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摘要:
针对运载火箭助推段预测校正制导方法中在线快速轨迹预测困难的问题，提出一种适用于大攻角剖面的运载火箭助推段高精度弹道解析解。推导得到以质量为自变量的纵平面简化动力学模型，并创新性地将攻角的正弦值设计为关于质量的多项式。因为大攻角机动的影响，简化动力学模型依旧是高度非线性的，无法直接求得解析解。通过受力分析，构造近似多项式替代原方程中非线性强但模值较小的项，并将真实值与近似值之差作为摄动小量。根据摄动理论可以对动力学模型进行分阶以获得可以解析求解的子系统。对子系统积分可以得到速度、弹道倾角、航程和高度的解析解。仿真实验表明：在大攻角条件下，所提出解析解的精度比现有解析解至少提升了85%。
Abstract:To rapidly predict the ascent trajectory of launch vehicles, high-accuracy analytical solutions for ascent trajectory were proposed under high angle-of-attack (AOA) conditions. First, a simplified longitudinal-plane dynamics model with mass as the independent variable was developed. The sine of AOA was innovatively expressed as a polynomial of mass. Due to the high AOA, the simplified dynamics model remains highly nonlinear, which prevents it from being analytically solved directly. Approximate polynomials are introduced to replace the strongly nonlinear but relatively small terms in the original equations through force analysis. Furthermore, the difference between the true values and the approximations was treated as a minor perturbation. The dynamics model was divided into analytically solvable subsystems according to the perturbation theory. Analytical solutions for velocity, flight-path angle, downrange, and altitude were derived by solving the subsystems. Simulation results confirm that the proposed solutions are at least 85% more accurate than existing solutions under high AOA conditions.
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Key words:
- analytical solution /
- ascent trajectory /
- high angle of attack /
- launch vehicle /
- perturbation method
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图 3 算例一解析解精度对比
Figure 3. Analytical solution results and trajectory simulation for case 1
表 1 算例一解析解终端误差及计算时间
Table 1. Terminal results and computing time for case 1
速度/(m·s−1) 速度相对误差1/% 速度相对误差2/% 弹道倾角/(°) 弹道倾角
相对误差1/%弹道倾角
相对误差2/%数值仿真 ${x^{\left( 0 \right)}}$ ${x^{\left( 0 \right)}} + {x^{\left( 1 \right)}}$ 数值仿真 ${x^{\left( 0 \right)}}$ ${x^{\left( 0 \right)}} + {x^{\left( 1 \right)}}$ 1 609.5 1 607.6 1 607.9 −0.120 2 −0.098 9 41.728 4 43.475 6 41.435 4 4.187 0 −0.702 3 航程/km 航程相对
误差1/%航程相对
误差2/%高度/km 高度倾
角相对
误差1/%高度倾
角相对
误差2/%计算时间/s 数值仿真 ${x^{\left( 0 \right)}}$ ${x^{\left( 0 \right)}} + {x^{\left( 1 \right)}}$ 数值仿真 ${x^{\left( 0 \right)}}$ ${x^{\left( 0 \right)}} + {x^{\left( 1 \right)}}$ 数值仿真 ${x^{\left( 0 \right)}}$ ${x^{\left( 0 \right)}} + {x^{\left( 1 \right)}}$ 24.507 8 24.995 0 24.819 7 1.988 0 1.272 8 37.833 5 39.671 7 37.869 5 4.858 6 0.095 0 0.819 4 1.563 4×10−5 3.472 0×10−5 注：相对误差1是x(0)相对数值仿真的误差；相对误差2是x(0)+x(1)相对数值仿真的误差。 
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