A symplectic pseudo-spectral successive convex optimization method for trajectory planning of ascent stage of exo-atmosphere launch vehicle
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摘要:
为完成对运载火箭入轨飞行段轨迹的优化，提高算法的计算效率及收敛性，从拉格朗日力学出发，在离散力学与最优控制(DMOC)计算方法的基础上结合伪谱法高精度的优点，推导出伪谱离散拉格朗日方程，并结合序列凸优化方法提出了基于保辛伪谱序列凸优化的轨迹优化方法。保辛伪谱序列凸优化方法可使离散动力学系统保留原连续系统的结构特征，同时使离散系统状态变量的维数大幅度降低，有效提升收敛性及计算效率。仿真结果表明：相比于经典的伪谱序列凸优化方法，保辛伪谱序列凸优化方法在不损失精度的情况下大幅度提高了计算效率，并且对初值扰动具有很好的适应性。
Abstract:In order to improve the computational efficiency and convergence of the algorithm, this paper derives the pseudo-spectral discrete Lagrange equation based on Lagrange mechanics, discrete mechanics and optimal control (DMOC) calculation method, combining with the advantages of high accuracy of the pseudo-spectral method. In conjunction with the successive convex optimization technique, a symplectic pseudo-spectral successive convex optimization-based trajectory optimization method is suggested. By significantly reducing the dimension of the discrete system’s state variables, the symplectic pseudo-spectral sequential convex optimization method can considerably increase convergence and computational efficiency while maintaining the structural features of the original continuous system. In contrast to the classical pseudo-spectral successive convex optimization method, the simulation results demonstrate that the symplectic pseudo-spectral successive convex optimization method has a good adoptability to the initial disturbance and can significantly increase computational efficiency without sacrificing accuracy.
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表 1 仿真参数
Table 1. Simulation parameters
参数 数值 $ {a_{{\text{tg}}}}/{\text{m}} $ 7218871.485 ${e_{{\text{tg}}}}$ 0.001503 ${i_{{\text{tg}}}}$/(°) 42 $ {{\varOmega }_{{\mathrm{tg}}}}$/(°) 288.5286 ${\omega _{{\text{tg}}}}$/(°) 155.7384 $P/{\text{kN}}$ 150 ${I_{{\text{sp}}}}/{\text{s}}$ 342.7864 ${m_0}/{\text{kg}}$ 29125.718 ${R_{\rm{e}}}/{\text{m}}$ 6378140 ${g_0}/\left( {{\text{m}} \cdot {{\text{s}}^{{{ - 2}}}}} \right)$ 9.8066 $\mu /\left( {{{\text{m}}^3} \cdot {{\text{s}}^{{{ - 2}}}}} \right)$ 3.986×1014 
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表 2 计算效率
Table 2. Calculate efficiency
计算方法 迭代次数 ECOS求解器
总耗时/ms计算
总时间/ms伪谱序列凸优化 6 110.7 116.6 保辛伪谱序列凸优化 6 58.7 63.1
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表 3 入轨误差
Table 3. Orbital insertion
计算方法 $\Delta a/{\text{m}}$ $\Delta e$ $\Delta i{/ (^\circ) }$ $\Delta {\varOmega }{/ (^\circ) }$ $\Delta \omega {/(^\circ) }$ 伪谱序列凸优化 − 0.1958 $ 2.991\;0 \times {10^{ - 8}} $ $ - 6.146\;8 \times {10^{ - 8}} $ $ - 4.184\;0 \times {10^{ - 8}} $ 0.2219 保辛伪谱序列凸优化 − 0.3226 $ 2.233\;2 \times {10^{ - 8}} $ $ - 4.110\;5 \times {10^{ - 8}} $ $ - 8.650\;0 \times {10^{ - 8}} $ 0.2210
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