本論文以搭載阿波羅 11 號任務的農神五號運載火箭（Saturn V，載具編號 AS-506）為研究對象，求解火箭從起飛到第三節引擎第一次燃燒結束、進入地球停泊軌道為止，燃料消耗最少的推力向量控制上升軌跡。升空過程以三自由度質點模型描述，依第一節分離、發射逃逸系統拋棄與第二節分離三個事件劃分為四個飛行階段，考慮大氣阻力並加入地球扁率於重力模型中進行模擬。
最佳軌跡數值求解分為兩個步驟：先以 Hermite-Simpson 直接配置法，將連續的最佳控制問題離散化為非線性規劃問題，再用 MATLAB 最佳化工具箱的序列二次規劃法求解。為了提升求解效率，本論文推導目標函數的解析梯度，並分析限制函數雅可比矩陣的稀疏結構，配合行著色技術，使每次評估雅可比矩陣所需的函數呼叫次數固定為常數，不隨問題規模增加而增加。初始猜測值則按飛行階段分段建立：第一節由兩參數重力轉彎的邊界值問題產生，第二節與第三節改用線性正切律的邊界值問題，求解器即以此組猜測值開始疊代。
求解出的最佳化軌跡，先與阿波羅 11 號飛行評估報告比對，以確認建立的模擬環境能重現真實飛行：在第一節外側發動機關機、第二節外側發動機關機與第三節第一次關機三個時間點，逐項檢視火箭到地心的距離、相對速度與飛行路徑角，數值量級皆與報告相符。接著，本論文比較推力向量偏轉上限 ±15°、±13° 與 ±12°（三節共用同一上限）三組設定下的解，觀察控制權限的影響；取 ±12° 為最小的一組，是因為上限再降低（±11.5°）時已求不出滿足收斂標準的解；更大的 ±20° 與 ±25° 僅列於表中以顯示趨勢。這三組解在大氣層內的機動幅度都遠大於實際飛行；三組解的入軌質量雖分別比實測值高出 0.8%、0.5% 與 0.2%，但這些節省量是在不考慮風與性能擾動的理想化條件下所得，只能視為模型內的理論值。若改採與實際飛行同屬小攻角的飛行方式——即將第一節偏轉限制在萬向節規格 ±5° 以內（對照 Apollo 11 實際測量值最大攻角 1.8°）——所得的最佳結果，和 Apollo 11 實際送入軌道的質量相比，低了約 1.1%；即使把第二、三節（近真空飛行段）的偏轉上限一路放寬到 20°，此差距仍無法補回——在本文的氣動模型下，第一節維持小攻角飛行本身就多消耗了 2,200 kg 的燃料。此約 1% 的差距，可能與質點模型本身的建模誤差有關，且 Apollo 11 實際採用的疊代導引，已相當接近最省推進劑的最佳解。最後，選取組中控制權限餘裕最大的 ±15° 解作為參考軌跡，在 Simulink 中建立含追蹤式抗飽和的比例—積分—微分控制器進行閉迴路追蹤。模擬結果顯示，加入風場擾動後，追蹤誤差雖然變大，但控制器仍能穩定跟隨參考軌跡並成功入軌，驗證這條軌跡在實作上確實可行。

This thesis computes the fuel-optimal thrust-vector-control (TVC) ascent trajectory of the Apollo 11 Saturn V launch vehicle (AS-506), from liftoff to the end of the S-IVB first burn at insertion into an Earth parking orbit. The ascent is modeled as a three-dimensional point mass over a rotating oblate Earth and is divided into four phases by S-IC separation, launch-escape-system (LES) jettison, and S-II separation. Hermite-Simpson direct collocation transcribes the optimal control problem into a nonlinear program, which is solved by sequential quadratic programming (SQP) with analytic gradients, a sparse constraint Jacobian, and phase-by-phase initial guesses built from boundary value problems. The optimized trajectory is validated against the Apollo 11 flight evaluation report. Solutions under deflection limits of ±15°, ±13°, and ±12° deliver about 0.8%, 0.5%, and 0.2% more mass to orbit than the actual flight — theoretical savings obtained by maneuvering far more aggressively inside the atmosphere than the actual vehicle did. When the first stage is instead confined to the small-angle-of-attack regime of the actual mission, the optimum falls about 1.1% below the measured insertion mass, which supports the assessment that the iterative guidance flown on Apollo 11 was already close to fuel-optimal. The ±15° solution is tracked closed-loop in Simulink by a proportional–integral–derivative (PID) controller under the measured AS-506 wind profile, confirming that the trajectory is practical to implement.

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