sensors Article A Robust Observer-Based Control Strategy for n-DOF Uncertain Robot Manipulators with Fixed-Time Stability Anh Tuan Vo 1 , Thanh Nguyen Truong 1 , Hee-Jun Kang 1, * 1 2 *



 Citation: Vo, A.T.; Truong, T.N.; and Mien Van 2 Department of Electrical, Electronic and Computer Engineering, University of Ulsan, Ulsan 44610, Korea; voanhtuan2204@gmail.com (A.T.V.); thanhnguyen151095@gmail.com (T.N.T.) School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast BT7 1NN, UK; m.van@qub.ac.uk Correspondence: hjkang@ulsan.ac.kr; Tel.: +82-52-259-2207 Abstract: In this paper, a robust observer-based control strategy for n-DOF uncertain robot manipulators with fixed-time stability was developed. The novel fixed-time nonsingular sliding mode surface enables control errors to converge to the equilibrium point quickly within fixed time without singularity. The development of the novel fixed-time disturbance observer based on a uniform robust exact differentiator also allows uncertain terms and exterior disturbances to be proactively addressed. The designed observer can accurately approximate uncertain terms within a fixed time and contribute to significant chattering reduction in the traditional sliding mode control. A robust observer-based control strategy was formulated, according to a combination of the fixed-time nonsingular terminal sliding mode control method and the designed observer, to yield global fixed time stability for n-DOF uncertain robot manipulators. The proposed controller proved definitively that it was able to obtain global stabilization in fixed time. The approximation capability of the proposed observer, the convergence of the proposed sliding surface, and the effectiveness of the proposed control strategy in fixed time were fully confirmed by simulation performance on an industrial robot manipulator. Kang, H.-J.; Van, M. A Robust Observer-Based Control Strategy for n-DOF Uncertain Robot Manipulators Keywords: uniform robust exact differentiator; nonsingular terminal sliding mode control; fixed-time control; robot manipulators with Fixed-Time Stability. Sensors 2021, 21, 7084. https://doi.org/ 10.3390/s21217084 1. Introduction Academic Editor: Andrey V. Savkin Received: 15 September 2021 Accepted: 24 October 2021 Published: 26 October 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Over the past decade, robot manipulators have drawn major attention in academia and across industries. The potential applications of the robot are wide-ranging. Currently, robots can be found working in many fields, such as deburring, welding, automotive industry, bomb detection, ocean exploration, polishing, surgery, agriculture, and so on. In these applications, robots run in components of physical interaction with the working environment. It is well-known that nonlinearities and uncertain dynamics occur widely in robot manipulators, and include unstructured uncertainties and structured uncertainties. Furthermore, exterior disturbances, payload variations, and sensor noise cannot be prevented. These issues can reduce the control performance, stability, safety, and reliability of robots. Hence, more attention should be focused on proposing efficient controllers with robust anti-uncertainty ability, fast convergence rates, small overshoot, and high accuracy. To test the effectiveness of the control methods, motion tracking control of robot manipulators is a popular topic in engineering and science. In recent years, several different control algorithms were proposed for robot manipulators. They mostly included linear control strategies and nonlinear strategies, such as the proportional–integral–derivative (PID) control [1,2], linear quadratic regulator (LQR) [3], computed torque control (CTC) [4], backstepping control [5], model predictive control (MPC) [6], and sliding mode control (SMC) [7,8], which were integrated into the motion control of the robot manipulators. Linear control strategies are strictly limited to a limited domain, leading to difficulties in scaling to most real-time applications. Nonlinear control Sensors 2021, 21, 7084. https://doi.org/10.3390/s21217084 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 7084 2 of 19 strategies could improve stability and expand the operation domain. Therefore, nonlinear control strategies frequently attract more attention than linear control strategies in controlling robotic manipulators. However, most of the mentioned nonlinear methods are highly sensitive to uncertainty terms, or require precise model parameters. These intrinsic weaknesses can be handled by applying SMC. SMC is not sensitive towards external disturbances and parametric uncertainties. It can effectively compensate for the effects of uncertain terms. Therefore, SMC has been widely implemented in real robot applications. Unfortunately, SMC only provides exponential convergence, while the control inputs involve undesirable oscillation. For the exponential convergence, the trajectory of the control errors only reaches zero once time goes to infinity. To achieve higher performance, faster convergence performance is required to match real systems. Hence, asymptotic convergence seems to be unsuitable for applications requiring high accuracy. Furthermore, oscillation also known as chattering, leads to undesirable mechanical stress on actuators and the structure of robot manipulators [9]. A great deal of effort has been devoted to the finite-time convergence guarantee of the system states. The control methods that could provide finite-time stability include the higherorder sliding mode control (HOSMC) [10,11], the terminal SMC (TSMC) [12,13], the nonsingular TSMC (NTSMC) [14,15], fast TSMC (FTSMC) [16,17], global FTSMC (GFTSMC) [18,19], the nonsingular fast TSMC (NFTSMC) [20,21], and the finite-time TSMC (FnTSMC) [22,23]. HOSMC is capable of providing finite-time stability, and chattering reduction can also be achieved by regularization of switching functions and by considering (virtual) actor dynamics as input low-pass filters. Most of the mentioned TSMC-based methods performed so far have not rigorously solved problems such as chattering or slow convergence in finite time control when the initial starting point of the system’s trajectories has a large value. In addition, these methods involve a trade-off between chattering behavior and control performance. Due to dependence on initial conditions, convergence time rises unlimitedly once those conditions go to infinity, in the theory of finite-time controllers. To minimize that dependency, fixed-time control methods were proposed [24,25]. The main advantage of fixed-time controllers is that the convergence time can be pre-computed by setting appropriate design constants, which are bounded. These controllers often exhibit excellent performance and powerful disturbance cancellation. Therefore, they are increasingly applicable to robotic systems [26,27]. It is well-known that the existence of uncertain terms in the robotic system is inevitable. Therefore, it is necessary to enhance the robustness and durability of the controllers under the influence of uncertain terms. In the literature, numerous observer-based control strategies were proposed. For example, an observer-based control strategy was proposed for fault-tolerant control (FTC) of the robot manipulators [28], and the extended observer-based synchronous SMC scheme for FTC of the robot manipulators was developed [29]. However, these observer-based controls only ensure asymptotic stability. In addition, there are a few more proposed observers, such as the high-gain observer (HGO) [30] and the third-order sliding mode observer (TOSMO) [31]. While HGO only ensures asymptotic convergence in the study [30], the proof of finite-time convergence has not been fully yielded in the study [31]. The Kalman filter (KF) is one of the most extensively applied approaches for monitoring and estimation. KF’s advantages include observability, simplicity, controllability, smoothing, optimality, and robustness [32–34]. Unfortunately, using KF for nonlinear systems can face many difficulties and problems. To apply the traditional linear Kalman filter to nonlinear systems, the most common method is to employ an Extended Kalman Filter (EKF), which simply linearizes all nonlinear models. As mentioned above, linear approaches are strictly limited to a limited domain, leading to difficulties in scaling to most real-time applications. Therefore, the application of KF to the design of robot control seems to be unsuitable. In recent years, to further enhance the accuracy and speed of perturbation estimates, SMC methods based on finite-time disturbance observers (FnDOs) [35,36] or fixed-time disturbance observers (FxDOs) [26,27] were proposed. In the studies [35,36], because the convergence time of FnDOs relies on the initial conditions, it increased indefinitely Sensors 2021, 21, 7084 3 of 19 as those conditions went to infinity. In the study [26], the authors developed an FxDO to estimate uncertain terms of nonlinear systems. However, FxDO will not effectively estimate perturbation when the data from the accelerometer is obtained to be degraded. In the works [37,38], robust exact differentiators combined with SMC were introduced to improve performance, including estimation accuracy and robustness against measurement errors and chattering phenomena reduction. Nevertheless, once the norm of the initial conditions rose unlimitedly, the convergence time of the observers/differentiators tended toward infinity. It should be noted that the observer’s convergence property in a fixed time is important for separation-like properties in robot manipulators. It implies that the observer’s estimation errors reach zero before the real trajectories of the robot have flowed to infinity. To achieve both estimation accuracy and robustness in fixed time, and to remove the dependence of the initial conditions, a uniform robust exact differentiator (URED) was proposed [39]. An arbitrary order differentiator was further developed in the study [40]. Most of the observer-based control strategies introduced so far guarantee that the estimation errors or the tracking error will approach to equilibrium point within finite time. Numerous methods achieve asymptotic convergence of both types of the mentioned errors. Some observers/differentiators only focus on estimating the unmeasurable states, and ignore the effect of uncertainty or disturbance under time-varying impacts on the robot manipulators. Moreover, because chattering is a key weakness of the SMC methods, we also need to focus on this problem. Based on the stated goal, our paper developed an observer-based control algorithm for n-DOF robot manipulators under the existence of uncertain terms. This was developed with the important contributions below, which facilitated the proposed work for real-time implementation. • • • • • The novel fixed-time nonsingular terminal sliding mode (FxNTSM) surface was proposed to quickly obtain a fixed-time convergence of the control errors without singularity. To proactively deal with uncertain terms and exterior disturbances, the FxDO was developed based on a URED. The designed FxDO accurately approximated uncertain terms within a fixed time and contributed to significantly reduced chattering in the traditional SMC. In addition, the proposed FxDO removed the requirements for measuring acceleration, as presented in high-order sliding mode (HOSM) observers [26,41]. The proposed controller had a simple design suitable for extension to actual robots. It was formed according to a combination of the fixed-time nonsingular terminal sliding mode control (FxNTSMC) method and the designed FxDO, to offer global fixed-time stability for robot manipulators. The convergence time was able to be pre-computed by setting appropriate design constants, which were bounded. The proposed controller obtained high tracking accuracy, small overshoot, chattering reduction, robust anti-uncertainty ability, and fast convergence of both the tracking errors and the estimation errors within fixed time. The proposed FxNTSMC proved definitively that it was able to obtain global stability in fixed time using the Lyapunov criteria. The arrangement of the article is presented as follows. Following the introduction, the assumptions, basic definitions, lemmas, and problem formulations are described in Section 2. The control design in Section 3 describes a novel FxNTSM surface, a novel FxDO based on a URED, and a novel FxNTSMC strategy. In Section 4, a 3-DOF industrial robot system simulated under the existence of uncertain terms is used to investigate the control performance of the suggested control strategy. Finally, notable conclusions from the proposed theory and simulation results are summarized in Section 5. To assist readers, the list of notations and nomenclature is given in Table 1. Sensors 2021, 21, 7084 4 of 19 Table 1. Notations and nomenclature. Notation Rn R+ Rn×m v ·T |·| k·k p . p .. p M(p) M̂(p) ∆M(p)
. C p, p
. Ĉ p, p
. ∆C p, p G(p) Ĝ(p) ∆G(p
) . fr p τ τd (t) A(v) Z(v) δ(v, ∆, τd ) . .. pd , pd , pd e . e1 , e2 , e2 s ̺i , ̺i , κ1 , κ2 , σ, K ℮ RMSE(·) Description the real n-dimensional space the set of positive real numbers the set of m by n real matrices given vector or matrix the transpose of absolute value of Euclidean norm of vector of joint angular position, p ∈ Rn×1 . vector of joint angular velocity, p ∈ Rn×1 .. vector of joint angular acceleration, p ∈ Rn×1 positive–definite and symmetric matrix of inertia parameters, M(p) ∈ Rn×n estimated part of M(p), M̂(p) ∈ Rn×n uncertain dynamic part of M(p), ∆M(p) ∈ Rn×n
. matrix of the Coriolis and centripetal forces, C p, p ∈ Rn×n .
.
estimated part of C p, p , Ĉ p, p
∈ Rn×n
. . uncertain dynamic part of C p, p , ∆C p, p ∈ Rn×n vectors of the gravitational force, G(p) ∈ Rn×1 estimated part of G(p), Ĝ(p) ∈ Rn×1 uncertain dynamic part of G(p),
∆G(p) ∈ Rn×1 . vectors of the friction force, fr p ∈ Rn×1 vector of the control input torque, τ ∈ Rn×1 unknown time-varying external disturbance, τd (t) ∈ Rn×1 lumped nominal part of the robot a smooth function lumped unknown uncertainty . .. desired trajectory, the first and second derivative of pd under varying time, pd , pd , pd ∈ Rn×1  T  T vector of control errors, e = e1 e2T ∈ R2n control errors, the first and second derivative of e1 under time-varying,  T  T . e1 = e11 , . . . , e1n , e2 = e21 , . . . , e2n , e2 ∈ Rn×1 . e2 is the time derivative of e1 n × 1 vector of FxNTSM surface, s ∈ R positive constants Euler’s number Root-mean-square error 2. Problem Statement, Basic Definitions, and Assumptions 2.1. Description of Robot Manipulators’ Dynamic Model A description of an n-DOF uncertain robot manipulators’ dynamic model is presented along with disturbance, as follows: .. .
. .
M(p)p + C p, p p + G(p) + fr p = τ − τd (t). (1) In fact, the dynamics of the robot involve uncertain terms with high nonlinearity, such as wear, Coulomb friction, varying payload, etc. For complete consideration, the terms .
of dynamical uncertainty are described as follows: M p = M̂ p + ∆M p , C p, p = ( ) ( ) ( ) .
.
Ĉ p, p + ∆C p, p , and G(p) = Ĝ(p) + ∆G(p). h i  T .T T Set v = v1T , v2T = pT , p and u = τ; accordingly, the dynamic model of the robot (1) is depicted in state space by:  . v1 = v2 . , v2 = Z(v)u + A(v) + δ(v, ∆, τd ) (2)
.
. −1 −1 −1 where A(v) = −M̂ (p) Ĉ
p, p p + Ĝ(p) ,
Z(v) = M̂ (p), and δ(v, ∆, τd ) = −M̂ (p)
. .. . . fr p + ∆M(p)p + ∆C p, p p + ∆G(p) + τd . Sensors 2021, 21, 7084 5 of 19 . . . .. .. Define e1 , p − pd , e2 , p − pd , and e2 , p − pd , so, the system (2) can be formulated with the form involved the control errors:  . e1 = e2 . , (3) e2 = Z(v)u + H(v) + δ(v, ∆, τd ) T  −1 ∈ R2n indicates vector of control errors and H(v) = −M̂ (p) where e = e1T e2T

.. . . Ĉ p, p p + Ĝ(p) − pd represents the smooth nonlinear function. 2.2. Basic Definitions and Assumptions Lemmas, definitions, and assumptions are necessary for the design procedure of the proposed controller and proof of convergence and stability in finite time or fixed time. The sign(·) function is described with the following expression:   1 if v > 0 . sign(v) = 0 if v = 0  −1 otherwise It can be clearly confirmed that as ϕ ≥ 0  sig(v) ϕ = |v| ϕ sign(v) ϕ ϕ −1 . . d v dt sig( v ) = ϕ | v | Assumption 1: The system states of Equation (1) for controls are bounded for all time. Assumption 2. Assume that the lumped unknown uncertainty at each joint is bounded by: |δi (v, ∆, τd )| < ̺i , (4) where ̺i is a positive constant. Assumption 3 ([42]). Assume that the first derivative of the lumped unknown uncertainty at each joint is bounded by: . δi (v, ∆, τd ) < ̺i , (5) where ̺i is a positive constant. Let us consider autonomous system as follows: . v(t) = f(v(t)), v(0) = v0 , (6) where v ∈ Rn and f: Rn → Rn is a nonlinear function. Let us assume that the origin is an equilibrium point of Equation (6). Definition 1 ([27]). The equilibrium point of Equation (6) is considered to be a finite-time stable equilibrium if the origin is Lyapunov stable, and any solution v(t) starting from vo satisfies lim v(t, v0 ) = 0 for all t ≥ T (v0 ), where T: Rn → R+ is called the settling time function. v→∞ Definition 2 ([27]). The equilibrium point of Equation (6) is considered to be a fixed-time stable equilibrium if it is globally finite-time stable and its bounded convergence time T (v0 ) < Tmax , where Tmax > 0 is a positive number. Lemma 1 ([43]). Let us consider the scalar differential equation, as follows: . L(v) = −κsig(L(v))α , (7) Sensors 2021, 21, 7084 6 of 19 where κ > 0, 0 < α < 1. Then, the origin is a finite-time-stable equilibrium of Equation (7), and the settling time T is satisfied by the following inequality: T1 ≤ 1 L1−α (v(0)). κ (1 − α ) (8) Lemma 2 ([27]). Let us consider a scalar differential equation, as follows: . L(v) = −Z1 sig(L(v))| − Z2 sig(L(v))m , (9) where Z1 , Z2 > 0,| > 1, and 0 < m < 1. Then, the system (7) is globally fixed-time stable and the convergence time T2 is given by: T2 < 1 1 1 1 + . Z1 | − 1 Z2 1 − m (10) 3. Robust Observer-Based Control Strategy for n-DOF Uncertain Robot Manipulators with Fixed-Time Stability In this section, a robust observer-based control strategy for n-DOF uncertain robot manipulators with fixed-time stability is developed. Firstly, the novel FxNTSM surface is proposed to quickly obtain a fixed-time convergence of the control errors without singularity. Secondly, to proactively deal with uncertain terms and exterior disturbances, the FxDO is developed based on a URED. The designed FxDO accurately approximates uncertain terms within a fixed time and contributes to significant chattering reduction in the traditional SMC. Finally, a robust observer-based control strategy is formed according to a combination of the FxNTSMC method and the FxDO, to offer global fixed-time stability for n-DOF uncertain robot manipulators. 3.1. Proposal of the FxNTSM Surface To attain the fixed-time convergence of the control errors in system (3) without singularity, the novel FxNTSM surface was developed as: γ s = ς + Γe2 , (11)

γ γ T 2 γ , Γ = diag Γ1 , . . . , Γn , Γi > 0, and ς = [ 1 + e11 e21 , . . . , e2n  
2 γ arctan( e ) ] T . γ is a number that can change according to the arctan(e11 ), . . . 1 + e1n 1n following relation:     η q q η γ = 0.5 + + 0.5 − + sign(|e1i | − 1). (12) q η q η ( q η | e1i | > 1 It is noted that γ = in which η and q are positive odd integers and they η q | e1i | ≤ 1 γ where e2 =  η γ are chosen along with the condition 1 < q < 2, hence, e2 ∈ R ∀e2 ∈ R. This precludes the generation of complex values. As a result, the proposed sliding surface has no singularity. Theorem 1. Applying the novel FxNTSM surface in Equation (11), the trajectories of the control errors e1i , (i = 1, · · · , n) will be approached to zero in fixed time ts . Sensors 2021, 21, 7084 7 of 19 Proof of Theorem 1. Once the sliding motion of the proposed FxNTSM surface in Equation (11) occurs and satisfies the condition si = 0, (i = 1, · · · , n) , a set of the following differential equations is also attained:  2 1 + e1i γ γ arctan(e1i ) + Γi e2i = 0, i = 1, . . . , n. (13) For i = 1, . . . , n, Equation (13) can be rewritten as:   1 1 . 2 1 + e1i arctan(e1i ) γ = −Γiγ e1i . (14)   −1 . . 2 e1i . x = 1 + e1i (15) Let x = arctan(e1i ), hence, the derivative of x is:
. . 2 x, Equation (14) yield: With e1i = 1 + e1i 1 1 . x γ + Γiγ x = 0. (16) The below two cases are considered. q The first case: |e1i | > 1 → γ = η : The initial starting point of the system’s trajectories is set far from the equilibrium point |e1i | > 1, Equation (16) becomes: η −q . x = − Γi η xq. (17) Lyapunov function L1 = 0.5x2 is considered. The first-order time derivative of Lyapunov function, L1 , is now calculated along with the obtained result in Equation (17) as: . . η −q . L1 = x x = − Γ i η xq +1 . (18) . Obviously, L1 > 0 and L1 < 0. As a result, x and x asymptotically stabilize to the equilibrium point. In stage e1i (0) → |e1i | = 1 then, x (0) → | x | = π4 . Hence, the sliding motion takes place in the following computation time: η R x (0) − η q x q d(| x |) dt = Γ π 0 i 4 η R x (0) − η q t1si = Γi π x q d(| x |) 4  η 
1− η x (0) q q q = Γi q−η x − π4 R t1si η q ≤ Γi q q−η  η
η π 1− q 2 π 4 − η π 1− q 4  (19) . The second case: |e1i | ≤ 1 → γ = q : The initial starting point of the system’s paths is near the designated path |e1i | ≤ 1, so Equation (16) is written as: . q −η x = − Γi q xη . (20) Sensors 2021, 21, 7084 8 of 19 Selecting Lyapunov function L2 = 0.5x2 , the first-order derivative of L2 according to time combining with Equation (20) results:   q q . − L2 = x − Γ i η x η − q q +1 (21) = − Γi η x η q √ ηq +1 − ηq ( η 2+1) Γ i L2 =− 2 .  η q η q q η +1 
The term is chosen along with the condition 1 < < 2/ As a result, 2 ∈ 34 , 1 . With the statement presented in Lemma 1, we conclude that x can reach origin in finite-time. Due to x = arctan(e1i ), e1i can approach zero in finite-time with computation time below: q η t2si ≤  Γi 1− q η  |arctan(e1i )| q 1− η q η ≤  Γi 1− q η   π 1− ηq 2 . (22) From Equations (19) and (21), the convergence time that occurs in sliding motion is given below: ts = max {tsi } ≤ max {t1si + t2si }. (23) 1≤ i ≤ n 1≤ i ≤ n The value ts stated in Equation (23) only relates to the design constants. Consequently, the control errors e1i will surely attain the equilibrium point in fixed time. This proof is completed.  3.2. Design of a Fixed-Time Disturbace Observer The lumped uncertainty is approximated by an FxDO. This observer is designed based on a URED, as follows:    δ. 0 = v̂2 − v2 v̂2 = Z(v)u + A(v) + δ̂ − κ1 ψ1 (δ0 ) , (24)   . δ̂ = −κ2 ψ2 (δ0 ) where v̂2 indicates an approximated value of v2 , and κ1 and κ2 are observer gains. The terms ψ1 (δ0 ) and ψ2 (δ0 ) are designed based on URED in [39], as follows: ( 3 1 ψ1 (δ0 ) = |δ0 | 2 sign(δ0 ) + ̟ |δ0 | 2 sign(δ0 ) . (25) ψ2 (δ0 ) = 12 sign(δ0 ) + 2̟δ0 + 23 ̟ 2 |δ0 |2 sign(δ0 ) . Theorem 2. Applying the proposed observer in Equation (24), when the term δi (v, ∆, τd ) < ̺i in Assumption 2 is satisfied, then the estimation error of the proposed observer will converge to zero in fixed time, independent of the initial condition and exterior disturbances. Proof of Theorem 2. Differentiating the first order of δ0 , one obtains: . δ0 . . = v̂2 − v2 = δ̂ − δ − κ1 ψ1 (δ0 ) = δ1 − κ1 ψ1 (δ0 ), where δ1 = δ̂ − δ represents the estimation error and δ1 = by a known constant |δ1i | ≤ B , i = 1, · · · , n. (26)  δ11 , · · · , δ1n T is limited Sensors 2021, 21, 7084 9 of 19 Taking the first-order time derivative of δ1 and referring to Equation (24), we can gain: . δ1 H= ( . . = δ̂ − δ . = −κ2 ψ2 (δ0 ) − δ. (27) The observer gains κ1 and κ2 are ) selected in the set, as follows: n o 2 2 √ √ κ 4λ (28) (κ1 , κ2 ) ∈ R2 0 < κ1 ≤ 2 λ , κ2 > 1 + 2 ∪ (κ1 , κ2 ) ∈ R2 κ1 > 2 λ , κ2 > 2λ , 4 κ1  in which λ = max ̺1 , ̺2 , · · · , ̺n . Referring to Equation (3) in the study [39], it is seen that Equations (26) and (27) are given . the same form. In addition, the estimated value δ is bounded by Assumption 2 and corre.. sponds to f 0 in the study [39], as stated in Equation (3). As a result, the proposed observer will exactly estimate the lumped uncertainty when we can achieve δ1i = 0, (i = 1, · · · , n) within the fixed time, T0 as observe in [39], and the convergence time of the proposed observer is calculated by assigning κ1 , κ2 , and ̟ for any initial conditions (readers can refer to Equations (5)–(9) and Appendix A in [39]). Consequently, we can conclude that using the proposed observer in Equation (24) with the suitable conditions, we can exactly estimate the lumped uncertainty in fixed-time, independent of the initial condition, and despite disturbances. This proof is completed.  3.3. Design of a FxNTSMC Method Computing the first-order derivative of FxNTSM surface according to time, we gain:  . . . γ −1 γ −1 s = ς + γΓdiag e21 e2 , . . . , e2n   . γ −1 γ −1 (29) = ς + γΓdiag e , . . . , e (Z(v)u + H(v) + δ(v, ∆, τd )) . 2n 21 = ς + γΦ(Z(v)u + H(v) + δ(v, ∆, τd )),   γ −1 γ −1 where Φ = diag(Φ1 , · · · , Φn ) = Γdiag e21 . , . . . , e2n Then, the control torques of FxNTSMC are designed based on the proposed FxNTSM Surface in Section 3.1 and the proposed FxDO in Section 3.2, as follows:     u = −Z−1 (v) H(v) + F + δ̂ − Z−1 (v) K|s| β sign(s) + B sign(Φs) , (30) where β > 2, F=  2 ) (1+e11 Γ1 γ γ −1 2− γ (1 + 2γe11 arctan(e11 ))e21 , ..., 2 ) (1+e1n Γn γ γ −1 2− γ (1 + 2γe1n arctan(e1n ))e2n T . Theorem 3. If the control torques of FxNTSMC are designed for robot manipulators (1) based on the proposed FxNTSM Surface in Equation (11) and the proposed FxDO in Equation (24) which is given in Equation (30) then the proposed controller offers global fixed-time stability for robot manipulators. Proof of Theorem 3. Inserting the control input (30) into Equation (29) obtains   . s = γΦ −K|s| β sign(s) − B sign(s) − δ1 . A set of the differential equations from Equation (31) is described as:   . si = γΦi −K|si | β sign(si ) − B sign(si ) − δ1i , i = 1, · · · , n. (31) (32) Sensors 2021, 21, 7084 10 of 19 The Lyapunov function candidate is defined as L3i = 0.5s2i i = 1, 2, . . . , n. Then, differentiating Lyapunov function gives: . L3i . = si si   = si γΦi −K|si | β sign(si ) − B sign(si ) − δ1i   = γ −K Φi |si | β+1 − B Φi |si | − Φi si δ1i (33) ≤ −γK Φi |si | β+1 − γ(B − |δ1i |)Φi |si | ≤ −γK Φi |si | β+1 − µ|si | ≤ 0, where µ = γ(B − |δ1i |)Φi > 0. The suggested FxNTSM surface will be attained the equilibrium point in finite-time tr . It means that the convergence and stability of the designed control strategy are guaranteed in finite time t = tr + ts . We will prove that Equation (33) is fixed-time stable. Therefore, Equation (33) is 𝑡 rewritten as follows: 𝑡 = 𝑡 + 𝑡 . L3i ≤ −2 β +1 2 β +1 1 1 β +1 1 γK Φi L3i2 − 2 2 µL3i2 = −Z1 L3i2 − Z2 L3i2 , β +1 (34) 1 𝛾𝒦𝛷 −𝒵 ℒ ≤ 0−2 Φi > , Z2 = 2 2 µℒ> 0.− 2 𝜇ℒ = −𝒵 ℒ where Z1 = 2 2 γK ℒ β +1 Due to Z1 , Z2 > 0 , and 2 > 1 , based on Lemma 2, the convergence time of the > 0 by 𝒵 =2 𝜇>0 = 2 is𝛾𝒦𝛷 𝒵 phase reaching bounded 𝒵 ,𝒵 > 0 >1 2 2 1 tri ≤ . (35) + Z1 β − 1 Z2 𝑡 ≤ + 𝒵 𝒵 From Equations (23) and (35), it can be concluded that the proposed control system can also obtain convergence and stability within the following fixed time: t = tr + ts = max {tri } + max {t1si + t2si }. i≤n 𝑡 1≤imax ≤n 𝑡 +1≤max +𝑡 𝑡 =𝑡 +𝑡 = This proof is completed. □  The design procedure of the proposed controller is briefly explained in Figure 1. Figure 1. Block diagram of the proposed control system. (36) Sensors 2021, 21, 7084 11 of 19 4. Illustrative Example Figure 2 shows a 3D Description of a 3-DOF manipulator based on SOLIDWORKS. Figure 2. 3D Description of a 3-DOF manipulator based on SOLIDWORKS. The kinematic design and dynamic computation of the robot system were conducted based on the PUMA 560 manipulator [44,45]. To facilitate the presentation of simulation performance, the manipulator is designed with three degrees of freedom (DOF). In this paper, SOLIDWORKS was used to design the robot manipulator parts, the structure of the robot, the addition of the coordinate system, the measuring devices, and the direction of gravitational force. Each mechanical component of the robot system was constructed separately and assembled using suitable joints. Using the Simscape Multibody Link Tool from SOLIDWORKS, we created two types of files. The XML file included important parameters of the robot’s mechanical components and parameters of the coordinate system of the assembly environment, such as the center of mass, length of link, mass, inertia moment, etc. The detailed design parameters of the robot can be found in Table 2. The STEP files comprised the 3-D computer-aided design (CAD) model of the mechanical parts. To achieve a realistic model when performing simulations, both file types were included in the MATLAB/Simulink environment via Simscape Multibody Link. Furthermore, the lumped uncertainty, including uncertain dynamics, exterior disturbances, and friction forces was assumed to add to the robot manipulator. The mechanical model of the robot in SOLIDWORKS was the same as the real robot model. In addition, the simulated environment of the robot was considered to be the same as the real conditions. Therefore, it was determined that the SOLIDWORKS model of the robot was able to be used to validate the control performance effectively. Table 2. The detailed design parameters of the robot. 𝑚 33.429 𝑚 34.129 Description Symbol 15.612 𝑚 m1 250 𝑙 m2 Mass of each link 𝑙 700 m3 600 𝑙 l1 Length l2 0,0, −74.610 𝑙 ,of 𝑙 link ,𝑙 𝑙 ,𝑙 ,𝑙 ,𝐼 ,𝐼 Center 𝑙 of , 𝑙 mass ,𝑙 𝐼 𝐼 ,𝐼 Inertia ,𝐼 Value 33.429 34.129 15.612 250 700 600 l3   T 347.7,0,0 [0, 0, −74.610] T l ,l ,l  c1x c1y c1z  T 314.2,0,0 [347.7, 0, 0] T l ,l ,l  c2x c2y c2z  T [314.2, 0, 0] T lc3x , lc3y0.7486,0.5518,0.5570 , lc3z  T [0.7486, 0.5518, 0.5570] T I , I 0.3080,2.4655,2.3938 ,I  1xx 1yy 1zz  T [0.3080, 2.4655, 2.3938] T I ,I ,I  2xx 2yy 2zz  T [0.0446, 0.7092, 0.7207] T I3xx , I3yy , I3zz kg kg kg mm mm mm mm mm mm kg. m kg. m Unit kg kg kg mm mm mm mm mm mm kg·m2 kg·m2 kg·m2 Sensors 2021, 21, 7084 12 of 19 As we know, there exists a discrepancy between the real model
and the calculated . model. To simulate these model errors, we included ∆M(p), ∆C p, p , ∆G(p). Throughout the
simulation, the unidentified dynamics were assumed to be ∆M(p) = 0.2M(p), . .
∆C p, p = 0.2C p, p , and ∆G(p) = 0.2G(p). To test the robustness of the proposed control strategy, the friction force and exterior .
. .
τd1 (t) = 0.1sign p1 + 2 p1 + 2.5 sin(10t disturbance were assumed at each joint as fr1 p +

. . . . . − 20) p1 −
2.2p31 (N·m), fr2 p + τ
d2 (t) = 0.1sign p2 + 2 p2 + 2.3 sin(10t − 20) p2 − 4.2p32 (N·m), . . . . and fr3 p + τd3 (t) = 0.1sign p3 + 2 p3 + 3.5 sin(10t − 20) p3 + 3.2p33 (N·m). To evaluate the motion control of the robotic manipulator when approaching and maintaining a specified path, the configuration of the trajectory was designed in the form of a circle in XYZ coordinate system, as follows: X = 0.85 − 0.01t (m), Y = 0.2 + 0.2 sin(0.5t) (m), Z = 0.7 + 0.2 cos(0.5t) (m), and t ≤ 20 s. The selected reference trajectory was a circle in three-dimensional spaces. This meant that the amplitude in the YZ direction of this reference trajectory changed over time for a given periodicity, and the amplitude in the X direction of this reference trajectory changed linearly over time. Therefore, it served as a general trajectory for verifying tracking control. In addition, to check aspects of any initial conditions (|e1i | > 1 and |e1i | ≤ 1), the manipulator was configured with the initial starting points at each joint as: p1 = −1.6 (rad), p2 = −1 (rad), and p3 = −0.5 (rad). In comparison, other state-of-the-art controllers, including NFTSMC1, based on the method of [46], and NFTSMC2, based on the method of [47], have been considered to compare with the proposed controller in aspects such as: convergence rate, chattering, robustness to cope with uncertain terms, and accuracy in tracking control. The control torques of NFTSMC1 were constructed for the manipulator as:                 . s = e + ϕ1 e + φ1 |e|ω1 sign(e) . , u = −Z−1 (v)(H(v) + ( ϕ1 + φ1 ω1 |e|ω1 −1 )e + K1 s + (̺ + B)sign(s)) where ϕ1 , φ1 , K1 are the design positive constants, 0 < ω1 < 1. The control torques of NFTSMC2 were designed for the manipulator as: . 2φ 2ϕ2 e + 1+℮σ2 (|e2 |−ε ) |e|ω2 sign(e) s = e+ 2 (|ei |−ε 2 ) 2 i 1+℮−θ  . . 2ϕ2 θ2 e sign(e)℮−θ2 (|ei |−ε 2 ) 2ϕ2 2φ2 ω2 ω2 − 1 . e e e + H( v + e + ) | |   2   1+℮−θ2 (|ei |−ε 2 ) 1+℮σ2 (|ei |−ε 2 ) 1+℮−θ2 (|ei |−ε 2 )   u = − Z−1 (v )  . σ (|e |−ε )  2φ2 σ2 e℮ 2 i 2 ω2   − + K s + (̺ + B) sign(s)  |e| 1+℮σ2 (|ei |−ε 2 ) 2 (37) (38) 2   1 ( 1 − ω2 ) φ where ϕ2 , φ2 , θ2 , σ2 , K2 are the design positive constants, 0 < ω2 < 1, ε 2 = ϕ22 . Three control systems were applied to stabilize the manipulator (1); their control parameters are presented in Table 3. The proposed FxNTSMC was developed based on the proposed FxDO, hence, the FxDO parameters are also reported in Table 3. Table 3. Design parameters of the three control systems. Control Method Control Parameter NFTSMC1 ϕ1 = 5, φ1 = 5, ω1 = 0.8, K1 = 5, ̺ = 13, B = 0.1 NFTSMC2 Proposed FxNTSMC ϕ2 = 5, φ2 = 5, θ2 = 0.9, σ2 = 1.2 ω2 = 0.8, K2 = 5, ̺ = 13, B = 0.1 q = 3, η = 5, Γ = diag(0.4, 0.4, 0.4), K = 5, B = 0.1 κ1 = diag(18, 18, 18), κ2 = diag(180, 180, 180), ̟ = diag(2, 2, 2) Sensors 2021, 21, 7084 13 of 19 For the convenience of accuracy comparison, the root-mean-square errors were calculated from the 2nd sec to the 20th s, as described in Table 4. Table 4. Root-mean-square errors. Control Method Root-Mean-Square Errors from the 2nd s to the 20th s
RMSE(ez ) RMSE(e1 ) RMSE(e2 ) RMSE(e3 ) 0.1 × 10−3 0.11 × 10−2 0.3744 × 10−3 0.3 × 10−3 0.1582 × 10−3 0.7 × 10−3 0.6240 0.3 × 10−3 0.1 × 10−3 0.3310 × 10−3 0.0753 × 10−7 0.1363 × 10−7 0.1553 × 10−7 0.1595 × 10−7 0.1597 × 10−7 0.1476 × 10−7 RMSE(e x ) RMSE ey NFTSMC1 0.6241 0.9 × 10−3 NFTSMC2 Proposed FxNTSMC Remark 1. For an optimal choice of control parameters, while guaranteeing fairness between control strategies, several choice methods were applied to attain good tracking performance for all three control strategies in the aspects of fast convergence speed, high tracking precision, stability, and chattering reduction. The control parameter selection of the proposed sliding surface ensured the conditions presented below Equations (11) and (12) attained the fixed-time convergence of the control errors in system (3) without singularity. The observer gains κ1 and κ2 were selected in the set, as stated in Equation (28). The formula (̺ + B) is the sliding gain of the reaching control law in Equations (37) and (38). These parameters are assigned a value greater than the upper-bound value of the lumped unknown uncertainty. Therefore, this condition guarantees asymptotic stabilization for the control system. With the selection of other control parameters of all three controllers, the reader can easily find instructions or conditions presented below the equations of the control signal. Furthermore, the selection of control parameters is performed by repeated experiments to get the optimal control parameters. The effectiveness of the proposed FxDO is firstly considered in order to evaluate its approximation capability. As shown in Figure 3, the trajectory of the observed velocity completely coincided with the trajectory of the measured velocity from the sensor at the initial time, and remained until the end of the simulation time. In Figure 4, we note that the proposed FxDO exactly approximated the supposed value of the lumped uncertainty at each joint in two aspects: amplitude and frequency. The estimation errors of the proposed FxDO converged to zero within the fixed time. The convergence property of the observer in a fixed time is important for separation-like properties in the robot manipulator. It implies that the estimation errors of the observer reach zero before the real trajectories of the robot have flowed to infinity. Consequently, it provides timely and accurate information about the uncertain terms to the control system, and this plays a major role in enhancing robustness against uncertainty and reducing the dynamic computation burden. The control performances of the three different control strategies for a 3-DOF uncertain robot manipulator are shown in Figures 5–7. In Figure 5, it can be seen that the initial point of the end effector of the robot was designed far from the designated path for investigating fixed-time convergence with arbitrary initial conditions. The control performances in Figure 5 show that all three controllers guaranteed high tracking accuracy for the robot, while the actual path under the suggested control strategy reached the designated path with the greatest rapidity, due to dynamic coefficients designed in the FxNTSM surface that could be adjusted to the control errors, as stated in Section 3.1. Sensors 2021, 21, 7084 14 of 19 Figure 3. Measured value of velocity 𝑥x𝑥2 and observed value of velocity x̂𝑥2𝑥at each Joint. Figure 4. Supposed value of the lumped uncertainty and observed value of the lumped uncertainty at each Joint. Sensors 2021, 21, 7084 15 of 19 Figure 5. Specified path and actual path of the end effector of the robot under three control strategies in 3-dimensional space (XYZ). Figure 6. Path of the control errors in 3-dimensional space (XYZ). Sensors 2021, 21, 7084 16 of 19 Figure 7. Path of the control errors at each joint. Performing a detailed comparison of control errors in Figures 6 and 7, and using a quantification method for the root-mean-square errors, as reported in Table 4, it can be easily observed that NFTSMC2 offered better tracking accuracy than NFTSMC1. It is noteworthy that the suggested observer-based control algorithm with robust anti-uncertainty ability provided the highest tracking accuracy compared to the two remaining control strategies; it greatly improved the control performance with excellent accuracy and small overshoot. The control errors in the proposed observer-based controller converged fastest to the equilibrium point in fixed time. NFTSMC1 and NFTSMC2 were accorded same the sliding value to cope with the effects of the lumped uncertainty. Therefore, both controllers provided nonsmooth control signals with high-frequency oscillation. Meanwhile, by feeding the information of the uncertain terms accurately to the control loop from the proposed FxDO, the performance of the controller was not only significantly improved, but the chattering phenomenon was also effectively reduced, as shown in Figure 8. Sensors 2021, 21, 7084 17 of 19 Figure 8. Control torque of the three different control strategies and comparison of chattering phenomena at each Joint. 5. Conclusions Our paper developed an observer-based control algorithm for n-DOF uncertain robot manipulators with important contributions as follows: The proposed FxNTSM surface guaranteed that it obtained fixed-time convergence of the control errors without singularity. The designed FxDO based on a URED accurately approximated uncertain terms within a fixed time, and contributed to a significant chattering reduction in the traditional SMC. In addition, the proposed FxDO removed the requirements for measuring acceleration. The proposed controller has a simple design suitable for application in actual robots. The design was formulated according to a combination of the FxNTSMC method and the designed FxDO to offer global fixed-time stability for robot manipulators. The convergence time was bounded, and it could be pre-computed by setting appropriate design constants. Through the quality evaluation of the control performance and comparisons, the proposed controller obtained high tracking accuracy, small overshoot, chattering reduction, robust anti-uncertainty ability, and fast convergence of both the tracking errors and the estimation errors within fixed time. In addition, the proposed FxNTSMC was proven to obtain global stability in fixed time using the Lyapunov criteria. Following this work, we plan to propose an FTC for robotic manipulators, which will consider faults in the measuring sensors. In addition, the proposed controller will also be applicable in real robot manipulators. Author Contributions: Conceptualization, methodology, validation, writing—original draft preparation, and writing—review and editing, A.T.V.; software, visualization, and resources, T.N.T. and M.V.; supervision, funding acquisition, and project administration, H.-J.K.; formal analysis, investigation, and data curation, M.V., T.N.T. and H.-J.K. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Ministry of Education (NRF-2019R1D1A3A03103528). Sensors 2021, 21, 7084 18 of 19 Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data sets generated and/or analyzed during the current study are available from the corresponding author on reasonable request. 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