fuel-optimal ascent trajectory problem for launch vehicle
TRANSCRIPT
Research ArticleFuel-Optimal Ascent Trajectory Problem for Launch Vehicle viaTheory of Functional Connections
Shiyao Li ,1 Yushen Yan ,1 Kun Zhang ,2 and Xinguo Li 1,3
1School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China2Xi’an Modern Control Technology Research Institute, Xi’an 710065, China3National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi’an 710072, China
Correspondence should be addressed to Shiyao Li; [email protected]
Received 3 June 2021; Revised 16 August 2021; Accepted 25 August 2021; Published 13 September 2021
Academic Editor: Erdal Kayacan
Copyright © 2021 Shiyao Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, we develop a method based on the Theory of Functional Connections (TFC) to solve the fuel-optimal problem in theascending phase of the launch vehicle. The problem is first transformed into a nonlinear two-point boundary value problem(TPBVP) using the indirect method. Then, using the function interpolation technique called the TFC, the problem’sconstraints are analytically embedded into a functional, and the TPBVP is transformed into an unconstrained optimizationproblem that includes orthogonal polynomials with unknown coefficients. This process effectively reduces the search space ofthe solution because the original constrained problem transformed into an unconstrained problem, and thus, the unknowncoefficients of the unconstrained expression can be solved using simple numerical methods. Finally, the proposed algorithm isvalidated by comparing to a general nonlinear optimal control software GPOPS-II and the traditional indirect numericalmethod. The results demonstrated that the proposed algorithm is robust to poor initial values, and solutions can be solved inless than 300ms within the MATLAB implementation. Consequently, the proposed method has the potential to generateoptimal trajectories on-board in real time.
1. Introduction
With the recent development in space exploration, launchvehicles are very important as they are the only means forhumans to explore space from the earth. In general, a launchvehicle mission has been planned over a long period, and thetrajectory was designed in advance, and it cannot be updatedduring flight, which means it is not robust or flexible. The fastlaunch and trajectory reconstruction are the main research ofthe guidance system, and both need rapid trajectory planningtechnology. Rapid trajectory planning can shorten the launchmission cycle and quickly update the trajectory in case ofthrust failure during the flight of the vehicle to ensure thesuccess of the mission.
The primary aim of the trajectory planning algorithm isto solve the optimal control problem that is generally basedon nonlinear dynamics, which achieves specific performanceindicators under the constraints of state and control vari-ables. The solution of such problems is mainly achieved
using the indirect method [1–3] and the direct method[4–6]. The direct method transforms the optimal controlproblem of continuous space into a nonlinear programmingproblem and uses a numerical method to directly optimizethe performance index [7–9]. Although the direct method,represented by a sequential quadratic programming algo-rithm with the pseudospectral discrete, has advanced a lotover a period of time, it still has considerable issues betweenthe on-board application. The algorithm for the generalnonlinear programming, for example, the famous sequentialquadratic programming algorithm, had low algorithm effi-ciency and low sensitivity to the initial value and was unableto guarantee convergence in the past, but now, these issueshave easily resolved by using a convex optimization method.Ralph Rockafellar, a renowned mathematician, pointed outthat the key determinant of the performance of a numericaloptimization algorithm is neither the linearity nor nonline-arity of the problem, but the convexity or nonconvexity ofthe problem [10]. In 2007, JPL proposed lossless convexity
HindawiInternational Journal of Aerospace EngineeringVolume 2021, Article ID 2734230, 13 pageshttps://doi.org/10.1155/2021/2734230
technology for the dynamic descent guidance of the Marslander [11]. After that, a systematic summary of the researchand development of lossless convexity technology is pre-sented in [12]. The applicability of the lossless convexitymethod is extended to general linear systems with multiplestate constraints using the concepts of control theory.Unfortunately, only a few nonconvex constraints can beused for lossless convexification, and there is no analyticalconvexification method for the system dynamic constraintsin the trajectory optimization problem of aircraft. A methodbased on the Newton–Kantorovich/pseudospectral andsequence convexification is used for the ascending phase ofthe launch vehicle [13]. However, the sequential convexoptimization algorithm is a convexification method basedon linearization, which increases the dependence on the ref-erence trajectory. This not only offers higher requirementsfor the reference trajectory but also annihilates the advan-tage of the convex optimization algorithm that does not relyon the good initial value. Nevertheless, considering therapidity of the convex optimization algorithm in solvingconvex problems, in recent years, trajectory planning basedon the convex optimization algorithm, such as planetarylanding [14–16], rocket ascent guidance [17], and entryguidance [18], has been widely studied.
The indirect method solves the optimal control problemby using the classical variational method and the PontryaginMinimum Principle, derives the first-order necessary condi-tions of the optimal control, and transforms the optimalcontrol problem into a two-point boundary value problem(TPBVP) [19] that is comprised of initial conditions,Hamiltonian differential equations, optimal conditions, andterminal boundary conditions (including terminal transver-sal conditions and terminal constraints). However, sincethe convergence radius of the indirect method is small, andthe convergence of the numerical iteration is extremely sen-sitive to the initial value estimation, which requires a higheraccuracy of the initial value estimation, determination of theinitial value is highly difficult. To overcome this problem, thedeep learning algorithm is used to obtain a higher accuracyof initial value estimation and a better target shootingsuccess rate [20–22]. In general, the higher sensitivity ofTPBVP to the initial value makes the problem difficult tosolve. Therefore, although the indirect method yields a moreaccurate solution, it is rarely used in practice.
Recently, a mathematical framework called the Theoryof Functional Connections (TFC) has been proposed in[23] to derive the expressions with embedded constraints.The expressions, called constrained expressions, arecomposed of functionals and functions of functions. Theconstrained expression is written as
y tð Þ = g tð Þ + 〠k=1
n
Φk tð Þpk tð Þ, ð1Þ
where gðtÞ is a free function and ΦkðtÞ are the switch func-tions composed of the support function skðtÞ with unknowncoefficients αk. The support function is a set of linearly inde-pendent functions. If one of the switch functions is equal to
1, the constraint it is referencing is evaluated; otherwise, thatis, if it is equal to 0, all other constraints are evaluated. Theswitch functions can be expressed asΦkðtÞ = skðtÞαk. The pro-jection functionals pkðtÞ are derived by constraint functions.The constrained optimization problem is transformed intoan unconstrained one using the constrained expressions,which reduces the search space of the solution to the admis-sible solutions that satisfy all constraints. Finally, using com-mon basis functions such as Chebyshev polynomials orLegendre polynomials to express the free function and thenusing the least-square method to find its unknown coefficient,the solution of the problem can be found. In [24, 25], the TFCalgorithm quickly solves the nonlinear differential equationsand obtains high-precision solutions. In [26, 27], it is appliedto fixed-time asteroid landing and optimal energy landing.The results show that the solution time is basically less than100ms, which proves that the algorithm has real-time appli-cation potential.
This paper is organized as follows. Section 2 gives a briefdescription of the TFC mathematical framework to solvingTPBVP. In Section 3, the fuel-optimal ascent trajectoryproblem is described in detail, and the necessary conditionsare derived. In Section 4, the fuel-optimal problem in theascending phase of the launch vehicle is formulated usingthe TFC framework. Finally, the result and discussion areprovided in Section 5.
2. Theory of Functional Connections
In this section, we present an outline of the TFC mathe-matical framework and a method for solving second-order TPBVP with TFC.
2.1. TFC for TPBVP. In general, trajectory optimization prob-lems are second-order TPBVP [28], which is expressed as
F t, y tð Þ, _y tð Þ, €y tð Þð Þ = 0 subject to :
y t0ð Þ = y0,
y t f� �
= yf ,
_y t0ð Þ = _y0,
_y t f� �
= _yf ,
8>>>>><>>>>>:
ð2Þ
where t0 and t f represent the initial time and the terminaltime, respectively, and y0, yf , _y0, _yf are the initial and terminalconstraints, respectively. As mentioned earlier, the constraintsare expressed by (1), and then, (2) is simply rewritten as
y tð Þ = g tð Þ +Φ1 tð Þp1 +Φ2 tð Þp2 +Φ3 tð Þp3 +Φ4 tð Þp4, ð3Þ
where pkðtÞ is expressed as
p1 t, gi tð Þð Þ = y0i − gi t0ð Þ,p2 t, gi tð Þð Þ = yf i − gi t f
� �,
p3 t, gi tð Þð Þ = _y0i − _gi t0ð Þ,p4 t, gi tð Þð Þ = _yf i − _gi t f
� �:
ð4Þ
2 International Journal of Aerospace Engineering
The algorithm to derive the term of the constrainedexpression is given as follows:
(1) Choose skðtÞ, which are k linearly independentsupport functions
(2) Calculate switching functions ΦkðtÞ as a linear com-bination of the support functions with k unknowncoefficients
(3) Formulate a system of equations to solve for theunknown coefficients based on ΦkðtÞ
The support functions are defined as skðkÞ = tk−1,according to [28]. The switching function is then obtainedby solving equations. When the first switch function isactivated, the equations are
Φ1 t0ð Þ = 1,
Φ1 t f� �
= 0,
Φ1′ t0ð Þ = 0,
Φ1′ t f� �
= 0:
8>>>>><>>>>>:
ð5Þ
The equations of the first switch functions are com-bined into the matrix form as
1 t0 t20 t30
1 t f t2f t3f
0 1 2t0 3t200 1 2t f 3t2f
2666664
3777775
α11
α21
α31
α41
2666664
3777775 =
1
0
0
0
2666664
3777775: ð6Þ
Similarly, application of this method to the other threeswitch functions yields the following matrix:
1 t0 t20 t30
1 t f t2f t3f
0 1 2t0 3t200 1 2t f 3t2f
2666664
3777775
α11 α21 α31 α41
α21 α22 α32 α42
α31 α23 α33 α43
α41 α24 α34 α44
2666664
3777775 =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
2666664
3777775:
ð7Þ
The coefficient αk is obtained by matrix inversion, andthe expression of the switch functions is obtained as
Φ1 tð Þ = 1t f − t0� �3 −t2f 3t0 − t f
� �+ 6t0t f t − 3 t0 + t f
� �t2 + 2t3
� �,
Φ2 tð Þ = 1t f − t0� �3 −t20 t0 − 3t f
� �− 6t0t f t + 3 t0 + t f
� �t2 − 2t3
� �,
Φ3 tð Þ = 1t f − t0� �2 −t0t
2f + t f 2t0 + t f
� �t − t0 + 2t f� �
t2 + t3� �
,
Φ4 tð Þ = 1t f − t0� �2 −t20t f + t0 t0 + 2t f
� �t − 2t0 + t f� �
t2 + t3� �
:
ð8Þ
The boundary conditions of (2) are effectively embed-ded within the constrained expression by substituting (4)and (8) into (3). Then, by substituting (3) into the vectordifferential equation Fðt, yðtÞ, _yðtÞ, €yðtÞÞ, the constrainedTPBVPs are transformed into an unconstrained problem.According to (3), yðtÞ is replaced by gðtÞ; thus, the origi-nal vector differential equation Fðt, yðtÞ, _yðtÞ, €yðtÞÞ istransformed into F̂ðt, gðtÞ, _gðtÞ, €gðtÞÞ, which is only afunction of t, the free functions gðtÞ, and their derivatives
F̂ t, g tð Þ, _g tð Þ, €g tð Þð Þ = 0: ð9Þ
As mentioned earlier, F̂ðt, gðtÞ, _gðtÞ, €gðtÞÞ is uncon-strained because the boundary is represented by the switchfunction ΦkðtÞ and the projection function pkðtÞ.
After determining the switch function and the projectionfunction, we next discuss the free function gðtÞ.2.2. Definition of the Free Function. In selecting a free func-tion, we are essentially looking for the best function approx-imator. A natural choice for the free function is a linearcombination of basis functions, as this is capable of spanningthe entire function space that the basis spans, as the numberof basis functions approaches infinity. The free function isexpressed as
g tð Þ = ξTh, ð10Þ
where ξ are m × 1 unknown coefficients and h are m basisfunctions.
Next, the problem domain t is mapped to the domain ofthe basis functions z, and Chebyshev and Legendre polyno-mials are commonly used, the domains of which are definedin ½−1, 1�. To implement the basis functions, a map betweent and z is defined as
z = z0 +zf − z0t f − t0
t − t0ð Þ↔ t = t0 +t f − t0zf − z0
z − z0ð Þ: ð11Þ
By using (11), the derivatives of gðtÞ are computed as
dkgdtk
= ckξTdkh zð Þdzk
, where k ∈ 0, n½ �, ð12Þ
where c≔ dz/dt = ðzf − z0Þ/ðt f − t0Þ.2.3. Domain Discretization. For solving TPBVPs numeri-cally, the domain t ∈ ½t0, t f � must be discretized by N + 1points. The common method is uniform distribution, butthe advantage of Chebyshev–Gauss–Lobatto collocationpoints is that when the number of basis functions increases,the condition number should also increase slowly, which isuseful for improving computational efficiency. The Cheby-shev–Gauss–Lobatto collocation points are defined as
zk = − coskπN
� � for k = 1, 2,⋯,N: ð13Þ
3International Journal of Aerospace Engineering
Thus, the new vector differential equation F̂ðt, gðtÞ, _gðtÞ, €gðtÞÞ becomes ~Fðz, ξÞ, where the unknown coefficient ξis the variable that needs to be solved. ~Fðz, ξÞ is expressedin the form of loss functions at each discrete point
L i ξð Þ =
~Fi z0, ξð Þ⋮
~Fi zd , ξð Þ⋮
~Fi zN , ξð Þ
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;: ð14Þ
By setting L i = 0, the unknown coefficient ξ is solvedusing optimization schemes such as iterative least-squares.
To solve the nonlinear least-square problem, we need theJacobian matrix of the loss function, which is written as
J ξð Þ =
∂~Fi z0, ξð Þ∂ξ1
∂~Fi z0, ξð Þ∂ξ2
⋯∂~Fi z0, ξð Þ
∂ξj⋮ ⋮ ⋮
∂~Fi zd , ξð Þ∂ξ1
∂~Fi zd , ξð Þ∂ξ2
⋯∂~Fi zd , ξð Þ
∂ξj⋮ ⋮ ⋮
∂~Fi zN , ξð Þ∂ξ1
∂~Fi zN , ξð Þ∂ξ2
⋯∂~Fi zN , ξð Þ
∂ξj
26666666666666664
37777777777777775
: ð15Þ
The estimation is updated by
ξk+1 = ξk − Δξ, ð16Þ
where Δξ = ðJðξÞT JðξÞÞ−1 JðξÞTLðξÞ.The iterative process stops when the convergence toler-
ance is met:
L2 L ξð Þ½ � < ε, ð17Þ
where ε is the stopping criterion that is defined by theuser.
Figure 1 shows the outline of the TFC framework.Recently, the position of numerical calculation in the
current guidance and control field is emphasized in [29],which, based on numerical calculation, is called as the Com-putational Guidance and Control (CG&C). The CG&Creplaces offline planning and closed-loop guidance withon-board computing, which is more robust, more accurate,and more flexible and can adapt to more complex environ-ments and missions, but offers high requirements for com-putational efficiency. As mentioned earlier, the algorithmsused in the CG&C are basically divided into two: directand indirect. In this study, we used the indirect method,because the optimal control problem is transformed intoTPBVP, and then, the TFC method is used to transformand solve the problem.
We found that the TFC method is quite similar to thecollocation method in which all three methods using orthog-onal polynomials over the global domain, according to thedifferent method. In the collocation method, the states andcostates are expanded by using orthogonal polynomials,and the boundary conditions are considered as part of theoptimization scheme [30]. It is similar that the pseudospec-tral method used orthogonal polynomials like Chebyshev orLegendre polynomials to approximate the states and cost-ates, and the boundary conditions are also considered as partof the optimization scheme. The TFC method may use asimilar operation, but the fundamental difference betweenthe TFC method and the other two methods lies in handlingthe constraints of the problem. The TFC method usesorthogonal polynomials to expand the free function gðtÞ ina constrained expression and then expresses the problem con-straints analytically step by step as mentioned above, whichcan reduce the search space of the solution and thus improvethe computational efficiency. In fact, the advantages of theTFC method are presented in [27]; the results in [27] showthat the TFC method is two orders of magnitude faster thanthe pseudospectral method in a fixed time optimal controlproblem and one order of magnitude faster than the pseudos-pectral method in a free time optimal control problem.
3. Fuel-Optimal Problem in the AscendingPhase of the Launch Vehicle
In this section, the problem is transformed into a TPBVP,and the first-order necessary conditions and transversal con-ditions of the problem are derived using the PontryaginMinimum Principle.
3.1. Dynamical Model. In this section, the last stage of thelaunching vehicle is studied. The dimensionless equationsof motion of a three-dimensional (3-D) launch vehicle canbe expressed in the Earth Center Inertial Coordinate Systemas follows:
_r = v,
_v =Tm
+ ag,
_m =T
Ispg0,
ð18Þ
where r is the inertial position, which is normalized by theradius of the Earth R0 = 6378145m. v is the velocity, whichis normalized by
ffiffiffiffiffiffiffiffiffiffiR0g0
p, in which g0 = 9:81m/s2 represents
the gravitational acceleration magnitude on the surface ofthe Earth. The mass of the launch vehicle is denoted by m.The thrust is denoted by T = TIb, where Ib is the unit vectorof the body axis satisfying
Ibk k ≡ 1: ð19Þ
For most launch vehicles, the mass flow is uncontrollable;thus, the thrustmagnitude T = kTk is constant and uncontrol-lable during the same flight phase. The gravity acceleration
4 International Journal of Aerospace Engineering
ag = −r/r3, where r = krk. The specific impulse of the engine isdenoted by Isp. The differentiation of the equations in (18) is
with respect to dimensionless time normalized byffiffiffiffiffiffiffiffiffiffiffiR0/g0
p.
As the mass flow is constant, the optimal ascent problemis treated as a minimum-time problem:
min J = t f , ð20Þ
subject to
_r = v, _v =Tm
+ ag, _m =T
Ispg0,
r 0ð Þ = r0, v 0ð Þ = v0, m 0ð Þ =m0,
r t f� �
= rf , v t f� �
= vf :
ð21Þ
3.2. First-Order Necessary Conditions. On the basis of thePontryagin Minimum Principle, the mass of the vehicle istreated as a prescribed function of time instead of a statevariable. The Hamiltonian function is written as
H = 1 + λTr v + λTvTmIb −
rr3
� �, ð22Þ
where Ib = −λv/λv is called the primer’s vector, according toLawden’s theory [31]. Thus, (22) is rewritten as
H = 1 + λTr v −Tm
λvk k − λTv rr3
: ð23Þ
The first-order necessary conditions for optimality thengive the differential equations of the costate variables:
_λr = −∂H∂r
=λTv rr5
,
_λv = −∂H∂v
= −λr:
ð24Þ
The transversality condition is expressed as
H tf� �
= 0: ð25Þ
4. Solution via TFC
4.1. TPBVP in TFC Framework. As mentioned in the previ-ous section, to find the optimal state, the following nonlinearTPBVPs must be solved:
_r = v, ð26Þ
_v = −Tmλvλv
−rr3, ð27Þ
_m =T
Ispg0, ð28Þ
_λr = −∂H∂r
=λTv rr5
, ð29Þ
_λv = −∂H∂v
= −λr , ð30Þ
Projection function pk (t)Support function sk (t)Switch function 𝛷k (t)
Free functiong (t) = 𝜉T h (z)
𝜉 = 𝜉k
𝜉k
dzdt
zf – z0tf – t0
c : = =
TFC
N
Y
L2 𝜀(𝜉k)
i (𝜉) = 0
F (z, 𝜉) = 0
y (t) = g (t) + 𝛴𝛷k (t)pk (t)k=1
n
F t, y(t), y(t), y(t) = 0
Figure 1: TFC framework.
5International Journal of Aerospace Engineering
H tf� �
= 1 + λTr t f� �
v t f� �
−Tm
λv t f� � − λTv t f
� �r t f� �
r3 t f� � = 0,
ð31Þwhere (26) and (27) are subject to
r 0ð Þ = r0, v 0ð Þ = v0, m 0ð Þ =m0,
r t f� �
= r f , v t f� �
= vf :ð32Þ
Additionally, there is a redundant equation and can beremoved by the TFC constraints. The derivative of rðtÞ isexactly the function vðtÞ because TFC constraints are analyt-ical expressions; thus, (26) can be disregarded. The problemis now reduced. The new equations are expressed as
La tð Þ = a − _λr −λTv rr5
,
Lr tð Þ = _λr −λTv rr5
,
Lv tð Þ = _λv + λr ,
LH = 1 + λTr v −Tm
λvk k − λTv rr3
:
ð33Þ
To solve the above equations, the TFC constraints withrðtÞ, λrðtÞ, λvðtÞ need to be constructed by the TFC method.The unknown coefficients in the TFC constraint expressionsare expressed as ξa, ξλr , ξλv , and vðtÞ, aðtÞ, _λrðtÞ, _λvðtÞ can beobtained by taking the derivative of the TFC constraintexpression rðtÞ, λrðtÞ, λvðtÞ, respectively.
The initial and terminal constraints of the problem dis-cussed in this paper are position and velocity constraints,respectively. The TFC constraint expressions of rðtÞ, vðtÞ, aðtÞ are written as
r tð Þ = g tð Þ +Φ1 tð Þp1 +Φ2 tð Þp2 +Φ3 tð Þp3 +Φ4 tð Þp4,v tð Þ = g tð Þ + _Φ1 tð Þp1 + _Φ2 tð Þp2 + _Φ3 tð Þp3 + _Φ4 tð Þp4,a tð Þ = g tð Þ + €Φ1 tð Þp1 + €Φ2 tð Þp2 + €Φ3 tð Þp3 + €Φ4 tð Þp4,
ð34Þ
where the projection function is written as
p1 tð Þ = r0 − g t0ð Þ,p2 tð Þ = r f − g t f
� �,
p3 tð Þ = v0 − _g t0ð Þ,p4 tð Þ = vf − _g t f
� �:
ð35Þ
Next, consider constructing a free function. According to(10)–(12), the time domain is mapped to the Chebyshevdomain. However, it should be noted that in the time-freeTPBVP, the parameter c is a function of t f . Combined withthe TFCmethod, the new unknown variable ξt is used to repre-sent the parameter c, and the optimal time is obtained by solv-
ing ξt. In addition, to ensure the solved final time is positive, b isused instead of b, where b2 = c. Then, rðtÞ is rewritten as
r tð Þ = r zð Þ =Φ1 zð Þr0 +Φ2 zð Þr f +Φ3 zð Þ v0b2
+Φ4 zð Þ vfb2
+ h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′� �T
ξa,
v tð Þ = b2v zð Þ = b2 Φ1′ zð Þr0 +Φ2′ zð Þrf +Φ3′ zð Þ v0b2
+Φ4′ zð Þ vfb2
+ h′ zð Þ −Φ1′ zð Þh0 −Φ2′ zð Þhf −Φ3′ zð Þh0′ −Φ4′ zð Þhf′� �T
ξa
�,
a tð Þ = b4r zð Þ = b4 Φ1″ zð Þr0 +Φ2″ zð Þrf +Φ3″ zð Þ v0b2
+Φ4″ zð Þ vfb2
+ h″ zð Þ −Φ1″ zð Þ0 −Φ2″ zð Þhf −Φ3″ zð Þh0′ −Φ4″ zð Þhf′� �T
ξa
�,
ð36Þ
where h0 = hðz0Þ and hf = hðzf Þ.The expression of the switch function is similar to (8):
Φ1 zð Þ = 1zf − z0� �3 −z2f 3z0 − zf
� �+ 6z0zf z − 3 z0 + zf
� �z2 + 2z3
� �,
Φ2 zð Þ = 1zf − z0� �3 −z20 z0 − 3zf
� �− 6z0zf z + 3 z0 + zf
� �z2 − 2z3
� �,
Φ3 zð Þ = 1zf − z0� �2 −z0z
2f + zf 2z0 + zf
� �z − z0 + 2zf� �
z2 + z3� �
,
Φ4 zð Þ = 1zf − z0� �2 −z20zf + z0 z0 + 2zf
� �z − 2z0 + zf� �
z2 + z3� �
:
ð37Þ
Next, the TFC constraint expressions of the costates areconstructed by the same steps as above:
λr tð Þ = λr zð Þ = h zð ÞTξr ,_λr tð Þ = b2λr′ zð Þ = b2h′ zð ÞTξr ,λv tð Þ = λv zð Þ = h zð ÞTξv,_λv tð Þ = b2λv′ zð Þ = b2h′ zð ÞTξv:
ð38Þ
Substitution of the above TFC constraint expression intothe loss function yields the loss function with respect tounknown coefficient ξ. Then, the solution of the problemis obtained using the nonlinear least-square method. Theunknown coefficient ξ is expressed as
ξ = ξTa1 ξTa2ξTa3 ξ
Tλr1
ξTλr2 ξTλr3
ξTλv1 ξTλv2
ξTλv3 ξt
n oT∈ℝ9m+1:
ð39Þ
The loss function can be expressed as
L = LTa1LT
a2LT
a3LT
λr1LT
λr2LT
λr3LT
λv1LT
λv2LT
λv3LH
n oT:
ð40Þ
6 International Journal of Aerospace Engineering
To use the nonlinear least-square method, the partialderivative of the loss function needs to be calculated as
a tð Þ = b4r tð Þ = b4 Φ1′′ zð Þr0 +Φ2′′ zð Þr f +Φ3′′ zð Þ v0b2
+Φ4′′ zð Þ vfb2
+ h″ zð Þ −Φ1′′ zð Þh0 −Φ2′′ zð Þhf −Φ3′′ zð Þh0′ −Φ4′′ zð Þhf′� �T
ξa
�:
ð41Þ
The terms of (41) are defined by
Ja,ξa =
∂La1
∂ξa1
∂La1
∂ξa2
∂La1
∂ξa3∂La2
∂ξa1
∂La2
∂ξa2
∂La2
∂ξa3∂La3
∂ξa1
∂La3
∂ξa2
∂La3
∂ξa3
26666666664
37777777775, Ja,ξv =
∂La1
∂ξv1
∂La1
∂ξv2
∂La1
∂ξv3∂La2
∂ξv1
∂La2
∂ξv2
∂La2
∂ξv3∂La3
∂ξv1
∂La3
∂ξv2
∂La3
∂ξv3
26666666664
37777777775,
Ja,ξt =∂La1
∂ξt
∂La2
∂ξt
∂La3
∂ξt
�T,
Jλr ,ξa =
∂Lr1
∂ξa1
∂Lr1
∂ξa2
∂Lr1
∂ξa3∂Lr2
∂ξa1
∂Lr2
∂ξa2
∂Lr2
∂ξa3∂Lr3
∂ξa1
∂Lr3
∂ξa2
∂Lr3
∂ξa3
26666666664
37777777775, Jλr ,ξr =
∂L r1
∂ξr10 0
0∂Lr2
∂ξr20
0 0∂Lr3
∂ξr3
26666666664
37777777775,
Jλr ,ξv =
∂Lr1
∂ξv10 0
0∂Lr2
∂ξv20
0 0∂Lr3
∂ξv3
26666666664
37777777775, Jλr ,ξb =
∂Lr1
∂ξt∂Lr2
∂ξt∂Lr3
∂ξt
2666666664
3777777775,
Jλv ,ξr =
∂Lv1
∂ξr10 0
0∂Lv2
∂ξr20
0 0∂Lv3
∂ξr3
26666666664
37777777775, Jλv ,ξv =
∂Lv1
∂ξv10 0
0∂Lv2
∂ξv20
0 0∂Lv3
∂ξv3
26666666664
37777777775,
Jλv ,ξt =∂Lv1
∂ξt
∂Lv2
∂ξt
∂Lv3
∂ξt
�T,
JH,ξa =∂LH
∂ξa1
∂LH
∂ξa2
∂LH
∂ξa3
�,
JH,ξr =∂LH
∂ξr1
∂LH
∂ξr2
∂LH
∂ξr3
�,
JH,ξv =∂LH
∂ξv1
∂LH
∂ξv2
∂LH
∂ξv3
�,
JH,ξt =∂LH
∂ξt, ð42Þ
where all partial derivatives are provided in theAppendix.
4.2. Initialization. When using the iterative least-squaremethod to calculate, some parameters need to be estimatedreasonably, so that the iterative process of the algorithmcan go on without violating the basic mathematical princi-ples. The simplest initialization is to set them equal to zero.This is equivalent to connecting the boundary value problemwith the simplest interpolating expression. However, the λvis related to the thrust direction; initialization of λv = 0 willcause issues in the TFC method because Ib = −λv/λv and ∥Ib∥≡1. Thus, the coefficient ξv is initialized using
ξv0 =v0v0k k , ξvf = −
r0r0k k : ð43Þ
In addition, the first guess of ξr , ξa is set equal to zero,and for setting the first guess of ξt , an estimate of the finaltime t f is needed; in this paper, the initial t f is set to be300, and the initial ξt can be expressed as
ξt =ffiffiffiffiffiffiffiffiffiffiffiffi2
t f − t0
s: ð44Þ
These are uniformly discrete according to the number ofpolynomials m.
5. Simulations
In this section, we apply the proposed algorithm to theascent problem of the launch vehicle to verify the feasibilityof the algorithm, and the results are compared with those ofthe GPOPS-II and classical indirect method solutions of theother two algorithms. All numerical results are obtained on adesktop with Intel Xeon E3-1230 3.4GHz. Table 1 lists theparameters of the launch vehicle and mission in the numer-ical simulations.
Table 2 shows the initial and terminal parameters ofthe experiment, where the orbital elements correspondingto the terminal position and velocity are also given becausethe launch vehicle generally uses the orbital elements forthe target.
To prove the validity and effectiveness of the algorithmproposed in this study, the results of the algorithm pro-posed in this paper are compared with those obtained bythe traditional indirect method and GPOPS-II. The finaltime calculated by GPOPS-II is 300.97 s, that obtained bythe single shooting method is 301.01 s, and the final timeobtained by TFC is 301.25 s. The locations of the vehicleand the velocity vector are provided in Figures 2 and 3,respectively. Figure 4 shows the thrust vector of the launchvehicle. In the bottom part of Figure 4, the purple linerepresenting the sum of squares of thrust directions isequal to 1, which also indicates the validity of the TFCsolution, and the other three lines are the vector of thebody axis. Considering that the pitch angle and yaw angleare generally used as the guidance command of the launch
7International Journal of Aerospace Engineering
vehicle, the results of pitch angle and yaw angle are alsogiven here. Figures 5 and 6 show the results of the heightand the velocity, respectively. It can be seen that the results
solved by the three methods are basically the same, whichagain shows the validity of the TFC algorithm proposed inthis study. To quantify the accuracy of the TFC method,
Table 1: Parameters for launch vehicle.
Parameter Value
Longitude of launch point (°) 110.95
Latitude of launching point (°) 19.61
Launch azimuth (°) 90
The specific impulse, Isp (m/s) 3365
Thrust, T (N) 2843599.98
Mass rate, _m (kg/s) 845.052
Table 2: Parameters of the boundary conditions.
Parameter Value
Initial position, r0 (m) [371973.739, 6493779.849, -13899.978]
Initial velocity, v0 (m/s) [3652.033, 556.843, -2.666]
Initial mass, m0 (kg) 350306
Semimajor (m) 6595487
Eccentricity 0.0053
Inclination (°) 20.009
Ascending node (°) 17.250
Argument of perigee (°) 32.390
True anomaly (°) 77.969
Terminal position, rf (m) [1912866.558, 6304148.648, 2551.256]
Terminal velocity, vf (m/s) [7457.930, -2220.619, 178.661]
0 50 100 150 200 250 300 350
4000
6000
8000
v x (m
/s)
TFCGpops-IISingle shooting
0 50 100 150 200 250 300 350
–2000
–1000
0
v y (m
/s)
0 50 100 150 200 250 300 350Time (s)
0
100
200
v z (m
/s)
Figure 2: Vector of velocity solution by TFC, GPOPS-II, and single shooting.
8 International Journal of Aerospace Engineering
Figure 7 shows the residual of acceleration, and it can beseen that the residual of TFC is about 10−14 or less forthe whole solution domain.
In this simulation, the cost time of the proposed methodis 0.23 s and those of GPOPS-II and the single shootingmethod are 3.88 and 0.34 s, respectively. It is also known that
0 50 100 150 200 250 300 3500
1
2
r x (m
)
×106
×106
0 50 100 150 200 250 300 3506.3
6.4
6.5r y
(m)
0 50 100 150 200 250 300 350Time (s)
–10000
–5000
0
r z (m
)
TFCGpops-IISingle shooting
Figure 3: Vector of position solution by TFC, GPOPS-II, and single shooting.
0 50 100 150 200 250 300 350
0
20
40
Pitc
h an
gle (
°)
0 50 100 150 200 250 300 350
-2-101
Yaw
angl
e (°)
0 50 100 150 200 250 300 350Time (s)
0
0.5
1
Thur
st ve
ctor
x-axisy-axisz-axis
1
Figure 4: Vector of thrust solution by TFC.
9International Journal of Aerospace Engineering
the MATLAB programming language is 10 times slowerthan C++; thus, the algorithm proposed in this study hasonline application potential. In terms of solution accuracy,the TFC method is not dominant among the three methods,but combined with the analysis of calculation efficiency, itshows that the proposed method is an effective method.
The above results show the comparison between theresults of the proposed algorithm, GPOPS-II, and singleshooting method, which verifies the validity of the proposedalgorithm. Next, the effect of the number of discrete pointsand polynomials on the algorithm is studied. Table 3 showsthat the excessive number of discrete points will not onlyreduce the calculation efficiency, but also reduce the accu-racy of the solution. In addition, the selection of the number
of polynomials is also analyzed in this paper. In [28], thenumber of state polynomials and costate polynomials is thesame and not studied separately. In our simulation, theresults show that the selection of the number of state poly-nomials and costate polynomials can be different, and abetter result can be obtained. If the number of the state poly-nomials is selected too much, then not only the calculationperformance will be degraded but also the accuracy of thesolution will be greatly affected. It is seen from Table 3 thatwhen the number of costate polynomials is as large as thenumber of state polynomials, the allowed iteration numberis reached and the iteration progress will not converge; itmeans the solution of the problem cannot be solved. In addi-tion, appropriately increasing the number of costate polyno-mials can increase the accuracy of the solution, but it willreduce the computational efficiency. Thus, the number ofdiscrete points and polynomials should be select carefully.
0 50 100 150 200 250 300 350Time (s)
120
130
140
150
160
170
180
190
200
210
220
Hei
ght (
km)
TFCGPOPS-IISingle shooting
Figure 5: Height solution by TFC, GPOPS-II, and single shooting.
0 50 100 150 200 250 300 350Time (s)
0
0.5
1
1.5
2
2.5
3
Resid
uals
×10–14
Lax
Lay
Laz
Figure 7: Residual by TFC.
0 50 100 150 200 250 300 350Time (s)
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
Velo
city
(m/s
)
TFCGPOPS-IISingle shooting
Figure 6: Velocity solution by TFC, GPOPS-II, and single shooting.
Table 3: Solutions of different discrete points and polynomialnumbers.
Discretepoints
State and costatepolynomial numbers
First-orderoptimality
Costtime(s)
Iterationnumber
100 10, 4 0.414 0.15 61
100 30, 4 0.00138 0.16 14
100 60, 4 2.09e-06 0.23 15
100 80, 4 0.00526 0.29 14
100 80, 10 2.13e-05 0.42 29
100 60, 60 50.3 — Max
50 10, 4 0.349 0.16 54
50 30, 4 5.94e-05 0.23 45
50 50, 4 0.000555 0.25 26
10 International Journal of Aerospace Engineering
6. Conclusion
In this study, we proposed a new approach to solve the fuel-optimal problem in the ascending phase of the launchingvehicle using TFC. The main conclusions can be summa-rized as follows:
(1) The first-order necessary condition of the optimiza-tion problem is constructed by the indirect method;the problem’s constraints are embedded in theexpression by using the TFC method
(2) The constrained optimization problem is trans-formed into an unconstrained optimization problemby using the TFC method, which reduces the searchspace of the solution, and a simple root-finding algo-
rithm can be used to obtain the solution of theproblem
(3) The residual of the solution is about 10−14 or less; forobtaining more accurate numerical solutions, thenumber of discrete points and polynomials shouldbe selected carefully
(4) The proposed algorithm has the potential for onlineapplication. The calculation time of the algorithm iswithin 300ms with MATLAB programming
Appendix
Partial Derivative of Loss Functions
The partial derivative of loss functions is
∂Lai
∂ξv j=
Tm
hTv zð Þλvk k −
hTv zð Þλ2v jλvk k3
!, i = j,
−hTv zð Þλviλvj
λvk k3, i ≠ j,
8>>>>><>>>>>:
ðA:1Þ
∂Lai
∂ξt= 4b3 h″ zð Þ −Φ1″ zð Þh0 −Φ2″ zð Þhf −Φ3″ zð Þh0′ −Φ4″ zð Þhf′
� �Tξai
+ 2b Φ3″ zð Þv0i +Φ4″ zð Þvf i� �
−−2T
m0 − z + 1ð Þ _m/b2� �2 z + 1ð Þ _m
b3
+−2 Φ3 zð Þv0i +Φ4 zð Þvf i� �
b3 rk k3+6 rk k Φ3 zð Þv0i +Φ4 zð Þvf i
� �b3 rk k5
,
ðA:2Þ
∂Lri
∂ξvj=
hTvr1rk k3 , i = j,
0, i ≠ j,
8<:
∂Lri
∂ξt= 2bh′Tr ξri , ðA:3Þ
∂Lvi
∂ξr j=
h′Tr , i = j,
0, i ≠ j,
(
∂Lvi
∂ξvj=
b2h′Tv , i = j,
0, i ≠ j,
(ðA:5Þ
∂Lvi
∂ξt= 2bh′Tv ξvi ,
∂LH
∂ξri= hTrf vif , ðA:6Þ
∂LH
∂ξvi= −
T
m0 − zf − z0� �
/b2� �
_m
λvifλvf −
rf
r f 3
!hTvf , ðA:7Þ
∂LH
∂ξt= −2T λvf
−2 _mb3 m0 − zf − z0
� �/b2
� �_m
� �2 , ðA:8Þ
∂LH
∂ξai= −λv
h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′� �T
rk k3
−3r2i h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′� �T
rk k5 ,
ðA:8Þ
11International Journal of Aerospace Engineering
Data Availability
The data used to support the findings of this study areincluded within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
References
[1] P. Lu, H. Sun, and B. Tsai, “Closed-loop endoatmosphericascent guidance,” Journal of Guidance, Control, and Dynamics,vol. 26, no. 2, pp. 283–294, 2003.
[2] P. Lu and B. Pan, “Highly constrained optimal launch ascentguidance,” Journal of Guidance, Control, and Dynamics,vol. 33, no. 2, pp. 404–414, 2010.
[3] M. J. Grant and R. D. Braun, “Rapid indirect trajectory optimi-zation for conceptual design of hypersonic missions,” Journalof Spacecraft and Rockets, vol. 52, no. 1, pp. 177–182, 2014.
[4] F. Fahroo and I. M. Ross, “Advances in pseudospectralmethods for optimal control,” in AIAA Guidance, Navigationand Control Conference and Exhibit, Honolulu, Hawaii, 2008.
[5] M. A. Patterson and A. V. Rao, “GPOPS-II: a MATLAB soft-ware for solving multiple-phase optimal control problemsusing hp-adaptive Gaussian quadrature collocation methodsand sparse nonlinear programming,” Acm Transactions onMathematical Software, vol. 41, no. 1, pp. 1–37, 2014.
[6] I. M. Ross and M. Karpenko, “A review of pseudospectral opti-mal control: from theory to flight,” Annual Reviews in Control,vol. 36, no. 2, pp. 182–197, 2012.
[7] H. Peng, X. Wang, M. Li, and B. Chen, “An hp symplecticpseudospectral method for nonlinear optimal control,” Com-munications in Nonlinear Science and Numerical Simulation,vol. 42, pp. 623–644, 2017.
[8] X. Wang, H. Peng, S. Zhang, B. Chen, and W. Zhong, “A sym-plectic pseudospectral method for nonlinear optimal control
problems with inequality constraints,” ISA Transactions,vol. 68, pp. 335–352, 2017.
[9] H. Peng, X. Wang, S. Zhang, and B. Chen, “An iterative sym-plectic pseudospectral method to solve nonlinear state-delayed optimal control problems,” Communications in Non-linear Science and Numerical Simulation, vol. 48, pp. 95–114,2017.
[10] R. T. Rockafellar, Convex Analysis, Princeton University Press,1970.
[11] B. Acikmese and S. R. Ploen, “Convex programming approachto powered descent guidance for Mars landing,” Journal ofGuidance, Control, and Dynamics, vol. 30, no. 5, pp. 1353–1366, 2007.
[12] M. W. Harris, Lossless convexification of optimal control prob-lems, Doctor of Philosophy, The University of Texas at Austin,2014, https://repositories.lib.utexas.edu/handle/2152/24904.
[13] X. Cheng, H. Li, and R. Zhang, “Efficient ascent trajectory opti-mization using convex models based on the Newton–Kantor-ovich/pseudospectral approach,” Aerospace Science andTechnology, vol. 66, pp. 140–151, 2017.
[14] F. F. Al-Bakri, A. F. Al-Bakri, and C. A. Kluever, “Approachand landing guidance for an unpowered gliding vehicle,” Jour-nal of Guidance, Control, and Dynamics, vol. 43, no. 12,pp. 2366–2371, 2020.
[15] Z. Y. Song, C. Wang, S. Theil et al., “Survey of autonomousguidance methods for powered planetary landing,” Frontiersof Information Technology & Electronic Engineering, vol. 21,no. 5, pp. 652–674, 2020.
[16] B. Acikmese, J. M. Carson, and L. Blackmore, “Lossless con-vexification of nonconvex control bound and pointing con-straints of the soft landing optimal control problem,” IEEETransactions on Control Systems Technology, vol. 21, no. 6,pp. 2104–2113, 2013.
[17] Y. Li, B. Pang, C. Wei, N. Cui, and Y. Liu, “Online trajectoryoptimization for power system fault of launch vehicles via con-vex programming,” Aerospace Science and Technology, vol. 98,article 105682, 2020.
∂Lai
∂ξaj=
b4 h″ zð Þ −Φ1″ zð Þh0 −Φ2″ zð Þhf −Φ3″ zð Þh0′ −Φ4″ zð Þhf′� �
+h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′� �T
rk k3 −3r2i h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′� �T
r5, i = j,
−3rir j h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′
� �Trk k5 , i ≠ j,
8>>>>>>><>>>>>>>:
ðA:9Þ
∂Lri
∂ξaj=
λTvi
h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′� �T
rk k3 ,
264
−3r2i h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′� �T
rk k5
375,
i = j,
−λTvi3rir j h zð Þ −Φ1 zð Þh0 −Φ2 zð Þhf −Φ3 zð Þh0′ −Φ4 zð Þhf′
� �Trk k5 , i ≠ j:
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
ðA:10Þ
12 International Journal of Aerospace Engineering
[18] Z. Wang and M. J. Grant, “Constrained trajectory optimiza-tion for planetary entry via sequential convex programming,”Journal of Guidance, Control, and Dynamics, vol. 40, no. 10,pp. 2603–2615, 2017.
[19] A. E. Bryson and Y. Ho, Applied Optimal Control, Techno-metrics, 1975.
[20] L. Cheng, F. Jiang, Z. Wang, and J. Li, “Multi-constrained real-time entry guidance using deep neural networks,” IEEE Trans-actions on Aerospace and Electronic Systems, vol. 57, no. 1,pp. 325–340, 2021.
[21] D. Izzo and E. Öztürk, “Real-time guidance for low-thrusttransfers using deep neural networks,” Journal of Guidance,Control, and Dynamics, vol. 44, no. 2, pp. 315–327, 2021.
[22] L. Cheng, H. Li, Z. Wang, and F. Jiang, “Fast solution contin-uation of time-optimal asteroid landing trajectories using deepneural networks,” Acta Astronautica, vol. 167, pp. 63–72, 2020.
[23] D. Mortari, “The theory of connections: connecting points,”Mathematics, vol. 5, no. 4, p. 4, 2017.
[24] D. Mortari, “Least-squares solutions of linear differential equa-tions,” 2017, http://arxiv.org/abs/1702.08437.
[25] H. Johnston and D. Mortari, “Least-squares solutions ofboundary-value problems in hybrid systems,” Journal of Com-putational and Applied Mathematics, vol. 393, article 113524,2021.
[26] E. Schiassi, A. D’Ambrosio, A. Scorsoglio, R. Furfaro, andF. Curti, Class of optimal space guidance problems solved viaindirect methods and physics-informed neural networks, 2021.
[27] E. Schiassi, A. D’Ambrosio, H. Johnston, R. Furfaro, F. Curti,and D. Mortari, Complete energy optimal landing on smalland large planetary bodies via theory of functional connections,2020.
[28] K. Drozd, R. Furfaro, E. Schiassi, H. Johnston, and D. Mortari,“Energy-optimal trajectory problems in relative motion solvedvia theory of functional connections,” Acta Astronautica,vol. 182, pp. 361–382, 2021.
[29] D. Mortari, H. Johnston, and L. Smith, “High accuracy least-squares solutions of nonlinear differential equations,” Journalof Computational and Applied Mathematics, vol. 352,pp. 293–307, 2019.
[30] L. Ping, “Introducing computational guidance and control,”Journal of Guidance Control & Dynamics, vol. 40, no. 2,pp. 193–193, 2017.
[31] D. E. Rutherford and D. F. Lawden, “Optimal trajectories forspace navigation,” The Mathematical Gazette, vol. 48,no. 366, p. 478, 1964.
13International Journal of Aerospace Engineering