Adaptive Synchronization Sliding Mode Control for an Uncertain Dual-Arm Robot with Unknown Control Direction

Abstract

In this paper, an adaptive synchronization sliding mode control is proposed for a dual-arm robot against parameter variations, external disturbance, and unknown control directions. The proposed control is designed by using cross-coupling error and sliding mode control to guarantee the position synchronization of the dual-arm manipulator. The control objective of the proposed control is to synchronize the movement of both arms beside the trajectory tracking issue. In order to manage the lumped uncertainties caused by the parameter variations, external disturbance, and unknown control directions, an extended state observer is used in the proposed control. It enhances the stability of the controlled system against uncertainties. Additionally, a Nussbaum gain function is integrated into the control algorithm to deal with the issue of unknown control direction. Lyapunov stability theory is used to demonstrate the stability of the controlled system. Finally, some simulations are implemented in MATLAB Simulink with a dual 3-DOF manipulator system. The results of the proposed control are compared to other controllers to verify its effectiveness.

1. Introduction

Nowadays, many robotic applications are employed beyond the working capacity of a single robotic manipulator. Some applications [1] concerning manufacturing and automotive applications, for example, assembly, packaging, painting, and welding, usually require high flexibility and maneuverability. A single robot cannot manage these tasks well because of limitations of time and space in the single robot. A multi-robotic system gives the potential ability to handle them well. Some other motivations [2] for developing multi-robot systems are the following: (1) distributed properties; (2) more time/energy efficiency; (3) fewer system requirements; (4) strong robustness and adaptivity; and (5) high scalability and flexibility. The challenges in multi-robot systems are communication, synchronization, and tracking problems. The synchronization problem in multi-robot manipulators is a significant challenge that has attracted many researchers worldwide [3]. It ensures that multiple robots follow the same motion sequence simultaneously. In practice, both synchronous and tracking objectives are required in control design.

In 2002, a synchronized controller [4] based on a cross-coupling approach and adaptive strategy was developed for a two-manipulator system. The multiple manipulators obtain both tracking and synchronous objectives by driving the cross-coupling error to zero. The effectiveness of the proposed method was verified by both simulation and experiments. In 2010, a sliding mode control [5] was designed for a dual-arm robotic system. The objectives of this study were safe load handling, transportation, and trajectory realization. In 2011, a fuzzy sliding mode control [6] was proposed for the cooperative motion of a dual-arm robot against parameter variations and external disturbances in load handling and transport applications. In 2014, a finite-time synchronous control [7] was developed for multiple manipulators under the presence of sensor saturation. The effectiveness of this approach was demonstrated by theory and simulation results. In 2014, nonlinear dynamics and synchronization controllers [8] were designed for a manipulator system against joint frictions and parameter variations. The four synchronous controllers are neuron synchronization controllers, an improved OPCL synchronization controller, an MRAC-PD synchronization controller, and an improved adaptive synchronization control. In 2015, a decentralized robust fuzzy adaptive control [9] was constructed for grasping an object of a dual-arm robot with unknown dead-zone input. The proposed control makes the motion and internal force track a reference trajectory against the parametric uncertainties and external disturbances. In 2015, an adaptive synchronous PI-type sliding mode control [10] was designed for two cooperative robotic manipulators to handle a lightweight beam application against uncertainties and external disturbances. The proposed control forced both tracking and synchronization errors to reach zero simultaneously. In 2016, an adaptive synchronized tracking control [11] was developed for multiple robotic manipulators under the presence of uncertain kinematics and dynamics. The tracking and coordinating objectives in the multiple robotic manipulator systems are guaranteed. In 2017, an adaptive neural network sliding mode control [12] was applied for a dual-arm manipulator (DAM) against external disturbance and parametric uncertainties. The radial basis function network is constructed to estimate the unknown dynamic model in the DAM. In 2019, an adaptive cooperative image-based approach [13] was constructed for coordinating manipulators with time-varying actuator constraints and uncertain dynamics. The challenges in this study are kinematic and dynamic uncertainties, uncalibrated camera models, and actuator constraints. A Nussbaum-type gain was conducted to manage the time-varying actuator constraints. In 2020, an adaptive neural-network-based dynamic surface control [14] was developed for a dual-arm robot against parameter variation, external disturbance, and actuator nonlinearities. The neural network was used to approximate the uncertainties in the robotic system. In 2021, a consensus sliding mode control [15] was applied for multiple manipulators against external disturbances. The proposed approach was developed by combining graph theory and sliding mode control. In 2022, a synchronous control [16] was proposed for a 2-DOF master–slave manipulator against the asymmetric nonlinear dead-zone characteristics. The dead-zone issues were managed by the recursive least squares method. In 2023, a multi-objective synchronization control [17] was proposed for a dual-robot to obtain high-accuracy motion and force compliance behavior simultaneously. The proposed method was developed based on an integral of the past synchronization force error and the integral of the future predicted synchronization motions error for designing the impedance-based force feedback law and motion feedback law, respectively. In 2023, an adaptive synchronous sliding mode control [18] was designed for a dual-arm robot against parameter variations. The approach used an adaptive law to adapt the robust gains in the sliding mode control to manage uncertainties effectively. Additionally, a cross-coupling error in the control design guarantees synchronization between two robots. Based on the above analysis, synchronous control in multiple manipulators takes into account external disturbance, modeling error, and actuator issues such as dead zone, saturation, and friction. They were usually considered as lumped uncertainties which are estimated by universal approximations such as neural networks, fuzzy logic systems, and adaptive laws.

In order to apply neural networks (NNs) and fuzzy logic systems (FLSs) as universal approximators in practice, it requires the practitioner’s considerable knowledge to choose the suitable parameters relating to the structure of the NNs and FLSs, fuzzy rules, membership functions, learning rates, initial weighting vector, etc. In addition, some intensive computations are also required when these approximators are used. Notably, the extended state observer (ESO), introduced by Han [19] in 2009, has been conducted to estimate both the unmeasured state and lumped uncertainties in nonlinear systems [20,21,22] in an infinite time. In [20], an ESO was designed to approximate external disturbances such as nonlinear loads and parameter perturbations in voltage source converter stations. In [21], an ESO was developed to reject the lumped disturbance, such as external disturbances and model parameter perturbation, and estimate the unmeasured velocity states in underactuated underwater vehicles. In [22], Tran et al. used the ESO to estimate not only the unmeasured states but also the lumped uncertainties, including the input dead zone, the unknown frictions, and the external disturbances.

In practice, the sign of the control gain is often unknown, also named the control direction. When this issue occurs, it can lead to system instability. The Nussbaum gain technique, which was first introduced in [23] has been widely investigated to manage unknown control directions with nonlinear systems [24], nonlinear switched stochastic systems [25], nonlinear MIMO systems [26], and robotic manipulators [27]. To the authors’ knowledge, synchronization control for multiple robots with unknown control directions and uncertainties has not been fully conducted in the existing literature.

Based on the above discussions, a novel adaptive synchronous sliding mode control is proposed for a dual-arm system against modeling error and unknown control directions. In order to guarantee both tracking and synchronous objectives in the robotic manipulators, the proposed control is designed by using sliding mode control and cross-coupling error. By forcing the cross-coupling error toward zero, tracking and synchronous errors converge to zero. Because an extended state observer is utilized to estimate and compensate for the modeling error in the robotic system, the accuracy of the tracking goal is enhanced significantly. Additionally, a Nussbaum gain function is integrated into the control algorithm to deal with the issue of unknown control direction. In order to demonstrate the stability and robustness of the controlled system, a Lyapunov function is utilized. Some simulations are implemented in MATLAB Simulink with a dual 3-DOF manipulator system. Finally, to ensure the proposed control’s effectiveness, a comparison was conducted between its results and those of other controllers. The main merits of this paper are presented as follows:

(1)

Compared with some existing works for a dual robotic arm [4,5,6,7,8,9,10,11,12,13,14,15,16,17], an unknown control direction is taken care of besides the other uncertainties. Both theory and simulation are used to analyze the effectiveness of the proposed method.

(2)

Elaborate analysis and simulations are conducted by the Lyapunov approach and MATLAB Simulink with different working conditions. The results show that the proposed control can suitably handle with the modeling error, external disturbances, and wrong control directions in the robotic system to obtain both tracking and synchronous objectives.

The structure of this study is as follows. In Section 2, the kinematic and dynamic formulas are presented. The challenges in the study are also introduced. In Section 3, the proposed control based on synchronous sliding mode control, extended state observer, and Nussbaum function is described, and its stability and robustness are also analyzed by using the Lyapunov approach. In Section 4, some numerical simulations are conducted to verify the effectiveness, performance, and feasibility of the proposed control. Finally, in Section 5, some conclusions are presented.

2. Problem Formulations

2.1. System Description

The dynamic equations of each 3-DOF manipulator in joint space can be expressed as follows:

$$M_i\left(q_i\right){\ddot{q}}_i+C_i\left(q_i,{\dot{q}}_i\right){\dot{q}}_i+G_i\left(q_i\right)+d_i={\Pi }_i{\tau }_i$$

(1)

where $i\left(1\le i\le 2\right)$ presents the ith robot index in the network and p is the total number of the individual elements. Furthermore, $q_i={\left[q_{i1},q_{i2},q_{i3}\right]}^T\in R^{3\times 1}$, ${\dot{q}}_i={\left[{\dot{q}}_{i1},{\dot{q}}_{i2},{\dot{q}}_{i3}\right]}^T\in R^{3\times 1}$, ${\ddot{q}}_i={\left[{\ddot{q}}_{i1},{\ddot{q}}_{i2},{\ddot{q}}_{i3}\right]}^T\in R^{3\times 1}$ derive the vector of position, velocity, and acceleration vector in the joint space of the ith robot, respectively; $M_i\left(q_i\right)\in R^{3\times 3}$, $C_i\left(q_i,{\dot{q}}_i\right){\dot{q}}_i\in R^{3\times 1}$ and $G_i\left(q_i\right)\in R^{3\times 1}$ are the known symmetric and uniformly positive definite inertia matrix, the known Coriolis and centrifugal matrix, and the known gravitational force vector, respectively; $d_{{}_i}=J_i^T{F}_{exti}\in R^{3\times 1}$ presents the lumped uncertainties including parametric variation and external disturbances in the ith robot coordinates; $J_i\in R^{3\times 3}$ is a Jacobian matrix of the ith robot; $F_{exti}\left(t\right)\in R^{3\times 1}$ is an external force vector at the end-effector of the ith robot; ${\tau }_i\in R^{3\times 1}$ ${\tau }_i\in R^{3\times 1}$ presents the input torque vectors in the ith robot coordinates; and ${\Pi }_i\in R^{3\times 3}$ is the control direction vector of the ith robot.

The properties of the manipulator dynamics are presented as follows [28]:

Property 1:

Matrix $2C_i\left(q_i,{\dot{q}}_i\right)-{\dot{M}}_i\left(q_i\right)$ is a skew-symmetric matrix, defined as $x^T\left(2C_i\left(q_i,{\dot{q}}_i\right)-{\dot{M}}_i\left(q_i\right)\right)x=0$.

Property 2:

The inequalities $\Vert C_i\left(q_i,{\dot{q}}_i\right)\Vert \le {\lambda }_{ci},\left(i=1,2\right)$ and $\Vert G_i\left(q_i\right)\Vert \le {\lambda }_{gi},\left(i=1,2\right)$ are satisfied, where ${\lambda }_c$ and ${\lambda }_g$ are known positive constants.

Property 3:

The inequality ${\lambda }_{M_{1i}}\le \Vert M_i\left(q_i\right)\Vert \le {\lambda }_{M_{2i}}$ is satisfied where ${\lambda }_{Mij}\left(i,j=1,2\right)$ are known positive constants.

Assumption 1:

The lumped uncertainties, including parametric variation and external disturbance, $d_i$, are differentiable and bounded.

Figure 1 illustrates the system of a dual-arm model using $\left\{O_W\right\}$ as the world coordinate system; $\left\{O_{bi}\right\}$ $\left(i=1,2\right)$ is the base frame of the ith robot; $q_{ij}$ is the rotation angle of the jth joint of the ith robot; and $c_1$ and $c_2$ are vectors of each robot’s end-effector reference to $\left\{O_W\right\}$ with $n(c_1)$ and $n(c_2)$ as the rotation matrix, respectively. ${}_{b1}{}^{W}T(.)$ and ${}_{b2}{}^{W}T(.)$ are correspondingly the transformation matrix from the base frame of the ith robot, $\left\{O_{bi}\right\}$ $\left(i=1,2\right)$, to the world coordination system, $\left\{O_W\right\}$.

Let define $x_{i1}=q_i\in R^{3\times 1}$, and $x_{i2}={\dot{q}}_i\in R^{3\times 1}$ with $i=1,2$. The state space form of the dual arms is presented as follows:

$$\begin{array}{l}{\dot{x}}_{i1}=x_{i2}\\ {\dot{x}}_{i2}=M_i^{-1}\left(x_{i1}\right)\left({\tau }_i-C_i\left(x_{i1},x_{i2}\right)x_{i2}-G\left(x_{i1}\right)-d_i\right)\end{array}$$

(2)

Let is define the position tracking error as follows:

$$e=x-x_d\in R^{6\times 1}$$

(3)

where $x={\left[x_{11}^T,x_{21}^T\right]}^T\in R^{6\times 1}$ and $x_d={\left[x_{11d}{}^T,x_{21d}{}^T\right]}^T\in R^{6\times 1}$ present a position vector and the desired position vector, respectively.

The position synchronization between two robotic arms: synchronization errors are evaluated by the synchronous errors, $e_{is}\left(i=1,2\right)$, which are defined as

$$\begin{array}{l}{e}_{1s}=e_1-e_2\in R^{3\times 1}\\ e_{2s}=e_2-e_1\in R^{3\times 1}\end{array}$$

(4)

The relationship between the tracking error and synchronous error in (4) can be rewritten as

$$e_s={\rm T}e$$

(5)

where $T=\left[\begin{array}{cc}{I}_{3\times 3}& -I_{3\times 3}\\ -I_{3\times 3}& I_{3\times 3}\end{array}\right]$ is the cross-coupling matrix, $e_s={\left[e_{1s}^T,e_{2s}^T\right]}^T\in R^{6\times 1}$ is the synchronization error vector, and $I\in R^{3\times 3}$ is an identity matrix.

A cross-coupling error is defined in (6) to guarantee that both the position synchronization between two robotic arms and the tracking objective in the proposed control are obtained.

$$e_c=e+\alpha e_s=\left(I+\alpha T\right)e\in R{ }^{6\times 1}$$

(6)

where $\alpha =diag\left[{\alpha }_1,\dots ,{\alpha }_6\right]\in R^{6\times 6}$ is a positive-definite diagonal matrix. When $\alpha$ is chosen properly, $\left(I+\alpha T\right)$ can become positive definite with full rank. As a result, when the cross-coupling error, $e_c$, is driven to zero, the synchronization error, $e_s$, and the tracking error, $e$, will reach zero instantaneously. Then, the control goals will be achieved.

2.2. Nussbaum Function

In order to manage the unknown control direction, the Nussbaum gain approach is investigated in this study. A Nussbaum-type function possesses the following properties:

$$\begin{array}{l}\underset{s\to \infty }{\lim }\sup {\displaystyle \underset{s_0}{\overset{s}{\int }}N\left(\zeta \right)d\zeta }=+\infty \\ \underset{s\to \infty }{\lim }\inf {\displaystyle \underset{s_0}{\overset{s}{\int }}N\left(\zeta \right)d\zeta }=-\infty \end{array}$$

(7)

Nussbaum functions are commonly used as $k^2\sin \left(k\right)$, $k^2\cos \left(k\right)$, and $\exp \left(k^2\right)\cos \left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.k\right)$. In this study, the Nussbaum function $\exp \left({\zeta }^2\right)\cos \left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\zeta \right)$ is exploited.

Lemma 1

[29,30,31]: if $V\left(t\right)$ and ${\zeta }_i\left(t\right)$ are smooth functions defined on $\left[0,t_f\right)$ with $V\left(t\right)>0$, $\forall t\in \left[0,t_f\right)$, if the following inequality satisfies:

$$V\left(t\right)<c_0+e^{-C_{1}t}{\displaystyle \underset{0}{\overset{t}{\int }}{\displaystyle \sum _{i=1}^n\left(g_{i}N\left({\zeta }_i\right)\pm 1\right){\dot{\zeta }}_i{e}^{C_{1}t}d\tau }},\forall t\in [0,t_f)$$

(8)

where $g_i$ is a non-zero constant, $C_1$ > 0, and $c_0$ expresses some appropriate constants, then $V\left(t\right)$, ${\zeta }_i\left(t\right)$ and ${\displaystyle {\int }_0^t{\displaystyle \sum _{i=1}^n\left(g_{i}N\left({\zeta }_i\right)\pm 1\right)}}{\dot{\zeta }}_{i}d\tau$ must be bounded on $\left[0,t_f\right)$.

Definition 1:

Given a nonlinear system, $\dot{x}=f\left(x,u\right)\in R^{n\times 1}$ with the equilibrium $x_e$ is semi-globally uniformly ultimately bounded (SGUUB) if for any $\mathsf{\Omega }$, a compact subset of $R^n$ and all $x\left(t_0\right)=x_0\in \mathsf{\Omega }$, there exists a $\mu >0$ and a number $T\left(\mu ,x_0\right)$ such that $\Vert x\left(t\right)\Vert <\mu$ for all $t>t_0+T$.

2.3. Extended State Observer

By employing the extended state observer (ESO) referring to [32] with u and x as inputs and state variables, the observed system variable can be defined as follows:

$$\begin{array}{l}{\dot{\widehat{x}}}_{i1}={\widehat{x}}_{i2}+{\kappa }_{i1}\left(x_{i1}-{\widehat{x}}_{i1}\right)\\ {\dot{\widehat{x}}}_{i2}={\widehat{x}}_{i3}+{\kappa }_{i2}\left(x_{i1}-{\widehat{x}}_{i1}\right)+u+{\widehat{f}}_i\\ {\dot{\widehat{x}}}_{i3}={\kappa }_{i3}\left(x_{i1}-{\widehat{x}}_{i1}\right)\end{array}$$

(9)

where $\widehat{x}={\left[{\widehat{x}}_{i1},{\widehat{x}}_{i2},{\widehat{x}}_{i3}\right]}^T\in R^{3\times 1}$ and ${\kappa }_{ij}$ with $i,j=1,2,3$ are observer gains to be specified. The gains are selected as:

$$\left[{\kappa }_{i1},{\kappa }_{i2},{\kappa }_{i3}\right]=\left[3{\omega }_0,3{\omega }_0^2,{\omega }_0^3\right]$$

(10)

where ${\omega }_0>0$ is the observer bandwidth.

By defining ${\tilde{x}}_{ij}=x_{ij}-{\widehat{x}}_{ij}$ for $i,j=1,2,3$, the observer estimation error can be presented as follows:

$$\begin{array}{l}{\dot{\tilde{x}}}_{i1}={\tilde{x}}_{i2}-3{\omega }_0{\tilde{x}}_{i1}\\ {\dot{\tilde{x}}}_{i2}={\tilde{x}}_{i3}-3{\omega }_0^2{\tilde{x}}_{i1}+{\tilde{f}}_i\\ {\dot{\tilde{x}}}_{i3}=-{\omega }_0^3{\tilde{x}}_{i1}+{\delta }_i\end{array}$$

(11)

where ${\delta }_i$ is the differential of the lumped uncertainties.

Let ${\chi }_{ij}=\frac{{\tilde{x}}_{ij}}{{\omega }_0^{j-1}},\left(i=1,2;j=1,2,3\right)$; then, the observer estimation error in (11) can be represented as follows:

$${\dot{\chi }}_i={\omega }_{0}A{\chi }_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}$$

(12)

where ${\chi }_i={\left[{\chi }_{i1},{\chi }_{i2},{\chi }_{i3}\right]}^T\in R^{3\times 1}$, $A=\left[\begin{array}{ccc}-3& 1& 0\\ -3& 0& 1\\ -1& 0& 0\end{array}\right]$, $B={\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T$, $C={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T$.

Assumption 2:

the function, $f_0$, is a Lipschitz function.

3. Proposed Control Design

3.1. Control Design

As presented in Figure 2, the proposed control is constructed by synchronous sliding mode control, Nussbaum gain function, and extended state observer to deal with the challenges in the uncertain dual-arm robot. The synchronous sliding mode control can maintain both movement synchronization and tracking objective in the dual arm because it is developed by cross-coupling errors and the sliding mode control technique. The extended state observer is used to estimate the lumped uncertainties consisting of parameter variation and external disturbance in each robotic arm. Additionally, the Nussbaum gain function helps to avoid the misconnection control signal. As a result, the proposed method will inherit all superiorities of these techniques.

The sliding variable is defined as follows:

$$s={\dot{e}}_c+\lambda e_c$$

(13)

where $\lambda \in R^{6\times 6}$ is a positive definite diagonal matrix.

The time derivative of the sliding variable (13) is expressed as follows:

$$\dot{s}={\ddot{e}}_c+\lambda {\dot{e}}_c=\left(I+\alpha T\right)\left(\ddot{e}+\lambda \dot{e}\right)=\left(I+\alpha T\right)\left(\ddot{x}-{\ddot{x}}_d+\lambda \dot{e}\right)$$

(14)

The second differential of the state variable, $x$, is calculated as follows:

$$\ddot{x}=M^{-1}\left(x\right)\left(\Pi \tau -C\left(x,\dot{x}\right)\dot{x}-G\left(x\right)-d\right)$$

(15)

where $x={\left[x_{11}^T,x_{21}^T\right]}^T\in R^{6\times 1}$, $\dot{x}={\left[x_{12}^T,x_{22}^T\right]}^T$, $\ddot{x}={\left[\begin{array}{cc}{\dot{x}}_{12}^T& {\dot{x}}_{22}^T\end{array}\right]}^T$, $M\left(x\right)=\left[\begin{array}{cc}{M}_1\left(x_{11}\right)& 0\\ 0& M_2\left(x_{21}\right)\end{array}\right]\in R^{6\times 6}$, $C\left(x,\dot{x}\right)=\left[\begin{array}{cc}{C}_1\left(x_{11},x_{12}\right)& 0\\ 0& C_2\left(x_{12},x_{22}\right)\end{array}\right]$, $G\left(x\right)=\left[\begin{array}{l}{G}_1\left(x_{11}\right)\\ G_2\left(x_{21}\right)\end{array}\right]\in R^{6\times 1}$, $d={\left[\begin{array}{cc}{d}_1^T& d_2^T\end{array}\right]}^T\in R^{6\times 1}$.

By substituting (15) into (14), its result is presented as follows:

$$\dot{s}=\left(I+\alpha T\right)(M^{-1}\left(x\right)\left(\Pi \tau -G\left(x\right)-C\left(x,\dot{x}\right)\dot{x}-d\right)-{\ddot{x}}_d+\lambda \dot{e})$$

(16)

The proposed control law is selected as follows:

$$\tau =N\left(\zeta \right){\tau }_0$$

(17)

where ${\tau }_0=M\left(x\right)\left({\ddot{x}}_d-\lambda \dot{e}\right)+C\left(x,\dot{x}\right)\dot{x}+G\left(x\right)-Ks-\eta sign\left(s\right)+M\left(x\right){\widehat{x}}_3\in R^{6\times 1}$, $N\left(\zeta \right)=diag\left(\left[N\left({\zeta }_1\right),\dots ,N\left({\zeta }_6\right)\right]\right)$, $N\left({\zeta }_i\right)=e^{{\zeta }_i^2}\cos \left(\frac{\pi }{2}{\zeta }_i\right)$, ${\dot{\zeta }}_i={\sigma }_i{s}_i{\tau }_{0i},\left(i=1,\dots ,6\right)$. where ${\sigma }_i\left(i=1,\dots ,6\right)$ are positive constants.

Theorem 1.

When we apply a controller defined by (17) and an extended state observer (9) for the dual-arm system (2), then the sliding variable, $s$, the cross-coupling error, $e_c$, the synchronous error, $e_s$, and the tracking error, $e$, are bounded. Consequently, the controlled system is demonstrated to be semi-globally uniformly and ultimately bounded (SGUUB). Otherwise, the dual-arm system simultaneously achieves both tracking objective and motion synchronization between two arms under the presence of parametric variation, external disturbance, and unknown control direction.

3.2. Proof of Stability

The stability of the entire system is demonstrated through the use of the Lyapunov function (18), and the analysis is carried out as follows:

$$V=\frac{1}{2}{s}^{T}s+\frac{1}{2}{\displaystyle \sum _{i=1}^2{\tilde{\chi }}_i^T{\tilde{\chi }}_i}$$

(18)

The time derivative of the Lyapunov function (19) is expressed as follows:

$$V=\frac{1}{2}{s}^{T}s+\frac{1}{2}{\displaystyle \sum _{i=1}^2{\tilde{\chi }}_i^T{\tilde{\chi }}_i}$$

(19)

By substituting the time derivative of the sliding variable, $s$, in (16) and the observer error vector (12) into (19), its result is exhibited as follows:

$$\dot{V}=s^T(\left(I+\alpha T\right)(M^{-1}\left(x\right)(\Pi \tau -G\left(x\right)-C\left(x,\dot{x}\right)\dot{x}-d)+\lambda \dot{e}-{\ddot{x}}_d)+{\displaystyle \sum _{i=1}^2\left({\omega }_0{\tilde{\chi }}_i^{T}A{\tilde{\chi }}_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}$$

(20)

By replacing the proposed control (17) into (19), the time derivative of the Lyapunov function is presented as follows:

$$\begin{array}{cc}\hfill \dot{V}& =s^T(\left(I+\alpha T\right)(M^{-1}\left(x\right)(\Pi N\left(\zeta \right){\tau }_0-G\left(x\right)-C\left(x,\dot{x}\right)\dot{x}-d)+\lambda \dot{e}-{\ddot{x}}_d)\hfill \\ & +{\displaystyle \sum _{i=1}^2\left({\omega }_0{\tilde{\chi }}_i^{T}A{\tilde{\chi }}_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}\hfill \end{array}$$

(21)

When we add and remove a term, ${\tau }_0$, into (21), its result is represented as follows:

$$\begin{array}{cc}\hfill \dot{V}& =s^T\left(I+\alpha T\right)(M^{-1}\left(x\right)(\Pi N\left(\zeta \right){\tau }_0-{\tau }_0+{\tau }_0-G\left(x\right)-C\left(x,\dot{x}\right)\dot{x}-d)+\lambda \dot{e}-{\ddot{x}}_d)\hfill \\ & +{\displaystyle \sum _{i=1}^2\left({\omega }_0{\tilde{\chi }}_i^{T}A{\tilde{\chi }}_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}\hfill \\ & =s^T\left(I+\alpha T\right)(M^{-1}\left(x\right)(\left(\Pi N\left(\zeta \right)-I\right){\tau }_0+C\left(x,\dot{x}\right)\dot{x}+G\left(x\right)+M\left(x\right)\left({\ddot{x}}_d-\lambda \dot{e}\right)\hfill \\ & -Ks-\eta sign\left(s\right)+M\left(x\right){\widehat{x}}_3-G\left(x\right)-C\left(x,\dot{x}\right)\dot{x}-d)+\lambda \dot{e}-{\ddot{x}}_d)\hfill \\ & +{\displaystyle \sum _{i=1}^2\left({\omega }_0{\tilde{\chi }}_i^{T}A{\tilde{\chi }}_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}\hfill \\ & =s^T\left(I+\alpha T\right)(M^{-1}\left(x\right)(\left(\Pi N\left(\zeta \right)-I\right){\tau }_0-Ks-\eta sign\left(s\right)+M\left(x\right){\widehat{x}}_3-d))\hfill \\ & +{\displaystyle \sum _{i=1}^2\left({\omega }_0{\tilde{\chi }}_i^{T}A{\tilde{\chi }}_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}\hfill \\ & =-s^T\left(I+\alpha T\right)M^{-1}\left(x\right)\left(I-\Pi N\left(\zeta \right)\right){\tau }_0-s^T\left(I+\alpha T\right)M^{-1}\left(x\right)Ks\hfill \\ & +s^T\left(I+\alpha T\right)M^{-1}\left(x\right)\left(\epsilon -\eta sign\left(s\right)\right)+{\displaystyle \sum _{i=1}^2\left({\omega }_0{\tilde{\chi }}_i^{T}A{\tilde{\chi }}_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}\hfill \\ & \le -s^T\left(I+\alpha T\right)M^{-1}\left(x\right)\left(I-\Pi N\left(\zeta \right)\right){\tau }_0-s^T\left(I+\alpha T\right)M^{-1}\left(x\right)Ks\hfill \\ & +{\displaystyle \sum _{i=1}^2\left({\omega }_0{\tilde{\chi }}_i^{T}A{\tilde{\chi }}_i+B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}\hfill \\ & \le -c_{0}V-c_1\left(I-\Pi N\left(\zeta \right)\right)\dot{\zeta }+\delta \hfill \end{array}$$

(22)

where $c_0=\min \left({\lambda }_{\min }\left(K\right)\left(I+\alpha T\right)M^{-1}\left(x\right),{\lambda }_{\min }\left(-{\omega }_0{A}_0\right)\right)$, $c_1={\lambda }_{\min }\left(\sigma \left(I+\alpha T\right)M^{-1}\left(x\right)\right)$$\delta ={\Vert {\displaystyle \sum _{i=1}^2\left(B\frac{{\tilde{f}}_i}{{\omega }_0}+C\frac{{\delta }_i}{{\omega }_0^2}\right)}\Vert }_{\infty }$.

By multiplying two sides of (21) with $e^{ct}$ and integrating over $\left[0,t_f\right]$, the result is presented as follows:

$$0\le V\left(t\right)\le \frac{\delta }{c_0}+\left(V\left(0\right)-\frac{\delta }{c_0}\right)e^{-c_{0}t}+c_1{e}^{-c_{0}t}{\displaystyle \sum _{i=1}^6{\displaystyle {\int }_0^{t_f}\left({\mathsf{\Pi }}_{i}N\left({\zeta }_i\right)-1\right){\dot{\zeta }}_i{e}^{c_0\tau }d\tau }}$$

(23)

From Lemma 1 and (26), we can state that $V\left(t\right)$, ${\zeta }_i$, and ${\displaystyle {\int }_0^{t_f}\left(1-\mathsf{\Pi }N\left({\zeta }_i\right)\right){\dot{\zeta }}_i{e}^{c_0\tau }d\tau }$ are bounded and semi-globally uniformly ultimately bounded (SGUUB).

Remark 1:

Because this sliding mode control processes two properties: (1) it reaches the sliding surface and (2) the equilibrium point asymptotically. When the sliding variable of the controlled system and the cross-coupling errors converge to zero asymptotically. as a result, the synchronous and tracking objectives are achieved with the proposed control.

4. Numerical Simulation

In order to demonstrate the efficiency of the proposed method on the dual-arm robotic system, a dual 3-DOF manipulator, as presented in Figure 3, is designed for simulation. This section will describe the simulation environment, evaluation criteria, and result discussion.

4.1. Simulation Description

For simulation, a dual-3 DOF manipulator with the dynamics and its parameters presented in [18] was simulated on MATLAB Simulink with a sampling time of 10 ms and the type of auto solver. The simulation period was divided into different scenarios for 40 s. The first scenario takes place in the first 10 s when only the modeling errors in the dual arm exist. Each robot has a different modeling error. The second scenario happens in the second 10 s when an external force is applied at the end-effector of the second robotic arm. The third scenario is the third 10 s when the first actuator in the first arm raises the wrong control direction. The last scenario is the final 10 s when the wrong control direction occurs in the first actuator at the second manipulator.

The reference trajectories in the joint space of each manipulator are given as

$$\begin{array}{l}{q}_{1d}\left(t\right)=30\sin \left(0.2\pi t\right)+90\left(\mathrm{deg}.\right)\\ q_{2d}\left(t\right)=30\sin \left(0.2\pi t\right)\left(\mathrm{deg}\right)\\ q_{3d}\left(t\right)=30\sin \left(0.2\pi t\right)\left(\mathrm{deg}\right)\end{array}$$

(24)

The modeling error is the unknown friction, including viscous and Coulomb frictions. Its equation is expressed as follows:

$${\tau }_{fric}\left(t\right)=v\dot{q}+csign\left(\dot{q}\right)\left(Nm\right)$$

(25)

where $v=0.2I_{6\times 6}$ is viscous friction gain and $c=0.3I_{6\times 6}$ is the Coulomb parameter.

The external disturbance is an external force applied along the z-axis at the end-effector of the second robot with an amplitude of 10N.

$$F_{ext1}={\left[0,0,10\right]}^T\left(N\right)$$

(26)

The efficiency of the actuators in normal cases is presented with respect to $\Pi =I_{6\times 6}$. The wrong control direction of the first actuator in the second robot occurs and the input matrix $\Pi =diag\left(\left[1,1,1,-1,1,1\right]\right)\in R^{6\times 6}$. In the fourth scenario, when the wrong control direction happens in the first actuator of the first robot, the input matrix is presented as $\Pi =diag\left(\left[-1,1,1,-1,1,1\right]\right)\in R^{6\times 6}$.

4.2. Simulation Results

In order to show the superiorities of the proposed control, a synchronous sliding mode control (SSMC) with the Nussbaum function (27) was conducted on the dual robotic arm system on MATLAB Simulink with the same working conditions. The controlled system in MATLAB Simulink is presented in Figure 3. The input torques of two robots are limited by a saturation block from −100 to +100 (Nm). In advance, the dynamics of each robot are separated into different MATLAB function blocks, and two-step blocks set up the control directions in each robot.

$${\tau }_1=M\left(x\right)\left({\ddot{x}}_d-\lambda \dot{e}\right)+C\left(x,\dot{x}\right)\dot{x}+G\left(x\right)-Ks-\eta sign\left(s\right)$$

(27)

Remark 2:

In order to guarantee equality in the comparisons, the control parameters of the proposed control are inherited from the SSMC. Because adjusting the parameters of the controllers is a complex task, the trial–error approach is used to select the appropriate parameters of the synchronous sliding mode control. The parameters of these controllers are presented in

Table 1. Parameters of the synchronous sliding mode control and the proposed control.

Controllers	Parameters
Synchronous sliding mode control	${\alpha }_i{}_{\left(i=1,\dots ,6\right)}=1$,${\sigma }_i{}_{\left(i=1,\dots ,6\right)}=2\times {10}^{-3}$ $\lambda =diag\left(\left[9.5,9,8,9.5,9,8\right]\right)$, $K=diag\left(\left[3,1.4,0.8,3,1.4,0.8\right]\right)$ $\eta =0.5I_{6\times 6}$
Proposed control	${\alpha }_i{}_{\left(i=1,\dots ,6\right)}=1$,${\sigma }_i{}_{\left(i=1,\dots ,6\right)}=2\times {10}^{-3}$ $\lambda =diag\left(\left[9.5,9,8,9.5,9,8\right]\right)$, $K=diag\left(\left[3,1.4,0.8,3,1.4,0.8\right]\right)$ $\eta =0.5I_{6\times 6}$ ${\omega }_0=100I_{6\times 1}$

.

Figure 4 and Figure 5 present the output response in the joint space of two robotic arms. The black, blue, and red lines correspondingly describe the references and output responses of the SSMC and the proposed control. In the first period, although the two robots had different modeling errors, the SSMC and proposed control are suitable against these issues. In the second period, an external force was added at the end-effector of the second robot. The proposed control with ESO coped well with this challenge. However, the SSMC could not handle it; the output responses in three joints were degraded significantly. Because the SSMC was developed based on the cross-coupling error between two robots, the output responses of the first robot were also degraded. At the 20th and 30th seconds, the wrong control direction arises in the two robots. Although the Nussbaum functions were used to deal with these problems, they cannot manage instantly. As a result, the impulses arose in these instances. Consequently, the results show that these responses track the predefined references under the presence of friction, external force, and wrong input connections in two robotic arms.

In order to evaluate the efficiency of the proposed control clearly, the tracking errors in the joint space of two robotic arms are presented in Figure 6 and Figure 7 with the black lines and red lines of the SSMC and the proposed control, respectively. In the first scenario, when the main challenge is the unknown frictions, the tracking errors in both robotic arms of the proposed control are better than the SSMC. In the second scenario, an external force is applied at the end-effector of the second robotic arm along the y-axis. Consequently, the tracking errors in the joints of the second robot increased significantly with the SSMC. Because the SSMC used the cross-coupling error to form the relationship between two respective joints in the dual arm, the influences on the second robot also affect the first robot. As a result, the tracking errors in the first robot are also degraded, similar to the second one with the SSMC. In the proposed control, because the ESO perfectly estimated the lumped uncertainties caused by the unknown friction and external force, the tracking accuracy in both robots is maintained like the first scenario. In the third and fourth scenarios, the wrong control direction issue arises in each robotic arm’s first actuator. The stability in two robotic arms with two controllers is guaranteed because the Nussbaum function was used for designing these controllers. The Nussbaum function calculates the sign of the control input at each robotic arm joint. It is one with the right connection and minus one with the wrong connection. The results of the Nussbaum functions in the two robots are presented in Figure 8. The lumped uncertainties in these scenarios are still unknown friction and modeling error. The proposed control with the ESO gives the accuracy in each joint better than the SSMC.

Figure 9 presents the synchronous error between the respective joints in the two robotic arms with the SSMC with the black lines and the proposed control with the red lines. The results show that the cross-coupling approach manages well with the modeling error and friction in the two controllers. The results of the SSMC in the second scenario show that the cross-coupling approach cannot handle the external disturbance to maintain the synchronous objectives. On the contrary, the proposed control with the ESO handles the modeling error, friction, and external disturbance well. Therefore, the ESO rejects the influence of the lumped uncertainties.

Figure 10 presents the estimated errors between the estimated angles of the ESO and the joint angles of two robots with black, blue, and red lines of joint 1, joint 2, and joint 3, respectively. The estimated errors dramatically increase at the 20th and 30th seconds when the wrong control direction happens in two robots.

Figure 11 and Figure 12 show the estimated lumped uncertain responses of the ESO and the lumped uncertainties at each joint of the two robots. The black and red lines in the figures present the lumped uncertainties and estimated results, respectively. As presented in Figure 11, the results proved that the ESO approximates the Coulomb and static frictions well in the first robot. Figure 12 shows that the ESO can also work well with the external disturbance that arises from the 10th to the 20th second in the second robot. Although the Nussbaum function can estimate the practical control direction well, the influence of the wrong control direction generated a high lumped uncertainty at the 20th second and 30th second. Thus, the ESO is not fast enough to deal with this lumped uncertainty in these instances. It just approximates the smooth lumped uncertainties in the two robots well.

Figure 13 presents the control inputs of the proposed control with black, blue, and red lines for joint 1, joint 2, and joint 3, respectively. The control inputs dramatically increase when the wrong control directions occur in the two robots.

5. Conclusions

This paper proposed an adaptive synchronous sliding mode control for a dual-arm robot under the presence of unknown control direction, parametric uncertainties, and external disturbance. The proposed control was constructed by using cross-coupling error, sliding mode control, extended state observer, and Nussbaum function. By defining a cross-coupling mistake to integrate both tracking and synchronous objectives, when the proposed control drives the cross-coupling error forward to zero, both tracking and synchronous errors converge to zero simultaneously. The Nussbaum function managed the unknown control direction. The external state observer was also used to estimate and compensate for the lumped uncertainties, including parametric uncertainties and external disturbance. The stability and robustness of the controlled system were analyzed by the Lyapunov approach in theory. Finally, some simulations on MATLAB Simulink were conducted with a dual 3-DOF manipulator under the above challenges. The results demonstrated that the tracking and synchronous objectives of the dual arm were achieved with the proposed control under the presence of unknown friction, external disturbance, and unknown control direction.

Although the Nussbaum function can instantly detect the wrong control direction when they suddenly happen in the robot system, because the wrong control direction is a serious problem, the lumped uncertainties still increase dramatically. As a result, they can affect the stability of the whole system. Although, the ESO was designed in this study to estimate the lumped uncertainties whose time derivatives are bounded. The ESO effectively worked in this case because of the uncertainty caused by the unknown control direction. Then, the control signals of the proposed control dramatically increase before they are stable again. In future work, advanced approaches will be proposed to manage this issue, and the proposed control will be verified on a practical test bench.