Quadrotor Control By Policy Iteration With Signed Derivative Conrado S. Miranda 1 and Janito V. Ferreira 1 Abstract— Proven stable algorithms, such as backstepping, use control constants that may be hard to tune, and either model’s parameters or complex adaptive laws. However, prac- tical applications tend to use simpler controllers that are easier to understand and adjust, such as PID and LQR, although the tunning process may be cumbersome. Based on these simpler controllers, this work presents a quadrotor controller that doesn’t require any vehicle’s parameters knowledge, demanding only an initial parameter set to stabilize the system. These parameters are then adjusted to minimize a given cost function, automating the tunning process for each particular system. Results show that the quadrotor is able to hover and follow a circular trajectory for a wide range of parameters. The technique’s limitations and methods to improve performance are discussed, and future extensions are proposed. I. I NTRODUCTION In the area of aerial vehicles, quadrotors have been the focus of many research topics [1], [2] due to their under- actuated dynamics and miniaturization capabilities [3]. To provide appropriate system’s behaviour, a good controller must be used, and most of them can be classified in two categories. The first one comprises controllers with strong theoret- ical stability guarantees for tracking position and heading references. [4] uses a feedback linearization controller to transform the quadrotor into a linear model, where classi- cal techniques can be used. [5] builds a controller using backstepping, which is extended as an adaptative controller by [6] to allow the quadrotor’s mass to be unknown. [7] presents another backstepping controller with added integral terms for robustness, but considering small angles approxi- mation. These techniques usually require knowledge of many system’s parameters, which may be hard to measure, while ignoring aerodynamic and motor effects, and demanding user chosen parameters, which may be difficult to tune. In some particular cases, robust controllers have been developed to compensate for external disturbances and model uncertain- ties [8], at the cost of introducing more parameters and increasing the controller complexity. Despite the inherent problems caused by model assumptions not being true, such as unmodeled dynamics which may render the system unstable even though the simplified model’s controller has theoretical stability proof, the difficulty in defining their parameters’ values is frequently used as rationale not to use these controllers. The other category is composed of well known traditional controllers originally designed for linear systems control, *This work was supported by FAPESP through the process 2012/01511-6. 1 Conrado S. Miranda and Janito V. Ferreira are with School of Mechanical Engineering, University of Campinas, Campinas, SP, Brazil {cmiranda,janito}@fem.unicamp.br which are used on quadrotors with the assumption that the errors are small and the linear approximation is reasonable. [9] provides a comparison between PID and LQ controllers for quadrotor control, and PIDs are used on other works [10], [11], suggesting that simple controllers may be able to successfully control such systems. Indeed these controllers have been successfully used for many years in projects such as the Paparazzi 2 , OpenPilot 3 , and AeroQuad 4 . One justifi- cation for PIDs use is that they are simpler to understand, making the parameter tunning process more intuitive while time-consuming. Besides these two categories, some algorithms based on machine learning have been developed over the years mainly focused on policy iteration, where a controller’s parameters are modified to minimize a given cost. [12] shows that policy iteration can be used to control a helicopter, although a model must be used to simulate the real system. To solve this problem, [13] introduced the idea of approximating the gradient compute by the policy iteration using an approxi- mate system’s behaviour called signed derivative. In this paper, the signed derivative is used to adjust param- eters for two biased PD controllers, one for the position and another for the attitude, from initial stabilizing controllers. The parameters are adjusted according to a user defined quadratic cost function, so that previous knowledge from de- signing LQ controllers can be used. This paper’s contribution is that the parameter tunning, usually performed by hand, is made online and automatic, requiring no user interaction or vehicle’s parameters. This automatic adjustment allows for much finer tunning, and adaptation to vehicle’s changes while flying. The performance increase if the nominal propeller parameters are known, as they are usually available from the manufacturer and not subject to modifications, is also investigated. The sections are organized as follows. Section II describes the complete quadrotor model used in the simulations to validate the controller. Section III explains the underlying controller used to track the desired trajectory. The signed derivative algorithm is summarized in Sec. IV, and its use on the quadrotor is elaborated in Sec. V. Section VI describes how the experiments are performed, including parameter generation and learning sequence, and Sec. VII shows the results obtained for hovering and following circular trajec- tories. Finally, Sec. VIII outlines the conclusions from the experiments, and provides future research directions. 2 paparazzi.enac.fr 3 www.openpilot.org 4 www.aeroquad.com
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Quadrotor Control By Policy Iteration With Signed Derivative
Conrado S. Miranda1 and Janito V. Ferreira1
Abstract— Proven stable algorithms, such as backstepping,use control constants that may be hard to tune, and eithermodel’s parameters or complex adaptive laws. However, prac-tical applications tend to use simpler controllers that are easierto understand and adjust, such as PID and LQR, although thetunning process may be cumbersome. Based on these simplercontrollers, this work presents a quadrotor controller thatdoesn’t require any vehicle’s parameters knowledge, demandingonly an initial parameter set to stabilize the system. Theseparameters are then adjusted to minimize a given cost function,automating the tunning process for each particular system.Results show that the quadrotor is able to hover and followa circular trajectory for a wide range of parameters. Thetechnique’s limitations and methods to improve performanceare discussed, and future extensions are proposed.
I. INTRODUCTION
In the area of aerial vehicles, quadrotors have been the
focus of many research topics [1], [2] due to their under-
actuated dynamics and miniaturization capabilities [3]. To
provide appropriate system’s behaviour, a good controller
must be used, and most of them can be classified in two
categories.
The first one comprises controllers with strong theoret-
ical stability guarantees for tracking position and heading
references. [4] uses a feedback linearization controller to
transform the quadrotor into a linear model, where classi-
cal techniques can be used. [5] builds a controller using
backstepping, which is extended as an adaptative controller
by [6] to allow the quadrotor’s mass to be unknown. [7]
presents another backstepping controller with added integral
terms for robustness, but considering small angles approxi-
mation. These techniques usually require knowledge of many
system’s parameters, which may be hard to measure, while
ignoring aerodynamic and motor effects, and demanding user
chosen parameters, which may be difficult to tune. In some
particular cases, robust controllers have been developed to
compensate for external disturbances and model uncertain-
ties [8], at the cost of introducing more parameters and
increasing the controller complexity. Despite the inherent
problems caused by model assumptions not being true,
such as unmodeled dynamics which may render the system
unstable even though the simplified model’s controller has
theoretical stability proof, the difficulty in defining their
parameters’ values is frequently used as rationale not to use
these controllers.
The other category is composed of well known traditional
controllers originally designed for linear systems control,
*This work was supported by FAPESP through the process 2012/01511-6.1Conrado S. Miranda and Janito V. Ferreira are with School of
Mechanical Engineering, University of Campinas, Campinas, SP, Brazilcmiranda,[email protected]
which are used on quadrotors with the assumption that the
errors are small and the linear approximation is reasonable.
[9] provides a comparison between PID and LQ controllers
for quadrotor control, and PIDs are used on other works
[10], [11], suggesting that simple controllers may be able to
successfully control such systems. Indeed these controllers
have been successfully used for many years in projects such
as the Paparazzi2, OpenPilot3, and AeroQuad4. One justifi-
cation for PIDs use is that they are simpler to understand,
making the parameter tunning process more intuitive while
time-consuming.
Besides these two categories, some algorithms based on
machine learning have been developed over the years mainly
focused on policy iteration, where a controller’s parameters
are modified to minimize a given cost. [12] shows that policy
iteration can be used to control a helicopter, although a
model must be used to simulate the real system. To solve
this problem, [13] introduced the idea of approximating the
gradient compute by the policy iteration using an approxi-
mate system’s behaviour called signed derivative.
In this paper, the signed derivative is used to adjust param-
eters for two biased PD controllers, one for the position and
another for the attitude, from initial stabilizing controllers.
The parameters are adjusted according to a user defined
quadratic cost function, so that previous knowledge from de-
signing LQ controllers can be used. This paper’s contribution
is that the parameter tunning, usually performed by hand, is
made online and automatic, requiring no user interaction or
vehicle’s parameters. This automatic adjustment allows for
much finer tunning, and adaptation to vehicle’s changes while
flying. The performance increase if the nominal propeller
parameters are known, as they are usually available from
the manufacturer and not subject to modifications, is also
investigated.
The sections are organized as follows. Section II describes
the complete quadrotor model used in the simulations to
validate the controller. Section III explains the underlying
controller used to track the desired trajectory. The signed
derivative algorithm is summarized in Sec. IV, and its use
on the quadrotor is elaborated in Sec. V. Section VI describes
how the experiments are performed, including parameter
generation and learning sequence, and Sec. VII shows the
results obtained for hovering and following circular trajec-
tories. Finally, Sec. VIII outlines the conclusions from the
experiments, and provides future research directions.
To avoid large control values due to non-zero initial error
and to test the performance on inconsistent trajectories, the
values derived from the trajectory used by the controller are
given by
v′ref =
vrefttw, if t ≤ tw
vref , otherwise
where tw is the time window to reach the original trajectory,
such that larger values provide a smoother transition but the
trajectory is inconsistent for longer time periods, and v is
any trajectory derived value used by the controller, such as...p.
VII. RESULTS
There are two different sets of results in this paper. The
first one assumes that the nominal noisy propeller parameters
κ′T and κ′Q are known, so that the matrix M in Eq. (6)
is closer to the real one. The second set assumes that
these parameters aren’t available, and are replaced by their
approximations O(κi).The legend t corresponds to learning using St, while
t′ corresponds to using St′ , where the difference between
the two is discussed in Sec. V-C. The heading angle ψ is
computed from the attitude q considering only a rotation
around z, while the heading error is given by eψ = ψ−ψref .
A. Known propeller parameters
Figure 2 shows the quadrotor performance during hover.
It’s clear that in the first few seconds the controller is
getting used to the dynamics, adjusting the parameters ag-
gressively. Nonetheless, the position stayed within reasonable
boundaries, suggesting that this first learning can happen
in any available space. The signed derivative St had faster
0 20 40 60 80 100 120−4
−2
0
2
4
6
8
tt′
ψ(
)
Time (s)0 20 40 60 80 100 120
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
tt′
px
(m)
Time (s)
0 20 40 60 80 100 120−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
tt′
py
(m)
Time (s)0 20 40 60 80 100 120
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
tt′
pz
(m)
Time (s)
Fig. 2: Upper and lower bounds while hovering with correct parameters, and two options of S.
0 20 40 60 80 100 120−200
−100
0
100
200
0 20 40 60 80 100 120−20
−15
−10
−5
0
5
10
tt′
ψ(
)eψ
()
Time (s)
0 20 40 60 80 100 120−3
−2
−1
0
1
2
3
0 20 40 60 80 100 120−1.5
−1
−0.5
0
0.5
1
1.5
tt′
px
(m)
epx
(m)
Time (s)
0 20 40 60 80 100 120−3
−2
−1
0
1
2
3
0 20 40 60 80 100 120−1.5
−1
−0.5
0
0.5
1
1.5
tt′
py
(m)
epy
(m)
Time (s)0 20 40 60 80 100 120
−1
−0.5
0
0.5
1
1.5
2
tt′
pz
(m)
Time (s)
Fig. 3: Upper and lower bounds for circular trajectory with rT = 2, ωT = 2π4 , tw = 10, correct
parameters, and two options of S.
convergence than its counterpart S′t on some cases, but both
performed similarly most of the time.
It’s worth noting that the boundaries for x and y are
practically constant, presenting minor oscillations. Although
the error is small, it shows a flaw in the controller that must
be kept in mind: the signed derivative in Eq. (23) doesn’t
0 20 40 60 80 100 120−200
−100
0
100
200
0 20 40 60 80 100 120−10
−8
−6
−4
−2
0
2
4
tt′
ψ(
)eψ
()
Time (s)
0 20 40 60 80 100 120−4
−2
0
2
4
6
0 20 40 60 80 100 120−2
−1.5
−1
−0.5
0
0.5
1
1.5
tt′
px
(m)
epx
(m)
Time (s)
0 20 40 60 80 100 120−6
−4
−2
0
2
4
0 20 40 60 80 100 120−2
−1.5
−1
−0.5
0
0.5
1
1.5
tt′
py
(m)
epy
(m)
Time (s)0 20 40 60 80 100 120
−1
−0.5
0
0.5
1
1.5
2
2.5
tt′
pz
(m)
Time (s)
Fig. 4: Upper and lower bounds for circular trajectory with rT = 4, ωT = 2π8 , tw = 20, correct
parameters, and two options of S.
0 20 40 60 80 100 120−5
−4
−3
−2
−1
0
1
2
3
4
5
tt′
ψ(
)
Time (s)0 20 40 60 80 100 120
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
tt′
px
(m)
Time (s)
0 20 40 60 80 100 120−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
tt′
py
(m)
Time (s)0 20 40 60 80 100 120
−1
−0.5
0
0.5
1
1.5
tt′
pz
(m)
Time (s)
Fig. 5: Upper and lower bounds while hovering with approximate parameters and two options of
S.
express any relation between Fz and the position on x and
y due to limitations discussed in Sec. V-C. Therefore, the
controller seem to be able to compensate for lateral wind
only by applying some torque to prevent the quadrotor from
0 20 40 60 80 100 120−200
−100
0
100
200
0 20 40 60 80 100 120−30
−20
−10
0
10
20
tt′
ψ(
)eψ
()
Time (s)
0 20 40 60 80 100 120−4
−2
0
2
4
0 20 40 60 80 100 120−4
−3
−2
−1
0
1
2
tt′
px
(m)
epx
(m)
Time (s)
0 20 40 60 80 100 120−6
−4
−2
0
2
4
0 20 40 60 80 100 120−3
−2
−1
0
1
2
3
tt′
py
(m)
epy
(m)
Time (s)0 20 40 60 80 100 120
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
tt′
pz
(m)
Time (s)
Fig. 6: Upper and lower bounds for circular trajectory with rT = 2, ωT = 2π5 , tw = 10, approximate
parameters, and two options of S.
0 20 40 60 80 100 120−200
−100
0
100
200
0 20 40 60 80 100 120−8
−6
−4
−2
0
2
4
6
tt′
ψ(
)eψ
()
Time (s)
0 20 40 60 80 100 120−5
0
5
0 20 40 60 80 100 120−4
−2
0
2
4
tt′
px
(m)
epx
(m)
Time (s)
0 20 40 60 80 100 120−5
0
5
0 20 40 60 80 100 120−2
−1
0
1
2
3
4
5
tt′
py
(m)
epy
(m)
Time (s)0 20 40 60 80 100 120
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
tt′
pz
(m)
Time (s)
Fig. 7: Upper and lower bounds for circular trajectory with rT = 4, ωT = 2π10 , tw = 20, approximate
parameters, and two options of S.
drifting. However, as the errors in pz are significantly higher,
due to its higher sensitivity to wind, and every position’s
component has the same weight, the controller focus most
of its effort on the altitude.
Figures 3 and 4 show the controller following two different
circular trajectories described by Eq. (24) after learning the
hovering parameters. Even though the underlying controller
is simple, being composed basically of PD controllers with
no angular feedforward, the signed derivative doesn’t have
correct columns scaling, and the propeller parameters may
not be adequate, the learnt parameters allow the vehicle to
follow the trajectory reasonably well.
B. Approximate propeller parameters
Figure 5 shows the controller learning to hover a quadrotor
without any parameter knowledge. Although the errors dur-
ing the first few seconds are higher than the previous case,
the performance after the parameters have nearly converged
is similar to the one presented in Fig. 2. This occurs because,
during hover, the main control effort used is the force
Fz , with the torques near zero, so that the rotors’ speed
coupling isn’t strong. In this situation, O(M) is clearly a
good approximation.
However, the approximation degrades the performance
during highly coupled trajectories, such as the circular ones
in Figs. 6 and 7. In these, the controller isn’t much capable
of dealing with the trajectory inconsistencies during the
initial window, leading to large transient errors. The rotation
speed was lowered from the previous case as the transient
inconsistency on the original speed destabilized the system.
Even though the steady lateral errors also increase, the
vehicle follows the trajectory close enough for some appli-
cations, and is kept stable even with the gradient direction
considerably distorted. Although not presented here, using
unmodified propeller parameters, i.e., setting β = 0 for the
values in Tab. II, the errors are reduced in half, indicating
that the values assigned may not be used in real systems and
the real error may be considerably smaller with appropriate
components choice.
From the position px in Fig. 7 at t = 40, 50, 60, . . .,it’s obvious that the errors aren’t consistent each turn. This
is a result of the incorrect gradient, as the controller tries
to minimize the total error and may end up increasing it.
However, the trajectory is aggressive enough that a static
controller, with the parameters learnt during hover, isn’t able
to stabilize the system. As the learning algorithm searches for
the locally optimal solution, it’s able to change its parameter
settings based on the local error, thus forming a dynamic
system on itself.
VIII. CONCLUSION
This paper presented a simple PD-based controller to stabi-
lize a quadrotor without parameters knowledge using signed
derivative policy iteration. Two approaches were analysed,
where they differed on whether the controller knows only the
nominal propeller parameters or an approximation is used.
Even though the gradient is distorted due to the signed
derivative having incorrect columns scaling, which is a
limitation of the technique, the controller ignores rotor
dynamics and most aerodynamic effects, and doesn’t use
angular feedforward, the vehicle achieved stable hovering
and was able to follow a circular trajectory. However, if
the propeller parameters are unknown, the error on fast
trajectories may be too large for some applications. To the
best of the authors’ knowledge, this is the first quadrotor
controller proposal that requires no parameter knowledge or
hand tunning whatsoever.
If the nominal propeller parameters are known, the per-
formance may be further improved if the position signed
derivative is aware of how the thrust affects the x and ypositions, while keeping the torque knowledge. For the case
where no parameters are known, knowing the ratio between
the propeller parameters may also boost performance. As
the signed derivative isn’t able to learn these parameters,
an approach currently being studied is the use of another
learning algorithm to learn these scalings online.
Simulations have shown that the quadrotor’s transient
behaviour may distance significantly from the desired tra-
jectory. As safety is a major concern for these systems [22],
one may be able to apply reachability sets [23] to disable
learning and switch to a safe controller if necessary. Also,
if the task performed is repetitive, other approaches such as
iterative learning control [24] and trajectory corrections [25],
[26] can also be integrated to compensate for errors caused
by policy iteration limitations, although the effects of this
simultaneous learning are still being analysed.
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