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Prioritized Control Allocation for Quadrotors Subject to Saturation

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This paper deals with the problem of actuator saturation for INDI (Incremental Nonlinear Dynamic Inversion) controlled flying vehicles. The primary problem that arises from actuator saturation for quadrotors, is that of arbitrary control objective realization. We have integrated the weighted least squares control allocation algorithm into INDI, which allows for prioritization between roll, pitch, yaw and thrust. We propose that for a quadrotor, the highest priority should go to pitch and roll, then thrust, and then yaw. Through an experiment, we show that through this method, and the appropriate prioritization, errors in roll and pitch are greatly reduced when applying large yaw moments. Ultimately, this leads to increased stability and robustness.
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Prioritized Control Allocation for Quadrotors Subject to
Saturation
E.J.J. Smeur
, D.C. H ¨oppener, C. De Wagter
Delft University of Technology, Kluyverweg 1, Netherlands
ABS TRAC T
This paper deals with the problem of actuator
saturation for INDI (Incremental Nonlinear Dy-
namic Inversion) controlled flying vehicles. The
primary problem that arises from actuator sat-
uration for quadrotors, is that of arbitrary con-
trol objective realization. We have integrated the
weighted least squares control allocation algo-
rithm into INDI, which allows for prioritization
between roll, pitch, yaw and thrust. We propose
that for a quadrotor, the highest priority should
go to pitch and roll, then thrust, and then yaw.
Through an experiment, we show that through
this method, and the appropriate prioritization,
errors in roll and pitch are greatly reduced when
applying large yaw moments. Ultimately, this
leads to increased stability and robustness.
1 IN TRODUC TIO N
Control allocation is often described as the problem of
distributing control effort over more actuators than the num-
ber of controlled variables [1, 2, 3]. This is something that
occurs in traditional aircraft as well as drones, such as hexaro-
tors and octorotors. What sometimes does not receive a lot of
attention, is that the problem of how to deal with actuator sat-
uration is also part of the control allocation topic and, in some
cases, can be very important.
Especially for aerial vehicles with coupled control effec-
tors, such as quadrotors, actuator saturation may lead to unde-
sired, or if occuring for longer timespans, even catastrophic
behaviour. It may be that the desired thrust, and/or control
moments in roll, pitch and yaw, can not be achieved due to
actuator saturation. In absence of an adequate control alloca-
tion algorithm, it is left to chance which part of the control
objective will suffer, it may be the thrust, roll, pitch, or yaw.
However, for the flight stability of multirotor vehicles, it
is far more important to apply the right roll and pitch control
moments than to apply the right yaw moment, since the thrust
vector is indifferent to the yaw in body axis. Therefore, we
would like the control allocation algorithm to prioritize the
control objective of roll and pitch over that of yaw, and to
calculate the control inputs accordingly.
In previous research, we have developed an Incremen-
tal Nonlinear Dynamic Inversion (INDI) controller for Micro
Email address: e.j.j.smeur@tudelft.nl
Air Vehicles (MAV) [4, 5]. We have shown that this control
method is very good at disturbance rejection and needs little
model information. Moreover, we presented a method to in-
clude the effects of propeller inertia, yielding faster and more
accurate yaw control. This aggressive yaw control can easily
lead to saturation of multiple actuators, especially when com-
manding large yaw changes. These saturations often lead to
errors in roll and pitch angles and in the thrust, causing the
vehicle to lose control of its position and potentially crash.
But also external moments, such as wind disturbances,
or actuator faults can lead to saturation. This is why a con-
trol allocation method needs to be added to the INDI control
structure. Multiple control allocation algorithms have been
proposed, some of which do not adequately address prioriti-
zation: ganging, redistributed pseudo-inverse, direct control
allocation; and some of which do: linear programming and
quadratic programming [6]. In this paper, we will consider a
quadratic cost function, and the corresponding quadratic op-
timization problem. A solution to this problem can be found
in a straightforward way using the active set method, as has
been shown by H ¨arkeg˚ard [7].
In this paper, we integrate the Weighted Least Squares
(WLS) control allocation algorithm into the INDI attitude
controller. Further, we show through an experiment that pri-
oritization of roll and pitch over yaw leads to stability im-
provements. The structure of this paper is as follows: first,
the INDI control law is introduced in Section 2. Second, Sec-
tion 3 elaborates on the WLS method and how it integrates
with INDI attitude control. Third, the experimental results
are presented in Section 4, and we end with conclusions and
future work in Section 5.
1.1 Related Work
As opposed to our approach of prioritization, some re-
search has focused on the preservation of control direction
[8, 3]. This means that in case of saturation, a solution for the
actuator inputs is sought that corresponds to a linear scaling
of the original control objective. This approach may be useful
for systems where all axes are equally important. However,
for a quadrotor, if a large yawing moment is needed, the actu-
ators can easily saturate due to the low control effectiveness
in this axis. Scaling the desired control moments will make
the roll and pitch control suffer, which may lead to instability.
Recently, Faessler et al. implemented a heuristics based
algorithm for priority management [9]. They showed that pri-
oritizing roll and pitch over yaw can lead to stability improve-
l
Z
X
Y
b
M
2
M
3
M
4
M
1
Figure 1: Axis definitions.
ments. However, the suggested algorithm resembles the Re-
distributed Pseudo Inverse method (RPI), which is known in
some cases to not find the control solution even if the control
objective is achievable [10]. Furthermore, the scheme is par-
ticularly constructed for quadrotors, and does not generalize.
The WLS approach is much more general, as it does not
depend on a certain configuration of actuators. The method
has been suggested for quadrotors by Monteiro et al. [11],
but was only implemented in simulation. Furthermore, the
weighting matrix, that determines the priorities in the cost
function, is not discussed.
2 IN CRE MEN TAL NON LIN EAR DYNAMIC IN VER SIO N
AND ACT UATOR SATU RATION
In previous work [4], we derived INDI control for MAVs.
A detailed derivation is beyond the scope of this paper, but
the main feature of the controller is its incremental way of
controlling angular accelerations. The basic idea is that the
current angular acceleration is caused by the combination of
inputs and external moments. In order to change the angular
acceleration, all that is needed is to take the previous inputs
and increment them, based on the error in angular accelera-
tion and the control effectiveness.
A distinction is made between two types of forces and
moments: those that are produced by inputs, and those that
are produced by changes in inputs. The forces and moments
produced due to the propeller aerodynamics fall in the first
category, and the torque it takes to spin up a propeller falls
in the second category. Both need to be accounted for in dif-
ferent ways, which is why the control effectiveness matrix is
split up in two parts: G=G1+G2, where G2accounts
for the propeller spin up torque. Though the algorithms pre-
sented here have broad applicability, we will, in order to pro-
mote clarity, consider the quadrotor shown in Figure 1, with
the illustrated axis definitions. We define the angular rotation
vector , its derivative ˙
and the angular rate of the propellers
ω. Then, if we assume a linear control effectiveness and that
gyroscopic effects of the vehicle can be neglected [4], the sys-
tem equation in incremental form is
˙
˙
0+G2L(ωω0) = (G1+G2)(ωω0),(1)
subject to
ωmin ωωmax,(2)
where Lis the lag operator, e.g. ω(k1) = (k). Note that
the angular acceleration needs to be obtained by deriving it
from gyroscope measurements through finite difference. This
signal can be quite noisy, and will need appropriate filtering.
In order to synchronize all signals with subscript 0, they all
need to be filtered with this same filter.
Equation 1 can be turned into a control law using the ma-
trix inverse or the pseudo-inverse:
ω=ω0+ (G1+G2)1(ν˙
0+G2L(ωcω0)),(3)
but calculating the control input like this does not guarantee
satisfying Equation 2. If the control inputs exceed the bounds,
simply clipping them will result in different control moments
than desired.
Instead, Equation 3 is replaced with a method that cal-
culates the control inputs while respecting the limits and pri-
oritization. This can be done with a weighted least squares
(WLS) optimization. Since our system description (Equa-
tions 1 and 3) is in incremental form, we will first write it
as a standard least squares problem through a change of vari-
ables:
v=Gu (4)
subject to
umin uumax.(5)
where the control objective is v=˙
˙
0+G2L(ωω0),
and the input is u=ωω0. The limits umin and umax follow
from these definitions and Equation 2.
3 US ING T HE AC TIV E SET MET HOD TO SOLVE T HE
CON STR AINED ALLOC ATION PROBLEM
Though in this paper we will apply the algorithm to a
quadrotor, for the control allocation we will also consider
over-actuated systems. This means that we have to include
a cost for actuator usage in the cost function, such that there
is only one optimum. This will make the derived methodol-
ogy easily applicable to other systems, like multirotors with
more than four rotors, or some over-actuated hybrid systems
like the Quadshot [12].
In most cases, we would like to formulate the control allo-
cation problem as a sequential least squares problem. Primar-
ily, we want to minimize the error between the control objec-
tive and the angular acceleration increment produced by the
calculated control increment. This can be captured in a first
cost function. Secondly, given the inputs that minimize the
primary cost function, we would like the actuators to spend
the lowest amount of energy possible. If Ghas full rank, the
secondary cost function can be omitted, as the primary cost
function will only have one solution. However, when there
are more actuators than control objectives, the second cost
function will minimize expended energy and avoid actuators
steering in opposite directions.
The sequential least squares problem is more difficult to
solve than a least squares problem with a single cost func-
tion. This is why we adopt the WLS problem formulation
from H¨arkeg˚ard [7], where the cost for errors in the control
objective is combined with a cost for applying inputs:
C(u) = kWu(uud)k2+γkWv(Gu v)k2
=
γ1
2WvG
Wuuγ1
2Wvv
Wuud
2
,(6)
where Wvis the diagonal weighting matrix for the control
objective, and Wuis the diagonal weighting matrix for the
inputs. The distinction between the primary and secondary
objective is made by the scale factor γ >> 1. For conve-
nience, we define
A=γ1
2Wv(G1+G2)
Wuand b=γ1
2Wvv
Wuud.(7)
Now that the problem is formulated as a regular quadratic
programming problem, it can be solved using the well known
active set method [7, 13, 14], to find the inputs that minimize
the cost function. The algorithm divides the inputs into a free
set and an active set, which correspond to the inputs that are
not saturated and to the actuators that are saturated respec-
tively. The method disregards the inequality constraints for
the free set, and for the active set Wtreats the constraints as
equality constraints. At every iteration, it is evaluated if the
division between active and free set is correct. For complete-
ness, we explain our implementation of the active set method
in Algorithm 1.
The algorithm stops when the solution is optimal, or a
maximum number of iterations is reached. Though the algo-
rithm is guaranteed to find the optimum in a finite number
of iterations, we may impose a maximum number of itera-
tions that can be executed in a real time application. If the
algorithm stops because the maximum number of iterations
is reached, the solution will not be optimum. However, since
the value of the cost function decreases at each iteration [14],
the result will be better than at the start of the algorithm.
3.1 Particularities for WLS applied to INDI
Since we are applying the WLS control allocation scheme
to the INDI controller, the inputs are incremental. This means
that the bounds on the input (increment) change every time
step, and the solution for the increment at one time may not
be feasible the next time step. The initial guess for the input,
u0, can therefore not be the solution of the previous time,
as is often done for non-incremental controllers [7, 13, 6].
Instead, we take as initial input the mean of the maximum
and minimum input increment:
u0=1
2(umax umin).(13)
Additionally, if we consider an over-actuated system, the
choice of the preferred increment upbecomes important, as
Algorithm 1: Active set method for WLS problem
Initialization:
W={∅},u0= (umax umin)/2,d=bAu0,
S= []
for i= 0,1,2, .., nmax do
Determine the free columns in A:
Af=A(:, h), h /W(8)
Determine the optimal perturbation by solving the
following least squares problem for pf:
d=Afpf(9)
Now pis constructed from pfwith zeros for the
elements that are in W.
if ui+pis feasible then
ui+1 =ui+pand: d=dAfpf
The gradient and Lagrange multipliers are
computed with:
=ATdand: λ=S(10)
if all λ0then
The solution ui+1 is optimal u=ui+1 ;
else
The constraint associated with the most
negative λhas to be removed from the
active set W. Re-iterate with this active
set.
else
The current solution violates a constraint
which is not in W. Determine the maximum
factor αsuch that αp is a feasible
perturbation, with 0α < 1. Update the
residual dand the solution ui+1 :
ui+1 =ui+αp (11)
d=dAfαpf(12)
Finally, update the active set and store the
sign of the constraint: Sjj =sign (pj)with j
the index of the new active constraint.
there is some degree of freedom in choosing the inputs that
will produce the required forces and moments. Some of these
combinations may require more energy than what is optimal,
for instance if two actuators counteract each other in order
to produce a net zero output. Clearly, this can be achieved
more efficiently by giving zero input to both actuators. For
non-incremental controllers, this means that that upis a zero
vector. For an incremental controller, this means that up=
umin, assuming that the actuators produce zero force/moment
at umin.
3.2 Choice of Weighting Matrices
As for any optimization, the result entirely depends on the
choice of the cost function. In this case, we have the freedom
to choose Wv,Wuand γ.
For Wv, we choose the diagonal elements to be 1000,
1000, 1 and 100 for roll, pitch, yaw and thrust respectively.
The reason that we give roll and pitch a higher priority than
thrust, is because the thrust can only be applied in the right
direction if the vehicle has the right attitude. As an example,
suppose that the quadrotor is inverted. With the thrust vector
pointing down, it will lose altitude fast. The controller will
have to flip the airframe, and increase thrust to climb. How-
ever, if priority would be put on the thrust, the vehicle could,
in the extreme case, never change the attitude, as all motors
would have to give full thrust.
In general, it appears that satisfying (part of) the roll and
pitch objectives, will lead to a reduction of said objectives in
the short term, as it typically does not take long to rotate to
a desired attitude. On the other hand, satisfying (part of) the
thrust objective, might not lead to a reduction of this objective
in the short term, as the thrust vector may be pointing in the
wrong direction or a large continuous thrust may be needed
over a long period of time. Therefore to the authors, priori-
tizing pitch and roll over thrust seems to be the most stable
configuration. However, for a specific quadrotor, the best pri-
oritization scheme may depend on the mission profile.
We choose γ1
2= 10000 and for Wuwe take the identity
matrix, since all actuators are ’equal’. Do note that the rel-
ative scaling of the signals uand vplays a role here. Also
note that, even though we give a lower weighting to some
signals, they can still become dominant in the cost function
if no bounds are applied. As an example, consider a quadro-
tor that has to climb five kilometers. In case of a simple PD
controller without bounds, an enormous thrust will be com-
manded, leading to more cost in Equation 6.
3.3 Computational Complexity
The computational complexity of the active set algorithm
scales with the number of actuators in two ways. First, each
additional actuator will add a row and a column to the matrix
A, and therefore increase the computational complexity of
solving the quadratic problem each iteration of the active set
algorithm. Additionally, if there are more actuators, more
actuators can saturate in different combinations. This may
lead to more iterations on average, as well as more iterations
in a worst case scenario.
An analysis of the performance of the active set algorithm
on a benchmark problem set, with control objectives in R3
was done by Petersen and Bodson [13]. They report that the
method is efficient in case of few actuators, but that it does not
scale well with the problem size. Specifically, for 15 actuators
or more, an interior point method is more efficient. Since our
control objective is in R4, this point can be somewhere else.
Clearly, it is very beneficial for the computational per-
formance to have few actuators. If computational time is a
problem, it might be an option to combine several actuators
into single ’virtual’ actuators, often referred to as ’ganging’.
However, we are able to run the WLS scheme on an
STMF4 microprocessor, which is equipped with a floating
point unit, for four actuators at 512 Hz without any problem.
Our implementation uses single precision floating point vari-
ables.
4 EX PER IMENTS
As mentioned in the introduction, actuator saturation of-
ten occurs due to yaw commands, as the yaw moment gen-
eration of the actuators is relatively weak. Without proper
priority management, this is a case where instability can oc-
cur. In order to demonstrate the ability of the WLS control
allocator to improve stability of the vehicle through priority
management, an experiment is performed.
The hypothesis is that the WLS control allocation scheme,
with the prioritization as defined in section 3.2, improves the
tracking of pitch and roll when large yaw moments are re-
quired, as compared to calculating the inputs with the pseudo-
inverse and clipping the result.
To test for this hypothesis, the hovering drone will be
given an instant step in its heading reference of 50 degrees.
This is enough to cause severe actuator saturation. The drone
is controlled by a pilot, who will bring the drone back to the
hovering position after each maneuver. During the maneuver,
the pilot does not give any commands.
4.1 Experimental Setup
The test is performed using a Bebop 1 quadrotor from
Parrot, running the Paparazzi open source autopilot software.
The Bebop is equipped with an internal RPM controller,
which accepts commands between 3000 and 12000 RPM. In
practice, we found that in static conditions the motors satu-
rate well before 12000 RPM. To avoid commands above the
saturation limit that will not have any effect, the limit in the
software is put at 9800 RPM.
Again, for details on the INDI control algorithm em-
ployed, we refer to our previous papers [4, 5]. However, we
will list the parameters used for the experiment. Prior to the
experiment, the following control effectiveness matrices were
identified through test flights:
G1=
18 18 18 18
11 11 11 11
0.7 0.70.7 0.7
0.40.40.40.4
·103(14)
G2=
0 0 0 0
0 0 0 0
65 65 65 65
0 0 0 0
·103(15)
The filter that is used for the angular acceleration is a sec-
ond order Butterworth filter with a cutoff frequency of 5 Hz.
4.2 Results
Figure 2 shows the results of the experiment for the
pseudo-inverse on the top and for WLS on the bottom. From
the left, the first three figures on each row show the Euler
angles for 15 and 12 repetitions of the experiment for the
pseudo-inverse and WLS respectively. For WLS, two repe-
titions were rejected, because the pilot steered during the yaw
step. The last figure on each row shows the inputs to the ac-
tuators during the first repetition.
First, from the plot of the ψangle it can be observed that
with WLS there is no overshoot, but the rise time is longer.
The longer rise time can be explained, because for WLS, the
inputs are not saturated the whole time the vehicle is moving
towards the reference. Because of this, for WLS, the angular
velocity does not become as high and the quadrotor is able to
reduce the angular velocity without saturating the actuators.
For the pseudo-inverse, the situation can be compared with
integrator windup. The quadcopter builds up so much angular
velocity while the actuators are saturated, that when it has to
reduce this angular velocity, the actuators saturate in the other
direction and the vehicle overshoots.
Though now it may seem that WLS solves this problem,
this is not the case. The figure merely shows that due to the
prioritization, the vehicle can not accelerate as fast in the yaw
axis, which is why the overshoot does not occur. For larger
heading changes, when the vehicle will accumulate angular
velocity in the yaw axis over a longer time, overshoot is also
observed.
However, the plots of pitch and roll show the merit of the
WLS control allocation (note the different scale). To con-
dense this information, we consider the maximum deviation
of the roll and pitch angle from zero as a measure of the per-
formance for each repetition. The mean and standard devia-
tion of this maximum error per repetition is presented in Table
4.2.
Clearly there is a very significant improvement in the
tracking of the pitch and roll angles. We therefore conclude
the hypothesis, that WLS improves the tracking of pitch and
roll when producing large yaw moments, to be true.
Finally, from Figure 2 it does become apparent that there
still is some small cross coupling between roll and pitch mo-
φ θ
µ σ µ σ
Pseudo-inverse 12.2 4.8 22.8 9.7
WLS 0.9 0.2 0.5 0.4
Table 1: Mean and standard deviation of the maximum pitch
and roll error in degrees.
ments and the yaw moment for WLS. The exact cause is be-
yond the scope of this paper, and may be a topic of future
research, but there are possible explanations. For instance,
the controller takes into account a linear control effective-
ness, while this can be expected to be a quadratic one. Es-
pecially for large input changes, as is the case here, some
error may be expected. Furthermore, we may consider the
fact that for WLS, everything is combined into one cost func-
tion. This means that putting more weight on roll and pitch
may reduce the error in tracking these angular accelerations,
but will never bring it to zero. To improve this, the sequential
formulation may be a solution.
5 CO NCL USION
In this paper we have applied the WLS control allocation
scheme to incremental nonlinear dynamic inversion control.
We propose the following prioritization of controlled forces
an moments: first roll and pitch, then thrust, then yaw. This
ensures the capability of the vehicle to come back to a stable
situation from any attitude. Through an experiment we show
that the WLS control allocation with these priorities improves
the stability when applying large yaw moments.
The algorithm is readily applicable to other types of
MAVs for which priorities in controlled axes can be defined,
such as hexacopters, or even hybrid aircraft such as the Cy-
clone [15]. Future research will focus on how constraints in
the guidance loop should be taken into account, and how this
is affected by limits in the inner loop. Finally, given the strong
disturbance rejection properties of the INDI controller, this
control allocation scheme is expected to also increase the ro-
bustness against faults.
ACK NOWLEDGEM ENT S
The authors would like to thank Anton Naruta for his help
with the implementation. This work was supported by the
Delphi Consortium.
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... Considering w τ r → 0 and the solution's continuity with respect to w τ r , ∀ > 0, ∃ δ > 0, such that |f ss+ − f * ss+ | < , |τ p,ss+ − τ * p,ss+ | = |Δ τ p | < and |τ q,ss+ − τ * q,ss+ | = |Δ τ q | < hold with |w τ r | < δ, where Δ τ p and Δ τ q are seen as the result of the perturbations of p * and q * , respectively. According to (23) and (24), there are KΔ p = Δ τ p and KΔ q = Δ τ q , where K = J xx k rotor k ω . That is |Δ p | < /K and |Δ q | < /K. ...
... For system (15) with controller (19), we have |W (t, t 0 )| → 0 as Δ τ p → 0 and Δ τ q → 0, namely w τ r → 0. Therefore, h 1 → h * 1 and h 2 → h * 2 as w τ r → 0. As [23] introduces, the QP problem (25) is able to be solved by a microprocessor for the quadcopter. In practice, the QP problem (25) is expanded and solved through the active-set method in the ...
... The DOC, defined in [27], of system (37) about k r , η, and r ss is shown in Fig. 9. In practice, η is able to be adjusted by adding −k couple ω × Jω on controller (23) so that it has η = 1 + k couple . The calculation method of DOC is given as ρ g 2 = n/tr((Q T c Q c ) −1 ). ...
Article
This article provides a uniform fault-tolerant controller for a quadcopter without controller switching in case one rotor fails completely. The fault-tolerant control adopts control allocation based on an optimization method as the core of control. It will abandon the yaw channel when a rotor fails completely, for which the motion characteristics of the free-rotation quadcopter and the degree of controllability are analyzed. For practical implementation, the rotor dynamics have been further taken into consideration as first-order dynamics and then a compensator is used to improve the rotor response. To validate the proposed controller, first, digital simulation studies are conducted to verify the control logic. Second, the hardware-in-the-loop simulation highlights the feasibility and high efficiency of the fault-tolerant control method on an embedded system. Finally, an outdoor experiment with onboard sensors and GPS demonstrates that the fault-tolerant controller has the ability to make quadcopters fly safely when an arbitrary rotor fails completely.
... Effectively handling control input limits is a remaining challenge for non-predictive methods, including DFBC. So far, existing methods have prioritized the position tracking over heading using various approaches, such as redistributed pseudo inversion [29], weighted-least-square allocation [30], controlprioritization method [31,13], and constrained-quadratic- 3 programming allocation [32]. While these methods can mitigate the actuator saturation effect when the trajectories are dynamically feasible, its performance in tracking dynamically infeasible trajectories is still questionable. ...
... I) is advantageous to prevent quadrotor loss-of-control when motor saturations are inevitable (e.g., tracking dynamically infeasible trajectories). If the solution is originally within control bounds, the result is the same as the direct-inversion allocation from (30). As for the implementation details, we solve this quadratic programming problem using an Active-Set Method from the qpOASES solver [42]. ...
... Note that, for the DFBC method,T d andα B are different from those derived from (14) and (28) if the optimal cost of (31) is non-zero. Then from INDI, we can get the desired body torque (see (30) and (31) of [13] for detailed derivations) ...
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Article
Accurate trajectory-tracking control for quadrotors is essential for safe navigation in cluttered environments. However, this is challenging in agile flights due to nonlinear dynamics, complex aerodynamic effects, and actuation constraints. In this article, we empirically compare two state-of-the-art control frameworks: the nonlinear-model-predictive controller (NMPC) and the differential-flatness-based controller (DFBC), by tracking a wide variety of agile trajectories at speeds up to 20 m/s (i.e.,72 km/h). The comparisons are performed in both simulation and real-world environments to systematically evaluate both methods from the aspect of tracking accuracy, robustness, and computational efficiency. We show the superiority of NMPC in tracking dynamically infeasible trajectories, at the cost of higher computation time and risk of numerical convergence issues. For both methods, we also quantitatively study the effect of adding an inner-loop controller using the incremental nonlinear dynamic inversion (INDI) method, and the effect of adding an aerodynamic drag model. Our real-world experiments, performed in one of the world’s largest motion capture systems, demonstrate more than 78% tracking error reduction of both NMPC and DFBC, indicating the necessity of using an inner-loop controller and aerodynamic drag model for agile trajectory tracking.
... properly becomes a critical problem [77,63]. This problem is generally referred as "Control Allocation(CA)", and as a general solution in literature, rotational channels are prioritized over translational channels in case of actuator saturation [35,77]. ...
... properly becomes a critical problem [77,63]. This problem is generally referred as "Control Allocation(CA)", and as a general solution in literature, rotational channels are prioritized over translational channels in case of actuator saturation [35,77]. In this way, the stability of the aircraft is guaranteed via proper CA design. ...
... Therefore, actuator saturation might cause catastrophic stability problems. As a common approach, CA prioritizes rotational axis over the translational axis to track the rotational acceleration commands accurately in case of actuator saturation [77,56]. If the control channels/axis and control effectors/actuators can be decoupled, then relating each other becomes straightforward and CA can be designed easily to resolve the actuator saturation related problems. ...
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Thesis
On-demand urban air mobility (UAM) has become very popular in recent years with the introduction of the electric vertical take-off and landing (eVTOL) aircraft concept. Thanks to the key advantages of electric propulsion (e.g., very low noise and zero carbon-emission), short/medium range eVTOL "air-taxi" concept emerged as a feasible solution considering the requirements of the on-demand UAM. With this motivation, flight control problems of a novel eVTOL air-taxi are discussed and a unified flight controller is designed considering the full flight envelope. The air-taxi has a fixed-wing surface to have aerodynamically efficient forward flight, and uses only tilting electric propulsion units (i.e. the pure thrust vector control) to achieve full envelope flight control. The aircraft does not have any conventional control and stability surfaces such as aileron, elevator, rudder, horizontal/vertical tail. Therefore, the unified controller design becomes more challenging compared to the conventional aircraft configuration. The flight dynamics model of the air-taxi does not exist in literature since the air-taxi has a novel configuration. First, a preliminary flight dynamics model is generated using the component build-up approach for hover and high speed forward flight. Then, the hover and forward flight models are merged to simulate the transition dynamics. Two main challenges regarding the flight control are the severe nonlinearities in the flight dynamics during the transition flight and deterioration of the controller's performance in specific flight conditions due to the limited control authority (i.e., the actuator saturation). The first challenge is resolved via designing a sensor-based nonlinear controller for the entire flight envelope using the Incremental Nonlinear Dynamic Inversion (INDI) method. The INDI approach has improved robustness to modeling errors compared to the classical nonlinear dynamic inversion (NDI) methods. Therefore, the INDI based controller design fits very well to the problem considering the severe nonlinearities in the flight dynamics model. The INDI controller is formulated specifically considering the highly coupled pure thrust vector control approach. For the second problem, an online optimization-based Control Allocation (CA) algorithm is designed and integrated into the INDI controller. Resolving the actuator saturation related problems requires special attention due to the thrust vector control's coupled nature. The CA prioritizes the rotational channels over the translational channels to adequately allocate the limited control authority in case of actuator saturation. Various nonlinear simulation tests are performed considering the full envelope flight control, disturbance rejection characteristics at limited control authority and criticality of the CA design, robustness to model parameters, etc. Simulation results show that the controller has satisfactory performance, disturbance rejection characteristics, and significant robustness to the modeling errors. Moreover, it is observed that the CA plays a vital role in guaranteeing stable flight in case of severe actuator saturation.
... Many works have highlighted the issue of saturating rotors [22][23][24][25][26][27][28][29][30]. In [22] variable pitch rotors are suggested for increasing the total control effort available to the plant; however, this can be expensive and impractical for many smaller/mid sized multi-rotors. ...
... One common solution is the use of an optimal version of CA: this was shown to be effective in both [23] and [24] to ensure that the rotor control signals remained within their limits through solving a series of constrained on-line optimisation problems, whereas the work in [25] proposes a redistributed pseudo-inverse approach. The works in [26,27] propose versions of optimised CA which prioritises performance in the height, roll and pitch tracking (i.e., those channels that are most vital to system stability) in favour of poorer yaw tracking. For any of these methods to be effective (in removing the performance degradation associated with rotor saturation), the assumption needs to be made that the 'virtual' control (or at least some of its components in the case of [26,27]) remains achievable with respect to the rotor's health and saturation limits. ...
... The works in [26,27] propose versions of optimised CA which prioritises performance in the height, roll and pitch tracking (i.e., those channels that are most vital to system stability) in favour of poorer yaw tracking. For any of these methods to be effective (in removing the performance degradation associated with rotor saturation), the assumption needs to be made that the 'virtual' control (or at least some of its components in the case of [26,27]) remains achievable with respect to the rotor's health and saturation limits. Such an assumption becomes difficult to justify when presented with extreme fault/failure scenarios, aggressive manoeuvres and disturbances/uncertainty (or a mixture of all three). ...
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Article
Abstract In this paper, a quasi‐linear parameter varying sliding mode control allocation law is proposed for the fault tolerant control of an octorotor. In the event of rotor faults/failures, an allocation law redistributes the control effort among the remaining healthy rotors. The sliding mode control law is designed to guarantee asymptotic tracking of a reference model which is tuned on‐line, through an interpolated feedback gain, to ensure that the control signals remain within their saturation limits. A method for designing the parameterised feedback gain is proposed which is shown to maximise a defined stability criteria whilst preventing undesirable performance characteristics in the reference model. The proposed scheme is tested on a non‐linear octorotor model in the presence of severe rotor failures and uncertainty/disturbances.
... Fixed-wing aircraft have great endurance thanks to their wing-induced lift. [9][10][11] Rotorcraft on the other hand, like designs by Luukkonen, 12 Zhiqiang et al. 13 and Smeur et al., 14 are much more versatile since they can hover, take off and land vertically. They are also inexpensive to produce, mechanically simple and their control has been well solved. ...
... 28 However effective, NDI highly relies on detailed and accurate models of the vehicle it controls. A variation on this approach provides a solution to this problem and is called Incremental Non-linear Dynamic Inversion, or INDI, and was demonstrated in flight by Smeur et al. 27,14 While it still relies on an actuator model, instead of using a vehicle model to predict its angular and linear accelerations as a result of its states, it uses inertial measurement data to observe these accelerations. And the control effectiveness model does not need to be as accurate, since the controller will compensate for any unexpected effects of the actuators by incrementing. ...
... Since the calculation of UAV control demands typically needs to be performed several hundred times per second, the optimization used in an INCA controller needs to be as efficient as possible. Based on control allocation research performed by Stolk 26 and Smeur et al., 14 the optimization method selected for this research is the Active Set Method. This method requires similar amounts of computing power as e.g. the Redistributed Pseudo-Inverse method and the Fixed-Point algorithm, yet yields more accurate solutions. ...
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Article
Hybrid UAVs have gained a lot of interest for their combined vertical take-off & landing (VTOL) and efficient forward flight capabilities. But their control is facing challenges in over-actuation and conflicting requirements depending on the flight phase which can easily lead to actuator saturation. Incremental Non-linear Control Allocation (INCA) has been proposed to solve the platform’s control allocation problem in the case of saturation or over-actuation by minimizing a set of objective functions. This work demonstrates INCA on quadplanes, an in-plane combination between a quadrotor and a conventional fixed-wing, and proposes an extension to control the outer loop. The novel controller is called Extended INCA (XINCA) and adds the wing orientation as a force-generating actuator in the outerloop control optimization. This leads to a single controller for all flight phases that avoids placing the wing at negative angles of attack and minimizes the load on hover motors. XINCA has low dependence on accurate vehicle models and requires only several optimization parameters. Flight simulations and experimental flights are performed to demonstrate the performance.
... Efficient allocation schemes maximize the use of the control volume or optimize secondary objectives such as thrust or aerodynamic drag [3,4]. The existing approaches can be divided into pseudo-inverse-based methods [5][6][7], direct allocation [8], or linear and quadratic programming methods [3,4,[9][10][11]]. An extensive overview of aircraft control allocation methods can be found in [12]. ...
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Article
Existing control allocation approaches usually minimize an L1 or L2 norm of the actuator commands that is loosely related to the thrust produced or the aerodynamic drag. This paper proposes a novel incremental control allocation method that directly minimizes the total power required for an over-actuated, propeller-driven transition aircraft. The minimum power cost function is derived in a convex form and leads to a closed-form algebraic solution for an additional command increment in the null space of the local control effectiveness matrix. The complete control allocation procedure consists of a two-step approach that is direction-preserving in the presence of actuator saturations and nonlinear due to the incremental formulation. The proposed control allocation is particularly advantageous in the case of a transition aircraft, where the tradeoff between using aerodynamic lift and vertical thrust can be solved naturally by including incremental attitude commands as virtual inputs in the minimum power control allocation. The resulting control allocation scheme allows to both maximize control authority and find minimum power trim solutions. Flight tests are performed on an over-actuated transition aircraft, showcasing the minimum power trim and control allocation without the use of flight mode switching or blending functions throughout all flight regimes.
... This change is crucial in keeping the UAV's stability when large commands are given with fixed attitude strategies. Many such strategies have been introduced for underactuated UAVs and can be easily modified for the fully-actuated vehicles (e.g., see [26,46,178]). In our implementation, we modified the Airmode functionality of the PX4 firmware and prioritized the moments around theX B andŶ B axes over the thrusts and the moment around theẐ B axis. ...
Preprint
The physical interaction of aerial robots with their environment has countless potential applications and is an emerging area with many open challenges. Fully-actuated multirotors have been introduced to tackle some of these challenges. They provide complete control over position and orientation and eliminate the need for attaching a multi-DoF manipulation arm to the robot. However, there are many open problems before they can be used in real-world applications. Researchers have introduced some methods for physical interaction in limited settings. Their experiments primarily use prototype-level software without an efficient path to integration with real-world applications. We describe a new cost-effective solution for integrating these robots with the existing software and hardware flight systems for real-world applications and expand it to physical interaction applications. On the other hand, the existing control approaches for fully-actuated robots assume conservative limits for the thrusts and moments available to the robot. Using conservative assumptions for these already-inefficient robots makes their interactions even less optimal and may even result in many feasible physical interaction applications becoming infeasible. This work proposes a real-time method for estimating the complete set of instantaneously available forces and moments that robots can use to optimize their physical interaction performance. Finally, many real-world applications where aerial robots can improve the existing manual solutions deal with deformable objects. However, the perception and planning for their manipulation is still challenging. This research explores how aerial physical interaction can be extended to deformable objects. It provides a detection method suitable for manipulating deformable one-dimensional objects and introduces a new perspective on planning the manipulation of these objects.
... Thereby, for the latter, a settling time lower than 3 seconds is reasonable. Furthermore, it was shown, in [28], that an aggressive yaw control can easily lead to saturation of multiple actuators, especially when commanding large yaw changes, and, consequently, affect the control performance. Regarding the overshoot and the static error, the angular responses should fulfill the same requirements established for the position responses. ...
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Article
This paper proposes a novel control architecture for quadrotors that relies twice on the Feedback Linearization technique. The solution comprises a tracking inner-loop resulting from applying the mentioned method to the attitude and altitude dynamics. The horizontal movement, and, thereby, the zero dynamics, are stabilized without linearizing nor simplifying it by resorting to the same nonlinear technique. Linear quadratic controllers with integral action are implemented to the resulting chain of integrators of the inner and outer loops. As a result, the inner-loop dynamics asymptotically track the desired attitude and altitude over a broad region of the state-space, and the outer-loop yields a tracking system that is input-to-state stable and exponentially stable in the absence of external inputs. The stability of the proposed inner-outer loop control architecture is studied, leading to the proof of asymptotic stability in an extensive region of the state-space. Trajectory tracking, the capacity to overcome significant deviations on the mass and inertia values, and the robustness to external disturbances are evaluated using a simulation model, in which measurement noise and saturation limits are considered. In addition, comparisons regarding the performance in trajectory tracking of the proposed strategy and the results obtained with similar solutions from the literature are established. Experimental tests were conducted using a commercially available drone, equipped with an Inertial Measurement Unit, a compass, and an altimeter. A motion capture system gives the inertial position of the drone. The results obtained allow the validation of the modeling and control system solution.
... The constrained control allocation problem has been addressed in the literature of underactuated MAVs [10,27,28]. Their methods have been formulated as the optimization of some quadratic cost, subject to both equality and inequality constraints representing, respectively, the control allocation equation and the rotor bounds. ...
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Article
The present work deals with the optimal control allocation of fully actuated multirotor aerial vehicles (MAVs) equipped with fixed (non-vectoring and constant-pitch) rotors. To tackle the problem, a cascaded control architecture is considered in which the control allocation is separated from the control law itself. The latter provides the resulting control efforts (three-dimensional force and torque) from the desired state trajectory, while the former is entrusted to distribute the resulting control efforts among the available actuators. The control allocation is formulated as a convex optimization problem, which, on the one hand, unifies the previous methods and, on the other hand, extends the literature by rigorously considering the rotors' dynamics and bounds, thus resulting in a novel constrained optimal control allocation algorithm suitable for quite general fixed-rotor fully actuated MAVs. Moreover, a control-allocation feasibility analysis based on the control allocator admissible set is presented. It provides a necessary and sufficient condition for the existence of a solution to the control allocation problem. We argue that this condition can be explicitly used in the design of the control law, thus improving its synergy with the control allocator. The proposed control allocation method is mathematically analyzed and widely illustrated by the computer simulation of two non-planar omnidirectional MAVs.
... A control allocation method that directly considers the physical limits of the rotors is employed in Monteiro et al. (2016); Smeur et al. (2017); Kirchengast et al. (2018). It consists of solving a quadratic optimization that aims to minimize the rotor commands magnitude while assuming those bounds as inequality constraints. ...
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Conference Paper
The present work deals with the control allocation of multirotor aerial vehicles (MAVs). We propose an optimal method that directly considers, as a hard equality constraint, the structural relation (control allocation equation) between the rotor commands and the control variable, besides respecting the physical or operational limits of the rotors. The proposed approach aims to exactly allocate the desired control efforts, thus ensuring the designed control features (e.g., performance and stability) for the closed-loop system. The method is compared with three control allocators most used in the aerospace literature: the pseudo-inverse, the direct allocation, and the conventional optimal allocator. Through computational simulations, we can attest that the proposed control allocator performs quite similar to the best existing approaches, but, different from the others, it also guarantees an exact control allocation.
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Conference Paper
This study presents the development of the transitioning vehicle Cyclone, which has been specifically designed for meteorological and agricultural applications. The mission requirements demand takeoff and landing from a small area and the ability to cope with high wind speeds. In contrast with recent suggestions, our proposed design aims to be closer to a fixed-wing airplane rather than a rotary wing. In particular, the design focuses on a tilt-body style transitioning vehicle with blown-wing concept. The propeller wing interaction is calculated using a semi-empirical method. The total wing span and wing surface area are decided according to the mission performance requirements. For the control of the vehicle, incremental nonlinear dynamic inversion is used. This control method does not need the modeling of external forces or moments and is able to counteract the strong aerodynamic forces and moments acting on the vehicle through the feedback of its angular acceleration. Together with the design phases and manufacturing process, several test flights are presented. Particular difficulties of the proposed design are discussed, including lack of providing sufficient pitch-up moment and control reversal during descent. The test flights demonstrate the vertical takeoff and landing capabilities of the vehicle, as well as its transitioning into forward flight from hovering, and vice versa, for an efficient mission performance.
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Article
Quadrotors are well suited for executing fast maneuvers with high accelerations but they are still unable to follow a fast trajectory with centimeter accuracy without iteratively learning it beforehand. In this paper, we present a novel body-rate controller and an iterative thrust-mixing scheme, which improve the trajectory-tracking performance without requiring learning and reduce the yaw control error of a quadrotor, respectively. Furthermore, to the best of our knowledge, we present the first algorithm to cope with motor saturations smartly by prioritizing control inputs which are relevant for stabilization and trajectory tracking. The presented body-rate controller uses LQR-control methods to consider both the body rate and the single motor dynamics, which reduces the overall trajectory-tracking error while still rejecting external disturbances well. Our iterative thrust-mixing scheme computes the four rotor thrusts given the inputs from a position-control pipeline. Through the iterative computation, we are able to consider a varying ratio of thrust and drag torque of a single propeller over its input range, which allows applying the desired yaw torque more precisely and hence reduces the yaw-control error. Our prioritizing motor-saturation scheme improves stability and robustness of a quadrotor’s flight and may prevent unstable behavior in case of motor saturations. We demonstrate the improved trajectory tracking, yaw-control, and robustness in case of motor saturations in real-world experiments with a quadrotor.
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Conference Paper
Micro Aerial Vehicles (MAVs) are limited in their operation outdoors near obstacles by their ability to withstand wind gusts. Currently widespread position control methods such as Proportional Integral Derivative control do not perform well under the influence of gusts. Incremental Nonlinear Dynamic Inversion (INDI) is a sensor-based control technique that can control nonlinear systems subject to disturbances. This method was developed for the attitude control of MAVs, but in this paper we generalize this method to the outer loop control of MAVs under gust loads. Significant improvements over a traditional Proportional Integral Derivative (PID) controller are demonstrated in an experiment where the drone flies in and out of a fan's wake. The control method does not rely on frequent position updates, so it is ready to be applied outside with standard GPS modules.
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Incremental nonlinear dynamic inversion is a sensor-based control approach that promises to provide high-performance nonlinear control without requiring a detailed model of the controlled vehicle. In the context of attitude control of micro air vehicles, incremental nonlinear dynamic inversion only uses a control effectiveness model and uses estimates of the angular accelerations to replace the rest of the model. This paper provides solutions for two major challenges of incremental nonlinear dynamic inversion control: how to deal with measurement and actuator delays, and how to deal with a changing control effectiveness. The main contributions of this article are 1) a proposed method to correctly take into account the delays occurring when deriving angular accelerations from angular rate measurements; 2) the introduction of adaptive incremental nonlinear dynamic inversion, which can estimate the control effectiveness online, eliminating the need for manual parameter estimation or tuning; and 3) the incorporation of the momentum of the propellers in the controller. This controller is suitable for vehicles that experience a different control effectiveness across their flight envelope. Furthermore, this approach requires only very coarse knowledge of model parameters in advance. Real-world experiments show the high performance, disturbance rejection, and adaptiveness properties. Read More: http://arc.aiaa.org/doi/abs/10.2514/1.G001490
Conference Paper
This paper considers the problem of control allocation for a modified quadrotor helicopter unmanned system based on reliability analysis. As one of the objectives is to increase the overall system's reliability, the conventional quadrotor helicopter which is equipped with four actuators is first upgraded by adding four additional actuators. Second, control allocation is applied to the upgraded system for a better redundancy management while considering information on actuator failure rates. With respect to such a reliability consideration, more control duties are allocated to the most reliable actuators and less control duties are allocated to the least reliable ones. Experimental flight tests show how the consideration of reliability analysis in the control allocation affects the generated control inputs for a better control efforts distribution with consideration of a higher reliability in actuator components and also overall quadrotor system. Moreover, it is shown how the system's global reliability and consequently its availability are improved. © 2012 by the American Institute of Aeronautics and Astronautics, Inc.
Conference Paper
This paper presents the Quadshot, a novel aerial robotic platform with Vertical Take-Off and Landing (VTOL) capability. Highly dynamic maneuverability is achieved via a combination of differential thrust and aerodynamic surfaces (elevons). The relaxed stability, flying wing, tail-sitter configuration, Radio Controlled (RC) airframe is actively stabilized by onboard controllers in three complementary modes of operation, i.e. hover, horizontal flight and aerobatic flight. In hover mode the vehicle flies laterally, similar to a quadrotor helicopter, can maintain accurate position for aiming payload and land with pinpoint accuracy when equipped with a GPS unit. In horizontal and aerobatic modes it flies like an airplane to cover larger distances more rapidly and efficiently. Dynamic modeling and control algorithms have been discussed before for quadrotors [1]-[4] and classical aircraft configurations, as have other VTOL concepts such as tilt-rotors (eg. the V-22 Osprey) and tail-sitters (eg. the Sydney Univ. T-wing and the Convair XFY-1 Pogo) [5]-[6]. The important contributions of this paper are the combined use of differential thrust in multiple axes and aerodynamic surfaces for flight control, the assisted transition between hover and forward flight control modes with pitch rotation of the entire airframe and the elimination of failure-prone mechanisms for thruster tilting. The development and use of highly extensible Open Source Software and Hardware from the Paparazzi project in a transitioning vehicle is also novel. The vehicle is made highly affordable for both researchers and hobbyists by the use of the Paparazzi Open Source Software [16] and its Lisa embedded avionics suite. Careful attention to the mechanical design promotes large scale manufacturing and easy assembly, further bringing down the cost. The materials selected create a highly durable airframe, which is still inexpensive. Modular airframe design enables quick modification of actuators and electron- cs, allowing a greater variety of missions. The electronics are also designed to be extensible, supporting the addition of extra sensors and actuators. Custom designed airfoils provide good payload capacity while maintaining 3D aerobatic flight capability; the wing design ensures adequate stability for manual glide control in non-normal situations. This paper covers the software, mechanical and electronic hardware design, control algorithms and aerodynamics associated with this airframe. Experimental flight control results and the design lessons learned are discussed.
Article
The control algorithm hierarchy of motion control for over-actuated mechanical systems with a redundant set of effectors and actuators commonly includes three levels. First, a high-level motion control algorithm commands a vector of virtual control efforts (i.e. forces and moments) in order to meet the overall motion control objectives. Second, a control allocation algorithm coordinates the different effectors such that they together produce the desired virtual control efforts, if possible. Third, low-level control algorithms may be used to control each individual effector via its actuators. Control allocation offers the advantage of a modular design where the high-level motion control algorithm can be designed without detailed knowledge about the effectors and actuators. Important issues such as input saturation and rate constraints, actuator and effector fault tolerance, and meeting secondary objectives such as power efficiency and tear-and-wear minimization are handled within the control allocation algorithm. The objective of the present paper is to survey control allocation algorithms, motivated by the rapidly growing range of applications that have expanded from the aerospace and maritime industries, where control allocation has its roots, to automotive, mechatronics, and other industries. The survey classifies the different algorithms according to two main classes based on the use of linear or nonlinear models, respectively. The presence of physical constraints (e.g input saturation and rate constraints), operational constraints and secondary objectives makes optimization-based design a powerful approach. The simplest formulations allow explicit solutions to be computed using numerical linear algebra in combination with some logic and engineering solutions, while the more challenging formulations with nonlinear models or complex constraints and objectives call for iterative numerical optimization procedures. Experiences using the different methods in aerospace, maritime, automotive and other application areas are discussed. The paper ends with some perspectives on new applications and theoretical challenges.
Article
The details of a computationally efficient method for calculating near-optimal solutions to the three-objective, linear, control-allocation problem are described. The control-allocation problem is that of distributing the effort of redundant control effectors to achieve some desired set of objectives. The optimal solutions is that which exploits the collective maximum capability of the effectors within their individual physical limits. Computational efficiency is measured by the number of floating-point operations required for solution. The method presented returned optimal solutions in more than 90% of the case examined; nonoptimal solutions returned by the method were typically much less than 1% different from optimal. The computational requirements of the method presented varied linearly with increasing numbers of controls and were much lower than those of previously described facet-searching methods, which increase in proportion to the square of the number of controls.
Article
The performance and computational requirements of optimization methods for control allocation are evaluated. Two control allocation problems are formulated: a direct allocation method that preserves the directionality of the moment and a mixed optimization method that minimizes the error between the desired and the achieved moments as well as the control effort. The constrained optimization problems are transformed into linear programs so that they can be solved using well-tried linear programming techniques such as the simplex algorithm. A variety of techniques that can be applied for the solution of tire control allocation problem in order to accelerate computations are discussed. Performance and computational requirements are evaluated using aircraft models with different numbers of actuators and with different properties. In addition to the two optimization methods, three algorithms with low computational requirements are also implemented for comparison: a redistributed pseudoinverse technique, a quadratic programming algorithm, and a fixed-point method. The major conclusion is that constrained optimization tan be performed with computational requirements that fall within an order of magnitude of those of simpler methods. The performance gains of optimization methods, measured in terms of the error between the desired and achieved moments, are found to be small on the average but sometimes significant. A variety of issues that affect the implementation of the various algorithms in a flight-control system are discussed.