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Aerospace Science and Technology 119 (2021) 107092
Contents lists available at ScienceDirect
Aerospace Science and Technology
www.elsevier.com/locate/aescte
A Control Allocation approach to induce the center of pressure
position and shape the aircraft transient response
Carmine Varriale a,∗, Mark Voskuijl b
aFaculty of Aerospace Engineering, Delft University of Techno log y, Delft, the Netherlands
bFaculty of Military Sciences, Netherlands Defence Academy, Den Helder, the Netherlands
a r t i c l e i n f o a b s t r a c t
Article history:
Received 17 December 2020
Received in revise d form 14 July 2021
Accepted 31 Augu st 2021
Avail abl e online 9 September 2021
Communicated by Christian Circi
Keywords:
Flight mechanics
Control Allocation
Direct Lift Control
Box-wing
This paper presents a Control Allocation formulation aimed at altering the dynamic transient response
of an aircraft by exclusive means of the aerodynamic effectiveness of its control effectors. This is done,
for a given Flight Control System architecture and, optionally, closed-loop performance, by exploiting the
concept of Control Center of Pressure, i.e. the center of pressure due to only aerodynamic control forces.
Two formulations are proposed, and their advantages and disadvantages presented. The first is based on
the straightforward augmentation of the control effectiveness matrix, the second on a weighting matrix to
prioritize control effectors. The latter is implemented in three application studies on a box-wing aircraft
configuration with redundant control surfaces: a simple pull-up maneuver, a trajectory tracking task,
and an altitude holding task in turbulent atmosphere. Results show that the proposed formulation can
significantly impact performance metrics that are closely related to the aircraft transient response. In the
best case scenario, the aircraft is able to completely cancel the non-minimum phase behavior typical of
pitch dynamics, hence achieving a sharp initial response to longitudinal commands. If compared to a
standard Control Allocation algorithm, the proposed formulation results in improved tracking precision,
better disturbance rejection, and a measurably improved feeling of comfort on board.
©2021 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY
license (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Disruptive aircraft configurations are becoming increasingly
popular in modern research studies, from future commercial con-
cepts to unmanned aircraft systems applications. This is due to
the inherent performance benefits that they may allow to obtain,
but also to their potential capability of reshaping the aeronauti-
cal sector in a more profound way. Examples are represented by
the Blended Wing Body configuration [1], the more recent Flying-
V concept [2], and a wide range of non-planar wing geometries.
While the concepts proposed in this paper are applicable to any
aircraft configuration, the present investigation focuses on a tran-
sonic commercial transport box-wing aircraft model, referred to as
the PrandtlPlane (PrP) and shown in Fig. 1.
Brought to fame by an intuition of Ludwig Prandtl [3], the box-
wing has been proven to generate the least induced drag for a
given span and lift [4]. This property has constituted the scientific
ground of several engineering research efforts, aimed at integrating
its complex geometry in complete aircraft and compound rotorcraft
*Corresponding author.
E-mail addresses: C.Varriale@tudelft.nl (C. Varriale),
M.Voskuijl@mindef.nl
(M. Voskuijl).
architectures [5–7]. The unique aerodynamic properties of the box-
wing allow the PrP to be competitive in the modern commercial
aviation market [8]. Additionally, the possibility to install redun-
dant control surfaces, inboard and outboard on both the front and
rear wings, allows the PrP to make use of unconventional piloting
techniques such as Direct Lift Control (DLC) [9,10].
DLC is defined as the capability to use control effectors to di-
rectly control the aircraft lift. This is a common technique for
helicopter pilots, for example, who are able to control vertical
dynamics directly through the collective command. Vertical con-
trol of conventional airplanes, instead, revolves around the use of
a tail elevator to generate a small, dislocated control lift. While
this lift contribution is generally small, it produces a significant
pitch moment and gives raise to some angle of attack dynamics.
This very indirect control technique results in the classic, unde-
sired non-minimum phase behavior of pitch dynamics, with the
initial aircraft response (due to control effectors dynamics) being
opposite to the much larger steady-state response (due to angle of
attack dynamics).
With respect to symmetric motion in the vertical plane, the
most complete fundamental analysis about DLC shows that its per-
formance mainly depends on the longitudinal position of the Con-
trol Center of Pressure (CCoP) [11]. This is the center of pressure of
https://doi.org/10.1016/j.ast.2021.107092
1270-9638/©2021 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
Nomenclature
Symbols
ageneric acceleration........................... m/s
2
αangle of attack .................................. rad
breference wingspan............................... m
Bcontrol effectiveness matrix.................. 1/rad
βangle of sideslip................................. rad
cmean aerodynamic chord ........................ m
C(·)generic non-dimensional coefficient............... -
δ,δcontrol effectors actual displacements.......... rad
ggravitational acceleration...................... m/s
2
γflight path angle................................. rad
haltitude........................................... m
Iidentity matrix..................................... -
Jyy moment of inertia about the yBaxis........ kgm
2
Kcontroller gain..................................... -
κmatrix condition number.......................... -
L,M,Nroll, pitch, yaw moments....................... Nm
Mgeneralized forces and moments......... N or Nm
mmass.............................................. kg
MMach number...................................... -
nload factor......................................... -
Ngeneric quantity................................... -
ν,νgeneric control allocation objectives
p,q,rroll, pitch, yaw rotational speeds ............. rad/s
q∞asymptotic dynamic pressure................. N/m
2
Sreference surface area........................... m
2
ttime................................................ s
θangle of elevation............................... rad
u,ucontrol effectors ideal displacements........... rad
Vairspeed......................................... m/s
Wuweighting matrix.................................. -
xB,yB,zBlongitudinal, lateral, normal body axes
Znormal force in body axes........................ N
Subscripts and superscripts
cr cruise
des desired
MTO maximum at take-off
pax passengers
pk peak
ref reference
tr trim
Abbreviations
AMS Attainable Moment Set
CA Control Allocation
DA Direct Allocation
CCoP Control Center of Pressure
CG Center of Gravity
DLC Direct Lift Control
FCS Flight Control System
ICR Instantaneous Center of Rotation
LAMS Largest Attainable Moment Set
NDI Non-linear Dynamic Inversion
PI Pseudo Inverse
PrP PrandtlPlane
RMS Root Mean Squared
WPI Weighted Pseudo Inverse
Fig. 1. The Prandtlplane aircraft configuration with deflected control surfaces.
the aerodynamic forces generated solely by displacing the aircraft
control effectors. Without focusing on any specific aircraft config-
uration or practical implementation of DLC, the derivation in [11]
shows the theoretical variation of the load factor time response as
a function of the position of CCoP. For example, conventional pitch
control, obtained with a single control force very far aft the air-
craft Center of Gravity (CG), is characterized by the CCoP roughly
coinciding with the location of the control effector itself. As the
CCoP moves fore of the aircraft aerodynamic center, the initial
and steady-state load factor responses are concordant in sign. In
particular, if the CCoP is fore of the aerodynamic center by the
same distance the maneuver point is aft of the CG, the load factor
steady-state response is theoretically equal to the initial one. This
is referred to as “Pure DLC” [11], since the capability to generate
lift with angle of attack dynamics is not exploited in this case.
In the most extreme case of Pure DLC, it should be easy to
understand how enabling this control technique would allow the
pilot to have precise and almost instantaneous control of the air-
craft lift. Even if at the cost of limited control power, DLC can then
be beneficial in all flight scenarios that require maneuver accu-
racy and response quickness, such as precision landing or obstacle
avoidance tasks, for example.
Existing applications of some forms of DLC can be traced to
the use of spoilers and flaps [12–14], although the latter cannot
be used as control effectors and hence have no role in maneuver-
ing flight. With a more original approach, an interesting research
study evaluates DLC performance by means of classic and newly
proposed handling qualities criteria, for two conventional aircraft
models, as a function of the gearing ratios used to gang control
effectors together [15].
Gearing and ganging control effectors is the most straightfor-
ward way to constrain their relative motion. This clearly has an
impact on the position of the CCoP, and hence on the type of tran-
sient response that can be achieved in maneuvering flight. On the
other hand, gearing ratios and ganging matrices need to be se-
lected a priori and somewhat arbitrarily, and usually need to be
optimized for different flight scenarios. This clearly hinders the po-
tential range of achievable aircraft dynamic responses.
A more advanced approach to calculate the control effectors
position required to perform a given maneuvering task is repre-
sented by Control Allocation (CA) methods [16]. These methods
exploit the aerodynamic effectiveness of each effector to achieve
a given control objective while satisfying some assigned optimality
criterion. Among the many CA formulations available in literature,
two well-known approaches, with very different characteristics and
performance, are the Weig hted Pseudo Inverse (WPI) method and
the Direct Allocation (DA) method.
2
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
Tabl e 1
Top-le ve l design parameters of the PrandtlPlane.
b36.0 m
c4.31 m
S266.7 m2
hcr 11 km
Mcr 0.79
Npax 308
mMTO 122 ×103kg
Many research works have already focused on evaluating the
practical implications of performance differences between these,
and other, CA algorithms. A detailed evaluation of the numeri-
cal performance of four classic formulations is presented in [17].
A comprehensive study, relying on a high-fidelity wind-tunnel
database for a Blended Wing Body aircraft model, estimates the
impact of classic CA methods on trim drag and control surface
design [18,19]. A recent, innovative research work defines and ex-
ploits the concept of robust Attainable Moment Set (AMS), with
applications to fighter aircraft with uncertainties in their control
effectiveness [20]. In light of its independence from any itera-
tive procedure, the WPI method has been implemented in a dis-
tributed CA scheme for spacecraft attitude stabilization [21]. Sev-
eral research studies have also proposed modifications of classic CA
methods to solve specific engineering problems: from a multi-step
DA method to minimize drag [22], to the inclusion of aerodynamic
interactions among the effectors [23]; from a multi-objective CA
formulation aimed at minimizing structural loads due to control
efforts [24], to gust load alleviation by means of CA of the lift and
pitch moment coefficients [25].
The objective of the present work is to develop a CA formula-
tion which is able to alter the dynamic transient response of an
aircraft. This is achieved for a given Flight Control System (FCS) ar-
chitecture and, optionally, tuning, by exclusive means of the aero-
dynamic effectiveness of the aircraft control effectors.
The following Section 2.1 starts by illustrating the aircraft and
flight mechanics model implemented for the present study. Sec-
tion 2.2 goes into more detail on the employed FCS architecture, as
well as on the procedure implemented to tune it. A brief technical
overview of the classic CA problem is then outlined in Section 2.3,
and the proposed novel CA formulation is then presented in Sec-
tion 2.4. Three relevant application studies are presented in Sec-
tion 3, with results and discussion. Lastly, conclusions are drawn
in Section 4, with an outlook on future research possibilities.
2. Methodology
2.1. Aircraft model
The PrP concept under consideration in the present article has
been designed for short and medium range flights within the PAR -
SIFAL (Prandtlplane ARchitecture for the Sustainable Improvement
of Future AirpLanes) research project. Some relevant design param-
eters are reported in Table 1.
Its aerodynamic characteristics have been obtained using the
commercial off-the-shelf panel code VSAERO [26]. Each of the
aerodynamic forces and moments acting on the aircraft is ex-
pressed as a tabular function of flight parameters and control
surface deflections δ. The resulting aerodynamic model, assuming
partial superposition of effects, is reported in Equation (1).
CM=CM0(α,β,M,δ=0)
steady, clean
+
Nδ
i=1
CM(α,β,M,δ
i)
steady, control effectors
+
ω=p,q,r
CMω(α,β,M,δ=0)ω
unsteady, clean
(1)
The propulsive model of its two engines has been generated with
an in-house, physics-based, simulation toolbox [27]. It expresses
thrust and fuel consumption as a tabular function of altitude, Mach
number and throttle, and has been validated in a previous study
about mission performance of the PrP [8].
Togethe r with a FCS architecture, which is described in more
detail in the following section, the aerodynamic, propulsive and
mass databases are merged into a consistent flight mechanics
model, within the framework of the Performance, Handling Qual-
ities and Load Analysis Toolbox (PHALANX). This software suite,
written in MATLAB®and Simulink®, revolves around a Simscape
Multibody Dynamics core to perform non-linear flight simulations.
It relies on a modular, physics-based and configuration-agnostic
architecture, which is able to operate consistently with different
levels of input fidelity. Thanks to ts capability to support automatic
aircraft design workflows, PHALANX has been used in a number
of research studies and applications on different aircraft configu-
rations [8–10,28–31]. A block-scheme overview of the toolbox is
shown in Fig. 2.
For the present study, symmetric flight has been imposed by
constraining the aircraft model to have only three degrees of free-
dom in the vertical plane. The normal load factor in body axes has
been defined as
nzB=−ZB
mg (2)
in order to have nzB≈1in straight and level flight.
2.2. Flight Control System architecture
The implemented FCS architecture is reported in Fig. 3. It con-
sists of a simple airspeed hold, employing the throttle command,
and a longitudinal control law based on Non-linear Dynamic In-
version (NDI). The latter technique allows to neatly separate the
control law from the CA components, and to use classic methods
from linear control theory for tuning the controller gains. Control
inputs for the latter are provided either from the pilot stick, with
a pitch rate response type, or from an altitude controller.
The altitude channel employs a series of linear controllers and
transformations to achieve stable and robust augmented dynam-
ics [32]. The transformation between the reference vertical acceler-
ation
¨
hdes and the normal load factor assumes there is no variation
in airspeed and is expressed in Equation (3).
¨
h=gnzBcos θ−1(3)
The transformation between the load factor and the desired an-
gle of attack αdes makes use of the linear approximation of the
lift curve in body axes at trim conditions, and is reported in
Equation (4). The commanded angle of attack is clipped between
−5deg and 5deg to prevent the run-time values of αfrom ex-
ceeding the boundaries of the underlying aerodynamic dataset.
q∞SCtr
Z0+Ctr
Zαα−αtr=−
mgnzB(4)
Lastly, the transformation between the reference angle of attack
rate ˙
αdes and the commanded pitch rate qcmd is expressed in Equa-
tion (5).
q=˙
α−g
Vcos θ−ntr
zB(5)
3
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
Fig. 2. Block-scheme overview of PHALANX.
a) Altitude control channel
b) Main flight control law
Fig. 3. Block scheme overview of the chosen FCS architecture.
Fig. 4. Baseline architecture for all controllers shown in Fig. 3.
Each of the controllers shown in Fig. 3has the baseline archi-
tecture reported in Fig. 4, consisting of two cascaded proportional-
integral control loops, with a parallel feedthrough branch for im-
proved tracking response [32]. For each controller, the feedthrough
gain fraction Kfis assigned manually in order to obtain desired
closed-loop characteristics. The proportional gains K1and K2are
left to be determined by an automatic tuning procedure. This is
formulated as an optimization problem, with the objective to find
the values of the proportional gains that minimize the difference
between the closed-loop dynamics of the linear aircraft model and
an assigned reference model. Without loss of generality, the latter
has been arbitrarily chosen as follows:
•the reference altitude dynamics is a critically damped second
order system with a time constant of 2s;
•the reference airspeed dynamics is a first order system with a
time constant of 3s.
The proprietary Control System Tuner algorithm by MathWorks®
has been used to solve the optimization problem and find the
tuned values of the gains for every case study presented in the
present paper.
In the case of symmetric flight with three-degrees of freedom,
the classic NDI of aircraft rotational dynamics resolves to the sim-
ple scalar equation reported in Equation (6), where νqis the non-
dimensional pitch control moment required to obtain the desired
pitch acceleration ˙
qdes.
Jyy ˙
qdes =Mtr
q∞S
c+νq(6)
The pitch control moment νq, together with null control mo-
ments about the roll and yaw axes, is then allocated to the effec-
tors by solving an appropriate CA problem based on Equation (7).
Bu=ν⇔⎡
⎣
BL
BM
BN
⎤
⎦u=⎧
⎨
⎩
0
νq
0
⎫
⎬
⎭
(7)
The notation, explained in the following Equation (8), has been
chosen to highlight to contribution of each row of the Bmatrix,
and is going to be used in the next section, covering the proposed
CA formulation.
BH=∂CH
∂u1
∂CH
∂u2
··· ∂CH
∂unfor H=L,M,N(8)
2.3. Overview of the classic Control Allocation problem
The baseline, generic CA problem consists in finding the value
of the control effectors displacements uwhich solves the following
Equation (9).
Bu=ν(9)
In this simple relation, νis a vector of objectives to be achieved by
means of displacement of the control effectors u. Reference values
for νare typically prescribed by the FCS and can represent control
forces and moments or rotational rates, in the most common appli-
cations. Bis the control effectiveness matrix, which is not square
4
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
and cannot be inverted in case of redundant effectors. CA meth-
ods define an analytic or algorithmic function f, which allows to
express the control effectors displacements as
u=f(B,ν,...). (10)
The WPI method finds the required effectors displacements by
solving the following optimization problem:
min
u
Wu(u−udes)2
s.t. Bu−ν=0
(11)
where Wuis a weighting matrix used to prioritize the effectors,
and udes is a preferred effectors displacement. This formulation ad-
mits the following analytic solution [16]
u=udes +W−1
uBTBW−1
uBT−1
(ν−Budes)(12)
which makes the WPI method very robust in every applica-
tion. However, this comes at the price of sub-optimal allocation
performance, as the AMS of the WPI method is significantly
less extended than the aircraft Largest Attainable Moment Set
(LAMS) [33]. This means that a large set of prescribed objectives
ν, which would be theoretically achievable in light of the control
effectiveness of the aircraft B, are not practically attainable, be-
cause the WPI algorithm is not capable of mapping them to an
admissible set of control effectors positions u.
On the other hand, the DA method relies on the geometric rep-
resentation of the AMS itself, and is hence capable of achieving all
of the prescribed objectives within the LAMS [33]. DA is usually
formulated as the optimization problem shown in Equation (13),
making use of the auxiliary variable w≡u.
max
ρ,wρ
s.t. Bw=ρν
umin <w<umax
⎫
⎪
⎪
⎬
⎪
⎪
⎭
⇒u=w/ρ,if ρ>1
u=w,if ρ≤1(13)
In its most computationally efficient formulation, DA is cast as
a Linear Programming problem, hence relying on an iterative al-
gorithm for its solution [17]. The DA formulation takes into ac-
count effectors saturation limits and preserves direction in Mo-
ment Space for unattainable desired moments [33], but does not
allow any prioritization of effectors via weighting matrices.
2.4. Novel Control Allocation formulation
The novel CA formulation revolves around the mathematical
definition of the CCoP. In the scope of symmetric flight in a ver-
tical plane, its non-dimensional longitudinal position in body axes
is calculated as in Equation (14)[11].
xδ=xδ
c=−
Nδ
i=1
CMi
Nδ
i=1
CZi
≈−
Nδ
i=1
CMδiδi
Nδ
i=1
CZδiδi
=−BMδ
BZδ(14)
For an alternative interpretation of this quantity from a flight dy-
namics perspective, it is useful to elaborate on the previous expres-
sion in the case of a single control surface, e.g. an elevator. In this
case,
xδcan be simply expressed as the ratio between the pitch
moment and normal force control derivatives, and coincides with
the position of the elevator itself. Additionally, it is also strictly re-
lated to the Instantaneous Center of Rotation (ICR) of the aircraft,
which is calculated as in Equation (15)in the scope of a linear
dynamic formulation [34].
xicr =Jyy
m
c2
CZδe
CMδe
=− Jyy
m
c2
1
xδ
(15)
In light of this, it should be evident how the position of the
CCoP is capable to substantially affect the flying qualities of the
aircraft in the pitch axis [35]. For a linear dynamic model, it can be
derived from the previous equation that the product between the
positions of the CCoP and the ICR must be constant. If the elevator
is very far aft of the aircraft CG, the ICR falls relatively close to it.
In other words, the impact of the normal control force is negligi-
ble when compared to the pitch control moment, and the aircraft
motion resembles a pure rotation about its CG. On the other hand,
if the CCoP tends to the aircraft CG, the ICR moves infinitely away
from it, and the aircraft motion tends to a pure normal translation.
These types of deductions, obtained in the simplified case of
a single control effector and linear dynamics, motivate the effort
presented in the current paper. The objective of the present inves-
tigation is to shape the transient response of the aircraft by means
of CA methods. This is achieved by driving the CCoP towards a pre-
scribed reference location
xref , for which the transient response is
known to have desired characteristics, and translates into the fol-
lowing Equation (16).
xδ=−BMδ
BZδ→
xref ⇔BZ
xref +BMδ
BZδ→0 (16)
Assuming that there exists at least a combination of
xref and δ
which verifies this limit, i.e. that control effectors can physically
drive the CCoP to its desired location, Equation (16) becomes an
equality. Furthermore, for any realistic maneuvering scenario, i.e.
assuming that δ= 0, the latter reduces to its numerator as shown
in Equation (17).
BZ
xref +BMδ=0 (17)
The last equation can be re-written in matrix form as
xref 1BZ
BMδ=XB∗δ=0 (18)
where X=[
xref 1]and B∗=BZBMT.
From a geometric point of view, Equation (18)identifies a hy -
perplane in Control Space, which passes through the origin, and
whose orientation depends on the value of
xref . This linear con-
straint obviously maps to a much smaller AMS than the one ob-
tained when all effectors are free to move independently. From a
more technical standpoint, Equation (18)can be regarded in two
alternative ways, which give life to two different approaches to ex-
ploit it.
2.4.1. Control effectiveness matrix augmentation
The first, and probably most straightforward, approach sparks
out of the interpretation of Equation (18)as a CA problem itself,
where the prescribed objective is equal to zero, and the desired
position of the CCoP acts as a weight to prioritize the generation
of control lift over control pitch, or vice-versa. The combined ef-
fectiveness XB∗can then be used to augment the Bmatrix of any
standard CA problem formulation, as shown in the following Equa-
tion (19).
Bu=ν−→ B
XB∗u=ν
0(19)
5
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
−5−4−3−2−1012345
10
15
20
25
30
xref
κ
XB∗u=0
BZ
xref u=−νq
Fig. 5. Condition number of the augmented control effectiveness matrix, for two
augmentation approaches.
In this way, Equation (18) has been directly injected into the CA
problem, and has been given the same dignity as the other equa-
tions for the allocation of prescribed objectives. This approach can
be applied to any existing CA method, including DA, as it only
needs the presence of a baseline control effectiveness matrix in
the problem formulation.
On the other hand, this approach presents a major drawback:
the condition number κof the augmented effectiveness matrix
increases abruptly as the CCoP tends to the aircraft CG, until di-
verging completely at the latter position. This is due to the fact
that the pitch moment row BMand the extra row XB∗of the
augmented effectiveness matrix become more and more similar as
xref approaches zero. From a physical point of view, this happens
because the pitch effectiveness of the control effectors is used to
allocate both the demanded pitch moment νqand the null extra
objective required to prioritize the effectors.
The numerical conditioning of the augmented effectiveness ma-
trix can be slightly improved by additionally assuming that the
CA algorithm always converges, i.e. that the equality for the pitch
moment BMu=νqis verified at all times. This is not true for com-
manded control moments outside of the AMS, for example. With
this additional hypothesis, Equation (17)can be re-written as in
Equation (20), and the augmented problem takes the shape pre-
sented in Equation (21).
BZ
xref u=−νq(20)
Bu=ν−→ B
BZ
xref u=ν
−νq(21)
As shown in Fig. 5, also in this case the augmented effec-
tiveness matrix becomes ill-conditioned for
xref →0. In light of
this, this approach would not be suitable for any practical im-
plementation of the proposed CA formulation. It is therefore not
implemented in any of the applications presented in the remain-
der of this paper. The more robust weighted prioritization method
illustrated in the next section does not present any numerical in-
stabilities, and has been used to obtain the results presented in the
remainder of the article.
2.4.2. Weig hted prioritization
Equation (18)can also be interpreted as a particular case of
the classic effector prioritization expression, shown in the follow-
ing Equation (22), used in all CA methods based on quadratic-
programming algorithms.
XB∗u⇔Wu(u−udes )(22)
The equivalence can be easily achieved by imposing the preferred
effectors position as null, and placing the combined effectiveness
vector on the diagonal of the weighting matrix, as shown in Equa-
tion (23).
udes =0Wu=diag XB∗(23)
This approach does not present any numerical conditioning issues.
On the other hand, it can only be applied to CA problems which
allow some form of effectors prioritization, such as those employ-
ing the WPI method. In case
xref coincides with the CG, control
effectors are simply prioritized according to their pitch moment
effectiveness BM.
A WPI method based on this particular formulation is going to
be employed for all the applications proposed in the following Sec-
tion 3. The control surface deflections solving such CA problem are
obtained by substituting the expressions reported in Equation (23)
into Equation (12).
3. Applications and results
Three study cases have been performed to explore the flight
mechanics possibilities of the PrP. In all cases, the performance of
the novel CA formulation presented in the previous Section 2.4.2
is evaluated as a function of the prescribed position of the CCoP
and compared against the standard Pseudo Inverse (PI) formula-
tion. The latter is simply equivalent to the WPI approach with the
effectors not being weighted, i.e. Wu=I.
As explained in the next section, the aircraft is trimmed using
an iterative methodology based on DA and the concept of AMS. For
all the subsequent applications, the control effectiveness matrix B
is calculated at trim conditions and is held constant throughout
each flight simulation. Actuators are modeled as first order sys-
tems with a time constant of 0.1 s and a rate limit of ±45 deg/s.
Control surface deflections are saturated at ±30 deg. The FCS is
re-tuned for each prescribed value of
xref , using the automatic pro-
cedure described in Section 2. However, if the FCS is tuned using
the standard PI method and then left unaltered when using the
modified CA approach, results are substantially not affected and
conclusions unhindered.
The reference trim condition is briefly illustrated in the follow-
ing Section 3.1. Section 3.2 presents a simple pull-up maneuver
with detailed analysis of time histories of the normal load factor
and control surface deflections. In Section 3.3, an altitude tracking
maneuver is analyzed. This is performed by closing the altitude
channel switch in Fig. 3b and prescribing a reference altitude pro-
file. In a similar fashion, Section 3.4 presents an altitude holding
task in turbulent atmosphere, with the estimation of a quantita-
tive index of the comfort level on board.
3.1. Trim condition
The aircraft model is trimmed in straight and level flight using
the CA-based methodology presented in [10]. With this approach,
the resulting control surface deflections are not constrained by
an imposed ganging and gearing kinematic chain, but are inde-
pendently set to obtain the maximum balanced control authority
about the lift and pitch axes. The standard DA method of Equa-
tion (13)is
used in this case, because of its properties concerning
the AMS. The Linear Programming formulation provided by [17]is
implemented. The AMS geometry at trim, and the combination of
trim control forces generated by the effectors are shown in Fig. 6a.
Trim control surface deflections are reported in Fig. 6b. The aircraft
is trimmed at Vtr =170 m/s at sea level altitude.
6
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
−0.4−0.3−0.2−0.10 0.10.20.30.4
−1
−0.5
0
0.5
1
CZ
CM
AMS boundary
Trim targe t
Trim point
a) Control forces and AMS geometry
Inner
Front
Port
Inner
Front
Starb.
Outer
Front
Port
Outer
Front
Starb.
Inner
Rear
Port
Inner
Rear
Starb.
Outer
Rear
Port
Outer
Rear
Starb.
−30
−25
−20
−15
−10
−5
0
5
δtr (deg)
b) Control surface deflections
Fig. 6. Reference trim condition: Vtr =170 m/s, αtr =−0.46 deg. Control surface deflections are bounded in the [−30,+30]deg interval.
11.21.41.61.82
1
1.1
1.2
1.3
1.4
1.5
PI
−5.0
−3.0
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+3.0
+5.0
t(s)
nzB
a) Time histories
PI
-5
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
5
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
xref
∂nzB/∂t|t=1s (1/s)
b) Time derivative at t=1s
Fig. 7. Normal load factor response after an impulsive pull-up maneuver, for different values of the reference CCoP location and the standard PI approach.
Trim control surface deflections, positive if with trailing edge
down, are all symmetric and contained in magnitude, apart from
the outboard ones on the rear wing, which are partially sacrificed
to obtain a very small trim angle of attack αtr =−0.46 deg. Rud-
ders have been explicitly excluded from the CA problem, as they
have a significant cant angle which makes them suitable for pitch
control as well. This was judged to be undesired in a conventional
flight control scenario.
3.2. Pull-up maneuver
A step pull-up maneuver is performed by prescribing a constant
pitch rate command through the pilot longitudinal control channel
δlon, for the duration of 10 s, with the altitude channel switch left
open, as shown in Fig. 3b. The initial instants of the arising load
factor response are reported in Fig. 7a, for several values of
xref
and the standard PI formulation. Similarly, the time derivative of
the load factor response at the start of the pilot maneuver is re-
ported in Fig. 7b and used as an indication of the sharpness of the
transient response.
The standard PI method results in the typical non-minimum
phase behavior of conventional aircraft configurations. Such re-
sponse corresponds to a CCoP far aft the aircraft CG, which would
be equivalent, in the modified CA formulation, to a value of
xref ap-
proximately equal to −2. A similar behavior is observed with the
modified CA approach, for values of
xref ranging from −5to about
−0.9.
By advancing the prescribed location of the CCoP, i.e. increasing
the value of
xref from −5to about 0.8, the initial decrease in load
factor is progressively reduced, neutralized and converted into a
sharp initial increase. The maximum response sharpness appears
to plateau for 0.4 <
xref <1, where the transient response clearly
shows the typical characteristics of DLC [11]. By further increasing
xref , the trend is reversed, and for
xref =5a small initial decrease
in the load factor is again observed.
7
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
11.52
−30
−20
−10
0
10
20
30
PI
t(s)
δ(deg)
11.52
−30
−20
−10
0
10
20
30
xref =−5
t(s)
Inner Front Outer Front Inner Rear Outer Rear
11.52
−30
−20
−10
0
10
20
30
xref =−0.8
t(s)
11.52
−30
−20
−10
0
10
20
30
xref =0.6
t(s)
δ(deg)
11.52
−30
−20
−10
0
10
20
30
xref =0.8
t(s)
11.52
−30
−20
−10
0
10
20
30
xref =1
t(s)
Fig. 8. Control surface deflections time histories after an impulsive pull-up maneuver, for different values of the reference CCoP location and the standard PI approach.
These results are consistent for different amplitudes and du-
rations of the commanded input, which may be representative of
different levels of pilot aggressiveness in performing the maneuver.
In this specific application, for 0 <
xref <1, the aircraft response is
so sharp that the commanded pitch rate is achieved before the end
of the pilot maneuver. This forces control surfaces to be deflected
abruptly back, to some extent, before reaching a steady state value,
and results in the acceleration cusp visible in the corresponding
curves of Fig. 7a.
The time histories of control surface deflections are shown in
Fig. 8for notable values of
xref . First, it can be seen how the
standard PI allocation mainly relies on the use of control surfaces
8
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
PI
-5
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
5
0.4
0.6
0.8
1
1.2
xref
t=t2−t1(s)
Level 1
a) Rise time
PI
-5
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
5
0.04
0.06
0.08
0.1
0.12
0.14
xref
t1(s)
Level 1
Level 2
b) Equivalent time delay
Fig. 9. Time-domain flying qualities of the short period pitch rate response q, for different values of the reference CCoP location and the standard PI approach.
on the rear wing. As expected, both the inner and outer rear ef-
fectors are deflected upwards in order to perform the prescribed
pull-up maneuver, while the inner front effectors are only slightly
activated downwards. This is in light of their lower pitch effective-
ness, which is due to their smaller distance to the aircraft CG w.r.t .
their counterpart on the rear wing. The outer front effectors are
left completely untouched.
Similar observations hold for the modified approach, in the case
of
xref =−5. Such high, negative value places even more emphasis
on the rear wing effectors, and in particular on their lift effective-
ness. In fact, outer rear effectors are now prioritized w.r. t. inner
ones, since they are closer to the CG and their pitch effectiveness
is lower. The former are deflected upwards until saturation, while
the latter are now deflected less than in the previous case. All front
effectors are unused.
In the case of
xref =−0.8, all effectors are used in a balanced
way, resulting in a smooth initial load factor response. Front ef-
fectors are deflected downwards, while rear effectors are deflected
upwards, so that the combined deflection generates a pure torque
about the aircraft CG, and the maneuver is substantially started by
α-generated lift.
For 0.6 <
xref <1, all the control effort is placed on front ef-
fectors, while rear ones are almost completely ignored. Both inner
and outer effectors on the front wing are deflected significantly
downwards, with the outer ones exhibiting larger deflections in
light of their smaller effectiveness. For
xref =0.8, inner effectors
are deflected less than in the case of
xref =0.6, while outer ones
are deflected up almost to the saturation limit. Both
xref =0.6 and
xref =0.8result in the highest response sharpness for the present
application, with the former value being able to obtain a larger
initial load factor excursion thanks to the more substantial use of
inner front effectors.
For
xref =1, inner front effectors are deflected more than in the
case of
xref =0.8, outer front effectors reach the same final value
as in the previous case, while a very slight upward deflection of
all the rear effectors is once again noticeable. By further advanc-
ing the reference CCoP location, control effectors deflections result
similar to the ones obtained for the
xref negative value that ex-
hibits a similar response, e.g.
xref =−0.6 and
xref =3, or
xref =−1
and
xref =5, as shown in Fig. 7a.
The rise time and equivalent time delay of the pitch rate re-
sponse are shown in Fig. 9. These are two classic flying qualities
metrics, used to characterize the short period pitch rate response
of the aircraft to pilot commands [35]. Time delay t1is defined as
the lapse between the step command and the instant when the
tangent line at the maximum pitch rate slope intersects the time
axis. Rise time t=t2−t1is defined as the lapse between t1and
the instant when the maximum pitch rate slope line intersects the
steady state value of the pitch rate for the first time. For the pre-
sented application, these metrics have been evaluated on the basis
of non-linear flight dynamics simulations.
As it can be seen in the figure, the metrics show a some-
what complementary behavior as a function of
xref . For basically
all values of
xref , the modified CA approach determines a consis-
tent improvement of the rise time with respect to the standard PI
method. All values are well within the Level 1 flying quality rat-
ing, and the minimum rise time is achieved for
xref =0.6. On the
other hand, a slight deterioration of the time delay can be seen for
nearly all values of
xref , with a peak increase for
xref =0.6. The lat-
ter is the only case for which the flying quality criterion results in
a Level 2 rating.
The selected metrics are deemed appropriate to characterize
the dynamic behavior of the augmented aircraft model, although
the criteria which prescribe their limit values have been devel-
oped for more conventional aircraft configurations. In particular,
concerning their suitability for the present application, it must be
reported that “several questions remain unresolved [...]. Effects of
pilot location and blended direct lift control have been observed
and need to be accounted for” [35]. This may constitute the object
of interesting future research studies.
3.3. Altitude shift maneuver
The present application has the objective of estimating the im-
pact of the modified CA approach on a practical performance met-
ric such as tracking precision. A 10 m square wave altitude profile
href(t)is prescribed for the aircraft to track. This is achieved by
closing the altitude channel switch in Fig. 3b and leaving the pilot
input unaltered. The resulting altitude time histories are reported
in Fig. 10 for different values of
xref and the standard PI approach.
9
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
0 102030405060708090100110120
−10
−5
0
5
10
t(s)
h(m)
href
Fig. 10. Altitude time histories for the altitude shift task, for different valu es of the reference CCoP location and the standard PI approach. Trajectories are fundamentally
indistinguishable and therefore have not been labeled individually.
Apart from a very small difference during climb and descent
phases, the trajectories are basically indistinguishable at the scale
of the full maneuver duration. This is also in light of the fact that
the FCS is re-tuned for every value of
xref , using the same perfor-
mance requirements within the automatic procedure presented in
Section 2.
Nevertheless, the alteration of the dynamic response achieved
with the modified CA approach has a small but noticeable impact
on trajectory tracking precision. The Root Mean Squared (RMS) de-
viation
hof each altitude time history w.r.t. the reference one is
reported in Fig. 11a. A clear trend is visible in the chart, where the
cancellation of the non-minimum phase behavior clearly leads to
better tracking precision due to a faster transient response. While
very negative values of
xref lead to a deterioration of tracking per-
formance, the best tracking precision is obtained for 0.2 <
xref <1,
as expected in light of the results obtained in the previous Sec-
tion 3.2.
The same conclusions can be drawn by observing the trend in
the agility quickness performance metric, reported in Fig. 11b. This
parameter, calculated as the ratio between the peak load factor
and the maximum excursion of the flight path angle, has been
previously proposed as a measure of short-term agility for rotor-
craft maneuvering in forward flight [36]. In the opinion of the
authors, it is also well suited for interpreting the dynamic perfor-
mance of aircraft capable of DLC. The highest agility quickness is
achieved for
xref =0.8, in line with results from all previous anal-
yses.
3.4. Altitude hold in turbulent atmosphere
In the present application, the aircraft is required to hold its ini-
tial altitude while flying in a turbulent air field. The von Karman
turbulence model has been implemented, and a moderate turbu-
lence intensity level has been selected for the test case [37]. In the
same way as before, the task is performed exclusively by means of
the altitude control channel shown in Fig. 3, with no input from
the pilot.
Altitude time histories are shown in Fig. 12. In this case, differ-
ent features of the trajectories can be identified for different values
of the prescribed position of the CCoP. As expected in light of
Tabl e 2
Guidelines for the interpretation of the overall frequency-weighted RMS ac-
celeration as an indication of the level of comfort on board [39].
Acceleration magnitude
a(m/s2)
Comfort level indication
<0.315 Not uncomfortable
0.315 – 0.63 A little uncomfortable
0.5 – 1 Fairly uncomfortable
0.8 – 1.6 Uncomfortable
1.25 – 2.5 Very uncomfortable
>2 Extremely uncomfortable
previous results, extreme values of
xref, as well as the classic PI ap-
proach, result in lower frequency oscillations of greater amplitude.
On the other hand, due to the cancellation of the non-minimum
phase behavior, values of
xref between 0 and +1.0result in higher
frequency altitude oscillations of smaller amplitude, i.e. an over-
all faster and sharper response to the external disturbance. These
observations are once again confirmed by the RMS deviation w.r.t .
the reference initial altitude href (t)=0, reported in Fig. 13a. The
trend of
has a function of
xref is completely analogous to the
one seen in the previous application.
As a final analysis on these simulations, the overall frequency-
weighted RMS acceleration
aperceived by a passenger seated at
the aircraft CG location has been estimated. The methodology de-
scribed in [38] has been implemented, with reference to the ISO
2631-1 standard [39]. Results are reported in Fig. 13b as a function
of the prescribed position of the CCoP. While it is hard to compare
the absolute numeric values to similar studies on conventional air-
craft of similar category and in similar flight scenarios, the chart
once again highlights a trend analogous to all previous cases. The
ISO 2631-1 standard also provides guidelines to interpret the nu-
merical values of
aas a quantitative measure of comfort on board.
These are graphically represented on the left side of Fig. 13b and
additionally reported in Table 2. As it can be seen, the overall
acceleration obtained with the standard PI approach is classified
as either “uncomfortable” or “fairly uncomfortable”. On the other
hand, the proposed CA approach is able to obtain a consistent
improvement, resulting in a clear classification as “little uncom-
fortable” in the best case.
10
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
PI
-5
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
5
3.58
3.59
3.6
3.61
3.62
3.63
xref
h(m)
a) Altitude RMS deviation
PI
-5
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
5
1.8
2
2.2
2.4
xref
npk
zB/γmax (1/deg)
b) Agility quickness in the first climb time span (t<30 s)
Fig. 11. Derived performance metrics for the altitude shift task, for different values of the reference CCoP location and the standard PI approach.
0510 15 20 25 30 35 40 45 50
−2
−1
0
1
2
3
PI
xref ∈[−5,−0.4]
xref ∈[3,5]
xref ∈[−0.2,1]
t(s)
h(m)
Fig. 12. Altitude time histories for the altitude hold task in turbulent atmosphere, for different values of the reference CCoP location and the standard PI approach. Legend
labels have been aggregated, colors match Fig. 7a.
4. Conclusions
A novel Control Allocation (CA) approach has been proposed
with the objective of shaping the aircraft transient response by ex-
ploiting the concept of Control Center of Pressure (CCoP), i.e. the
center of pressure due to only aerodynamic control forces. First, a
formulation based on the straightforward augmentation of the con-
trol effectiveness matrix has been presented. This can be used to
modify any classic CA method already existing, but may result in
an ill-conditioned effectiveness matrix in some limit cases. Another
formulation, based on a weighting matrix to prioritize effectors,
has been outlined and implemented in three applications featur-
ing a box-wing aircraft configuration: a simple pull-up maneuver,
a trajectory tracking task, and an altitude holding task in turbulent
atmosphere.
The performance of the proposed CA formulation is studied as a
function of the prescribed position of the CCoP, and compared to a
classic Pseudo Inverse (PI) CA method. Results show that, with the
same closed-loop characteristics, the proposed approach can sig-
nificantly impact performance metrics that are closely related to
the aircraft transient response, such as delay due to non-minimum
phase behavior, tracking precision, and capability of disturbance
rejection. In the best case scenario, the aircraft is able to com-
pletely cancel the non-minimum phase behavior typical of pitch
dynamics, hence achieving a sharp and more agile initial response
to longitudinal commands. This results in improved tracking pre-
cision, better disturbance rejection, and ultimately in an improved
feeling of comfort on board.
Although the presented CA method is applicable to any air-
craft configuration, the obtained results reflect, to some extent, the
flight mechanics potential of the box-wing geometry. Thanks to the
presence of redundant control surfaces, both fore and aft of the air-
craft center of gravity, this configuration allows a large excursion of
the CCoP, which is probably infeasible for more conventional archi-
tectures. Further research could be therefore devoted to assessing
the benefits of the proposed CA approach on conventional aircraft
configurations. From a more fundamental standpoint, improving
the control effectiveness matrix conditioning problem could make
11
C. Varriale and M. Voskuijl Aerospace Science and Technology 119 (2021) 107092
PI
-5
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
5
0.2
0.4
0.6
0.8
1
1.2
xref
h(m)
a) Altitude RMS deviation
PI
-5
-3
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
3
5
0.4
0.5
0.6
0.7
0.8
0.9
1
xref
a(m/s2)
A little uncomfortable
Fairly uncomfortable
Uncomfo rtable
b) Overall frequency weighted RMS acceleration perceived at the air-
craft CG [39]
Fig. 13. Derived performance metrics for the altitude hold task, for different values of the reference CCoP location and the standard PI approach.
the proposed formulation applicable to a wider range of already
available CA methods. Lastly, the development of specific flying
qualities metrics for DLC longitudinal response remains an open
challenge.
Funding
The research presented in this paper has been carried out in the
framework of the PARSIFAL (Prandtlplane ARchitecture for the Sus-
tainable Improvement of Future AirpLanes) research project, which
has been funded by the European Union within the Horizon 2020
Research and Innovation Program (Grant Agreement No. 723149).
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
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