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摘要:
针对常值推力下航天器面内轨道转移燃耗最省的轨道优化问题，利用极大值原理导出了最优轨迹下推力方向角应满足的控制方程，结合动力学方程建立了一种求解航天器面内最优转移轨道的改进间接法，及其在推力方向角调节能力受限条件下的应用方法。由于避免了协态变量微分方程组的求解，改进间接法相对于传统间接法降低了初值猜测的难度和计算量；与采用Gauss伪谱法求解相比，所建立的改进间接法求解结果精度更高，数值光滑性更好。仿真算例表明：推力方向角调节能力受限会改善推力方向角变化规律，降低推力方向角变化范围；就燃耗而言，推力越大燃耗越多，优化轨道节省燃耗更加显著。
Abstract:The transfer of a spacecraft between coplanar orbits under continuous constant thrust with minimum fuel consumption was investigated. A control equation of the optimal trajectory, which the steering angle must satisfy, was derived by using maximum principle. Combining the control equation with dynamic equations, we establish an improved indirect method to design optimal coplanar transfer orbit and propose an approximate application method for the condition when thrust direction adjustment ability was limited. Due to avoiding solving the Lagrange costate differential equations, the improved indirect method reduces difficulty of initial value estimation and calculation a lot than traditional indirect method. Compared with Gauss pseudospectral method, the improved indirect method can get higher precision and better numerical smoothness. Simulations show that a limitation on the magnitude of the second derivative of thrust angle can improve change law of thrust angle and reduce change range of thrust angle; as to the fuel consumption, the greater the thrust magnitude is, the more the fuel consumption is, and the optimal transfer orbit can save fuel consumption a lot under a certain large magnitude of thrust.
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Key words:
- orbit transfer /
- optimal control /
- constant thrust /
- maximum principle /
- coplanar orbit
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图 4 不同推力加速度下推力方向角变化曲线
Figure 4. Thrust angle vs time under different thrust acceleration
图 6 |ϕ″|≤0.008 (°)/s2时推力方向角变化规律
Figure 6. Change law of thrust angle when |ϕ″|≤0.008 (°)/s2
表 1 不同条件下的转移轨道燃耗
Table 1. Fuel consumption of orbit transfer under different conditions
推力加速度/
(10-3m·s-2)速度变化量/(m·s-1) 切向推力转移轨道 优化轨道 Gauss伪谱法求解不限ϕ″ Gauss伪谱法求解|ϕ″|≤0.008 (°)/s2 改进间接法求解不限ϕ″ 改进间接法求解|ϕ″|≤0.008 (°)/s2 0.02 13.205 12.757 12.757 12.757 12.757 0.10 14.862 14.005 14.007 14.005 14.005 0.15 12.858 12.713 12.713 12.713 12.713 0.20 14.995 14.254 14.257 14.254 14.254 0.40 22.588 20.645 20.722 20.645 20.670 1.00 40.315 34.428 34.447 34.309 34.421 4.00 100.270 72.974 73.007 72.856 72.937 10.00 185.780 117.660 117.770 117.560 117.690 
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