By applying a lossless convexification technique, the nonconvex control constraints are convexified, and an equivalent problem is formed. The equivalence.
JOURNAL OF SPACECRAFT AND ROCKETS
Optimization of Minimum-Time Low-Thrust Transfers Using Convex Programming Zhenbo Wang∗ and Michael J. Grant† Purdue University, West Lafayette, Indiana 47907-2045
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DOI: 10.2514/1.A33995 In this paper, a convex optimization method for the numerical solution of the minimum-time low-thrust orbit transfer problem is presented. The main contribution is the transformation of the free-final-time low-thrust trajectory optimization problem into a sequence of convex optimization problems. First, a new independent variable is introduced to rewrite the equations of motion, and a nonlinear optimal control problem is obtained. Then, the nonlinearity in the dynamics is reduced through a change of variables. By applying a lossless convexification technique, the nonconvex control constraints are convexified, and an equivalent problem is formed. The equivalence of the relaxation and the existence of the solution to the relaxed problem are proved. Based on the linearization of the dynamics, a successive convex approach is developed, and in each iteration, a second-order cone programming problem is solved efficiently by state-of-the-art interior-point methods. The effectiveness of the proposed method is verified through numerical simulations of an Earth-to-Mars low-thrust transfer problem. Furthermore, the performance of this convex approach is demonstrated by comparing with a general-purpose optimal control solver for transfers with multiple revolutions.
Nomenclature A, B H Isp k m m0 N N rev p R0 r T T max t u u ur ut V0 ve vr vt x z γ δ ε η θ λ μ
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
I.
matrices describing the linearized dynamics Hamiltonian of the system specific impulse for thrusters iteration number mass of the spacecraft initial mass of the spacecraft (mass unit) number of discretized nodes number of revolutions costate vector initial radial distance (distance unit) radial distance thrust magnitude maximum thrust (force unit) time of transfer thrust acceleration control vector radial component of the thrust acceleration tangential component of the thrust acceleration initial velocity (velocity unit) exhaust velocity radial component of the velocity tangential component of the velocity state vector natural logarithm of mass rate of convergence trust-region radius tolerance value for convergence thrust angle transfer angle Lagrange multiplier vector gravitational constant
Received 3 June 2017; revision received 16 October 2017; accepted for publication 27 October 2017; published online 23 November 2017. Copyright © 2017 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0022-4650 (print) or 1533-6794 (online) to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. *Ph.D. Candidate, School of Aeronautics and Astronautics; wang2351@ purdue.edu. Student Member AIAA. † Assistant Professor, School of Aeronautics and Astronautics; mjgrant@ purdue.edu. Senior Member AIAA.
T
Introduction
RAJECTORY optimization for low-thrust transfers has received a lot of attention during the past few decades due to the advantages of electric propulsion over chemical propulsion in rendezvous, orbit maintenance, orbit transfers, and interplanetary missions. Despite the highly appealing efficiency of low-thrust propulsion, the resulting trajectory optimization problem is challenging to solve due to several reasons. First, a large number of revolutions will be needed to complete the transfers when the initial and terminal orbits are widely spaced. Second, computational challenges might exist due to the long duration of the orbit transfers [1]. The optimal orbit transfer problem has been posed as a nonlinear constrained optimal control problem, and there are essentially two approaches: the indirect method and direct method [2]. The indirect approach solves the optimal control problem by explicitly deriving the necessary conditions of optimality based on the calculus of variations and the minimum principle. The resulting two-point boundary-value problem can be solved numerically [3–5]. However, indirect methods are extremely sensitive to the initial guess for the adjoint variables, which makes the convergence of these methods quite difficult. In addition, the primer vector theory has been studied in the past few decades and applied to solve low-thrust transfer problems by leveraging calculus of variations and parameter optimization [6–8]. In contrast, complicated and lengthy mathematical derivations can be avoided for the direct approach. Instead, the direct approach transforms the original continuous-time optimal control problem into a parameter optimization problem, which can be solved by nonlinear programming (NLP) algorithms, such as sequential quadratic programming [9–12]. Despite the resulting large-scale NLP problem with a large number of variables and constraints, the direct approach has received a great deal of interest for many years with the advances in the development of NLP techniques. During recent years, convex optimization has been introduced to solve optimal control problems in aerospace due to the advantages that convex optimization methods offer compared to the NLP algorithms. These advantages include the low complexity of the converted convex problems, the polynomial solution time and global convergence of convex solvers, and the lack of a need for a user-supplied initial guess [13–15]. As such, convex optimization approaches are very promising for onboard applications. In [16–18], a second-order cone programming (SOCP) approach was developed to solve the fuel-optimal powered descent guidance problem for Mars pinpoint landing. Based on these results, a customized interior-point method (IPM) algorithm was developed in [19] and demonstrated on eight flight tests in [20]. Motivated by this research, successive convex approaches were proposed and used for the fuel-optimal trajectory optimization of
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near-field rendezvous and proximity operations in [21–23] and hypersonic entry trajectory optimization in [24–26]. However, all of the preceding problems were fixed-final-time problems with an exception of [16], where a line-search technique was applied to solve for the optimal time of flight (TOF). It is more challenging to solve free-final-time problems because new independent variables that are monotonically increasing need to be introduced to rewrite the equations of motion in a manner that can be readily discretized into convex optimization problems. For example, the equations of motion for hypersonic entry flight were reformulated with respect to energy in [25], and the TOF could be free. To address the free-time closed-loop optimal guidance for constrained impact and reduce the nonlinearity in the dynamics, the range was used as the independent variable for the kinematic equations of motion in [27]. Additionally, to solve the optimal guidance design problem for aerodynamically controlled missiles, a new independent variable was defined to rewrite the equations of motion in [28] by assuming that the new independent variable is monotonically increasing. The resulting problems were solved by a successive SOCP method. A convex approach was proposed in our previous research work in [29] to solve a fuel-optimal low-thrust transfer problem with a fixed TOF. The solutions showed that if both the thrust magnitude and thrust direction are unconstrained, the spacecraft will begin the transfer with a short burn, then coast along the trajectory before reaching the target orbit with another short burn. The time-optimal trajectory optimization is another large class of problems for low-thrust transfers, and the thrust magnitude is expected to employ maximum thrust for the entire trajectory. Unfortunately, the existing convex methods cannot be directly applied to address this type of problems. Instead, these problems need to be reformulated into a new form that can ultimately be converted and solved using convex optimization. Motivated by the preceding research, a new problem formulation is first developed by introducing a new independent variable with a monotonically increasing profile. Then, the nonlinearity of the dynamics is reduced through a change of variables, and a series of convexifications are implemented via successive linearization of the dynamics and relaxations of the control constraints. Finally, a sequential convex programming (SCP) method is developed to find an approximate optimal solution to the original problem. The convergence and computational performance of this approach are demonstrated by numerical solutions of an Earth-to-Mars time-optimal transfer problem with multiple revolutions.
II.
Problem Formulation
A. Equations of Motion
In this paper, we focus on low-thrust planar orbit transfer problems, and the polar coordinates are chosen to describe the motion of the spacecraft shown in Fig. 1. For simplicity, only the gravitational effects of the central body are considered in this paper. For the case of general planet-to-planet orbit transfer, we assume that the spacecraft is far from other gravitational bodies for the bulk of the trajectory, and the trajectory begins and ends at the boundary of the spheres of influence. Thus, for interplanetary
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Table 1 Scale Distance unit Velocity unit Time unit Mass unit Force unit
Scales for normalization Value Initial radial distance Rp 0 Initial circular velocity V 0 μ∕R0 R0 ∕V 0 Initial mass m0 Maximum thrust T max
transfers, the gravitational effects of the origin and target planets are neglected. The equations of motion are as follows [30]:
v_ r
r_ vr
(1)
v θ_ t r
(2)
v2t μ T sin η − m r r2
(3)
vr vt T cos η m r
(4)
T ve
(5)
v_ t −
_ − m
where r is the radial distance from the central body to the spacecraft; θ is the transfer angle measured from r0 at the start point; vr is the radial component of velocity; vt is the tangential component of velocity; and m is the mass of the spacecraft. The thrust is assumed to be noncontinuous, and its direction is controllable. As such, the control variables are the thrust magnitude T and thrust direction angle η, which is defined as the angle between the thrust vector and the local horizon. The variable μ is the gravitational constant of the central body, and ve is the exhaust velocity of the engine. The preceding differential equations are with respect to time t. To avoid an ill-conditioned problem formulation and improve the convergence of the solution method, a series of constants are introduced in Table 1 as reference values to normalize the variables and parameters in the problem. After applying the normalization, the nondimensional equations of motion take the following form:
v_r
r_ vr
(6)
v θ_ t r
(7)
v2t 1 cT sin η − m r r2
(8)
vr vt cT cos η r m
(9)
cT ve
(10)
v_t −
_ − m
Fig. 1
Polar coordinates for low-thrust orbit transfers.
where c T max R0 ∕m0 V 20 is constant. Note that the gravitational constant μ becomes 1 after normalization. The angles θ and η are not normalized because they are in radians and close to unity over most of the domain of interest. Additionally, the state and control constraints that will be discussed later are also well-scaled based on this simple normalization method.
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B. Change of Independent Variable
A free-final-time optimal control problem associated with low-thrust transfers is considered in this paper. In fact, a popular way to solve free TOF problems is to normalize the time to τ such that τ ∈ 0; 1. Then, it can be solved as a fixed-flight-time problem, in which the actual flight time tf is also optimized. However, this approach further increases the nonlinearity of the dynamics because we must multiply the dynamics by the tf parameter. When the time is normalized in this manner, the dynamics will be more difficult to be convexified if we want to implement convex optimization methods. In this section, we will develop an alternative approach that is compatible with convex optimization approaches by introducing a new independent variable and reformulating the equations of motion. Theoretically, any variable with a monotonically increasing trend and fixed boundary conditions can achieve this goal. From Eqs. (1) and (2), we can see that both radial distance r with positive vr and transfer angle θ with positive vt can be chosen as the new independent variable; however, the simulations show that the variable with smoother and more linearly increasing profile achieves better convergence performance. Numerical simulations in Sec. V will show that θ has advantages over r in this respect. Additionally, invalid linearization of the dynamics may occur when choosing r as the independent variable because infinitely large terms will emerge when vr tends to zero at the endpoint. As such, θ is selected as the new independent variable for the transfer problem considered in this paper. To begin with, we divide Eqs. (6) and (8–10) by Eq. (7) to make θ as a new independent variable by assuming that vt is always positive. This ensures that θ is monotonically increasing during the transfer. Then, the new equations of motions become r0
vr0 vt −
rvr vt
1 crT sin η rvt mvt
vt0 −vr
crT cos η mvt
crT m0 − ve vt t0
r vt
(11)
(12)
(13)
where rθ0 is the radius of the initial orbit. Based on the circular orbit assumptions, the radial and tangentialpcomponents of the initial velocity are vr θ0 0 and vt θ0 μ∕rθ0 , respectively. The initial mass mθ0 is specified, and the initial time is simply chosen as tθ0 0. The variable rθf is the radius of the target orbit. For the p circular target orbit, vr θf 0 and vt θf μ∕rθf . Because a minimum-time trajectory optimization problem is considered in this paper, tθf is free, and the constraint on mθf is not enforced. Based on the assumption of a monotonic increase of θ, the tangential component of velocity vt should remain positive during the transfer. In addition, the other state variables should be limited in certain ranges, and the thrust should not exceed the maximum value. As such, the following state and control constraints are used: 2 3 2 3 2 3 0.1 10 r 6 7 6 7 6 7 6 −10 7 6 vr 7 6 10 7 6 7 6 7 6 7 6 7 6 7 6 7 6 1e − 5 7 ≤ 6 vt 7 ≤ 6 10 7 (19) 6 7 6 7 6 7 6 7 6 7 6 7 6 0.1 7 6 m 7 6 1 7 4 5 4 5 4 5 t
0 "
0 −π
# ≤
" # T η
∞
≤
" # 1 π
(14)
(20)
where both the state and control variables are bounded for practical reasons, and the lower and upper bounds are also nondimensional. For example, the lower bounds of r, vt , and m should be 0; however, it will cause singularities in the dynamics. As such, more realistic values are chosen for the lower and upper bounds of these variables. Similarly, lower and upper bounds are also defined for vr, t, T, and η. Because the TOF is to be optimized, no upper bound is needed for t; however, for a specific amount of fuel onboard, the maximum TOF should be defined without changing the results of the solution method discussed in this paper. Based on the preceding transformations, the objective to minimize the time of fight is defined as follows: J tθf
(21)
Then, a minimum-time low-thrust transfer problem can be formulated and posed as an optimal control problem shown next. Problem 1:
(15)
Minimize:Eq: 21 T;η
where the differentiations are now with respect to θ, and Eq. (15) is a reformulation of Eq. (7). In this manner, the time t becomes a state variable in the new dynamics. As such, the state vector of a generic planar orbit transfer can be denoted as x r; vr ; vt ; m; t, and the equations of motion defined in Eqs. (11–15) can be represented as a state-space formulation as follows: x 0 fx; u
3
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(16)
where u T; η is the control vector. As can be seen, the control vector is highly coupled with the states through the thrust terms of crT sin η∕mvt , crT cos η∕mvt , and crT∕ve vt . C. Optimal Control Problem
For the low-thrust transfer problem considered in this paper, the spacecraft is assumed to start from an initial circular orbit to a target circular orbit within a specific range of θ ∈ θ0 ; θf . Therefore, the initial and terminal conditions can be defined as follows:
Subject to: Eqs:16–20 where the optimal thrust magnitude profile Tθ and thrust angle profile ηθ that minimize the performance index in Eq. (21) are found and satisfy the dynamics in Eq. (16), boundary conditions in Eqs. (17) and (18), and state and control constraints in Eqs. (19) and (20). Note that the control constraints defined in Eq. (20) are originally convex; however, the dynamics in Eqs. (11–15) are nonlinear, and the states and controls are highly coupled through trigonometric and reciprocal terms, which will bring difficulties to the convergence of both NLP algorithms and the convex optimization method that will be developed later in this paper. To convert and solve problem 1 using convex optimization methods, a series of relaxation techniques are applied, and the main technical results will be presented in the following sections.
III.
Problem Convexification
xθ0 rθ0 ; vr θ0 ; vt θ0 ; mθ0 ; tθ0
(17)
A. Change of Variables
xθf rθf ; vr θf ; vt θf
(18)
To reduce the coupling of the states and controls in the dynamics, a change of variables is introduced to reformulate the problem.
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2
To begin with, the following two new variables are defined: T u m
(22)
z ln m
(23)
From Eq. (14), we have z0
m0 crT cr − − u m mve vt ve vt
(24)
Then, the equations of motion can be written as rvr vt
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r0
1 cr u sin η rvt vt
vr0 vt −
vt0 −vr
cr u cos η vt
cr z0 − u ve vt t0
r vt
(26)
(27)
(28)
(30)
where ur and ut are the radial and tangential components of the control variable u, respectively, and the following constraint must hold: u2r u2t u2
6 7 6 vt − 1∕rvt 7 6 7 6 7 7; fx 6 −v r 6 7 6 7 6 7 0 4 5 r∕vt 2 3 0 0 0 6 7 6 cr∕vt 7 0 0 6 7 6 7 6 7 Bx 6 0 cr∕vt 0 7 6 7 6 0 0 −cr∕ve vt 7 4 5 0
(29)
ut u cos η
(31)
rv r r vt 1 cr u vr0 vt − rvt vt r vt0 −vr
z0 −
t0
cr u vt t
cr u ve vt
r vt
(32)
0
xθ0 rθ0 ; vr θ0 ; vt θ0 ; zθ0 ; tθ0
(39)
xθf rθf ; vr θf ; vt θf
(40)
where zθ0 ln mθ0 0 is the initial value of z, and zθf is free. All the other values are the same as those defined in Eqs. (17) and (18). Meanwhile, the state and control constraints become 2
0.1
3
2
r
3
2
10
3
6 7 7 7 6 6 −10 7 6 7 vr 7 6 6 7 6 6 10 7 7 6 6 7 6 7 6 7 6 1e − 5 7 ≤ 6 vt 7 ≤ 6 10 7 6 7 6 7 6 7 6 7 7 7 6 6 ln 0.1 7 6 6 0 7 z 5 4 4 5 4 5 t 0 ∞ −10 −10
#
" ≤
ur ut
#
" ≤
10
(33)
(34)
(35)
(36)
10
(42)
(43)
where the lower and upper bounds of z are enforced based on the constraint on m in Eq. (19). From the constraint on T defined in Eq. (20), we can get 0 ≤ u ≤ 1∕m; thus, we have Eqs. (42) and (43). Notice that all of the preceding state and control constraints are convex except for Eq. (43). We will apply a first-order Taylor series expansion to convexify the right part of inequality (43) and obtain 0 ≤ u ≤ e−z 1 − z − z
(44)
where z θ is a given history of zθ. Next, we will show that a solution that satisfies the constraint in Eq. (44) also satisfies the constraint in Eq. (43). The upper bound of u has been approximated by a first-order Taylor series expansion as shown in Eq. (44). To prove the preceding statement, we will approximate e−z using a Taylor series expansion up to quadratic terms around z and obtain [16]
(37)
where x r; vr ; vt ; z; t is the new state vector, and the control vector becomes u ur ; ut ; u. The column vectors fx ∈ R5 and Bx ∈ R5×3 are shown next:
(41)
#
0 ≤ u ≤ e−z
The corresponding state-space formulation of the preceding equations of motion takes the following form: x 0 fx Bxu
0
(38)
It can be seen that the nonlinearity and the coupling of the states and control have been reduced in the new dynamics, which is helpful in improving the convergence of the sequential convex method developed in the following section. However, a nonconvex control constraint in Eq. (31) is introduced along with the change of variables, but it will be relaxed to a convex form later in this section. With the new variables, the initial and terminal conditions become
"
Then, the dynamics become 0
3
rvr ∕vt
(25)
To further reduce the nonlinearity of the dynamics, another two variables are introduced: ur u sin η and
/ WANG AND GRANT
e−z e−z 1 − z − z e−^z
z − z 2 2
where z^ ∈ z ; z. Because the term e−^z z − z 2 ∕2 ≥ 0, we can get
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e−z 1 − z − z ≤ e−z which proves that the upper bound in Eq. (44) will not exceed the upper bound in Eq. (43), and a feasible solution that satisfies Eq. (44) will also satisfy Eq. (43). The objective will remain the same as before, and an approximate optimal control problem of problem 1 can be obtained in terms of the new control variables and updated constraints as follows. Problem 2: Minimize: Eq:21 ur ;ut ;u
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Subject to: Eqs: 31; 37; 39–42; 44 Remark 1: The dynamics in Eq. (37), boundary conditions in Eqs. (39) and (40), and state and control constraints in Eqs. (31), (41), and (42) are directly reformulated from problem 1 through a change of variables. Thus, they are equivalent to those in problem 1. The only difference between problem 1 and problem 2 is the approximation of constraint (43) by Eq. (44), and we have proved that the feasible set defined by Eq. (44) is contained in the feasible set defined by Eq. (43). As such, problem 2 is an effective approximation of problem 1, and if there exists a feasible solution to problem 2, then this solution also defines a feasible solution to problem 1. B. Relaxation of Nonconvex Control Constraint
Problem 2 is a nonlinear optimal control problem with the equality quadratic constraint in Eq. (31) and the nonlinear dynamics in Eq. (37). In this subsection, we will handle the nonconvex control constraint first. The control space determined by Eq. (31) represents the surface of a second-order cone, which is nonconvex because the interior of the cone does not belong to the domain of the control. This feature will add difficulties in achieving rapid convergence for NLP algorithms and cannot be handled by convex optimization. One option to address this constraint is to linearize it, but the resulting linear approximation will depend on the control, and high-frequency jitters will appear in the solutions. Another convexification technique is to relax the nonconvex constraint to a convex one by expanding its feasible set [16,21,25]. In this section, we will relax the equality nonconvex constraint in Eq. (31) into a solid second-order cone represented by an inequality constraint shown next: u2r u2t ≤ u2
(45)
An illustration of this control constraint convexification is depicted in Fig. 2. As such, the nonconvex control set represented by the surface of a cone is mapped into a convex set defined by a solid convex cone. With the relaxation of the control constraint, a new optimal control problem can be formulated as follows.
Fig. 2
Convexification of control constraint.
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Problem 3: Minimize: Eq: 21 ur ;ut ;u
Subject to: Eqs: 37; 39–42; 44; 45 Proposition 1: The solution of the relaxed problem 3 is identical to the solution of problem 2. That is to say, if x θ; ur θ; ut θ; u θ is a solution to problem 3 over a fixed interval [θ0 , θf ], then it is also a solution to problem 2, and ur θ2 ut θ2 u θ2 . Proof: See Appendix A. Proposition 2: Let Γ represent the set of all feasible states and control for problem 3 (i.e., x⋅; u⋅ ∈ Γ), which implies that, for θ ∈ θ0 ; θf , xθ and uθ define a feasible state history and control signal for problem 3. If Γ is nonempty, then there exists an optimal solution to problem 3. Proof: See Appendix B. C. Handling Nonlinear Dynamics
After replacing the nonlinear control constraint in Eq. (31) with the relaxed one in Eq. (45), the only nonconvex component in problem 3 is the nonlinear dynamics defined in Eq. (37). Because convex optimization requires all the equality constraints to be linear, the nonlinear dynamics should be converted from the form of Eq. (37) into linear dynamics, which can be discretized into linear equality constraints. A successive small-disturbance-based linearization method is used to approximate the dynamics, and a sequential convex method will be developed in Sec. IV. Suppose that x θ is a fixed state history, which could be the solution from the kth iteration, xk θ, in the successive method that will be discussed later. Then, the dynamics in Eq. (37) can be approximated as follows: x 0 fx Ax x − x Bx u
(46)
with 2
vr ∕vt
r∕vt
−rvr ∕v2t
6 6 1∕r2 vt 0 1 1∕rv2t 6 6 ∂fx 6 Ax 0 −1 0 ∂x xx 6 6 6 0 0 0 4 1∕vt 0 −r∕v2t 2 3 0 0 0 6 7 6 cr∕vt 7 0 0 6 7 6 7 7 0 cr∕v Bx 6 0 t 6 7 6 7 6 0 7 0 −cr∕v v e t 5 4 0
0
0
0 0
3
7 0 07 7 7 0 07 7 7 0 07 5 0 0 xx
xx
where the nonlinear term fx is linearized, and Bx is replaced by Bx . A variety of linearization methods can be found in rendezvous and proximity studies, which can be classified in different ways according to the choice of coordinate frames, linearization parameters, and nominal orbits [31–33]. However, the linearized dynamics in Eq. (46) is different from the conventional linearization methods because the nominal transfer orbit is chosen as the solution from the previous iteration, and we do not linearize the term Bxu. This feature will eliminate the high-frequency jitters in the control profiles and improve the convergence of the SCP method because the information of u is not required for Eq. (46), and the optimal control from the iteration k, uk , will not affect the current iteration for the successive solution procedure developed in the following section. In addition, a trust-region constraint is enforced along with the linearization of dynamics and is shown as follows:
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kx − x k ≤ δ
(47)
where the variable δ defines the trust-region radius. This technique has been demonstrated to be very effective in improving the convergence of the sequential methods developed for various types of problems. With the approximation of the nonlinear dynamics in Eq. (37) by Eqs. (46) and (47), problem 3 can be converted into problem 4 as follows, which is a continuous-time convex optimal control problem with respect to a given state history x. Problem 4: Minimize: Eq: 21 ur ;ut ;u
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k−1
0 ≤ u ≤ e−z
u2r u2t ≤ u2 kx − xk−1 k ≤ δ; kx − xk−1 k ≤ γkxk−1 − xk−2 k;
Sequential Convex Programming Method
Similar to the successive solution methods developed for near-filed rendezvous and hypersonic entry trajectory optimization [21,24,25], a sequential convex programming method is devised in this section to approximately solve problem 1. In this successive approach, a sequence of convex optimal control subproblems defined by problem 4 are formed using the solution from the previous iteration. 1) Set k 0. Define the initial states xθ0 rθ0 ; vr θ0 ; vt θ0 ; zθ0 ; tθ0 . Propagate the equations of motion in Eqs. (32–36) with these initial conditions and a specific control to provide an initial trajectory x0 for the solution procedure. 2) For k ≥ 1, solve the following optimal control problem (similar to problem 4) to find a solution pair {xk , uk }. For convenience, the problem is written next with an additional constraint shown in Eq. (57). Minimize the objective functional: J tθf
(48)
Subject to x_ fxk−1 Axk−1 x − xk−1 Bxk−1 u
(49)
xθ0 rθ0 ; vr θ0 ; vt θ0 ; zθ0 ; tθ0
(50)
xθf rθf ; vr θf ; vt θf
(51)
3
2
r
3
2
10
3
6 7 7 7 6 6 −10 7 6 7 vr 7 6 6 10 7 6 7 6 7 6 6 7 7 6 7 6 6 1e − 5 7 ≤ 6 vt 7 ≤ 6 10 7 6 7 7 6 7 6 6 7 7 7 6 6 6 ln 0.1 7 6 7 z 0 4 5 5 4 5 4 t ∞ 0
k1 γ ∈ 0;1 and k > 1
(56) (57)
10 −10 ur ≤ ≤ ut 10 −10
(52)
(53)
k>1
(58)
where ε is a prescribed tolerance value for convergence. If the condition in Eq. (58) is satisfied, go to step 4; otherwise, set k k 1 and go to step 2. 4) The solution of the problem is found to be x xk and u uk . Remark 3: For each k ≥ 1, a nonlinear optimal control problem is defined by Eqs. (48–57). Because only linear time-varying dynamics, affine equality constraints, and second-order cone inequality constraints are included in each subproblem, it can be discretized into an SOCP problem and solved efficiently by IPM. Note that, except for the trust-region constraint introduced in Eq. (56), a convergence technique is imposed in Eq. (57) to guarantee the convergence of the solution procedure as long as it can be satisfied in each iteration. This constraint will form a Cauchy sequence through the successive process and enforce convergence to a final solution. Remark 4: To find the numerical solution to each continuous-time subproblem defined by Eqs. (48–57), a trapezoidal discretization method is applied to convert each subproblem to a finite-dimensional parameter optimization problem [24]. With the enforcement of the trust-region constraint in Eq. (56) and the Cauchy sequence in Eq. (57), the convergence of the solution procedure is expected. Our simulation results also show a strong evidence of convergence of the SCP method.
V.
Numerical Simulations
In this section, an interplanetary minimum-time transfer problem from Earth to Mars is used to assess the effectiveness and performance of our proposed method. First, the convergence and optimality of the successive SOCP approach are verified using a half-revolution transfer case. Then, the effectiveness and computational performance of the SCP method is demonstrated through comparisons with a state-of-theart optimal control solver for the cases with a greater number of revolutions. In the simulations, a xenon ion propulsion system is used for the low-thrust orbit transfer problem, and the specific impulse is assumed to be constant over the entire thrust range. The parameters used for simulations are provided in Table 2.
Table 2
0.1
(54) (55)
sup kxk − xk−1 k ≤ ε;
θ0 ≤θ≤θf
Remark 2: After a series of transformations and relaxations, our goal is to find an optimal solution to problem 1 by solving for an optimal solution to problem 3. To implement convex optimization, the nonlinear dynamics are convexified to a linear time-varying system, and a convex optimal control problem 4 is obtained. However, we cannot solve a single problem 4 to get an optimal solution to problem 1. Instead, a successive approach can be designed to find an approximate optimal solution to problem 1. The linearization of dynamics discussed previously will be applied in the next section to formulate a sequential SOCP approach to the low-thrust transfer problem, and numerical results in Sec. V will show that the convergence can be achieved very quickly.
2
1 − z − zk−1
3) Check the convergence condition:
Subject to: Eqs: 39–42; 44–47
IV.
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Parameters for simulations
Parameter Gravitational constant of the sun (μ) Earth–sun distance RE [rθ0 ] Mars–sun distance RM [rθf ] Initial transfer angle θ0 Initial radial velocity vr θ0 Initial tangential velocity vt θ0 Initial mass mθ0 Initial time tθ0 Terminal radial velocity vr θf Terminal tangential velocity vt θf Terminal mass mθf Time of flight (tf ) Specific impulse Isp
Value 1.3271244e20 m3 ∕s2 1.49597870e11 m (1 AU) 2.2793664e11 m (1.52 AU) 0 0 p μ∕RE 1000 kg 0 0 p μ∕RM Free Free 2000 s
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A. Demonstration of Convergence
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A low-thrust half-revolution (N rev 0.5) transfer with the maximum thrust magnitude T max 0.5 N and a specified terminal transfer angle θf π is considered to demonstrate the convergence of the SCP method. The methodology described in this paper has been implemented in ECOS, an interior-point solver that applies a standard primal–dual Mehrotra predictor–corrector method with Nesterov– Todd scaling and self-dual embedding techniques for SOCP problems [34]. In the simulations, the trajectory is discretized into N 300 nodes. The trust-region size in Eq. (56) and the stopping criteria in Eq. (58) are selected as 5e10 m 2000 m∕s 2000 m∕s 50 days T ; ; ; 50 kg; δ R0 V0 V0 R0 ∕V 0 1000 m 1e − 3 m∕s 1e − 3 m∕s 1e − 5 days T ε ; ; ; 1e − 5 kg; R0 V0 V0 R0 ∕V 0 The results are obtained and shown in Figs. 3–7 and Table 3. Figure 3 shows that the successive method converges, and the value of the objective function (time of flight) decreases to the converged solution in 12 iterations for the specified tolerance value. The minimum time for transfer is 230.84 days with an arrival mass of
Fig. 5
Convergence of radial distance profile with Nrev 0.5.
Fig. 6
Fig. 3
Convergence of mass profile with Nrev 0.5.
Convergence of the objective with Nrev 0.5.
Fig. 7 Convergence of thrust magnitude profile with Nrev 0.5.
Fig. 4
Convergence of Δr between consecutive steps with Nrev 0.5.
491.72 kg. The corresponding changes of r between consecutive iterations are shown in Fig. 4. Figures 5–7 present the convergence of the radial distance profiles, mass profiles, and the thrust magnitude
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Table 3
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Iteration number 1 2 3 4 5 6 7 8 9 10 11 12
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Difference of the states between consecutive iterations with Nrev 0.5
jΔrj, m 1.0001e 10 2.3535e 09 3.9335e 08 1.0269e 08 2.2862e 07 4.9971e 06 1.0926e 06 2.3682e 05 5.1221e 04 1.1051e 04 2.3797e 03 511.8855
jΔvr j, m∕s 621.5617 558.3815 107.4143 24.6566 5.4709 1.1785 0.2550 0.0549 0.0118 0.0025 5.4567e − 04 1.1721e − 04
profiles, respectively. To make the progression of the convergence clearer, the profiles for all iterations are depicted by curves from the initial guess to the converged solution in Figs. 5–7. As can be seen from the zoomed-in views, the solutions are quite close after about four iterations. To further demonstrate the convergence performance of the successive procedure, more quantitative results are reported in Table 3, which summarizes the difference of each state variable between consecutive steps for all 12 iterations. The difference of the states shown in the table is defined as jΔyj ≔ Maxjyk θi − yk−1 θi j, i 1; 2; : : : ; N, where y could be any state variable.
jΔvt j, m∕s 963.8598 780.3895 207.0004 49.0732 10.2827 2.1326 0.4441 0.0933 0.0198 0.0042 9.0593e − 04 1.9476e − 04
jΔmj, kg 23.7088 7.4633 1.6842 0.3794 0.0830 0.0480 0.0128 0.0039 8.3405e − 04 1.7948e − 04 3.8584e − 05 8.2898e − 06
jΔtj, day 11.0594 3.6347 0.7203 0.1751 0.0375 0.0082 0.0018 3.7874e − 04 8.1496e − 05 1.7520e − 05 3.7642e − 06 8.0840e − 07
solutions are compared to the results obtained by GPOPS-II. GPOPS is a general-purpose Matlab software program for solving multiple-phase optimal control problems, and it employs a Legendre–Gauss–Radau quadrature orthogonal collocation method to convert the continuoustime problem to a large sparse NLP problem [35]. An adaptive mesh refinement method is implemented to determine the required number of nodes and the degree of the approximating polynomial to achieve a specified accuracy. The default NLP solver used by GPOPS is
B. Verification of Feasibility and Optimality
Next, we will verify the feasibility and optimality of the converged solution obtained in Sec. V.A for the proposed SCP approach. First, based on the obtained optimal controls u θ ur θ; ut θ; u θ, 2 2 the values of u2 r θ ut θ − u θ are calculated and shown in 2 2 Fig. 8. We can see that the value of ur θ u2 t θ − u θ is close to zero during the entire transfer, which indicates that the relaxed control constraint in Eq. (45) remains active along the optimal trajectory. As such, proposition 1 is numerically validated. Second, to demonstrate the optimality of the obtained solution, the optimal control histories from SCP are used to propagate the equations of motion in Eqs. (32–36) from the same initial condition. The results are shown in Figs. 9–11. The circle profiles represent the optimal solution from SCP, and the solid curves are the propagated state histories. We can see that the propagated states are in excellent agreement with the SCP states, which demonstrates the optimality and accuracy of the converged solution. C. Comparison with GPOPS
To further demonstrate the effectiveness and accuracy of the SCP method proposed for the time-optimal low-thrust transfer problem, the
Fig. 8 Equivalence of control constraint relaxation with Nrev 0.5.
Fig. 9
Comparisons to the propagated state profiles with Nrev 0.5.
Fig. 10 Comparisons to the propagated state profiles with Nrev 0.5.
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Fig. 11 Comparison to the propagated time profile with Nrev 0.5.
Fig. 14 Comparison of time profiles with Nrev 0.5.
Fig. 12 Comparisons of state profiles with Nrev 0.5.
Fig. 15 Comparisons of thrust magnitude and thrust angle profiles with Nrev 0.5.
Fig. 13 Comparisons of state profiles with Nrev 0.5.
Fig. 16 Minimum-time transfer by SCP with Nrev 0.5.
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Fig. 17 Minimum-time transfer by GPOPS with Nrev 0.5.
Fig. 18 CPU time cost by SCP with Nrev 0.5.
IPOPT [36]. In the simulations, GPOPS solves the optimal control problem defined by problem 3, with the only exception that the control constraint in Eq. (44) is replaced by Eq. (43), to show the accuracy of the linearized constraint in Eq. (44) used for the successive convex approach. The results obtained from both SCP and GPOPS are compared and shown in Figs. 12–17 and Table 4. The dash-dotted curves are the SCP solutions, and the solid profiles are the results from GPOPS. Figures 12–14 show that the histories of the nondimensional state variables match extremely well under both methods with the transfer angle θ as the independent variable. The radius vector transitions from 1 to 1.52 astronomical units (AU) very smoothly, with the transfer angle increasing to a final value of π. The mass profiles have the same trend, and both decrease to similar final values. The radial component of velocity decreases to negative values in the beginning, then increases to a maximum value before decreasing to achieve a zero arrival radial velocity. The tangential component of velocity increases to 1.065 in the beginning before decreasing to match Mars’ nondimensional circular velocity of 0.81. Figure 15 shows the control histories during the transfer. The SCP and GPOPS controls are in excellent agreement for both the thrust magnitude and thrust direction angle. As expected, the engines employ the maximum thrust for the minimum-time transfer during the entire flight, and the constant-thrust solution is well known for the minimum-time problem considered in this paper. The heliocentric transfer orbits are shown in Figs. 16 and 17 for SCP and GPOPS, respectively. The viewpoint is from the north pole of the solar ecliptic plane. The arrows are oriented with the thrust direction and scaled to the magnitude of the thrust vector. The time-optimal trajectories can be observed from these two heliocentric views. The engines operate during the entire transfer. The thrust angle switches between inward directions and outward directions. In addition, the optimal solutions from these two methods are reported in Table 4, which indicates that the converged solution of the SCP method is very similar to the optimal solution by GPOPS. All the terminal constraints are satisfied. The CPU time costs under both approaches are recorded as well. In general, solving low-thrust planar transfer problems takes considerably less computational time than solving more complicated problems such as hypersonic trajectory optimization due to the fewer number of states and less complicated dynamics in planar low-thrust transfers. In our simulations, it takes
GPOPS more than 1 min to converge for the half-revolution transfer problem with default parameter settings. In contrast, Fig. 18 shows that it takes less than 0.5 s to solve each subproblem and about 5 s to converge for the SCP method in Matlab on a MacBook Pro with a 64-bit Mac OS and an Intel Core i5 2.5 GHz processor. If smaller tolerances are chosen, then more iterations and CPU time will be required; however, the proposed SCP method still remains more computationally efficient than the NLP method used for comparisons in this paper. For example, if we reduce the tolerance on the radial distance from 1000 to 10 m, it will take about 15 iterations and 7 s for the SCP method to converge on the half-revolution transfer problem. Similar phenomena can be observed when different numbers of discretized nodes are used in the simulations. For example, it takes less than 2 s for the SCP method to converge under the tolerance defined in Sec. V.A with 100 nodes used. More CPU time will be required when larger numbers of nodes are chosen; however, the SCP method still remains faster than GPOPS when the solutions are obtained with the similar accuracy. The following subsection will show more evidence on the benefit of the proposed SCP method in solving multirevolution transfers with larger numbers of discretized nodes.
Table 4
D. Multirevolution Transfers
Compared to classical orbit transfers, the low-thrust trajectories not only require a much longer TOF but also tend to perform many revolutions around the central body. As such, we consider the cases requiring many revolutions to further demonstrate the computational performance of the proposed SCP method by comparing against the performance of GPOPS. To this end, low-thrust transfers with various low levels of thrust T max and different values of the number of revolutions (N rev ) are considered. In all of the cases, we will use the same normalization scales in Table 1 and the same parameter settings in Table 2. Table 5 provides the details of the performance of both SCP and GPOPS for five different cases, and Figs. 19–21 depict the minimumtime transfer trajectories of the cases with N rev 5, N rev 20, and N rev 50, respectively. The minimum TOF for each thrust level is provided in Table 5. The solutions under these two approaches are very close, and as expected, the TOF increases for lower thrust values and higher number of revolutions. As can be seen, GPOPS is much more time-consuming than the SCP method for transfers with higher number
Comparison of solutions with Nrev 0.5
Method
rθf , m
vr θf , km∕s
vt θf , km∕s
mθf , kg
tθf , day
CPU time, s
SCP GPOPS
2.2794e11 2.2794e11
−8.6437e − 11 0
24.1293 24.1293
491.7253 491.8123
230.8407 230.8019
5.38 69.71
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Table 5
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N rev 2 5 10 20 50
T max , N 0.3 0.1 0.02 0.01 0.005
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Comparison of the performance for different thrust levels with different Nrev TOF (SCP), day 561.5 1460.9 4100.8 8194.7 19020.4
TOF (GPOPS), day 561.1 1460.7 4100.7 8194.6 19020.4
Fig. 19 Minimum-time transfer by SCP with Nrev 5.
CPU time (SCP), s 14.3 28.7 42.8 79.5 182.3
Fig. 21 Minimum-time transfer by SCP with Nrev 50.
VI.
Fig. 20 Minimum-time transfer by SCP with Nrev 20.
CPU time (GPOPS), s 125.4 365.7 874.6 2135.5 6378.6
Conclusions
In this paper, a direct convex optimization approach has been proposed to solve the problem of minimum-time low-thrust orbit transfers. By introducing a new independent variable, the time-optimal orbital transfer problem is reformulated as a constrained nonlinear optimal control problem. Through a change of variables and relaxation of the control constraints, the nonlinearity of the problem is reduced, and a sequential convex programming method is developed to solve the problem. The convergence of the proposed SCP method is demonstrated by an Earth-to-Mars interplanetary low-thrust transfer problem, and its accuracy and computational performance are also studied by comparing with GPOPS for the cases with lower levels of thrust and higher number of revolutions. This work is not intended to study high-fidelity low-thrust orbit transfer problems. The goal is to demonstrate that convex optimization can be successfully applied to the optimization of minimum-time many-revolution transfers. The simulation results indicate that the SCP method is capable of generating optimal solutions for low-thrust orbit transfer problems with relatively stable convergence and fast computational speed. Consequently, the proposed method has great potential for onboard applications.
Appendix A: Proof of Proposition 1 of revolutions. The reason is that it becomes more challenging for the NLP algorithms when solving problems with larger scales and longer durations of orbit transfers. In addition, it is known that, for problems with longer TOF and higher number of revolutions, computational difficulty might be encountered. In our simulations, the convergence of GPOPS is unpredictable, even when solving the same problem. In contrast, the SCP method is generally more stable and faster than GPOPS. As the thrust magnitudes become smaller and the number of revolutions increases, the benefit of using our proposed method becomes noticeable. As such, the SCP method developed in this paper has great potential of onboard applications for low-thrust orbit transfer problems.
We can prove the statement using the minimum principle. The Hamiltonian for problem 3 is as follows: rv 1 cr cr Hx; u; p p1 r p2 vt − ur p3 −vr ut rvt vt vt vt cr r u − p5 (A1) p4 − ve vt vt where x r; vr ; vt ; z; t is the state vector, u ur ; ut ; u is the control vector, and p p1 ; p2 ; p3 ; p4 ; p5 is the costate vector. Suppose that x θ is the optimal state solution to problem 3, and
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p θ is the corresponding costate vector; the optimal control u θ ur θ; ut θ; u θ is determined by pointwise minimization of H in Eq. (A1) with respect to ur , ut , and u [37]: Hx θ; u θ; p θ min Hx θ; uθ; p θ; θ ∈ θ0 ; θf u∈U
(A2) where U fur ; ut ; u:u2r u2t ≤ u2 ; 0 ≤ u ≤ e−z 1 − z − z ;
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In this paper, θf is fixed, and the dynamics are autonomous; thus, H is constant along the optimal trajectory. For each u, the minimization of H with respect to ur ; ut over the following convex set can be performed: Uur ;ut fur ; ut :u2r u2t ≤ u2 ; −10 ≤ ur ≤ 10; −10 ≤ ut ≤ 10g To derive the Karush–Kuhn–Tucker (KKT) conditions for minimizing H over the preceding constraint, we need to introduce the Lagrangian L:R3 × R5 → R as follows: Lu; λ Hx ; u; p λ1 u2r u2t − u2 λ2 ur − 10 (A3)
where λ λ1 ; λ2 ; λ3 ; λ4 ; λ5 is the Lagrange multiplier vector. The KKT conditions for minimizing H with respect to ur ; ut are shown next: ∂L cr p2 2λ1 ur λ2 − λ3 0 ∂ur vt
(A4)
∂L cr p3 2λ1 ut λ4 − λ5 0 ∂ut vt
(A5)
where (
5) Consider the following set Q θ; x for problem 3: rv 1 cr Q θ;x a1 ;a2 ;a3 ;a4 ;a5 :a1 r ;a2 vt − ur ; rvt vt vt cr cr r u;a5 ;0≤u≤e−z 1−z−z ; a3 −vr ut ;a4 − vt ve vt vt u2r u2t ≤u2 ;−10≤ur ≤10;−10≤ut ≤10 which indicates that a1 ; a2 ; a3 ; a4 ; a5 ∈ Q θ; x if and only if the following conditions are satisfied:
− 10 ≤ ur ≤ 10; −10 ≤ ut ≤ 10g
λ3 −ur − 10 λ4 ut − 10 λ5 −ut − 10
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2 2 λ1 0 if u2 r ut < u 2 2 λ1 > 0 if u2 r ut u
This condition is known as complementary slackness, which roughly means that the ith optimal Lagrange multiplier is zero unless the ith constraint is active at the optimum [13]. The optimality conditions in Eqs. (A4) and (A5) show that a unique solution for optimal ur ; ut exists only when λ1 ≠ 0; thus, λ1 > 0 and 2 2 u2 r ut u . Consequently, the constraint in Eq. (45) must be 2 2 active along the optimal trajectory (i.e., u2 r ut u ), which proves that the solutions to problem 2 and problem 3 are identical.
Appendix B: Proof of Proposition 2 An existence theorem from [38] is used to complete the proof. 1) Because θf is fixed for the time-optimal low-thrust transfer problem considered in this paper and the fuel onboard is limited, lower and upper bounds are enforced in Eq. (41) on the states. Thus, there should exist a compact set K such that, for all feasible trajectories x⋅, we have θ; xθ ∈ K for all θ ∈ θ0 ; θf . 2) Both the initial state xθ0 and terminal state xθf are specified in Eqs. (39) and (40). Therefore, the set of [xθ0 , θf , xθf ] is closed and bounded. 3) The control set defined by Eqs. (42), (44), and (45) is also compact. 4) All the differential equations in Eq. (37) are continuous functions.
vv 0 ≤ − e t a4 ≤ e−z 1 − z − z cr 1 2 a3 vr 2 ≤ −ve a4 2 a2 − vt rvt v 1 v −10 ≤ t a2 − vt ≤ 10 −10 ≤ t a3 vr ≤ 10 rvt cr cr For each θ; xθ ∈ K, because all the preceding equalities are linear and the inequalities are either linear or second-order cone constraints, the set Q θ; x for problem 3 is convex with respect to (a1 , a2 , a3 , a4 , a5 ). Based on the preceding observations and the existence theorem in [38], we can draw the conclusion that problem 3 has an optimal solution if its feasible set is nonempty. More descriptions of the existence theorem and its proof can be found in [38].
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M. Xin Associate Editor
