This document describes a Chebyshev pseudospectral method for directly solving optimal control problems. Some key points: - The method approximates the state and control variables using Lagrange polynomials with values at Chebyshev-Gauss-Lobatto points as coefficients. This yields a nonlinear programming problem with the state/control values at points as unknowns. - Numerical examples show the Chebyshev method provides more accurate results than traditional collocation methods. - Advantages of the Chebyshev method include high accuracy of pseudospectral approximations due to choice of node points, and closed-form evaluation of the node points, allowing rapid optimization without advanced linear algebra techniques.

1. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
Vol. 25, No. 1, January– February 2002
Direct Trajectory Optimization by a Chebyshev
Pseudospectral Method
Fariba Fahroo¤ and I. Michael Ross†
Naval Postgraduate School, Monterey, California 93943
We present a Chebyshev pseudospectral method for directly solving a generic Bolza optimal control problem
with state and control constraints. This method employs Nth-degree Lagrange polynomial approximationsfor the
state and control variables with the values of these variables at the Chebyshev– Gauss– Lobatto (CGL) points as
the expansion coef cients. This process yields a nonlinear programming problem (NLP) with the state and control
values at the CGL points as unknown NLP parameters. Numerical examples demonstrate that this method yields
more accurate results than those obtained from the traditional collocation methods.
I. Introduction cients. When the properties of the Chebyshev polynomials are used,
I N recent years, direct solution methods have been used exten- the state equations,performanceindex, and the boundary conditions
sively in a variety of trajectory optimization problems.1;2 Their are converted to algebraic or transcendental equations in terms of
advantage over indirect methods, which rely on solving the nec- these unknown coef cients. In this manner, conversion of the orig-
essary conditions derived from the Pontryagin et al. minimum inal optimal control problem to an NLP is achieved.
principle,3 is that direct method algorithms have better convergence Recently, Elnagar et al.8 and Fahroo and Ross9;12;13 have used
properties.In addition,because the necessaryconditionsdo not have the idea of expansion of the state and control variables in terms
to be derived, the direct methods can be quickly used to solve a num- of Lagrange polynomials at suitably chosen points, the Legendre–
ber of practical trajectory optimization problems. Gauss– Lobatto (LGL) nodes. This idea is the basis for pseudospec-
Direct methods are based on discretizing the control time his- tral methods, which are used extensivelyfor solving uid dynamics
tory and/or state variable time history at the nodes of discretization, problems.14;15 In approximatingoptimal control problems, with this
transforming the optimal control problem to a nonlinear program- choice of node points and properties of the Lagrange polynomials,
ming problem (NLP), and then solving the resulting NLP. In direct the state equations and the possible state and control constraints are
collocation schemes, which discretize both the control and state readily transformed to algebraic equations in terms of values of the
variables, the unknowns are the values of the controls and the states state and control variables at the nodes. Differential constraints are
at these nodes. The cost function and the state equations can be ex- imposed only at the LGL points by way of a differentiation opera-
pressed in terms of these values. An interpolation scheme is used tor (matrix). In this sense, the pseudospectral methods are different
to obtain the time histories of both the control and the state vari- from other collocation schemes,1;2;4 where higher-order numerical
ables. In the traditional collocation schemes, piecewise-continuous integration techniques are used to approximate the differential con-
polynomials such as linear or cubic splines are used as the inter- straints not only at the nodes but at points in between (midpoints,
polating polynomials over each time segment.4;5 Satisfaction of the for example, as in Hermite– Simpson).
state differential equations is achieved by using some form of inte- In this paper, we present a Chebyshev pseudospectral method
gration scheme over each subinterval, which results in formulation and consider differential constraints given in terms of controlled
of defects. One of the popular schemes is the Hermite– Simpson differentialinclusions,differential algebraic equations (DAEs), and
rule, which amounts to Simpson’s quadrature rule over the subin- ordinarydifferentialequations(ODEs). Our method applies without
tervals (see Ref. 4). Recently, Gauss– Lobatto quadrature rules such change to all of these types of constraints and, thus, encompasses a
as trapezoidal, Simpson’s, or higher-order rules with Jacobi poly- wider range of problems than the usual collocation methods. Note
nomials as the interpolant have also been used for collocation (see that, while revising this paper, the parallel work of Elnagar and
Refs. 6 and 7). Razzaghi16 came to our attention. We pay homage to their work
In literature,the choice of polynomialspace for expansionof state and cite some key differences. The authors of Ref. 16 also propose
and control variables is not limited to piecewise-continuouspolyno- a Chebyshev pseuodospectral method for solving a variety of op-
mials and in fact globally orthogonalpolynomials such as Legendre timal control problems. One major difference between their work
and Chebyshev polynomials or Lagrange interpolatingpolynomials and ours is the way the integral cost function is approximated.They
have been used for approximating the control and state variables in employ a cell-averaging technique, whereas we use the Clenshaw–
nonlinear optimal control problems.8¡11 In general, there are two Curtis quadrature scheme (see Refs. 17 and 18) for discretizing the
ways to construct a polynomial approximation to the solution y.t /: cost function. The focus of their paper is mostly on handling the
One is to use an interpolatingpolynomialbetween the values y.t j / at discontinuousfunctions, which indeed establishes and validates the
some points t j : The other is to use a series expansion in terms of or- power and range of applicability of pseudospectral methods to op-
thogonal polynomials. In Refs. 10 and 11, the latter idea is pursued, timal control problems with discontinuous controls. In our effort to
and both the states and the controlvariablesare expandedin terms of show the power of pseudospectral methods, we apply it to a more
Chebyshev polynomials with unknown generalized Fourier coef - general class of problems including DAEs and differential inclu-
sions. Thus, in this sense, our paper is complementary to that of
Received 24 July 2000; revision received 17 May 2001; accepted for Ref. 16. It should be noted that an earlier version of this present
publication 22 May 2001. This material is declared a work of the U.S. paper appeared in Ref. 19.
Government and is not subject to copyright protection in the United States. One notable advantage of using Chebyshev pseudospectral
Copies of this paper may be made for personal or internal use, on condition method over other direct collocation methods is the high degree of
that the copier pay the $10.00 per-copy fee to the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code
accuracythat pseudospectralapproximationsoffer.17 One reason for
0731-5090/02 $10.00 in correspondence with the CCC. this is the choice of node points as the Chebyshev– Gauss– Lobatto
¤ Associate Professor, Department of Mathematics, Code MA/Ff; (CGL) points that result in interpolatingpolynomialsthat are closest
ffahroo@nps.navy.mil. Senior Member AIAA. to the optimal polynomial in the max-norm approximation of any
† Associate Professor, Department of Aeronautics and Astronautics, Code function. Also the derivatives of these interpolating polynomials
AA/Ro; imross@nps.navy.mil. Associate Fellow AIAA. can be calculated exactly at these nodes by a differentiation matrix.
160

2. FAHROO AND ROSS 161
Aside from their popularity in engineering applications,Chebyshev
µ ¶
2
polynomials have the added computational advantage in that their fl · f x.t /; x.t /; u.t /; ¿ .t/ · fu
P (10)
corresponding CGL points can be evaluated in closed form. Thus, ¿ f ¡ ¿0
the Chebyshev pseudospectralmethod offers the possibility of rapid (11)
Ãl · Ã[x.¡1/; x.1/; ¿ f ¡ ¿0 ] · Ãu
trajectory optimization because it does not require the use of ad-
vanced numerical linear algebra techniques that are necessary for gl · g[u.t /; x.t/; ¿ .t/] · gu (12)
the calculation of LGL points and weights.20 In the three numeri-
cal examples presented in this paper, the results clearly show that To re ect transformation of the time domain, one must use differ-
Chebyshev polynomials are very effective in direct optimization ent symbols. However, for the purpose of brevity we retain these
techniques and offer superior results to those of the existing collo- symbols. Thus, in this context, one must view x.t /; for example, as
cation methods.
x[¿ .t/] D xf[.¿ f ¡ ¿0 /t C .¿ f C ¿0 /]=2g
II. Problem Formulation
Consider the following optimal control problem. Determine the III. Chebyshev Pseudospectral Method
control function u.¢/ and the correspondingstate trajectory x.¢/ that The Chebyshev pseudospectral method is one special case of a
minimize the Bolza cost function: more general class of spectral methods.15 The basic formulation
Z ¿f of these methods involves two essential steps: One is to choose a
J [x.¢/; u.¢/; ¿ f ] D M[x.¿ f /; ¿ f ] C L[x.¿ /; u.¿ /; ¿ ] d¿ (1) nite-dimensional space (usually a polynomial space) from which
¿0 an approximationto the solution of the differentialequationis made.
The other step is to choose a projection operator, which imposes the
where ¿ 2 R, x 2 R n , u 2 R m , M : R n £ R ! R, and L : R n £ R m £
differential equation in the nite-dimensional space. One impor-
R ! R. We assume the dynamic constraints to be given by the
tant feature of spectral methods, which distinguishes them from
following system of controlled differential inclusions:
nite element or nite difference methods, is that the underlying
fl · f [P .¿ /; x.¿ /; u.¿ /; ¿ ] · fu ;
x ¿ 2 [¿0 ; ¿ f ] (2) polynomial space is spanned by orthogonal polynomials that are
in nitely differentiableglobal functions. Among examples of these
where f : R n £ R n £ R m £ R ! R n : This formulation generalizes orthogonalpolynomials are Legendre and Chebyshev polynomials,
and uni es much of the previous apparently distinct formula- which are orthogonal on the interval [¡1; 1]; with respect to an ap-
tions.7;21 For example, if we set the lower and upper bounds equal propriate weight function [w.t / D 1 for Legendre polynomials, and
p
to zero, we get a controlled DAE, w.t/ D 1= .1 ¡ t 2 / for Chebyshev polynomials of the rst kind].
In pseudospectral methods, the functions are expanded in terms of
f [P .¿ /; x.¿ /; u.¿ /; ¿ ] D 0;
x ¿ 2 [¿0 ; ¿ f ] (3) interpolating polynomials so that the expansion coef cients are the
values of the functionat the node points. Because an arbitrarychoice
If, in addition, we assume that the Jacobian @f =@ x is nonsingular,
P
of node points can give very poor results in interpolation, different
then Eq. (3) may be reduced to a controlled ODE,
Gauss quadrature points are chosen to give the best accuracy in
x.¿ / D f [x.¿ /; u.¿ /; ¿ ];
P ¿ 2 [¿0 ; ¿ f ] (4) interpolation of a function. The derivatives of these interpolating
polynomials at these node points are given exactly by a differenti-
Furthermore, for u 2 U , the following hodograph transformation ation matrix. These two important aspects (choice of interpolating
may be used to transform the controlled ODE to an ordinary differ- polynomials as the trial functions and Gauss quadrature points) of
ential inclusion: pseudospectral methods separate them from the other collocation
methods. In the Chebyshev pseudospectral method, the interpola-
P
x.¿ / 2 f y.¿ / j y.¿ / D f[x.¿ /; u.¿ /; ¿ ]; u 2 U g; ¿ 2 [¿0 ; ¿ f ] tion points are given in closed form,
(5)
tk D cos.¼ k = N /; k D 0; : : : ; N (13)
We elaborateon these formulations further in the numerical section.
In addition, we assume the boundary conditions to be given by the These points lie in the interval [¡1; 1] and are the extrema of the
relations N th-order Chebyshev polynomial TN .t /. The j th-order Chebyshev
(6) polynomial is expressed by
Ãl · Ã[x.¿0 /; x.¿ f /; .¿ f ¡ ¿0 /] · Ã u
where à : Rn£ Rn
£R! Rp
and Ãl and à u 2 Rp
are constantvec- T j .t / D cos. j cos¡1 t /; j D 0; : : : ; N (14)
tors representing the lower and upper bounds of these inequalities.
Possible state and control constraints are formulated as mixed con- which yields
straints, T j .tk / D cos.¼ k j = N /
gl · g[x.¿ /; u.¿ /; ¿ ] · gu ; g : Rn £ R m £ R ! Rr (7)
Thus, T0 .t / D 1; T1 .t / D t; T2 .t / D 2t 2 ¡ 1; T3 .t / D 4t 3 ¡ 3t , and
where gl and gu 2 R r denote the lower and upper bounds of g. so on. These node points have the distribution property that they
Clearly, an equality constraint may be obtained by simply setting cluster around the endpoints of the interval. This clustering results
the lower and upper bounds to be equal.Thus, this formulationis not in avoidance of the Runge phenomenon, which is the divergence at
only more general, but is also more elegant than the one that sepa- the endpointsof interpolationon equispacedinterpolationpoints. In
rately identi es equality and inequality constraintson the states and fact, it has been shown that interpolationat the CGL nodes gives the
control variables. closest result to the optimal polynomial approximation to a given
In the numerical approximation of the optimal control problem, function. This slight error from optimality is given by the Lebesgue
because the node points (CGL) lie in the computational interval constant, which is the smallest for the CGL nodes distribution (see
[¡1; 1]; the problem is transformed to this interval by the linear Ref. 17).
transformation for t 2 [t0 ; t N ] D [¡1; 1]: For approximatingthe continuousequations,we seek polynomial
approximations of the form
¿ D [.¿ f ¡ ¿0 /t C .¿ f C ¿0 /]=2 (8)
X
N
resulting in the following reformulationof Eqs. (1), (2), (6), and (7): x N .t/ D x j Á j .t / (15)
jD0
J [x.¢/; u.¢/; ¿ f ] D M[x.1/; ¿ f ]
´Z 1 X
N
¿ f ¡ ¿0 u N .t/ D u j Á j .t / (16)
C L[x.t /; u.t /; ¿ .t/] dt (9)
2 ¡1 jD0

3. 162 FAHROO AND ROSS
where, for j D 0; 1; : : : ; N , For the off-diagonal elements, from Eq. (20) we can easily see that
´
.¡1/ j C1 t2P
.1 ¡ / TN .t/ ck .¡1/ j C k ck .¡1/ j C k
Á j .t/ D Dk j D D
N 2c j t ¡ tj c j tk ¡ t j cj .¡t N ¡ k / ¡ .¡t N ¡ j /
are the Lagrange interpolating polynomials of order N (see the Ap- ´
ck .¡1/ j C k Q
pendix for a derivation), with D D ¡ Dk j (22)
» cj ¡.tk ¡ tQj /
Q
2; j D 0; N
cj D Therefore, we have
1; 1· j · N ¡1
Q
Dk j D ¡Dk j (23)
From its interpolating property, the Lagrange polynomials satisfy
the condition Additionally,the original matrix D has another symmetry property,
»
1; if j Dk which can be easily shown by the preceding relationships22 :
Á j .tk / D ± jk D
0; if j 6D k Dk j D ¡D N ¡ k;N ¡ j (24)
Hence, it follows that
Using Eqs. (23) and (24) and denoting the values of the approximate
x N .tk / D xk ; u N .tk / D uk (17) states at the sorted CGL points by
In other words, the node points are the interpolating points. To x N ¡ k D x N .t N ¡ k / D x N .tQk / D xk
Q Q
express the derivative x N .t/ in terms of x N .t/ at the node points tk ;
P
we differentiate Eq. (15), which results in a matrix multiplicationof we have
the following form14;15 :
X
N X
N
X
N X
N x N .tk / D
P Dk j x N .t j / D Q Q PN
D N ¡ k;N ¡ j x N .tQN ¡ j / D x .tQN ¡ k /
Q
P N
dk D x .tk / D P
x j Á j .tk / D Dk j x j (18) jD0 jD0
j D0 j D0 (25)
or more succinctly, we have
where Dk j are entries of the .N C 1/ £ .N C 1/ differentiation ma-
trix D P PN
dk D x N .tk / D x .tQN ¡ k /
Q (26)
8 £ ¯ ¤
jCk
>.ck =c j / .¡1/
> ¯ ¡
.tk ¡ t j / ; j 6D k
This result shows that the sorting of the CGL points does not affect
>
< ¢
2
¡tk 2 1 ¡ tk ; 1· j Dk · N ¡1 the underlying de nitions and formulation of the pseudospectral
D :D [Dk j ] :D method. One can now proceed with the formulation of the NLP
>.2N 2 C 1/=6;
> j DkD0
>
: with the sorted CGL points. In the formulation to come, we drop
¡.2N 2 C 1/=6; jDkDN the tilde notation with the understanding that the sorted points are
used as the node points in the differentiation matrix and elsewhere.
(19)
Multiplication by this matrix, transforms a vector of the state vari- B. NLP Formulation
ables at the CGL points to the vector of approximate derivatives at The state equationsand the initialand terminalstate conditionsare
these points. discretized by rst substituting Eqs. (15), (16), and (18) in Eq. (10)
and collocating at the CGL nodes tk , The state equations are trans-
A. Modi cation of the Chebyshev Differentiation Matrix formed into the following algebraic inequalities:
The de nitions of the CGL points and the given differentiation
matrix are the standardones in the literatureon spectralmethods.15;17 fl · f [2=.¿ f ¡ ¿0 / dk ; xk ; uk ; ¿k ] · fu ; k D 0; : : : ; N
However, in our implementation of this method for discretizing op-
timal control problems, it is more convenient to modify these def- where dk is as de ned in Eq. (26). The initial and terminal state
initions. The CGL points as de ned by Eq. (13) are from 1 to ¡1, conditions are given by
that is, the initial point is 1 and the nal point is ¡1. For trajec-
tory optimization problems, it is more convenient to take the ini- Ãl · Ã[x0 ; x N ; .¿ f ¡ ¿0 /] · Ã u
tial point at ¡1 and the nal point at 1. This sorting modi es the
de nition of the differentiation matrix in the following sense: Us- The control inequality constraints are approximated by
ing the de nition of Eq. (19), we will show that the new matrix with gl · g.xk ; uk ; ¿k / · gu
Q
the sorted points, D is of the opposite sign of the matrix with the
unsorted points, D. To see this, note that in the de nition of ma- Next, the cost function in Eq. (9) is discretized: We use the
trix D, the points t0 D 1; t1 ; : : : ; t N ¡ 1 ; t N D ¡1 are used, and for Clenshaw– Curtis quadrature scheme (see Ref. 17, chapter 12) to
Q
matrix D, the sorted points tQ0 D ¡1; : : : ; tQN D 1; are used, that is, discretize the integral part of the cost function to a nite sum. The
t N ¡ k D tQk ; k D 0; : : : ; N . For the CGL points, we have the following idea is based on nding optimal weights wk for a given set of CGL
symmetry relationship: points tk such that the approximation
tk D ¡t N ¡ k D ¡tQk (20) Z 1 X
N
p.t / dt D p.tk /wk ; p.t / 2 P N
which yields
¡1 kD0
tk ¡ t j D ¡.t N ¡ k ¡ t N ¡ j / D ¡.tQk ¡ tQj /
for P N ; the space of polynomials of degree ·N , is exact. For N
For the elements D00 and D NN ; it can be easily shown that even, the weights are17;18
Q
D00 D D NN D ¡D00 ; Q
D NN D D00 D ¡D NN 1
w0 D w N D (27)
N2 ¡ 1
2
For the other diagonal elements, Dkk D ¡tk 2.1 ¡ tk /,
= from
N =2 00
Eq. (20) we have 4X 1 2¼ j s N
ws D w N ¡ s D cos ; s D 1; : : : ;
tN ¡ k Q
t N 1 ¡4j 2 N 2
j D0
Dkk D ¡ ¢ D ¡ k 2 ¢ D ¡ Dkk
Q (21)
2
2 1 ¡ tN ¡ k 2 1 ¡ tQk (28)

4. FAHROO AND ROSS 163
For N odd, the weights are given by17 Table 1 Comparison of cost functions for various methods
Ã
and formulations for N = 11
1
w0 D w N D 2 (29) Method Ji j Ji ¡ Jana j Np
N
00
Analytic solution 1.253314
4 X
.N ¡ 1/=2 Collocation (Simpson) 1.253005 0.000309 34
1 2¼ j s
ws D w N ¡ s D cos Collocation ( fth-degree GL) 1.253183 0.000131 84
N 1 ¡ 4 j2 N Pseudospectral (CGL) 1.253309 0.000005 34
jD0
Pseudospectral (DAE) 1.253366 0.000052 23
N ¡1
(30)
s D 1; : : : ;
2
In the preceding summations, the double prime means that the rst Let X , Y , and 2 representthe solution vectorto this problemat the
and the last elements have to be halved. CGL nodes. Then the NLP for this problem is simply to minimize
Therefore, the cost function in terms of the coef cients ¿ f subject to the equality constraints
p
X D .x0 ; x1 ; : : : ; x N /; U D .u0 ; u1 ; : : : ; u N / .2=¿ f /D ¤ X ¡ 2gY cos 2 D 0 (39)
p
can be discretized as .2=¿ f /D ¤ Y ¡ 2gY sin 2 D 0 (40)
¿ f ¡ ¿0 X
N
and
J .X; U; ¿ f / ¼ M.x N ; ¿ f / C L.xk ; uk ; ¿k /wk (31)
2 k D0 X .0/ D Y .0/ D 0; X .N / ¡ 0:5 D 0
Thus, the optimal control problemin Eqs. (9– 12) is approximated All operations in the preceding equations are array operations, that
by the following nonlinear optimization problem: Find coef cients is, term-by-term,whereas the asteriskis used to denote matrix multi-
plication. With this notation, the striking resemblance of the NLP to
X D .x0 ; x1 ; : : : ; x N /; U D .u0 ; u1 ; : : : ; u N / the originalproblemis immediately apparent.The analyticsolutions
to this problem are the equations of a cycloid,
and possibly the nal time ¿ f to minimize
x.¿ / D .g¿ f =¼ /f¿ ¡ .¿ f =¼ / sin[¼.1 ¡ ¿ =¿ f /]g (41)
¿ f ¡ ¿0 X
N
¡ ¯ ¢
J N .X; U; ¿ f / D M.x N ; ¿ f / C L.xk ; uk ; ¿k /wk y.¿ / D 2g¿ 2 ¼ 2 cos2 [.¼ =2/.1 ¡ ¿ =¿ f /] (42)
2 kD0
f
(32) The optimal control (angle) is given by the following expression:
subject to (43)
µ .¿ / D .¼ =2/.1 ¡ ¿ =¿ f /
fl · f[2=.¿ f ¡ ¿0 /dk ; xk ; uk ; ¿k ] · fu ; k D 0; : : : ; N (33)
From x.¿ f / D 0:5; Eq. (41), and g D 1, the minimum time calculated
p
gl · g.xk ; uk ; ¿k / · gu ; k D 0; : : : ; N (34) to six decimal places is ¿ f D .0:5¼ / D 1:253314:
The Chebyshev pseudospectral method was used for discretiza-
Ãl · Ã[x0 ; x N ; .¿ f ¡ ¿0 /] · Ã u (35) tion of the problemand NPSOL 24 was utilized as the NLP solver.The
implementation was performed in MATLAB ® via the use of MEX
From the preceding equations, one can see the simplicity of the les. In Table 1, we compare the minimum cost function obtained
method, which retains much of the structure of the continuous from our method to the ones from the Simpson collocation method
problem. and a fth-degree Gauss– Lobatto method reported in Ref. 7. All
O O
reported results are for N D 11 nodes, where N D N C 1: The vari-
IV. Numerical Examples able N p in the last column of Table 1 stands for the total number
of NLP variables, which includes the free nal time as well. Note
In this section, we present three numerical examples that are in- that for the fth-degree Gauss– Lobatto (GL) collocation method
tended to illustrate separate points. In the rst example, we choose (done the traditional way), the number of NLP variables are signif-
the benchmark brachistochrone problem discussed in Ref. 7 and icantly higher than the pseudospectralmethod for the same number
demonstrate the accuracy of our technique. In the second example, of nodes. In any case, it is evident that even for a low number degree
we reformulate the brachistochrone problem as a DAE and show of discretization (and number of NLP variables), the Chebyshev
how our method remains unchanged.In the third and nal problem, pseudospectral method gives results superior to either of the other
we solve the moon-landing problem23 formulated as a differential collocation methods.
inclusion by a hodograph transformation. In Figs. 1 and 2, we show the time historiesof the state and control
O
variables against the analytic solutions for N D 11. The solid lines
A. Example 1: Brachistochrone as an ODE representthe analytic solutionsand ¤ is for the solutionsfrom the nu-
In this well-known problem, the control problem is formulated merical method. The graphs clearly demonstrate the accuracy of the
as nding the shape of a wire so that a bead sliding on the wire results from the Chebyshev pseudospectralmethod. Note that from
will reach a given horizontal displacement in minimum time. No Ref. 7 the collocation solutions are generally accurate to O.10¡4 /;
frictional forces are considered, and the gravity force is uniform. which is about the same order of accuracy for the Chebyshev pseu-
The problem is then to minimize ¿ f subject to the equations dospectral method for this number of nodes. However, the order of
p accuracy in the control for the other collocation methods is about
xD
P 2gy cos µ (36) O.10¡2 / [with a maximum pointwise error at the rst node of about
p O.10¡1 /], whereas for our method the maximum pointwise error is
yD
P 2gy sin µ (37) of order O.10¡3 /.
with boundary conditions B. Example 2: Brachistochrone as a DAE
Here we eliminate the control variable µ and rewrite Eqs. (36)
x.0/ D y.0/ D 0; x.¿ f / D 0:5 (38) and (37) in terms of a single fully implicit DAE [see Eq. (3)]:
The control, angle µ ; is the slope of the wire as a function of time. x 2 C y 2 D 2gy
P P (44)

5. 164 FAHROO AND ROSS
subject to the equations of motion
dh
Dv (47)
d¿
dv T
D ¡g C (48)
d¿ m
dm T
D¡ (49)
d¿ Isp g
where the state variablesh; v, and m are altitude,speed,and mass, re-
spectively.The control is providedby the thrust T , which is bounded
by
0 · T · Tmax
The other parameters in the problem are g; the gravity of moon (or
any planet without an atmosphere), and Isp , the speci c impulse of
the propellent.Given any set of initial conditions h 0 ; v0 , and m 0 ; the
normalized parameters for the problem were arbitrarily chosen as
Tmax =m 0 g D 1:1; Isp g =v0 D 1; h.0/= h 0 D 0:5
v.0/=v0 D ¡0:05; m.0/= m 0 D 1
Fig. 1 Time history of x and y for the brachistochrone problem.
Therefore, we have the following normalized initial conditions:
h.0/ D 0:5; v.0/ D ¡0:05; m.0/ D 1:0 (50)
For soft landing, we must have
h.¿ f / D 0; v.¿ f / D 0 (51)
In addition, for a physically meaningful trajectory, we must have
m.¿ f / > 0 (52)
The discrete differential inclusion version of these equations for
the variables H D [h.t0 /; h.t1 /; : : : ; h.t N /]T , V D [v.t0 /; v.t1 /; : : : ;
v.t N /]T , and M D [m.t0 /; m.t1 /; : : : ; m.t N /]T can be formulated as
.2=¿ f /D ¤ H ¡ V D 0 (53)
M[.2=¿ f /D ¤ V C g] C Isp g[.2=¿ f /D ¤ M] D 0 (54)
0 · ¡Isp g[.2=¿ f /D ¤ M ] · Tmax (55)
Fig. 2 Time history of control µ for the brachistochrone problem.
Thus, there is a reduction in the number of NLP variables by N ,O
the number of CGL points. The DAE constraint is discretized at the
CGL points resulting in
¡ ¯ 2¢ ¡ ¯ ¢
4 ¿ f .D ¤ X /2 C 4 ¿ 2 .D ¤ Y /2 ¡ 2gY D 0
f (45)
From Table 1, we see that, although the degree of accuracy for
the cost function is not as high as the original formulation, it is
still better than the results from the other collocation methods. The
accuracy of the solutions is still of the order of O.10¡4 /. Therefore,
the pseudospectral method can handle formulation of the problem
as an DAE without sacri cing accuracy.
C. Example 3: Moon-Landing Problem as a Differential Inclusion
The control problem is formulated as maximizing the nal mass
and, hence, minimizing
J D ¡m.¿ f / (46) Fig. 3 Phase portrait of h vs v for the moon-landing problem.

6. FAHROO AND ROSS 165
De ning
Y
N
w.t / D .t ¡ tm /
mD0
we easily obtain
w 0 .tk / D .tk ¡ t0 /.tk ¡ t1 /; : : : ; .tk ¡ tk ¡ 1 /.tk ¡ tk C 1 /; : : : ; .tk ¡ t N /
Y
N
D .tk ¡ tm /
m D0
m 6D k
Therefore, we have
w.t / 1
Ák .t / D
.t ¡ tk / w 0 .tk /
From the de nition of the CGL points that are the extrema of TN
0
or the zeros of TN .t/, and t0 D 1; t N D ¡1; we can write
0
TN .t/ D .t ¡ t1 /.t ¡ t2 /; : : : ; .t ¡ t N ¡ 1 /
Y
N
w.t/ D .t ¡ tm / D .t ¡ t0 /.t ¡ t1 /; : : : ; .t ¡ t N ¡ 1 /.t ¡ t N /
Fig. 4 Time history of m and control T for the moon-landing problem. m D0
D .t 2 ¡ 1/TN .t /
0
Figure 3 shows the phase portrait projected into the v – h space
For w 0 .tk /, we use that the Chebyshev polynomials are the eigen-
for 32 CGL points. This problem has at most one switch in the
functions of
control variable (Ref. 22), and the results are shown in Fig. 4. The
£ ¤
control was calculated from Eq. (49). It is clear that the method has .1 ¡ t 2 /TN ¡ t TN C N 2 TN .t/ D 0
00 0
adequately captured the optimal bang– bang structure of the control
with one switch. From this expression and the equation for w.t /, we have
£ ¤0
V. Conclusions w 0 .t/ D ¡.1 ¡ t 2 /TN
0
D N 2 TN .t/ C t TN
0
(A2)
The simplicity, ef ciency, and versatility of the Chebyshev pseu-
dospectral method allows one to perform rapid and accurate tra- Evaluating the preceding expressionat t D tk , k 6D 0, N , and using
0
jectory optimization. A low degree of discretization appears to be that TN .tk / D 0, for k D 1; : : : ; N ¡ 1, and TN .tk / D cos.¼ k N = N / D
suf cient to generate good results. Because the CGL points and .¡1/k , k D 1; : : : ; N ¡ 1, we have
weights can be quickly evaluated, it suggests that the technique has
a high potential for use in optimal guidance algorithms that require w 0 .tk / D .¡1/k N 2 ; k D 1; : : : ; N ¡ 1
a corrective maneuver from the perturbed trajectory. In any case,
reference optimal paths can be easily generated, and the numerical For k D 0; N , we use the following identities:
examples indicate that the converged solutions are indeed optimal.
Further tests and analysis are necessary to investigate the stability TN .t0 D 1/ D 1; TN .1/ D N 2
0
and accuracy of the method.
TN .t N D ¡1/ D .¡1/ N ; TN .¡1/ D .¡1/ N N 2
0
Appendix: Derivation of Lagrange Polynomials which give us from Eq. (A2)
for CGL Nodes
We derive the following expressionfor the Lagrange polynomials w 0 .1/ D 2N 2 ; w0 .¡1/ D 2.¡1/ N N 2
of order N interpolating a function at the CGL nodes:
Finally, we have
P
.¡1/k C 1 .1 ¡ t 2 / TN .t/
Ák .t/ D w.t/ 1 .t 2 ¡ 1/TN .t / .¡1/k
0
N k
2c .t ¡ tk / Ák .t / D D
t ¡ tk w k
0 .t / .t ¡ tk / N 2 ck
where 0
» .1 ¡ t 2 /TN .t / .¡1/k C 1
k D 0; N D
2; .t ¡ tk / N 2 ck
ck D (A1)
1; 1· k · N ¡1 where ck is as de ned in Eq. (A1).
TN .t/ is the Chebyshev polynomialof degree N , and tk are the CGL References
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