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本文是对2014年全国大学生数学建模竞赛A题的解答.本文利用动力学模型,基于变推力发动机结合螺旋搜索等算法解决了嫦娥三号软着陆预测及优化问题.在问题一中,首先建立物理学模型,解出嫦娥三号着陆准备轨道近、远月点速度大小;然后在主减速阶段,建立了主减速变推力动力学模型,对主减速阶段嫦娥三号所经过的极角θ进行求解,加以考虑月球自转带来的影响,确定着陆准备轨道近月点和远月点的位置以及嫦娥三号在近、远月点的速度方向.在问题二中,建立变推力发动机燃耗优化模型对于问题一中的主减速阶段进行优化;以安全半径与速度增量为指标建立综合评价模型,求出粗避障式安全的降落范围;精避障阶段以保证安全为前提,建立评价指标为不平坦程度与平均坡度的综合评价模型确定精确降落区域.最后根据对每个阶段建立的力学模型,进行求解.最后综合起来确定嫦娥三号的着陆轨道和在6个阶段的最优控制策略.在问题三中,建立误差模型对设计的着陆轨道和控制策略做相应的误差分析,并且利用单因素敏感性分析对主减速阶段与剩余4阶段进行自变量与因变量敏感度计算.
This article is a solution to the question A of National Undergraduate Mathematical Contest in Modeling in 2014. In this paper, we first solve the problem of the prediction and optimization of Chang’e-3 soft landing based on the dynamic model, the algorithm of variable thrust engine combined with spiral search, etc. In question one, Physics model to solve the Chang’e III landing preparation orbit near and far point of the size of the point velocity; and then in the main deceleration phase, established a main deceleration variable thrust dynamics model, the main deceleration phase Chang’e III through the polar angle θ To solve the problem, the influence of lunar rotation should be taken into consideration to determine the position of the lunar orbit and lunar month of the landing preparation track and the speed direction of Chang’e-3 near and far moon.In the second problem, The model is optimized for the main deceleration phase of the first problem. The comprehensive evaluation model is established based on the index of safety radius and speed increment, and the landing range of rough obstacle avoidance is obtained. In order to ensure the safety, the evaluation index The exact landing area is determined for the comprehensive evaluation model of unevenness and average slope.At last, the mechanical model established for each stage is used to solve the problem. Together to determine the landing trajectory of Chang’e III and the optimal control strategy in six phases.In the third problem, the error model was established to make corresponding error analysis on the designed landing trajectory and control strategy, and the single factor sensitivity analysis Main deceleration phase and the remaining four stages of independent variable and dependent variable sensitivity calculation.