This is an open access article under the CC BY license : Al-Khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal ISSN (printed): 1818 – 1171, ISSN (online): 2312 – 0789 Vol. 21, No. 3, September, (2025), pp. 66- 75 Studying the Effect of Rotor’s Coefficients on a Y-shaped Tri-copter Abdullah Mohammed Yahia1*, Ahmed Alkamchi2 and Ergun Erçelebi3 1,2 Department of Mechatronics, Al-Khwarizmi College of Engineering, University of Baghdad, Baghdad, Iraq 3 Electrical and Electronic Department, College of Engineering, Gaziantep University, Turkey *Corresponding Author’s Email: abd.yahia2102m@kecbu.uobaghdad.iq (Received 23 May 2024; Revised 21 October 2024; Accepted 28 December 2024; Published 1 September 2025) https://doi.org/10.22153/kej.2025.12.002 Abstract Unmanned aerial vehicles (UAVs) are small yet highly capable aircraft widely used in many applications. UAV designs and shapes vary depending on their intended usage. Tri-copters, known for their agility, have three propellers. However, one of their main challenges is yawing, which occurs when the UAV rotates around its z-axis (yaw angle). This yawing issue is a result of the asymmetry in the number of propellers. Unlike quadcopters, which have an even number of propellers, the aerodynamic drag torque produced by the tri-copter’s propellers’ does not cancel out. In this paper, a Y-shaped tri-copter model is tested, modified and compared to address the yawing problem. Aiming to mitigate yawing, two propellers are set to rotate clockwise, whilst the third propeller rotates clockwise. In the first configuration, the force coefficient is set equally for all propellers. In the second configuration, the force coefficient of the counter-clockwise rotating propeller is doubled. The UAV model is controlled by six PIDs associated with feedback linearisation for both configurations—three PIDs for attitude control and three for altitude control. The PIDs are tuned using a genetic algorithm, and the system is simulated in MATLAB Simulink. The proposed configuration demonstrates lower integral time absolute error (ITAE) values, indicating improved UAV performance. The average ITAE values for the Y-shaped model are 0.1553 for the first case and 0.1017 for the second case. The second case shows remarkable tracking in the desired output, with no violation of design limitations (servo angles and motor speeds). Keywords: PID; genetic algorithm; UAV; thrust-vectoring; feedback-linearisation. 1. Introduction Unmanned aerial vehicles (UAVs) are small aircraft that can be controlled remotely or by an onboard navigation unit. UAVs have a wide range of applications across various fields, leading to numerous configurations (including bi-copters, tri- copters and quad-copters), each with different frame shapes and motor configurations [1]. Bi-copters are amongst the simplest in terms of the number of actuators. They typically have only two to four actuators: two main propellers and one or two additional servo motors to control yaw and roll motions[2, 3]. The control systems of bi-copters are considered challenging due to the limited number of actuators, making them under-actuated. Thus, bi- copters are governed by a set of non-linear equations and generally offer less stability [4]. Quad-copters are amongst the most reliable UAVs, utilising four propellers (under-actuated)[5]. By adding two to four servo motors, these UAVs can be fully actuated[6] or even over actuated[7], simplifying control. However, one of the main challenges for UAVs is power consumption, which can limit their capabilities and flight time [8]. In contrast, tri-copters are known for their power efficiency because they use only three rotors [9]. Aiming to enhance agility and manoeuvrability [10], a thrust-vectoring mechanism can be added. The main issue with tri-copters is yawing, which mailto:abd.yahia2102m@kecbu.uobaghdad.iq https://doi.org/10.22153/kej.2025.12.002 Abdullah Mohammed Al-Khwarizmi Engineering Journal, Vol. 21, No.3, pp.66- 75 (2025) 67 arises from the asymmetry in rotor numbers and their rotational directions. With three rotors, the aerodynamic forces acting on the blades of each rotor do not cancel out, leading to yaw instability. This issue can be addressed by using coaxial rotors [11], implementing specific control strategies[12] or employing tiltable rotors[13]. Tilt mechanisms can be integrated into different configurations to meet desired application demands, as shown in [14], [15] and [16]. For system control, various methods have been verified; for example, the linear quadratic regulator is used in [17], H-infinity is used in [18] and a proportional integral derivative (PID) is applied in [19]. The PID control, when associated with optimisation techniques to maximise system performance, can be a very practical approach to UAV control [20]. Various optimisation techniques, such as grey wolf optimisation, have been employed for this purpose [21]. 2. Y Tri-coper Mathematical Model In this section, the mathematical model of the UAV is presented for two different motor configurations (two cases). The UAV parameters are referenced from [22], as shown in Table 1. Table 1, UAV parameters Value Description parameter 1.448 𝑘𝑔 UAV Mass 𝑚 0.33 𝑚 Arm length 𝑎 0.1035 𝑘𝑔. 𝑚2 Body inertia of x axis 𝛪𝑥𝑥 0.1003 𝑘𝑔. 𝑚2 Body inertia of y axis 𝛪𝑦𝑦 0.1709 𝑘𝑔. 𝑚2 Body inertia of z axis 𝛪𝑧𝑧 1.084 × 10−5 𝑘𝑔. 𝑚 Force coefficient 𝑘𝑓 1.726 × 10−7 𝑘𝑔. 𝑚2 Torque coefficient 𝑘𝑡 The primary model and dynamic equations are sourced from [23], along with the feedback linearisation equation, incorporating a modification to the original motor rotation directions. In both cases, the motor rotation directions are consistent: the front motors rotate clockwise, whilst the rear motor rotates counter-clockwise, as shown in Figure 1. In this context, α represents the servo angle, with the subscripts referring to the corresponding arm (right [r], left [l] and back [l]), and ‘a’ denotes the arm length, which is equal for all arms. Fig. 1. UAV motor configuration The motor coefficients will only differ in the second case, where the back motor coefficients are doubled. When this modification is implemented, certain adjustments to the original model will be required, as shown below. 2.1. First Case In this case, the force coefficient for all motors will remain the same. The only change will be the direction of the rear motor, which will affect the mathematical representation of the drag torque, as shown below: 𝜏𝑑1 = [ 0.866𝑘𝑡𝜔𝑚𝑟 2 𝑐𝑜𝑠(𝛼𝑟) − 0.866𝑘𝑡𝜔𝑚𝑙 2 𝑠𝑖𝑛(𝛼𝑙 ) −𝑘𝑡𝜔𝑚𝑟 2 𝑠𝑖𝑛(𝛼𝑟) + 0.5𝑘𝑡𝜔𝑚𝑙 2 𝑠𝑖𝑛(𝛼𝑙)−0.5𝑘𝑡𝜔𝑚𝑏 2 𝑠𝑖𝑛(𝛼𝑏) −𝑘𝑡𝜔𝑚𝑟 2 𝑐𝑜𝑠(𝛼𝑟) − 𝑘𝑡𝜔𝑚𝑙 2 𝑐𝑜𝑠(𝛼𝑙) + 𝑘𝑡𝜔𝑚𝑏 2 𝑐𝑜𝑠(𝛼𝑏 ) ] …(1) where 𝜏𝑑( ) is the drag torque for the corresponding case, 𝑘𝑡 is the torque coefficient, 𝛼( ) is the servo angle for the corresponding arm (b for back, r for right and l for left) and 𝜔( ) 2 is the squared angular velocity for the corresponding arm. 2.1. Second Case In this case, the drag torque will remain the same as in case one because the motor’s rotation directions are set identically. The only difference is that the torque coefficient for the back motor is doubled. CC W a a a Abdullah Mohammed Al-Khwarizmi Engineering Journal, Vol. 21, No.3, pp.66- 75 (2025) 68 𝜏𝑑2 = [ 0.866𝑘𝑡𝜔𝑚𝑟 2 𝑐𝑜𝑠(𝛼𝑟) − 0.866𝑘𝑡𝜔𝑚𝑙 2 𝑠𝑖𝑛(𝛼𝑙 ) −𝑘𝑡𝜔𝑚𝑟 2 𝑠𝑖𝑛(𝛼𝑟) + 0.5𝑘𝑡𝜔𝑚𝑙 2 𝑠𝑖𝑛(𝛼𝑙) + 2(−0.5𝑘𝑡)𝜔𝑚𝑏 2 𝑠𝑖𝑛(𝛼𝑏) −𝑘𝑡𝜔𝑚𝑟 2 𝑐𝑜𝑠(𝛼𝑟) − 𝑘𝑡𝜔𝑚𝑙 2 𝑐𝑜𝑠(𝛼𝑙 ) + 2𝑘𝑡𝜔𝑚𝑏 2 𝑐𝑜𝑠(𝛼𝑏) ] …(2) By doubling the force coefficient, the force and torque equations are modified as shown below: 𝑓 = [ −0.866𝑘𝑓 𝜔𝑚𝑙 2 𝑠𝑖𝑛(𝛼𝑙) + 0.866(2𝑘𝑓)𝜔𝑚𝑏 2 𝑠𝑖𝑛(𝛼𝑏 ) 𝑘𝑓𝜔𝑚𝑟 2 𝑠𝑖𝑛(𝛼𝑟) + −0.5𝑘𝑓 𝜔𝑚𝑙 2 𝑠𝑖𝑛(𝛼𝑙) + 2(−0.5𝑘𝑓)𝜔𝑚𝑏 2 𝑠𝑖𝑛(𝛼𝑏) 𝑘𝑓𝜔𝑚𝑟 2 𝑐𝑜𝑠(𝛼𝑟) + 𝑘𝑓𝜔𝑚𝑙 2 𝑐𝑜𝑠(𝛼𝑙) + 2𝑘𝑓𝜔𝑚𝑏 2 𝑐𝑜𝑠(𝛼𝑏) ] …(3) 𝜏 = [− 0.866𝑘𝑓 𝜔𝑚𝑙 2 𝑐𝑜𝑠(𝛼𝑙) − 0.866(2𝑘𝑓)𝜔𝑚𝑏 2 𝑐𝑜𝑠(𝛼𝑏) 𝑘𝑓𝜔𝑚𝑟 2 𝑐𝑜𝑠(𝛼𝑟) + 0.5𝑘𝑓𝜔𝑚𝑙 2 𝑐𝑜𝑠(𝛼𝑙) + 2(0.5𝑘𝑓)𝜔𝑚𝑏 2 𝑐𝑜𝑠(𝛼𝑏 ) 𝑘𝑓𝜔𝑚𝑟 2 𝑠𝑖𝑛(𝛼𝑟) + 𝑘𝑓𝜔𝑚𝑙 2 𝑠𝑖𝑛(𝛼𝑙 ) + 2𝑘𝑓𝜔𝑚𝑏 2 𝑠𝑖𝑛(𝛼𝑏) ] …(4) where 𝑘𝑓 is the force coefficient, 𝑓 is the total force and 𝜏 is the total torque. 3. Proposed Controller The tri-rotor equations exhibit strong output coupling, necessitating the use of PID controllers in combination with feedback linearisation. As referenced in Section 1, the feedback linearisation method is also adopted from previous work. A total of six PID controllers are used to control the UAV: three for position control (x, y, z) and three for attitude control (, , ). This control approach remains the same for both cases, as shown in Figure 2. Fig. 2. System block diagram 3.1 Proportional Integral Divertive Controller (PID) After linearising the system, a linear PID controller can be applied. The PID controller is considered an optimal choice due to its simple implementation and high efficiency [24]. The standard PID formula used to compute the error signal is referenced from [25]. 𝑢(𝑡) = 𝐾𝑃𝑒(𝑡) + 𝐾𝐼 ∫ 𝑒(𝑡)𝑑𝑡 + 𝐾𝐷�̇�(𝑡) …(5) where 𝐾𝑃 is the proportional gain, 𝐾𝐼 is the integral gain, 𝐾𝐷 is the derivation gain and 𝑒(𝑡) represents the error signal. The system uses six PID controllers, divided between position and attitude control. Each controller is separately tuned using the genetic algorithm (GA) optimisation method, with the integral time absolute error (ITAE) used as the cost function. 3.2 PID Tunning The PIDs are tuned using the GA, a method introduced by John Holland in 1975 [26]. Based on the concepts of evolution, GA operate by choosing the most adapted individuals (elite members) form a population to reproduce and generate a new generation. This process involves selecting individuals, evaluating their performance, combining their solutions through genetic crossover, applying random mutations and modifying the population with the best-performing solutions. This approach, which mirrors the concept of natural selection, is effective for minimising specific cost functions (in this case, the system error). The GA is used to tune the PID controllers for both cases, implemented using a built-in MATLAB function. The selected cost function for evaluating performance is the ITAE [27]. ∫ 𝑡|𝑒(𝑡)|𝑑𝑡 ∞ 0 …(6) Figure 3 shows the genetic algorithm flow chart. Abdullah Mohammed Al-Khwarizmi Engineering Journal, Vol. 21, No.3, pp.66- 75 (2025) 69 Genetic algorithm working steps flow chartFig. 3. [28] 4. Simulation Results The PID parameters are tuned by setting the desired outputs for position and altitude to 1 m and 10deg for attitude. In this section, the system responses for both cases will be represented, along with their corresponding ITEA values and a simulated flight scenario. Each case will be evaluated based on its tracking performance and control efficiency. 4.1. First Case The PID gain and the ITAE values for the first case are shown in Table 2. Table 2 PID parameters, first case ITEA kd ki kp parameter 0.0421 3.63 0.397 14.64 Phi 0.0241 4.15 0.033 19.48 Theta 0.0252 4.72 0.03 19.77 Psi 0.3591 4.21 16.06 18.58 x 0.3605 4.02 14.49 19.85 y 0.1211 4.58 0.026 17.9 z Figures 4 and 5 show the tuning responses for the altitude and attitude, respectively, during the tuning process for the first case. The performance metrics of case one are as follows: altitude settle time: (1.7 s), X overshoot: (38%), Y overshoot: (37%) and Z overshoot: (10%). For case one, attitude settle time is (2.1 s), Phi overshoot is 17%, Theta overshoot is 16 % and Psi overshoot is 11%. Fig. 4. UAV altitude response for the first case Fig. 5. UAV attitude response for the first case 4.2. Second Case The second case showed lower ITAE values after doubling the back motor’s coefficients, as shown in Table 3. Table 3 PID parameters, second case ITEA kd ki kp parameter 0.0307 4.97 0.595 19.98 Phi 0.0167 5.67 0.015 19.26 Theta 0.0368 3.7 0.403 19.63 Psi 0.0889 5.83 0.018 19.82 x 0.3447 4.45 18.03 18.57 y 0.0924 5.46 0.016 19.81 z Figures 6 and 7 illustrate the tuning responses for altitude and attitude, respectively, during the tuning process of the second case. The performance metrics of case two are as follows: altitude settle time: (1.7 s), X overshoot: (3%), Y overshoot: (40%) and Z overshoot: (5%). For Case two, attitude Abdullah Mohammed Al-Khwarizmi Engineering Journal, Vol. 21, No.3, pp.66- 75 (2025) 70 settle time is 2 s, Phi overshoot is 10%), Theta overshoot is 11% and Psi overshoot is 22%. Fig. 6. UAV altitude response for the first case Fig. 7. UAV attitude response for the first case The second case demonstrated lower ITAE values; thus, a flight scenario was then applied to further evaluate the UAV performance. The UAV is initially positioned at (0,1,0), ascending vertically and then follows a spiral shape. This approach is achieved using a circular equation for the x and y coordinates, a ramp function for the z coordinate and setting all attitude variables to zero. Figure 8 illustrates the flight scenario. Fig. 8. UAV helical flight scenario The UAV smoothly tracked due to its enhanced manoeuvrability provided by the thrust vectoring mechanism. The motor forces remained balance due to the back motor having increased force and torque coefficients. Aiming to thoroughly evaluate the system’s performance, the responses of altitude, attitude, motor speeds and servo angles were captured during the flight. Figure 9 shows the altitude response. Fig. 9. UAV helical flight scenario altitude response The altitude responses closely followed the desired trajectory without any issues, demonstrating the UAV’s effective manoeuvrability. Figure 10 displays the attitude response. Abdullah Mohammed Al-Khwarizmi Engineering Journal, Vol. 21, No.3, pp.66- 75 (2025) 71 Fig. 10. UAV helical flight scenario attitude response The attitude responses remained zero throughout the entire flight, indicating that the UAV maintained good balance. Figures 11 and 12 show the responses of the motors and servo motors, respectively. The motor speed limits are set between 0 and 3600 rpm, whilst the servo angle limits range from −90° to 90°. Fig. 11. UAV helical flight scenario motors response Fig. 12. UAV helical flight scenario servo motors response The motors speed values increased as the UAV took off, then settled at a specific value and did not exceed the design boundaries. The back motor shows lower speed values because it has larger coefficients. The responses of the servo angles change continuously to maintain the UAV on the desired path. The second case showed lower ITAE values; therefore, a disturbance test and an uncertainty test were then applied to further evaluate the UAV’s performance. The uncertainty test responses and ITAE values after changing the mass to 1.8 kg and the arm length to 0.2 m are shown in Table 4. Table 4 Uncertainty ITAE values, second case ITEA uncertainty Parameter 0.03 Phi 0.016 Theta 0.035 Psi 0.09 x 0.347 y 0.093 z Figure 13 shows the uncertainty test responses for altitude and attitude. All responses stabilise within approximately 2 s. (a) (b) Fig. 13. Uncertainty altitude & attitude responses for Y-Model: (a) Altitude responses, (b) Attitude responses Figure 14 shows an impulse disturbance applied along the z-axis. The disturbance has a magnitude of 5 N and lasts for 0.25 s, starting at T = 4 s. The system regains stability approximately 1 s after the disturbance ends. Abdullah Mohammed Al-Khwarizmi Engineering Journal, Vol. 21, No.3, pp.66- 75 (2025) 72 Fig. 14. Disturbance on z axis Figure 15 shows disturbances on the phi rotation angle. The disturbance has a magnitude of 10 N.m. This disturbance starts at T = 2.5 s and ends at T = 2.75 s. The system regains stability approximately 2 s after the disturbance ends. Fig. 15. Disturbance on phi 5. Conclusion This research aims to study the effects of rotor coefficients on UAV performance. A previously developed model is modified by changing the rotation direction of the rotors: the front two propellers are set to rotate clockwise, and the back propeller rotates counter-clockwise. The modified model is then divided into two cases. Case one includes the model with the changed rotation direction and no further modifications, whilst case two involves doubling the coefficients of the back motor. Both cases are linearised using feedback linearisation to address the output coupling problem and are controlled by six PID controllers. The PID controllers are tuned using a GA, with the ITAE as the cost function. The average ITAE values for the Y- shaped model are 0.1553 and 0.1017 for cases one and two, respectively. Case two showed improved tracking of the desired output in the applied scenario without exceeding design limitations (servo angles and motor speeds). Aiming to further investigate the effects of rotor coefficients, implementation on a model without thrust vectoring is recommended. Notations x x- axis, m y y- axis, m z f z- axis, m Force, N a Length, m kf Force coefficient, 𝑁. 𝑠2 kt Torque coefficient, 𝑁. 𝑚. 𝑠2 m Mass, kg Ixx Body inertia of x-axis, 𝑘𝑔. 𝑚2 Iyy Body inertia of y-axis, 𝑘𝑔. 𝑚2 Izz Body inertia, of z-axis 𝑘𝑔. 𝑚2 e Error Kp Proportional PID gain Ki integral PID gain Kd Derivative PID gain Greek symbols  Servo angle, deg  Roll angle, deg  Pitch angle, deg 𝜓 Yaw angle, deg  Torque, 𝑁. 𝑚  Angular velocity, rpm References [1] M. H. Sabour, P. Jafary, and S. 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( 2025) 66-75، صفحة 3، العدد21مجلة الخوارزمي الهندسية المجلد عبد هللا محمد يحيى 75 Yدراسة تأثير معامالت الدوار على مروحية ثالثية الشكل على شكل حرف 3اركون ارچلبي 2احمد محروس راغب* *1عبد هللا محمد يحيى العراق بغداد، قسم هندسة الميكاترونكس، كلية الهندسة الخوارزمي، جامعة بغداد، 1،2 يا ترك ، جامعة غازي عنتاب ، قسم الهندسة الكهربائية وااللكترونية 3 abd.yahia2102m@kecbu.uobaghdad.edu.iqالبريد االلكتروني: * لمستخلص ا وتستخدم على نطاق واسع في العديد من التطبيقات. هناك العديد من التصاميم بكفاءة الطائرات بدون طيار هي طائرات صغيرة ولكنها قادرة على العمل ألنها تحتوي على ثالثة ؛ واألشكال للطائرات بدون طيار اعتمادًا على استخدامها. تعد الطائرات ثالثية المراوح واحدة من أكثر الطائرات بدون طيار رشاقة (. يحدث ذلك بسبب عراج)زاوية االن zالطائرة بدون طيار وهو دوران الطائرة بدون طيار حول محورها عراجمراوح فقط. ومع ذلك، فإن إحدى مشاكلها هي ان رات المراوح كما هو الحال في طائرة رباعية المراوح وح. بمعنى آخر، لن يتم إلغاء عزم السحب الناتج عن الديناميكا الهوائية لشفاالمر عدادعدم التماثل في أ يتم و، عراج لتقليل تأثير مشكلة االن ؛ وتعديله ومقارنته Yتم اختبار نموذج طائرة ثالثية المراوح على شكل حرف عمل)عدد زوجي من المراوح(. في هذه ال . ى األول حالةوح في الضبط مروحتين في اتجاه عقارب الساعة، بينما يتم ضبط المروحة الثالثة في االتجاه المعاكس. يتم ضبط معامل القوة بالتساوي لجميع المرا يتم التحكم في النموذج من خالل ستة معّرفات ف. ةالثاني حالةثم يتم مضاعفة معامل القوة للمروحة الواحدة التي تدور في اتجاه عكس اتجاه عقارب الساعة لل PID وثالثة لالرتفاع. يتم ضبط معّرفات ، كوينين ثالثة للموقفمرتبطة بالخطية الراجعة لكال التPID باستخدام خوارزمية وراثية ويتم محاكاة النظام في MATLAB Simulink يظهر التكوين المقترح قيم .ITAE أقل )خطأ مطلق زمني متكامل( مما يشير إلى أداء أفضل للطائرات بدون طيار. متوسط قيم ITAE للنموذج على شكلY ( للحالة األولى0.1553هو )، ( للحالة الثانية. أظهرت الحالة الثانية تتبعًا كبيًرا لإلخراج المطلوب في السيناريو 0.1017و ) المطبق ولم تتجاوز قيود التصميم )زوايا المؤازرة وسرعات المحركات(.