Bang-bang Control Design by Combing Pseudospectral Method with a novel Homotopy Algorithm
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AIAA Guidance, Navigation, and Control Conference AIAA 2009-5955
10 - 13 August 2009, Chicago, Illinois
Bang-bang Control Design by Combing Pseudospectral
Method with a novel Homotopy Algorithm
Xiaoli Bai∗, James D. Turner†, John L. Junkins‡
Texas A & M University, College Station, Texas, 77843-3141, USA
The bang-bang type of control problem for spacecraft trajectory optimization is solved
by using a hybrid approach. First, a pseudospectral method is utilized to generate approxi-
mate switching times, control structures, and initial co-states. Second, a homotopy method
is used to solve the two-point boundary value problems derived from the Euler-Lagrangian
equations. The unknown variables in the homotopy method include both switching times
and the unknown initial states and co-states. The homotopy algorithm is made robust to
the nonlinearity of the problems by enforcing the constraint satisfaction along the homo-
topy path. The optimization variables are treated as continuous variables and the final
solutions have the same accuracy as the ordinary differential equation solvers. An orbit
transfer problem is presented to show the advantages of this hybrid methodology.
I. Introduction
A classical subject in optimal control fields is the bang-bang type of control problems1, .2 These problems
often arise when the constrained control appears linearly in both the state differential equations and the
performance function while the final time can be either free or fixed. For example, the solution of time
optimal three-axis reorientation of a rigid body is usually a bang-bang controller3, .4 The low thrust spacecraft
trajectory design with the aim to minimize the fuel consumption, which is equivalent to maximizing the final
mass, also frequently leads to bang-bang controls, or thrusting and coasting type of controls5, .6 The bang-
bang control for low thrust trajectory optimization is of particular interest in this paper.
The computational techniques to solve optimal control problems are either indirect shooting or direct
shooting.7 The direct methods introduce a parametric representation of the control variables (and frequently
the state variables as well), and then resort to optimizers such as ‘fmincon’ in MATLAB, SNOPT,8 or SOCS9
to solve the resulting nonlinear programming problems.10 With the increasing power of these optimizers, it
is possible to discretize the continuous system by using very small step. For example, Betts and Erb10 used
a collocation or direct transcription method to design an optimal low thrust trajectory to the moon through
SOCS software.9 Their final nonlinear programs include 211031 variables and 146285 constraints. Usually
the direct approach is robust to the initial guess for the problem. Since there is no need to derive for the
Euler-Lagrange equations,2 it is easy to automate the direct transcription process so this direct method has
special interest in industry, leading to some example software such as POST and GTS.7 Although the direct
approaches have been very attractive to solve orbit transfer problems with impulses burns,11 for low thrust
propulsion where the thrust level is low relative to the spacecraft mass, the integrated trajectory using the
interpolated control from the direct approach can drift from the optimal trajectory and the optimality is
difficult to guarantee.
Indirect approaches are based on the calculus of variations. Necessary conditions are derived from
Pontryagin’s principles.2 A simple shooting or multiple shooting method is usually used to solve the resulting
two-point value problems, with the goal to find the unknown initial states and co-states. This method is not
popular in industry because of the difficulty encountered in automating the process to translate the original
∗ GraduateResearch Assistant, Department of Aerospace Engineering, 3141-TAMU, Student Member AIAA
† ResearchProfessor, Director of Operations, Consortium for Autonomous Space Systems, Department of Aerospace Engi-
neering, 3141-TAMU, Associate Fellow AIAA
‡ Regents Professor, Distinguished Professor of Aerospace Engineering, Holder of the Royce E. Wisenbaker ’39 Chair in
Engineering, Department of Aerospace Engineering, 3141-TAMU, Fellow AIAA.
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American Institute of Aeronautics and Astronautics
Copyright © 2009 by Xiaoli Bai, James Turner, and John Junkins. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.problem to a two-point boundary value problem. However, mathematical programming languages such as
AMPL and the automatic differentiation techniques have made automation of this process possible12,13, .14
The solutions obtained from the indirect approach assure the optimality and accuracy, which is the main
reason why this method has been very popular for low thrust trajectory design5,15,16, .17 The greatest
difficulty encountered with this approach is that it is very sensitive to the initial guess; this problem is
because that for the state equation and co-state equations resulting from the Euler-Lagrange equations, one
of them is stable to integrate forward while the other one is stable only if it is integrated backward from the
final time.18
The small convergence domain issue becomes even more difficult for solving bang-bang type of control
problems using the indirect approach for two reasons. First, the control is non-differentiable, creating the
possibility that the Jacobian matrix, which is required to compute when gradient or Newton’s based meth-
ods1 are used to solve the two-point boundary values problems, may become singular on a large domain.
Second, discontinuous control makes it difficult for most available ordinary differential equation solvers to
generate high accuracy solutions. Additionally, as mentioned Bai and Junkins4 and Bskens,19 the current
knowledge about the second order sufficient conditions for these bang-bang control problems are still limited.
Maurer and Osmolovskii20 provided a systematic numerical method to verify the second order conditions. In
the case of one or two switches, the tests are very easy to implement. The authors precluded simultaneous
switching bang-bang control structures when they derived the second order conditions. However, simultane-
ous switching cases are found quite often for the three axis rigid body maneuver.4 Bertrand and Epenoy6
used new smoothing techniques to solve such bang-bang problems. The authors studied different type of
perturbation terms that can be added to the objective function to improve the convergence domain of the
Newton’s method to solve such problems. For an Earth to Venus problem, where a possible global optimal
trajectory includes six switches, using 100 starting points, the reported results show that the convergence
rate is less than 10%.
Because of the pros and cons of both direct and indirect approaches, combing them to solve complicated
problems has been very successful4,21, .22 Usually, the co-states and control structure information is first
extracted from a nonlinear programming approach. The solutions are refined by using an indirect shooting
method. Although hybrid approaches are usually very effective to expand the convergence domain for the
indirect methods and increase the accuracy for the direct methods, both efficient direct algorithm and quality
indirect method are required to solve complicated problems.
A Legendre pseudospectral method is chosen as the direct algorithm in this paper. Pseudospectral
methods were initially used widely in fluid dynamics,23 and has become a very active research field in
recent years24,25,26, .27 Comparing with Chebyshev polynomial methods,24 Legendre pseudospectral method
with Legendre-Gauss-Lobatto (LGL) nodes27 are chosen in the paper since the well established co-vector
mapping theorem provides the proper connection to commute dualization with discretization.27 Ross et al.26
claimed that the pseudospectral method is able to solve low thrust trajectory optimization problems with
high accuracy. However, we believe there is no guarantee that the integrated trajectory using the control
obtained from the pseudospectral methods through interpolation is the real optimal solution. This issue is
addressed in the next section and further demonstrated in the application section.
The accuracy of the direct solutions is improved through an indirect approach, by introducing a novel
homotopy method in the paper. To solve optimization problems when using homotopy method , researchers
either construct a continuation algorithm or use the probability-one homotopy algorithm. 28,29,30 The
probability-one homotopy algorithm parameterizes both the state variables and the homotopy variable as
functions of arc length such that the homotopy variable can both increase and decrease. Previous published
homotopy strategy translates the problem into a one-parameter chain of problems. The starting reference
problem is easy to solve, and its solution serves as the initial guess for the next problem. By changing the
marching parameter variable(the homotopy variable), this process is continued until the objective problem is
reached and solved. This strategy is discussed in detail when Bulirsch, Montrone and Pesch used it to solve
a complicated control problem of abort landing of a passenger aircraft in the presence of windshear .31,32
The homotopy algorithm utilized in this paper was developed by Bai, Junkins, and Turner.33,34 Instead of
solving a chain of problems, the proposed homotopy method solves just one problem. The homotopy method
was demonstrated to solved several algebraic optimization problems which are beyond the capabilities of
‘fmincon’33 first. They further designed unconstrained optimal thrust direction for an Earth to Apophis
rendezvous problem.34 For the cases that can not be solved using SNOPT, their homotopy algorithm
encounters no problems. This current paper extends the pervious two papers to solve bang-bang control
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American Institute of Aeronautics and Astronauticsproblems using the homotopy methodology.
The organization of this paper is as follows. Section II briefly describes the pseudospectral method and
discusses the problems that one may encounter if this method is used to generate high accuracy solutions,
especially for low thrust problems. Section III first presents the mathematical equations of the optimal control
problems, which are formulated for solutions by the homotopy method. The homotopy algorithm is presented
for rigorously tracking equality constraints. An orbit rendezvous problem is presented in Section IV. The
procedures to solve the problem and simulation results are discussed. Conclusion remarks follow in Section V.
II. Approximate Solution by Using a Psedospectral Method
Comparing with traditional collocation methods,7 psedospectral methods use global orthogonal polyno-
mials to describe both states and controls. Detailed discussions about psedospectral methods can be found
in24,25,26, .27 One problem with psedospectral methods is that the psedospectral nodes are constructed in a
mathematical formulation such that they are dense at the end points and are sparse in the middle. But for
orbit transfer, especially for multi-revolution trajectory, it may be desired to have a decent number of nodes
for each revolution. This issue can be solved by using the psedospectral knotting method.35 We focus on
some other fundamentals that explain why it is difficult for psedospectral methods to generate high accuracy
solutions by themselves, especially for bang-bang control problems, which have received little attention in
the literature.
A. Interpolation polynomial in the Lagrange form
Lagrange polynomials are used by the psedospectral methods to construct the solutions of the problems. For
a given set of N + 1 data points t0 , t1 , t2 , · · · , tN , the Lagrangian basis polynomials are
k=N
Y t − tk
φi (t) = , i = 0, 1, · · · , N (1)
ti − tk
k=0,k6=i
Using LGL nodes, ti , 1 ≤ i ≤ N are chosen as the zeros of the derivative of the Legendre polynomials of
order N with t0 = −1 and tN = 1. The state variables are approximated by using N th order interpolation
polynomial in the Lagrange form, which is linearly expanded as
i=N
X
x(t) = xi φi (t) (2)
i=0
Similarly, the control is assumed to have the form
i=N
X
u(t) = ui φi (t) (3)
i=0
Since φi (ti ) = 1 and φi (tj ) = 0 for i 6= j, we have x(tk ) = xk , u(tk ) = uk
Notice as we increase the number of nodes, high order polynomials are used to approximate the solution
behavior. Unfortunately, if the order of the polynomials is high, the connected states and controls can have
big oscillations between the nodes.
B. Differentiation Matrix
The derivative of the state variables x(t) in the psedospectral methods is given by
i=N
X i=N
X
ẋ(tk ) = xi φ̇i (tk ) = Dki xi (4)
i=0 i=0
where φ̇i (tk ) = Dki are the entries of the (N + 1) × (N + 1) differentiation matrix D, which has the following
form23 LN (tk ) 1
LN (ti ) tk −ti , k 6= i
N (N +1)
− 4 ,k = i = 0
D := Dki = N (N +1) (5)
,k = i = N
4
0, otherwise
3 of 14
American Institute of Aeronautics and Astronauticswhere LN (t) is the N th order Legendre polynomials. Unlike Chebyshev polynomials, there is no closed form
solution to either solve for the LGL nodes or calculate the differentiation matrix. The code we implemented
later is based on the method discussed by Canuto.23 Notice this matrix is exact only if the state variable x is
a polynomial of degree at most N .36 However, for the bang-bang type of control problems, the interpolation
polynomials are oscillating at the discontinuous points because of the Gibbs phenomenon. Furthermore, the
derivatives are oscillating, yielding poor approximations at the LGL points where the differentiation matrix
is formulated for the exact differentiation at these points.37
Although the mesh refinement techniques27 or removing the Gibbs phenomenon by some filtering proce-
dures23 can relieve these problems to some extend, to guarantee the accuracy and optimality of the solutions,
we utilize a robust homotopy method to find the optimal solution through an indirect approach.
III. Indirect Solution by Using a Novel Homotopy Method
A. Problem Formulation
The mathematical equations of the optimal control problems are formulated for solutions by the homotopy
method. The equations of motion for a general dynamic system with control appearing linearly can be
written as
ẋ = f (t) = A(x, t)x + B(x, t)u + C(f ∈ Rn , u ∈ Rm ) (6)
where A and B can be both time-varying and dependent on x(t); C is a constant force vector. Using
perturbation methods,38 the solution of Eq. 6 is given by
Z t
x(t) = φ(t, 0)x(0) + φ(t, 0) φ−1 (τ, 0)(AQ + Bu)dτ + Q (7)
0
with
Q̇ = C (8)
φ(t, 0) is the state transition matrix; its dynamic equation is
∂B
φ̇(t, 0) = (A + u)φ(t, 0) (9)
∂x
and the initial condition is φ(0, 0) = I, where I is the identity matrix with the same order as the state
equations.
For a bang-bang type of control, we assume that the optimal control seeks to minimize the performance
function J = g(x(t0 ), x(tf ), t), which is not dependent on the control directly. Furthermore, we assume the
control is bounded by
|u| ≤ ub (10)
The Hamiltonian equation is formulated as
H = λT f = λT (A(x, t)x + B(x, t)u + C) (11)
Through Pontryagin’s principle, the stationary condition leads to the bang-bang type of optimal control
as
u = −sign(λT B)ub (12)
T
where λ B denotes the switching function and the sign function is defined as
1, S ≥ 0
sign(S) = undef ined, S = 0 (13)
−1, S ≤ 0
We only study strict bang-bang control problems in this paper, such that S = 0 is not valid for a finite time
of intervals.
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American Institute of Aeronautics and AstronauticsB. Homotopy Algorithm
In the homotopy algorithm, we combine the dynamic differential equations and the co-state differential
equations, leading to
Ẋ = h(X, u, t) = Ā(x, t)x + B̄(x, t)u1 + C̄ + D̄(X, u2 , t) (14)
where X consists of both the states x and the co-states λ.
Notice this equation has a more general form than Eq. 6, with two terms for controls. The differential
equation is linearly in u1 so that u1 will have a bang-bang form. The new term D̄(X, u2 , t) represents other
controls that will not have a bang-bang structure. The structure in Eq. 14 is often encountered for low thrust
design, where the thrust magnitude has a bang-bang form while the thrust angle has a continuous form.
In the following derivations, we assume we know the optimal control will switch k times with the approx-
imate switching times and control structures. Notice for switched system, these will usually be known before
optimization, especially for the switched linear system.39,40,41 For example, the speeding up of an automobile
power train requires switches from 1-4. The aim for this case is to find the optimal switching times. For a
more general dynamic system, direct methods can always be used to provide a good initial guess for how
many switches are required4, .21 A pseudospectral method is used to provide an initial solution in the paper.
The boundary conditions are formulated as
" #
λT (ti )B̄(ti , x)
Z= = 0, i = 1, 2, . . . , k (15)
κ(t0 , tf , xt0 , xtf )
where λT (ti )B(ti ) consists of k switching conditions and κ(t0 , tf ) are the standard boundary conditions
from the first order optimality conditions, which usually include the boundary conditions for the states and
co-states and possibly some conditions on the Hamiltonian. The goal is to find the unknown variables χ,
which consists of switching times ti and unknown initial states xn (t0) and co-states λn (t0). Explicitly, we
have
ti , i = 1, 2, · · · , n
χ= xn (t0) (16)
λn (t0)
The solution for χ is obtained by using the homotopy algorithm.
Notice the standard indirect approach does not shoot for the switching times since these times are
not independent but are implicitly dependent on the initial states and co-states with the optimal control.
However, since the control is discontinuous in this case, most ordinary differentiation equation integration
methods need to restart every time when the control switches to maintain the accuracy of the solutions. To
recover the switching times explicitly, they are defined as shooting variables to solve this problem. In fact,
similar ideas have been used by Bskens et al.19
Next we present a novel homotopy method to solve Eq. 15. Define the constraint residuals as
g = Z(χ̂) (17)
where χ̂ is the current integral solution for some χ0 . Define a homotopy path z as
z = g − γg0 = 0 (18)
where g0 = Z(χ̂0 ) and χ̂0 is the initial guess for χ. Notice in Eq. 18, when γ = 1, z = 0 is satisfied
automatically; when γ = 0, z = g = 0 = Z(χ̂) and an exact solution χ is found.
The homotopy path integral solution algorithm is discussed by Bai et al.34 The main step is discussed
here for completeness. Using the arc length based homotopy approach,28 Eq. 18 is solved along a homotopy
path from γ = 1 to γ = 0, where γ is allowed to both increase and decrease such that turning points are not
a problem in this approach.
To track the zero curve of Eq. 18, the derivative of z with respect to arc length s is calculated through
" #
dz h ∂g i dχ̂
ds
= ∂ χ̂ −g0 dγ =0 (19)
ds ds
5 of 14
American Institute of Aeronautics and Astronauticsh iT
dχ̂ dγ
Thus the direction of the homotopy path ds ds
lies in the null space of matrix M defined by
h i
∂g
M= ∂ χ̂ −g0 (20)
While tracking the zero curve of z, we constrain the marching variable s to be the real arc length by using
µ ¶0.5
dγ dχ̂ dγ dχ̂1 2 dχ̂2 2 dχ̂n 2
k , k2 = ( )2 + ( ) +( ) + ... + ( ) =1 (21)
ds ds ds ds ds ds
The final value of s is not known a prior. The stopping condition γ(s) = 0 is used to terminate the homotopic
tracking process.
C. Nonstandard sensitivity formulations
From Eq. 15 and 17, we can see that the homotopy path derivative ∂∂gχ̂ includes the standard boundary
conditions (when using the indirect methods) and the additional switching conditions (when using the direct
methods) with respect to both the unknown switching times and the unknown initial states and costates.
The partial derivative of the states appearing in Eq. 15 with respect to X(0) can be obtained from the
direction cosine matrix. We emphasize that the system dynamic equations can have different forms for
different time periods between the interval of [ti , ti+1 ], which happens often for the switched linear systems.
The general form of the state transition matrix is
−
∂X(ti ) ∂X(ti ) ∂X(ti−1 ) ∂X(t−
1)
= + + ... = φi (ti , ti−1 ) . . . φ1 (t1 , t0 ) (22)
∂X(0) ∂X(ti−1 ) ∂X(ti−2 ) ∂X(t0)
Since the state transition matrix in this case is not the standard one,42 we outline the computational
procedures to solve for the state transition matrix.
We integrate Eq. 14 to obtain Z t
X(t) = X(0) + h(X, u, t)dt (23)
0
where X(0) consists of both the known and the unknown initial states and co-states.
Taking the derivatives of Eq. 23 with respect to X(0)
Z tµ ¶
∂X(t) ∂h ∂X(t) ∂h ∂u2 ∂X(t)
=I+ + dt (24)
∂X(0) 0 ∂X(t) ∂X(0) ∂u2 ∂X(t) ∂X(0)
we obtain the differential equations for the state transition matrix by taking the time derivative, yielding
µ ¶
d ∂h ∂h ∂u2 ∂X
(∂X(t)/∂X(0)) = + (25)
dt ∂X(t) ∂u2 ∂X(t) ∂X0
Notice since u1 is accounted for in the bang-bang form, the partial derivatives we calculate here only
involve the non-bang-bang controls. Additionally, in Eq. 25, we assume the optimal control is an implicit
function of the states. The numerical expression for ∂u2 /∂X(t) is obtained from the Hamiltonian. The
stationary condition provides the form for the optimal control as
∂H
Hu = =0 (26)
∂u2
where H is the Hamiltonian function. Taking the derivative of Eq. 26 with respect to X(t), we obtain
∂Hu ∂Hu ∂Hu ∂u2
= + =0 (27)
∂X(t) ∂X(t) ∂u2 ∂X(t)
thus
∂u2 ∂Hu
= −(Huu )−1 (28)
∂X(t) ∂X(t)
6 of 14
American Institute of Aeronautics and Astronauticswhere Huu = ∂H∂u2 .
u
∂X
Using Eq. 28, Eq. 25 is propagated using an identity matrix as the initial condition for ∂X 0
.
Another non-standard term is the partial derivative of the states with respect to some switching time ti .
Following Eq. 7, this partial derivative can be solved by
Z t
∂X(t) ∂u(τ )
= φ(t, 0) φ−1 (τ, 0)B(τ ) dτ
∂ti 0 ∂ti
Z t
= φ(t, 0) φ−1 (τ, 0)B̄(τ )aδ(τ − ti )dτ
0
= aφ(t, 0)φ−1 (ti , 0)B(ti ) (29)
where a is dependent on the jump condition of the control and is known once we know the bang-bang control
structures. For example, if the control switches from −1 to 1, we have a = −2.
Equations 25 and 29 need to be solved for all the solutions searched along the homotopy path. They are
used to track the direction of the homotopy path from Eq. 19.
This homotopy strategy differs from penalty based methods when no penalty terms are defined. It also
differs from sequential quadratic programming(SQP).43 SQP usually linearizes the performance function
and uses second order expansions to approximate the constraint functions. The SQP approximation does
not guarantee that the final solution remains on the constraint surface. For high nonlinear problems, these
approximations will cause numerical difficulties if the initial guess is far way from the right solution. Without
approximating or linearizing either the constraints or performance function, the proposed algorithm achieves
its robustness by enforcing the satisfaction of the dynamic equation constraints along the homotopy path. In
this way, we enlarge the convergence domain of the initial guess for the two-point boundary value problems.
IV. An Orbit Rendezvous Example Problem
This example is a simplified version of a subproblem from the 4th Global Trajectory Optimisation Com-
petition.44 The problem is to design an planar optimal trajectory starting from one asteroid and intercepting
with another asteroid in a fixed time. We require the spacecraft on a circular orbit at both the initial time
and the final intercepting time.
A. Mathematical Formulations
The spacecraft is assumed to be a point mass with a variable mass m and only the gravity force from the Sun
is considered. The position of the spacecraft in a solar-centric polar coordinate is (r, θ) as shown in Fig. 1,
where r is the distance of the spacecraft to the Sun and θ is the phase angle with respect to some inertial
axis. u is the velocity along the radial direction and v is the velocity along the local horizontal direction.
The angle between the thrust direction and the local horizontal is the control variable represented by β.
The dynamic equations for the spacecraft are
ṙ = u (30)
θ̇ = v/r (31)
2 2
u̇ = v /r − µ/r + T /m sin β (32)
v̇ = −uv/r + T /m cos β (33)
where the thrust magnitude is bounded
0 ≤ T ≤ 0.135N (34)
The mass equation is
T
ṁ = − (35)
g0 Isp
The spacecraft has a constant specific impulse Isp as 3000sec. The initial mass is 1500kg and the standard
acceleration due to gravity g0 is 9.80665m/s2 . The transfer time is 240days. The optimization goal is to
minimize the fuel cost, which is equivalent to maximize the final mass.
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American Institute of Aeronautics and Astronauticsy
T
v
β u
S/C
r
θ
0 x
Sun
Figure 1. Frame, State and Control Definition
The co-state differential equations obtained through Pontryagin’s principle are
λ̇r = −λu (−v 2 /r2 + 2/r3 ) − λv uv/r2 + λθ v/r2 (36)
λ̇θ = 0 (37)
λ̇u = −λr + λv v/r (38)
λ̇v = −2λu v/r + λv u/r − λθ /r (39)
p
λ̇m = −T /m2 λ2u + λ2v (40)
The optimal thrust direction is
β = atan2(−λu , −λv ) (41)
where atan2() is the four-quadrant inverse tangent function.
The switching function for the thrust T is
p
(λu (t)2 + λv (t)2 )gIsp
S(t) = − − λm (t) (42)
m(t)
thus it is thrusting when S(t) is less than zero and it is coasting whenever S(t) is greater than zero.
The two-point boundary conditions are
r(0) = 1 (43)
θ(0) = 0 (44)
v(0) = 0 (45)
m(0) = 1 (46)
r(tf ) = 1.05242919219003 (47)
θ(tf ) = 3.99191781862267 (48)
u(f ) = 0 (49)
v(f ) = 0.97477314754443 (50)
λm (f ) = −1 (51)
where all the distance variables have been non-dimensionalized by 1AU = 1.495978706910000 × 1011 m; all
the variables involving time have been non-dimensionalized by 1T U = 5.022642890912782 × 106 sec; the
mass is non-dimensionalized by the initial mass 1500kg. All these non-dimensionalizations lead to that the
maximum thrust magnitude is 0.01517685201253; the maximum mass flow rate is ṁ = 0.01536501116359;
the transfer time is 4.12850374800020T U .
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American Institute of Aeronautics and AstronauticsB. Approximate Solutions from Pseudospectral Method
Although the homotopy method is robust when compared with the standard gradient based methods, for
bang-bang type of problems, decent control structures, and switching times are required for its successes. We
obtain these information from a pseudospectral method. The code is implemented in MATLAB. SNOPT.8
is the nonlinear programming solver that provides the major feasibility tolerance, minor feasibility tolerance,
and major optimality tolerance as 1e − 6.
The initial guess is chosen as the states integrated by using half of the maximum thrust and sampled at
64 LGL points. Figure 2 shows the control solutions. We can see that the bang-bang structure is captured
very well, yet quite a few nodes are not correct.
thrust
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
time(TU)
Figure 2. Control from pseudospectral
To verify the accuracy of the control, both linear interpolation and piecewise cubic hermite interpolation
are used to represent the control histories, which are shown in Fig. 3. The time scale is normalized to the
range [−1, 1] where Legendre polynomials are defined. The final position and velocity errors are shown in
Table 1. Notice although these numbers are small when looking at the nondimensionalized unit, they are
often not acceptable for high fidelity trajectory design.
Interpolated thrust with PS solution
0.016
0.014 PS
linear
0.012 Piecewise Cubic Hermite
0.01
Thrust
0.008
0.006
0.004
0.002
0
−1 −0.5 0 0.5 1
Normalized time
Figure 3. Interpolated thrust with PS solution
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American Institute of Aeronautics and AstronauticsFinal Errors Linear interpolation Piecewise Cubic hermite Interpolation
4r(AU ) 0.12583599381699 × 10−3 0.11586689900422 × 10−3
4u(AU/T U ) 0.18681358039823 × 10−3 0.16111720421912 × 10−3
4v(AU/T U ) 0.37652130891286 × 10−3 0.38315737654437 × 10−3
4θ(rad) 0.01071562338772 × 10−3 −0.04031962569595 × 10−3
Table 1. Final Errors
C. Optimal Solution from Homotopy Method
Using the initial co-states obtained from the pseudospectral methods and the approximate switching times
as the initial guess for the homotopy method, we find accurate switching times and initial co-states with
the optimal trajectory. The initial co-states and switching times from the pseudospectral methods and the
homotopy solutions are listed in Table 2.
The homotopy tracking histories are shown in Figs. 4 and 5, which is the path the homotopy method
Initial conditions and switching times pseudospectral homotopy
λr (t0 ) −0.47908295213219 −0.48433902798013
λφ (t0 ) −0.16643708862167 −0.16754426885400
λu (t0 ) 0.33263137092118 0.31568214760216
λv (t0 ) −0.85778071767051 −0.87711007733394
λm (t0 ) −0.97740428340329 −0.96820223538411
t1 0.2 0.18436102665154
t2 0.925 1.02830990909980
t3 3.35 3.25065284139823
Table 2. Initial co-states and switching times
marches along, starting with the solutions from the psedospectral method and ending at the accurate optimal
solutions. The optimal trajectory is shown in Fig. 6. The optimal state history is shown in Fig. 7 with the
mass history in Fig. 8. The switching function and the thrust magnitude are shown in Fig. 9, which clearly
shows that the obtained control satisfies the Pontryagin’s principles. Notice for illustration purpose the
thrust magnitude has been multiplied by ten. The final errors, which include the boundary condition in Eqs.
from 47 to 51 and the three switching functions S(ti ) = 0, i = 1, 2, 3 in Eq. 42, are less than 10−11 . The final
mass is 0.9735, which means 2.645547298022% mass was consumed during the transfer.
V. Conclusions and Future Works
A novel homotopy algorithm has been combined with a pseudospectral method to generate high accu-
racy and optimality guaranteed solutions for bang-bang type of optimal control problems, with particular
applications to the low thrust trajectory optimization. The pseudospectral method is used to generate the
initial guess of the solution to start the homotopy algorithm. The homotopy method solves the optimal con-
trol problems through an indirect shooting method. Although more computationally intense than gradient
based shooting methods, the homotopy algorithm is more robust to the nonlinearly of the problems and high
accuracy solutions are obtained. The efficient utilization of the state transition matrix, the sensitivity of the
boundary conditions with respect to the switching times, and the fact that the homotopy path only marches
along the problem’s constrained surface, are the three keys for its success. Since the current homotopy
algorithm only handles equality constraints, sometimes it fails to find the solutions since the inequality con-
straints on the switching times are not included. Under this case, we find a better bang-bang shape solution
from the pseudospectral method and restart the homotopy algorithm. The ongoing work includes extending
the homotopy method to solve inequality constrained problems, thus both ordering switching times and state
constrained problems are expected to be solved.
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American Institute of Aeronautics and AstronauticsInitial lambda tracking history
0.4
λ (0)
r
λφ(0)
0.2
λu(0)
λv(0)
0
λm(0)
−0.2
−0.4
−0.6
−0.8
−1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Tracking arch length (s)
Figure 4. Co-state tracking history
Switching time tracking history
3.5
t1
t2
3
t3
2.5
2
1.5
1
0.5
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Tracking arch length (s)
Figure 5. Switching time tracking history
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