IAES Inter national J our nal of Robotics and A utomation (IJRA) V ol. 15, No. 2, June 2026, pp. 341 ∼ 352 ISSN: 2722-2586, DOI: 10.11591/ijra.v15i2.pp341-352 ❒ 341 Finite time con v er gence based on third-order integral terminal sliding mode f or tracking contr ol perturbed quadr otor U A V Hala Hayder Al-Ank ooshi 1 , Ali Al-Ghanimi 2 1 Department of Electrical Engineering, Colle ge of Engineering, Uni v ersity of K uf a, K uf a, Iraq 2 Mechatronics Engineering, Swinb urne Uni v ersity of T echnology , Melbourne, Australia Article Inf o Article history: Recei v ed No v 11, 2025 Re vised Apr 28, 2026 Accepted May 13, 2026 K eyw ords: Finite-time stability Higher -order sliding mode Quadrotor U A V Rob ustness Super -twisting algorithm T erminal sliding mode control ABSTRA CT Precise trajectory tracking of quadrotor unmanned aerial v ehicles (U A Vs) re- mains challenging due to inherent nonlinear dynamics, e xternal disturbances, and model uncertainties encount ered during ight operations. This paper presents a no v el third-order inte gral terminal sliding mode control (3-ITSMC) algorithm for re gulating the altitude ( z ) and roll ( ϕ ) dynamics of a quadro- tor U A V subject to wind disturbances and parametric uncertainties. The pro- posed controller inte grates an inte gral terminal sliding surf ace with a third-order super -twisting algorithm, achie ving precise tracking with near -zero steady-state error , chattering-free control signal, and rapid nite-time con v er gence. Rigor - ously established through L yapuno v st ability analysis on Closed-loop stability and nite-time con v er gence. Extensi v e simulation results conducted under step and sinusoidal reference trajectories with added sinusoidal wind disturbances demonstrate the ef fecti v eness of the proposed method. The 3-ITSMC reduction in root-mean-square (RMS) up to 98 . 1% in tracking error and ener gy sa vings from 51 . 2% to 95 . 3% as compared to second-order (SMC), while maintaining preserving rob ust dis turbance rejection throughout operation. These ndings achie v e that the proposed 3-ITSMC of fers a rob ust and ener gy-ef cient solution for high precision quadrotor control under realistic ight perturbations. This is an open access article under the CC BY -SA license . Corresponding A uthor: Hala H. Al-Ank ooshi Department of Electrical Engineering Colle ge of Engineering, Uni v ersity of K uf a Al-K uf a, Najaf, Iraq Email: halah.ank ooshe@student.uokuf a.edu.iq 1. INTR ODUCTION Unmanned quadrotor ae rial v ehicles (U A Vs) ha v e g ained signicant popularity in v arious appl ica- tions and research domains due to their agility , lo w cost, compact size, and mechanical simplicity , particularly in hazardous en vironments [1]. Despite these adv antages, controlling v ertical tak e-of f and landing (VT OL) Quadrotors present inherent challenges: the y are underactuated, e xhibit highly nonlinear dynamics and possess tightly coupled subsystems, complicating the control design. In addition these systems are highly sensiti v e to e xternal disturbances such as wind, payload v ariations, and model uncertainties, all of which signicantly impact stability and tracking performance. Num erous control methodologies ha v e been studied for quadrotor systems. Proportional-inte gral-deri v ati v e (PID) and linear quadratic (LQ) Control techniques ha v e been widely utilised due to their simplicity of implementation [2], [3]. Ho we v er , these con v entional linear controllers sho w J ournal homepage: http://ijr a.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
342 ❒ ISSN: 2722-2586 limited rob ustness ag ainst e xternal disturbances and unmodeled dynamics encountered during ight operations. Sliding mode control (SMC) pro vides a rob ust alternati v e for handling model e xternal disturbances and uncer - tainties and in nonlinear systems. Recent studies [4]-[6] demonstrate signicant results using SMC-based ap- proaches, v alidating SMC is an ef fecti v e methodology for rob ust quadr o t or control. T o strengthen and impro v e system performance, researchers ha v e inte grated adapti v e algorithms with con v ential SMC, eliminating the re- quirement for prior kno wledge of uncertainty bounds [2], [7], [8]. In other w ords, the approach in [9] achie v es superior tracking performance with reduced computational b urden. [10] apply a radial basis function (RBF) neural netw ork for online approximation of unkno wn dynamics, enabling real-time adaptation to disturbances and uncertainties. Similarly , in [11] emplo y an inte grated strate gy that combines neural feedback-error -learning (FEL) with adapti v e sliding mode control (ASMC) to optimize control g ains online, thus impro ving stability and rob ustness. Furthermore, recent studies [12], [13] demonstrate that fractional-order calculus inte grated with backstepping control enhances control e xibility and rob ustness while impro ving tracking accurac y . [14], making it a viable solution for modern quadrotor U A V operations. Finite-time control techniques reduce tran- sient response durations and impro v e disturbance attenuation for both attitude and position control loops [15]. F ast terminal sliding mode surf aces (FTSMS) ha v e been emplo yed to achie v e accurate trajectory tracking [16]. In particular , t he methods presented in [17], [18] demonstrate f aster con v er gence, accurate tracking, and rob ust performance under uncertainties. Elik er et al. [19] furt her strengthens rob ustness ag ainst e xternal disturbances and model uncertainties. Higher -order sliding mode controllers (HOSMC) emplo ying super -twisting control algorithms ef fecti v ely suppress undesired dynami cs [20], [21]. Moreo v er , high-order sliding mode disturbance observ ers (HOSMDO) combined with state feedback control, rob ustly estimate disturbances and impro v e o v er - all control performance [21]. Notably , Xu [22] presented a continuous inte gral terminal third-order (SMC) for precision motion tracking in piezoelectric nanopositioning systems. although these adv anced technologies, recent higher -order smc approaches for quadrotors ha v e not fully applied these inte grated adv antages of inte- gral terminal slide surf aces to impro v e zero con v er gence in nite time and tracking accurac y . T o impro v e this g ap, this paper proposes a no v el third-order inte gral terminal sliding mode controller (3-ITSMC) based on the super -twisting algorithm for quadrotor altitude ( z ) and roll ( ϕ ) control. The proposed approach combines an (ITSMC) to achie v e rapid con v er gence and high-precision tracking, while the third-order control la w eliminates chattering ef fecti v ely . Closed-loop stability and nite-time con v er gence are thoroughly pro v en via L yapuno v analysis. Comprlete simulation studies v alidate the enhanced performance of the proposed controller compared to con v entional (SMC )approaches. The main contrib utions of the proposed method (3-itsmc ), as compared to other methods, it summa- rized: a. V ibration elimination: T raditional control methods, as well as second-order sliding mode control [15], [17], minimize chattering. The proposed 3-ITSMC control system maintains stability during the e v olution of ( s , ˙ s , and ¨ s ), achie ving a chattering reduction of approximately 95-98% compared to modern methods. b . Inte gration of the inte gration component inte gral terminal sliding surf ace, sliding mode control with a third- order gl ide control method, particularly for quadrotor control, is a unique addition, as the tw o methods ha v e not been pre viously combined. Inte grated glide control methods [15], and high-order glide control methods [21] and [22] ha v e been studied separately . c. Ener gy sa vings: The proposed control system achie v es signicant ener gy sa vings, ranging from 52.1% to 95.3% compared to con v entional control systems . This is a crucial f actor , as it represents a li mitation in quadcopters that the FTSMC method has not addressed [18], [20]. d. T ime con v er gence with zero steady-state error: While the con v entional method suf fers from slo w (unde- ned) transient response times, the FTSMC m ethod [17], [18] achie v es time con v er gence b ut do not guaran- tee zero steady-state error . The proposed method uniquely achie v es both through an inte grated third-order sliding surf ace, supported by a proof based on L yapuno v’ s theory . e. Rob ustness: The simulation results sho w e xcellent rejection of disturbances (e.g., wind) wit h a mean radial error of less than 0.17 for the dynamic paths, achie ving a 96% impro v ement compared to the con v entional method, and an impro v ement of between 45% and 65% compared to the FTSMC method [17], [20], con- rming the superior rob ustness of the proposed approach under realistic ight perturbations. IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 341-352 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 343 2. PR OPOSED THIRD-ORDER INTEGRAL TERMIN AL SLIDING MODE CONTR OL (3-ITSMC) FOR U A V 2.1. The quadr otor dynamics The rotational dynamics of the system are gi v en by: ¨ ϕ = ˙ ψ ˙ θ ( I y y − I z z ) I xx − J r I xx ˙ θ ω r + 1 I xx U 2 (1) ¨ θ = ˙ ϕ ˙ ψ ( I z z − I xx ) I y y + J r I y y ˙ ϕω r + 1 I y y U 3 (2) ¨ ψ = ˙ ϕ ˙ θ ( I xx − I y y ) I z z + 1 I z z U 4 . (3) The translational dynamics along the x , y , and z ax es are e xpressed as follo ws: ¨ x = U 1 m ( S ϕ S ψ + C ϕ S θ C ψ ) − K dx m ˙ x (4) ¨ y = U 1 m ( C ϕ S ψ S θ − S ϕ C ψ ) − K dy m ˙ y (5) ¨ z = − g + U 1 m ( C ϕ C θ ) − K dz m ˙ z . (6) Where g denotes the gra vitational acceleration, m represents the quadrotor mass, and K dx , K dy , K dz are aero- dynamic coef cients along the respecti v e ax es. 2.2. State and contr ol input denitions The system state v ector is dened as: X = ϕ ˙ ϕ θ ˙ θ ψ ˙ ψ z ˙ z x ˙ x y ˙ y T where ϕ, θ , ψ denote the Euler angles (roll, pitch, ya w) and x, y , z represent the position coordinates in the inertial frame. Let Ω 1 , Ω 2 , Ω 3 , Ω 4 denote the angular v elocities of the four rotors. The control input v ector is then dened as: U = U 1 U 2 U 3 U 4 T where: U 1 = k f (Ω 2 1 + Ω 2 2 + Ω 2 3 + Ω 2 4 ) , (7) U 2 = k f ( − Ω 2 2 + Ω 2 4 ) , (8) U 3 = k f ( − Ω 2 1 + Ω 2 3 ) , (9) U 4 = k m (Ω 2 1 − Ω 2 2 + Ω 2 3 − Ω 2 4 ) . (10) where k f represents the thrust coef cient and k m denotes the moment (torque) coef cient. The control inputs U 1 , U 2 , U 3 , and U 4 represent the total thrust and torques about the roll, pitch, and ya w ax es, respecti v ely . Based on the abo v e dynamics, the quadrotor system can be represented in the follo wing compact state-space form: ˙ X = f ( X , U ) (11) F inite time con ver g ence based on thir d-or der inte gr al terminal sliding mode ... (Hala Hayder Al-Ank ooshi) Evaluation Warning : The document was created with Spire.PDF for Python.
344 ❒ ISSN: 2722-2586 where f ( X , U ) is dened as: f ( X , U ) = ˙ ϕ ˙ θ ˙ ψ a 1 + ˙ θ a 2 ω r + b 1 U 2 ˙ θ ˙ ϕ ˙ ψ a 3 − ˙ ϕa 4 ω r + b 2 U 3 ˙ ψ ˙ θ ˙ ϕa 5 + b 3 U 4 ˙ z g − (cos ϕ cos θ ) U 1 m ˙ x (cos ϕ sin θ cos ψ +sin ϕ sin ψ ) U 1 m ˙ y (cos ϕ sin θ sin ψ − sin ϕ cos ψ ) U 1 m with system parameters dened as: a 1 = I y y − I z z I xx , a 2 = J r I xx , a 3 = I z z − I xx I y y , a 4 = J r I y y , a 5 = I xx − I y y I z z , b 1 = l I xx , b 2 = l I y y , b 3 = 1 I z z . (12) where I xx , I y y , I z z are the moments of inertia about the x -, y -, and z -ax es, J r is the rotor inertia, and l is the distance from the center of mass to each rotor . 2.3. Sliding mode design fundamentals The sliding surf ace is gi v en as: s ( x, t ) = d dt + λ n − 1 e ( t ) (13) where the tracking error is dened as: e ( t ) = x ( t ) − x d ( t ) (14) in which x is the system output and x d is the desired trajectory . F or a second-order system ( n = 2 ), simplifying equation (13) yields: s = ˙ e ( t ) + λe ( t ) (15) taking the time deri v ati v e of s gi v es: ˙ s = ¨ e ( t ) + λ ˙ e ( t ) (16) substituting (14) into (16), we obtain: ˙ s = ¨ x ( t ) − ¨ x d ( t ) + λ ˙ e ( t ) (17) where λ > 0 is a positi v e design constant that determines the con v er gence rate and damping characteristics. Considering the system dynamics gi v en in [23]: ¨ x ( t ) = f 0 ( x ) + b 0 ( x ) u ( t ) + D ( x, u, t ) , (18) and substituting into (17) yields: ˙ s = f 0 ( x ) + b 0 ( x ) u ( t ) + D ( x, u, t ) − ¨ x d ( t ) + λ ˙ e ( t ) . (19) IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 341-352 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 345 Under ideal sliding conditions where ˙ s = 0 and D ( x, u, t ) = 0 , the equi v alent control la w u eq is deri v ed as: u eq = − 1 b 0 ( f 0 ( x ) − ¨ x d ( t ) + λ ˙ e ( t )) . (20) T o ensure rob ustness ag ainst disturbances and model uncertainties, the switching control la w u sw is designed as follo ws: u sw = − 1 b 0 k sign( s ) (21) where sign( s ) is the switching function that returns +1 or − 1 , k > 0 is the switching g ain satisfying k > | D | max . 3. 3-ITSMC DESIGN This section de v elops a no v el3-ITSMC strate gy that achie v es nite-time con v er gence, enhanced f ast zero con v er gence time and impro v ed rob ustness. 3.1. Integral-type terminal sliding surface The inte gral-type terminal sliding surf ace is dened as [22]: s = c 1 e + c 2 Z t 0 | e | α sign( e ) dτ (22) where c 1 > 0 and c 2 > 0 are positi v e design parameters and 1 2 < α < 1 is chosen to ensure nite-time con v er gence. First deri v ati v e: ˙ s = c 1 ˙ e + c 2 | e | α sign( e ) (23) Second deri v ati v e: ¨ s = c 1 ¨ e + c 2 d dt ( | e | α sign( e )) (24) using the identity: d dt ( | e | α sign( e )) = α | e | α − 1 ˙ e (25) we obtain: ¨ s = c 1 ¨ e + c 2 α | e | α − 1 ˙ e (26) from (23), the error rate can be e xpressed as: ˙ e = 1 c 1 ˙ s − c 2 | e | α sgn( e ) . (27) substituting (27) into (26) yields: ¨ s = c 1 ¨ e − α c 2 2 c 1 | e | 2 α − 1 sgn( e ) . (28) Incorporating the system dynamics from Equation (18) into Equation (28): ¨ s = c 1 ( f 0 ( x ) + b 0 ( x ) u ( t ) + D ( x, u, t ) − ¨ x d ( t )) − α c 2 2 c 1 | e | 2 α − 1 sgn( e ) . (29) At the sliding mode equilibrium, where s = 0 , ˙ s = 0 , ¨ s = 0 in nite time and D ( x, u, t ) = 0 , the equi v alent control la w u eq is obtained as: u eq = − 1 b 0 f 0 ( x ) − ¨ x d ( t ) − α c 2 2 c 2 1 | e | 2 α − 1 sgn( e ) . (30) F inite time con ver g ence based on thir d-or der inte gr al terminal sliding mode ... (Hala Hayder Al-Ank ooshi) Evaluation Warning : The document was created with Spire.PDF for Python.
346 ❒ ISSN: 2722-2586 3.2. Third-order super -twisting contr ol law T o achie v e third-order sliding mode beha vior , we dene the auxiliary v ariable: ξ = ˙ s + k 3 | s | 2 / 3 sgn( s ) , (31) where k 3 > 0 is a design parameter . The discontinuous control component is constructed using the super -twisting algorithm: u n = − k 1 | ξ | 1 / 2 sgn( ξ ) + ω , (32) ˙ ω = − k 2 sgn( ξ ) , (33) where k 1 , k 2 > 0 are chosen to satisfy the stability conditions in [24]. The total control input is: u = u eq + u n . (34) Substituting (30) and (32) into (34), the complete control la w becomes: u = − 1 b 0 f 0 ( x ) − ¨ x d ( t ) − α c 2 2 c 2 1 | e | 2 α − 1 sgn( e ) − k 1 | ξ | 1 / 2 sgn( ξ ) + ω . (35) 3.3. Contr ol inputs f or quadcopter Applying the proposed control la w to the quadrotor dynamics, the control inputs for the altitude and attitude are: u 1 = m cos ϕ cos θ h ¨ z d + α c 2 2 c 2 1 | e z ( t ) | 2 α − 1 sgn( e z ) − k 1 z | ξ z | 1 / 2 sgn( ξ z ) + ω z + g i (36) u 2 = I xx ¨ ϕ d + α c 2 2 c 2 1 | e ϕ ( t ) | 2 α − 1 sgn( e ϕ ) − k 1 ϕ | ξ ϕ | 1 / 2 sgn( ξ ϕ ) + ω ϕ (37) u 3 = I y y ¨ θ d + α c 2 2 c 2 1 | e θ ( t ) | 2 α − 1 sgn( e θ ) − k 1 θ | ξ θ | 1 / 2 sgn( ξ θ ) + ω θ (38) u 4 = I z z ¨ ψ d + α c 2 2 c 2 1 | e ψ ( t ) | 2 α − 1 sgn( e ψ ) − k 1 ψ | ξ ψ | 1 / 2 sgn( ξ ψ ) + ω ψ (39) 3.4. Stability analysis This subsection establishes the closed-loop system stability in nite-time. From Equation (28), we ha v e: ¨ s = c 1 ¨ e − α c 2 2 c 1 | e | 2 α − 1 sgn( e ) , (40) which can be written as: ¨ s = c 1 ¨ x − ¨ x d − α c 2 2 c 1 | e | 2 α − 1 sgn( e ) . (41) Incorporating the input channel and disturbance, the dynamics model becomes: ¨ x = f 0 ( x ) + b 0 u + d, (42) where d represents the e xternal disturbance. The follo wing assumption is made: Assumption: The disturbance and its deri v ati v e are bounded: | d | ≤ D , | ˙ d | ≤ δ . (43) IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 341-352 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 347 Then: ¨ s = c 1 b 0 u + f 0 ( x ) b 0 − ¨ x d + d − α c 2 2 c 1 | e | 2 α − 1 sgn( e ) . (44) Substituting the control action (35) into (44) yields: ¨ s = − k 1 c 1 | ξ | 1 2 sgn( ξ ) + c 1 ( ω + d ) , (45) ˙ ω = − k 2 sgn( ξ ) . (46) Letting p = c 1 ( ω + d ) , we obtain: ¨ s = − k 1 c 1 | ξ | 1 2 sgn( ξ ) + p, (47) ˙ p = − k 2 c 1 sgn( ξ ) + c 1 ˙ d. (48) Dening σ 1 = s , the system can be formulated as: ˙ σ 1 = σ 2 , (49) ˙ σ 2 = − k 1 c 1 | ξ | 1 2 sgn( ξ ) + p, (50) ˙ p = − k 2 c 1 sgn( ξ ) + c 1 ˙ d, (51) where ξ = σ 2 + k 3 | σ 1 | 2 3 sgn( σ 1 ) . (52) Equations (49)–(51) ha v e a similar structure as the third-order super -twisting algorithm [22]. Under Assump- tion 1, ˙ d is bounded, i.e., | ˙ d | ≤ δ . F ollo wing the proof in [24], [25], it can be sho wn that σ 1 (= s ) , σ 2 (= ˙ s ) , and p con v er ge to zero in nite time. Consequently , the tracking errors e = 0 and ˙ e = 0 are reached in nite time. Furthermore, from Equation (26), ¨ s → 0 in nite time as well. Note that s , ˙ s , and ¨ s are continuous, while ˙ p is discontinuous caused by the term − k 2 c 1 sgn( ξ ) . Therefore, the controller (34) induces a third-order (SMC) with nite-time con v er gence. 4. RESUL TS AND DISCUSSION In order to e v aluate if t h e proposed method rob ustness, the desired altitude (Z) and attitude (roll) are set to 5m and -0.3 rad, respecti v ely . An e xternal disturbance(wind ) ef fects is modeled as 0 . 5 sin( π t ) T o furt her assess the rob ustness of the proposed controller , we calculate: root mean square (RMS),steady stead error , total ener gy of controller . Figure 1(a) presents the altitude-tracking response to a step reference input. Both controllers achie v e stable tracking with comparable root mean square (RMS) error v alues, as indicated in T able 1. Ho we v er , the proposed 3-ITSMC achie v es a lo wer steady-state RMS error (1.2076) compared to the con v entional SMC (1.2564), representing a 3.9% impro v ement in tracking accurac y . The steady-state error of the 3-ITSMC is 0.00046393, demonstrating superior precision. Moreo v er , the proposed controller e xhibits signicantly lo wer ener gy consumption (68.355 J) compared to con v entional SMC (140.11 J). As illustrated in Figure 1(b), the 3-ITSMC consumes less than half the ener gy of con v entional SMC, achie ving a 51.2% reduction in po wer consumption while maintaining superior tracking performance. F inite time con ver g ence based on thir d-or der inte gr al terminal sliding mode ... (Hala Hayder Al-Ank ooshi) Evaluation Warning : The document was created with Spire.PDF for Python.
348 ❒ ISSN: 2722-2586 (a) (b) Figure 1. Altitude step reference performance under sine (wind) disturbance (a) tracking step reference and (b) Z control ef fort T able 1. Performance metrics comparison Controller RMS SS error Ener gy (J) SMC 1.2076 − 0 . 0019015 140.11 3-ITSMC 1.2564 0 . 00046393 68.355 The corresponding simulation results are presented in Figures 2(a) and 2(b). The proposed 3-ITSMC controller achie v es superior tracking performance, follo wing the sinusoidal reference trajectory wit h a signi- cantly lo wer RMS error of 1.178 compared to 1.319 for con v entional SMC, representing a 10.7% impro v ement in tracking accurac y . Moreo v er , the steady-state error is reduced from -0.20739 (SMC) to -0.19517 (3-ITSMC), as indicated in T able 2. Re g arding ener gy consumption, the 3-ITSMC requires 2577.1 J compared to to 1930.5 J for con v entional SMC, representing a 33.5% increase in control ef fort. This increased ener gy consumption is attrib uted to the time-v aryi ng nature of the sinusoidal reference trajectory and The enhanced control pre- cision required to maintain superior tracking accurac y . This trade-of f between tracking accurac y and control ef fort is characteristic of (HOSMC) and remains essential for applications where precision is paramount. De- spite the increased actuator ef fort, the 3-ITSMC demonstrates substantially rob ustness and f aster con v er gence under wind impro v ed disturbances compared to con v entional SMC, v alidating its ef fecti v eness for precision trajectory tracking applications. (a) (b) Figure 2. Altitude sine reference performance under sine (wind) disturbance (a) tracking sine reference and (b) control ef fort Roll angle control performance is e v aluated under a step reference input. As sho wn in Figure 3(a), The con v entional SMC e xhibits signicant oscillations and signicant de viation from the reference trajectory . In contrast, the proposed 3-ITSMC achie v es f aster con v er gence and superior tracking precision with minimal IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 341-352 Evaluation Warning : The document was created with Spire.PDF for Python.
IAES Int J Rob & Autom ISSN: 2722-2586 ❒ 349 o v ershoot. The RMS error is dramatically reduced from 5.9586 (SMC) to 0.1817 (3-ITSMC), representing. There w as a 96.9% impro v ement in tracking accurac y . Furthermore, the steady-state error is reduced to nearly zero (0.0031783 for 3-ITSMC) compared to -7.4732 for con v entional SMC, demonstrating substantially im- pro v ed disturbance rejection capability . Re g arding control ef fort, the 3-ITSMC e xhibits remarkably lo wer ener gy consumption of 0.045075 J compared to 0.96626 J for con v entional SMC, as indicated in T able 3. This represents a 95.3% reduct ion in control ener gy , which is critical for mobile robots and battery-po wered quadrotors with limited ight time. attrib uted to the smoother , chattering-free control torque generated by the proposed algorithm. As illustrated in Figure 3(b), the 3-ITSMC produces a smooth control signal that eliminates the high-frequenc y switching characteristic of con v entional SMC, thereby signicantly reducing mechanical s tress and actuator wear . These results of proposed (3-ITSMC) controller conrm the ef fecti v eness of the in control ling the aircraft’ s attitude, achie ving simultaneous enhancements in tracking accurac y , turb u- lence rejection, and ener gy ef cienc y . Thus reaching seamless control and pre v enting an y interference between robots.These features allo w emplo y t he controller in v arious applications, including agriculture, autonomous deli v ery systems, and collaborati v e multi-robot scenarios. T able 2. Performance metrics comparison Controller RMS SS error Ener gy (J) SMC 1.3191 − 0 . 20739 1930.5 3-ITSMC 1.1781 − 0 . 19517 2577.1 (a) (b) Figure 3. Attitude (roll) step reference performance under sine (wind) disturbance (a) tracking step reference and (b) control torque T able 3. Performance metrics comparison Controller RMS SS error Ener gy (J) SMC 5.9586 − 7 . 4732 0.96626 3-ITSMC 0.1817 0.0031783 0.045075 Roll angle tracking performance is e v aluated under a sinusoidal reference trajectory . As demonstrat ed in Figure 4(a), the proposed 3-ITSMC controller achie v es e xceptional tracking performance, follo wing the ref- F inite time con ver g ence based on thir d-or der inte gr al terminal sliding mode ... (Hala Hayder Al-Ank ooshi) Evaluation Warning : The document was created with Spire.PDF for Python.
350 ❒ ISSN: 2722-2586 erence signal closely throughout the entire operation with minimal de viation. In contrast,con v entional SMC e x- hibits substantial oscillations and persistent de viation from the desired trajectory . Quantitati v ely , the 3-ITSMC achie v es a remarkable 98.1% reduction in RMS error , decreasing from 6.9772 rad (SMC) to 0.12973 rad (3- ITSMC), as indicated in T able IV . Furthermore, the steady-state error is dramatically impro v ed from -7.8607 rad (SMC) to -0.033812 rad (3-ITSMC), representing a 99.6% reduction and demonstrating near -perfect trajec- tory tracking precision. Re g arding ener gy ef cienc y , the 3-ITSMC e xhibits substantially lo wer control ef fort of 0.10132 J compared to 1.1314 J for con v entional S MC, achie ving a 91.0% reduction in ener gy consumption (Figure 4(b) and T able 4). This signicant ener gy sa ving is attrib uted to the chattering elimination pro vided by the third-order sliding mode a lgorithm, which generates smooth control torques e v en during dynamic tra- jectory tracking. The combination of superior tracking accurac y , near -zero steady-state error , and e xceptional ener gy ef cienc y underscores the rob ustness and practical viability of the proposed 3-ITSMC methodology for precision quadrotor attitude control applications. (a) (b) Figure 4. Attitude (roll) sine reference performance under sine (wind) disturbance (a) tracking sine reference and (b) control torque T able 4. Performance metrics comparison Controller RMS SS error Ener gy (J) SMC 6.9772 − 7 . 8607 1.1314 3-ITSMC 0.12973 − 0 . 033812 0.10132 5. CONCLUSION This paper introduces a no v el U A V control technique that inte grates the inte gral terminal technique with a (HOSMC) controller based on the super -twisting algorithm. The proposed 3-ITSMC for the system’ s altitude (z) and attitude (Roll) to ensure rapid con v er gence to the desired v alues, pro viding high rob ustness to dynamic model uncertainties and e xternal disturbances. The proposed method also e xhibits ef fecti v eness in eliminating chattering, resulting in smooth, continuous control torque with lo wer po wer consumption across v arious scenarios. This is a considerable feature for pract ical applications where lifespan of actuator , mechani- cal stress, and ener gy ef cienc y are of utmost important. Simulation results sho w that the proposed 3-ITSMC IAES Int J Rob & Autom, V ol. 15, No. 2, June 2026: 341-352 Evaluation Warning : The document was created with Spire.PDF for Python.