When the two spacecraft are relatively distant, the accuracy of the image feature extraction is low, and the depth information of the feature points to the target spacecraft can be ignored. The camera model can be approximated by a simplified perspective projection model [
29‐
31]. Consequently, we obtain
$$z_{i}^{c} = t_{z} ,\quad i = 1,2,3,4.$$
(10)
Simplified perspective projection refers to the projection on a plane parallel to the imaging plane through the origin of the target spacecraft. Therefore, it ignores the depth of the target spacecraft point relative to the origin of the target spacecraft. When the two spacecraft are relatively distant, the neglect error is insignificant. From Eq. (
8), we have
$$\begin{gathered} k_{1} = \frac{{(u_{2} - u_{1} )}}{2af}{, }\,\,k_{2} = \frac{{(v_{2} - v_{1} )}}{2af}{, }\,\,k_{3} = \frac{{(u_{3} + u_{4} - u_{2} - u_{1} )}}{2bf}, \hfill \\ k_{4} = \frac{{(v_{3} + v_{4} - v_{2} - v_{1} )}}{2bf}{, }\,\,k_{5} = \frac{{(u_{2} + u_{1} )}}{2df}{, }\,\,k_{6} = \frac{{(v_{2} - v_{1} )}}{2df}, \hfill \\ \end{gathered}$$
(11)
where
ki can be calculated by image points. Equation (
7) contains nine variables and six equations. Thus, it cannot be solved directly. The rotation matrix
R has the following constraints:
$$\left\{ \begin{aligned} &r_{11}^{2} + r_{12}^{2} + r_{13}^{2} = 1, \hfill \\ &r_{21}^{2} + r_{22}^{2} + r_{23}^{2} = 1, \hfill \\ &r_{31}^{2} + r_{32}^{2} + r_{33}^{2} = 1, \hfill \\ &r_{11} r_{21} + r_{12} r_{22} + r_{13} r_{23} = 0, \hfill \\ &r_{31} r_{21} + r_{32} r_{22} + r_{33} r_{23} = 0, \hfill \\& r_{11} r_{31} + r_{12} r_{32} + r_{13} r_{33} = 0. \hfill \\ \end{aligned} \right.$$
(12)
From the first, third, and sixth equations of Eq. (
12), we can obtain
$$(r_{11} r_{21} - r_{12} r_{22} )^{2} - (r_{11}^{2} + r_{12}^{2} + r_{21}^{2} + r_{22}^{2} ) + 1 = 0.$$
(13)
From Eqs. (
8) and (
13), we obtain
$$(k_{1} k_{3} - k_{2} k_{4} )^{2} t_{z}^{4} - (k_{1}^{2} + k_{2}^{2} + k_{3}^{2} + k_{4}^{2} )t_{z}^{2} + 1 = 0.$$
(14)
Eq. (
14) is a quartic equation. Therefore, the number of roots is four. Two negative roots are removed according to the relationship between roots and coefficients, and two positive roots meet the following conditions:
$${\text{Condition 1}}: t_{z1}^{2} \le \frac{{k_{1}^{2} + k_{3}^{2} }}{{(k_{1} k_{3} - k_{2} k_{4} )^{2} }},$$
(15)
$${\text{Condition 2}}:t_{z2}^{2} \ge \frac{{k_{2}^{2} + k_{4}^{2} }}{{(k_{1} k_{3} - k_{2} k_{4} )^{2} }}.$$
(16)
Condition 2 can only be satisfied when the rotation angle is greater than 60°; therefore, the result of applying Condition 1 is selected.
Rotation matrix
R can be described by four quaternion parameters
\((q_{0} ,q_{1} ,q_{2} ,q_{3} )\):
$${\varvec{R}} = \left[ {\begin{array}{*{20}c} {q_{0}^{2} + q_{1}^{2} - q_{2}^{2} - q_{3}^{2} } & {2(q_{1} q_{2} + q_{3} q_{0} )} & {2(q_{1} q_{3} - q_{2} q_{0} )} \\ {2(q_{1} q_{2} - q_{3} q_{0} )} & {q_{0}^{2} - q_{1}^{2} + q_{2}^{2} - q_{3}^{2} } & {2(q_{2} q_{3} + q_{1} q_{0} )} \\ {2(q_{1} q_{3} + q_{2} q_{0} )} & {2(q_{2} q_{3} - q_{1} q_{0} )} & {q_{0}^{2} - q_{1}^{2} - q_{2}^{2} + q_{3}^{2} } \\ \end{array} } \right] \, {.}$$
(17)
Assumed that
$$\begin{aligned} \beta &= \frac{1}{2}(r_{32} - (r_{12} r_{31} - r_{11} r_{32} )) \hfill \\ &= \frac{1}{2}(k_{4} t{}_{y} - k_{3} k_{2} t_{z}^{2} - k_{1} k_{4} t_{z}^{2} ) = 2q_{1} q_{2} . \hfill \\ \end{aligned}$$
(18)
Form Eq. (
14), we obtain
$$\left\{ \begin{aligned} &q_{1}^{2} + q_{2}^{2} = \frac{1}{2}(1 + r_{11} ), \hfill \\ &q_{1}^{2} - q_{2}^{2} = - \frac{1}{2}\sqrt {(1 + r_{11} )^{2} - 4\beta^{2} } , \hfill \\ &q_{1} r_{12} + q_{2} r_{31} = 2q_{4} (q_{2}^{2} - q_{1}^{2} ), \hfill \\& q_{2} r_{12} + q_{1} r_{31} = 2q_{3} (q_{2}^{2} - q_{1}^{2} ). \hfill \\ \end{aligned} \right.$$
(19)
As a result, we have
$$\left\{ \begin{aligned} &q_{1} = \frac{1}{2}\sqrt {1 + k_{1} t_{z} - \sqrt {(1 + k_{1} t_{z} )^{2} - 4\beta^{2} } } , \hfill \\ &q_{2} = \frac{\beta }{{2q_{1} }}, \hfill \\& q_{3} = \frac{{q_{2} k_{3} + q_{1} k_{2} }}{{2(q_{2}^{2} - q_{1}^{2} )}}t_{z} , \hfill \\ &q_{4} = \frac{{q_{1} k_{3} + q_{2} k_{2} }}{{2(q_{2}^{2} - q_{1}^{2} )}}, \hfill \\& t_{x} = \frac{{t_{z} }}{2f}(u_{1} (1 + \varepsilon_{1} ) + u_{2} (1 + \varepsilon_{2} )) - dr_{13} , \hfill \\& t_{y} = \frac{{t_{z} }}{2f}(v_{1} (1 + \varepsilon_{1} ) + v_{2} (1 + \varepsilon_{2} )) - dr_{33} , \hfill \\ \end{aligned} \right.$$
(20)
$$\left\{ \begin{aligned} t_{x} = \frac{{t_{z} }}{2f}(u_{1} (1 + \varepsilon_{1} ) + u_{2} (1 + \varepsilon_{2} )) - dr_{13} , \hfill \\ t_{y} = \frac{{t_{z} }}{2f}(v_{1} (1 + \varepsilon_{1} ) + v_{2} (1 + \varepsilon_{2} )) - dr_{33} . \hfill \\ \end{aligned} \right.$$
(21)