Constrained Trajectory Optimization for Planetary

AIAA 2016-3241 AIAA Aviation 13-17 June 2016, Washington, D.C. AIAA Atmospheric Flight Mechanics Conference

Constrained Trajectory Optimization for Planetary Entry via Sequential Convex Programming

Downloaded by PURDUE UNIVERSITY on July 18, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-3241

Zhenbo Wang1 and Michael J. Grant2 Purdue University, West Lafayette, Indiana, 47907-2045, USA

The main contribution of this research is the formulation of highly nonlinear constrained entry trajectory optimization problems as a sequence of convex problems from a new perspective. By introducing new state and control variables to the original three-dimensional equations of motion, the new control is naturally decoupled from the states without any additional assumptions, and the non-convex control constraint can be avoided in the new system dynamics. Therefore, the fidelity of the problem is retained. The non-convex objective function and path constraints are convexified by first-order Taylor series expansions, and the nonlinear dynamics are approximated by linear, time-varying systems. Subsequently, a sequential solution procedure is developed to approximate the original problem. The existence and convergence of successive solutions are discussed. There are efficient convex solvers with deterministic convergence properties for the sub-problems. This rapid trajectory optimization methodology is verified and compared to GPOPS by numerical solutions of minimum terminal velocity and minimum heat load entry trajectory optimization problems. This sequential method converges to the optimal solutions within a few seconds using Matlab, which demonstrates its potential real-time application for computational guidance.

Nomenclature Aref A(x) B CD CL D g0 L m n q Q! R0 r t u V x α γ

θ φ

1 2

= = = = = = = = = = = = = = = = = = = = = =

reference area system matrix control matrix drag coefficient lift coefficient drag acceleration acceleration of gravity at sea level lift acceleration mass of vehicle normal load dynamic pressure heat rate radius of Earth distance from Earth’s center flight time control Earth-relative speed state vector angle of attack flight-path angle longitude latitude

Graduate Student, School of Aeronautics and Astronautics, AIAA Student Member, [email protected] Assistant Professor, School of Aeronautics and Astronautics, AIAA Senior Member, [email protected] 1 American Institute of Aeronautics and Astronautics

Copyright © 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

ψ ρ σ


T

= heading angle = atmospheric density = bank angle

I. Introduction

HIS research was mainly motivated by the requirement of solution methods with guaranteed convergence and potential onboard application for highly nonlinear constrained planetary entry trajectory optimization problems. These problems have been generally solved as nonlinear optimal control problems, and most of these approaches find the optimal solution by solving general nonlinear programming problems.1 However, these methods have undesirable properties. First, there is no known bound on the solution time. Second, there exists no guarantee on the convergence to an optimal or even a feasible solution in a predetermined number of iterations. Finally, good initial guesses are often required for nonlinear optimization algorithms, and in many cases, these guesses must be supplied by a human. Consequently, such approaches may not be appropriate for onboard applications, and it is essential to exploit more effective structures of the problem to design algorithms with guaranteed convergence to the optimum with deterministic convergence criteria. Convex optimization is such a kind of method that might work for these problems. If the problem is formulated in a convex optimization framework, specifically formulated as a subclass of a convex optimization problem, such as Linear Programming (LP), Quadratic Programming (QP), Second-Order Cone Programming (SOCP), or SemiDefinite Programming (SDP), the problem will have low complexity and can be solved in polynomial time.2-4 There exist algorithms, such as interior-point methods,5 which compute an optimal solution with deterministic stopping criteria and with a prescribed level of accuracy. This implies that the optimum can be computed to any given accuracy with a deterministic upper bound on the number of iterations needed for convergence. In addition, primaldual interior-point algorithms employ a self-dual embedding technique that allows the algorithm to start from a selfgenerated feasible point without the need for any user-supplied initial guesses.6 All of these traits offer the kind of advantages not matched by other direct or indirect optimization methods. As such, convex optimization approaches are very promising for onboard applications. One of the earliest applications of convex optimization for nonlinear optimal control problems in aerospace is the study of powered descent guidance for Mars pinpoint landing.7-10 The powered descent guidance problem was firstly formulated as a propellant optimal trajectory optimization problem with state and control constraints. The primary non-convex constraint in this problem is the thrust magnitude, which has a lower nonzero bound that defines a non-convex feasible region in the control space. A slack variable was introduced to relax this non-convex constraint and approximate the relaxed optimal control problem as a finite-dimensional convex optimization problem via discretization, which leads to a SOCP problem. Similar methods are also used in trajectory optimization for rendezvous and proximity operations.11-13 First, the rendezvous problem was formulated as a nonlinear optimal control problem with various state and control constraints on interior points and terminal conditions. Then, the original problem was relaxed and approximated by a successive solution process, in which the solutions of a sequence of constrained SOCP sub-problems with linear, time-varying dynamics were sought. Trajectory optimization for hypersonic cases is potentially more complicated than these problems because of the existence of highly nonlinear dynamics, aerodynamic forces, and path constraints. Recently, the sequential SOCP method for rendezvous was extended to address entry trajectory optimization.14 First, the equations of motion (EoM’s) were reformulated with respect to energy, which simplifies the problem by replacing the differential equation of velocity with an approximate algebraic function. Then, the original highly constrained optimal control problem was formulated and solved by SOCP with a combination of successive linearization and relaxation techniques. However, there are some drawbacks of this approach. First, the velocity was approximated by an algebraic function of energy, instead of using the original velocity differential equation. After discretizing the energy domain, a fixed sequence of velocity was obtained and used through the whole solution process. This results in a large error between the real velocity and the approximate velocity, which has significant influence on the accuracy of the solution. Second, an equality non-convex control constraint was introduced and relaxed to a second-order cone inequality constraint; however, the proof of this relaxation technique is challenging. Several assumptions are made to ensure the satisfaction of the original control constraint at the solution point. Third, regularization term is also required to guarantee the equivalence of the relaxed control constraint to the original control constraint. Without this term, the proof will not hold, and the solutions of the relaxed problem will be infeasible for the original problem. Finally, in order to accommodate the constraints on terminal longitude and terminal latitude, a penalty term was added to the performance index, which further decreases the accuracy of the solution method. This method 2 American Institute of Aeronautics and Astronautics


might be accurate enough for the conceptual study of hypersonic entry mission design. For onboard applications, however, the fidelity of the problem formulation and the solution method should be improved. This paper addresses entry trajectory optimization problems using the Sequential Convex Programming (SCP) method from a new perspective. The focus of this investigation is on gliding entry trajectory optimization problems. Compared with the energy approach in Ref. 14, the original system dynamics is used. By introducing new state and control variables, the non-convex control constraint will be avoided, and no approximations are needed to decouple the control from the states. All common path constraints on heat rate, dynamic pressure, and load factor are included via sequential linearization, along with a trust-region constraint. The terminal constraints and control constraint can be satisfied easily within the SCP framework without adding any regularization or penalty terms to the performance index. As such, the fidelity of the problem formulation is highly improved. This paper is organized as follows: Sec. II presents a continuous-time optimal control problem for 3-D entry flight with augmented system dynamics, control constraints, terminal constraints, and path constraints. The feasibility of the solution to the converted problem is demonstrated as well. Small-perturbation-based linear approximation is used in Sec. III to convexify the nonlinear dynamics, objective, and path constraints, following which the existence of the optimal solution is proved for the convexified problem. A SCP method is developed in Sec. IV to find an approximate solution to the original problem. The convergence of the sequential method is analyzed. Simulation studies are conducted and shown in Sec. V to demonstrate the performance of the proposed approach. The main work of this research is summarized in Sec. VI.

II. Problem Formulation A. Entry Trajectory Optimization Problem The energy-based EoM’s of entry vehicles have been applied and improved over the years, especially for entry guidance design of the space shuttle based on drag acceleration.15 In this paper, however, the original EoM’s are considered. The dimensionless EoM’s of 3-D unpowered flight for an entry vehicle over a spherical, nonrotating Earth can be expressed in the wind-relative frame as follows.16 (1) r! = V sin γ θ! = V cos γ sinψ r cos φ (2)

(

)

φ! = V cos γ cosψ r V! = −D − sin γ r 2

(

)

γ! = Lcos σ V + V −1 r cos γ (Vr ) 2

ψ! = Lsin σ (V cos γ ) +V cos γ sinψ tan φ r

(3) (4) (5) (6)

where r is the radial distance from Earth’s center to the vehicle, which is normalized by the radius of the Earth R0 = 6378 km. θ and φ are longitude and latitude, respectively. The Earth-relative velocity, V , is normalized by

R0 g0 , where g0 = 9.81 m/s2. γ is the flight-path angle, and ψ is the heading angle of the velocity vector, measured clockwise in the local horizontal plane from the north. The dimensionless lift and drag accelerations L and D are normalized by g 0 , and the simplified functions are shown below. The differentiation of EoM’s (1-6) is with respect to dimensionless time normalized by

R0 g0 .

L = R0 ρV 2 Aref C L D = R0 ρV 2 Aref C D

(2m) (2m)

where m and Aref are the dimensional mass and dimensional reference area of the vehicle, respectively.

ρ = ρ0 exp ( −h / hs ) is the dimensional atmospheric density, which is a function of altitude, h . CL and CD are

aerodynamic lift and drag coefficients, respectively, which are given in tabulated data as functions of Mach number and angle of attack, α . Traditionally, the profile of α is pre-specified with respect to velocity based on the considerations of thermal protection system and maneuverability requirements. Consequently, the bank angle, σ , is the only control variable for entry trajectory optimization in Eqs. (1)-(6). With the entry EoM’s in Eqs. (1)-(6), a typical entry trajectory optimization problem is considered and defined as follows. 3 American Institute of Aeronautics and Astronautics

Problem 0:

( )

(

t

)

f min J = ϕ ⎡ x t f ⎤ + ∫ g x,σ dt ⎣ ⎦ t0 x,σ

(

s.t. x! = f x,σ

( )

( )

(7)

)

(8)

x t0 = x0 , x t f = x f

(9)

x∈ ⎡⎣ xmin , xmax ⎤⎦

(10)

σ min ≤ σ ≤ σ max

(11)

( ) q = f ( r,V ) = 0.5ρV ≤ q n = f ( r,V ) = L + D ≤ n

Q! = f1 r,V = kQ ρV 3.15 ≤ Q! max

(12)

2

2

2

2

3

(14)

max

where x = ⎡⎣ r; θ ; φ ;V ; γ ;ψ ⎤⎦ is a six-dimensional state vector. x0 is the initial state, and x f is the desired terminal state. xmin and xmax are the lower and upper bounds of the states, respectively. σ min and σ max are the minimum and maximum values of bank angle, respectively. Q! max , qmax , and nmax are the dimensionless maximum values of heat rate Q! , dynamic pressure q , and normal load n . Instead of propagating Eq. (4), the following algebraic equation was used in Ref. 14 to compute V from the given energy e by approximating r ≈ 1 .

(

V = 2 1− e

)

(15)

However, numerical simulation shows that large error could result from this approximation. If we use the parameters in Section V to propagate the energy-form EoM’s from the given initial condition and control history, a large error is observed in Fig. 1. Furthermore, since all path constraints are nonlinear functions of r and V , the approximate velocity would further influence the satisfaction of path constraints.

Altitude (km)

100 Approximate Velocity Real Velocity

80 60 40 20

Velocity error (m/s)


(13)

max

0

1000

2000

3000 4000 5000 Velocity (m/s)

6000

7000

8000

400 300 200 100 0.5

0.6

0.7

0.8

0.9

1

Energy

Fig. 1 Velocity calculation error using energy approach. As such, the original EoM’s in Eqs. (1)-(6) are used for the entry trajectory optimization problem considered in this paper. The velocity is calculated not from the approximate algebraic function shown in Eq. (15) but from the propagation of the original velocity differential equation (4), which will improve the accuracy of the solution method. 4 American Institute of Aeronautics and Astronautics

B. Choice of New Control As can be seen, Eqs. (1)-(6) are highly nonlinear in both state variables and control, σ . When applying successive linearization to the dynamics, high-frequency jitters will emerge, as will be seen in Section V. One reason for these jitters is the coupling of states and control in the dynamics. A similar phenomenon has also been observed when the problem is solved by nonlinear programming solvers such as the sequential quadratic programming method.17 In order to address this problem, a pair of new controls was introduced in Ref. 14 as follows. u1 = cos σ , u2 = sin σ and the energy-based system dynamics would be in the following form. x! = f x + B x u

( )

( )

(16)

where x = ⎡⎣ r; θ ; φ ; γ ;ψ ⎤⎦ is the state vector and u = ⎡⎣u1; u2 ⎤⎦ is the control vector, from which the following constraint must be satisfied.


u12 + u22 = 1

(17)

Eq. (17) is a quadratic non-convex equality constraint and cannot be directly addressed by convex programming. It was relaxed to a second-order cone constraint u12 + u22 ≤ 1 in Ref. 14, and a large amount of work was done to prove that the relaxed inequality second-order cone constraint remains active at the optimal solution. Adding further complication, a regularization term is required for the proof. The control matrix of energy-based dynamics (16) is as follows. ⎡ ⎤ 03×1 03×1 ⎢ ⎥ ⎢ ⎥ 2 B x = ⎢ L DV 0 ⎥ ⎢ ⎥ 2 0 L DV cos γ ⎥ ⎢ ⎣ ⎦ Note that the control u is coupled with multiple states, including flight-path angle through cos γ in the B-matrix. To decouple this state from the control, a small flight-path angle was assumed in Ref. 14, which further reduces the fidelity of the model. In fact, there exist easier approaches to address this coupling problem and to obtain more accurate solutions with the same rate of convergence. To this end, we define the bank angle rate, σ! , as the new control input and add an additional state to the 3-D EoM’s, (18) σ! = u

(

( )

where u is the scalar control normalized by

)

(

)

g0 R0 and must be selected from some control domain U ∈ℜ which

could be convex. By adding the rate dynamics (18) to the original EoM’s in Eqs. (1)-(6), the augmented EoM’s take the form x! = f x + Bu (19)

( )

( )

where we arrive to a seven-dimensional state vector x = ⎡⎣ r; θ ; φ ;V ; γ ;ψ ; σ ⎤⎦ and a new control variable, u . f x and B are 7 × 1 vectors shown below.

⎡ ⎤ V sin γ ⎢ ⎥ ⎢ ⎥ V cos γ sinψ r cos φ ⎢ ⎥ V cos γ cosψ r ⎢ ⎥ ⎢ ⎥ 2 f x =⎢ −D − sin γ r ⎥ 2 ⎢ ⎥ Lcos σ V + V −1 r cos γ Vr ⎢ ⎥ ⎢ Lsin σ V cos γ +V cos γ sinψ tan φ r ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎣ ⎦

(

( )

(

(

)

)

)

B=⎡ 0 0 0 0 0 0 1 ⎤ ⎣ ⎦

(20)

( )

T

5 American Institute of Aeronautics and Astronautics

(21)

The control matrix B in Eq. (21) is constant and independent of the states, x , so the control, u , is naturally decoupled from the states in the new model. This feature is significant to the successive linearization of dynamics and to the convergence of the sequential method developed in the following section. As such, the non-convex control constraint encountered in Ref. 14 will be avoided. The smoothness of bank angle, σ , can be enhanced by enforcing a constraint on the new control, u . Furthermore, numerical simulation will show that no regularization terms are needed to ensure the satisfaction of control constraints and the convergence of the sequential method. Thus, the new approach developed in this paper is potentially robust and more accurate for entry trajectory optimization.


C. Reformulated Optimal Control Problem With the original EoM’s in Eq. (8) replaced by the augmented dynamics in Eqs. (19)-(21), Problem 0 can be reformulated into a new optimal control problem as follows. Since the bank angle, σ , is treated as a new state variable, the new control, u , is not included in the objective functional. Problem 1:

( )

( )

t

f min J = ϕ ⎡ x t f ⎤ + ∫ g x dt ⎣ ⎦ t0 x,u

(22)

s.t. x! = f x + Bu

(23)

x t0 = x0 , x t f = x f

( )

(24)


(25)

u ≤ umax

(26)

( )

( )

( ) q = f ( r,V ) = 0.5ρV ≤ q n = f ( r,V ) = L + D ≤ n

Q! = f1 r,V = kQ ρV 3.15 ≤ Q! max 2

2

max

2

2

3

max

(27) (28) (29)

where Eq. (26) is the constraint on the new control and umax is the maximum bank angle rate. Lemma 1: If there exists an optimal solution for Problem 1 given by ⎡⎣ x ∗ ,u∗ ⎤⎦ = ⎡⎣ r ∗ ,θ ∗ ,φ ∗ ,V ∗ ,γ ∗ ,ψ ∗ ,σ ∗ ,u∗ ⎤⎦ . Then, ⎡⎣ r ∗ ,θ ∗ ,φ ∗ ,V ∗ ,γ ∗ ,ψ ∗ ,σ ∗ ⎤⎦ is a feasible solution of Problem 0. Proof: Problem 1 is obtained by introducing a new state variable and a new control variable to the original system dynamics in Eqs. (1)-(6). The original control constraint on σ becomes a boundary condition on the new state σ , which is included in Eq. (25). An additional constraint in Eq. (26) is introduced as a new control constraint on the bank angle rate. Thus, the set of feasible controls of Problem 1 is strictly contained in the set of feasible controls of Problem 0. Therefore, since Problem 0 is merely a relaxation of Problem 1, an optimal solution of Problem 1 satisfies all the constraints of Problem 0 and clearly defines a feasible solution of Problem 0. However, a feasible solution of Problem 0 does not necessarily define a feasible solution of Problem 1. In addition, the equivalence of the dynamics with and without σ! = u has also been demonstrated by numerical simulations. □ Remark 1: Lemma 1 presents a relationship between the solutions of Problem 0 and Problem 1. It implies that the non-convex constraint shown in Eq. (17) in the energy approach is avoided by introducing an additional state variable (18) and replacing the original control constraint (11) in Problem 0 with a new control constraint (26) on the bank angle rate in Problem 1. Consequently, a feasible solution to Problem 0 can be obtained by solving for the optimal solution of Problem 1 with convex control constraints as well as decoupled control and states.

III. Convexification Problem 1 is a highly nonlinear optimal control problem with nonlinear dynamics and path constraints, which adds complexity when compared to the Mars pinpoint landing problem and rendezvous problem. In many cases, the objective function is nonlinear as well. Fortunately, the boundary conditions in Eq. (24), and state constraints in Eq. (25) as well as the control constraint in Eq. (26) are all linear, convex constraints. However, the nonlinear dynamics, nonlinear objective, and nonlinear path constraints must be converted to tractable formulations for convex programming. A successive small-disturbance-based linearization method is used to relax Problem 1 into a sequence 6 American Institute of Aeronautics and Astronautics

of convex optimization problems, specifically to a sequence of SOCP problems. Numerical simulations in Section V demonstrate that the successive linearization approximation of Problem 1 is very accurate. A. Linear Relaxation First, a general objective is considered, and many objectives, such as minimum heat load, fall into the integral form shown in Eq. (22). The integrand can be linearized by a first-order Taylor series expansion about a given state history x∗ t as shown in Eq. (30).

()

( )

( )

g x ≈ g x∗ +

( )

∂g x ∂x

(x − x )

(30)

∗

x= x∗

In addition, the terminal cost in Eq. (22) could also be non-convex. A linear relaxation shown in Eq. (31) can be

( )

applied to relax it about a reference terminal point x∗ t f . As such, after discretization, the objective will be a linear Downloaded by PURDUE UNIVERSITY on July 18, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-3241

combination of the states at all nodes.

( )

( )

ϕ ⎡ x t f ⎤ ≈ ϕ ⎡ x∗ t f ⎤ + ⎣ ⎦ ⎣ ⎦

( ) ( )

∂ϕ ⎡ x t f ⎤ ⎣ ⎦ ∂x t f

( ) ( )

( )

( )

⎡x t − x t ⎤ ∗ f ⎦ ⎣ f

(31)

x t f = x∗ t f

As mentioned before, the control, u , has been decoupled from the states, x , by introducing an additional state

( )

variable. Consequently, the nonlinear term in Eq. (23) is f x , given by Eq. (20), and could be linearized about the

()

fixed state history, x∗ t , as well. The linearization of the dynamics is as follows.

( ) ( )(

)

x! ≈ f x∗ + A x∗ x − x∗ + Bu

(32)

where

( )

A x∗ =

a21 = −

( )

∂f x ∂x

x= x∗

⎡ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

0

sin γ

V cos γ

0

a21 0 a23

a24

a25

a26

a31 0

0

a34

a35

a36

a41 0

0

a44

a45

0

a51 0

0

a54

a55

0

a61 0 a63

a64

a65

a66

0

0

0

0

0

0

0

0

0 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ a57 ⎥ ⎥ a67 ⎥ ⎥ 0 ⎥ ⎦ x= x∗

V cos γ sinψ V cos γ sinψ sin φ cos γ sinψ V sin γ sinψ V cos γ cosψ , a23 = , a24 = , a25 = − , a26 = 2 2 r cos φ r cos φ r cos φ r cos φ r cos φ

V cos γ cosψ cos γ cosψ V sin γ cosψ V cos γ sinψ 2sin γ , a34 = , a35 = − , a36 = − , a41 = −Dr + 2 r r r r r3 cos γ cos σ V cos γ 2cos γ cos σ Lcos σ cos γ cos γ a44 = −DV , a45 = − 2 , a51 = Lr − + 3 , a54 = LV − + + 2 2 2 V V r r r rV V2 rV ⎛ 1 V⎞ Lcos σ sin σ V cos γ sinψ tan φ V cos γ sinψ Lcos σ a55 = ⎜ 2 − ⎟ sin γ , a57 = − , a61 = Lr − , a63 = , a67 = 2 2 r V V cos γ V cos γ ⎝r V ⎠ r r cos φ a31 = −

a64 =

sin σ Lsin σ cos γ sinψ tan φ Lsin σ sin γ V sin γ sinψ tan φ V cos γ cosψ tan φ L − + , a65 = − , a66 = V cos γ V V 2 cos γ r r r V cos2 γ

R2 ρV 2 Aref C D R ρVAref C L R R ∂D ∂D R0 ρVAref C D =− 0 = − 0 D, DV = = , Lr = − 0 L, LV = 0 ∂r 2mhs hs ∂V m hs m In addition, the nonlinear path constraints in Eqs. (27)-(29) are functions of r and V and can be linearized with Dr =

(

)

respect to r and V about a fixed state r∗ ,V∗ . The linearized path constraints have the following form. 7 American Institute of Aeronautics and Astronautics

( )

(

)

(

)

f i r,V ≈ f i r∗ ,V∗ + f i ′ r∗ ,V∗ ⋅ ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ , i = 1,2,3

(33)

where

(

⎡ ∂f ∂f ⎤ ⎡ f1′ r∗ ,V∗ = ⎢ 1 , 1 ⎥ = ⎢ −0.5R0 kQ V R0 g0 ⎣ ∂r ∂V ⎦ ⎣

(

)

)

3.15

ρ hs , 3.15kQ

(

)

⎡ ∂f ∂f ⎤ ⎡ f 2′ r∗ ,V∗ = ⎢ 2 , 2 ⎥ = ⎢ −0.5R0 ρ V R0 g0 ⎣ ∂r ∂V ⎦ ⎣

(

2

hs ,

(

R0 g0

R0 g0

)

3.15

) ρV ⎤⎥⎦(

⎤ V 2.15 ρ ⎥ ⎦( r∗ ,V∗ )

2

r∗ ,V∗

)

⎡ ∂f ∂f ⎤ f 3′ r∗ ,V∗ = ⎢ 3 , 3 ⎥ = ⎡ −0.5R02 Aref C L2 + C D2 ρV 2 mhs , R0 Aref C L2 + C D2 ρV m ⎤ ⎢ ⎦⎥( r∗ ,V∗ ) ⎣ ∂r ∂V ⎦ ⎣ With the relaxation of the nonlinear dynamics in Eq. (23) by Eq. (32), the nonlinear objective functional in Eq. (22) by Eqs. (30) and (31), and the nonlinear path constraints in Eqs. (27)-(29) by Eq. (33), Problem 1 can be converted into Problem 2, which is a continuous-time optimal control problem with a convex objective as well as convex state and control constraints. Problem 2:

(


)

(

)

( )

{ ()

( )}

( ) ( )

( ) x! = A ( x ) x + Bu + f ( x ) − A ( x ) x x (t ) = x , x (t ) = x

( )

t

( )

f min J = ϕ ⎡ x∗ t f ⎤ + ∂ϕ ⎡ x∗ t f ⎤ ⎡ x t f − x∗ t f ⎤ + ∫ ⎡⎣ ∂g x∗ x + g x∗ − ∂g x∗ x∗ ⎤⎦ dt ⎣ ⎦ ⎣ ⎦⎣ ⎦ t0 x,u

s.t.

∗

∗

0

0

f

∗

(34) (35)

∗

(36)

f


(37)

u ≤ umax

(38)

f i r∗ ,V∗ + f i ′ r∗ ,V∗ ⋅ ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ ≤ f i, max , i = 1,2,3

(39)

x − x∗ ≤ δ

(40)

(

)

(

)

Note that a trust-region constraint is enforced in Eq. (40), which is a second-order cone constraint. δ is the radius of the trust region. This technique is very important to ensure the validity of linearization in Eqs. (34), (35), and (39). B. Existence of an Optimal Solution After discretization, Problem 2 can be converted into a SOCP problem for a given x∗ , and Problem 1 will be solved as a sequence of such SOCP problems in Section IV. Therefore, an important theoretical issue is the existence of an optimal solution to Problem 2. Assumption 1: Let Γ represent the set of all feasible state trajectories and control signals for Problem 2, i.e., ⎡ x • ,u • ⎤ ∈Γ , which implies that for t ∈ ⎡t0 ,t f ⎤ , x t and u t define a feasible state trajectory and control ⎣ ⎦ ⎣ ⎦ signal for Problem 2. For this problem, we assume that Γ is nonempty. Theorem 1: Consider the following general optimal control problem:18

() ()

()

( ( ) ( ))

()

( )

t

( )

f min J x • ,u • = ϕ ⎡ x t f ⎤ + ∫ g x,u dt ⎣ ⎦ t0 x,u

()

( )

s.t. x! t = f x,u

( )

( )

x t0 ∈ X 0 , x t f ∈ X f

()

()

x t ∈ X , u t ∈U ,

∀ t ∈ ⎡⎣t0 ,t f ⎤⎦

Suppose that

()

()

1) There exists a compact set K such that for all feasible trajectories x • , we have ⎡⎣t, x t ⎤⎦ ∈K for all

t ∈ ⎡⎣t0 ,t f ⎤⎦ .

( )

( )

( )

( )

2) The set of all feasible ⎡ x t0 ,t f , x t f ⎤ ∈M is closed, such that x t0 ∈ X 0 and x t f ∈ X f . ⎣ ⎦ 8 American Institute of Aeronautics and Astronautics

3) The control set U is compact. 4) Both g and f are continuous functions on X ×U .

()

( ) (t, x ) = {( z , z ) : z

5) For each ⎡⎣t, x t ⎤⎦ ∈K , the set Q + t, x defined below is convex.

Q+

1

2

1

( )

( )

≥ g x,u , z2 = f x,u , u ∈U

Then there exists a solution pair ⎡⎣ x ∗ ,u∗ ⎤⎦ in Γ such that

(

) ( ( ) ( ))

() ()

∀ ⎡⎣ x • ,u • ⎤⎦ ∈Γ Additional descriptions of the above existence theorem and its proof can be found in Ref. 18. Proposition 1: Under Assumption 1, i.e., if the set of all feasible states and controls for Problem 2 is nonempty, then there exists an optimal solution to Problem 2. Proof: We will prove the existence of the optimal solution by showing that all conditions in Theorem 1 can be satisfied by Problem 2. 1) For Problem 2, the time of entry flight, t f , considered in this paper is fixed, and both the position vector J x ∗ ,u∗ ≤ J x • ,u •


}

⎡⎣ r; θ ; φ ⎤⎦ and velocity vector ⎡⎣V ; γ ;ψ ⎤⎦ are constrained in compact sets as shown in Eq. (37). There are also lower and upper bounds on the new state σ . Thus, we have ⎡⎣t, x t ⎤⎦ ∈K with t ∈ ⎡⎣t0 ,t f ⎤⎦ for all feasible state trajectories and Condition 1 is satisfied. 2) The satisfaction of Condition 2 can be guaranteed as well since both the initial condition, x t0 ∈ X 0 , and

()

( )

( )

( )

( )

terminal condition, x t f ∈ X f , are specified. Therefore, ⎡ x t0 ,t f , x t f ⎤ ∈M is closed and bounded. ⎣ ⎦ 3) The control set U defined by Eq. (38) is also compact, which leads to the third condition. 4) Condition 4 is obviously satisfied since both g and f are continuous functions on X ×U regardless if they are linearized or not. 5) The set Q + t, x in Condition 5 for Problem 2 is as follows.

( )

( ) {( z , z ) : z

Q + t, x =

1

2

1

z2

( ) ( ) ( ) = A ( x ) x + Bu + f ( x ) − A ( x ) x ,

≥ ∂g x∗ x + g x∗ − ∂g x∗ x∗ , ∗

z1 ∈ℜ,

∗

z2 ∈ℜ , u ≤ umax 7

( )

}

∗

(

∗

)

( )

Assume that A x∗ is invertible almost everywhere for t ∈ ⎡⎣t0 ,t f ⎤⎦ , thus z1 , z2 ∈Q + t, x if and only if the following conditions are satisfied.

( ) ( )

( ) ( ) ( ) z = A ( x ) x + Bu + f ( x ) − A ( x ) x ⎡ z − A( x ) x − f ( x ) + A( x ) x ⎤ ≤ u ⎣ ⎦

( )

z1 ≥ ∂g x∗ A−1 x∗ ⎡⎣ z2 − Bu − f x∗ + A x∗ x∗ ⎤⎦ + g x∗ − ∂g x∗ x∗ ∗

2

B

T

2

∗

∗

∗

∗

∗

∗

∗

max

Since the above equality is linear and the inequalities are either linear or second-order cone equations, the set Q + t, x for Problem 2 is convex, and Condition 5 is satisfied.

( )

According to Theorem 1, we can draw the conclusion that Problem 2 has an optimal solution when its feasible set is nonempty. □

IV. Sequential Convex Programming Method The only difference between Problem 1 and Problem 2 is the approximations of the objective functional (22) by Eq. (34), dynamics (23) by Eq. (35), and path constraints (27)-(29) by Eq. (39), so the solution to Problem 2 is potentially a good approximation to the corresponding solution to Problem 1 if the reference trajectory, x∗ t , for

()

()

linearization is close enough to the real trajectory, x t . In this section, we attempt to find the solution to Problem 1 by solving a sequence of convex optimal control problems defined by Problem 2. 9 American Institute of Aeronautics and Astronautics

A. Solution Procedure Similar to the successive solution method developed for the rendezvous problem,11-14 a sequential convex programming method is devised in this section to approximate and solve Problem 1. In this approach, a sequence of convex optimal control sub-problems are formed using the state from the previous iteration. This process is described as follows. 1) Set k = 0 . Initialize the states r t0 = r0 , θ t0 = θ 0 , φ t0 = φ0 , V t0 = V0 , γ t0 = γ 0 , ψ t0 = ψ 0 , and

( )

( )

( )

( )

( )

( )

σ (t0 ) = σ 0 . Propagate EoM’s (19)-(21) with these initial conditions and a specific constant control to

0 provide an initial trajectory x ( ) for the solution procedure.

{

Given the state equations:

( )

( ) ( )

k −1 k −1 k −1 k −1 x! = A x ( ) x + Bu + f x ( ) − A x ( ) x ( ) ,


}

k k For k ≥ 1 , solve the following optimal control problem to find the solution pair x ( ) , u( ) .

2)

(41)

Minimize the objective functional: k −1 k −1 k −1 min J = ϕ ⎡⎢ x ( ) t f ⎤⎥ + ∂ϕ ⎡⎢ x ( ) t f ⎤⎥ ⎡⎢ x t f − x ( ) t f ⎤⎥ x,u ⎣ ⎦ ⎣ ⎦⎣ ⎦ tf k −1 k −1 k −1 k −1 ( ) ( ) ( ) ( ) ⎤ dt + ∫ ⎡ ∂g x x+g x − ∂g x x ⎢ t0 ⎣ ⎦⎥ Subject to:

{

( )

( ) ( )

( )

( ) ( )

(43)


(44)

u ≤ umax

(45)

) (

)

x tf = xf

k −1 k −1 k −1 k −1 k −1 k −1 f i r ( ) ,V ( ) + f i ′ r ( ) ,V ( ) ⋅ ⎡⎢ r − r ( ) ; V −V ( ) ⎤⎥ ≤ f i, max , i = 1,2,3 ⎣ ⎦ ( k −1) x−x ≤δ

3)

(42)

( )

( )

x t0 = x0 ,

(

( )}

(46) (47)

Check the convergence condition k k −1 sup x ( ) − x ( ) ≤ ε ,

t0 ≤t≤t f

k >1

(48)

where ε is a prescribed tolerance value for convergence. If the above condition is satisfied, go to Step 4; otherwise set k = k + 1 and go to Step 2. k k The solution of the problem is found to be x ∗ = x ( ) and u∗ = u( ) .

4) Remark 2: For each k ≥ 1 , a nonlinear optimal control problem is defined by Eqs. (41)-(47). Since only linear, time-varying dynamics, affine equality constraints, and second-order cone inequality constraints are included in these sub-problems, they can be easily discretized into SOCP problems and solved efficiently by convex solvers as described in the next subsection. The conventional linearization techniques are used for the relaxation of Problem 1 based on small perturbations and the accuracy of the successive linearization method can be guaranteed by enforcing a trust-region constraint in Eq. (47). B. Discretization The sub-problems defined by Eqs. (41)-(47) are still infinite-dimensional optimal control problems. To find their numerical solutions, a trapezoidal discretization is applied to convert the infinite-dimensional optimization problems to finite-dimensional ones.11 To do this, the time domain is discretized into N equal time intervals, and the

(

constraints are imposed at N + 1 nodes. The step size is Δt = t f − t0

{t ,t ,t ,...,t ,t } with t sequences { x , x , x ,..., x 0

1

2

N −1

N

0

1

2

i

)

N , and the discretized nodes are denoted by

= ti−1 + Δt, i = 1,2,..., N . The corresponding state and control are discretized into the

N −1

, xN

} and {u ,u ,u ,...,u 0

1

2

N −1

}

,uN . Then, the dynamics can be integrated numerically as

follows. 10 American Institute of Aeronautics and Astronautics

(49) ) ( ) = A ( x ) , B is constant, and f = f ( x ) . After rearrangement, Eq. (49) can be written as ⎞ ⎛ ⎞ Δt Δt Δt Δt (50) ⎟⎠ x − ⎜⎝ I + 2 A ⎟⎠ x − 2 Bu − 2 Bu = 2 ( f − A x + f − A x )

xi = xi−1 +

()

where xik = x k ti , Aik

k i

⎛ Δt k −1 ⎜⎝ I − 2 Ai If we choose

(

Δt ⎡ k −1 k −1 k −1 k −1 A x + Bui−1 + fi−1 − Ai−1 xi−1 + Aik −1 xi + Bui + fi k −1 − Aik −1 xik −1 ⎤ ⎦ 2 ⎣ i−1 i−1

1

k −1 i−1

i

{ x , x , x ,..., x 0

k

2

N −1

, xN

(

}

i−1

and

i

{u ,u ,u ,...,u 0

1

k −1 i−1

i−1

2

optimization vector will be y = x0 , x1 ,..., x N ,u0 ,u1 ,...,uN


k i

i

N −1

)

,uN

}

k −1 k −1 i−1 i−1

k −1

i

k −1 k −1 i i

as the design variables, then the augmented

and the state equation (41) is converted into linear

equality constraints on y defined by Eq. (50). Note that the linearized objective functional in Eq. (42) can be discretized into a linear function of y as well. All the constraints defined in Eqs. (43)-(47) will be enforced at each node and become linear equality constraints and second-order cone inequality constraints. After discretization, each optimal control sub-problem defined in Eqs. (41)-(47) is converted into a large-scale nonlinear programming problem as follows. Problem 3: (51) min c T y y

s.t.

M i y + pi ≤ qiT y + ri ,

i = 0,1,2,..., N n

where Eq. (52) is a general description of the discretized constraints. c ∈ℜ y , M i ∈ℜ

(52) ni ×n y

n

n

, pi ∈ℜ i , qi ∈ℜ y , and

{

}

k −1 k k k ri ∈ℜ are calculated based on the solution x ( ) from the previous iteration before solving y ( ) = x ( ) , u( ) in the

current iteration. n y denotes the dimension of the optimization vector y after discretization. Note that the linear inequality constraints are included in Eq. (52) since each linear constraint is a special case of the second-order cone constraint. Additionally, linear equality constraints can be expressed by two linear inequality constraints, which are also included in Eq. (52). Thus, the discretized problem is a nonlinear, but SOCP, problem, and the discretization is accurate for a sufficiently large N . Assumption 2: We assume that the existence of the optimal solution to Problem 2 and each sub-problem (which has been proved in the previous section) is guaranteed after the discretization process. Finally, we can solve a sequence of Problem 3 to obtain an approximate numerical solution to Problem 1. C. Convergence Analysis Since the existence of the optimal solution to each of the sub-problems has been demonstrated in the proceeding section by Proposition 1, then based on Assumption 2, our attention turns to the convergence of the sequential solution procedure of Problem 3 and the equivalence of the converged solution to the solution to Problem 1. A complete proof of the convergence of the sequential method for entry trajectory optimization problems is still unreachable. However, some characteristics of the problem can be used to provide confidence of convergence in the context of preceding research in sequential methods.19-21 1. Sequential Linear-Quadratic Approximations The work in Ref. 20 is a close resemblance to the method in this paper, in which the convergence of successive approximations has been proved for nonlinear optimal control problems with “pseudo-linear” systems and a standard quadratic cost functional. The nonlinear dynamics considered take the following form. x! = A x x + B x u, x t0 = x0

( )

( )

( )

together with a standard quadratic cost functional 1 1 tf J = x T t f Fx t f + ∫ ⎡⎣ x T Q x x + uT R x u ⎤⎦ dt 2 2 t0 where F ≥ 0 , Q x ≥ 0 , and R x > 0 for all x . The solution to this problem can be found by solving a sequence

( )

( )

( ) ( )

( )

of time-varying linear-quadratic approximations as follows. k k −1 k k −1 k x! ( ) = A x ( ) x ( ) + B x ( ) u ( ) ,

( )

( )

( )

( )

k x ( ) t0 = x0


( )

( )

( )

( )

1 kT 1 tf k k kT k −1 k kT k −1 k J ( ) = x ( ) t f Fx ( ) t f + ∫ ⎡ x ( ) Q x ( ) x ( ) + u ( ) R x ( ) u ( ) ⎤ dt ⎥⎦ 2 2 t0 ⎢⎣ Since each approximating problem has a quadratic performance index and linear, time-varying dynamics, the optimal control can be solved explicitly. Assume that A x and B x satisfy the following conditions.

( ( ))

1) µ A x = lim+

( )

( )

I − hA x −1 h

h→0

( ) ( ) B( x ) − B( x ) ≤ c x − x B ( x ) ≤ b ∀x ∈ℜ .

≤ µ0

∀x ∈ℜn ,

2) A x1 − A x2 ≤ c1 x1 − x2

∀x1 , x2 ∈ℜn ,

3)

∀x1 , x2 ∈ℜn ,

4)

1

2

2

1

2

( )

n

0


where µ0 , c1 , c2 , and b0 are some constants. µ denotes the logarithmic norm of the matrix. Under the above boundary conditions and local Lipschitz continuity conditions, the convergence of the sequential approximation method for the previous linear-quadratic problem can be proved.20 In our problem, matrix B in Eq. (21) is constant, thus B is Lipschitz continuous and Conditions 3 and 4 are met. However, the objective functional in Eq. (22) is not generally in standard quadratic form, and the entry dynamics in Eq. (23) do not naturally fall into the “pseudo-linear” form. Additionally, there are various constraints in Eqs. (24)-(29) for our problem. As such, a complete theoretical proof of the successive solution method in this paper is much more difficult than for the linear-quadratic problem. Even if the convexified system in Eq. (35) is considered, the properties of matrix A x∗ and its satisfaction of Conditions 1 and 2 cannot be determined

( )

explicitly. However, some similar characteristics can be identified. Consider the system in Eq. (23) subject to the constraint in Eq. (26). For a given control u , there exist appropriate constants rmin , θ min , φ min , Vmin , γ min , ψ min , σ min , rmax , θ max , φ max , Vmax , γ max , ψ max , σ max such that for any fixed t f , the following conditions are satisfied for any admissible control.

() ≤ γ (t ) ≤ γ

()

()

()

rmin ≤ r t ≤ rmax , θ min ≤ θ t ≤ θ max , φ min ≤ φ t ≤ φ max , Vmin ≤ V t ≤ Vmax

γ min

max

()

()

, ψ min ≤ ψ t ≤ ψ max , σ min ≤ σ t ≤ σ max , ∀t ∈ ⎡⎣t0 ,t f ⎤⎦

The above conditions are equivalent to the state constraints defined in Eq. (25), based on which all the differential equations in Eq. (23) have maximum and minimum values on a compact set. Thus, the linearized system matrix A x∗ is expected to be bounded and satisfy Condition 1. In addition, A x∗ is continuously differential on a

( )

( )

closed interval, so the Lipschitz continuous Condition 2 is expected to be satisfied as well. Remark 3: Because of the complex structure of A x∗ shown in Eq. (32), explicit expressions for the

( )

( )

logarithmic norm of A x∗ and the left hand side of the Lipschitz continuity Condition 2 are hard to derive;

( )

k −1 however, the satisfaction of Conditions 1 and 2 by the system matrices A x ( ) of the sequential SOCP sub-

problems could be verified numerically for the entry trajectory optimization problems considered in this paper. 2. Sequential SOCP Approximations Another sequential convergence study can be found in Ref. 13. Nonlinear optimal control problems with concave state inequality constraints and nonlinear terminal equality constraints were considered there. In this section, the following optimal control problem is considered first. Problem 4:

( )

( ) s.t. x! = A (t ) x + B (t ) u x (t ) = x , x (t ) = x t

f min J = ϕ ⎡ x t f ⎤ + ∫ g x,u,t dt ⎣ ⎦ t0 x,u

0

0

f

f


(53) (54) (55)

M i y + pi ≤ qiT y + ri ,

( )

s j x,t ≤ 0,

i = 1,2,...,l

(56)

j = 1,2,...,m

()

(57)

()

where a linear, time-varying system dynamics is defined. A t and B t

( )

are continuous. y = x;u is an

augmented vector used to define all linear and second-order cone constraints shown in Eq. (56). The integrand g x,u,t in the performance index is assumed to be second-order cone representable, which can always be

(

)

discretized into a linear cost function with second-order cone constraints. The only non-convex part of Problem 4 is the inequality constraints in Eq. (57), which is assumed to be concave. When the concave constraints are linearized about a reference point, Problem 4 can be discretized into the form of Problem 3. The solution of Problem 4 can be obtained by solving a sequence of Problem 3.13 k Let y ( ) be a sequence of solutions obtained by solving Problem 3 successively. Assume that the quadraticDownloaded by PURDUE UNIVERSITY on July 18, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-3241

{ }

cone constraints in each Problem 3 are strictly feasible and there exists a unique solution for each Problem 3. Under these assumptions, the convergence of the sequential solution method is proved in Ref. 13. That means there exists a k solution sequence y ( ) , which forms a non-increasing objective function. The converged solution y ∗ satisfies the

{ }

Karush-Kuhn-Tucker (KKT) condition for Problem 4 and y ∗ is at least a local minimum of Problem 4. One prerequisite required for the existence and convergence of the sequential solutions is the concavity of the inequality constraints in Eq. (57). Without this condition, we cannot prove that any feasible solution for Problem 3 is also feasible for Problem 4, and the sequential method may fail. For the entry trajectory optimization problems considered in this paper, we have the following assurance about the feasibility of the solutions to the relaxed Problem 2. Proposition 2: If a feasible state trajectory x of Problem 2 satisfies the constraints in Eq. (39), then it also satisfies the original path constraints in Eqs. (27)-(29) based on the linearization defined by Eq. (33). Proof: Let x be any feasible state of Problem 2, i.e., f i r∗ ,V∗ + f i ′ r∗ ,V∗ ⋅ ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ ≤ f i, max , i = 1,2,3 (58)

( )

(

( )

)

(

)

(

)

where r,V ∈ x and f i r,V from Eqs. (27)-(29) are all linearized with respect to the reference point r∗ ,V∗ ∈ x∗ . The Jacobian f i ′ can be found in Eq. (33).

( )

Now, we can further represent f i r,V using the second-order Taylor series expansion. T 1 f i r,V = f i r∗ ,V∗ + f i ′ r∗ ,V∗ ⋅ ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ + ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ ⋅ ∇ 2 f i r∗ ,V∗ ⋅ ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ 2 2 where ∇ f i is the Hessian of f i .

( )

(

)

(

)

(

)

(59)

By substituting Eq. (58) into Eq. (59), we have T 1 (60) f i r,V ≤ f i, max + ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ ⋅ ∇ 2 f i r∗ ,V∗ ⋅ ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ , i = 1,2,3 2 The Hessian ∇ 2 f i can be easily obtained by taking the second derivatives of Eqs. (27)-(29). Let’s take the heat rate

( )

(

)

constraint as an example with its Hessian as follows.

⎡ 0.25R2V 3.15 −1.575R0V 2.15 ⎤ 0 ⎢ ⎥ 3.15 hs ⎢ ⎥ hs2 R0 g0 ρ⎢ ⎥ 2.15 ⎢ −1.575R0V ⎥ 1.15 6.7725V ⎢ ⎥ h s ⎢⎣ ⎥⎦ 2 Note that ∇ f1 r,V is symmetric and negative definite for all r,V ∈ x . Similarly, the Hessian matrices ∇ 2 f 2 and ⎡ ∂2 f 1 ⎢ ⎢ ∂r 2 2 ∇ f1 r,V = ⎢ 2 ⎢ ∂ f1 ⎢⎣ ∂V ∂r

( )

( )

∂ 2 f1 ⎤ ⎥ ∂r∂V ⎥ = kQ ∂ 2 f1 ⎥⎥ ∂V 2 ⎥⎦

(

)

( )

∇ f 3 for dynamic pressure and normal load constraints are all negative definite. Thus, the following inequalities 2

hold from Eq. (60), which are exactly the constraints in Eqs. (27)-(29). f i r,V ≤ f i, max , i = 1,2,3

( )


Therefore, the original path constraints are all satisfied by the feasible state trajectory x of Problem 2.

( )

□


Remark 4: Without the concavity of the second-order Taylor series expansion of f i r,V , there is no guarantee that any feasible state trajectory x satisfying the linearized path constraints in Eq. (39) will also satisfy the original path constraints in Eqs. (27)-(29). Thus, the feasibility to the original path constraints in Problem 1 can be guaranteed by the feasible solutions of the successive sub-problems. Remark 5: Although the existence of the optimal solution for each relaxed problem can be proved for our specific entry applications and the feasibility of the path constraints can be preserved by the sequential solutions, a complete theoretical proof of the convergence of the SCP method for generic entry flight is much more difficult than the problems considered in the literature because of the following reasons. First, the objective functional in Eq. (22) is not in quadratic or general convex form for most cases. As such, it cannot be directly discretized into a linear function. Second, strong nonlinear terms, such as aerodynamic forces and an exponential atmospheric model, are naturally included in the entry dynamics in Eq. (23). All of these reasons make a general proof of the existence of the solution to Problem 0 or Problem 1 rather elusive, let alone the convergence and equivalence of the sequential solution method. Remark 6: A complete theoretical proof of the convergence of the sequential method is still an open challenge; however, our numerical simulations show a strong evidence of convergence in solving hypersonic entry trajectory optimization problems, and the accuracy of the successive approximation method can be guaranteed as well. A faster rate of convergence is expected when a more accurate initial trajectory could be provided. In addition, the optimality of the converged solution to the original problem can be demonstrated numerically by simply substituting the converged optimal control into the original system and verifying the similarity between the propagated trajectory and the optimal trajectory obtained by the sequential method. Satisfaction of the original constraints can be verified as well.

V. Numerical Demonstration The methodology described in this paper has been implemented in SDPT3, a MATLAB-based convex optimization solver that uses a primal-dual interior-point algorithm for linear, quadratic, and semi-definite programming problems.22 For each fixed k , the CPU time taken by SDPT3 to solve each of the sub-problems in Sec. IV on a typical desktop computer is 0.3 to 0.5 seconds. The execution time is expected to be considerably shorter if the simulation environment is changed that can take advantage of compiled or parallel programs. A Reusable Launch Vehicle (RLV) model used for numerical demonstration is much like the configuration examined in,23 which is a vertical-takeoff, horizontal-landing, winged-body vehicle. For entry flight, the mass of the vehicle is set to be m = 104,305 kg. The reference area Aref = 391.22 m2. For the conceptual entry trajectory optimization in this paper, the aerodynamic coefficients are approximated by fitting the data in hypersonic regime with24 C L = −0.041065 + 0.016292α + 0.0002602α 2 (61)

C D = 0.080505 − 0.03026C L + 0.86495C L2

(62)

where α is in degrees and is scheduled with respect to velocity as shown below.

⎧⎪ 40, V > 4570 m s (63) α =⎨ 2 2 ⎪⎩ 40 − 0.20705(V − 4570) 340 , otherwise A nominal profile of angle of attack and profiles of aerodynamic coefficients are shown in Fig. 2. For simplicity, we only consider fixed-flight-time entry and t f = 1600 s , however, the approach developed in this paper is also applicable for free-flight-time entry problems. The time domain ⎡⎣t0 , t f ⎤⎦ is discretized into 500 intervals, and the maximum number of iterations is set to be 100. The simulation parameters are shown in Table 1. Based on the data, this entry mission has an initial downrange of 7792 km and right cross-range of 1336 km from an altitude of 100 km and velocity of 7450 m/s at entry interface. The desired terminal altitude is 25 km, the terminal flight-path angle must be -10 deg, and the terminal heading angle must be 90 deg, pointing to the east. The limit on the magnitude of bank angle rate is 10 deg/s. The upper and lower bounds of bank angle and the values of limits on the path constraints are also given in Table 1. To verify the performance of the SCP method developed in this paper, numerical results are compared to the solutions obtained by GPOPS-II, which is a general-purpose MATLAB software program for solving multiple-phase 14 American Institute of Aeronautics and Astronautics

optimal control problems.25 GPOPS employs a Legendre-Gauss-Radau quadrature orthogonal collocation method to convert the continuous-time optimal control problem to a large sparse Nonlinear Programming (NLP) problem. An adaptive mesh refinement method is implemented to determine the number of nodes required and the degree of the approximating polynomial to achieve a specified accuracy. The default NLP solver used by GPOPS is IPOPT.26

θ0 φ0

0 deg

V0

7450 m/s

γ0 ψ0 σ0

-0.5 deg

hf

25 km

θf

12 deg

φf γf

70 deg

0 deg

0 deg 0 deg

-10 deg

Angle of attack (deg)

ψf umax σ min σ max

90 deg 10 deg/s 0 deg 80 deg

Q! max

1500 kW/m2

qmax nmax

18000 N/m2 2.5 g

50 40 30 20 10 0

0

5

10

15

20

25

Mach 3 CL, CD, L/D


Table 1 Parameters for entry flight Parameter Value h0 100 km

CL CD L/D

2 1 0

0

5

10

15 20 25 Angle of attack (deg)

30

35

40

Fig. 2 Nominal angle of attack and aerodynamic coefficients. 15 American Institute of Aeronautics and Astronautics

A. Example 1: Minimum Terminal Velocity Entry With the given initial and terminal conditions, we firstly considered the minimum terminal velocity entry with specific terminal constraints and path constraints. The objective function is shown in Eq. (64), which is a linear function of V . Thus, no linearization is needed to convexify the objective function.

( )

(64)

Figure 3 shows that the values of objective function decreases to the converged solution and the sequential method converges in 21 steps without any initial guess for each sub-problem. If looser tolerance is used, fewer steps will be needed for convergence. In each step, SDPT3 solves a convex problem in about 0.35 second CPU time. The trajectories for all iterations are shown in Fig. 4, and, as can be seen, the trajectories are quite close after 12 steps. The trajectory profiles by the SCP method and GPOPS are shown in Figs. 5-8, and the optimal solutions and CPU time consumption are summarized in Table 2. As can be seen from these figures, the solutions of SCP and GPOPS are very similar. The profiles of bank angle and the control histories have the same trend as well; however, high-frequency chattering exists in the bank angle history and the control history obtained by GPOPS. In contrast, these high-frequency jitters are eliminated, and the profiles are smoother with the problem formulation developed in this paper due to the decoupling of the states and control. All the constraints are satisfied under these two approaches. In addition, the small differences in terminal velocities could be caused by several potential reasons such as the linearization and relaxation of dynamics, different discretization strategies used by the two approaches, etc. Generally speaking, the NLP solver GPOPS is much slower than the SCP method, good initial guesses are required for GPOPS to converge, and the time cost of NLP solvers are unpredictable. Table 2 Comparison of solutions for minimum terminal velocity entry Method Terminal velocity (m/s) CPU time (sec) SCP GPOPS

622.48 629.10

7.35 45.04

1000 950

Objective function (m/s)


J =V tf

900 850 800 750 700 650 600

0

5

10 15 Step number

20

25

Fig. 3 Optimal values of objective function for minimum terminal velocity entry using SCP.


100 Optimal trajectory 90 80

Altitude (km)

70 60 50

30 20

0

1000

2000

3000 4000 5000 Velocity (m/s)

6000

7000

8000

Fig. 4 Convergence of altitude v.s. velocity profiles for minimum terminal velocity entry using SCP.

Altitude (km)

100 SCP GPOPS

80 60 40 20

0

1000

2000

3000 4000 5000 Velocity (m/s)

6000

7000

8000

80

Latitude (deg)


40

60 40 20 0

0

2

4

6 8 Longitude (deg)

10

12

14

Fig. 5 Comparison of trajectories for minimum terminal velocity entry.


Flight-path angle (deg) Heading angle (deg)

0 -5 -10 -15

SCP GPOPS 0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

100

50

0

Fig. 6 Comparison of flight-path angle and heading angle profiles for minimum terminal velocity entry.

Bank angle (deg)

80 SCP GPOPS

60 40 20 0

Bank angle rate (deg/s)


5

0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

10 5 0 -5 -10

Fig. 7 Comparison of bank angle and optimal bank angle rate profiles for minimum terminal velocity entry.


SCP GPOPS

1000 0

g-loading (g) Dynamic presure [N/m 2]

Heat rate (kW/m 2) Downloaded by PURDUE UNIVERSITY on July 18, 2016 | http://arc.aiaa.org | DOI: 10.2514/6.2016-3241

2000

0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

104

2 1 0

4 2 0

Fig. 8 Comparison of path constraints for minimum terminal velocity entry. B. Example 2: Minimum Heat Load Entry For the second example, the minimum heat load entry is considered. It is more complicated than the minimum terminal velocity entry, since the objective becomes a highly nonlinear integral function shown in Eq. (65). In order to investigate the efficiency of the proposed method, the constraints on terminal longitude and latitude are relaxed to

( )

ranges with 1 deg variation in longitude and 2 deg variation in latitude, i.e., θ t f − θ f ≤ 1 deg ,

( )

φ t f − φ f ≤ 2 deg . All the other parameters and constraints are the same as Example 1.

( )

{ (

)

(

}

)

tf tf tf J = ∫ Q! dt = ∫ f1 r,V dt ≈ ∫ f1 r∗ ,V∗ + f1′ r∗ ,V∗ ⋅ ⎡⎣ r − r∗ ; V −V∗ ⎤⎦ dt t0 t0 t0

(65)

The values of the performance index are shown in Fig. 9. The trajectories for all iterations are depicted in Fig. 10. Figures 11-14 present the trajectory profiles obtained by SCP and GPOPS. For the minimum heat load problem, the sequential method converges in 20 steps. In each step, it takes SDPT3 about 0.5 seconds CPU time to solve a sub-problem. Figure 10 shows that the trajectories are very close after 8 steps. The profiles for each state are quite similar under these two approaches. One noteworthy item observed in Fig. 11 is that the terminal latitudes achieved under SCP and GPOPS are 68.0015 deg and 68.0000 deg, respectively, which means the vehicle is controlled to fly to the lower bound of the desired terminal latitude to reduce the heat load. The control profiles in Fig. 13 show that high-frequency jitters are reduced by adding additional constraints on the bank angle rate, and the corresponding bank angle profile obtained by SCP is much smoother, which has the same trend as that of GPOPS. Other constraints are all satisfied under the optimal control. In addition, small differences in heat load are reported in Table 3, which further demonstrates the optimality and accuracy of the SCP method. Table 3 Comparison of solutions for minimum heat load entry Method Heat load (kWh/m2) CPU time (sec) SCP GPOPS

1314.06 1308.74

8.75 145.68


1370

1350

1340

1330

1320

1310

0

5

10 Step number

15

20

Fig. 9 Optimal values of objective function for minimum heat load entry using SCP.

100 Optimal trajectory 90 80

Altitude (km)


Objective function (kWh/m 2)

1360

70 60 50 40 30 20

Fig. 10

0

1000

2000

3000 4000 5000 Velocity (m/s)

6000

7000

8000

Convergence of altitude v.s. velocity profiles for minimum heat load entry using SCP.


100 SCP GPOPS

Altitude (km)

80 60 40 20

0

1000

2000

3000 4000 5000 Velocity (m/s)

6000

7000

8000


Latitude (deg)

80 60 40 20 0

0

Heading angle (deg)

Flight-path angle (deg)

Fig. 11

Fig. 12

2

4

6 8 Longitude (deg)

10

12

14

Comparison of trajectories for minimum heat load entry.

5 0 -5 -10

SCP GPOPS 0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

100

50

0

Comparison of flight-path angle and heading angle profiles for minimum heat load entry.


Bank angle (deg)

80

40 20

Bank angle rate (deg/s)

0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

10 5 0 -5 -10

Comparison of bank angle and optimal bank angle rate profiles for minimum heat load entry.

2000 SCP GPOPS

1000 0

g-loading (g) Dynamic presure [N/m 2]

Heat rate (kW/m 2)


0

Fig. 13

SCP GPOPS

60

0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

0

200

400

600

800 1000 Time (s)

1200

1400

1600

10

2

4

1 0 4 2 0

Fig. 14

Comparison of path constraints for minimum heat load entry.

VI. Conclusion In this paper, a sequential convex programming method is developed for 3-D planetary entry trajectory optimization problems. This method is designed to solve the highly nonlinear optimal control problems associated with hypersonic entry via convex programming methods. Our main contribution is the decoupling of states and control by defining new state and control variables and the formulation of the trajectory optimization problem with nonlinear dynamics and nonlinear path constraints as a sequence of finite-dimensional convex optimization problems, specifically as a sequence of SOCP programming problems. The original equations of motion are used, and no assumptions or approximations are made to calculate velocity. There exist convex programming solution algorithms, such as interior-point algorithms, which converge to a global optimum very efficiently with a 22 American Institute of Aeronautics and Astronautics

deterministic stopping criteria and a prescribed level of accuracy. This reliable and rapid trajectory generation is attractive for planetary entry problems because a genetic nonlinear programming algorithm cannot guarantee a deterministic convergence. Consequently, this method has potential for onboard implementation. The existence and convergence of the sequential solutions is partially proved where currently possible, and the proposed method has been demonstrated by numerical solutions to minimum terminal velocity and minimum heat load entry trajectory optimization problems through comparisons with GPOPS. The methodology proposed in this paper is potentially applicable to solve general nonlinear optimal control problems.

References


1

Betts, J. T., “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998, pp. 193–207. 2 Boyd, S., and Vandenberghe, L., Convex Optimization, Cambridge Univ. Press, Cambridge, England, 2004. 3 Vandenberghe, L., and Boyd, S., “Semidefinite Programming,” SIAM Review, Vol. 38, No. 1, 1995, pp. 49–95. 4 Nesterov, Y., Introductory Lectures on Convex Optimization, Kluwer, Boston, 2004. 5 Nesterov, Y., and Nemirovsky, A., Interior-Point Polynomial Methods in Convex Programming, SIAM, Philadelphia, PA, 1994. 6 Wright, S. J., Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA, 1997. 7 Acikmese, B., and Ploen, S. R., “Convex Programming Approach to Powered Descent Guidance for Mars Landing,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 5, 2007, pp. 1353–1366. 8 Blackmore, L., Acikmese, B., and Scharf, D. P., “Minimum Landing Error Powered Descent Guidance for Mars Landing Using Convex Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 33, No. 4, 2010, pp. 1161–1171. 9 Acikmese, B., and Blackmore, L., “Lossless Convexification for a Class of Optimal Problems with Nonconvex Control Constraints,” Automatica, Vol. 47, No. 2, 2011, pp. 341–347. 10 Acikmese, B., Carson, J. M., and Blackmore, L., “Lossless Convexification of Nonconvex Control Bound and Pointing Constraints of the Soft Landing Optimal Control Problem,” IEEE Transactions on Control Systems Technology, Vol. 21, No. 6, 2013, pp. 2104–2113. 11 Lu, P., and Liu, X., “Autonomous Trajectory Planning for Rendezvous and Proximity Operations by Conic Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 2, 2013, pp. 375–389. 12 Liu, X., and Lu, P., “Robust Trajectory Optimization for Highly Constrained Rendezvous and Proximity Operations,” AIAA Paper 2013-4720, Aug. 2013. 13 Liu, X., and Lu, P., “Solving Nonconvex Optimal Control Problems by Convex Optimization,” Journal of Guidance, Control, and Dynamics, Vol. 37, No. 3, 2014, pp. 750–765. 14 Liu, X., Shen, Z., and Lu, P., “Entry Trajectory Optimization by Second-Order Cone Programming,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 2, 2016, pp. 227–241. 15 Roenneke, A. J., and Markl, A., “Reentry Control of a Drag vs. Energy Profile,” Journal of Guidance, Control, and Dynamics, Vol. 17, No. 5, 1994, pp. 916–920. 16 Lu, P., “Entry Guidance: A Unified Method,” Journal of Guidance, Control, and Dynamics, Vol. 37, No. 3, 2014, pp. 713– 728. 17 Shaffer, P. J., Ross, I. M., Oppenheimer, M. W., Doman, D. B., and Bollino, K. B., “Fault-Tolerant Optimal Trajectory Generation for Reusable Launch Vehicles,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 6, 2007, pp. 1794–1802. 18 Berkovitz, L. D., Optimal Control Theory, Springer–Verlag, Berlin, 1975. 19 Palacios-Gomez, F., Lasdon, L., and Engquist, M., “Nonlinear Optimization by Successive Linear Programming,” Management Science, Vol. 28, No. 10, 1982, pp. 1106–1120. 20 Banks, S. P., and Dinesh, K., “Approximate Optimal Control and Stability of Nonlinear Finite- and Infinite-Dimensional Systems,” Annals of Operations Research, Vol. 98, 2000, pp. 19–44. 21 Tomas-Rodriguez, M., and Banks, S.P., Linear, Time-varying Approximations to Nonlinear Dynamical Systems with Applications in Control and Optimization, Springer-Verlag, London, 2010. 22 Toh, K. C., Todd, M. J., and Tutuncu, R. H., “On the Implementation and Usage of SDPT3 - a MATLAB Software Package for Semidefinite-Quadratic-Linear Programming, version 4.0,” Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications, 2010. 23 Stanley, D. O., Engelund,W. C., Lepsch, R. A., McMillin,M.,Wurster, K. E., Powell, R. W., Guinta, T., and Unal, R., “Rocket-Powered Single-Stage Vehicle Configuration Selection and Design,” Journal of Spacecraft and Rockets, Vol. 31, No. 5, 1994, pp. 792–798. 24 Lu, P., “Entry Guidance and Trajectory Control for Reusable Launch Vehicle,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 1, 1997, pp. 143–149. 25 Patterson, M. A., and Rao, A. V., “GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear Programming,” ACM Transactions on Mathematical Software (TOMS), Vol. 41, No. 1, 2014, pp. 1–37. 26 Wachter, A., and Biegler, L. T., “On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming,” Mathematical Programming, Vol. 106, No. 1, 2006, pp. 25–57.


Constrained Trajectory Optimization for Planetary

Constrained Trajectory Optimization for Planetary

Suggest Documents

Reentry Vehicle Constrained Trajectory Optimization

Trajectory Optimization for Constrained UAVs: a Virtual Target Vehicle

constrained differential evolution for constrained optimization

Navigation-based Constrained Trajectory Generation for Advanced

INVERSION BASED CONSTRAINED TRAJECTORY ... - CiteSeerX

Particle Swarm Optimization Method for Constrained Optimization ...

Are Landscapes for Constrained Optimization

Methods for Constrained Optimization - CiteSeerX

Energy-constrained Modulation Optimization for

Research in Constrained Optimization

Distributed Constrained Optimization - NASA

Constrained Optimization and

Distributed Constrained Optimization - NASA

Spacecraft Trajectory Optimization - Basque Center for Applied

Coupled attitude and trajectory optimization for a

longitude trajectory optimization for autonomous ...

Multi-Objective 4D Trajectory Optimization for ...

Trajectory Optimization for Target Localization ... - Semantic Scholar

longitude trajectory optimization for autonomous ...

Trajectory Optimization Procedures for Rotorcraft ...

Trajectory Optimization Procedures for Rotorcraft Vehicles ... - CiteSeerX

Direct trajectory optimization framework for vertical ...

A fast ascent trajectory optimization method for

Spatio-Temporal Constrained Human Trajectory ... - Semantic Scholar