Abstract : This paper proposes a discrete sliding mode variable structure controller with a variable rate reaching law, based on the characteristics of the mathematical model of a parallel robot driven by a hydraulic servo system. Simulation results show that this control scheme avoids the chattering phenomenon inherent in conventional variable structure control, and the system exhibits strong robustness to parameter perturbations and external disturbances. Keywords: Hydraulic servo system, Parallel robot, Discrete sliding mode, Variable structure control, Variable rate reaching law KEY WORDS : Hydraulic SelWO system, Parallel robot, Discrete sliding mode, VSC, Variational ratereaching law 0 Introduction Compared to serial robots, parallel robots have advantages such as high rigidity, strong load-bearing capacity, and no cumulative error. Therefore, in recent years, parallel robots have received widespread attention both domestically and internationally, and are becoming an important branch of international robotics research. The research object of this paper is a large-scale laboratory prototype—a six-degree-of-freedom parallel robot—developed by our university, which has advanced domestic standards. Its geometric structure is shown in Figure 1. It consists of a motion platform capable of spatial position changes, a fixed base, and six structurally identical support rods with adjustable lengths connecting the motion platform and the base. A manipulator is mounted on the motion platform, and the rods are connected to the platform using Hooke hinges. [align=center] Figure 1 Schematic diagram of the parallel robot structure[/align] The support rods of this prototype are all driven by an electro-hydraulic servo system. From a mechanical theory perspective, these six rods are designed as independent, uncoupled units. However, under actual loads, the links are weakly coupled, and their corresponding system models and parameters are not entirely the same, exhibiting certain time-varying nonlinearities and random disturbances. Therefore, finding a control law that can overcome parameter perturbations and external disturbances while simultaneously achieving high dynamic and static performance is crucial for parallel robots to move from the laboratory to practical applications. Variable structure control is a robust control method that can handle linear and nonlinear systems. Because the sliding mode in a variable structure system is invariant—that is, it is independent of system perturbations and external disturbances—this ideal robustness has attracted great attention in the control community and has seen significant development. With the rapid development of computer technology and the practical needs of fields such as industrial automation, control algorithms are frequently implemented using digital computers. Therefore, discrete variable structure control has even broader applications. This paper, from the above perspective, designs a discrete sliding mode variable structure controller with a variable speed reaching law, specifically targeting the characteristics of the mathematical model of a hydraulic servo-driven parallel robot. Simulation results show that the control effect is satisfactory. 1 Design of Discrete Sliding Mode Variable Structure Controller Consider the discrete state equation of the controlled object as: X(k+1)=(A+△A)X(k)+(B+△B)u(k)+d(k) (1) Where: the state vector x∈Rn, the input vector U∈R1, A and B are the corresponding dimension matrices, and it is assumed that (A, B) is controllable. △A and △B are parameter changes, and d(k) is the external disturbance, all of which are unknown. Here, the influence of the unknown terms △A, △B and d(k) is equivalent to an external disturbance φ(k), φ(k)=△AX(k)+△Bu(k)4-d(k)=X(k+1)-Ax(k)-Bu(k) (2) φ(k) is unknown. The sliding surface equation of this discrete system is described as: S(k)=CX(k) (3) Where: c∈R1xn is a constant matrix. In sliding mode control, the determination of the covariance matrix c must ensure system stability, that is, the eigenvalues ​​of the controlled system are within the unit range. The choice of control law must make the motion of the state trajectory tend toward the sliding surface. If the convergence condition is met, the state trajectory will be forced to tend toward the sliding surface. Here, the design of the switching function uses the discrete approach law: S(k+1)=(1—δT)S(k)-εTsgnS(k) (4) where 0<δT<1, E>0, and T is the sampling period. The variable structure controllability condition CB≠O holds, and the variable structure control can be obtained as: U(k)=U[sup]±[/sup] (k)=-(CB)[sup]-1[/sup][CAX(k)-(1—δT)s(k)+ εTsgnS(k)+ Cφ(k)] (6) Since φ(k) is unknown, this control cannot be realized. When the dynamic characteristics of the uncertain part of the system are much slower than the sampling frequency, the equivalent disturbance φ(k) at t=kT can be considered to be close to the value φ(k-1) at t=(k-1)T: φ(k-1)=X(k)-Ax(k-1)-Bu(k-1) (7) φ(k-1) is known, so we have: u(k)=U[sup]±[/sup](CB)(k)[CAX(k)(1-δT)S(k)+εTsgnS(k)+Cφ(k)] (8) As can be seen from reference [3], the determination of the switching function S(k)=S(x(k))=CX(k) or the ideal quasi-sliding mode and its stability are exactly the same as when there is perturbation and disturbance. When the perturbation and disturbance are bounded and their bounds are not large, the closed-loop control system obtained by using this controller is asymptotically stable. However, according to reference [4], once the system state trajectory crosses the sliding surface, it will repeatedly cross the sliding surface at each consecutive sampling time. This means that in the quasi-sliding mode, sgn(S(k+2)) = -sgn(S(k+1)) = sgn(S(k)), and the state trajectory remains within the quasi-sliding mode band in the state space. The width of the quasi-sliding mode band is given by the coefficients δ and ε, which can be selected. The smaller E is selected, the smaller the thickness of the switching band, and the value of the system state is very close to the equilibrium point 0. It can be seen that in the discrete approach law, the smaller the coefficient ε will correspondingly reduce the system's oscillation. However, if the value of ε is too small, it will affect the approach speed of the system to the switching surface. Therefore, the ideal value of E should be time-varying, that is, the value of E should be larger at the beginning of the system motion and gradually decrease as time increases. When E = 0, the system will eliminate high-frequency oscillation, and the system state will reach the equilibrium point 0. Based on this idea, this paper takes ε=ρllXll[sub]1[/sub] as the proportionality coefficient, which is the state norm, that is, ε is a variable value that is proportional to the state norm. The improved variable structure control with variable speed approach law is obtained as follows: u(k) = U[sup]±[/sup](k)≈-(CB)[sup]-1[/sup][CAX(k)(1-δT)S(k)+ρllXll[sub]1[/sub]TsgnS(k)+Cφ(k)] （9） 2 Mathematical model of single channel of parallel robot driven by hydraulic servo Considering the similarity of the six parallel channel structures of the parallel robot, the mathematical model of a single channel is given as follows: Where: K[sub]1[/sub]=K[sub]a[/sub]，K[sub]av[/sub]， is the amplifier gain, K[sub]av[/sub]，K[sub]2[/sub] is the servo valve flow gain; K[sub]2[/sub]=1/A，A is the hydraulic cylinder plunger area; ω[sub]n[/sub] is the natural frequency of the hydraulic cylinder; δn is the damping ratio; x is the cylinder displacement output; KF(TFs+1) is the force disturbance transfer function, and F is the external force. Here, ωn and δn are parameters that vary within a certain range. The error vector is defined as the expected output value of the system in response to a step input. Equation (11) can be written as the error state equation by replacing x with E in equation (11). The sampling period is then taken as T=5ms, and equation (11) is discretized to obtain the error discrete state equation of the controlled object: 3 Simulation Study For the state model of the single-channel system of the hydraulic servo-driven parallel robot described by equation (12). The control law designed according to the method described in section 1 is as follows: Considering the limited output current capability of the servo valve in practical applications, the output quantity U of the controller has a maximum positive and negative limit value of ±u[sub]m[/sub] (U[sub]m[/sub]=5V in this system). Through Maflab simulation optimization, the proportional coefficient ρ in the variable speed reaching law is taken as 10.6, and the parameter δ is taken as 5.5. [align=center] [/align] Figure 2 shows the simulation results of the discrete sliding mode variable structure control proposed in this paper under no external disturbance and no parameter perturbation. Figure 3 shows the control quantity and sliding mode curve of the discrete sliding mode control with 40% parameter perturbation. Figure 4 shows the control quantity, sliding mode curve and error response curve of the discrete sliding mode control with 5% random external disturbance. As can be seen from the simulation results, the discrete variable structure controller with variable speed reaching law given in this paper can effectively weaken or even eliminate chatter, and has strong robustness to perturbation of system parameters and random disturbances. 4 Conclusion This paper proposes a design method for a discrete sliding mode variable structure controller with a variable speed reaching law, based on the characteristics of the mathematical model of a hydraulic servo-driven parallel robot. The variable speed reaching law is adopted, and time delay technology is used to estimate disturbances online, which can effectively reduce or eliminate chatter. Simulation results show that this discrete sliding mode control not only ensures the speed and stability of the system, but also has strong robustness to perturbations and random disturbances of system parameters. This is of particular importance for the control of robots driven by hydraulic servo systems. In summary, this discrete variable structure control has certain practical value. References 1 Fumta, K. Sliding mode control of a discrete system. Systemz Control Letters, 1990, 14, 145-152 2 Gao Weibing. Variable structure control of discrete time systems. Acta Automatica Sinica, 1995, 21(2): 154-161 3 Gao Weibing. Theory and design method of variable structure control. Beijing: Science Press, 1996. 278-299 4 Zhai Changlian, Wu Zhiming. Variable structure control design for uncertain discrete-time systems. Acta Automatica Sinica, 2000, 26(2). 184-191 5 Wang Hongrui et al. Research on variable structure control for trajectory tracking of parallel robots. Robot, 17(2). 65-69 6 Sun Fuehun, Sun Zengqi. Sliding mode control for discrete-time systems and its application to the control of a discrete-control system. Control Theory and Applications, 1997, 14(4), 467-472