J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

Journal of Mechatronics, Electrical Power, 

and Vehicular Technology 
 

e-ISSN:2088-6985 

p-ISSN: 2087-3379 
 

 

 

www.mevjournal.com 
 

© 2015 RCEPM - LIPI All rights reserved. Open access under CC BY-NC-SA license. Accreditation Number: 633/AU/P2MI-LIPI/03/2015. 

doi: 10.14203/j.mev.2015.v6.19-30 

NONLINEAR DYNAMIC MODELING OF A FIXED-WING UNMANNED 

AERIAL VEHICLE: A CASE STUDY OF WULUNG  
 

Fadjar Rahino Triputra
a,c,

*, Bambang Riyanto Trilaksono
a
, Trio Adiono

a
, 

Rianto Adhy Sasongko
b
, Mohamad Dahsyat

c
 

a
School of Electrical Engineering and Informatics, Institut Teknologi Bandung 

Jl. Ganesha 10, Bandung, Indonesia 
b
Faculty of Mechanical and Aerospace Engineering, Institut Teknologi Bandung 

Jl. Ganesha 10, Bandung, Indonesia 
c
Agency for the Assessment and Application of Technology (BPPT) 

Komplek Puspiptek Serpong, Tangerang, Indonesia 

 
Received 13 February 2015; received in revised form 30 June 2015; accepted 30 June 2015 

Published online 30 July 2015 

 

Abstract 
Developing a nonlinear adaptive control system for a fixed-wing unmanned aerial vehicle (UAV) requires a mathematical 

representation of the system dynamics analytically as a set of differential equations in the form of a strict-feedback systems. This 

paper presents a method for modeling a nonlinear flight dynamics of the fixed-wing UAV of BPPT Wulung in any conditions of 

the flight altitude and airspeed for the first step into designing a nonlinear adaptive controller. The model was formed into 10-

DOF differential equations in the form of strict-feedback systems which separates the terms of elevator, aileron, rudder, and 

throttle from the model. The model simulation results show the behavior of the flight dynamics of the Wulung UAV and also 

prove the compliance with the actual flight test results. 

 

Keywords: fixed-wing UAV; nonlinear flight dynamics; strict-feedback systems. 

 

I. INTRODUCTION 
A fixed-wing unmanned aerial vehicle (UAV) 

has many advantages to fulfill important missions 

in a wide range of territory. Agency for the 

Assessment and Application of Technology 

(BPPT) developed a fixed-wing UAV called 

Wulung as shown in Figure 1, to provide a 

solution of the country problem for keeping 

critical assets in a wide territory such as ocean 

and forest from disaster, illegal fishing, and 

illegal logging. This UAV has the proficiency to 

carry a 20 kg payload and the flight range of 200 

km from the home base for various missions such 

as surveillance, aerial photography, search and 

rescue, and weather modification. However, to 

meet the needs of concerned missions, an 

adaptive flight control is needed to drive this 

fixed-wing UAV to the mission locations 

autonomously. Hence, a nonlinear dynamic 

model of Wulung UAV is also needed as part of 

the flight control design to manage the UAV 

flying to the mission destinations reliably and 

safely. Afterwards, some modeling studies of 

Wulung UAV have been conducted to formulate 

the flight control design. 

Formerly, the flight dynamics modeling of a 

fixed-wing UAV in the case of BPPT Wulung 

UAV [1] have been conducted using the linear 

systems approach [2] in which the analytical 

model aerodynamic coefficients are calculated 

using DATCOM software [3]. Flight test data of 

Wulung UAV has also been obtained to identify 

a linear model using grey-box method [4] that 

involves the analytical linear model. However, 

the flight dynamics model in the form of linear 

state-space cannot handle the changes in altitude 

and airspeed because the differential equations of 

flight dynamics use DATCOM aerodynamic 

coefficients which only can be applied for a 

specific altitude and airspeed. These studies led 

to a conclusion to develop a nonlinear Wulung 

UAV model in any condition of the flight altitude 

and  airspeed for the first step into designing a 

nonlinear adaptive controller. Consequently, a * Corresponding Author: Tel: +62-8158000730 
E-mail: fadjar.rahino@bppt.go.id 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

20 

practical nonlinear flight dynamic model is 

required for a chosen adaptive control systems. 

Subsequently, an adaptive control systems of 

an integrator back stepping, that have the 

advantage to handle the nonlinear model well, 

have been described by Krstic [5]. Nevertheless, 

a complicated analytical model derivation must 

be solved in designing this control systems. 

Successively, command filtered back stepping 

(CFBS) has been proposed by Farrell [6][7] that 

eliminated the requirement of analytical model 

derivation and simplified its control design. 

However, introducing the Wulung UAV model to 

this controller needs a nonlinear model in the 

form of strict-feedback systems [8][9] that 

separates the terms of the fixed-wing UAV 

control variables as shown in Figure 2 and 

expressed as the following model: 

𝑥 =  𝑓 𝑥 + 𝑔 𝑥 𝑢 (1) 

where x and u are vectors of state and control 

variables respectively. 

This paper presents the development of a 

nonlinear flight dynamics model in any 

conditions of the flight altitude and  airspeed by 

calculating aerodynamic coefficients as functions 

of altitude and  airspeed. The program source of 

DATCOM software [3] are used as the basis to 

build analytical equations for aerodynamic 

coefficients using basic aerodynamics of lifting 

surfaces [10]. The advantage of this proposed 

nonlinear model is to eliminate the use of third-

party software to obtain the aerodynamic 

coefficients, thus the geometry characteristic of 

the fixed-wing UAV such as wing span and wing 

chord can be directly applied. Later, we 

constructed 10 degrees of freedom (DOF) of 

differential equations in the form of strict-

feedback systems to represent a non-linear 

dynamic model [2][10][11] using the model of 

the proposed forces and moments that have input 

parameters of state variables including the 

altitude and the  airspeed. Simulations of BPPT 

Wulung UAV in longitudinal and lateral dynamic 

have been conducted using this model, then their 

results have been matched with the actual flight 

test data to prove the compliance of this 

nonlinear model. 

 

II. FIXED-WING UAV MODELING  
Figure 3 shows the movement components of 

fixed-wing UAV consisting of the attitude 

𝛯 =  𝜙 𝜃 𝜓 ⊺, velocity 𝑉 =  𝑈 𝑉 𝑊 ⊺, angular 
rate 𝛺 =  𝑝 𝑞 𝑟 ⊺, forces 𝐹 =  𝑋 𝑌 𝑍 ⊺, and 

moments 𝑀 =  𝐿𝜙 𝑀𝜃 𝑁𝜓  
⊺
 in the vehicle 

coordinate frame  𝑥𝑣 , 𝑦𝑣 , 𝑧𝑣 . Based on the 

Newton's motion equations of the rigid body, the 

forces and moments [2][10] are defined as 

follows: 

𝐹 = 𝑚𝑉 + 𝛺 × 𝑉 (2) 

𝑀 = 𝐽𝛺 + 𝛺 × 𝐽𝛺 (3) 

where 𝑚  is the mass and J  is the moment of 

inertia. 

 

Figure 1. A prototype of BPPT Wulung UAV 

 

 

Figure 2. Strict-feedback systems model 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

21 

Hence, the model of fixed-wing UAV is then 

written in the following equation: 

𝑉 =
𝐹

𝑚
− 𝛺 × 𝑉 (4) 

𝛺 = 𝐽−1 𝑀 − 𝛺 × 𝐽𝛺  (5) 

The velocity is generally obtained from inertia 

sensors such as GPS/INS that provides calculated 

velocity in an inertia coordinate frame. So these 

components can be converted into a vehicle 

coordinate frame as follows: 

𝑉 = 𝑅𝑒
𝑣 𝑉𝑔𝑝𝑠  (6) 

where 𝑉𝑔𝑝𝑠 =  𝑉𝑛𝑜𝑟𝑡 𝑕 𝑉𝑒𝑎𝑠𝑡 𝑉𝑑𝑜𝑤𝑛  
⊺ is inertia velocity 

from GPS/INS and R𝑒
𝑣  is direct cosine matrix 

(DCM) as defined as follows: 

𝑅𝑒
𝑣 = 𝑅𝜙

𝑣 𝑅𝜃
𝑣 𝑅𝜓

𝑣
 (7) 

𝑅𝜙
𝑣 =  

1 0 0
0 𝑐𝑜𝑠 𝜙 𝑠𝑖𝑛 𝜙
0 − 𝑠𝑖𝑛 𝜙 𝑐𝑜𝑠 𝜙

  (8) 

𝑅𝜃
𝑣 =  

𝑐𝑜𝑠 𝜃 0 −𝑠𝑖𝑛 𝜃
0 1 0

𝑠𝑖𝑛 𝜃 0 𝑐𝑜𝑠 𝜃
  (9) 

𝑅𝜓
𝑣 =  

𝑐𝑜𝑠 𝜓 𝑠𝑖𝑛 𝜓 0
−𝑠𝑖𝑛 𝜓 𝑐𝑜𝑠 𝜓 0

0 0 1

  (10) 

Beside angular rate of equation (5), the 

calculation of the Euler angular rate Ξ  that has a 

link with the following angular rate is required: 

𝛺 =  
𝜙 

0
0

 + 𝑅𝜙
𝑣  

0
𝜃 

0
 + 𝑅𝜙

𝑣 𝑅𝜃
𝑣  

0
0
𝜓 
  (11) 

Hence, the Euler angular rate is written as 

follows: 

𝛯 = 𝑅𝛺𝛺 (12) 

where, 

𝑅𝛺 =  

1 𝑠𝑖𝑛 𝜙 𝑡𝑎𝑛 𝜃 𝑐𝑜𝑠 𝜙 𝑡𝑎𝑛 𝜃
0 𝑐𝑜𝑠 𝜙 − 𝑠𝑖𝑛 𝜙

0
𝑠𝑖𝑛 𝜙

𝑐𝑜𝑠 𝜃

𝑐𝑜𝑠 𝜙

𝑐𝑜𝑠 𝜃

  (13) 

In addition, the altitude rate can be written as 

follows: 

𝑕 = −𝑉𝑑𝑜𝑤𝑛 = 𝑅𝑕 𝑉 (14) 

where, 

𝑅𝑕 =  𝑠𝑖𝑛 𝜃 − 𝑠𝑖𝑛 𝜙 𝑐𝑜𝑠 𝜃 − 𝑐𝑜𝑠 𝜙 𝑐𝑜𝑠 𝜃  (15) 

Therefore, a nonlinear fixed-wing UAV 

model is obtained using equations (4), (5), (12), 

and (14) as 10-DOF of differential equations. 

Thereafter, the forces and the moments due to the 

influence of vehicle aerodynamics, propeller 

thrust, and gravity must be described as follows: 

𝐹 = 𝐹𝑎 + 𝐹𝑝 + 𝐹𝑔 (16) 

𝑀 = 𝑀𝑎 + 𝑀𝑝  (17) 

where 𝐹𝑎  and 𝑀𝑎 respectively are aerodynamic 

forces and moments, 𝐹𝑝  and 𝑀𝑝  respectively are 

propeller thrust forces and moments, and 𝐹𝑔 is 

gravity forces. 

 

A.  Aerodynamic Forces and Moments 
Figure 4 shows three frames of coordinate, 

namely the vehicle coordinate frame  𝑥𝑣 , 𝑦𝑣 , 𝑧𝑣 , 

the stability coordinate frame  𝑥𝑠 , 𝑦𝑠, 𝑧𝑠 , and the 

wind coordinate frame  𝑥𝑤 , 𝑦𝑤 , 𝑧𝑤  . Aerodynamic 

forces and moments, that occurs in fixed-wing 

UAV, generally are caused by three kind of 

forces in the wind coordinate frame, i.e. lift force 

𝐿, drag force 𝐷, and side force 𝑆. Then the 

aerodynamic forces in vehicle coordinate frame 

is written as follows: 

𝐹𝑎 = 𝑅𝑠
𝑣 𝑅𝑤

𝑠  −𝐷 𝑆 −𝐿 ⊺ (18) 

where, 

𝑅𝑠
𝑣 =  

𝑐𝑜𝑠 𝛼 0 −𝑠𝑖𝑛 𝛼
0 1 0

𝑠𝑖𝑛 𝛼 0 𝑐𝑜𝑠 𝛼
  (19) 

𝑅𝑤
𝑠 =  

𝑐𝑜𝑠 𝛽 𝑠𝑖𝑛 𝛽 0
−𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 𝛽 0

0 0 1

  (20) 

 

Figure 3. Vehicle coordinate frame components 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

22 

Whereas the aerodynamic moments in vehicle 

coordinate frame is also written as follows: 

𝑀𝑎 =  𝐿𝜙𝑎 𝑀𝜃𝑎 𝑁𝜓𝑎  
⊺
 (21) 

In order to form strict-feedback system, the 

forces and moments must be separated into two 

kind of forces and moments, i.e. affected by the 

control surfaces (elevator 𝛿𝑒 , aileron 𝛿𝑎 , and 

rudder 𝛿𝑟 ) or not affected. Then equations (18) 

and (21) are rewritten as follows: 

𝐹𝑎 = 𝐹𝑎𝑣 + 𝐶𝐹𝑎𝛿
 𝛿𝑒 𝛿𝑎 𝛿𝑟  

⊺
 (22) 

𝑀𝑎 = 𝑀𝑎𝑣 + 𝐶𝑀𝑎𝛿
 𝛿𝑒 𝛿𝑎 𝛿𝑟  

⊺
 (23) 

Hence, the lift, drag, and side forces in wind 

coordinate frame must also be separated as it is 

done to aerodynamic forces in vehicle coordinate 

frame, then equations (22) and (23) are broken 

down into the following equation: 

𝐹𝑎𝑣 = 𝑅𝑠
𝑣 𝑅𝑤

𝑠  −𝐷𝑣 𝑆𝑣 −𝐿𝑣 
⊺
 (24) 

𝐶𝐹𝑎𝛿
= 𝑅𝑠

𝑣 𝑅𝑤
𝑠  

−𝐶𝐷𝑒 0 −𝐶𝐷𝑟
0 0 𝐶𝑆𝑟

−𝐶𝐿𝑒 0 0
  (25) 

𝑀𝑎𝑣 =  
𝐿𝜙𝑣 𝑀𝜃𝑣 𝑁𝜓𝑣  

⊺
 (26) 

𝐶𝑀𝑎𝛿
=  

0 𝐶𝐿𝜙 𝑎
𝐶𝐿𝜙 𝑟

𝐶𝑀𝜃𝑒
0 0

0 𝐶𝑁𝜓 𝑎
𝐶𝑁𝜓 𝑟

  (27) 

Afterwards, the lift, drag, and side forces are 

calculated as follows: 

𝐿𝑣 =
1

2
𝜌𝑉𝑎

2𝑆𝑤  𝐶𝐿 0 + 𝐶𝐿𝛼 𝛼  (28) 

𝐶𝐿𝑒 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝐶𝐿𝛿𝑒
 (29) 

𝑆𝑣 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝐶𝑆𝛽 𝛽 (30) 

𝐶𝑆𝑟 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝐶𝑆𝛿𝑟
 (31) 

𝐷𝑣 =
1

2
𝜌𝑉𝑎

2𝑆𝑤  𝐶𝐷 0 + 𝐶𝐷𝛼 𝛼 + 𝐶𝐷𝛽 𝛽  (32) 

𝐶𝐷𝑒 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝐶𝐷𝛿𝑒
 (33) 

𝐶𝐷𝑟 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝐶𝐷𝛿𝑟
 (34) 

In addition, the rolling, pitching, and yawing 

moments are calculated as follows: 

𝐿𝜙𝑣 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤 𝐶𝐿𝜙 𝛽
𝛽 (35) 

𝐶𝐿𝜙 𝑎
=

1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤 𝐶𝐿𝜙 𝛿𝑎
 (36) 

𝐶𝐿𝜙 𝑟
=

1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤 𝐶𝐿𝜙 𝛿𝑟
 (37) 

𝑀𝜃𝑣 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤  𝐶𝑀𝜙 0
+ 𝐶𝑀𝜙 𝛼

𝛼  (38) 

𝐶𝑀𝜃 𝑒
=

1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤 𝐶𝑀𝜃 𝛿𝑒
 (39) 

𝑁𝜓𝑣 =
1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤 𝐶𝑁𝜓 𝛽
𝛽 (40) 

𝐶𝑁𝜓 𝑎
=

1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤 𝐶𝑁𝜓 𝛿𝑎
 (41) 

𝐶𝑁𝜓 𝑟
=

1

2
𝜌𝑉𝑎

2𝑆𝑤 𝑏𝑤 𝐶𝑁𝜓 𝛿𝑟
 (42) 

where 𝜌 is the air density at altitude 𝑕, 𝑉𝑎  is the 

true airspeed, 𝑆𝑤  is the main wing surface area, 

and 𝑏𝑤  is the main wingspan. 

 

Figure 4. Aerodynamic and thrust forces 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

23 

In linear system approach, all of those coeffi-

cients 𝐶∗ in equations (28) until (42) are assumed 

as constants. However, those coefficients are also 

depend on the true airspeed 𝑉𝑎  and the altitude 𝑕. 

Hence, all of the coefficients are calculated into 

functions of 𝐶∗ 𝑉𝑎 , 𝑕 . Note that ∗ denotes any 
character. Subsequently, the aerodynamics 

coefficients are written as follows: 

𝐶𝐿 0 = 𝐶𝐿𝑤𝛼
 𝜂𝑤 − 𝛼𝑤0  + 𝐶𝐿𝑕𝛼

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
  

  
𝑑

𝑑𝛼
 𝛼𝑤0 − 𝜂𝑤  − 𝛼𝑕0 + 𝜂𝑕   (43) 

𝐶𝐿𝛼 = 𝐶𝐿𝑤𝛼
+ 𝐶𝐿𝑕𝛼

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
 1 −

𝑑

𝑑𝛼
  (44) 

𝐶𝐿𝛿𝑒
= 𝐶𝐿𝑕𝛿𝑒

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
 (45) 

𝐶𝑆𝛽 = 𝐶𝑆𝑣𝛽
𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤
 (46) 

𝐶𝑆𝛿𝑟
= 𝐶𝑆𝑣𝛿𝑟

𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤
 (47) 

𝐶𝐷 0 = 𝐶𝐷𝑤𝑝
+ 𝐶𝐷𝑕𝑝

𝑞𝑕
𝑞∞

𝑆𝑕
𝑆𝑤

+ 𝐶𝐷𝑣𝑝

𝑞𝑕
𝑞∞

𝑆𝑣
𝑆𝑤

 

+
 𝐶𝐿𝑤𝛼

 𝜂𝑤 − 𝛼𝑤0   
2

𝜋𝐴𝑤 𝑒𝑤
 

 +
 𝐶𝐿𝑕𝛼

 𝛼𝑕  0 − 𝛼𝑕0   
2

𝜋𝐴𝑕 𝑒𝑕

𝑞𝑕
𝑞∞

𝑆𝑕
𝑆𝑤

 

𝐶𝐷       0  +
2𝐶𝐿 𝑕𝛼

2 𝛼𝑕  0 −𝛼𝑕 0  

𝜋𝐴𝑕 𝑒𝑕

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
𝜂𝑕  (48) 

𝐶𝐷𝛼 =
2𝐶𝐿 𝑤 𝛼

2 α+𝜂𝑤 −𝛼𝑤 0  

𝜋𝐴𝑤 𝑒𝑤
  

𝐶𝐷𝛼 =
2𝐶𝐿𝑕𝛼

2
 𝛼𝑕  𝛼 − 𝛼𝑕0  

𝜋𝐴𝑕 𝑒𝑕

𝑞𝑕
𝑞∞

𝑆𝑕
𝑆𝑤

 

   1 −
𝑑

𝑑𝛼
  (49) 

𝐶𝐷𝛽 =
2𝐶𝑆𝑣𝛽

2 𝛽−𝛽𝑣0  

𝜋𝐴𝑣𝑒𝑣

𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤
 (50) 

𝐶𝐷𝛿𝑒
=

2𝐶𝐿 𝑕𝛼
 𝛼𝑕  𝛼 −𝛼𝑕 0  

𝜋𝐴𝑕 𝑒𝑕
𝐶𝐿𝑕𝛿𝑒

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
 (51) 

𝐶𝐷𝛿𝑟
=

2𝐶𝑆𝑣𝛽
 𝛽−𝛽𝑣0  

𝜋𝐴𝑣𝑒𝑣
𝐶𝑆𝑣𝛿𝑟

𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤
 (52) 

𝐶𝐿𝜙 𝛽
= 𝐶𝐿𝜙 𝑤 𝛽

+ 𝐶𝐿𝜙 𝑕𝛽

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤

𝑏𝑕

𝑏𝑤
  

𝐶       𝐿𝜙 𝛽
+ 𝐶𝐿𝜙 𝑣𝛽

𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤

𝑏𝑣

𝑏𝑤
 (53) 

𝐶𝐿𝜙 𝛿𝛼
= 𝐶𝐿𝜙 𝑤 𝛿𝛼

 (54) 

𝐶𝐿𝜙 𝛿𝑟
= 𝐶𝑆𝑣𝛿𝑟

𝑧MAC v
𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤 𝑏𝑤
 (55) 

 𝐶𝑀𝜙 0
= 𝐶𝑀𝜙 𝑤 ac

+ 𝐶𝑀𝜙 𝑓0
 

+𝐶𝐿𝑤𝛼
 𝜂𝑤 − 𝛼𝑤0    

+𝐶𝐿𝑤𝛼
 𝜂𝑤 − 𝛼𝑤0  

𝑥ac 𝑤𝑓
−𝑥cg

𝑐 
  

 −𝐶𝐿𝑕𝛼
𝑥cg −𝑥ac 𝑕

𝑐 

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
 

 
𝑑

𝑑𝛼
 𝛼𝑤0 − 𝜂𝑤  − 𝛼𝑕0   (56) 

𝐶𝑀𝜙 𝛼
= 𝐶𝐿𝑤𝛼

𝑥ac 𝑤𝑓
−𝑥cg

𝑐 
  

     𝐶𝑀𝜙 𝛼
− 𝐶𝐿𝑕𝛼

𝑥cg −𝑥ac 𝑕

𝑐 

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
 

 1 −
𝑑

𝑑𝛼
  (57) 

𝐶𝑀𝜃 𝛿𝑒
= −𝐶𝐿𝑕𝛿𝑒

𝑥cg −𝑥ac 𝑕

𝑐 

𝑞𝑕

𝑞∞

𝑆𝑕

𝑆𝑤
 (58) 

𝐶𝑁𝜓 𝛽
= −𝐶𝑆𝑣𝛽

𝑥cg −𝑥ac 𝑣

𝑐 

𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤
 (59) 

𝐶𝑁𝜓 𝛿𝑎
= 𝐶𝑁𝜓 𝑤 𝛿𝑎

 (60) 

𝐶𝑁𝜓 𝛿𝑟
= −𝐶𝑆𝑣𝛿𝑟

𝑥cg −𝑥ac 𝑣

𝑐 

𝑞𝑕

𝑞∞

𝑆𝑣

𝑆𝑤
 (61) 

where 𝜂𝑤  and  𝜂𝑕  respectively are wing and hori-

zontal tail plane (HTP) rigging angles, 𝐴𝑤 , 𝐴𝑕 , 𝐴𝑣 

respectively are wing, HTP, and vertical tail 

plane (VTP) aspect ratios, 𝛼𝑤0 , 𝛼𝑕0 , 𝛽𝑣0  

respectively are wing, HTP, and VTP angles of 

attack at zero-lift, 
𝑞𝑕

𝑞∞
 is HTP dynamic pressure 

ratio, 
𝑑

𝑑𝛼
 is downwash gradient, 𝑒𝑤 , 𝑒𝑕 , 𝑒𝑣 

respectively are wing, HTP, and VTP Oswald 

coefficient,  𝑆𝑕  and 𝑆𝑣 respectively are HTP and 

VTP surface area, 𝑏𝑕  and 𝑏𝑣 respectively are HTP 

and VTP span. 𝛼𝑕  𝛼  is HTP angle of attack due 

to the vehicle angle of attack that is formulated as 

follows: 

𝛼𝑕  𝛼 = 𝛼 + 𝜂𝑕 − 𝜂𝑤 −
𝑑

𝑑𝛼
 α − 𝛼𝑤0   (62) 

𝑐 is wing mean aerodynamic chord (MAC) length, 

𝑥cg  is center position of gravity, 𝑥ac 𝑤𝑓 , 𝑥ac 𝑕 , 𝑥ac 𝑣  

respectively are wing-fuselage, HTP, and VTP 

aerodynamic center (a.c.) positions, and 𝑧MAC v  is 

VTP MAC normal position. 𝐶𝐷𝑤𝑝
, 𝐶𝐷𝑕𝑝

, 𝐶𝐷𝑣𝑝
 

respectively are wing, HTP, and VTP parasite 

drag. 𝐶𝑀𝜙 𝑤 ac
 and 𝐶𝑀𝜙 𝑓0

 respectively are wing and 

fuselage rolling moment coefficient at 𝛼 = 0. 

 𝐶𝑁𝜓 𝑤 𝛿𝑎
is wing yawing moment coefficient due to 

aileron. 

Furthermore, the wing, HTP, and VTP lift 

coefficients respectively are defined as follows 

[10]: 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

24 

𝐶𝐿𝑤𝛼
=

2𝜋𝐴𝑤

2+

 
 
 
 
 
 
 
 
 
 
 
 
 

𝐴𝑤  1− 
𝑉𝑎
𝑎

 
2
 

 

𝑐𝑙𝑤 𝛼 0
2𝜋

 

2

 
 
 

 
 

1+

 tan 𝛬 1
2𝑤

 

2

1− 
𝑉𝑎
𝑎

 
2

 
 
 

 
 

+4

 (63) 

𝐶𝐿𝑕𝛼
=

2𝜋𝐴𝑕

2+

 
 
 
 
 
 
 
 
 
 
 
 
 

𝐴𝑕  1− 
𝑉𝑎
𝑎

 
2
 

 

𝑐𝑙𝑕𝛼 0
2𝜋

 

2

 
 
 

 
 

1+

 tan 𝛬 1
2𝑕

 

2

1− 
𝑉𝑎
𝑎

 
2

 
 
 

 
 

+4

 (64) 

𝐶𝑆𝑣𝛽
= −

2𝜋𝐴𝑣

2+

 
 
 
 
 
 
 
 
 
 
 
 
 

𝐴𝑣 1− 
𝑉𝑎
𝑎

 
2
 

 

𝑐𝑙𝑣𝛼 0
2𝜋

 

2

 
 
 

 
 

1+

 tan 𝛬 1
2𝑣

 

2

1− 
𝑉𝑎
𝑎

 
2

 
 
 

 
 

+4

 (65) 

where 𝑎 is the speed of sound at altitude 𝑕, 𝑐𝑙 𝑤𝛼 0
, 

𝑐𝑙 𝑕𝛼 0
, 𝑐𝑙 𝑣𝛼 0

 respectively are wing, HTP, and VTP 

airfoil lift coefficients, 𝛬1
2𝑤

, 𝛬1
2𝑕

, 𝛬1
2𝑣

 respectively 

are wing, HTP, and VTP half sweep angles. 

In addition, the elevator, aileron, and rudder 

coefficients respectively are defined as follows 

[10]: 

 𝐶𝐿𝑕𝛿𝑒
=

𝑐𝑙𝑕𝛿𝑒0

 1− 
𝑉𝑎
𝑎

 
2
 (66) 

 𝐶𝐿𝜙𝑤 𝛿𝑎
=

𝑐𝑙𝜙 𝑕 𝛿𝑎 0

 1− 
𝑉𝑎
𝑎

 
2
 (67) 

𝐶𝑆𝑣𝛿𝑟
=

𝑐𝑙𝑣𝛿𝑟0

 1− 
𝑉𝑎
𝑎

 
2
 (68) 

where 𝑐𝑙 𝑕𝛿𝑒0
, 𝑐𝑙 𝜙𝑕 𝛿𝑎 0

, 𝑐𝑙 𝑣𝛿𝑟0
 respectively are 

elevator, aileron, and rudder airfoil lift 

coefficients which are not affected by altitude 

and airspeed. 

The wing, HTP, and VTP rolling moment are 

defined as follows [10]: 

𝐶𝐿𝜙 𝑤 𝛽
= −𝐶𝐿𝑤𝛼

 𝜂𝑤 − 𝛼𝑤0  
𝑦MAC 𝑤

𝑏𝑤
  

     𝐶𝐿           sin 2𝛬𝑤LE − 𝛤𝑤 𝐶𝐿𝑤𝛼
𝑦MAC 𝑤

𝑏𝑤
 (69) 

𝐶𝐿𝜙 𝑕𝛽
= −𝐶𝐿𝑕𝛼

 𝜂𝑕 − 𝛼𝑕0  
𝑦MAC 𝑕

𝑏𝑕
  

                sin 2𝛬𝑕LE − 𝛤𝑕 𝐶𝐿𝑕𝛼
𝑦MAC 𝑕

𝑏𝑕
 (70) 

𝐶𝐿𝜙 𝑣𝛽
= 𝐶𝑆𝑣𝛽

𝑧MAC 𝑣

𝑏𝑣
 (71) 

where 𝑦MAC 𝑤  and 𝑦MAC 𝑕  respectively are wing and 

HTP MAC side positions, 𝛤𝑤  and 𝛤𝑕  respectively 

are wing and HTP dihedral angles, 𝛬𝑤LE  and 𝛬𝑕LE  

respectively are wing and HTP leading-edge 

sweep angles. 

 

B. Propeller Thrust Forces and Moments 
The relation between the engine power and 

the thrust generally use the power and thrust 

coefficients [12][11] that are defined respectively 

as follows: 

𝐶𝑝 𝐽 =
𝑃

𝜌𝑛prop
3 𝐷prop

5  (72) 

𝐶𝑡 𝐽 =
𝑇

𝜌𝑛prop
2 𝐷prop

4  (73) 

where 𝑃 is engine power, 𝐷prop  is propeller 

diameter, 𝑛prop  is propeller rotation per second, 

and  𝐽 is rate of advance that is also defined as 
follows [12][11]: 

𝐽 =
𝑉𝑎

𝑛prop 𝐷prop
 (74) 

Figure 5a and Figure 5b show the power and 

thrust coefficient for Wulung UAV model that 

can be represented as polynomial equations as 

follow: 

𝐶𝑝 𝐽 = 𝑎𝐶𝑝 1
𝐽3 + 𝑎𝐶𝑝 2

𝐽2 + 𝑎𝐶𝑝 3
 

              +𝑎𝐶𝑝 4
 (75) 

 𝐶𝑡 𝐽 = 𝑎𝐶𝑡1
𝐽3 + 𝑎𝐶𝑡2

𝐽2 + 𝑎𝐶𝑡3
𝐽 

               +𝑎𝐶𝑡4
 (76) 

Then, the relation between the throttle and 

engine power is proposed as follows: 

𝑃 𝛿𝑡 = 𝑎𝑃1 𝛿𝑡
3 + 𝑎𝑃2 𝛿𝑡

2 + 𝑎𝑃3 𝛿𝑡 + 𝑎𝑃4  (77) 

Figure 5c shows the relation of equation (77) 

for Wulung UAV. Therefore, a solution for the 

following equation must be performed to obtain 

propeller rotation 𝑛. 

𝑃 𝛿𝑡 = 𝑎𝑛1 𝑛
3 + 𝑎𝑛2 𝑛

2 + 𝑎𝑛3 𝑛 + 𝑎𝑛4  (78) 

where 𝑎𝑛1 = −𝑎𝐶𝑝 4
𝜌𝐷𝑝𝑟𝑜𝑝

3 , 𝑎𝑛2 = −𝑎𝐶𝑝 3
𝜌𝑉𝑎 𝐷𝑝𝑟𝑜𝑝

2 , 

𝑎𝑛3 = −𝑎𝐶𝑝 2
𝜌𝑉𝑎

2𝐷𝑝𝑟𝑜𝑝 , and 𝑎𝑛4 = −𝑎𝐶𝑝 1
𝜌𝑉𝑎

3. 

Hence, the propeller thrust can be calculated 

as follows: 

𝑇 = 𝐶𝑡  
𝑉𝑎

𝑛𝑝𝑟𝑜𝑝 𝐷𝑝𝑟𝑜𝑝
 𝜌𝑛2𝐷𝑝𝑟𝑜𝑝

4
 (79) 

Next, the thrust as shown in Figure 5d is 

linearized as follows: 

𝑇 = 𝑇0 + 𝐶𝑇𝛿𝑡
𝛿𝑡  (80) 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

25 

Similar to aerodynamic coefficients 

calculation, 𝑇0  and 𝐶𝑇𝛿𝑡
 are depend on the true 

airspeed 𝑉𝑎  and the altitude 𝑕. Hence, the forces 

and moments caused by the thrust in vehicle 

coordinate frame is written as follows: 

𝐹𝑝 = 𝐹𝑝𝑣 + 𝐶𝐹𝑝 𝛿
𝛿𝑡  (81) 

𝑀𝑝 = 𝑀𝑝𝑣 + 𝐶𝑀𝑝 𝛿
𝛿𝑡  (82) 

Similar to aerodynamic forces and moment, 

the equations (81) and (82) is then broken down 

as follows: 

𝐹𝑝𝑣 =  𝑇0 𝑐𝑜𝑠 𝜏 0 −𝑇0 𝑠𝑖𝑛 𝜏 
⊺
 (83) 

𝐶𝐹𝑝 𝛿
=  𝐶𝑇𝛿𝑡

𝑐𝑜𝑠 𝜏 0 −𝐶𝑇𝛿𝑡
𝑠𝑖𝑛 𝜏 

⊺
 (84) 

𝑀𝑝𝑣 = 𝐹𝑝𝑣 × 𝑃𝑝𝑟𝑜𝑝  (85) 

𝐶𝑀𝑝 𝛿
= 𝐶𝐹𝑝 𝛿

× 𝑃𝑝𝑟𝑜𝑝  (86) 

where 𝑃𝑝𝑟𝑜𝑝 =  𝑥𝑝𝑟𝑜𝑝 0 𝑧𝑝𝑟𝑜𝑝  
⊺ is the position 

of the propeller in vehicle coordinate frame and 𝜏 

is propeller rigging angle. 

 

C. Gravitational Forces 
Figure 6b shows the forces of gravitation in 

inertia coordinate frame, while Figure 6a shows 

the forces in vehicle coordinate frame. Thus, we 

write the gravitational forces in vehicle 

coordinate frame as follows: 

 

(a) 

 

(b) 

 

(c) 

 

(d) 

Figure 5. Wulung propeller model: (a) Power coefficient; (b) Thrust coefficient; (c) Power vs throttle; (d) Thrust vs throttle 

 

 

(a) (b) 

Figure 6. Gravitational forces: (a) Vehicle frame; (b) Inertia frame 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

26 

𝐹𝑔 = 𝑅𝑒
𝑣  0 0 𝑚𝑔 ⊺  

    = 𝑚𝑔 −𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜙 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 𝜙 ⊺ (87) 

where 𝑚 is mass of the fixed-wing UAV and 𝑔 is 

gravitational constant. 

 

D. Nonlinear Flight Dynamic Model 
The control input 𝑢 and the state variable 𝑥 are 

defined as follows: 

𝑢 =  𝛿𝑒 𝛿𝑎 𝛿𝑟 𝛿𝑡 
⊺ (88) 

𝑥 =  𝑉⊺ 𝛺⊺ 𝛯⊺ 𝑕 
⊺ (89) 

Hence, from equations (12), (14), (24)-(27), 

(83)-(87), the nonlinear functions f x  and g x  of 

equation (1) can be written as follows: 

𝑓 𝑥 =

 
 
 
 
 

𝐹𝑎 𝑣 +𝐹𝑝𝑣 +𝐹𝑔

𝑚
− 𝛺 × 𝑉

𝐽−1 𝑀𝑎𝑣 + 𝑀𝑝𝑣 − 𝛺 × 𝐽𝛺 

𝑅𝛺𝛺
𝑅𝑕 𝑉  

 
 
 
 

 (90) 

𝑔 𝑥 =

 
 
 
 

𝐶𝐹𝑎 𝛿

𝑚

𝐶𝐹𝑝𝛿

𝑚

𝐽−1𝐶𝑀𝑎 𝛿
𝐽−1𝐶𝑀𝑝 𝛿

04×3 04×1  
 
 
 
 (91 

 

III. FLIGHT DYNAMICS RESPONSES 
Wulung UAV profiles in Table 1 are used as 

model parameters for simulating the responses of 

the flight dynamics. Two scenarios of disturbance 

are done by providing doublet angles of elevator 

and aileron to show the longitudinal and lateral 

dynamics responses of Wulung UAV. 

Before performing simulation, the steady state 

of flight must be found in the condition that the 

UAV cruises without changing the altitude by 

trimming the elevator and throttle. Therefore, if 

the altitude, throttle, and elevator are respectively 

set to 𝑕 = 3000 feet, 𝛿𝑡 = 67%, and 𝛿𝑒 =

−3.3 deg, the steady state condition of Wulung 

UAV model will be occurred at pitching angle 

𝜃 = 1.87 deg, axial velocity 𝑈 = 58.75 knots, and 

normal velocity 𝑊 = 1.21 knots. Wulung UAV 

model is then simulated using 10-DOF 

differential equation (90) and (91). After the 

calculation, the velocity, angular rate, attitude, 

and altitude of Wulung UAV are obtained for 

each simulation scenario. 

 

A. Longitudinal Dynamics Responses 
In this simulation scenario, the UAV moves 

westward with the throttle, altitude, velocity, and 

 

Table 1. 

Wulung UAV profiles 

Parameter Unit 

Mass, 𝑚 120 (kg) 

Wing area, 𝑆𝑤  3.9718 (m
2
) 

Wingspan, 𝑏𝑤  6.355 (m) 

HTP area, 𝑆𝑕  0.819 (m
2
) 

HTP span, 𝑏𝑕  1.95 (m) 

VTP area, 𝑆𝑣 0.519 (m
2
) 

VTP span, 𝑏𝑣 0.845 (m) 

Wing rigging angle, 𝜂𝑤  6 

HTP rigging angle, 𝜂𝑕  -3 

Wing dihedral, 𝛤𝑤  3 

c.g. position, 𝑥cg  -1.576 (m) 

Wing-fuselage a.c. position, 𝑥ac 𝑤𝑓  -1.497 (m) 

HTP a.c. position, 𝑥ac 𝑕  -4.136 (m) 

VTP a.c. position, 𝑥ac 𝑣  -3.97 (m) 

Wing MAC side position, 𝑦MAC 𝑤  1.589 (m) 

HTP MAC side position, 𝑦MAC 𝑕  0.488 (m) 

VTP MAC normal position, 𝑧MAC v  0.294 (m) 

𝐽xx  79.045 (kgm
2
) 

𝐽yy  103.473 (kgm
2
) 

𝐽𝑧𝑧  159.541 (kgm
2
) 

𝐽𝑥𝑧  19.131 (kgm
2
) 

Engine max power 20 (HP) 

Propeller diameter, 𝐷𝑝𝑟𝑜𝑝  32 (inch) 

 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

27 

attitude as well as steady state condition. 

Therefore, the elevator angle is set to 𝛿𝑒 =

6.7 deg at time 𝑡 = 10 sec, 𝛿𝑒 = −13.3 deg at time 

𝑡 = 10.5 sec, and 𝛿𝑒 = −3.3 deg at time 𝑡 = 11 sec 

as shown in Figure 7a. 

Figure 7a also shows the responses of the 

pitch and attack angles. Once elevator trailing 

edge down (positive), decreasing the both angles 

of pitch and attack, and after elevator trailing 

edge up (negative), increasing the both angles of 

pitch and attack. Thereafter, the angle of attack 

immediately return to normal angle less than 3 

second, but the pitching angle takes time of 

damped oscillation to back to steady states. In 

addition, Figure 7b shows the responses of the 

velocity and altitude. It seems clear that the 

Wulung UAV suffered Phugoid motion with 2 

minutes of damping. This simulation denotes to a 

conclusion that the longitudinal dynamics 

characteristic of Wulung UAV is stable without a 

hard control effort that indicated by the 

convergence of states. 

B. Lateral Dynamics Responses 
Similar to longitudinal simulation scenario, 

the fixed-wing UAV moves westward in steady 

state condition. Therefore, the aileron angle is set 

to 𝛿𝑎 = 5 deg at time 𝑡 = 10 sec, 𝛿𝑎 = −5 deg at 

time 𝑡 = 10.5 sec, and aileron angle back to zero 

again at time 𝑡 = 11 sec as shown in Figure 8a. 

Figure 8a also shows the responses of the roll, 

pitch, and sideslip angles. Once left aileron 

trailing edge down (positive), increasing bank 

angle (roll to right) at high roll rate but 

decreasing the sideslip angle slightly, and after 

aileron trailing edge up (negative), still increasing 

the bank angle at weakened roll rate and also 

increasing the sideslip angle. Thereafter, the 

sideslip angle takes time about 2 minutes of 

damped oscillation to back to the steady state. In 

contrast, the bank angle decreases to minimum 

negative angle (roll to left) about 30 seconds and 

then slowly rises gradually towards steady state 

in a long time. The aileron doublet disturbance is 

also reacting on the pitching angle that oscillates 

for about 3 minutes damping. Figure 8b shows 

the responses of the heading and altitude. It 

seems clear that Wulung UAV bank angle did not 

immediately return to the steady state in a long 

time after aileron doublet disturbance, thus 

causing the vehicle has a tendency to turn. This 

 
(a) 

 
(b) 

Figure 7. Wulung longitudinal dynamics responses: (a) 

Elevator doublet and the responses; (b) The velocity and 

altitude 

 
(a) 

 
(b) 

Figure 8. Wulung lateral dynamics responses: (a) Aileron 

doublet and the responses; (b) The heading and altitude 

 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

28 

simulation denotes to a conclusion that the lateral 

dynamics characteristic of Wulung UAV is stable 

but still need a lateral control to restore the bank 

angle to the steady state immediately. Wulung 

UAV also suffered phugoid motion for about 2 

minutes that is indicated by pitching angle 

damped oscillation. 

 

IV.  SIMULATION AND FLIGHT TEST 
A flight test of Wulung UAV has been 

conducted and the results are compared with 

model simulation. The elevator doublet of flight 

test as shown in Figure 9b is mimicked into 

model simulation of longitudinal dynamic as 

shown in Figure 9a. The difference of physical 

angles may be caused by initial steady state and 

the windy conditions when the flight test 

conducted. Further, both results are normalized 

so the steady state is same and compared as 

shown in Figure 9c and Figure 9d.  

Figure 9c shows that both responses of the 

pitching angle and the angle of attack between 

model simulation and flight test approached 

similarity. While Figure 9d shows that pitch rate 

responses are nearly similar between model 

simulation and the flight test. Hence, these results 

demonstrate conformity of model simulation with 

the actual flight of longitudinal dynamic. The 

model is better than the system identification 

method using grey-box [1] because it is 

compliance with any conditions of altitude and 

airspeed. Figure 10 shows the response of 

normalized altitude and velocity so the initial 

steady state is same. The difference between the 

simulation and the flight test may be caused by 

the windy conditions when the flight test 

conducted. 

 

      

 (a) (b) 

   

 (c) (d) 

Figure 9. Model simulation and flight test results of longitudinal dynamics responses; (a) Model simulation results; (b) Flight test 

results; (c) Pitching and attack angles responses; (d) Pitch rate responses 

 



F.R. Triputra et al. / J. Mechatron. Electr. Power Veh. Technol 06 (2015) 19–30 

 

29 

V. CONCLUSION 
The longitudinal and lateral dynamics 

responses of Wulung UAV are good enough 

concerning to the outcome of model simulations 

that show the stability of flying without the hard 

control effort. Our proposed model is also 

compliance with flight test results of Wulung 

UAV. Furthermore, our nonlinear model of the 

fixed-wing UAV has the advantage to calculate 

flight dynamics for all conditions of altitude and 

airspeed that is important to build a controller 

that adapt to the UAV altitude and airspeed. As 

future works, this model will be used for an 

adaptive nonlinear controller using command 

filtered backstepping method. This model also 

will be employed for building the hardware in the 

loop simulation (HILS) systems that very useful 

for testing the hardware controller module before 

being used in the field. 

 

ACKNOWLEDGEMENT 
The authors express thanks to UAV team of 

BPPT Technology Center for Defense and 

Security Industry, for the support and the 

providing of required facilities and conducive 

conditions for this research. 

 

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Figure 10. Altitude and velocity responses 



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