2.1 Scenario definition

This paper studies the orbital pursuit-evasion games problem of optical space surveillance satellites. The primary consideration is that for the important asset satellite whose space moving target is close to its own side, the observation satellite needs to observe the moving target and continuously monitor it. The intention of the moving target can be judged by means of load identification and orbit prediction, which can provide support for further defensive actions.

Three types of spacecrafts are defined: important asset spacecraft (Asset), such as space station, relay satellite, etc., running on geosynchronous orbit (GEO); the target satellite (Target), a non-cooperative target satellite close to the asset, runs near GEO; the chasing satellite (Chaser), our monitoring satellite carrying optical payload, is designed to approach the target satellite and observe it at close range. As shown in Fig. 1, assuming that the chasing satellite finds that the target satellite is close to the important asset spacecraft, the chasing satellite needs to apply multiple pulses to reach the target satellite’s solar-illuminated area in advance.

Figure 1.

The orbital pursuit-evasion game.

The safe approach region is defined with the target satellite as the vertex, encompassing the conical solar-illuminated area between the target satellite and the sun. Within this region, the camera field of view of the target satellite experiences a sunlit background, creating an approximate blind spot, while the chasing satellite’s camera field of view is optimal. This optimal field of view lays the foundation for subsequent assessments of the target satellite’s intentions and the subsequent response. The target satellite also carries optical payloads. Upon detecting the approach of the chasing satellite, the target satellite should undertake specific evasion manoeuvers based on the chasing satellite’s position and velocity. This is done to prevent the chasing satellite from entering its own solar-illuminated area. Simultaneously, the target satellite, in order to maintain its mission of observing asset spacecraft, needs to preserve its original orbital position and orientation as much as possible.

This paper considers close-range pursuit-evasion tasks with a dynamic model based on the two-body problem centred on Earth. During the orbital manoeuver process, the relative distance between the chasing satellite and the target satellite is much smaller than the orbital radius of GEO. The Clohessy–Wiltshire (CW) equations [Reference Clohessy and Wiltshire17] are employed to approximate the relative orbital motion of the two satellites. These equations are expressed as:

(1) \begin{align}\left\{ \begin{array}{l}\ddot x = 2\omega \dot y + 3{\omega ^2}x + {f_x}\\[5pt] \ddot y = - 2\omega \dot x + {f_y}\\[5pt] \ddot z = - {\omega ^2}z + {f_z}\end{array} \right.\end{align}

where $\omega = \sqrt {\mu /a_0^3} $ represents the average orbital angular velocity of the reference spacecraft.

Equation 1 is a second-order differential equation, and its solution is:

(2) \begin{align}\left[ {\begin{array}{*{20}{c}}{{\boldsymbol{r}}(t)}\\[5pt] {{\dot{\boldsymbol{r}}}(t)}\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c}{{{\boldsymbol{\varPhi }}_{rr}}(\tau )} {}& {{{\boldsymbol{\varPhi }}_{rv}}(\tau )}\\[5pt] {{{\boldsymbol{\varPhi }}_{vr}}(\tau )} {}& {{{\boldsymbol{\varPhi }}_{vv}}(\tau )}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{\boldsymbol{r}}({t_0})}\\[5pt] {{\dot{\boldsymbol{r}}}({t_0})}\end{array}} \right] + \int {\left[ \begin{array}{l}{{\boldsymbol{\varPhi }}_{rv}}(\tau )\\[5pt] {{\boldsymbol{\varPhi }}_{vv}}(\tau )\end{array} \right]} f(s)ds\end{align}

where ${\boldsymbol{r}}(t) = {\left[ {\begin{array}{*{20}{c}}{x(t)} {}{y(t)} {}{z(t)}\end{array}} \right]^T}$ is the relative position vector, ${\dot{\boldsymbol{r}}}(t) = {\left[ {\begin{array}{*{20}{c}}{\dot x(t)} {}{\dot y(t)} {}{\dot z(t)}\end{array}} \right]^T}$ is the relative velocity vector, $\tau = t - {t_0}$ is the flight time, and the state transition matrices are:

(3) \begin{align}{{\boldsymbol{\varPhi }}_{rr}}(\tau ) = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{4 - 3\cos \omega \tau } {}& 0 & {}0\\[5pt] {6(\sin \omega \tau - \omega \tau )} & {}1 {}& 0\\[5pt] 0 {}& 0& {}{\cos \omega \tau }\end{array}} \right]\end{align}

(4) \begin{align}{{\boldsymbol{\varPhi }}_{rv}}(\tau ) = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{\dfrac{1}{\omega }\sin \omega \tau } & {}{\dfrac{2}{\omega }(1 - \cos \omega \tau )} {}&0\\[5pt] {\dfrac{2}{\omega }(\cos \omega \tau - 1)}& {}{\dfrac{1}{\omega }\left( {4\sin \omega \tau - 3\omega \tau } \right)} {}&0\\[5pt] 0 {}& 0 {}& {\dfrac{1}{\omega }\sin \omega \tau }\end{array}} \right]\end{align}

(5) \begin{align}{{\boldsymbol{\varPhi }}_{vr}}(\tau ) = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{3\omega \sin \omega \tau }& {}0& {}0\\[5pt] {6\omega (\cos \omega \tau - 1)} {}&0& {}0\\[5pt] 0 {}&0& {}{ - \omega \sin \omega \tau }\end{array}} \right]\end{align}

(6) \begin{align}{{\boldsymbol{\varPhi }}_{vv}}(\tau ) = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{\cos \omega \tau } {}& {2\sin \omega \tau }& {}0\\[5pt] { - 2\sin \omega \tau }& {}{4\cos \omega \tau - 3} {}&0\\[5pt] 0& {}0& {}{\cos \omega \tau }\end{array}} \right]\end{align}

Considering impulse-type orbital manoeuvers, the expression for the solution to Equation 1 becomes:

(7) \begin{align}\left[ {\begin{array}{*{20}{c}}{{\boldsymbol{r}}(t)}\\[5pt] {{\dot{\boldsymbol{r}}}(t)}\end{array}} \right] = {\boldsymbol{\varPhi }}(t,{t_0})\left[ {\begin{array}{*{20}{c}}{{\boldsymbol{r}}({t_0})}\\[5pt] {{\dot{\boldsymbol{r}}}({t_0})}\end{array}} \right] + \sum\limits_{i = 1}^N {{\boldsymbol{\varPhi }}(t,{t_i})} \left[ {\begin{array}{*{20}{c}}0\\[5pt] E\end{array}} \right]\Delta {v_i}\end{align}

where t _i is the time of the impulse orbital manoeuver, N is the number of manoeuvers that occurred before time t, and Δv _i is the impulse velocity increment vector corresponding to the moment t _i .

Geometrically defining the safe approach region, let r _S denote the position of the sun in the Earth’s equatorial inertial frame. In the calculations, obtained based on ephemerides, the position vector from the target satellite to the sun at time t is given by:

(8) \begin{align}{\boldsymbol{L}}\left( t \right){\rm{ = }}{\boldsymbol{r}_S} - {\boldsymbol{r}_T}\left( t \right)\end{align}

2.2 Objective function

In the pursuit-evasion game, the chasing satellite needs to optimise its strategy to approach the safety proximity region of the target satellite, while the target satellite simultaneously optimises its strategy to prevent the chasing satellite from entering its own safety proximity region. Constructing the game model:

(9) \begin{align}\begin{array}{l}{{\dot x}_C} = f({x_C},{u_C},t)\\[5pt] {{\dot x}_T} = f({x_T},{u_T},t)\end{array}\end{align}

x _C , x _T represent the states of the chasing and target satellites, where C represents the chasing satellite, T represents the target satellite, and u denotes a pair of strategies.

The initial states constitute the initial boundary conditions for the game:

(10) \begin{align}\begin{array}{l}{x_C}({t_0}) = {x_{C0}}\\[5pt] {x_T}({t_0}) = {x_{T0}}\end{array}\end{align}

Terminal time constraint:

(11) \begin{align}t \lt {t_f}\end{align}

To ensure that the chasing satellite enters the safety proximity region of the target satellite at the terminal time, there are terminal position constraints:

(12) \begin{align}\left\{ \begin{array}{l}||\boldsymbol{r}_{CT}^o({t_f})||_{}^{} \le {\boldsymbol{r}_{\lim }}\\[5pt] {\theta _T}({t_f}) \lt {\theta _{\lim }}\end{array} \right.\end{align}

The objective of the target satellite is to perform orbital manoeuvers to ensure that, at the terminal time, the chasing satellite is not within its safety proximity region:

(13) \begin{align}\left\{ \begin{array}{l}||\boldsymbol{r}_{CT}^o({t_f})||_{}^{} \gt {\boldsymbol{r}_{\lim }}\\[5pt] or\\[5pt] {\theta _T}({t_f}) \gt {\theta _{\lim }}\end{array} \right.\end{align}

Simultaneously, the total fuel constraint is expressed through velocity increments:

(14) \begin{align}\left\{ \begin{array}{l}||\Delta {V_C}||_{}^{} \le \Delta {V_{C\max }}\\[5pt] ||\Delta {V_T}||_{}^{} \le \Delta {V_{T\max }}\end{array} \right.\end{align}

The objective function ${u_C}^{\ast},{u_T}^{\ast}$ for a set of optimal saddle point strategies is:

(15) \begin{align}\begin{array}{l}{\eta _T}(u_C^*,u_T^*) = \left\{ \begin{array}{l}\max ||\boldsymbol{r}_{CT}^o({t_f})||\\[5pt] \min ||\Delta {V_T}||\end{array} \right.\\ \\ {\eta _C}(u_C^*,u_T^*) = \left\{ \begin{array}{l}\min ||\boldsymbol{r}_{CT}^o({t_f})||\\[5pt] \min ||\Delta {V_C}||\end{array} \right.\end{array}\end{align}

3.1 Multi-granularity onboard relative orbit determination model

This study contemplates using an optical camera and lidar data for relative orbit determination. The onboard relative orbit determination algorithm utilises multiple images and distance measurements within the instrument coordinate system to obtain state parameters for various relative position points. Employing fitting calculations at different computational granularities, the algorithm determines the initial relative position, thereby deriving the orbital parameters of the target. Subsequently, the orbit is extrapolated to facilitate the computation of proximity manoeuvers in the subsequent stages.

Due to pixel constraints in optical cameras, the relative positions r _x , r _y in the instrument coordinate system are determined by the centre position of the pixel block where the target is located in the camera’s field of view. For instance, if the actual relative position of the target with respect to the camera’s instrument centre is (+30.3, −20.7) m, and each pixel block on the camera represents a spatial distance of 1m, then the measured values from the camera would be (+30, −21) m at this distance. The lidar measurement r _z is subject to systematic errors and is directly proportional to the actual distance. The velocities v _x , v _y , v _z are obtained through interval sampling and interpolation.

The state equation and observation equation are defined as follows, with input and output state variables shown in Fig. 3.

(16) \begin{align}X(t) = \left[ {\begin{array}{*{20}{c}}{{\boldsymbol{r}}(t)}\\[5pt] {{\dot{\boldsymbol{r}}}(t)}\end{array}} \right],{X_{Det}} = \left[ {\begin{array}{*{20}{c}}{{{\boldsymbol{r}}_{Det}}}\\[5pt] {{{{\dot{\boldsymbol{r}}}}_{Det}}}\end{array}} \right]\end{align}

Figure 3.

The state variables for relative orbit determination.

Selecting state parameter ${X_i},i = 1, \cdots, N$ , the motion state equation for the studied object can be expressed in the following form of first-order differential equations:

(17) \begin{align}{\rm{d}}X(t)/{\rm{d}}t = \,f(X) + G(t) \cdot W(t)\end{align}

where $W(t)$ is a random disturbance, with a mean value $E[W(t)] = 0$ .

The correlation function is:

(18) \begin{align}\begin{array}{l}COV[W(t),W(\tau )] = Q(t)\delta (t - \tau )\\[5pt] \delta (t - \tau ) = \left\{ \begin{array}{l}1\,\,\,\,\,\,t = \tau \\[5pt] 0\,\,\,\,\,\,t \ne \tau \end{array} \right.\end{array}\end{align}

where $Q(t)$ is the variance of $W(t)$ .

The initial estimate of the state vector is:

(19) \begin{align}X({t_0}) = {X_0}\end{align}

The observation quantity ${Y_j},j = 1, \cdots, M$ expresses the observation equation in vector form.

(20) \begin{align}Y{\rm{(}}t{\rm{)}} = h{\rm{(X)}} + V{\rm{(t)}}\end{align}

where $V(t)$ is a random disturbance, with a mean value $E[V(t)] = 0$ .

The correlation function is:

(21) \begin{align}COV[V(t),V(\tau )] = R(t)\delta (t - \tau )\end{align}

where $R(t)$ is the variance of $V(t)$ .

Assume that $W(t)$ and $V(t)$ are mutually independent.

(22) \begin{align}COV[W(t),V(\tau )] = 0\end{align}

Both the medium-grained OD and fine-grained OD models utilise the least squares fitting method. Assuming that at time t0, the state of the studied object is X0, the state at time t is obtained by solving the state equation using numerical methods:

(23) \begin{align}{X_{\rm{t}}} = {X_0} + \,\int_{{{\rm{t}}_0}}^{\rm{t}} {f(X) + G({\rm{t}}) \cdot W({\rm{t}})} \end{align}

The deviation between observed measurement values and calculated values

(24) \begin{align}\Delta Y = {Y_{\rm{M}}} - Y({X_t})\end{align}

The change in state variables after one iteration is defined as $\Delta X = \mathop {{X_0}}\limits^ \wedge - {X_0}$

Using the numerical finite difference method, calculate the partial derivative matrix of the observations with respect to the improved state.

(25) \begin{align}H(\Delta X) = \frac{{\partial Y}}{{\partial \Delta X}} = \frac{{Y({X_0} + \Delta X) - Y({X_0})}}{{\Delta X}}\end{align}

So, the new estimate after one iteration is

(26) \begin{align}\mathop {{X_0}}\limits^ \wedge = {X_0} + {\left( {{H^T}WH} \right)^{ - 1}}{H^T}W\Delta Y\end{align}

Repeatedly iterate until convergence to obtain the value of ${X_{Det}}$ .

The following introduces three granularity levels of orbit determination models.

1. (1) Coarse-grained OD

At the coarse-grained level, only a small number of observations are calculated. The orbit is propagated by the observation time interval, and pairwise averages are taken. This granularity level has the smallest computational load but the poorest fitting effect.

1. (2) Medium-grained OD

At the medium-grained level, the least squares fitting algorithm is employed with fewer iterations, resulting in a slightly better fitting effect. At this granularity level, adopting a small number of observations and a larger convergence error helps the algorithm quickly exit the iteration, reducing computation time.

1. (3) Fine-grained OD

At the fine-grained level, a larger number of observations, a smaller convergence error and a larger maximum iteration count are adopted to achieve higher fitting accuracy and obtain better initial orbit fitting results. However, the corresponding fitting time is also the longest.

3.2 Multi-granularity onboard impulse orbit manoeuver model

The orbital manoeuver model employs a variable-timing multi-impulse orbital manoeuver design. To mitigate the risk of the chasing satellite adopting a two-impulse strategy, where the first impulse concludes before the target executes an evasive manoeuver, resulting in the need for additional fuel to correct the orbit after the initial impulse, a multi-impulse orbital manoeuver approach is employed. The multi-impulse orbital manoeuver, after a certain degree of optimisation, generally results in a smaller first impulse compared to the first impulse of a two-impulse manoeuver. Subsequent orbital manoeuvers incur reduced fuel consumption. A representative three-impulse scenario is illustrated in Fig. 4.

1. (1) Coarse-grained OM

Figure 4.

The three-impulse orbital manoeuver.

The numerical solution coarse optimisation granularity initially employs numerical approximation to calculate and obtain multiple pulse values, swiftly yielding usable initial values. Utilising the numerical solution as a starting point, a narrow-range optimisation is performed in the vicinity of the numerical solution. This approach involves fewer iterations and computational steps, resulting in the fastest computational speed. However, the optimisation effect on fuel consumption is comparatively worse, often converging to a local saddle point.

Using the orbit determination result ${X_{Det}}$ as the input parameter at time t0.

From the state transition matrix, the relationship between the position and velocity states at time t ₀ and t _f is given by:

(27) \begin{align}\left[ {\begin{array}{*{20}{c}}{{\boldsymbol{r}}({t_f})}\\[5pt] {{\dot{\boldsymbol{r}}}({t_f})}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{{\boldsymbol{\varPhi }}_{rr}}(\tau )} {}{{{\boldsymbol{\varPhi }}_{rv}}(\tau )}\\[5pt] {{{\boldsymbol{\varPhi }}_{vr}}(\tau )} {}{{{\boldsymbol{\varPhi }}_{vv}}(\tau )}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{\boldsymbol{r}}({t_0})}\\[5pt] {{\dot{\boldsymbol{r}}}({t_0})}\end{array}} \right]\end{align}

The straight vector ${\boldsymbol\rho _{f,0}}$ from ${\boldsymbol{r}}({t_f})$ to ${\boldsymbol{r}}({t_0})$ represents the expected approach orbit. Let ${\boldsymbol{r}}(t)$ be the relative position vector of the satellite to the straight vector ${\boldsymbol\rho _{f,0}}$ at time t. Obviously, ${t_0} \le t \le {t_f}$ , and the boundary conditions for $\boldsymbol\rho (t)$ are:

(28) \begin{align}\left\{ \begin{array}{l}\boldsymbol\rho ({t_0}) = {\boldsymbol\rho _{f,0}}\\[5pt] \boldsymbol\rho ({t_f}) = 0\end{array} \right.\end{align}

Using N(N ≥ 3) velocity impulses, the chasing satellite moves from the initial position ${\boldsymbol{r}}({t_0})$ (i.e., $\boldsymbol\rho ({t_0}) = {\boldsymbol\rho _{f,0}}$ ) to the final position ${\boldsymbol{r}}({t_f})$ (i.e., $\boldsymbol\rho ({t_f}) = 0$ ) within the time T. The straight vector ${\boldsymbol\rho _{f,0}}$ is decomposed into N-1 equally spaced segments. A velocity impulse is applied at the beginning and end of each segment. The time interval between the two impulses in each segment is the same, i.e., Δt = T(N-1).

Positions along the manoeuvers at each instance from ${\boldsymbol\rho _{f,0}}$ :

(29) \begin{align}\Delta \boldsymbol\rho = \frac{{{\boldsymbol\rho _{f,0}}}}{{N - 1}}\end{align}

For the kth impulse,

(30) \begin{align}\begin{array}{l}{\boldsymbol{r}}({t_{k - 1}}) = {\boldsymbol{r}}({t_f}) + {\boldsymbol\rho _{f,k - 1}}\\[5pt] {\boldsymbol{r}}({t_k}) = {\boldsymbol{r}}({t_f}) + {\boldsymbol\rho _{f,k}}\end{array}\end{align}

The initial velocity after the kth manoeuver can be obtained.

(31) \begin{align}{\dot{\boldsymbol{r}}}(t_{k - 1}^ + ) = {G_{rr}}{\boldsymbol{r}}({t_{k - 1}}) + {G_{rv}}{\boldsymbol{r}}({t_k})\end{align}

Similarly, the final velocity before the kth manoeuver.

(32) \begin{align}{\dot{\boldsymbol{r}}}(t_{k - 1}^ - ) = {G_{vr}}{\boldsymbol{r}}({t_{k - 2}}) + {G_{vv}}{\boldsymbol{r}}({t_{k - 1}})\end{align}

where:

(33) \begin{align}\begin{array}{l}{G_{rr}} = - {\boldsymbol{\varPhi }}_{rv}^{ - 1}(T){{\boldsymbol{\varPhi }}_{rr}}(T),\quad \quad \quad \quad \quad \quad {G_{rv}} = {\boldsymbol{\varPhi }}_{rv}^{ - 1}(T)\\[5pt] {G_{vr}} = {{\boldsymbol{\varPhi }}_{vr}}(T) - {{\boldsymbol{\varPhi }}_{vv}}(T){\boldsymbol{\varPhi }}_{rv}^{ - 1}(T){{\boldsymbol{\varPhi }}_{rr}}(T),\quad {G_{vv}} = {{\boldsymbol{\varPhi }}_{vv}}(T){\boldsymbol{\varPhi }}_{rv}^{ - 1}(T)\end{array}\end{align}

The velocity increment required for the kth impulse is

(34) \begin{align}{\boldsymbol{v}_{k,k - 1}} = {\dot{\boldsymbol{r}}}(t_{k - 1}^ + ) - {\dot{\boldsymbol{r}}}(t_{k - 1}^ - )\end{align}

1. (2) Medium-grained OM

At the medium-grained level, particle swarm optimisation [Reference Marini and Walczak18] is used for the solution. In this granularity, a smaller number of particles, a smaller upper limit of iteration count, and a larger iteration termination error are employed. The optimisation effect is better, but the calculation time is longer compared to coarse-grained.

Using the sum of N impulse velocity increments as the optimisation objective.

(35) \begin{align}J = \sum\limits_{i = 1}^N {|\Delta {v_i}|} \end{align}

From Equation 34, the fuel optimisation function is constructed and expressed as follows:

(36) \begin{align} &\textrm{min}\; J = \sum\limits_{i = 1}^N {|\Delta {v_i}|}\nonumber \\[5pt] &s.t.\nonumber \\[5pt] &\quad \left\{ \begin{array}{l@{\quad}l}{t_0} \le {t_i} \le {t_0} + T, & i = 1 \cdots N\\[5pt] {t_i} \le {t_j}, & i \lt j\\[5pt] |\Delta {v_i}| \geqslant 0, & i = 1 \cdots N - 2\end{array} \right.\end{align}

The optimisation variables for the multi-pulse orbital manoeuver problem are the pulse moments for each pulse. Given the intricate nature of the constraints, the particle swarm optimisation algorithm can effectively solve the global optimal solution.

1. (3) Fine-grained OM

At the fine-grained level, using a larger number of particles, a larger upper limit of iteration count and a smaller iteration termination error for global optimisation results in the best fuel optimisation effect. However, the corresponding computation time is the longest.

3.3 Multi-granularity onboard decision module

In this section, manoeuvering game strategies were designed for the chasing satellite using a multi-granularity observation model. Simultaneously, corresponding manoeuvering game strategies were also designed for the target satellite that did not adopt a multi-granularity observation model. The orbital manoeuver strategies for the chaser and target are illustrated in Fig. 5.

Figure 5.

The orbital manoeuver strategies for the chaser and target.

The multi-granularity game process for the chasing satellite is outlined as follows:

1. (1) Selects an appropriate orbit determination granularity for determining the orbit of the target. Refers to the orbit determination model in Section 3.1 and proceeds to step (2).

2. (2) The orbit determination results from (1) are used to calculate the minimum distance between the target and the asset $\boldsymbol{d}_{minTargetToAsset}$ , as well as the time to reach the minimum distance $\boldsymbol{t}_{minDisCha}$ . Proceed to step (3).

3. (3) Check whether the minimum distance between the target and the asset is less than the threat distance of the asset $\boldsymbol{d}_{TreatAsset}$ . If true, proceed to step (4).

4. (4) If the minimum distance between the target and the asset is less than the threat distance of the asset, the chaser selects an appropriate orbit manoeuver granularity to calculate the orbit manoeuver impulse for approaching the target. It also stores the orbit determination parameters, time of reaching the minimum distance, manoeuver time and manoeuver components. Proceed to step (5).

   If the minimum distance between the target and the asset is greater than the threat distance of the asset, return to step (1) to re-determine the orbit for the target.

5. (5) Check whether the manuoeuver time has been reached based on the stored manoeuver time. If the recorded time has been reached, execute the corresponding impulse manoeuver. If the manoeuver time has not been reached, proceed to step (6).

6. (6) Continues to select an appropriate granularity to determine the orbit of the target. Both the current orbit determination result and the stored previous orbit determination result are propagated to the stored time of reaching the minimum distance. Compares the propagated results of the two orbit determinations, denoted as $\boldsymbol{d}_{LastTarToAssTerminal}$ and $\boldsymbol{d}_{TarToAssTerminal}$ .

   If the difference is less than the set manoeuver detection threshold, it is considered that the target has not made an avoidance manoeuver. Return to step (6) to re-determine the orbit of the target. If the difference is greater than the set manoeuver detection threshold, it is considered that the target has made an avoidance manoeuver. Proceed to step (7).

7. (7) Further checks whether the propagated result of the current orbit determination is less than the threat distance of the asset. If the result is less than the threat distance, it is considered that the target has manoeuvered and has not changed its threat intent. Clears the stored minimum distance, time of reaching the minimum distance, manoeuver time, and manoeuver components. Return to step (1) to re-determine the orbit for the target and recalculate the minimum distance and time of reaching the minimum distance.

   If the result is greater than the threat distance of the asset, it is considered that the target has manoeuvered, but the end position does not pose a threat to the asset. There are two possibilities: either the target has abandoned the mission of threatening the asset, or the target executes the initial pulses of a multi-pulse orbit manoeuver and is on a transitional orbit where the threat cannot be determined. In either case, the target intended to threaten the asset. Therefore, the chasing satellite adopts a strategy of approaching the asset for both possibilities. Simultaneously, it selects an appropriate orbit manoeuver granularity, solves and stores the manoeuver time and impulse components. Proceed to step (5).

The orbital game process for the target is as follows:

The target does not have multi-granularity computing capability. As referenced in Section 3.1, the orbit determination model fixedly selects one of the granularities, and the orbit manoeuver model, as referenced in Section 3.2 Multi-granularity onboard impulse orbit manoeuver model, fixedly selects one.

1. (1) Selects an appropriate orbit determination granularity for determining the orbit of the target. Refers to the orbit determination model in Section 3.1 and proceeds to step (2).

2. (2) The target determines its orbit as the chaser and proceeds to step (3).

3. (3) Based on the orbit determination results from (2), solve for the minimum distance between the chaser and the target $\boldsymbol{d}_{minChaserToTarget}$ , as well as the time to reach the minimum distance, and proceed to step (4).

4. (4) If the minimum distance between the chaser and the target is less than the threat distance of the target, proceed to step (5).

5. (5) If the minimum distance between the chaser and the target is less than the threat distance of the target, the target uses a fixed granularity for orbit determination to calculate the orbital manoeuver pulse to evade the chaser. Additionally, store the orbit determination parameters, time of reaching the minimum distance, manoeuver time, and manoeuver components. Proceed to step (6). If the minimum distance exceeds the threat distance, go back to step (1) and re-determine the orbit for the chaser.

   In this process, the target’s evasion manoeuver is depicted in Fig. 6. The chaser floats past the target at the minimum distance, dividing the circle with the target as the centre and the target threat distance as the radius into two halves: one containing the minimum distance and the other without. The target moves towards the half-circle without the minimum distance by a distance equal to the target threat distance. The target must ensure it stays within the threat distance of the asset satellite. In this plan, the target performs a small-distance pulse manoeuver.

6. (6) Check whether the manoeuver time has been reached based on the stored manoeuver time. If the recorded time has been reached, execute the corresponding impulse manoeuver. If the manoeuver time has not been reached, proceed to step (2).

Figure 6.

The target’s evasion manoeuver.

The orbit determination algorithm and the orbit manoeuver algorithm adopt an iterative optimisation method. The initial iteration value and direction are stochastic. Therefore, when the algorithm jumps out of the iteration, the number of cycles is not necessarily the same, resulting in uncertainty in the decision-making time cost of the multi-granularity observation model. The output of the orbit determination algorithm is the relative position vector and the relative velocity vector at the end of the orbit determination. An input of the orbit manoeuver algorithm is to allow the manoeuver time of the first pulse. If the actual decision end time exceeds the calculated first pulse manoeuver time due to uncertainty, the first pulse cannot be applied as planned and needs to be recalculated, even leading to mission failure.

Chaser employing the Absolute Robust Criterion follows the decision-making steps outlined below:

1. (1) Determine the set of potential scenarios for statistical uncertainty variables.

2. (2) Identify the set of viable options.

3. (3) Compute the benefits or costs for each feasible option under each potential scenario.

4. (4) Determine the minimum benefit or maximum cost for each option along with its corresponding worst-case scenario. In this context, the benefits or costs associated with each option represent the decision time cost, so calculate the maximum cost for each option.

5. (5) Compare the costs under the worst-case scenarios for each option and identify the option with the minimum cost.

In the context of the uncertainty problem associated with the decision time cost for the same decision granularity scheme in this section, the Absolute Robust Criterion method is applied. This involves statistically analysing the actual decision time costs for each scheme under different conditions. The maximum decision time cost across all conditions is then used as the basis for both scheme selection and execution.

In the context of the space dynamic target tracking game considered in this paper, there is inherent uncertainty in the parameters used for each orbit determination, leading to uncertainties in the observed orbit determination errors. As the distance between the chaser and the target diminishes, the spatial scale corresponding to the one-pixel unit decreases gradually. The positional errors caused by pixel units will reduce from the meter level to the centimeter level. To address such dynamic uncertainties, the paper introduces the concept of sequential three-way decision-making.

This section establishes a sequential three-way decision granularity selection method framework for the space dynamic target pursuit-evasion game, as illustrated in Fig. 7.

Figure 7.

The sequential three-way decision granularity selection method framework.

The framework relies on the multi-level granularity structure defined earlier, which is divided into coarse-grained, medium-grained and fine-grained layers. Each layer utilises three-way decisions to partition the gaming scenarios into three categories. The decision to move to the next granularity layer is based on the requirements of the gaming scenario. Each layer is divided into three regions: positive, negative and boundary. The positive and negative regions adopt acceptance and rejection strategy schemes, respectively. The boundary region employs a delay decision strategy to gather more information to further support decision-making.

Let $g(S)$ be the relative orbit determination algorithm for each granularity layer, encompassing two outputs, denoted as ${g_1}(S),{g_2}(S)$ . $h(S)$ represents the pulse orbital manoeuver algorithm for each granularity layer.

(37) \begin{align}\begin{array}{l}{g_1}(S) = {\boldsymbol{d}_{ChaserToTarget}}\\[5pt] {g_2}(S) = {\boldsymbol{d}_{minTargetToAsset}}\\[5pt] h(S) = \boldsymbol{J}\end{array}\end{align}

1. (1) Firstly, for each granularity selection problem, enter the coarse-grained layer initially:

   In the coarse-grained layer, divide the solution space ${\textbf{U}_{Coarse}}$ into three regions ${\textbf{R}_{Coarse, POS}}, {\textbf{R}_{Coarse, BND}}, {\textbf{R}_{Coarse, NEG}}$ using the boundary region threshold ${\boldsymbol\alpha _{Coarse}}$ , positive and negative region thresholds ${\boldsymbol\beta _{Coarse}}$ , and evaluation function ${g_{Coarse}}(S),{h_{Coarse}}(S)$ :

   For the OD process, coarse-grained models are used for calculations. Given a coarse-grained distance threshold to divide the positive and negative regions and the boundary region, when the distance between the chaser and target is less than the coarse-grained distance threshold ${\boldsymbol\alpha _{CoarseOD}}$ , it is classified into the coarse-grained layer’s boundary region, and the decision is delayed. When ${\boldsymbol{d}_{ChaserToTarget}}$ is greater than ${\boldsymbol\alpha _{CoarseOD}}$ , it is classified into the positive or negative region based on the calculation results. If the minimum distance ${\boldsymbol{d}_{minTargetToProperty}}$ obtained from the orbit determination is less than the threat distance to the asset ${\boldsymbol{d}_{TreatProperty}}$ , it is classified into the positive region. If ${\boldsymbol{d}_{minTargetToProperty}}$ is greater than ${\boldsymbol{d}_{TreatProperty}}$ it is classified into the negative region.

   (38) \begin{align}\begin{array}{l}\left\{ \begin{array}{l}{\textbf{R}_{Coarse OD, POS}} = \{ S \in {\textbf{U}_{CoarseOD}}|{g_1}(S) \leq {\boldsymbol\alpha _{CoarseOD}}\;,\;{g_2}(S) \le {\boldsymbol\beta _{CoarseOD}}\} \\[5pt] {\textbf{R}_{Coarse OD, BND}} = \{ S \in {\textbf{U}_{CoarseOD}}|{g_1}(S) \gt {\boldsymbol\alpha _{CoarseOD}}\} \\[5pt] {\textbf{R}_{Coarse OD, NEG}} = \{ S \in {\textbf{U}_{CoarseOD}}|{g_1}(S) \leq {\boldsymbol\alpha _{CoarseOD}}\;,\;{g_2}(S) \gt {\boldsymbol\beta _{CoarseOD}}\} \end{array} \right.\\[5pt] \left\{ \begin{array}{l}{\textbf{R}_{Coarse OM, POS}}\,\, = \{ S \in {\textbf{U}_{CoarseOM}}|{g_1}(S) \leq {\boldsymbol\alpha _{CoarseOM}}\;,\;h(S) \le {\boldsymbol\beta _{CoarseOM}}\} \\[5pt] {\textbf{R}_{Coarse OM, BND}} = \{ S \in {\textbf{U}_{CoarseOM}}|{g_1}(S) \gt {\boldsymbol\alpha _{CoarseOM}}\} \\[5pt] {\textbf{R}_{Coarse OM, NEG}} = \{ S \in {\textbf{U}_{CoarseOM}}|{g_1}(S) \leq {\boldsymbol\alpha _{CoarseOM}}\;,\;h(S) \gt {\boldsymbol\beta _{CoarseOM}}\} \end{array} \right.\end{array}\end{align}

   For the OM process, if ${\boldsymbol{d}_{ChaserToTarget}}$ is less than ${\boldsymbol\alpha _{CoarseOM}}$ , it is classified into the boundary region of the coarse-grained layer, and the decision is delayed. For fuel solving in orbit manoeuver, if the fuel consumption J is less than the remaining fuel upper limit for the chaser, it is classified into the positive region. If J is greater than the remaining fuel upper limit, it is classified into the negative region.

   According to the three-decision rule, the following three decision rules are obtained:

   If classified into the positive region, take action to accept the calculation results of this granularity layer.

   If classified into the boundary region, take action to acquire more observation information and enter the next granularity layer for calculation.

   If classified into the negative region, take action to reject the calculation results of this granularity layer, obtain observation information again, and enter the current granularity layer for calculation.

2. (2) Enter the medium-grained layer for the problem classified into the boundary region of the coarse-grained layer in the previous layer:

   Similar to the logic of the coarse-grained layer, For the OD process,

   (39) \begin{align}\left\{ \begin{array}{l}{\textbf{R}_{Medium OD, POS}} = \{ S \in {\textbf{U}_{MediumOD}}|{g_1}(S) \leq {\boldsymbol\alpha _{MediumOD}}\;,\;{g_2}(S) \le {\boldsymbol\beta _{MediumOD}}\} \\[5pt] {\textbf{R}_{Medium OD, BND}} = \{ S \in {\textbf{U}_{MediumOD}}|{g_1}(S) \gt {\boldsymbol\alpha _{MediumOD}}\} \\[5pt] {\textbf{R}_{Medium OD, NEG}} = \{ S \in {\textbf{U}_{MediumOD}}|{g_1}(S) \leq {\boldsymbol\alpha _{MediumOD}}\;,\;{g_2}(S) \gt {\boldsymbol\beta _{MediumOD}}\} \end{array} \right.\end{align}

   For the OM process,

   (40) \begin{align}\left\{ \begin{array}{l}{\textbf{R}_{Medium OM, POS}}\,\, = \{ S \in {\textbf{U}_{MediumOM}}|{g_1}(S) \leq {\boldsymbol\alpha _{MediumOM}}\;,\;h(S) \le {\boldsymbol\beta _{MediumOM}}\} \\[5pt] {\textbf{R}_{Medium OM, BND}} = \{ S \in {\textbf{U}_{MediumOM}}|{g_1}(S) \gt {\boldsymbol\alpha _{MediumOM}}\} \\[5pt] {\textbf{R}_{Medium OM, NEG}} = \{ S \in {\textbf{U}_{MediumOM}}|{g_1}(S) \leq {\boldsymbol\alpha _{MediumOM}}\;,\;h(S) \gt {\boldsymbol\beta _{MediumOM}}\} \end{array} \right.\end{align}

3. (3) Enter the fine-grained layer for the problem classified into the boundary region of the medium-grained layer in the previous layer:

   The fine-grained layer is the bottom layer of the sequential three-decision method and does not further partition the boundary region.

   For the OD process,

   (41) \begin{align}\left\{ \begin{array}{l}{\textbf{R}_{Fine OD, POS}}\,\, = \{ S \in {\textbf{U}_{FineOD}}|\;{g_2}(S) \le {\boldsymbol\beta _{FineOD}}\} \\[5pt] {\textbf{R}_{Fine OD, NEG}} = \{ S \in {\textbf{U}_{FineOD}}|\;{g_2}(S) \gt {\boldsymbol\beta _{FineOD}}\} \end{array} \right.\end{align}

   For the OM process,

   (42) \begin{align}\left\{ \begin{array}{l}{\textbf{R}_{Fine OM, POS}}\,\, = \{ S \in {\textbf{U}_{FineOM}}|\;h(S) \le {\boldsymbol\beta _{FineOM}}\} \\[5pt] {\textbf{R}_{Fine OM, NEG}} = \{ S \in {\textbf{U}_{FineOM}}|h(S) \gt {\boldsymbol\beta _{FineOM}}\} \end{array} \right.\end{align}

4. (4) If there is only one round of decision-making, the process of gradually refining the solution from the coarse-grained layer to the fine-grained layer is unquestionable. This stepwise abstraction of the problem aligns with the natural thought process of human problem-solving. However, the game problem considered in this section involves multiple rounds of interaction between the two parties. It requires making decisions on granularity selection multiple times. If, during each decision-making iteration, the process iterates from coarse-grained to fine-grained, the computational burden associated with the sequential three-way decision-making, particularly as the number of decision rounds and the granularity layer divisions increase, becomes significantly pronounced.

Therefore, in the multi-round iterative decision-making process considered in this section, a decision-making approach that inherits prior knowledge is adopted. Once a granularity layer has been evaluated in a previous iteration, it is not reassessed from the coarsest-grained layer in the subsequent iteration. In the same gaming scenario, the granularity layer accepted in the previous round serves as the starting point for the next round. For instance, if the medium-grained layer was accepted in the previous round, the next round can directly start the division from the medium-grained, eliminating the need to solve and assess from the coarse-grained again.