Dynamic Smooth Sliding Control Applied to UAV Trajectory Tracking

This paper proposes a sliding mode controller with a smooth control signal for a class of linear plants with nonlinear input state-dependent disturbance. The proposed controller is obtained by allowing some constant parameters of the earlier Smooth Sliding Control (SSC) to vary as a function of the output tracking error and its time derivative, improving the control chattering alleviation in practical implementations. Furthermore, during the sliding mode, the new scheme can synthesize a range of controllers, such as fixed gain PI controllers and approximations of the Super-Twisting Algorithm (STA). A complete closed-loop stability analysis is provided, leading to global stability properties, exact output regulation, and practical output tracking. In addition, realistic simulation results with an Unmanned Aerial Vehicle (UAV) model, incorporating aerodynamic effects and internal closed-loop controllers, are obtained and validated via experiments with a commercial hexacopter.


I. INTRODUCTION
The number of UAV applications has been increasing in the last decade, mainly due to their maneuvering capability, allowing a variety of remote sensing and monitoring or inspection tasks [1], [2], [3].In particular, the development of UAV trajectory tracking robust controls is crucial for the detection and identification of oil leaks in offshore environments, or for the extraction of traffic data via aerial video images [4], [5], [6], where the wind influence cannot be disregarded and an optimal desired trajectory in general is required to reduce the battery consumption, for instance.
Several control systems were proposed for the trajectory tracking problem of unmanned aerial vehicles.Classic techniques such as the Proportional-Integral-Derivative (PID) control have been applied to most existing flight control systems, because of their simple design and implementation, when is reasonable to approximate the vehicle dynamics to a linear model [7], [8], [9], with uniformly norm-bounded The associate editor coordinating the review of this manuscript and approving it for publication was Huiyan Zhang .
or almost constant disturbances.When the linearization process [10] is not valid, non-linear control methods come into play, such as optimization strategies for path planning [11], linear quadratic regulation control (LQR) [12], [13], feedback linearization control schemes [14], [15] and robust strategies via sliding modes [16], [17].In addition, the performance of classic techniques is, in general, limited by the presence of uncertainties, disturbances, and unmodelled dynamics.
In contrast, it is well-known that sliding mode-based controllers are robust concerning external disturbances (eventually state/output dependent), parameter uncertainties, and unmodelled dynamics, but suffer from the chattering phenomenon.In this context, aiming to attenuate chattering, sliding mode control based on the Super-Twisting Algorithm has been widely applied [18], [19], [20], [21], [22], [23], [24].Solutions for the trajectory tracking problem of UAVs have been pursued in the literature [25].In [26] and [27], the altitude control of a quadrotor is based on a combined STA and high-order sliding mode (HOSM) observer, and in [28], a similar combination is addressed to estimate linear and angular velocities, and unknown lumped disturbance, including experimental evaluation with the quadcopter DJI M100.To be able to cover inspection tasks with UAVs, manipulators have been attached, such as in [29], where an STA with gain adaptation law is designed in the presence of a disturbance caused by the manipulator dynamics.
Additionally, beyond affecting UAV's dynamics by adding manipulators, UAV geometric parameters can be variable over time [30] and the pick-and-place task can generate mass variation [31], [32].In [30], a Fast Terminal Sliding Mode Controller was applied to guarantee the flight stability and rapid convergence of the states in finite time with a reconfigurable UAV.In [31], a sliding mode technique was proposed allowing the UAV control to adapt to the altered mass without re-tuning the controller and, in [32], the landing problem on a moving platform was solved together with a pick-and-place task, by using the so-called SSC scheme.
The SSC was presented in [33], as an alternative modification in the Variable Structure Model Reference Control (VS-MRAC) [34], [35], to provide a smooth control effort, since the VS-MRAC is a strategy with discontinuous control effort.It introduces an averaging filter to obtain a continuous control signal.To compensate for the phase lag added by the averaging filter, an internal prediction loop was employed so that the ideal sliding mode could be preserved, leading to chattering avoidance and robustness concerning unmodelled dynamics [36].Recently, this strategy was generalized for plants with time-varying control gain and applied to the autonomous landing problem in a moving platform [32].More recently, in [37] was presented a new SSC scheme with the averaging filter time constant being updated via the tracking error for a real UAV trajectory tracking application.
In this paper, a novel extension of the results of [37] is proposed.As the main contributions of the paper, we consider: (i) the development of the new (dynamic) SSC scheme, named Dynamic Smooth Sliding Control (DSSC), where the original fixed parameters of the SSC averaging filter and the predictor are replaced by functions that are updated via the tracking error; (ii) a complete closed-loop stability analysis of the DSSC algorithm for the considered class of linear plants with nonlinear disturbance; (iii) a connection of the synthesized DSSC during sliding mode with the Variable Gain Super-Twisting Algorithm (VGSTA) and the standard STA, by selecting appropriate functions in the DSSC; and (iv) experimental evaluation of the DSSC for trajectory tracking in a real-world scenario with the commercial DJI M600 Pro hexacopter.

A. NOTATIONS AND TERMINOLOGIES
A mixed time-domain and frequency-domain notation will be adopted to avoid clutter.In this manner, a rational function G(s) will denote either an operator, where s is the differential operator, or a transfer function, where s is the Laplace complex frequency variable.Therefore, the time and frequency dependencies of the signals will be mostly omitted.In general, for a scalar composite function f (e(t), σ (t), t), where e(t) and σ (t) are scalar functions of the time instant t ≥ 0 (time-varying functions), we perform along the paper the abuse of notation f (t) = f (e(t), σ (t), t).

B. MOTIVATION FOR THE CONSIDERED CLASS OF PLANTS
In this paper, the trajectory tracking control problem of a UAV is addressed.The proposed control scheme considers that an internal Kalman filter provides full state estimation, and inaccessible inner controllers are responsible for decoupling the four UAV degrees of freedom, corresponding to the UAV's linear velocity and to the yaw angle rate, see Figure 1, allowing to perform a cascade control strategy.The UAV dynamic model is given by: where ∈ IR 3 is the angular velocity represented in the body frame, v = v x v y v z T ∈ IR 3 is the UAV's linear velocity vector (in the inertial frame), R represents the rotation matrix from the body frame to the inertial frame, J R is the Jacobian Euler angles representation, v ψ = ψ is the yaw angle velocity, g is the gravity acceleration, J is the (diagonal) inertia matrix, M ∈ IR is the UAV mass, e 3 = 0 0 1 T , M ∈ IR 3 is the net moment and i=1 f i is the net thrust magnitude, where f i ∈ IR is the i-th propeller's thrust magnitude and n r is the number of rotors.The drag terms are incorporated in F drag and τ drag and τ dist is a torque disturbance depending on the propellers' angular velocity and acceleration, see details in the Appendix A. The inner controllers generate the torques and forces commands via propellers' thrusts for tracking the velocity references (u x , u y , u z or u ψ ) generated by the outer controller (DSSC), which in turn is designed for tracking position and orientation, i.e., to assure that the UAV position p x , p y , p z track the desired positions p x d , p y d , p z d and that the yaw angle ψ tracks the desired angle ψ d .For the dynamics of each degree of freedom (DOF), the inner controllers and the Kalman filter can be treated as unmodelled dynamics for the outer controller (DSSC), so that a kinematic model from velocities commands to actual velocities is considered.
We consider the complete model for the UAV based on the following premises and motivations: 1) Low/medium velocity profiles are considered so that the motors and the motors drivers (ESCs) dynamic can be neglected, as well as, the internal Kalman Filter dynamics of the UAV leading to the availability of the full UAV state vector.
2) The proposed DSSC scheme acts as an outer controller that provides velocity commands as references for the inner velocity control loops.These inner velocity control loops are designed so that a reasonable control performance is achieved.1As a motivation for the class of plants considered in this paper, let us describe shortly the resulting closed-loop dynamics with a given inner controller for the altitude degree of freedom.The same idea can be extended to the other degrees of freedom.
Here, the control objective is to assure that the altitude velocity ν tracks the desired velocity command u, at least, with some residual error.From the last component of the ν-dynamics Mν = −Mge 3 + Rfe 3 + F drag , see Figure 1, one possible control law is given by a composition of a feedback linearization control term f := M(U + g), which is parameter dependent, and a PI-control law , where the roll and pitch angles are considered small.The closed-loop dynamic behavior from the velocity command input u to the actual velocity ν is given by where D = e T 3 F drag /M is a disturbance due to the aerodynamic drag F drag , see Figure 1.Thus, by choosing the control gains appropriately one can impose an acceptable closed-loop performance and conclude that, for low acceleration commands (ü ≈ 0) and for low aerodynamic drag, e(t) := ν − u tends to zero, as t → ∞.For any eventual residual error in the inner control loop, the outer position control loop (DSSC) can compensate for it.Moreover, a more elaborate inner controller could be considered [38], but this is not the focus of this paper.
The closed-loop system (2) can be represented by Note that G(s) is a relative degree one and minimum phase (k p , k i > 0) transfer function from u + d to ν.This system can be written in state-space in the normal form as where η ∈ IR is the zero dynamics state vector (for details, see Appendix B).The disturbance d could also incorporate other eventually remaining terms due to any mismatch parameters in the feedback linearization control term.
In what follows, motivated by ( 3)-( 4), a class of relative degree one and minimum phase plants, of arbitrary order, is considered for the linear position and yaw angle tracking control problem formulation.

II. PROBLEM FORMULATION: ONE DEGREE OF FREEDOM
Consider the following class of uncertain plants given by where u p ∈ IR is the control input, y ∈ IR is the plant output, η ∈ IR n−2 is the inverse system (zero dynamics) state vector, d ∈ IR is regarded as a matched input disturbance, k p > 0 is the uncertain high-frequency gain (HFG), a p is an uncertain parameter, and is the state vector.Without loss of generality, consider that the inverse system has a state-space realization (A η , B η , C η ), with (A η , B η ) in the canonical controllable form with B η = 0 . . .0 1 T ∈ IR n−2 .We assume that A η is a Hurwitz matrix (minimum phase assumption) and η is unavailable for feedback.The uncertain function d(y, ẏ, t) is assumed piecewise continuous in t and locally Lipschitz continuous in the other arguments.For each solution of ( 5)-( 7), there exists a maximal time interval of definition given by [0, t M ), where t M may be finite or infinite.Thus, finite-time escape is not precluded a priori.Regarding the application covered in this paper, in ( 7)-( 8), y is a generic output representing a UAV's degree of freedom (p x , p y , p z or ψ) and u p is the corresponding generic velocity command input (u x , u y , u z or u ψ ), see Figure 1.
Remark 1 Plant Input Disturbance: UAV's Application: In ( 5)- (8), the input disturbance d(y, ẏ, t) represents the coupling between degrees of freedom, the wind influence, and possible nonlinearities remaining due to some unmatched parameters in the inner velocity loop.Moreover, it is considered that the wind velocity has low-frequency components or can be represented by piecewise functions with jump discontinuities where the discontinuity points have zero measure.

A. CONTROL OBJECTIVE
The aim is to achieve global convergence properties in the sense of uniform signal boundedness and asymptotic output practical tracking.The control objective is to design a control law u p (t) for the uncertain plant ( 5)-( 8) such that y(t) tracks a bounded desired trajectory y m (t) as close as possible, i.e., the tracking error converges to zero as t → +∞, or at least, to the neighborhood of zero (practical tracking).The desired trajectory y m (t) is assumed to be smooth enough so that ẏm and ÿm are well defined available signals.
For the sake of simplicity and to focus on the novelty of the proposed controller (DSSC) in comparison to the previous version (SSC), we assume that y and ẏ are available for feedback.Furthermore, as mentioned before, the application considered in the paper endorses this assumption.
Let the relative degree one output variable σ (y, ẏ, t) : IR 3  → IR be defined by The main idea is to design u p so that σ tends to zero as t → +∞, or at least, to the vicinity of zero, despite the input disturbance d(y, ẏ, t).Thus, the convergence of the tracking error e(t) to a residual set is assured by setting l 0 > 0, according to (10).
The extension to the case where only y is available for feedback can be obtained by using the approximation ẏ ≈ y f := s (τ f s+1) y, with τ f > 0 being a design constant, and extension to systems with an arbitrary relative degree can be obtained by using a linear lead filter, a high gain observer (HGO) [39], or a robust exactly differentiator (RED) [40], as in [36], [41], and [42].

B. ERROR DYNAMICS
Let the control signal be composed of two terms where the control effort u is the DSSC robust control effort and u n is a nominal control law, both to be defined later on.
In practical applications there exists some level of knowledge of the plant parameters and, in general, a nominal control based on this knowledge can be applied to reduce the magnitude of the robust action (here being the DSSC), which is designed to deal with disturbances and/or parameter uncertainties.

III. DYNAMIC SMOOTH SLIDING CONTROL (DSSC)
In Figure 2, it is illustrated the DSSC block diagram, which has the same structure as the original SSC [33], [43], [44].In comparison to the SSC, the Dynamic SSC (DSSC) differs in one main aspect: the averaging filter time constant τ av , the predictor time constant τ m and the predictor gain k o are allowed to vary with the time t and/or the plant signals σ (t) and e(t).The DSSC law is given by with a time-varying averaging filter where τ av (t) = τ av (σ (t), e(t), t) > 0 (∀σ, e, t) and is the predictor's discontinuous term, with modulation function ϱ(t).In the DSSC, the sliding variable σ is defined as where σ is the output of the predictor

A. EXISTENCE OF IDEAL SLIDING MODE
With σ defined in (17), the σ -dynamics in (13) and the smooth control law (14), one has that Moreover, by using the predictor dynamics in (18) and the relationship σ = σ − σ , one can further obtain where 2 Despite that, the original SSC [33] can be applied for a broader class of plants with arbitrary relative degree [43] and [44], we focus on the case where y and ẏ are available for feedback.However, the DSSC scheme can also deal with arbitrary relative degree plants, by using linear lead filters to estimate output time derivatives.We restrict ourselves to the case of relative degree one which is the simplest case amenable by pure Lyapunov design.General DSSC block diagram for arbitrary relative degree case and with generic state-dependent functions τ m (σ (t ), e(t ), t ), k o (σ (t ), e(t ), t ) and τ av (σ (t ), e(t ), t ).The predictor is given in (18) and depends on k o and τ m , while the averaging filter is given in (15) and depends on τ av .For the class relative degree one plants considered here with ẏ available for feedback, one can set τ f = 0, so that σ f = σ with σ in (10).
with d1 := k p u n − a p ẏ + k p (d − C η η/k p ) being an uncertain term and d2 := σ/τ m + l 0 ẏ − σm being a known signal that could be directly canceled by redefining the control term u 0 in (16).For simplicity, at the cost of being more conservatism, we treat d2 as an uncertain term too.
As in the original SSC, sliding mode occurs at σ ≡ 0 so that σ converges to zero in some finite time t s ∈ [0, t M ), i.e., σ (t) = σ (t), ∀t ∈ [t s , t M ), provided that the modulation function ϱ (in the discontinuous term u 0 ) satisfies ϱ > d 0 /k o , modulo vanishing terms due to initial conditions, i.e., being designed to dominate the norm of the total disturbance d 0 /k o faced by u 0 in the (19), after some initial transient.The proof of the sliding mode existence and the avoidance of finite time escape (mainly due to the unboundedness observability property of the closed-loop system) are provided later on in Theorem 1.

B. MAIN FEATURES OF THE PROPOSE DSSC
The main features of the new DSSC scheme can be summarized as follows: (i) the algorithm presents robustness w.r.t.unmodelled dynamics and global/semi-global stability properties (the full proof is provided), and (ii) it is observed an improvement w.r.t. the previous SSC, where a better control chattering alleviation is obtained.
The complete closed-loop stability analysis is one of the main contributions of this paper and it is provided later on for the class of plants ( 5)- (8).In addition, the robustness of the proposed control scheme (DSSC) w.r.t.unmodelled dynamics is verified in the experiments with the DJI M600 and via numerical simulations with the full coupled dynamics model, including the inner controllers and aerodynamic effects, corroborating that a relative degree one is fair enough for representing each of the four DOF, mainly for low/medium velocity profiles.Any residual coupling between the DOFs is embedded in an input disturbance.
Moreover, with this modification in the original SSC, one can observe an improvement in control chattering alleviation in practical implementations where discretization, for instance, generates numerical chattering even for the relative degree one case.Let us explain this improvement when the averaging filter has a pass-band inversely proportional to |e(t)|.First, we point out that in both the SSC and the DSSC, sliding mode occurs in an internal variable different from the output tracking error e.In general, sliding mode takes place from the beginning, when |e| can be large, as well as, the modulation function ϱ.Second, in the SSC, practical tracking is obtained in the sense that the tracking error converges to a residual set O(τ av ) proportional to the fixed averaging filter time constant τ av .In the DSSC, in contrast, the residual set is proportional to the steady-state value τ * av of τ av (e(t)).Thus, to obtain high tracking precision in the SSC, the fixed time constant should be small, while in the DSSC, only the steady-state value τ * av should be small.It means that, if one set τ av (e(t)) ∝ |e(t)| + O(τ * av ), then the DSSC filters out the high frequencies of the switching control more than the SSC, exactly in the time interval where the modulation function and the tracking error are large.When e is small, the filter can allow high frequencies to pass, since the modulation function is small.The final result is a smoother control action when compared with the original SSC, maintaining at least the same level of tracking precision.

IV. DESIGN OF THE STATE-DEPENDENT FUNCTIONS OF THE DSSC
As mentioned in Section III-A, the modulation function ϱ is designed to overcome the disturbance in the σ -dynamics (19) to assure ideal sliding mode at the manifold σ = 0.After achieving the ideal sliding mode, an equivalent synthesized controller results.This synthesized controller depends on the state-dependent functions τ av , τ m and k o .At this point, we have freedom to choose the DSSC functions.

A. GENERAL SYNTHESIZED EQUIVALENT CONTROLLER DURING SLIDING MODE
Now, let us find a general formulation for the synthesized equivalent DSSC control law during sliding mode.First, denote ūav 0 = u av 0 as the solution of ( 15), when the discontinuous control u 0 is replaced by the equivalent control 44310 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
k o , directly obtained from the σ -dynamics (19).This is the so called reduced dynamics τ av (t) uav 0 = −ū av 0 + u eq .Also replace u 0 by u eq in the predictor dynamics (18), leading to . Since, during sliding mode at σ = 0 one has σ = σ , then one can further write leading to the time derivative of the synthesized DSSC law ū := ūav 0 given by3 Let us rewrite the terms in square brackets of ( 22) in a more convenient form for selecting the DSSC functions k o (t), τ av (t) and τ m (t), in terms of σ, e and t.One can find appropriate functions g 1 (t) = g 1 (σ (t), e(t), t) and The functions g 1 (σ, e, t) > 0 and g 2 (σ, e, t) > 0 must be chosen sufficiently smooth and so that k o , τ av > 0 and τ m > 0 are well-defined for all finite values of σ, e and t.Then, ( 22) can be rewritten as Therefore, by integrating both sides of (24), the synthesized DSSC law can be written as: ∀t ≥ t s , where C s := ū(t s ) + g 1 (t s )σ (t s ) is a constant 4 and g 1 (t) and g 2 (t) are nonlinear gains that should be designed so that the functions k o (t), τ m (t) and τ av (t) be positive.Depending on the choices for the nonlinear gains g 1 and g 2 the resulting synthesized controller has different structures and properties.In fact, as illustrated in Appendix C, one can arrive at synthesized controllers starting from a simple PI controller, passing through an approximation for the standard STA [16], [18], to approximation for the variable gain STA (VGSTA) [19], [23].
A simple PI control or the standard STA (with constant gains) can deal with second-order plants, without inverse dynamics.However, as in [19] and [23], plants with inverse dynamics require the STA's gains to be state-dependent, leading to the VGSTA.
Our focus is on the choice for g 1 and g 2 which leads to an approximation for the VGSTA since the plant ( 5)- (7) has inverse dynamics, which is considered in the complete stability analysis given later on in Theorem 1.
In fact, the variable gains κ 1 (σ, e, t) > 0 and κ 2 (σ, e, t) > 0 must be designed so that τ av > 0 and τ m > 0 are well-defined for all finite values of σ, e.The DSSC's state-dependent functions and the design guidelines of the corresponding control parameters are given in Appendix F and summarized in Table 1.
The synthesized equivalent controller approaches the VGSTA, far from the origin of the state-space (σ, e), and acting like a reduced gain version of the VGSTA, near the origin.

V. THE CASE STUDIED: STABILITY RESULTS, NUMERICAL SIMULATIONS, AND EXPERIMENTAL RESULTS
In what follows, we provide the main stability results with the DSSC functions in Table 1, the corresponding numerical simulations with the full UAV's dynamic model (1), including the aerodynamic effects and the inner control loops, and with the model ( 5)-( 8), and the corresponding experimental results obtained with the commercial DJI M600 Pro hexacopter.

A. CLOSED-LOOP STABILITY RESULTS
To obtain the stability result and perform the full stability analysis, some assumptions must be considered.The plant parameters k p and a p in ( 7) are assumed uncertain with known bounds and we consider a class of input disturbances that can be partitioned as The following assumption is considered: where u n p (t) = u n p (e(t)) and u n d (t) = u n d (σ (t)).For simplicity and without loss of generality, we restrict the nominal control to have terms that satisfy the following additional assumption: (A4) There exist non-negative constants c σ , c e1 , c e2 , c iσ , c ie and c m such that It must be highlighted that the nominal control is not regarded as a disturbance and can be disregarded when the plant uncertainty is large.Remark 2 Modulation Function Design: To satisfy ϱ > d 0 /k o , modulo vanishing terms due to initial conditions, the modulation function can be chosen as: where is an available norm bound for the sum d1 + d2 and δ ρ > 0 is an arbitrary small constant.In (34) where π e is an exponentially decaying term depending on the initial conditions and this residual set does not depend TABLE 1. DSSC's dynamic functions and parameters.The free parameters are: l 0 , φ a , ϵ, δ > 0, and ε i > 0 (i = 1, 2, 3).
on the initial conditions.In addition, all closed-loop signals remain uniformly bounded, finite-time escape is avoided and the sliding variable becomes identically null after some finite time t s ≥ 0. Proof: For this particular case, where the functions τ av (t), τ m (t) and k o (t) are chosen according to ( 28) and ( 29), an approximation for VGSTA is synthesized during the sliding mode, and the main idea of the proof is as follows.Firstly, we prove that finite-time escape cannot occur before σ (t) = 0, i.e., before sliding mode takes place.Secondly, once σ = 0 enters in sliding motion in finite time, then the proof follows the main steps of the proof given in [19], [23] and [45].The main difference is the introduction of the Small-Gain Theorem to deal with the approximation of the VGSTA.See Appendix D for the complete proof.

1) ADDITIONAL COMMENTS
First, we recall that φ b ̸ = 0 in Theorem 1 allows obtaining global results.However, semi-global stability results can be achieved, when φ b = 0, since one can perform similar stability analysis as in Theorem 1, where the main difference is that the gains κ 1 and κ 2 can be designed constant around the origin of (σ, e), depending on the initial conditions.
Second, in the regulation mode case, one has that βm and ẏm are zero in the small-gain based analysis (Appendix D), since k d5 = 0 and α d2 = α d3 = 0 when a constant disturbance is under consideration.Thus, in this case, the tracking error e converges to zero, exponentially, and a constant disturbance is rejected.
Third, additionally to Theorem 1, prescribed finite-time convergence for the residual set can be assessed.Following the proof in Appendix D, the term −2ϵ {P} , in (55), is responsible to assure that σ and the tracking error e reach a residual set in a prescribed finite-time.

2) UNMODELLED DYNAMICS ROBUSTNESS ANALYSIS
Assume that an unmodelled dynamic represented by a transfer function of the form where W µ (µs) is stable and strictly proper, is now in series with the plant input u p = u + u n , in (7), i.e.
modulo exponentially decaying terms due to the unmodelled dynamics initial conditions.This extra term d µ can be regarded as an additional input disturbance and incorporated in the input disturbance d, in (7).Recalling that u = −u av 0 and τ av uav 0 = −u av 0 + u 0 , then one can write As some examples for the unmodelled dynamics transfer function, one has: Thus, this additional disturbance d µ is a filtered version of the averaging control u av 0 and the discontinuous control u 0 , via a proper and stable transfer function of order O(µ/τ av ).Thus, for µ/τ av sufficiently small and despite some parasitic dynamics µ, the ideal sliding mode can still be enforced after some finite time, for the appropriate design of the modulation function.
To explain the main idea, for simplicity, consider that: (i) u n = 0, (ii) τ av is a constant and τ m = (|σ | 1/2 +δ)/κ 1 , with κ 1 and δ being positive constants, (iii) the system has order two (no zero dynamics) and is perfectly known (a n p = a p and k n p = k p ), and (iv) the nominal control is given by k n p u n := −(l 0 − a n p )ẏ + σm .So, d σ and the σ -dynamics, both in (13), become respectively, where we have replaced u by the DSSC's control law u = −u av 0 , with u av 0 in (15).Now, one can subsequently conclude that: (i) the disturbance term d1 , in (20) Now, an approximated analysis can be carried out for understanding the superior performance of the DSSC in comparison to the STA, in the presence of unmodelled dynamics.
Since the averaging control ūav 0 is an approximation of the equivalent control u eq , for τ av sufficiently small [46], one has that u eq ≈ ūav 0 implies With u av 0 = ūav 0 in (36), the closed-loop σ -dynamics can be approximated by for δ > 0 and small.Thus, the closed-loop σ -dynamics with the DSSC law approaches the closed-loop σ -dynamics with the standard STA, without the presence of input disturbance (with κ 2 = 0).Finally, since d incorporates the equivalent disturbance d µ generated by the unmodelled dynamics, it becomes evident that the DSSC should outperform the corresponding STA.

B. NUMERICAL SIMULATION
The UAV's dynamic model is developed for low-velocity profiles.It means that the dynamics of the motors and the motors' drivers (ESC's) and the internal Kalman Filter dynamics (full state feedback) can be neglected, while the more relevant effects are due to the aerodynamic forces and torques.
The inner control was considered as simple as possible to be representative of the unavailable internal control loops in the DJI M600, without putting any effort into stability analysis or tuning control parameters methodologies.The consistency of the inner control loops developed here was verified first with the DJI Assistant 2 Simulator5 and then with experimental data.
In what follows, we first present simulation results with the full UAV's dynamic model, including the aerodynamic effects and the inner control loops, and with the model ( 5)- (8).
The aerodynamics parameters (see Appendix A), extracted from the literature [47], are as follows: the thrust aerodynamic coefficient k T i = 0.0024, in Ns 2 /rad, the aerodynamic torque coefficient c τ = 0.57, in mrad/s 2 , the matrix coefficient K F d = diag 0.03 0.03 0.015 of the drag force on the structure, in Ns 2 /m 2 , and the matrix coefficient K F di = diag 1 1 1 (8 × 10 −6 ) of the propeller drag force, in Ns 2 /(mrad).
To simplify the control allocation, we consider a quadrotor with the same weight, size, and geometry as the DJI M600.The UAV's parameters can be summarized as follows: the number of rotors n r = 4, the directions of rotation All initial conditions were set at zero except the drone position p x (0) = 40m, p y (0) = 10m and p z (0) = 10m, and yaw angle ψ(0) = (π/4)rad (45deg).The desired trajectories are: p x d (t) = 20 sin(2π/40t), p y d (t) = 20 cos(2π/40t), p z d (t) = 3 sin(2π/60t) + 5, and ψ d (t) = −(π/4) sin(2π/40t) + π/4.For all four degrees of freedom, the DSSC algorithm is implemented with τ av constant and with the state-dependent functions where δ = 1 and κ m = 4.0166.Moreover, for the x and y subsystems, were selected the parameters κ o = 110.651,τ av = 0.03 and ϱ = 1.5.For the z-subsystem, were selected κ o = 55.3255,τ av = 0.06 and ϱ = 0.5.For the ψ-subsystem, were selected κ o = 55.3255,τ av = 0.06 and ϱ = 0.15.The other DSSC's parameter is l 0 = 0.2, for all subsystems.Now, we consider two cases.Case 1: the full UAV dynamic (1), see also Figure 1, with a constant wind velocity v w = 8 −8 8 T , in m/s, added after t = 20s.This effect can be observed only for the full UAV's model which incorporates the aerodynamic drag terms (blue lines).Case 2: the simplified UAV model ( 6)-( 8), without inverse dynamics and with a constant input disturbance d added in each subsystem ( 5)-( 8), after t = 20s: d = −0.8,for the x-subsystem; d = 0.8, for the y-subsystem; d = 0.2, for the z-subsystem; and d = 0.1, for the ψ-subsystem.For simplicity, we set a n p = a p = 1 and k n p = k p = 1 for the four subsystems in Case 2. This lead to a simplified nominal control, which can be chosen as In the left column of Figure 3, one can see the velocity command reactions to compensate for the disturbances, after t = 20s.Recall that the disturbances are different for the full UAV dynamic (wind disturbance) and for the UAV model dynamics ( 5)-( 8) (input disturbance d).However, before the disturbances (t < 20), both tracking errors' behavior (right column) is very similar, except for a residual oscillation in the DSSC control signal (left column), when applied to the full UAV model case (blue line), due to the inner control loops (unmodelled dynamics effect).
Figure 4 illustrates the time-varying behavior of the state-dependent functions k o (t) and τ m (t), where both increase when the disturbance acts after t = 20s.
In what follows, we study the robustness of the DSSC scheme w.r.t.unmodelled dynamics in comparison to the STA.Consider the STA and the DSSC applied to the full UAV model, without wind disturbance.For the x and y subsystems, were selected the parameters τ av = 0.03 and ϱ = 0.375.For the z-subsystem, were selected τ av = 0.06 and ϱ = 0.125.For the ψ-subsystem, were selected τ av = 0.06 and ϱ = 0.0375.The other DSSC's parameters (for all subsystems) are: δ = 2 for k o , δ = 0.0025 for τ m , and l 0 = 0.2.
For the x, y, and ψ subsystems, the STA's gain κ 2 = 100 provokes significant oscillations when compared with the DSSC equivalent implementation, see Figure 5.For the z subsystem, in particular, this effect is attenuated since the inner control performs perfect decoupling and feedback

C. EXPERIMENTAL RESULTS
The desired trajectory for the commercial DJI M600 Pro hexacopter was created to be executed in the field next to the laboratory (a soccer field), which is free of obstacles and barriers, at the Federal University of Rio de Janeiro.The path was obtained by using the Path Sketch Interface (PSI), a Python interface with a satellite image from the area of interest that allows the users to choose the desired points, see Figure 6.Then, a Matlab script converts the georeferenced points to the east-north-up (ENU) reference system and generates a smooth trajectory version using a differentiable parametric curves approach.Finally, the controllers are developed using the Robotic Operation System (ROS) and C++.The ROS control node runs on an onboard Raspberry Pi 4 and loads the trajectory information generated by the Matlab script to execute the mission.Our main purpose here is to experimentally evaluate the DSSC scheme in a real environment with the presence of real wind disturbances while ratifying that its closed-loop behavior during sliding mode approaches the STA.
It must be highlighted that the same code implemented for all control laws works for the real-time implementation embedded in the UAV computer, as well as, in the simulator developed based on the full UAV model ( 1) and in the DJI Assistant 2 Simulator.The controllers were satisfactory tested with wind disturbance in the DJI Assistant 2 considering the wind speed of 8 m/s for the x − y axes and 2 m/s for the z axis.The results are omitted due to lack of space.
After the test in the simulator, the controllers were tested in a representative environment within the Federal University of Rio de Janeiro, a soccer field (Figure 6), on the same day (April 20, 2022) and with the same wind conditions, i.e., a moderate wind with speed ranging from 5 m/s to 8 m/s, according to the anemometer installed in the field.
It was assumed that the nominal values for the uncertain parameters are (for all channels): a n p = k n p = 2.The same DSSC's control parameters, as well as, the STA's parameters are used in all subsystems (x, y, x, and ψ).It was verified that a constant modulation function (ϱ(t) = 4) was enough to deal with the uncertainties and the relative degree one output variable σ , in (10), was implemented with l 0 = 2.
The STA control was tuned to ensure acceptable performance in the real scenario, resulting in κ 2 = 0.035 and κ 1 = 0.075.The DSSC was implemented with Figure 7 shows the closed-loop tracking performance for both controls.To put in evidence the influence of the STA's gains, we have left the gain of the STA altitude control (z axis) at the same level as in the simulation.This effect is observed at the bottom of Figure 7, where the tracking error using the STA (red line) is significantly greater than the error using the DSSC scheme (blue line).The left-bottom xy plot appearing in Figure 6 illustrates the path tracking in the xy axes.The corresponding control efforts are very similar, ratifying that the DSSC closed-loop behavior during sliding mode approaches the STA, but the curves are not shown to save space.

VI. CONCLUSION
A modification in the previous Smooth Sliding Control was proposed, named Dynamic Smooth Sliding Control (DSSC), which incorporates functions in the predictor and the averaging filter depending on the output tracking error e and its time derivative.The internal predictor and averaging filter are responsible for ensuring ideal sliding mode (in an internal variable) and smooth control effort, respectively, leading to chattering avoidance.
Global stability properties were achieved, leading to exact output regulation and exponential convergence of the output tracking error to a residual set depending on the steady-state value of the smooth filter pass-band (practical tracking).
Experimental evaluation with a commercial hexacopter corroborated that the new design based on the class of plants considered here, in fact, is robust to unmodelled dynamics and is feasible to be digitally implemented with a regular sampling rate.

A. DISCUSSION AND FUTURE WORKS
It was verified that the DSSC can be designed so that the synthesized control law during sliding mode generates a family of controllers, in particular, approximations for the standard and variable gain STA.This has motivated the complete stability analysis of the DSSC inspired by the STA analysis.
Improvement was obtained in the practical implementation where discretization, for instance, generates numerical chattering even for the relative degree one case.Since sliding motion can occur even with still large output error, so that the modulation function is also large, a time-varying passband (inversely proportional to e) resulted in a practical smoother control action when compared with the original SSC, maintaining at least the same level of tracking precision.The robustness concerning unmodelled dynamics (parasitic) was inherited from the intrinsic robustness with respect to the smooth filter.Moreover, numerical simulations with a UAV dynamic model including aerodynamics effects and inner controllers and with the DJI Assistant 2 simulator, also illustrated that the class of plants fits the application.
Investigation of alternative state-dependent functions for the DSSC, closed-loop stability analysis in the presence of parasitic or unmodelled dynamics, and a methodology to adapt the actual UAV simulation model for other types of UAVs are under development.

APPENDIX A UAV MODEL: AERODYNAMIC TERMS AND DISTURBANCES
In the UAV dynamic model (see also Figure 1), the term τ dist and the parameter J are defined as The following terms incorporate all drag effects on the UAV: where p bi is the position vector of the origin of the propeller frame relative to the origin of the body frame (represented in the body frame), and for the control allocation, M := n r i=1 (p bi × T i ) + n r i=1 τ di is the net moment, and f = n r i=1 f i is the net thrust magnitude.The aerodynamic terms are defined in what follows.The Propeller Aerodynamic Thrust is given by T i (body frame) with the thrust magnitude f i := k T i θ2 i proportional to the rotor spin rate square via Rayleigh's equation, where k T i > 0 is the thrust aerodynamic constant and θi is the i-th propeller spin rate.The Propeller Aerodynamic Drag Torque is denoted by τ di (body frame), with magnitude |τ di | = c τ k T i θ2 i , torque direction s i = sgn(τ di ) = −sgn( θi ) and aerodynamic torque constant c τ > 0. The Propeller Aerodynamic Drag Force is given by is the propeller air-relative velocity, v i is the linear velocity of the i-th propeller frame, v w is the wind velocity, both represented in the inertial frame and K F di > 0 is the propeller aerodynamic drag force matrix coefficient.Finally, F d := −RK F d R T v r ∥v r ∥ is the UAV Aerodynamics Drag Force on the Structure, where v r := v − v w is the air-relative velocity, and K F d > 0 is the structure aerodynamic drag force matrix coefficient.

APPENDIX B NORMAL FORM
In this section, the development of the normal form (3)-( 4) is provided.From ( 2), one has that or, equivalently, Reminding that d = s Now, with the change of coordinate where the last two equalities comes from (40).In addition, from (41) one can write with a η := − k i k p and obtain (3), with b η = 1.Finally, one can further write from (41) that which is precisely (4), with a p =

APPENDIX C SOME CASES OF DSSC'S SYNTHESIZED CONTROLLERS
To illustrate some of those possibilities, consider the four cases illustrated in Table 2, where for simplicity, we let k o τ av , τ m , g 1 and g 2 be functions of σ , only.Thus, ∂g 1 ∂e = ∂g 1 ∂t = 0 and, from (23), one has TABLE 2. Some special cases of synthesized controllers (PI, standard STA, and two approximations for the standard STA).For the standard STA case, since k o τ av and τ m tend to zero as σ → 0, then the DSSC's averaging filter and predictor dynamic become very fast, leading to an undesired stiff problem.
From Table 2, for the plant ( 5)-( 7), the synthesized DSSC can result in an approximation for the STA [16], [18].This approximation acts like a gain reducer near the origin of the error system state-space (σ, e), thus, improving the robustness concerning unmodelled dynamics.In what follows, we point out some remarkable features of the synthesized DSSC.
From a theoretical point of view, this synthesized approximation becomes exactly the standard STA, as δ → 0. In addition, when the pass-band of the averaging filter tends to infinity, the closed-loop dynamics with the synthesized DSSC law approaches the closed-loop dynamics with the STA, in the absence of unmodelled dynamics, as described in the approximated analysis in Section V-A2.
On the other hand, from a practical point of view, small values for δ are enough to obtain similar results as the standard STA, far away from the origin, while assuring acceptable input disturbance rejection capabilities near the origin.
In addition, it should be highlighted that the initial value of the DSSC's control effort can be arbitrary chosen (e.g., at zero) by setting the initial condition of the averaging filter.

APPENDIX D PROOF OF THEOREM 1
The proof is carried out in two parts: before and after sliding mode takes place.
Finally, one can conclude that sliding mode occurs before any closed-loop signal escapes in finite time.However, finite-time escape is not precluded after sliding mode takes place.To complete the proof, we will evoke the Small Gain Theorem.

PART B: ANALYSIS IN SLIDING MODE
From Part (a), there exists a finite time t s ∈ [0, t M ) such that, ∀t ∈ [t s , t M ), sliding mode occurs, i.e., the sliding variable σ (t) becomes identically null.
During sliding mode, the synthesized DSSC law is given by ū = ûvgsta + C s , with C s := ū(t s ) + κ 1 (t s ) φ1 (t s ) and ûvgsta in (27), leading to Now, in what follows, consider the signal and the norm bounds given in Appendix E.Then, with u = ū, defining the auxiliary variable with σ a in (64) and σa in (65), the closed-loop system during sliding mode can be written as (∀t with β 1 in (63), β 2 in (68), β e in (67), β m in (66), where we have used the fact that C s is a constant.As in [23], an additional transformation will be useful for the convergence analysis and gains design.Defining and noting that ζ1 = φ′ 1 σ and φ2 = φ′ 1 φ1 , we rewrite ( 48) and ( 49 where β e in (67), β m in (66) and α 1 and α 2 in (50).Similarly to [23], consider the Lyapunov function candidate where γ , ϵ > 0 are design constants and k p is one of the uncertain plant parameter (thus, P is an uncertain matrix).Then, one can obtain where Q := − A T P + PA .The variable gains are designed (Table 1) to assure that matrix Q − 2ϵI is positive definite, see Appendix F. Now, with Q − 2ϵI > 0 and reminding that where , φ a , δ > 0 , and the Rayleigh quotient was applied.To simplify the analysis at the cost of being more conservative and losing the capability of achieving the prescribed finite-time convergence for a residual set, we disregard this negative term In addition, from (71) and ( 70 Now, reminding that . From (58) and (60), one has the following pair of inequalities: where and we use the fact that |e|, ∥η∥ ≤ ∥x η ∥ ≤ W η 1/2 min {P η }.Now, let Wv and Wη be the solutions of the differential equations corresponding to the equalities in (61)-(62), with initial conditions Wv (t s ) = W v (t s ) and Wη (t s ) = W η (t s ).Thus, by using the Comparison Lemma [39], one has W v ≤ Wv and Now, the proof follows by using the Small-Gain Theorem [49] applied to the pair of differential equations corresponding to the equalities in (61)-(62).From Appendix G, for φ b sufficiently large so that

APPENDIX E AUXILIARY SIGNALS AND NORM BOUNDS
By considering the partitions (30) and σ a is the auxiliary signal The time derivative of the auxiliary signal σ a is σ -dependent, but can be decomposed in three signals: (i) β 2 , which is σ -dependent; (ii) β e , which is e-dependent; and β m , which is an exogenous uniformly norm-bounded timevarying signal.

APPENDIX F GAIN FUNCTIONS DESIGN
The variable gains κ 1 and κ 2 are designed so that the matrix Q, appearing in (54), satisfies Q − 2ϵI > 0. One possibility is to set which leads to One has that Q − 2ϵI is positive definite for every value of (t, e, σ ) if It is clear that inequality (74) holds if the following one is valid with γ satisfying γ k p − 4ϵ 2 > 0, k p being considered as an uncertain parameter and ρ 1 and ρ 2 being known norm bounds for α 1 and α 2 , respectively, obtained in what follows by using the available norm bounds for β 1 and β 2 .

DSSC STATE-DEPENDENT FUNCTIONS
Now, we will provide some additional restrictions to the parameters κ a , κ b and φ b , so that the DSSC's state-dependent functions k o τ av > 0 and τ m > 0 are well-defined for all finite values of σ, e.Two sufficient conditions for that are Finally, one can guarantee that the DSSC's dynamics functions are well-defined, by selecting φ b > l 0 ϵ and κ a > κ b l 0 .This is summarized in Table 1.

FIGURE 1 .
FIGURE 1.The inner and outer control topology for UAV trajectory tracking.The inner controllers are PI controller with feedback linearization and feedforward terms.The DSSC outer controller is provided in Table1.Ideally, the decoupling/coupling blocks are constant matrices depending only on the UAV geometry such that M = M c and f = f c .

FIGURE 2 .
FIGURE 2. General DSSC block diagram for arbitrary relative degree case and with generic state-dependent functions τ m (σ (t ), e(t ), t ), k o (σ (t ), e(t ), t ) and τ av (σ (t ), e(t ), t ).The predictor is given in(18) and depends on k o and τ m , while the averaging filter is given in(15) and depends on τ av .For the class relative degree one plants considered here with ẏ available for feedback, one can set τ f = 0, so that σ f = σ with σ in(10).

FIGURE 3 .
FIGURE 3. Simulations of the DSSC with the full UAV dynamic model (blue line) and with the simplified model (red line).The control efforts are in the left column, while the tracking errors are given in the right column.

FIGURE 4 .
FIGURE 4. Simulations of the DSSC with the full UAV dynamics model (solid line) and with the simplified model (dash line).The time history of k o (t ) and τ m (t ) are illustrated for the 4 degree of freedom.

FIGURE 5 .
FIGURE 5. Simulations of the DSSC (blue line) and the STA (red line) with the full UAV dynamic model.

FIGURE 6 .
FIGURE 6. Desired trajectory obtained via the developed Path Sketch Interface (PSI).

k o = 10
and δ = 0.1, for all subsystems.The control gains of the STA and the DSSC's parameters were increased in the experiments in comparison to the gains used in the DJI Assistant 2 simulator.

FIGURE 7 .
FIGURE 7. Field Test.Trajectory tracking performance under STA (dash blue) and DSSC (dot red) and the trajectory error along the three axes.The desired trajectory is illustrated in black.

J
= diag n r I xy + I bx n r I xy + I by n r I z + I bz , where the inertia tensor of the i-th propeller hub (propeller plus motor), represented in E i (the i-th propeller frame), is a diagonal matrix I i = diag I xi I yi I zi , with I xi = I yi = I xy and I zi = I z (∀i); and the inertia tensor of the UAV's structure, represented in E b , is a diagonal matrix I b = diag I bx I by I bz .

2 min
{P η } one can, subsequently, conclude that: |z| converges exponentially to a residual set of order O(1/φ b ), |σ | and |e| converges exponentially to a residual set of order O(1/φ 2 b ) and finitetime escape is avoided in all closed-loop signals.
| ≥ φ b |σ |, the one has that |z| converges to a residual set of order O(1/φ b ) and |σ | (and |e|) converges to a residual set of order O(1/φ2 b ).ACRONYMS DSSC Dynamic Smooth Sliding Control PID Proportional-Integral-Derivative SSC Smooth Sliding Control STA Super-Twisting Algorithm UAV Unmanned Aerial Vehicle VGSTA Variable Gain Super-Twisting Algorithm 44322 VOLUME 12, 2024

Table 1 .
Then, for φ b (from Table1) sufficiently large, the output tracking error is globally exponentially convergent w.r.t. a small residual set of order O(1/φ 2 , we have used a norm observer for the inverse system state norm to generate η > ∥C η η∥, modulo vanishing terms due to initial conditions.We have also considered the available norm bound function α d for the plant input disturbance d, given in (A1), and the available upper bounds kp and āp for k p and a p , respectively, both given in (A0).b ), satisfying the inequality