# Optimal Control Approach to [PERSON]'s Problem and Gibbs' Method
## 5 Numerical Simulations
### Extended Lambert's Problem  
A numerical simulation was performed to demonstrate the validity of the proposed optimal control approach to the extended Lambert's problem. The data used in this simulation were taken from [6]. In the first simulation, the energy-specified [PERSON]'s problem was considered. The initial position of an Earth satellite was determined to be:  
\[\overset{\rightarrow}{r}(t_{1})=5000\ \dot{t}_{1}+10,000\ \dot{t}_{2}+2100\ \dot{t}_{3}\ ( km) \tag{42}\]  
and after some time, the final position vector was determined to be:\(\overrightarrow{r}(t_{2})\ m)\ 10,000\(n\) earth satellite is determined to be, and after one hour the position vector is  
\[=-14,600\ t_{1}+2500\ t_{2}+7000\ t_{3}\ (\mathrm{km}) \tag{43}\]  
where \(\hat{t}_{1},\hat{t}_{2},\hat{t}_{3}\) are the unit direction vectors of the ECI frame. The orbital energy was kept as:  
\[\mathcal{E}_{set}=-9.963549\ \mathrm{km}^{2}/\mathrm{s}^{2}\]  
which corresponds to an elliptical orbit. The exact time of flight was determined as:  
\[\Delta t=t_{2}-t_{1}=3600\ \sec\]  
and the exact initial velocity was determined as:  
\[\overrightarrow{v}(t_{1})=-5.992495\ \hat{t}_{1}+1.925364\ \hat{t}_{2}+3.245637\ \hat{t}_{3}\ ( \mathrm{km}/\mathrm{s}) \tag{44}\]  
This extended [PERSON]'s problem was solved by the optimal control approach using LPM with the NLP solver 'fmincon' in Matlab with 12 LGL points. The time of flight was calculated as:  
\[\Delta\hat{t}=3600.00023\ \sec\]  
and the average orbital energy was calculated as:  
\[\hat{\mathcal{E}}_{\mathrm{true}}=-9.963598\ \mathrm{km}^{2}/\mathrm{s}^{2}\]  
which shows that the proposed optimal control approach calculates the time of flight and the orbital energy very accurately. Figure 2 shows the time history of the orbital energy estimate error, \(e_{\mathcal{E}}=\mathcal{E}_{set}-\mathcal{E}(t)\). The initial velocity was calculated as:  
\[\widehat{v}(t_{1})=-5.992494\ \hat{t}_{1}+1.925376\ \hat{t}_{2}+3.245451\ \hat{t}_{3}\ ( \mathrm{km}/\mathrm{s})\]  
and the error was \(e_{v}=|\overrightarrow{v}(t_{1})-\widehat{v}(t_{1})|=1.857\times 10^{-4}\ ( \mathrm{km}/\mathrm{s})\) which shows that the proposed optimal control approach accurately produces the [PERSON]'s solution. Figure 2: Time history of the orbital energy estimate error. Other simulations were performed with the same initial and final positions, but with different energy levels, such as \(\mathcal{E}_{set}=0\) km\({}^{2}\)/s\({}^{2}\) and \(\mathcal{E}_{set}=+9.96355\) km\({}^{2}\)/s\({}^{2}\), which correspond to parabolic and hyperbolic orbits, respectively. In the case of a parabolic orbit, the time of flight was calculated as \(\Delta t=2761.3743\) sec, while for a hyperbolic orbit, it was \(\Delta t=2357.0746\) sec. Simulation results are presented in Figure 3, which shows elliptical, parabolic, and hyperbolic orbits in the ECI frame with the same initial and final position, respectively. These simulation results show that the energy-specified [PERSON]'s problem can provide an alternative orbit design tool in which the orbit connects two points with a specified orbital energy level or a specified orbit shape. In the next simulation, the velocity-specified [PERSON]'s problem was solved. In this simulation, the initial position of an Earth satellite was determined to be:  
\[\overrightarrow{r}(t_{1})=5000\ \hat{t}_{1}+10,\!000\ \hat{t}_{2}+2100\ \hat{t}_{3}\ ( \text{km})\]  
and after one hour the velocity vector was determined to be:  
\(\overrightarrow{v}(t_{2})\ m)\ 10,\!000n\) earth satellite is determined to be, and after one hour the position vector is  
\(=-3.312460\ \hat{t}_{1}-4.196617\ \hat{t}_{2}-0.385288\ \hat{t}_{3}\ ( \text{km}/\text{s})\)  
The exact finial position was given by:  
\(\overrightarrow{r}(t_{2})\ m)\ 10,\!000n\) earth satellite is determined to be, and after one hour the position vector is  
\(=-14,\!600\ \hat{t}_{1}+2500\ \hat{t}_{2}+7000\ \hat{t}_{3}\ (\text{km})\). This problem was also solved by the optimal control approach using LPM with 12 LGL points.