GEO Spiral Cruising Orbit Design Using Relative Orbit Elements

Abstract

A new method of round-trip spiral cruising orbit design for GEO object surveillance is proposed using relative orbit elements (ROEs). The relationship between the in-plane configuration of spiral cruising orbits and ROEs is described. After that, in-plane impulse control strategies are shown for round-trip spiral cruising orbits, based on the relative motion control laws. Along-track maneuvers are performed on both sides of the cruise region in the strategies. Finally, the impulse control strategies are simulated, and the required velocity increment and performance of the cruising orbit are analyzed. The results indicate that the proposed configuration and control strategies are effective for round-trip spiral cruising orbit design. The required velocity impulse increases with cruising velocity while being free from the cruising period. Only a few velocity increments are needed for closely observing objects in the cruise region multiple times.

1. Introduction

The geosynchronous Earth orbit (GEO) is a highly valuable orbital resource. With the increasing number of GEO resident objects, the orbital environment for GEO spacecraft is becoming increasingly complex. Monitoring and identifying GEO targets on-orbit is of great significance for enhancing collision warning capability, ensuring spacecraft operational safety, and improving space situational awareness. There exists a series of methods for monitoring GEO targets. For example, GEO targets can be observed through ground-based optical telescopes [1]. However, the altitude of GEO above Earth is nearly 36,000 km, making it difficult to detect small targets. Thus, space-based observation methods have attracted much interest, such as the geostationary rendezvous [2], the collocation technique [3] (where the observer satellite operates very close to a target in the same geosynchronous slot), and the GEO monitoring constellation deployed in eccentric orbits around the GEO belt [4].

A spiral cruising orbit is a type of relative trajectory defined with respect to a designated target orbit or a specific arc segment of that orbit. A single spacecraft, referred to as the cruising spacecraft, can perform spiral cruising maneuvers near the target orbit to achieve close-range and high-precision observations of all resident objects and the space environment along the designated arc. By applying orbital maneuvers at the boundaries of the defined GEO arc segment, the cruising spacecraft can reverse its spiral direction, thus enabling repeated round-trip inspections of multiple space targets within a specific GEO arc.

Several studies have been conducted on spiral cruising orbit design and control. Using Clohessy–Wiltshire (C-W) equations [5], a widely used relative motion model that describes relative positions and velocity evolutions, Zhang, Y. et al. [6] proposed a set of descriptive parameters for spiral cruising orbits and developed methods for designing such orbits and controlling round-trip spiral maneuvers. Later, Zheng, L. et al. [7] applied a particle swarm optimization algorithm to study the fuel optimization problem for multi-target spiral cruising in GEO. Zhou H.J. et al. [8] derived GEO relative motion equations based on a linearized first-order approximation of classical orbital element differences. Xu Y.L. et al. [9] developed a description method for GEO spiral cruising orbits based on station-keeping orbital elements and analyzed long-term perturbation effects from Earth’s J2 and J22 terms. Recently, Chen, N. et al. [10] proposed an orbit design method for traverse inspection of GEO-belt targets. Moreover, a method for patrolling inclined geosynchronous orbit (IGSO) targets is presented based on crossing points and spiral rings [11].

Considering that the spiral cruising orbit can be regard as a type of spacecraft relative motion, another relative motion model, namely relative orbital elements (ROEs), have attracted widespread attention in recent years due to the clear geometric relationship for describing relative motion [12]. This provides an opportunity for designing the spiral cruising orbit conveniently. The concept of ROEs can be traced back to He, Q. [13], who established relative motion equations based on ROEs for near-circular reference orbits and proposed both in-plane and out-of-plane ROE control strategies. Subsequently, Han, C. [12] and Yin, J. [14] provided a strict definition of ROEs with clearer geometric interpretation, facilitating a configuration design of relative motion. Based on ROEs, Bai, S. et al. [15] proposed a fly-around formation design method using a parallelogram configuration, and multistage, constant-vector, thrust control strategies [16,17,18] were utilized for formation control. Recently, ROEs have been applied for exoatmospheric intercept guidance [19], bounded hovering formation control in elliptical orbits [20], and relative orbit transfer [18]. Although ROEs have been successfully used for satellite formation design and control, difficulties arise when they are utilized for spiral cruising orbit description and control. First, ROEs only describe general characteristics of relative motion; the intuitive geometric parameters for describing spiral cruising orbits still need to be determined. Next, the relationships between ROEs and the geometric parameters of spiral cruising orbits have not been studied. Finally, an efficient control method for these geometric parameters is still unclear.

The main objective of this paper is to develop a set of intuitive geometric parameters for GEO spiral cruising orbit design and an effective spiral cruising orbit control method. First, the in-plane descriptive parameters for spiral cruising orbits are presented, and the mapping relationship between these parameters and ROEs is derived. Next, based on the relative motion control equations for near-circular orbits, we propose three types of in-plane impulsive control strategies for the spiral cruising orbit: cruising speed control, combined control of cruising speed and radial cruising radius, and in-plane, full-parameter control.

This paper makes two main contributions. First, a new set of geometric parameters for describing GEO spiral cruising orbits is proposed. In previous works [6,7,8,9], the geometric interpretation of the relationship between the spiral cruising orbit and orbit states remains unclear and not intuitive, thus preventing a fast spiral cruising orbit design in engineering application. The proposed geometric parameters are intuitive, making it simple for the spiral cruising orbit design in engineering practice. Second, analytical impulsive control methods for the proposed geometric parameters are developed. In previous work [9], cruising orbit design relied on numerical optimizations, which are computationally expensive. In this paper, the proposed geometric parameters associate the spiral cruising motion with ROEs, and the analytical control strategy of the ROEs can be directly utilized for cruising orbit control; thus, they are efficient and suitable for on-board application in space missions.

The rest of the article is organized as follows: Section 2 describes the geometric parameters of the GEO spiral cruising orbit, as well as the relationships between these parameters and ROEs. Section 3 presents the control strategies for the geometric parameters of the round-trip spiral cruising orbit. In Section 4, the effectiveness of the proposed geometric parameters, as well as their control methods, are validated through numerical examples. Finally, the conclusions are summarized in Section 5.

2. Description Method of Spiral Cruising Orbits

The spiral cruising orbit essentially represents the relative trajectory of an observer spacecraft with respect to a reference spacecraft. To describe the geometric characteristics of the spiral cruising orbit more intuitively, this paper introduces a close-range relative motion model based on ROEs.

2.1. Definition of Relative Orbit Elements

To characterize the relative motion between two spacecrafts (denoted as the observer and reference satellites), reference [8] adopted a spherical geometry approach to strictly define the ROEs, namely the relative drift rate $D$, the relative eccentricity vector $\Delta e=\left[\Delta e_x,\Delta e_y\right]$, the relative inclination vector $\Delta i=\left[\Delta i_x,\Delta i_y\right]$, and the initial relative mean latitude $\Delta M^{\prime }$. Specifically, $D$ represents the difference in the mean motion of the observer and reference satellite. $\Delta e=e_{\mathrm{obs}}^{\prime }-e_{\mathrm{ref}}$ describes the orbital shape of the observer satellite with respect to the reference satellite, where $e_{\mathrm{ref}}$ is the eccentricity vector of the reference satellite’s orbit, and $e_{\mathrm{obs}}^{\prime }$ represents the projection of the eccentricity vector of the observer satellite’s orbit onto the orbital plane of the reference satellite. $\Delta i$ is expressed as the unit inclination vector of the observer satellite projected onto the orbital plane of the reference satellite. In addition, $\Delta M^{\prime }$ is the difference in the mean arguments of latitude between the two satellites and is used to measure their phase error. The detailed definition and explanations for ROEs are given by [12]. Under close-range conditions, the observer satellite’s ROEs with respect to the reference satellite is given by [12], as follows:

$$\left\{\begin{array}{l}D=n_{\mathrm{obs}}-n_{\mathrm{ref}}\hfill \\ \Delta e_x=e_{\mathrm{obs}}\cos {\omega }_{\mathrm{ref}}-e_{\mathrm{obs}}\cos {\omega }_{\mathrm{ref}}-\left(e_{\mathrm{ref}}\sin {\omega }_{\mathrm{ref}}\right)\Delta \Omega \cos i_{\mathrm{obs}}\hfill \\ \Delta e_y=e_{\mathrm{obs}}\sin {\omega }_{\mathrm{obs}}-e_{\mathrm{ref}}\sin {\omega }_{\mathrm{ref}}+\left(e_{\mathrm{ref}}\cos {\omega }_{\mathrm{ref}}\right)\Delta \Omega \cos i_{\mathrm{obs}}\hfill \\ \Delta i_x=\left({\Omega }_{\mathrm{obs}}-{\Omega }_{\mathrm{ref}}\right)\sin i_{\mathrm{ref}}\hfill \\ \Delta i_y=\left(i_{\mathrm{obs}}-i_{\mathrm{ref}}\right)\hfill \\ {\Delta M}^{\prime }=\left({\omega }_{\mathrm{ref}}-{\omega }_{\mathrm{obs}}\right)+\left(M_{\mathrm{ref}}-M_{\mathrm{obs}}\right)+\left({\Omega }_{\mathrm{ref}}-{\Omega }_{\mathrm{obs}}\right)\cos i_{\mathrm{obs}}\hfill \end{array}\right.$$

(1)

where $\left(a_{\mathrm{obs}},e_{\mathrm{obs}},i_{\mathrm{obs}},{\Omega }_{\mathrm{obs}},{\omega }_{\mathrm{obs}},M_{\mathrm{obs}}\right)$ and $\left(a_{\mathrm{ref}},e_{\mathrm{ref}},i_{\mathrm{ref}},{\Omega }_{\mathrm{ref}},{\omega }_{\mathrm{ref}},M_{\mathrm{ref}}\right)$ are the classical orbital elements of satellites, the observer satellite and the reference satellite, respectively. Here, $a$, $e$, $i$, $\Omega$, $\omega$, and $M$ denote the semi-major axis, eccentricity, inclination, right ascension of ascending node, argument of perigee, and mean anomaly, respectively. And the subscripts “obs” and “ref” represent the observation and reference satellites, respectively. In addition, $n_{\mathrm{obs}}=\sqrt{\mu /{a_{\mathrm{obs}}}^3}$ and $n_{\mathrm{ref}}=\sqrt{\mu /{a_{\mathrm{ref}}}^3}$ are the mean motion of the observer satellite and the reference satellite, respectively, where $\mu =\mathrm{398,600} {\mathrm{km}}^3/{\mathrm{s}}^2$ is the central-body gravitational constant.

2.2. Relative Motion Equations

Reference [13] provides the relative motion equations of an observer spacecraft in a near-circular reference orbit, expressed in the reference-centered orbital coordinate frame, as follows:

$$\left\{\begin{array}{l}x=a_{\mathrm{ref}}\left(\Delta M^{\prime }+2(\Delta e_x\sin u-\Delta e_y\cos u)\right)+a_{\mathrm{ref}}Dt\hfill \\ y=a_{\mathrm{ref}}(\Delta i_x\cos u+\Delta i_y\sin u)\hfill \\ z=a_{\mathrm{ref}}\left(\Delta e_x\cos u+\Delta e_y\sin u+{\displaystyle \frac{2D}{3n_{\mathrm{ref}}}}\right)\hfill \end{array}\right.$$

(2)

$$\left\{\begin{array}{l}{v}_x=2a_{\mathrm{ref}}{n}_{\mathrm{ref}}(\Delta e_x\cos u+\Delta e_y\sin u)+a_{\mathrm{ref}}D\hfill \\ v_y=a_{\mathrm{ref}}{n}_{\mathrm{ref}}(\Delta i_x\sin u-\Delta i_y\cos u)\hfill \\ v_z=a_{\mathrm{ref}}{n}_{\mathrm{ref}}(-\Delta e_x\sin u+\Delta e_y\cos u)\hfill \end{array}\right.$$

(3)

Here, $u={\omega }_{\mathrm{ref}}+M_{\mathrm{ref}}$ denotes the argument of latitude of the reference satellite and $\left(x,y,z\right)$ denotes the position of the observer spacecraft in the orbital-centered coordinate frame of the reference spacecraft, where the $x$, $y$, and $z$ directions correspond to the along-track, cross-track, and radial directions, respectively; $\left(v_x,v_y,v_z\right)$ denotes the velocity of the observer spacecraft in the same frame; and $\left[D,\Delta e_x,\Delta e_y,\Delta i_x,\Delta i_y,\Delta M^{\prime }\right]$ represents the ROEs at time $t=0$.

From Equations (2) and (3), it is evident that the along-track and radial component motion is decoupled from the cross-track direction. Since the X-Z plane is the orbital plane of the reference satellite, the in-plane and out-of-plane motions can be considered independently. This paper focuses on in-plane motion only.

2.3. Analysis of In-Plane Relative Motion Trajectories

We define the phase angle of the relative eccentricity vector as

$$\phi =\mathrm{atan}2\left(\Delta e_x,\Delta e_y\right)$$

(4)

The definition of function $\mathrm{atan}2$ is as follows:

$$\mathrm{atan}2\left(x,y\right)=\left\{\begin{array}{ll}\mathrm{arc}\tan \left(x/y\right),\hfill & y>0\hfill \\ \mathrm{arc}\tan \left(x/y\right)+\pi x\ge 0,\hfill & y<0\hfill \\ \mathrm{arc}\tan \left(x/y\right)-\pi x<0,\hfill & y<0\hfill \\ \pi /2x>0,\hfill & y=0\hfill \\ -\pi /2x<0,\hfill & y=0\hfill \\ x=0,\hfill & y=0\hfill \end{array}\right.$$

(5)

Accordingly, we have

$$\left\{\begin{array}{c}\Delta e_x=‖\Delta e‖\sin \phi \\ \Delta e_y=‖\Delta e‖\cos \phi \end{array}\right.$$

(6)

where $‖\Delta e‖$ is the 2-norm of the relative eccentricity vector.

Substituting Equation (6) into Equations (2) and (3) yields

$$\left\{\begin{array}{l}x=a_{\mathrm{ref}}\left(\Delta M^{\prime }-2‖\Delta e‖\cos \left(\phi +u\right)+Dt\right)\hfill \\ z=a_{\mathrm{ref}}\left(‖\Delta e‖\sin \left(\phi +u\right)+2D/\left(3n_{\mathrm{ref}}\right)\right)\hfill \\ v_x=a_{\mathrm{ref}}\left(2n_{\mathrm{ref}}‖\Delta e‖\sin \left(\phi +u\right)+D\right)\hfill \\ v_z=a_{\mathrm{ref}}{n}_{\mathrm{ref}}‖\Delta e‖\cos \left(\phi +u\right)\hfill \end{array}\right.$$

(7)

According to the first two formulas of Equation (7), the terms $\cos \left(\phi +u\right)$ and $\sin \left(\phi +u\right)$ can be expressed as

$$\left\{\begin{array}{l}\cos \left(\phi +u\right)={\displaystyle \frac{x-a_{\mathrm{ref}}\left(\Delta M^{\prime }+Dt\right)}{2a_{\mathrm{ref}}‖\Delta e‖}}\\ \sin \left(\phi +u\right)={\displaystyle \frac{z-2a_{\mathrm{ref}}D/\left(3n_{\mathrm{ref}}\right)}{a_{\mathrm{ref}}‖\Delta e‖}}\end{array}\right.$$

(8)

Accordingly, since ${\cos }^2\left(\phi +u\right)+{\sin }^2\left(\phi +u\right)=1$, we have

$${\displaystyle \frac{{\left(x-a_{\mathrm{ref}}\left(\Delta M^{\prime }+Dt\right)\right)}^2}{{\left(2a_{\mathrm{ref}}‖\Delta e‖\right)}^2}}+{\displaystyle \frac{{\left(z-2a_{\mathrm{ref}}D/\left(3n_{\mathrm{ref}}\right)\right)}^2}{{\left(a_{\mathrm{ref}}‖\Delta e‖\right)}^2}}=1$$

(9)

From Equation (9), it can be seen that the in-plane relative trajectory is an ellipse whose center drifts along the along-track direction. At each moment, the observer satellite lies on a so-called instantaneous ellipse. When $D>0$, the relative motion trajectory is shown in Figure 1, as follows:

As shown in Figure 1, $S$ is the reference satellite, $A$ is the observer satellite, and $C$ is the instantaneous ellipse center; the semi-major and semi-minor axes of the instantaneous ellipse are $2a_{\mathrm{ref}}‖\Delta e‖$ and $a_{\mathrm{ref}}‖\Delta e‖$, respectively. The center of the ellipse drifts along-track with the speed of $V_D=a_{\mathrm{ref}}D$, and the coordinate of the center’s z-component is given by

$$h=2a_{\mathrm{ref}}D/\left(3n_{\mathrm{ref}}\right)$$

(10)

Define $\phi$ as the in-plane phase angle of the observer satellite on the instantaneous ellipse as follows:

$$\varphi =u+\phi ,\varphi \in \left[0,2\pi \right)$$

(11)

$P$ is defined as the vertex of the spiral ring, and it is the point on the instantaneous ellipse where the relative velocity direction is opposite to $V_D$. Then, when the observer satellite arrives at the point $P$, we have

$$\left\{\begin{array}{l}0>(v_x-D)D\hfill \\ v_z=0\hfill \end{array}\right.$$

(12)

Accordingly, substituting Equation (7) into Equation (12) yields

$$\left\{\begin{array}{l}0>-{a_{\mathrm{ref}}}^2\left(2n_{\mathrm{ref}}‖\Delta e‖\sin \varphi \right)D\hfill \\ a_{\mathrm{ref}}{n}_{\mathrm{ref}}‖\Delta e‖\cos \varphi =0\hfill \end{array}\right.$$

(13)

Denote the in-plane phase angle of point P as ${\varphi }_P$. It satisfies the following equation:

$${\varphi }_P=\left\{\begin{array}{ll}{\displaystyle \frac{3\pi }{2}},\hfill & D>0\hfill \\ {\displaystyle \frac{\pi }{2}},\hfill & D<0\hfill \end{array}\right.$$

(14)

2.4. In-Plane Geometric Parameter of Spiral Cruising Orbit

The in-plane parameters used to describe the geometry of a spiral cruising orbit are illustrated in Figure 2.

As shown in Figure 2, these geometric parameters include the cruising velocity $V_D$, the cruising radius $R_r$, the location of the spiral vertex $L$, and the initial in-plane phase angle ${\varphi }_0$. Here, $V_D$ denotes the along-track drift rate of the instantaneous ellipse center and represents the drift distance of the observer satellite in each orbit revolution; $R_r$ represents the maximum distance between the spiral trajectory and the reference orbit within the orbital plane; $L$ represents the along-track coordinate of the vertex of the spiral ring; and ${\varphi }_0$ is the initial phase angle of the observer spacecraft in the instantaneous ellipse. Subsequently, the relationship between the geometric parameters and ROEs are derived as follows.

• Cruising Velocity

$$V_D=D{a}_{\mathrm{ref}}$$

(15)

The along-track drift distance of the observer satellite with respect to the reference satellite in one orbit revolution is

$$DistP=2\pi V_D/n_{\mathrm{ref}}$$

(16)

The cruising angular velocity of the GEO spiral cruising orbit is

$$V_A=V_D/a_{\mathrm{ref}}$$

(17)

For GEO reference orbits, the relationship between the cruising velocity and parameters such as the relative drift rate is shown in Figure 3.

2.

Cruising Radius

As shown in Figure 2, the cruising radius is given by

$$R_r=h+a_{\mathrm{ref}}‖\Delta e‖={\displaystyle \frac{2a_{\mathrm{ref}}\left|D\right|}{3n_{\mathrm{ref}}}}+a_{\mathrm{ref}}‖\Delta e‖$$

(18)

It is evident that when the spiral loop exists, $‖\Delta e‖>0$, the radial cruising radius satisfies the condition $R_r>{\displaystyle \frac{2a_{\mathrm{ref}}\left|D\right|}{3n_{\mathrm{ref}}}}$.

3.

Initial In-Plane Phase Angle

Let $u\left(t_0\right)=u_0$ be the argument of latitude of the reference satellite at the initial time; the in-plane phase angle ${\varphi }_0$ of the observer satellite is

$${\varphi }_0=u_0+\phi$$

(19)

where $\phi$ is given by Equation (4).

4.

Spiral Vertex Location

Substituting the Equations (11) and (14) into Equation (7), the along-track coordinate of the spiral vertex is given by

$$L=x_P=a_{\mathrm{ref}}\left(\Delta M^{\prime }+D\left(t_P-t_0\right)\right)$$

(20)

where $t_P$ is the time when the observed satellite $A$ arrives at the vertex $P$ in the spiral ring, and the latitude argument of the reference satellite at this time is given by

$$u_P={\varphi }_P-\phi$$

(21)

For near-circular orbits, the following approximation holds:

$$t_P=t_0+{\displaystyle \frac{u_P-u_0}{n_{\mathrm{ref}}}}$$

(22)

Therefore, the along-track location of the spiral vertex is given by

$$L=a_{\mathrm{ref}}\left(\Delta M^{\prime }+D\left(\left(1+{\displaystyle \frac{D}{2\left|D\right|}}\right)\pi -\phi -u_0\right)/n_{\mathrm{ref}}\right)$$

(23)

Given the classical orbital elements of the reference satellite, and once the spiral cruising parameters are specified (cruising velocity $V_D$, radial cruising radius $R_r$, initial in-plane phase angle $\varphi$, and spiral vertex location $L$), the corresponding in-plane ROEs—$D$, $\Delta M^{\prime }$, $\Delta e_x$ and $\Delta e_y$—can be determined by jointly solving the derived Equations (6), (15), (18), (20) and (23). Since the relative inclination vector $\Delta i=\left[\Delta i_x,\Delta i_y\right]$ is unrelated to in-plane motion, we may assign $\Delta i_x=0$ and $\Delta i_y=0$. This yields a complete set of ROEs compatible with the desired spiral cruising geometry.

$$\left\{\begin{array}{l}D=V_D/a_{\mathrm{ref}}\hfill \\ \Delta e_x={\displaystyle \frac{1}{a_{\mathrm{ref}}}}\left(R_r-{\displaystyle \frac{2\left|V_D\right|}{3n_{\mathrm{ref}}}}\right)\sin \left({\varphi }_0-u_0\right)\hfill \\ \Delta e_y={\displaystyle \frac{1}{a_{\mathrm{ref}}}}\left(R_r-{\displaystyle \frac{2\left|V_D\right|}{3n_{\mathrm{ref}}}}\right)\cos \left({\varphi }_0-u_0\right)\hfill \\ \Delta M^{\prime }=L/a_{\mathrm{ref}}-D\left(\left(1+{\displaystyle \frac{D}{2\left|D\right|}}\right)\pi -{\varphi }_0\right)/n_{\mathrm{ref}}\hfill \\ \Delta i_x=0\hfill \\ \Delta i_y=0\hfill \end{array}\right.$$

(24)

The absolute orbital elements of the spiral cruising spacecraft can then be calculated by applying the known transformation from ROEs to absolute elements.

3. Control Strategies for Round-Trip Spiral Cruising Orbits

A round-trip spiral cruising orbit refers to a type of relative trajectory that allows the observer spacecraft to perform repeated close-range inspections of multiple targets within a designated arc segment of a GEO [6]. This is achieved by applying in-plane impulsive maneuvers at the boundaries of the cruising region to adjust the geometric parameters of the spiral cruising orbit, thereby reversing the cruising direction.

In this section, the cruising region is defined in terms of the sub-satellite point longitudes in GEO. Virtual reference satellites are placed at the boundaries of the cruising region. The control of spiral cruising orbit geometric parameters is then transformed into the control of ROEs. Based on the impulsive control equations for relative motion in near-circular orbits derived in [13], control strategies under various constraint conditions are formulated and analyzed. These constraints include the following:

• Cruising velocity constraint;
• Combined constraint on cruising velocity and cruising radius;
• In-plane full-parameter control.

3.1. Reference Satellite Placement Strategy

A schematic diagram of the round-trip spiral cruising orbit is shown in Figure 4.

As can be seen, the observer satellite A performs a round-trip spiral motion around the geostationary orbit. The relative trajectory of the observer satellite is defined with respect to the reference satellite S in GEO. Let $lon1$ and $lon2$ denote the longitudes of the cruising region boundary. Note that the reference satellite S is used solely for visualization of the spiral cruising orbit and does not influence the actual control of relative motion of the observer satellite. This is because the large along-track range of the round-trip cruising orbit may result in significant radial errors due to the nonlinearity of the relative motion, thus leading to significant errors in cruising orbit control. To address this problem, virtual reference satellites $S_1$ and $S_2$ are placed at GEO corresponding to the cruising region boundaries $lon1$ and $lon2$, respectively. These virtual reference satellites serve as local reference points to ensure the linearity of the relative motion for control maneuvers.

When the observer satellite A cruises forward and arrives at the longitude $lon2$, the virtual reference satellite $S_2$ is taken as the reference for cruising orbit control. The ROEs of the observer satellite with respect to $S_2$ are evaluated, and the cruising orbit geometric parameters of the reverse spiral cruising orbit are defined accordingly. Subsequently, tangential impulsive maneuvers are executed to transition the observer satellite A into the reverse cruising orbit so that it can cruise backward to the other boundary $lon1$. And at the point $lon1$, the virtual reference satellite $S_1$ is taken as the reference for cruising orbit control to ensure the observer satellite cruising forward to the boundary $lon2$ again. The control strategy of the cruising orbit at each boundary is described in detail in the following subsection.

3.2. In-Plane, Along-Track Impulsive Maneuver Equation

In-plane impulsive control can be categorized into along-track and radial direction impulsive controls. Along-track impulses are more efficient in eliminating deviations in the relative eccentricity vector and initial mean latitude offset, requiring less velocity increment than the radial impulses. Therefore, the control strategy in this paper adopts in-plane along-track impulsive maneuvers.

To derive the along-track impulse maneuver equation, assume that the along-track velocity impulse is given by $\Delta v_x$ (the radial velocity impulse $\Delta v_z=0$). Let $D_{-}$, $\Delta e_{x-}$, $\Delta e_{y-}$ and $\Delta {M^{\prime }}_{-}$ denote the ROEs of the observer satellite before the impulsive maneuver moment, $x_{-}$ and $z_{-}$ denote the relative position components, and $v_{x-}$ and $v_{z-}$ denote the relative velocity components before the maneuver moment. In addition, $D_{+}$, $\Delta e_{x+}$, $\Delta e_{y+}$, and ${\Delta M^{\prime }}_{+}$ denote the ROEs after the maneuver moment, and $x_{+}$, $z_{+}$, $v_{x+}$, and $v_{z+}$ denote the relative position and velocity components after the maneuver moment. From Equations (2) and (3), we have

$$\left\{\begin{array}{l}{x}_{-}=a_{\mathrm{ref}}\left(\Delta {M^{\prime }}_{-}+2(\Delta e_{x-}\sin u-\Delta e_{y-}\cos u)\right)+a_{\mathrm{ref}}{D}_{-}t\\ z_{-}=a_{\mathrm{ref}}\left(\Delta e_{x-}\cos u+\Delta e_{y-}\sin u+2D_{-}/(3n_{\mathrm{ref}})\right)\\ v_{x-}=2a_{\mathrm{ref}}{n}_{\mathrm{ref}}(\Delta e_{x-}\cos u+\Delta e_{y-}\sin u)+a_{\mathrm{ref}}{D}_{-}\\ v_{z-}=a_{\mathrm{ref}}{n}_{\mathrm{ref}}(-\Delta e_{x-}\sin u+\Delta e_{y-}\cos u)\end{array}\right.$$

(25)

$$\left\{\begin{array}{l}{x}_{+}=a_{\mathrm{ref}}\left(\Delta {M^{\prime }}_{+}+2(\Delta e_{x+}\sin u-\Delta e_{y+}\cos u)\right)+a_{\mathrm{ref}}{D}_{+}t\\ z_{+}=a_{\mathrm{ref}}\left(\Delta e_{x+}\cos u+\Delta e_{y+}\sin u+2D_{+}/(3n_{\mathrm{ref}})\right)\\ v_{x+}=2a_{\mathrm{ref}}{n}_{\mathrm{ref}}(\Delta e_{x+}\cos u+\Delta e_{y+}\sin u)+a_{\mathrm{ref}}{D}_{+}\\ v_{z+}=a_{\mathrm{ref}}{n}_{\mathrm{ref}}(-\Delta e_{x+}\sin u+\Delta e_{y+}\cos u)\end{array}\right.$$

(26)

Noting that the relative velocity can be changed instantaneously by the impulsive maneuver and the relative position is unchanged at the maneuver moment, we have

$$\left\{\begin{array}{l}0=x_{-}-x_{+}\\ 0=z_{-}-z_{+}\\ \Delta v_x=v_{x+}-v_{x-}\\ \Delta v_z=0=v_{z+}-v_{z-}\end{array}\right.$$

(27)

Accordingly, by substituting Equations (25) and (26) into Equation (27), the in-plane along-track impulsive control equations are as follows [9]:

$$\left\{\begin{array}{l}0=d\Delta M^{\prime }+2\left(d\Delta e_x\sin u-d\Delta e_y\cos u\right)+dDt\hfill \\ 0=\left(d\Delta e_x\cos u+d\Delta e_y\sin u\right)+2dD/\left(3n_{\mathrm{ref}}\right)\hfill \\ \Delta v_x=2a_{\mathrm{ref}}{n}_{\mathrm{ref}}\left(d\Delta e_x\cos u+d\Delta e_y\cos u\right)+a_{\mathrm{ref}}dD\hfill \\ 0=a_{\mathrm{ref}}{n}_{\mathrm{ref}}\left(-d\Delta e_x\sin u+d\Delta e_y\cos u\right)\hfill \end{array}\right.$$

(28)

Here, $dD=D_{+}-D_{-}$, $d\Delta e_x=\Delta e_{x+}-\Delta e_{-}$, $d\Delta e_y=\Delta e_{y+}-\Delta e_{-}$, and $d\Delta M^{\prime }=\Delta {M^{\prime }}_{+}-\Delta {M^{\prime }}_{-}$ represent the changes in $D$, $\Delta e_x$, $\Delta e_y$, and $\Delta M^{\prime }$ before and after the maneuver, respectively. Solving Equation (28) yields

$$\left\{\begin{array}{l}dD=-{\displaystyle \frac{3}{a_{\mathrm{ref}}}}\Delta v_x\hfill \\ d\Delta e=\left[\begin{array}{c}d\Delta e_x\\ d\Delta e_y\end{array}\right]={\displaystyle \frac{2\Delta v_x}{a_{\mathrm{ref}}{n}_{\mathrm{ref}}}}\left[\begin{array}{c}\cos u\\ \sin u\end{array}\right]\hfill \\ d\Delta M^{\prime }={\displaystyle \frac{3\Delta v_x}{a_{\mathrm{ref}}}}t\hfill \end{array}\right.$$

(29)

It can be seen that a single along-track velocity impulse can only eliminate one deviation component while inevitably affecting the others. Therefore, to independently correct a single ROE deviation while minimizing changes to others, a two-impulse strategy is preferred. The details of the strategy for correcting ROEs using along-track impulses are provided in Appendix A.

3.3. Cruising Velocity Control Strategy

When only the cruising velocity constraint is considered, let the forward and reverse cruising velocities of the round-trip orbit be denoted as $V_{D1}$ and $V_{D2}$, respectively. At the transition point from forward to reverse cruising, the required change in cruising velocity is

$$\Delta V_D=V_{D2}-V_{D1}$$

(30)

This change corresponds to a required adjustment in the relative drift rate, as follows:

$$dD={\displaystyle \frac{\Delta V_D}{a_{\mathrm{ref}}}}$$

(31)

According to the impulsive control laws, applying a single tangential impulse at either cruising boundary to adjust the relative drift rate will also result in a change in the relative eccentricity vector, thereby altering the cruising radius. To avoid this, two velocity impulses with equal magnitude and the same direction can be used. This method allows for an adjustment of the relative drift rate without affecting the eccentricity vector; that is, it enables control of the cruising velocity without changing the cruising radius.

Denote the two velocity impulses as $\Delta v_{xD1}$ and $\Delta v_{xD2}$, where $\Delta v_{xD1}=\Delta v_{xD2}$. Then, based on Equation (31) and Appendix A, the first impulse is

$$\Delta v_{xD1}=-{\displaystyle \frac{\Delta V_D}{6}}$$

(32)

This impulse is applied when the observer satellite reaches the cruising boundary. Considering two boundary controls procedure in one round-trip cruising cycle, the total impulse required for one round-trip cruising cycle is

$$\Delta v=2\left(\left|\Delta v_{xD1}\right|+\left|\Delta v_{xD2}\right|\right)={\displaystyle \frac{2\left|\Delta V_D\right|}{3}}={\displaystyle \frac{2\left|V_{D1}-V_{D2}\right|}{3}}$$

(33)

Since $V_{D1}$ and $V_{D2}$ have opposite signs, $\Delta v$ is given by

$$\Delta v={\displaystyle \frac{2\left(\left|V_{D1}\right|+\left|V_{D2}\right|\right)}{3}}$$

(34)

3.4. Combined Cruising Velocity and Cruising Radius Control Strategy

In the case that both the cruising velocity and the cruising radius are constrained, denote the forward and reverse cruising velocities as $V_{D1}$ and $V_{D2}$, respectively, and the corresponding cruising radius as $R_{r1}$ and $R_{r2}$, respectively.

From Equation (24), the changes in the relative drift rate and the magnitude of the relative eccentricity vector are governed by

$$\left\{\begin{array}{l}dD={\displaystyle \frac{V_{D2}-V_{D1}}{a_{\mathrm{ref}}}}\hfill \\ d‖\Delta e‖={\displaystyle \frac{1}{a_{\mathrm{ref}}}}\left(R_{r2}-R_{r1}-{\displaystyle \frac{2\left(\left|V_{D2}\right|-\left|V_{D1}\right|\right)}{3n_{\mathrm{ref}}}}\right)\hfill \end{array}\right.$$

(35)

From Equation (29), the change in the relative eccentricity vector resulting from a single tangential impulse is

$$‖d\Delta e‖={\displaystyle \frac{2\left|\Delta v_x\right|}{a_{\mathrm{ref}}{n}_{\mathrm{ref}}}}$$

(36)

given that

$$d‖\Delta e‖=‖\Delta e+d\Delta e‖-‖\Delta e‖$$

(37)

and

$$‖d\Delta e‖\ge ‖\Delta e+d\Delta e‖-‖\Delta e‖$$

(38)

The equal sign holds only when $\Delta e$ and $d\Delta e$ are collinear. Combining Equations (36)–(38), it is known that when $\Delta e$ and $d\Delta e$ are collinear, the minimum velocity pulse is required to control the magnitude of the relative eccentricity vector change. From Equation (35), the composite control of cruising velocity and cruising radius yields a relative orbital element change as follows:

$$\left\{\begin{array}{l}dD={\displaystyle \frac{V_{D2}-V_{D1}}{a_{\mathrm{ref}}}}\hfill \\ d\Delta e_x=d‖\Delta e‖\cdot {\displaystyle \frac{\Delta e_x}{‖\Delta e‖}}\hfill \\ d\Delta e_y=d‖\Delta e‖\cdot {\displaystyle \frac{\Delta e_y}{‖\Delta e‖}}\hfill \end{array}\right.$$

(39)

At the moment when the sub-satellite longitude of the observer spacecraft reaches either cruising boundary, a pair of tangential impulses are applied to simultaneously eliminate the deviations in the relative eccentricity vector and the relative drift rate. According to Section 3.2 and the derivations in Appendix A, the combined control strategy for cruising velocity and radial cruising radius is summarized in

Table 1. Combined control strategy for cruising velocity and radial cruising radius.

Pulse Sequence	Impulse	Calculation Formula	Application Timing
1	$\Delta v_{x1}$	$\Delta v_{xe1}+\Delta v_{xD1}$	$t_1=t{|}_{u=u_1}$
2	$\Delta v_{x2}$	$\Delta v_{xe2}+\Delta v_{xD2}$	$t_2=t_1+{\displaystyle \frac{\pi }{n_{\mathrm{ref}}}}$

.

Here, $u_1=\mathrm{a}\tan 2\left(\Delta e_y,\Delta e_x\right)$ or $u_1=\mathrm{a}\tan 2\left(\Delta e_y,\Delta e_x\right)+\pi$. Each impulse component can be expressed as

$$v_{xe1}=\left\{\begin{array}{l}{\displaystyle \frac{a_{\mathrm{ref}}{n}_{\mathrm{ref}}}{4}}d‖\Delta e‖u_1=\mathrm{a}\tan 2\left(\Delta e_y,\Delta e_x\right)\hfill \\ -{\displaystyle \frac{a_{\mathrm{ref}}{n}_{\mathrm{ref}}}{4}}d‖\Delta e‖u_1=\mathrm{a}\tan 2\left(\Delta e_y,\Delta e_x\right)+\pi \hfill \end{array}\right.$$

(40)

$$\Delta v_{xe2}=-\Delta v_{xe1}$$

(41)

$$\Delta v_{xD1}=-{\displaystyle \frac{V_{D2}-V_{D1}}{6}}$$

(42)

$$\Delta v_{xD2}=\Delta v_{xD1}$$

(43)

Thus, the speed pulse size for one cycle cruising boundary control is

$$\Delta v=2\left(\left|\Delta v_{x1}\right|+\left|\Delta v_{x2}\right|\right)$$

(44)

3.5. In-Plane, Full-Parameter Control Strategy

In the case of in-plane, full-parameter control, the round-trip cruising orbit is subject to simultaneous constraints on cruising velocity $V_D$, cruising radius $R_r$, initial in-plane phase angle ${\varphi }_0$, and spiral vertex position $L$. Denote the orbit parameters before and after control as $V_{D1}$, $R_{r1}$, ${\varphi }_{01}$, $L_1$ and $V_{D2}$, $R_{r2}$, ${\varphi }_{02}$, $L_2$, respectively. According to Equation (24), a set of relative orbital elements before and after control can be obtained as follows: $\left[D_1,\Delta e_{x1},\Delta e_{y1},\Delta i_{x1},\Delta i_{y1},{\Delta M_1}^{\prime }\right]$ and $\left[D_2,\Delta e_{x2},\Delta e_{y2},\Delta i_{x2},\Delta i_{y2},{\Delta M_2}^{\prime }\right]$. Then, the change in ROEs due to control is

$$\left\{\begin{array}{c}dD=D_2-D_1\\ d\Delta e_x=\Delta e_{x2}-\Delta e_{x1}\\ d\Delta e_y=\Delta e_{y2}-\Delta e_{y1}\\ d\Delta M^{\prime }=\Delta {M_2}^{\prime }-\Delta {M_1}^{\prime }\end{array}\right.$$

(45)

Subsequently, the strategy of comprehensively eliminating the deviation of ROEs using three along-track pulses can be adopted, and the control strategy can be calculated according to

Table A1. Three-impulse tangential ROE correction strategy.

Impulse Sequence	Impulse	Calculation Formula	Execution Time
1	$\Delta v_{x1}$	$\Delta v_{xe1}+\Delta v_{xD1}+\Delta v_{xM1}$	$t_1=t{|}_{u=u_1}$
2	$\Delta v_{x2}$	$\Delta v_{xe2}+\Delta v_{xD2}$	$t_2=t_1+{\displaystyle \frac{\pi }{n_{\mathrm{ref}}}}$
3	$\Delta v_{x3}$	$\Delta v_{xM2}$	$t_3=t_2+{\displaystyle \frac{\pi }{n_{\mathrm{ref}}}}$

in Appendix A.

4. Numerical Examples

In this section, the effectiveness of the proposed control strategies for round-trip spiral cruising orbits are validated through numerical simulations, including the cruising velocity control strategy, combined control of cruising velocity and cruising radius, and the in-plane, full-parameter control strategy. Moreover, the efficiency of the proposed control method is compared with previous works [7].

4.1. Simulation Setup

Set the epoch time as 20 August 2021 04:00:00.000 UTC, corresponding to the simulation’s initial time. The geostationary reference satellite is assumed to have a fixed sub-satellite longitude of −101° and an inclination of 0°. In addition, the initial orbital elements of the observer satellite are listed in

Table 2. Initial orbital elements of the observer satellite.

Parameter	Value
$a$/km	42,143.0073
$e$	6.8393 × 10−4
$i$/deg	0
$\Omega$/deg	0
$\omega$/deg	227.2776
$M$/deg	359.8645

.

The cruising region is defined by the following boundary longitudes: $lon1=$−101.7 deg, $lon2=$−100.3 deg. The simulation ends when the cruising satellite completes a cycle of the round trip within the defined cruising region, returning to its initial sub-satellite longitude.

4.2. Simulation of Cruising Velocity Control Strategy

In this subsection, the cruising velocity control strategy is validated. Only the cruising velocity constraint is considered; the parameters of the round-trip cruising orbit are given in

Table 3. Parameters of the round-trip cruising orbit.

Parameter	Forward Cruising	Backward Cruising
$V_D$/(km/day)	200	−200
Boundary longitude	−101.7 deg	−100.3 deg

.

According to the two-impulse control strategy described in Section 3.3, the trajectory of the observer satellite is depicted in Figure 5. The corresponding time and velocity impulses at each boundary for the cruising velocity control are given in

Table 4. Velocity impulse strategy for cruising velocity control.

Boundary Longitude (deg)	Time (s)	Impulse (m/s)
−100.3	260,925.2	0.772
	304,007.3	0.772
−101.7	736,786.6	−0.772
	779,868.6	−0.772

. As can be seen in Figure 5, the blue line represents the relative trajectory of the observer satellite with respect to the reference satellite (the black dot mark), and the triangle marks represent the position where the velocity impulses are conducted. The star marks represent the virtual reference satellite at the cruising boundaries. The distance between two boundaries is about 1000 km. It can be seen that the observer satellite first cruises forward to the boundary longitude of −100.3 deg. Accordingly, two velocity impulses are implied at this boundary so that the observer satellite can cruise backward to the other boundary longitude of −101.7 deg. Subsequently, two velocity impulses are conducted again around −101.7 deg, and the cruising direction turns forward again.

Moreover, as can be seen from Table 4, the magnitude of each velocity impulse is only 0.772 m/s, and the total velocity increment in a cycle of round-trip cruising is only 3.088 m/s. The simulation results demonstrate that the cruising trajectory can be effectively controlled through the proposed cruising velocity strategy. And only a small velocity increment of 3.088 m/s is required to enable the observer to cruise backwards and forwards in a GEO region of about 1000 km.

4.3. Simulation of Combined Velocity and Radius Control Strategy

In this subsection, the combined cruising velocity and radius control strategy is validated. The cruising velocity $V_D$ and cruising radius $R_r$ for the round-trip spiral cruising orbit are listed in

Table 5. Composite control parameters for cruising velocity and cruising radius.

Parameter	Forward Cruising	Backward Cruising
$V_D$/(km/day)	200	−200
$R_r$/km	50	70
Boundary longitude	−101.7 deg	−100.3 deg

.

Based on the combined control strategy detailed in Section 3.4, the trajectory of the observer satellite is depicted in Figure 6. It can be seen that the observer satellite cruises between the two boundary longitudes (−100.3 deg and −101.7 deg) after two velocities at each boundary. In addition, the cruising radius of the forward and backward trajectory satisfies the constraints of 50 and 70 km, respectively, revealing that the proposed method is effective for controlling both the cruising velocity and radius.

Moreover, the velocity impulses and the corresponding time and positions at each boundary for the control strategy are given in

Table 6. Velocity impulse strategy for combined cruising velocity and radius control.

Boundary Longitude (deg)	Time (s)	Impulse (m/s)
−100.3	301,379.5	0.407
	344,461.6	1.136
−101.7	861,830.7	−1.137
	904,912.7	−0.407

. As can be seen, the two velocity impulses at each boundary longitude are not equal to each other due to the combined control of cruising velocity and radius. In addition, for both control boundaries, the time duration between two impulses is about 43,000 s, which is about half of the orbital period. The total velocity increment in a round-trip cruise cycle is about 3.086 m/s. The simulation results demonstrate that only a small velocity increment is required to enable the observer to travel between the boundaries with a specific cruising velocity and radius.

4.4. Simulation of In-Plane, Full-Parameter Control Strategy

In this subsection, the in-plane, full-parameter control strategy is validated. The geometric parameters for the round-trip spiral cruising orbit are presented in

Table 7. Composite control parameters for in-plane, full-parameter control.

Parameter	Forward Cruising	Backward Cruising
$V_D$/(km/day)	200	200
$R_r$/km	50	70
${\varphi }_0$/rad	0	π/3
$L$/km	0	20
Boundary longitude	−101.7 deg	−100.3 deg

.

According to the in-plane, full-parameter control strategy outlined in Section 3.5, the trajectory of the observer satellite is shown in Figure 7. As can be seen in Figure 7, the observer satellite cruises between the two boundary longitudes after three velocities at each boundary. Moreover, the corresponding impulse conduction details are provided in

Table 8. Pulse application strategy (in-plane, full-parameter control).

Boundary Longitude (deg)	Time (s)	Impulse (m/s)
−100.3	276,436.8	1.552
	319,518.8	0.171
	362,600.9	0.179
−101.7	696,466.4	0.694
	739,548.5	0.394
	782,630.5	0.455

. Take the impulse strategy at the boundary of −100.3 deg as an example. The first velocity impulse of 1.552 m/s is implied at the time of 276,436.8 s. Accordingly, the second impulse of 0.171 m/s is conducted after about half of the orbital period with respect to the first impulse. Then, the third impulse of 0.179 is conducted continually after half of the orbital period. The four cruising parameters are successfully controlled by the three impulses. And the total velocity increment required in one round-trip cycle is only 3.445 m/s.

Subsequently, the control process at the two cruising boundaries (−101.7 and −100.3 deg) is visualized in Figure 8 and Figure 9, respectively. Taking Figure 9 as an example, when the sub-satellite longitude of the observer satellite arrives at the boundary −100.3 deg, a virtual reference satellite (the black dot in Figure 9) is created. Based on the backward cruising orbit parameters in Table 7 ($V_D$ = 200 km/day, $R_r$ = 70 km, ${\varphi }_0=\pi /3$, $L$ = 20 km), the designed backward cruising trajectory is generated. As can be seen, the green dashed line represents the instantaneous ellipse, and the blue ‘x’ mark represents the position corresponding to the initial phase angle of the observer satellite in the instantaneous ellipse, which is consistent with the constraint ${\varphi }_0=\pi /3$. In addition, the triangle marks represent the positions where the control velocity impulses are executed, and the blue solid line and red dash-dotted line represent the actual and designed cruising trajectory of the observer satellite. In can be seen that after three velocity impulses, the two trajectories are finally aligned with each other, which means that the observer satellite is inserted into the desired cruising orbit. These results verify the effectiveness of the in-plane, full-parameter control strategy.

4.5. Comparison with Previous Works

In this subsection, to further evaluate the effectiveness and efficiency of the proposed cruising orbit design and control method, it is compared with the particle swarm optimization cruising orbit design method (PSOCODM) by Zheng, L. et al. [7] for multiple GEO target satellite inspections.

The simulation parameters are presented as follows: The initial sub-satellite longitude of the observer satellite is $l_{\mathrm{obs}}$ = 90.0 deg. Three GEO satellites are taken as the targets which are inspected by the observer satellite through the cruising orbit. Denote the three target satellites as targets 1, 2, and 3, respectively. The corresponding sub-satellite longitudes are $l_1$ = 90.2, $l_2$ = 90.5, and $l_3$ = 90.8 deg, respectively. In addition, the closest range between the observer and target satellites is $d_{\min }$= 10 km, and the simulation duration is $t_{\mathrm{total}}$ = 10.9 days.

In this paper, the idea for determining the cruising orbit of the observer satellite is as follows: First, the inspection time for the three target satellites are given by $t_i={\displaystyle \frac{l_i-l_{\mathrm{obs}}}{l_3-l_{\mathrm{obs}}}}\cdot t_{\mathrm{total}}$. Then, three cruising orbits are designed for the inspection of the three target satellites, namely nominal cruising orbits 1, 2, and 3, respectively. For the i-th nominal cruising orbit, assuming that the minimum range constraint is satisfied at time $t_i$, the corresponding geometric parameters (including the cruising velocity $V_D$, the cruising radius $R_r$, the location of the spiral vertex $L$, and the initial in-plane phase angle ${\varphi }_0$) are determined according to the position of the target satellites. Next, for each nominal cruising orbit, the corresponding ROEs can be obtained according to their geometric parameters and Equation (24). Finally, the three-impulse maneuver strategy in Section 3.5 is utilized for the observer satellite to travel between the nominal cruising orbits. As a result, the three target satellites can be finally inspected by the observer satellite.

Subsequently, the cruising trajectories of the observer satellite based on the proposed method and PSOCODM are shown in Figure 10. The corresponding velocity impulse execution details are provided in

Table 9. Pulse application strategy comparison.

Result	This Paper		PSOCODM
Maneuver Strategy	Time (s)	Impulse (m/s)	Time (s)	Impulse (m/s)
	24,148.8	−0.604	24,178.2	−0.069
	67,348.8	0.391	61,048.2	−0.387
	110,548.8	0.242	175,248.2	0.250
	261,798.1	−0.078	230,798.3	−0.303
	304,998.1	0.155	378,598.3	−0.141
	348,198.1	−0.078	412,014.3	0.238
	611,547.9	−0.078	625,647.4	−0.111
	654,747.9	0.155	675,827.4	0.248
	697,947.9	−0.078	686,527.4	−0.136
Total Impulse (m/s)	1.859		1.783
Run time (s)	0.01		1.2

. From Figure 10, it can be seen that the cruising trajectories obtained by the two methods are basically similar, and both trajectories satisfy the minimum range constraint $d_{\min }$ = 10 km. From Table 9, it can be seen that the total velocity impulses of the proposed method and PSOCODM are 1.859 and 1.783 m/s, respectively, and the corresponding CPU run time is 0.01 and 1.2 s, respectively. The relative deviation between the impulses of the two methods is only 4.3%, while the run time of the proposed method is only 0.83% of that of PSOCODM. These results demonstrate the efficiency advantage of the proposed cruising orbit design and control method with respect to PSOCODM.

5. Conclusions

This paper presents a novel spiral cruising orbit design method based on a set of in-plane geometric parameters and establishes a map between these parameters and the ROEs. Moreover, three analytical and efficient control strategies are developed for round-trip spiral cruising orbit control, including the cruising velocity control strategy, combined cruising velocity and radius control strategy, and the in-plane, full-parameter control strategy. Simulation results demonstrate that the geometry of the GEO spiral cruising orbit can be intuitively described and designed using the proposed in-plane geometric parameters. In addition, only a velocity impulse of 1.5 m/s is required in the boundary longitude to enable the observer satellite to travel in the GEO region at a velocity of about 200 km/day. Moreover, the proposed control strategies are effective for the round-trip spiral cruising orbit control, while reducing over 99% of the run time compared with previous works. Due to their intuitiveness and simplicity, the proposed spiral cruising orbit design and control methods have significant application value for future GEO target monitoring and space situational awareness.