基于序列凸优化的多目标规避方法
doi: 10.11728/cjss2026.01.2025-0048 cstr: 32142.14.cjss.2025-0048
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周敬博 男, 2000年出生, 硕士研究生, 研究方向为轨道动力学与空间安全防护. E-mail: zhoujingbo200005@126.com -
李克行 男, 1977年出生, 研究员, 研究方向为航天器导航制导与控制技术. E-mail: lkh345@126.com
作者简介:
Muti-Collision Avoidance Method Based on Sequential Convex Optimization
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摘要: 随着近地轨道航天器与空间碎片数量激增, 航天器同空间碎片发生的交会事件不断增多, 航天器可能同时面对多个碎片的碰撞威胁, 因此航天器需具备对多个空间碎片的规避能力. 针对多个空间碎片短期交会的情况, 以航天器推力约束与碰撞概率约束为依据, 提出了基于序列凸优化的多目标规避方法. 将连续推力控制问题转化为脉冲推力的规划问题, 进而将凸优化问题的目标函数与非线性约束进行凸化处理, 采用序列凸优化方法求解该规划问题. 在对多目标的规避问题上, 该方法既能有效降低航天器与空间碎片的碰撞风险, 又能保证较低的燃料消耗, 能够适用于低推力航天器长时间规避机动规划. 同时, 序列凸优化问题的求解速度较快, 可以满足自主计算的需求.Abstract: As the number of spacecraft and space debris in Near-Earth Orbit (NEO) increases, the number of encounters between spacecraft and space debris continues to rise. It is imperative that spacecraft are equipped with the capacity to evade multiple collisions with space debris, as they may face such threats concurrently. A multi-collision avoidance method based on sequential convex optimization, which is designed to achieve short-term rendezvous of multiple space debris objects while considering the constraints of spacecraft thrust and collision probability, has been proposed. First, the continuous thrust control problem is transformed into a planning problem for impulse thrust. Then the relative dynamics and constraints are convexified to solve the planning problem using the sequential convex optimization method. The proposed method has been demonstrated to be effective in reducing the risk of spacecraft collision with space debris in the avoidance problem for multiple targets. It can also carry out long-time avoidance maneuver planning for low-thrust spacecraft and ensure lower fuel consumption. Furthermore, the solution to the sequence convex optimization problem has been shown to have a fast solution speed, making it suitable for autonomous computation.
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图 1 主要物体与次要物体初始轨道
Figure 1. Schematic diagram of initial orbit of primary and secondary objects
图 2 主要物体与次要物体在b平面上的相对位置矢量
Figure 2. Relative position vectors of primary and secondary objects in the b-plane
表 1 主要物体初始时刻轨道根数
Table 1. Orbital elements at the initial moment of primary objects
$ a $/km $ e $ $ i $/(º) Ω/(º) $ \omega $/(º) $ {f}_{0} $/(º) 7158 $ {0.001\;45} $ $ \text{86.4} $ $ \text{0} $ $ \text{0} $ $ \text{90} $ 
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表 3 主要物体与次要物体协方差
Table 3. Covariance of primary and secondary objects
主要物体协方差/$ \mathrm{km}^2 $ $ {9.317\;009\;058\;875\;35}\times {{10}}^{-5} $ $ {-2.623\;398\;113\;500\;550}\times {{10}}^{-4} $ $ {2.360\;382\;173\;935\;300}\times {{10}}^{-5} $ $ {-2.623\;398\;113\;500\;550}\times {{10}}^{-4} $ $ {1.777\;964\;542\;795\;11}\times {{10}}^{-2} $ $ {-9.331\;225\;387\;386\;501}\times {{10}}^{-5} $ $ {2.360\;382\;173\;935\;300}\times {{10}}^{-5} $ $ {-9.331\;225\;387\;386\;501}\times {{10}}^{-5} $ $ {1.917\;372\;231\;880\;040}\times {{10}}^{-5} $ 次要物体协方差/$ \mathrm{km}^2 $ $ {6.346\;570\;910\;720\;371}\times {{10}}^{-4} $ $ {-1.962\;292\;216\;245\;289}\times {{10}}^{-3} $ $ {7.077\;413\;655\;227\;660}\times {{10}}^{-5} $ $ {-1.962\;292\;216\;245\;289}\times {{10}}^{-3} $ $ {8.199\;899\;363\;150\;306}\times {{10}}^{-1} $ $ {1.139\;823\;810\;584\;350}\times {{10}}^{-3} $ $ {7.077\;413\;655\;227\;660}\times {{10}}^{-5} $ $ {1.139\;823\;810\;584\;350}\times {{10}}^{-3} $ $ {2.510\;340\;829\;074\;070}\times {{10}}^{-4} $
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算法1 序列凸化算法 输入的CDM信息: $ \boldsymbol{x}'_{\text{p,CA}},\boldsymbol{x}'_{{\text{s}}_{l},\text{CA}} $,$ \boldsymbol{\varSigma }'_{\text{p,RTN}},\boldsymbol{\varSigma }'_{{\text{s}}_{l},\text{RTN}} $,$ t'_{l,\text{CA}} $,$ {R}_{\text{HBR}} $ 指定: $ {t}_{0} $,$ \Delta t $,$ {t}_{\text{end}} $,$ N $,$ \Delta {v}_{\max } $,$ {P}_{\max } $,$ to{l}_{\text{m}} $,$ to{l}_{\text{M}} $ 反向从$ t'_{\text{CA}} $推广到$ {t}_{0} $, 获得$ \boldsymbol{x}'_{\text{p}}\left(0\right),\boldsymbol{x}'_{{\text{s}}_{l}}\left(0\right) $,$ \boldsymbol{\varSigma }'_{\text{p,ECI}}\left(0\right),\boldsymbol{\varSigma }'_{{\text{s}}_{l},\text{ECI}}\left(0\right) $ 分段时间: $ {t}_{0}\colon \Delta t\colon {t}_{\text{end}} $ $ j\leftarrow 0,{\boldsymbol{X}}^{0}\leftarrow ,t_{\mathrm{l},\text{CA}}^{0}\leftarrow t'_{l,\text{CA}},\Delta \boldsymbol{r}_{l,\text{CA}}^{1,0}\leftarrow \Delta \boldsymbol{r}'_{l,\text{CA}},\boldsymbol{\varSigma }_{l,b2}^{0}\leftarrow \boldsymbol{\varSigma }'_{l,b2} $ while $ \left(j=0\right)\text{or}{\left|\left|{\boldsymbol{X}}^{j}-{\boldsymbol{X}}^{j-1}\right|\right|}_{\mathrm{\infty }}\geq {e}_{\text{M}} $ $ j\leftarrow j+1 $ 使用$ {\boldsymbol{X}}^{j-1} $作为输入, 解微分方程(6)获得参考轨迹 通过式(45)、式(48)、式(54)更新$ N_{l,\text{CA}}^{j},t_{l,\text{CA}}^{j},\Delta \boldsymbol{r}_{l,\text{CA}}^{j},\boldsymbol{B}_{l,\text{CA}}^{j}, $ $ \boldsymbol{\varSigma }_{\mathrm{l},b2}^{j},d_{l,\text{CA},\max }^{j} $ $ k\leftarrow 0,\Delta \boldsymbol{r}_{\text{CA}}^{j,0}\leftarrow \Delta \boldsymbol{r}_{\text{CA}}^{j} $ while $ \left(k=0\right)\text{or}\underset{l=1\cdots M}{\text{max}}\left({\left|\left|\Delta \boldsymbol{r}_{l,\text{CA}}^{j,k}-\Delta \boldsymbol{r}_{l,\text{CA}}^{j,k-1}\right|\right|}_{2}\right)\geq {e}_{\text{m}} $ $ k\leftarrow k+1 $ 解$ M $个子凸优化问题式(42)得到$ z_{l}^{k} $ 解机动规划凸优化问题式(55)得到$ {\boldsymbol{X}}^{j,k} $ $ \Delta \boldsymbol{r}_{l,\text{CA}}^{j,k}\leftarrow \Delta \boldsymbol{r}_{l,\text{CA}}^{j,k-1}+\boldsymbol{B}_{l,\text{CA}}^{j}\left({\boldsymbol{X}}^{j,k}-{\boldsymbol{X}}^{j,k-1}\right) $ end while $ {\boldsymbol{X}}^{j}\leftarrow {\boldsymbol{X}}^{j,k} $ end while
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