A New ShapeBased MultipleImpulse Strategy for Coplanar Orbital Maneuvers
Abstract
A new shapebased geometric method (SBGM) is proposed for generation of multiimpulse transfer trajectories between arbitrary coplanar oblique orbits via a heuristic algorithm. The key advantage of the proposed SBGM includes a significant reduction in the number of design variables for an impulse orbital maneuver leading to a lower computational effort and energy requirement. The SBGM generates a smooth transfer trajectory by joining a number of confocal elliptic arcs such that the intersections share common tangent directions. It is proven that the wellknown classic Hohmann transfer and its bielliptic counterpart between circular orbits are special cases of the proposed SBGM. The performance and efficiency of the proposed approach is evaluated via computer simulations whose results are compared with those of optimal Lambert maneuver and traditional methods. The results demonstrate a good compatibility and superiority of the proposed SBGM in terms of required energy effort and computational efficiency.
keywords:
Orbital maneuver, Impulsive transfer, Shapebased geometric method, Coplanar orbits1 Introduction
Multipleimpulse orbital maneuvers have been widely utilized in many space missions. The most wellknown multiple impulsive maneuver for coplanar orbits was first proposed by Walter Hohmann (01) in year 1925. Then, Lion et al. (02) have utilized the primer vector for optimal impulsive timefixed approaches. The primer vector has been implemented for both circular and elliptical orbits (03; 04). In addition, the effect of path and thrust constraints on impulsive maneuvers have been investigated (05; 06), and the optimal impulsive transfer in presence of time constraints is driven (07). Further, various heuristic optimization algorithms such as the genetic algorithm (GA) (08), the particle swarm optimization (PSO) (09; 09p), and the simulated annealing (09pp) have been proposed to design an optimal impulsive trajectory (also the reader can refer to (09ppp) and references therein for more details). However, the Lambertâs approach has traditionally been utilized for conic trajectories between any two spatial points in space within a specified time interval. Due to vast applicability of the Lambert method (10), multiplerevolution (11; 12; 13), perturbed (14; 15) and optimized Lambert solutions (16) have been subsequently developed for the multiple impulsive maneuvers. Relative Lambert solutions are also developed (17) and utilized for impulsive rendezvous of spacecraft (18). Moreover, the effect of orbital perturbations including the Earth oblateness (19; 20) and the third body (21) on the impulsive trajectories has been investigated to make more efficient transfers. The impulsive orbital transfers have also been addressed against the lowthrust trajectories (22), and it is shown that the impulsive transfer can be viewed as the limit of the lowthrust transfer when its number of revolutions increases.
Impulse vectors can be applied perpendicular to the orbit plane for plane change maneuvers where in (22p), analytic fuel optimal solutions are provided. Impulsive maneuvers sometimes are applied tangentially at the origin and/or destination points. Altman et al. (23) have proposed the hodograph theory to generate tangent maneuvers, which can improve the rendezvous safety (24). Zhang et al. (25) have analytically analyzed the existence of tangent solutions between elliptical orbits, followed by further study on utility of twoimpulse tangent orbital maneuvers (26; 27). In addition, the reachable domain of spacecraft with just a single tangent impulse is studied in (28; 29).
Shapebased approaches for orbital maneuvers are based on the trajectory geometry. In continuous propulsion systems, some candidate spiral functions can be considered to define the spacecraft trajectory (30; 31; 32). Utilizing these functions makes the resultant optimization problem have fewer parameters in comparison to the main problem. The parameters are selected such that satisfy the problem constraints. An efficient function for the trajectory shaping is obviously a smooth curve that passes through the initial and final positions and also avoids large control efforts.
The current paper presents a novel shapebased geometric method (SBGM) to form a smooth multipleimpulse coplanar transfer trajectory between any two arbitrary spatial points in space. The smoothness property is guaranteed via solving a system of nonlinear equations. Smoothness constraint reduces the number of required design variables for the resultant optimization problem indeed, and lead to a near optimal solution in terms of the energy requirement. Unlike the continuous propulsion case, the proposed approach in the current work provides a final smooth trajectory via joining a number of piecewise elliptic arcs, while observing the continuity requirements. The proposed SBGM reduces the number of design variables and enhances the ability of seeking for global optimum in a bounded region. The applicability and efficiency of the proposed SBGM are verified through comparison with the traditional methods such as the optimal Hohmann (33; 34) and Lambert solutions as well as the single impulse maneuver. In this sense, the main contributions of the current work can be summarized as follows:

The SBGM can construct a smooth trajectory between any pair of coplanar obliqueelliptic orbits with any number of impulses;

The SBGM reduces the number of required design variables in comparison with the existing methods such that it is computationally more efficient;

The optimal solution of SBGM results in significantly lower control effort in comparison to the methods with similar number of design variables.
The remaining parts of this paper are organized as follows: Basic geometric formulation as well as the shapebased impulsive maneuver algorithm are introduced in Section 2. Subsequently, Section LABEL:S:3 is devoted to analyze circletocircle maneuvers for two and threeimpulse maneuvers. Applications of the proposed SBGM to the other types of orbital maneuvers are reported in Section LABEL:S:4. An extension of the SBGM for small variations is formulated in Section LABEL:S:4p. Finally, concluding remarks are presented in Section LABEL:S:5.
2 Geometric Formulation and Requirements
The purpose of this section is to develop the mathematical formulation needed to form an overall smooth trajectory constructed by joining a number of cofocal elliptical arcs. Since the satellite orbits are assumed in a twobody context, these elliptic arcs construct the satellite transfer trajectory after and before each impulse.
Definition 2.1.
A trajectory curve is called smooth, if it has and continuities, i.e., the curves intersect and share identical tangent direction at the intersection points (35).
Remark 2.2.
A smooth trajectory does not necessarily share a common center of curvature at the intersection points ( continuity), unless the eccentricity of the connecting arcs is the same.
The following assumptions are further stipulated for the intended trajectories:
Assumption 2.3.
The trajectory segments are elliptic, with the Earth center as a focus.
Assumption 2.4.
The trajectory segments are smooth at the intersection points.
Considering an Earth centered inertial (ECI) reference coordinate system with the and axes as defined in Fig. 1, an elliptic arc can be defined in depicted polar coordinate system as:
(1) 
where is the local radius, denotes the semimajor axis, and represents the orbital eccentricity. As shown in Fig. 1, the angle defines the rotation of the preapsis with respect to , and represents the orbital true anomaly. In this sense, a set of orbital elements can be introduced as a complete description of the spacecraft position on a planar trajectory. In turn, orbital elements can be constructed just with the use of the radius vector , and its corresponding velocity vector in the reference ECI coordinate system as follows (36):
(2) 
(3) 
(4) 
(5) 
where stands for the Euclidean norm, , , , and , respectively. The slope of a tangent line to the polar curve given in Eq. (1) is
(6) 
In general we assume that a complete orbital maneuver uses impulses to transfer from a point on initial orbit to a final destination point on the final orbit. The initial orbit is indexed by and the final orbit is denoted by . Consequently, the sequential arcs are indexed from to . In addition, it is assumed that the initial and the final orbital elements, i.e., , , , and as well as , , , and are known beforehand.
Obviously, for an impulse maneuver, arcs need to be designed. At the intersection of any two sequential arcs, such as and ,the following equations need to be held to satisfy Assumptions 2.3 and 2.4:
(7) 
(8) 
(9) 
(11) 
As there are intersections, consequently equations will be developed, . While, the unknown parameters are , , and for , plus for , there would be unknowns to be determined. Since, the number of unknowns are more than the number of equations for , some additional feasible assumptions on adjustable parameters, are required. Assume that parameters are adjustable, then the number of unknowns and the equations will be equal that makes a feasible solution.
Proposition 2.5.
Proof.
Suppose that . Therefore, from Eq. (10):
(12) 
(13) 
is satisfied if , , or . If , Eq. (11) leads to and consequently no impulses need to be applied, that is not acceptable. In addition, or , means that the impulse should be exerted at the perigee or the apogee of the first orbit, respectively. Moreover, is satisfied if , , or . If , Eq. (11) leads to that is similarly not acceptable. Finally, or , means that the impulse should be exerted at the perigee or apogee of the third orbit, respectively. This principle can be extended to other adjacent orbits, i.e., , , and as well. ∎
Remark 2.6.
In the numerical computations of the SBGM algorithm, singularities may occur. If the th orbit, , is circular, the determinant of the Jacobian matrix of , required for numerical solution utilizing classical methods like Newton, becomes zero and then singularity would arise. In this sense, the initial guess should not contain zero eccentricities in a Newtonbased solution. However, using other computational methods like GA can rescue the solution process from singularities.
Remark 2.7.
It is convenient for impulsive maneuvers to apply the first impulse at the perigee since the semimajor axis is more sensitive to the perigee velocity. In this case, for a twoimpulse maneuver, without loss of generality one can assume and . Consequently, according to Eq. (10), can be eliminated from the system of equations. In addition, Eq. (11) for reduces to that can be substituted in Eq. (11) for in order to eliminate and finally arrive at the following equation:
(14) 
Eq. (10) for reduces to:
(15) 
In this sense, Eqs. (14) and (15) construct a system of two equations which can be used to solve two scalar unknowns and . If however, the final orbit () is circular (ellipsetocircle maneuver), , such that , the solutions of Eqs. (14) and (15) reduce to:
(16) 
The interesting feature of the proposed geometric method is reducing the number of variables need to be adjusted. Table 2 shows the design (or optimization) variables required to construct an impulse orbital maneuver in the conventional methods and compares them against the proposed SBGM.
Initial Impulse Location  Final Impulse Location  Maneuver Time  No. of Design Variables  Related Ref. 
F  F  F  (05)  
F  F  NF  (07)  
F  NF  F  (02)  
F  NF  NF  (07)  
NF  NF  F  (02)  
NF  NF  NF  (04)  
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