Multi-mode Sliding Mode Control of Four-Cable Parallel Robot Based on Wind Disturbance Observation

Abstract

Aiming at the problem of accurate trajectory tracking of four-cable parallel robot under wind disturbance in the process of material handling in buildings, a multi-mode sliding mode control (MSMC) method based on wind disturbance observer is proposed. Firstly, the wind field model is established, and the comprehensive wind speed model is introduced into the accurate dynamic model as a wind disturbance factor. Secondly, the wind disturbance observer is introduced to estimate the total disturbance of wind disturbance error in real time, which effectively reduces the switching gain, thus effectively reducing the vibration and improving the control accuracy of the system. Combined with the dynamic performance of multimode sliding mode controller, it can be switched at will to reduce chattering. Compared with the traditional sliding mode control method SMC, the anti-interference ability of MSMC is verified. The results show that the designed multimode sliding mode controller can effectively suppress the influence of wind disturbance on the vibration of the end effector.

1 Introduce

The parallel robot with flexible cables drives the hanging object to move through four or more flexible cables, and can move freely in the direction in the building space. The flexible cable driving device is arranged above or around the construction site, and the transported hanging objects are placed in the three-dimensional space through a plurality of driving flexible cables. This kind of parallel robot with flexible cables moves fast, and only needs remote control by people, which can ensure the safety of people in the process of lifting objects. This robot can take multi-mode shots in many occasions [1, 2]. In Ref. [3], a robust switching integral sliding mode control method is proposed for mismatched uncertain systems, which makes each subsystem robust and stable to uncertain disturbances. References [4,5,6] all use sliding mode control based on disturbance observer to compensate the mismatch uncertainty. However, the above two methods all rely on the assumption that the mismatch uncertainty satisfies the time-invariant or slow-time-varying assumption. For the more general uncertainty of the function model, the output of the control system cannot converge to zero strictly, but only to the neighborhood near zero. Different from the traditional second-order sliding mode and high-order sliding mode control methods, In Ref. [7], the upper bound function type of mismatched uncertainty in mismatched uncertain systems is extended from constant type to more general positive function type, and the corresponding Second ordersliding mode (SOSM) control law is designed based on this type of disturbance boundary, but the chattering phenomenon cannot be well suppressed. Literature [8] Nonlinear Extended State Observer (NLESO) provides good tracking performance for uncertain systems, but the selection of nonlinear functions has no clear theoretical basis, and in most case it depends on the empirical judgment of researchers. Therefore, a four-cable parallel machine based on multi-mode sliding mode control is designed to meet the demand of efficient load and suppress buffeting caused by wind disturbance [9, 10].

2 Mathematical Model of Four-Cable Parallel Robot

2.1 Wind Speed Model

The basic wind selects the average wind speed value, which is 5 m/s.

$$ V_{{\text{b}}} = K $$

(1)

Gust is used to describe the sudden change of wind speed:

$$ V_{g} = \left\{ \begin{gathered} 0,(t < t_{1} ) \hfill \\ \frac{{V_{g\max } }}{2}\left\{ {1 - \cos \left[ {2\pi \left( {\frac{{t - t_{1} }}{{T_{g} }}} \right)} \right]} \right\},(t_{1} \le t \le t_{1} + T_{g} ) \hfill \\ 0,(t > t_{1} + T_{g} ) \hfill \\ \end{gathered} \right. $$

(2)

Gradual wind is used to describe the characteristics of wind speed gradient:

$$ V_{r} = \left\{ \begin{gathered} 0,(t < t_{r1} {\text{ or }}t > t_{r2} + t_{r3} ) \hfill \\ V_{r\max } \frac{{t - t_{r1} }}{{t_{r2} - t_{r1} }},(t_{r1} \le t \le t_{r2} ) \hfill \\ V_{r\max } ,(t_{r2} < t \le t_{r2} + t_{r3} ) \hfill \\ \end{gathered} \right. $$

(3)

Random wind speed is used to describe the randomness of wind speed in wind field;

$$ V_{n} = V_{n\max } R_{am} ( - 1,1)\cos (\omega_{n} t + \varphi_{n} ) $$

(4)

Based on the wind speed descriptions in the above four parts, the simulated wind speed acting on the impeller can be obtained from Eqs. (1)–(4) as follows: integrating the wind shape to (see Fig. 1).

Fig. 1

Graph of measured total wind speed

2.2 Dynamic Model of Four Cable Parallel Robot

The working range of the four cables parallel robot motion platform is a three-dimensional space, which has the function of moving up, down, left, right, and can also achieve three motion directions simultaneously. The overall geometric model of the four cable parallel robot system has been established, as shown in Fig. 2.

Fig. 2

Geometric model of four-cable parallel robot

In the Cartesian coordinate system, the length L of each flexible cable can be solved by the position vector of the end effector and the position vector of the pulley. The position vector of the end effector, the position vector of the pulley, k = 1, 2, 3, 4, and the length of any cable can be calculated according to the following equation:

$$ L_{k} = P - B_{k} $$

(5)

Under the Lagrange function of the reference frame, a four cables parallel robot can be constructed:

$$ M\left( q \right)\ddot{q} + C\left( {q,\dot{q}} \right)\dot{q} + G\left( q \right) = F_{d} - F $$

(6)

$$ \left\{ \begin{gathered} F_{d} = \omega pA \hfill \\ p = \frac{1}{2}rV_{w}^{2} \hfill \\ V_{w} = V_{b} + V_{g} + V_{r} + V_{n} \hfill \\ \end{gathered} \right. $$

(7)

The wind coefficient is 1.2, P is the calculated wind pressure, r is the air density of 1.25 kg/m, V is the calculated wind speed, and A is the windward area of the load. The dynamic model of a four cables parallel robot can be obtained:

$$ M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = J^{T} T + F_{d} $$

(8)

3 Four Cable Parallel Multi-mode Sliding Mode Control and Stability Analysis

3.1 Design of Disturbance Observer

Based on the previously obtained dynamic model of a four cables parallel human:

$$ M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = \tau + F_{d} $$

(9)

Among them is the Jacobian matrix of the robot system, where the cable force of the T-drive parallel robot is the control signal vector. According to the mathematical model observer of the four cables parallel robot:

$$ \dot{Z} = - L(q,\dot{q})z + L(q,\dot{q})(C(q,\dot{q})\dot{q} + G(q) - \tau - p(q,\dot{q})) $$

(10)

$$ \hat{F}_{d} = Z + p\left( {q,\dot{q}} \right) $$

(11)

After adding a disturbance observer, the disturbance torque of the system decreases from F_d to observation error, indicating that adding a disturbance observer can effectively reduce disturbance momentum.

3.2 Design and Stability Analysis of Multimodal Sliding Mode Control

The so-called multimodality refers to the design of a sliding mode motion path connected by multiple sliding mode regions with s(0) = 0 and s(0) ≠ 0, allowing the system state points to move from one mode to another along a determined path. The final approaching target point SMC consists of two parts: ISMC and SMC. Switching control may exhibit chattering due to the discontinuous sign of the term. Sign is replaced by a smooth function, and stability is demonstrated using Lyapunov [11].

Set $q_{d}$ as a reference signal, defined as:

$$ e = q_{d} - q $$

(12)

Construct a sliding surface, called a sliding function, and select the sliding surface:

$$ \left\{ {\begin{array}{*{20}l} {s_{1} = \dot{e} + \mu e} \hfill \\ {\dot{s}_{1} = \ddot{e} + \mu \dot{e}} \hfill \\ \end{array} } \right. $$

(13)

$u_{k}$ Satisfy the arrival condition and implement sliding motion control when manifold s = 0. Set a high gain saturation function to approximate the sign function to reduce chattering. Set the control law:

$$ u_{1} = m\ddot{q}_{d} + c_{1} \dot{q} + k_{1} q + \mu m\dot{e} + d_{1} {\text{sign}} \left( {s_{1} } \right) + d_{2} s_{1} $$

(14)

If the initial condition is not on the sliding surface and due to wind disturbance, a controller is proposed to drive the output to the sliding surface to be 0 in a finite time. Under this condition, the output will reach the sliding surface. The Lyapunov function represents:

$$ \begin{gathered} V_{1} (t) = \frac{1}{2}s_{1}^{2} ,\quad V_{1} (t) > 0,\quad {\text{and s}}_{1} \ne 0 \dot{V_{1} (t)} = s_{1} (t) \hfill \\ \dot{s_{1} } (t) = s_{1} \left[ {\ddot{q_{d} } - \frac{1}{m}\left( {u_{1} (t) - c_{1} \dot{q} - d_{1} q + \mu \dot{e}} \right)} \right] = \frac{1}{m}s_{1} \left[ { - d_{1} {\text{sign}} \left( {s_{1} } \right) - d_{2} s_{1} } \right] \hfill \\ \end{gathered} $$

(15)

When the error gradually decreases, the integration method is introduced into the SMC to eliminate the error, thereby improving the tracking performance of the system. The integration part is added to the SMC, making the sliding surface an integral augmented SMC. In the figure, when the error is less than or equal to the threshold e, the control system will adopt ISMC.

$$ \left\{ {\begin{array}{*{20}l} {s_{2} = \dot{e} + \mu_{1} e + \mu_{2} \int\limits_{0}^{t} e (\tau )d\tau } \hfill \\ {\dot{s_{2} } = \ddot{e} + \mu_{1} \dot{e} + \mu_{2} e} \hfill \\ \end{array} } \right. $$

(16)

The control law is:

$$ \begin{aligned} u_{2}& = m\ddot{q_{d} } + c_{1} \dot{q} + k_{1} q + \mu m\dot{e} + \mu_{2} me \\ & \quad +d_{3} \frac{{s_{2} }}{{\left| {s_{2} } \right| + \delta }} + d_{4} s_{2} \quad \left( {d_{3} > 0,d_{4} > 0} \right) \hfill \\ \end{aligned} $$

(17)

When the error threshold is greater than, u₁ is used, and when the error threshold is less than or equal to, u₂ is used as the control input u.

4 Simulation Analysis

In the construction of MATLAB/SIMULINK, this chapter constructs a system simulation model and writes the S function in MATLAB language to better describe the system, control algorithms, and improve system performance. Use the above method to control the trajectory of the four cables parallel robot. For the four cables parallel control platform, the length a = 20 m, width b = 12 m, and the mass of the end effector is m = 5 kg, g = 9.8 m/s. a₁ = a₂ = 0.1, d₁ = d₂ = d₃ = d₄ = 0.01. In order to verify the performance advantages of the multimodal sliding mode control based on wind disturbance observer proposed in this article, simulation analysis will be conducted from two aspects: speed and displacement tracking observation results, and compared with the classical sliding mode control method SMC. The classic sliding mode controller used is:

$$ \begin{gathered} e = qd - q\quad T \hfill \\ de = dqd - dq \hfill \\ s = ce + de \hfill \\ u = m*(n*sign(s) + p*sS + c*(dqd - dq) \hfill \\ \quad \quad + ddqd + (k/m)*q) \hfill \\ \end{gathered} $$

(18)

The simulation diagrams of velocity and displacement for comparison between MSMC and SMC controllers, including x, y, and z, as well as the local comparison diagrams of velocity tracking and displacement tracking, as shown in Figs. 3 and 4.

Fig. 3

Speed tracking x, y, z y and z comparison chart

Fig. 4

x, y, z y and z comparison diagram of displacement tracking

From the simulation results, it can be seen that under the influence of wind disturbance observer compensation, the speed and displacement are approximately close, and there is a slight vibration in the speed. After being compensated by disturbance observer, the tracking effect is very good through the action of multimodal sliding mode controller. Compared with traditional sliding mode control methods, the displacement and velocity are closer to the expected trajectory, and the vibration is reduced, resulting in better control effect (see Figs. 5 and 6).

Fig. 5

Speed tracking diagram

Fig. 6

Displacement tracking diagram

5 Conclusion

This article establishes a refined wind field synthesis model for a four cables parallel robot system with wind disturbance. A wind disturbance observer is introduced to estimate and compensate for the changes in the total disturbance in real time, and a (MSMC) controller is designed in combination with a multimodal sliding mode control with small chattering and fast response speed. The simulation results show that the control algorithm proposed in this paper can achieve precise trajectory control after compensating for wind disturbance by the wind disturbance observer. The numerical simulation results show that the position optimization rate of the four cables parallel robot system is closer to the numerical value and better than that of the traditional SMC controller under wind disturbance, indicating that the system has strong anti-interference ability and robustness. It has high reference value for the design of the actual intelligent storage flexible cable handling mechanism controller.