Improvements to a Hybrid Algorithm for Rapid Generation of 3-D Optimal Launch Vehicle Ascent Trajectories

Diplomarbeit von Peter F. Gath

Abstract:

This thesis presents improvements to FLOAT, a hybrid analytical/numerical algorithm for rapid generation of three dimensional, optimal launch vehicle ascent trajectories. Improvements have been made to the terminal constraints, which are now available in a more general form to allow for an optimal attachment point to the target orbit.The existing algorithm also has been extended with logic that allows for vehicles with low thrust to weight ratios in the upper stage and successful convergence of problems with path constraints for normal force and angle of attack Another major extension made to the code is the introduction of coasting arcs. Coasting arcs are implemented using a completely analytical solution for the prediction of states and costates as well as for the required sensitivity matrix. This allows for a very fast and accurate calculation even with long coasting arcs.

Finally, an approach for the optimization of start and end time of coast arcs is presented.This approach was implemented and the results of a test case compare very well with results generated with OTIS for the same case.

At the end, suggestions for future development are made.

Table of Contents:

	Summary	i
	Acknowledgements	ii
	Contents	iii
	Nomenclature	v
	Figures	viii
	Introduction	1
1.	Problem description	3
1.1	Describing the final orbit	3
1.2	Coordinate frame	5
1.3	Dynamic system	6
1.4	Initial conditions	7
1.5	Path constraints	7
1.6	Performance index	7
1.7	Terminal constraints	8
1.8	Solution method	8
1.9	Non-dimensionalization of the variables	9
2.	Solving the two-point boundary value problem	10
2.1	Vacuumsolution	10
2.1.1	Simplified model equations	10
2.1.2	Optimal control for vacuum solution	11
2.1.3	Thrust integrals and closed form solution for ascent in vacuum	12
2.2	Atmospheric solution	13
2.2.1	Dynamic system and collocation variables	13
2.2.2	Optimality condition to solve for 1b	14
2.2.3	Differential equations for the costate variables	16
2.3	Terminal constraints	16
2.3.1	Attaching at perigee	17
2.3.2	Free attachment point	17
2.4	Transversality conditions	18
2.4.1	Final costates for attaching at perigee	18
2.4.2	Final costates for free attachment point	19
2.4.3	Equatorial orbits	22
2.5	Adjusting final time	22
2.6	Computation procedure	23
2.7	Numerical results	24
3.	Low thrust upper stages	27
3.1	Typical low thrust case	27
3.2	Problems with low thrust upper stages	28
3.3	Upper stage modification	30
3.4	Advantage of free attachment point for low thrust upper stage vehicles	32
4.	Path constraints	33
4.1	Axial acceleration constraint	33
4.2	Normal force constraint	34
4.3	Angle of attack constraint	35
4.4	Additional homotopy phase to introduce constraints	37
5	Coast arcs	38
5.1	Propagating states	38
5.2	Propagating costates	43
5.3	Adjusting coast arc position and duration	45
5.3.1	Optimizing coast arc duration	46
5.3.2	Optimizing start and end time of coast arc	48
5.3.3	Optimizing n optimal burn arcs	49
5.3.4	Inserting additional coast arcs	51
6.	Numerical results with coast arc and suggestions for future improvements	53
6.1	Ascent to equatorial orbit	53
6.2	Behavior of Hamiltonian	60
6.3	Comparison to OTIS results	61
6.4	Suggestions to improve convergence	62
7.	Conclusions and recommendations for future work	67
	References	70
	Appendix A - Obtaining the derivatives for the free attachment point constraint	72
	Appendix B - Obtaining the derivatives for the inclination constraint	75
	Appendix C - Calculation of the Jacobian	78
	Appendix D - Calculation of df and dA1 through dA4	81
	Appendix E - Example vehicles	84