As the number of space objects (SO) increases, collision avoidance problem in the rendezvous tasks or re-constellation of satellites with SO has been paid more attention, and the dangerous area of a possible collision should be derived. In this paper, a maneuvering method is proposed for avoiding collision with a space debris object in the phasing orbit of the initial optimal solution. Accordingly, based on the plane of eccentricity vector components, relevant dangerous area which is bounded by two parallel lines is formulated. The axises of eccentricity vector system pass through the end of eccentricity vector of phasing orbit in the optimal solution, and orientation of axis depends on the latitude argument where a collision will occur. The dangerous area is represented especially with the graphical dialogue, and it allows to find a compromise between the SO avoiding and the fuel consumption reduction. The proposed method to solve the collision avoidance problem provides simplicity to calculate rendezvous maneuvers, and possibility to avoid collisions from several collisions or from “slow” collisions in a phasing orbit, when the protected spacecraft and the object fly dangerously close to each other for a long period.

Currently, Spacecraft rendezvous has been considered as one of the key technologies for on-orbit service, including space debris removing, on-orbit assembly, spacecraft repairing, and on-orbit refueling [

Also, sliding mode control is widely used in spacecraft rendezvous considering the collision avoidance. Kasaeian et al. [

Considering the optimal problem of fuel consumption, Cao et al. [

Since the calculation of special single impulse maneuver which thrust the protected spacecraft out of the dangerous area is quite simple, the main attention in these work is paid to determining the size of the dangerous area in which could be a collision [

The difference of this work is that it does not consider special avoidance maneuvers, but seeks such a solution to the rendezvous problem in order to evade a possible collision in a phasing orbit if a danger exists. The second difference is that the hazardous area (the position of which are considered known, since that they can be determined by the algorithms given in the above papers) is considered not in the traditional coordinate space, but on the plane of the components of the eccentricity vector. In addition, the proposed method to solve the problem makes it easy to calculate rendezvous maneuvers that provide avoidance from several collisions or from “slow” collisions in a phasing orbit, when the protected spacecraft and the SD object fly dangerously close to each other for a long time.

The need seeking for such a solution to the rendezvous problem can be explained by several reasons. For example, at the spacecraft “Soyuz” and “Progress”, after performing the first two maneuvers (in 2–3 orbits) from a series of four connected maneuvers with a two-day rendezvous scheme, beginning from deaf turns where it is impossible to conduct maneuvers. If the phasing orbit formed by the first two maneuvers is dangerous in terms of collision with SD objects on it, then changing it with a special velocity and thereby avoiding the collision is no longer possible. Considering that, the only way out for solving this problem can be described as follows: firstly, the orbit after applied the first two impulses should be predicted under the optimal scheme, if in this orbit there is a possibility of a collision with SD, constraints should be added to find a solution to the rendezvous that the spacecraft will not pass through the danger zone after the first two impulses.

Moreover, even if it is predicted that no collision will occur, recalculation for rendezvous maneuvers in order to form a safe phasing orbit allows to reduce both the number of maneuvers performed and the total fuel costs.

The paper is organized as follows: In Section 2, mathematical description of the problem to be solved is introduced. The specific methodology and solution is presented in Section 3, which provide the optimal solutions in plane of eccentricity vector in different maneuvers. Conclusions are summarized in the end of the paper.

Assume that the chasing spacecraft and the target spacecraft are located in close near-circular orbits (or the target spacecraft as a given point in the final orbit to which the chasing spacecraft should be transferred during the formation of satellite systems or groups). Then we can use the linearized equations to calculate the parameters of maneuvers, influence of earth’s non-spherical perturbation and atmospheric drag on the spacecraft here are not considered, and instant change in spacecraft velocity is assumed. Using the famous iterative method [

where _{x} = _{f}_{y} = _{f}_{f}_{0})/_{0}, _{f}_{0}), _{0}/_{0}, _{0}/_{0}, _{0}, _{0}, _{0}.

Here, “_{f}_{0} are the eccentricities of the final orbit and initial orbit; _{f}_{0} are the semi-major axis of orbits; _{x}_{f}_{0} is the time, when spacecraft moving in initial orbit, the projection of radius vector on the plane of final orbit passing through the given point; _{0} is the deviation distance between the initial orbit and the final orbit plane of the spacecraft at time _{0}; _{z0} is the lateral relative velocity at this moment; _{0}, _{0} (_{0} = _{f}

A multi-turn flight scheme is also assumed in this paper. This scheme has several advantages as shown below: Firstly, when the total characteristic velocity (TCV) of the rendezvous problem is equal to the TCV of the solution to transition problem (there is no restriction on flight time.), in this case a fairly wide range of the initial phase can be considered (phase is the difference in the latitude argument

where _{1}, _{N}

For ensuring rendezvous problems of spacecraft by maneuvers, it is necessary to determine the parameters of impulses _{ri}, _{ti},

Under the constraints defined by

In this process, the protected area ranges are also determined. Since the rendezvous problem is solved and it is impossible to avoid collisions by changing position along the orbit, the minimum required deviation along the radius

Here it is advisable to choose a four-impulses scheme to solve this problem, two impulses are applied at the first maneuvering interval, and two impulses are applied at the second maneuvering interval. The proposed approach is described in detail in the following section.

The magnitude of semi-major axis of the phasing orbit, which should be formed by the impulses of the first maneuvering interval, is uniquely determined from

where

Thus, since the semi-major axis of phasing orbit is fixed, it is possible to obtain a safe orbit only by deliberately changing the eccentricity of phasing orbit.

On the _{x}_{y}_{0}, argument of the orbit pericenter is _{0}, and a collision with SD occurs at the argument of latitude _{0}. The radius of orbit at this point is calculated by the formula as follows:

In order to avoid collision with SD on the latitude argument _{0}, the radius of orbit at this moment should be increased to

Outer and inner boundary of the prohibited region can be built by

In _{0} = 10,000 km, eccentricity _{0} = 0.3, and latitude argument _{0} = 30

As is shown in _{0},

_{0} = 100_{0} = 210

As noted above, the actual value of _{0} = 6660 km, eccentricity _{0} = 0.005, latitude argument _{0} = 358_{0} = 135_{x0}, _{y0})), boundary of the prohibited area is approximately linear. Gap of prohibited region is 2

Based on this scheme, analyses about the position of prohibited region depending on the argument of latitude at which collision occurs can be carried out. In the same coordinate plane, several regions under study that correspond to the possibility of collisions at angles _{0} = 0_{0} changes, prohibited region rotates around point X (_{x0}, _{y0}).

Consider an example for the Russian spacecraft “Soyuz” and orbit station “Progress” based on the results obtained previously. In order to ensure a flight to the station vicinity for these spacecraft, a four impulses rendezvous problem is initially solved. All the impulses have transversal components, and lateral components are performed in the first two impulses.

For near circular orbit motion, as described follows from _{ti}_{x}_{y}_{ti}_{x}

From _{z}_{zi}_{z}

Thus, in multi-impulse solution of the problem in coordinates _{x}_{y}_{z}_{t1}, _{y2} , _{z1} are positive, _{z2} is negative.

The angle of impulse application is optimal for correcting the deviation of eccentricity vector. The angle of application of the lateral impulse component is optimal for correcting the deviation in the orientation of orbit plane (the impulse is applied on the line of nodes). These angles are calculated by the following formulas:

A four-impulse solution corresponds to a four-link broken line starting in point _{tI}_{1} = 2|_{tI}_{2} = 2|_{tII}_{tII}

At the optimal initial phase for the optimal transfer flight between non-intersected orbits with radius _{1} = 2|_{tI}_{2} = 2|_{tII}_{tI}_{tII}_{1} and _{2} circles (see

Solution set with the same

With the adopted maneuvering scheme, it is possible to effectively avoid collision if the approaching point is close to the ascending or descending node of the orbit (_{0} = 0_{0} = 180_{x0}, _{y0}) corresponding to the eccentricity vector of the waiting orbits, obtained after the second impulse (see

In order to obtain a safe solution, the eccentricity vector of the phasing orbit must be derived from the prohibited region, which is easy to realize by changing the angles of application of the first two impulses; and in order to obtain a solution in which _{1} and _{2}. A solution that satisfies both requirements is shown in

Consider a more complicated case, when the point of possible collision is as far from the equator as possible (_{0} = 90_{0} = 270

Without changing the maneuvering scheme (changing the angle of application of the third or fourth pulses), it is impossible to avoid a collision, because the end of the eccentricity vector of the phasing orbit (_{x0}, _{y0}) should belong to a straight line passing through the middle of the prohibited area.

As shown in

The area corresponding to collisions at different latitude arguments can also be considered as an option, when in a phasing orbit a collision with several SD objects at different points of the orbit. In slow collisions, the possibility of a collision does not exist at a point, but in a significant range of latitude arguments. The prohibited zone will be restricted to the boundaries line between the two extreme points.

In the slow rendezvous situation, the possibility of collision does not exist at a point, but over a wide range of latitude arguments. The prohibited lane will be limited by straight lines which are the areas boundaries for the two extreme points of the range of possible collisions. An approximate view of the danger area for such a case when a dangerous approach is possible in the latitude range from 60

It should be noted that if the eccentricity vector is derived from this area, then the risk of collision in the area from 240

The solution to the multi-impulse rendezvous problem for a given class of orbits is usually sought using the simplex method. We can offer two other methods for solving this problem, which have shown their effectiveness [

The angles of application of the first two impulses of velocity

It is most efficient to obtain the desired solution using a graphical dialogue with a problem. In this scheme, based on the analysis of the above presented, angles of application of any of the velocity impulses can be changed, and thus a compromise solution that displays the eccentricity vector of phasing orbit from the danger area can be obtained. Thus, a compromise solution can be obtained that removes the eccentricity vector of the phasing orbit from the dangerous region, but leaves it in the region of optimal solutions. A solution can be found in which point

In this paper a fundamentally new approach to calculate maneuvers for avoiding a collision with space debris is proposed, an effective algorithm is proposed for solution changes to the rendezvous problem, which avoids the collision in phasing orbit of the initial optimal solution. It is demonstrated that the protected area from object must be derived is built on the plane of eccentricity vector components. The prohibited region is bounded by two parallel lines, the axis of this region passes through the end of the eccentricity vector of the phasing orbit of the optimal solution. The orientation of the axis depends on the latitude argument at which the collision may occur. The sizes of this region are determined, depending on

The prohibited region and the region of optimal solutions are depicted on the same plane, which allows using a graphical dialogue when solving the rendezvous problem to find a compromise between the desire to avoid the dangerous object as far as possible and to reduce the delta costs that grow with distance from the optimal solution. The proposed approach to solving the problem also makes it quite simple to calculate rendezvous maneuvers, which also provide evasion from several collisions or from “slow” collisions in a phasing orbit, when the protected spacecraft and the SD object fly dangerously close to each other for a long time. The use of the constructed area will allow solving the problem of collision avoidance of a satellite group (Satellite Formation Flying) with a group of space objects.