Cassini 2
Here we consider a different model for the Cassini trajectory with respect to the "Cassini1" case: deep space maneuvers are allowed between each one of the planets. This leads to a higher dimensional problem and to a higher complexity. We also consider, in the objective function evaluation, the final DV as a rendezvous rather than as an orbital insertion as in the "Cassini1" case.
The code
- MATLAB: use the function cassini2.m and pass to it the MGAproblem variable contained in cassini2.mat
- C++: call the function "double cassini2(const std::vector & x)" provided in the GTOPtoolbox.
- C++ (PaGMO): use the class pagmo::problem::cassini_2
- Python 2.7 (PyGMO): use PyGMO.problem.cassini_2().obj_fun(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 | -1000 | 0 | MJD2000 |
| x(2) | Vinf | 3 | 5 | km/sec |
| x(3) | u | 0 | 1 | n/a |
| x(4) | v | 0 | 1 | n/a |
| x(5) | T1 | 100 | 400 | days |
| x(6) | T2 | 100 | 500 | days |
| x(7) | T3 | 30 | 300 | days |
| x(8) | T4 | 400 | 1600 | days |
| x(9) | T5 | 800 | 2200 | days |
| x(10) | eta1 | 0.01 | 0.9 | n/a |
| x(11) | eta2 | 0.01 | 0.9 | n/a |
| x(12) | eta3 | 0.01 | 0.9 | n/a |
| x(13) | eta4 | 0.01 | 0.9 | n/a |
| x(14) | eta5 | 0.01 | 0.9 | n/a |
| x(15) | r_p1 | 1.05 | 6 | n/a |
| x(16) | r_p2 | 1.05 | 6 | n/a |
| x(17) | r_p3 | 1.15 | 6.5 | n/a |
| x(18) | r_p4 | 1.7 | 291 | n/a |
| x(19) | b_incl1 | -pi | pi | rad |
| x(20) | b_incl2 | -pi | pi | rad |
| x(21) | b_incl3 | -pi | pi | rad |
| x(22) | b_incl4 | -pi | pi | rad |
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
Author | Affiliation | Date | Obj Fun | Solutions | Algo |
|---|---|---|---|---|---|
| T., Vinko, D., Izzo | European Space Agency | March, 2008 | 8.924 km/s | x=[-815.144, 3, 0.623166, 0.444834, 197.334, 425.171, 56.8856, 578.523, 2067.98, 0.01, 0.470415, 0.01, 0.0892135, 0.9, 1.05044, 1.38089, 1.18824, 76.5066, -1.57225, -2.01799, -1.52153, -1.5169] | DE |
| B. Addis, A. Cassioli, M. Locatelli, F. Schoen | Global Optimization Laboratory, University of Florence and University of Turin | May, 2008 | 8.530 km/s | x=[-775.073192377, 3.34939840763, 0.519859871615, 0.377648835192, 164.514496917, 420.857826262, 54.7953357438, 587.550012334, 2172.83783929, 0.666969680984, 0.437098773474, 0.0143681690338, 0.0151496118993, 0.0852702838402, 1.54714942102, 1.18980793974, 1.25955058062, 71.3030467945, -1.62797088705, -1.98297404025, -1.53976626419, -1.51425029677] | MBH |
| M. Vasile, E. Minisci | University of Glasgow | July, 2008 | 8.419 km/s | x=[ -778.047305631216, 3.18312649956441, 0.517194165210629, 0.399454532008061, 164.47695637194, 423.970795573978, 54.7479052540502 589.990666953636, 2199.60570034135, 0.762963004566719, 0.550121138102629, 0.0126601160795708, 0.17223077205906, 0.270111519882677, 1.42837324938403, 1.1774879453091, 1.26425574056075, 70.1906989621161, -1.59765413874108, -1.98140734837078, -1.53978935969435, -1.51329954805784] | DE |
| M. Vasile, E. Minisci, M. Locatelli | University of Glasgow, University of Turin | July, 2008 | 8.409 km/s | x=[-779.060197373242, 3.32046443745595, 0.531333503613675, 0.376218447342955, 168.685775870437, 422.672656805198, 53.3360098337041, 589.777827855018, 2200, 0.718720247401635, 0.532962541494841, 0.159170896444411, 0.470495109020601, 0.0986526263521857, 1.46946051297954, 1.05138706406598, 1.30594027188689, 69.8194077461197, -1.60160853231321, -1.9600386515463, -1.55445003054861, -1.51343200828766] | MBH |
| B. Addis, A. Cassioli, M. Locatelli, F. Schoen | Global Optimization Laboratory, University of Florence and University of Turin | Sept., 2008 | 8.405 km/s | x=[-778.765203484, 3.23018448427, 0.523118701494, 0.392764786131, 165.698711849, 424.757237926, 53.7855982056, 589.617865592, 2197.32875485, 0.733366262496, 0.545182500335, 0.02342601953, 0.07119604834, 0.0508270625102, 1.31540232722, 1.09024682281, 1.29246247343, 70.037231829, -1.58673415524, -1.96631114594, -1.54982899287, -1.51341338598] | MBH |
| M. Vasile, E. Minisci, M. Locatelli | University of Glasgow, University of Turin | July, 2008 | 8.385 km/s | x=[-779.046753814506, 3.25911446832345, 0.525976214695235, 0.38086496458657, 167.378952534645, 424.028254165204, 53.2897409769205, 589.766954923325, 2200, 0.769483451363201, 0.513289529822621, 0.0274175362264024, 0.263985256705873, 0.599984695281461, 1.34877968657176, 1.05, 1.30730278372017, 69.8090142993495, -1.5937371121191, -1.95952512232447, -1.55498859283059, -1.5134625299674] | -- |
| M. Schlueter, J. Fiala, M. Gerdts | University of Glasgow, University of Birmingham | May, 2009 | 8.383 km/s | x=[-779.046753814506, 3.25911446832345, 0.525976214695235, 0.38086496458657, 167.378952534645, 424.028254165204, 53.2897409769205, 589.766954923325, 2200, 0.769483451363201, 0.513289529822621, 0.0274175362264024, 0.263985256705873, 0.599984695281461, 1.34877968657176, 1.05, 1.30730278372017, 69.8090142993495, -1.5937371121191, -1.95952512232447, -1.55498859283059, -1.5134625299674] | Found by MIDACO within the project "Non-linear mixed-integer-based Optimisation Technique for Space Applications" co-funded by ESA Networking Partnership Initiative, Astrium Limited (Stevenage, UK) and the School of Mathematics, University of Birmingham, UK. |