Flexible arms are widely used in spacecraft, flexible robots, and other fields due to their advantages such as light weight and compact structural design. However, their end effectors are prone to vibration during operation, which seriously affects their working efficiency and positioning accuracy, and even endangers the safety of the entire system. Domestic and international researchers have proposed control methods such as PID control, fuzzy control, adaptive control, sliding mode variable structure control, and optimal control to address the vibration suppression problem of flexible arms, and have explored feedback control methods for vibration suppression. Input shaping technology, first proposed by Singer and Seering, has been widely used for vibration suppression in flexible structures. This method belongs to feedforward control and is also a unique vibration suppression method. Youmin et al. applied input shaping technology to bridge cranes, suppressing vibration while improving the crane's transport efficiency; Alsaibie et al. applied input shaping to suppress the sloshing of liquids during transport; Dhanda et al. improved the input shaper, designing an optimal input shaper to effectively suppress the residual vibration of the crane; Pradhan et al. combined input shaping with adaptive control to control the lateral swaying of swinging loads. However, while input shaping technology suppresses vibration in flexible systems, it also leads to a delay in system response time, severely reducing the system's working efficiency.
This paper addresses the response time delay problem caused by input shaping technology. Based on optimal control theory, an optimal input shaper is designed to reduce the system response time. Furthermore, the optimal input shaper is combined with a fuzzy PID controller to suppress the vibration of the flexible arm.
Flexible arm dynamics model
The mechanical structure and physical model of the flexible arm rotating around the servo motor shaft are shown in Figure 1. In Figure 1(a), the servo motor shaft is connected to the turntable, one end of the flexible arm is connected to the motor shaft through the turntable clamp, the mass block is attached to the other end, and the base is used to fix the entire experimental platform.
In Figure 1(b), let the inertial frame XOY and the body coordinate system xOy be established with the center O of the motor shaft as the origin. The elastic modulus of the flexible arm is E, the moment of inertia of the cross section about the neutral axis is I, the density is , the cross-sectional area is A, the length is l, the mass of the end mass block is ml, the sum of the rotational inertia of the servo motor shaft and the turntable is J0, the rigid body rotation angle corresponding to the movement of the flexible arm is θ(t), and the input control torque of the shaft is u(t).
Figure 1. Mechanical structure and physical model of the flexible arm
Assuming the lateral vibration of the flexible arm is much greater than its axial vibration, and the lateral vibration is relatively small, according to the principles of vibration dynamics, the flexible arm can be considered as an Euler-Bernoulli beam. Let P(X,Y) be the coordinates of any point on the flexible arm at time t, and w(x,t) be the lateral elastic vibration displacement of point P in the coordinate system xOy. According to the vibration theory of Euler-Bernoulli beams, the differential equation for the bending free vibration of a flexible arm of uniform material and uniform cross-section is:
(1)
By the assumed modal method, let
(2)
In the formula, Wi(x) is the mode function or modal function of the flexible arm at x, and qi(t) is the amplitude of the corresponding mode, called the modal coordinate.
Defined based on the four boundary conditions of the flexible arm.
Figure 9 Experimental platform for vibration suppression of flexible arm
