# The Two-Point Boundary-Value Problem for Rocket Trajectories

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Three Methods of Calculation of Gravity Turn

#### 2.1. Method III: Iterative Use of Initial Conditions in Taylor Series

- (i)
- the variation of mass with time, Equation (2b), corresponds to a constant, Equation (2a), propellant rate:$$\begin{array}{cc}\hfill c& \equiv -\frac{\mathrm{d}m}{\mathrm{d}t}:\hfill \end{array}$$$$\begin{array}{cc}\hfill m\left(t\right)& ={m}_{0}-c(t-{t}_{0});\hfill \end{array}$$
- (ii)
- the acceleration of gravity is assumed to be uniform, and the altitude coordinate z is taken opposite to it; (iii) the thrust matrix, Equation (3a) is constant for thrust T, angle-of-attack $\alpha $ and angle $\epsilon $ of the thrust with the rocket axis:$$\begin{array}{cc}\hfill {T}_{ij}& =T\left[\begin{array}{cc}cos(\alpha +\epsilon )& -sin(\alpha +\epsilon )\\ sin(\alpha +\epsilon )& \phantom{-}cos(\alpha +\epsilon )\end{array}\right]\hfill \end{array}$$$$\begin{array}{cc}\hfill {F}_{ij}& =\left[\begin{array}{cc}cos\alpha & \phantom{-}sin\alpha \\ sin\alpha & -cos\alpha \end{array}\right]\left[\begin{array}{cc}{C}_{\mathrm{D}}& \phantom{-}{C}_{\mathrm{L}}\\ {C}_{\mathrm{L}}& -{C}_{\mathrm{D}}\end{array}\right],\hfill \end{array}$$
- (iii)
- the aerodynamic matrix (3b) is constant for constant angle-of-attack $\alpha $ and lift ${C}_{\mathrm{L}}$ and drag ${C}_{\mathrm{D}}$ coefficients; (iv) the aerodynamic forces are proportional to the square of the velocity and to the atmospheric mass density; (v) the latter decays exponentially (4b) on the scale height ℓ for an isothermal atmosphere with scale height (4a), viz.:$$\begin{array}{cc}\hfill \ell & \equiv R\theta /g,\hfill \end{array}$$$$\begin{array}{cc}\hfill \rho \left(z\right)& ={\rho}_{0}exp\left[-(z-{z}_{0})/\ell \right],\hfill \end{array}$$

#### 2.2. Method I: Mass Fraction of Burned Fuel as the Time Variable

#### 2.3. Method II: Ratio of Atmospheric Mass Densities as Altitude Variable

## 3. General Approach to the Two-Point Boundary-Value Problem (TPBVP)

#### 3.1. Smooth Matching of Ascending and Descending Trajectories

- (i)
- an ascending solution for increasing time starting at lift-off up to burn-out$${t}_{0}\u2a7dt<{t}_{1}:\phantom{\rule{2.em}{0ex}}X\left(t\right)=\left\{x\left(t\right),z\left(t\right),u\left(t\right),w\left(t\right),{x}_{0},{z}_{0},{u}_{0},{w}_{0}\right\};$$
- (ii)
- a descending solution for decreasing time starting at burn-out$${t}_{1}\u2a7et>{t}_{0}:\phantom{\rule{2.em}{0ex}}\overline{X}\left(t\right)=\left\{\overline{x}\left(t\right),\overline{z}\left(t\right),\overline{u}\left(t\right),\overline{w}\left(t\right),{x}_{1},{z}_{1},{u}_{1},{w}_{1}\right\}.$$

#### 3.2. Feasibility of Desired Burn-Out Condition for Payload Launch

#### 3.3. Performance Envelope at Burn-Out Condition for Payload Launch

- (i)
- the ascending solutions are needed not only for the downrange, Equation (49a), and altitude, Equation (49b), but also for horizontal, Equation (52a), and vertical, Equation (52b), velocities:$$\begin{array}{cc}\hfill u\left(t\right)& \equiv \frac{\mathrm{d}x}{\mathrm{d}t}={v}_{\u2605}\sum _{n=2}^{\infty}{X}_{n}n{\tau}^{n-1},\hfill \end{array}$$$$\begin{array}{cc}\hfill w\left(t\right)& \equiv \frac{\mathrm{d}z}{\mathrm{d}t}={v}_{\u2605}{e}^{-z/\ell}\sum _{n=2}^{\infty}{Z}_{n}n{\tau}^{n-1},\hfill \end{array}$$
- (ii)
- it also appears in the descending trajectory, Equations (51a) and (51b), when the horizontal, Equation (53a), and vertical, Equation (53b), velocities are calculated:$$\begin{array}{cc}\hfill \overline{u}\left(t\right)& \equiv \dot{\overline{x}}=-{v}_{\u2605}\sum _{n=0}^{\infty}{P}_{n}(n+1){(1-\tau )}^{n},\hfill \end{array}$$$$\begin{array}{cc}\hfill \overline{w}\left(t\right)& \equiv \dot{\overline{z}}=-{v}_{\u2605}{e}^{\overline{z}/\ell}\sum _{n=0}^{\infty}{Q}_{n}(n+1){(1-\tau )}^{n};\hfill \end{array}$$
- (iii)
- (iv)
- substitution of the dimensionless burn-out time ${\tau}_{1}$ in the descending solution, Equations (51a) and (51b), (53a) and (53b), specifies four dimensionless constants:$$\begin{array}{cc}\hfill {x}_{1}/\ell & =\sum _{n=0}^{\infty}{P}_{n}{({m}_{1}/{m}_{0})}^{n+1}\equiv {I}_{1},\hfill \end{array}$$$$\begin{array}{cc}\hfill {e}^{-{z}_{1}/\ell}& =\sum _{n=0}^{\infty}{Q}_{n}{({m}_{1}/{m}_{0})}^{n+1}\equiv {I}_{2},\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{1}/{v}_{\u2605}& =-\sum _{n=0}^{\infty}{P}_{n}(n+1){({m}_{1}/{m}_{0})}^{n}\equiv {I}_{3},\hfill \end{array}$$$$\begin{array}{cc}\hfill ({w}_{1}/{v}_{\u2605}){e}^{-{\overline{z}}_{1}/\ell}& =-\sum _{n=0}^{\infty}{Q}_{n}(n+1){({m}_{1}/{m}_{0})}^{n}\equiv {I}_{4},\hfill \end{array}$$

## 4. Trajectory for a Given Horizontal Velocity at Burn-Out

#### 4.1. Matching Time as a Root of the Series Solution

#### 4.2. Rocket Data Required for Trajectory Calculation

#### 4.3. Combination of Methods I and II for the TPBVP

## 5. Second Alternative Set of Matching Conditions for the TPBVP

#### 5.1. Alternative Choices of Ascent Trajectories for Matching up to Burn-Out

- (i)
- in Equations (79a) and (79b):$$\left[\begin{array}{c}2{R}_{2}\\ 2{S}_{2}+g\end{array}\right]=\frac{T}{{m}_{0}}\left[\begin{array}{cc}cos(\alpha +\epsilon )& -sin(\alpha +\epsilon )\\ sin(\alpha +\epsilon )& \phantom{-}cos(\alpha +\epsilon )\end{array}\right]\left[\begin{array}{c}cos{\gamma}_{0}\\ sin{\gamma}_{0}\end{array}\right],$$
- (ii)
- in Equations (80) and (79b),$$\left[\begin{array}{c}2{R}_{3}\\ 2{S}_{3}\end{array}\right]=\frac{c}{3{m}_{0}}\left[\begin{array}{c}2{R}_{2}\\ 2{S}_{2}+g\end{array}\right]=\frac{cT}{3{m}_{0}^{2}}\left[\begin{array}{cc}cos(\alpha +\epsilon )& -sin(\alpha +\epsilon )\\ sin(\alpha +\epsilon )& \phantom{-}cos(\alpha +\epsilon )\end{array}\right]\left[\begin{array}{c}cos{\gamma}_{0}\\ sin{\gamma}_{0}\end{array}\right].$$

#### 5.2. Matching Distinct Ascending Solutions to the Same Descending Solution

#### 5.3. Trajectory Matching outside the Burn Range

## 6. Six Trajectories of the TPBVP Using Three Pairs of Solutions

#### 6.1. Matching of the Third Pair of Solutions in Two Forms

#### 6.2. Third Set of Matching Condition for TPBVP Trajectories

#### 6.3. Third Trajectory with Specified Horizontal Velocity at Burn-Out

- (i)
- by method III using Equation (77a) with $n=2$ leading to Equation (102c)$$\begin{array}{cc}\hfill {\ddot{x}}_{0}& =2{R}_{2}:\hfill \end{array}$$$$\begin{array}{cc}\hfill {\ddot{x}}_{0}& =\underset{t\to 0}{lim}\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^{2}}=\ell {\left(\frac{\mathrm{d}\tau}{\mathrm{d}t}\right)}^{2}\underset{\tau \to 0}{lim}\frac{{\mathrm{d}}^{2}\chi}{\mathrm{d}{\tau}^{2}}=2{X}_{2}\ell {\left(\frac{c}{{m}_{0}}\right)}^{2}=\frac{2{X}_{2}{v}_{\u2605}c}{{m}_{0}};\hfill \end{array}$$
- (ii)

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lawden, D.F. Rocket trajectory optimization—1950–1963. J. Guid. Control Dyn.
**1991**, 14, 705–711. [Google Scholar] [CrossRef] - Manohar, D.R.; Krishnan, S. Trajectory reconstruction during thrusting phase of rockets using differential corrections. J. Guid. Control Dyn.
**1985**, 8, 406–408. [Google Scholar] [CrossRef] - Buchanan, G.; Wright, D.; Hann, C.; Bryson, H.; Snowdon, M.; Rao, A.; Slee, A.; Sültrop, H.P.; Jochle-Rings, B.; Barker, Z.; et al. The Development of Rocketry Capability in New Zealand–World Record Rocket and First of Its Kind Rocketry Course. Aerospace
**2015**, 2, 91. [Google Scholar] [CrossRef] - da Cás, P.L.K.; Veras, C.A.G.; Shynkarenko, O.; Leonardi, R. A Brazilian Space Launch System for the Small Satellite Market. Aerospace
**2019**, 6, 123. [Google Scholar] [CrossRef][Green Version] - Bryson, H.; Sültrop, H.P.; Buchanan, G.; Hann, C.; Snowdon, M.; Rao, A.; Slee, A.; Fanning, K.; Wright, D.; McVicar, J.; et al. Vertical Wind Tunnel for Prediction of Rocket Flight Dynamics. Aerospace
**2016**, 3, 10. [Google Scholar] [CrossRef] - Messineo, J.; Shimada, T. Theoretical Investigation on Feedback Control of Hybrid Rocket Engines. Aerospace
**2019**, 6, 65. [Google Scholar] [CrossRef][Green Version] - Silveira, G.d.; Carrara, V. A Six Degrees-of-Freedom Flight Dynamics Simulation Tool of Launch Vehicles. J. Aerosp. Technol. Manag.
**2015**, 7, 231–239. [Google Scholar] [CrossRef][Green Version] - Trevisi, F.; Poli, M.; Pezzato, M.; Iorio, E.D.; Madonna, A.; Bressanin, N.; Debei, S. Simulation of a sounding rocket flight’s dynamic. In Proceedings of the 2017 IEEE International Workshop on Metrology for AeroSpace (MetroAeroSpace), Padua, Italy, 21–23 June 2017; pp. 296–300. [Google Scholar]
- Lawden, D.F. Calculation of singular extremal rocket trajectories. J. Guid. Control Dyn.
**1992**, 15, 1361–1365. [Google Scholar] [CrossRef] - Zhang, K.; Yang, S.; Xiong, F. Rapid ascent trajectory optimization for guided rockets via sequential convex programming. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**2019**, 233, 4800–4809. [Google Scholar] [CrossRef] - Xiong, J.; Tang, S.; Guo, J.; Wu, X. Rapid ascent trajectory optimization for a solid rocket engine vehicle. In Proceedings of the 2013 IEEE Third International Conference on Information Science and Technology (ICIST), Yangzhou, China, 23–25 March 2013; pp. 142–145. [Google Scholar]
- Palaia, G.; Pallone, M.; Pontani, M.; Teofilatto, P. Accurate Modeling and Heuristic Trajectory Optimization of Multistage Launch Vehicles. In Proceedings of the 3rd IAA Conference on Dynamics and Control of Space Systems, Moscow, Russia, 30 May–1 June 2017; pp. 809–824. [Google Scholar]
- Azimov, D.M. Active Rocket Trajectory Arcs: A Review. Autom. Remote Control
**2005**, 66, 1715. [Google Scholar] [CrossRef] - Betts, J.T. Survey of Numerical Methods for Trajectory Optimization. J. Guid. Control Dyn.
**1998**, 21, 193–207. [Google Scholar] [CrossRef] - Ross, I.M. An analysis of first-order singular thrust-arcs in rocket trajectory optimization. Acta Astronaut.
**1996**, 39, 417–422. [Google Scholar] [CrossRef] - Azimov, D.M. Analytic solutions for intermediate-thrust arcs of rocket trajectories in a Newtonian field. J. Appl. Math. Mech.
**1996**, 60, 421–427. [Google Scholar] [CrossRef] - Martinon, P.; Bonnans, F.; Laurent-Varin, J.; Trelat, E. Numerical Study of Optimal Trajectories with Singular Arcs for an Ariane 5 Launcher. J. Guid. Control Dyn.
**2009**, 32, 51–55. [Google Scholar] [CrossRef] - Gath, P.F.; Well, K.H.; Mehlem, K. Initial Guess Generation for Rocket Ascent Trajectory Optimization Using Indirect Methods. J. Spacecr. Rocket.
**2002**, 39, 515–521. [Google Scholar] [CrossRef] - Tsuchiya, T.; Mori, T. Optimal Conceptual Design of Two-Stage Reusable Rocket Vehicles Including Trajectory Optimization. J. Spacecr. Rocket.
**2004**, 41, 770–778. [Google Scholar] [CrossRef] - Kiforenko, B.N. Singular Optimal Controls of Rocket Motion (Survey). Int. Appl. Mech.
**2017**, 53, 237–286. [Google Scholar] [CrossRef] - Campos, L.M.B.C.; Gil, P.J.S. On Four New Methods of Analytical Calculation of Rocket Trajectories. Aerospace
**2018**, 5, 88. [Google Scholar] [CrossRef][Green Version] - Culler, G.J.; Fried, B.D. Universal Gravity Turn Trajectories. J. Appl. Phys.
**1957**, 28, 672–676. [Google Scholar] [CrossRef] - Thomson, W.T. Introduction to Space Dynamics, 2nd ed.; Dover: Mineola, NY, USA, 1986. [Google Scholar]
- Miele, A. Flight Mechanics; Addison-Wesley: Boston, MA, USA, 1962; Volume 2. [Google Scholar]
- Connor, M.A. Gravity turn trajectories through the atmosphere. J. Spacecr. Rocket.
**1966**, 3, 1308–1311. [Google Scholar] [CrossRef] - Sotto, E.D.; Teofilatto, P. Semi-Analytical Formulas for Launcher Performance Evaluation. J. Guid. Control Dyn.
**2002**, 25, 538–545. [Google Scholar] [CrossRef] - Rutherford, D.E. Classical Mechanics, 2nd ed.; Oliver and Boyd: Edinburgh, UK, 1967. [Google Scholar]
- Miele, A. Method of particular solutions for linear, two-point boundary-value problems. J. Optim. Theory Appl.
**1968**, 2, 260–273. [Google Scholar] [CrossRef] - Miele, A.; Pritchard, R.E.; Damoulakis, J.N. Sequential gradient-restoration algorithm for optimal control problems. J. Optim. Theory Appl.
**1970**, 5, 235–282. [Google Scholar] [CrossRef] - Ariane 5. Available online: https://en.wikipedia.org/wiki/Ariane_5 (accessed on 10 June 2020).
- Ariane 5-VA226-Launch Profile. Available online: https://spaceflight101.com/ariane-5-va226/ariane-5-va226-launch-profile/ (accessed on 10 June 2020).

**Figure 1.**Forces on a rocket in flight (thrust T, weight W, drag D and lift L) and relevant angles (angle-of-attack $\alpha $, flight path angle $\gamma $, and thrust vector angle $\epsilon $).

**Figure 2.**Ascent trajectory 1 from lift-off Equation (41a) and descent trajectories (2a, 2b, 2c) from burn-out Equation (41b) in the two cases when Equation (42), the two-point boundary-value problem (TPBVP), has: (a) no solution, and the trajectories do not cross because Equation (42) cannot be satisfied for any matching time t

_{2}; (b) has at least one solution, because Equation (42) has solutions for matching time t

_{2}corresponding to the crossing of trajectories. At the matching time the trajectories are tangent because both the horizontal and vertical coordinates and velocities are continuous.

Symbol | Meaning | Value | Unit |
---|---|---|---|

Environment | |||

g | acceleration of gravity | 9.81 | $\mathrm{m}/\mathrm{s}$ |

${\rho}_{0}$ | sea level mass density | 1.225 | $\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ |

ℓ | atmospheric scale height | $2.6\times {10}^{4}$ | $\mathrm{m}$ |

Propulsion | |||

T | thrust | $1.555\times {10}^{7}$ | $\mathrm{k}\mathrm{g}\mathrm{m}/{\mathrm{s}}^{2}$ |

c | propellant flow rate | $2.029\times {10}^{3}$ | $\mathrm{k}\mathrm{g}/\mathrm{s}$ |

${m}_{0}-{m}_{1}$ | propellant mass | $2.841\times {10}^{5}$ | $\mathrm{k}\mathrm{g}$ |

${t}_{1}-{t}_{0}$ | burn time | $1.4\times {10}^{2}$ | $\mathrm{s}$ |

Aerodynamics | |||

${m}_{0}$ | initial mass | $7.77\times {10}^{5}$ | $\mathrm{k}\mathrm{g}$ |

S | cross-sectional area | 3.76 x 10 | ${\mathrm{m}}^{2}$ |

${C}_{\mathrm{D}}$ | drag coefficient | 0.15 | * |

${C}_{\mathrm{L}}$ | lift coefficient | 0.10 | * |

Calculated parameters | |||

${m}_{*}$ | reference mass | $1.198\times {10}^{6}$ | $\mathrm{k}\mathrm{g}$ |

a | weight parameter | 5.533 × 10 | * |

b | thrust parameter | $1.129\times {10}^{2}$ | * |

f | aerodynamic parameter | $7.706\times {10}^{-1}$ | * |

${t}_{*}-{t}_{0}$ | reference time | $3.829\times {10}^{2}$ | $\mathrm{s}$ |

${v}_{*}$ | reference velocity | 6.789 × 10 | $\mathrm{m}/\mathrm{s}$ |

Calculated matrices | |||

trust | aerodynamic | ||

${b}_{11}$ | 1.128 × 10^{2} | ${f}_{11}$ | 0.118 |

${b}_{12}$ | −3.94 | ${f}_{12}$ | 0.073 |

${b}_{21}$ | $3.94$ | ${f}_{21}$ | $-0.073$ |

${b}_{22}$ | 1.128102 | ${f}_{22}$ | 0.118 |

Pair of Methods * | |
---|---|

I + II (Section 4) | III + II (Section 5) |

${X}_{2}=$ 3.85 × 10^{1} | ${R}_{0}=0={R}_{1}$ |

${P}_{0}=3.78$ | ${S}_{0}=0={S}_{1}$ |

${Q}_{0}=-6.12$ | ${R}_{2}=6.82\mathrm{m}/{\mathrm{s}}^{2}$ |

${P}_{1}=$ −1.80 × 10^{2} | ${S}_{2}=2.41\mathrm{m}/{\mathrm{s}}^{2}$ |

${Q}_{1}=$ 2.17 × 10^{2} | ${R}_{3}=$ 5.94 × 10^{−3} $\mathrm{m}/{\mathrm{s}}^{3}$ |

${I}_{3}=$ 7.88 × 10^{1} | ${S}_{3}=$ 6.37 × 10^{−3} $\mathrm{m}/{\mathrm{s}}^{3}$ |

${u}_{11}=$ −1.88 × 10^{4} $\mathrm{m}/\mathrm{s}$ | ${u}_{13}=$ −1.88 × 10^{4} $\mathrm{m}/\mathrm{s}$ |

${u}_{12}=$ −7.98 × 10^{3} $\mathrm{m}/\mathrm{s}$ | ${u}_{14}=$ −7.98 × 10^{3} $\mathrm{m}/\mathrm{s}$ |

^{3}$\mathrm{m}/\mathrm{s},\phantom{\rule{1.em}{0ex}}{m}_{1}/{m}_{0}=0.634,\phantom{\rule{1.em}{0ex}}{\gamma}_{0}=\pi /4$.

Expression | Value | Unit |
---|---|---|

${u}_{1}$ | $5.35\times {10}^{3}$ | $\mathrm{m}/\mathrm{s}$ |

${u}_{1}/{v}_{\u2605}={J}_{3}$ | $7.88\times {10}^{1}$ | |

$1-{m}_{1}/{m}_{0}$ | 0.366 | |

$2{X}_{2}(1-{m}_{1}/{m}_{0})$ | $2.81\times {10}^{1}$ | |

${u}_{15}={v}_{\u2605}{J}_{3}$ | $5.35\times {10}^{3}$ | $\mathrm{m}/\mathrm{s}$ |

$2{X}_{2}{v}_{\u2605}$ | $5.23\times {10}^{3}$ | $\mathrm{m}/\mathrm{s}$ |

$2{R}_{2}{m}_{0}/c$ | $5.23\times {10}^{3}$ | $\mathrm{m}/\mathrm{s}$ |

${u}_{16}$ | $5.35\times {10}^{3}$ | $\mathrm{m}/\mathrm{s}$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

M. B. C. Campos, L.; Gil, P.J.S.
The Two-Point Boundary-Value Problem for Rocket Trajectories. *Aerospace* **2020**, *7*, 131.
https://doi.org/10.3390/aerospace7090131

**AMA Style**

M. B. C. Campos L, Gil PJS.
The Two-Point Boundary-Value Problem for Rocket Trajectories. *Aerospace*. 2020; 7(9):131.
https://doi.org/10.3390/aerospace7090131

**Chicago/Turabian Style**

M. B. C. Campos, Luís, and Paulo J. S. Gil.
2020. "The Two-Point Boundary-Value Problem for Rocket Trajectories" *Aerospace* 7, no. 9: 131.
https://doi.org/10.3390/aerospace7090131