248
A. Riccardi et al.
The TPBVP defined by Eq. (7.9), coupled with the conditions (7.10) and (7.11),
has no analytical closed-form solution for complex problems Hence, numerical
methods shall be employed. However, further information can be obtained by
exploitation of the problem’s first integrals. If the functions L and f defined in
the System (7.6) do not depend explicitly on the independent variable t, then the
Hamiltonian is a first integral of the TPBVP along an optimal trajectory [71]. In
general, if a first integral is found, the redundant information that it generates can
be exploited to eliminate one adjoint equation, formally transforming the original
TPBVP into another one of lower dimension, by following the procedure shown by
Visser [67].
7.2.4.2 Direct Methods
A direct method does not require the derivation of the necessary conditions needed
by indirect methods. On the contrary, it aims to find a sequence of profiles which
progressively reduce the non-augmented performance index J and the constraint’s
violation. Direct methods require a parametrisation of the control functional form
over trajectory’s arcs. This is generally achieved by two conceptually different
methods [71]:
• A grid at different times where the control parameters are to be found and the
values within an interval are computed through interpolation.
• A set of orthogonal basis of mathematical functions dependent on time. Usually
Fourier series, Legendre polynomials or the Chebyshev ones.
The goal is then to determine the values of the specified free parameters, either
control values at fixed times in the grid form or the coefficients of the series in the
second case, able to minimise the objective index and to respect the constraints. In
this step, the number of free parameters is reduced from infinite degrees of freedom
to a finite number of parameters, depending on the chosen parametrisation. This
passage could seem a limitation of the direct methods when compared to the indirect
ones. However, as already stated in the previous section, a numerical procedure
is necessary also for indirect methods when dealing with complex cases such as
low-thrust trajectory optimisation. These numerical methods require a so-called
transcription to convert the infinite-dimension optimal problem into a solvable
finite-dimension one. Hence, what seemed a limitation of the direct methods is a
required passage of any technique nonetheless.
A direct method’s solution is generally not an optimal solution itself, i.e.
not a local minimum of the performance index, but just an approximation as
a consequence of the discretisation or interpolation steps. Hence, the necessary
conditions (7.9) and (7.11) can be used as an indicator of how close the found
solution is to the real local optimum [72].
A. Riccardi et al.
The TPBVP defined by Eq. (7.9), coupled with the conditions (7.10) and (7.11),
has no analytical closed-form solution for complex problems Hence, numerical
methods shall be employed. However, further information can be obtained by
exploitation of the problem’s first integrals. If the functions L and f defined in
the System (7.6) do not depend explicitly on the independent variable t, then the
Hamiltonian is a first integral of the TPBVP along an optimal trajectory [71]. In
general, if a first integral is found, the redundant information that it generates can
be exploited to eliminate one adjoint equation, formally transforming the original
TPBVP into another one of lower dimension, by following the procedure shown by
Visser [67].
7.2.4.2 Direct Methods
A direct method does not require the derivation of the necessary conditions needed
by indirect methods. On the contrary, it aims to find a sequence of profiles which
progressively reduce the non-augmented performance index J and the constraint’s
violation. Direct methods require a parametrisation of the control functional form
over trajectory’s arcs. This is generally achieved by two conceptually different
methods [71]:
• A grid at different times where the control parameters are to be found and the
values within an interval are computed through interpolation.
• A set of orthogonal basis of mathematical functions dependent on time. Usually
Fourier series, Legendre polynomials or the Chebyshev ones.
The goal is then to determine the values of the specified free parameters, either
control values at fixed times in the grid form or the coefficients of the series in the
second case, able to minimise the objective index and to respect the constraints. In
this step, the number of free parameters is reduced from infinite degrees of freedom
to a finite number of parameters, depending on the chosen parametrisation. This
passage could seem a limitation of the direct methods when compared to the indirect
ones. However, as already stated in the previous section, a numerical procedure
is necessary also for indirect methods when dealing with complex cases such as
low-thrust trajectory optimisation. These numerical methods require a so-called
transcription to convert the infinite-dimension optimal problem into a solvable
finite-dimension one. Hence, what seemed a limitation of the direct methods is a
required passage of any technique nonetheless.
A direct method’s solution is generally not an optimal solution itself, i.e.
not a local minimum of the performance index, but just an approximation as
a consequence of the discretisation or interpolation steps. Hence, the necessary
conditions (7.9) and (7.11) can be used as an indicator of how close the found
solution is to the real local optimum [72].