Soft Comput (2017) 21:4583–4592
Reentry trajectory optimization for hypersonic vehicle based on
improved Gauss pseudospectral method
· Duolin Zhang
· Lu Wang
Published online: 1 June 2016
© Springer-Verlag Berlin Heidelberg 2016
Abstract Over the past decade, Gauss pseudospectral met-
hod (GPM) has been widely used to deal with the reentry
trajectory optimization problems for hypersonic vehicle.
However, for the trajectory generated by GPM, all constraints
can only be satisﬁed at each Gauss point. To overcome this
problem, in this paper, an improved GPM is proposed. Two
kinds of newly deﬁned breaks are applied to divide the tra-
jectory into multiple segments, based on the distribution
character of the Gauss points; the waypoints are transformed
into Gauss points and dense Gauss points can locate near
the appointed areas such as the no-ﬂy zones; in this way,
all the positions can meet the requirements of all the com-
plex constraints. Furthermore, because the function of the
breaks is to alter the distribution of the whole Gauss points
and the number of them does not increase, the convergence
speed advantage of GPM is not changed. Simulation results
demonstrate that the proposed method can rapidly generate
a trajectory with high precision.
Keywords Hypersonic vehicle · Reentry trajectory
optimization · Gauss pseudospectral method · Waypoint ·
Communicated by A. Di Nola.
School of Air and Missile Defense, Air Force Engineering
University, Xi’an 710051, China
Various hypersonic and reentry vehicle technologies are
being pursued to support the US Air Force’s global reach con-
cept (Woolf 2007). Because of the hypersonic vehicle (HV)’s
ability of reentering and gliding without power through the
atmosphere to attack the ground target depending on aerody-
namic control (George 1999), reentry trajectory optimization
method for HV, which plays a key role in steering an efﬁcient
and safety ﬂight in the complex environment (Zhao et al.
2014), has been widely concerned.
Of many numerical methods, Gauss pseudospectral met-
hod (GPM) has been demonstrated as one of the most
convenient tools and is widely used to deal with the ascent
or reentry trajectory optimization problem (Tian and Zong
2012; Zong et al. 2010; Zhang et al. 2012; Rao and Clarke
2002; Darby et al. 2011; Zhang and Li 2013; Tang et al.
2013; Cao and Ge 2013) subjecting to many “hard” con-
straints such as dynamic pressure, aerodynamic load and
heating rate (Zhao et al. 2014). However, with the environ-
ment in modern warfare more complicated, more complex
practical factors are inevitable to be taken into account, such
as waypoints, which are positions to ﬂyover to satisfy pay-
load delivery or reconnaissance mission requirements, and
the no-ﬂy zones, which are regions with a boundary that the
vehicle may contact, but must not violate for threat avoidance
(Xie et al. 2013).
In GPM, a series of discrete points called Gauss points are
used to discretize the optimal control problem (OCP) (Huang
et al. 2012), so the ﬁnal solution meets the requirements of
all constraints at each Gauss point, but its feasibility in other
positions cannot be sure. As a result, when the waypoint and
no-ﬂy zone constraints are taken into consideration, some
problems appear as below.