research article steady glide dynamic modeling and

Research ArticleSteady Glide Dynamic Modeling and Trajectory Optimization forHigh Lift-to-Drag Ratio Reentry Vehicle

Liang Yang, Wanchun Chen, Xiaoming Liu, and Hao Zhou

School of Astronautics, Beihang University, Beijing 100191, China

Correspondence should be addressed to Xiaoming Liu; liuxiaoming [email protected]

Received 13 March 2016; Accepted 30 June 2016

Academic Editor: Christian Circi

Copyright © 2016 Liang Yang 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.

Steady glide trajectory optimization for high lift-to-drag ratio reentry vehicle is a challenge because of weakly damped trajectoryoscillation. This paper aims at providing a steady glide trajectory using numerical optimal method. A new steady glide dynamicmodeling is formulated via extending a trajectory-oscillation suppressing scheme into the three-dimensional reentry dynamicswitha spherical and rotating Earth.This scheme comprehensively considers all factors acting on the flight path angle and suppresses thetrajectory oscillation by regulating the vertical acceleration in negative feedback form and keeping the lateral acceleration invariant.Then, a study on steady glide trajectory optimization is carried out based on this modeling and pseudospectral method. Twoexamples with and without bank reversal are taken to evaluate the performance and applicability of the newmethod. A comparisonwith the traditional method is also provided to demonstrate its superior performance. Finally, the feasibility of the pseudospectralsolution is verified by comparing the optimal trajectorywith integral trajectory.The results show that thismethod not only is capableof addressing the case which the traditional method cannot solve but also significantly improves the computational efficiency. Moreimportantly, it provides more stable and safe optimal steady glide trajectory with high precision.

1. Introduction

Entry guidance plays an important role in generating thesteering command to guide the vehicle from its initial condi-tion to reach the destination safely and accurately. In general,traditional reentry guidance is divided into two parts. Thefirst part is the generation of a feasible reference trajectory.The second part is the tracking of this reference trajectory[1]. This paper focuses on generating a feasible steady glidereference trajectory, especially for high lift-to-drag ratioreentry vehicle, using numerical optimal method. Previousresearches in reentry trajectory optimization are summa-rized as follows. Scott applied the Legendre Pseudospectralmethod into the trajectory optimization of reentry vehicles.In Josselyn and Ross’s work [2], covectormapping theorem ofLegendre Pseudospectral method was used to verify the first-order optimality condition arising in the path constraintstrajectory optimization. Rao and Clarke [3] also studied theproblem of reentry trajectory optimization using LegendrePseudospectral method. The key features of the optimal tra-jectory and quality of trajectory obtained from the Legendre

Pseudospectral method were discussed. Jorris and Cobb andZhao and Zhou [4, 5] employed the Gauss Pseudospectralmethod to optimize the 2D and 3D reentry trajectory forCommon Aero Vehicle (CAV), in which waypoint and no-flyzone constraints were considered as inner-point constraintsin optimal process. Rahimi et al. [6] applied the particleswarm optimization into spacecraft reentry optimization.High-order polynomials were used to approximate the angleof attack and bank angle in problem formulation. The coeffi-cients of both polynomials were considered as input variablesin optimal process. It should be noted that, because of notconsidering the trajectory-oscillation suppressing scheme,the optimal trajectories generated from above methods arenaturally oscillatory. In steady glide, the heating rate will notchange sharply and the steady state will greatly release theburden of control system. Therefore, steady glide trajectoryis the best reference trajectory for the reentry guidance.Actually, quasi-equilibrium-glide condition (QEGC) is awell-known “soft” path constraint that makes the trajec-tory change monotonously. However, complicated reentrydynamics, especially for high lift-to-drag ratio vehicle, are so

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2016, Article ID 3527460, 14 pageshttp://dx.doi.org/10.1155/2016/3527460

2 International Journal of Aerospace Engineering

sensitive to the “soft” path constraint that it is very difficultfor numerical optimizationmethod to converge when QEGCis considered in optimal process. Generally speaking, thisconstraint is suitable for the trajectory planning in which oneor two parameters are searched by secant method so as togenerate a feasible trajectory [7–9]. Therefore, steady glidetrajectory optimization for the high lift-to-drag ratio vehicleis always a challenge for numerical optimization.

The objective of this paper is to investigate the steadyglide dynamic modeling and trajectory optimization for thehigh lift-to-drag ratio vehicle. A new steady glide dynamicmodeling is formulated by extending a trajectory-oscillationsuppressing scheme, which is presented by Yu and Chenin [10], into three-dimensional reentry dynamics. Firstly, aspecial fight path angle which is able to keep the vehicleflying in a steady glide is calculated from the commandangle of attack, command bank angle, and current state.Then, the trajectory oscillation is suppressed via regulatingthe longitudinal acceleration in negative feedback form andkeeping the lateral acceleration invariant. It should be notedthat the negative feedback signal is the deviation betweenthe special flight path angle and actual flight path angle.Simulation result shows that this scheme performs well insuppressing trajectory oscillation and guides the vehicle intosteady glide as soon as possible. Additionally, a study onsteady glide trajectory optimization is investigated based onthis new modeling. The derivatives of command angle ofattack and bank angle are chosen as the control variables.And the performance index is the weighted squares sumof those derivatives. The limits on actual angle of attackand bank angle are considered as the path constraints. Infact, steady glide trajectory optimization is a typical optimalcontrol problem whose solutions change rapidly in certainregions. Therefore, Hp-adaptive Gaussian quadrature collo-cation method [11], which performs well in dealing with thiskind of problem, is chosen to transfer the optimal controlproblem into a standard nonlinear programming problemand solve it. The notable difference from the Yu’s methodis that the scheme is suitable for the three-dimensionalreentry dynamics. Another notable difference is that thescheme comprehensively considers all factors (includingthe derivatives of reference angle of attack and referencebank angle) acting on the flight path angle. That makes iteasy to integrate into the motion dynamics by the choiceof those derivatives as the control variables. Two classicalnumeric optimal examples (with and without bank reversal)are taken to evaluate the performance of the steady glidetrajectory optimization. In order to demonstrate the superiorperformance in applicability and computational efficiency,a comparison with the traditional method is also provided.Furthermore, a comparison between optimal trajectory andintegral trajectory is carried out to verify the feasibility ofthe pseudospectral solution. The results show that the newmethod not only significantly improves the computationalefficiency of trajectory optimization since using fewer nodeswill achieve a higher accuracy for the steady glide reentrytrajectory but also has an extensive applicability in consider-ing more final constraints even with the bank reversal. Mostimportantly, it is capable of providing more stable and safe

optimal steady glide trajectory with high precision, whichwould be a better choice for tracking guidance.

This paper is organized as follows: entry dynamicsincluding entry trajectory constraints and vehicle model aredescribed in Section 2; three-dimensional reentry trajectory-oscillation suppressing scheme is presented in Section 3;steady glide dynamic modeling and trajectory optimizationare presented in detail in Section 4.

2. Dynamics and Vehicle Description

2.1. Entry Dynamics. The 3-DOF point mass dynamics of thereentry vehicle over a spherical, rotating Earth is described asfollows [12]:

𝑟 = 𝑉 sin 𝛾, (1)

𝜃 =

𝑉 cos 𝛾 sin𝜓𝑟 cos𝜑

, (2)

�� =

𝑉 cos 𝛾 cos𝜓𝑟

,(3)

𝑉 = −

𝐷

𝑚

− 𝑔 sin 𝛾 + 𝜔

2𝑟 cos𝜑 (sin 𝛾 cos𝜑

− cos 𝛾 sin𝜑 cos𝜓) ,(4)

�� =

1

𝑉

(

𝐿 cos𝜎𝑚

+ (

𝑉

2

𝑟

− 𝑔) cos 𝛾 + 2𝜔𝑉 cos𝜑 sin𝜓

+ 𝜔

2𝑟 cos𝜑 (cos 𝛾 cos𝜑 + sin 𝛾 sin𝜑 cos𝜓)) ,

(5)

𝜓 =

1

𝑉

(

𝐿 sin𝜎𝑚

+

𝑉

2

𝑟

cos 𝛾 sin𝜓 tan𝜑

− 2𝜔𝑉 (cos𝜑 tan 𝛾 cos𝜓 − sin𝜑)

+

𝜔

2𝑟

cos 𝛾sin𝜑 cos𝜑 sin𝜓) ,

(6)

where 𝑟 is the radial distance from the center of the Earth tothe vehicle. In the latter, ℎ denotes the altitude. The radius ofthe Earth is 6378145m. 𝜃 and 𝜑 are the longitude and latitude,respectively. 𝑉 is the Earth relative velocity. 𝛾 is the flightpath angle of the Earth relative velocity. 𝜓 is the azimuthangle of the Earth relative velocity. 𝑚 is the mass of thevehicle. 𝑔 = 𝜇/𝑟

2 is the gravity acceleration, where 𝜇 is Earth’sgravitational constant. 𝜔 denotes the Earth self-rotation rate.The aerodynamic lift 𝐿 and drag𝐷 are given as follows:

𝐿 =

1

2

𝜌𝑉

2𝐶

𝑙𝑆ref ,

𝐷 =

1

2

𝜌𝑉

2𝐶

𝑑𝑆ref ,

(7)

where 𝜌 = 𝜌

0exp(−ℎ/𝐻) is the atmospheric density, where

𝜌

0is the standard atmospheric pressure from the sea level.

𝑆ref is the reference area of the vehicle. 𝐶𝑙and 𝐶

𝑑are lift

and drag coefficients which are dependent on the vehicleconfiguration.

International Journal of Aerospace Engineering 3

2.2. Entry Trajectory Constraints. Typical entry trajectoryinequality path constraints for hypersonic vehicle include

𝑘√𝜌𝑉

3.15≤

𝑄max, (8)

|𝐿 cos𝛼 + 𝐷 sin𝛼| ≤ 𝑛max, (9)

1

2

𝜌𝑉

2≤ 𝑞max, (10)

where (8) is the heating rate at a stagnation point on thesurface of the vehicle. Equation (9) is the aerodynamic accel-eration in the body-normal direction. Equation (10) is thedynamic pressure. The heating limit value is

𝑄max, the loadfactor limit value is 𝑛max, and the dynamic pressure limit valueis 𝑞max.These are dependent on the vehicle configuration andmission. Those three constraints are considered to be “hard”constraints that should be enforced strictly.

2.3. Vehicle Description and Model Assumption. CommonAero Vehicle (CAV) is one of the most representative hyper-sonic entry vehicles with high lift-to-drag ratio. Relying onaerodynamic control, this vehicle is able to glide withoutpower through the atmosphere. There are two types of CAVin the report of Phillips [13], the low-lift CAV and the high-lift CAV.The high-lift CAV, namely, CAV-H, is modeled hereto extend the trajectory optimization. The weight of CAV-His 907 kg, the area reference is 0.4839m2, and the maximumlift-to-drag ratio is about 3.5. In order to make derivationanalysis more intuitive and easier to follow, it is assumed thatthe lift and drag coefficients are only dependent on the angleof attack. They can be expressed in the form as

𝐶

𝑙= 𝑘

𝑙1𝛼 + 𝑘

𝑙2,

𝐶

𝑑= 𝑘

𝑑1𝛼

2+ 𝑘

𝑑2𝛼 + 𝑘

𝑑3,

(11)

where 𝑘𝑙1= 0.04675, 𝑘

𝑙2= −0.10568, 𝑘

𝑑1= 0.000508, 𝑘

𝑑2=

0.004228, and 𝑘

𝑑3= 0.0161. Moreover, because the angle of

attack having the maximum lift-to-drag ratio is about 10 deg.,the scope of angle of attack is extended to [5∘, 20∘]. The scopeof bank angle is limitedwithin [−60∘, 60∘].The limiting valuesof heating rate, dynamic pressure, and normal load factor are400W/cm2, 60 kpa, and 2 g, respectively.

3. Trajectory-Oscillation Suppressing Scheme

The objective of trajectory-oscillation suppressing scheme isto make the vehicle flying in a steady glide so that the heatingrate will not change sharply, which also will significantlyrelease the burden of control system. In this section, atrajectory-oscillation suppressing scheme using the flightpath angle feedback is extended. Firstly, the command angleof attack and bank angle are used to calculate the special flightpath angle that keeps the two-order derivative of flight pathangle zero. Then, the trajectory oscillation is suppressed viaregulating the longitudinal acceleration in negative feedbackform and keeping the lateral acceleration invariant. Thenegative feedback signal is the deviation between the specialflight path angle and actual flight path angle. Simulation

results show that the scheme is capable of guiding the vehicleinto a steady condition in which the command angle of attackand bank angle can generate enough vertical lift to sustain thevehicle glide.

3.1. Generic Theory for the Oscillation Suppressing Scheme.Let us pay attention to (5). While the flight path angle issmall and varies relatively slow, it is assumed that cos 𝛾 = 1

and sin 𝛾 = 𝛾. Therefore, (5) without considering the earthrotation is formulated as

�� =

1

𝑉

(

𝐿 cos𝜎𝑚

+ (

𝑉

2

𝑟

− 𝑔)) , (12)

where 𝐿 = (1/2)𝜌𝑉

2𝐶

𝑙𝑆ref and 𝑔 = 𝜇/𝑟

2. Substitute them into(12), then, (13) is the derivative of (12) with respect to time.Consider

�� =

1

𝑉

𝑑 (𝜌𝑉

2𝐶

𝑙𝑆ref cos𝜎/2𝑚 + (𝑉

2/𝑟 − 𝜇/𝑟

2))

𝑑𝑡

+ (

𝐷

𝑚

+ 𝑔𝛾)(

𝜌𝐶

𝑙𝑆ref cos𝜎2𝑚

+ (

1

𝑟

−

𝑔

𝑉

2)) ,

(13)

where

𝑑 (𝜌𝑉

2𝐶

𝑙𝑆ref cos𝜎/2𝑚 + (𝑉

2/𝑟 − 𝜇/𝑟

2))

𝑑𝑡

=

𝑉

2𝐶

𝑙𝑆ref cos𝜎2𝑚

�� +

𝜌𝑉𝐶

𝑙𝑆ref cos𝜎𝑚

𝑉

+

𝜌𝑉

2𝑆ref

2𝑚

(

𝐶

𝑙cos𝜎 − 𝐶

𝑙sin𝜎��) + 2𝑉

𝑟

2

𝑉

−

𝑉

2

𝑟

2𝑟 + 2

𝜇

𝑟

3𝑟.

(14)

Substitute (1), (4), and the derivative of the atmosphericdensity into (14).Then, substitute (14) into (13).The two-orderderivative of flight path angle can be rewritten as

�� = −

𝜌𝑉

2𝐶

𝑙𝑆ref cos𝜎

2𝑚𝐻

𝛾 −

𝜌𝐶

𝑙𝑆ref cos𝜎𝐷2𝑚

2

−

𝜌𝐶

𝑙𝑆ref cos𝜎𝑔2𝑚

𝛾

+

𝜌𝑉𝑆ref2𝑚

(

𝐶

𝑙cos𝜎 − 𝐶

𝑙sin𝜎��) − 𝐷

𝑟𝑚

−

𝑉

2

𝑟

2

𝛾

−

𝐷𝑔

𝑚𝑉

2+

𝑔

𝑟

𝛾 −

𝑔

2

𝑉

2𝛾,

(15)

where the derivative of lift coefficient is formulated as follows:

𝐶

𝑙=

𝜕𝐶

𝑙

𝜕𝛼

⋅

𝜕𝛼

𝜕𝑡

+

𝜕𝐶

𝑙

𝜕𝑀𝑎

⋅

𝜕𝑀𝑎

𝜕𝑟

⋅

𝜕𝑟

𝜕𝑡

.(16)


For reentry vehicle, the Mach number is much largerthan five. Therefore, the aerodynamic coefficients are oftenassumed to be not dependent on theMach number.Then, thelast term in (16) can be neglected.

Now, consider that the vehicle is in a certain conditionof reentry process; if the command angle of attack andbank angle are given, the special flight path angle that keepsthe two-order derivative of flight path angle zero can beformulated as follows:

𝛾

𝑚=

𝐷

∗

−𝑚𝑉

2− 𝑚𝑔 − 2𝑚

2𝑉

2/𝑟

2𝜌𝐶

𝑙∗𝑆ref cos𝜎 + 2𝑚

2𝑔/𝑟𝜌𝐶

𝑙∗𝑆ref cos𝜎 − 2𝑚

2𝑔

2/𝑉

2𝜌𝐶

𝑙∗𝑆ref cos𝜎

+

(

𝐶

𝑙∗ cos𝜎 − 𝐶

𝑙∗ sin𝜎��)

𝑉𝐶

𝑙∗ cos𝜎/𝐻 + 𝐶

𝑙∗ cos𝜎𝑔/𝑉 + 2𝑚𝑉/𝑟

2𝜌𝑆ref − 2𝑚𝑔/𝑟𝜌𝑉𝑆ref + 2𝑚𝑔

2/𝑉

3𝜌𝑆ref

+

𝐷

∗

−𝜌𝑉

2𝐶

𝑙∗𝑆ref cos𝜎𝑟/2𝐻 − 𝜌𝐶

𝑙∗𝑆ref cos𝜎𝑔𝑟/2 − (𝑉

2/𝑟)𝑚 + 𝑚𝑔 − 𝑔

2𝑟𝑚/𝑉

2

+

𝐷

∗

−𝜌𝑉

4𝐶

𝑙∗𝑆ref cos𝜎/2𝐻𝑔 − 𝜌𝑉

2𝐶

𝑙∗𝑆ref cos𝜎/2 − 𝑚𝑉

4/𝑟

2𝑔 + 𝑚𝑉

2/𝑟 − 𝑚𝑔

,

(17)

where superscript ∗ denotes the lift and drag coefficientsdominated by the command angle of attack and bank angle.The notable difference from Yu’s work is that the specialfight path angle is derived from the three-dimensionalreentry dynamics and comprehensively considers all factorsincluding the derivatives of command angle of attack andbank angle. In the later section, the special flight pathangle is easily calculated in the trajectory optimization whenconsidering the derivatives of command angle of attack andbank angle as control variables. Then, trajectory oscillationis suppressed via regulating the longitudinal acceleration innegative feedback form and keeping the lateral accelerationinvariant. The flight path angle also tends to the special flightpath angle as soon as possible. Consider

𝐶

𝑙2

cos𝜎2= 𝐶

𝑙1

cos𝜎1+ 𝐾 (𝛾 − 𝛾

𝑚) ,

𝐶

𝑙2

sin𝜎2= 𝐶

𝑙1

sin𝜎1,

(18)

where 𝐶𝑙2

is the actual lift coefficient and 𝐶𝑙1

is the commandlift coefficient dominated by the command angle of attack.𝜎

2is the actual bank angle. 𝜎

1is the command bank angle.

𝐾 is a negative feedback gain; it should be noted that abetter 𝐾 will perform well in suppressing the trajectoryoscillation. Then, a numerical simulation of entry processis carried out to evaluate the performance of the proposedscheme.

3.2. Performance of the Trajectory-Oscillation SuppressingScheme. In this section, the proposed scheme is appliedinto reentry simulation with constant command angle ofattack and bank angle. The simulation model, aerodynamicdata, and some key parameters are stated in Section 2.3. Thecommand angle of attack is 10 deg.The command bank angleis also 10 deg. If the assumptionsmentioned in Section 2.3 are

held, the actual angle of attack and bank angle suppressing thetrajectory oscillation can be calculated as follows:

𝜎

2= arctan(

𝐶

𝑙1

sin𝜎1

𝐶

𝑙1


𝑚)

) ,

𝛼

2=

1

𝑘

𝑙1

(

𝐶

𝑙1


𝑚)

cos𝜎2

− 𝑘

𝑙2) .

(19)

The initial states are ℎ

0= 60 km, 𝜃

0= 0 deg., 𝜑

0=

0 deg., 𝑉0

= 7100m/s, 𝛾0

= 0 deg., and 𝜓

0= 90 deg.

All programs run on a personal computer with a 3.3 Ghzprocessor andMATLAB2008b.The solver of integral isODE-45. The simulation stops when the altitude reduces to 30 km.Another worthy note is the negative feedback gain. Aftersome attempts, it is easy to find that the trajectory oscillationwill be suppressed perfectly when 𝐾 is −16. Figure 1 showsthe three-dimensional view of reentry trajectories. Note thatthe reference trajectory is the integral trajectory using theconstant angle of attack and bank angle. It is obvious that thevehicle glides through the atmosphere in a great performance.Moreover, the glide trajectory is similar to the referencetrajectory except that the trajectory oscillation is suppressed.

Figures 2 and 3 show the time histories of angle of attackand bank angle, respectively. The actual angle of attack andbank angle change substantially at the beginning of the flightand converge into the command angle of attack and bankangle quickly. Therefore, it is easy to conclude that applyingthe proposed scheme will guide the vehicle into a specialcondition in which the command angle of attack and bankangle will keep the vehicle steady glide. Another point shouldbe noted is that the strict mathematical proof of convergencefor this proposedmethod is absent.The futureworkwill focuson this point.


0 50 100 150

0

1020

3

4

5

6

Longitude (deg.)

Latitude (deg.)

Alti

tude

(m)

−10

×104

Glide trajectoryReference trajectory

Longitude-latitudeLongitude-altitude

Figure 1: Three-dimensional view of trajectory.

4. Steady Glide Dynamic Modeling andTrajectory Optimization

In this section, the steady glide trajectory optimization iscarried out to find the optimal controls satisfying path andfinal constraints. Firstly, a new steady glide dynamic mod-eling is formulated via integrating the trajectory-oscillationsuppressing scheme into the motion dynamics while theintrinsic properties remain. Secondly, the steady glide trajec-tory optimization is formulated based on this new modeling.The derivatives of command angle of attack and bank angleused to determine the special flight path angle are consideredas control variables, and a performance index used to providethe stable controls is selected. Because steady glide trajectoryoptimization is a typical control problem whose solutionschange rapidly in certain regions or are discontinuous,the Hp-adaptive Gaussian quadrature collocation method isnaturally selected to transfer it into a nonlinear programmingproblem. Finally, two numeric examples with and withoutbank reversal are used to evaluate the performance andapplicability of the new method. In order to demonstrate itssuperior performance in computational efficiency, a compar-ison with the traditional method is also provided. Moreover,a comparison between the optimal trajectory and integraltrajectory is carried out to further verify the feasibility of thesolution provided by the pseudospectral solver.

4.1. Steady Glide Dynamic Modeling. In previous section,the trajectory-oscillation suppressing scheme is presentedin Section 3.1. The performance of this scheme is eval-uated by the simulation in Section 3.2. Now, the steadyglide dynamic modeling is formulated via integrating thetrajectory-oscillation suppressing scheme into the three-dimensional reentry dynamics in Section 2.1. Consider

𝑟 = 𝑉 sin 𝛾;

𝜃 =

𝑉 cos 𝛾 sin𝜓𝑟 cos𝜑

;

�� =

𝑉 cos 𝛾 cos𝜓𝑟

,

𝑉 = −

𝐷

2

𝑚

− 𝑔 sin 𝛾 + 𝜔

2𝑟 cos𝜑 (sin 𝛾 cos𝜑

− cos 𝛾 sin𝜑 cos𝜓) ,

�� =

1

𝑉

(

𝐿

2cos𝜎2

𝑚

+ (

𝑉

2

𝑟

− 𝑔) cos 𝛾

+ 2𝜔𝑉 cos𝜑 sin𝜓

+ 𝜔

2𝑟 cos𝜑 (cos 𝛾 cos𝜑 + sin 𝛾 sin𝜑 cos𝜓)) ,

𝜓 =

1

𝑉

(

𝐿

2sin𝜎2

𝑚

+

𝑉

2

𝑟

cos 𝛾 sin𝜓 tan𝜑

− 2𝜔𝑉 (cos𝜑 tan 𝛾 cos𝜓 − sin𝜑)

+

𝜔

2𝑟

cos 𝛾sin𝜑 cos𝜑 sin𝜓) ,

𝜎

2= atan(

𝐶

𝑙1

sin𝜎1

𝐶

𝑙1


𝑚)

) ;

𝛼

2=

1

𝑘

𝑙1

(

𝐶

𝑙1


𝑚)

cos𝜎2

− 𝑘

𝑙2) ,

𝛾

𝑚= 𝑓 (𝑟, 𝑉, 𝛼

1, 𝜎

1, ��

1, ��

1) ;

��

1= 𝑢

1;

��

1= 𝑢

2,

(20)

where 𝐿

2is the actual aerodynamic lift dominated by the

command angle of attack, ��

1, ��

1are the derivatives of

command angle of attack and bank angle, and 𝛼

1, 𝜎1, and 𝜎

2

are defined in previous sections. If ��1and ��

1are given, there

exists a specific steady glide trajectory determined by thesteady glide dynamic modeling. Thus, steady glide trajectoryoptimization is continued based on this modeling in thenext section so as to find the optimal control satisfying pathand final constraints. Another point that should be noted isthat there is existing analytical solution for the actual angleof attack and bank angle because of the simplification ofaerodynamic coefficients. However, if complex aerodynamiccoefficients are considered in the steady glide trajectoryoptimization, it only needs to regard the complex function ofaerodynamic coefficients as the equality constraint in optimalprocess.

4.2. Hp-Adaptive Gaussian Quadrature Collocation Method.Hp-adaptive Gaussian quadrature collocation method is anefficient tool for solving multiple-phase optimal controlproblems using variable-order Gaussian quadrature collo-cation methods [14, 15]. Because an adaptive mesh refine-ment scheme [16, 17] is implemented to achieve a specified


0 500 1000 1500 2000 2500 3000 35009

9.5

10

10.5

11

Time (sec)

Ang

le o

f atta

ck (d

eg.)

Command attack-angleReal attack-angle

(a)

0 500 1000 1500 2000 2500 3000 35008.5

9

9.5

10

10.5

11

Time (sec)

Bank

angl

e (de

g.)

Command bank angleReal bank angle

(b)

Figure 2: Histories of angle of attack and bank angle.

accuracy, this method performs well in solving the OCPwhose solutions change rapidly in certain regions or arediscontinuous. In each mesh interval, the dynamics andconstraints are transferred into a set of nonlinear algebraicconstraints by employing a Legendre-Gauss-Radau quadra-ture collocation method. Therefore, the continuous-timeoptimal control problem is transferred to a large sparsenonlinear programming problem that is solved using well-established techniques. This method takes the advantage ofthe exponential convergence in regions where the solution issmooth and places mesh points only near potential disconti-nuities or in regions where the solution changes rapidly.

Steady glide trajectory optimization problem for highlift-drag ratio reentry vehicle is a complex constrainedcontinuous-time optimal control problem. What is more,there exist several regions where the state and control vari-ables change rapidly. At the beginning of the reentry, becausethe vehicle has no enough control capacity at high altitude,the altitude decreases rapidly in this phase. When the vehicledecreases to an appropriate height in which the vehicle hasenough vertical lift to sustain the vehicle glide, the state andcontrol variables also change rapidly. At the end of flying, itonly takes a short time to steer the vehicle to meet the finalrequirements. The state and control variables also changerapidly. However, the vehicle glides in a steady and smoothcondition for most of the flight duration. Naturally, Hp-adaptive Gaussian quadrature collocation method is suitablefor solving the steady glide trajectory optimization problem.The later numerical results also show that Hp-adaptiveGaussian quadrature collocation method performs well invarious glide trajectory optimizations.

4.3. Numerical Example of Trajectory Optimization withoutBank Reversal. In this subsection, Hp-adaptive Gaussianquadrature collocation method is applied to solve the steadyglide trajectory optimization problem.Meanwhile, a compar-ison with the traditional entry trajectory optimization using

the dynamics of motion in (1)∼(6) is also provided so as tofurther demonstrate the superior performance of the newmethod. Some of the simulation parameters are the same asthosementioned in above sections. For steady glide trajectoryoptimization, the motion dynamics are formulated in (18).In optimization process, the derivatives of command angleof attack and bank angle are chosen as the control variables,and the angle of attack and bank angle are considered asprocedure variables. In order to make the trajectory stateand optimal control as smooth and stable as possible, theperformance index is selected as follows:

𝐽 = ∫

𝑡𝑓

𝑡0

𝐾

1��

2

1+ 𝐾

2��

2

1𝑑𝑡, (21)

where𝐾1and𝐾

2are control weighting coefficients.The limits

on actual angle of attack and actual bank angle are consideredas path constraints and presented as follows:

𝜎min ≤ arctan(𝐶

𝑙1

sin𝜎1

𝐶

𝑙1


𝑚)

) ≤ 𝜎max,

𝛼min ≤

1

𝑘

𝑙1

(

𝐶

𝑙1


𝑚)

cos𝜎2

− 𝑘

𝑙2) ≤ 𝛼max.

(22)

For the traditional trajectory optimization, the control vari-ables are the derivatives of angle of attack and bank angle,and the performance index is the weighted square sumof those derivatives. The initial and final conditions forboth methods are listed in Table 1. It should be noted thatthe traditional trajectory optimization will fail to convergedue to considering too many final constraints. So the finallongitude is set to be free. However, the new method iscapable of addressing the problem with all final constraints.Therefore, the final longitude for such optimization is fixedat the optimal longitude provided by the traditional method.Another point that should be noted is that both optimalresults come from the same initial guess.


0 1000 2000 3000 40003

4

5

6

7

Time (sec)

Alti

tude

(m)

×104

Traditional method New method

(a) Altitude histories

0 50 100 150 2000

5

10

15

20

25

30

Longitude (deg.)

Latit

ude (

deg.

)


(b) Ground tracks

0 1000 2000 3000 40002000

3000

4000

5000

6000

7000

Time (sec)

Velo

city

(m/s

)


(c) Velocity histories

0 1000 2000 3000 4000

0

0.5

1

Time (sec)

Flig

ht p

ath

angl

e (de

g.)

−0.5

−1


(d) Flight path angle histories

0 1000 2000 3000 400060

80

100

120

140

160

180

Time (sec)

Azi

mut

h an

gle (

deg.

)


(e) Azimuth angle histories

0 1000 2000 3000 40006

8

10

12

14

16

Time (sec)

Ang

le o

f atta

ck (d

eg.)


(f) Azimuth angle histories

Figure 3: Continued.


0 1000 2000 3000 4000

0

20

40

60

Time (sec)

Bank

angl

e (de

g.)

−20

−40


(g) Bank angle histories

0 1000 2000 3000 40000

100

200

300

400

500

Time (sec)


Hea

ting

rate

(W/c

m2)

(h) Heating rate histories

0 1000 2000 3000 40000

1

2

3

4

5

6

Time (sec)

Dyn

amic

pre

ssur

e (N

/m2)

×104


(i) Dynamic pressure histories

0 1000 2000 3000 40000

0.5

1

1.5

Time (sec)

Load

fact

or


(j) Load factor histories

Figure 3: Numeric results of Steady Glide Trajectory Optimization without Bank Reversal.

Table 1: Initial and final conditions for different methods.

Initial condition𝜃

0(deg.) 𝜑

0(deg.) ℎ

0(m) 𝑉

0(m/s) 𝛾

0(deg.) 𝜓

0(deg.)

Traditional method 0 0 70000 6900 0 65New method 0 0 70000 6900 0 65

Final condition𝜃

𝑓(deg.) 𝜑

𝑓(deg.) ℎ

𝑓(m) 𝑉

𝑓(m/s) 𝛾

𝑓(deg.) 𝜓

𝑓(deg.)

Traditional method — 0 30000 2400 0 —New method 156.6 0 30000 2400 0 —

In the procedure of optimization, finite differencemethodis used to provide the derivatives of nonlinear programmingproblem for both optimizations. Because the flight path angleis so sensitive, it is necessary to set the step of finite differencemethod to be a small value. The step is then set at 10−9. In

addition, the feasible and optimal tolerances are set at 10−8,respectively.

Numeric results for the case without bank reversal arepresented in Figure 3. As seen from those figures, it is appar-ent that a perfect steady glide optimal trajectory satisfying all


Table 2: Optimal result by SNOPT.

Mesh iteration Node Feasible Optimal Maximum relative error for each mesh Time consumption in each iteration Time (sec)Traditional method

1 41 9.2𝐸 − 15 6.8𝐸 − 9 0.1483900 37.17

222.042 287 3.4𝐸 − 15 3.2𝐸 − 9 1.8234𝐸 − 5 88.023 473 2.3𝐸 − 15 6.7𝐸 − 9 5.6194𝐸 − 7 82.584 498 2.3𝐸 − 15 7.5𝐸 − 9 9.3151𝐸 − 9 14.27

New method1 41 5.1𝐸 − 15 3.7𝐸 − 9 0.0193110 1.61

82.062 209 2.5𝐸 − 15 4.5𝐸 − 9 0.0015244 24.913 278 2.2𝐸 − 15 8.7𝐸 − 9 3.9624𝐸 − 7 41.564 285 1.9𝐸 − 15 1.0𝐸 − 9 9.0488𝐸 − 9 13.98

constraints is provided via the new method, around whichthe entry trajectory provided by the traditionalmethod showssome damped oscillationwith a decreasing period.Moreover,the angle of attack and bank angle provided by the newmethod are very smooth and stable except at the beginningand end of the flight. The angle of attack and bank angleprovided by the traditional method also appear to oscillatearound that provided by the new method. It should be notedthat because the angle of attack, bank angle, and the flightpath angle are plotted over a large time scale (it is from 0to 4000 sec), they seem to vary very sharply at the end offlight. In fact, those angles vary very slowly and completelymeet the control requirements. The derivative of flight pathangle is less than 1 deg./s. Another significant phenomenondepicted in Figures 3(h), 3(i), and 3(j) is that the pathconstraints generated by the traditional method are muchlarger than that provided by the new method due to thetrajectory oscillation. It is obvious that the heating rate forthe traditional method is much larger than the limit (themaximum is up to 430W/cm2), but the one for the newmethod is within the safe range. The distribution of meshnodes for both methods is shown in Figure 4. It is clear thatthe initial mesh nodes are the same for both methods. Afterseveral iterations, the mesh nodes for the traditional methodare distributed with a high density, but the ones for the newmethod are distributed sparsely.

Table 2 shows the statistics of optimal results by the NLPsolver (SNOPT [18]). It is apparent that it only takes 4 times ofmesh refinement to achieve the required feasible and optimaltolerances for both methods. However, the number of totalnodes for the traditional method is much more than that forthe new method, which is 498 for the traditional methodbut 285 for the new method. Moreover, because the timeconsumption will increase exponentially with the increasein the number of total nodes, the computational efficiencyhas been significantly improved for the new method. It onlycosts 82.06 sec to finish the optimization for the newmethod,which is only one-third of that for traditional method.Therefore, it is concluded that the new method not only iscapable of solving the problem that the traditional methodcannot solve but also has high computational efficiencysince it will use less nodes to achieve a higher accuracy.Importantly, it will also provide more stable and safe optimal

0 0.5 10

1

2

3

4

Mesh nodes

Itera

tion

−1 −0.5

Figure 4: Distribution of mesh nodes (blue: new method, red:traditional method).

entry trajectory. It should be noted that it only takes half thetime if “forward-difference” or “back-difference” is used inoptimization. Applying more efficient resources is anotherway to reduce the computing time.

4.4. Numerical Example of Trajectory Optimization with BankReversal. Generally speaking, the objective of steady glidetrajectory optimization is to provide reference trajectory fortraditional entry guidance such as linear-quadratic regulatortracking laws.Those laws focus on tracking only the referencelongitudinal profile. The bank reversal is used to null thelateral errors. Meanwhile, reentry flight with bank reversalwill significantly increase the trajectory shaping capabilityfor the vehicle and provide more smooth control commandsexcept when the bank reversal happened. Therefore, in orderto provide the available reference trajectory, steady glidetrajectory optimization with bank reversal is considered inthis subsection. It is assumed that there is only one bankreversal during the reentry flight. Therefore, the optimalcontrol problem is divided into two phases. And the signsof the bank angles before and after the bank reversal areopposite. The interior point constraints used to maintain


Table 3: Different terminal states for various cases.

Cases Longitude (deg.) Latitude (deg.) Velocity (m/s) Flight path angle (deg.)Case 1 155 0 2400 0Case 2 165 0 2400 0Case 3 175 0 2400 0Case 4 185 0 2400 0Case 5 195 0 2400 0

Table 4: Optimal results by SNOPT.

Mesh iterations Total node Feasible Optimal Maximum relative error for each mesh Time (sec)7 179 2.6𝐸 − 15 7.8𝐸 − 9 9.5388𝐸 − 7 170.777 188 2.5𝐸 − 15 9.2𝐸 − 8 6.8891𝐸 − 7 168.967 196 3.4𝐸 − 15 6.1𝐸 − 8 1.9260𝐸 − 7 204.704 192 3.2𝐸 − 15 6.4𝐸 − 8 8.0681𝐸 − 7 96.565 187 2.6𝐸 − 15 8.0𝐸 − 8 6.5696𝐸 − 7 135.68

continuity in the state at the phase boundaries are presentedas

𝑡

+= 𝑡

−,

𝑟 (𝑡

+) = 𝑟 (𝑡

−) ,

𝜃 (𝑡

+) = 𝜃 (𝑡

−) ,

𝜑 (𝑡

+) = 𝜑 (𝑡

−) ,

V (𝑡+) = V (𝑡−) ,

𝛾 (𝑡

+) = 𝛾 (𝑡

−) ,

𝜓 (𝑡

+) = 𝜓 (𝑡

−) ,

𝛼 (𝑡

+) = 𝛼 (𝑡

−) ,

𝜎 (𝑡

+) = −𝜎 (𝑡

−) ,

(23)

where 𝑡 denotes the switching time for two phases. Theperformance index is stated in (21), and themotion dynamicsare also stated in (20). All simulation environments are thesame as that in Section 4.1. Initial states for glide vehicle areℎ

0= 70000m, 𝜃

0= 0 deg., 𝜑

0= 0 deg., 𝑉

0= 7100m/s,

𝛾

0= 0 deg., and 𝜓

0= 90 deg. In addition, various cases for

different cross-ranges are done to evaluate the applicability ofthe new method. The final longitude, latitude, velocity, andflight path angle are listed in Table 3.

Numeric results for the cases with bank reversal are pre-sented in Figure 5. Obviously, the new method is applicablefor the cases with the bank reversal in different cross-ranges.All trajectories are optimal steady glide trajectories and meetall constraints. And it is very easy to find the bank reversalin the time history of bank angle. And another point thatshould be noted is that the magnitudes of the bank anglevarymore stably and linearly than that without bank reversal.The meshes for various cases are placed perfectly in regionswhere the state and control change substantially. Additionally,the maximum heating rate in case 1 reaches the limit. The

statistics for optimal results are presented in Table 4. It hasbeen seen that the numbers of mesh iterations are 7 for case1, case 2, and case 3, 4 for case 4, and 5 for case 5. The totalnode for various cases is about 190. All tolerances satisfy thestop criteria that are set at the beginning of optimization.From the statistics of CPU time, it only takes 2-3min tosuccessfully finish the steady glide optimization with highnumerical precision.

4.5. Verification of Feasibility for the Pseudospectral Solution.According to the results obtained above, it is not difficult toconclude that the proposed scheme not only performs wellin suppressing the trajectory oscillation but also has highefficiency in three-dimensional steady glide optimizationwith and without bank reversal. However, there are threereasons that an elementary check should be conducted tovalidate the solution provided by the pseudospectral method.First, the feasible tolerance listed in Table 3 is for NLP[19]. Second, according to the research in [20], Hp-adaptiveGaussian quadrature collocationmethod has some deficiencythat it is unable to solve some special simple problems [20].Third, the solution provided by the pseudospectral methodmeets dynamics only at a limited number of nodes.Therefore,a comparison between the optimal trajectories and integraltrajectories is carried out to verify the feasibility of theresulting optimal trajectories.

Figure 6 shows the results comparing between the opti-mal trajectories provided by the pseudospectral solver andintegral trajectories generated by integrating the equations ofmotion using the optimal controls presented in Section 4.3with variable step (the min step size is 0.001 sec and the maxstep size is 0.1 sec). The comparison includes altitude, flightpath angle, velocity, and ground footprint. It is apparent thatthose trajectories are practically the same even though theflight path angle varies within the range from −0.5 to 0.1 deg.The statistic on final errors for various integral trajectoriesis presented in Table 5. The maximal error for altitudeis 2.5193m, the maximal error for velocity is 0.1714m/s,the maximal error for flight path angle is −0.003006 deg.,


0 1000 2000 3000 40003

4

5

6

7

Time (sec)

Alti

tude

(m)

Longitude 155Longitude 165Longitude 175

Longitude 185Longitude 195

×104


0 50 100 150 2000

5

10

15

20

Longitude (deg.)

Latit

ude (

deg.

)



(b) Ground tracks

0 1000 2000 3000 40002000

3000

4000

5000

6000

7000

8000

Time (sec)

Velo

city

(m/s

)




0 1000 2000 3000 4000

0

2

Time (sec)

Flig

ht p

ath

angl

e (de

g.)

−2

−4

−6

−8

−10

×10−3




0 1000 2000 3000 40001

1.5

2

2.5

3

3.5

Time (sec)

Azi

mut

h an

gle (

deg.

)



(e) Azimuth angle histories

0 1000 2000 3000 40008

9

10

11

12

13

14

Time (sec)

Ang

le o

f atta

ck (d

eg.)



(f) Angle of attack histories

Figure 5: Continued.


0 1000 2000 3000 4000

0

50

Time (sec)

Bank

angl

e (de

g.)



−50

(g) Bank angle histories

0 1000 2000 3000 40000

100

200

300

400

500

Time (sec)

Hea

ting

rate

(W/c

m2)



(h) Heating rate histories

0 1000 2000 3000 40000

1

2

3

4

5

6

Time (sec)

Dyn

amic

pre

ssur

e (N

/m2)

×104



(i) Dynamic pressure histories

0 1000 2000 3000 40000

0.5

1

1.5

Time (sec)

Load

fact

or



(j) Load factor histories

Figure 5: Numeric results of Steady Glide Trajectory Optimization with Bank Reversal.

Table 5: Final errors for various integral trajectories.

Cases Altitude (m) Velocity (m/s) Flight path angle (deg.) Longitude (deg.) Latitude (deg.)Case 1 0.2858 0.1398 −0.001026 0.002742 0.002329Case 2 2.5193 0.1714 0.001847 0.002007 0.001763Case 3 1.3820 0.1538 −0.001858 0.002772 0.001763Case 4 2.1876 0.0058 0.001411 0.000452 0.001763Case 5 −1.1173 0.1686 −0.003006 0.002534 0.001690

the maximal error for longitude is 0.002742 deg., and themaximal error for latitude is 0.002329 deg. Obviously, thesteady glide trajectory optimal control problem formulatedin this paper can be solved by Hp-adaptive pseudospectralsolver with a very high efficiency. And the solution is of highprecision that it is suitable to be considered as the referencetrajectory for the tracking reentry guidance law.

5. Conclusion

In this paper, a new steady glide dynamic modeling for highlift-to-drag ratio reentry vehicle is presented via extendingthe trajectory-oscillation suppressing scheme into three-dimensional reentry dynamics. Simulation shows that thisscheme performs well in trajectory-oscillation suppressing.


0 1000 2000 3000 40002

3

4

5

6

7

Time (sec)

Alti

tude

(m)

Integral trajectoryOptimal trajectory

×104


0 50 100 150 200

0

5

10

15

20

Longitude (deg.)

Latit

ude (

deg.

)

−5


(b) Ground tracks

0 1000 2000 3000 40002000

3000

4000

5000

6000

7000

8000

Time (sec)

Velo

city

(m/s

)



0 1000 2000 3000 4000

0

0.1

0.2

Time (sec)

Flig

ht p

ath

angl

e (de

g.)

−0.1

−0.2

−0.3

−0.4

−0.5



Figure 6: Comparison between numeric optimal trajectories and integral trajectories.

Based on this new modeling, a study on steady glidetrajectory optimization with multiple final constraints isinvestigated. The derivatives of command angle of attackand bank angle are chosen as the control variables. And theweighted square sum of those derivatives is selected as theperformance index so as to achieve the smoothest controls.Then, because the steady glide trajectory optimization isa typical optimal control problem whose solutions changerapidly in certain regions or are discontinuous, Hp-adaptiveGaussian quadrature collocation method is selected to solveit. Two classical optimization examples (with and withoutbank reversal) are conducted to show that the new methodperforms well in providing optimal steady glide trajectoryeven considering multiple final constraints. A comparisonwith the traditional method is done to demonstrate that the

new method significantly improves the computational effi-ciency since the steady glide trajectory can be discretizedwithfewer nodes. And a comparison between optimal trajectoryand integral trajectory is also carried out to verify that thesolution provided by the pseudospectral method is of veryhigh accuracy.

Competing Interests

The authors declare that they have no competing interests.

References

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[18] P. E. Gill, W. Murray, and M. A. Saunders, “SNOPT: an SQPalgorithm for large-scale constrained optimization,” SIAMReview, vol. 47, no. 1, pp. 99–131, 2005.

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