Adaptive second-order fast nonsingular terminal sliding mode control for robotic manipulators

ISA Transactions 90 (2019) 41–51

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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Adaptive second-order fast nonsingular terminal sliding mode control for robotic manipulators Shanchao Yi, Junyong Zhai

∗

Key Laboratory of Measurement and Control of CSE, Ministry of Education, School of Automation, Southeast University, Nanjing, Jiangsu 210096, China

highlights • A SOFNTSM controller is designed to ensure fast convergence and trajectory tracking. • Actual control law after integral is continuous and thus system is chattering-free. • Without prior knowledge of uncertainties, adaptive laws are designed to estimate them.

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Article history: Received 20 September 2018 Received in revised form 4 December 2018 Accepted 27 December 2018 Available online 25 January 2019 Keywords: Adaptive tuning Second-order fast nonsingular terminal sliding mode Robotic manipulator Trajectory tracking Lyapunov stability

a b s t r a c t This paper presents an adaptive chattering-free sliding mode controller for trajectory tracking of robotic manipulators in the presence of external disturbances and inertia uncertainties. To achieve fast convergence and desirable tracking precision, a second-order fast nonsingular terminal sliding mode (SOFNTSM) controller is designed to guarantee system performance and robust stability. Chattering is eliminated using continuous control law due to high-frequency switching terms contained in the first derivative of actual control signals. Meanwhile, uncertainties are compensated by introducing the adaptive technique, whose prior knowledge about upper bound is not required. Finally, simulation results validate the effectiveness of the proposed control scheme. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction As the robotic manipulator becomes increasingly significant in various sophisticated tasks like drilling, welding and painting, it has gained extensive interest owing to the high performance in these works requiring perfect precision for trajectory tracking. However, it is great challenging for implementation of controllers due to the dynamics of robotic manipulators with uncertainties and disturbances such as payload variation, backlash, time-varying friction and coupled terms. Therefore, a variety of advanced control techniques like decentralized control [1], feedback linearization [2], PID control [3], model predictive control [4], fuzzy control [5], robust control [6], neural network control [7] and sliding mode control (SMC) [8–13] have been applied to tracking tasks of robotic manipulator systems. Because of its special attributes such as strong robustness, fast transient response and simplicity [14,15], SMC has obtained ∗ Corresponding author. E-mail address: [email protected] (J. Zhai). https://doi.org/10.1016/j.isatra.2018.12.046 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

tremendous popularity. However, the linear SMC can only guarantee the asymptotic convergence of states, which implies that high gains are required to obtain the fast convergence and may result in the rapid saturation of actuators. Considering this issue, Man, Paplinski et al. [16] developed the terminal sliding mode (TSM) control based on nonlinear sliding surfaces in order to achieve finite-time convergence without the cost of a large control input. By choosing suitable fractional powers in control laws, a nonsingular terminal sliding mode (NTSM) is proposed to eliminate the singularity resulting from TSM in [17]. Then, fast nonsingular terminal sliding mode (FNTSM) control is investigated to accelerate the speed of convergence in [18–22]. Lu and Xia [19] designed a novel fast nonsingular terminal sliding mode surface to achieve finitetime tracking including the advantages of the linear SMC and the NTSM together. In [21], adaptive FNTSM controller is established to obtain finite-time convergence of two subsystems in wheeled mobile robots. Boukattaya, et al. [22] employed an effective FNTSM algorithm with adaptation techniques to ensure that the position and the velocity errors can be stabilized to zero in finite time for robotic manipulators. On the other hand, the disadvantage of SMC

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S. Yi and J. Zhai / ISA Transactions 90 (2019) 41–51

is the chattering phenomenon due to high control gains for disturbance compensation and infinite switching action on the sliding surface, which may excite unmodeled high frequency dynamics and even result in instability. The problem has been tried to solve with boundary layer technique [23], which replaces the discontinuous sign function with continuous saturation function or sigmoid function. In [24], a boundary layer with saturation function is introduced to smooth out the control discontinuity to alleviate the chattering. But the method degrades the system performance and robustness to some extent. A wide boundary layer may cause the steady-state error while a narrow boundary layer may not eschew the chattering problem. Another effective control scheme is known as high-order sliding mode control (HOSMC) including twisting algorithm, super-twisting control [25–27], integral sliding mode control [28,29] and second-order SMC (SOSMC) [30–32]. Ashtiani and Mobayen [33] presented a robust super-twisting decoupled terminal sliding mode control scheme with the parameter-tuning adaptive law to establish the chattering-free performance and finite-time convergence for a class of fourth-order systems. In [34], an adaptive second-order terminal sliding mode controller is proposed to track precisely the reference trajectory, which is also able to eliminate the chattering by the integral action. In this paper, motivated by the above discussion, a novel adaptive second-order fast nonsingular terminal sliding mode (SOFNTSM) controller is proposed to track the n-link rigid robotic manipulator in the presence of model uncertainties and external disturbances. The main contributions of this paper are characterized as follows: (1) Compared with the existing sliding mode control such as [10–13,16–21,29–31,33,34] that cannot guarantee both global fast convergence and chattering avoidance, by combining the advantages of FNTSM and SOSMC, a second-order fast nonsingular terminal sliding manifold is proposed to design the control law, which can ensure the fast convergence and better tracking accuracy. (2) Unlike [22–24] which employed the boundary layer technique to alleviate chattering at the cost of degrading system robustness, the discontinuous sign function is included in the time derivative of the proposed control law. Therefore, actual control signal after integral is continuous and thus system is chatteringfree without affecting the tracking performance. (3) Compared with [12,16,17,31] whose bound of the system uncertainty and disturbances are usually required to be known in advance, an adaptive tuning law is designed to estimate unknown uncertainty without prior knowledge about the upper bound of system uncertainty and external disturbance. The remainder of this paper is organized as follows. Section 2 introduces mathematical model of robotic manipulators and some relevant assumptions. The adaptive SOFNTSM controller designed for trajectory tracking of robotic manipulators with Lyapunov stability analysis is presented in Section 3. Simulation studies performed on a two-link rigid robotic manipulator are provided in Section 4. Finally, conclusions are offered in Section 5.

where Fd (q, q˙ , q¨ ) = τd − ∆M(q)q¨ − ∆C (q, q˙ )q˙ − ∆G(q) is the lumped disturbance, which includes external disturbance and the system perturbations. The objective of this paper is to design a robust controller to realize fast convergence and better tracking precision. To solve the problem, the following conditions are imposed on system (2).

2. Mathematical model of robotic manipulators

s = e˙ + α e,

The dynamic equation of n-link robotic manipulator can be expressed as[16]

where s = [s1 , . . . , sn ]T , α = diag(α11 , . . . , αnn ) is a positive definite matrix. In order to guarantee s converge to zero in finite time, a fast nonsingular terminal sliding manifold is chosen as

M(q)q¨ + C (q, q˙ )q˙ + G(q) = τ + τd ,

(1)

n

where q, q˙ , q¨ ∈ R represent the vectors of joint positions, velocities and accelerations, respectively. M(q) = M0 (q) + ∆M(q) is a positive definite inertia matrix, C (q, q˙ ) = C0 (q, q˙ ) + ∆C (q, q˙ ) is the centripetal Coriolis matrix, G(q) = G0 (q) + ∆G(q) is the gravitational vector, τ is the joint torque input vector and τd is the external disturbance torque vector. Here, M0 (q), C0 (q, q˙ ), G0 (q) denote the nominal values and ∆M(q), ∆C (q, q˙ ), ∆G(q) stand for the system perturbations. Then, system (1) can be rewritten as M0 (q)q¨ + C0 (q, q˙ )q˙ + G0 (q) = τ + Fd (q, q˙ , q¨ ),

(2)

Assumption 2.1. Inertia matrix M(q) is bounded for all q even in the presence of the uncertainty. Assumption 2.2. The lumped disturbance Fd (q, q˙ , q¨ ) is bounded by

∥Fd (q, q˙ , q¨ )∥ < a0 + a1 ∥q∥ + a2 ∥˙q∥2 ,

(3)

where a0 , a1 , a2 are unknown positive constants and ∥ · ∥ denotes the Euclidean norm of a vector. Remark 2.1. According to the characteristics of industrial robotic manipulators, the above assumptions are reasonable, such as [17, 22,34]. In addition, ∥F˙d ∥ is limited by the maximum change rate of external disturbance and system perturbations, whose accurate value does not be known precisely, such as [26,27,32]. 3. Adaptive SOFNTSM controller design Let qd be the desired position vector. The tracking error and its derivatives ... ... are given as e = q − qd , e˙ = q˙ − q˙ d , e¨ = q¨ − q¨ d and ... e = q − q d . From (2), one has e¨ = M0−1 (q) τ − (C0 (q, q˙ )q˙ + G0 (q)) + Fd (q, q˙ , q¨ ) − q¨ d .

(

)

(4)

Then, by a direct calculation, it yields ( ) ... ˙ −1 (q) τ − (C0 (q, q˙ )q˙ + G0 (q)) + Fd (q, q˙ , q¨ ) e =M 0

) ... ( d + M0−1 (q) τ˙ − (C0 (q, q˙ )q˙ + G0 (q)) + F˙d (q, q˙ , q¨ ) − q d dt ( ) ˙ −1 (q) τ − (C0 (q, q˙ )q˙ + G0 (q)) + M −1 (q) =M 0 0 ( ) d × τ˙ − (C0 (q, q˙ )q˙ + G0 (q)) dt

...

− q d + F (q, q˙ , q¨ ),

(5)

˙ −1 (q)Fd (q, q˙ , q¨ ) + M −1 (q)F˙d (q, q˙ , q¨ ). where F (q, q˙ , q¨ ) = M 0 0 Under Assumptions 2.1 and 2.2, one has ∥F (q, q˙ , q¨ )∥ < v0 + v1 ∥q∥ + v2 ∥˙q∥2 ,

(6)

where v0 , v1 and v2 are unknown positive constants. In the following, we will design a robust controller for system (2) to track the reference trajectory. First of all, consider a linear sliding surface as (7)

σ = s˙ + K1 s + K2 λ(s),

(8)

where Kj = diag(kj1 , . . . , kjn ), j = 1, 2 are positive definite matrices. λ(s) = [λ(s1 ), . . . , λ(sn )]T is governed by

λ(si ) =

{

q/p

si ,

γ1 si + γ2 sign(si )s2i ,

if σ¯ i = 0 or σ¯ i ̸ = 0, |si | ≥ µ if σ¯ i ̸ = 0, |si | < µ

,

(9)

where µ is a sufficiently small positive constant, σ¯ i = s˙i + k1i si + q/p k2i si , i = 1, . . . , n. p and q are positive odd integers which satisfy 1 < pq < 1. γ1 = (2 − pq )µq/p−1 and γ2 = ( pq − 1)µq/p−2 . 2

S. Yi and J. Zhai / ISA Transactions 90 (2019) 41–51

43

Fig. 1. Control diagram of the designed scheme.

The time derivative of (8) is

By a direct calculation, the time derivative of V is

σ˙ = s¨ + K1 s˙ + K2 λ¯ (s, s˙),

(10)

) ( =σ T s¨ + K1 s˙ + K2 λ¯ (s, s˙) + (vˆ 0 − v0 )v˙ˆ 0 + (vˆ 1 − v1 )vˆ˙ 1 ξ2 (vˆ 2 − v2 )v˙ˆ 2 .

where λ¯ (s, s˙) = [λ¯ (s1 , s˙1 ), . . . , λ¯ (sn , s˙n )]T is given by

λ¯ (si , s˙i ) =

⎧ ⎨ q sq/p−1 s˙ ,

if σ¯ i = 0 or σ¯ i ̸ = 0, |si | ≥ µ

⎩γ s˙ + 2γ sign(s )s s˙ , 1 i 2 i i i

if σ¯ i ̸ = 0, |si | < µ.

p

i

i

,

The joint torque control law τ consists of the equivalent control ueq and the auxiliary control uau , that is t

∫

(

u˙ eq (ζ ) + u˙ au (ζ ) dζ ,

)

(12)

0

where u˙ eq can be obtained by (10) with σ˙ (t) = 0 and F (q, q˙ , q¨ ) = 0, the auxiliary control uau is introduced to deal with the uncertainty. Then u˙ eq and u˙ au are designed as u˙ eq =

u˙ au

d dt

... (C0 (q, q˙ )q˙ + G0 (q)) + M0 (q) q d − α M0 (q)e¨

( ) ˙ −1 τ − (C0 (q, q˙ )q˙ + G0 (q)) − M0 M 0 ) ( − M0 (q) K1 s˙ + K2 λ¯ (s, s˙) ( ) = − M0 (q) βσ + (vˆ 0 + vˆ 1 ∥q∥ + vˆ 2 ∥˙q∥2 )sign(σ ) ,

v˙ˆ 1 = ∥σ ∥∥q∥,

v˙ˆ 2 = ∥σ ∥∥˙q∥2 .

(13)

(14)

Since the switching term sign(σ ) is contained in (13), the actual control law τ is continuous after integration and chattering phenomenon in the system can be eliminated. The control diagram of the designed scheme is shown in Fig. 1. Theorem 3.1. Considering the nonlinear uncertain system (2), the tracking error dynamics can reach the sliding mode surface in finite time, and then asymptotically converge to zero via the second-order fast nonsingular terminal sliding mode (8), the control law (12) and the adaptive law (14). Proof. Define the adaption error as v˜ i = vi − vˆ i , i = 0, 1, 2 and consider the Lyapunov function candidate as V =

1( 2

) σ T σ + v˜ 02 + v˜ 12 + v˜ 22 .

(

˙ −1 (q) τ − C0 (q, q˙ )q˙ + G0 (q) V˙ =σ T M 0

(

))

( )) ... d( + M0−1 (q) τ˙ − C0 (q, q˙ )q˙ + G0 (q) − q d dt ) + F (q, q˙ , q¨ ) + α¨e + K1 s˙ + K2 λ¯ (s, s˙) + (vˆ 0 − v0 )v˙ˆ 0 + (vˆ 1 − v1 )v˙ˆ 1 + (vˆ 2 − v2 )v˙ˆ 2 . Applying (13) gives V˙ =σ T −βσ − (vˆ 0 + vˆ 1 ∥q∥ + vˆ 2 ∥˙q∥2 )sign(σ ) + F (q, q˙ , q¨ )

(

)

+ (vˆ 0 − v0 )v˙ˆ 0 + (vˆ 1 − v1 )v˙ˆ 1 + (vˆ 2 − v2 )v˙ˆ 2 ( ) ≤σ T F (q, q˙ , q¨ ) − (vˆ 0 + vˆ 1 ∥q∥ + vˆ 2 ∥˙q∥2 )sign(σ ) + (vˆ 0 − v0 )vˆ˙ 0 + (vˆ 1 − v1 )vˆ˙ 1 + (vˆ 2 − v2 )vˆ˙ 2 ≤ ∥F (q, q˙ , q¨ )∥ ∥σ ∥ − (vˆ 0 + vˆ 1 ∥q∥

where β = diag(β11 , . . . , βnn ) is a positive definite matrix, and vˆ i , i = 0, 1, 2 are the estimates of vi , whose adaptive laws are given by

v˙ˆ 0 = ∥σ ∥,

(16)

Substituting (5) and (7) into (16), it yields

(

(11)

τ = ueq + uau =

V˙ =σ T σ˙ + (vˆ 0 − v0 )v˙ˆ 0 + (vˆ 1 − v1 )v˙ˆ 1 + (vˆ 2 − v2 )v˙ˆ 2

(15)

+ vˆ 2 ∥˙q∥2 ) ∥σ ∥ + (v0 + v1 ∥q∥ + v2 ∥˙q∥2 ) ∥σ ∥ − (v0 + v1 ∥q∥ + v2 ∥˙q∥2 ) ∥σ ∥ + (vˆ 0 − v0 ) ∥σ ∥ + (vˆ 1 − v1 ) ∥σ ∥ ∥q∥ + (vˆ 2 − v2 ) ∥σ ∥ ∥˙q∥2 ≤ − ρ ∥σ ∥ , where ρ = (v0 + v1 ∥q∥ + v2 ∥˙q∥2 ) − ∥F (q, q˙ , q¨ )∥. According to (6), one has ρ > 0. It can be concluded that σ and v˜ i are both bounded under the control law (12) and the adaptive law (14). Take the Lyapunov function V0 = 21 σ T σ . From the above inequality, it is known that V˙ 0 = V˙ + v˜ 0 v˙ˆ 0 + v˜ 1 v˙ˆ 1 + v˜ 2 v˙ˆ 2

≤ −ρ ∥σ ∥ + (v0 − vˆ 0 )v˙ˆ 0 + (v1 − vˆ 1 )v˙ˆ 1 + (v2 − vˆ 2 )v˙ˆ 2 .

(17)

Case 1: When all vˆ i (0) < vi , i = 0, 1, 2, the analysis is composed of two steps. Step 1: With the aid of v˙ˆ i ≥ 0, suppose that there is the finite time T1 ≥ 0 such that all vˆ i (t) ≥ vi , t ≥ T1 , which gives rise to V˙ 0 (t) ≤ −ρ ∥σ ∥, t ≥ T1 . Step 2: Otherwise, there exists another finite time T2 such that at least one of v˙ˆ i = 0 and vˆ i (t) < vi , t ≥ T2 . Without loss of generality, assume that v˙ˆ 2 (t) = 0 and vˆ 2 (t) < v2 , t ≥ T2 , which yields V˙ 0 (t) ≤ −ρ ∥σ ∥ + (v0 − vˆ 0 )v˙ˆ 0 + (v1 − vˆ 1 )v˙ˆ 1 , t ≥ T2 . As for vˆ 0 and vˆ 1 , the analysis follows the same way as the

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S. Yi and J. Zhai / ISA Transactions 90 (2019) 41–51

Remark 3.1. The proposed method will be applicable when ∥σ ∥ = 0 is reachable. Practically, ∥σ ∥ cannot become exactly zero for all the times due to measurement noise and switching delay [35]. As a result, the adaptive parameters vˆ 0 , vˆ 1 and vˆ 2 may increase slowly and boundlessly. A promising method to overcome this drawback is to modify the adaptation law by utilizing the dead zone technique as

{ ∥σ ∥ , if ∥σ ∥ ≥ ε , v˙ˆ 0 = 0, if ∥σ ∥ < ε { ∥σ ∥ ∥q∥ , if ∥σ ∥ ≥ ε , v˙ˆ 1 = 0, if ∥σ ∥ < ε { ∥σ ∥ ∥˙q∥2 , if ∥σ ∥ ≥ ε , v˙ˆ 2 = 0, if ∥σ ∥ < ε Fig. 2. Configuration of a two-link robotic manipulator.

Step 1 or Step 2. Therefore, we can get V˙ 0 (t) ≤ −ρ ∥σ ∥ in some finite time T3 . Case 2: When at least one of vˆ i (0) ≥ vi , i = 0, 1, 2, one has vˆ i (t) ≥ vi , t ≥ 0, that is, (vi − vˆ i )v˙ˆ i ≤ 0. As for vˆ j (0) < vj , j ̸ = i, similar to the analysis of Case 1, it yields that V˙ 0 (t) ≤ −ρ ∥σ ∥ in finite time thus the finite-time convergence of σ can be guaranteed. In the view of Case 1 and Case 2, for any initial condition (σ (0), vˆ i (0)), there always exist a positive constant ρ and a finite time constant T such that

√

1

V˙ 0 (t) ≤ −ρ ∥σ ∥ ≤ − 2ρ V02 , t ≥ T .

(19) (20) (21)

where ε is a small positive constant. The gains vˆ i , i = 0, 1, 2 go on increasing during ∥σ ∥ ≥ ε due to the adaptation laws (19)–(21) up to a value large enough to counteract the bounded uncertainty with unknown bounds till the real sliding mode starts. The adaptive parameters will maintain a constant value such that finite-time convergence to a neighborhood of σ can be guaranteed as long as the real sliding mode is reached, which allows a given accuracy of σ -stabilization. Remark 3.2. The scalar ε is quite important for the adaptation law. If ε is chosen to be too small, ∥σ ∥ will never stay lower than ε . It yields that vˆ i is increasing, which induces larger oscillation. On the other hand, if ε is too large, ∥σ ∥ is evolving around ε , and it follows that controller accuracy is not as good as possible. Considering all this, ε should rather be larger than smaller.

(18) 4. Simulation studies

Hence, according to the Lyapunov stability criterion, the FNTSM manifold σ = 0 can be established in finite time. On the other hand, once σ reaches zero, it will stay zero thereafter. Thus, the sliding variable s will converge to zero along the FNTSM manifold in finite time. As s = s˙ = 0, along the linear sliding manifold, the tracking error of the robotic manipulator e(t) = q − qd can asymptotically converge to zero. This completes the proof. □

In this section, the simulations are conducted for demonstrating the effectiveness of the proposed control scheme and compared with the existing controllers in [22,23]. The designed adaptive SOFNTSM controller is applied for the trajectory tracking of a two-link rigid robotic manipulator shown in Fig. 2. The dynamic of robotic manipulator in (1) and (2) by Lagrangian equation is

Fig. 3. Position tracking responses of joint 1 and 2.

S. Yi and J. Zhai / ISA Transactions 90 (2019) 41–51

45

Fig. 4. Velocity tracking responses of joint 1 and 2.

Fig. 5. Position and velocity tracking errors of joint 1 and 2.

ˆ 1 and m ˆ 2 determine the terms analysis. Note that the known m M0 (q), C0 (q, q˙ ) and G0 (q) used in the proposed controller. The initial angle displacements and velocities are q(0) = [0.2, 2.1]T rad and q˙ (0) = [0, −0.1]T rad/s, respectively. The reference signals are selected as

represented as m2 )l21

⎡

m2 l22

(m1 + + M(q) = ⎣ +2m2 l1 l2 cos(q2 ) + J1 m2 l22

m2 l22 + m2 l1 l2 cos(q2 )

⎤ ⎦,

m2 l22

+ m2 l1 l2 cos(q2 ) + J2 [ ] −m2 l1 l2 sin(q2 )q˙ 2 −m2 l1 l2 sin(q2 )(q˙ 1 + q˙ 2 ) C (q, q˙ ) = , m2 l1 l2 sin(q2 )q˙ 1 0 [ ] (m1 + m2 )l1 g cos(q1 ) + m2 l2 g cos(q1 + q2 ) G(q) = , m2 l2 g cos(q1 + q2 )

[

1.45 − (7/5)e−t + (3/5)e−4t

qd =

1.25 + e−t − (1/2)e−4t

] .

(23)

The following external disturbances are applied to the manipulator (22)

where q1 (t) and q2 (t) are the angular positions of revolute joints 1 and 2, τ = [τ1 , τ2 ]T is the input torque. Moreover, the two-link robotic manipulator has four inner states x1 (t) = q1 (t), x2 (t) = q˙ 1 (t), x3 (t) = q2 (t), x4 (t) = q˙ 2 (t), two outputs y1 (t) = q1 (t), y2 (t) = q2 (t) and two inputs u1 (t) = τ1 (t), u2 (t) = τ2 (t). The frictional effects in the joints are ignored. Table 1 enlists the physical attributes of the manipulator used for the simulation

[ τd =

]

2 sin(t) + 0.5 sin(200π t) cos(2t) + 0.5 sin(200π t)

.

(24)

Furthermore, considering that the sudden load variation may be involved in the running robotic manipulators such as suddenly picking an object up, we can assume that the mass of joint 2 increases to 2.0 kg after t ≥ 2 s. The tuning parameters required for adaptive SOFNTSM controller are listed in Table 2.

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Fig. 6. Control input torques of joint 1 and 2.

Fig. 7. Sliding surfaces versus time. Table 1 Physical parameters of the two-link robotic manipulator.

Table 2 Controller parameters for manipulator system.

Symbol

Definition

Value

Parameters p q µ

ˆ1 m ˆ2 m m1 m2 l1 l2 J1 J2 g

Nominal mass of link 1 Nominal mass of link 2 Mass of link 1 Mass of link 2 Length of link 1 Length of link 2 Moment of inertia of link 1 Moment of inertia of link 2 Gravitational constant

0.4 kg 1.2 kg 0.5 kg 1.5 kg 1m 0.85 m 5 kg m2 5 kg m2 9.81 m/s2

Values

Simulation results are presented in Figs. 3–8. The positions and velocities of joints 1 and 2 in comparison with the desired paths are illustrated in Figs. 3 and 4, respectively. Tracking error signals are shown in Fig. 5. It is observed that favorable trajectory tracking responses are obtained by using the proposed control law. Fig. 6

K1

K2

α

β

vˆ 0 (0) vˆ 1 (0) vˆ 2 (0) ε

5 3 0.005 diag(3, 3) (3, 3) (5, 5) (50, 50) 0

0

0

0.2

exhibits the control torques, where one can clearly see that the input signals are continuous and chattering-free. The time responses of the sliding surfaces are plotted in Fig. 7, which confirms that they can converge to zero quickly. Fig. 8 displays the convergence of the adaptive parameters. Considering the impact of measurement noise on the actual position angles, a Gaussian white noise with zero mean value, variance of 0.001 and sampling time of 0.01 s is applied. The position tracking responses, sliding surfaces and control input torques of joint 1 and 2 are illustrated in Figs. 9–11. Note that sliding surfaces and control torques have a slight chattering because of the measurement noise, while position tracking signals contain almost no

S. Yi and J. Zhai / ISA Transactions 90 (2019) 41–51

47

Fig. 8. Adaptation of parameters.

Fig. 9. Position tracking responses of joint 1 and 2 with measurement noise.

chattering under the designed controller, which is quite similar to the result in Fig. 3. These simulation results verify the good tracking performance and the strong robustness of the designed control approach against the model uncertainties, external disturbances, sudden load variation and measurement noise. To better demonstrate the superiority of the proposed method, two other types of sliding mode controllers are also considered in simulations for the purpose of comparison, which are adaptive nonsingular fast terminal sliding mode control of Boukattaya et al. [22] and adaptive super-twisting global nonlinear sliding mode control of Mobayen et al. [23]. The sliding manifold s, the control law τ and the adaptive law designed by Boukattaya [22] are given as

M0 (q)

|˙e|2−η2 (1 + η1 k1 |e|η1 −1 )sign(e˙ ) ( ) τau = −M0 (q) k · s + (aˆ 0 + aˆ 1 ∥q∥ + aˆ 2 ∥˙q∥2 + ξ )sign(s) a˙ˆ 0 = λ0 ∥s∥ ∥˙e∥η2 −1 , aˆ˙ 1 = λ1 ∥s∥ ∥˙e∥η2 −1 ∥q∥ , −

η2 k2

a˙ˆ 2 = λ2 ∥s∥ ∥˙e∥η2 −1 ∥˙q∥2 ,

(25)

where 1 < η2 < 2, η1 > η2 and k1 , k2 , k, ξ , λi are positive constants. Here, similar to Assumption 2.2, aˆ 0 + aˆ 1 ∥q∥+ aˆ 2 ∥˙q∥2 is the estimate of the upper bound of the unknown system uncertainty. The sliding manifold s, the control law τ and the adaptive law developed by Mobayen [23] are presented as with r = e˙ + λe

s = e + k1 |e|η1 sign(e) + k2 |˙e|η2 sign(e˙ )

s = H r(t) − r(0) exp(−υ t)

τ = τeq + τau τeq = M0 (q)q¨ d + C0 (q, q˙ )q˙ + G0 (q)

τ = τeq + τau ( ) τeq = M0 (q)q¨ d + C0 (q, q˙ )q˙ + G0 (q) − M0 (q) λ˙e + υ r(0) exp(−υ t)

(

)

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S. Yi and J. Zhai / ISA Transactions 90 (2019) 41–51

Fig. 10. Control input torques of joint 1 and 2 with measurement noise.

Fig. 11. Sliding surfaces versus time with measurement noise.

( ) 0.5 τau = −M0 (q) k · sH T + µ0 |sH T | sign(sH T ) + υa with υ˙ a = Γˆ sign(sH T )   Γˆ˙ = κ sH T  ,

Table 3 Control parameters of two existing controllers. Controllers

(26)

where H is a constant row vector and λ, υ , k, µ0 , κ are positive constants. Here, Γˆ is the estimate of the upper bound of the unknown system uncertainty. The comparative simulations are conducted with the same initial conditions given above in the presence of model uncertainties, external disturbances and sudden load variation. In addition, for quantitative analysis and comparison, the∫integral of the absolute tf value of the error (IAE) given by IAE = 0 |e(t)| dt, the integral of the time multiplied ∫ tf by the absolute value of the error (ITAE) given by ITAE = 0 t · |e(t)| dt and the integral of the square value (ISV) of the control input given by ISV

=

∫ tf 0

τ 2 dt are

Tuning parameters

Boukattaya et al. controller [22] η1 = 2, η2 = 5/3, k1 = 1, k2 = 1, k = 50, ξ = 0.5, λ0 = λ1 = λ2 = 1 Mobayen et al. controller [23] H = [1.3, 1.2], λ = 30, υ = 4, k = 50, µ0 = 5, κ = 1

introduced [36], where tf represents the total running time. The parameters required for implementing these two controllers are listed in Table 3. The position tracking performance and control input torques are shown in Figs. 12–14. It is observed from Figs. 12 and 13 that the proposed controller has faster global convergence speed and less steady state errors than the two other controllers. Moreover,

S. Yi and J. Zhai / ISA Transactions 90 (2019) 41–51

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Fig. 12. Comparison of position tracking responses under three types of controllers.

Fig. 13. Comparison of position tracking errors under three types of controllers.

in Fig. 14, the proposed scheme is chattering-free as well as the controller in [23] thanks to the HOSMC while the controller in [22] cannot eschew the chattering problem. Table 4 exhibits the comparison of the performance indices, where one can clearly see that the proposed scheme gives lower IAE and ITAE values than the existing control methods with almost the same magnitude of control torques, which is verified through the ISV values. Consequently, the simulation results reveal that the designed method can provide faster global convergence rate, high-precision tracking, chattering avoidance, strong robustness and effective adaptive control. The superior performance of the proposed adaptive SOFNTSM controller is validated. 5. Conclusion In this paper, an adaptive second-order fast nonsingular terminal sliding mode controller is proposed for trajectory tracking of

Table 4 Comparison of the performance indices. Controllers

Joints

IAE (rad)

ITAE (rad s)

ISV (N m)2

Proposed controller

Joint 1 Joint 2 Joint 1 Joint 2 Joint 1 Joint 2

0.1254 0.0720 0.2021 0.1441 0.1394 0.1119

0.0293 0.0156 0.0735 0.0482 0.0626 0.0580

705 825 682 852 921 656

Boukattaya et al. controller [22] Mobayen et al. controller [23]

n-link rigid robotic manipulators. Combining the linear SMC and TSM, the FNTSM is presented to ensure the rapidity and accuracy of tracking control. Furthermore, the actual control is continuous since there exists the switching term containing the sign function in the derivative control signal and thus chattering is avoided by integrating the first derivative of control input. The adaptive tuning

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Fig. 14. Comparison of input torques under three types of controllers.

law is introduced to estimate the unknown uncertainties while the prior knowledge about the upper bound of system uncertainty and external disturbance is not required. Based on the Lyapunov function method, it shows that the tracking error asymptotically converges to zero. Finally, the designed SOFNTSM controller is applied to control a two-link robotic manipulator in simulation and results prove the validity of the offered control scheme. It should be noted that the presented method can be also adopted to control other highly complicated nonlinear uncertain systems.

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