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A. Riccardi et al.
the following techniques are applicable to both indirect and direct approaches. A
complete review of the common methods, with a focus on low-thrust trajectory
optimisation, has been compiled by Betts [76], whereas in this section two major
classes will be addressed.
Single Shooting
Typically, the single shooting method does not actually find application in the
field of complex non-linear optimal control. However, it is useful to introduce
the notation and several concepts shared by its extension, the multiple shooting
method. The discretisation grid is composed by only two points, the initial and
final times. Initially, the n free parameters in y T = [¯ x 1 , . . . , ¯
u n c ], composed by
the initial conditions and the control parameters, are guessed. Hence, the trajectory
is propagated forward (or equivalently backward) from the starting to the end time,
leading to the final state:
x
p
f = x 0 +
t f
t 0
f(t, x, u)dt
In general the propagated state x
p
f will not coincide with the required final one x F .
Hence, the difference between these two quantities becomes a constraint to nullify.
In literature, this constraint is generally labelled as defect:
c(y) = x
p
f − x f
(7.12)
The numerical values of the violation of the boundary conditions can be exploited
to iteratively adjust the control parameters with NLP algorithms in order to finally
solve the constrained minimisation.
The advantage of this basic method is that the NLP subproblem has only a small
number of variables to optimise, i.e. the initial state guess and control parameters.
However, for long time-scales and non-linear dynamics, even small changes in the
parameters can result in very large defects change, leading to hypersensitivity with
respect to the free parameters.
Multiple Shooting
In order to overcome the drawback of parameter sensitivity, it is possible to segment
the overall time interval into a set of m−1 smaller steps discretising the interval at m
grid points t 0 < t 1 < t 2 < · · · < t f . Then, each of the segments can be treated as an
independent single shooting method, with continuity constraints added. Therefore,
first guesses of the n s state variables for each intermediate segment are now needed.
The first guess trajectory is usually found by fast and low-fidelity methods. The
A. Riccardi et al.
the following techniques are applicable to both indirect and direct approaches. A
complete review of the common methods, with a focus on low-thrust trajectory
optimisation, has been compiled by Betts [76], whereas in this section two major
classes will be addressed.
Single Shooting
Typically, the single shooting method does not actually find application in the
field of complex non-linear optimal control. However, it is useful to introduce
the notation and several concepts shared by its extension, the multiple shooting
method. The discretisation grid is composed by only two points, the initial and
final times. Initially, the n free parameters in y T = [¯ x 1 , . . . , ¯
u n c ], composed by
the initial conditions and the control parameters, are guessed. Hence, the trajectory
is propagated forward (or equivalently backward) from the starting to the end time,
leading to the final state:
x
p
f = x 0 +
t f
t 0
f(t, x, u)dt
In general the propagated state x
p
f will not coincide with the required final one x F .
Hence, the difference between these two quantities becomes a constraint to nullify.
In literature, this constraint is generally labelled as defect:
c(y) = x
p
f − x f
(7.12)
The numerical values of the violation of the boundary conditions can be exploited
to iteratively adjust the control parameters with NLP algorithms in order to finally
solve the constrained minimisation.
The advantage of this basic method is that the NLP subproblem has only a small
number of variables to optimise, i.e. the initial state guess and control parameters.
However, for long time-scales and non-linear dynamics, even small changes in the
parameters can result in very large defects change, leading to hypersensitivity with
respect to the free parameters.
Multiple Shooting
In order to overcome the drawback of parameter sensitivity, it is possible to segment
the overall time interval into a set of m−1 smaller steps discretising the interval at m
grid points t 0 < t 1 < t 2 < · · · < t f . Then, each of the segments can be treated as an
independent single shooting method, with continuity constraints added. Therefore,
first guesses of the n s state variables for each intermediate segment are now needed.
The first guess trajectory is usually found by fast and low-fidelity methods. The