ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] Computers and Electrical Engineering 0 0 0 (2017) 1–18 Contents lists available at ScienceDirect Computers and Electrical Engineering journal homepage: www.elsevier.com/locate/compeleceng Backstepping terminal sliding mode control of robot manipulator using radial basis functional neural networksR M. Vijay, Debashisha Jena∗ Department of Electrical and Electronics Engineering, National Institute of Technology Karantaka, Surathkal, Mangalore 575025, India a r t i c l e i n f o Article history: Received 30 April 2017 Revised 28 October 2017 Accepted 7 November 2017 Available online xxx Keywords: Robot manipulator Neural network (NN) Backstepping terminal sliding mode control Adaptive control Position tracking Disturbance rejection a b s t r a c t This paper examines an observer-based backstepping terminal sliding mode controller (BTSMC) for 3 degrees of freedom overhead transmission line de-icing robot manipulator (OTDIRM). The control law for tracking of the OTDIRM is formulated by the combination of BTSMC and neural network (NN) based approximation. For the precise trajectory tracking performance and enhanced disturbance rejection, NN-based adaptive observer backstepping terminal sliding mode control (NNAOBTSMC) is developed. To obviate local minima problem, the weights of both NN observer and NN approximator are adjusted off-line using particle swarm optimization. The radial basis function neural network-based observer is used to estimate tracking position and velocity vectors of the OTDIRM. The stability of the proposed control methods is verified with the Lyapunov stability theorem. Finally, the robustness of the proposed NNAOBTSMC is checked against input disturbances and uncertainties. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Transmission line icing is one of the most significant factors that affects safety and reliability of a power system. The adverse impact of icing on transmission lines can result in tower collapse, unnecessary tripping and power outages. Robot manipulator de-icing technology can mitigate icing problems without interrupting power supply on transmission lines. The exact dynamics of the robot manipulator are not available to design a perfect controller. This is due to the nonlinearities, model uncertainties, elasticity, cross coupling and frictional effects that increase the system complexity. One of the main objectives of a robotic manipulator controller is to generate a control signal that helps to track the reference trajectory [1,2]. Nonlinear control techniques, such as sliding mode control (SMC), artificial intelligence based adaptive control techniques help to enhance the performance of conventional control techniques. Many researchers have analyzed that SMC is one of the best nonlinear controllers that provides fast response in terms of trajectory tracking and disturbance rejection [3–6]. In the conventional sliding mode control, the convergence of the state is usually asymptotic due to the linearity of the switching plane. However, this convergence can only be achieved in infinite time, although the SMC parameters can be adjusted to make convergence faster. For high-precision control systems, faster convergence is the priority and can only be achieved at large control inputs. These large control inputs can lead to saturation of the actuator. The terminal sliding mode control (TSMC) includes nonlinear function in the outline of the sliding hyperplane. By using a nonlinear sliding surface, R ∗ Reviews processed and recommended for publication to the Editor-in-Chief by Associate Editor Dr. A. Chaudhary. Corresponding author. E-mail addresses: mokenapalli.vijay@nitk.edu.in (M. Vijay), djena.2490ee@nitk.edu.in (D. Jena). https://doi.org/10.1016/j.compeleceng.2017.11.007 0045-7906/© 2017 Elsevier Ltd. All rights reserved. Please cite this article as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot manipulator using radial basis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 2 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 TSMC enables the rapid convergence of the state without the need for an extensive control action. A nonsingular terminal sliding mode manifold is used to design a chattering free adaptive control scheme for the robot manipulator [7]. In [8], an adaptive TSMC regulated a 2-link robot manipulator to track the variable signals. However, the significant drawback of the TSMC is chattering phenomenon that includes high-frequency oscillations due to discontinuous control signals. The combinations of artificial intelligence with SMC techniques to control the nonlinear systems are addressed in [9–12]. In [13], a new adaptive backstepping SMC (ABSMC) scheme with fuzzy monitoring technique is discussed for the desired trajectory tracking control of nonlinear systems. The neural network (NN) is utilized to approximate or estimate uncertainties and disturbances of the unknown model in [14,15]. Authors in [16], have developed control methodology based on outputfeedback to keep up the predefined control execution for a class of uncertain MIMO nonlinear systems. In [17], a versatile control based radial basis neural network (RBFNN) is designed for a variable-pitch and variable-speed wind turbine. Fuzzy logic controller (FLC) based methodologies have been developed for the robot manipulator in the presence of structured and unstructured disturbance conditions [18,19]. The propulsive positioning and on-line levitated balancing of a hybrid magnetic levitation are achieved by fuzzy neural based backstepping SMC scheme in [20]. Several researchers proposed disturbance observer (DOB) for SMC to mitigate the chattering phenomenon and retain its desirable control behavior under the presence of mismatched uncertainties [21–23]. In this paper, we discuss different control methods of the 3 degrees of freedom (DOF) overhead transmission line de-icing robot manipulator (OTDIRM) to eliminate the effects of disturbance and uncertainty associated with direct measurements. The designing of a new controller considers the combination of modified backstepping TSMC (BTSMC) with NN identifier and NN observer. The optimal weights of the NN observer, NN identifier, and the BTSMC parameters are obtained with the help of particle swarm optimization (PSO). Estimated position and velocity vectors of the RBFNN based observer are fed to another RBFNN based identifier to approximate the auxiliary control input torque to the de-icing robot manipulator. The chattering effect is mitigated by utilizing boundary layer phenomenon, and the Lyapunov stability test ensures the stability of the anticipated control strategy. In [5], the authors discussed conventional SMC based on the chemical reaction optimization and radial basis functional link net (CRLSMC) for the de-icing robot manipulator to achieve desired trajectory tracking performance under various operating conditions. In [12], authors have designed a conventional SMC scheme based on NN for better trajectory tracking of the mobile robot manipulator. The position control of the 2-link robot manipulator is designed by considering the combination of conventional SMC and NN based observer [14]. In [15], the wavelet neural network (WNN) based controller is designed for the de-icing robot manipulator without consideration of an observer-based control structure. This paper mainly differs from [5], [12], [14] and [15], by replacing conventional SMC with modified backstepping TSMC scheme in combination with PSO based observer and identifier. Several performance methods are examined to show the effectiveness of the proposed control technique. The structure of this paper is dealt with as follows: Section 2 addresses the description of the robot manipulator. It also demonstrates the design of BTSMC and NNBTSMC with stability analysis. NN based versatile observer framework and control designs are discussed in Section 3. Section 4 presents simulation verification. Finally, Section 5 concludes the paper. 2. Description of controlling sytems Dynamic equation of the robot manipulator with n-DOF can be characterized as: D(q )q̈ + C (q, q˙ )q˙ + G(q ) + F (q, q˙ ) + τd = τ (1) where q, q˙ and q̈ ∈ Rn are the link position, velocity and acceleration vectors respectively, D(q) ∈ Rn × n is the symmetric positive definite inertia matrix, C (q, q˙ )q˙ ∈ Rn×n is the Coriolis or centrifugal forces, G(q) ∈ Rn × 1 consolidates the gravitational force, F (q, q˙ ) ∈ Rn×1 incorporates the friction terms and τ d represents external disturbances [24]. Dynamic equation (1) can be composed as: q̈ = D−1 (q )[τ − (C (q, q˙ )q˙ + G(q ) + F (q, q˙ ) + τd )] (2) 2.1. Design of BTSMC The backstepping methodology is a nonlinear scheme generally utilized as a part of controller design. The mathematical model of the robot manipulator is expressed in (3)–(5) as: x˙ 1 = x2 (3) x˙ 2 = q̈ = D−1 (q )[τ − (C (q, q˙ )q˙ + G(q ) + F (q, q˙ ) + τd )] (4) y = x1 (5) where x1 and x2 are the position and velocity vectors of the robot manipulator. The tracking error of the position is given as: (6) e1 = qd − q Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 3 The stabilizing function is characterized as α1 = λ1 e1 + 1 λ2 1  (7) e1 2  where λ1 , λ2 and 1 , 2 (1 < 1 < 2) are positive odd numbers. The tracking error of the velocity has upgraded with 2 stability function. It is characterized in Eq. (8) and appeared as: e2 = e˙ 1 + α1 (8) The primary Lyapunov stability function is characterized as: V1 = 1 2 e 2 1 (9) V˙ 1 = e1 e2 − λ 2 1 e1 − 1   +  1 λ2 2 2 e1 (10) From (2) and (8), we get e˙ 2 = q̈d − D−1 (q )    −  1 2   1  τ − C (q, q˙ )q˙ + G(q ) + F (q, q˙ ) + τd + λ1 e˙ 1 + e1 2 e˙ 1 λ2 2 (11) The second Lyapunov stability function is described as: V = V1 + 1 T s s 2 (12) The satisfactory condition, which gives the affirmation that the tracking error will make an elucidation from achieving stage to sliding stage, is called the achieving condition and given in (13). V˙ < 0, s = 0 (13) The sliding surface ‘s’ is characterized as: (14) s = e1 + e2 The derivative of the second Lyapunov stability function is given in Eq. (15). V˙ = e1 e2 − λ1 e21 − 1 λ2   +  1 e1 2 2   −  1 2 1  + s (e˙ 1 (1 + λ1 ) + e1 2 e˙ 1 + q̈d − q̈ ) λ2 2 T (15) The total control torque (τ ) to the robot manipulator is characterized as: τ = τ0 + τc (16) By substituting (16) in (15), we get V˙ = e1 e2 − λ1 e21 − 1 λ2  e1 1 +2 2  +s T  e˙ 1 (1 + λ1 ) + − (C (q, q˙ )q˙ + G(q ) + F (q, q˙ ) + τd ))  1 e λ2 2 1 1 −2 2  e˙ 1 + q̈d − D−1 (q )( (τ0 + τc )  (17) The arrangement of s˙ = 0 gives the control signal, which is known as equivalent control law and it is denoted by ‘τ 0 ’. This equivalent control law is essential to fulfill the execution of favored trajectory tracking without considering disturbances and uncertainties (i.e., τd = 0).   −  1 2  1 2 τ0 = D(q ) e˙ 1 (1 + λ1 ) + e e˙ 1 + q̈d + C (q, q˙ )q˙ + G(q ) + F (q, q˙ ) λ2 2 1  (18) An extra control exertion is needed to wipe out the unpredictable disturbances and uncertainties as equivalent control torque (τ 0 ) is lacking to provide the favored tracking performance. Ultimately, the tracking error dies out asymptotically, which means the sliding surface becomes stable. To exhibit the stability of the created control framework for robot manipulator, the Lyapunov-like Lemma is utilized. From (17) and (18) we arrive at an expression for V˙ as follows: V˙ = e1 e2 − λ1 e21 − Please cite this nipulator using 1 λ2 article radial   +  1 e1 2 2 + sT (−D−1 (q )τc ) (19) as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 4 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 To meet the Lyapunov stability condition, the corrective control law (τ c ) is defined as τc   1 = D (q ) T s e1 e2 − 1 λ2  e1 1 +2 2  + KW sign(s ) (20) By substituting τ c in (19) and yields: V˙ = −λ1 e21 − Kw sT sign(s ) (21) where Kw is the sliding gain. V˙ ≤ −λ1 |e21 | − Kw |s| (22) sT sign(s ). where |s| = The chattering effect on the control input signal is introduced by the signum function (‘sign’), which is utilized as a part of (21), so as to diminish or dispense with this impact, the signum function (‘sign’) is substituted by the hyperbolic tangent function (‘tanh ’) as given in (23). V˙ ≤ −λ1 e21 − Kw sT tanh(s ) (23) ‘sT tanh (s)’ The term in (23) is constantly positive so that whole condition gets to be negative (i.e., or s < 0). The achieving control signal (τ c ) is modified as: τc   1 = D (q ) T s e1 e2 − 1 λ2  e1 1 +2 2  sT tanh(s) > 0 if either s > 0 + Kw tan h(s ) (24) Where Kw = diag{Kw1 , Kw2 , . . . , Kwn } is control gain matrix. The saturation function with boundary layer (φ ) is designed to eliminate discontinuous portions in control input toque (τ c ) is given as: tanh s sat = φ s , φ s for φs ≥ 1; φ , (25) for φs < 1. The BTSMC control law is defined as: τc = D ( q )   1 sT e1 e2 − 1 λ2  e1 1 +2 2  + K tanh s (26) φ where K = D(q )Kw . When s = 0 is reached at t = tr , e1 = 0 becomes terminal surface, i.e., e1 + e2 = 0. The effect of terminal surface will take the state of e1 from e1 (tr ) = 0 to e1 (tr + ts )=0 with finite time ts given by ts = 1 l n (1 + λ1 )e1 + 1 (1 + λ1 )(  − 1) 2 1 λ2 1  e1 2 − 1 l n ( e1 ) 2 (27) 2.2. Design and stability analysis of NNBTSMC In this section, the RBFNN is utilized to build up the control plans for the robot manipulator to track the desired trajectories under unknown dynamics of the system. Fig. 1 demonstrates the schematic representation of NNBTSMC. The approximation function is f(X): Rq → R3 and defined as: f (X ) = W T σ (X ) + ǫ (X ) (28) ( X )] T where σ (X ) = [σ1 (X ), σ2 (X ), σ3 (X ), . . . , σm is the Gaussian RBF function, ‘ǫ ’ is the approximation error. ‘X’ , and ‘m’ (m > 1) are input vector, weight matrix and number of neurons respectively. The RBF function is defined as: σi (X ) = exp − ||X − Ci ||2 2b2i , i = 1, 2, . . . , m W ∈ Rm × 3 , (29) where ‘Ci ’ is the center and ‘bi ’ is the width of the ith neuron of RBF neural network respectively. Tracking error of position is defined as: (30) e1 = qd − q q˙ r = q˙ d + (1 + λ1 )e1 + Please cite this nipulator using article radial 1 λ2 1  e1 2 (31) as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 5 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 Fig. 1. Block diagram of NNBTSMC scheme. where ‘q˙ r ’ is auxiliary signal. The modified sliding surface (s) is characterized as: s = q˙ r − q˙ = (1 + λ1 )e1 + 1 λ2 1  e1 2 + d e1 dt (32) From (1), (31) and (32), we get D(q )s˙ = −C (q, q˙ )s + f (X ) + τd − τ (33) where ‘τ d ’ represents external torque disturbances. f (X ) = D(q )q̈r + C (q, q˙ )q˙ r + F (q, q˙ ) + G(q ) (34) [q̈Td , q˙ Td , qTd , q˙ T , qT ]T . where X= The approximation of f(X) is defined in (35). ˆ T σ (X ) f (X ) = W (35) ˆ ’ is the NN adjustment law. Now (33) can be revised as: where ‘σ (X)’ is the basis function and ‘W ˜ T σ (X ) + W ˆ T σ ( X ) + ǫ ∗ + τd − τ D(q )s˙ = −C (q, q˙ )s + W (36) ˜ = W∗ − W ˆ , W∗ and ǫ ∗ are ideal weights matrix and approximation error respectively. where W Theorem 1. Consider the robot manipulator is demonstrated by (1), if the total control torque is expressed as τ = τ0 + τc + τNN , ˆ T σ (X ). The evaluated adaptive in which the updated control law of BTSMC is characterized as τc = k1 s + K tanh( φs ) and τNN = W law for the NN identifier is characterized as: ˜˙ T = Bσ (X )sT ˆ˙ T = −W W (37) where ‘B’ is a positive definite matrix and the tracking errors of position and velocity (i.e., e and e˙ ) of the system asymptotically converge to zero as t → ∞. Proof. NNBTSMC stability function is given in (38). V = 1 1 T ˜ T B−1W ˜) s D(q )s + tr (W 2 2 (38) V˙ = 1 T ˜˙ ) ˜ T B−1W s D˙ (q )s + sT D(q )s˙ + tr (W 2 (39) By substituting (36) in (39), we get V˙ = s 1 T ˜˙ ) ˜ T σ (X ) + ǫ ∗ − τd − k1 s − K tanh ˜ T B−1W s [D˙ (q ) − 2C (q, q˙ )]s + sT [W ] + tr (W 2 φ Since D˙ (q ) − 2C (q, q˙ ) is a skew symmetric matrix, the first term in (40) becomes zero (i.e., (23) and (37), V˙ can be rewritten as: V˙ ≤ −k1 sT s + ||s||δ0 Please cite this nipulator using article radial 1 T ˙ 2 s [D ( q ) (40) − 2C (q, q˙ )]s = 0), from (41) as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 6 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 where ‘δ 0 ’ is the upper bound of ||ǫ ∗ − τd ||. The term ||s||δ 0 is bounded by ||s||δ0 ≤ 21 (||s||2 + ρδ02 ), where ρ > 0 is picked ∞ to such an extent that 0 ρ dt < ∞ and after that substituted in (41) to get stability criteria with finite boundedness. 1 (||s||2 + ρδ02 ) 2 V˙ ≤ −k1 ||s||2 + (42) Integrating both sides of (42) from t = 0 to T, yields V ( T ) − V ( 0 ) ≤ − ( k1 − Since V(T) ≥ 0 and lim sup T →∞ 1 T  ∞ 0 T 1 ) 2 T  0 T 1 2  V (0 ) + 1 2 δ 2 0 ||s||2 dt + δ02 ( ∀ k1 > 2 ) ρ dt, (43) 0 ρ dt < ∞ holds, ||s||2 dt ≤ 0 1 1 2 ( k1 − )   ∞ 0  1 ρ dt lim sup = 0 T →∞ (44) T From (43) and (44), we can get s → 0 as t → ∞. Subsequently, we can conclude, from (32), the tracking errors of position and velocity (i.e., e and e˙ ) asymptotically converges to zero as t → ∞. Henceforth, the stability criteria is fulfilled by the proposed NNBTSMC scheme.  Remark 1. The addition of BTSMC and NN identifier provides the robust controller and such errors caused by extrinsic disturbances and uncertainties. The NN identifier can be remunerated to give better trajectory tracking and intensify the disturbance rejection under different disturbance conditions. 3. An adaptive observer-based control of Robot Manipulator 3.1. Design of neural network adaptive observer (NNAO) In this segment, the NNAO is designed to estimate link trajectories of the robot manipulator. The velocity of the link is characterized as: q˙ = p (45) where ‘q’ is position vector and ‘p’ is velocity vectors of the robot manipulator. Dynamic equation (1) is redefined as: p˙ = H (q, p) − D−1 (q )τd + D−1 (q )τ (46) −D−1 (q )[C (q, where H (q, p) = p) p + G(q ) + F (q, p)]. As indicated by the estimation property of NN, the approximation function of H(q, p) is characterized as: H (q, p) = Wo∗T σo (q, p) + ǫo∗ (q, p), ‘Wo∗T ’ ||ǫo∗ (q, p)|| ≤ ǫ0N (47) ‘ǫo∗ (q, p)’ and ‘ǫ 0N ’ are approximation error, and it’s upper bound limit respectively. where is an optimal weight matrix, The NNAO estimates q as qˆ and p as pˆ, estimation errors of the NNAO are q˜ = q − qˆ and p˜ = p − pˆ. Then, the approximation function of (47) in terms of qˆ and pˆ can be given by (48). ˆ oT σo (qˆ, pˆ ) Hˆ (q, p) = W (48) An adaptive observer is defined as: qˆ˙ = pˆ + ˜ 1q (49) ˆ oT σo (qˆ, pˆ ) + D−1 (q )τ + pˆ˙ = W where 1, 2 and q˜˙ = p˜ − ˜ 1q 3 ˜ 2q + ˙ ˜ 3q (50) are positive design constants. (51) p˜˙ = p˙ − pˆ˙ (52) ˆ oT σo (qˆ, pˆ ) − D−1 (q )τd − p˜˙ = Wo∗T σo (q, p) − Wo∗T σo (qˆ, pˆ ) + Wo∗T σo (qˆ, pˆ ) − W ˜ oT σo (qˆ, pˆ ) − D−1 (q )τd + ǫo∗ − p˜˙ = Wo∗T σ˜ o (q, p) + W ˜ 2q − ˙ ˜ 3q ˜ 2q − ˙ ˜ 3q (53) (54) By substituting (51) in (54), then we get ˜ oT σo (qˆ, pˆ ) − D−1 (q )τd + ǫo∗ − p˜˙ = Wo∗T σ˜ o (q, p) + W ˜o = where σ˜ o (q, p) = σo (q, p) − σo (qˆ, pˆ ), W Please cite this nipulator using article radial Wo∗ ˆ o and −W ˜ 3p 2 = (55) 1 3, p˜ = q˜˙ . as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 7 Fig. 2. Block diagram of the NNAOBTSMC scheme. Theorem 2. Let NN observer is characterized by (49) and (50) respectively and the evaluated adaptation law for the observer is designed as ˆ˙ o = −W ˜˙ o = Boσo (qˆ, pˆ ) p˜T W (56) then the estimated errors of observer q˜ and p˜ are asymptotically converge to zero. Proof. The NNAO stability function is given in (57). V = 1 1 T T ˜ p˜ p˜ + tr (W˜ o B−1 o Wo ) 2 2 (57) V˙ = p˜T {Wo∗T σ˜ o (q, p) − D−1 (q )τd + ǫo∗ } − V˙ ≤ − ˜T 3p T ˜˙ ˆ, pˆ ) p˜T )} p˜ + tr{W˜ o (B−1 o Wo + σo (q ˜T 3p p˜ + || p˜||β0 (58) (59) where ‘β 0 ’ is an upper bound of ||Wo∗T σ˜ o (q, p) − D−1 (q )τd + ǫo∗ ||. To induce stability criteria with finite boundedness, the term || p˜||β0 is bounded by || p˜||β0 ≤ 21 (|| p˜||2 + ρβ02 ) and substituted in (59). V˙ ≤ − ˜T 3p p˜ + 1 (|| p˜||2 + ρβ02 ) 2 (60) Integrating both sides of (60) from t = 0 to T, yields  V (T ) − V (0 ) ≤ − Since V(T) ≥ 0 and lim sup T →∞ 1 T  ∞ 0 ∞ 3 − 1 2  T 0 || p˜||2 dt + β02  T ρ dt, (∀ 3 > 2) (61) 0 ρ dt < ∞ holds, || p˜||2 dt ≤ 0 1 ( 3 − 12 )  V (0 ) + 1 2 β 2 0  0 ∞  1 ρ dt lim sup = 0 T →∞ T (62) From (61) and (62), we can obtain p˜ → 0 as t → ∞. Therefore, we can conclude that both q˜ → 0 and p˜ → 0 i.e., the proposed NNAO satisfies the stability criteria.  3.2. Design of NNAOBTSMC The NNAOBTSMC is developed with the help of NNAO for precise desired trajectory tracking and improve the disturbance rejection under different working conditions (i.e., external disturbances and uncertainties) of the robot manipulator. Fig. 2 demonstrates the structure of the NNAOBTSMC. The evaluated error of the link position is characterized as: eˆ1 = qd − qˆ Please cite this nipulator using (63) article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 8 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 The estimated sliding surface ‘sˆ’ is designed as: 1 (1 + λ1 )eˆ1 + λ12 eˆ12 + sˆ = φ d eˆ dt 1 (64) Theorem 3. Considered the robot manipulator described by (1). Let the NNAO estimates q and p as qˆ and pˆ, estimation errors are q˜ = q − qˆ and p˜ = p − pˆ. Then, the approximation function of f(X) described in (34) is conveyed as: T fˆ(Xˆ ) = Wˆ c σ (Xˆ ), Xˆ = [q̈Td , q˙ Td , q˙ Td , pˆT , qˆT ]T (65) where ‘Wˆ c ’ is estimated adaptation matrix. The adaptation law is defined as: Wˆ˙ c = −W˜˙ c = Bc σ (Xˆ )sˆT (66) The updated control law of NNAOBTSMC is characterized as: 3 sˆ + τ= K tanh  sˆ  T + Wˆ c σ (Xˆ ) φ (67) T where ‘ 3 sˆ + K tanh( φsˆ )’ is the boundary sliding mode control law and ‘Wˆ c σ (Xˆ )’ is approximation function to the robot manipulator, then the estimation errors q˜ and p˜ are asymptotically converge to zero. Proof. The Lyapunov stability candidate function for NNAOBTSMC is characterized as: (68) V = Vo + Vc where ‘Vo ’ and ‘Vc ’ are the observer stability and controller stability candidate functions respectively. 1 ˜ oT B−1 ˜ tr (W o Wo ) 2 (69) 1 1 T ˜ cT B−1 ˜ s D(q )s + tr (W c Wc ) 2 2 (70) Vo = p˜T p˜ + Vc = The derivative of (70) becomes V˙ c = 1 T ˜˙ ˜ cT B−1 s D˙ (q )s + sT D(q )s˙ + tr (W c Wc ) 2 (71) Since D˙ (q ) − 2C (q, p) is a skew symmetric matrix and s = sˆ + q˜˙ + ˆ cT σ˜ (Xˆ ) + ǫ ∗ − τd } + p˜T {W˜ c V˙ c = sˆT {W T ˜ 1q = sˆ + p˜, then we get σ (Xˆ ) + Wˆ cT σ˜ (Xˆ ) + ǫ ∗ − τd } −  sˆ  ˜ cT (B−1 ˜˙ ˆ T −K sˆT tanh + tr{W c Wc + σ (X )sˆ )} φ T 3 sˆ sˆ − ˜T sˆ 3p (72) T ˆ cT σ˜ (Xˆ ) + ǫ ∗ − τd }, corresponding to this ‘ζ ’, ˆ cT σ˜ (Xˆ ) + ǫ ∗ − τd || and ζ = {W˜ c σ (Xˆ ) + W where ‘γ 0 ’ is maximum limit of ||W there exists a positive constant ‘ζ 0 ’ such that ||ζ || ≤ ζ 0 . V˙ c ≤ ||sˆ||γ0 + || p˜||ζ0 − The terms ||sˆ||γ0 , || p˜||ζ0 and − 3 (|| p˜||2 || + ˜T sˆ 3p T 3 sˆ sˆ − 3 p˜T sˆ (73) are bounded by ||sˆ||γ0 ≤ 1 ˆ 2 2 (||s|| + ργ02 ), || p˜||ζ0 ≤ 1 ˜ 2 2 (|| p|| + ρζ02 ) and − ˜T sˆ ≤ 3p ||sˆ||2 ) respectively. To induce stability criteria with finite boundedness, the constrained terms are substituted 2 in (73). The expression for V˙ c gets to be: V˙ c ≤ − 3 2 − 1 2 3 ||sˆ||2 + 2 + 1 2  1  || p˜||2 + ρ γ02 + ζ02 , 2 ∀ ( > 2) 3 (74) The derivative of (69) and from (58), we will get V˙ o = p˜T {Wo∗T σ˜ o (q, p) − D−1 (q )τd + ǫo∗ } − Since ||Wo∗T σ˜ o (q, p) − D−1 (q )τd + ǫo∗ || ˜T 3p T ˜˙ ˆ, pˆ ) p˜T )} p˜ + tr{W˜ o (B−1 o Wo + σo (q ≤ β0 and || p˜||β0 is bounded by || p˜||β0 ≤ 1 ˜ 2 2 (|| p|| + 1 V˙o ≤ − 3 p˜T p˜ + (|| p˜||2 + ρβ02 ) 2 ˙V = V˙ o + V˙ c V˙ ≤ − 1 2 3 Please cite this nipulator using −2  1 || p˜||2 − article radial 2 3 −1 (75) ρβ02 ), then (76) (77)  2 1  2  ||sˆ|| + ρ γ0 + ζ02 + β02 , 2 ∀ ( 3 > 2) (78) as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 9 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 Fig. 3. Architecture of 3 DOF OTDIRM (Ref [5]). Integrating both sides from t = 0 to T, yields V (T ) − V (0 ) ≤ − Since V(T) ≥ 0 and ∞ 0 1 2 3 −1  T ||sˆ||2 dt − 0 ρ dt < ∞ holds, 1 2 3 −2  T || p˜||2 dt + 0 3 −1 2  0  T ρ dt (79) 0 T     T ||sˆ||2 dt + 3 − 2 || p˜||2 dt ≤ 2 V (0 ) 0 0 T →∞ T  ∞  1 2 1 2 2 + (γ0 + ζ0 + β0 ) ρ dt lim sup = 0 lim sup  1   1 2 γ + ζ02 + β02 2 0 T →∞ (80) T From (79) and (80), it is clear that p˜ → 0, sˆ → 0 as t → ∞. That concludes q˜ → 0 and p˜ → 0. Thus, the proposed NNAOBTSMC fulfills the stability criteria.  4. Results and discussions In this paper, MATLAB/SIMULINK tool has been used to carry out simulations for the 3 DOF OTDIRM. Fig. 3 demonstrates the architecture of a de-icing robot manipulator. It has two revolute joints and one prismatic joint. The description of the Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 10 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 Table 1 Physical parameters of the de-icing robot manipulator [5]. Sl. No Symbol Definition Value 1 2 3 4 5 6 m1 m2 m3 L1 L2 g Mass of the link 1 Mass of the link 2 Mass of the link 3 Length of the link 1 Length of the link 2 Gravitational constant 3(kg) 2(kg) 2.5(kg) 0.14(m) 0.32(m) hspace0.4 cm 9.81(m/s2 ) system, from Eq. (1) can be depicted as follows: D (q ) =  D11 D21 D31 D12 D22 D32 D13 D23 D33  D11 = 1 9 m1 L1 + m2 c2 L2 + L21 + L1 L2 [c12 − s21 ] + m3 (c2 L − 22 + L22 + 2c2 L1 L2 ) 4 4 D22 = 1 4 m2 L22 + m3 L22 + m1 L21 4 3   D23 = D32 = m3 c2 L2 D33 = m3 D13 = D31 = D12 = D21 = 0  C11 C (q, q˙ ) = C21 C31 C12 C22 C32 C13 C23 C33 C11 = −8m2 L1 L2 c1 s1 q˙ 1 −  1 2  m2 L22 c2 s2 + 2m3 [L1 L2 s2 + L22 c2 s2 ] q˙ 2 C22 = −m3 L2 s2 q˙ 3 C23 = −2m3 L2 s2 q˙ 2 C32 = −m3 L2 s2 q˙ 2 C33 = C12 = C13 = C31 = 0 ⎡ G(q ) = ⎣  1 c c L 2 1 2 2  + c1 L1 m2 g ⎤ ⎦ (81) − 21 s1 s2 L2 m2 + c2 L2 m3 g where s1 = sin(q1 ), s2 = sin(q2 ), c1 = cos(q1 ) and c2 = cos(q2 ). The link position and velocity vectors are q = [q1 , q2 , q3 ]T and q˙ = [q˙ 1 , q˙ 2 , q˙ 3 ]T respectively. The desired reference trajectories for de-icing robot manipulator and essential parameters that influence the control activity of the robotic systems (i.e., uncertainties and external disturbances) are considered according to literature Ref [5]. The physical parameters of the OTDIRM are listed in Table 1. Initial conditions are q(0 ) = [0.9, 0.1, 0.1]T and q˙ (0 ) = [0, 0, 0]T . The reference tracking signals are qd1 (t ) = sin(t ), qd2 (t ) = cos(t ) and qd3 (t ) = cos(t ). Considered friction terms are F (q1 , q˙1 ) = 20q˙ 1 + 0.8sign(q˙ 1 ), F (q2 , q˙2 ) = 4q˙ 2 + 0.16sign(q˙ 2 ) and F (q3 , q˙3 ) = 4q˙ 3 + 0.16sign(q˙ 3 ). The external disturbances are τd1 = 5 sin(5t ), τd2 = 0.5 sin(5t ) and τd3 = 0.5 sin(5t ). BTSMC parameters are λ1 = 980, λ2 = 100, 1 =5, 2 =3, Kw1 = 440, Kw2 = 440, Kw3 = 440 and φ = 0.1. The finite times (ts) for proposed method (i.e., NNAOBTSMC) are calculated as 0.0422 sec, 0.0405 sec and 0.0438 sec for link 1, Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 11 Fig. 4. Position tracking of link 1. Table 2 MSE examination of CRLSMC [5], NNBTSMC and NNAOBTSMC. Sl. No MSE CRLSMC [5] NNBTSMC NNAOBTSMC 1 2 3 link 1 link 2 link 3 2.732e-06 3.526e-05 4.864e-06 2.451e-15 2.007e-14 9.321e-16 4.118e-17 3.924e-20 1.44e-20 Fig. 5. Position tracking of link 2. link 2 and link 3 respectively. The simulated responses of the OTDIRM are compared with those presented in Ref [5] are shown in figures from 4 to 15. Output performance: To evaluate output performance, the mean square error (MSE) of tracking positions are computed. The total sampling time is ‘T’, desired trajectory is ‘qdi ’ and estimated trajectory is ‘qˆi ’ of the ith link, tracking position mean square error (MSE) is given in (82). MSEi = T 1 [qdi − qˆi ]2 , T i = 1, 2, 3 (82) t=1 The MSE values of the 3 DOF OTDIRM trajectories under different control methodologies, such as CRLSMC in [5], NNBTSMC and NNAOBTSMC are presented in Table 2. From the table, it is ascertained that NNAOBTSMC provides least MSE values in comparison with other methods (i.e., NNAOBTSMC system has better tracking trajectory performance). Figs. 4–6 indicate tracking positions, Figs. 7–9 indicate tracking errors and Figs. 10–12 show control torque. From the Figs. 10–12, it is clear that the control torque is smoother for NNAOBTSMC compared with existing CRLSMC [5]. This indicates that the proposed method requires less control effort for same trajectory tracking. The MSE response plots of the OTDIRM link trajectories exhibited from Figs. 13–15. Input performance: In order to evaluate the manipulated input usage, we calculated total variation (TV)[7] of the input u(t) is calculated as TV = ∞  ||u j+1 − u j || (83) j=1 Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 12 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 Fig. 6. Position tracking of link 3. Fig. 7. Position tracking error of link 1. Fig. 8. Position tracking error of link 2. Fig. 9. Position tracking error of link 3. Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 13 Fig. 10. Control input for link 1. Fig. 11. Control input for link 2. Fig. 12. Control input for link 3. This should be as small as possible. The total variation is a good measure of the signal “smoothness”. A large value of TV means more excessive input usage or more complex controllers [7]. The energy of the input signal is calculated by using the 2-Norm method. The control energy is expected to be as small as possible. The output and input performances are calculated for the period from 0 to 30 s with a sampling time of 0.0 0 01 s. From Tables 3 and 4, it clear that the values of TV and 2-Norm of input for proposed methods are very small as compared to existing method CRLSMC [5]. Figs. 16–18 show the box plot of control torque for link 1, link 2 and link 3 with the mean, median, ± 25% quartiles (notch boundaries), ± 75% quartiles (box ends), ± 95% bounds and the outliers. From the size of the boxes shown, it is clear that the NNAOBTSMC control strategy experiences minimum variation than others. Comparing the box plot of NNBTSMC and NNAOBTSMC, it observed that NNAOBTSMC has less variation in control input torque for link 1, link 2 and link 3 of the OTDIRM. Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 14 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 Fig. 13. Mean square error of link 1 position tracking. Fig. 14. Mean square error of link 2 position tracking. Fig. 15. Mean square error of link 3 position tracking. Table 3 Total variance examination of CRLSMC [5], NNBTSMC and NNAOBTSMC . Please cite this nipulator using article radial Sl. No Total variance (TV) CRLSMC [5] NNBTSMC NNAOBTSMC 1 2 3 link 1 link 2 link 3 6.1354 4.8902 20.8775 1.8337 3.5510 5.1885 1.4024 3.4074 4.0198 as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 15 Table 4 2-Norm of input examination of CRLSMC [5], NNBTSMC and NNAOBTSMC . Sl. No 2-Norm of input CRLSMC [5] NNBTSMC NNAOBTSMC 1 2 3 link 1 link 2 link 3 2.4778e3 3.4905e3 1.3727e4 2.2119e3 3.4535e3 1.3550e4 2.1856e3 3.4452e3 1.3530e4 Control torque for link 1(Nm) 10 8 6 4 2 0 -2 CRLSMC [5] NNBTSMC Control strategy NNAOBTSMC Fig. 16. Box plot representation of control torque for link 1. Control torque for link 2 (Nm) 10 8 6 4 2 0 CRLSMC [5] NNBTSMC Control strategy NNAOBTSMC Control torque for link 3 (Nm) Fig. 17. Box plot representation of control torque for link 2. 30 25 20 15 CRLSMC [5] NNBTSMC Control strategy NNAOBTSMC Fig. 18. Box plot representation of control torque for link 3. Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 16 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 5. Conclusions In this paper, different control methodologies are proposed for the 3 degrees of freedom overhead transmission line de-icing robot manipulator. The proposed control approaches are based on the backstepping terminal sliding mode control, neural network based approximator and observer to provide robustness to external disturbances and parameter uncertainties. In general, the results illustrate that the proposed observer-based controller offers a superior tracking performance and smoother control input compared to other existing methods. The derivation of the control law guarantees the convergence of the tracking error. Several performance methods are examined to support the strength of the proposed and existing control approaches. The output and control input performances are calculated for the period from 0 to 30 s with a sampling time of 0.0 0 01 s. The box plots of control torques clearly show that the observer-based control scheme has less variation in control input compared to other methods. Appendix A. PSO parameters Cognitive parameter (c1 ) = 2, Social parameter (c2 ) = 2, itermax = 100, Construction factor (C) = 1 and Population size = 100. The minimum and maximum values of the weight factors (inertial coefficients) of PSO are taken as wmin = 0.217 and wmax = 0.9 respectively. Appendix B. NN parameters for OTDIRM Parameters for neural network based observer design: Wo =  0.867 1.321 2.321 0.343 0.8732 0.863  3.343 0.8754 0.1542 1.232 0.763 1.872  2.432 1.098 1.092 0.822 1.0983 1.0923  Co = 0.342 0.543 0.259 0.987 0.654 0.632  0.639 0.764 0.824 0.154 0.872 bo = 0.984  Parameters for neural network based NN identifier: 0.987 ⎢ 2.322 ⎢0.0983 Wc = ⎢ ⎢ 0.698 ⎣ 0.0983 1.8633 ⎡ Cc =  bc =  0.109 0.187 0.312 0.451 0.243 0.987 0.0983 1.143 0.465 0.543 1.764 0.0983 0.8764 0.987 0.154 1.098 0.098 0.273 0.201 0.0432 0.672 0.132 1.982 0.231 0.987 0.873 0.432 0.098 0.432 1.232 0.314 0.098 0.152 0.134 0.1092 0.443 0.0253 2.321 0.0983 0.762 0.781 1.322 0.098 0.0987 0.873 0.987 0.874 0.786 0.462 0.987 0.414 0.873 2.123 0.0453 0.0144 0.654 0.098 0.211 0.432 0.872 0.098 0.092 0.162 0.82 0.414 0.125 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 0.0982 0.123  0.167  References [1] Zhang W, Bae J, Tomizuka M. Modified preview control for a wireless tracking control system with packet loss. IEEE/ASME Trans Mechatron 2015;20(1):299–307. [2] Heck D, Saccon A, de Wouw N V, Nijmeijer H. Guaranteeing stable tracking of hybrid positionforce trajectories for a robot manipulator interacting with a stiff environment. Automatica 2016;63. 235-47. [3] Vaidyanathan S. Advances and applications in nonlinear control systems. Springer; 2016. [4] Soltanpour MR, Khooban MH, Khalghani MR. An optimal and intelligent control strategy for a class of nonlinear systems: adaptive fuzzy sliding mode. J Vib Control 2016;22(1):159–75. [5] Van Tran T, Wang Y, Ao H, Truong TK. Sliding mode control based on chemical reaction optimization and radial basis functional link net for de-icing robot manipulator. J Dyn Syst Meas Control 2015;137(5). 051009. [6] Vijay M, Jena D. Intelligent adaptive observer-based optimal control of overhead transmission line de-icing robot manipulator. Adv Rob 2016;30(17–18):1215–27. [7] Mondal S, Mahanta C. Adaptive second order terminal sliding mode controller for robotic manipulators. J Franklin Inst 2014;351(4):2356–77. [8] Li P, Ma J, Zheng Z. Robust adaptive sliding mode control for uncertain nonlinear MIMO system with guaranteed steady state tracking error bounds. J Franklin Inst 2016;353(2):303–21. [9] Tran MD, Kang HJ. Adaptive terminal sliding mode control of uncertain robotic manipulators based on local approximation of a dynamic system. Neurocomputing 2017;228:231–40. [10] Li H, Yu J, Hilton C, Liu H. Adaptive sliding-mode control for nonlinear active suspension vehicle systems using TS fuzzy approach. IEEE Trans Ind Electron 2013;60(8):3328–38. [11] Dinh HT. Dynamic neural network-based robust control methods for uncertain nonlinear systems. University of Florida; 2012. [12] Rossomando FG, Soria C, Carelli R. Sliding mode neuro adaptive control in trajectory tracking for mobile robots. J Intell Rob Syst 2014;74(3–4):931. Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 17 [13] Song Z, Sun K. Adaptive backstepping sliding mode control with fuzzy monitoring strategy for a kind of mechanical system. ISA Trans 2014;53(1):125–33. [14] Sun T, Pei H, Pan Y, Zhou H, Zhang C. Neural network-based sliding mode adaptive control for robot manipulators. Neurocomputing 2011;74(14):2377–84. [15] Wei S, Wang Y, Zuo Y. Wavelet neural networks robust control of farm transmission line deicing robot manipulators. Comput Standards Interfaces 2012;34(3):327–33. [16] Sun G, Li D, Ren X. Modified neural dynamic surface approach to output feedback of MIMO nonlinear systems. IEEE Trans Neural Netw Learn Syst 2015;26(2):224–36. [17] Jafarnejadsani H, Pieper J, Ehlers J. Adaptive control of a variable-speed variable-pitch wind turbine using radial-basis function neural network. IEEE Trans Control Syst Technol 2013;21(6):2264–72. [18] Ho HF, Wong YK, Rad AB. Adaptive fuzzy sliding mode control with chattering elimination for nonlinear SISO systems. Simul Modell Pract Theory 2009;17(7):1199–210. [19] Chaoui H, Gueaieb W, Biglarbegian M, Yagoub MC. Computationally efficient adaptive type-2 fuzzy control of flexible-joint manipulators. Robotics 2013;2(2):66–91. [20] Wai RJ, Chen MW, Yao JX. Observer-based adaptive fuzzy-neural-network control for hybrid maglev transportation system. Neurocomputing 2016;175:10–24. [21] Chang JL. Sliding mode control design for mismatched uncertain systems using output feedback. Int J Control Autom Syst 2016;14(2):579–86. [22] Sun H, Guo L. Neural network-based DOBC for a class of nonlinear systems with unmatched disturbances. IEEE Trans Neural Netw Learn Syst 2017;28(2):482–9. [23] Zhang L, Li Z, Yang C. Adaptive neural network based variable stiffness control of uncertain robotic systems using disturbance observer. IEEE Trans Ind Electron 2017;64(3):2236–45. [24] Spong MW, Vidyasagar M. Robot dynamics and control. John Wiley & Sons; 2008. Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007 ARTICLE IN PRESS JID: CAEE [m3Gsc;November 14, 2017;20:57] 18 M. Vijay, D. Jena / Computers and Electrical Engineering 000 (2017) 1–18 M Vijay has received Engineering degree in Electrical Engineering (2009) from Osmania University and Masters (M.Tech) degree (2012) in Power and Energy Systems from NITK Surathkal, India. He is currently working towards to his Ph.D. degree in the Electrical Engineering, NITK Surathkal, India. His research interests include control of nonlinear systems and predictive modeling/behavior modeling methods for nonlinear systems. Debashisha Jena has received Engineering degree (1996) from University College of Engineering, Burla, India. He obtained his Master (M.Tech) degree (2004) in Electrical Engineering and Ph.D. Engineering, NIT, Rourkela, India in 2010. Currently, he is an assistant professor at NITK Surathkal, India. His research interests include evolutionary computation, system identification and neuro-evolutionary computation application to the power system. Please cite this nipulator using article radial as: M. Vijay, D. Jena, Backstepping terminal sliding mode control of robot mabasis functional neural networks, Computers and Electrical Engineering (2017), https://doi.org/10.1016/j.compeleceng.2017.11.007