Complex traffic conditions and engine rotation cause shocks and multi-dimensional vibrations to vehicles, which can severely impact the performance of onboard equipment. Vibrations caused by vehicle movement cannot be completely eliminated; therefore, reducing the transmission of vibration energy from the vehicle body to onboard equipment through vibration isolation devices is the most effective approach. Vehicle vibration is multi-dimensional, and according to the principle that the degrees of freedom of the vibration isolation system should match those of the vibration source, the degrees of freedom of the vibration isolation system should be the same as those of the vehicle body. Since multi-dimensional vibration isolation is achieved by adding vibration isolation units to the vibration isolation mechanism, its performance is entirely determined by the vibration isolation mechanism and its control strategy.
Parallel mechanisms, with their advantages of high static stiffness, low inertia, and high flexibility, are excellent candidates for the main structure of multidimensional vibration isolation systems. The Gouig-Stewart platform and the Hexapod platform have been widely used in six-dimensional vibration isolation systems for precision systems. Multidimensional vibration isolation systems based on parallel mechanisms have seen rapid development in recent years, especially systems based on MR (Magnetorheological) dampers. In such vibration isolation systems, the control strategy is a crucial factor affecting system performance. Over the past 20 years, numerous theoretical and experimental studies have shown that semi-active control strategies offer good vibration isolation effects and relatively low costs; however, their isolation performance is significantly influenced by the control strategy. Generally, semi-active control strategies must consider both obtaining the optimal control force and the actual output force of the MR damper. This paper employs a fuzzy model based on a genetic algorithm to calculate the input current of the MR damper to avoid solving highly nonlinear models. Based on the vibration characteristics of on-board equipment, a three-dimensional vibration isolation system based on a 3-RPC parallel mechanism is proposed. The kinematic and dynamic models of the system are introduced, and a semi-active control strategy based on the MR damper is proposed. Finally, an experimental prototype is fabricated, and the effectiveness of the vibration isolation system is verified through experiments.
Vibration isolation system model
The vibration isolation system proposed in this paper is used to isolate vibration signals transmitted from the vehicle body to onboard equipment (such as onboard robot systems). Based on the vibration characteristics of onboard equipment, the proposed vibration isolation system has three translational degrees of freedom, and the main mechanism of the system is a 3-RPC parallel mechanism. The system consists of a lower platform, an upper platform, and three independent branches (see Figure 1(a)). Each branch contains a spring-damper vibration isolation subunit, as shown in Figure 1(b), where k is the spring stiffness coefficient and c is the damping coefficient of the damper. The branches are denoted by A and B. The bottom end of each branch is connected to the lower platform at point A via a revolute joint, while the upper end is connected to the upper platform at point B via a cylindrical joint. A and B are symmetrically arranged in their respective planes, and the axes of the revolute joint and cylindrical joint of each branch are parallel to each other. According to the screw theory, each branch of the 3-RPC mechanism... Two coplanar constraint couples can be provided, with their axes perpendicular to the axis of the revolute joint. Since the constraint couples provided by the three branches restrict the spatial rotational degrees of freedom of the upper platform, it only has three translational degrees of freedom. To better achieve the goal of three-dimensional vibration isolation, the system's geometry was optimized by minimizing the maximum sum of the natural frequencies of the mechanism across all degrees of freedom in the workspace, while considering practical constraints such as workspace, dexterity, and the rotational range of the kinematic pairs: PB is 150 mm, OA is 400 mm, and the initial height of the plane is 245 mm.
Kinematic and dynamic modeling
In Figure 1(b), 0'(X', Y', Z') is a Cartesian coordinate system established on the ground, 0'3X,Y,Z} is the coordinate system of the lower platform, and the moving coordinate system P{x,y,z} is located at the geometric center point P of the upper platform. To simplify the modeling process without loss of generality, the axes of the three coordinate systems are initially parallel to each other.
1. Jacobian matrix
The kinematic analysis of parallel mechanisms includes inverse kinematics, velocity and acceleration analysis, etc. Since the kinematic analysis of parallel mechanisms is relatively simple, it will not be described in detail here; only the Jacobian matrix of the system will be given. The loop equation method is used to establish the Jacobian matrix. As shown in Figure 2, the inverse Jacobian matrix of the mechanism can be obtained according to the loop equation method as follows:
2. Dynamic Modeling
The closed-loop dynamic equations of the system are established using the Newton-Euler equations. Since the dynamic modeling process is quite complex, this paper will not describe it in detail but will only provide the system's dynamic model. For a more detailed modeling process, please refer to relevant literature.
Ignoring the rotational inertia and second-order velocity of the link during modeling, the closed-loop dynamic model of the mechanism can be obtained as follows:
3. Semi-active control of multi-dimensional vibration isolation systems
(1) Control model of vibration isolation system
If the initial deformation of the spring can offset all the static gravity in the dynamic model expressed by equation (2), then after eliminating the static equilibrium term in the dynamic model, the vibration model of the system can be simplified to:
In the formula: Xp and Xb are the displacement vectors of the upper and lower platforms; C and K are the damping matrix and stiffness matrix, respectively, and C=diag(c1czc3) and K=diag(k1kzk3).
Based on the vibration model, the system's control model can be obtained, where the translational acceleration of the upper platform serves as the system's control objective, and the translational velocity of the lower platform serves as the system's disturbance signal. Since state variable feedback can comprehensively reflect the internal characteristics of the control system, this paper uses state feedback to achieve vibration control of the isolation platform. In this paper, the relative displacement between the upper and lower platforms and the velocity of the upper platform are used as state feedback quantities. Additionally, the inverse model of the MR damper requires the relative velocity signal across the damper, which can be obtained by differentiating the relative displacement of the upper and lower platforms and then solving for the Jacobian matrix. In actual control, the output force of the MR damper is limited; therefore, the control signal of the MR damper should also be subject to reasonable constraints. The variables and disturbances in the control system are:
The control system model, composed of the system's dynamic state equations and the controlled output equations, can be expressed as:
In addition, since the output force of the MR damper is limited, the following constraints also need to be considered during the controller design process:
(2) Semi-active control strategy for vibration isolation system
The semi-active control algorithm for the vibration isolation platform designed in this paper includes three steps: calculating the optimal control force, calculating the possible output force of each branch MR damper, and calculating the input current of each MR damper. Considering these three requirements and the actual structure of the vibration isolation platform in this paper, a fuzzy optimal control strategy is proposed to achieve vibration control of the platform.
In this strategy, the optimal control force in each branch is obtained through the H-state feedback control strategy. Then, based on the motion state in the branch and the working mechanism of the MR damper, the achievable output force of the MR damper in each branch is obtained. Finally, the input current of each MR damper is obtained through a fuzzy model. The specific control flow is shown in Figure 3.
In calculating the input current of the MR damper, an evolutionary Takagi-Sugeno fuzzy model based on a genetic algorithm was employed. Because the dynamic model of the MR damper is highly nonlinear, directly solving the inverse dynamic model of the MR damper to obtain the input current from the motion state and output force would result in a significant increase in computational complexity and low efficiency due to solving strongly nonlinear equations. Addressing highly nonlinear problems is a key feature of fuzzy control; this control method can quickly approximate the inverse model of the MR damper while avoiding control force overshoot. Due to space limitations, this paper will not elaborate on the fuzzy model-based method for identifying the MR damper input current; further details can be found in the relevant literature.
Experiments and Analysis
As a system designed for engineering applications, experimentation is the most reliable way to verify its performance. Therefore, this paper designs and manufactures an experimental prototype and establishes a hardware and software control system. Then, under existing experimental conditions, vibration response experiments were conducted on the system with random and sinusoidal signal inputs from different directions. The experimental results show that the system has good vibration isolation performance.
1. Experimental Design
Due to limitations in experimental conditions, the vibration isolation performance of the system in both the horizontal and vertical directions was verified in this paper. The experimental scheme mainly includes the following steps:
(1) Unidirectional sinusoidal frequency sweep experiment.
The lower platform of the vibration isolation platform was connected to a shaking table, and sinusoidal signal frequency sweep experiments were conducted in the X and Z directions, respectively, and the accelerations of the upper and lower platforms were recorded. This experiment mainly aims to find the natural frequencies of the system in various directions and verify the vibration isolation performance of the platform in the frequency domain.
(2) Unidirectional random vibration experiment.
The lower platform of the vibration isolation platform was connected to a vibration table, and random signal vibration experiments were conducted in the X and Z directions, recording the acceleration of both the upper and lower platforms. This experiment was conducted in two modes: passive control and semi-active control. The main purpose of this experiment was to verify the vibration isolation effect of the platform under random signal excitation and to compare the effects of passive and semi-active control.
(3) Unidirectional fixed-frequency sine wave experiment.
Experiments were conducted using fixed-frequency sinusoidal signals in the X and Z directions, with the excitation frequencies being the platform's natural frequencies in those directions. The experiments were performed in both passive and semi-active control modes. This experiment primarily verifies the vibration isolation capability of the vibration isolation platform when encountering signals with their natural frequencies.
2. Vibration Experiment and Result Analysis
The experimental procedure is shown in Figure 4. Figure 4(a) shows the horizontal vibration experiment, and Figure 4(b) shows the vertical vibration experiment.
(1) Unidirectional sinusoidal frequency sweep experiment
The main purpose of the sinusoidal sweep frequency experiment is to find the natural frequencies of the vibration isolation system in various directions. Therefore, the MR damper was not controlled in this experiment; instead, the experiment was conducted under zero-field damping of the MR damper. Since the first-order natural frequency of a car is generally below 5Hz, the sweep frequency range of the vibration table used in the experiment was 1-0Hz. A constant amplitude sweep frequency of 20mm was used in the 1-5Hz range, and a constant acceleration sweep frequency of 2g was used in the 5-20Hz range, with a sweep rate of 1OCT/min.
During the experiment, it was found that when the vibration test table was excited in the X-axis or Z-axis directions, motion signals could be measured in other non-excitation directions. This was mainly due to the coupled motion of the parallel mechanism. In the vibration model given in equation (3), the mass matrix, damping matrix, and stiffness matrix of the system are all off-diagonal matrices. Therefore, the translational motion of the system in each direction is coupled together. When sinusoidal sweep vibration experiments were carried out in the X and Z directions, the acceleration frequency domain characteristic curves of the prototype platform were collected as shown in Figure 5. As can be seen from Figure 5, the vibration curves of the vibration table in the two directions are...
The trends are basically the same: at the beginning of the frequency sweep, the acceleration of the upper platform is basically the same as the excitation acceleration; as the frequency of the frequency sweep signal increases, the acceleration of the upper platform suddenly rises, and the system resonates; after the frequency sweep frequency crosses the resonance zone, the vibration acceleration of the upper platform is smaller than the excitation acceleration, indicating that the vibration isolation system plays a role in isolating the transmission of vibration.
(2) Unidirectional random vibration experiment
Random vibration experiments are an important method for verifying the actual vibration isolation capability of a vibration isolation platform. Although the vibration performance of the vibration isolation system differs in different directions within the horizontal plane, the theoretical analysis and simulations presented earlier show that the basic trends of vibration characteristics in each direction are roughly the same. Therefore, this paper only selected the X-axis and Z-axis directions for random vibration experiments.
Based on the road power spectral density function given by the national standard, assuming that the car is driving on a Class B road, the frequency range of the vibration signal used in the experiment is 1-20Hz, and the acceleration power spectral density is 0.05(m/sz)Z/Hz.
(3) Unidirectional sinusoidal constant frequency vibration experiment
This experiment aims to verify the performance of the vibration isolation platform under disturbance from signals with the platform's natural frequencies in various directions. In the experiment, the amplitude of the sinusoidal signal output by the vibration table was 10 mm, and the frequency of the disturbance signal was the first-order natural frequency of the system in each direction obtained from the frequency sweep experiment. Figure 6 shows the response acceleration of the upper platform when the vibration isolation platform is disturbed by a sinusoidal signal with its natural frequencies. As can be seen from the figure, in passive mode, the platform will reach resonance, and the response acceleration will be amplified by approximately three times. However, when using the semi-active control strategy proposed in this paper, although the response acceleration is still larger than the disturbance acceleration, it is still significantly greater than...
There is a significant decrease when using passive vibration isolation.
