Analysis of Repetitive Positioning Accuracy of Parallel Mechanisms Based on Monte Carlo Method

Based on the inverse kinematics model of the Stewart parallel mechanism, a motion platform pose error model was constructed through differential transformation. The error model shows that the motion platform pose error comes from the length error of 6 struts and the position error of 12 hinge centers, with a total of 42 error terms. Monte Carlo simulation was used to analyze the effects of struts length error on the pose error of the moving platform. The results show that when the length error of the struts follows a normal distribution, the posture error of the moving platform approximately follows a normal distribution. Monte Carlo simulation is a highly effective error analysis method that can analyze the error distribution pattern of the mechanism before processing and assembly, which provides a method for error analysis of Stewart parallel mechanisms.


Introduction
Precise active adjustment devices can be used for precise adjustment of relative pose in optoelectronic load systems, such as space camera.By using a spatial multi-DOF adjustment device, the pose relationships of various components of the optical detection system can be returned to an ideal state, thus meeting the application requirements of high-resolution imaging in the system.
In terms of engineering implementation, parallel mechanisms are commonly used to achieve highprecision 6-DOF pose adjustment.The traditional 6-DOF adjustment device consists of four main parts, a moving platform, 6 retractable active struts, 12 Hook hinges, and a static platform.The static platform is fixed, the dynamic platform is connected to the load, and the platform is connected to the active drive struts through hinges.The support strut is driven by an electric motor for telescopic motion, achieving the translation of the moving platform along the x, y, and z axes and the rotation of RX, RY, and RZ around the x, y, and z axes [1][2][3][4][5].
Compared with series mechanisms, Stewart parallel mechanisms have advantages such as high stiffness, high motion accuracy, superior dynamic performance, and easy real-time control.They are widely used in flight simulators, motion simulators, parallel machine tools, precision positioning platforms, vibration isolation platforms, and other fields.In recent years, with the improvement of the accuracy indicators of various environmental experimental simulation systems, the accuracy of Stewart parallel mechanisms has attracted widespread attention.Therefore, it is necessary to improve the platform accuracy through reasonable modeling and analysis of the pose error of the Stewart parallel mechanism platform [6][7][8][9][10].

Error Source
As a typical 6-DOF parallel mechanism, the structural parameter error of the 6-DOF adjustment mechanism includes the axial error of six struts, as well as the positioning error of the hinge centres of 12 hinges along the x, y, and z directions, totalling 42 errors.The establishment of pose error model adopts vector differentiation method.
In order to reduce the complexity of the model, after comprehensively analyzing the impact of various error sources on the pose error of the parallel mechanism, only 42 independent mechanism parameter errors, including hinge point position error and struts length error, were considered for error mathematical modeling.When the system error is given, the upper platform pose error can be calculated using the error model.
The commonly used error models only consider geometric errors, and the complexity of parallel mechanism models does not have a significant impact on the motion accuracy of parallel mechanisms.Therefore, when establishing the Stewart platform error model, non-geometric error factors are ignored, and only system errors such as installation error, manufacturing error, and strut length error are considered.
Consider the following points for system error: (1) Installation error: For Stewart parallel mechanisms, there are a total of 36 independent parameters for the installation error of the hinge fixed on the platform, including the installation error of the 12 hinge points cj of the Hooke hinge on the upper and lower platforms.
(2) Manufacturing error: The hinge itself is limited by machining and installation accuracy, and its rotational axes cannot be perfectly perpendicular, and the motion centres cannot coincide at a single point, all of which affect the motion accuracy of the hinge.
(3) Strut length error: The driving struts is affected by the machining accuracy during the machining process, and its initial length may also have errors (i = 1, 2, ... , 6), with a total of 6 independent length error parameters.The installation and manufacturing errors of the hinge ultimately result in positional deviation of the hinge rotation hinge point, which is the hinge point position error.There are a total of 36 hinge point position errors: dcj( ) and ( ) (i = 1, 2, ... , 6).Therefore, for general Stewart platforms, in order to reduce the complexity of calibration models, only hinge joints are considered in the process of error modeling

Error Model
For the i th support strut, there is an inverse solution model: Among them, is the length of the i-th support strut, is the unit vector of the i th support strut, is the coordinate of the i th upper hinge point in the moving platform , are coordinate of the i th lower hinge point in a static platform , is rotation matrix, represent the position of the origin of the upper platform or secondary mirror moving coordinate system relative to the static coordinate system .
For equation (1), perform full differentiation to obtain Equation ( 2) multiply both sides by at the same time to obtain Since is a unit vector, , ，substituting into equation ( 3) yields

Because
, where are the Euler rotation angle of the upper platform， .Equation ( 4) can be transformed into Equation ( 5) is organized into a matrix form as (6) Parallel mechanisms should avoid singularity in their configuration in the workspace, so is reversible.Therefore, equation ( 6) can be organized into the form of the relationship between input error and output error (7) where is the pose error of the parallel mechanism, , is the axial length error of the driving rod, , is the positioning error of the upper and lower hinge points, , ) is the Jacobian matrix for error propagation, , is the matrix of structural error coefficients, When the system error, which can be simplified to the 42 error sources mentioned above, is given, the upper platform pose error can be comprehensively calculated using equation (7).
After calculating the design configuration parameters and accuracy index parameters of the 6-DOF adjustment mechanism using equation ( 7), a strut accuracy of ±0.2 µm that meets the accuracy index requirements is obtained.

Verification of Monte Carlo Simulation Method
Randomly generate 10000 sample points lj (j = 1, 2, ..., 10000) based on the strut accuracy allocation criteria.The distribution of 6 struts errors is shown in figure 1, with the horizontal axis being the struts error in nm and the vertical axis being the number of points within a certain error range.In the case of adjusting the initial pose of the mechanism, due to the random error of each strut with a mean square error of 0.2 µm, the corresponding actual strut length is L0+lj.The actual error distribution of the initial pose of the adjusting mechanism's moving platform is obtained through forward kinematics, as shown in figure 2.

Conclusion
This article applies the Monte Carlo method to analyze the repetitive positioning accuracy of parallel mechanisms, and analyzes the repetitive positioning accuracy of a mirror parallel platform.The analysis results show that: (1) The pose error of the Stewart parallel mechanism platform is influenced by 42 factors, including the length error of the support strut and the position error of the hinge center.When the length error of the support strut follows a normal distribution, the pose error of the moving platform approximately follows a normal distribution; (2) Monte Carlo simulation is a highly effective error analysis method that can analyze the error distribution pattern of the mechanism before processing and assembly.
normal distribution.When the expected value E=0 and the mean square error of the struts σ= 200 nm, a normal distribution of 200nm, the pose error of the secondary mirror platform is x, y, z ≤ ± 1μm, RX, RY, RZ ≤ ±1 ''.

Figure 2 .
Figure 2. The actual pose error distribution of the moving platform.