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era (Tang et al., 2018), (Guo and Leang, 2020). Get-
ting a direct measurement of the payload position is
a reasonable option. Even with a relatively low mea-
surement rate compared to the low-level controller, it
allows making aggressive flights (Tang et al., 2018).
However, cameras suffer from several problems like
brightness (close to shadow), flickering (while flying
over the sea), or fogging (in a humid area), making
their use standalone unsafe for reliable industrial ap-
plications. The issue has also been revealed by (Lee
and Kim, 2017), where the proposed solution con-
sists of adding force sensors to the system. Equiva-
lent sensing solutions have been adopted by (Lv et al.,
2021), with the addition of a universal joint between
the drone and the cable for the second one. Recently,
(Panetsos et al., 2022) used four different sensors to
get accurate cable state estimation, while (Outeiro
et al., 2023) proposed an adaptive geometric control
method with asymptotic tracking stability.
A similar approach to ours is described by (Kauf-
mann et al., 2020) and (Cioffi et al., 2022) to perform
agile maneuvers. The use of internal sensing in those
papers is called sensorimotor, but for the unification
of the terms in robotics, we will call it proprioception,
like it has been done in (Lee et al., 2020). Even if we
do not have access to direct measurement of the motor
speed, the PWM (Pulse-Width Modulation) command
gives a fair proportional estimation. A neural network
(NN) is implemented to control the drone. In partic-
ular, the work done by (Cioffi et al., 2022) learns in-
ertial odometry and gets accurate position estimation
without using any visual perception. However, the
positioning tracking has been done with previously
known trajectories. In contrast, for safety reasons, we
implemented a neural network trained by supervised
learning on the estimation stage of the tethered pay-
load controller.
One of the most advanced research on payload
state estimation using quadrotor proprioception is for
parameter estimation (Prka
ˇ
cin et al., 2020). Re-
cently, such a work has been improved by (Prka
ˇ
cin
et al., 2021) with the implementation of an extended
Kalman filter (EKF). However, estimation of load pa-
rameters remains challenging with the employed fast
Fourier transform technique, getting only off-line re-
sults. Unfortunately, even if the research looks in an
interesting direction with a minimal drone setup, real-
time performance has not been reached. Real-world
experiments showed poor results in tracking the an-
gles of the payload. The system is non-linear, and
the IMU is noisy while drones fly at high velocity.
This makes classical EKF implementation unusable
for real flights application. To tackle real-time, we use
the concept of a neural observer (Chen et al., 2018)
with a recurrent NN to access indirectly measured
data (Habtom and Litz, 1997).
1.2 Contribution
In this paper, we prove the feasibility of real-time
position estimation of a cable-suspended end-effector
using only inertia sensors onboard a standard quadro-
tor. Contrary to the classic EKF approach, our method
does not need to define any parameter a priori. We
used a supervised learned network, making the posi-
tion estimation of the suspended load for a standard
quadrotor attitude controller. The neural estimator is
trained in a simulated environment, with domain ran-
domization, and runs the software controller in the
loop. We reached zero-shot generalization of the net-
work for load position estimation. After being trained
on our dataset, the network has the capability to pre-
dict previously unseen perturbations, in particular, di-
rect injection of energy into the end-effector. This
shows the consistency of the estimator implementa-
tion, as well as its generalization capabilities.
2 MATERIALS AND METHODS
Making payload pose estimation is not a trivial task,
and the classical filter approach showed accuracy lim-
itations (Prka
ˇ
cin et al., 2021). In this section, we first
define our model, which is needed for the simulation,
and implement an attitude controller. Then, we focus
on data processing to measure our features to train
the network. Finally, to smooth the training, we grad-
ually increase the measurement domain exploration
with progressively more sophisticated trajectories.
2.1 Model
For the quadrotor, we consider the dynamic model
with the Euler angles (Ollero and Siciliano, 2019).
Here we use FLU (Front-Left-Up) convention to de-
fine the body frame B with axes {x, y, z}, and the
world frame W with axes {x
w
, y
w
, z
w
}. We define the
drone position and its attitude as p
b
= [x
b
, y
b
, z
b
]
⊤
∈
R
3
and η
b
= [φ
b
, θ
b
, ψ
b
]
⊤
∈ R
3
, respectively, with
their time derivatives ˙p
b
and
˙
η
b
. The attitude can also
be defined with R
b
∈ SO(3), the rotation matrix from
B to W , the special orthogonal group of dimension
three, from which the roll-pitch-yaw angles φ
b
, θ
b
,
and ψ
b
, respectively, can be extracted. These give the
linear and angular accelerations of the base as,
¨p
b
= g e
3
+
1
m
u
T
R
b
e
3
, (1)
Neural-Network for Position Estimation of a Cable-Suspended Payload Using Inertial Quadrotor Sensing
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