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在激光操控空间碎片的运动姿态中,是采用小光斑激光、点覆盖碎片表面局部一点情况下进行的,激光烧蚀力产生对碎片质心的力矩,所以,激光烧蚀力的力矩影响碎片运动姿态,并且激光烧蚀力还影响目标的运动轨道。在给出有无激光烧蚀力作用下的轨道分析方法的基础上,以典型圆柱体碎片为例,提出激光烧蚀力作用下的空间碎片运动姿态分析方法,并建立了天基激光平台消旋碎片目标分析方法,进一步分析典型碎片姿态运动和轨道运动的变化规律。
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图10所示为圆柱体碎片的底面半径为
$R$ ,高度为$H$ ,质心为$C$ ,在体固联坐标系${X_b}{Y_b}{Z_b}$ 中(以质心为原点),碎片平台位置矢量为${{{r}}_{\rm{DS},}}_{{X_b}}$ ,激光辐照点为${{{r}}_{b0}} = {({x_{b0}},{y_{b0}},{z_{b0}})^{\rm{T}}}$ ,由于碎片存在初始角速度${\omega _{xb,0}}$ ,利用激光烧蚀力施加反向力矩以达到降低转动角速度的目的。图 10 圆柱体碎片的激光辐照方向与激光烧蚀力方向
Figure 10. Laser irradiation direction and laser ablation force direction with cylindrical debris
激光辐照方向矢量
${{{L}}_{R,{X_b}}}$ 为:$${{{L}}_{R,{X_b}}} = {{{r}}_{{\rm{DS}},}}_{{X_b}} + {{{r}}_{{b0}}}$$ (1) 在赤道惯性坐标系
$XYZ$ 中,在任意时刻,空间碎片位置矢量为${{{r}}_{{\rm{deb}},X}} = {({r_{{\rm{deb}},x}},{r_{{\rm{deb}},y}},{r_{{\rm{deb}},z}})^{\rm{T}}}$ ,空间平台位置矢量为${{{r}}_{{\rm{sta}},X}} = {({r_{{\rm{sta}},x}},{r_{{\rm{sta}},y}},{r_{{\rm{sta}},z}})^{\rm{T}}}$ ,碎片平台位置矢量为:$${{{r}}_{{\rm{DS}},X}} = {{{r}}_{{\rm{deb}},X}} - {{{r}}_{{\rm{sta}},X}} = \left[ {\begin{array}{*{20}{c}} {{r_{{\rm{deb}},x}} - {r_{{\rm{sta}},x}}} \\ {{r_{{\rm{deb}},y}} - {r_{{\rm{sta}},y}}} \\ {{r_{{\rm{deb}},z}} - {r_{{\rm{sta}},z}}} \end{array}} \right]$$ (2) 在体固联坐标系
${X_b}{Y_b}{Z_b}$ 中,碎片平台位置矢量${{{r}}_{{\rm{DS}},}}_{{X_b}}$ 为:$${{{r}}_{{\rm{DS}},{X_b}}} = {{{Q}}_{{X_b}X}}{{{r}}_{{\rm{DS}},X}}$$ (3) 式中:
${{{Q}}_{{X_b}X}}$ 为用欧拉角表示的坐标旋转变换矩阵。激光辐照方向矢量
${{{L}}_{R,{X_b}}}$ 为:$${{{L}}_{R,{X_b}}} = {{{r}}_{{\rm{DS}},{X_b}}} + {{{r}}_{b{\rm{0}}}} = {{{Q}}_{{X_b}X}}{{{r}}_{{\rm{DS}},X}} + {{{r}}_{b{\rm{0}}}}$$ (4) 在激光辐照点
${{{r}}_{b0}} = {({x_{b0}},{y_{b0}},{z_{b0}})^{\rm{T}}}$ 处,圆柱体表面法向单位矢量${{{\hat n}}_{{X_b}}} = {({\hat n_{xb}},{\hat n_{yb}},{\hat n_{zb}})^{\rm{T}}}$ 为:$${{{\hat n}}_{{X_b}}} = {({x_{b0}}/R,{y_{b0}}/R,0)^{\rm{T}}}$$ (5) 激光辐照方向矢量为
${{{L}}_{R,{X_b}}} = {({L_{R,xb}},{L_{R,yb}},{L_{R,zb}})^{\rm{T}}}$ ,激光能够辐照点${{{r}}_{b0}} = {({x_{b0}},{y_{b0}},{z_{b0}})^{\rm{T}}}$ 的条件为:$${{{\hat n}}_{{X_b}}} \cdot {{{L}}_{R,{X_b}}} = \dfrac{{{x_{b0}}}}{R}{L_{R,xb}} + \dfrac{{{y_{b0}}}}{R}{L_{R,yb}} < 0$$ (6) 单脉冲激光烧蚀冲量为:
$${{{F}}_{L,{X_b}}}{\tau '_L} = - {C_m}{I_L}{\tau _L}{A_L}\left| {\cos ({{{{\hat n}}}_{{X_b}}},{{{L}}_{R,{X_b}}})} \right|{{{\hat n}}_{{X_b}}}$$ (7) 式中:
${{{F}}_{L,{X_b}}}$ 为单脉冲平均激光烧蚀力;Cm为碎片材料的冲量耦合系数;IL为激光功率密度;τL为激光脉宽;τ′L为激光烧蚀力的作用时间;rL为激光光斑半径;激光光斑的横截面面积为 ${A_L} = \pi r_L^2$ ;R、H分别为圆柱体的半径和高,并且${r_L} < < R$ 和${r_L} < < H$ 。碎片目标单位质量的激光烧蚀冲量为:
$${{{f}}_{L,{X_b}}}{\tau '_L} = - \dfrac{{{C_m}{I_L}{\tau _L}{A_L}\left| {\cos ({{{{\hat n}}}_{{X_b}}},{{{L}}_{R,{X_b}}})} \right|}}{{{\text{π}}{R^2}H\rho }}{{{\hat n}}_{{X_b}}}$$ (6) 式中:
${{{f}}_{L,{X_b}}}$ 为碎片目标单位质量的激光烧蚀力;$\rho $ 为碎片材料密度。单脉冲激光烧蚀冲量矩(激光烧蚀冲量
${{{F}}_{L,{X_b}}}{\tau '_L}$ 的冲量矩)为:$${{{L}}_{{X_b}}}{\tau '_L} = {{{r}}_{b0}} \times ({{{F}}_{L,{X_b}}}{\tau '_L})$$ (9) 式中:
${{{L}}_{{X_b}}} = {{{r}}_{b0}} \times {{{F}}_{L,{X_b}}}$ 表示单脉冲激光烧蚀力矩。 -
已知空间碎片轨道参数为
$(a,e,i,\varOmega ,\omega ,M)$ ,分别为轨道半长轴、偏心率、倾角、升交点赤经、近地点幅角、平近角,在小偏心率条件下,以轨道参数$(a,i,\varOmega ,\xi = e\sin \omega ,\eta = e\cos \omega ,\lambda = M + \omega )$ 表示。在坐标系$STW$ 中,空间碎片的单位质量激光烧蚀力为${{{f}}_{L,S}} = {({f_{L,S}},{f_{L,T}},{f_{L,W}})^{\rm{T}}}$ ,激光烧蚀力作用时间为${\tau '_L}$ ,碎片目标单位质量激光烧蚀产生的速度增量为:$$\left\{ {\begin{array}{*{20}{c}} {\varDelta {v_{L,S}} = {f_{L,S}}{{\tau '}_L}} \\ {\varDelta {v_{L,T}} = {f_{L,T}}{{\tau '}_L}} \\ {\varDelta {v_{L,W}} = {f_{L,W}}{{\tau '}_L}} \end{array}} \right.$$ (10) 式中:碎片单位质量的激光烧蚀冲量为
${{{f}}_{L,S}}{\tau '_L} = {({f_{L,S}}{\tau '_L},{f_{L,T}}{\tau '_L},{f_{L,W}}{\tau '_L})^{\rm{T}}}$ 。如果认为激光烧蚀力瞬间作用,在该时刻轨道参数改变量为:
$$\Delta a = \dfrac{2}{{n\sqrt {1 - {e^2}} }}\left[ {\Delta {v_{L,S}}(e\sin f) + \Delta {v_{L,T}}\left( {\dfrac{p}{r}} \right)} \right]$$ (11) $$\Delta i = \dfrac{{r\cos u}}{{n{a^2}\sqrt {1 - {e^2}} }}\Delta {v_{L,W}}$$ (12) $$\Delta \varOmega = \dfrac{{r\sin u}}{{n{a^2}\sqrt {1 - {e^2}} \sin i}}\Delta {v_{L,W}}$$ (13) $$\begin{split} & \Delta \xi = - \eta \cos i\Delta \varOmega + \\ & \dfrac{{\sqrt {1 - {e^2}} }}{{na}}\left[ \begin{array}{l} - \Delta {v_{L,S}}\cos u + \Delta {v_{L,T}}(\sin u + \sin \tilde u) \\ + \Delta {v_{L,T}}\dfrac{{\eta (e\sin E)}}{{\sqrt {1 - {e^2}} (1 + \sqrt {1 - {e^2}} )}} \\ \end{array} \right] \\ \end{split} $$ (14) $$\begin{split} & \Delta \eta = \xi \cos i\Delta \varOmega + \\ & \dfrac{{\sqrt {1 - {e^2}} }}{{na}}\left[ \begin{array}{l} \Delta {v_{L,S}}\sin u + \Delta {v_{L,T}}(\cos u + \cos \tilde u) \\ - \Delta {v_{L,T}}\dfrac{{\xi (e\sin E)}}{{\sqrt {1 - {e^2}} (1 + \sqrt {1 - {e^2}} )}} \\ \end{array} \right] \\ \end{split} $$ (15) $$\begin{split} \Delta \lambda =& n{{\tau '}_L} - \cos i\Delta \varOmega - \dfrac{{2r}}{{n{a^2}}}\varDelta {v_{L,S}} + \\ &\dfrac{{\sqrt {1 - {e^2}} }}{{na(1 \sqrt {1 - {e^2}} )}}[ - \varDelta {v_{L,S}}(e\cos f) + \\ &\Delta {v_{L,T}}\left( {1 + \dfrac{r}{p}} \right)(e\sin f)] \\ \end{split} $$ (16) 式中:
$p = a(1 - {e^2})$ ;$u = \omega + f$ ;$\tilde u = \omega + E$ ;n为平均角速度;E为偏近点角。 -
只有地球中心引力场作用,无激光烧蚀力作用时,根据碎片轨道摄动方程,可得:
$$\dfrac{{{\rm{d}}\lambda }}{{{\rm{d}}t}} = n$$ (17) 即轨道参数
$(a,i,\varOmega ,\xi = e\sin \omega ,\eta = e\cos \omega ,\lambda = M + \omega )$ 中,只有$\lambda $ 变化,具体为:$$\lambda = {\lambda _0} + n(t - {t_0})$$ (18) 式中:
$t = {t_0}$ 时,初始条件为$\lambda = {\lambda _0}$ 。在给定
$(\xi ,\eta ,\lambda )$ 条件下,根据开普勒方程迭代求解$\tilde u$ ,并进而计算$\sin \tilde u$ 和$\cos \tilde u$ 。采用迭代方法求解开普勒方程时可令:$$\tilde u = \lambda + \eta \sin \tilde u - \xi \cos \tilde u$$ (19) -
在赤道惯性坐标系
$XYZ$ 中,空间碎片目标位置矢量为${{{r}}_{{\rm{deb}},X}} = {({r_{{\rm{deb}},x}},{r_{{\rm{deb}},y}},{r_{{\rm{deb}},z}})^{\rm{T}}}$ ,空间平台速度矢量为${{{v}}_{{\rm{sta}},X}} = {({v_{{\rm{sta}},x}},{v_{{\rm{sta}},y}},{v_{{\rm{sta}},z}})^{\rm{T}}}$ ,空间平台位置矢量为${{{r}}_{{\rm{sta}},X}} = ({r_{{\rm{sta}},x}}, {{r_{{\rm{sta}},y}},{r_{{\rm{sta}},z}})^{\rm{T}}}$ ,碎片与平台位置矢量为${{{r}}_{{\rm{DS}},X}} = {{{r}}_{{\rm{deb}},X}} - {{{r}}_{{\rm{sta}},X}}$ ,激光操控窗口和判据为:$${r_{{\rm{DS}},X}} \leqslant {r_{L,\max }}$$ (20) $$\left\{ \begin{array}{l} {\gamma _{L,\max }} \geqslant \arccos \dfrac{{{{{v}}_{{\rm{sta}},X}} \cdot {{{r}}_{{\rm{DS}},X}}}}{{\left| {{{{v}}_{{\rm{sta}},X}}} \right|\left| {{{{r}}_{{\rm{DS}},X}}} \right|}}\; \\ \dfrac{{{{{v}}_{{\rm{sta}},X}} \cdot {{{r}}_{{\rm{DS}},X}}}}{{\left| {{{{v}}_{{\rm{sta}},X}}} \right|\left| {{{{r}}_{{\rm{DS}},X}}} \right|}} \geqslant 0 \\ \end{array} \right.$$ (21) $${r_{{\rm{DS}},X}} \geqslant {r_{{\rm{DS}},\min }}$$ (22) 式中:
${r_{L,\max }}$ 为最大激光作用距离,$0 \leqslant {\gamma _{L,\max }} < {\text{π}}/2$ 为最大激光发射角(激光辐照方向与平台当地速度方向之间夹角)。天基平台操控空间目标需要满足以下条件[19-20]:(1) 探测、捕获、跟踪、瞄准、发射等综合能力的要求;(2) 碎片在平台前方运动且在激光发射角以内的要求;(3) 碎片与天基平台防止碰撞的要求。 -
假设天基平台激光的重频为10 Hz,脉宽为10 ns,激光烧蚀力作用时间为100 ns,激光功率密度为
${10^{13}}\;{\rm{W/}} {{\rm{m}}^{\rm{2}}}$ (${10^9}\;{\rm{W/c}}{{\rm{m}}^{\rm{2}}}$ )。空间目标在天基平台前方,两者之间的相对距离为425 m。以典型铝质碎片为例,当碎片目标在激光小光斑辐照下时,其冲量耦合系数受激光功率密度影响,典型数值来源于参考文献[21]。圆柱体碎片的底面半径为R,高度为H(轴线方向尺寸),则主轴转动惯量为:$$\left\{ \begin{array}{l} {I_{\rm{bx}}} = {I_{\rm{by}}} = \dfrac{1}{4}\left( {{R^2} + \dfrac{{{H^2}}}{3}} \right)M \\ {I_{\rm{bz}}} = \dfrac{{{R^2}}}{2}M \\ \end{array} \right.$$ (23) 式中:
$M = {\text{π}}{R^2}H\rho $ 。假设薄壁圆筒碎片目标,其尺寸为
$(R,r,H) = (25,24,50)$ (单位:cm),在体固联坐标系${X_b}{Y_b}{Z_b}$ 中,碎片的初始角速度为${\omega _{{{{xb}},0}}} = 1\;{\rm{rad/s}}$ 。远场激光光斑半径为${r_L} = 1\;{\rm{cm}}$ ,激光器平均功率为$3.141\;593 \times {10^2}\;{\rm{W}}$ (激光单脉冲能量为$3.141\;593 \times 10\;{\rm{J}}$ )。按照激光烧蚀消旋策略,对薄壁圆筒碎片施加激光烧蚀力矩。薄壁圆筒转动惯量为外圆柱体转动惯量与内圆柱体转动惯量之差。碎片初始欧拉角和角速度为:$$({\varphi _0},{\theta _0},{\psi _0},{\dot \varphi _0},{\dot \theta _0},{\dot \psi _0}) = ({\text{π}}/4,{\text{π}}/4,{\text{π}}/4,1,0,0)$$ (24) 碎片和天基平台轨道高度为400 km,碎片相对平台同向运动,轨道倾角、升交点赤经和近地点幅角,分别为:
$${i_{{{\rm{deb}},0}}} = {i_{{{\rm{sta}},0}}} = {\text{π}}/2,\;{\varOmega _{{{\rm{deb}},0}}} = {\varOmega _{{{\rm{sta}},0}}} = {\text{π}}/2$$ (25) $${\omega _{{{\rm{sta}},0}}} = {\text{π}}/2,\;{\omega _{{{\rm{deb}},0}}} = {\text{π}}/2 + \Delta {\omega _{{{\rm{deb}},0}}}$$ (26) 式中:
${i_{{{\rm{deb}},0}}}$ ,${i_{{{\rm{sta}},0}}}$ 分别为碎片目标和天基平台的初始轨道倾角;${\varOmega _{{{\rm{deb}},0}}}$ ,${\varOmega _{{{\rm{sta}},0}}}$ 分别为初始升交点赤经;${\omega _{{{\rm{deb}},0}}}$ ,${\omega _{{{\rm{sta}},0}}}$ 分别为碎片目标和平台的初始近地点幅角,$\Delta {\omega _{{{\rm{deb}},0}}}$ 表示碎片在平台的前方运动。近距离伴飞、可辨识碎片姿态运动的距离为${r_{\rm{DS,iden}}}$ ,则有:$$\Delta {\omega _{{{\rm{deb}},0}}} = \dfrac{{{r_{\rm{DS,iden}}}}}{{{R_0}}}$$ (27) 式中:R0为地球半径。图11所示为薄壁圆筒碎片角速度
${\omega _{xb}}$ 随着时间的变化,由于反向激光烧蚀力矩的作用,碎片角速度${\omega _{xb}}$ 不断减小,最后减小为$1.{\rm{547\;134}} \times {10^{{\rm{ - }}4}}\;{\rm{rad/s}}$ ,耗时450.2 s,此时,${\omega _{yb}} = {\omega _{zb}} = 0$ 。可以看出,由于天基平台在近距离下烧蚀消旋碎片目标,相比较于两者的轨道高度而言,两者之间的相对距离变化较小,因此远场激光功率密度基本保持不变,同时由于烧蚀反喷方向为烧蚀表面法向方向,因此产生的烧蚀力矩方向保持不变。综上,产生的激光消旋烧蚀力及力矩基本保持不变,因此角速度变化率基本保持不变。图 11 碎片尺寸为25 cm/24 cm/50 cm下碎片角速度ωxb的变化
Figure 11. Changes of angular velocity ωxb of debris with the size of 25 cm/24 cm/50 cm
图12所示为薄壁圆筒碎片欧拉角φ随着时间的变化,由于反向激光烧蚀力矩的作用,碎片转动周期不断增大,欧拉角约为
$\varphi \approx {4^{\rm{o}}}$ 时${\omega _{xb}} = 0$ ,此时,$\theta = \psi = {45^{\rm{o}}}$ 不变。图 12 碎片尺寸为25 cm/24 cm/50 cm下碎片欧拉角φ的变化
Figure 12. Changes of Euler angle φ of the debris with the size of 25 cm/24 cm/50 cm
在整个碎片消旋过程中,激光能量的能耗为
${\rm{1}}{\rm{.414\;345}} \times {10^5}\;{\rm{J}}$ 。图13所示为薄壁圆筒碎片半长轴、远地点和近地点半径的变化,近地点半径基本不变(蓝线),远地点半径(红线)和半长轴(黑线)有所变化,整个激光操控过程结束时,远地点半径增大约500 m,半长轴增大约250 m。图 13 碎片尺寸为25 cm/24 cm/50 cm下半长轴、远/近地点半径的变化
Figure 13. Variation trend of semimajor axis, apogee and perigee with the debris size of 25 cm/24 cm/50 cm
图14所示为碎片平台距离和碎片矢径差的变化,整个激光操控过程结束时,碎片平台距离变化小于40 m,碎片矢径差(碎片矢径与操控前矢径之差)变化为–1.25 ~ 2 m。
图 14 碎片尺寸为25 cm/24 cm/50 cm下碎片与天基平台距离和矢径差的变化
Figure 14. Variation trend of the distance and the radius vector between the space-based laser platform and the debris with the debris size of 25 cm/24 cm/50 cm
图15为碎片升交点赤经、轨道倾角和偏心率的变化,碎片升交点赤经(上方黑线)增大3×10–4 (°),轨道倾角基本不变(下方黑线),偏心率增大为4×10–5。可以看出,升交点赤经、轨道倾角有一定的折现变化,主要是因为激光累积作用的结果,由于累积的激光烧蚀力的作用,使升交点赤经、倾角等参数在一定时间累积后出现了显著变化。
Concept of laser de-tumbling and its space-based application for space-based spinning target
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摘要: 大量废弃的空间目标在失控状态下,由于残余角动量以及光压等摄动影响会处于翻滚状态,对其直接捕获之前进行消旋控制是较为安全的方式。介绍了国内外典型空间目标的消旋操控技术,提出了空间目标激光小光斑近距离辐照操控消旋的概念,分析了激光消旋操控的主要特点,并基于脉冲激光烧蚀冲量耦合效应,完成了典型柱状空间目标天基激光消旋操控应用分析。分析结果表明:激光操控方式可精确控制空间目标的运动姿态,有效降低翻滚空间目标的旋转角速度,所提出的天基激光操控方式可为翻滚空间目标消旋控制提供一种新的解决方法。Abstract: As a large number of abandoned space objects are out of control, they will be in a tumbling state due to the perturbation effects, such as residual angular momentum, light pressure and so on. De-tumbling control of the space objects is probably a safer way before directly capturing them. De-tumbling control techniques for typical space objects at home and abroad was introduced, the concept of de-tumbling by close irradiation of laser with a small spot was proposed, and the main characteristics of laser de-tumbling control were also analyzed. Moreover, based on the pulsed laser ablation impulse coupling effect, the concept of space-based laser de-tumbling analysis was carried out with the typical cylindrical space targets. The results indicate that the laser control method can precisely control the motion posture of the space target. The space-based laser control method proposed in this paper can provide a new method for the de-tumbling control of the spinning space target.
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Key words:
- laser de-tumbling /
- space object /
- space-based laser /
- laser irradiation
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