913100699, 1.513622824139136, 2.593053687091726, 1.754040819378354, 1.568520102392375] | M. Schlueter, (found by MIDACO solver) | 24/04/2014 |
| 1.959 | x = [2037.8595972244, 4.0500001697, 0.5567269199, 0.6347532625, 451.6575153013, 224.6939374104, 221.4390510408, 266.0693628875, 357.9584322778, 534.1038782374, 0.6378086222, 0.7293472066, 0.6981836705, 0.7407197230, 0.8289833176, 0.9028496299, 1.8337484775, 1.1000000238, 1.0499999523, 1.0499999523, 1.0499999523, 2.7481808788, 1.5952416573, 2.6241779073, 1.6276418577, 1.6058416537] | M. Schlueter, Mohamed Wahib and Naoya Maruyama (found by MXHPC, which is a parallelization framework for MIDACO) | 15/09/2014 |