 Open Access
 Total Downloads : 1222
 Authors : Myunggon Yoon
 Paper ID : IJERTV5IS010073
 Volume & Issue : Volume 05, Issue 01 (January 2016)
 Published (First Online): 02012016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Transfer Function Model of Thrust Dynamics for MultiRotor Helicopters
Myunggon Yoon
Department of Precision Mechanical Engineering GangneungWonju National University,
South Korea
AbstractThis paper proposes a transfer function model for a dynamic thrust of a motordriven propeller widely used for small multirotor helicopters. From frequency responses of a propeller speed with respect to a PWM (pulse width modulation) duty ratio of a motor speed controller, it is found that a thrust dynamics can be accurately modelled as a firstorder transfer function. This frequencydomain result is compared to a time domain method which is based on a step response of a thrust.
KeywordsThrust Dynamics, Transfer Function, Multirotor Helicopter

INTRODUCTION
Obtaining a reliable mathematical model of a thrust force
is one of the most important and difficult steps in a controller synthesis for a multirotor helicopter.
There are fundamental difficulties in modeling a dynamic thrust of a propeller, as addressed in [1]. First, a commercial ESC (electronic speed controller) driving a BLDC (blushless direct current) motor itself is a microprocessorbased digital system and as such it has its own operational parameters but information on those parameters is usually undisclosed. Furthermore, with a custom firmware for an ESC, those parameters and other settings can be also modified. Second, an aerodynamic relation between a propeller speed and a thrust is highly nonlinear and complicated. Furthermore, depending on an operating condition, the relation can be significantly changed.
Because of the aforementioned difficulties, for a reliable characterization of a thrust force, it is generally unavoidable to depend on experimental procedures. This situation motivated us to develop a simple experimental method for a characterization of both a static and dynamic thrust in [2]. The proposed experimental procedure of [2] uses a loadcell type force sensor for a thrust measurement and an optical sensor for a speed measurement. The manual procedure performed in [2] for a static thrust is completely automatized with a microprocessorbased thrust measurement system in [3].
However an identification of a dynamic thrust is much harder compared to that of a static thrust from the following reason. A loadcell sensor for a thrust measurement uses a highgain analog instrument amplifier. This highgain amplification combined with a continual switching of large currents in a BLDC motor which needs to be installed near the loadcell (see Fig. 1), make a thrust sensor signal extremely vulnerable to electrical noises. This noise is significant and cannot be easily removed especially when a propeller speed
changes abruptly. As a consequence of this, a precise measurement of a dynamic thrust is challenging in general.
Nevertheless, we have obtained a rough characterization of a dynamic thrust in [2], assuming that the dynamic relation between an ESC PWM command and a thrust can be described as a firstorder transfer function. Parameters of the firstorder transfer function were estimated from a step response of a thrust with respect to a step PWM command in [2].
However a fundamental limitation in the dynamic thrust model of [2] is that the firstorder dynamic relation between a thrust and an ESC command is presumed, without a sufficient justification. In fact the same firstorder model was also adopted in [4] for an example.
A key contribution of this paper is to experimentally substantiate the presumed firstorder dynamics. To be concrete, we obtained a frequency response between an ESC command and a thrust force, and confirmed that the dynamic thrust can be rather precisely modelled as a firstorder system. Furthermore, our identification method based on a frequency response was compared to the timedomain counterpart in [2].

DYAMIC MODELING OF THRUST

Experimental Setup
Fig.1 from [3] shows our thrust experiment system composed of an ESC, a BLDC motor, a loadcell for a thrust measurement and an optical sensor for a speed measurement. Components of our thrust measurement system in Fig.1 have the technical specifications in Table 1, cited from [1].
Figure 1 Sensor Configuration [3]
TABLE I. COMPONENTS SPECIFICATION [1]
BLDC Motor
Motor Outer Diameter
58.5 mm
Stator Diameter
50.0 mm
Speed per Volt
340 RPM /V
Stator Number
12
Motor Poles
14
Weight
168 g
Propeller
Length
18 inches
Pitch
5.5 inches
Material
carbon fiber
Blade Root Thickness
3.3 mm
Load Cell
Capacity
5 kg
Resistance
1000
Material
Aluminum
Nonlinearity
0.05 %
ESC
Output (continuous)
40 A
Battery
Type
LiPo
Capacity
10000 mAh
Nominal Voltage
22.2 V
Discharging Rate
25C

TimeDomain Method
In this section we will apply the timedomain identification method for a dynamic thrust proposed in [2], to our case study model.
Our ESC has a custom firmware provided by BLheliSuite
14.2.0.1 [5]. This firmware allows us to use a 4 kHz PWM signal as a command signal. Details on this custom firmware and its driving signals can be found in [2].
From experiments, correlations between a propeller angular speed, a static thrust force and a PWM duty ratio were found to be as shown in Fig. 24. In addition, the stepresponse of a thrust with respect to step PWM command given in Fig. 5 is cited from [1]. The data with a label 4kHz is to be used.
It is generally accepted that the relation between a propeller speed and a thrust is static. This static relation in our case could be found from a quadratic interpolation of the data in Fig. 4 as
= 1.18 Ã— 106 2 (1)
where denote the thrust in Newton and denotes the speed in RPM unit.
Figure 2 Speed versus Duty Ration
Figure 3 Thrust versus Duty Ratio
Figure 4 Thrust versus Speed
Figure 5 Thrust Step Response [1]
From the static relation (1), in principle, we have only to identify the dynamic response of either a thrust or a speed with respect to a PWM command. However the measurement of a speed is much more robust to electrical noises, compared to that of a thrust, and therefore we will investigate the dynamic property of a propeller speed first.
As a first step, we chose an operating point of the duty ratio of an ESC command input as = 30 %. Then from Fig. 2 the corresponding operating point of a propeller speed is around = 2400.
From a linearization
= + , = + ,
we need to identify two unknown parameters (, ) in the firstorder transfer function
()
() = + 1/ (2)
where (), () denote the Laplace transform of the input () and the output (), respectively.
In(2), the unknown parameter can be determined from the step response shown in Fig. 5. Specifically, by reading a rising time 0.16 (sec) from Fig. 5, we obtain = 0.16.
As a second step, for the unknown parameter , we note that the DC (static) gain

Freqency Response Method
In this section, without resorting to the assumption that the transfer function from a PWM dutyratio command to a thrust is a firstorder system, we measure the frequency response of a thrust and characterize its transfer function representation.
The frequency response of a thrust, with respect to a PWM dutyratio, was obtained with a dynamic signal analyzer Agilent 35670A [6].
As our signal analyzer is an analog device, it was necessary to convert an analog input signal from the signal analyzer to a digital PWM signal for an ESC, and a digital pulse train from a photo sensor for a speed measurement into an analog output signal whose voltage is proportional to a propeller speed. In addition, in order to handle negative voltages from the analyzer, an operational amplifier circuit was designed for a voltage shift. Furthermore, the zero voltage of both input and output signals of the analyzer were mapped to an operating point in the previous section.
(0) =
=
(0) 1/
should be equal to the slop 60.8 (rpm/duty (%)) of a tangential line at = 30 % as illustrated in Fig. 2. This gives = 60.8 = 380.0.
In summary, around an operating point ( , ) = (30, 2400) , the transfer function between a dutyratio command and a propeller speed is given as
()
380.0
() = + 6.67 (
% ) (3)
Figure 6 A Schematic Diagram
Note that at the operating point the thrust force (1) can be
linearized as
The analogdigital conversion, digitalanalog conversion
= + 2 + 2
= 6.80+0.0057 ( = 1.18 Ã— 106)
(4)
and the mapping between voltage signals and an operating point were implemented with a microprocessor (Arduino Due
From this result, the transfer function between a duty command and a thrust at an operating point ( , ) = (30, 6.80) is given
() 2.17
Â©)based thrust measurement board described in [3]. A schematic diagram of various signal conditioning of our experiment are illustrated in Fig. 5.
Note that in Fig. 5 the ADC has a DC gain = 10 (%) and
() = + 6.67 (
) . (5)
y 1
%
We note that in our experimental data in Fig. 4, the nominal
the DAC has a gain =
denote the output (in
1000
( ) where () and ()
(output) of the signal
thrust is around 6.34 (N) which is slightly different from the estimated value 6.80 in (4).
put) and input
analyzer (thrust system, respectively). As a result, the relation
() Y()
() = 100 U() (6)
holds where the transfer function Y()
U()
corresponds to a
frequency response that the signal analyzer will estimate.
The signal analyzer under the configuration of Fig. 6 gave the frequency response in Fig. 7. The frequency span was [0.1,10] (Hz) and frequency responses at 401 different frequencies were measured with a swiping sinusoidal signal.
A critical implication from both the magnitude and phase plots in Fig. 7 is that, qualitatively, the transfer function between a PWM command and a propeller speed (and a thrust, too) is a typical firstorder transfer function.
This frequencydomain observation justifies our previous assumption of the firstorder transfer function.
For a quantitative analysis of the frequency response data, we made a comparison between the transfer function in (3) which is based on a stepresponse of a thrust, and the frequency response data. To be concrete, in Fig. 8, the dashed blue line is
the Bode plot of the transfer function 1 () and the solid red
100 ()
line is the experimental data in Fig. 7. The scaling factor 1
100
comes from the relation (6).
The two frequency responses in Fig. 8 show a surprising agreement in overall, even though our choice of the rising time = 0.16 (sec) from Fig. 5 was not precise. This result also suggests that the timedomain approach in [2] for a thrust transfer function is reliable.
Finally, Fig. 9 shows an example of inputoutput signals of the signal analyzer during a frequency response experiment. The frequency span in this figure is around [2.5, 5.5] (Hz).
Figure 7 Speed Frequency Response
Figure 8 Comparison of Frequency Responses
Figure 9 Input and Output Signals of Analyzer


CONCLUSION
A frequency response of a dynamic thrust with respect to a PWM dutyratio command for an electronic speed controller was obtained with a dynamic signal analyzer. It was found that the transfer function between a PWM command and a thrust can be precisely modelled as a firstorder transfer function. This transfer function gives a good agreement with another transfer function estimated from a step response of a thrust.
REFERENCES

M. Yoon, On Driving Signal of Electronic Speed Controller for Small MultiRotor Helicopter, International Journal of Engineering Research and Technology, 4 (11), pp. 456459, 2015.

M. Yoon, Experimental Identification of Thrust Dynamics for a Multi rotor Helicopter, International Journal of Engineering Research and Technology, 4 (11), pp. 206209, 2015.

M. Yoon, An Automatic Thrust Measurement System for Multirotor Helicopters, International Journal of Engineering Research and Technology, 4 (12), pp. 346350, 2015.

G. M. Hoffmann, H. Huang, S. L. Waslander and C. J. Tomlin Precision flight control for a multivehicle quadrotor helicopter testbed Control Engineering Practice 19, pp. 10231036, 2011.

BLHeliSuite Available at: https://blhelisuite.wordpress.com/ [Accessed 16 November 2015].

Keysight Technology Available at: http://www.keysight.com/en/pd 1000001335%3Aepsg%3Apropn35670A/fftdynamicsignal analyzerdc1024khz?cc=US&lc=eng [Accessed 29 December 2015].