An Intelligent Position-Tracking Controller for Constrained Robotic Manipulators Using Advanced Neural Networks

Abstract

Nowadays, robots have become a key labor force in industrial manufacturing, exploring missions as well as high-tech service activities. Possessing intelligent robots for such the work is an understandable reason. Adoptions of neural networks for excellent control accuracies of robotic control systems that are restricted in physical constraints are practical challenges. This chapter presents an intelligent control method for position tracking control problems of robotic manipulators with output constraints. The constrained control objectives are transformed to be free variables. A simple yet effective driving control rule is then designed to force the new control objective to a vicinity around zeros. To suppress unexpected systematic dynamics for outstanding control performances, a new neural network is employed with a fast-learning law. A nonlinear disturbance observer is then used to estimate the neural estimation error to result in an asymptotic control outcome. Robustness of the closed loop system is guaranteed by the Lyapunov theory. Effectiveness and feasibility of the advanced control method are validated by comparative simulation.

1. Introduction

The world is now passing the Industry Revolution 4.0 in which robots have played a crucial role in industrial, manufacturing, discovering, rescuing and day-life activities. Excellent position controllers are required in most of industrial robots [1, 2]. However, in reality, it is not easy to achieve outstanding control precision with simple control structures due to unexpected influences of internal uncertain nonlinearities and unpredictable external disturbances in systematic dynamics [3, 4, 5, 6]. Nevertheless, most real-life robot joints are restricted in certain physical ranges. Note that, few danger issues could be activated if the joints went over such the boundaries [4, 5]. To deal with the strict control problems, many research outcomes have been recently reported for both fully actuated and underactuated robotic systems [7, 8]. To realize control objective in predefined constraints, backstepping-based controllers are favorite approaches for developers [9, 10]. Barrier Lyapunov functions are employed as core-stones to implement the nonlinear control procedures [11, 12]. Such the advanced state-interfered techniques could cope with both static and dynamical practical constraints of robotic systems [11, 13]. As comparing to the backstepping-based methods, sliding-mode-control (SMC) approaches are also potential control solutions for output-constraint control problems thanks to the simpler design and implementation [14, 15]. Furthermore, the SMC ones could be upgraded with employment of soft boundaries to result in Prescribed-Performance Control (PPC) remedies which could maintain the control objectives within predefined control accuracies [16, 17].

To reach excellent control performances, the nonlinear behaviors of the robotic systems need to be compensated during the control process [14, 15, 16, 17, 18]. The uncertain functionalities could be modeled with classical approaches such as basic force/torque transformation or optimal-energy solutions or decomposition analyses [18, 19]. Such the classical methods seem to be effective with simple robotic systems since they highly depend on the system structure [7, 20]. To enhance the modeling performances, fast-estimation approaches were studied in the past few years based on time-delay estimation (TDE) technologies [21, 22]. lumped dynamics of the system are simply approximated from information of acceleration signals and input gain matrices selected [23, 24]. Owing to the simplicity in deployment, a vast of real-time applications have been developed using such the TDE algorithm [24, 25]. Since the acceleration signals are normally computed from the position signals using high-order time derivatives, measurement noises could be amplified reducing the estimation effect [26, 27]. In fact, to learn the systematic behaviors in a model-free manner, intelligent methods are also great solutions [28, 29]. Thanks to the ability of universal approximation, the system dynamics could be learnt under black-box models using Radial-basis function (RBF) networks [30, 31, 32] or Fuzzy-hybrid-networks [33, 34, 35]. Once the neural networks are integrated in the control process, the control error could be adopted as main excitation signals of the learning process.

Since the networks require abundant excitation signals to activate the learning processes, the intelligent controllers would sacrifice unexpected transient time to reach the excellent steady-state control outcomes [30, 36]. As a result, high learning rates were normally adopted in the classical learning rules to speed up the estimation processes. Note that, the conventional learn laws only ensure boundedness of the control errors instead of the learning errors [5, 37]. To create certain bounds of neural weighting coefficients, the networks were modified by integrating linear-leakage functions in their adaptation mechanisms [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Since the adaptation rules of free channels were not properly deactivated, the convergence processes of the overall systems are slower than those of the conventional ones. Note that although the nonlinear dynamics of the robotic systems could be efficiently compensated by the advanced neural networks, to yield outstanding transient control precision, the neural estimation errors need to be tackled [11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. Integration of both neural networks and disturbance observers in nonlinear controllers have been proven to be excellent solutions for modern robotic systems [39, 40, 41]. Indeed, such the nonlinear integration was shown promising control results in stiffness-control robots, and in an exoskeleton, or in cooperative robots [42, 43, 44]. However, intelligent controllers using linear leakage functions in the neural adaptation rules require large robust signals to attain asymptotic control errors. With the combination of the neural network and disturbance in the intelligent approaches, the transient performances were remarkably improved, but the control errors were not still driven to zero in a smooth manner.

This chapter presents a new intelligent high-performance motion controller for robotic manipulators with output constraints. To deal with such the constraint problem, the control objective is first converted to a free variable using a new nonlinear transformation function. The indirect control objective is next driven to a certain vicinity using a sliding-mode-like control signal. A nonlinear neural network and disturbance observer are combined in a special fashion to construct a new closed-loop system in which both the estimation and control errors are pushed to zero in infinite time. The proposed controller possesses the following contributions:

• A novel nonlinear controller is proposed to stabilize the control objective inside arbitrary vicinity of zero without violation of the physical constraints.

• A nonlinear learning law of the neural network is developed to effectively estimate uncertain nonlinearities in the system model.

• To result in the asymptotic control performance of the overall system the neural estimation error is finally tackled by a nonlinear disturbance observer integrated.

• Working performances of the proposed control method is rigorously analyzed by an integral Lyapunov approach and extensive simulation results.

Outline of the paper is structured as follows. Section 2 presents the modeling of the studied systems and problem statements. Section 3 shows the design procedure of the proposed control algorithm with new neural disturbance estimation techniques and the stability analysis. Section 4 discusses the validation results obtained from comparative simulations. The conclusions are finally drawn in Section 5.

2. System model and problem statements

Behaviors of a general nDOF robot can be expressed using the following dynamics [19, 20]:

$$\mathbf{M}⌊\mathbf{q}⌋\ddot{\mathbf{q}}\mathbf{+}\mathbf{C}⌊\mathbf{q},\dot{\mathbf{q}}⌋\dot{\mathbf{q}}\mathbf{+}\mathbf{g}⌊\mathbf{q}⌋\mathbf{+}\mathbf{f}⌊\dot{\mathbf{q}}⌋\mathbf{+}{\mathbf{\tau }}_{\mathbf{d}}\mathbf{=}\mathbf{\tau }$$E1

where $\mathbf{\tau }\in {\Re }^n$ is the vector of the control torques generated by joint actuators, $\mathbf{q}\in {\Re }^n$ is the vector of joint position or the system output, $\mathbf{M}⌊\mathbf{q}⌋\in {\Re }^{n\times n}$ is the positive-definite mass matrix, $\mathbf{C}⌊\mathbf{q},\dot{\mathbf{q}}⌋\dot{\mathbf{q}},\mathbf{g}⌊\mathbf{q}⌋\mathbf{,}\mathbf{f}⌊\dot{\mathbf{q}}⌋\mathbf{,}{\mathbf{\tau }}_{\mathbf{d}}\in {\Re }^n$ are the Centripetal/Coriolis vector, the gravitational, frictional, and the external disturbance torques, respectively.

Remark 1: The main control objective here is to derive a proper control signal $\left(\mathbf{\tau }\right)$ to drive the system output $\left(\mathbf{q}\right)$ following a desired trajectory $\left({\mathbf{q}}_d\right)$.

Before designing the expected control approach, the following assumptions are consolidated.

Assumption 1 [45, 46]: The disturbance $\left({\mathbf{\tau }}_{\mathbf{d}}\right)$ is bounded and Lipschitz continuous.

Assumption 2: the reference profile $\left({\mathbf{q}}_{\mathbf{d}}\right)$ is known, bounded and twice continuously differentiable.

Assumption 3: The system states $\left(\mathbf{q},\dot{\mathbf{q}}\right)$ are measurable.

Remark 2: The robotic system Eq. (1) is a passivity model with bounded time-derivative states [19, 22, 45]. For practical systems, the robot joints $\left(\mathbf{q}\right)$ are limited in physical ranges:

$$\underset{¯}{\mathbf{q}}\le \mathbf{q}\le \overline{\mathbf{q}}$$E2

where $\left(\underset{¯}{\mathbf{q}}\text{and}\overline{\mathbf{q}}\right)$ are respectively the lower and upper bounds of the system output $\left(\mathbf{q}\right)$.

In reality, unexpected impacts from physical collisions could make the system danger.

Remark 3: To obtain an excellent controller for the stated problem, one needs a proper control strategy that could deal with dynamical nonlinear behaviors of the robotic system Eq. (1) in complying with the physical constraint and be able to drive the control objective to zero as fast as possible. Furthermore, the controller is also expected to be robust and model-free.

3. Intelligent nonlinear constrained controller

A robust adaptive controller is designed in this section based on a new constrained sliding mode framework and new learning mechanism of a basic neural network and nonlinear disturbance observer. Stability of the closed control system is then investigated by Lyapunov theories.

3.1 Constrained sliding mode control with neural network

We first define the following control error as the main control objective:

$$\mathbf{e}=\mathbf{q}-{\mathbf{q}}_{\mathbf{d}}$$E3

The error is in fact allowed to vary in the following range that is constructed by the constraint (2).

$$\left\{\begin{array}{l}\underset{¯}{\mathbf{e}}\le \mathbf{e}\le \overline{\mathbf{e}}\\ \overline{\mathbf{e}}\equiv \overline{\mathbf{q}}-{\mathbf{q}}_{\mathbf{d}}>0\\ \underset{¯}{\mathbf{e}}\equiv \underset{¯}{\mathbf{q}}-{\mathbf{q}}_{\mathbf{d}}<0\end{array}\right.$$E4

where $\left(\underset{¯}{\mathbf{e}}\text{and}\overline{\mathbf{e}}\right)$ are the lower and upper physical bounds of the control error$\left(\mathbf{e}\right)$, respectively.

The following transformation function is next proposed to map the constrained error $\left(\mathbf{e}\right)$ to a new free space:

$$y_{i\mid i=1..n}=\frac{e_i}{\left({\overline{e}}_i-e_i\right)\left(e_i-{\underset{¯}{e}}_i\right)}$$E5

where $\mathbf{y}={\left[y_1,y_2,\dots ,y_n\right]}^T$ is the transformed error, $e_i$ is a specific entry of the control error vector $\mathbf{e}={\left[e_1,e_2,..e_i.,e_n\right]}^T$.

A sliding manifold is defined as an indirect control objective of the studied system:

$$\mathbf{s}=\dot{\mathbf{e}}+{\mathbf{K}}_0\mathbf{y}$$E6

where ${\mathbf{K}}_0=diag⌊{\mathbf{k}}_0⌋=diag⌊\left[k_{01};\dots ;k_{0n}\right]⌋$ is a positive-definite diagonal gain matrix.

The time derivative of the manifold Eq. (6) under the dynamics Eq. (1) is expressed

$$\dot{\mathbf{s}}=-\mathbf{v}+{\overline{\mathbf{M}}}^{-1}\mathbf{\tau }-{\ddot{\mathbf{q}}}_{\mathbf{d}}+{\mathbf{K}}_{\mathit{0}}\dot{\mathbf{y}}$$E7

where, $\mathbf{v}=-{\mathbf{M}}^{-1}\left(\mathbf{C}\dot{\mathbf{q}}+\mathbf{g}+\mathbf{f}+{\mathbf{\tau }}_{\mathbf{d}}\right)+\left({\mathbf{M}}^{-1}-{\overline{\mathbf{M}}}^{-1}\right)\mathbf{\tau }\in {\Re }^n$ is defined as a systematic-deviation term that is composited from both the internal dynamics and external disturbances, and $\overline{\mathbf{M}}=diag⌊\left[{\overline{m}}_1,{\overline{m}}_2,\dots ,{\overline{m}}_m\right]⌋$ is a nominal positive-definite mass matrix selected.

Based on the manifold system Eq. (7), the final control signal is structured from a dynamical control term $\left({\mathbf{\tau }}_{\mathbf{DYN}}\right)$, error-driving term $\left({\mathbf{\tau }}_{\mathbf{DRI}}\right)$, and robust control term $\left({\mathbf{\tau }}_{\mathbf{ROB}}\right)$, as follows:

$$\mathbf{\tau }\mathbf{=}\overline{\mathbf{M}}\left({\mathbf{\tau }}_{\mathbf{DYN}}+{\mathbf{\tau }}_{\mathbf{DRI}}+{\mathbf{\tau }}_{\mathbf{ROB}}\right)$$E8

The dynamical signal $\left({\mathbf{\tau }}_{\mathbf{DYN}}\right)$ is used to compensate for the internal dynamics $\left(\mathbf{v}\right)$ in Eq. (7). With robotic manipulators, the lumped dynamics $\left(\mathbf{v}\right)$ are bounded [19, 42] but very complicated and not easy to derive [20]. To study such the complex behaviors, a neural network could be thought of a reasonable tool. The dynamics $\mathbf{v}={\left[v_1,v_2,\dots ,v_n\right]}^T$ can be modeled using the following universal linear combination:

$$v_{i\mid i=1..n}={\mathbf{w}}_i^T{\mathbf{r}}_i⌊\mathbf{q},\dot{\mathbf{q}},\mathbf{\tau }⌋+{\delta }_i$$E9

where ${\mathbf{w}}_i,{\mathbf{r}}_i⌊\mathbf{q},\dot{\mathbf{q}},\mathbf{\tau }⌋,{\delta }_i$ are optimal weight vectors, neural regression vectors, and the modeling error, respectively.

Hence, the signal is structured as follows:

$${\mathbf{\tau }}_{\mathbf{MOD}}=\widehat{\mathbf{v}}+{\ddot{\mathbf{q}}}_{\mathbf{d}}-{\mathbf{K}}_{\mathit{0}}\dot{\mathbf{y}}$$E10

where, the approximation ${\widehat{v}}_i$ is estimate of the dynamics $v_i$, and is designed as [17, 42]

$${\widehat{v}}_{i\mid i=1..n}={\widehat{\mathbf{w}}}_i^T{\mathbf{r}}_i⌊\mathbf{q},\dot{\mathbf{q}},\mathbf{\tau }⌋$$E11

in which, ${\widehat{\mathbf{w}}}_i$ is estimate of the weight vector ${\mathbf{w}}_i$.

By employing the dynamical control signal Eq. (10), the dynamics Eq. (7) become

$$\dot{\mathbf{s}}=\stackrel{\sim }{\mathbf{v}}\mathbf{+}{\mathbf{\tau }}_{\mathbf{DRI}}+{\mathbf{\tau }}_{\mathbf{ROB}}$$E12

where $\stackrel{\sim }{\mathbf{v}}={\left[{\stackrel{\sim }{v}}_1,{\stackrel{\sim }{v}}_2,\dots ,{\stackrel{\sim }{v}}_n\right]}^T=\widehat{\mathbf{v}}-\mathbf{v}\in {\Re }^n$ is an estimation-error vector.

Since the role of the control signal $\left({\mathbf{\tau }}_{\mathbf{DRI}}\right)$ is to drive the sliding manifold to around zero from an arbitrary initial position, it is selected as

$${\mathbf{\tau }}_{\mathbf{DRI}}=-{\mathbf{K}}_1\mathbf{s}$$E13

where ${\mathbf{K}}_1=diag⌊{\mathbf{k}}_1⌋=diag⌊\left[k_{11};\dots ;k_{1n}\right]⌋$ is a diagonal positive-definite gain matrix.

Since the robust control signal ${\mathbf{\tau }}_{\mathbf{ROB}}$ is adopted to suppress the estimation error $\left(\mathbf{\delta }\right)$, it is designed as

$${\mathbf{\tau }}_{\mathbf{ROB}}=-{\mathbf{K}}_{2}sgn⌊\mathbf{s}⌋$$E14

where ${\mathbf{K}}_2=diag⌊{\mathbf{k}}_2⌋=diag⌊\left[k_{21};\dots ;k_{2n}\right]⌋$ is a diagonal positive-definite gain matrix.

The manifold dynamics Eq. (12) is now expressed as

$$\dot{\mathbf{s}}=\stackrel{\sim }{\mathbf{v}}-{\mathbf{K}}_1\mathbf{s}-{\mathbf{K}}_{2}sgn⌊\mathbf{s}⌋$$E15

Remark 3: The dynamics Eq. (15) indicate that the closed-loop system is bounded stable if the estimate $\left(\widehat{\mathbf{v}}\right)$ is bounded. In theoretical aspects, the asymptotic control performance would be resulted in if the robust gain $\left({\mathbf{k}}_2\right)$ is selected satisfying a condition of $\left({\mathbf{k}}_2>\mathbf{\delta }\right)$. However, with such a big robust control gain, it could activate chattering phenomena. In contrast, a small robust gain could reduce the control precision.

To approximate the nonlinear dynamics $\left(\mathbf{v}\right)$, the network Eq. (11) is activated using the control information of the sliding manifold under following rule:

$${\dot{\widehat{\mathbf{w}}}}_i=-diag⌊{\mathbf{a}}_{1i}⌋diag⌊{\mathbf{r}}_i^2⌋\frac{s_i^2}{1+s_i^2}{\widehat{\mathbf{w}}}_i-b_i{s}_i{\mathbf{r}}_i$$E16

where $\mathbf{B}=diag⌊\left[b_1,b_2,\dots ,b_n\right]⌋\text{and}diag{⌊{\mathbf{a}}_{1i}⌋}_{i=1..n}$ are diagonal positive-definite constant matrices.

The control performance of the neural-constrained sliding mode system is investigated by the following statements.

Theorem 1: By employing the robust control rule Eqs. (3)–(14) and the neural learning law Eqs. (11), Eq. (16) to control the robotic system Eq. (1) under the output constraint Eq. (2), the closed-loop system is asymptotically stable if the control gains comply with

$$\left\{\begin{array}{l}{k}_{1i\mid i=1..n}>\frac{1}{4{\epsilon }_i}{\mathbf{r}}_i^{T}diag⌊{\mathbf{r}}_i⌋diag⌊{\mathbf{a}}_{1i}⌋{\left|{\mathbf{w}}_i^2\right|}^T\\ k_{2i}>{\left|{\delta }_i\right|}_{\max }\end{array}\right.$$E17

The proof of Theorem 1 is given in Appendix A.

Remark 4: As seen in Eq. (15), once the real dynamics $\left(\mathbf{v}\right)$ are well estimated with an arbitrary small accuracy, small robust gains would yield good control performances. Note that, approximation by the neural network is a multi-channel learning work. The learning rule Eq. (16) is hence designed to increase the neural updating effect.

4. Disturbance-observer integration

The neural-constrained nonlinear control structure provides the excellent control performances with stationary trajectory signals. In high-speed working frequencies, the estimation error $\left(\mathbf{\delta }\right)$ becomes large and could degrade the control accuracy. Adoption of an additional control term based on the disturbance-observer technology could be an understandable solution. The following assumption could be taken into account:

Assumption 4: The error $\left(\mathbf{\delta }\right)$ and its time derivative are bounded. It could be thus modeled as a first-order system:

$$\dot{\mathbf{\delta }}=-diag⌊\mathbf{\alpha }⌋\mathbf{\delta }+\mathbf{\zeta }$$E18

where $diag⌊\mathbf{\alpha }⌋=diag⌊\left[{\alpha }_i,{\alpha }_2,\dots ,{\alpha }_n\right]⌋$ is a diagonal positive-definite constant matrix.$\mathbf{\zeta }={\left[{\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n\right]}^T$ is a virtual bounded disturbance vector.

To effectively compensate for the estimation error $\left(\mathbf{\delta }\right)$, the robust control signal Eq. (14) is updated with a disturbance-estimation term, as follows:

$${\mathbf{\tau }}_{\mathbf{ROB}}=-{\mathbf{K}}_{2}sgn⌊\mathbf{s}⌋+\widehat{\mathbf{\delta }}$$E19

at which $\widehat{\mathbf{\delta }}={\left[{\widehat{\delta }}_1,{\widehat{\delta }}_2,\dots ,{\widehat{\delta }}_n\right]}^T$ is estimate vector of the neural modeling error $\left(\mathbf{\delta }\right)$ . It is computed from the following learning rule:

$$\dot{\widehat{\mathbf{\delta }}}=-diag⌊\mathbf{\alpha }⌋\widehat{\mathbf{\delta }}-{\mathbf{BP}}^{-1}\mathbf{s}-{\mathbf{K}}_{3}sgn⌊\mathbf{s}⌋$$E20

Here, $\mathbf{P}=diag⌊\left[p_1,p_2,\dots ,p_n\right]⌋,{\mathbf{K}}_3=diag⌊\left[k_{31},k_{32},\dots ,k_{3n}\right]⌋$ are diagonal positive-definite constant matrices.

Validation results in previous work [46, 47] confirmed the learning efficiency of the disturbance observer Eq. (20) for simple systems. To connect the disturbance observer with the neural sliding mode control scheme, the adaptation rule of the network is improved as

$$\begin{array}{l}{\dot{\widehat{\mathbf{w}}}}_{i\mid i=1..n}=-diag⌊{\mathbf{a}}_{1i}⌋diag⌊{\mathbf{r}}_i^2⌋\frac{s_i^2}{1+s_i^2}{\widehat{\mathbf{w}}}_i-diag⌊{\mathbf{a}}_{2i}⌋diag⌊{\mathbf{r}}_i^2⌋{\widehat{\mathbf{w}}}_i\\ -\left(b_i{s}_i+p_i{k}_{3i}sgn⌊s_i⌋\right){\mathbf{r}}_i\end{array}$$E21

where $diag{⌊{\mathbf{a}}_{2i}⌋}_{i=1..n}$ is a diagonal positive-definite constant matrix.

The stability of the closed-loop system is validated by the following statement.

Theorem 2: By employing the robust control rule Eqs. (3)–(14) combining with the neural learning law Eqs. (11), Eq. (21) and disturbance observer Eqs. (19), (20) to control the robotic system Eq. (1) under the output constraint Eq. (2), the closed-loop system is asymptotically stable if the control gains comply with

$$\left\{\begin{array}{l}{k}_{1i\mid i=1..n}>\frac{1}{4b_i}{\mathbf{r}}_i^{T}diag⌊{\mathbf{r}}_i⌋diag⌊{\mathbf{a}}_{1i}⌋{\left|{\mathbf{w}}_i^2\right|}^T\\ k_{2i}>0\\ k_{3i}>{\left|{\varsigma }_i\right|}_{\max }+\frac{1}{4p_i{k}_{2i}}{\mathbf{r}}_i^{T}diag⌊{\mathbf{r}}_i⌋diag⌊{\mathbf{a}}_{2i}⌋{\left|{\mathbf{w}}_i^2\right|}^T\end{array}\right.$$E22

The proof of Theorem 2 is discussed in Appendix B.

Remark 5: From Theorem 2, it can be seen that the robust control gain $\left({\mathbf{k}}_2\right)$ could be selected with a small value for a high control accuracy. Obviously, the robustness of the closed-loop system is undertaken by a large value of the disturbance-observer gain$\left({\mathbf{k}}_3\right)$.Remark 6: After the sliding manifold $\left(\mathbf{s}\right)$ converges to zero, the control error $\left(\mathbf{e}\right)$ will approach to origin under the sliding phase [25, 27]. Adoption of the nonlinear synthetization (9) could speed up the convergence time of the sliding process [17, 18]. The detailed block diagram of the proposed controller is presented in Figure 1.

Figure 1.

Structure of the proposed controller.

5. Verification results

Validation results of the developed controller in various testing conditions are discussed are discussed in this section. To provide the competitive evaluation, a classical Proportional-Integral-Derivative (PID) controller and linear neural-disturbance-observer (LND) controller were also implemented to control the same system in the same working conditions. The LND algorithm was referred from previous and is re-expressed in Appendix C.

The controllers were employed for motion control of a 3DOF robot, as depicted in Figure 2. Detailed dynamics of the 3DOF robot were derived based on the Lagrange method [4, 19, 47], as formulated in Appendix D. The neural network had 9 inputs ${\left(q_i,{\dot{q}}_i,{\tau }_i\right)}_{\mid i=1,2,3}$ and 730 neurons with the logsig activation function in the hidden layer [42, 45]. All of the initial values of the weight vectors $\left({\widehat{\mathbf{w}}}_{i\mid i=1,2,3}\right)$ were set to be zero. Other simulation parameters of the dynamics and the controllers are shown in Tables 1 and 2, respectively. The control results obtained by the controllers are intensively discussed in the following subsections.

Figure 2.

Configuration of the simulation 3DOF robot.

Description	Parameters	Values	Unit
Link length 1	l1, l2, l3	0.1, 0.2, 0.2	m
Gravitational Accel	g	9.81	m/s^2
Friction coefficient	a1, a2, a3	20, 20, 20	N.s
Mass of links	m1, m2, m3	5, 3, 2	kg

Table 1.

Detailed parameters of the simulation model.

Description	Parameters	Values
LND Controller [45]
Nominal mass matrix	$\overline{\mathbf{M}}$	${\mathbf{I}}_3$
Control gains	${\mathbf{K}}_{c0},{\mathbf{K}}_{c1}$	$\begin{array}{l}diag\left(\left[10;100;10\right]\right),\\ diag\left(\left[200;100;10\right]\right)\end{array}$
Disturbance gain	${\mathbf{K}}_{c3}$	$30{\mathbf{I}}_3$
Learning rate	${\mathrm{\Gamma }}_{i\mid i=\mathrm{1..3}}$	$500{\mathbf{I}}_3$
Learning rates	${\mu }_{i\mid i=\mathrm{1..3}}$	$0.002{\mathbf{I}}_3$
PID
Control gains	KP, KI, KD	$\begin{array}{l}diag\left(\left[700;900;500\right]\right),\\ diag\left(\left[50;10;10\right]\right),\\ diag\left(\left[10;10;10\right]\right),\end{array}$
Proposed Controller
Nominal mass matrix	$\overline{\mathbf{M}}$	${\mathbf{I}}_3$
Control gains	${\mathbf{K}}_0,{\mathbf{K}}_1,{\mathbf{K}}_2,\text{and}{\mathbf{K}}_3$	$\begin{array}{l}diag\left(\left[100;100;10\right]\right),\\ diag\left(\left[40;100;2\right]\right),\\ 0.1{\mathbf{I}}_3,200{\mathbf{I}}_3.\end{array}$
Leakage rates	${\mathbf{a}}_{1i\mid i=\mathrm{1..3}},{\mathbf{a}}_{2i}$	$10{\mathbf{I}}_3,10{\mathbf{I}}_3$
Excitation rates	$\mathbf{B},\mathbf{P}$	$200{\mathbf{I}}_4,10{\mathbf{I}}_4$
Disturbance gain	${\alpha }_{i\mid i=\mathrm{1..3}}$	2

Table 2.

Selected parameters of the controllers.

6. Simulation results

In the first simulation, the desired profiles were sinusoidal signals with different frequencies (0.1 (Hz), 0.3 (Hz), and 0.5 (Hz)), as plotted in Figure 3. Physical ranges of the robot joints were set to be $\left[-0.5\pi ;0.5\pi \right]$. The control results obtained are compared in Figures 4–6.

Figure 3.

The desired profiles of the robot joints in the first simulation.

Figure 4.

System responses of the controllers obtained in the first simulation.

Figure 5.

Comparative control errors of the controller in the first simulation.

Figure 6.

Learning performances of the proposed controller in the first simulation.

As shown in Figure 5, stability of the closed-loop system could be maintained by the PID controller and with good control errors: 1 (deg), 1.39 (deg) and 5.6 (deg) for joints 1, 2, and 3, respectively. However, as carefully observed in the response of joint 2 in Figure 4, the physical constraints were violated by the PID control in the transient time. To void the unexpected collision and provide high control performances both in the transient and steady-state phases, a combination of neural network, disturbance-observer learning techniques and the constrained backstepping control signal was adopted in the LND controller. Indeed, outstanding control precision was resulted in by the LND one: the control precision at joints 1,2, and 3 were 0.094 (deg), 0.105 (deg), and 1.85 (deg), respectively. The output-constraint control problem could be also dealt with by the constrained control algorithm designed. Moreover, the nonlinear dynamics of the robotic system were eliminated well by the proposed neural-disturbance learning method. As a result, higher control performances were delivered by the studied controller: the control accuracies at joints 1 and 2 were 0.11 (deg) and 0.108 (deg), respectively. The control results in Figure 5 imply that although the control performances of the LND and proposed controllers were almost same in low-speed work conditions, they were clearly different in the high-frequency trajectory-tracking control. To this end, the proposed control algorithm was employed the new nonlinear learning rule Eq. (21) to improve the estimation effect, which are revealed from estimation data presented in Figure 6.

The controllers were continuously challenged with new various frequencies of the sinusoidal trajectories in the second test. The new frequencies at joints 1, 2, and 3 were selected to be 1 (Hz), 0.3 (Hz), and (0.7 Hz), respectively. Figure 7 presents pieces of the new reference signals with respect to time. Applying the same controllers to the robotic system, the results obtained are shown in Figure 8.

Figure 7.

The desired profiles of the robot joints in the second simulation.

Figure 8.

Comparative control errors in the second simulation.

Control results in Figure 8 indicate that the PID control accuracies were seriously degraded in arduous testing conditions: the control errors were increased to 9.32 (deg) and 8.18 (deg) at joints 1 and 2, respectively. The LND control method could however maintain acceptable control performances thanks to the merging linear learning algorithm: control precision at joints 1 and 3 was slightly increased to be 0.7 (deg) and 4.1 (deg), respectively. Note that as discussed in previous work [45], it is difficult to result in asymptotic control outcomes by the LND approach. This drawback was well overcome by the new learning rule proposed, in which the nonlinear network and disturbance observer were properly combined with an arbitrary small robust gain to ensure the asymptotic convergence of the closed-loop system. The convergences of the control errors obtained by the proposed control algorithm, as demonstrated in Figure 8, show that the uncertain nonlinearities and external disturbances in the system dynamics were well estimated by the collaborative nonlinear adaptation laws. The control and learning effectiveness of the new control approach was confirmed by the validation results achieved.

7. Additional discussion

By comparing the control results obtained by the two intelligent controllers, as presented in Figures 5 and 8, it can be seen that their control performances would be same in the steady-state phases but really different in the transient phases. The nonlinear learning integration led to the faster learning effect and higher control precision.

Estimation data illustrated in Figures 7 and 9 imply that the neural network played as a crucial role in approximating the system dynamics, and the estimation error was then learnt by the nonlinear disturbance observer. Furthermore, with the merging control technique proposed, one only needs an arbitrary small robust signal to result in asymptotic control outcomes, that ensures the smooth control behaviors as presented in Figure 10.

Figure 9.

Learning performances of the proposed controller in the second simulation.

Figure 10.

Control signals generated by the controllers in the second simulation.

Table 3 summarizes the maximum absolute (MA) and root-mean-square (RMS) values of the control errors for a specific working time (75 s to 85 s). As seen in the table, the best RMS errors were always provided by the designed controller even though its MA values were not the highest one in some cases. Here, we propose a ratio of RMS/MA values to deeply evaluate the control performances of the controllers in which those of the PID, LND and proposed controllers were in range of 0.64, 0.41, and 0.31, respectively. The smaller factors imply that the internal deviation and external disturbances were effectively eliminated by the corresponding controllers in better manners. The superior control performances of the proposed controller over the previous control methods are thus confirmed again by the intensive analyses based on the obtained results.

Control Error		Joint 1		Joint 2		Joint 3
		MA	RMS	MA	RMS	MA	RMS
The 1st case	PID	1.02	0.69	3.9	2.5	5.8	4.1
	LND	0.096	0.051	0.12	0.044	1.94	1.21
	PRO	0.1.	0.008	0.11	0.025	0.41	0.041
The 2nd case	PID	9.3	6.3	3.9	2.3	8.2	5.5
	LND	0.7	0.44	0.13	0.02	4.1	2.6
	PRO	0.16	0.04	0.12	0.02	0.83	0.22

Table 3.

Statistical control errors of the comparative controllers.

8. Conclusions

This chapter presents a new intelligent control method for high-performance motion control of robotic manipulators with output constraints. The controller is built based on a new neural-disturbance constrained sliding mode structure. A nonlinear sliding-mode control signal is first derived to strictly stabilize the control objective within predefined output constraints. The control accuracies are next improved by eliminating the nonlinear uncertainties and external disturbances in the system dynamics using a new nonlinear neural network. The estimation error is then compensated by proper integration of a nonlinear disturbance observer. By adoption of this neural-disturbance mechanism and a minor robust signal, an asymptotic control outcome is resulted in. The effectiveness of the overall control system is investigated by the rigorous theoretical proofs and comparative simulation results in various working conditions.

The following Lyapunov function is first considered:

$$L_1=0.5\sum _{i=1}^n\left(b_i{s}_i^2+{\stackrel{\sim }{\mathbf{w}}}_i^T{\stackrel{\sim }{\mathbf{w}}}_i\right)$$E23

By substituting the dynamics Eqs. (15) and (16) into the time derivative of the function (_A1), we next have

$$\begin{array}{l}{\dot{L}}_1=-\sum _{i=1}^n\left(b_i{k}_1{s}_i^2+\left|s_i\right|\left(k_2+{\delta }_i\right)\right)-\sum _{i=1}^n\left(\frac{s_i^2}{1+s_i^2}{\stackrel{\sim }{\mathbf{w}}}_i^{T}diag⌊{\mathbf{a}}_{1i}⌋diag⌊{\mathbf{r}}_i^2⌋{\widehat{\mathbf{w}}}_i\right)\\ \le -\sum _{i=1}^n\left(b_i{k}_1{s}_i^2+\left|s_i\right|\left(k_2+{\delta }_i\right)\right)-\sum _{i=1}^n\left(\frac{s_i^2}{1+s_i^2}{\stackrel{\sim }{\mathbf{w}}}_i^{T}diag⌊{\mathbf{a}}_{1i}⌋diag⌊{\mathbf{r}}_i^2⌋{\stackrel{\sim }{\mathbf{w}}}_i\right)\\ +\sum _{i=1}^n\left(\frac{s_i^2}{1+s_i^2}{\left|{\stackrel{\sim }{\mathbf{w}}}_i\right|}^{T}diag⌊{\mathbf{a}}_{1i}⌋diag⌊{\mathbf{r}}_i^2⌋\left|{\mathbf{w}}_i\right|\right)\end{array}$$E24

From the condition Eq. (17), there always exist two positive constants ${\lambda }_{i1\mid i=1..n},{\lambda }_{i2}$ for the following constraint:

$${\dot{L}}_1\le -\sum _{i=1}^n\left({\lambda }_{i1}{b}_i{k}_1{s}_i^2+{\lambda }_{i2}\frac{s_i^2}{1+s_i^2}{\stackrel{\sim }{\mathbf{w}}}_i^{T}diag⌊{\mathbf{a}}_{1i}⌋diag⌊{\mathbf{r}}_i^2⌋{\stackrel{\sim }{\mathbf{w}}}_i\right)$$E25

It means that the proof of Theorem 1 has been completed.

Dynamics of the subsystems Eqs. (15), (19), (20) in element-wise forms are presented as follows:

$${}_{i=1..n}\left\{\begin{array}{l}{\dot{s}}_i={\stackrel{\sim }{\mathbf{w}}}_i^T{\mathbf{r}}_i-k_1{s}_i-k_{2i}sgn⌊s_i⌋+{\stackrel{\sim }{\delta }}_i\\ {\dot{\stackrel{\sim }{\delta }}}_i=-{\alpha }_i{\stackrel{\sim }{\delta }}_i-\frac{b_i}{p_i}{s}_i-k_{3i}sgn⌊s_i⌋-{\varsigma }_i\end{array}\right.$$E26

A new integral-type Lyapunov function is investigated:

$$L_2=L_1+\sum _{i=1}^n\left(0.5p_i{\stackrel{\sim }{\delta }}_i^2+\underset{s_i⌊0⌋}{\overset{s_i}{\int }}\left(k_{3i}sgn⌊s_i⌋+{\varsigma }_i\right){\mathit{ds}}_i+L_{20i}\right)$$E27

where $L_{20i\mid i=1..n}$ is a positive constant selected as [46]:

$$L_{20i}=\frac{{\left(k_{3i}+{\left|{\varsigma }_i\right|}_{\max }\right)}^2}{2b_i}+\left(k_{3i}+{\left|{\varsigma }_i\right|}_{\max }\right)s_i⌊0⌋$$E28

The time derivative of the function (_B2) under the dynamical behaviors Eqs. (_B1) and (21) is constrained in the following inequality:

$$\begin{array}{l}{\dot{L}}_2\le -\sum _{i=1}^n\left(b_i{s}_i\left(k_1{s}_i+k_{2i}sgn⌊s_i⌋\right)+{\alpha }_i{p}_i{\stackrel{\sim }{\delta }}_i^2\right)\\ -\sum _{i=1}^n\left({\stackrel{\sim }{\mathbf{w}}}_i^T\left(diag⌊{\mathbf{a}}_{1i}⌋\frac{s_i^2}{1+s_i^2}+diag⌊{\mathbf{a}}_{2i}⌋\right)diag⌊{\mathbf{r}}_i^2⌋{\stackrel{\sim }{\mathbf{w}}}_i\right)\\ -\sum _{i=1}^n\left(p_i\left(k_1{s}_i+k_{2i}sgn⌊s_i⌋\right)\left(k_{3i}sgn⌊s_i⌋+{\varsigma }_i\right)\right)\\ +\sum _{i=1}^n\left({\left|{\stackrel{\sim }{\mathbf{w}}}_i\right|}^T\left(diag⌊{\mathbf{a}}_{1i}⌋\frac{s_i^2}{1+s_i^2}+diag⌊{\mathbf{a}}_{2i}⌋\right)diag⌊{\mathbf{r}}_i^2⌋\left|{\mathbf{w}}_i\right|\right)\end{array}$$E29

If the gains selected satisfying Eq. (22), there always exist another constant ${\lambda }_{i3\mid i=1..n}$ for the following inequality:

$$\begin{array}{l}{\dot{L}}_1\le -\sum _{i=1}^n\left({\lambda }_{i1}{b}_i{k}_1{s}_i^2+{\alpha }_i{p}_i{\stackrel{\sim }{\delta }}_i^2\right)\\ -\sum _{i=1}^n\left({\stackrel{\sim }{\mathbf{w}}}_i^T\left({\lambda }_{i2}\frac{s_i^2}{1+s_i^2}diag⌊{\mathbf{a}}_{1i}⌋+{\lambda }_{i3}diag⌊{\mathbf{a}}_{2i}⌋\right)diag⌊{\mathbf{r}}_i^2⌋{\stackrel{\sim }{\mathbf{w}}}_i\right)\end{array}$$E30

It leads to the proof of Theorem 2.

From a previous work [45], A linear neural-disturbance-observer backstepping (LND) controller is re-designed here for validation. Note that, the previous controller is developed in the single system space. From the control error Eq. (3), a virtual control signal $u_{i\mid i=1..n}$ and virtual control error $z_{i\mid i=1..n}$ are chosen as

$${}_{i=1..n}\left\{\begin{array}{l}{u}_i=-k_{c0i}e+{\dot{q}}_{\mathit{di}}\\ z_i={\dot{q}}_i-u_i\end{array}\right.$$E31

where $k_{c0i\mid i=1..n}$ are positive control gains.

The final control signal of the system is then selected as

$${\tau }_{i\mid i=1..n}=-e_i-k_{c1i}{z}_i-{\widehat{\phi }}_i+u_i+{\widehat{\mathbf{w}}}_i^T{\mathbf{\psi }}_i⌊\mathbf{q},\dot{\mathbf{q}},\dot{\mathbf{z}},\dot{\mathbf{u}}⌋$$E32

where $k_{c1i}$ are positive control gains, ${\mathbf{\psi }}_{i\mid i=1..n}$ are the regression vectors of the neural network. ${\widehat{\mathbf{w}}}_{i\mid i=1..n}$ are estimates of the weight vector ${\mathbf{\psi }}_{i\mid i=1..n}$, and are updated by:

$${\dot{\widehat{\mathbf{w}}}}_i=-{\mathrm{\Gamma }}_i\left({\mathbf{\psi }}_i{z}_i+{\mu }_i{\widehat{\mathbf{w}}}_i\right)$$E33

where ${\mu }_i$ are positive leakage rates, and ${\mathrm{\Gamma }}_i$ are diagonal positive-definite matrices.

${\widehat{\phi }}_{i\mid i=1..n}$ are estimates of systematic disturbances, and are computed throughout an auxiliary variable ${\widehat{\varphi }}_{i\mid i=1..n}$, that is estimated by the following learning mechanism:

$$\left\{\begin{array}{l}{\widehat{\phi }}_i={\widehat{\varphi }}_i+k_{c2i}z\\ {\dot{\widehat{\varphi }}}_i=-k_{c2i}{\overline{m}}_i^{-1}\left({\tau }_i-{\dot{q}}_i+{\widehat{\phi }}_i\right)\end{array}\right.$$E34

where, $k_{c2i}$ is a positive disturbance gain selected.

The dynamics (1) of the robot whose configuration is presented in Figure 2, can be derived in detail using the Euler–Lagrange method as follows:

$$\left\{\begin{array}{l}\mathbf{M}⌊\mathbf{q}⌋=\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& m_{23}\\ 0& m_{32}& m_{33}\end{array}\right]\\ m_{11}=m_1{l}_1^2+m_2{\left(l_1+l_{2}cos\left(q_2\right)\right)}^2\\ +m_3{\left(l_1+l_2{\mathrm{c}}_2+l_{3}cos\left(q_2+q_3\right)\right)}^2\\ m_{22}=m_2{l}_2^2+m_3\left(l_2^2+l_3^2+2l_2{l}_{3}cos\left(q_3\right)\right)\\ m_{23}=m_{32}=m_3\left(l_2{l}_{3}cos\left(q_3\right)+l_3^2\right)\\ m_{33}=m_3{l}_3^2\end{array}\right.$$E35

$$\left\{\begin{array}{l}\mathbf{C}⌊\mathbf{q},\dot{\mathbf{q}}⌋\dot{\mathbf{q}}={\left[c_1;c_2;c_3\right]}^T\\ c_1=-2m_2\left(l_1+l_{2}cos\left(q_2\right)\right)l_{2}sin\left(q_2\right){\dot{q}}_2{\dot{q}}_1\\ -2m_3\left(l_1+l_{2}cos\left(q_2\right)+l_{3}cos\left(q_2+q_3\right)\right)\left(l_2{\dot{q}}_{2}sin\left(q_2\right)+\left({\dot{q}}_2+{\dot{q}}_3\right)l_{3}sin\left(q_2+q_3\right)\right){\dot{q}}_1\\ c_2=-2m_3{l}_2{l}_3{s}_3{\dot{q}}_2{\dot{q}}_3-m_3{l}_2{l}_3{s}_3{\dot{q}}_3^2+l_2{s}_2\left(l_1+l_2{c}_2\right)m_3{\dot{q}}_1^2\\ -m_3\left(-l_2{s}_2-l_3{s}_{23}\right)\left(l_1+l_2{c}_2+l_3{c}_{23}\right){\dot{q}}_1^2\\ c_3=m_3\left(l_2{l}_{3}sin\left(q_3\right){\dot{q}}_3\right){\dot{q}}_2+m_3\left(l_1+l_{2}cos\left(q_2\right)+l_{3}cos\left(q_2+q_3\right)\right)l_{3}sin\left(q_2+q_3\right){\dot{q}}_1^2\\ +m_3{l}_2{l}_{3}sin\left(q_3\right){\dot{q}}_2^2+m_3{l}_2{l}_{3}sin\left(q_3\right){\dot{q}}_2{\dot{q}}_3\end{array}\right.$$E36

$$\mathbf{g}⌊\mathbf{q}⌋=-g_0{\left[0;\left(2l_{2}cos\left(q_2\right)+l_{3}cos\left(q_2+q_3\right)\right);l_{3}cos\left(q_2+q_3\right)\right]}^T$$E37

$$\mathbf{f}⌊\dot{\mathbf{q}}⌋={\left[a_1{\dot{q}}_1;a_2{\dot{q}}_2;a_3{\dot{q}}_3\right]}^T$$E38

where $q_i,l_i,m_i\text{and}{a}_{i\mid i=1,2,3}$ are joint positions, link lengths, link masses and frictional coefficients, respectively; g₀ is the absolute gravitational-acceleration value.