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1. 哈尔滨工业大学 航天学院, 哈尔滨 150001;
2. Politecnico di Milano, Department of Aerospace Science and Technology, Milano 20156

Rendezvous and docking of spacecraft with single thruster: Path planning and tracking control
GENG Yuanzhuo1, LI Chuanjiang1, GUO Yanning1, James Douglas BIGGS2
1. School of Astronautics, Harbin Institute of Technology, Harbin 150001, China;
2. Department of Aerospace Science and Technology, Politecnico di Milano, Milano 20156, Italy
Abstract: A novel path planning and tracking control approach is proposed for rendezvous and docking of spacecraft with a single thruster. Firstly, since the thruster is fixed along the X axis of the spacecraft, the transfer trajectory of the spacecraft from the initial position to the desired position is designed as a helix whose parameters are calculated to ensure that the initial point and final velocity of the trajectory are in accordance with those of the spacecraft. The curvature of the trajectory can be minimized by the proposed helical motion. Secondly, to reduce the difficulty in trajectory tracking and decrease the amplitude of the control torque, this paper proposes an improved helical motion by rotating the traditional helical line with appropriate angles in 3-D space. In this way, the initial direction of the trajectory can be aligned with the X axis of the spacecraft, while the curvature integral of the curve is minimized. Furthermore, to track the planned trajectory and the desired force direction, a sliding mode based Control Lyapunov Function (CLF) method is presented. When the angle between the X axis and the desired control force direction is large, the standard CLF law is adopted. Then the control law switches to sliding mode control in the case that the states are near the sliding mode surface. Simulations are conducted to show the superiority of the proposed rotated helical motions to traditional approaches.
Keywords: single thruster spacecraft    rendezvous and docking    trajectory planning    helical motion    CLF

1 航天器相对位置及姿态模型 1.1 相对位置模型

 图 1 交会对接示意图 Fig. 1 Sketch of rendezvous and docking

 $\left\{ {\begin{array}{*{20}{l}} {\ddot x - 2{\omega _0}\dot y - 3\omega _0^2x = \frac{{{f_{{\rm{d}}x}}}}{{{m_{\rm{c}}}}}}\\ {\ddot y + 2{\omega _0}\dot x = \frac{{{f_{{\rm{d}}y}}}}{{{m_{\rm{c}}}}}}\\ {\ddot z + \omega _0^2z = \frac{{{f_{{\rm{d}}z}}}}{{{m_{\rm{c}}}}}} \end{array}} \right.$ （1）

 ${\mathit{\boldsymbol{\ddot r}}_{{\rm{tc}}}} = \mathit{\boldsymbol{M}}{\mathit{\boldsymbol{r}}_{{\rm{tc}}}} + \mathit{\boldsymbol{N}}{\mathit{\boldsymbol{\dot r}}_{{\rm{tc}}}} + \frac{{{\mathit{\boldsymbol{f}}_{\rm{d}}}}}{{{m_{\rm{c}}}}}$ （2）

 $\mathit{\boldsymbol{M}} = \left[ {\begin{array}{*{20}{c}} {3\omega _0^2}&0&0\\ 0&0&0\\ 0&0&{ - \omega _0^2} \end{array}} \right];\mathit{\boldsymbol{N}} = \left[ {\begin{array}{*{20}{c}} 0&{2{\omega _0}}&0\\ { - 2{\omega _0}}&0&0\\ 0&0&0 \end{array}} \right]$
1.2 姿态误差描述

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{e}}_\theta } = {{[{e_{\theta x}},{e_{{\theta _y}}},{e_{\theta z}}]}^{\rm{T}}}}\\ {{e_{\theta x}} = {\rm{cos}}{\kern 1pt} {\kern 1pt} {\theta _x} - 1 = t_x^{\rm{b}} - 1 = {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}(1,z){\mathit{\boldsymbol{t}}^{\rm{I}}} - 1} \end{array}\\ \begin{array}{*{20}{l}} {{e_{\theta y}} = {\rm{cos}}{\kern 1pt} {\kern 1pt} {\theta _y} = t_y^{\rm{b}} = {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}(2,:){\mathit{\boldsymbol{t}}^{\rm{I}}}}\\ {{e_{\theta z}} = {\rm{cos}}{\kern 1pt} {\kern 1pt} {\theta _z} = t_z^{\rm{b}} = {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}(3,:){\mathit{\boldsymbol{t}}^{\rm{I}}}} \end{array} \end{array} \right.$ （3）

 ${\mathit{\boldsymbol{e}}_\theta } = {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}{\mathit{\boldsymbol{t}}^{\rm{I}}} - {[1,0,0]^{\rm{T}}}$ （4）

eθ求导，可得

 $\begin{array}{l} {{\mathit{\boldsymbol{\dot e}}}_\theta } = {{\mathit{\boldsymbol{\dot R}}}_{{\rm{bI}}}}{\mathit{\boldsymbol{t}}^{\rm{I}}} + {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}{{\mathit{\boldsymbol{\dot t}}}^{\rm{I}}} = - \omega _{{\rm{bI}}}^ \times {R_{{\rm{bI}}}}{\mathit{\boldsymbol{t}}^{\rm{I}}} + {R_{{\rm{bI}}}}{{\mathit{\boldsymbol{\dot t}}}^{\rm{I}}} = {\mathit{\boldsymbol{t}}^{\rm{b}}} \times {\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}{{\mathit{\boldsymbol{\dot t}}}^{\rm{I}}} = \left[ {\begin{array}{*{20}{c}} 0&{ - {e_{\theta z}}}&{{e_{\theta y}}}\\ {{e_{\theta z}}}&0&{ - 1 - {e_{\theta x}}}\\ { - {e_{\theta y}}}&{1 + {e_{\theta x}}}&0 \end{array}} \right]{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}{{\mathit{\boldsymbol{\dot t}}}^{\rm{I}}} = \mathit{\boldsymbol{F}}({\mathit{\boldsymbol{e}}_\theta }){\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}{{\mathit{\boldsymbol{\dot t}}}^{\rm{I}}} \end{array}$ （5）

 ${\mathit{\boldsymbol{\dot \omega }}_{{\rm{bI}}}} = - \mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^ \times {\mathit{\boldsymbol{J}}_{\rm{s}}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + \mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}({\mathit{\boldsymbol{\tau }}_{\rm{c}}} + \mathit{\boldsymbol{d}})$ （6）

2 交会对接轨迹规划

2.1 不考虑初始速度方向

 $\left\{ {\begin{array}{*{20}{l}} {{x_{\rm{d}}}(t) = r{\rm{cos}}(At + B) + {D_1}}\\ {{y_{\rm{d}}}(t) = r{\rm{sin}}(At + B) + {D_2}}\\ {{z_{\rm{d}}}(t) = {C_0} + {C_1}t} \end{array}} \right.$ （7）

 图 2 转移轨迹示意图 Fig. 2 Sketch of transfer trajectory

 $\left\{ {\begin{array}{*{20}{l}} {{x_{\rm{d}}}(t) = r{\rm{cos}}(At + B) + {D_1}}\\ {{y_{\rm{d}}}(t) = r{\rm{sin}}(At + B) + {D_2}}\\ {{z_{\rm{d}}}(t) = {C_0} + {C_1}t} \end{array}} \right.$ （8a）

 $\left\{ {\begin{array}{*{20}{l}} {{x_{\rm{d}}}(t) = r{\rm{cos}}(At + B) + {D_1}}\\ {{y_{\rm{d}}}(t) = r{\rm{sin}}(At + B) + {D_2}}\\ {{z_{\rm{d}}}(t) = {C_0} + {C_1}{t_1} + 0.5{a_z}{{(t - {t_1})}^2}} \end{array}} \right.$ （8b）

 $\left\{ \begin{array}{l} {x_{\rm{d}}}(t) = 0\\ {y_{\rm{d}}}(t) = r{\rm{sin}}(A{t_2} + B) + {D_2} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} rA{\rm{cos}}(A{t_2} + B)(t - {t_2})\\ {z_{\rm{d}}}(t) = 0 \end{array} \right.$ （8c）

 $\left\{ {\begin{array}{*{20}{l}} {{D_2} = {y_{20}},{D_1} = \frac{{x_0^2 + y_0^2 - 2{y_0}{y_{20}} + y_{20}^2}}{{2{x_0}}}}\\ {r = \sqrt {{{({x_0} - {D_1})}^2} + {{({y_0} - {D_2})}^2}} }\\ {B = {\rm{arctan}}\frac{{{y_0} - {D_2}}}{{{x_0} - {D_1}}}}\\ {A = \frac{{{k_2}\pi - B}}{{{t_2}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_2} = 0, \pm 1, \pm 2, \cdots }\\ {{C_0} = {z_0},{C_1} = \frac{{ - 2{C_0}}}{{{t_1} + {t_2}}}}\\ {{a_z} = \frac{{ - {C_1}}}{{{t_2} - {t_1}}}} \end{array}} \right.$ （9）

 ${t_f} = {t_2} + \frac{{{y_{20}}}}{{rA{\rm{cos}}(A{t_2} + B)}}$ （10）

 $A = \pm \frac{{|{y_{20}}|}}{{r({t_{\rm{f}}} - {t_2})}}$ （11）

 $\left\{ {\begin{array}{*{20}{l}} {{{({x_0} - {D_1})}^2} + {{({y_0} - {D_2})}^2} = D_1^2}\\ {r = |{D_1}|}\\ {B = {\rm{arctan}}\frac{{{y_0} - {D_2}}}{{{x_0} - {D_1}}},A = \pm \frac{{{D_2}}}{{{D_1}({t_{\rm{f}}} - {t_2})}}}\\ {{C_0} = {z_0},{C_1} = \frac{{2{C_0}}}{{{t_2} - 3{t_1}}}}\\ {{a_z} = \frac{{ - {C_1}}}{{{t_2} - {t_1}}}} \end{array}} \right.$ （12）

2.2 考虑初始速度方向

2.1节中，虽然给出了螺旋线形式的转移轨迹，但是并没有考虑追踪星初始Xcb方向。为了产生转移轨迹切向的控制力，追踪星在初始时刻需要进行姿态机动。此时，转移轨迹曲率积分最小的优越性将失去意义，因为追踪星需要跟踪上转移轨迹，而对于只安装有小推力、单方向推力器的追踪星，完成轨迹跟踪有较大挑战性，其需要较长时间从初始位置和速度转移到转移轨迹上。

 图 3 考虑初始速度方向的交会对接示意图 Fig. 3 Sketch of rendezvous and docking considering initial velocity direction

 $\left\{ {\begin{array}{*{20}{l}} {{x_{\rm{d}}}(0) = {x_0}}\\ {{y_{\rm{d}}}(0) = {y_0},}\\ {{z_{\rm{d}}}(0) = {z_0}} \end{array}\left\{ {\begin{array}{*{20}{l}} {{x_{\rm{d}}}({t_1}) = 0}\\ {{y_{\rm{d}}}({t_1}) = {y_{20}}}\\ {{z_{\rm{d}}}({t_1}) = 0} \end{array}} \right.} \right.$ （13a）

 $\left\{ {\begin{array}{*{20}{c}} \begin{array}{l} {[{{\dot x}_{\rm{d}}}(0),{{\dot y}_{\rm{d}}}(0),{{\dot z}_{\rm{d}}}(0)]^{\rm{T}}} = {k_{{\rm{v1}}}}{\mathit{\boldsymbol{x}}_{{\rm{cb}}}}\\ {\left[ {{{\dot x}_{\rm{d}}}({t_1}),{{\dot y}_{\rm{d}}}({t_1}),{{\dot z}_{\rm{d}}}({t_1})} \right]^{\rm{T}}} = \end{array}\\ {{k_{{\rm{v2}}}}{\mathit{\boldsymbol{v}}_{\rm{f}}} = {k_{{\rm{v2}}}}{{[0,1,0]}^{\rm{T}}}} \end{array}} \right.$ （13b）

 $\left[ {\begin{array}{*{20}{l}} {{x_{\rm{d}}}(t)}\\ {{y_{\rm{d}}}(t)}\\ {{z_{\rm{d}}}(t)} \end{array}} \right] = \mathit{\boldsymbol{R}}(\psi ,\theta ,\varphi )\left[ {\begin{array}{*{20}{c}} {r{\rm{cos}}(At + B) + {D_1}}\\ {r{\rm{sin}}(At + B) + {D_2}}\\ {{C_0} + {C_1}t + {C_2}{t^2}} \end{array}} \right]$ （14）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{R}}(\varphi ) = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&{{\rm{cos}}\varphi }&{{\rm{sin}}\varphi }\\ 0&{ - {\rm{sin}}\varphi }&{{\rm{cos}}\varphi } \end{array}} \right]}\\ {\mathit{\boldsymbol{R}}(\theta ) = \left[ {\begin{array}{*{20}{c}} {{\rm{cos}}\theta }&0&{ - {\rm{sin}}\theta }\\ 0&1&0\\ {{\rm{sin}}\theta }&0&{{\rm{cos}}\theta } \end{array}} \right]}\\ {\mathit{\boldsymbol{R}}(\psi ) = \left[ {\begin{array}{*{20}{c}} {{\rm{cos}}\psi }&{{\rm{sin}}\psi }&0\\ { - {\rm{sin}}\psi }&{{\rm{cos}}\psi }&0\\ 0&0&1 \end{array}} \right]} \end{array}} \right.$ （15）

 图 4 旋转后的螺旋线示意图 Fig. 4 Sketch of rotated helical motion

 $\left\{ \begin{array}{l} {t_1} + \left| {\frac{{{y_{20}}}}{{{v_y}({t_1})}}} \right| = {t_{\rm{f}}}\\ {v_y}({t_1}) = rA{\kern 1pt} {\kern 1pt} {\rm{cos}}{\kern 1pt} {\kern 1pt} \varphi {\kern 1pt} {\kern 1pt} {\rm{cos}}(A{t_1} + B) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({C_1} + 2{C_2}{t_1}){\rm{sin}}{\kern 1pt} {\kern 1pt} \varphi \end{array} \right.$ （16）

 $\left[ {\begin{array}{*{20}{l}} {f_{{\rm{d}}x}^\prime }\\ {f_{{\rm{d}}y}^\prime }\\ {f_{{\rm{d}}z}^\prime } \end{array}} \right] = \mathit{\boldsymbol{R}}(\psi ,\theta ,\varphi )\left[ {\begin{array}{*{20}{l}} {{f_{{\rm{d}}x}}}\\ {{f_{{\rm{d}}y}}}\\ {{f_{{\rm{d}}z}}} \end{array}} \right]$ （17）

 $\left\{ {\begin{array}{*{20}{l}} {{f_{{\rm{d}}x}} = - {m_{\rm{c}}}(r{A^2} + 2{\omega _0}rA + 3\omega _0^2r){\rm{cos}}(At + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} B) - {m_{\rm{c}}}3\omega _0^2{D_1}}\\ {{f_{{\rm{d}}y}} = - {m_{\rm{c}}}(r{A^2} + 2{\omega _0}rA){\rm{sin}}(At + B)}\\ {{f_{{\rm{d}}z}} = 2{m_{\rm{c}}}{C_2} + {m_{\rm{c}}}\omega _0^2({C_0} + {C_1}t + {C_2}{t^2})} \end{array}} \right.$ （18）

3 轨迹跟踪控制

3.1 位置跟踪控制

 $\left\{ {\begin{array}{*{20}{l}} {{{\mathit{\boldsymbol{\dot e}}}_x} = {\mathit{\boldsymbol{e}}_v}}\\ {{{\mathit{\boldsymbol{\dot e}}}_v} = \mathit{\boldsymbol{M}}({\mathit{\boldsymbol{r}}_{{\rm{tc}}}}){\mathit{\boldsymbol{r}}_{{\rm{tc}}}} + \mathit{\boldsymbol{N}}{{\mathit{\boldsymbol{\dot r}}}_{{\rm{tc}}}} + \frac{{{\mathit{\boldsymbol{f}}_{\rm{d}}}}}{{{m_{\rm{c}}}}} - {{\mathit{\boldsymbol{\ddot r}}}_{{\rm{tcd}}}}} \end{array}} \right.$ （19）

 $\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{f}}_{\rm{d}}} = - {m_{\rm{c}}}(\mathit{\boldsymbol{M}}({\mathit{\boldsymbol{r}}_{{\rm{tc}}}}){\mathit{\boldsymbol{r}}_{{\rm{tc}}}} + \mathit{\boldsymbol{N}}{{\mathit{\boldsymbol{\dot r}}}_{{\rm{tc}}}}) + {m_{\rm{c}}}{{\mathit{\boldsymbol{\ddot r}}}_{{\rm{tcd}}}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {m_{\rm{c}}}({k_{\rm{p}}}{\mathit{\boldsymbol{e}}_{\rm{x}}} + {k_{\rm{d}}}{\mathit{\boldsymbol{e}}_v})} \end{array}$ （20）

 ${\mathit{\boldsymbol{f}}^{\rm{b}}} = {\left[ {\begin{array}{*{20}{l}} {c({\theta _x})\left\| {{\mathit{\boldsymbol{f}}_{\rm{d}}}} \right\|,0,0} \end{array}} \right]^{\rm{T}}}$ （21）

3.2 姿态跟踪控制

 $\mathit{\boldsymbol{\dot x}} = \mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) + \mathit{\boldsymbol{g}}(\mathit{\boldsymbol{x}})\mathit{\boldsymbol{u}}$ （22）

 $\mathit{\boldsymbol{x}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{e}}_\theta }}\\ {{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}} \end{array}} \right],\mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{F}}({\mathit{\boldsymbol{e}}_\theta }){\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + {\mathit{\boldsymbol{R}}_{{\rm{bI}}}}{{\mathit{\boldsymbol{\dot t}}}^{\rm{I}}}}\\ { - \mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^ \times {\mathit{\boldsymbol{J}}_{\rm{s}}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}} \end{array}} \right]$
 $\mathit{\boldsymbol{g}} = \left[ {\begin{array}{*{20}{c}} {{{\bf{0}}_{3 \times 3}}}\\ {\mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}} \end{array}} \right],\mathit{\boldsymbol{u}} = {\mathit{\boldsymbol{\tau }}_{\rm{c}}}$

 $\mathit{\boldsymbol{V}} = \frac{a}{2}\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}{\mathit{\boldsymbol{e}}_\theta } + b\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{J}}_{\rm{s}}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + \frac{c}{2}\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}{\mathit{\boldsymbol{J}}_{\rm{s}}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}$ （23）

 $\mathit{\boldsymbol{V}} = \left[ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}}&{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\frac{a}{2}\mathit{\boldsymbol{I}}}&{\frac{b}{2}{\mathit{\boldsymbol{P}}^{\rm{T}}}{\mathit{\boldsymbol{J}}_{\rm{s}}}}\\ {\frac{b}{2}{\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{P}}}&{\frac{c}{2}{\mathit{\boldsymbol{J}}_{\rm{s}}}} \end{array}} \right]\left[ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{e}}_\theta }}\\ {{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}} \end{array}} \right]$ （24）

 $\frac{{\partial {\mathit{\boldsymbol{V}}^{\rm{T}}}}}{{\partial \mathit{\boldsymbol{x}}}} = [a\mathit{\boldsymbol{e}}_\theta ^{\rm{T}} + b\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}{\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{P}}\quad b\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{J}}_{\rm{s}}} + c\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}{\mathit{\boldsymbol{J}}_{\rm{s}}}]$ （25）

 $\left\{ {\begin{array}{*{20}{l}} {{L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}} = \frac{{\partial {\mathit{\boldsymbol{V}}^{\rm{T}}}}}{{\partial \mathit{\boldsymbol{x}}}}\mathit{\boldsymbol{g}} = b\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}\mathit{\boldsymbol{F}} + c\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}}\\ {{L_\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{V}} = \frac{{\partial {\mathit{\boldsymbol{V}}^{\rm{T}}}}}{{\partial \mathit{\boldsymbol{x}}}}\mathit{\boldsymbol{f}}(\mathit{\boldsymbol{x}}) = (a\mathit{\boldsymbol{e}}_\theta ^{\rm{T}} + b\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}{\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{P}})\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (b\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}\mathit{\boldsymbol{F}} + c\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}})\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^ \times {\mathit{\boldsymbol{J}}_{\rm{s}}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}} \end{array}} \right.$ （26）

 $b\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}\mathit{\boldsymbol{F}} + c\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}} = b\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}{\mathit{\boldsymbol{P}}^{\rm{T}}} + c\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}} = {{\bf{0}}_{1 \times 3}}$ （27）

 $\begin{array}{l} {L_\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{V}} = (a\mathit{\boldsymbol{e}}_\theta ^{\rm{T}} + b\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}{\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{P}})\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} = - \frac{{ac}}{b}\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {b\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^{\rm{T}}{\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{PF}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} \le - \frac{{ac}}{b}{{\left\| {{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}} \right\|}^2} + b{{\left\| {{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}} \right\|}^2} \cdot }\\ {\left\| {{\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{PF}}} \right\| = \left( { - \frac{{ac}}{b} + b\sqrt {{\lambda _{{\rm{max}}}}({\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{PF}}{\mathit{\boldsymbol{F}}^{\rm{T}}}{\mathit{\boldsymbol{P}}^{\rm{T}}}{\mathit{\boldsymbol{J}}_{\rm{s}}})} } \right) \cdot }\\ {{{\left\| {{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}}} \right\|}^2}} \end{array} \end{array}$ （28）

λmax(JSPFFTPTJS)=η，对于∀ωbI≠0如果ac>b2η，则LfV＜0。

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{u}}^*} = \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{u}}_1^*}&{{L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}} \ne {\bf{0}}}\\ {\mathit{\boldsymbol{u}}_2^*}&{{L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}} = {\bf{0}}} \end{array}} \right.}\\ {\mathit{\boldsymbol{u}}_1^* = - {\mathit{\boldsymbol{R}}^{ - 1}} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{L_\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{V}} + \sqrt {{{({L_\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{V}})}^2} + l(\mathit{\boldsymbol{x}})({L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}}){\mathit{\boldsymbol{R}}^{ - 1}}{{({L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}})}^{\rm{T}}}} }}{{({L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}}){\mathit{\boldsymbol{R}}^{ - 1}}{{({L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}})}^{\rm{T}}}}} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{({L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}})}^{\rm{T}}}}\\ {\mathit{\boldsymbol{u}}_2^* = {\bf{0}}} \end{array}} \right.$ （29）

l(x)=xTQx，指标函数

 $J = \int_0^\infty {(l(} \mathit{\boldsymbol{x}}) + {\mathit{\boldsymbol{u}}^{\rm{T}}}\mathit{\boldsymbol{Ru}}){\rm{d}}t$ （30）

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{s}} = b\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{e}}_\theta } + c{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} = {\bf{0}}}\\ {{s_1}:{\omega _x} = 0}\\ {{s_2}:b{e_{\theta z}} + c{\omega _y} = 0}\\ {{s_3}: - b{e_{\theta y}} + c{\omega _z} = 0} \end{array}} \right.$ （31）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{u}}^*} = \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{u}}_1^*}&{\left\| {{L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}}} \right\| \ge {\delta _1}}\\ {\mathit{\boldsymbol{u}}_2^*}&{\left\| {{L_\mathit{\boldsymbol{g}}}\mathit{\boldsymbol{V}}} \right\| < {\delta _1}} \end{array}} \right.}\\ {\mathit{\boldsymbol{u}}_2^* = \mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^ \times {\mathit{\boldsymbol{J}}_{\rm{s}}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} - \frac{b}{c}{\mathit{\boldsymbol{J}}_{\rm{s}}}\mathit{\boldsymbol{PF}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} - \frac{{{k_1}}}{c}{\mathit{\boldsymbol{J}}_{\rm{s}}} {\rm{sig}} {{(\mathit{\boldsymbol{s}})}^{\frac{m}{n}}}} \end{array}} \right.$ （32）

 $\left\{ \begin{array}{l} {\varOmega _1}:\{ \mathit{\boldsymbol{s}}|\left\| \mathit{\boldsymbol{s}} \right\| \le {\delta _1}\} \\ {\varOmega _2}:\{ \mathit{\boldsymbol{s}}|\left\| \mathit{\boldsymbol{s}} \right\| < {\delta _2}\} ,{\delta _2} = {[\kappa k_1^{ - 1}{(1 - \varepsilon )^{ - 1}}]^{\frac{n}{{m + n}}}},\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa = c\bar d{\delta _1}\left\| {\mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}} \right\|\\ {\varOmega _3}:\left\{ {{\mathit{\boldsymbol{e}}_\theta }|\left\| {\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{e}}_\theta }} \right\| < \frac{{c{\delta _2}}}{b}} \right\} \end{array} \right.$ （33）

 ${V_1} = \frac{1}{2}{\mathit{\boldsymbol{s}}^{\rm{T}}}\mathit{\boldsymbol{s}}$ （34）
 $\begin{array}{l} \begin{array}{*{20}{l}} {{{\dot V}_1} = {\mathit{\boldsymbol{s}}^{\rm{T}}}[b\mathit{\boldsymbol{PF\omega }} + c\mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}( - \mathit{\boldsymbol{\omega }}_{{\rm{bI}}}^ \times {\mathit{\boldsymbol{J}}_{\rm{s}}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} + \mathit{\boldsymbol{u}}_2^* + \mathit{\boldsymbol{d}})] = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{s}}^{\rm{T}}}( - {k_1} {\rm{sig}} {{(\mathit{\boldsymbol{s}})}^{\frac{m}{n}}} + c\mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}\mathit{\boldsymbol{d}}) = - {k_1}\sum\limits_{i = 1}^3 | {s_i}{|^{\frac{{m + n}}{n}}} + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} c{\mathit{\boldsymbol{s}}^{\rm{T}}}\mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}\mathit{\boldsymbol{d}} \le - {k_1}{{\left\| \mathit{\boldsymbol{s}} \right\|}^{\frac{{m + n}}{n}}} + c\bar d\left\| \mathit{\boldsymbol{s}} \right\|\left\| {\mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}} \right\| = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \left\| \mathit{\boldsymbol{s}} \right\|({k_1}{\left\| \mathit{\boldsymbol{s}} \right\|^{\frac{m}{n}}} - c\bar d\left\| {\mathit{\boldsymbol{J}}_{\rm{s}}^{ - 1}} \right\|) \end{array}$ （35）

 ${\dot V_1} \le - {k_1}{\left\| \mathit{\boldsymbol{s}} \right\|^{\frac{{m + n}}{n}}} + c\bar d{\delta _1}\left\| {\left. {J_{\rm{s}}^{ - 1}} \right|} \right. = - {k_1}{2^{\frac{{m + n}}{{2n}}}}V_1^{\frac{{m + n}}{{2n}}} + \kappa$ （36）

 ${V_2} = \frac{1}{2}\mathit{\boldsymbol{e}}_\theta ^{\rm{T}}{\mathit{\boldsymbol{e}}_\theta }$ （37）

 $\begin{array}{*{20}{l}} {{{\dot V}_2} = \mathit{\boldsymbol{e}}_\theta ^{\rm{T}}\mathit{\boldsymbol{F}}{\mathit{\boldsymbol{\omega }}_{{\rm{bI}}}} = \mathit{\boldsymbol{e}}_\theta ^{\rm{T}}\mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{s}} - \frac{b}{c}\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{e}}_\theta }} \right) \le }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\| {\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{e}}_\theta }} \right\|\left( {{\delta _2} - \frac{b}{c}\left\| {\mathit{\boldsymbol{P}}{\mathit{\boldsymbol{e}}_\theta }} \right\|} \right)} \end{array}$ （38）

4 仿真分析

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{J}}_{\rm{s}}} = 0.01\left[ {\begin{array}{*{20}{c}} {100.9}&0&0\\ 0&{25.1}&0\\ 0&0&{91.6} \end{array}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}}}\\ {{m_{\rm{c}}} = 22.82{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{kg}}} \end{array}} \right.$

 $\begin{array}{*{20}{l}} {\mathit{\boldsymbol{d}} = {{10}^{ - 4}} \times [{\rm{sin}}(0.001t) - {\rm{cos}}(0.001t)}\\ {{\rm{sin}}(0.001t){]^{\rm{T}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{N}} \cdot {\rm{m}}} \end{array}$

 轨道相关变量 追踪星 目标星 轨道倾角i/(°) 10.001 10 升交点赤经Ω/(°) 60 60 近地点幅角ω/(°) 80 80 偏心率e 0 0 轨道半径r/km 15 999 16 000 初始真近点角f0/(°) -0.003 0

 控制 不考虑初始速度方向 考虑初始速度方向 位置控制 kp=4×10-6, kd=0.002 3 kp=4×10-6, kd=0.002 3 姿态控制 a=0.2, b=0.12, c=1.2, k1=1, m=9, n=11 a=0.2, b=0.12, c=1.2, k1=1, m=9, n=11

4.1 不考虑初始速度方向

 图 5 期望转移轨迹及追踪星实际轨迹 Fig. 5 Desired and real trajectories of rendezvous and docking

 图 6 追踪星位置跟踪误差(不考虑初始速度方向) Fig. 6 Position tracking errors of chaser (without considering initial speed direction)
 图 7 追踪星与目标星的相对位置(不考虑初始速度方向) Fig. 7 Relative position between chaser and target (without considering initial speed direction)
 图 8 期望轨迹速度以及追踪星速度(不考虑初始速度方向) Fig. 8 Desired and real velocities of chaser(without considering initial speed direction)
 图 9 转移轨迹所需控制力(不考虑初始速度方向) Fig. 9 Desired force of trajectory(without considering initial speed direction)

 图 10 位置跟踪所需控制力(不考虑初始速度方向) Fig. 10 Desired force of trajectory tracking(without considering initial speed direction)
 图 11 追踪星本体实际控制力(不考虑初始速度方向) Fig. 11 Control force of chaser (without considering initial speed direction)

 图 12 追踪星Xb轴与期望推力方向夹角(不考虑初始速度方向) Fig. 12 Angle between Xb axis and desired force (without considering initial speed direction)
 图 13 追踪星角速度(不考虑初始速度方向) Fig. 13 Angular velocity of chaser (without considering initial speed direction)
 图 14 追踪星控制力矩(不考虑初始速度方向) Fig. 14 Control torque of chaser (without considering initial speed direction)
4.2 考虑初始速度方向

 图 15 考虑初始指向下期望转移轨迹及追踪星实际轨迹 Fig. 15 Desired and real trajectories of rendezvous and docking considering initial direction of chaser
 $\left\{ {\begin{array}{*{20}{l}} {A = - 6.234,B = - 6.332,r = 230.604}\\ {{D_1} = - 230.605,{D_2} = 15.599}\\ {{C_0} = - 1{\kern 1pt} {\kern 1pt} {\kern 1pt} 333.172,{C_1} = 1{\kern 1pt} {\kern 1pt} {\kern 1pt} 487.797}\\ {{C_2} = - 167.141} \end{array}} \right.$

 图 16 追踪星位置跟踪误差 Fig. 16 Position tracking errors of chaser

 图 17 期望轨迹速度以及追踪星速度 Fig. 17 Desired and real velocities of chaser
 图 18 位置跟踪所需控制力 Fig. 18 Required control force for position tracking
 图 19 追踪星与目标星的相对位置 Fig. 19 Relative position between chaser and target
 图 20 追踪星控制力矩 Fig. 20 Control torque of chaser
 图 21 追踪星Xb轴与期望推力方向夹角 Fig. 21 Angle between Xb axis and desired force
 图 22 滑模变量 Fig. 22 Sliding mode surface

 图 23 控制切换标志 Fig. 23 Flag of control switches
5 结论

1) 基于螺旋线设计了位置转移轨迹，通过引入姿态旋转矩阵，将传统螺旋线进行三维旋转，增加了转移轨迹的自由度，使其同时满足初末位置要求以及初末速度方向要求，有效降低了位置跟踪误差和位置跟踪所需控制力。

2) 为了使得追踪星本体推力方向与期望控制力方向重合，本文设计了基于CLF的姿态跟踪滑模控制策略。当姿态误差较大时，采用CLF，有效降低姿态控制能量消耗；当姿态误差较小时，切换为滑模控制，提升控制精度。

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http://dx.doi.org/10.7527/S1000-6893.2020.23880

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#### 文章信息

GENG Yuanzhuo, LI Chuanjiang, GUO Yanning, James Douglas BIGGS

Rendezvous and docking of spacecraft with single thruster: Path planning and tracking control

Acta Aeronautica et Astronautica Sinica, 2020, 41(9): 323880.
http://dx.doi.org/10.7527/S1000-6893.2020.23880