Robust adaptive dual layer sliding mode controller: Methodology and application of uncertain robot manipulator

Abstract

This paper presents a novel robust adaptive dual layer sliding mode control (ADLSMC) for the problem of high accuracy tracking trajectory of robot manipulator in the presence of uncertainties and external disturbances. This new control scheme has a dual layer structure. The first layer drives the robot manipulator system reaches the global nonlinear sliding surface in finite time, and the second layer tackles the values of the two control gains overestimation problem. Moreover, compared with the traditional super-twisting with time delay estimation algorithm, the proposed controller can realize not only set-point tracking but also dynamic tracking, which is very widely used in practice. The stability of the closed–loop system and the finite time convergence are analyzed using Lyapunov techniques. The effectiveness of the proposed method is demonstrated by simulations and experimental studies.

Keywords

Adaptive tuning gains dual layer sliding mode robot manipulator tracking controller robust control

Introduction

In the last two decades, robotic manipulators have attracted a great deal of attention because of their extensive use in various sophisticated tasks, such as chirurgical ones, industrial ones and others (Hu et al., 2012; Huang et al., 2016; Mehdi and Boubaker, 2011; Van et al., 2019; Yang et al., 2018; Zhang et al., 2018). Meanwhile, the requirements for high control performance are also increasing, including fast convergence, robust control, and high trajectory tracking accuracy. How to deal with the complexity of the internal robot model dynamics and the presence of external disturbance has always been a challenge for robotic manipulator (Golestani et al., 2016; Zhang et al., 2018).

Therefore, to achieve high tracking performance in the presence of uncertainties for robotic manipulator becomes inevitable, which needs further study and improvement (Cuong and Wang, 2016). Until now, to cope with the trajectory tracking control of these complex systems, lots of advanced nonlinear control strategies have been widely employed, such as a backstepping control (BC) (Hu et al., 2012; Van et al., 2019), finite time control (FTC) (Galicki, 2016; Golestani et al., 2016; Yang et al., 2018; Zhang et al., 2018; Zhao et al., 2010), neural network control (NNC) (Chen et al., 2020; Cuong and Wang, 2016; Sun et al., 2011), observer-based control (OBC) (Chen et al., 2012; Chen et al., 2019; Sun and Liu, 2020; Yang et al., 2012), adaptive control (AC) (Ankur and Akhilesh, 2017; Mobayen et al., 2017; Utkin and Poznyak, 2013; Hu et al., 2020), and active disturbance rejection control (ADRC) (Castaneda and Juarez, 2015). These control schemes are conceptually straightforward and have obtained great popularity theoretically as well as practically.

One potential control method to appear, due to its simplicity structure and robustness against system uncertainties, is sliding mode control (SMC) (Ankur and Akhilesh, 2017; Feng et al., 2002; Mobayen et al., 2017; Moreno and Osorio, 2008; Shtessel et al., 2010; Utkin and Poznyak, 2013; Yu et al., 2005). Conventional sliding mode controller is called the first-order sliding mode controller (1-SMC). Its design is based on a chosen slide surface function in system state space and the system trajectories to converge to the chosen surface by a high gain switching law. It should be noted that the high gains of the SMC controllers are usually chosen to be larger than the upper-bound of uncertainties (Guo et al., 2017; Guo, 2018). This excessive choice can ensure fast convergence and superior robustness but may cause the sever control chattering phenomenon, which is the major drawback of SMC. Consequently, in order to eliminate chattering phenomenon, so-called second order sliding mode controller (2-SMC) appeared. In particular, the super twisting is a popular second order sliding mode control algorithm since its inception (Edwards and Shtessel, 2014, 2015, 2016a, 2016b).

Recently, there is more interest in the adaptive super twisting algorithm (ASTA) and new researches have been proposed (Barth et al., 2016; Kali et al., 2018; Mobayen et al., 2017; Yassine et al., 2018). The amplitude of chattering is proportional to the magnitude of switching gain. From the perspective of eliminating chattering, one possible adaptation scheme is to reduce this amplitude to the minimum admissible value dictated by the conditions for guaranteeing a sliding motion (Kobayshi and Furuta, 2007; Plestan et al., 2010). However, in a succession of promising papers, Lyapunov methods have been discovered successfully to analyze the properties of the super twisting algorithm for uncertain systems for the first time (Moreno and Osorio, 2008; Shtessel et al., 2010). The existing literature on adaptive approaches can be broadly split into two broad categories. The first one most extensively researched is involved in developing adaptive terms to increase the switching control gains so that a second order sliding is established. Numerous publications relating to such control structures can be found (Alwi and Edwards, 2013; Shtessel et al., 2012). The second category of research papers tries to find the smallest gain values whilst guaranteeing a sliding motion. This idea has been subsequently pursued by Guo et al. (2017), Guo (2018), and Edwards and Shtessel (2014, 2015, 2016a, 2016b) through a so-called adaptive dual concept.

Most of them presented are only simulation results and, generally, the control methods are based on simplified models without modeling errors and external disturbances. In this paper, the design methodology is motivated by Edwards and Shtessel (2014), a novel robust adaptive dual layer sliding mode control (ADLSMC) for the problem of high accuracy tracking trajectory of robot manipulator in the presence of uncertainties and external disturbances is proposed. The main contribution of this paper in comparison with the related are listed as follows:

A novel robust ADLSMC for the problem of high accuracy tracking trajectory of robot manipulator in the presence of uncertainties and external disturbances is proposed. The control scheme allows the magnitude and rate of change of the two gains to adapt whilst guaranteeing a sliding motion.

By introducing the equivalent control technique, the proposed ADLSMC method can guarantee the adaptive gain minimization, which is quite different from holding the constant value gain like other existing adaptive sliding mode controls.

A new control technique has been suggested to establish strong robust performance and finite time convergence, the proposed controller can realize not only set-point tracking, but also dynamic tracking, which is very widely used in practice.

The remainder of this article is organized as follows. Section 2 introduces mathematical modeling of n-link robotic manipulator and problem formulation. In Section 3, the proposed ADLSMC controller is designed and applied in robotic manipulator. The closed-loop stability analysis of robotic manipulator system is also presented in this section. In Section 4, numerical simulations on a 2-Degree of Freedom (DOF) robot manipulator are proposed. Besides, practical implementation strategy are performed on 2-DOF robot arm and compared with the classical STA with time delay to demonstrate the proposed controller efficacy. Finally, concluding remarks are drawn in Section 5.

Mathematical modeling and control objective

Mathematical model of robot manipulator

The dynamic equation of n-DOF robotic manipulator with external disturbances is given by the following general equations (Chen et al., 2019)

M (q) \overset{\cdot\cdot}{q} + C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G (q) + F (\overset{\cdot}{q}) = τ + τ_{d}

(1)

where $q, \overset{\cdot}{q}, \overset{\cdot\cdot}{q} \in R^{n}$ are the vectors of joint positions, velocities, and accelerations, respectively. $M (q) \in R^{n \times n}$ is the bounded positive-definite inertia matrix, $C (q, \overset{\cdot}{q}) \in R^{n \times n}$ is centrifugal and Coriolis matrix, $G (q) \in R^{n}$ indicates the gravitational vector, $F (\overset{\cdot}{q}) \in R^{n}$ is the vector of viscous friction forces, $τ \in R^{n}$ is the control input torque and $τ_{d} \in R^{n}$ denotes the external disturbances torque. Practically, the value of these physical parameters cannot be obtained precisely because of modeling error, payload variations, external disturbance and other inherent factors. Hence, the uncertainties resulting from parameter error system can be denoted as

{\begin{matrix} \begin{matrix} M (q) = M_{0} (q) - Δ M (q) \\ C (q, \overset{\cdot}{q}) = C_{0} (q, \overset{\cdot}{q}) - Δ C (q, \overset{\cdot}{q}) \end{matrix} \\ G (q) = G_{0} (q) - Δ G (q) \\ F (\overset{\cdot}{q}) = F_{0} (\overset{\cdot}{q}) - Δ F (\overset{\cdot}{q}) \end{matrix}

(2)

where the notations $M_{0} (q)$ , $C_{0} (q, \overset{\cdot}{q})$ , $G_{0} (q)$ , $F_{0} (\overset{\cdot}{q})$ are the nominal matrices, and $Δ M (q)$ , $Δ C (q, \overset{\cdot}{q})$ , $Δ G (q)$ , $Δ F (\overset{\cdot}{q})$ represent the system uncertain parts.

Consequently, the dynamic robot manipulator equation (1) can be rewritten as

\begin{matrix} M_{0} (q) \overset{\cdot\cdot}{q} + C_{0} (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G_{0} (q) + F_{0} (\overset{\cdot}{q}) = τ + τ_{d} + Δ M (q) \overset{\cdot\cdot}{q} + \\ Δ C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + Δ G (q) + Δ F (\overset{\cdot}{q}) \end{matrix}

(3)

Then, in the presence of the uncertainties and disturbances, the dynamic equation (3) can be expressed as

\begin{matrix} \overset{\cdot\cdot}{q} = M_{0} {(q)}^{- 1} {- C_{0} (q, \overset{\cdot}{q}) \overset{\cdot}{q} - G_{0} (q) - F_{0} (\overset{\cdot}{q}) + τ + τ_{d} \\ + Δ M (q) \overset{\cdot\cdot}{q} + Δ C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + Δ G (q) + Δ F (\overset{\cdot}{q}) \end{matrix}}

(4)

From equation (4), the simplified dynamical model can be written as

\overset{\cdot\cdot}{q} = f_{0} (q, \overset{\cdot}{q}) + M_{0} {(q)}^{- 1} τ (t) + D (t)

(5)

where $D (t) = M_{0} {(q)}^{- 1} {τ_{d} + Δ M (q) \overset{\cdot\cdot}{q} + Δ C (q, \overset{\cdot}{q}) \overset{\cdot}{q} + Δ G (q) + Δ F (\overset{\cdot}{q})}$ is the total lumped system uncertainty and disturbances, and $f_{0} (q, \overset{\cdot}{q}) = - M_{0} {(q)}^{- 1} {C_{0} (q, \overset{\cdot}{q}) \overset{\cdot}{q} + G_{0} (q) + F_{0} (\overset{\cdot}{q})}$ is the bounded known nonlinear function.

Remark 1: From equation (1), it is noted that the robotic manipulators are not only highly coupled, but also suffer from time varying dynamic parameters that could further lead to robot system instability, such as unknown parameters, fraction and unmodeled nonlinearities, payload variations, and highly coupled and gear backlash can existed and in the robotic manipulator. In practice, it is very difficult to obtain mathematical model precisely, but it is feasible to obtain partial model knowledge. Hence, prior knowledge of the mathematical model should be used properly for model-based controller design rather than totally discarded, as proposed in Chen et al. (2019).

Problem formulation

The control objective is to design a robust ADLSMC for high accuracy tracking a desired known trajectory of robot manipulator in the presence of uncertainties and external disturbances

lim_{t \to t_{f}} | e | = lim_{t \to t_{f}} | (q - q_{d}) | = 0

(6)

where $q_{d} \in R^{n}$ is a double differentiable desired trajectory; $e = q - q_{d}$ is trajectory tracking error; $t_{f}$ is finite time. To this end, the controller will be designed and its stability analysis will be carried out by some basic assumptions and lemmas as follows.

Assumption 1: The joint position $q$ and velocity $\overset{\cdot}{q}$ states are measurable via the robot sensors.

Assumption 2: The total lumped uncertainty $ψ D (t)$ satisfies $| ψ D (t) | ⩽ ω_{0}$ and $| ψ \overset{\cdot}{D} (t) | ⩽ ω_{1}$ , where $ψ$ is a constant diagonal matrix, which will be given later. The constant $ω_{0}$ and $ω_{1}$ are positive but are unknown.

Lemma 1: (Guo et al., 2017). Suppose that, the time invariant system below

\dot{z} = f (z), f (0) = 0

(7)

Consider system (6), suppose there is a continuous function $V (z) : D \to R$ defined on a neighborhood $U \subseteq D$ of the origin such that the following conditions hold

(1) is positive definite function on $D \subseteq R^{n}$ ;

(2) There exist real numbers $k > 0$ and $λ \in (0, 1)$ such that

\overset{\cdot}{V} (z) + k V^{λ} (z) ⩽ 0, z \in U / {0}

(8)

then system (6) is locally finite time stable. If $D = R^{n}$ and $V (z)$ is also unbounded, system (6) is globally finite time stable.

Robust ADLSMC

In this section, a robust ADLSMC will be designed to ensure the finite time convergence of the error to zero and tracking a desired known trajectory even in the presence of uncertainties and external disturbances. Figure 1 show the architecture of the closed-loop system.

Design of global nonlinear sliding surface

The global nonlinear sliding surface for system (5) can be defined as

σ (t) = ψ (θ (t) - θ (0) e^{- β t})

(9)

with

θ (t) = \overset{\cdot}{e} (t) + λ e (t)

(10)

where $e (t) = q - q_{d}$ is the joint space tracking position error with $q_{d} \in R^{n}$ is the known desired position trajectory, $\overset{\cdot}{e} (t) = \overset{\cdot}{q} - {\overset{\cdot}{q}}_{d}$ is the joint space tracking velocity error, ${\overset{\cdot}{q}}_{d}$ is the desired velocity trajectory, $ψ$ is a constant diagonal matrix, and $ψ = diag (ψ_{1}, ψ_{2}, \dots, ψ_{n})$ , $λ$ and $β$ are the positive coefficients. Taking the derivate of equation (9) with respect to time yields

\begin{matrix} \overset{\cdot}{σ} (t) = ψ (\overset{\cdot}{θ} (t) + β θ (0) e^{- β t}) \\ = ψ (\overset{\cdot\cdot}{e} (t) + λ \overset{\cdot}{e} (t) + β θ (0) e^{- β t}) \end{matrix}

(11)

where considering $\overset{\cdot\cdot}{e} (t) = \overset{\cdot\cdot}{q} - {\overset{\cdot\cdot}{q}}_{d}$ , one can deduce from equations (5) and (11) that

\begin{matrix} \dot{σ} (t) = ψ (\overset{\cdot\cdot}{q} - {\overset{\cdot\cdot}{q}}_{d} + λ \overset{\cdot}{e} (t) + β θ (0) e^{- β t}) \\ = ψ (f_{0} (q, \overset{\cdot}{q}) - {\overset{\cdot\cdot}{q}}_{d} + λ \overset{\cdot}{e} (t) + β θ (0) e^{- β t} + M_{0} {(q)}^{- 1} τ (t) + D (t)) \end{matrix}

(12)

The above equation can be rewritten as

\overset{\cdot}{σ} (t) = f (t) + h (t) τ (t) + ω (t)

(13)

where $f (t)$ , $h (t)$ , $ω (t)$ are listed as follows

{\begin{matrix} f (t) = ψ (f_{0} (q, \overset{\cdot}{q}) - {\overset{\cdot\cdot}{q}}_{d} + λ \overset{\cdot}{e} (t) + β θ (0) e^{- β t}) \\ h (t) = ψ M_{0} {(q)}^{- 1} \\ ω (t) = ψ D (t) \end{matrix}

(14)

Design of adaptive dual layer sliding mode

Then, the classical super twisting algorithm is formulated as (Levant, 2003, 2006)

{\begin{matrix} \overset{\cdot}{σ} (t) = - l_{1} (t) Λ (σ (t)) sign (σ (t)) + ϖ (t) \\ \overset{\cdot}{ϖ} (t) = - l_{2} (t) sign (σ (t)) + η (t) \end{matrix}

(15)

where $Λ (σ (t)) = diag ({| σ_{1} (t) |}^{1 / 2}, \dots, {| σ_{n} (t) |}^{1 / 2})$ , $l_{1} = diag (l_{11}, \dots, l_{1 n})$ and $l_{2} = diag (l_{21}, \dots, l_{2 n})$ are diagonal positive matrices that will be fixed later, $η (t)$ is the unknown bounded uncertainties that correspond to Assumption 2, that is, $| η (t) | ⩽ ω_{0}$ , $| \overset{\cdot}{η} (t) | ⩽ ω_{1}$ where $ω_{0}$ and $ω_{1}$ are a fixed constant. Additionally, $sign (σ (t)) = {[sign (σ_{1} (t)), \dots, sign (σ_{n} (t))]}^{T}$ with

sign (σ_{i} (t)) = {\begin{matrix} 1 if σ_{i} (t) > 0 \\ 0 if σ_{i} (t) = 0 \\ - 1 if σ_{i} (t) < 0 \end{matrix}

(16)

If the boundary $ω_{0}$ is known, we can select $l_{1} = diag (1.5 \sqrt{ω_{0}}, \dots, 1.5 \sqrt{ω_{0}})$ and $l_{2} = diag (1.1 ω_{0}, \dots, 1.1 ω_{0})$ , then it can be shown that $σ (t) = \overset{\cdot}{σ} (t) = 0$ in finite time (Edwards and Shtessel, 2016a). In this case, it is important to note that both $l_{1}$ and $l_{2}$ are constants, meanwhile the gain $l_{2}$ must be larger than the boundary $ω_{0}$ so that the switching term $l_{2} (t) sign (σ (t))$ can dominate the uncertainty term $η (t)$ , Then, the control gains may be overestimated, which yields increased chattering. Trying to find a good way to design the adaption of control gains $l_{1}$ and $l_{2}$ properly is highly significant.

Equivalent control $u_{eq} (t)$ on the sliding surface means average control (Utkin et al., 2013). Once the system reaches the sliding surface, equivalent control $u_{eq} (t)$ occurs, that is, $u_{eq} (t) = l_{2} (t) sign (σ (t)) |_{eq} = η (t)$ . The equivalent control is used to make the system moving on the sliding surface and the sliding mode gain must be larger than the upper bound of the lump uncertainties. Apparently, the system (15) moving on the sliding surface is discontinuous and switching, while the essence of control is continuous and equivalent approximation. The approximate value of the equivalent control can be get by a first order low-pass filter, which can be made arbitrary close to the actual value.

Introduce a new variable

s (t) = K (t) - \frac{1}{μ β_{0}} | u_{eq} (t) | - ϵ

(17)

where $0 < μ < 1 / β_{0} < 1$ and $ϵ > 0$ is a small positive scalar and $u_{eq} (t)$ is equivalent control on the sliding surface. The proposed adaptive control law $K (t)$ is defined as

K (t) = k_{0} + k (t)

(18)

where $k_{0}$ is a positive constant, and $k (t)$ satisfies

\overset{\cdot}{k} (t) = - ϱ (t) sign (s (t))

(19)

The time varying scalar $ϱ (t)$ is by definition

ϱ (t) = r_{0} + r (t)

(20)

where $r_{0}$ is a positive constant and $r (t)$ is given by

\overset{\cdot}{r} (t) = γ | s (t) |

(21)

where $γ$ is a fixed positive constant.

Then, a robust ADLSMC is proposed in the following theorem.

Theorem 1: Consider the uncertain robot manipulator given in equation (5). The dual layer adaptation scheme in (17)–(21) ensures the convergence of the selected global nonlinear sliding surface (9) to zero in a finite time smaller than

T_{f} = \frac{2 V^{1 / 2} (0) λ_{\max} {P}}{λ_{\min} {\bar{Q}} λ_{\min}^{1 / 2} {P}}

(22)

if the gains $l_{1} (t) = diag (l_{11} (t), \dots, l_{1 i} (t))$ and $l_{2} (t) = diag (l_{21} (t), \dots, l_{2 i} (t))$ for $i = 1, \dots, n$ satisfy

l_{1 i} (t) = 0.9 K {(t)}^{1 / 2} l_{2 i} (t) = 1.4 K (t)

(23)

l_{1 i} (t) > 0, l_{2 i} (t) > \frac{3 ω_{0} + 2 {ω_{0}}^{2}}{{l_{1 i}}^{2} (t)}

(24)

whilst the adaptive law (18)–(21) can make the adaptive gain $K (t)$ larger than the uncertainty, that is, $K (t) > | η (t) |$ , when the system (15) is moving on the sliding surface.

Proof: Frist of all, for simplicity, let us decompose the system (15) into n subsystems as follows

{\begin{matrix} {\overset{\cdot}{σ}}_{i} = - l_{1 i} (t) Λ (σ_{i}) sign (σ_{i}) + ϖ_{i} \\ \dot{ϖ_{i}} = - l_{2 i} (t) sign (σ_{i}) + η_{i} \end{matrix}

(25)

Then, the proof is split into three steps (Guo et al., 2017; Moreno and Osorio, 2008).

Step 1: If the gain satisfies $K (t) < ω_{0}$

When the system is on the reaching phase, we can obtain $| u_{eq} (t) | = l_{2 i} (t)$ because of $| sign (s (t)) |$ =1, which leads to

s (t) = K (t) - \frac{1}{μ β_{0}} l_{2 i} (t) - ϵ < 0

(26)

Hence

\overset{\cdot}{k} (t) = - ϱ (t) sign (s (t)) = ϱ (t) > 0

(27)

The gain $K (t) = k_{0} + k (t)$ is increasing on the reaching phase, so after a certain period of time $K (t) > ω_{0}$ is satisfied.

Step 2: If the gain satisfies $K (t) > ω_{0} > | η (t) |$ .

In this step, for the stability analysis, a Lyapunov function in quadratic form is selected

V (t) = Θ {(t)}^{T} P Θ (t)

(28)

where $Θ (t) = [Θ_{1} (t) Θ_{2} (t)] = {[{| σ_{i} (t) |}^{0.5} sign (σ_{i} (t)) ϖ_{i} (t)]}^{T}$ and $P$ is a positive definite symmetrical matrix as follows

P = \frac{1}{2} [\begin{matrix} l_{1 i}^{2} + 4 l_{2 i} & - l_{1 i} \\ - l_{1 i} & 2 \end{matrix}]

(29)

then, let us calculate the time derivative of $V (t)$

\overset{\cdot}{V} (t) = - \frac{1}{{| σ_{i} (t) |}^{0.5}} Θ {(t)}^{T} Q Θ (t) + η_{i} (t) q^{T} Θ (t)

(30)

where

Q = \frac{1}{2} [\begin{matrix} l_{1 i}^{2} + 2 l_{2 i} & - l_{1 i} \\ - l_{1 i} & 1 \end{matrix}], q^{T} = [- l_{1 i} 2]

(31)

From the above condition $ω_{0} > | η (t) |$ , we can get

\overset{\cdot}{V} (t) ⩽ - \frac{1}{{| σ_{i} (t) |}^{0.5}} Θ {(t)}^{T} \bar{Q} Θ (t)

(32)

where

\bar{Q} = \frac{l_{1 i}}{2} [\begin{matrix} l_{1 i}^{2} + 2 l_{2 i} - 2 ω_{0} & - l_{1 i} - \frac{2 ω_{0}}{l_{1 i}} \\ - l_{1 i} - \frac{2 ω_{0}}{l_{1 i}} & 1 \end{matrix}]

(33)

Using equation (24),the obtained matrix $\bar{Q}$ is symmetrical positive definite i.e.,

$\bar{Q} > 0$ , therefore, $\overset{\cdot}{V}$ is negative definite, so $\overset{\cdot}{V} < 0$ holds, and the system will converge to zero, then

\overset{\cdot}{V} (t) ⩽ - \frac{1}{{| σ_{i} (t) |}^{0.5}} Θ {(t)}^{T} \bar{Q} Θ (t) ⩽ - \frac{1}{{| σ_{i} (t) |}^{0.5}} λ_{\min} (\bar{Q}) 11 Θ (t) 11_{2}^{2}

(34)

One has

λ_{\min} {P} 11 Θ (t) 11_{2}^{2} ⩽ \overset{\cdot}{V} (t) ⩽ λ_{\max} {P} 11 Θ (t) 11_{2}^{2}

(35)

where $11 Θ (t) 11_{2}^{2}$ is the Euclidean norm of $Θ (t)$ , $λ_{\min} {P}$ and $λ_{\max} {P}$ denoting, respectively, the maximum and the minimum eigenvalues of $P$ , and it follows that

{| σ_{i} (t) |}^{\frac{1}{2}} ⩽ 11 Θ (t) 11_{2} ⩽ \frac{V^{\frac{1}{2}} (t)}{λ_{\min}^{\frac{1}{2}} {P}}

(36)

Combining with equation (34), we can obtain

V (t) ⩽ - c V^{\frac{1}{2}} (t)

(37)

where

c = \frac{λ_{\min} {\bar{Q}} λ_{\min}^{\frac{1}{2}} {P}}{λ_{\max} {P}}

(38)

From Lemma1 $V (t)$ can reach the equilibrium in finite time, and the converge time is given by

T_{f} ⩽ \frac{2 V^{\frac{1}{2}} (0) λ_{\max} {P}}{λ_{\min} {\bar{Q}} λ_{\min}^{\frac{1}{2}} {P}}

(39)

Step 3: On the sliding surface

Introduce an error variable $e (t)$ is expressed as

e (t) = \frac{ω_{1}}{μ β_{0}} - r (t)

(40)

where $ω_{1}$ and $μ β_{0}$ are known and the time derivative of $e (t)$ is

\overset{\cdot}{e} (t) = - γ | s (t) |

(41)

There always exists $σ (t) = \overset{\cdot}{σ} (t)$ =0 due to the system moves on the sliding surface, from equation (15), the equivalent control $u_{eq} (t)$ equals to the external disturbance $η (t)$ . Taking the derivative of $s (t)$ on the sliding surface is

\overset{\cdot}{s} (t) = \overset{\cdot}{K} (t) - \frac{1}{μ β_{0}} \frac{d}{dt} | u_{eq} (t) | = \overset{\cdot}{k} (t) - \frac{1}{μ β_{0}} \frac{d}{dt} | η (t) |

(42)

The following Lyapunov function candidate is considered for the system (17) and (40)

V (s, e) = \frac{1}{2} s^{2} (t) + \frac{1}{2 γ} e^{2} (t)

(43)

The derivate of the above equation is obtained as follows

\overset{\cdot}{V} (s, e) = s (t) \overset{\cdot}{s} (t) + \frac{1}{γ} e (t) \overset{\cdot}{e} (t)

(44)

In order to prove $\overset{\cdot}{V} (s, e) ⩽ 0$ , the proof is split into two steps.

Step 1: Derive the expression of $s (t) \overset{\cdot}{s} (t)$

From equation (42), we can get

\begin{matrix} s (t) \overset{\cdot}{s} (t) = s (t) (\overset{\cdot}{k} (t) - \frac{1}{μ β_{0}} \frac{d}{dt} | η (t) |) \\ = s (t) \overset{\cdot}{k} (t) - s (t) \frac{1}{μ β_{0}} \frac{d}{dt} | η (t) | \end{matrix}

(45)

where $0 < μ < 1 / β_{0} < 1$ , $| \overset{\cdot}{η} (t) | ⩽ ω_{1}$ . Therefore, from equation (45) it follows that

s (t) \overset{\cdot}{s} (t) ⩽ s (t) \overset{\cdot}{k} (t) - s (t) \frac{ω_{1}}{μ β_{0}}

(46)

It is easy to get $\frac{ω_{1}}{μ β_{0}} > 0$ , then we need to analyze the value range of $s (t)$ and this problem will be discussed on three cases:

If $s (t) > 0$

From equation (46), the following inequality holds

s (t) \overset{\cdot}{s} (t) ⩽ s (t) \overset{\cdot}{k} (t) - s (t) \frac{ω_{1}}{μ β_{0}} < s (t) \overset{\cdot}{k} (t) + | s (t) | \frac{ω_{1}}{μ β_{0}}

(47)

If $s (t) = 0$

It follows that

s (t) \overset{\cdot}{s} (t) = 0

(48)

If $s (t) < 0$

Similarly, from equation (46), the following inequality holds

\begin{matrix} s (t) \overset{\cdot}{s} (t) ⩽ s (t) \overset{\cdot}{k} (t) - s (t) \frac{ω_{1}}{μ β_{0}} \\ = s (t) \overset{\cdot}{k} (t) + | s (t) | \frac{ω_{1}}{μ β_{0}} \end{matrix}

(49)

Combining equation (47), equation (48) and equation (49), we can get the following important inequality

s (t) \overset{\cdot}{s} (t) ⩽ s (t) \overset{\cdot}{k} (t) + | s (t) | \frac{ω_{1}}{μ β_{0}}

(50)

Step 2: Derive the expression of $e (t) \overset{\cdot}{e} (t)$

e (t) \overset{\cdot}{e} (t) = e (t) (- γ | s (t) |)

(51)

The proof is complete.

Substituting equation (49) and equation (51) into equation (44) yields

\begin{matrix} \overset{\cdot}{V} (s, e) = s (t) \overset{\cdot}{s} (t) + \frac{1}{γ} e (t) \overset{\cdot}{e} (t) \\ ⩽ s (t) \overset{\cdot}{k} (t) + | s (t) | \frac{ω_{1}}{μ β_{0}} - e (t) | s (t) | \\ = - ϱ (t) | s (t) | + | s (t) | \frac{ω_{1}}{μ β_{0}} - e (t) | s (t) | \\ = - r_{0} (t) | s (t) | - r (t) | s (t) | + | s (t) | \frac{ω_{1}}{μ β_{0}} - e (t) | s (t) | \\ = - r_{0} (t) | s (t) | + (\frac{ω_{1}}{μ β_{0}} - r (t)) | s (t) | - e (t) | s (t) | \\ = - r_{0} (t) | s (t) | + e (t) | s (t) | - e (t) | s (t) | \\ = - r_{0} (t) | s (t) | \\ ⩽ 0 \end{matrix}

(52)

From the above equation, we can see that both $s (t)$ and $e (t)$ are bounded due to $\overset{\cdot}{V} (s, e)$ ⩽ 0. Furthermore, according to the LaSalle’s invariance principal (Shtessel et al., 2010), $s (t) \to 0$ as $t \to \infty$ . Consequently, from equation (17), we have the following inequality

| s (t) | = | K (t) - \frac{1}{μ β_{0}} | u_{eq} (t) | - ϵ | ⩽ ϵ / 2

(53)

hence

K (t) - \frac{1}{μ β_{0}} | u_{eq} (t) | - ϵ ⩾ - ϵ / 2

(54)

and therefore

K (t) ⩾ \frac{1}{μ β_{0}} | u_{eq} (t) | + \frac{ϵ}{2} > | u_{eq} (t) | + \frac{ϵ}{2} > | η (t) |

(55)

This completes the proof of Theorem 1.

Remark 2: The adaptive gain $K (t)$ is bounded instead of infinite. According to equation (45), we can get

| K (t) | < | s (t) | + \frac{1}{μ β_{0}} | u_{eq} (t) | + ϵ ⩽ | s (t) | + \frac{ω_{0}}{μ β_{0}} + ϵ

(56)

Remark 3: Although the ADLSMC controller is intuitive and simple, but it is not practical. The equivalent control $u_{eq} (t)$ is a theoretical control and cannot be obtained in practice. For practical implementation the approximate value ${\bar{u}}_{eq} (t)$ introduced by a first order low pass filter and the method proposed as

δ {\dot{\bar{u}}}_{eq} = l_{2} (t) sign (σ (t)) - {\bar{u}}_{eq} (t)

(57)

where $δ$ is a small positive constant so that ${\bar{u}}_{eq} (t)$ can be close to the real value $u_{eq} (t)$ arbitrary.

Based on the above results and analysis, the proposed robust ADLSMC of the robot manipulator is presented as follows

{\begin{matrix} τ (t) = h^{- 1} (t) (- f (t) - l_{1} (t) {| σ (t) |}^{1 / 2} sign (σ (t)) + υ (t)) \\ \overset{\cdot}{υ} (t) = - l_{2} (t) sign (σ (t)) \end{matrix}

(58)

where the adaptive gains $l_{1} (t)$ and $l_{2} (t)$ are selected by Theorem 1. $f (t)$ and $h (t)$ are given in equation (14).

Analysis of closed loop stability of robot manipulator

The stability of the closed loop of robot manipulator system is analyzed in the following Theorem 2.

Theorem 2: Consider the robot manipulator system (1) satisfies Assumptions 1 and 2, and the control law is proposed as equation (50) that can guarantee the finite time convergence of the error $e (t)$ to zero and track a desired known trajectory $q_{d}$ even in the presence of uncertainties and external disturbances asymptotically.

Proof: Substituting the control law equation (50) into the system (13) yields

{\begin{matrix} \overset{\cdot}{σ} (t) = - l_{1} (t) Λ (σ (t)) sign (σ (t)) + ω (t) + υ (t) \\ \overset{\cdot}{υ} (t) = - l_{2} (t) sign (σ (t)) \end{matrix}

(59)

Let $φ (t) = ω (t) + υ (t)$ and $η (t)$ = $\overset{\cdot}{ω} (t)$ , the above equation can be rewritten as

{\begin{matrix} \overset{\cdot}{σ} (t) = - l_{1} (t) Λ (σ (t)) sign (σ (t)) + φ (t) \\ \overset{\cdot}{φ} (t) = - l_{2} (t) sign (σ (t)) + η (t) \end{matrix}

(60)

From Assumption 2, we can obtain $| η (t) | = | \overset{\cdot}{ω} (t) | ⩽ ω_{1}$ , the system (15) converges $σ (t) = \overset{\cdot}{σ} (t) = 0$ in finite time. In particular, let $σ (t) = 0$ , then from equation (9), we get

θ (t) = θ (t) (0) \exp (- β t)

(61)

where the above equation is the solution of the following first order differential equation

\overset{\cdot}{θ} (t) + β θ (t) = 0

(62)

Since $β > 0$ , $θ (t)$ converges to zero asymptotically (Khalil, 1992), so we can deduce $θ (t) \to 0$ when $t \to \infty$ , and that leads to $e (t) \to 0$ and $q \to q_{d}$ . This completes the proof for Theorem 2.

Simulation and experiment

Simulation results

To prove the effectiveness of the proposed control scheme, simulations are conducted on a robot manipulator, as shown in Figure 2. The simulation is divided into two studies. Case 1 verifies the set-point tracking, and Case 2 demonstrates the dynamic tracking.

The dynamic equation of robot manipulator is given as (Feng et al., 2002)

\begin{matrix} M (q) = [\begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix}], C (q, \overset{\cdot}{q}) \overset{\cdot}{q} = [\begin{matrix} C_{1} \\ C_{2} \end{matrix}], \\ G (q) = [\begin{matrix} G_{1} \\ G_{2} \end{matrix}], τ_{d} = [\begin{matrix} τ_{d 1} \\ τ_{d 2} \end{matrix}] \\ m_{11} = l_{2}^{2} m_{2} + 2 l_{1} l_{2} m_{2} \cos (q_{2}) + l_{1}^{2} (m_{1} + m_{2}) + J_{1} \\ m_{12} = m_{21} = m_{2} l_{2}^{2} + m_{2} l_{1} l_{2} \cos (q_{2}) \\ m_{22} = l_{2}^{2} m_{2} + J_{2} \\ C_{1} = - l_{1} l_{2} m_{2} \sin (q_{2}) {\overset{\cdot}{q}}_{1}^{2} - 2 l_{1} l_{2} m_{2} \sin (q_{2}) {\overset{\cdot}{q}}_{1} {\overset{\cdot}{q}}_{2} \\ C_{2} = l_{1} l_{2} m_{2} \sin (q_{2}) {\overset{\cdot}{q}}_{2}^{2} \\ G_{1} = l_{2} m_{2} g \cos (q_{1} + q_{2}) + (m_{1} + m_{2}) l_{1} g \cos (q_{1}) \\ G_{2} = l_{2} m_{2} g \cos (q_{1} + q_{2}) \\ τ_{d 1} = 2 \sin (t) + 0.05 \sin (200 π t) \\ τ_{d 2} = \cos (2 t) + 0.05 \sin (200 π t) \\ Δ M_{q} = [\begin{matrix} 0.5 \sin (q_{2}) & 0.5 \cos (q_{1}) \\ 0.5 \sin (q_{1}) & 0.5 \cos (q_{2}) \end{matrix}] \\ Δ C (q, \overset{\cdot}{q}) + Δ G (q) \\ = [\begin{matrix} 0.5 \cos (q_{2}) + \sin (q_{1} + q_{2}) + 0.5 {\overset{\cdot}{q}}_{1} {\overset{\cdot}{q}}_{2} \sin (q_{2}) \\ 0.5 \cos (q_{1} + q_{2}) + {\overset{\cdot}{q}}_{1} {\overset{\cdot}{q}}_{2} \cos (q_{2}) \end{matrix}] \end{matrix}

(63)

The parameters of the robotic manipulator are listed in Table 1.

Table 1.

Parameters of 2-DOF robot manipulator.

Symbol	Definition	Value
m ₁	Nominal mass of link 1	0.4kg
m ₂	Nominal mass of link 2	1.2kg
l ₁	Length of link 1	1.0m
l ₂	Length of link 2	0.85m
J ₁	Inertia tensor of link 1	2kg.m²
J ₂	Inertia tensor of link2	2kg.m²
g	Gravity acceleration	9.80m/s²

The control objective aims to design a controller to enable the angles of the joints to track desired reference trajectory well while time extends infinitely. The effectiveness of the proposed robust ADLSMC scheme is shown through simulations and comparisons with a super twisting algorithm with time delay estimation control in Yassine et al. (2018). The simulations are performed by ignoring $F (\overset{\cdot}{q})$ . Detailed simulation parameters for both controllers are given in Table 2.

Table 2.

Simulation parameters for both controllers.

Controller	Tuning parameters
ADLSMC	$μ β_{0} = 0.95$ , $ϵ = 0.01$ , $k_{0} = 3.5$
	$r_{0} = 3.5$ , $γ = 6$ , $δ = 0.005$ , $λ = 3$
	$β = 2$ , $ψ = diag {3, 2}$
STA with TDE	$\bar{M} = diag (0.07, 0.04)$ , $λ = diag (4, 11)$
	$K_{1} = diag (4, 4)$ , $K_{2} = (3.2, 3.2)$
	$L = 1 ms$ , $T_{s} = 1 ms$
Initial condition	$q_{0} = [q_{1}, q_{2}] = {[0 0]}^{T}$ , ${\overset{\cdot}{q}}_{0} = [{\overset{\cdot}{q}}_{1}, {\overset{\cdot}{q}}_{2}] = {[0 0]}^{T}$

Case 1. The desired trajectory is defined by

\begin{matrix} q_{d} = {[q_{d 1}, q_{d 2}]}^{T} = \\ {[1.25 - (5 / 7) e^{- t} + (7 / 20) e^{- 4 t}, 1.25 + e^{- t} - (1 / 4) e^{- 4 t}]}^{T} \end{matrix}

The simulation results are given in Figure 3. The two used controllers guarantee high trajectory tracking accuracy even in the presence of uncertainties and disturbances. However, it is noteworthy that the chattering phenomenon is significantly larger with super twisting algorithm with TDE. As we can see, both the proposed control scheme and the control method in Yassine et al. (2018) demonstrate good dynamic performances, in the presence of external disturbances. However, the significant difference is that the proposed method can find the smallest gain values that will maintain sliding and counteract uncertainties. Generally speaking, larger controller gains can lead to larger chattering, therefore it has advantage of reducing the chattering. This proves the superiority of the proposed control scheme.

Case 2. The desired trajectory is selected dynamic tracking as follows

\begin{matrix} q_{d} = {[q_{d 1}, q_{d 2}]}^{T} = \\ {[(1 - e^{- t}) \sin (π t) + 3.2, (1 - e^{- t}) \sin (π t) + 1.7]}^{T} \end{matrix}

Under the dynamic tracking conditions, the trajectory tracking simulation results and corresponding analysis are as follows.

Figure 4 shows the tracking errors under super twisting algorithm with TDE controller. As we can see, the positions of joints 1 and 2 cannot be tracking the known desired trajectory. Furthermore, the tracking errors do not converge to a neighborhood around the zero as well, and large oscillations of the tracking errors appear. Obviously, the chattering in STA with TDE algorithm is larger than that in the proposed ADLSMC controller, which may lead to the robotic manipulator system instability and poor performance. The proposed controller in Figure 4 also demonstrates the positions of joints 1 and 2 in comparison with the STA with TDE algorithm. It is noted that using the proposed controller, the positions of joints 1 and 2 track the desired trajectory quickly and achieve efficient tracking performance in contrast with the super twisting algorithm with TDE controller. In general, larger control gains can lead to faster transient response and robustness against uncertainties and disturbances, which may also result in larger chattering phenomenon. The control method presented in this paper can find the smallest control gain values whilst still will guarantee a sliding motion. That is to say, it has advantage from the chattering mitigation viewpoint. As a result, the robotic system under the proposed controller has better control accuracy and dynamic property.

Experiment results

In this section, to further prove the effectiveness of the proposed controller, experiments are implemented in real time using joint 2 and joint 3 of the QJR6S industrial manipulator. The rated torques are 1.9 N.m and 1.3 N.m, and gear reduction rations are 80.8008:1 and 81.0008:1 for joints 1 and joint 2, respectively. But the other joints are locked during the experiments (Chen et al., 2019). Joint positions can be obtained by using the encoders. The servo drivers can provide estimate of joint velocities by means of its internal algorithm. We used an ARM-A8 control system with Texas Instruments operating system (TIOS) to conduct the experiments. The TIOS can improve the real time performance and the stability of control system. The ADLSMC controller is written in C++ under Windows 10 operating system. The control system utilizes EtherCAT to transfer data between the controller and servo drivers. The layout of the experimental system is demonstrated in Figure 5.

To verify the advantages of the proposed ADLSMC controller, super-twisting with time delay estimation algorithm is still tested for comparison in experiments. The desired known trajectory $q_{d}$ is given different from the section of simulation results. According to the desired known trajectory designed by the user, then $q_{d}$ generates discrete data points and stores them in txt file. The control system reads $q_{d}$ data in the txt file and further controls the current position of the robot manipulator tracking. A sampling time $T_{s} = 0.002 s$ has been selected for the experiments. Detailed experiment parameters for ADLSMC and STA with TDE algorithm are the same as in Table 2. The nominal parameters of the robot manipulator can be obtained by parameter identification, which is omitted here to save space.

From Figure 6, we can see that the proposed ADLSMC controller ensures the finite time convergence of measured trajectory to the desired trajectory with high precision due to the good performance overcomes the system uncertainties. The control torque with our designed controller can tracking the actual control torque appropriately after 2.8 seconds without chattering phenomenon. The values of the control torque are acceptable for our robot manipulator. Comparing the obtained results shown in Figure 6, it is obvious that the proposed ADLSMC controller gave a better performance than the STA with TDE algorithm in the presence or absence of uncertainties and external disturbances rejection and chattering reduction.

To evaluate the performance of the four controllers and make a quantitative comparison, we employ two performance indices as follows. Integral of the time multiplied by the absolute value of the error (ITAE)

ITAE = \int_{0}^{t_{r}} t | e (t) | dt

(64)

Integral of the square value (ISV) of the control input

ISV = \int_{0}^{t_{r}} τ^{2} dt

(65)

ITAE is an error criterion to evaluate the entire error track tracking performance, where t_r is the total running time. ITAE penalizes errors that last for a long time because time is an important factor in it. ISV represents energy consumption. The performance indices obtained from simulations are shown in Table 3.

Table 3.

Comparison of the controller performance.

Controller	Joint	ITAE (rad)	ISV (10³N²m²s)
ADLSMC	Joint 1	0.0314	3.126
	Joint 2	0.0138	2.638
STA with TDE	Joint 1	0.0732	6.301
	Joint 2	0.0426	4.972

Obviously, the robotic system under the proposed ADLSMC controller has better control accuracy, and less energy consumption than STA with TDE method.

Conclusion

In this paper, we investigate a novel robust ADLSMC with global nonlinear sliding surface for the problem of high accuracy tracking trajectory of robot manipulator in the presence of uncertainties and external disturbances. This new control scheme has a dual layer structure. The first layer drives the robot manipulator system to reach the global nonlinear sliding surface in finite time, and the second layer tackles the values of the two control gains overestimation problem. Moreover, compared with the traditional super-twisting with time delay estimation algorithm, the proposed controller can realize not only set-point tracking, but also dynamic tracking, which is very widely used in practice. The stability of the closed –loop system and the finite time convergence are analyzed using Lyapunov techniques. The effectiveness of the proposed method is demonstrated by simulations and experimental studies. It should be noted that the proposed technique can be applied to a large class of nonlinear control problems even in the presence of uncertainties and external disturbances.

Footnotes

Acknowledgements

The authors wish to thank the Editor and anonymous reviewers for their time and their constructive comments and suggestions to review and improve the quality of this paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research project was supported by Tianjin Science and Technology Support Key Project, China, under Grant No. 19YFZCSN00360; China Postdoctoral Science Foundation, under Grant No. 2020M680445; Postdoctoral Science Foundation of Beijing Academy of Agriculture and Forestry Sciences of China, under Grant No. 2020-ZZ-001.

ORCID iD

YA-Jun Ma

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