{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:13:04Z","timestamp":1760238784071,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,2]],"date-time":"2020-09-02T00:00:00Z","timestamp":1599004800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/50022\/2020"],"award-info":[{"award-number":["UIDB\/50022\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Aerospace"],"abstract":"<jats:p>The two dimensional gravity turn problem is addressed allowing for the effects of variable rocket mass due to propellant consumption, thrust and thrust vector angle, lift and drag forces at an angle-of-attack and atmospheric mass density varying with altitude; Coriolis and centrifugal forces are neglected. Three distinct analytical solutions are obtained for constant: propellant flow rate, thrust, thrust vector angle, angle-of-attack and acceleration of gravity; the lift and drag are assumed to be proportional to the square of velocity, and the mass density is assumed to decrease exponentially with altitude. The method III uses power series of time for the horizontal (downrange) and vertical (altitude) coordinates; the method II replaces the altitude as variable by the atmospheric mass density and method I by its inverse. Thus the three solutions have distinct properties, e.g., I and III converge best close to lift-off and II close to burn-out. The three solutions: I, II, III, can be applied in isolation (or matched in combination) to the single-point boundary-value problem (SPBVP) of finding the trajectory with given initial conditions at launch. They can also be used as pairs in six distinct ways (I + II, I + III, II + III or reverse orders) to solve the two-point boundary-value problem (TPBVP), viz.: from given conditions at launch achieve one (not more) specified condition at burn-out, e.g., \u00e3 desired horizontal velocity for payload release. Each of the six distinct combinations of methods of addressing the TPBVP shares three features: (i) it can determine if there is a solution, viz. if the rocket has enough performance to reach the desired burn-out condition; (ii) if the desired burn-out condition is achievable it can calculate the complete trajectory from launch to burn-out; (iii) it can determine the range of achievable burn-out conditions, e.g., the minimum and maximum possible horizontal velocity at burn-out for given initial conditions at launch.<\/jats:p>","DOI":"10.3390\/aerospace7090131","type":"journal-article","created":{"date-parts":[[2020,9,2]],"date-time":"2020-09-02T09:29:28Z","timestamp":1599038968000},"page":"131","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Two-Point Boundary-Value Problem for Rocket Trajectories"],"prefix":"10.3390","volume":"7","author":[{"given":"Lu\u00eds","family":"M. B. C. Campos","sequence":"first","affiliation":[{"name":"CCTAE, IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2183-6221","authenticated-orcid":false,"given":"Paulo J. S.","family":"Gil","sequence":"additional","affiliation":[{"name":"CCTAE, IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"705","DOI":"10.2514\/3.20703","article-title":"Rocket trajectory optimization\u20141950\u20131963","volume":"14","author":"Lawden","year":"1991","journal-title":"J. Guid. Control Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"406","DOI":"10.2514\/3.19996","article-title":"Trajectory reconstruction during thrusting phase of rockets using differential corrections","volume":"8","author":"Manohar","year":"1985","journal-title":"J. Guid. Control Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Buchanan, G., Wright, D., Hann, C., Bryson, H., Snowdon, M., Rao, A., Slee, A., S\u00fcltrop, H.P., Jochle-Rings, B., and Barker, Z. (2015). The Development of Rocketry Capability in New Zealand\u2013World Record Rocket and First of Its Kind Rocketry Course. Aerospace, 2.","DOI":"10.3390\/aerospace2010091"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"da C\u00e1s, P.L.K., Veras, C.A.G., Shynkarenko, O., and Leonardi, R. (2019). A Brazilian Space Launch System for the Small Satellite Market. Aerospace, 6.","DOI":"10.3390\/aerospace6110123"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bryson, H., S\u00fcltrop, H.P., Buchanan, G., Hann, C., Snowdon, M., Rao, A., Slee, A., Fanning, K., Wright, D., and McVicar, J. (2016). Vertical Wind Tunnel for Prediction of Rocket Flight Dynamics. Aerospace, 3.","DOI":"10.3390\/aerospace3020010"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Messineo, J., and Shimada, T. (2019). Theoretical Investigation on Feedback Control of Hybrid Rocket Engines. Aerospace, 6.","DOI":"10.3390\/aerospace6060065"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"231","DOI":"10.5028\/jatm.v7i2.433","article-title":"A Six Degrees-of-Freedom Flight Dynamics Simulation Tool of Launch Vehicles","volume":"7","author":"Silveira","year":"2015","journal-title":"J. Aerosp. Technol. Manag."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Trevisi, F., Poli, M., Pezzato, M., Iorio, E.D., Madonna, A., Bressanin, N., and Debei, S. (2017, January 21\u201323). Simulation of a sounding rocket flight\u2019s dynamic. Proceedings of the 2017 IEEE International Workshop on Metrology for AeroSpace (MetroAeroSpace), Padua, Italy.","DOI":"10.1109\/MetroAeroSpace.2017.7999584"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1361","DOI":"10.2514\/3.11397","article-title":"Calculation of singular extremal rocket trajectories","volume":"15","author":"Lawden","year":"1992","journal-title":"J. Guid. Control Dyn."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4800","DOI":"10.1177\/0954410019830268","article-title":"Rapid ascent trajectory optimization for guided rockets via sequential convex programming","volume":"233","author":"Zhang","year":"2019","journal-title":"Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng."},{"key":"ref_11","unstructured":"Xiong, J., Tang, S., Guo, J., and Wu, X. (2013, January 23\u201325). Rapid ascent trajectory optimization for a solid rocket engine vehicle. Proceedings of the 2013 IEEE Third International Conference on Information Science and Technology (ICIST), Yangzhou, China."},{"key":"ref_12","unstructured":"Palaia, G., Pallone, M., Pontani, M., and Teofilatto, P. (June, January 30). Accurate Modeling and Heuristic Trajectory Optimization of Multistage Launch Vehicles. Proceedings of the 3rd IAA Conference on Dynamics and Control of Space Systems, Moscow, Russia."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1007\/s10513-005-0207-x","article-title":"Active Rocket Trajectory Arcs: A Review","volume":"66","author":"Azimov","year":"2005","journal-title":"Autom. Remote Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"193","DOI":"10.2514\/2.4231","article-title":"Survey of Numerical Methods for Trajectory Optimization","volume":"21","author":"Betts","year":"1998","journal-title":"J. Guid. Control Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/S0094-5765(96)00105-1","article-title":"An analysis of first-order singular thrust-arcs in rocket trajectory optimization","volume":"39","author":"Ross","year":"1996","journal-title":"Acta Astronaut."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/S0021-8928(96)00053-6","article-title":"Analytic solutions for intermediate-thrust arcs of rocket trajectories in a Newtonian field","volume":"60","author":"Azimov","year":"1996","journal-title":"J. Appl. Math. Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"51","DOI":"10.2514\/1.37387","article-title":"Numerical Study of Optimal Trajectories with Singular Arcs for an Ariane 5 Launcher","volume":"32","author":"Martinon","year":"2009","journal-title":"J. Guid. Control Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"515","DOI":"10.2514\/2.3864","article-title":"Initial Guess Generation for Rocket Ascent Trajectory Optimization Using Indirect Methods","volume":"39","author":"Gath","year":"2002","journal-title":"J. Spacecr. Rocket."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"770","DOI":"10.2514\/1.1082","article-title":"Optimal Conceptual Design of Two-Stage Reusable Rocket Vehicles Including Trajectory Optimization","volume":"41","author":"Tsuchiya","year":"2004","journal-title":"J. Spacecr. Rocket."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10778-017-0809-3","article-title":"Singular Optimal Controls of Rocket Motion (Survey)","volume":"53","author":"Kiforenko","year":"2017","journal-title":"Int. Appl. Mech."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Campos, L.M.B.C., and Gil, P.J.S. (2018). On Four New Methods of Analytical Calculation of Rocket Trajectories. Aerospace, 5.","DOI":"10.3390\/aerospace5030088"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"672","DOI":"10.1063\/1.1722828","article-title":"Universal Gravity Turn Trajectories","volume":"28","author":"Culler","year":"1957","journal-title":"J. Appl. Phys."},{"key":"ref_23","unstructured":"Thomson, W.T. (1986). Introduction to Space Dynamics, Dover. [2nd ed.]."},{"key":"ref_24","unstructured":"Miele, A. (1962). Flight Mechanics, Addison-Wesley."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1308","DOI":"10.2514\/3.28652","article-title":"Gravity turn trajectories through the atmosphere","volume":"3","author":"Connor","year":"1966","journal-title":"J. Spacecr. Rocket."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"538","DOI":"10.2514\/2.4914","article-title":"Semi-Analytical Formulas for Launcher Performance Evaluation","volume":"25","author":"Sotto","year":"2002","journal-title":"J. Guid. Control Dyn."},{"key":"ref_27","unstructured":"Rutherford, D.E. (1967). Classical Mechanics, Oliver and Boyd. [2nd ed.]."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1007\/BF00937371","article-title":"Method of particular solutions for linear, two-point boundary-value problems","volume":"2","author":"Miele","year":"1968","journal-title":"J. Optim. Theory Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/BF00927913","article-title":"Sequential gradient-restoration algorithm for optimal control problems","volume":"5","author":"Miele","year":"1970","journal-title":"J. Optim. Theory Appl."},{"key":"ref_30","unstructured":"(2020, June 10). Ariane 5. Available online: https:\/\/en.wikipedia.org\/wiki\/Ariane_5."},{"key":"ref_31","unstructured":"(2020, June 10). Ariane 5-VA226-Launch Profile. Available online: https:\/\/spaceflight101.com\/ariane-5-va226\/ariane-5-va226-launch-profile\/."}],"container-title":["Aerospace"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2226-4310\/7\/9\/131\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:05:56Z","timestamp":1760177156000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2226-4310\/7\/9\/131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,2]]},"references-count":31,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["aerospace7090131"],"URL":"https:\/\/doi.org\/10.3390\/aerospace7090131","relation":{},"ISSN":["2226-4310"],"issn-type":[{"type":"electronic","value":"2226-4310"}],"subject":[],"published":{"date-parts":[[2020,9,2]]}}}