Dynamic Synovial Control Method of Four-Cable-Driven Parallel Robot Based on Interference Observer

Abstract

Aiming at the problem of high-precision trajectory tracking of four-cable-driven parallel robot, a dynamic non-singular fast terminal sliding mode surface based on the estimation output of interference observer is proposed. Firstly, based on the combination of traditional fast terminal sliding mode control and non-singular terminal synovial control, the estimation value of interference observer is introduced, and a new nonlinear sliding mode surface is established to suppress the influence of unknown disturbance, and at the same time track the desired trajectory from unstable state to steady state. Based on the Lyapunov stability theory of the controller, the effectiveness of the control scheme is proved. Finally, in the Matlab/Simulink environment, a comparative simulation with the existing linear synovial film and non-singular fast terminal sliding mode is carried out, and the designed control method realizes high-precision fast tracking of the desired trajectory of the end grab under the uncertainty of modeling error, external interference and joint friction, which verifies the effectiveness and feasibility of the proposed control method.

1 Introduction

With the development of construction machinery manufacturing industry, the characteristics of building types becoming more and more complex have become increasingly prominent. For the capture, identification and handling of target building materials in the construction environment, most of them are still carried out manually. As a typical parallel robot that realizes the movement of the end effector through the retracting and releasing control of the flexible cable, the flexible rope parallel drive robot has many advantages such as large working space, small dynamic inertia and fast movement speed [1], and is widely used in many fields such as astronomical observation [2], field video [3], 3D printing [4] and so on. However, the soft rope is also exposed to the external disturbance caused by the flexibility, elasticity, nonlinearity and unidirectional constraint characteristics of the soft cable. Therefore, ensuring the high-precision trajectory tracking of the end effector of the robot driven in parallel has become an urgent problem to be solved.

Many scholars at home and abroad have conducted research on motion control of soft rope parallel robot. Yan-Lin Wang et al. [5] proposed a new controller consisting of an inner loop fuzzy sliding mode controller and an outer loop variable admittance controller to control the cable-driven lower limb rehabilitation robot. Ji, Y.F., et al. [6] established an eight-rope traction robot in a low-speed wind tunnel, and verified the phenomenon of aerodynamic retardation through experiment. The above literature adopts various control strategies for high-precision. Peng Liu et al. [7] proposed a robust adaptive fuzzy control strategy, which proved the reliability of trajectory tracking of the soft-cable-driven picking robot through Simula trajectory tracking of soft cable-driven robots. An, B., et al. [8] proposed an adaptive terminal sliding mode based on interference observer in the aircraft model, which combines the interference signal with the terminal synovial membrane to construct and eliminate the singularity in the system. Zhang, Q. W., et al. [9] proposed a non-singular terminal sliding mode control method for redundant drive parallel mechanism system, which realized high-precision trajectory tracking control. In this paper, based on the combination of traditional fast terminal sliding mode control and non-singular terminal synovial control, the estimation value of interference observer is introduced, a new nonlinear sliding mode surface is established to suppress the influence of unknown disturbance, and enable the system to track the expected trajectory from an unstable state to a stable state.

2 Kinematics and Dynamics Models of Rope-Driven Robots

2.1 Rope-Driven Robot Kinematics

In this paper, the principle of vector closure is used to establish the kinematic model of cable length space and end material gras** space of four-cable robot, and the end grab is regarded as an abstract point P(x, y, z). Establish a Cartesian coordinate system O-XYZ, OX horizontal direction, OY direction determined according to the right-hand rule, OZ perpendicular to the corresponding platform. a, b and h represent the spatial structure dimensions of the four-cable robot, respectively. The center of the pulley set in the coordinate system is represented as $A_{i} (x_{i} ,y_{i} ,z_{i} )$, i = 1,2,3,4. Then each length of the cord can be expressed as:

$$L_{i} = \sqrt {\left( {x - x_{i} } \right)^{2} + \left( {x - y_{i} } \right)^{2} + \left( {x - z_{i} } \right)^{2} } \left( {i = 1,2,3,4} \right)$$

(1)

The kinematics diagram of the four-cable driven robot was analyzed (see Fig. 1).

Fig. 1

Schematic diagram of kinematics of a four-cable parallel robot

Equation (1) describes the map** relationship between the end grab space and the cable length space of the robot. In robot kinematics, the Jacobi matrix represents the linear map** of the robot's operational space velocity and joint space velocity. In a cable-driven parallel robot, it can also be expressed between the running speed of the end grab and the speed at which the cord changes, as shown in Eq. (2)

$$\dot{L}_{i} = JV$$

(2)

Formula $\dot{L}_{i}$ = $[\dot{L}_{1} \quad \dot{L}_{2} \quad \dot{L}_{3} \quad \dot{L}_{4} ]^{T}$, V represents the velocity vector of the end grab, V = $[\dot{x}\quad \dot{y}\quad \dot{z}]^{T}$. J represents the Jacobi matrix. J = $[\frac{{\partial L_{i} }}{\partial x}\quad \frac{{\partial L_{i} }}{\partial y}\quad \frac{{\partial L_{i} }}{\partial z}]$ ( i = 1,2,3,4).

2.2 Rope-Driven Robot Dynamics

In this paper, the end grab of the four-cable robot is the research object, and the driving force of each soft cable applied by the facility to the end grab is $T_{{\text{i}}}$, and the force is shown in Fig. 1. Thus, its dynamic model can be obtained by the Euler–Lagrange method:

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

(3)

Among them, $q,\dot{q},\ddot{q}, \in R^{n}$ is the motion position, speed and acceleration of the end gripper. $M({\text{q}}) \in R^{n \times n}$ is the positively definite symmetric mass inertia matrix, $C\left( {{\text{q}},{\dot{\text{q}}}} \right) \in R^{n \times n}$ is the Coriolis force coupling matrix, $G({\text{q}}) \in R^{n \times n}$ are the gsravitational vectors, $\tau_{d} ,\tau_{f} \in R^{n}$ is the external disturbance and the friction force vector, T is the driving force of each rope, and J is the Jacobi structure Considering the modeling error of the Flexo robot, there is $M(q) = M_{0} (q) + \Delta M(q);$$C(q,\dot{q}) = C_{0} (q,\dot{q}) + \Delta C(q,\dot{q});$$G(q) = G_{0} (q) + \Delta G(q)$. Corresponding estimate is $M_{0} (q),C_{0} (q,\dot{q}),G_{0} (q)$. System modeling error is $\Delta M(q),\Delta C(q,\dot{q}),\Delta G(q)$. Definition $d = \Delta M(q)\ddot{q} + \Delta C(q,\dot{q})\dot{q} + \Delta G(q) + \tau_{f} + \tau_{d}$.

Therefore, The kinetic model can be simplified to:

$$M_{0} (q)\ddot{q} + C_{0} (q,\dot{q})\dot{q} + G_{0} (q) + d = J^{T} T$$

(4)

3 Controller Design

3.1 Interference Observer Design

Aiming at the problem of composite interference in the control system of the flexible drive robot, a nonlinear interference observer is used to numerically estimate the interference signal, and then use it as a feedback signal to adjust the control torque of the system to reduce the influence of the interference signal on the system.

Definition $\hat{d}$ is the estimate of d, then the estimated error F of interference is defined as $F = d - \hat{d}$. The interference observer can be temporarily set to

$$\dot{\hat{d}} = L(q,\dot{q})(d - \hat{d})$$

(5)

Formula, $L(q,\dot{q})$ is the gain matrix to be designed in the interference observer.

Assuming that the disturbance d has a very small or negligible dynamic change rate compared to a nonlinear observer ($\dot{d} = 0$).it can be obtained: $\dot{F} = - L(q,\dot{q})F$. As the estimation error can be exponentially converged by designing a properly designed gain matrix. Since the acceleration signal of the end grab is not easy to obtain in practice, it is also necessary to define new auxiliary variables to build a new observer

$$z = \hat{d} - p(q,\dot{q})$$

(6)

Formula, $z \in R^{2}$; $p(q,\dot{q})$ is the nonlinear function to be designed. Moreover, The gain matrix $L(q,\dot{q})$ and nonlinear functions $p(q,\dot{q})$ need to be satisfied:

$$L(q,\dot{q})M(q)\ddot{q} = \frac{{dp(q,\dot{q})}}{dt}$$

(7)

The combined formula (3) ~ (7) can obtain a nonlinear interference observer:

$$\left\{ \begin{gathered} \dot{z} = - L(q,\dot{q})z + L(q,\dot{q})[C(q,\dot{q})\dot{q} + G(q) - T - p(q,\dot{q})] \hfill \\ \hat{d} = z + p(q,\dot{q}) \hfill \\ \end{gathered} \right.$$

(8)

3.2 Dynamic Non-Singular Fast Terminal Sliding Mode Surface Constructed Based on the Estimated Output of Interference Observer

When constructing the traditional terminal sliding mode controller, it is faced with the problem of singularity, and in order to overcome the problem of singularity. Presented by References [10]. it is proposed that a dynamic non-singular sliding mode surface be constructed on this basis

$${\text{s}} = {\text{e}} + \frac{1}{\alpha }\left| e \right|^{\gamma + 1} {\text{sgn}} e + \frac{1}{\beta }(\dot{e} + \int {M^{ - 1} \hat{d}} )^{\frac{p}{q}}$$

(9)

Formula, $\alpha$、$\beta$、$\gamma > 0$ is a design constant; p、 q is positive odd and satisfied $1 < p/q < 2$; e is the difference between the desired trajectory and the actual trajectory $e = q - q_{d}$; $\hat{d}$ is an estimate of the interference. The synovial control law is proposed:

$$\begin{aligned} T & = (J^{T} )^{ - 1} ( - \frac{\beta q}{p}M(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(2 - \frac{p}{q})}} {\kern 1pt} - ML{\text{sgn}} (s) + d - \hat{d} + g \\ & \quad + M\ddot{q}_{d} {\kern 1pt} - \frac{\beta q}{p}M(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(1 - \frac{p}{q})}} (\frac{\gamma + 1}{\alpha }\left| e \right|^{\gamma } \dot{e})) \\ \end{aligned}$$

(10)

Formula, $L > 0$ is the constant to be designed. This is obtained by Eq. (10).

$$\begin{aligned} \dot{s} & = {\dot{\text{e}}} + \frac{\gamma + 1}{\alpha }\left| e \right|^{\gamma } \dot{e} + \frac{p}{\beta q}(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(\frac{p}{q} - 1)}} (\ddot{e} + M^{ - 1} \hat{d}) \\ & = {\dot{\text{e}}} + \frac{\gamma + 1}{\alpha }\left| e \right|^{\gamma } \dot{e} + \frac{p}{\beta q}(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(\frac{p}{q} - 1)}} (M_{0}^{ - 1} (q)(J^{T} T \\ & - C_{0} (q,\dot{q})\dot{q} - G_{0} (q) - d) - \ddot{q}_{d} + M^{ - 1} \hat{d}) = - \frac{p}{\beta q}(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(\frac{p}{q} - 1)}} L{\text{sgn}} (s) \\ \end{aligned}$$

(11)

3.3 Lyapunov Proved

Consider the Lyapunov function as follows

$$V(t) = \frac{1}{2}s^{2}$$

(12)

Derive the above equation:

$$\begin{aligned} \dot{V}(t) & = s^{T} \dot{s} = s^{T} [ - \frac{p}{\beta q}(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(\frac{p}{q} - 1)}} L{\text{sgn}} (s)]\frac{{\partial^{2} \Omega }}{{\partial u^{2} }} \\ & \quad \le - \left| s \right|\frac{p}{\beta q}(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(\frac{p}{q} - 1)}} L \\ \end{aligned}$$

(13)

Formula, $\beta ,q,p,L > 0$. Hypothesis $\dot{\tilde{e}} = \dot{e} + \int {M^{ - 1} \hat{d}}$. when $\dot{\tilde{e}} \ne 0$, From $0 < \frac{p}{q} - 1 < 1$ you get $(\dot{e} + \int {M^{ - 1} \hat{d}} )^{{(\frac{p}{q} - 1)}} > 0$. , according to Reference [11], When a sufficiently large $L > 0$ is satisfied, When $\dot{\tilde{e}} \le - L$ is then $s > 0$, The opposite is the same, Therefore, $\dot{\tilde{e}} = 0$ has little effect on the stability of the sliding surface, and it can also be said that $\left| {\dot{\tilde{e}}} \right| < \delta$ exists near the $\dot{\tilde{e}} = 0$ range. So when $s > 0$ is then $\tilde{e} \le - L$, The opposite is the same, Therefore, the intersection of dynamic synovial trajectories when $s > 0$ is $\tilde{e} = \delta$ from $\tilde{e} = - \delta$. So it can be launched

$$\dot{V}(t) = s^{T} \dot{s} \le 0$$

(14)

4 Simulation Experiments

The simulation of all the examples in this simulation was completed under MATLAB2023aSimulink. The simulation object selects the four-cable parallel drive robot model. The matrix parameters in the kinetic model (3) are: $M(q) = diag(m,m,m)$; $C(q,\dot{q}) = diag(0,0,0)$; $G(q) = [mg;0;0]^{T}$. Four-cable drive parallel robot structure parameters a = 10 m,b = 5 m,h = 5 m,m = 1 kg,g = 9.8N/kg. Take the perturbation item $d = [0.25\sin (t);0.25\sin (t);0.25\sin (t)]^{T}$. In the interference observer $p(q,\dot{q}) = \rho \dot{q},\rho = 0.5$. The initial position of the end gripper of the four-cable-driven parallel robot is set q = [0;0;0];The simulated motion trajectory is as follows: $x = 2\sin (0.5\pi t) + 1;$$y = \sin (0.5\pi t) + 1;$$z = 0.5\sin (0.5\pi t) + 1$. When the expected trajectory of the end grab of the four-cable parallel drive robot is the above trajectory, the synovial control rate is used to track and control the motion trajectory of the end grab, and it can be seen from the figure that the three-dimensional diagram of the expected spiral trajectory and the actual trajectory is obtained for simulation. Taking values on the sliding surface: p = 7, q = 5, α = 3, β = 1000, γ = 0.5.

From the observation of the above figure (see Fig. 2), it can be found that the dynamic non-singular fast terminal sliding mode control strategy constructed based on the estimated output of the interference observer has a good trajectory tracking effect on the motion trajectory of the four-cable parallel drive robot end grab.

Fig. 2

Trajectory tracking

As shown in the figure (see Fig. 3) above, the speed tracking trajectory change diagram of the end gripper in the three directions of x, y, and z can clearly see that the overall fluctuation is not large and is in a stable change. Among them, the speed error in the x direction is absolutely within 3.3%, the velocity error in the y direction is within 2.3%, and the speed error in the z direction is within 2.1%.

Fig. 3

Velocity error

As shown in the figure (see Fig. 4) above, the trajectory error of the proposed controller and the fast terminal synovial membrane controller is compared, indicating that the trajectory error of this observer is smaller, stable and faster.

Fig. 4

Track tracking comparison

5 Conclusion

Aiming at the interference such as external disturbance and modeling error of the four-cable parallel drive robot, a dynamic model is established based on the Newton–Euler method, and a dynamic non-singular fast terminal sliding mode surface based on the estimated output construction of the interference observer is designed. Lyapunov was used to prove the stability of the design control system, and the trajectory tracking effect was good and the error was within the acceptable range through simulation, and it was compared with the fast terminal synovial controller to prove that it had a better tracking trajectory and tracking error.