Adaptive Sliding Mode Control Anticipating Proportional Degradation of Actuator Torque in Uncertain Serial Industrial Robots

Abstract

The paper focuses on faulty actuator problems in an industrial robot using servomotors, and provides an adaptive sliding mode control law to overcome this circumstance. Because of multifarious reasons, robot actuators can undergo a variety of failures, such as locked or stuck joints, free-swinging joints, and partial or total loss of actuation effectiveness. The robot behavior will become worsen if the system controller has not been designed with adequate faulty tolerance. The proportional degradation of actuator torque at unknown degrees of loss, which is one type of partial loss of actuation effectiveness, is considered in this study to design a suitable controller. The robot model is constructed with uncertain parameters and unknown friction, whereas the controller uses only the approximate parameters. Symmetry and skew-symmetry give important contributions in robot modeling and transformation, as well as in the process of proving the system stability. An adjustable coefficient vector of the proposed controller can adaptively reach the upper bounds of an uncertain parametric vector, which guarantees the criterion of Lyapunov stability. In the numerical simulation stage, the selected industrial robot is a Serpent 1 robot with three degrees of freedom. A quasi-physical model based on MATLAB/Simscape Multibody for the robot is built and used in order to increase the reliability of the simulation performance closer to reality. Simulation results illustrate the efficiency of the proposal control methodology in the presence of the mentioned failure. The controller can still deliver satisfactory responses to the robot system under reasonable levels of actuator torque degradation.

1. Introduction

Robots have long been used in many fields and have been developing to become the core implementation component of Industry 4.0 [1,2,3]. Currently, the growing demand for the reliability of robotic systems in production lines leads researchers and manufactures to keep enhancing the performance of robot in some fault situations. Robotic systems typically respond to a faulty state by shutting down the entire system. The downtime due to the failed parts will reduce productivity and increase costs. In a study about defective industrial robots [4], almost all common failures can be detectable by using a proposed training criteria. Many studies have shown that several certain failure kinds of robotic systems can be overcome by reasonable strategies. One of them is an unexpected decrease in robotic actuator torque, which is investigated in this paper. Generally, some attractive research topics related to drive failure are: (i) detection and location, (ii) analysis and diagnosis, and (iii) failure tolerance control. Many diverse studies related to the topics (i) and (ii) have been published. Robot drive failures are detected in [5,6] by Support-Vector–Machines and a Partial-Least-Squares-based statistical technique, respectively. A non-linear observer in [7] and a failure identification method [8] are developed to recognize faulty states in robotic actuators. In [9], a discrete time framework is established and performed to analyze robot failures by utilizing a decision-making system. Failures in both actuators and sensors of a robot system can be identified by an adaptation-based methodology in [10]. Several analyses regarding manipulator responses in state of a free-moving joint, a locked joint, and actuator torque attenuation are studied in [11,12,13,14], respectively.

Under attraction of the topic (iii), many related investigations have been provided. In [15], an adaptive scheme is designed to counteract the effects of actuator torque degradation on a robot system. In [16], a fault tolerance controller using boundary estimation and an adaptation technique is developed for dealing with partial loss of robot actuators. The result in [17] gives an adaptive control law synthesized with a backstepping scheme for robotic manipulators in case of joint constraints and actuator faults. The sliding mode methodology has been exploited and developed in many various engineering fields, such as angle estimations for salient-pole wound rotor synchronous generators [18], 7-DOF full-car suspension systems [19], position control of an uncertain underwater vehicle [20], fault tolerance control based on an adaptive sliding mode scheme for a quadcopter UAV [21], attitude control of quadrotor UAVs [22], etc. Some diverse interesting applications could be given in more detail in the followings. A novel adaptive PID-type fast terminal sliding mode controller is synthesized for a class of non-linear systems with disturbances [23]. Hyper-chaotic systems can be synchronized by using an integral terminal sliding mode approach [24]. The study in [25] contributes a PID-type terminal sliding surface strategy to enhance the performance of PMSM wind energy conversion systems. For saturated uncertain non-linear systems, a barrier function based adaptive sliding mode scheme is proposed without using the information of disturbance bounds [26]. A sliding mode controller with a leakage-type adaptation law is provided to handle an uncertain Euler–Lagrange system under actuator saturation [27]. The authors in [28] developed a super twisting fractional-order terminal sliding mode method for a type of wind energy conversion systems with non-linearities and disturbances.

Following that strong development trend, the sliding mode methodology is also applied in the robotic control discipline. Robot actuator failures can be rebuilt in a finite time by developing a fast terminal sliding mode observer based fashion [29]. Reference [30] presents a fault tolerance controller using a sliding mode approach to overcome the kind of robot actuator failures: additive time-varying and constant torque. Various studies handle with a mix of robotic joint faults: identification, diagnosis, analysis, and control. Some of them are: [31] for flexible-joint manipulators, [32] for robots subject to free-moving joints, and [33] for locked joints occurring in a specific type of manipulators.

On the other hand, the actuator fault problems in general mechanical systems have been received considerable interest up to now. The study in [34] provides an LMI-based robust adaptive fault-tolerant controller to stabilize a Lipschitz non-linear system. An adaptive controller combined with a sliding mode method is proposed in [35] for uncertain multi-input systems under disturbances and actuation failures. In [36], an overview of direct adaptive compensation control for faults in redundant actuators is given. For a class of uncertain SISO non-linear systems, a partial failed-actuator condition is dealt with a robust adaptive algorithm proposed in [37]. For an unknown MISO non-linear system suffering from actuator faults, some compensation fashions are: adaptive control using a backstepping technique with uncertain degrees of actuator loss [38], mix of a neural-network adaptation strategy and an automatic switching procedure [39], and failure compensation using event-triggering control and an adaptation method with the boundary estimation of failed-actuator parameters [40].

Inspired by the above research, this paper addresses an adaptive sliding mode strategy to anticipate multiple failed actuators for a non-redundant serial industrial robot with uncertain parameters. The investigated kind of failures is unknown proportional degradation of actuator torque. The reaching/switching gains of the proposed sliding mode control method are alterable and regulated by an adaptation law. The effects of actuator torque degradation can be reduced, and system responses are maintained at acceptable qualities under reasonable degrees of loss. The system stability is confirmed by a Lyapunov-based approach. The next parts of this paper are as follows. Section 2 provides the expression of the proportional degradation of the actuator torque in a robot dynamic model. Section 3 contributes to the design of an adaptive sliding mode controller anticipating the mentioned fault. In Section 4, a quasi-physical robot model is constructed as a virtual plant. Section 5 shows the efficiency of the proposed controller through several numerical simulations, and Section 6 gives some important remarks.

2. Proportional Degradation of the Actuator Torque in a Robot Model

Industrial robots considered in this study belong to a class of n-DOF serial manipulators with revolute joints. The matrix form of dynamic equations for that robot kind is:

$$\mathbf{M}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}+\mathbf{G}\left(\mathbf{q}\right)+{\mathsf{\tau }}_f={\mathsf{\tau }}_a$$

(1)

where $\mathbf{q}={[q_1,q_2,...,q_n]}^T\in {\mathbb{R}}^{\mathrm{n}\times 1}$ is the joint angle variable, ${\mathsf{\tau }}_a={[{\mathsf{\tau }}_{1a},{\mathsf{\tau }}_{2a},...,{\mathsf{\tau }}_{na}]}^T\in {\mathbb{R}}^{\mathrm{n}\times 1}$ is the actual torque produced by actuators, $\mathbf{M}\left(\mathbf{q}\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ is the general inertia matrix, $\mathbf{C}\left(\mathbf{q},\dot{\mathbf{q}}\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ is the Coriolis/centrifugal matrix, $\mathbf{G}\left(\mathbf{q}\right)\in {\mathbb{R}}^{\mathrm{n}\times 1}$ is the gravity vector, and ${\mathsf{\tau }}_f\in {\mathbb{R}}^{\mathrm{n}\times 1}$ is the rotational friction torque. In Equation (1), torque ${\mathsf{\tau }}_a$ generated by robot actuators suffering the proportional degradation of torque (PDT) can be represented as:

$${\mathsf{\tau }}_a=\mathsf{\Lambda }{\mathsf{\tau }}_c$$

(2)

where ${\mathsf{\tau }}_c={[{\mathsf{\tau }}_{1c},{\mathsf{\tau }}_{2c},...,{\mathsf{\tau }}_{nc}]}^T\in {\mathbb{R}}^{\mathrm{n}\times 1}$ is the control torque, $\mathsf{\Lambda }=\mathrm{diag}\left(\lambda \right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ is the diagonal matrix representing the coefficient degradation of actuator torque, $\lambda ={\left[{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\right]}^T$, ${\lambda }_i\in \left(0,1\right]$, and ($i=1,2,\dots ,n$). Actuator i is in the normal operation state if ${\lambda }_i=1$ or in a PDT faulty state if $0<{\lambda }_i<1$. The total loss of actuator torque (${\lambda }_i=0$) will lead to either the free-swinging joint failure or the locked-joint failure if the fail-safe brake is activated. Therefore, the total torque loss is not considered in this investigation. Since matrix $\mathbf{M}\left(\mathbf{q}\right)$ is a positive definite and has the symmetry property, it is invertible. Thus, after substituting (2) into (1), Equation (1) can be rewritten in the second-order differential equation as:

$$\ddot{\mathbf{q}}={\mathbf{M}}^{-1}\left(\mathbf{q}\right)[-\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}-\mathbf{G}\left(\mathbf{q}\right)-{\mathsf{\tau }}_f]+{\mathbf{M}}^{-1}\left(\mathbf{q}\right)\mathsf{\Lambda }{\mathsf{\tau }}_c$$

(3)

Compactifying (3) gives:

$$\ddot{\mathbf{q}}=\mathbf{k}+\mathsf{\Delta }{\mathsf{\tau }}_c$$

(4)

where:

$$\begin{array}{c}\mathbf{k}:={\mathbf{M}}^{-1}\left(\mathbf{q}\right)[-\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}-\mathbf{G}\left(\mathbf{q}\right)-{\mathsf{\tau }}_f]\hfill \\ \mathsf{\Delta }:={\mathbf{M}}^{-1}\left(\mathbf{q}\right)\mathsf{\Lambda }\hfill \end{array}$$

(5)

Because $\mathbf{M}$, $\mathbf{C}$, $\mathbf{G}$, ${\mathsf{\tau }}_f$, and $\mathsf{\Lambda }$ are not achieved precisely, $\mathbf{k}$ and $\mathsf{\Delta }$ in (4) and (5) are also not obtained exactly. However, it is possible to interpret $\mathbf{k}=\widehat{\mathbf{k}}+\tilde{\mathbf{k}}$ and $\mathsf{\Delta }=\widehat{\mathsf{\Delta }}+\tilde{\mathsf{\Delta }}$, where $\widehat{\mathbf{k}}$ and $\widehat{\mathsf{\Delta }}$ are the nominal values of $\mathbf{k}$ and $\mathsf{\Delta }$, respectively. Estimation errors $\tilde{\mathbf{k}}$ and $\tilde{\mathsf{\Delta }}$ are assumed to be matched uncertainties. In the following section, under the presence of uncertainties, an adaptive sliding mode control method is designed to make joint variable $\mathbf{q}$ follow a given desired joint trajectory ${\mathbf{q}}_d\in {\mathbb{R}}^{\mathrm{n}\times 1}$ as well as tracking error $\mathbf{e}={\mathbf{q}}_d-\mathbf{q}$ converge to zero. Besides the symmetry characteristic of matrix $\mathbf{M}\left(\mathbf{q}\right)$, the skew-symmetry property in robot dynamics always exists itself, but its expression depends on the structure of the Coriolis/centrifugal matrix. By exploiting the results provided in [41], the generalized inertia matrix $\mathbf{M}\left(\mathbf{q}\right)$ can be obtained by:

$$\mathbf{M}\left(\mathbf{q}\right)=\sum _{i=1}^n\left(m_i{\left({\mathbf{J}}_{T_i}^0\right)}^T{\mathbf{J}}_{T_i}^0+{\mathbf{J}}_{R_i}^T{\mathbf{I}}_i{\mathbf{J}}_{R_i}\right)$$

(6)

and the Coriolis/centrifugal matrix $\mathbf{C}\left(\mathbf{q},\dot{\mathbf{q}}\right)$ satisfying the mentioned skew-symmetry property is formulated as:

$$\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})=\frac{1}{2}\left[\frac{\partial \mathbf{M}}{\partial \mathbf{q}}\left({\mathbf{1}}_n\otimes \dot{\mathbf{q}}\right)+\frac{\partial \mathbf{M}}{\partial \mathbf{q}}\left(\dot{\mathbf{q}}\otimes {\mathbf{1}}_n\right)-{\left(\frac{\partial \mathbf{M}}{\partial \mathbf{q}}\left(\dot{\mathbf{q}}\otimes {\mathbf{1}}_n\right)\right)}^T\right]$$

(7)

The vector of the gravitational term $\mathbf{G}\left(\mathbf{q}\right)$ is:

$$\mathbf{G}\left(\mathbf{q}\right)={\left(\frac{\partial P}{\partial \mathbf{q}}\right)}^T$$

(8)

where $m_i$ is the mass of robotic body i, ${\mathbf{J}}_{R_i}\in {\mathbb{R}}^{3\times \mathrm{n}}$ is the rotational Jacobian matrix, ${\mathbf{J}}_{T_i}^0\in {\mathbb{R}}^{3\times \mathrm{n}}$ is the translational Jacobian matrix, symmetric matrix ${\mathbf{I}}_i\in {\mathbb{R}}^{3\times 3}$ is the inertia tensor of body i with respect to the frame attached at the centroid of body i and parallel to the attached-body frame i, ${\mathbf{1}}_{\mathrm{n}}$ is the n-by-n identity matrix, ⊗ signifies Kronecker product operator, and P is the potential energy of the whole robot body. Hence, the parametric vectors and matrices in robot model (1) can be calculated by (6)–(8).

3. Adaptive Sliding Mode Controller

Usually, the robust ability of a sliding mode controller will be gained by an appropriate choice of reaching/switching gains if the uncertain system parameters are constrained by known certain bounds. However, the boundaries of system uncertainties are mostly unknown in reality. Therefore, an adaptive sliding mode controller (ASMC) could be designed to fine-tune the controller parameters. The sliding variable vector $\mathbf{s}={\left[s_1,s_2,\dots ,s_n\right]}^T$ is selected as:

$$\mathbf{s}=\mathbf{Be}+\dot{\mathbf{e}}$$

(9)

where $\mathbf{B}=\mathrm{diag}\left(\mathbf{b}\right)$ is the n-by-n diagonal matrix with $\mathbf{b}={\left[b_1,b_2,\dots ,b_n\right]}^T$, $b_i>0$ ($i=1,2,...,n$) is selected such that the dynamics of sliding manifold $s_i=b_i{e}_i+{\dot{e}}_i=0$ is stable. Taking the time derivative for both sides of (9) yields:

$$\begin{array}{cc}\hfill \dot{\mathbf{s}}& =\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-\ddot{\mathbf{q}}\hfill \\ \hfill & =\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-(\mathbf{k}+\mathsf{\Delta }{\mathsf{\tau }}_c)\hfill \\ \hfill & =\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-(\widehat{\mathbf{k}}+\tilde{\mathbf{k}}+(\widehat{\mathsf{\Delta }}+\tilde{\mathsf{\Delta }}){\mathsf{\tau }}_c)\hfill \\ \hfill & =\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-(\widehat{\mathbf{k}}+\widehat{\mathsf{\Delta }}{\mathsf{\tau }}_c)-(\tilde{\mathbf{k}}+\tilde{\mathsf{\Delta }}{\mathsf{\tau }}_c)\hfill \\ \hfill & =\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-(\widehat{\mathbf{k}}+\widehat{\mathsf{\Delta }}{\mathsf{\tau }}_c)-\mathbf{w}\hfill \end{array}$$

(10)

where $\mathbf{w}:=\tilde{\mathbf{k}}+\tilde{\mathsf{\Delta }}{\mathsf{\tau }}_c={\left[w_1,w_2,\dots ,w_n\right]}^T$ is the vector of uncertainty combination. Coefficient ${\lambda }_i>0$; therefore, $\mathsf{\Lambda }$, $\mathsf{\Delta }$, and $\widehat{\mathsf{\Delta }}$ are all invertible. The adaptive sliding mode controller is designed in the form as follows:

$$\begin{array}{cc}\hfill {\mathsf{\tau }}_{sl}& ={\widehat{\mathsf{\Delta }}}^{-1}(\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-\widehat{\mathbf{k}})\hfill \\ \hfill {\mathsf{\tau }}_{sw}& ={\widehat{\mathsf{\Delta }}}^{-1}(\widehat{\mathbf{Z}}+\mathsf{\Sigma })\mathrm{sgn}\left(\mathbf{s}\right)\hfill \\ \hfill {\mathsf{\tau }}_c& ={\mathsf{\tau }}_{sl}+{\mathsf{\tau }}_{sw}\hfill \\ \hfill & ={\widehat{\mathsf{\Delta }}}^{-1}[\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-\widehat{\mathbf{k}}+(\widehat{\mathbf{Z}}+\mathsf{\Sigma })\mathrm{sgn}\left(\mathbf{s}\right)]\hfill \end{array}$$

(11)

where ${\mathsf{\tau }}_{sl}$ is the sliding control term, ${\mathsf{\tau }}_{sw}$ is the switching control term using adaptive gains, $\widehat{\mathbf{Z}}=\mathrm{diag}\left(\widehat{\mathbf{z}}\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ is the diagonal matrix with $\widehat{\mathbf{z}}={\left[{\widehat{z}}_1,{\widehat{z}}_2,\dots ,{\widehat{z}}_n\right]}^T$, ${\widehat{z}}_i$ is the adjustable coefficient, $\mathsf{\Sigma }=\mathrm{diag}\left(\sigma \right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$, $\sigma ={\left[{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right]}^T$, ${\sigma }_i>0$ is the arbitrary additive positive parameter, and $\mathrm{sgn}\left(\mathbf{s}\right):={\left[\mathrm{sgn}\left(s_1\right),\mathrm{sgn}\left(s_2\right),\dots ,\mathrm{sgn}\left(s_n\right)\right]}^T$, $\mathrm{sgn}\left(.\right)$ is the signum function. Substituting (11) into (10) gives:

$$\begin{array}{cccccc}\hfill & \dot{\mathbf{s}}\hfill & \hfill & =\hfill & \hfill & \mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & -\{\widehat{\mathbf{k}}+\widehat{\mathsf{\Delta }}{\widehat{\mathsf{\Delta }}}^{-1}[\mathbf{B}\dot{\mathbf{e}}+{\ddot{\mathbf{q}}}_d-\widehat{\mathbf{k}}+(\widehat{\mathbf{Z}}+\mathsf{\Sigma })\mathrm{sgn}\left(\mathbf{s}\right)]\}-\mathbf{w}\hfill \\ \hfill & \hfill & \hfill & =\hfill & \hfill & -(\widehat{\mathbf{Z}}+\mathsf{\Sigma })\mathrm{sgn}\left(\mathbf{s}\right)-\mathbf{w}\hfill \end{array}$$

(12)

Assumption 1.

Vector $\mathbf{w}$ is changeable but bounded by an unknown constant vector $\mathbf{z}={\left[z_1,z_2,\dots ,z_n\right]}^T$ such that $\left|w_i\right|\le z_i$, $i=1,2,\dots ,n$.

Under Assumption 1, although $z_i$ is undetermined, the criterion of sliding mode can be guaranteed if an adaptation law can fine-tune ${\widehat{z}}_i$ convergent to $z_i$. The adaptation law is defined as:

$${\dot{\widehat{z}}}_i={\psi }_i\left|s_i\right|={\psi }_i{s}_i\mathrm{sgn}\left(s_i\right)$$

(13)

where ${\psi }_i>0$ ($i=1,2,...,\mathrm{n}$) is the selected positive factor which specifies the adaptation speed. The larger ${\psi }_i$, the faster matching convergence. By denoting $\mathbf{Z}=\mathrm{diag}\left(\mathbf{z}\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$, the adaptation error matrix and vector are $\tilde{\mathbf{Z}}=\mathbf{Z}-\widehat{\mathbf{Z}}$ and $\tilde{\mathbf{z}}=\mathbf{z}-\widehat{\mathbf{z}}$, respectively. In the Lyapunov stability theory, there are some types of symmetric, positive definite parameter matrices often used in a chosen Lyapunov function that supports the proof of the Lyapunov criterion. Let us give a Lyapunov function candidate as:

$$V=\frac{1}{2}{\mathbf{s}}^T\mathbf{s}+\frac{1}{2}{\tilde{\mathbf{z}}}^T{\mathsf{\Psi }}^{-1}\tilde{\mathbf{z}}$$

(14)

where $\mathsf{\Psi }=\mathrm{diag}\left(\mathsf{\psi }\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ are the diagonal matrices with $\mathsf{\psi }={\left[{\psi }_1,{\psi }_2,\dots ,{\psi }_n\right]}^T$, ${\mathsf{\Psi }}^{-1}=\mathrm{diag}\left({\left[\psi {}_1^{-1},\psi {}_2^{-1},\dots ,\psi {}_n^{-1}\right]}^T\right)>0$. Taking the time derivative of V in (14) we obtain:

$$\dot{V}={\mathbf{s}}^T\dot{\mathbf{s}}+{\tilde{\mathbf{z}}}^T{\mathsf{\Psi }}^{-1}\dot{\tilde{\mathbf{z}}}$$

(15)

After inserting (12) into (15) we obtain:

$$\dot{V}={\mathbf{s}}^T[-(\widehat{\mathbf{Z}}+\mathsf{\Sigma })\mathrm{sgn}\left(\mathbf{s}\right)-\mathbf{w}]+{\tilde{\mathbf{z}}}^T{\mathsf{\Psi }}^{-1}\dot{\tilde{\mathbf{z}}}$$

(16)

Vector $\mathbf{z}$ is constant according to Assumption 1. Therefore, $\dot{\tilde{\mathbf{z}}}=-\dot{\widehat{\mathbf{z}}}$, and substituting it into (16) leads to:

$$\dot{V}=-{\mathbf{s}}^T(\widehat{\mathbf{Z}}+\mathsf{\Sigma })\mathrm{sgn}\left(\mathbf{s}\right)-{\mathbf{s}}^T\mathbf{w}-{(\mathbf{z}-\widehat{\mathbf{z}})}^T{\mathsf{\Psi }}^{-1}\dot{\widehat{\mathbf{z}}}$$

(17)

From the adaptive law (13), the vector type of the adaptive law is:

$$\dot{\widehat{\mathbf{z}}}=\mathsf{\Psi }\left|\mathbf{s}\right|=\mathsf{\Psi }\mathbf{S}\mathrm{sgn}\left(\mathbf{s}\right)$$

(18)

where $\mathbf{S}=\mathrm{diag}\left(\mathbf{s}\right)\in {\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$. Substituting (18) into (17) yields:

$$\begin{array}{cccccc}\hfill & \dot{V}\hfill & \hfill & =\hfill & \hfill & -{\mathbf{s}}^T(\widehat{\mathbf{Z}}+\mathsf{\Sigma })\mathrm{sgn}\left(\mathbf{s}\right)-{\mathbf{s}}^T\mathbf{w}-{(\mathbf{z}-\widehat{\mathbf{z}})}^T{\mathsf{\Psi }}^{-1}\mathsf{\Psi }\mathbf{S}\mathrm{sgn}\left(\mathbf{s}\right)\hfill \\ \hfill & \hfill & \hfill & =\hfill & \hfill & -{\mathbf{s}}^T\widehat{\mathbf{Z}}\mathrm{sgn}\left(\mathbf{s}\right)-{\mathbf{s}}^T\mathsf{\Sigma }\mathrm{sgn}\left(\mathbf{s}\right)-{\mathbf{s}}^T\mathbf{w}\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & -{\mathbf{z}}^T\mathbf{S}\mathrm{sgn}\left(\mathbf{s}\right)+{\widehat{\mathbf{z}}}^T\mathbf{S}\mathrm{sgn}\left(\mathbf{s}\right)\hfill \end{array}$$

(19)

Due to matrices $\mathbf{S}$, $\mathbf{Z}$, and $\widehat{\mathbf{Z}}$ being diagonal, the commutative property of the matrix multiplication is satisfied. Hence:

$${\widehat{\mathbf{z}}}^T\mathbf{S}=\sum _{i=1}^n\left({\widehat{z}}_i{s}_i\right)={\mathbf{s}}^T\widehat{\mathbf{Z}}$$

(20)

$${\mathbf{z}}^T\mathbf{S}=\sum _{i=1}^n\left(z_i{s}_i\right)={\mathbf{s}}^T\mathbf{Z}$$

(21)

Substituting (20) and (21) into (19) gives:

$$\begin{array}{cc}\hfill \dot{V}& =-{\mathbf{s}}^T\mathsf{\Sigma }\mathrm{sgn}\left(\mathbf{s}\right)-{\mathbf{s}}^T\mathbf{Z}\mathrm{sgn}\left(\mathbf{s}\right)-{\mathbf{s}}^T\mathbf{w}\hfill \\ \hfill & =\sum _{i=1}^n\left(-{\sigma }_i{s}_i\mathrm{sgn}\left(s_i\right)-z_i{s}_i\mathrm{sgn}\left(s_i\right)-s_i{w}_i\right)\hfill \\ \hfill & =\sum _{i=1}^n\left(-{\sigma }_i\left|s_i\right|-z_i\left|s_i\right|-s_i{w}_i\right)\hfill \\ \hfill & \le \sum _{i=1}^n\left(-{\sigma }_i\left|s_i\right|-z_i\left|s_i\right|+\left|s_i\right|\left|w_i\right|\right)\hfill \\ \hfill & =\sum _{i=1}^n\left(-{\sigma }_i-z_i+\left|w_i\right|\right)\left|s_i\right|\hfill \end{array}$$

(22)

Applying Assumption 1 to (22) gives:

$$\dot{V}\le -\sum _{i=1}^n{\sigma }_i\left|s_i\right|\le 0$$

(23)

Equation (23) expresses that the global stability of the robot system using the proposal adaptive sliding mode controller (11) is guaranteed by the Lyapunov criterion. The convergence to zero of sliding variable $s_i$ within finite time is proven. Consequently, tracking error $e_i$ converges to zero.

4. Quasi-Physical Model of the Serpent 1 Robot

The effectiveness of the designed ASMC can be verified through simulation by using a conventional mathematical robot model as a virtual plant. However, in order to increase the reliability of the results, a quasi-physical robot model will be used instead of the mathematical robot model. In this section, a Serpent 1 robot is selected as an illustrative object, and its quasi-physical model will be built. The SCARA Serpent 1 robot has three rotary joints using servomotors. The robot end-effector can be moved vertically by a pneumatic cylinder. This study concentrates on the problem of actuator torque loss; therefore, the pneumatic cylinder can be inactive. In fact, this neglect does not affect the position and orientation of the end-effector (EE) projected on the working plane. The initial configuration of the Serpent 1 robot with attached-body frames is depicted in Figure 1, and the D-H parameters are described in

Table 1. The D-H parameters of the Serpent 1 robot.

Joint i	${\mathit{\theta }}_{\mathit{i}}\left(\mathbf{rad}\right)$	${\mathit{d}}_{\mathit{i}}\left(\mathbf{m}\right)$	${\mathit{a}}_{\mathit{i}}\left(\mathbf{m}\right)$	${\mathit{\alpha }}_{\mathit{i}}\left(\mathbf{rad}\right)$
1	$q_1$	$d_1=0$	$a_1=0.25$	${\alpha }_1=0$
2	$q_2$	$d_2=0$	$a_2=0.15$	${\alpha }_2=0$
3	$q_3$	$d_3=0$	$a_3=0.070$	${\alpha }_3=+\pi$

.

In Figure 1, besides the base frame, which is $O_0{x}_0{y}_0{z}_0$, another fixed frame $O_w{x}_w{y}_w{z}_w$ (called the world frame) is also depicted with $a_0=0.15$ and $d_0=0.3725$. Based on the three-dimensional CAD assembly files of the Serpent 1 robot depicted in Figure 2, the approximated values of mass $m_i$ (kg), body centroid ${\mathbf{r}}_{C_i}$ (m) with respect to the attached-body coordinate system, and inertia tensor ${\mathbf{I}}_i$ (kg m²) can be obtained by any professional mechanical design software. In this way, the nominal values of robot parameters have been achieved and are shown in (24):

$$\begin{array}{cccccc}\hfill & \mathrm{Base}\hfill & \hfill & :\hfill & \hfill & m_0=7.645\hfill \\ \hfill & \mathrm{Link}1\hfill & \hfill & :\hfill & \hfill & m_1=3.529,{\mathbf{r}}_{C_1}^T=\left[-85.088,0,69.331\right]\times {10}^{-3},\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & {\mathbf{I}}_1=\left[\begin{array}{ccc}57.219& 0& -20.2\\ 0& 88.702& 0\\ -20.2& 0& 33.877\end{array}\right]\times {10}^{-3}\hfill \\ \hfill & \mathrm{Link}2\hfill & \hfill & :\hfill & \hfill & m_2=1.397,{\mathbf{r}}_{C_2}^T=\left[-80.403,0,-3.111\right]\times {10}^{-3},\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & {\mathbf{I}}_2=\left[\begin{array}{ccc}2.220& 0& -0.381\\ 0& 6.817& 0\\ -0.381& 0& 5.433\end{array}\right]\times {10}^{-3}\hfill \\ \hfill & \mathrm{Link}3\hfill & \hfill & :\hfill & \hfill & m_3=1.315,{\mathbf{r}}_{C_3}^T=\left[8.101,0,-244.765\right]\times {10}^{-3},\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & {\mathbf{I}}_3=\left[\begin{array}{ccc}48.294& 0& 1.831\\ 0& 48.480& 0\\ 1.831& 0& 0.423\end{array}\right]\times {10}^{-3}\hfill \end{array}$$

(24)

The gravity acceleration is chosen as 9.807 m/s². Because the robot parameters are not perfectly exact, the controller uses nominal values (24), whereas the virtual robot model (quasi-physical model) is set up with 10% deviation from its nominal values. Figure 3 depicts a quasi-physical model of the Serpent 1 robot based on MATLAB/Simscape Multibody similar to the approach presented in [42]. Every joint in the quasi-physical model is affected by rotational friction with parameters described in

Table 2. Rotational friction parameters (Reprinted/adapted with permission from [42]. 2020, Le Ngoc Truc. See [42] for detail descriptions of notations).

Joint i	${\mathit{\tau }}_{\mathit{brki}}\left(\mathbf{Nm}\right)$	${\mathit{\omega }}_{\mathit{brki}}(\mathbf{rad}/\mathbf{s})$	${\mathit{\tau }}_{\mathit{Ci}}\left(\mathbf{Nm}\right)$	${\mathit{k}}_{\mathit{vi}}(\mathbf{Nm}/(\mathbf{rad}/\mathbf{s}\left)\right)$
1	$30\times {10}^{-3}$	$0.01$	$29\times {10}^{-3}$	$0.001$
2	$23\times {10}^{-3}$	$0.01$	$20\times {10}^{-3}$	$0.001$
3	$10\times {10}^{-3}$	$0.01$	$9\times {10}^{-3}$	$0.001$

.

5. Simulation Results

The initial robot configuration in joint-space is $\mathbf{q}\left(0\right)=\dot{\mathbf{q}}\left(0\right)={\left[0,0,0\right]}^T$, and in the base frame $O_0{x}_0{y}_0{z}_0$ is point O = ${\left[0.4,0,d_3\right]}^T$ with three orientation vectors ($\mathbf{n}={\left[1,0,0\right]}^T$, $\mathbf{o}={\left[0,-1,0\right]}^T$, and $\mathbf{a}={\left[0,0,-1\right]}^T$). Vectors $\mathbf{n}$, $\mathbf{o}$, and $\mathbf{a}$ are the unit vectors of axes $x_3$, $y_3$, and $z_3$ with respect to the base frame, respectively. In the base frame, consider two points A and B with attached coordinates ${\left[0.36,0,d_3\right]}^T$ and ${\left[0,0.36,d_3\right]}^T$, respectively. The given task of the robot is to move its end-effector from point A to point B along the straight line AB (Figure 4). The motion trajectory is divided into two phases. Firstly, the robot end-effector slides along the $x_0$-axis from point O to point A (phase 1), and subsequently moves from point A to point B (phase 2). According to robot forward kinematics, position vector $\mathbf{p}$ and three orientation vectors of the end-effector are calculated in (25):

$$\begin{array}{cc}\hfill \mathbf{p}& :={[p_x,p_y,p_z]}^T={[a_1{c}_1+a_2{c}_{12},a_1{s}_1+a_2{s}_{12},d_3]}^T\hfill \\ \hfill \mathbf{n}& :={[n_x,n_y,n_z]}^T={[c_{123},s_{123},0]}^T\hfill \\ \hfill \mathbf{o}& :={[o_x,o_y,o_z]}^T={[s_{123},-c_{123},0]}^T={[n_y,-n_x,0]}^T\hfill \\ \hfill \mathbf{a}& :={[a_x,a_y,a_z]}^T={[0,0,-1]}^T\hfill \end{array}$$

(25)

where $s_1:=sin\left(q_1\right)$, $s_{12}:=sin\left(q_{12}\right)=sin\left(q_1+q_2\right)$, $s_{123}:=sin\left(q_{123}\right)=sin(q_1+q_2+q_3)$, $c_1:=cos\left(q_1\right)$, $c_{12}:=cos\left(q_{12}\right)=cos\left(q_1+q_2\right)$, and $c_{123}:=cos\left(q_{123}\right)=cos\left(q_1+q_2+q_3\right)$.

The requirement for the end-effector orientation during the movement on trajectory AB is that vector $\mathbf{n}$ is kept perpendicular to line AB. This implies that the angle $q_{123}$ described in Figure 4 (i.e., the angle between vector $\mathbf{n}$ and the $x_0$-axis) is maintained at $\pi /4$ rad through phase 2. Let us call the angle $q_{123}$ the “angle of vector $\mathbf{n}$”, and the “angle of vector $\mathbf{o}$” is named for the angle ($-\pi /2+q_{123}$). In each phase, the speed and acceleration along the desired trajectory in the base frame are mainly converted to velocity $v_x$ and acceleration $ac{c}_x$ along the $x_0$-axis. The velocity and acceleration components along the $y_0$-axis are calculated as $v_y=-v_x$ and $ac{c}_y=-ac{c}_x$, respectively. The $z_0$-components $v_z=0$ and $ac{c}_z=0$, since there is no movement along the $z_0$-axis. The motion trajectory along the $x_0$-axis is designed by utilizing trapezoidal acceleration profiles (Figure 5 and Figure 6). The design of corresponding orientation also uses that technique with the same time-frame (Figure 7 and Figure 8). Then, the desired position trajectories combined with the corresponding orientation can be converted to the desired joint variable trajectories.

Remark 1.

A sliding surface (9) is selected with larger coefficient $b_i$, then error $e_i$ more rapidly converges to zero. Saturation function $sat\left(s_i/h_i\right)$ can be implemented instead of the sign function in (11) and (13) for chattering reduction. The greater the value of $h_i$ is, the lesser the amplitude of the chattering phenomenon is; however, this will prolong the time to reach the sliding surface. The larger the coefficient ${\psi }_i$ in (13), the faster the convergence of ${\widehat{z}}_i$ to $z_i$; however, actuators will be required to provide more torque in the beginning of operation. The increase of factor ${\sigma }_i$ not only improves the convergence speed of sliding variable $s_i$ to zero but also raises the risk of the chattering phenomenon.

The value range of the design parameters is quite wide and depends on specific plants. After some simulations have been tested, the controller parameters are set as an illustration as follows: sliding surface (9) with $b_1=b_2=b_3=15$; saturation function $\mathrm{sat}\left(s_i/h_i\right)$ with $h_1=h_2=h_3=0.1$; adaptive speed parameters in (13) are ${\psi }_1={\psi }_2={\psi }_3=60$; additive positive parameters in (11) are ${\sigma }_1={\sigma }_2={\sigma }_3=1$. Simulations are performed in three cases:

Case 1: Parameters in (24) are assumed to be ideal and set in the quasi-physical model of the robot (Figure 3). In this case, rotational friction at joints is not considered in the quasi-physical model. All actuators work in normal operation $\lambda ={\left[1,1,1\right]}^T$.

Case 2: Assume that the deviation of parameters in (24) is of $10\%$, and consider that the rotational friction components (Table 2) exist in the respective joints. These parameters are set in the quasi-physical model of the robot (Figure 3). Every robot actuator suffers $10\%$ loss in PDT, corresponding to $\lambda ={\left[0.9,0.9,0.9\right]}^T$.

Case 3: The conditions of the quasi-physical robot model are initiated as in Case 2. However, the level of torque loss increases to $50\%$ corresponding to $\lambda ={\left[0.5,0.5,0.5\right]}^T$.

To evaluate the performance qualities of the end-effector position and orientation in the work-space, we judge the magnitude of the position error vector ${\mathbf{e}}_{\mathbf{p}}={\left[e_{px},e_{py},e_{pz}\right]}^T$ and the absolute error $e_{\mathbf{n}}$ between the desired and response angle of vector $\mathbf{n}$ that are defined in (26) and (27), respectively:

$$∥{\mathbf{e}}_{\mathbf{p}}∥=\sqrt{e_{px}^2+e_{py}^2+e_{pz}^2}=\sqrt{{(p_{xd}-p_x)}^2+{(p_{yd}-p_y)}^2+{(p_{zd}-p_z)}^2}$$

(26)

$$\left|e_{\mathbf{n}}\right|=\left|\mathrm{atan}2(n_{yd},n_{xd})-\mathrm{atan}2(n_y,n_x)\right|$$

(27)

where ${\mathbf{p}}_d={[p_{xd},p_{yd},p_{zd}]}^T$ is the desired vector of $\mathbf{p}$ and ${\mathbf{n}}_d={[n_{xd},n_{yd},0]}^T$ is the desired vector of $\mathbf{n}$. Simulation results for aforementioned cases are shown below.

5.1. Position Trajectory Response Analysis

When the model is correct and all robot actuators work in the normal operating range (Figure 9), the position response of the end-effector gives excellent quality (Figure 10) with a maximum error $∥{\mathbf{e}}_{\mathbf{p}}∥$ of 1.1$\times 10^{-5}$ m. In the remaining two cases, the end-effector finally follows the desired trajectory (Figure 11 and Figure 12), despite the existence of model parameter deviations, joint friction, and actuator loss in PDT being 10% (Figure 13), 50% (Figure 14) when implementing the ASMC controller. In case 2, the largest position deviation of the end-effector $∥{\mathbf{e}}_{\mathbf{p}}∥$ is $6.6\times 10^{-4}$ m from the initial time and $4.6\times 10^{-4}$ m counts from 1 s. For case 3, these maximum position deviations are $2.8\times 10^{-3}$ m and $4.2\times 10^{-4}$ m. The moving trajectories along the $x_0$, $y_0$, and $z_0$ axes of the end-effector are $p_x$, $p_y$, and $p_z$ reflected in Figure 15, Figure 16 and Figure 17, respectively, for cases 1, 2, and 3. The deviation in the $z_0$-axis is zero because the end-effector of the Serpent 1 robot moves only in the $O_0{x}_0{y}_0$ plane. The tracking errors $e_x$ and $e_y$ along the axes $x_0$ and $y_0$ are tiny for all three cases. Case 1 gives the best quality, and the other two cases also have small errors within $\pm 5\times 10^{-4}$ m after 1 s and the errors are always ensured to converge to zero.

5.2. Translational Velocity and Acceleration Response Analysis

Case 1 (Figure 18) shows that the translational velocity along the $x_0$ and $y_0$ axes of the end-effector has met the requirements for the motion stages with tiny errors (below $1\times 10^{-4}$ m/s). In Cases 2 and 3, the translational velocity still follows the design velocity, but the deviations gradually appear more obvious, especially during acceleration and deceleration (Figure 19 and Figure 20). Moreover, the uncertainties, friction, and actuator torque loss also have certain influences on the uniform motion phase. The $y_0$ axial velocity largely fluctuates during the first 1 s of acceleration (ranges approximately $\pm 5\times 10^{-3}$ m/s case 1, and approximately $\pm 0.02$ m/s in case 2). However, these deviations occur only in a short time and are quickly eliminated to zero. The translational acceleration responses are also shown in Figure 21, Figure 22 and Figure 23 for cases 1, 2, and 3, respectively. Actuator loss in torque has a major influence on acceleration. The greater the degree of attenuation, the greater the effect. Similar to velocity, the acceleration also fluctuates during the first 1 s, and overshoot happens at several transitions between the different profiles of the acceleration set (curved and straight profiles). It can be seen that the ASMC controller still shows reliable results in limiting these effects and making the acceleration follow the desired acceleration profile.

5.3. Orientation Responses Analysis

The desired orientation of the end-effector in the base coordinate is represented in Figure 24. The Serpent 1 robot is considered with motion in the horizontal plane; hence, vector $\mathbf{a}$ is always parallel and opposite to axis $z_0$. The orientation responses of the end-effector in the three cases are shown in Figure 25, Figure 26 and Figure 27, respectively. The projections on plane $O_0{x}_0{y}_0$ partly illustrate that the orientation of the end-effector has met the requirements in the process of moving along the trajectory. However, for a more detailed assessment of the orientation response quality, Figure 28, Figure 29 and Figure 30, which reflect the angle of vector $\mathbf{n}$ (i.e., $q_{123}$) and the angle of vector $\mathbf{o}$ (i.e., $q_{123}-\pi /2$), are also considered.

In case 1, we can see that angular errors of vectors $\mathbf{n}$, $\mathbf{o}$ are ensured to be zero when the robot move uniformly on OA and AB (Figure 28). The greatest error $\left|e_{\mathbf{n}}\right|$ is $4.1\times 10^{-5}$ rad only exists on OA when accelerating (in the first 1 s) and decelerating (from time 2 s to time 3 s). Furthermore, the direction requirement of vector $\mathbf{n}$ is maintained at $q_{123}=\pi /4$ rad.

In case 2 (Figure 29), in about the first 1 s, maximum overshoot of tracking error $e_{\mathbf{n}}$ is witnessed at the amplitude of 0.095 rad $=5.4^0$ and then, this error converges to zero. In the motion along AB (from time 4 s to time 10 s), $e_{\mathbf{n}}$ is kept in a vicinity of zero with the greatest deviation of $\left|e_{\mathbf{n}}\right|$ is 0.0026 rad $=0.15^0$ and decreases while approaching B.

In case 3, the orientation response is relatively similar to case 2 but with worse control quality due to higher torque loss in actuators. (Figure 30). Vector $\mathbf{n}$ takes 1.2 s to track the reference angular trajectory. In the transient time period of 1.2 s, the amplitude of error $e_{\mathbf{n}}$ has the greatest value of 0.106 rad $=6.07^0$. In the motion along AB, error $e_{\mathbf{n}}$ slowly varies around the origin with the greatest amplitude of 0.0032 rad $=0.18^0$ and converges to zero while approaching B.

6. Conclusions

This paper proposed a fault-tolerant controller for a class of serial industrial robots with uncertainties and proportional torque loss in actuators. The proposed adaptive sliding mode controller (ASMC) is synthesized to have the fault-tolerant ability and reduce the effect of this situation. The parameters of the control algorithm are chosen based on the Lyapunov criterion to guarantee the global stability of the robot system. The effectiveness of the proposed ASMC is verified by applying on a 3-DOF Serpent 1 robot. Simulation results show that the robot is completely able to track the reference trajectory with infinitesimal errors when a $10\%$ torque loss exists in actuators. Requirements for positions, velocities, accelerations, and orientations are satisfied. It is undeniable that the response of the system worsens as the decline degree of actuator torque increases. Nevertheless, when the torque loss is up to $50\%$, the controller still ensures the stability of the robot system and brings an acceptable response. Theoretically, the proposal control algorithm is proven to be able to handle the study problem, and simulation results have shown its effectiveness. However, when it is applied for a real robot system, there are some arising problems to face, such as adjustment of control parameters to minimize the chattering magnitude, measurement noises, signal processing techniques, effects of sampling time and transient time of inner control loops embedded inside motor-drive boards, etc. Additionally, in this study, the reduction in the chattering phenomenon is just the usage of saturation functions instead of signum functions, and external disturbances have not been taken into account. This will be our focus in the next experimental phase.