# Solving Coplanar Power-Limited Orbit Transfer Problem by Primer Vector Approximation Method

Authors: Weijun Huang
Journal: International Journal of Aerospace Engineering
 (2012)

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## Abstract

The coplanar orbit transfer problem has been an important topic in astrodynamics
since the beginning of the space era. Though many approximate solutions for
power-limited orbit transfer problem have been developed, most of them rely on simplifications
of the dynamics of the problem. This paper proposes a new approximation
method called primer vector approximation method to solve the classic power-limited
orbit transfer problem. This method makes no simplification on the dynamics, but
instead approximates the optimal primer-vector function. With this method, this paper
derives two approximate solutions for the power-limited orbit transfer problem.
Numerical tests show the robustness and accuracy of the approximations.


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## Body

## 1. Introduction

Most trajectory optimization problems are nonlinear problems with no analytic solutions. However, to the coplanar power-limited orbit transfer in the classical inverse-square gravity field, many researchers have proposed approximate solutions, for example, Edelbaum [1–4], Zee [5], Marinescu [6], Marec and Vinh [7], Haissig et al. [8], Kechichian’s [9], and Casalino and Colasurdo’s [10]. These proposed solutions are built on assumptions about the transfer scenarios. For example, the solution of [6] assumes the transfer is in a close range, the solutions of [2, 4–6] assume the transfer happens in a long duration, and the solutions of [1, 3, 7–11] assume the admissible control to be within a rather limited set.An approximate solution to an optimal control problem implies both the approximation of the control policy and the approximation of the dynamics. However, we automatically approximate the optimal control policy as well, when we approximate the dynamics. The mentioned references more or less approximate the dynamics. One potential problem with approximating dynamics is that, once the assumptions are violated the obtained control will be infeasible. From a software point of view, an infeasible control produces unexpected results and might cause the software to crash. Therefore, a more robust approximation method for optimal control problem should be to purely make approximations on the optimal control, not the dynamics.A popular method for generating feasible transfer trajectories is the shape-based method [12–15]. In its essence, this method generates feasible control without any compromise on the dynamics. Though there are different variates of the shape-based method, as far as the author knows, there is no theoretical research to address the connection between the real optimal solution and the trajectory generated by the shape-based method.This paper proposes an innovative approximation method—the primer vector approximation method, which combines the advantages of the approximation method and the shape-based method and uses feasible control to approximate the optimal control. The method reformulates the classic transfer problem with a nonlinear transformation from Carter and Humi [16]. The purpose of this reformulation is to put all the nonlinear terms to the coefficients of the control variables. Thus the optimal control vector, called primer vector, of the new formulation can be analyzed and approximated without affecting the dynamics. To demonstrate this method, this paper derives two approximate solutions for both low-thrust close-range transfer and low-thrust long-duration transfer.Four transfer scenarios are designed to test the two approximate solutions numerically. The tests show that the approximations are close to the optimal solutions when the scenarios are within the assumptions of the approximations. More importantly, the solutions are feasible even when the scenarios are far away from the assumptions of the approximations. Theoretically, both approximations can generate feasible transfer control between any two kinds of orbits, including hyperbolic and parabolic orbit.The remaining part of this paper is organized in the following way. The second section is about the formulations of the coplanar power-limited orbit transfer problem. The third section introduces the primer vector approximation method and derives two approximate solutions. The fourth section numerically tests the two solutions. A conclusion is made in the fifth section.


## 2. Formulations of Coplanar Power-Limited Orbit Transfer Problem

### 2.1. Polar Coordinate Formulation

The polar coordinate system used in this paper is presented in Figure1. The meanings of the symbols are given in Nomenclature.Figure 1
Definition of the polar coordinate.Mathematically, the dynamics of a controlled satellite in an inverse-square gravity field can be described as(1)r̈-rθ̇2=-μr2+ur,θ̈r+2θ̇ṙ=uθ.Setvr=dr/dt and vθ=θ̇r. The two second-order ordinary differential equations (ODEs), (1), are equivalent to the following four first-order ODEs:
(2)ṙ=vr,θ̇=vθr,v̇r=vθ2r-μr2+ur,v̇θ=-vθvrr+uθ.Setu=(ur,uθ)T. If a transfer start from time t0 to time tf, the objective function of the coplanar power-limited orbit transfer problem is
(3)J(u)=12∫t0tfuTudt.
The initial orbit at t0 is defined by a vector x0=(r0,θ0,vr0,vθ0)T and the final orbit at tf is defined by a vector xf=(rf,θf,vrf,vθf)T. A free-time rendezvous problem is to find the optimal control function u*, which minimizes the objective function, (3), and transfers a spacecraft from the original position x0 at time t0 to the final position xf at the optimal arriving time tf*. This problem is the simplest problem to introduce the primer vector approximation method. A more common power-limited transfer problem unbounded thrust is the fixed-time  rendezvous problem, the solution to which can be found by adding a time constraint on the approximate solution to the free-time rendezvous problem. We can adopt the method in Novak’s [15] paper to use a solution to the free-time rendezvous problem as a basis for the solution to the related fixed-time rendezvous problem. This paper, however, focuses only on the free-time rendezvous problem.To numerically solve the free-time rendezvous problem, we can use the indirect shooting method, which first transforms the problem into a two-point boundary value problem (TPBVP) and then solves the TPBVP with the shooting technique. To setup the TPBVP, we write down the augmented Hamiltonian of the optimal problem as follows:(4)H=-12uTu+λrvr+λθvθr+λvr(vθ2r-μr2+ur)+λθ(-vθvrr+uθ).The adjoint ODEs of the problem are(5)λ̇r=λθvθr2+λvr(vθ2r2-2μr3-λvθvrvθr2),λ̇θ=0,λ̇vr=-λr+λvθvθr,λ̇vθ=1r(λvθvr-λθ-2λvrλθ).The optimal control isu*=p, where p=(λvr,λvθ)T and is called primer vector. A common shooting method will shoot the optimal arrival time tf* and the values of the four variables λr(t0), λθ(t0), λvr(t0), and λvθ(t0) until the shooting functions x(tf*)=xf and transversality condition H(tf*)=0 are satisfied. At each iteration of the shooting process, there are eight ODEs, (2) and (5), to solve. The shooting process will be time-consuming if the initial guess is poor or the number of the transfer revolutions is large.In this paper, the indirect shooting method with the polar coordinate formulation will be used to generate optimal solutions for the four tests in the fourth session.


### 2.2. A New Formulation

The polar formulation of the power-limited orbit transfer problem has nonlinear terms in the dynamic equations and adjoint equations. A new formulation of the power-limited orbit transfer problem will be derived here with a nonlinear transformation from Carter and Humi [16]. This transformation will transfer the state variables (r,θ,vr,vθ) with respect to time t to new state variables (y1,y2,y3) with respect to polar angle θ.Use(·)′ and (·)′′ to represent the first and second derivative operations with respective to the polar angle θ. Set h=θ̇r2. Thus θ̇=h/r2 and t′=dt/dθ=r2/h. The new state variables are defined as y1=1/r, y2=-r′/r2, and y3=μ/h2. The mapping from the new variables (y1,y2,y3) back to the variables in the polar coordinate is(6)r=1y1,vr=-y2μy3,vθ=y1μy3,t=θ0+∫θ0θ1y12y3μdθ.From the chain rule, we have(7)θ̈=[h′r2-2hr3r′]hr2,ṙ=r′hr2,r̈=[hr′′+r′h′-2h(r′)2r]hr4.Therefore, with (7) and (1), the dynamic equations of the new variables (y1,y2,y3) are(8)y1′=y2,y2′=-y1+y3-y3μy12ur-y2y3μy13uθ,y3′=-2y32μy13uθ.For further simplification, write (8) in matrix form, set y=(y1,y2,y3)T and define(9)A=(010-101000),B=(001001),Bs(y)=(-y3μy12-y2y3μy130-2y32μy13),ν=Bs(y)u.Thus, state space representation (SSR) of the ODEs in (8) is(10)y′=Ay+Bν.The new boundaries of a transfer in the new dynamic equations will be defined by the initial polar angleθ0, the final polar angle θ1 and two constant vectors of the new states(11)y(θ0)=y0,y(θf)=yf.Under the new state vectory and new control vector ν, the objective function, (3), becomes (t′(y)=1/y12y3/μ)(12)J(u(y,ν))=J(y,ν)=12∫θ0θt′(y)νT(Bs(y)Bs(y)T)-1νdθ.The objective function, (12), dynamic equations, (10) and the boundary conditions, (11), compose the new formulation of the coplanar power-limited orbit transfer problem.Mathematically, a coplanar power-limited orbit transfer problem can be described as(13)find:ν*=argmin⁡νJ(y(ν),ν)=argmin⁡ν12∫θ0θt′νT(Bs(y)Bs(y)T)-1νdθsubjectto:y′=Ay+Bν;y(θ0)=y0;y(θf)=yf.To solve this problem, setK to be the new augmented Hamiltonian and κ to be the new adjoint vector corresponding to the new state variable y(14)K=-12t′(y)νT(Bs(y)Bs(y)T)-1ν+κT(Ay+Bν).Apply the theory of optimal control and the optimal control of the new formulation becomesν*=l, in which l is the primer vector of the new formulation and(15)l=1t′Bs(y)Bs(y)TBTκ.The mapping from the new primer vectorl to the original primer vector p is(16)p=Bs(y)-1l.Then the adjoint equations are(17)∂K∂y=-κ′=-12∂(t′(y)νT(Bs(y)Bs(y)T)-1ν)∂yT+ATκ.In conclusion, the transfer problem given by (13) is an alternative formulation of the coplanar power-limited orbit transfer problem. The optimal solution for this new formulation needs to satisfy a TPBVP defined by the dynamic equation, (10), adjoint equation, (17), the boundary conditions, (11).


### 2.3. Properties of the New Formulation

The new formulation has unique properties that facilitate analysis. Firstly, the system can be solved explicitly when there is no control (ν=0) on the state equation, (10). The solutions are
(18)y(θ)=Φ(θ-θ0)y(θ0),κ(θ)=ΦT(θ0-θ)κ(θ0),Φ(θ)=eAθ.Secondly, the nonlinear term in the adjoint vector equation (17) has the following interesting property.Theorem 1.
For an optimal trajectory, the solution for the adjoint vector has the following form:(19)κ(θ)=Φ(θ0-θ)T[Q(θ)+I3×3]κ(θ0),
where θ0 is the initial polar angle of the trajectory, I3×3 is a 3 by 3 unit matrix, and Q(θ-θ0) is a diagonal matrix function.Proof.
Since (17) is a linear nonhomogeneous equation with constant coefficients, its solution is the sum of the general solution for the related homogeneous equation and the particular integral. The solution can be written down as
(20)κ(θ)=Φ(θ0-θ)T∫θ0θ12∂(t′νT(Bs(y)Bs(y)T)-1ν)∂yTΦ(θ)Tdθ+Φ(θ0-θ)Tκ(θ0).∫θ0θ12∂(t′νT(Bs(y)Bs(y)T)-1ν)∂yT
Set
(21)(q1(θ),q2(θ),q3(θ))=∫θ0θ12∂(t′νT(Bs(y)Bs(y)T)-1ν)∂yTΦ(θ)Tdθ
and (use index notation for κ(θ0))
(22)Q(θ)=(q1(θ)(κ(θ0))1000q2(θ)(κ(θ0))2000q3(θ)(κ(θ0))3).
With (20), (21), and (22), we obtain (19) of Theorem 1.Up to this point, no assumption has been made about the new formulation of the transfer problem. It is, however, easy to see that, if the thrust level|ν| is very small, (q1(θ),q2(θ),q3(θ)) will become very small. Therefore, for a low-thrust transfer, the adjoint vector can be approximated by setting Q(θ)=03×3.


## 2.1. Polar Coordinate Formulation

The polar coordinate system used in this paper is presented in Figure1. The meanings of the symbols are given in Nomenclature.Figure 1
Definition of the polar coordinate.Mathematically, the dynamics of a controlled satellite in an inverse-square gravity field can be described as(1)r̈-rθ̇2=-μr2+ur,θ̈r+2θ̇ṙ=uθ.Setvr=dr/dt and vθ=θ̇r. The two second-order ordinary differential equations (ODEs), (1), are equivalent to the following four first-order ODEs:
(2)ṙ=vr,θ̇=vθr,v̇r=vθ2r-μr2+ur,v̇θ=-vθvrr+uθ.Setu=(ur,uθ)T. If a transfer start from time t0 to time tf, the objective function of the coplanar power-limited orbit transfer problem is
(3)J(u)=12∫t0tfuTudt.
The initial orbit at t0 is defined by a vector x0=(r0,θ0,vr0,vθ0)T and the final orbit at tf is defined by a vector xf=(rf,θf,vrf,vθf)T. A free-time rendezvous problem is to find the optimal control function u*, which minimizes the objective function, (3), and transfers a spacecraft from the original position x0 at time t0 to the final position xf at the optimal arriving time tf*. This problem is the simplest problem to introduce the primer vector approximation method. A more common power-limited transfer problem unbounded thrust is the fixed-time  rendezvous problem, the solution to which can be found by adding a time constraint on the approximate solution to the free-time rendezvous problem. We can adopt the method in Novak’s [15] paper to use a solution to the free-time rendezvous problem as a basis for the solution to the related fixed-time rendezvous problem. This paper, however, focuses only on the free-time rendezvous problem.To numerically solve the free-time rendezvous problem, we can use the indirect shooting method, which first transforms the problem into a two-point boundary value problem (TPBVP) and then solves the TPBVP with the shooting technique. To setup the TPBVP, we write down the augmented Hamiltonian of the optimal problem as follows:(4)H=-12uTu+λrvr+λθvθr+λvr(vθ2r-μr2+ur)+λθ(-vθvrr+uθ).The adjoint ODEs of the problem are(5)λ̇r=λθvθr2+λvr(vθ2r2-2μr3-λvθvrvθr2),λ̇θ=0,λ̇vr=-λr+λvθvθr,λ̇vθ=1r(λvθvr-λθ-2λvrλθ).The optimal control isu*=p, where p=(λvr,λvθ)T and is called primer vector. A common shooting method will shoot the optimal arrival time tf* and the values of the four variables λr(t0), λθ(t0), λvr(t0), and λvθ(t0) until the shooting functions x(tf*)=xf and transversality condition H(tf*)=0 are satisfied. At each iteration of the shooting process, there are eight ODEs, (2) and (5), to solve. The shooting process will be time-consuming if the initial guess is poor or the number of the transfer revolutions is large.In this paper, the indirect shooting method with the polar coordinate formulation will be used to generate optimal solutions for the four tests in the fourth session.


## 2.2. A New Formulation

The polar formulation of the power-limited orbit transfer problem has nonlinear terms in the dynamic equations and adjoint equations. A new formulation of the power-limited orbit transfer problem will be derived here with a nonlinear transformation from Carter and Humi [16]. This transformation will transfer the state variables (r,θ,vr,vθ) with respect to time t to new state variables (y1,y2,y3) with respect to polar angle θ.Use(·)′ and (·)′′ to represent the first and second derivative operations with respective to the polar angle θ. Set h=θ̇r2. Thus θ̇=h/r2 and t′=dt/dθ=r2/h. The new state variables are defined as y1=1/r, y2=-r′/r2, and y3=μ/h2. The mapping from the new variables (y1,y2,y3) back to the variables in the polar coordinate is(6)r=1y1,vr=-y2μy3,vθ=y1μy3,t=θ0+∫θ0θ1y12y3μdθ.From the chain rule, we have(7)θ̈=[h′r2-2hr3r′]hr2,ṙ=r′hr2,r̈=[hr′′+r′h′-2h(r′)2r]hr4.Therefore, with (7) and (1), the dynamic equations of the new variables (y1,y2,y3) are(8)y1′=y2,y2′=-y1+y3-y3μy12ur-y2y3μy13uθ,y3′=-2y32μy13uθ.For further simplification, write (8) in matrix form, set y=(y1,y2,y3)T and define(9)A=(010-101000),B=(001001),Bs(y)=(-y3μy12-y2y3μy130-2y32μy13),ν=Bs(y)u.Thus, state space representation (SSR) of the ODEs in (8) is(10)y′=Ay+Bν.The new boundaries of a transfer in the new dynamic equations will be defined by the initial polar angleθ0, the final polar angle θ1 and two constant vectors of the new states(11)y(θ0)=y0,y(θf)=yf.Under the new state vectory and new control vector ν, the objective function, (3), becomes (t′(y)=1/y12y3/μ)(12)J(u(y,ν))=J(y,ν)=12∫θ0θt′(y)νT(Bs(y)Bs(y)T)-1νdθ.The objective function, (12), dynamic equations, (10) and the boundary conditions, (11), compose the new formulation of the coplanar power-limited orbit transfer problem.Mathematically, a coplanar power-limited orbit transfer problem can be described as(13)find:ν*=argmin⁡νJ(y(ν),ν)=argmin⁡ν12∫θ0θt′νT(Bs(y)Bs(y)T)-1νdθsubjectto:y′=Ay+Bν;y(θ0)=y0;y(θf)=yf.To solve this problem, setK to be the new augmented Hamiltonian and κ to be the new adjoint vector corresponding to the new state variable y(14)K=-12t′(y)νT(Bs(y)Bs(y)T)-1ν+κT(Ay+Bν).Apply the theory of optimal control and the optimal control of the new formulation becomesν*=l, in which l is the primer vector of the new formulation and(15)l=1t′Bs(y)Bs(y)TBTκ.The mapping from the new primer vectorl to the original primer vector p is(16)p=Bs(y)-1l.Then the adjoint equations are(17)∂K∂y=-κ′=-12∂(t′(y)νT(Bs(y)Bs(y)T)-1ν)∂yT+ATκ.In conclusion, the transfer problem given by (13) is an alternative formulation of the coplanar power-limited orbit transfer problem. The optimal solution for this new formulation needs to satisfy a TPBVP defined by the dynamic equation, (10), adjoint equation, (17), the boundary conditions, (11).


## 2.3. Properties of the New Formulation

The new formulation has unique properties that facilitate analysis. Firstly, the system can be solved explicitly when there is no control (ν=0) on the state equation, (10). The solutions are
(18)y(θ)=Φ(θ-θ0)y(θ0),κ(θ)=ΦT(θ0-θ)κ(θ0),Φ(θ)=eAθ.Secondly, the nonlinear term in the adjoint vector equation (17) has the following interesting property.Theorem 1.
For an optimal trajectory, the solution for the adjoint vector has the following form:(19)κ(θ)=Φ(θ0-θ)T[Q(θ)+I3×3]κ(θ0),
where θ0 is the initial polar angle of the trajectory, I3×3 is a 3 by 3 unit matrix, and Q(θ-θ0) is a diagonal matrix function.Proof.
Since (17) is a linear nonhomogeneous equation with constant coefficients, its solution is the sum of the general solution for the related homogeneous equation and the particular integral. The solution can be written down as
(20)κ(θ)=Φ(θ0-θ)T∫θ0θ12∂(t′νT(Bs(y)Bs(y)T)-1ν)∂yTΦ(θ)Tdθ+Φ(θ0-θ)Tκ(θ0).∫θ0θ12∂(t′νT(Bs(y)Bs(y)T)-1ν)∂yT
Set
(21)(q1(θ),q2(θ),q3(θ))=∫θ0θ12∂(t′νT(Bs(y)Bs(y)T)-1ν)∂yTΦ(θ)Tdθ
and (use index notation for κ(θ0))
(22)Q(θ)=(q1(θ)(κ(θ0))1000q2(θ)(κ(θ0))2000q3(θ)(κ(θ0))3).
With (20), (21), and (22), we obtain (19) of Theorem 1.Up to this point, no assumption has been made about the new formulation of the transfer problem. It is, however, easy to see that, if the thrust level|ν| is very small, (q1(θ),q2(θ),q3(θ)) will become very small. Therefore, for a low-thrust transfer, the adjoint vector can be approximated by setting Q(θ)=03×3.


## 3. Primer Vector Approximation Method

### 3.1. Equivalent Optimal Control Problem

The primer vector approximation method aims to find an approximation for the optimal control (primer vector function), (15), without affecting the accuracy of the state dynamics. Thus, it is important to ensure that the approximate control will lead to a precise integration of the state equation, (10). This is the core of the primer vector approximation method.To introduce the strategy for primer vector approximation, first define a family of simpler optimal control problemsΞ(23)find:ν̂=argmin⁡ν̂Ĵ(ν̂)=argmin⁡ν̂12∫θ0θν̂TC(θ)-1ν̂dθsubjectto:ŷ′=Aŷ+Bν̂;ŷ(θ0)=y0;ŷ(θf)=yf;C(θ)∈χ,
where χ is a set of 2×2 matrix functions and all of its members are continuous while θ∈[θ0,θf]. The following theorem gives the relationship between the family Ξ and the optimal control problem, defined in (13).Theorem 2.
Given a transfer problem𝒜, defined by (13), among the family of optimal control problem Ξ, defined by (23), there is at least one member problem 𝒜̂ with C(θ)∈χ that has the same solution, that is, ν̂*(θ)=ν*(θ),ŷ*(θ)=y*(θ).Proof.
Suppose that the optimal solution for the given problemA, defined by (13), is known and set D(θ)=1/t′(y*(θ))Bs(y*(θ))Bs(y*(θ))T. With Theorem 1, the optimal primer vector function l can be written as an explicit function of θ(24)l=D(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]κ(θ0).
Set ν=l and plug it into the state equation of (13). We obtain the following relationships for an optimal solution to problem A:
(25)y*(θ)=Φ(θ-θ0)(y0+Ψ(θ)κ0),Ψ(θ)=∫θ0θ1Φ(θ0-θ)BD(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]dθ,κ0=(Φ(θ1-θ0)Ψ(θ1-θ0))-1(yf-Φ(θ1-θ0)y0).
Meanwhile, any member problem in Ξ can be solved by finding the solution for the following TPBVP:
(26)κ̂′=-ATκ̂,ŷ′=Aŷ+Bl̂,ŷ(θ0)=y0;ŷ(θf)=yf.
This solution is
(27)ŷ*(θ)=Φ(θ-θ0)(y0+Ψ̂(θ)κ̂0),l̂(θ)=C(θ)BTΦ(θ0-θ)Tκ̂(θ0),Ψ̂(θ)=∫θ0θ1Φ(θ0-θ)BC(θ0-θ)BTΦ(θ0-θ)Tdθ,κ̂0=(Φ(θ1-θ0)Ψ̂(θ1))-1(yf-Φ(θ1-θ0)y0).
Select a problem 𝒜̂∈Ξ with
(28)C(θ)=D(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]Φ(θ-θ0)TB.
Put (28) into (27). After simplification, we obtain
(29)Ψ̂(θ)=Ψ(θ),κ̂0=κ0.
Therefore, the optimal control and optimal trajectory of problems A and Â are the same, that is,
(30)ν̂*(θ)=ν*(θ),ŷ*(θ)=y*(θ).If𝒜̂∈Ξ and a transfer problem 𝒜 has the same solution, we call 𝒜̂ the equivalent optimal  control problem of the original problem 𝒜 in (13).Theorem 3.
If𝒜̂∈Ξ is an equivalent optimal control problem of a transfer problem 𝒜, defined by (13), then the optimal control l̂ of another member problem ℬ̂∈Ξ is a feasible control for the transfer problem 𝒜.Proof.
From the general solution equation (27), we know that no matter what functional form of C(θ) we pick, the obtained control l̂ will always generate a trajectory that satisfies the two end point boundaries. Thus, l̂ is a feasible control for the power-limited transfer problem.Theorem2 gives us a new way to solve any optimal power-limited transfer problem 𝒜. That is, we first find its equivalent problem in 𝒜̂∈Ξ, and then the optimal solution is given by y(θ)=ŷ(θ), l(θ)=l̂(θ), and (27). In fact, from the proof of Theorem 2, we can construct such a problem 𝒜̂∈Ξ with C(θ)=C*(θ),(31)C*(θ)=D(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]Φ(θ-θ0)TB,C*(θ0)=D(θ0).
Though we do not really know D(θ) until we solve the original problem 𝒜, we can always “shape” the matrix function C(θ)) based on our assumptions and knowledge of the transfer scenario. Suppose the “shaped” matrix function is C°(θ) and it corresponds to a problem ℬ̂∈Ξ. Theorem 3 indicates that, even though ℬ̂ is not exactly 𝒜̂, it gives a feasible solution for the original problem 𝒜 as long as ℬ̂∈Ξ. Thus, if we can find a proper C°(θ) satisfying C°(θ)≈C*(θ), then the solution for ℬ̂, (27), is a feasible approximate solution for 𝒜̂, essentially 𝒜. In other words, once we find C°(θ) (an approximation of C*(θ)), the approximate solution to 𝒜 is given by (27) with C(θ)=C°(θ). Since any C(θ)∈χ corresponds to a unique primer vector function l̂(θ) in (27), this process of finding an approximation of C*(θ) is named as primer vector approximation (PVA) in this paper. The following section uses this PVA to derive two explicit approximate solutions to the coplanar power-limited orbit transfer problem.


### 3.2. Primer Vector Approximation under the Low-Thrust Assumption

In this section, we assume the thrusting magnitude is very low (i.e.,|ν| is very small), such that Q(θ)≈03×3, C*(θ) becomes a symmetric matrix function and(32)C*(θ)≈1t′(y*(θ))Bs(y*(θ))Bs(y*(θ))T.The thrusting magnitude is usually very low in two kinds of transfers. One is the close range transfer and the other the long duration multirevolution transfer.


#### 3.2.1. Approximate Solution for Close Range Transfer

Because the transfer happens in the vicinity of the initial orbit, a reasonable approximation ofC*(θ) is(33)C°(θ)=1t′(y0(θ))Bs(y0(θ))Bs(y0(θ))T,y0(θ)=Φ(θ-θ0)y0,
where y0(θ) represents the initial orbit. Put (33) into (27). We obtain the approximate optimal solution for a close range transfer.Though (27) generally requires numerical integration to get Ψ, the approximate optimal trajectory y* and control ν* are still explicit functions of θ. Moreover, Ψ is independent of the target location y1. Therefore, for all possible close-range transfers of the initial orbit y0(θ), we only need to compute Ψ one time for one period of the initial orbit. This property is greatly useful when we schedule transfers for a satellite formation around an elliptic reference orbit specified by y0.When the initial orbit is circular, the computation ofΨ can be done analytically. Using index notation and setting Δθ=θf-θ0, c=cos(Δθ) and s=sin(Δθ), we have(34)(Ψ)11=12(y0)1(-16+3c)s+13Δθ,(Ψ)12=(Ψ)21=14(y0)1(-13+16c-3c2+3s2),(Ψ)13=(Ψ)31=4(y0)1(-s+Δθ),(Ψ)22=12(y0)1(-3c×s+5Δθ),(Ψ)23=(Ψ)32=(Ψ)33=4(y0)1(-1+c).
With (34), the approximate optimal solution, (27), becomes analytic.


#### 3.2.2. Approximate Solution for Long Duration Multirevolution Transfer

A long duration multirevolution transfer tends to build up its orbital energy and angular momentum monotonically. Thus, it is reasonable to approximate the characteristic matrix functionC*(θ) with a linear matrix function(35)C°(θ)=C°(θ0)+θ-θ0θf-θ0[C°(θf)-C°(θ0)],C°(θ0)=1t′(y0)Bs(y0)Bs(y0)T,C°(θf)=1t′(yf)Bs(yf)Bs(yf)T.With (35), the matrix function Ψ in (27) can be expanded analytically. With the analytic Ψ, the approximate optimal solution given by (27) is analytic. (The complete formula for Ψ is too lengthy to show here.)


## 3.1. Equivalent Optimal Control Problem

The primer vector approximation method aims to find an approximation for the optimal control (primer vector function), (15), without affecting the accuracy of the state dynamics. Thus, it is important to ensure that the approximate control will lead to a precise integration of the state equation, (10). This is the core of the primer vector approximation method.To introduce the strategy for primer vector approximation, first define a family of simpler optimal control problemsΞ(23)find:ν̂=argmin⁡ν̂Ĵ(ν̂)=argmin⁡ν̂12∫θ0θν̂TC(θ)-1ν̂dθsubjectto:ŷ′=Aŷ+Bν̂;ŷ(θ0)=y0;ŷ(θf)=yf;C(θ)∈χ,
where χ is a set of 2×2 matrix functions and all of its members are continuous while θ∈[θ0,θf]. The following theorem gives the relationship between the family Ξ and the optimal control problem, defined in (13).Theorem 2.
Given a transfer problem𝒜, defined by (13), among the family of optimal control problem Ξ, defined by (23), there is at least one member problem 𝒜̂ with C(θ)∈χ that has the same solution, that is, ν̂*(θ)=ν*(θ),ŷ*(θ)=y*(θ).Proof.
Suppose that the optimal solution for the given problemA, defined by (13), is known and set D(θ)=1/t′(y*(θ))Bs(y*(θ))Bs(y*(θ))T. With Theorem 1, the optimal primer vector function l can be written as an explicit function of θ(24)l=D(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]κ(θ0).
Set ν=l and plug it into the state equation of (13). We obtain the following relationships for an optimal solution to problem A:
(25)y*(θ)=Φ(θ-θ0)(y0+Ψ(θ)κ0),Ψ(θ)=∫θ0θ1Φ(θ0-θ)BD(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]dθ,κ0=(Φ(θ1-θ0)Ψ(θ1-θ0))-1(yf-Φ(θ1-θ0)y0).
Meanwhile, any member problem in Ξ can be solved by finding the solution for the following TPBVP:
(26)κ̂′=-ATκ̂,ŷ′=Aŷ+Bl̂,ŷ(θ0)=y0;ŷ(θf)=yf.
This solution is
(27)ŷ*(θ)=Φ(θ-θ0)(y0+Ψ̂(θ)κ̂0),l̂(θ)=C(θ)BTΦ(θ0-θ)Tκ̂(θ0),Ψ̂(θ)=∫θ0θ1Φ(θ0-θ)BC(θ0-θ)BTΦ(θ0-θ)Tdθ,κ̂0=(Φ(θ1-θ0)Ψ̂(θ1))-1(yf-Φ(θ1-θ0)y0).
Select a problem 𝒜̂∈Ξ with
(28)C(θ)=D(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]Φ(θ-θ0)TB.
Put (28) into (27). After simplification, we obtain
(29)Ψ̂(θ)=Ψ(θ),κ̂0=κ0.
Therefore, the optimal control and optimal trajectory of problems A and Â are the same, that is,
(30)ν̂*(θ)=ν*(θ),ŷ*(θ)=y*(θ).If𝒜̂∈Ξ and a transfer problem 𝒜 has the same solution, we call 𝒜̂ the equivalent optimal  control problem of the original problem 𝒜 in (13).Theorem 3.
If𝒜̂∈Ξ is an equivalent optimal control problem of a transfer problem 𝒜, defined by (13), then the optimal control l̂ of another member problem ℬ̂∈Ξ is a feasible control for the transfer problem 𝒜.Proof.
From the general solution equation (27), we know that no matter what functional form of C(θ) we pick, the obtained control l̂ will always generate a trajectory that satisfies the two end point boundaries. Thus, l̂ is a feasible control for the power-limited transfer problem.Theorem2 gives us a new way to solve any optimal power-limited transfer problem 𝒜. That is, we first find its equivalent problem in 𝒜̂∈Ξ, and then the optimal solution is given by y(θ)=ŷ(θ), l(θ)=l̂(θ), and (27). In fact, from the proof of Theorem 2, we can construct such a problem 𝒜̂∈Ξ with C(θ)=C*(θ),(31)C*(θ)=D(θ)BTΦ(θ0-θ)T[Q(θ)+I3×3]Φ(θ-θ0)TB,C*(θ0)=D(θ0).
Though we do not really know D(θ) until we solve the original problem 𝒜, we can always “shape” the matrix function C(θ)) based on our assumptions and knowledge of the transfer scenario. Suppose the “shaped” matrix function is C°(θ) and it corresponds to a problem ℬ̂∈Ξ. Theorem 3 indicates that, even though ℬ̂ is not exactly 𝒜̂, it gives a feasible solution for the original problem 𝒜 as long as ℬ̂∈Ξ. Thus, if we can find a proper C°(θ) satisfying C°(θ)≈C*(θ), then the solution for ℬ̂, (27), is a feasible approximate solution for 𝒜̂, essentially 𝒜. In other words, once we find C°(θ) (an approximation of C*(θ)), the approximate solution to 𝒜 is given by (27) with C(θ)=C°(θ). Since any C(θ)∈χ corresponds to a unique primer vector function l̂(θ) in (27), this process of finding an approximation of C*(θ) is named as primer vector approximation (PVA) in this paper. The following section uses this PVA to derive two explicit approximate solutions to the coplanar power-limited orbit transfer problem.


## 3.2. Primer Vector Approximation under the Low-Thrust Assumption

In this section, we assume the thrusting magnitude is very low (i.e.,|ν| is very small), such that Q(θ)≈03×3, C*(θ) becomes a symmetric matrix function and(32)C*(θ)≈1t′(y*(θ))Bs(y*(θ))Bs(y*(θ))T.The thrusting magnitude is usually very low in two kinds of transfers. One is the close range transfer and the other the long duration multirevolution transfer.


### 3.2.1. Approximate Solution for Close Range Transfer

Because the transfer happens in the vicinity of the initial orbit, a reasonable approximation ofC*(θ) is(33)C°(θ)=1t′(y0(θ))Bs(y0(θ))Bs(y0(θ))T,y0(θ)=Φ(θ-θ0)y0,
where y0(θ) represents the initial orbit. Put (33) into (27). We obtain the approximate optimal solution for a close range transfer.Though (27) generally requires numerical integration to get Ψ, the approximate optimal trajectory y* and control ν* are still explicit functions of θ. Moreover, Ψ is independent of the target location y1. Therefore, for all possible close-range transfers of the initial orbit y0(θ), we only need to compute Ψ one time for one period of the initial orbit. This property is greatly useful when we schedule transfers for a satellite formation around an elliptic reference orbit specified by y0.When the initial orbit is circular, the computation ofΨ can be done analytically. Using index notation and setting Δθ=θf-θ0, c=cos(Δθ) and s=sin(Δθ), we have(34)(Ψ)11=12(y0)1(-16+3c)s+13Δθ,(Ψ)12=(Ψ)21=14(y0)1(-13+16c-3c2+3s2),(Ψ)13=(Ψ)31=4(y0)1(-s+Δθ),(Ψ)22=12(y0)1(-3c×s+5Δθ),(Ψ)23=(Ψ)32=(Ψ)33=4(y0)1(-1+c).
With (34), the approximate optimal solution, (27), becomes analytic.


### 3.2.2. Approximate Solution for Long Duration Multirevolution Transfer

A long duration multirevolution transfer tends to build up its orbital energy and angular momentum monotonically. Thus, it is reasonable to approximate the characteristic matrix functionC*(θ) with a linear matrix function(35)C°(θ)=C°(θ0)+θ-θ0θf-θ0[C°(θf)-C°(θ0)],C°(θ0)=1t′(y0)Bs(y0)Bs(y0)T,C°(θf)=1t′(yf)Bs(yf)Bs(yf)T.With (35), the matrix function Ψ in (27) can be expanded analytically. With the analytic Ψ, the approximate optimal solution given by (27) is analytic. (The complete formula for Ψ is too lengthy to show here.)


## 3.2.1. Approximate Solution for Close Range Transfer

Because the transfer happens in the vicinity of the initial orbit, a reasonable approximation ofC*(θ) is(33)C°(θ)=1t′(y0(θ))Bs(y0(θ))Bs(y0(θ))T,y0(θ)=Φ(θ-θ0)y0,
where y0(θ) represents the initial orbit. Put (33) into (27). We obtain the approximate optimal solution for a close range transfer.Though (27) generally requires numerical integration to get Ψ, the approximate optimal trajectory y* and control ν* are still explicit functions of θ. Moreover, Ψ is independent of the target location y1. Therefore, for all possible close-range transfers of the initial orbit y0(θ), we only need to compute Ψ one time for one period of the initial orbit. This property is greatly useful when we schedule transfers for a satellite formation around an elliptic reference orbit specified by y0.When the initial orbit is circular, the computation ofΨ can be done analytically. Using index notation and setting Δθ=θf-θ0, c=cos(Δθ) and s=sin(Δθ), we have(34)(Ψ)11=12(y0)1(-16+3c)s+13Δθ,(Ψ)12=(Ψ)21=14(y0)1(-13+16c-3c2+3s2),(Ψ)13=(Ψ)31=4(y0)1(-s+Δθ),(Ψ)22=12(y0)1(-3c×s+5Δθ),(Ψ)23=(Ψ)32=(Ψ)33=4(y0)1(-1+c).
With (34), the approximate optimal solution, (27), becomes analytic.


## 3.2.2. Approximate Solution for Long Duration Multirevolution Transfer

A long duration multirevolution transfer tends to build up its orbital energy and angular momentum monotonically. Thus, it is reasonable to approximate the characteristic matrix functionC*(θ) with a linear matrix function(35)C°(θ)=C°(θ0)+θ-θ0θf-θ0[C°(θf)-C°(θ0)],C°(θ0)=1t′(y0)Bs(y0)Bs(y0)T,C°(θf)=1t′(yf)Bs(yf)Bs(yf)T.With (35), the matrix function Ψ in (27) can be expanded analytically. With the analytic Ψ, the approximate optimal solution given by (27) is analytic. (The complete formula for Ψ is too lengthy to show here.)


## 4. Numerical Test

### 4.1. Cases of Free-Time Rendezvous Problem

Without loss of generality, we can set the gravitation constantμ=1 and use normalized distance in the test cases.In Table1, four coplanar rendezvous cases are chosen to test the accuracy of the two analytic approximations. Cases A and B test the approximate solutions for close range transfers, with case A featuring a circular reference while case B featuring a high elliptic reference. Cases C and D test the approximate solutions for long duration transfers. Case C is a circular-to-elliptic transfer, while case D is a circular-to-circular transfer. We use “Approx. C.R.” to identify the approximate solution for close range transfer, (27) and (23), and use “Approx. L.D.” to identify the approximate solution for long range transfer, (27) and (35).Table 1
Numerical tests of approximate solutions.
CaseInitial Orb.Target Orb.Approx. C.R.Approx. L.D.Optimum(a0,e0,θ0)(a1,e1,θ1)TOFCostJ(u)TOFCostJ(u)TOFCostJ(u)A (<1 Rev)(1,0,0)(1,0.1,4)4.3640.00094744.3600.00094694.3590.0009424B (<1 Rev)(1,0.8,0)(1,0.85,4)6.0260.00052305.9620.0072686.0160.0005148C (=4 Revs)(1,0,0)(2,0.15,8π)41.480.00230142.260.00231441.280.002290D (=20 Revs)(1,0,0)(2,0,40π)208.20.0004287202.30.0004288205.10.0004277


### 4.2. Results and Discussions

The results of cases A and B show that “Approx. C.R” captures the close-range-transfer primer vector dynamics. Where the circular reference is used (case A), the percentage error of the cost and the time of flight (TOF) are about 0.5% and 0.1%, respectively. As the eccentricity of the reference orbit increases, the primer vector dynamics becomes more complicated. However, even when the eccentricity of the reference orbit is as high as 0.8 (case B), “Approx. C.R” still captures the primer vector dynamics well. The percentage error of the cost and the TOF are about 1.5% and 0.16%, respectively.The results of cases C and D show that “Approx. L.D” captures the long-duration-transfer primer vector dynamics. The percentage error of the cost and the time of flight (TOF) are around 1% and 2.3%, respectively, for case C, and around 0.26% and 1.4%, respectively, for case D.It is interesting to see that “Approx. C.R”, intended for close-range transfers, works very well for long-duration transfers too. It is, however, worth pointing out that, since “Approx. C.R” requires a numerical integration, when the initial orbit is elliptic, it is computationally slower than “Approx. L.D.” But when the initial orbit is circular, the analytic "Approx. C.R" actually becomes computationally faster than “Approx. L.D.”A significant advantage of the primer vector approximation method is that it precisely follows the dynamic equations of motion and gives a feasible solution. Table2 shows the missed target errors of case D, which are obtained by simulating Equations (2) with the generated control profiles of the two approximations. The results verify the advantage of the proposed method.Table 2
Errors induced by the approximate solutions (case D).
rθvrvθApprox. CR-1.51×10-7-9.77×10-6-1.02×10-67.66×10-7Approx. LD-4.47×10-7-7.13×10-6-8.00×10-71.73×10-7Figures2 and 3 show the thrust angle and thrust magnitude histories of case B, while Figures 5 and 6 show those of case D. Figures 4 and 7 present the trajectories for case B and case D, respectively. In the figures, the black solid line represents the optimal solution, while the black dashed line is from “Approx. LD” and the gray solid line from “Approx. CR”. Only the “Approx. CR” solution (gray line) and the true optimal solution are shown in Figure 3, because the “Approx. LD” solution is too far away from the optimum to effectively display.Figure 2
Thrust angle comparison of different trajectories in case B.Figure 3
Thrust magnitude comparison of different trajectories in case B.Figure 4
Visual comparison of different trajectories in case B.Figure 5
Thrust angle comparison of different trajectories in case D.Figure 6
Thrust magnitude comparison of different trajectories in case D.Figure 7
Visual comparison of different trajectories in case D.


## 4.1. Cases of Free-Time Rendezvous Problem

Without loss of generality, we can set the gravitation constantμ=1 and use normalized distance in the test cases.In Table1, four coplanar rendezvous cases are chosen to test the accuracy of the two analytic approximations. Cases A and B test the approximate solutions for close range transfers, with case A featuring a circular reference while case B featuring a high elliptic reference. Cases C and D test the approximate solutions for long duration transfers. Case C is a circular-to-elliptic transfer, while case D is a circular-to-circular transfer. We use “Approx. C.R.” to identify the approximate solution for close range transfer, (27) and (23), and use “Approx. L.D.” to identify the approximate solution for long range transfer, (27) and (35).Table 1
Numerical tests of approximate solutions.
CaseInitial Orb.Target Orb.Approx. C.R.Approx. L.D.Optimum(a0,e0,θ0)(a1,e1,θ1)TOFCostJ(u)TOFCostJ(u)TOFCostJ(u)A (<1 Rev)(1,0,0)(1,0.1,4)4.3640.00094744.3600.00094694.3590.0009424B (<1 Rev)(1,0.8,0)(1,0.85,4)6.0260.00052305.9620.0072686.0160.0005148C (=4 Revs)(1,0,0)(2,0.15,8π)41.480.00230142.260.00231441.280.002290D (=20 Revs)(1,0,0)(2,0,40π)208.20.0004287202.30.0004288205.10.0004277


## 4.2. Results and Discussions

The results of cases A and B show that “Approx. C.R” captures the close-range-transfer primer vector dynamics. Where the circular reference is used (case A), the percentage error of the cost and the time of flight (TOF) are about 0.5% and 0.1%, respectively. As the eccentricity of the reference orbit increases, the primer vector dynamics becomes more complicated. However, even when the eccentricity of the reference orbit is as high as 0.8 (case B), “Approx. C.R” still captures the primer vector dynamics well. The percentage error of the cost and the TOF are about 1.5% and 0.16%, respectively.The results of cases C and D show that “Approx. L.D” captures the long-duration-transfer primer vector dynamics. The percentage error of the cost and the time of flight (TOF) are around 1% and 2.3%, respectively, for case C, and around 0.26% and 1.4%, respectively, for case D.It is interesting to see that “Approx. C.R”, intended for close-range transfers, works very well for long-duration transfers too. It is, however, worth pointing out that, since “Approx. C.R” requires a numerical integration, when the initial orbit is elliptic, it is computationally slower than “Approx. L.D.” But when the initial orbit is circular, the analytic "Approx. C.R" actually becomes computationally faster than “Approx. L.D.”A significant advantage of the primer vector approximation method is that it precisely follows the dynamic equations of motion and gives a feasible solution. Table2 shows the missed target errors of case D, which are obtained by simulating Equations (2) with the generated control profiles of the two approximations. The results verify the advantage of the proposed method.Table 2
Errors induced by the approximate solutions (case D).
rθvrvθApprox. CR-1.51×10-7-9.77×10-6-1.02×10-67.66×10-7Approx. LD-4.47×10-7-7.13×10-6-8.00×10-71.73×10-7Figures2 and 3 show the thrust angle and thrust magnitude histories of case B, while Figures 5 and 6 show those of case D. Figures 4 and 7 present the trajectories for case B and case D, respectively. In the figures, the black solid line represents the optimal solution, while the black dashed line is from “Approx. LD” and the gray solid line from “Approx. CR”. Only the “Approx. CR” solution (gray line) and the true optimal solution are shown in Figure 3, because the “Approx. LD” solution is too far away from the optimum to effectively display.Figure 2
Thrust angle comparison of different trajectories in case B.Figure 3
Thrust magnitude comparison of different trajectories in case B.Figure 4
Visual comparison of different trajectories in case B.Figure 5
Thrust angle comparison of different trajectories in case D.Figure 6
Thrust magnitude comparison of different trajectories in case D.Figure 7
Visual comparison of different trajectories in case D.


## 5. Conclusions

This paper uses a new method to approximate solutions for a nonlinear optimal control problem. This method begins with a transformation to push all the nonlinearity to the coefficients of the control terms. Then it analyzes of the adjoint equations and embeds the process of finding solutions into the process of approximating the primer vector curve of the equivalent linear quadratic optimal control problem. This method is powerful and leads to an extremely simple and accurate explicit approximate solution. At the cases of circular close-range transfers and long-duration transfers, analytic approximate solutions exist.This paper focuses on the display of the primer vector approximation method itself and includes no numerical comparison of this new method with other approximation methods. However, as far as the author knows, the proposed method is quite unique and radically different from other approximation methods. This method has a precise integration of the state dynamics, while other approximation methods more or less approximate the state dynamics. As a result, even though the given transfer scenario largely violates the assumptions of the approximation, the obtained solutions are still feasible. This is a major advantage of using the primer vector approximation method.



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*Source: 480320-2012-04-02.xml*