Keywords

1 Introduction

Control of robot manipulators has gained more and more attention for its applications in industries, agricultures, and teleoperated surgeries. The difficulties in control of robot manipulators mainly include uncertainties and constraints in the position, velocity, and control input. On the one hand, uncertainties always exist in robot manipulator models due to modeling errors and disturbances. On the other hand, control input constraints always exist due to limited control powers, and motion constraints (e.g. position constraints and velocity constraints) are needed to avoid collision or injury to human beings, especially in human-robot interaction. Therefore, the control design for robot manipulators with uncertainties and constraints deserves more research.

Many robust control strategies have been developed for robot manipulators, including sliding mode control [1,2,3], neural network (NN) control [3,4,5,6,7,8], fuzzy control [9, 10], adaptive control [11], etc. However, sliding mode control usually suffers from chattering and the need of high-frequency bandwidth, adaptive control usually only handles structured uncertainties, and fuzzy control highly depends on the experiences of control engineers. Compared with other control approaches, NN control has its own advantages. NNs can approximate both structured and unstructured uncertainties due to their inherent function approximation abilities. The use of NNs estimators in control is possible to obtain desired control performances without high control gains.

Since constraints in robot manipulators need to be considered and ignoring constraints may deteriorate the control performance, some results have been obtained on control of constrained robot manipulators. Set-point regulation control and tracking control laws were designed in [12] and [13] for robot manipulators with velocity constraints, respectively. Quadratic programming-based kinematic control was developed in [14, 15] for velocity constrained redundant manipulators. Joint position constraints were considered and optimal control was designed based on adaptive dynamic programming in [16]. Recently, adaptive control was developed for robot manipulators where output or state constraints are tackled by bounding barrier Lyapunov functions (BLFs) in [17, 18]. Based on the above analysis, one can see that only position or joint velocity constraints are considered in existing robot manipulators control approaches.

In this paper, an adaptive NN control law is proposed for robot manipulators with uncertainties and constraints, including position, velocity and control constraints. The uncertainties are approximated by locally weighted adaptive NNs and compensated by the NN estimator in the control law. In locally weighted NNs, estimators composed of independently adjusted local models are used to reach the desired approximation accuracy. Thus, fewer neurons are needed to approximate smooth functions in the desired accuracy compared with other NNs. The system constraints are tackled by using BLFs in the backstepping control [19, 20] design for robot manipulators, which extends BLFs-based control for output and state constrained systems [17] to state and control constrained systems. It is demonstrated that uniform boundedness of all closed-loop signals is obtained while the constraints are not violated in theory.

2 Problem Statement

Consider a n-link robot manipulator with the following dynamics:

$$\begin{aligned} M(q){\ddot{q}}+V_m(q,{\dot{q}}){\dot{q}}+F{\dot{q}}+G(q)=\tau \end{aligned}$$
(1)

where \(q=q_1=[q_{11},q_{12},\cdots ,q_{1n}]^T\in R^n\) is a joint angle, \(q_2={\dot{q}}_1=[q_{21},q_{22}\), \(\cdots \), \(q_{2n}]^T\) is a joint velocity, \(M(q)\in R^{n\times n}\) is an inertia matrix, \(V_m(q,{\dot{q}})\in R^{n\times n}\) is a centripetal and Coriolis matrix, \(F{\dot{q}}\in R^{n}\) denotes a viscous friction torque, \(G(q)\in R^{n}\) denotes a gravitation torque, and \(\tau \in R^{n}\) denotes a control torque.

Assumption 1

M(q) satisfies the following inequalities:

$$\begin{aligned} m_1||x||^2\le x^TM(q)x\le m_2||x||^2,\; x\in R^n \end{aligned}$$
(2)

where \(m_1\), \(m_2\in R\) are positive constants.

Assumption 2

The uncertain function \(f(q,{\dot{q}})=M^{-1}(q)[V_m(q,{\dot{q}}){\dot{q}}+F{\dot{q}}+G(q)]\) is continuous.

Assumption 3

The reference trajectory is described as \(y_d(t)=[y_{d1},y_{d2}\), \(\cdots \), \(y_{dn}]^T \in R^{n}\) and satisfies \(|y_{di}|\le A_{i}\) and \(|{\dot{y}}_{di}|\le Y_i,i=1,\cdots , n.\)

The objective is to design an adaptive NN control law for the system (refeq1) to track desired trajectory \(q_d(t)\) and to satisfy the following constraints:

$$\begin{aligned} |q_{1i}|\le b_{1i}, ~ |q_{2i}|\le b_{2i}, ~|\tau _i|\le \tau _{di}, ~i=1,2,\cdots , n. \end{aligned}$$
(3)

3 BLF-Based Neural Control

3.1 Control Design

Let \(e_1=[e_{11},e_{12},e_{13}]^T=q_1-y_d\) be a tracking error. Consider BLFs as follows:

$$\begin{aligned} V_1=\frac{1}{2}\sum _{i=1}^n \frac{k_{1i}^2}{k_{1i}^2-e_{1i}^2} \end{aligned}$$
(4)

with \(k_{1i}\) to \(k_{1n}\) being positive design parameters. The time derivative of \(V_1\) is

$$\begin{aligned} {\dot{V}}_1=\sum _{i=1}^n \frac{e_{1i}}{k_{1i}^2-e_{1i}^2}(q_{2i}-{\dot{y}}_{di}). \end{aligned}$$
(5)

Design the following virtual control input:

$$\begin{aligned} \alpha _{1i}={\dot{y}}_{di}-\lambda _{1i}e_{1i},~i=1,2,\cdot ,n \end{aligned}$$
(6)

with \(\lambda _{1i},i=1,2,\cdots ,n\) being positive parameters.

Let \(e_2=[e_{21},\cdots , e_{2n}]^T=[q_{21}-\alpha _{11},\cdots ,q_{2n}-\alpha _{1n}]^T\). Then, one has

$$\begin{aligned} {\dot{V}}_1=-\sum _{i=1}^n\lambda _{1i}\frac{e_{1i}^2}{k_{1i}^2-e_{1i}^2}+\sum _{i=1}^n\frac{e_{1i}e_{2i}}{k_{1i}^2-e_{1i}^2}. \end{aligned}$$
(7)

Consider the following BLFs:

$$\begin{aligned} V_2&=V_1+\varLambda _1, \end{aligned}$$
(8)
$$\begin{aligned} \varLambda _1&=\sum _{i=1}^n \frac{1}{2}\log \frac{k_{2i}^2}{k_{2i}^2-e_{2i}^2} \end{aligned}$$
(9)

with \(k_{2i}\) to \(k_{2n}\) being positive design parameters. The time derivative of \(\varLambda _1\) is

$$\begin{aligned} {\dot{\varLambda }}_1&=\sum _{i=1}^n \frac{e_{2i}}{k_{2i}^2-e_{2i}^2}({\dot{q}}_{2i}-{\dot{\alpha }}_{1i}) \nonumber \\&=\xi ^T(f(q_1,q_2)+M^{-1}(q_1)\tau -[{\dot{\alpha }}_{11},\cdots ,{\dot{\alpha }}_{1n}]^T) \end{aligned}$$
(10)

where

$$\begin{aligned} \xi =\left[ \frac{e_{21}}{k_{21}^2-e_{21}^2},\cdots ,\frac{e_{2n}}{k_{2n}^2-e_{2n}^2}\right] ^T. \end{aligned}$$
(11)

Design the reference signal \(\tau _r\) for \(\tau \) as

$$\begin{aligned} \tau _r=[\tau _{r1},\cdots ,\tau _{rn}]^T=M(q_1)(-\lambda _2e_2-{\hat{f}}(q_1,q_2)-s) \end{aligned}$$
(12)

where \(\lambda _2\) is a positive design parameter, \({\hat{f}}(q_1,q_2)\) is an estimate of \(f(q_1,q_2)\), and

$$\begin{aligned} s =~&\frac{1}{2}\xi -[{\dot{\alpha }}_{11},\cdots ,{\dot{\alpha }}_{1n}]^T+[(k_{21}^2-e_{21}^2)e_{11}/(k_{11}^2-e_{11}^2), \nonumber \\&\cdots ,(k_{2n}^2-e_{2n}^2)e_{1n}/(k_{1n}^2-e_{1n}^2)]^T. \end{aligned}$$
(13)

Define \(e_{3}=[e_{31},\cdots ,e_{3n}]^T=\tau -\tau _r\). From (7)–(13), one obtains

$$\begin{aligned} {\dot{V}}_2=-\sum _{i=1}^n\lambda _{1i}\frac{e_{1i}^2}{k_{1i}^2-e_{1i}^2}-\sum _{i=1}^n\lambda _{2}\frac{e_{2i}^2}{k_{2i}^2-e_{2i}^2}+\xi ^T({\tilde{f}}+M^{-1}e_3)-\frac{1}{2}\xi ^T\xi \end{aligned}$$
(14)

Consider the following BLF:

$$\begin{aligned}&V_3=V_2+\varLambda _2, \end{aligned}$$
(15)
$$\begin{aligned}&\varLambda _2=\sum _{i=1}^n \frac{1}{2}\log \frac{k_{3i}^2}{k_{3i}^2-e_{3i}^2} \end{aligned}$$
(16)

with \(k_{3i}\) to \(k_{3i}\) being positive design parameters. Time derivative of \(\varLambda _2\) is

$$\begin{aligned} {\dot{\varLambda }}_2=\eta ^T({\dot{\tau }}-{\dot{\tau }}_r) \end{aligned}$$
(17)

where

$$\begin{aligned} \eta =[\frac{e_{31}}{k_{31}^2-e_{31}^2},\cdots ,\frac{e_{3n}}{k_{3n}^2-e_{3n}^2}]^T \end{aligned}$$
(18)

If the control law for the robot manipulator (1) is designed as follows:

$$\begin{aligned} \tau =-\lambda _3\int _0^t e_3(\sigma )d\sigma -\int _0^t [\text {diag}\{k_{3i}^2-e_{3i}^2\}M^{-1}(q_1)\xi ](\sigma ) d\sigma +\tau _r(t) \end{aligned}$$
(19)

where \(\lambda _3\) is a positive parameter, then one gets

$$\begin{aligned} {\dot{V}}_3=&-\sum _{i=1}^n\lambda _{1i}\frac{e_{1i}^2}{k_{1i}^2-e_{1i}^2}-\sum _{i=1}^n\lambda _{2}\frac{e_{2i}^2}{k_{2i}^2-e_{2i}^2} -\sum _{i=1}^n\lambda _{3}\frac{e_{3i}^2}{k_{3i}^2-e_{3i}^2}+\xi ^T{\tilde{f}}-\frac{1}{2}\xi ^T\xi . \end{aligned}$$
(20)

3.2 Locally Weighted Online NN Approximation

Let \(X=[X_1,\cdots ,X_{2n}]^T=[q_1^T,q_2^T]^T\) and \(D=\{X: |X_i|\le b_{1i}, |X_{n+i}|\le b_{2i}, i=1,\cdots , n \}\). The locally weighted NN approximation of f(X) is described by

$$\begin{aligned} {\hat{f}}(X)=\frac{\sum _{k=1}^{N}w_{k}(X){\hat{f}}_{k}(X)}{\sum _{k=1}^{N}w_{k}(X)} \end{aligned}$$
(21)

where \(w_k(X),k=1,\cdots ,N\) as weighted functions, and the local estimator \({\hat{f}}_{k}(X)\) is described as follows:

$$\begin{aligned} {\hat{f}}_{k}(X)=\theta _k^T\phi _{k}(X), ~\phi _{k}(X)=[1,(X-c_k)^T]^T. \end{aligned}$$
(22)

with \(c_k\) being the center of the k-th local estimator.

Assume \(D \subseteq \cup _{k=1}^{N}S_{k}\), where \(S_{k}=\{X:w_{k}\ne 0\},k=1,2,\cdots ,N\) are a series of compact sets. Define \(w_{k}(X)\) as follows:

$$\begin{aligned} w_{k}(X)= {\left\{ \begin{array}{ll} (1-(||X-c_{k}||/\mu _{k})^2)^2,&{} \text{ if }\ ||X-c_{k}||\le \mu _{k} \\ 0,&{} \text{ otherwise } \end{array}\right. } \end{aligned}$$
(23)

where \(\mu _{k}\) is the radius of \(S_k\). Let \({\bar{w}}_{k}(X)=w_{k}(x)/\sum _kw_{k}(X)\). Then, (20) can be equivalently expressed as follows:

$$\begin{aligned} {\hat{f}}(X)=\sum _{k=1}^{N}{\bar{w}}_{k}{\hat{f}}_{k}(X). \end{aligned}$$
(24)

Define the optimal parameter \(\theta _{k}^{*}\) for \(X\in S_{k}\) as follows:

$$\begin{aligned} \theta _k^{*}=\arg \min _{\theta _k}\left( \int _{X\in D} w_k(X)||f(X)-{\hat{f}}_k(X)||^2dX \right) . \end{aligned}$$
(25)

Also, define the error \(\epsilon _{k}\) as follows:

$$\begin{aligned}&\epsilon _{k}={\left\{ \begin{array}{ll} f(X)-{\hat{f}}_{k}(X),&{} \text{ on }\ {\bar{S}}_{k} \\ 0, &{} \text{ on }\ D-{\bar{S}}_{k} \end{array}\right. } \end{aligned}$$
(26)

and assume \(|\epsilon _{k}|\le \epsilon \) with \(\epsilon \) as a positive constant. Then, f(x) and its locally weighted NN approximation can be expressed to be

$$\begin{aligned}&f=\sum _{k=1}^{N}{\bar{w}}_{k}\theta _{k}^{*T}\phi _{k}+\sum _{k=1}^{N}{{\bar{w}}}_{k}\epsilon _{k}, \end{aligned}$$
(27)
$$\begin{aligned}&{\hat{f}}=\sum _{k=1}^{N}{\bar{w}}_{k}{\theta }_{k}^T\phi _{k}. \end{aligned}$$
(28)

It is obvious that \(|\sum _{k=1}^{N}{{\bar{w}}}_{k}\epsilon _{k}|\le \max (|\epsilon _{k}|)\sum _{k=1}^{N_i}{\bar{w}}_{k}\le \epsilon \).

Let \({\tilde{\theta }}_k=\theta _k^{*}-\theta _k\) and \(\varOmega _k\triangleq \{{\theta }_k:||{\theta }_k||\le c_{\theta k}\}\), and define

$$\begin{aligned} c=\max _{\theta _k^{*},{\theta }_k \in \varOmega _k} \sum _{k=1}^N {\tilde{\theta }}_k^T{\tilde{\theta }}_k/\eta \end{aligned}$$

with \(\eta \) being a positive design parameter. Design the update law of \(\theta _k\) to be

$$\begin{aligned} {\dot{\theta }}_k=\text {Proj}\left( \eta {\bar{w}}_k\phi _k\xi ^T \right) \end{aligned}$$
(29)

where \(\text {Proj(.)}\) is a projection operator given by

$$\begin{aligned} \text {Proj(.)}={\left\{ \begin{array}{ll} 0, &{} \text {if } \theta _k=-c_{\theta k} \text { and } .<0 \\ 0, &{} \text {if } \theta _k=c_{\theta k} \text { and } .>0 \\ ., &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
(30)

3.3 Stability Analysis

Theorem

Consider the system (1) with constraints (3). Assume Assumptions 1–3 hold and \(X(0)\in D, \tau (0)=0\), and the control law is designed as (19). Let

$$\begin{aligned}&A_{1i}=\max _{(e_{1i},y_{di})\in \varOmega _{1i}}|\alpha _{1i}(e_{1i},y_{di})|, \end{aligned}$$
(31)
$$\begin{aligned}&A_{ri}=\max _{(({\bar{e}}_{2i},{\bar{y}}_{di}))\in \varOmega _{ri}}|\tau _{ri}({\bar{e}}_{2i},{\bar{y}}_{di})|, \end{aligned}$$
(32)

where \({\bar{e}}_{2i}=[e_{1i},e_{2i}]^T\), \({\bar{y}}_{di}=[y_{di},{\dot{y}}_{di}]^T\), and

$$\begin{aligned}&\varOmega _{1i}=\{[e_{1i},y_{di}]: |e_{1i}|\le k_{1i},|y_{di}|\le A_i\}, \end{aligned}$$
(33)
$$\begin{aligned}&\varOmega _{\tau i}=\{[{\bar{e}}_{1i},{\bar{y}}_{di}]: |e_{1i}|\le k_{1i},|{e}_{2i}|\le k_{2i},|y_{di}|\le A_i,|{\dot{y}}_{di}|\le Y_i\}. \end{aligned}$$
(34)

If there exist \(\lambda _{1i},i=1,\cdots , n, \lambda _2, \lambda _3\) such that

$$\begin{aligned}&b_{1i}\ge k_{1i}+A_i, ~b_{2i}\ge k_{2i}+A_{1i}, ~\tau _{di}\ge k_{3i}+A_{ri},~i=1,\cdots ,n, \end{aligned}$$
(35)

then the constraints (3) are satisfied and the signals in the closed-loop control system are uniformly ultimately bounded.

Proof

Consider the following Lyapunov function:

$$\begin{aligned} V=V_3+\frac{1}{2\eta }tr\{\sum _{k=1}^N{\tilde{\theta }}_k^T{\tilde{\theta }}_k\} \end{aligned}$$
(36)

Based on (20) and (36), one obtains

$$\begin{aligned}&{\dot{V}} =-\sum _{i=1}^n\lambda _{1i}\frac{e_{1i}^2}{k_{1i}^2-e_{1i}^2}-\sum _{i=1}^n\lambda _{2}\frac{e_{2i}^2}{k_{2i}^2-e_{2i}^2} -\sum _{i=1}^n\lambda _{3}\frac{e_{3i}^2}{k_{3i}^2-e_{3i}^2} -\frac{1}{2}\xi ^T\xi \nonumber \\&\qquad +\xi ^T(\sum _{i=1}^N {\bar{w}}_k{\tilde{\theta }}_k^T\phi _k+\sum _{i=1}^N{\bar{w}}_k\epsilon _k)-\frac{1}{\eta }tr\{\sum _{k=1}^N{\tilde{\theta }}_k^T\dot{{\hat{\theta }}}_k\} \nonumber \\&=-\sum _{i=1}^n\lambda _{1i}\frac{e_{1i}^2}{k_{1i}^2-e_{1i}^2}-\sum _{i=1}^n\lambda _{2}\frac{e_{2i}^2}{k_{2i}^2-e_{2i}^2} -\sum _{i=1}^n\lambda _{3}\frac{e_{3i}^2}{k_{3i}^2-e_{3i}^2} -\frac{1}{2}\xi ^T\xi \nonumber \\&\qquad -\frac{1}{\eta }tr\{\sum _{k=1}^N{\tilde{\theta }}_k^T(\dot{{\hat{\theta }}}_k-\eta {\bar{w}}_k\phi _k\xi ^T)\} +\xi ^T\sum _{i=1}^N{\bar{w}}_k\epsilon _k. \end{aligned}$$
(37)

Substituting (29) into (37), one obtains

$$\begin{aligned} {\dot{V}}\le -\sum _{i=1}^n\lambda _{1i}\frac{e_{1i}^2}{k_{1i}^2-e_{1i}^2}-\sum _{i=1}^n\lambda _{2}\frac{e_{2i}^2}{k_{2i}^2-e_{2i}^2} -\sum _{i=1}^n\lambda _{3}\frac{e_{3i}^2}{k_{3i}^2-e_{3i}^2} +\frac{1}{2}\epsilon ^2 \end{aligned}$$
(38)

As \(\log [k_{ji}^2/(k_{ji}^2-e_{ji}^2)]\le e_{ji}^2/(k_{ji}^2-e_{ji}^2)\) for \(j=1,2,3.\) [21], one gets

$$\begin{aligned}&{\dot{V}}\le -2\lambda V_3+\frac{1}{2}\epsilon ^2 \le -2\lambda V+\beta \end{aligned}$$
(39)

where \(\lambda =\min \{\lambda _{1i},i=1,\cdots ,n,\lambda _2,\lambda _3\}\) and \(\beta =\lambda c+1/2\epsilon ^2\). It is concluded from (39) that V and all closed-loop signals are bounded. Then, based on the forms of BLFs \(V_i,i=1,2,3\) and \(X(0)\in D\), one gets \(|e_{1i}|\le k_{1i}\), \(|e_{2i}|\le k_{2i}\) and \(|e_{3i}|\le k_{3i}\). Since (35) holds, one concludes \(|q_{1i}|\le b_{1i}\), \(|q_{2i}|\le b_{2i}\) and \(|\tau _i|\le \tau _{di}\). According to (39), one also obtains

$$\begin{aligned} V(t)\le \exp (-\lambda t)(V(0)-\frac{\beta }{2\lambda })+\frac{\beta }{2\lambda }. \end{aligned}$$
(40)

Since \(\log \frac{k_{1i}^2}{k_{1i}^2-e_{1i}^2}\le 2V(t)\) for \(i=1,\cdots ,n\), one gets

$$\begin{aligned} \log \frac{k_{1i}^2}{k_{1i}^2-e_{1i}^2}\le 2\exp (-\lambda t)(V(0)-\frac{\beta }{2\lambda })+\frac{\beta }{\lambda } \end{aligned}$$
(41)

from which one obtains

$$\begin{aligned}&\limsup _{t\rightarrow \infty } \frac{k_{1i}^2}{k_{1i}^2-e_{1i}^2}\le \exp (\beta /\lambda ),\end{aligned}$$
(42)
$$\begin{aligned}&\limsup _{t\rightarrow \infty } |e_{1i}|\le k_{1i}\sqrt{1-\exp (\beta /\lambda )}. \end{aligned}$$
(43)

4 Simulation Results

To illustrate the effectiveness of the proposed BLFs-based locally weighted learning control law, simulations are carried out for a one-link robot manipulator with the reference trajectory \(y_d=0.5\cos (0.2t)\). The dynamics of the manipulator is given by

$$\begin{aligned} ml^2{\ddot{q}}+d{\dot{q}}+0.5mgl\cos (q)=\tau , \end{aligned}$$
(44)

where m = 1 kg, l = 1 m, g = 9.8 \({\text {m}}/{\text {s}}^2\), and \(d=1\,{\text {kg.m}}^2/{\text {s}}\). The constraints are \(|q_1|\le 1, |q_2|\le 1\) and \(|\tau | \le 10\) with \(q_1=q,q_2={\dot{q}}\). In the simulation, the initial system states are \(q_1(0)=0.2, q_2(0)=0.2\).

The control is designed as

$$\begin{aligned} \tau =-7\int _0^t e_3d\sigma -\int _0^t \frac{9^2-e_3^2}{0.6^2-e_2^2}e_2d\sigma + \tau _r \end{aligned}$$
(45)

where \(e_1=q_1-y_d,e_2=q_2-\alpha _1, e_3=\tau -\tau _r\) and are constrained in \(|e_1|\le 0.5\), \(|e_2|\le 0.6\) and \(|e_3|\le 7\), and the virtual control \(\alpha _1, \tau _r\) are described by

$$\begin{aligned}&\tau _r=-5 e_2+{\dot{\alpha }}_1-{\hat{f}}-0.5\frac{e_2}{0.6^2-z_2^2}-\frac{0.6^2-e_2^2}{0.5^2-e_1^2}e_1\\&\alpha _1=-2e_1+{\dot{y}}_d \end{aligned}$$

where \({\hat{f}}\) is a localized adaptive NN approximation of \(f=-q_2+9.8/2\cos (q_1)\). In the NN approximation, the centers location are chosen as \(c_1=[-1,1]^T, c_2=[0,1]^T,c_3=[1,1]^T,c_4=[-1,0]^T,c_5=[0,0]^T,c_6=[1,0]^T,c_7=[-1,-1]^T,c_8=[0,-1]^T,c_9=[1,-1]^T,c_{10}=[-0.5,0.5]^T,c_{11}=[0.5,0.5]^T,c_{12}=[-0.5,-0.5]^T\), \(c_{13}=[0.5,-0.5]^T\), \(c_{\theta k}=0.5\), \(\eta =100\), \(\mu _k=1.5\), and the basis functions are chosen as \(\phi _i=[1,q_1,q_2]^T-[0;c_i],i=1,\cdots , 13\).

Simulation results are presented in Fig. 1(a)–(c), where Fig. 1(a) shows the tracking errors \(e_1, e_2\) and \(e_3\), Fig. 1(b) shows the performance of the states \(q_1, q_2\) and the control input \(\tau \), and Fig. 1(c) shows the NN approximation error \(f-{\hat{f}}\). From Fig. 1(a), the tracking error is near to 0 after 2 s and the constraints satisfaction \(|e_1|\le 0.5, |e_2|\le 0.6, |e_3|\le 7\) and \(|q_1|\le 1, |q_2|\le 1, |\tau |\le 10\) is easily seen from Fig. 1(a)–(b). From Fig. 1(c), one sees that the approximation error \(f-{\hat{f}}\) converges to a small neighborhood of zero after 2 s. Therefore, the designed adaptive NN control law makes the system state and control input constraints fulfilled and the tracking error converge to a small neighborhood of 0.

Fig. 1.
figure 1

Control trajectories by the proposed controller. (a) The tracking errors \(e_1,e_2\) and \(e_3\). (b) The states \(q_1, q_2\) and control input \(\tau \). (c) The NN approximation error \(f-{\hat{f}}\).

Full size image

5 Conclusions

A BLFs-based adaptive NN control law was designed for robot manipulators with position, velocity and control constraints. The uncertainties were approximated by locally weighted adaptive NNs and the system constraints were tackled by using BLFs in the backstepping procedure. The control feasibility and uniform boundedness of all closed-loop signals were verified by theoretical analysis. From simulation results, we can see that under the proposed control the system constraints were never violated and absolute value of the tracking error converged to a small neighborhood of zero.