Payload Swing Estimation and Damping Without Payload Parameters for Multirotor UAVs

K. Taki; K. Umemoto

eess.SYAug 19, 2026

Payload Swing Estimation and Damping Without Payload Parameters for Multirotor UAVs

Authors: K. Taki, K. Umemoto

Organizations: Mechanical Design Engineering, Division of System and Infrastructure Engineering for Safe and Sustainable Society, Graduate School of Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui, 910-8507, Fukui, Japan

Abstract

Cable-suspended payload transport by multirotor UAVs is flexible but generates periodic swing disturbance that degrades tracking and risks instability. Existing anti-swing methods require additional sensors or precise identification of cable length and payload mass, limiting field deployment. We propose a swing-estimation and damping method using only the onboard IMU and throttle command, requiring no payload parameters. An extended Kalman filter extracts the periodic disturbance with the unknown pendulum frequency as an estimated state, and an active damping controller adds a correction angle to the attitude loop to dissipate pendulum energy. Flight experiments confirm robust damping across a tested range of cable-length and mass variations.

Figures & tables

Study	Oscillation Model	Additional Sensors	Prior Knowledge of Vehicle Params.	Prior Knowledge of Payload Mass	Prior Knowledge of Cable Length
Sreenath et al. 2013 [ 13 ]	Spherical pendulum (rigid body on S 2 )	Swing angle/ang. vel.	Moment of inertia, vehicle mass, thrust curve (designed as generalized forces)	Fixed, known	Fixed, known
Guerrero-Sánchez et al. 2017 [ 6 ]	Spherical pendulum (rigid body on S 2 )	Swing angle/ang. vel. (MoCap in experiments)	Moment of inertia, vehicle mass (designed as generalized forces)	Fixed, known	Fixed, known
Sreenath et al. 2019 [ 14 ]	Spherical pendulum (variable-length cable)	Swing angle/ang. vel. (required full state)	Moment of inertia, vehicle mass, pulley inertia/radius (designed as generalized forces)	Fixed, known	Variable, known
Jiroušek et al. 2025 [ 9 ]	Point-mass pendulum (2-DOF planar swing)	None	Moment of inertia, vehicle mass, FCU parameters 1	Fixed, known (robust to a bounded range)	Fixed, known
Bai et al. 2025 [ 2 ]	Point-mass pendulum (2-DOF planar swing)	Swing angle/ang. vel.	Moment of inertia, vehicle mass (designed as generalized forces)	Fixed, known	Fixed, known
Nascimento et al. 2026 [ 11 ]	Spherical pendulum (constrained dynamics)	None	Moment of inertia, vehicle mass, thrust constant $c_{t}$	Fixed, known	Fixed, known

Table 1: Detailed comparison with representative prior studies. Comparison of oscillation model, need for additional sensors, and required prior knowledge of vehicle parameters, payload mass, and cable length.

Figure 1: Control architecture: EKF observer integrated with ArduPilot cascade.

Item	Value
Vehicle mass	1.4 kg
Payload mass	0.378 kg
Nominal cable length $L$ (Gain–Damping / Online $\omega$ Estimation $+$ Damping / Mass Robustness / Zigzag Tracking)	$L=1.04$ m equivalent (FFT frequency approx. $0.64$ Hz)
Flight control	ArduPilot + Pixhawk 6C
Observer parameters	$Q_{W}=0.0005$ , $R_{MEAS}=46.0$ , frequency estimation fixed/online (per experiment)
Attitude ground truth	Motive motion capture

Table 2: Common vehicle and experimental conditions.

$Q_{W}$	FFT $d_{rel,x}$ [Hz]	EKF FreqX [Hz]	EKF–FFT diff. [%]	Convergence time [s]
0.0005 ( $\times 1$ )	0.4734	0.4715	0.40%	8.2
0.0008 ( $\times 1.6$ )	0.4943	0.4563	7.69%	3.9
0.001 ( $\times 2$ )	0.4964	0.4826	2.79%	3.2
0.005 ( $\times 10$ )	0.4520	0.4049	10.4%	12.3
0.10 ( $\times 200$ )	0.4712	0.4048	14.1%	64.8

Table 3:

Q_{W}

sweep results for EKF frequency convergence. FFT

d_{rel,x}

: FFT peak frequency of relative displacement

X

component measured by Motive motion capture. EKF FreqX: EKF-estimated pendulum frequency along

X

axis. EKF–FFT diff.: absolute difference between EKF and FFT frequencies [%]. Convergence time: time for EKF frequency to stay within

\pm 5\%

of FFT peak for 200 consecutive samples (2.0 s).

Q_{W}=0.0005

baseline, cable length

L=1.04

m, initial frequency offset

W_{INIT}=0.600

Hz, control OFF. Bold : best two values in each column.

Figure 2: Convergence time and predicted–estimated frequency difference

|f_{x}^{\text{pred}}-\hat{f}_{x}|

versus cable length

L

for EKF frequency convergence (variable cable length).

|f_{x}^{\text{pred}}-\hat{f}_{x}|

is the absolute difference between the EKF-predicted frequency

f_{x}^{\text{pred}}

and the EKF-estimated frequency

\hat{f}_{x}

, representing the innovation bias that drives EKF frequency updates.

Q_{W}=0.0005

fixed,

n=1

trial per cable length, control OFF.

$L$ [m] [Hz]	FFT $d_{rel,x}$ [Hz]	EKF FreqX [Hz]	EKF–FFT diff. [Hz]	Conv. time [s]
1.04	0.4734	0.4715	0.0019	8.24
0.74	0.5755	0.5617	0.0138	6.57
0.40	0.7024	0.7174	0.0150	2.16
0.28	0.7905*	0.8367	—	Not converged

Table 4: Variable cable length experiment results (

Q_{W}=0.0005

fixed). Conv.: convergence.

Figure 3: Relative displacement

d_{rel,x}

(solid lines) and exponential decay fits (dashed lines) for representative trials. Passive baseline (gain 0.0, blue;

\tau=6.41

s) exhibits slow natural decay, while gain 0.3 (optimal, red;

\tau=2.10

s) achieves rapid damping. Gain 0.6 (green;

\tau=8.01

s) exhibits initial behavioral instability.

Gain $k_{c}$	$\tau$ mean [s]	$\pm 1\sigma$ [s]	Min [s]	Max [s]	$R^{2}$
0.0	6.43	0.01	6.41	6.43	0.78
0.1	5.41	1.22	4.16	6.59	0.66
0.2	2.28	0.83	1.48	3.14	0.72
0.3	2.23	0.24	2.08	2.51	0.55
0.4	4.90	1.73	3.12	6.57	0.16
0.5	8.17	4.52	4.57	13.24	0.40

Table 5: Decay time constant

\tau

statistics across all gain conditions. Hilbert envelope + log-linear regression with residual C. Values are mean

\pm 1\sigma

with min/max range over

n=3

trials.

R^{2}

is the coefficient of determination of the exponential fit.

Figure 4: Relative displacement

d_{rel,x}

(solid lines) and exponential decay fits (dashed lines) for the first 20 s after control ON in the integrated online frequency estimation + active damping experiment (representative trials). Gain 0.0 (passive, blue;

\tau=6.45

s) shows slow natural decay, while gain 0.3 (active, red;

\tau=1.63

s) achieves rapid damping. Online frequency estimation enabled for both conditions.

Gain $k_{c}$	$\tau$ mean [s]	$\pm 1\sigma$ [s]	Min [s]	Max [s]	$R^{2}$
0.0	6.45	0.75	5.95	7.31	0.78
0.3	1.63	0.23	1.36	1.80	0.80

Table 6: Decay time constant

\tau

statistics for the integrated online frequency estimation + active damping experiment. Hilbert envelope

+

log-linear regression with residual

C

. Values are mean

\pm 1\sigma

with min/max over

n=3

trials.

Payload mass [kg]	$\tau$ gain 0.0 [s]	$\tau$ gain 0.3 [s]
0.134	$2.88\pm 0.92$	$1.74\pm 0.83$
0.378	$3.92\pm 0.81$	$0.79\pm 0.20$
0.421	$3.82\pm 0.68$	$2.23\pm 0.61$
0.471	$4.00\pm 0.48$	$6.47\pm 0.57$

Table 7: Decay time constant

\tau

vs. payload mass with active damping (gain 0.3) and passive baseline (gain 0.0).

\tau

obtained from Hilbert envelope log-linear fit of exponential decay model

env(t)=Ae^{-t/\tau}+C

with residual offset

C

correction. Values are mean

\pm 1\sigma

(

n=3

Figure 5: Relative displacement

d_{rel,x}

(solid lines) and exponential decay fits (dashed lines) for gain 0.0 (blue) and gain 0.3 (orange) across payload masses for representative trials. Panels: 0.134, 0.378, 0.421, and 0.471 kg.

\tau

values shown in legends.

Figure 6: Trajectories in the XZ (North–East) plane for zigzag waypoint tracking (representative trials). Blue: quadrotor body (body1). Orange: slung load (body2). Red dashed: reference path. Top: gain 0.0 (correction OFF); Bottom: gain 0.3 (correction ON). Online frequency estimation enabled.

Metric	Gain 0.0	Gain 0.3	Reduction
$\\|d_{rel}\\|_{RMS}$ [m]	$0.3157\pm 0.0195$	$0.2262\pm 0.0138$	28.3 %
$\\|d_{rel}\\|_{max}$ [m]	$0.5561\pm 0.0124$	$0.4082\pm 0.0088$	26.6 %

Table 8: Vehicle-to-payload relative displacement during flight with triangular-wave path (mean

\pm 1\sigma

n=3

\|d_{rel}\|=\sqrt{(p_{b1,x}-p_{b2,x})^{2}+(p_{b1,z}-p_{b2,z})^{2}}

denotes the horizontal deviation between vehicle and payload.

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This work proposes a novel control and estimation approach for aerial manipulation of a cable-suspended load using Unmanned Aerial Vehicles (UAVs). Common approaches in the state of the art have practical limitations, relying on direct load measurements and Lagrangian methods for dynamic modeling. The lack of a straightforward dynamic model of the system led us to propose adopting the Udwadia-Kalaba method to explicitly incorporate the cable's geometric constraints. This formulation allowed for the consistent derivation of the tension force and its direct integration into the NMPC prediction model. Additionally, we propose a sensorless load state estimation based on the same geometric constraints. Results from real-robot experiments demonstrated that the explicit inclusion of load dynamics in the optimization problem significantly reduces trajectory-tracking errors and yields better overall performance compared to strategies based on incomplete models.

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