MEV


 

 

Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 
 

Journal of Mechatronics, Electrical Power, 
and Vehicular Technology 

 
e-ISSN: 2088-6985  
p-ISSN: 2087-3379  

 

 

mev.lipi.go.id 

 
 

 
doi: https://dx.doi.org/10.14203/j.mev.2022.v13.101-112 
2088-6985 / 2087-3379 ©2022 National Research and Innovation Agency  
This is an open access article under the CC BY-NC-SA license (https://creativecommons.org/licenses/by-nc-sa/4.0/) 
MEV is Scopus indexed Journal and accredited as Sinta 1 Journal (https://sinta.kemdikbud.go.id/journals/detail?id=814) 
How to Cite: J. A. Prakosa et al., “Experimental studies of linear quadratic regulator (LQR) cost matrices weighting to control an accurate take-off 
position of bicopter unmanned aerial vehicles (UAVs),” Journal of Mechatronics, Electrical Power, and Vehicular Technology, vol. 13, no. 2, pp. 
101-112, Dec. 2022. 

Experimental studies of linear quadratic regulator (LQR) cost 
matrices weighting to control an accurate take-off position of 

bicopter unmanned aerial vehicles (UAVs) 

Jalu Ahmad Prakosa a, *, Hai Wang b, Edi Kurniawan a, Swivano Agmal c,  
Muhammad Jauhar Kholili c 

a Research Center for Photonics, National Research and Innovation Agency (BRIN) 
PUSPIPTEK, South Tangerang City, Indonesia 

b Discipline of Engineering and Energy, Murdoch University 
90 South Street, Murdoch, Western Australia, 6150, Australia 

c Research Center for Quantum Physics, National Research and Innovation Agency (BRIN) 
PUSPIPTEK, South Tangerang City, Indonesia 

 
Received 27 March 2022; 1st revision 22 May 2022; 2nd revision 29 May 2022; Accepted 30 May 2022; Published online 29 December 2022 

 
 

Abstract 

Controller design for airplane flight control is challenged to achieve an optimum result, particularly for safety purposes. 
The experiment evaluated the linear quadratic regulator (LQR) method to research the optimal gain of proportional-integral-
derivative (PID) to hover accurately the bicopter model by minimizing error. The 3 degree of freedom (DOF) helicopter facility 
is a suitable bicopter experimental simulator to test its complex multiple input multiple output (MIMO) flight control model to 
respond to the challenge of multipurpose drone control strategies. The art of LQR setting is how to search for appropriate cost 
matrices scaling to optimize results. This study aims to accurately optimize take-off position control of the bicopter model by 
investigating LQR cost matrices variation in actual experiments. From the experimental results of weighted matrix variation on 
the bicopter simulator, the proposed LQR method has been successfully applied to achieve asymptotic stability of roll angle, 
although it yielded a significant overshoot. Moreover, the overshoot errors had good linearity to weighting variation. Despite 
that, the implementation of cost matrices is limited in the real bicopter experiment, and there are appropriate values for 
achieving an optimal accuracy. Moreover, the unstable step response of the controlled angle occurred because of excessive 
weighting. 

©2022 National Research and Innovation Agency. This is an open access article under the CC BY-NC-SA license 
(https://creativecommons.org/licenses/by-nc-sa/4.0/). 

Keywords: experimental evaluation; cost matrices; LQR; bicopter; MIMO flight control. 

 
 

I. Introduction 
Research aircraft flight control [1] is complicated 

due to its dynamic behavior and uncertainties. The 
nonlinear behavior of plants also must be taken to 
account. Not only rotors [2], [3] characteristics but 
also aerodynamic behaviors of aircraft cause its 
uncertainty and dynamic. Besides, it must control an 
airplane's rotation angles in three dimensions: pitch, 
roll, and yaw. Interaxis coupling between angles 
adds a more complex task [4]. Because of these 
reasons, flight control is an exciting research topic. 

Flight control has a vital role in running unmanned 
aerial vehicles (UAVs) [5], [6] well, becoming famous 
today. The application of UAVs in remote sensing, 
surveillance, and disaster mitigation has gained 
more attention because of their relatively low cost, 
more accurate, and higher maneuvers possibility 
than on-board pilots [1]. The excellent development 
of UAVs flight control can ensure its application 
successfully, although the major use of DC motors 
provides challenges. 

The helicopter is a UAV type with excellent 
maneuverability and versatility for flight control 
studies. Moreover, this type has more advantages 
than fixed wings UAVs in smaller landing areas [7]. 
Bicopter is a helicopter-type with two rotors on both 

 
 
* Corresponding Author. Tel: +62-8888311358 

E-mail address: jalu.ahmad.prakosa@brin.go.id 

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J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 

 

102 

arms for its maneuver, which is famously 
implemented by the Chinook helicopter [8]. The 
plant of 3 DOF helicopters for the laboratory, which 
Quanser Consulting Inc. develops, is a useful 
experimental tool to test the flight control method 
strategies for the bicopter model [9]. Two propellers 
to generate thrust for lifting the helicopter are 
driven by both DC motors. The aircraft has free 
movement to pitch from its center on one side of the 
arm. Moreover, the arm allows the helicopter body 
to move in the elevation and yaw directions. Here is 
Figure 1, which describes the bicopter simulator. 

Figure 1 shows that a counterweight is used as a 
balancing from the propellers. Encoder sensors 
install some joints; therefore, the helicopter's 
angular rotation, namely pitch, roll, and yaw, can be 
measured accurately as feedback elements. The data 
acquisition is connected between the personal 
computer (PC) and motor through a driver and 
amplifier. The pitch, roll, and yaw measurement is 
delivered via its data acquisition to PC. The 
application software of MATLAB Simulink can be 
used to design the flight control technique due to its 
accessibility to the plant. Hence, the 3 DOF 
helicopter is a suitable experimental platform to test 
and validate new flight control strategies to deal 
with dynamic behavior, nonlinearities, and the 
helicopter's uncertainties as a UAV plant. 

Besides implementing adaptive control with LQR 
[10] to deal with dynamic behavior, nonlinearities, 
and uncertainties, the method of optimal control can 
also be used. Optimal control is a method to achieve 
the desired dynamic and deal with researching a 
control law system over a while to optimize an 
objective function. Some researchers applied many 
optimal control theories to the helicopter as an 
object. The application of the H-infinity optimal 
control [3] approach was analyzed to solve high-
order unmodeled dynamics on a helicopter. Sliding 
Mode Control [11] was simulated and studied for a 
laboratory helicopter of 3 DOF [12]. Robust LQR 
attitude control [13], [14] was built for aggressive 
maneuvers on 3 DOF helicopters, both theory and 
experiment [15]. LQR theory minimizes the cost 
function to solve dynamic system operation, 
commonly implemented in flight control [16]. 

The research of cost matrices variation of LQR 
algorithm for flight control [17] is not only exciting 
but also challenging to be conducted. The weighting 

of cost matrices [18] supplied by an engineer may 
have different behavior on another plant. The 
platform of 3 DOF helicopters should be used to 
study experimentally optimal flight control 
strategies for UAV type of bicopter [19]. This 
research aims to investigate the cost matrices 
variation of LQR theory to evaluate the most 
accurate flight control experimentally for the 
bicopter plant as an optimization strategy. The 
efficient weighting on cost matrices should be 
analyzed to achieve optimal flight control strategies 
[20]. The optimal desire can be obtained by 
minimizing its error or cost function in the LQR term. 
Therefore, the accurate position of the helicopter 
must be achieved to ensure its safety. In addition, it 
would succeed the application of UAV. Besides, the 
following points provide a summary of our 
contributions: 

• Demonstrate MIMO control system of the 
bicopter model through PID controller signal. 

• Test the strategies of optimal control theory on 
experimental tools of 3 DOF helicopters as the 
bicopter simulator.  

• Investigate cost matrices weighting variation 
of LQR optimization calculation. 

• Assess the accuracy of flight position on the 
LQR method through each error. 

• Develop cost matrices of LQR calculation to get 
minimal errors of angle control from bicopter 
experimental results. 

This activity is conducted as the following: 
Section II indicates the modeling of the bicopter 
system. Cost matrices variation of LQR calculation is 
investigated and discussed in Section III as 
experimental evaluation. Finally, Section IV presents 
the conclusion of this work. 

II. Materials and Methods 
A. Design and method 

The angles of rotation of 3-DOF helicopter is 
analogous with Chinook bicopter description; 
therefore, the pitch, roll, and yaw can be seen in 
Figure 2 as general aircraft flight control below:  

The set of the angle's rotation for flight control of 
Chinook's bicopter in Figure 2 is implemented to the 
3 DOF helicopter facilities. Its free body diagram is 
illustrated in Figure 3. 

 
Figure 1. The experimental platform of 3 DOF helicopter as bicopter simulator 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 
 

 

103 

The angles of rotation and the force illustration 
are described in Figure 3. The signs of each axis for 
angles of rotation are determined below: 

a) The roll angle lies in the direction of the X-axis 
rotation. The horizontal position of a 
helicopter if r(t)=0, then roll angle becomes 
positive, r(t)>0, a helicopter flies higher than 
counterweight. 

b) The yaw angle is in the direction of the Z-axis 
rotation. The yaw angle is positive, ɣ(t)>0, if it 
rotates counterclockwise (CCW) direction.  

c) The pitch angle is situated in the direction of 
the Y-axis rotation. The pitch angle increases 
positively, p(t)>0, when the front motor is 
higher than the back motor. 

The Newton’s Law I works when the helicopter 
hovers stationary or moves at a steady speed. 
Because of that reason, the whole force sum on the 
free body diagram of the helicopter is zero when it 
hovers. Those circumstances are illustrated in 
equations (1), (2), (3), and (4). 

0=−=∆
W

F
T

FF  (1) 

The total torque is also zero. 

0WT =τ−τ=τ∆  (2) 

gbmaLfmaLwmwLfKaLoV )(2 −−=  (3) 

fKaL

gbmaLfmaLwmwL
oV

2

)( −−
=  (4) 

The equation (4) shows the voltage (Vo) produced by 
the amplifier to drive the DC motor. The 
mathematical model of control systems design for 
analysis usually uses a state-space model that 
applies state variables to describe a system by a set 
of first-order differential or difference equations in 
equation (5) and Figure 4. 

 DuCxy;BuAxx +=+=
•

 (5) 

where: x = state vector; y = output vector; u = input 
vector; A = state matrix; B = input matrix; C = output 
matrix; D = feedforward matrix. 

The state vector is defined as equation (6): 





 γγ=

•••

prprx T  (6) 

On the other hand, the output vector can be written 
as (7): 

[ ]γ= pry T  (7) 

 
Figure 2. Angles rotation of bicopter 

 

Figure 3. Free body diagram of 3 DOF helicopter 

 
Figure 4. State-space model 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 

 

104 

Furthermore, the other state-space matrices are 
set in equation (8), (9), (10), and (11): 



























++

−−

=

0000
2

aL.fm.2
2

hL.fm.2
2

wL.wm

g).bm.aLfm.aLwm.wL(0

000000
000000
100000
010000
001000

A

 (8) 































−

++

=

00

2
hL.fm.2

fK
2

hL.fm.2

fK

2
aL.fm.2

2
wL.wm

fK.aL
2

aL.fm.2
2

wL.wm

fK.aL
00
00
00

B

 (9) 














=

000100
000010
000001

C  (10) 














=

00
00
00

D  (11) 

B. Proposed cost matrices variation of LQR 
algorithm 

The LQR theory applied a mathematical 
algorithm that minimizes a cost function by 
weighting factors supplied by an engineer. The sum 
of the deviations of critical measurements, such as 
pitch, roll, and yaw on flight control, is often called 
the cost function. These settings are usually 
implemented on machines or processes, particularly 
on airplanes. Further, the LQR algorithm is an 
automated method of finding an optimal state-
feedback controller of close loop systems [21] as 
shown in Figure 5. 

From the state-feedback law in Figure 5, the 
input vector is indicated in equation (12): 

Kxu −=  (12) 
State-feedback gains K becomes the optimal gain 

matrix to minimize the quadratic cost function. 

Input, u, provides the optimal control signal when 
the performance index function J is minimized on 
equation (13). Selecting appropriate weighting 
matrices Q and R that indicates an important 
relationship between the state error and control 
signal in the performance index function aims for 
the optimal controller design. 

dt
0

)NuTx2RuTuQxTx()u(J ∫
∞

++=  (13) 

Generally, for simple, N is set to 0 (N=0) and 
becomes equation (14). 

dt
0

)RuTuQxTx()u(J ∫
∞

+=  (14) 

Principally, the optimal gain of LQR can be 
determined by solving an algebraic Riccati equation. 
However, it is not efficient in computation time. 
Therefore, the calculation is held by the application 
program of MATLAB Simulink. The integrals of the 
roll and yaw states are included in equation (15) as 
augmented state vectors from equation (6). 







∫∫
•
γ

••
γ= prprprTx  (15) 

Because the updated state vector in equation (15) 
has eight columns, the cost matrix Q also has an 8x8 
matrix size in equation (16).  

































∫
∫

•
γ

•

•
γ

p0000000
0r000000
0000000

000p0000

0000r000

0000000
000000p0
0000000r

 (16) 

Then the weighting matrix, R, is written in equation 
(17). 









=

c0
0c

R  (17) 

The constant values, c, are chosen 0.01<c<1, for 
example c=0.05. 

State-feedback gains K are calculated by LQR 
calculation on equation (14) and state-space model 
of equation (5) to generate PID coefficient gains of 
motor voltage in equation (18).  













=

8,2
K

7,2
K

6,2
K

5,2
K

4,2
K

3,2
K

2,2
K

1,2
K

8,1
K

7,1
K

6,1
K

5,1
K

4,1
K

3,1
K

2,1
K

1,1
K

K
 (18) 

 
Figure 5. LQR close-loop diagram 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 
 

 

105 

The K matrix on 2x8 indicates the number of 
rows as output for two motor voltages, namely front 
motor voltage VF and back motor voltage VB. Further, 
the number of columns shows the PID gains, which 
are the third of the first columns for proportional 
coefficient KP, the following two columns for 
Derivative coefficient KD, and the last two columns 
for Integral coefficient KI. Therefore, equation (18) is 
developed from (15). This MIMO controller [22] of 3 
DOF Helicopter can be generally illustrated in 
Figure 6. 

The MIMO [23] diagram (Figure 6) explains 
clearly why the state-feedback gains K have two 
rows and eight columns as K2x8. The final outputs 
are voltages to drive both front and back motor from 
PID gains of K on equation (18) and motor voltage of 
Vo on equation (4) as following equation (19): 









+=









o

o
8,2

B

F

V
V

K
V
V

 (19) 

Investigation of cost matrices variation on LQR 
algorithm is proposed to research roll (r) weighting 
adjustment of Q matrix in the real experiment on 
bicopter simulator facilities because this angle is 
directly related to take-off position. The Stochastic 
Fractal Search (SFS) has been applied to optimize 
LQR through Q and R weighting matrices in 
quadcopter simulation [24]. On the other hand, this 
research wants to investigate weighting matrices 
variation in an actual experiment of bicopter to 
optimize its accuracy. 

III. Results and Discussions 
The variation of cost matrix Q on equation (16) 

was implemented by weighting adjustment to r 
angle due to the importance of helicopter take-off 
circumstances. The factory initially set the roll and 
yaw by 100 and 10, respectively. Further, the pitch 
was set to 1 then the other values of derivative and 
integral followed it. Attentively, the priority of a 
controlling factor had the heavier weight component 
due to LQR optimization theory. Hence, r: p: ɣ= 100: 
1:10 meant that the r angle control was more crucial 
100 times than p angle control. It also implied that 
the r angle control was more priority ten times than 
ɣ angle control. We only select yaw as the 
representative angle because of interaxis coupling 

between pitch and yaw angles. The prime concern of 
angle rotation is related to the take-off position of 
bicopter. The default of matrix Q from equation (16) 
was filled to equation (20): 





























=

1.00000000
010000000
00200000
00000000
00000000
000001000
00000010
0000000100

Q  (20) 

Because of that reason, the variation of r 
weighting is selected from 25 to 1000 scale. There 
were three measuring points of r angles at -10°, 0°, 
and 10°. These angles (Figure 7) were implemented 
as the take-off position of the helicopter. Time 
measurement was collected for 20 seconds. 

The comparison on more than one angle 
measuring point must be conducted to ensure the 
LQR optimal control algorithm's effectiveness on the 
angle control method. Due to the focus on r angle as 
the helicopter's take-off position, the other angles, 
namely p and ɣ, were set to 0°. Firstly, the 
experimental results by setting an angle of roll and 
yaw at -10° are shown in Figure 8 and Figure 9, 
respectively. 

The proposed LQR methods in equations (14), 
(16), (18), and Figure 6 have successfully controlled 
both roll and yaw angle at -10° measuring point. The 
variation of weighting on the Q cost matrix affected 
the stability of steps response of r angle at -10° 
setting point. The stability of angle control could be 
calculated by error, e, which was the difference 
between a measured angle and a set angle. Besides, 

 

Figure 6. The Overall MIMO control diagram of bicopter 

 
Figure 7. Three measuring points of r angle 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 

 

106 

the error could be analogous to the cost function, 
which should be minimized for control optimization 
in the LQR term. The performance index of the 
quadratic cost function, J, as in equation (14), was 
minimized as the error in this research to equation 
(21).  

dt
0

)RuTuQxTx()t(e ∫
∞

+=  (21) 

All symbols are explained in Table 1. The 
illustration of cost matrices scaling of LQR theory 
[18] is described in Figure 10, which aims to 
minimize error as a equation (21). In other words, 

the larger weight of cost matrices [25] was to 
achieve an optimal accuracy position of angle 
control. How to find appropriate cost matrices 
weighting to optimize desired was an art of LQR 
setting. Figure 8 showed that the proposed LQR 
method has successfully minimized errors to zero, 
which achieved asymptotic stability since errors 
closed to be zero as time ran in unlimited time 

( 0lim =
∞→

e
t

). Moreover, the excessive loading by 

1000 produced a huge overshoot helicopter position. 
It indicated that weighting on cost matrices was only 
effective for certain values in the real experiment, 
and if it is exaggerated on weighting, it would 
decrease the accuracy. 

 

Figure 8. Steps response of roll angle at -10° point 

 

Figure 9. Steps response of yaw angle at -10° point 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 
 

 

107 

Fascinating graphs occurred in Figure 9 which 
the r weighting variation evidently took effect to ɣ 
angle. The peaks of step response on ɣ angle before 
5 s followed the amount of r weighting, which 
suited the LQR objective. The smaller error at 20 s, 
e(t), tended to be caused by the larger proportion of 

r angle variation, except for the 1000 weight 
condition. Next, the larger control r angle at 0° and 
10° has already given the data in Figure 11, Figure 12, 
Figure 13, and Figure 14. 

Although the vast weighting of LQR calculation 
by 1000 has successfully gotten asymptotic stable at 
-10°, which errors tend to zero in the time domain, it 
was not stable for higher angle position by 0° and 
10°, respectively. The results characteristic at 10° 
was not different from the behaviour at 0°. Moreover, 
the overloading by 1000 scale conducted the 
unstable bicopter position on both angle points. This 
case indicated that the weighting factor in the LQR 
algorithm had limited effectiveness in the actual 
experiment. For instance, when roll angle was 
treated by significant cost matrices factor (10) and 
weighting variation, it should not produce a larger 
overshoot by heavier weight as the LQR algorithm 
[26]. The not optimal result would be achieved, but 
the worst outcome would be treated by excessive 
weighting. The control inputs in equation (12), for 
example in 0° point, should be observed further to 
analyze the cause of overshoot and unstable plant.  

Condition of control inputs from amplifier 
voltage could explain how the overshoots occurred 
in the bicopter plant in Figure 15. Moreover, an 
unstable plant might happen because of the 
saturated signal by 1000 scale. Here is Table 2, which 
resumed error for only yaw angle because whole roll 
position points, except in 1000 weight, have 
achieved zero errors in asymptotic stability for 20 s. 
The overshoot values and their errors should be 
investigated not only on cost weighting variation of 
LQR calculation but also by variation of measured 
control angle position, as tabulated in Table 3 and 
Table 4. 

Table 1. 
Symbols description 

Parameter Symbol Value (Unit) 

Roll angle  r(t) ° (degree) 

Pitch angle  p(t) ° 

Yaw angle  ɣ(t) ° 

Initial error  ei(t) ° 

Final error  ef(t) ° 

Error  e(t) ° 

Gravitation acceleration  g 9.81 m/s2 

Thrust force  FT N 

Thrust torque  τW N.m 

Weight  W N 

Torque because of 
gravitation  

τT N.m 

Counterweight mass  mw 1.87 kg 

Front motor mass  mf 0.575 kg 

Back motor mass  mb 0.575 kg 

Distance between yaw axis 
and helicopter body  

Lw 0.4699 m 

Distance between yaw axis 
and counterweight  

La 0.6604 m 

Distance between pitch axis 
and each motor  

Lh 0.1778 m 

Motor force-thrust constant  Kf 0.1188 N/Volt 

Gain constant in two rows 
and eight columns  

K2x8 Volt 

Quiescent voltage for motor  Vo Volt 

State vector  x - 

State matrix  A - 

Input matrix  B - 

Output matrix  C - 

Feedforward matrix  D - 

Time  t Second (s) 

State-feedback gain  K - 

Performance index function J - 

Cost matrix  Q - 

Weighting matrix  R - 

Weighting matrix  N - 

Constant in N matrix  c - 

Front motor voltage  VF Volt 

Back motor voltage  VB Volt 

Proportional gain 
coefficient  

KP - 

Integral gain coefficient KI - 

Derivative gain coefficient KD - 
 

 
Figure 10. LQR optimization calculation of cost matrices weighting 
to minimize error or maximize accuracy 

Table 2.  
Errors analysis of cost matrices variation on LQR algorithm 

Weighting variation 
Yaw control angle 

-10° 0° 10° 

25 4.92° 2.46° 1.27° 

50 4.17° 3.08° 1.76° 

75 4.17° 1.93° 3.08° 

100 4.61° 1.89° 2.15° 

200 4.44° 2.81° 1.63° 
 

Table 3.  
Overshoot evaluation of cost matrices variation on LQR algorithm 

Weighting variation 

Overshoot at measured control angle position 

r ɣ 

-10° 0° 10° -10° 0° 10° 

25 -2.01° 9.59° 21.98° 5.32° 9.59° 21.98° 

50 -2.01° 10.12° 22.69° 5.54° 10.12° 22.69° 

75 -1.66° 10.64° 23.13° 5.76° 10.64° 23.13° 

100 -1.84° 11.00° 23.56° 5.27° 11.00° 23.56° 

200 -0.69° 12.49° 24.88° 4.97° 12.49° 24.88° 

1000 2.21° 17.15° 27.06° 9.93° 17.15° 27.06° 
 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 

 

108 

Overshoot errors mean that difference between 
overshoot point and the set angle. mean square error 
(MSE) is the one of performances index to assess the 
accuracy of experimental results [27]. The purpose of 

LQR theory is to minimize the cost function, namely 
error in this case, which can be easier to be 
investigated by MSE equation (22). 

Table 4. 
Overshoot errors investigation 

Weighting variation 

Overshoot errors 

r ɣ 

-10° 0° 10° -10° 0° 10° 

25 7.99° 9.59° 11.98° 15.32° 9.59° 11.98 

50 7.99° 10.12° 12.69° 15.54° 10.12° 12.69 

75 8.34° 10.64° 13.13° 15.76° 10.64° 13.13 

100 8.16° 11.00° 13.56° 15.27° 11.00° 13.56 

200 9.31° 12.49° 14.88° 14.97° 12.49° 14.88 

1000 12.21° 17.15° 17.06° 19.93° 17.15° 17.06 
 

 
Figure 11. Steps response of roll angle at 0° point 

 
Figure 12. Steps response of yaw angle at 0° point 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 
 

 

109 

2))t(
n

1i
e(

n
1

MSE ∑
=

=  (22) 

In this case, the minimum cost function, which 
was errors, was required to achieve optimal accuracy 
because the measurement accuracy [21] was 
essential to assure flight safety. Interestingly, errors 
in yaw angle measurement in Table 2 indicated that 
the lower angle position got the higher errors. On 
the other hand, overshoot errors were larger in a 
higher angle position in Table 3, which is described 
for roll and yaw angles in Figure 16 and Figure 17, 
respectively. Furthermore, the MSE calculations have 

different results for various measuring points in 
Table 5. Nevertheless, the higher weighting tends to 
yield larger MSE values. 

A fascinating event occurred between overshoot 
errors and weighting, which had excellent linearity 
by R2 over 0.9. Additionally, the measuring point of 
0° achieved the best linearity because it got 
R2=0.9922 both in roll and yaw angles. The good 
linearity of both relations might link to the “Linear” 
term of LQR knowledge, particularly in equation (13). 
These findings can develop an accurate take-off 
position of bicopter UAVs according to their needs 
for future research. The possibility problem was in 
this case, which achieved the majority of asymptotic 

 

Figure 13. Steps response of roll angle at 10° point 

 

Figure 14. Steps response of yaw angle at 10° point 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 

 

110 

stability because roll angle control position directly 
corresponded to the primary disturbance of 
gravitation acceleration, g, as equation (4) (See 
Figure 3 and Figure 7). Nevertheless, yaw angle 
errors had an indirect connection to gravitation 

disturbance so that it produced different results, 
which took various errors. These observations could 
inspire building modification method on the LQR 
technique for accurately hovering bicopter. 

 
Figure 15. Control inputs 

 
Figure 16. Overshoot errors of roll angle relation to weighting variation in different angle position 

 

 
Figure 17. Overshoot errors of yaw angle relation to weighting variation in different angle position 



J.A. Prakosa et al. / Journal of Mechatronics, Electrical Power, and Vehicular Technology 13 (2022) 101-112 
 

 

111 

Table 5. 
Mean square error analysis 

Weighting variation 

Mean square error 

r ɣ 

-10° 0° 10° -10° 0° 10° 

25 12.79° 31.98° 62.01° 24.70° 28.55° 23.04° 

50 12.76° 31.86° 64.29° 24.45° 30.27° 32.56° 

75 12.56° 32.55° 63.89° 24.20° 34.06° 30.33° 

100 12.64° 31.93° 63.42° 23.42° 26.52° 26.63° 

200 12.97° 33.85° 64.78° 21.30° 28.26° 28.48° 

1000 16.52° 131.08° 155.96° 35.89° 1688.3° 1105° 

 

IV. Conclusion  
Experimental data from the bicopter simulator 

described that the proposed LQR algorithm has 
successfully controlled angle rotation through PID 
gains since the heavier weighting factor yielded a 
more accurate yaw angle. Moreover, the roll control 
angle achieved asymptotic stability, although it 
produced large overshoots. Overshoot errors tend to 
have a linear relation with the weighting variation of 
LQR cost matrices, particularly for higher angle 
positions. Despite that, the excessive weighting on 
controlling roll angle rotation even gave unexpected 
results, which the larger weight provided the less 
accurate plant. In addition, the higher weighting 
tends to produce the larger MSE values. The direct 
interference from gravitation may cause the 
difference in the LQR algorithm on flight control 
optimization between yaw and roll rotation angle. 
The modified control design and LQR algorithm 
challenge future research to optimize this UAV 
position control, especially accuracy optimization. 

Acknowledgements 

The authors are grateful to the management of 
the National Research and Innovation Agency of 
Republic Indonesia and Murdoch University for 
supporting this research. 

Declarations  
Author contribution  

J.A. Prakosa: Writing - Original Draft, Writing - Review 
& Editing, Conceptualization. H. Wang: Conceptualization, 
Review & Editing. E. Kurniawan: Formal analysis, 
Conceptualization, Investigation S. Agmal: Visualization, 
Validation, Data Curation. M.J. Kholili: Resources, 
Validation. 

Funding statement  

This research did not receive any specific grant from 
funding agencies in the public, commercial, or not-for-
profit sectors.  

Competing interest  

The authors declare that they have no known 
competing financial interests or personal relationships that 
could have appeared to influence the work reported in this 
paper.  

Additional information  

Reprints and permission: information is available at 
https://mev.lipi.go.id/. 

Publisher’s Note: National Research and Innovation 
Agency (BRIN) remains neutral with regard to jurisdictional 
claims in published maps and institutional affiliations. 

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	Introduction
	II. Materials and Methods
	A. Design and method
	B. Proposed cost matrices variation of LQR algorithm

	III. Results and Discussions
	IV. Conclusion 
	Acknowledgements
	Declarations 
	Author contribution 
	Funding statement 
	Competing interest 
	Additional information 

	References