Available via license: CC BY 4.0
Content may be subject to copyright.
Citation: Chen, H.; Ma, Z.; Wang, J.;
Su, L. Online Trajectory Optimization
Method for Large Attitude Flip
Vertical Landing of The Starship-like
Vehicle. Mathematics 2023,11, 288.
https://doi.org/10.3390/
math11020288
Academic Editors: Huawen Liu,
Chengyuan Zhang, Weiren Yu and
Chunwei Tian
Received: 12 December 2022
Revised: 28 December 2022
Accepted: 31 December 2022
Published: 5 January 2023
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
mathematics
Article
Online Trajectory Optimization Method for Large Attitude Flip
Vertical Landing of The Starship-like Vehicle
Hongbo Chen *, Zhenwei Ma , Jinbo Wang and Linfeng Su
School of Systems Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China
*Correspondence: chenhongbo@mail.sysu.edu.cn
Abstract:
A high-precision online trajectory optimization method combining convex optimization
and Radau pseudospectral method is presented for the large attitude flip vertical landing problem
of a starship-like vehicle. During the landing process, the aerodynamic influence on the starship-
like vehicle is significant and non-negligible. A planar landing dynamics model with pitching
motion is developed considering that there is no extensive lateral motion modulation during the
whole flight. Combining the constraints of its powered descent landing process, a model of the
fuel optimal trajectory optimization problem in the landing point coordinate system is given. The
nonconvex properties of the trajectory optimization problem model are analyzed and discussed,
and the advantages of fast solution and convergence certainty of convex optimization, and high
discretization precision of the pseudospectral method, are fully utilized to transform the strongly
nonconvex optimization problem into a series of finite-dimensional convex subproblems, which are
solved quickly by the interior point method solver. Hardware-in-the-loop simulation experiments
verify the effectiveness of the online trajectory optimization method. This method has the potential
to be an online guidance method for the powered descent landing problem of starship-like vehicles.
Keywords:
starship-like vehicle; large attitude flip vertical landing; online trajectory optimization;
convex optimization; Radau pseudospectral method
MSC: 37M10
1. Introduction
As the demand for human space exploration continues to expand, providing more reli-
able, economical, and fast spacecraft launch services is a significant development direction,
and one of the main challenges for the aerospace launch industry [
1
–
3
]. Reusable launch
vehicles (RLV) are an essential technological approach to meet this challenge, and have been
a research hotspot for the world’s major spacefaring nations [
4
,
5
]. Starting from the 1960s,
the United States has presented a variety of reusable launcher programs, and carried out a
large number of technology verification tests, the successful development and application
of the Space Shuttle being one of the high points [
6
]. Since entering the 21st century, SpaceX
has made a breakthrough in vertical takeoff and vertical landing (VTVL) technology. Based
on the continuous research and verification of many key technologies such as advanced
reusable liquid rocket motors and landing guidance control, a sub-stage of its Falcon 9
rocket has taken the lead in achieving sea/land fixed-point soft landing and multiple reuses.
The feasibility and reliability of relevant technologies were verified, and marking the entry
of VTVL reuse technology into the large-scale engineering application stage [7].
On this basis, for Mars one and other large-scale interplanetary exploration missions,
SpaceX proposed a fully reusable “Starship + Super Heavy” launch vehicle program [
8
].
Among them, the starship as the upper stage returns to re-entry at orbital velocity after
completing the delivery mission, resulting in more significant deceleration requirements,
and more complex aerodynamic [
9
]. Therefore, the flight profile of the starship return
Mathematics 2023,11, 288. https://doi.org/10.3390/math11020288 https://www.mdpi.com/journal/mathematics
Mathematics 2023,11, 288 2 of 17
landing process can be considered as a “combination” of the lifting body glide re-entry and
the Falcon 9 sub-stage landing. During the high-speed phase, the glide re-entry deceleration
is performed at a relatively small ballistic inclination, and this phase is controlled by a
combination of multiple aerodynamic rudders and a reaction control system (RCS), which
restarts the engine above the landing point, and performs an attitude flip close to 90 degrees,
followed by a vertical soft landing based on thrust vector control (TVC) technology [
10
].
After a series of flight test explorations, in May 2021, the starship technology prototype
SN15 achieved the above large attitude flip vertical landing (LAFVL) for the first time,
verifying the relevant guidance and control technologies, and laying the foundation for the
subsequent starship and super heavy combination tests [11].
The return-landing flight profile of Starship, and the new GNC technology towed by
it, are of high reference value for high-speed return and vertical landing flights such as
the upper stage recovery missions of reusable launch vehicles. In particular, the online
trajectory optimization problem of its unique “Belly Flop” LAFVL flight segment is of
high academic exploration and engineering application value. In this paper, we propose a
high-precision online trajectory optimization algorithm combining convex optimization
and Radau pseudospectral method (RPM) to provide methodological and technical support
for the research of related space exploration projects.
The vertical landing onboard real-time trajectory optimization problem must be solved
quickly, accurately and with high precision. Considering the in-flight deviations, especially
unexpected situations such as wind disturbances, the algorithm needs to converge without
good initial guesses, which is not satisfied by most conventional optimization methods.
The interior point method (IPM) can solve convex problems in polynomial time, without ini-
tial guesses from the user, which is attractive for aerospace applications [
12
]. However,
most practical aerospace problems are not convex optimization problems, and cannot
be solved directly by IPM. Therefore, most researchers focus on formulating nonconvex
optimization problems within a convex framework, called convexification [
13
]. Ideally,
the nonconvex trajectory optimization problem should be equivalently transformed into
a convex optimization problem, called lossless convexification (LCvx). In recent years,
LCvx has been successfully achieved by equivalence transformation of control variables
and constraint relaxation [
14
–
16
]. According to the optimal control theory, the equivalence
of the transformations can be proved rigorously. LCvx technology has been successfully
applied to powered Mars landings [
17
], launch vehicle ascent flights [
18
], and missiles [
19
].
However, most path and terminal constraints as well as nonlinear dynamic constraints
cannot be convexized by LCvx, then the method of successive convexification (SCvx) sub-
sequently emerged to convexify complex nonconvex and nonlinear constraints [
20
]. This is
an iterative procedure for solving the linearization convex subproblem until it converges
to an optimal solution [
21
,
22
]. At the same time, discretization techniques [
23
,
24
], lin-
earization strategies [
21
], and trust region constraints [
25
] need to be considered to ensure
the convergence of the method [
26
]. Convex optimization has been successfully applied
to UAVs [
27
], spacecraft [
28
], and high-speed atmospheric vehicles [
29
]. In summary,
the convex optimization algorithm has good computational efficiency and robustness.
In this paper, we present a SCvx-RPM algorithm for online trajectory optimization and
autonomous guidance of LAFVL of the starship-like vehicle (SLV). At the same time, the fol-
lowing studies are conducted to improve the optimization problem modeling, discretization
and convexification algorithm techniques:
(1)
The coupling relationship between pitch angle and engine nozzle swing angle of
a SLV, and the effect of nonlinear aerodynamic forces on the motion of the vehicle
are considered. Combining the characteristics of the LAFVL trajectory optimization
problem, a planar landing trajectory optimization model considering the pitch attitude
is developed. The model can describe the landing motion process of the SLV with
high granularity compared with 3-DOF problem [
30
,
31
], it significantly improves the
computational efficiency compared with 6-DOF problem [32,33].
Mathematics 2023,11, 288 3 of 17
(2)
Based on the above planning model, the research of the low-loss convexification
method is carried out to avoid direct linearization leading to large errors [
19
,
20
]. We
maximize the use of the LCvx method to pre-process nonconvex motion models in
order to improve the convergence efficiency and reliability of the subsequently pro-
posed SCvx algorithm. Based on the original SCvx method, an online update strategy
of the trust region is used to improve the speed of convergence of the SCvx algorithm.
(3)
Using RPM to discretize the continuous optimal control problem, and designing the
landing terminal moment as a special control variable to optimize together, which
improves the optimality of the moment value and the optimization precision of the
trajectory compared with the methods of fixed terminal moment and additional search
for the optimal moment [34–36].
The paper is organized as follows: a model of the fuel-optimal trajectory optimization
problem in the landing site coordinate system (LSCS) is given in Section 2; the nonconvex
trajectory optimization model is convexized and discretized, and a suitable SCvx algorithm
is designed in Section 3; an ANSI-C trajectory optimization algorithm is developed in
Section 4to verify the relevant modeling and analytical conclusions; and conclusions are
drawn in Section 5.
2. LAFAL Trajectory Optimization Problem for The SLV
Firstly, the dynamics model of the SLV is discussed. A 3-DOF rocket motion model is
commonly used to study the powered descent landing (PDL) problem. However, the 3-DOF
rocket motion model assumes that the rocket itself is a mass point, and focuses only on
the translational aspect of the PDL problem, which cannot describe the large attitude flip
process of a SLV. The 6-DOF rocket motion model is a more accurate problem formulation,
and can better characterize the rocket translational and center-of-mass motion during the
vertical landing of a SLV. However, there are no explicit structural properties of the solution
to the 6-DOF rocket motion model for the fuel-optimal trajectory optimization problem (e.g.,
Bang-Bang thrust properties for the 3-DOF problem). In addition, the nonlinear complexity
of the 6-DOF model and a large number of variables leads to a long computational time for
its trajectory optimization, which is not conducive to online implementation.
In this study, considering that there is no large-scale lateral motion modulation in the
final PDL phase of the SLV, in order to reduce the complexity of the model, a planar landing
flight motion model that considers the pitch attitude motion can describe the motion
characteristics of the LAFVL process of the SLV. Unlike the 3-DOF or 6-DOF motion models,
the planar landing motion model considering pitch attitude introduces pitch attitude and
angular velocity variables, and restricts the translational motion to the fixed
x-z
plane of the
LSCS, which can better characterize the large attitude flip motion process of the SLV than
the 3-DOF model, and has higher computational efficiency than the 6-DOF motion model.
As shown in Figure 1, the origin of the LSCS is located at the landing point, the
z
-axis
direction is pointed upward from the center of the earth, and the
x
-axis is perpendicular
to the
z
-axis, so that the initial position and velocity vectors in the LSCS are located in the
plane containing the landing point. Then, the equations of motion describing the planar
landing of the SLV are shown as follows.
Mathematics 2023,11, 288 4 of 17
˙
x=Vx,
˙
z=Vz,
˙
Vx=−Tsin(θ+δ)+Dx
m,
˙
Vz=Tcos(θ+δ)+Dz
m−g0,
˙
θ=w,
˙
w=(MT+MD)
J,
˙
m=−T
Ispg0
(1)
where
r=[x,z]T
is the position vector,
V=[Vx,Vz]T
is the velocity vector,
θ
is the
pitch angle,
δ
is the engine nozzle swing angle,
w
is the pitch angle rate,
m
is the SLV
mass,
T
is the engine thrust,
Isp
is the engine specific impulse, and
g0
is the Earth’s
gravitational acceleration.
Dx
and
Dz
are the total aerodynamic drag in the
x
and
z
directions, respectively, described as
Dx=−CLD ·qV2
x+V2
z·Vx,
Dz=−CLD ·qV2
x+V2
z·Vz
(2)
where
CLD
is the total drag coefficient generated by aerodynamic drag and lift.
MT
and
MD
are the moments generated by engine thrust and aerodynamic forces, respectively,
and
J
is the rotational inertia estimated from engineering experience about the position of
the vehicle’s center of mass, which is expressed as
J=1
12 m6r2
s+l2
s,
MT=−lcg ·T·sin δ,
MD=−lcp −lc g·(Dx·cos θ+Dz·sin θ)
(3)
where
rs
is the body radius,
ls
is the body height,
lcg
is the position of the vehicle center
of mass, and
lcp
is the position of the vehicle center of pressure. For this motion model,
the state variables are defined as
x=[x,z,Vx,Vz,θ,w,m]T∈R7
and the control variables
are u=[T,δ]T∈R2.
mg
T
A
©
δmax
x
z
Figure 1. Planar landing model considering pitch attitude motion.
Mathematics 2023,11, 288 5 of 17
The state and control constraints involved in the planar landing problem of the SLV
in the LSCS are discussed below. Although the landing mode of the SLV is vertical,
the deceleration mode is the same as that of the lift-type re-entry vehicle, which is mainly
aerodynamic until the large attitude flip maneuver is performed. After the large attitude
flip maneuver, the thrust of the SLV reaches its peak and decelerates under the combined
effect of thrust and aerodynamic force. The optimization model built in the LSCS has
a more definite dynamic pressure and heat flow density trajectory pattern and change
pattern as the powered descent phase begins, and is almost monotonically decreasing
to zero. Therefore, the model established in this section does not consider the dynamic
pressure and heat flow constraints, and only considers the initial state constraints and the
terminal state constraints that satisfy the fixed-point vertical soft landing as follows.
x(t0)=x0,x|1:6tf=xf,mtf≥mdry (4)
where
t0
and
tf
are the initial and terminal moments, respectively,
x0
is the initial state
navigation and sensor sampling value, and
xf
is the terminal state.
mdry
is the dry weight
of the rocket,
x|1:6tf,[xf,zf,Vx f ,Vzf ,θf,wf]T∈R6
. Since the landing point is the
origin of the LSCS, there are generally zero rocket states except for the rocket mass, which is
greater than the rocket dry weight. In order to characterize the large attitude flip maneuver
of the vehicle, θ0=π/2 is taken.
For the control constraints, there are thrust amplitude constraints and engine nozzle
swing angle amplitude constraints.
Tmin ≤T≤Tmax ,
−δmax ≤δ≤δmax
(5)
The objective function of the optimal control problem is defined as the fuel-optimal,
and its performance index can be expressed as
J=−mtf(6)
The nonconvex fuel-optimal trajectory optimization problem
P1
under the continuous
system is given by combining the above-mentioned SLV motion model and constraints as
follows. min J=−m(tf)
s.t. Dynamics : Equation (1)
Constraints : Equations (4)and (5)
(7)
where the optimization variables are
Xopt = [x
,
z
,
Vx
,
Vz
,
θ
,
w
,
m
,
T
,
δ
,
tf]T
, containing state
variables, control variables, and terminal time variable
tf
. It can be seen that the larger
nonconvexity of the optimization problem lies in the SLV motion model. Therefore, the fo-
cus in the next section is on convexification and discretization methods for nonconvex
system dynamics.
3. Convexization and Discretization of P1Problem
3.1. LCvx of P1Problem
In this subsection, the nonconvex trajectory optimization model is initially convexified.
Firstly, to improve the landing precision, the engine nozzle swing angle command is made
smoother, and the coupling of the state quantity pitch angle and the control quantity engine
nozzle swing angle is uncoupled. The engine nozzle swing angle rate is used as the control
Mathematics 2023,11, 288 6 of 17
quantity instead of engine nozzle swing angle, and an augmented SLV motion model
is proposed.
˙
x=Vx,˙
z=Vz,˙
Vx=−Tsin(θ+δ)+Dx
m,˙
Vz=Tcos(θ+δ)+Dz
m−g0,
˙
θ=w,˙
w=(MT+MD)
J,˙
m=−T
Isp g0
,˙
δ=χ
(8)
For this motion model, the state variables are defined as
x=[x,z,Vx,Vz,θ,w,m,δ]T∈
R8
and the control variables are
u=[T,χ]T∈R2
. Additionally, increasing the augmented
control quantity constraint on the angular rate of the engine nozzle swing angle.
−χmax ≤χ≤χmax (9)
where
χmax
is the upper limit of the angular rate of the engine nozzle swing angle. The aug-
mented nonconvex fuel-optimal trajectory optimization problem
P2
is obtained as follows.
min J=−m(tf)
s.t. Dynamics : Equation (8)
Constraints : Equations (4),(5)and (9)
(10)
Then, in order to reduce the degree of nonlinearity in the augmented dynamics,
the component of the engine thrust in the LSCS is introduced as a new control variable.
Tx=Tsin(θ+δ),Tz=Tcos(θ+δ)(11)
In turn, the trigonometric function term is substituted out of the system dynamics.
Before and after the transformation, the degrees of freedom of the system control variables
are two, but the variable substitution introduces new process constraints as follows.
T2
x+T2
y=T2(12)
arctanTx
Tz−(θ+δ)=0 (13)
Equation (12) is a natural derivation of the trigonometric relationship. The thrust
component trigonometric function is determined by two parameters, namely the pitch angle
and the engine nozzle swing angle, and the effect of applying the constraint Equation (13)
is to make its two relations equivalent to the original constraint form.
For the new nonconvex constraint introduced by the variable substitution,
Equation (12)
can be handled by borrowing the LCvx method, i.e., directly relaxing the equation constraint
to the inequality constraint.
T2
x+T2
z≤T2(14)
That is, the constraint is convexized by taking the form of a convex package, and the
equivalence of the two can be proved by the principle of maximal value. Equation (13),
however, still contains trigonometric terms, which are difficult to perform LCvx. In the
subsequent design, it will be sequential linearization.
The thrust-acceleration term in the velocity dynamics is still nonlinear due to the
time-varying mass of the SLV during the landing process. To convexify this nonlinear term,
a new variable is further introduced to substitute it.
σ,T
m,Txm ,Tx
m,Tzm ,Tz
m,v,ln m(15)
Mathematics 2023,11, 288 7 of 17
Then, the mass dynamics transformation is given by
˙
v=ασ (16)
where α=−1/Ispg0. The mass terminal constraint transformation is given by
v(tf)≥ln(mdry)(17)
The nonconvex constraint (13) transforms as
arctanTxm
Tzm −(θ+δ)=0 (18)
The convex constraint (14) and the first equation of (5) transform, respectively, as
qT2
xm +T2
zm ≤σ(19)
Tmin ·e−v≤σ≤Tmax ·e−v(20)
Then, the system dynamics is transformed as
˙
x=Vx,
˙
z=Vz,
˙
Vx=−Txm+Dxm ,
˙
Vz=Tzm +Dzm −g0,
˙
θ=w,
˙
w=LT·σ·sin δ+LD·(Dxm ·cos θ+Dzm ·sin θ),
˙
v=ασ,
˙
δ=χ
(21)
where Dxm =e−v·Dx,
Dzm =e−v·Dz,
LT=−12lcg /6r2
s+l2
s,
LD=−12lcp −lc g/6r2
s+l2
s
(22)
For this motion model, the state variables are defined as
x=[x,z,Vx,Vz,θ,w,v,δ]T∈
R8
and the control variables are
u=[Txm,Tzm,σ,χ]T∈R4
. Thereby, the fuel-optimal
trajectory optimization problem P3after LCvx is obtained as follows.
min J=−m(tf)
s.t. Dynamics : Equation (21)
Constraints : x(t0)=x0,x|1:6tf=xf,v(tf)≥ln(mdry)
−δmax ≤δ≤δmax,−χmax ≤χ≤χmax
arctan(Txm/Tzm )−(θ+δ)=0
qT2
xm +T2
zm ≤σ,Tmin ·e−v≤σ≤Tmax ·e−v
(23)
3.2. Discretization of Problem P3
The significance of using RPM to discretize the system dynamics is mainly reflected in
two aspects: high discretization precision and facilitation of handling the free time problem.
The technical details of the RPM and its precision and convergence speed are discussed in
Mathematics 2023,11, 288 8 of 17
detail in the literature [
23
–
25
]. Using the unique form of discretization time domain of RPM,
the terminal moment
tf
of the LAFVL process of the SLV is designed as a special control
variable for optimization, which is an important feature of this paper, and an effective
means to improve the optimality and precision of the results. In many similar problems,
in order to reduce the complexity of the optimization model, and ensure the convexity of
the problem, the terminal moment is determined offline or fixed, which may not be the
optimal shutdown point [
34
–
36
], and thus the optimality of the whole trajectory is not
guaranteed and the fuel consumption is not optimal.
The pseudospectral discretization of the system dynamics equations takes the follow-
ing form.
N+1
∑
j=1
Dij x(τi)−tf
2f(x(τi),u(τi)) = 0(24)
The terminal moments are regarded as special control variables and the above discrete
algebraic equation constraints are expressed as follows.
2
N+1
∑
j=1
Dij x(τi) + faug(yi) = 0(25)
where
faug(yi) = f(x(τi)
,
uaug(τi))
,
yi={x(τi)
,
uaug(τi)}
,
uaug = [u
,
tf]T
,
faug(y)
as a
function of the right-hand side of the augmented dynamics equation treating the terminal
moment as a special control variable.
3.3. SCvx of Discretization Optimization Problem
The preliminary convexification of the nonconvex trajectory optimization model is
performed in Section 3.1, and the nonconvex trajectory optimization model is completely
lossless by LCvx. The reason for giving the discretization model of the system dynamics
equations in Section 3.2 before the treatment of the SCvx method in this subsection is that
the discretization introduces the multiplication of the free time with the right-hand side
function of the system dynamics, which generates new nonlinear terms, requiring the con-
vexification method to transform them approximately. In this subsection, the discretization
free-time problem system dynamics equations and process constraints are approximated
linearly, the trust region constraints required by the SCvx algorithm are designed, and the
specific form of the discretization matrix required by the IPM solver format is given in
conjunction with other constraint models.
The dynamics constraints are first studied. In Equation (25), the nonlinear correlation
terms are concentrated in the function
faug(yi)
at the right end of the augmented dynamics
equation. The first-order Taylor expansion of faug(yi)takes the following form.
faug(yi) =Ai(xk,uk
aug)x(τi) + Bi(xk,uk
aug)uaug (τi) + wi(xk,uk
aug)(26)
where
A=∂faug
∂x
,
B=∂faug
∂uaug
, and
wi(xk
,
uk
aug) = faug-i(xk
,
uk
aug)−Ai(xk
,
uk
aug)xk(τi)−
Bi(xk
,
uk
aug)uk
aug(τi)
.
xk
and
uk
aug
are the reference points of the Taylor expansion in the
k
-th iteration, and
faug-i(xk
,
uk
aug)
denotes the value of the right-hand side function of the
augmented dynamics equation at the i-th discretization point out in the k-th iteration.
Secondly, for the nonlinear terms in the process constraints (18) and (20) of the tra-
jectory optimization problem, there is no LCvx methods for the time being, and their
nonlinearity is strong, so the above process constraints are similarly approximated by
sequential linearization, and the linearized expressions correspond to (27) and (28), respec-
tively, as follows.
ki
Tx Txm +ki
Tz Tzm −θ−δ=bi
T(27)
Tmine−v0·(1−(v−v0))≤σ≤Tmaxe−v0·(1−(v−v0))(28)
Mathematics 2023,11, 288 9 of 17
An important part of the above linearization process is the selection of the Taylor
expansion reference point. In the SCvx algorithm, the reference point for the 1st iteration
is provided by the coarse selection of the initial value, and the reference point in the sub-
sequent iterations is taken to be the optimal solution obtained in the previous iteration.
According to the characteristics of Taylor expansion, the linearized dynamics and state
constraints are a good approximation to the original nonlinear form only when the opti-
mization variables are taken near the reference point during the iterative process of SCvx.
Therefore, the following trust region constraints are added to the SCvx algorithm.
|xk+1−xk| ≤ εx,|uk+1
aug −uk
aug| ≤ εuaug (29)
where
εx
and
εuaug
are the trust region values designed for each state and control variable,
respectively, and become smaller as the number of iterations increases to improve the
convergence speed of the algorithm.
Ultimately, the linearization convex subproblem P4is obtained as follows.
min J=−m(tf)
s.t. Dynamics : Equation (26)
Constraints : x(t0)=x0,x|1:6tf=xf,v(tf)≥ln(mdry)
−δmax ≤δ≤δmax,−χmax ≤χ≤χmax
ki
Tx Txm +ki
Tz Tzm −θ−δ=bi
T,qT2
xm +T2
zm ≤σ
Tmine−v0·(1−(v−v0))≤σ≤Tmaxe−v0·(1−(v−v0))
(30)
3.4. SCvx Algorithm
For the convenience of programming the trajectory optimization algorithm, the equa-
tion constraints and inequality constraints for each iteration of the optimization problem
input to the IPM solver are transcribed as
A−Xopt =b
and
G−Xopt ≤h
standard forms,
respectively, to facilitate the implementation of the SCvx algorithm. Based on the above
discussion, the SCvx algorithm for LAFVL of a SLV is given in this subsection as follows.
Input: set the number of collocation points
N
; set the initial reference trajectory
X0
opt
;
set the initial value of the trust region constraint
εx
,
εuaug
; set the iteration termination
criterion parameter
ε>
0; set the maximum number of iterations
kmax
; set the number
of iterations k=1, 4X1
opt =1.
Step 1: Solve the linearization convex subproblem
P4
using the IPM solver and com-
pute the updates of the optimal variables Xk
opt.
Step 2: Check the convergence condition
|Xk
opt −Xk−1
opt |<ε
, if the convergence condi-
tion is satisfied, go to Step 3. Otherwise, set k=k+1, and return to Step 1.
Step 3: The optimization problem is solved, X∗
opt =Xk
opt.
4. Numerical Experiments
The numerical optimization simulation session is an important part of validating the
LAFVL guidance algorithm for the SLV. In this section, GPOPS validation programs for
the aforementioned
P1
,
P2
, and
P3
trajectory optimization problems for vertical landing,
and the C language program for the SCvx algorithm are developed. The IPM solution
of the subproblems in the C program can be called from the open source or commercial
packages such as ECOS, MOSEK, etc., and GPOPS is used as a benchmark for calibration
and verification of the algorithm. In this section, numerical simulation experiments are
conducted based on the above algorithm programs to verify the correctness, effectiveness,
and efficiency of the programs.
Section 4.1 prepares a trajectory optimization program based on the GPOPS-II package
for problems with different optimization contexts, followed by numerical optimization
Mathematics 2023,11, 288 10 of 17
simulation experiments on the Windows 10 operating system to verify the correctness of
the model and convexification methods.
In Section 4.2, a C-language program for optimizing the LAFVL trajectory of the
SLV based on the ECOS open source software package is developed for the
P4
trajectory
optimization problem under the premise that the GPOPS-II validation modeling and
lossless convexification methods are correct. At the same time, the C program is run on the
embedded guidance computer of the hardware-in-the-loop simulation system to simulate
the test operation conditions under the onboard computing platform. The correctness and
real-time performance of the SCvx guidance algorithm are effectively verified under the
Linux operating system.
To perform a SLV large attitude flip maneuver, the spacecraft speed needs to be
reduced by exposing the greater aerodynamic drag created by the wider surface. At an
altitude of about 600 m, the vehicle starts a large maneuver to change its attitude from
horizontal to vertical to perform a precision landing maneuver. The model parameters of
the SLV are shown in Table 1.
Table 1. SLV model parameters.
Variable Symbols Variable Name Numerical Value
m0Initial Mass 120,000 kg
mdry Dry Weight of The Vehicle 85,000 kg
lsVehicle Altitude 50 m
rsVehicle Radius 4.5 m
lcg Center of Mass Position 20 m
lcp Center of Pressure Position 22.5 m
δmax Maximum Engine Nozzle
Swing Angle 20 deg
χmax Maximum Engine Nozzle
Swing Angle Rate 20 deg/s
Tmax Maximum Engine Thrust 2210 kN
Tmin Minimum Engine Thrust 880 kN
Isp Engine Ratio Impulse 330 s
The boundary conditions to be satisfied are as follows.
x(t0)=100 m, z(t0)=600 m, Vx(t0)=0 m/s, Vz(t0)=−85 m/s
θ(t0)=π
2rad, w(t0)=0 rad/s, m(t0)=120,000 kg, xtf=0 m
ztf=0 m, Vxtf=0 m/s, Vztf=0 m/s, θtf=0 rad
wtf=0 rad/s, mtf≥85,000 kg
4.1. GPOPS Numerical Optimization Simulation Analysis
The main purpose of this subsection is to verify the correctness of the developed and
partially convexified models of the SLV planar landing trajectory optimization problem.
Based on the GPOPS-II software package, the trajectory optimization procedures for the
above
P1
,
P2
, and
P3
problems are prepared, and the analysis of the model characteristics
and the correctness of the convexification method is carried out based on numerical experi-
ments. The operating system environment of numerical simulation in this subsection is
Windows 10, intel i7-10710U CPU@1.1 GHz, and 16 GB RAM.
The free terminal time problem is considered, i.e., the terminal time is the discrete
optimization variable. Forty Radau collocation points are used in the preliminary optimiza-
tion experiments, i.e., a total of 41 discretization points. For this problem size, GPOPS-II is
invoked to solve the nonconvex fuel optimal trajectory optimization problem
P1
, the aug-
mented nonconvex fuel optimal trajectory optimization problem
P2
, and the nonconvex
fuel optimal augmented trajectory optimization problem
P3
after lossless convexification in
the above computer environment for comparison experiments. The optimization results
Mathematics 2023,11, 288 11 of 17
of the three problems are given below, as shown in Figure 2, and the position precision
and velocity precision of the vehicle landing by using the trajectory integration precision
comparison method are shown in Table 2.
0 2 4 6 8 10 12
Powered Landing Time, s
-20
0
20
40
60
80
100
X-directional Position, m
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-16
-14
-12
-10
-8
-6
-4
-2
0
2
X-directional Velocity, m/s
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-10
0
10
20
30
40
50
60
70
80
90
Pitch Angle, deg
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
1.14
1.15
1.16
1.17
1.18
1.19
1.2
1.21
Mass, kg
105
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
0
100
200
300
400
500
600
Z-directional Position, m
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Z-directional Velocity, m/s
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-25
-20
-15
-10
-5
0
5
10
Pitch Angle Rate, deg/s
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
800
1000
1200
1400
1600
1800
2000
2200
2400
Thrust, kN
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-15
-10
-5
0
5
10
15
20
Engine Swing Angle, deg
P1 Optimization Results
P2 Optimization Results
P3 Optimization Results
Figure 2. Comparison of optimization results for P1,P2and P3.
Mathematics 2023,11, 288 12 of 17
Table 2. Comparison of landing precision.
Optimization Problem Positional Precision Velocity Precision
P12.8273 m 0.42003 m/s
P20.42707 m 0.099739 m/s
P30.42708 m 0.09974 m/s
In summary, the preliminary analysis can be concluded as follows.
(1)
By invoking the GPOPS-II software package to solve the nonconvex fuel optimal
trajectory optimization problem
P1
, the augmented nonconvex fuel optimal trajec-
tory optimization problem
P2
and the nonconvex fuel optimal augmented trajectory
optimization problem
P3
after lossless convexification, the correctness of the current
trajectory optimization model design, transformation, and partial convexification is
initially verified.
(2)
The comparison of the optimization results of
P2
and
P3
problems shows that using
the engine nozzle swing angle rate instead of engine nozzle swing angle as the control
quantity uncouples the state quantity pitch angle and the control quantity engine
nozzle swing angle, makes the engine nozzle swing angle smoother, and effectively
improves the landing precision.
(3)
A comparison of the optimization results of the
P2
and
P3
problems shows that the
nonconvex constraint Equation (18) is equivalent to the original nonconvex constraint
Equation (11), which does not reduce its nonconvexity, but effectively reduces the con-
straint dimension without losing the additive characteristic relationship characterizing
the pitch angle of the vehicle and the swing angle of the engine nozzle.
The above conclusions indicate that the
P3
problem is the lowest nonconvex problem
representation before the pseudospectral discretization operation, and also has the condi-
tions to form a C program by discretization and sequential convexification processing in
the subsequent study.
4.2. Hardware in the Loop Simulation Analysis
In this subsection, the performance of the proposed SCvx algorithm is verified by a
hardware-in-the-loop simulation. The hardware-in-the-loop simulation system built based
on the Speedgoat real-time target computer also includes the real-time simulation host
computer, the onboard flight control computer, the simulated onboard guidance computer
and the nozzle actuation simulator, and the main equipment components, etc, in which
the GNC computers are analogous to those onboard the vehicle or at ground stations,
the navigation computer collects the current vehicle status, the guidance computer runs
the guidance algorithm and generates guidance commands based on the current vehicle
status, and the control computer generates control commands based on the guidance
information. The system architecture is shown in Figure 3. The test experiment uses a
self-developed dedicated guidance and control computer, which is based on the NVIDIA
Jetson Xavier NX motherboard as the core computing unit, the operating system is Ubuntu
18.04.5 LTS, the Visual Studio Code integrated development environment, using standard
C programming language, the CPU is NVIDIA Carmel Arm v8.2 64-bit CPU with 1.4 GHz,
6 cores and 16 GB RAM.
This subsection compares the SCvx algorithm proposed in this paper with the Matlab
GPOPS-II package, a RPM package that has been tested in a wide range of problem solving
and is a typical representative of software based on pseudospectral method and nonlinear
programming. Therefore, GPOPS-II is used in this paper to verify the correctness of
the algorithm. In this calculation example, the number of collocation points are set to 20.
The convex optimization procedure formed by the SCvx algorithm through C programming
uses the trajectory generated by linear interpolation after concatenating the initial and
terminal points as the initial trajectory, which is solved by the IPM solver ECOS.
Mathematics 2023,11, 288 13 of 17
Real-time Simulation
Host Computer
Control ComputerControl Computer
Rocket Nozzle
Simulator
Rocket Nozzle
Simulator
Navigation and Fault
Diagnosis Computer
Navigation and Fault
Diagnosis Computer
Real-time Simulation
Target Computer
Real-time Simulation
Target Computer
RS422
Host-Target Ethernet
Ethernet
CAN FD
Gigabit Ethernet Switch
Guidance ComputerGuidance Computer
Figure 3. Hardware-in-the-loop simulation system framework.
From the results in Figure 4and Table 3, it can be seen that the results obtained from
the optimization of the SCvx algorithm procedure are in basic agreement with Matlab
GPOPS-II. In terms of computational speed, the computational time consumed by the SCvx
algorithm procedure is 0.286 s, which satisfies the requirement of computational efficiency
for online trajectory optimization for powered landing. The number of iterations at the end
of the SCvx algorithm procedure is 5, which indicates that the proposed SCvx algorithm
can obtain the optimal solution by iteration without exact initial values. The large attitude
flip planar landing trajectory of the SLV is shown in Figure 5, which can more visually
reflect the vertical landing process of the vehicle under the SCvx guidance algorithm.
0 2 4 6 8 10 12
Powered Landing Time, s
-20
0
20
40
60
80
100
X-directional Position, m
X-directional Position Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
0
100
200
300
400
500
600
700
Z-directional Position, m
Z-directional Position Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-16
-14
-12
-10
-8
-6
-4
-2
0
2
X-directional Velocity, m/s
X-directional Velocity Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Z-directional Velocity, m/s
Z-directional Velocity Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
Figure 4. Cont.
Mathematics 2023,11, 288 14 of 17
0 2 4 6 8 10 12
Powered Landing Time, s
-20
0
20
40
60
80
100
Pitch Angle, deg
Pitch Angle Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-25
-20
-15
-10
-5
0
5
10
Pitch Angle Rate, deg/s
Pitch Angle Rate Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
1.14
1.15
1.16
1.17
1.18
1.19
1.2
1.21
Mass, kg
105Mass Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
800
1000
1200
1400
1600
1800
2000
2200
2400
Thrust, kN
Thrust Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
0 2 4 6 8 10 12
Powered Landing Time, s
-20
-15
-10
-5
0
5
10
15
20
Engine Swing Angle, deg
Engine Swing Angle Curve
Matlab GPOPS Optimization Results
SCvx Algorithm Optimization Results
Figure 4. Comparison of optimization trajectories.
-300 -200 -100 0 100 200 300 400
x, m
0
100
200
300
400
500
600
z, m
Starship Large Attitude Flip Planar Landing Trajectory
Figure 5. Starship large attitude flip planar landing trajectory.
Mathematics 2023,11, 288 15 of 17
Table 3. Optimization results.
Optimization
Procedure Terminal Moment Terminal Mass CPU Time
Consumption
SCvx Algorithm 11.5666 s 114,568.6 kg 0.286 s
Matlab GPOPS 11.5666 s 114,580.7 kg
The landing precision of the optimization results of the SCvx algorithm and Matlab
GPOPS-II are shown in Table 4. The two are comparable in magnitude, and the preci-
sion differs for different simulation conditions (e.g., number of discretization points, see
Table 5
), i.e., the SCvx algorithm proposed in this paper achieves the precision of a general
RPM-based nonlinear programming algorithm. When the number of discretization points
increases, the landing precision also increases, but the computational efficiency decreases.
By comparing the above optimization results, the correctness of the SCvx algorithm can
be verified. Meanwhile, the advantage of the SCvx algorithm in computational efficiency
makes it possible for onboard operation.
Table 4. Comparison of SCvx algorithm and Matlab GPOPS-II integration precision.
Optimization Procedure Positional Precision Velocity Precision
SCvx Algorithm 0.77127 m 0.30135 m/s
Matlab GPOPS 0.62173 m 0.33001 m/s
Table 5. Precision Comparison with Different Number of Collocation Points.
Optimization
Procedure
Number of
Collocation Points Positional Precision Velocity Precision
SCvx Algorithm
15 1.1597 m 0.29861 m/s
20 0.77127 m 0.30135 m/s
30 0.7097 m 0.23689m/s
Matlab GPOPS
15 1.1636m 0.39594 m/s
20 0.62173 m 0.33001 m/s
30 0.42617 m 0.1531 m/s
5. Conclusions
For the online trajectory optimization problem of LAFVL problem of SLV, a SCvx
algorithm combining RPM and convex optimization techniques is proposed, which has
high optimization precision and fast solving speed without exact initial values. The main
research work and conclusions obtained in this paper are as follows. (1) The problem
is discretized by the RPM with high precision, and the unique time-domain mapping of
the RPM is used to transform the powered soft-landing terminal moments into special
control variables, which is different from the fixed terminal moments method used in
similar literature, and improves the optimization precision and optimality of the trajectory
optimization results. (2) The nonconvex optimization model was convexified by combining
LCvx and SCvx techniques to obtain a sequential convex optimization problem equivalent
to the original problem. (3) The proposed SCvx algorithm is experimentally verified by
a hardware-in-the-loop simulation platform to verify the correctness, high precision and
fast convergence of the algorithm. In the future, we will further investigate the online
guidance problem of the 6DOF dynamics model of SLV and design solution algorithms
with faster computational efficiency in specific problems with more optimization variables
and larger dimensionality.
Author Contributions:
Methodology, Z.M.; Validation, L.S.; Data curation, H.C.; Writing—original
draft, Z.M.; Writing—review & editing, Z.M.; Supervision, H.C.; Project administration, J.W.; Funding
acquisition, J.W. All authors have read and agreed to the published version of the manuscript.
Mathematics 2023,11, 288 16 of 17
Funding: This research received no external funding.
Data Availability Statement: All data used during the study appear in the submitted article.
Conflicts of Interest:
The authors certify that there are no conflict of interest with any individ-
ual/organization for the present work.
References
1.
Chen, S.Q.; Huang, H.; Shao, Y.T.; Huang, B. Study on the requirement trend and development suggestion for China space
propulsion system. Astronaut. Syst. Eng. Technol. 2019,3, 62–70.
2.
Chen, S.Q.; Huang, H.; Zhang, Q.S.; Qin, X.D.; Rong, Y. Research on the development directions of Chinese launch vehicle liquid
propulsion system. Astronaut. Syst. Eng. Technol. 2020,4, 1–12.
3. Luo, M.; Chen, S.Q.; Li, D.P.; Pan, H. Characteristics of starship propulsion system and numerical simulation of propellant flow
during reentry. J. Nanjing Univ. Aeronaut. Astronaut. 2021,53, 9–16.
4.
Song, Z.Y.; Wang, C. Development of online trajectory planning technology for launch vehicle return and landing. Astronaut. Syst.
Eng. Technol. 2019,3, 1–12.
5.
Chen, Z.H.; Ning, L.; Wang, P. The development of launch vehicle booster recovery technology. Astronaut. Syst. Eng. Technol.
2021,5, 66–74.
6.
Hu, D.S.; Liu, N.; Liu, B.L.; Yan, N. Analysis on the development of reusable launch vehicles in the United States. Space Int.
2020
,
12, 38–45.
7. Yang, K.; Mi, X. Analysis of the development of SpaceX’s reusable launch vehicle. Space Int. 2020,9, 13–17.
8.
Yan, N.; Hu, D.S.; Hao, Y.X. Analysis of SpaceX’s “Super HeavyStarship” transportation system scheme. Space Int.
2020
,11, 11–17.
9. Long, X.D. A brief analysis of the super-heavy-starship transport system and its future impact. Aerosp. Technol. 2021,8, 32–35.
10.
Cantou, T.; Merlinge, N.; Wuilbercq, R. 3DoF simulation model and specific aerodynamic control capabilities for a SpaceX’s
Starship-like atmospheric reentry vehicle. In Proceedings of the 8nd European Conference for Aeronautics and Space Sciences,
Madrid, Spain, 1–5 July 2019.
11. Palmer, C. SpaceX starship lands on Earth, But manned missions to Mars will require more. Engineering 2021,7, 1345–1347.
12. Liu, X. Autonomous Trajectory Planning by Convex Optimization; Iowa State University: Ames, IA, USA, 2013.
13. Liu, X.; Lu, P.; Pan, B. Survey of convex optimization for aerospace applications. Astrodynamics 2017,1, 23–40.
14.
Açıkme¸se, B.; Blackmore, L. Lossless convexification of a class of optimal control problems with non-convex control constraints.
Automatica 2011,47, 341–347.
15.
Blackmore, L.; Açıkme¸se, B.; Carson, J.M. Lossless convexification of control constraints for a class of nonlinear optimal control
problems. Syst. Control. Lett. 2012,61, 863–870.
16.
Harris, M.W.; Açıkme¸se, B. Lossless convexification of non-convex optimal control problems for state constrained linear systems.
Automatica 2014,50, 2304–2311.
17.
Acikmese, B.; Ploen, S.R. Convex programming approach to powered descent guidance for mars landing. J. Guid. Control Dyn.
2007,30, 1353–1366.
18.
Cheng, X.; Li, H.; Zhang, R. Efficient ascent trajectory optimization using convex models based on the Newton–Kantorovich/
Pseudospectral approach. Aerosp. Sci. Technol. 2017 66, 140–151.
19.
Zhang, K.; Yang, S.; Xiong, F. Rapid ascent trajectory optimization for guided rockets via sequential convex programming. Proc.
Inst. Mech. Eng. Part G J. Aerosp. Eng. 2019,233, 4800–4809.
20.
Zhang, Z.; Li, J.; Wang, J. Sequential convex programming for nonlinear optimal control problems in UAV path planning. Aerosp.
Sci. Technol. 2018,76, 280–290.
21. Reynolds, T.P. Computational Guidance and Control for Aerospace Systems; University of Washington: Seattle, WA, USA, 2020.
22.
Malyuta, D. Convex Optimization in a Nonconvex World: Applications for Aerospace Systems; University of Washington: Seattle, WA,
USA, 2021.
23.
Garg, D. Advances in Global Pseudospectral Methods for Optimal Control; Massachusetts Institute of Technology: Cambridge, MA,
USA, 2011.
24.
Darby, C.L. hp-Pseudospectral Method for Solving Continuous-Time Nonlinear Optimal Control Problems; University of Florida:
Gainesville, FL, USA, 2011.
25.
Wang, J.B.; Cui, N.G.; Guo, J.F.; Xu, D.F. High precision rapid trajectory optimization algorithm for launch vehicle landing. Control
Theory Appl. 2018,35, 389–398.
26.
Wang, J.B.; Cui, N.G. A pseudospectral-convex optimization algorithm for rocket landing guidance. In Proceedings of the 2018
AIAA Guidance, Navigation, and Control Conference, Kissimmee, FL, USA, 8–12 January 2018; p. 1871.
27.
Szmuk, M.; Pascucci, C.A.; Dueri, D. Convexification and real-time on-board optimization for agile quad-rotor maneuvering and
obstacle avoidance. In Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS),
Vancouver, BC, Canada, 24–28 September 2017; pp. 4862–4868.
28.
Zhou, D.; Zhang, Y.; Li, S. Receding horizon guidance and control using sequential convex programming for spacecraft 6-DOF
close proximity. Aerosp. Sci. Technol. 2019,87, 459–477.
Mathematics 2023,11, 288 17 of 17
29.
Wang, Z.B. Optimal trajectories and normal load analysis of hypersonic glide vehicles via convex optimization. Aerosp. Sci.
Technol. 2019,87, 357–368.
30.
Wang, J.B.; Ma, H.J.; Li, H.X.; Chen, H.B. Real-time guidance for powered landing of reusable rockets via deep learning. Neural
Comput. Appl. 2022, 1–22. https://doi.org/10.1007/s00521-022-08024-4.
31.
Furfaro, R.; Scorsoglio, A.; Linares, R. Adaptive generalized ZEM-ZEV feedback guidance for planetary landing via a deep
reinforcement learning approach. Acta Astronaut. 2020,171, 156–171.
32.
Gaudet, B.; Linares, R.; Furfaro, R. Deep reinforcement learning for six degree-of-freedom planetary landing. Adv. Space Res.
2020
,
65, 1723–1741.
33.
Xu, X.; Chen, Y.; Bai, C. Deep Reinforcement Learning-Based Accurate Control of Planetary Soft Landing. Sensors
2021
,21, 8161.
34.
Dueri, D.; Açıkme¸se, B.; Scharf, D.P.; Harris, M.W. Customized real-time interior-point methods for onboard powered-descent
guidance. J. Guid. Control Dyn. 2017,40, 197–212.
35.
Dueri, D.; Zhang, J.; Açikmese, B. Automated custom code generation for embedded, real-time second order cone programming.
IFAC Proc. Vol. 2014, 47, 1605–1612.
36.
Ren, G.F.; Gao, A.; Cui, P.Y.; Luan, E.J. A rapid power descent phase trajectory optimization method with minimum fuel
consumption for Mars pinpoint landing. J. Astronaut. 2014,35, 1350–1358.
Disclaimer/Publisher’s Note:
The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
