Sliding-mode controller synthesis of robotic manipulator based on a new modified reaching law

1.   Introduction

In general, highly nonlinear characteristics exist in many practical plants, such as the manipulator. At present, robotic manipulators are widely used in mechanical manufacturing, medical treatment, space exploration, disaster rescue and other dangerous or high-precision scenes. As a result, it is critical to investigate manipulator tracking control at high speeds, and with high precision ^[1,2,3,4]. Due to modeling errors, external disturbances, etc., achieving control of the manipulator plant is a difficult task ^[5,6,7]. Sliding-mode control (SMC), as a typical robust control strategy ^[8,9,10,11], has the potential merits of simple structure, good adaptability and high robustness in the face of unknown nonlinear uncertainties ^{[12,13,14,15]}, which are also extensively applied to the manipulator.

Considering the tracking problem of robotic manipulator models, different control schemes have been put forward, such as backstepping control ^[16], neural network control ^[17], and SMC ^{[18,19,20,21]}. In ^[17], a new model-free neural network was presented; it could solve the learning-based control issue of a manipulator in a unified framework. In ^[18], a systematic approach to nonlinear disturbance observer (NDO) development was presented, and the NDO was utilized to design a computed-torque controller. A finite-time control strategy based on terminal SMC was proposed for the manipulator in ^[19]. In ^[20], based on time-delay estimation, a novel adaptive SMC scheme was developed, and the proposed adaptive SMC algorithm could achieve good tracking and reduce the chattering of the plant.

As seen, trajectory tracking control can be effectively achieved for the robotic manipulator via SMC ^[21]. However, there exists chattering near the equilibrium point of the system. It should be noted that severe chattering will degrade the system's performance. As such, the reaching law method is a common way to reduce chattering ^[22], as it can improve the dynamic performance of the system during the reaching phase of the sliding surface (SS) and alleviate the high-frequency chattering. Taking the exponential reaching law as an example, the exponential term ensures that the system state can approach to SS quickly when it is far away from the SS. While the state is close to the SS, the reaching speed will be decided mainly by the constant-rate term, which ensures the limited-moment reachability of the SS as expected.

In past years, a great amount of effort to modify the reaching law approach has been put forth by numerous scholars. Regarding permanent magnet synchronous motor control in ^[23], a new exponential reaching law (NERL) was proposed to improve the dynamic performance, where the NERL could reduce the chattering and accelerate the reaching speed of the SS. However, it is noted that, as system states become close to the equilibrium point, the effects of the NERL on the convergence and robustness of the system will be weaken. Thus, a novel reaching law (NRL) was provided in ^[24]; it decreased the reaching speed while approaching the SS, and the SMC applied to a 2-DOF (degrees of freedom) manipulator system was developed based on the NRL. Namely, if the system states are not located near the SS, the speed of the reaching phase according to the NRL is relatively slow; moreover, since the bounds of the switching gain are fixed, the reaching speed can not be updated dynamically; this further results in a longer reaching time and the degradation of the system performance to some degree. To this end, chattering reduction has always been an important control issue for the manipulator with SMC; the problem of how to further improve the reaching law-based MC method is still meaningful and open, and has thus formed the motivation of this study.

On the basis of the aforementioned studies, a novel adaptive control scheme for robotic manipulator dynamic models was developed in this study. The main contributions can be summarized as follows:

(1) An adaptive modified reaching law is provided; it can dynamically adjust the change of the sliding function (SF), accelerate the convergence speed and reduce the chattering to some extent.

(2) The devised switching gain is an adaptive value for the improvement of the control properties, and the novel adaptive SMC was designed for system tracking in conjunction with the disturbance observer (DO) to estimate the disturbance input; the hyperbolic tangent function is employed to make the control input smooth and weaken the chattering further.

(3) The proposed control scheme achieved the expected performance in simulation experiments; its advantage was further verified and analyzed from the transient and steady performance of the plant.

2.   Dynamic model of robotic manipulators

Taking the external disturbance and modeling error into consideration, an n-DOF robotic manipulator dynamic model is given by

where $q, \dot q, \ddot q \in {R^n}$ represent the position vector, velocity vector and acceleration vector, respectively.$M(q) \in {R^{n \times n}}$ stands for the positive definite inertia matrix of the plant; $C(q, \dot q) \in {R^{n \times n}}$ denotes the Coriolis-centripetal matrix; the gravitational torque vector is expressed as $G(q) \in {R^n}$; $u$ and $d$ are the control input vector and external disturbance, respectively. Generally, an accurate dynamic model of the manipulator during the actual control process may be difficult to solve; then, $M(q)$, $C(q, \dot q)$ and $G(q)$ are assumed to be as follows:

where ${M_0}(q)$, ${C_0}(q, \dot q)$ and ${G_0}(q)$ are the nominal values, and $\delta M(q)$, $\delta C(q, \dot q)$ and $\delta G(q)$ are the parameter differences between the nominal values and the actual values. Therefore, the dynamics of the robotic manipulator (1) can be expressed as

where ${\delta _\Delta } = - \delta M(q)\ddot q - \delta C(q, \dot q)\dot q - \delta G(q)$ represents the unknown uncertainties. Defining $x = {[\begin{array}{*{20}{c}} {{x_1}} & {{x_2}} \end{array}]^T}$, where ${x_1} = q$ and ${x_2} = \dot q$, the dynamical equation of the plant can be written as

where $Q(x) = - {M_0}^{ - 1}(q){C_0}(q, \dot q) - {M_0}^{ - 1}(q){G_0}(q)$, $P(x) = {M_0}^{ - 1}(q)$ and $F(t, x) = {M_0}^{ - 1}(q)(d + {\delta _\Delta })$; $F(t, x)$ is a lumped disturbance. In the position, the tracking error vector is then expressed as

where ${x_{1d}}, {x_{2d}} \in {R^n}$ are, respectively, the desired angular position and angular velocity. As can be seen, Figure 1 shows a two-joint manipulator model.

Lemma 1 ^[25]. Suppose that $V(x)$ is a smooth positive definite function defined on $D \subset {R^n}$; it satisfies $\dot V(x) + \mu {V^\eta }(x) \leqslant 0, \mu > 0, \eta \in (0, 1)$, where $\mu$ and $\eta$ are two positive constants. Then, for any specified ${t_0}$, there exists an area ${D_0} \subset {R^n}$ such that $V(x)$, which starts from ${D_0}$, can be $V(x) \equiv 0$ in finite time; moreover, the settling time ${t_r}$ satisfies ${t_r} \leqslant \frac{{{V^{1 - \eta }}\;\;({t_0})}}{{\mu (1 - \eta)}}$.

3.   Design of DO

According to the dynamics described by Eq. (4), we have

Hereafter, we refer to $\hat F(t, x)$ as the estimated value of the lumped disturbance. The aim is focused on correcting the difference between the estimated output and the actual output. Then, the basic DO can be written as

By substituting Eq. (6) into Eq. (7), the following can be obtained:

However, due to the observation noise in practice, it is not easy to obtain the acceleration signal by differentiating the velocity signal directly. For the robotic manipulators, it is also difficult to obtain the actual acceleration signal. Thus, we also introduce an auxiliary variable, as shown below.

By combining Eq. (4) and Eq. (9), it can be obtained that

Then, the devised DO is given as

in which the observation error $\tilde F(t, x)$ is defined as

Lemma 2 ^[26]. Considering the dynamics described by Eq. (4), the DO in Eq. (11) guarantees that the observation error $\tilde F(t, x)$ will converge to zero.

As can be observed, the DO does not need the information on the acceleration signal. The disturbance observation error can converge as expected, and the convergence accuracy is determined by $K$.

4.   Design of adaptive switch controller

In this section, by applying the provided DO and updating law, a new modified reaching law-based adaptive switch controller was synthesized to improve the dynamic performance of trajectory tracking and reduce the chattering; the control structure of the plant is shown in Figure 2.

4.1.   Development of modified reaching law

As is known, the reaching law strategy is extensively employed for SMC designs for manipulator systems, allowing them to achieves the dynamic performance of the reaching stage. A proper reaching law allows the system trajectories to reach the SS fast, and with low chattering. Firstly, the following Eq. (13) is the traditional constant-rate reaching law ^[22]:

Remark 1. It should be mentioned that the constant-rate reaching law in ^[22] results in a longer reaching time and more chattering. In ^[24], an NRL was also developed using the constant-rate reaching law approach; it could shorten the reaching time and reduce the chattering. A new modified adaptive reaching law has been established based on the NRL method by combining the adaptive method described in this paper to better alleviate the adverse factors of chattering and further speed up the tracking convergence.

The following is the proposed reaching law:

where $N(s) = {\partial _0} + (1 + {1 / {{{\left| s \right|}^\theta }}} - {\partial _0}){e^{ - \varepsilon {{\left| s \right|}^p}}}, {\partial _0} \in (0, 1)$, $p$ is a positive integer, $\varepsilon, \phi, \lambda \gt 0$ and $\theta \in (0, 1)$. ${K_1}$ is a part of the controller that, together with $N(s)$ and ${\partial _1}$, forms the switching gain; $\frac{{{K_1}}}{{{\partial _0}}} + {\partial _1}$ is the upper bound of the switching gain. ${\partial _1}$ is a small positive constant used to prove the reachability of the designed SS in finite time.

Remark 2. As can be ascertained from the reaching law presented in Eq. (14), $N(s)$ will tend to ${\partial _0}$ with increasing $\left| s \right|$; then, it has $\frac{{{K_1}}}{{N(s)}} > {K_1}$, and $\phi (1 - {e^{ - \lambda \left| s \right|}})$ becomes $\phi$. The system state will then approach the SS at a faster speed; in a like manner, as $\left| s \right|$ decreases, $N(s)$ will tend toward $1 + {1 / {{{\left| s \right|}^\theta }}}$, which is much bigger than 1. Thus, it follows that $\frac{{{K_1}}}{{N(s)}} \ll {K_1}$. Also, $\phi (1 - {e^{ - \lambda \left| s \right|}})$ will approach zero, and $\frac{{{K_1}}}{{N(s)}} + {\partial _1}$ will gradually decrease to a small value.

4.2.   SMC law synthesis

The SF is selected as follows:

In view of Eq. (5) and Eq. (15), the derivative of SF can be obtained as

Remark 3. In view of the modified reaching law given in Eq. (14), the parameter ${K_1}$ is presented as an unknown quantity that will be estimated by the adjusting parameter ${\hat K_1}$, which aims to change the convergence speed and further improve the control performance. Moreover, the boundary layer method is widely used for chattering reduction in SMC; the SF will not exactly reach zero, but it will arrive at a small vicinity of zero point because measurement noise and switching delay/imperfections may often occur in practice. Then, the following adaptive law with a small positive constant ${\sigma _0}$ is provided to avoid the infinite increase of the switching gain, as described below. (Here, ${\sigma _0}$ is set by a user based on the simulation results to control the effects in practical applications.)

where ${\sigma _0}$ is a small positive constant. Here, ${\tilde K_1}$ is the estimation error between ${K_1}$ and ${\hat K_1}$, which is defined as

Assumption 1. The estimation error of the DO is bounded and satisfies

According to Eq. (4), the devised adaptive switch controller for the considered manipulator plant, which was developed based on the proposed reaching law, is given as follows:

Theorem 1. Considering the adaptive law in Eq. (17), if the control law in Eq. (22) is applied to the system in combination with the DO in Eq. (11), the SF will approach zero in finite moments; then, the trajectory tracking error will converge to zero.

Proof. Consider the Lyapunov function as below.

Given Eq. (4), Eq. (16) and Eq. (22), the derivative of Eq. (23) is expressed as

By substituting the adaptive law of Eq. (17), it follows that

Since ${\dot {\hat K}_1} > 0\; (s > {\sigma _0})$, there exists an instant $T$, and for any $t > T$, one has ${\tilde K_1} \geqslant 0$; thus, it follows that $\frac{1}{\xi }{\tilde K_1}{\dot {\hat K}_1} \geqslant 0$. Let

It can be obtained that

From Lemma 1, the reaching time can be estimated as ${t_r} \leqslant \frac{{\sqrt 2 }}{{{\partial _1}}}{V^{0.5}}(0)$. Namely, the SF will equal zero in finite time. Due to the condition $s = 0$, it follows that ${e_1}$ and ${e_2}$ have opposite signs. Therefore, the tracking error will converge to zero. The proof is finished.

Remark 4 ^[27]. The signum function in the designed controller could be approximated by the hyperbolic tangent function described by Eq. (28) to make the switched control input continuous, and to further reduce the chattering of the plant.

where $\Lambda > 0$ is a constant, and $\frac{1}{\Lambda }$ is the boundary layer.

5.   Simulation results

The simulations of a 2-DOF robotic manipulator were carried out to verify the advantages and feasibility of the presented controller design. The dynamic model of the manipulator is expressed as

where ${m_{11}} = ({m_1} + {m_2}){r_2}^2 + 2{m_2}{r_1}{r_2}\cos {q_2} + {j_1}$; ${m_{12}} = {m_2}{r_2}^2 + {m_2}{r_1}{r_2}\cos {q_2}$; ${m_{21}} = {m_{12}}$; ${m_{22}} = {m_2}{r_2}^2 + {j_2}$; ${c_{11}} = - {m_2}{r_1}{r_2}\sin ({q_2}){\dot q_1}$; ${c_{12}} = - 2{m_2}{r_1}{r_2}\sin ({q_2}){\dot q_1}$; ${c_{21}} = 0$; ${c_{22}} = {m_2}{r_1}{r_2}\sin ({q_2}){\dot q_2}$; ${g_1} = [({m_1} + {m_2}){r_1}\cos {q_2}$$+ {m_2}{r_2}\cos ({q_1} + {q_2})]g$; ${g_2} = {m_2}{r_2}g\cos ({q_1} + {q_2})$.

For the simulation, the physical parameters of the manipulator described by Eq. (29) were set as ${m_1} = 0.5{\text{kg}}$, ${m_2} = 1.5{\text{kg}}$, ${r_1} = 1{\text{kg}}$, ${r_2} = 0.5{\text{kg}}$, $g = 9.8{\text{N}}/{\text{kg}}$ and ${j_1} = {j_2} = 5{\text{kg}} \cdot {\text{m}}$. Assume that the initial position ${q_0} = {[\begin{array}{*{20}{c}} 2 & 3 \end{array}]^T}$ and initial velocity ${\dot q_0} = {[\begin{array}{*{20}{c}} 0 & 0 \end{array}]^T}$ of the plant were set in advance.

Suppose that the plant uncertainties are $\delta M(q) = - 0.2{M_0}(q)$, $\delta C(q, \dot q) = - 0.2{C_0}(q, \dot q)$ and $\delta G(q) = - 0.2{G_0}(q)$. The external disturbance is supposed to be $d = {[18\cos (\pi t), 10\sin (\pi t)]^T}$. The simulation results obtained using the devised controller described by Eq. (22), which had the parameters $c = 3$, $K = 40$, ${\partial _0} = 0.5$, $\theta = 0.1$, $\varepsilon = 1$, $p = 1$, $\lambda = 1.6$, $\phi = 5$, $\Lambda = 70$, ${\partial _1} = 0.001$ and $\xi = 2.5$, are depicted in Figures 3–5. The goal was to control the manipulator joint to effectively follow the desired trajectory ${q_d} = {[\begin{array}{*{20}{c}} {1.25 - 1.4{e^{ - t}} + 0.35{e^{ - 4t}}} & {1.25 + {e^{ - t}} - 0.25{e^{ - 4t}}} \end{array}]^T}$.

It can be seen in Figures 3–5 that the position and velocity can be tracked well in spite of the lumped disturbance and unknown uncertainties. The tracking error reached and remained on the SS and asymptotically converged to 0 along the SS.

For comparison, the NERL in ^[23] and the NRL in ^[24] are described by Eq. (30) and Eq. (31).

where ${N_1}(s) = \partial + (1 + \partial){e^{ - {\varepsilon _1}{{\left| s \right|}^{{p_1}}}}}, \partial \in (0, 1)$. As can be seen in Figure 6, the estimated value of the adaptive parameter tended to ${\hat K_1} = 4.5$. Therefore, the simulation parameters of the reaching law in ^[24] were set as ${k_1} = {\hat K_1} = 4.5$, $\partial = 0.5$, ${\varepsilon _1} = 1$, ${p_1} = 1$, ${k_2} = 1$ and ${k_3} = 5$. Then, the comparative simulations are shown in Figures 7–9.

As can be seen, the comparison simulations of the position and velocity tracking errors of the two joints, which were achieved using the related methods, are depicted in Figures 7–8. It is clear that, as compared to the controller that used the NERL from ^[23] and the NRL from ^[24], the position tracking was faster and more efficient; furthermore, the velocity tracking chattering was reduced with the current controller design. Figure 9 shows the comparison results for the control inputs of the two joints; it shows that the chattering of the control input was reduced under the conditions of the modified reaching law.

Moreover, to further verify the advantages of the modified reaching law, the signum function of the reaching law in ^[24] was replaced by the hyperbolic tangent function. The results of the considered comparison simulations are shown in Figures 10–11, and Figure 12 provides the curves for the control input signals. It can be seen that the NRL method from ^[24] resulted in less chattering, but the effect achieved by the proposed modified reaching law was better.

Furthermore, the energy consumption (EC) of a plant is a key performance index in practice, and the formula of EC is as below.

Then, Table 1 shows the comparison results for the EC. The control method based on the current reaching law resulted in less energy loss than the methods proposed in ^[23] and ^[24].

Lastly, the following performance indexes, i.e., the integral squared error (ISE), integral absolute error (IAE) and integral time multiplied absolute error (ITAE) were also introduced from ^[28]. The values of these indexes, and the settling time (ST), are shown in Table 2; they were used to analyze the transient and steady performance of the manipulator.

where ${t_f}$ is the simulation time. From Table 2, it can be seen that the controller based on the modified adaptive reaching law more accurately tracked with a faster convergence.

6.   Conclusion

In this study, a new adaptive modified reaching law was developed to improve the tracking performance and reduce the chattering of the plant. Then, a novel adaptive SMC was developed for robot manipulators with external disturbances and unknown uncertainties to automatically adapt the plant parameters and improve the performance of the system; the DO and the modified reaching law approach were applied. Through simulation experiments, the developed adaptive reaching law-based controller was demonstrated to accelerate the convergence speed, reduce the chattering and EC and enhance the robustness of the plant.

In the future, finite-time DO-based SMC methods can be studied to enable the realization of fast compensation and the improvement of the transient performance of the plant; additionally, some complex issues in practical control systems, such as cases in which there is a dead zone, input saturation, etc., may also be explored by using the proposed method.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant number 61803217.

Conflict of interest

The authors declare that there is no conflict of interest.