# Performance Evaluation of Different Control Methods for an Underactuated Quadrotor Unmanned Aerial Vehicle (QUAV) with Position Estimator and Disturbance Observer

Authors: Ghulam E. Mustafa Abro; Zain Anwar Ali; Saiful A. Zulkifli; Vijanth Sagayan Asirvadam
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The main aim of this manuscript is to design and demonstrate the performance of different control algorithms with position estimator and disturbance observer to track the helical trajectory by an underactuated quadrotor craft under the influence of unmodelled dynamic factors and external disturbances. The manuscript consists of the derivations related to the kinematics and dynamics of quadrotor dully derived using the Newton Euler approach. It is one of the strenuous tasks to stabilize and control the quadrotor for helical trajectory tracking since it is an underactuated mechatronic system. In addition to this, with inclusion of unmodelled dynamic factors, it faces some of the serious transient and steady-state issues including Zeno noise. In this research manuscript, dual-loop single-dimension fuzzy sliding mode control (DLSDF-SMC) is proposed to improve the helical trajectory tracking performance, and to tackle the unmodelled dynamic factors, a state feedback controller is proposed consisting of a position estimator and disturbance observer design. The entire system is distributed into two subsystems such that within the angular subsystem, the attitude control is proposed using DLSDF-SMC, and for the translational subsystem, the paper proposes the position control design based on the hyperbolic function to avoid the gimbal lock issue. The overall stability of the proposed closed-loop control scheme is also proved. The simulation work for the proposed algorithm is performed using MATLAB and Simulink software and compared with the conventional sliding mode control (SMC) and fuzzy-based SMC control designs. This work demonstrates that the DLSDF-SMC control technique with position estimator and disturbance observer design in feedback not only improves the aggressive maneuvers while tracking the helical trajectory but also tackles the transient and steady-state issues.


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## Body

## 1. Introduction

The quadrotor unmanned aerial vehicle (QUAV) has four control inputs and six degrees of freedom (DOF). Since the number of control inputs is less than the DOF, that is,rankFq<dimq, where Fq is the input matrix and q is the state variables. This shows that the external forces are unable to exert the change in acceleration in an arbitrary direction in q; therefore, it is known as an underactuated system. Such mechatronic systems are very difficult to stabilize because of unmodelled dynamic factors and external disturbances [1–4]. Controlling and stabilizing, QUAV is one of the strenuous tasks, but if controlled properly with an intelligent control design, then it has several maneuvering capabilities to perform, that is, hovering in the air, path following, and vertical take-off and landing [5–9]. These quadrotor UAVs are used commonly in various domains [10]. Researchers have proposed a number of control strategies such that classical proportional integral derivative (PID) control for regulating [11] and evaluating the transient and steady-state factors [12]. Later, one may also see some of the robust and adaptive control designs to improvise these issues [13]. For the Cartesian position, researchers proposed several robust control designs such as backstepping control design [14]. All these control techniques were proposed to ensure better tracking performance, but, still there are some of the limitations observed, and this is all because of the impact of unmodelled dynamic factors [15]. Therefore, researchers proposed some of the adaptive control designs such that neural network-based control design that entertains the sudden changes that occur during the flight of quadrotor through online framework and have certain capabilities of tackling non-linear uncertainties [16].In literature, one may find the utilization of adaptive schemes such that fuzzy logic control (FLC) for similar issues [17], but due to sub-steps such as fuzzification, inference, and defuzzification, the quadrotor is unable to stimulate aggressive maneuvers. The trend of amalgamating more than one control design is also noticed. Researchers have hybridized the FLC with neural network for robust tracking performance but encounter deviation due to unmodelled dynamic factors and external disturbances [18]. One may notice several adaptive schemes as proposed in [19] where researchers tried to tackle the impact of external disturbances and unnecessary time delays in accelerations [20] using a robust attitude controller along with a state feedback scheme. Some of these algorithms are also proposed for hypersonic vehicles and unmanned aerial vehicles to compensate for the effect of unmodelled dynamic factors and time-varying uncertainties [21,22]. So far, all algorithms are proposed in an ideal or controlled environment where they have either upper or lower bounds of uncertainties. As per the best knowledge of the authors, the practical scenario is quite different from this controlled scenario, thus; very few researchers have focused on factors such as chattering noise that is a high number of oscillations experienced by the four rotors of an underactuated quadrotor during an aggressive maneuver. Researchers have focused either on transient and steady-state issues or the chattering phenomenon during an aggressive maneuver under unmodelled dynamic factors at one time [4]. The term robust control designs itself indicates that they are proposed for bounded values, that is, sliding mode control (SMC), which is proposed mostly for non-linear mechatronic systems. This is because of the insensitiveness to such external disturbances and factors on the manifold [23–29].Discussing the issues such that the strong dynamic coupling and singularity issue (gimbal lock), there are several research contributions that have proposed disturbance observers and estimators [30], state estimation using fuzzy scheme [31], approximation using neural network [32,33], and extended state observer [34]. Summarizing these schemes, one may find difficulty in tuning the parameters of disturbance observer, the low convergence rate for the neural network. This is one of the major reasons that people opted for extended state observer (ESO) because of its convergence rate over finite time index. Thus, the paper proposes an easy way to design the disturbance observer (DO) along with the position estimator to produce efficient results with less complexity for tuning the parameters for DO. If DO is studied properly, one may find it very easy to design as it requires less information about the model and has an ability to tackle the dynamic coupling with uncertainties over a large variation scale [35,36]. In terms of the research contribution, the paper provides a detailed study over the tracking issues with an inclusion of unmodelled dynamic factors. These factors are changing instantly; therefore, a novel control approach is amalgamated with a feedback control design that is divided into two sub-blocks, that is, position estimator and disturbance observer to estimate these bounded but instant changes. In addition to this, the reader may see the comparative analysis of the proposed technique with the conventional techniques. Mathematical proof is also provided to prove the stability analysis of the proposed control algorithm along with the proof that ensures the easy tuning of gains of observer design to achieve the faster convergence.Moreover, the manuscript is divided into six sections. One may find the basic introduction in Section1, explaining the background about previous control and estimation techniques. Section 2 describes the state-of-the-art approaches specifically and discusses the main limitations on which the solution in this paper is proposed. The mathematical derivation is well explained in Section 3, whereas Section 4 is all about control designs. This begins with the details of simple SMC, FSMC, and design related to dual-loop single-dimension fuzzy sliding mode control. Last but not the least, the discussion for position estimator and disturbance observer design is also included within this section. One can also find the stability proof for the proposed control algorithm within this section. In Section 5, the comparative results in terms of simulations in between three control algorithms, namely, SMC, FSMC, and DLSDF-SMC based on position estimator and disturbance observer designs are discussed. Finally, Section 6 presents a comprehensive conclusion for the entire research contribution.


## 2. Research Background

Studying some of the basic control designs such as proportional integral derivative (PID), sliding mode control (SMC), and linear quadratic regulator (LQR), it is noticed that SMC resolves some of the performance issues as compared to conventional LQR and PID control designs [36]. The only limitation with the SMC design is the high number of oscillations that appear on four rotors known as the chattering or Zeno effect [36]. To reduce this effect, one may hybridize the conventional SMC design with single-dimension fuzzy logic control (FLC) [37]. With such hybridization, it does not only reduce these oscillations but also enables an underactuated quadrotor unmanned aerial vehicle (QUAV) for aggressive maneuvering. By studying this approach thoroughly, one may identify a slow convergence rate. This is because of the instant variations in unmodelled dynamic factors. Therefore, this paper proposes the amalgamation of position estimator and disturbance observer to not only observe these changes but also estimate them to derive a faster convergence rate [18,38]. In this research paper, the authors derive a Newton–Euler dynamic model with hyperbolic function to address the singularity issue. Moreover, it suggests a dual-loop configuration of single-dimension-based fuzzy sliding mode control (DLSDF-SMC). This control design will be controlling and stabilizing the attitude and altitude of the quadrotor. However, the DO and PE will estimate the variations and provide significant feedback to control design. Once the feedback is received, the tuning parameters of single-dimension fuzzy sliding mode control will be tuned accordingly. This results in getting a faster convergence rate and stable helical trajectory tracking. In our work, the position estimator and disturbance observer observe and estimate the variations to tune the parameters of the DLSDF-SMC control design. Similar techniques have been used for air robots for path following with pigeon inspired optimization algorithm (PIO) [39–41] instead of simple position estimator and disturbance observer. The main purpose of introducing DO and PE here is to optimize the weighting matrix Q in SDF-SMC that will ease the design.This paper proposes dual-loop single-dimension fuzzy sliding model control design for helical trajectory tracking using position estimator and disturbance observer design because sliding mode control is insensitive to unmodelled dynamic factors, but it generates high oscillations [42]. To reduce this oscillatory effect, one may find the fusion of SMC with fuzzy logic control (FLC) [43]. This fusion is done to reduce the Zeno/chattering effect, but it consumes high power as well as processing time, which is one of the limitations of this technique. The reason for consuming high power and processing time is because of the intermediate steps of FLC, that are, fuzzification, inferences of rules, and defuzzification to compute the centre of gravity (CoG) [38]. There are some intelligent and hybrid versions of control that are proposed in [48], but they are inappropriate for underactuated quadrotor craft because of real-time tuning ability, and in most cases, they derive an impulsive behaviour [44]. A similar type of scheme is fuzzy H∞ output feedback control that produces chattering effect and time delay due to the impulsive behaviour [45–49]. Looking at the current literature, one may see the actuator disturbance rejection method [74]. This is presented to improve the overall attitude tracking performance, but in the end, one may notice the delay in accelerations. One may also see the issue of uncertainty is addressed by proposing a non-linear dynamic inversion estimator design [75,76]. At the time of introducing disturbance, the quadrotor may experience some chattering effect and may deviate from helical trajectory as well. Regarding the observer design, one must consider the dynamics of the brushless DC motor into account because this will increase the observer bandwidth and will ensure more stability [77]. Studying all about the control and observer designs, this paper presents the current state of the art approaches along with their main constraints in the next sub-section.


### 2.1. State of the Art Approaches

In this sub-section, the paper discusses the current state-of-the-art approaches that have been successful so far to address the tracking issues in the presence of unmodelled dynamic factors [18]. In the proposed work [18], researchers utilized the perks of disturbance observer (DO) to estimate the changes in unmodelled dynamic factors such as wind disturbance, payload variation, and the loss of rotor’s effectiveness. In this strategy, one may see two sub-control systems based on proportional integral (PI) that are responsible for compensating the positional errors whereas backstepping control design is the second one. This second sub-control design tackles the velocity error rates. The study states that there are some time delays in theoretical and predicted accelerations while an increment in thrust from 2,450 RPM to 3,320 revolution per minute (RPM). Moreover, there is a steady-state error of 2% in the presence of a payload mass of 200 grams. To tackle these issues, research proposed the inclusion of an integral term denoted as Cii=0.5 that resolves the issues but again produces some of the serious overshoots when the payload is released by quadrotor at any specific location. Discussing the case of rotor’s effectiveness loss, there is the same behaviour observed as it was in the case of payload variation. One may find the different versions of SMC that are either hybridized or used in different configurations such that dual-loop integral SMC (DLI-SMC) [48] with LESO. After critical analysis of theoretical results, one may see that the kinematics and dynamics of an underactuated quadrotor proposed in the paper have not considered the uncertainties. LESO used with this technique estimates the smooth changes so far. Thus, this paper suggests the use of DO and position estimator along with the hyperbolic function to tackle not only the smooth as well as non-smooth variations but also the issues such as chattering effect as well as steady-state error. In [48], one cannot understand the significant utilization of integral terms proposed with SMC in this dual-loop configuration. The second state-of-the-art approach is named as predictive optimal control design with the hybridization of disturbance observer (DO) [49]. The proposed strategy really works for smooth and non-smooth variations during its flight with a payload mass of 200 grams. In addition to this, one can see POC performs robust within the interval of ±25% of perturbations only. Thus, the performance of the proposed technique in [49] needs some more experimental work to be performed, whereas the disturbance observer (DO) whose parameters are normally very difficult to tune seems easy to adjust and tune. Thus, this paper adapts the same knowledge related to disturbance observers only from this research work.Third-order sliding mode control (TOSMC) is also one of the current approaches to address the issues of the chattering effect [50]. Here quaternion model is proposed to avoid the issues like gimbal lock. Still, one may observe the chattering effect that never reduced even after using high- and low-order filters. These are the current trending approaches, and their limitations are summarized in Table 1.Table 1
State-of-the-art approaches with limitations.
Sr.TechniquesLimitations1Proportional integral + backstepping control design with non-linear disturbance observer [18]There are several steady states and transient issues.One may notice the unnecessary delays in theoretical and predicted accelerations.2Dual-loop integral sliding mode control design with LESO [48]There is an estimation delay.One must compromise either over steady-state issues or transient issues.Chattering effect is observed on all four brushless DC motors.3Predictive optimal control using disturbance estimator [49]The results are robust but with the perturbations of±25% in the system.Only the payload case is discussed.4Third-order sliding mode control design with quaternion mathematics [50]It resolves the issue of gimbal lock.It does not discuss the impact of unmodelled dynamic factors.The stability proof is not provided.There is a delay in accelerations, chattering effect, and transient-steady-state issues.There are few articles that have considered the factors such as input saturation that most of the cases lead to the complexity of the control algorithm [51]. To address these constraints, researchers used the global sliding surface based on trajectory tracking error. For a fast convergence rate, researchers have proposed the virtual control input with an integral version, but this can bring some of the serious overshoots in case of payload variation. One may see the supertwisting version of sliding mode control (ST-SMC) that is proposed to achieve the finite time position and attitude stability for an underactuated quadrotor craft. The simulation results related to input delayand other factors such as wind disturbance and model uncertainty are addressed. The only limitation with this algorithm can be seen in the figures where the researchers mentioned the time histories of supertwisting switching surfaces. This can be one of the reasons for high oscillations known as Zeno phenomena over the four brushless DC (BLDC) motors later [52]. Researchers have proposed proportional integral derivative control with sliding mode control (SMC) with an adaptive flavour to ensure stability but if one may see the time responses of the control inputs even under known bounded disturbance, there is a chattering effect in the inputs [53].Moreover, the measured values obtained from several contributions are stated in Table2 to convey the main performance factors that must be in mind while performing theoretical or simulation solutions.Table 2
Measured versus required performance factors.
Sr.Performance factorsMeasured performanceRequired performance1Steady-state error2%–5%1%–2%2Settling time15.0<15.0 seconds3Overshoots10%–15%<10%4State estimation delay19.5 seconds<19.5 seconds5Zeno/chattering effect2.35–3.75%<2.375%6Tuning of disturbance observerDifficultEasy or moderate7Delay in accelerations17.5<17.5 secondsTable2 shows the seven significant performance factors that have been addressed by several researchers, but still, there is a need to improvise them. This is the only reason that the design of a control system for an underactuated quadrotor unmanned aerial vehicle (QUAV) is still one of the challenging problems in the presence of unmodelled dynamic factors. There is a need for an intelligent control design along with a predictive observer technique that can optimize the entire dynamic system to have fast, accurate, and stabilized trajectory tracking with better performance factors as indicated in Table 2.


### 2.2. Research Contribution

The single-dimension-based fuzzy sliding mode control in the simple configuration without any estimator and observer design has been proposed for the stability of various applications such that NPS autonomous vehicle [54], double pendulum-type overhead cranes [55], DSRV that stands for deep submergence rescue vehicle, and many other systems [56]. After studying the several control designs [57,58] one may conclude that sliding mode control provides better tracking performance when amalgamated with fuzzy logic control (FLC) [59]. By adding fuzzy mathematics, there is a slow convergence rate observed. This is because of intermediate steps of fuzzification, inference, and defuzzification. Thus, the manuscript proposes a novel dual-loop single-dimension-based fuzzy sliding mode control with a position estimator and disturbance observer to tackle the unmodelled dynamic factors and lumped uncertainties during the tracking of helical trajectory. In addition to this, the proposed algorithm enables our underactuated quadrotor craft to perform aggressive maneuvers. The main research contributions are mentioned as follows:(i)
The tracking issues are addressed in a better way than [60–62] along with an inclusion of unmodelled dynamic factors. In addition to this, the software simulations with different additional trajectory tracking prove the robustness of the proposed algorithm.(ii)
In the paper, the instant changes are observed and estimated using a feedback control that comprises a position estimator and disturbance observer, unlike the other proposed observers that estimate the bounded variations only [63–65].(iii)
In contrast with other control designs discussed in [13,66], the proposed dual-loop single-dimension-based fuzzy sliding mode control is an asymptotic stable, and one may see the stability proof of the proposed closed-loop control design provided within the paper.(iv)
Disturbance observer and position estimator proposed in this paper enable tuning of parameters for faster convergences so that the tracking errors may converge to zero as soon as possible.With the above research postulates, the proposed control strategy is robust, and one of its kinds, it surely improves the performance of all parameters as discussed in Table2.


## 2.1. State of the Art Approaches

In this sub-section, the paper discusses the current state-of-the-art approaches that have been successful so far to address the tracking issues in the presence of unmodelled dynamic factors [18]. In the proposed work [18], researchers utilized the perks of disturbance observer (DO) to estimate the changes in unmodelled dynamic factors such as wind disturbance, payload variation, and the loss of rotor’s effectiveness. In this strategy, one may see two sub-control systems based on proportional integral (PI) that are responsible for compensating the positional errors whereas backstepping control design is the second one. This second sub-control design tackles the velocity error rates. The study states that there are some time delays in theoretical and predicted accelerations while an increment in thrust from 2,450 RPM to 3,320 revolution per minute (RPM). Moreover, there is a steady-state error of 2% in the presence of a payload mass of 200 grams. To tackle these issues, research proposed the inclusion of an integral term denoted as Cii=0.5 that resolves the issues but again produces some of the serious overshoots when the payload is released by quadrotor at any specific location. Discussing the case of rotor’s effectiveness loss, there is the same behaviour observed as it was in the case of payload variation. One may find the different versions of SMC that are either hybridized or used in different configurations such that dual-loop integral SMC (DLI-SMC) [48] with LESO. After critical analysis of theoretical results, one may see that the kinematics and dynamics of an underactuated quadrotor proposed in the paper have not considered the uncertainties. LESO used with this technique estimates the smooth changes so far. Thus, this paper suggests the use of DO and position estimator along with the hyperbolic function to tackle not only the smooth as well as non-smooth variations but also the issues such as chattering effect as well as steady-state error. In [48], one cannot understand the significant utilization of integral terms proposed with SMC in this dual-loop configuration. The second state-of-the-art approach is named as predictive optimal control design with the hybridization of disturbance observer (DO) [49]. The proposed strategy really works for smooth and non-smooth variations during its flight with a payload mass of 200 grams. In addition to this, one can see POC performs robust within the interval of ±25% of perturbations only. Thus, the performance of the proposed technique in [49] needs some more experimental work to be performed, whereas the disturbance observer (DO) whose parameters are normally very difficult to tune seems easy to adjust and tune. Thus, this paper adapts the same knowledge related to disturbance observers only from this research work.Third-order sliding mode control (TOSMC) is also one of the current approaches to address the issues of the chattering effect [50]. Here quaternion model is proposed to avoid the issues like gimbal lock. Still, one may observe the chattering effect that never reduced even after using high- and low-order filters. These are the current trending approaches, and their limitations are summarized in Table 1.Table 1
State-of-the-art approaches with limitations.
Sr.TechniquesLimitations1Proportional integral + backstepping control design with non-linear disturbance observer [18]There are several steady states and transient issues.One may notice the unnecessary delays in theoretical and predicted accelerations.2Dual-loop integral sliding mode control design with LESO [48]There is an estimation delay.One must compromise either over steady-state issues or transient issues.Chattering effect is observed on all four brushless DC motors.3Predictive optimal control using disturbance estimator [49]The results are robust but with the perturbations of±25% in the system.Only the payload case is discussed.4Third-order sliding mode control design with quaternion mathematics [50]It resolves the issue of gimbal lock.It does not discuss the impact of unmodelled dynamic factors.The stability proof is not provided.There is a delay in accelerations, chattering effect, and transient-steady-state issues.There are few articles that have considered the factors such as input saturation that most of the cases lead to the complexity of the control algorithm [51]. To address these constraints, researchers used the global sliding surface based on trajectory tracking error. For a fast convergence rate, researchers have proposed the virtual control input with an integral version, but this can bring some of the serious overshoots in case of payload variation. One may see the supertwisting version of sliding mode control (ST-SMC) that is proposed to achieve the finite time position and attitude stability for an underactuated quadrotor craft. The simulation results related to input delayand other factors such as wind disturbance and model uncertainty are addressed. The only limitation with this algorithm can be seen in the figures where the researchers mentioned the time histories of supertwisting switching surfaces. This can be one of the reasons for high oscillations known as Zeno phenomena over the four brushless DC (BLDC) motors later [52]. Researchers have proposed proportional integral derivative control with sliding mode control (SMC) with an adaptive flavour to ensure stability but if one may see the time responses of the control inputs even under known bounded disturbance, there is a chattering effect in the inputs [53].Moreover, the measured values obtained from several contributions are stated in Table2 to convey the main performance factors that must be in mind while performing theoretical or simulation solutions.Table 2
Measured versus required performance factors.
Sr.Performance factorsMeasured performanceRequired performance1Steady-state error2%–5%1%–2%2Settling time15.0<15.0 seconds3Overshoots10%–15%<10%4State estimation delay19.5 seconds<19.5 seconds5Zeno/chattering effect2.35–3.75%<2.375%6Tuning of disturbance observerDifficultEasy or moderate7Delay in accelerations17.5<17.5 secondsTable2 shows the seven significant performance factors that have been addressed by several researchers, but still, there is a need to improvise them. This is the only reason that the design of a control system for an underactuated quadrotor unmanned aerial vehicle (QUAV) is still one of the challenging problems in the presence of unmodelled dynamic factors. There is a need for an intelligent control design along with a predictive observer technique that can optimize the entire dynamic system to have fast, accurate, and stabilized trajectory tracking with better performance factors as indicated in Table 2.


## 2.2. Research Contribution

The single-dimension-based fuzzy sliding mode control in the simple configuration without any estimator and observer design has been proposed for the stability of various applications such that NPS autonomous vehicle [54], double pendulum-type overhead cranes [55], DSRV that stands for deep submergence rescue vehicle, and many other systems [56]. After studying the several control designs [57,58] one may conclude that sliding mode control provides better tracking performance when amalgamated with fuzzy logic control (FLC) [59]. By adding fuzzy mathematics, there is a slow convergence rate observed. This is because of intermediate steps of fuzzification, inference, and defuzzification. Thus, the manuscript proposes a novel dual-loop single-dimension-based fuzzy sliding mode control with a position estimator and disturbance observer to tackle the unmodelled dynamic factors and lumped uncertainties during the tracking of helical trajectory. In addition to this, the proposed algorithm enables our underactuated quadrotor craft to perform aggressive maneuvers. The main research contributions are mentioned as follows:(i)
The tracking issues are addressed in a better way than [60–62] along with an inclusion of unmodelled dynamic factors. In addition to this, the software simulations with different additional trajectory tracking prove the robustness of the proposed algorithm.(ii)
In the paper, the instant changes are observed and estimated using a feedback control that comprises a position estimator and disturbance observer, unlike the other proposed observers that estimate the bounded variations only [63–65].(iii)
In contrast with other control designs discussed in [13,66], the proposed dual-loop single-dimension-based fuzzy sliding mode control is an asymptotic stable, and one may see the stability proof of the proposed closed-loop control design provided within the paper.(iv)
Disturbance observer and position estimator proposed in this paper enable tuning of parameters for faster convergences so that the tracking errors may converge to zero as soon as possible.With the above research postulates, the proposed control strategy is robust, and one of its kinds, it surely improves the performance of all parameters as discussed in Table2.


## 3. Mathematical Model

The main part to design single-dimension-based fuzzy sliding mode control begins with the proposition of a mathematical model for underactuated quadrotor craft. Thus, the coordinate transformation is discussed in this section. One may see the quadrotor in real time as a mechatronic system comprises four fixed rotors mounted at four ends of a cross frame as shown in Figure1(a). In most cases, these rotors are propelled by using brushless DC (BLDC) motors. This is because their revolution per minute (RPM) can easily be changed using an electronic speed controller (ESC). As we know, we have two types of coordinate systems such that body-fixed coordinate systems illustrated as oxyz and ground-fixed coordinate systems OXYZ. This is illustrated in Figure 1(b).Figure 1
(a) Underactuated quadrotor physical model and (b) body-fixed coordinate.
(a)(b)In Figure1(b), one may see ϕ,θ,ψ are roll, pitch, and yaw, respectively. These three angles are mainly controlled via four control input forces such as F1,F2,F3,F4. These forces can be changed by varying the RPMs of four BLDC motors. The weight of underactuated quadrotor craft is given as mg, where m is the mass of quadrotor and g is the acceleration due to gravity. One may opt the lifting forces and torques from the body-fixed coordinate system as illustrated oxyz. However, one may have to understand the exertion of these forces in the ground-fixed frame OXYZ. Thus, the transformation can be done as shown in Figure 2.Figure 2
Roll, pitch, and yaw angles: (a) roll, (b) pitch, and (c) yaw.
(a)(b)(c)One can obtain the transform matrix of each axis fromoxyz to OXYZ as follows:(1)Rx,ϕ=1000cϕ−sϕ0sϕcϕ,(2)Ry,θ=cθ0sθ010−sθ0cθ,(3)Rz,ψ=cψ−sψ0sψcψ0001,where the symbol c and s are cosine and sine functions. One may use the following equation to have interconversion between body-fixed coordinate oxyz and ground-fixed coordinate OXYZ:(4)Rϕ,θ,ψ=Rz,ψRy,θRx,ϕ.To simplify the model of the quadrotor, there are some assumptions to be followed as mentioned below:(i)
The physical structure of the quadrotor prototype is considered as one of the rigid and symmetrical structures, and the body-fixed coordinate origin must coincide with the centre of mass.(ii)
There is a proportional relationship between torque and DC voltages. Moreover, one may ignore the air resistance and gyroscopic effect, while the quadrotor is flying at low speed.(iii)
Consider that the attitude change is very small (i.e., less than 5°).The lifting forces are derived in body-fixed coordinate systems and are labelled asF1,F2,F3andF4. One may derive the value of net force as follows:(5)FB=FxFyFz=00F1+F2+F3+F4.The interconversion can be done using the following equation:(6)FG=FxFyFz=Rϕ,θ,ψ.FB=∑i=14Ficϕsθcψ+sϕsψcϕsθsψ−sϕcψcθcϕ.As per the equation of motion, one may try to deduce the following equations:(7)x¨=FXm,(8)y¨=Fym,(9)z¨=Fzm−g.Similarly, one may deduce the attitude change from the Newton–Euler formula as follows:(10)Jω˙+ω×Jω=τ,where the term J is defined as diagJθ,Jϕ,Jψ and Jθ,Jϕ,Jψ and are equivalent, respectively, with the moment of inertia about the axis for pitch, roll, and yaw. In addition to this, the term ω=θ˙,ϕ˙,ψ˙T, whereas the notation for torque is denoted as τ=τ1,τ2,τ3T. This torque term consists of two types of torques such that Mi that is the turning effect generated by the lifting forces and τi1 is the torque generated by self-rotation of the rotors of BLDC motors. The relation can be seen in the following equation:(11)τ=τ1τ2τ3=τ11τ21τ31+M1M2M3,whereasτ11=JγΩϕ˙,τ21=JγΩθ˙,τ31=JγΩ˙.The torque generated by the lifting forces is as follows:(12)M1=lF3−F1,M2=lF4−F2,M3=F2+F4−F1−F3.Moreover, the value of omega is as follows:(13)Ω=Ω1−Ω2+Ω3−Ω4.Studying equations (10) and (11), one may derive the following equation:(14)JγΩϕ¨=θ˙ψ˙Jϕ−Jψ+Jγθ˙Ω+M2,JγΩθ¨=ϕ˙ψ˙Jψ−Jθ+Jγϕ˙Ω+M1,JγΩψ¨=ϕ˙θ˙Jθ−Jϕ+JγΩ˙+Ω.The input variables of the underactuated quadrotor unmanned aerial vehicle (QUAV) is defined as follows:(15)U1=F1+F2+F3+F4=KtΩ12+Ω22+Ω32+Ω42,U2=F4−F2=KtΩ42−Ω22,U3=F3−F1=KtΩ32−Ω12,U4=−Ω=Kd−Ω12+Ω22−Ω32+Ω42,whereU1 is the main control variable for vertical speed and the control variable of roll angle is denoted by U2. Pitch angle is controlled using the control variable of U3, whereas the control variable of U4 is denoted for yaw angle. Furthermore, the variable Ωi is the rotating speed, where i=1,2,3and4 denoting the number of rotors. Lastly, the coefficients of lifting force and torques are given as Kt and Kd, respectively.Studying all equations from (6) to (15), the mathematical model for our proposed underactuated quadrotor is presented as follows:(16)x¨=cosϕsinθcosψ+sinϕsinψU1m,y¨=cosϕsinθsinψ−sinϕcosψU1m,z¨=cosθcosϕU1m−g,ϕ¨=θ˙ψ˙Jϕ−Jψ+Jγθ˙Ω+lU2Jθ,θ¨=ϕ˙ψ˙Jψ−Jθ+Jγϕ˙Ω+lU3Jϕ,ψ¨=ϕ˙θ˙Jθ−Jϕ+JγΩ˙+U4Jψ,where the notation Jγ is the equivalent moment of inertia about the motors, that is, Jθ,Jϕ,Jθ. Moreover, the angles such that roll, pitch, and yaw are denoted as ϕ,θ,andψ, respectively. The notation l is the distance between pivot to each BLDC motors (rotors) such that x¨, y¨andz¨ are accelerations about each axis. The term Ui is the equivalent input variable, the sum of the four rotors’ rotating speed is given by Ω, where the weight of proposed underactuated quadrotor is denoted by m and acceleration due to gravity is mentioned as g. These all parameters along with their definition are summarized in Table 3.Table 3
Summary of all parameters for the dynamics of underactuated quadrotor craft.
DefinitionParameterValueDistance between pivot and each BLDC motorslm0.250Equivalent inertia about the pitch axisJθ (kg.m2)0.0652Equivalent inertia about the roll axisJϕ (kg.m2)0.0652Equivalent inertia about the yaw axisJψ (kg.m2)0.120Coefficient of torquesKdN.m.s20.00748Coefficient of lifting forcesKtN.s20.0221Weight of quadrotormkg2.75Acceleration due to gravitymsec−29.8Opting the variables asX=x1,x2,x3,x4,x5,x6T=ϕ,ϕ˙,θ,θ˙,ψ,ψ˙T, whereas the input for the entire dynamic system is U=U1,U2,U3,U4T. Thus, equation (16) can be transformed as follows:(17)x˙1=x2,x˙2=x4x6Jϕ−JψJθ+Jγθ˙ΩJθ+lU2Jθ,x˙3=x4,x˙4=x2x6Jψ−JθJϕ+Jγϕ˙ΩJϕ+lU3Jϕ,x˙5=x6,x˙6=x2x6Jθ−JϕJψ+JγΩ˙Jϕ+U4Jϕ.For proposing the above-mentioned dynamic model, there are some assumptions to be considered as mentioned below:(i)
During flight mode, the accelerations in the body-fixed coordinate are approximately equal to the acceleration in the ground-fixed coordinate(ii)
However, the issue related to cross-coupling effects of angular speeds and gyroscopic effects are negligible. Thus, the state-space model can be derived using the standard definition as follows:(18)X=AX+BU,(19)Y=CX,(20)A=010000000000000100000000000001000000,B=000lJθ000000lJθ000000lJθ,C=100000010000001000000100000010000001.The values computed forl/Jθ,l/Jϕ,l/Jψ are 3.834, 3.834, and 8.334, respectively.


## 4. Control Design

This section discusses the design steps for dual-loop single-dimension-based fuzzy sliding mode control design along with an amalgamation of PE and DO. This proposed algorithm will make sure the better performance of an underactuated quadrotor craft while tracking the helical trajectory in the presence of external disturbances and unmodelled dynamic factors. Equation (16) shows all six control variables that are dully controlled using four control inputs such as U1,U2,U3,U4. To design the algorithm, this section has been divided into further sub-sections for the readers to understand better.


### 4.1. Dual-Loop Single-Dimension-Based Fuzzy Sliding Mode Control

This sub-section discusses the theory about sliding mode control design and its hybrid version single input based fuzzy sliding mode control design. The basic block diagram of sliding mode control SMC is illustrated in Figure3. This control design comprises a steering infinite quadrotor’s trajectory as a sliding manifold. This is designed to lead the underactuated quadrotor UAV coming all the way from any position to target to follow a specific trajectory and converge the states asymptotically to zero.Figure 3
Block diagram for sliding mode control design.The sliding surface for conventional sliding mode control design is as follows:(21)sf=e˙−ce.where the term sf is known as sliding surface, where the error rate is denoted by e and the rate of change of error rate is indicated as e˙. The gain is shown as c that can be tuned as per the requirement of the controller. Moreover, the derivative of the sliding surface is as follows:(22)sf˙=−ρsf−ksignsf,where the positive diagonal definite matrices are denoted as ρ and k. There are several research manuscripts where scholars provided the hybrid versions of SMC by amalgamating it with fuzzy logic control (FLC) design. This type of hybridization had been proposed to overcome the constraints, that is, Zeno or chattering effect that may damage the propellers and brushless DC (BLDC) motors [44,68]. This hybridization not only reduces the chattering but also enables the underactuated quadrotor to perform extraordinary aggressive maneuvers. The sliding surface depends on variables such as e and e˙ that can be computed using equations (21) and (22). The main purpose of fuzzy logic is to tune these two variables.(23)sf=ee˙k1.To design FLC, one should design the rule base system. In this manuscript, the rules are proposed with the help of vectoral distances between state trajectoryLsn and manifold Lo. Mathematically, it is possible to use the error and rate of change error as illustrated in the following equations:(24)Lsn=eQ˙+keQ˙1+k2,(25)Lo=N2−Lsn2.This derivation is well illustrated in Figure4, whereas the rules for fuzzy-based sliding mode control (FSMC) are defined in Table 4.Figure 4
Vectoral illustration forLsn and Lo.Table 4
Rules based on vectoral distance.
LoLsnNBNSNSZPSNSPBPBZPSPMPBPBPBPBNSNSZPSPMPBPBPBPSNMNSZPSPMPBPBZNBNMNSZPSPMPBNSNBNBNMNSZPSPMNSNBNBNBNMNSZPSNBNBNBNBNBNBNSZPB: positive big, PS: positive small, NB: negative big, NS: negative small, PM: positive medium, Z: zero, and NM: negative medium.The membership function type opted is the triangular type and is illustrated in Figure5. The output of FLC shall be singleton type based on the centre of gravity (CoG) as shown in Figure 6. Moreover, the scale defined for all input and output membership functions varies between ±0.6.Figure 5
Triangular-type input membership functions.Figure 6
Singleton-type output membership function based on CoG.The diagram shown in Figure7 is as same as the block diagram for a conventional SMC design illustrated in Figure 3. There is an addition of fuzzy logic control (FLC) and the sub-block shown in the red-dashed box. In this sub-block, the vectoral distance is computed and delivered later to FSMC Block to actuate the drone with the desired behaviour.Figure 7
Block diagram of fuzzy-based SMC design.This manuscript focuses on main issues such that tackling the unmodelled dynamic factors using position estimator and disturbance observer design. The reader may observe that the estimated values that are feedback to this block where it will tune the parameters for dual-loop single-dimension-based fuzzy sliding mode control (DLSDF-SMC). By doing this, DLSDF-SMC not only reduces the computation time but also has a faster convergence rate while tracking the trajectory from any position in the 3D plane. A single-dimension-based version of FSMC is proposed because of several perks that a simple FSMC cannot provide such as fast convergence rate with aggressive maneuvers. This is because of intermediate sub-processes such that fuzzification, inference of two-dimensional rules, and defuzzification for producing CoG-based output. Therefore, converting the 2D table for rules into a single dimension using the signed distance method [37,69] can be one of the methods to improve the convergence rate and obtain the aggressive maneuvering options. This can be possible and shown graphically in Figure 8.Figure 8
Graphical representation to deriveG variable.After studying the graphical illustration in Figure9, one may derive the mathematical relationship for the G variable as follows:(26)Lsn+αLo=0,(27)G=Lsn+αLo1+α2,where the variable G is a diagonal vector derived from Table 4 as illustrated in Figure 9.Figure 9
Deriving single-dimension input membership functions using the diagonal variableG.Table5 is reproduced using equations (26) and (27). This time, the input membership functions are triangular type but with different x− index values, whereas the output membership function will remain the same as discussed and shown in Figures 5 and 6. Having this entire information, one may now start designing the linear extended state observer.Table 5
Single-dimension rules usingG variable.
GGNBGNMGNSGZGPSGPMGPSG′NBNMNSZPSPMPB


### 4.2. Position Estimator and Disturbance Observer

To tackle the unmodelled dynamic factors, it is essential to propose a fine feedback structure comprised a position estimator and disturbance observer. This is possible by the deployment of various sensors to underactuated quadrotor craft. The most frequent sensors are inertial measurement unit (IMU), GPS, a computer vision system, optical flow sensor for the position alignment, and lastly the utilization of an ultrasonic or infrared sensor, that is, LiDAR v3 for altitude. While considering the data from these sensors, they all must be preprocessed to avoid the noise and extract only useful data [70]. This position estimator is designed to compute and measure the signals that cannot join several noisy signals to remove the noise. As the presence of such random noise can lead towards the deterioration of the final position with time for our proposed underactuated quadrotor unmanned aerial vehicle (QUAV). In the presence of unmodelled dynamic factors such as wind disturbance, our proposed disturbance observer ensures the performance of position tracking and the linear velocities by reducing the noise and effect of external disturbances. Discussing the position estimator, it computes the positional and velocity vectors for quadrotor UAV. These measure signals via IMU may contain an amplified noise while acquiring the velocity and position vectors. The frequent sensor for acquiring the positional vectors is the GPS sensor module which is commonly working at a 10 Hz frequency rate that means every new value will appear after every 100 ms. In Figure 10, it has been depicted that the main aim of the estimator is to acquire the data of position and velocity at the frequency rate of 100 Hz.Figure 10
Block diagram for proposed position estimator design.As per the above diagram, the measured variable is the actual positional and velocity vectors that are acquired from IMU. In addition to this, total thrust is also considered to observe and estimate the linear acceleration via a dynamic model of underactuated quadrotor UAV. The outcome is the linear estimated velocity, which is derived by amalgamating the velocity from estimated linear acceleration and velocity from the positional sensor as shown in Figure10. The entire calculation is depending on the number of states measured from the signal. In the case of transient issues, the positions are detected by estimated linear acceleration through double integration and rectified at every 100 ms. However, in case of steady-state performance, one has taken the signals directly.This was all about estimator design, whereas the disturbance observer is implemented to determine the exact amplitude of disturbances caused by unmodelled dynamic factors and external disturbances. Figure11 presents the detailed working of a disturbance observer that calculates the disturbance as a linear and angular acceleration via the dynamic model of an underactuated quadrotor unmanned aerial vehicle (QUAV). For the synchronization of estimated values with time during the flow of signals from the control block, a delay of motors is included. However, in the case of position, disturbance observer estimation is performed using the states from the accelerometer, and gravity acceleration is reduced using the gravity reduction sub-block as shown in Figure 11.Figure 11
Block diagram of proposed disturbance observer design.After studying all the equations, one may derive the observer error of each DO as shown below mathematically:(28)e^x˙=F1˙−F^˙1=Ax−LgxCxe^x+Exω1,(29)e^y˙=F2˙−F^˙2=Ay−LgyCye^y+Eyω2,(30)e^z˙=F3˙−F^˙3=Az−LgzCze^z+Ezω3,(31)e^ψ˙=F4˙−F^˙4=Aψ−LgψCψe^ψ+Eψω4.These equations can be represented in matrix form as follows:(32)e^˙=H^e^+Ed,where(33)e^=e^x,e^˙x,e^¨x,e^⋯x,e^⋯x.e^˙x,e^y,e^˙y,e^¨y,e^⋯y,e^⋯y.e^˙y,e^z,e^˙z,e^¨z,e^⋯z,e^⋯z.e^˙z,e^ψ,e^˙ψ,e^¨ψ,,(34)H^=H^1O5O5O3O5H^2O5O3O5O5H^3O3O5O5O4H^4,(35)Ed=0,0,0,0,ω1,0,0,0,0,ω2,0,0,0,0,ω3,0,0,ω4T,where H^ is the term defined as the roots of the matrix mentioned in equation (34), and it must be on the left half-plane because of the bandwidths, that is, Γxo,Γyo,Γzo, and Γψo are non-negative in values. These all-estimated error dynamics as defined in equations (28)–(31) are stable in nature provided by Lemma 1 [29,31].Lemma 1.
In equation (32), limt⟶∞e^t is considered as bounded if and only if one of the following two conditions are true such that first one. τi<a,i=1,2,3,4 for all time index t, whereas second one ωi<b,i=1,2,3,4 for all time t. If in case τi has a constant value, that is, ωi=0,i=1,2,3,4, then the estimated error dynamics shall be considered as asymptotically stable.


### 4.3. Stability of the Proposed Closed-Loop Control Design

The stability of the proposed closed-loop control system is described here within this sub-section. It starts all from state tracking error vector ofx− position:(36)erx=Fd1−X1,where the term Fd1=Fxd,F˙xd,F¨xd,F⋯xdT and X=F11,F12,F13,F14T. This leads to equation (37) as follows:(37)F˙d1=A1Fd1+B1F⋯xd.F˙xd,where A1=0000100001000010,B1=0001. In this way, the new feedback can be designed that is written in matrix form as follows:(38)X˙1=A1X1+B1vrx+τ1.Using the following equation, one may derive the feedback design as shown in equation (40):(39)e^r1=Fxd−F^11=Fxd−F11−F^11,(40)vrx=Kxerx+e^x+F⃛dx.F˙dx−τ^1.The variableKx in equation (40) is equal to Kx1,Kx2,Kx3,Kx4; thus, the state tracking error dynamics can be expressed more accurately as follows:(41)e˙r1=A1−B1Kxerx−B1Kxe^x−B1e^x.Hence, combining equations (28) and (41), one may obtain(42)e˙rxe^˙x=A1−B1Kx−B1KxB10Ax−LgxCxerxe^x+0Exω1.By using equation (42), one may find the closed-loop stability of the proposed underactuated quadrotor unmanned aerial vehicle (QUAV). This can be possible mathematically if one derives the eigenvalues of the terms, that is, A1−B1Kx and Ax−LgxCx, since the expressions such as A1,B1 and Ax,Cx are controllable and observable. Thus, the stability of such error dynamics for x− position given by equation (42) is ensured by opting for the suitable controller and observer poles. Similarly, the feedback for y,z, and ψ can also be designed.(43)e˙rye^˙y=A2−B2Ky−B2KyB1e^x0Ax−LgyCyerye^y+0Eyω2,(44)e˙rze^˙z=A3−B3Kz−B3KzB30Az−LgzCzerze^z+0Ezω3,(45)e˙rψe^˙ψ=A4−B4Kψ−B4KψB40Aψ−LgψCψerψe^ψ+0Eψω4.Moreover, in the above equations (41)–(45), A1=A2=A3, where B1=B2=B3, Ky=Ky1,Ky2,Ky3,Ky4,Kz=Kz1,Kz2,Kz3,Kz4, and lastly, Kψ=Kψ1,Kψ2. For this work, A4andB4 are mentioned as follows:(46)A4=0100,B4=01.The stability of the closed-loop systemy,z, and ψ can be obtained in the same way. After studying these all sub-sections, one may derive the intelligent control algorithm of dual-loop single-dimension fuzzy sliding mode control with position estimator and disturbance observer. The block diagram of the overall control algorithm and system is shown in Figure 12. In this figure, the entire working of dual-loop single-dimension fuzzy SMC is summarized. The important factor here is to opt for an appropriate control vector ut in a way that the given quadratic performance index mentioned in equation (47). It obtains the minimum value. It is later proved that it reaches the minimum value by means of the proposed control law equation as mentioned in (48).(47)ut=−KXt=R−1BTPXt,where P is the optimal matrix can be computed using the Riccati equation as follows:(48)ATP+PA−PBR−1BTP+Q=0.Figure 12
Block diagram of dual-loop single-dimension fuzzy SMC with PE and DO.In Figure12, the matrices Q and R are opted to have the form such as Q=diagq11,0,q33,0,q55,0 and R=diag1,1,1,1,1,1. The variables such as q11,q33 and q55 will be compensated using PE and DO propose with dual-loop single-dimension fuzzy slide mode control that will take care of the oscillatory response of the system. The implementation of the proposed algorithm follows below-mentioned steps in simple:(i)
Establishing the dynamics for underactuated quadrotor craft(ii)
Tuning the performance index and weightingQ and R matrices accordingly(iii)
Estimating the parameters for the proposed DLSDF-SMC control algorithm using PE and DO as shown in Figure12


## 4.1. Dual-Loop Single-Dimension-Based Fuzzy Sliding Mode Control

This sub-section discusses the theory about sliding mode control design and its hybrid version single input based fuzzy sliding mode control design. The basic block diagram of sliding mode control SMC is illustrated in Figure3. This control design comprises a steering infinite quadrotor’s trajectory as a sliding manifold. This is designed to lead the underactuated quadrotor UAV coming all the way from any position to target to follow a specific trajectory and converge the states asymptotically to zero.Figure 3
Block diagram for sliding mode control design.The sliding surface for conventional sliding mode control design is as follows:(21)sf=e˙−ce.where the term sf is known as sliding surface, where the error rate is denoted by e and the rate of change of error rate is indicated as e˙. The gain is shown as c that can be tuned as per the requirement of the controller. Moreover, the derivative of the sliding surface is as follows:(22)sf˙=−ρsf−ksignsf,where the positive diagonal definite matrices are denoted as ρ and k. There are several research manuscripts where scholars provided the hybrid versions of SMC by amalgamating it with fuzzy logic control (FLC) design. This type of hybridization had been proposed to overcome the constraints, that is, Zeno or chattering effect that may damage the propellers and brushless DC (BLDC) motors [44,68]. This hybridization not only reduces the chattering but also enables the underactuated quadrotor to perform extraordinary aggressive maneuvers. The sliding surface depends on variables such as e and e˙ that can be computed using equations (21) and (22). The main purpose of fuzzy logic is to tune these two variables.(23)sf=ee˙k1.To design FLC, one should design the rule base system. In this manuscript, the rules are proposed with the help of vectoral distances between state trajectoryLsn and manifold Lo. Mathematically, it is possible to use the error and rate of change error as illustrated in the following equations:(24)Lsn=eQ˙+keQ˙1+k2,(25)Lo=N2−Lsn2.This derivation is well illustrated in Figure4, whereas the rules for fuzzy-based sliding mode control (FSMC) are defined in Table 4.Figure 4
Vectoral illustration forLsn and Lo.Table 4
Rules based on vectoral distance.
LoLsnNBNSNSZPSNSPBPBZPSPMPBPBPBPBNSNSZPSPMPBPBPBPSNMNSZPSPMPBPBZNBNMNSZPSPMPBNSNBNBNMNSZPSPMNSNBNBNBNMNSZPSNBNBNBNBNBNBNSZPB: positive big, PS: positive small, NB: negative big, NS: negative small, PM: positive medium, Z: zero, and NM: negative medium.The membership function type opted is the triangular type and is illustrated in Figure5. The output of FLC shall be singleton type based on the centre of gravity (CoG) as shown in Figure 6. Moreover, the scale defined for all input and output membership functions varies between ±0.6.Figure 5
Triangular-type input membership functions.Figure 6
Singleton-type output membership function based on CoG.The diagram shown in Figure7 is as same as the block diagram for a conventional SMC design illustrated in Figure 3. There is an addition of fuzzy logic control (FLC) and the sub-block shown in the red-dashed box. In this sub-block, the vectoral distance is computed and delivered later to FSMC Block to actuate the drone with the desired behaviour.Figure 7
Block diagram of fuzzy-based SMC design.This manuscript focuses on main issues such that tackling the unmodelled dynamic factors using position estimator and disturbance observer design. The reader may observe that the estimated values that are feedback to this block where it will tune the parameters for dual-loop single-dimension-based fuzzy sliding mode control (DLSDF-SMC). By doing this, DLSDF-SMC not only reduces the computation time but also has a faster convergence rate while tracking the trajectory from any position in the 3D plane. A single-dimension-based version of FSMC is proposed because of several perks that a simple FSMC cannot provide such as fast convergence rate with aggressive maneuvers. This is because of intermediate sub-processes such that fuzzification, inference of two-dimensional rules, and defuzzification for producing CoG-based output. Therefore, converting the 2D table for rules into a single dimension using the signed distance method [37,69] can be one of the methods to improve the convergence rate and obtain the aggressive maneuvering options. This can be possible and shown graphically in Figure 8.Figure 8
Graphical representation to deriveG variable.After studying the graphical illustration in Figure9, one may derive the mathematical relationship for the G variable as follows:(26)Lsn+αLo=0,(27)G=Lsn+αLo1+α2,where the variable G is a diagonal vector derived from Table 4 as illustrated in Figure 9.Figure 9
Deriving single-dimension input membership functions using the diagonal variableG.Table5 is reproduced using equations (26) and (27). This time, the input membership functions are triangular type but with different x− index values, whereas the output membership function will remain the same as discussed and shown in Figures 5 and 6. Having this entire information, one may now start designing the linear extended state observer.Table 5
Single-dimension rules usingG variable.
GGNBGNMGNSGZGPSGPMGPSG′NBNMNSZPSPMPB


## 4.2. Position Estimator and Disturbance Observer

To tackle the unmodelled dynamic factors, it is essential to propose a fine feedback structure comprised a position estimator and disturbance observer. This is possible by the deployment of various sensors to underactuated quadrotor craft. The most frequent sensors are inertial measurement unit (IMU), GPS, a computer vision system, optical flow sensor for the position alignment, and lastly the utilization of an ultrasonic or infrared sensor, that is, LiDAR v3 for altitude. While considering the data from these sensors, they all must be preprocessed to avoid the noise and extract only useful data [70]. This position estimator is designed to compute and measure the signals that cannot join several noisy signals to remove the noise. As the presence of such random noise can lead towards the deterioration of the final position with time for our proposed underactuated quadrotor unmanned aerial vehicle (QUAV). In the presence of unmodelled dynamic factors such as wind disturbance, our proposed disturbance observer ensures the performance of position tracking and the linear velocities by reducing the noise and effect of external disturbances. Discussing the position estimator, it computes the positional and velocity vectors for quadrotor UAV. These measure signals via IMU may contain an amplified noise while acquiring the velocity and position vectors. The frequent sensor for acquiring the positional vectors is the GPS sensor module which is commonly working at a 10 Hz frequency rate that means every new value will appear after every 100 ms. In Figure 10, it has been depicted that the main aim of the estimator is to acquire the data of position and velocity at the frequency rate of 100 Hz.Figure 10
Block diagram for proposed position estimator design.As per the above diagram, the measured variable is the actual positional and velocity vectors that are acquired from IMU. In addition to this, total thrust is also considered to observe and estimate the linear acceleration via a dynamic model of underactuated quadrotor UAV. The outcome is the linear estimated velocity, which is derived by amalgamating the velocity from estimated linear acceleration and velocity from the positional sensor as shown in Figure10. The entire calculation is depending on the number of states measured from the signal. In the case of transient issues, the positions are detected by estimated linear acceleration through double integration and rectified at every 100 ms. However, in case of steady-state performance, one has taken the signals directly.This was all about estimator design, whereas the disturbance observer is implemented to determine the exact amplitude of disturbances caused by unmodelled dynamic factors and external disturbances. Figure11 presents the detailed working of a disturbance observer that calculates the disturbance as a linear and angular acceleration via the dynamic model of an underactuated quadrotor unmanned aerial vehicle (QUAV). For the synchronization of estimated values with time during the flow of signals from the control block, a delay of motors is included. However, in the case of position, disturbance observer estimation is performed using the states from the accelerometer, and gravity acceleration is reduced using the gravity reduction sub-block as shown in Figure 11.Figure 11
Block diagram of proposed disturbance observer design.After studying all the equations, one may derive the observer error of each DO as shown below mathematically:(28)e^x˙=F1˙−F^˙1=Ax−LgxCxe^x+Exω1,(29)e^y˙=F2˙−F^˙2=Ay−LgyCye^y+Eyω2,(30)e^z˙=F3˙−F^˙3=Az−LgzCze^z+Ezω3,(31)e^ψ˙=F4˙−F^˙4=Aψ−LgψCψe^ψ+Eψω4.These equations can be represented in matrix form as follows:(32)e^˙=H^e^+Ed,where(33)e^=e^x,e^˙x,e^¨x,e^⋯x,e^⋯x.e^˙x,e^y,e^˙y,e^¨y,e^⋯y,e^⋯y.e^˙y,e^z,e^˙z,e^¨z,e^⋯z,e^⋯z.e^˙z,e^ψ,e^˙ψ,e^¨ψ,,(34)H^=H^1O5O5O3O5H^2O5O3O5O5H^3O3O5O5O4H^4,(35)Ed=0,0,0,0,ω1,0,0,0,0,ω2,0,0,0,0,ω3,0,0,ω4T,where H^ is the term defined as the roots of the matrix mentioned in equation (34), and it must be on the left half-plane because of the bandwidths, that is, Γxo,Γyo,Γzo, and Γψo are non-negative in values. These all-estimated error dynamics as defined in equations (28)–(31) are stable in nature provided by Lemma 1 [29,31].Lemma 1.
In equation (32), limt⟶∞e^t is considered as bounded if and only if one of the following two conditions are true such that first one. τi<a,i=1,2,3,4 for all time index t, whereas second one ωi<b,i=1,2,3,4 for all time t. If in case τi has a constant value, that is, ωi=0,i=1,2,3,4, then the estimated error dynamics shall be considered as asymptotically stable.


## 4.3. Stability of the Proposed Closed-Loop Control Design

The stability of the proposed closed-loop control system is described here within this sub-section. It starts all from state tracking error vector ofx− position:(36)erx=Fd1−X1,where the term Fd1=Fxd,F˙xd,F¨xd,F⋯xdT and X=F11,F12,F13,F14T. This leads to equation (37) as follows:(37)F˙d1=A1Fd1+B1F⋯xd.F˙xd,where A1=0000100001000010,B1=0001. In this way, the new feedback can be designed that is written in matrix form as follows:(38)X˙1=A1X1+B1vrx+τ1.Using the following equation, one may derive the feedback design as shown in equation (40):(39)e^r1=Fxd−F^11=Fxd−F11−F^11,(40)vrx=Kxerx+e^x+F⃛dx.F˙dx−τ^1.The variableKx in equation (40) is equal to Kx1,Kx2,Kx3,Kx4; thus, the state tracking error dynamics can be expressed more accurately as follows:(41)e˙r1=A1−B1Kxerx−B1Kxe^x−B1e^x.Hence, combining equations (28) and (41), one may obtain(42)e˙rxe^˙x=A1−B1Kx−B1KxB10Ax−LgxCxerxe^x+0Exω1.By using equation (42), one may find the closed-loop stability of the proposed underactuated quadrotor unmanned aerial vehicle (QUAV). This can be possible mathematically if one derives the eigenvalues of the terms, that is, A1−B1Kx and Ax−LgxCx, since the expressions such as A1,B1 and Ax,Cx are controllable and observable. Thus, the stability of such error dynamics for x− position given by equation (42) is ensured by opting for the suitable controller and observer poles. Similarly, the feedback for y,z, and ψ can also be designed.(43)e˙rye^˙y=A2−B2Ky−B2KyB1e^x0Ax−LgyCyerye^y+0Eyω2,(44)e˙rze^˙z=A3−B3Kz−B3KzB30Az−LgzCzerze^z+0Ezω3,(45)e˙rψe^˙ψ=A4−B4Kψ−B4KψB40Aψ−LgψCψerψe^ψ+0Eψω4.Moreover, in the above equations (41)–(45), A1=A2=A3, where B1=B2=B3, Ky=Ky1,Ky2,Ky3,Ky4,Kz=Kz1,Kz2,Kz3,Kz4, and lastly, Kψ=Kψ1,Kψ2. For this work, A4andB4 are mentioned as follows:(46)A4=0100,B4=01.The stability of the closed-loop systemy,z, and ψ can be obtained in the same way. After studying these all sub-sections, one may derive the intelligent control algorithm of dual-loop single-dimension fuzzy sliding mode control with position estimator and disturbance observer. The block diagram of the overall control algorithm and system is shown in Figure 12. In this figure, the entire working of dual-loop single-dimension fuzzy SMC is summarized. The important factor here is to opt for an appropriate control vector ut in a way that the given quadratic performance index mentioned in equation (47). It obtains the minimum value. It is later proved that it reaches the minimum value by means of the proposed control law equation as mentioned in (48).(47)ut=−KXt=R−1BTPXt,where P is the optimal matrix can be computed using the Riccati equation as follows:(48)ATP+PA−PBR−1BTP+Q=0.Figure 12
Block diagram of dual-loop single-dimension fuzzy SMC with PE and DO.In Figure12, the matrices Q and R are opted to have the form such as Q=diagq11,0,q33,0,q55,0 and R=diag1,1,1,1,1,1. The variables such as q11,q33 and q55 will be compensated using PE and DO propose with dual-loop single-dimension fuzzy slide mode control that will take care of the oscillatory response of the system. The implementation of the proposed algorithm follows below-mentioned steps in simple:(i)
Establishing the dynamics for underactuated quadrotor craft(ii)
Tuning the performance index and weightingQ and R matrices accordingly(iii)
Estimating the parameters for the proposed DLSDF-SMC control algorithm using PE and DO as shown in Figure12


## 5. Simulation Results

In this section, simulation work is carried using MATLAB and Simulink software to show the comparative performance among SMC, FSMC, and dual-loop single-dimension fuzzy sliding mode control technique with a fusion of position estimator and disturbance observer design. The parameters for quadrotor craft used in the below-mentioned simulations arem=2.75kg,Jθ=0.0652kgm2,Jϕ=0.0652kgm2, Jψ=0.120kgm2,l=0.25m andg=9.81ms−2.


### 5.1. Sliding Mode Control Technique

During the execution of conventional sliding mode control for an underactuated quadrotor craft dynamic model, one observes the deviation at the 66th second, and it remains deviated from the helical trajectory. In Figure 13, one may see the impact of unmodelled dynamic factors.Figure 13
Helical trajectory tracking using conventional SMC.The unmodelled dynamic factors and unknown external disturbances have been introduced within the Simulink model using a random function generator at the 25th second in all positional vectors that can be seen in Figure 14. This not only affects the positional vectors but also deviates our quadrotor from its desired trajectory. After the inclusion of these factors at the 25th second, one may also notice the high number of oscillations that are known as the chattering or Zeno effect. This is one of the reasons that this paper also proposes to review the hybrid version of SMC with fuzzy logic control (FLC) design.Figure 14
Response of positional vectors using conventional SMC design.


### 5.2. Fuzzy-Based Sliding Mode Control

Since there were a high number of oscillations (chattering noise) while executing the SMC Design, the paper thus demonstrates the fusion of fuzzy logic control-based SMC design. One may see in Figure15, an underactuated quadrotor craft again starts deviating in the presence of unmodelled dynamics factors from its path at the 59th second and coming back again at the 63rd second.Figure 15
Helical trajectory tracking using fuzzy-based SMC (FSMC).Moreover, this fusion though reduced the chattering effect but also slower the convergence rate. This slow convergence rate is because of the intermediate steps within the fuzzy logic control (FLC) design. One may see the positional response produced through FSMC in Figure16.Figure 16
Response of positional vectors using FSMC design.After studying Figure16, one may see that the number of high oscillations (chattering/Zeno effect) is reduced now in these simulations, but there is still deviation of our proposed underactuated quadrotor UAV with inclusion of unmodelled dynamic factors.


### 5.3. DLSDF-SMC Control with PE and DO Design

Looking at the constraints such as high computation time, deviation, and slower convergence rate, one may conclude that the proposed UAV will not be able to perform the aggressive maneuvers. Thus, this paper proposes the pigeon-inspired optimization-based dual-loop single-dimension SMC control technique that improves the performance not only in terms of positional vectors but also over the limitations of FSMC and conventional SMC. In addition to this, it tackles the unmodelled dynamic factors very smartly. The demonstration for tracking the helical trajectory is shown in Figure17.Figure 17
Helical trajectory tracking using DLSDF-SMC with PE and DO Design.One can see the positional vectors in Figure18. This time, DLSDF-SMC design has tackled the mentioned issues very smartly. The proposed underactuated quadrotor craft does not deviate from the helical trajectory at any instant with a very low chattering effect.Figure 18
Response of positional vectors using DLSDF-SMC with PE and DO design.In these simulation results, there is an improvement in terms of tracking performance. There is no deviation from the helical trajectory and non-observable chattering effect.


#### 5.3.1. Performance Evaluation for Different Trajectories

To check the robustness of the proposed algorithm of DLSDF-SMC with PE and DO design, this paper presents two more trajectories tracking along with their responses for positional vectors. One can see this tracking performance illustrated in Figures19(a) and 19(b). Moreover, their linear velocity response is also illustrated over circular and diamond trajectories in Figures 20 and 21, respectively.Figure 19
Circular and diamond-shaped trajectory tracking using DLSDF-SMC with PE and DO.
(a)(b)Figure 20
Linear velocity response while circular trajectory tracking via DLSDF-SMC with PE + DO.Figure 21
Linear velocity response while diamond-shaped trajectory tracking via DLSDF-SMC with PE + DO.The performance of the proposed fusion of position estimator and disturbance observer is illustrated in Figure22. In this figure, the real signal is shown in blue; measured signal is shown in green; and finally, the estimated signal is shown in red colour. In the figure, real, measured, and estimated position, and velocity at z−axis is shown. Moreover, the attitude disturbance is also estimated and shown in Figure 23. In this figure, the disturbance observer estimates the angular acceleration from desired rotor angular velocities. It should be noted that the measured values are obtained using an inertial measurement unit.Figure 22
Performance of state estimation using PE and DO design.Figure 23
Identifying the disturbance from time instant 20th to 40th second.In Figure23, one may see that the disturbance observer trying to calculate the angular acceleration from angular velocities that are measured using an inertial measurement unit (IMU). The proposed estimating technique based on position estimator and disturbance observer starts estimating at time instant of 20th second and ends it at the 40th second. This is actually a correspondence to an equal force that tries to stabilize the underactuated quadrotor craft while tracking the helical trajectory. In the figure, real, measured, and estimated signals are shown.Simply, in the above figures, one may see the implementation of DLSDF-SMC with PE and DO technique on two different trajectories. The responses show the algorithm is valid for even circular and diamond trajectories as well. In the diagram, the red line shows the actual trajectory, whereas the blue colour shows the tracking of underactuated quadrotor craft. In addition to this, the entire summary for all three control algorithms is mentioned in Table6. The comparison is done in terms of few selected factors of concern such that execution time, chattering effect, control input energy, transient, and steady-state values. Among these factors, control input energy has been computed by acquiring the total power consumption by the four brushless DC motors. It is computed using an ordinary integral square method as shown in equation (49). In the same way, one may compute the chattering phenomena using the following equation:(49)ISE=∫0∞e2tdt.Table 6
Summary of SMC, FSMC, and DLSDF-SMC with PE and DO designs.
Control designsExecution time (sec)Chattering effect (%)Control input energy (CIE) (%)Steady-state errorSMC1004.366.544.0F-SMC1003.2756.544.0DLSDF-SMC (PE + DO)502.5434.753.2Table6 not only shares but also summarizes the overall performance of three control designs where our proposal algorithm outclasses the remaining two algorithms. These all results can also be derived using the simple command in MATLAB for the provided state-space model, that is, step(A, B, C, D), where A, B, C, and D are the state-space matrices. However, one may see that there are also some factors that have not been compared so far. This is because, at this moment, the algorithm is tested for these parameters. The research work is in progress, and there is still a need to do some mathematical amendments.


## 5.1. Sliding Mode Control Technique

During the execution of conventional sliding mode control for an underactuated quadrotor craft dynamic model, one observes the deviation at the 66th second, and it remains deviated from the helical trajectory. In Figure 13, one may see the impact of unmodelled dynamic factors.Figure 13
Helical trajectory tracking using conventional SMC.The unmodelled dynamic factors and unknown external disturbances have been introduced within the Simulink model using a random function generator at the 25th second in all positional vectors that can be seen in Figure 14. This not only affects the positional vectors but also deviates our quadrotor from its desired trajectory. After the inclusion of these factors at the 25th second, one may also notice the high number of oscillations that are known as the chattering or Zeno effect. This is one of the reasons that this paper also proposes to review the hybrid version of SMC with fuzzy logic control (FLC) design.Figure 14
Response of positional vectors using conventional SMC design.


## 5.2. Fuzzy-Based Sliding Mode Control

Since there were a high number of oscillations (chattering noise) while executing the SMC Design, the paper thus demonstrates the fusion of fuzzy logic control-based SMC design. One may see in Figure15, an underactuated quadrotor craft again starts deviating in the presence of unmodelled dynamics factors from its path at the 59th second and coming back again at the 63rd second.Figure 15
Helical trajectory tracking using fuzzy-based SMC (FSMC).Moreover, this fusion though reduced the chattering effect but also slower the convergence rate. This slow convergence rate is because of the intermediate steps within the fuzzy logic control (FLC) design. One may see the positional response produced through FSMC in Figure16.Figure 16
Response of positional vectors using FSMC design.After studying Figure16, one may see that the number of high oscillations (chattering/Zeno effect) is reduced now in these simulations, but there is still deviation of our proposed underactuated quadrotor UAV with inclusion of unmodelled dynamic factors.


## 5.3. DLSDF-SMC Control with PE and DO Design

Looking at the constraints such as high computation time, deviation, and slower convergence rate, one may conclude that the proposed UAV will not be able to perform the aggressive maneuvers. Thus, this paper proposes the pigeon-inspired optimization-based dual-loop single-dimension SMC control technique that improves the performance not only in terms of positional vectors but also over the limitations of FSMC and conventional SMC. In addition to this, it tackles the unmodelled dynamic factors very smartly. The demonstration for tracking the helical trajectory is shown in Figure17.Figure 17
Helical trajectory tracking using DLSDF-SMC with PE and DO Design.One can see the positional vectors in Figure18. This time, DLSDF-SMC design has tackled the mentioned issues very smartly. The proposed underactuated quadrotor craft does not deviate from the helical trajectory at any instant with a very low chattering effect.Figure 18
Response of positional vectors using DLSDF-SMC with PE and DO design.In these simulation results, there is an improvement in terms of tracking performance. There is no deviation from the helical trajectory and non-observable chattering effect.


### 5.3.1. Performance Evaluation for Different Trajectories

To check the robustness of the proposed algorithm of DLSDF-SMC with PE and DO design, this paper presents two more trajectories tracking along with their responses for positional vectors. One can see this tracking performance illustrated in Figures19(a) and 19(b). Moreover, their linear velocity response is also illustrated over circular and diamond trajectories in Figures 20 and 21, respectively.Figure 19
Circular and diamond-shaped trajectory tracking using DLSDF-SMC with PE and DO.
(a)(b)Figure 20
Linear velocity response while circular trajectory tracking via DLSDF-SMC with PE + DO.Figure 21
Linear velocity response while diamond-shaped trajectory tracking via DLSDF-SMC with PE + DO.The performance of the proposed fusion of position estimator and disturbance observer is illustrated in Figure22. In this figure, the real signal is shown in blue; measured signal is shown in green; and finally, the estimated signal is shown in red colour. In the figure, real, measured, and estimated position, and velocity at z−axis is shown. Moreover, the attitude disturbance is also estimated and shown in Figure 23. In this figure, the disturbance observer estimates the angular acceleration from desired rotor angular velocities. It should be noted that the measured values are obtained using an inertial measurement unit.Figure 22
Performance of state estimation using PE and DO design.Figure 23
Identifying the disturbance from time instant 20th to 40th second.In Figure23, one may see that the disturbance observer trying to calculate the angular acceleration from angular velocities that are measured using an inertial measurement unit (IMU). The proposed estimating technique based on position estimator and disturbance observer starts estimating at time instant of 20th second and ends it at the 40th second. This is actually a correspondence to an equal force that tries to stabilize the underactuated quadrotor craft while tracking the helical trajectory. In the figure, real, measured, and estimated signals are shown.Simply, in the above figures, one may see the implementation of DLSDF-SMC with PE and DO technique on two different trajectories. The responses show the algorithm is valid for even circular and diamond trajectories as well. In the diagram, the red line shows the actual trajectory, whereas the blue colour shows the tracking of underactuated quadrotor craft. In addition to this, the entire summary for all three control algorithms is mentioned in Table6. The comparison is done in terms of few selected factors of concern such that execution time, chattering effect, control input energy, transient, and steady-state values. Among these factors, control input energy has been computed by acquiring the total power consumption by the four brushless DC motors. It is computed using an ordinary integral square method as shown in equation (49). In the same way, one may compute the chattering phenomena using the following equation:(49)ISE=∫0∞e2tdt.Table 6
Summary of SMC, FSMC, and DLSDF-SMC with PE and DO designs.
Control designsExecution time (sec)Chattering effect (%)Control input energy (CIE) (%)Steady-state errorSMC1004.366.544.0F-SMC1003.2756.544.0DLSDF-SMC (PE + DO)502.5434.753.2Table6 not only shares but also summarizes the overall performance of three control designs where our proposal algorithm outclasses the remaining two algorithms. These all results can also be derived using the simple command in MATLAB for the provided state-space model, that is, step(A, B, C, D), where A, B, C, and D are the state-space matrices. However, one may see that there are also some factors that have not been compared so far. This is because, at this moment, the algorithm is tested for these parameters. The research work is in progress, and there is still a need to do some mathematical amendments.


## 5.3.1. Performance Evaluation for Different Trajectories

To check the robustness of the proposed algorithm of DLSDF-SMC with PE and DO design, this paper presents two more trajectories tracking along with their responses for positional vectors. One can see this tracking performance illustrated in Figures19(a) and 19(b). Moreover, their linear velocity response is also illustrated over circular and diamond trajectories in Figures 20 and 21, respectively.Figure 19
Circular and diamond-shaped trajectory tracking using DLSDF-SMC with PE and DO.
(a)(b)Figure 20
Linear velocity response while circular trajectory tracking via DLSDF-SMC with PE + DO.Figure 21
Linear velocity response while diamond-shaped trajectory tracking via DLSDF-SMC with PE + DO.The performance of the proposed fusion of position estimator and disturbance observer is illustrated in Figure22. In this figure, the real signal is shown in blue; measured signal is shown in green; and finally, the estimated signal is shown in red colour. In the figure, real, measured, and estimated position, and velocity at z−axis is shown. Moreover, the attitude disturbance is also estimated and shown in Figure 23. In this figure, the disturbance observer estimates the angular acceleration from desired rotor angular velocities. It should be noted that the measured values are obtained using an inertial measurement unit.Figure 22
Performance of state estimation using PE and DO design.Figure 23
Identifying the disturbance from time instant 20th to 40th second.In Figure23, one may see that the disturbance observer trying to calculate the angular acceleration from angular velocities that are measured using an inertial measurement unit (IMU). The proposed estimating technique based on position estimator and disturbance observer starts estimating at time instant of 20th second and ends it at the 40th second. This is actually a correspondence to an equal force that tries to stabilize the underactuated quadrotor craft while tracking the helical trajectory. In the figure, real, measured, and estimated signals are shown.Simply, in the above figures, one may see the implementation of DLSDF-SMC with PE and DO technique on two different trajectories. The responses show the algorithm is valid for even circular and diamond trajectories as well. In the diagram, the red line shows the actual trajectory, whereas the blue colour shows the tracking of underactuated quadrotor craft. In addition to this, the entire summary for all three control algorithms is mentioned in Table6. The comparison is done in terms of few selected factors of concern such that execution time, chattering effect, control input energy, transient, and steady-state values. Among these factors, control input energy has been computed by acquiring the total power consumption by the four brushless DC motors. It is computed using an ordinary integral square method as shown in equation (49). In the same way, one may compute the chattering phenomena using the following equation:(49)ISE=∫0∞e2tdt.Table 6
Summary of SMC, FSMC, and DLSDF-SMC with PE and DO designs.
Control designsExecution time (sec)Chattering effect (%)Control input energy (CIE) (%)Steady-state errorSMC1004.366.544.0F-SMC1003.2756.544.0DLSDF-SMC (PE + DO)502.5434.753.2Table6 not only shares but also summarizes the overall performance of three control designs where our proposal algorithm outclasses the remaining two algorithms. These all results can also be derived using the simple command in MATLAB for the provided state-space model, that is, step(A, B, C, D), where A, B, C, and D are the state-space matrices. However, one may see that there are also some factors that have not been compared so far. This is because, at this moment, the algorithm is tested for these parameters. The research work is in progress, and there is still a need to do some mathematical amendments.


## 6. Conclusion

The manuscript presents an underactuated quadrotor unmanned aerial vehicle (QUAV; UAV). It is underactuated because of the inequality between control inputs and the degree of freedom such that the control inputs are less than DOF. This research demonstrates the comparative simulation work between SMC, FSMC, and PIO-based DLSDF-SMC design to address the major limitations. First, this paper describes the conventional SMC control design as one of the highly sensitive control strategies for any sort of external disturbances and unmodelled dynamic factors. This is the reason behind the high number of oscillations that are experienced by four brushless DC motors of quadrotor UAV. Second, one may see fuzzy version of SMC design that reduces the chattering effect and control input energy (CIE).The convergence rate of all positional vectors is seen slower; thus, quadrotor craft with such rates cannot perform aggressive maneuvers. Therefore, this paper suggests dual-loop single-dimension fuzzy sliding mode control technique, which is fused with position estimator and disturbance observer. This not only reduces the chattering effect but also fastens the convergence rate so that quadrotor can perform some extraordinary maneuvers. In addition to this, the steady-state error is also improvised as compared to the other two control techniques, that is, SMC and FSMC. The simulation results for helical, circular, and diamond-shaped trajectories show the robustness of the proposed algorithm. The work is still in progress to perform some more simulations to compare the remaining performance parameters, whereas this proposed algorithm is to be implemented via a specific hardware configuration to validate the software results with hardware results. [67,71–73]



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*Source: 8791620-2021-12-07.xml*