Structural parameter optimization method of Stewart platform

Abstract

The invention discloses a structural parameter optimization method of a Stewart platform, and belongs to the technical field of parallel robots. The method comprises the following steps: firstly, establishing a mathematical model of a Stewart platform, carrying out degree-of-freedom analysis on the platform, and establishing a kinetic equation based on a positive and inverse kinematics relationship and a Lagrange method; then determining an optimization target which covers a working space, rigidity, motion precision and stability; on this basis, a particle swarm optimization (PSO) algorithm is adopted to carry out global optimization on key parameters such as branch length, platform layout, driver position and the like, and a fitness function is utilized to comprehensively evaluate platform performance; and finally, inputting an optimization result into finite element analysis (FEA) and multi-body dynamics simulation (MBD) software for verification. Experimental results show that the working space of the Stewart platform can be remarkably expanded, the motion precision and rigidity of the Stewart platform are improved, stress distribution is improved, and the overall stability is enhanced. The method breaks through the limitation of traditional empirical design and single-target optimization, has the advantages of high optimization efficiency, high global search capability and high convergence speed, is suitable for scenes such as flight simulators, precision machining and spaceflight test platforms, and has wide engineering application value.

Description

Technical Field

[0001] This invention relates to the field of parallel robot technology, specifically to a method for optimizing the structural parameters of the Stewart platform. This method is applied to the design and optimization of the Stewart platform, aiming to improve the platform's motion performance, stability, and control accuracy by optimizing its structural parameters. Background Technology

[0002] The Stewart platform, also known as a six-degree-of-freedom parallel mechanism, is widely used in precision motion control systems, such as flight simulators, robot work platforms, and spacecraft test platforms. It controls the platform's pose changes through six independent actuators, exhibiting high rigidity and a large load capacity. However, the performance of the Stewart platform is limited by its various structural parameters, such as branch length, actuator position, and platform layout. These parameters directly affect the platform's workspace, accuracy, and stability.

[0003] Currently, optimization research on the Stewart platform mainly focuses on its geometry and control algorithms. However, existing optimization methods mostly emphasize one aspect and lack a systematic optimization design that comprehensively considers the overall performance of the platform. Furthermore, how to improve the platform's dynamic performance and reduce errors and vibrations by rationally optimizing structural parameters while ensuring platform stability remains a significant design challenge.

[0004] Traditional Stewart platform design typically relies on experience and experimentation, resulting in a lack of efficiency and precision in the optimization process. Therefore, there is an urgent need for an innovative structural parameter optimization method that can comprehensively consider platform performance and stability, achieving all-round optimized design of the Stewart platform and improving its reliability and efficiency in practical applications. Summary of the Invention

[0005] The purpose of this invention is to provide a method for optimizing the structural parameters of the Stewart platform. By combining mathematical modeling, particle swarm optimization (PSO) algorithm, finite element analysis (FEA) and multibody dynamics (MBD) simulation, the method can optimize the configuration of key parameters such as branch length, platform layout and actuator position, thereby improving the platform's workspace and rigidity.

[0006] To achieve the above objectives, the specific technical solution of the Stewart platform structural parameter optimization method of the present invention is as follows: A method for optimizing the structural parameters of the Stewart platform includes the following steps: Step 1: Establish a mathematical model of the Stewart platform, clarifying the platform's degrees of freedom, actuator positions, and constraints; by modeling the geometric parameters of each component of the platform, obtain the platform's dynamic equations. This model considers the platform's kinematic and dynamic characteristics and can effectively describe the platform's pose changes and force conditions. Step 2: Determine the optimization objectives; the optimization objectives include the platform's workspace, rigidity, motion accuracy, and stability; the platform's workspace is defined as the maximum range of motion that the platform can cover, rigidity refers to the platform's ability to deform under external forces, and stability considers whether the platform is prone to resonance or instability during motion. Step 3: Optimize structural parameters using the Particle Swarm Optimization (PSO) algorithm; considering the structural characteristics of the Stewart platform, the target parameters for optimization include: (1) Branch length: Optimize the length of the branches to maximize the platform’s workspace and ensure that the platform maintains sufficient rigidity under various loads; (2) Platform layout: Adjust the relative positions of the upper and lower platforms to improve motion accuracy and control accuracy; (3) Actuator position: Optimize the position and angle of the actuator to reduce energy consumption and motion error when the platform performs complex actions; The PSO algorithm simulates the process of a group of particles "flying" continuously in the search space. Guided by individual and global optimal solutions, it gradually adjusts platform parameters to find the optimal structural configuration. The algorithm has strong global search capabilities, avoids local optima problems, and ensures the reliability and global optimality of the optimization results. Step 4: Perform simulation analysis on the optimized structural parameters. Use finite element analysis (FEA) and multibody dynamics (MBD) simulation software to verify the motion accuracy, stability, and mechanical properties of the optimized Stewart platform under different working conditions, ensuring that the optimization results meet the design requirements; Step 5: Adjust the optimization strategy based on the simulation results. If the simulation results do not meet the requirements, further adjust the parameters of the PSO algorithm, including the number of particles and the maximum number of iterations, and perform optimization iterations until the optimal parameter configuration is obtained.

[0007] Further, in step 1, establishing the mathematical model of the Stewart platform mainly includes the degree-of-freedom analysis, kinematic modeling, and dynamic modeling of the Stewart platform. The degree-of-freedom analysis primarily focuses on the fact that the Stewart platform has six degrees of freedom, including three translational degrees of freedom (movement in the X, Y, and Z directions) and three rotational degrees of freedom (rotation around the X, Y, and Z axes). These degrees of freedom are connected to the base and upper platform via six branches of the platform. The kinematic modeling mainly includes forward kinematics and inverse kinematics. The forward kinematics model primarily calculates the position and attitude of the platform using the branch lengths, i.e., based on the known branch length L. i Calculate the position and attitude matrix of the platform (Transformation matrix between the base and the upper platform): It is a homogeneous transformation matrix describing the transformation from the base to the upper platform, representing the position and orientation of the platform, L. i It is the length of the i-th branch; this transformation matrix is ​​usually calculated through geometric relationships, taking into account the branch lengths of the platform and the geometric relationships between the connection points.

[0008] The task of the inverse kinematics model is to solve for the length L of the branches based on the target position and orientation of the platform. i : The formula describes the transformation through a known matrix. The length of each branch is determined based on the target position and orientation. The dynamic modeling includes dynamic equations and torque balance equations. The dynamic equations for the Stewart platform can be derived using the Lagrangian method, a common method for describing the dynamics of multibody systems. This method derives the platform's dynamic equations based on the platform's generalized coordinates and generalized velocities. The platform's generalized coordinates are q, and its generalized velocity is... The platform's Lagrange equation can be expressed as: Furthermore, the optimization process of the particle swarm optimization (PSO) algorithm in step 3 specifically includes the following steps: Step 3.1, Initialize the particle swarm: Set the initial position and velocity of the particle swarm according to the initial structural parameters of the Stewart platform. The particle position represents the structural parameters of the platform, and the particle velocity represents the magnitude of parameter change. During initialization, set an appropriate number of particles and search space to ensure the global search capability of the algorithm. Step 3.2, Fitness Function Calculation: Based on the optimization objectives of the platform's workspace, rigidity, motion accuracy, and stability, a fitness function is designed to evaluate the fitness of the particles. The fitness function comprehensively considers factors such as the platform's motion accuracy, rigidity, load capacity, and stability, and uses this as the basis for optimization. Step 3.3, Velocity and Position Update: Based on the current particle's fitness, the global optimum, and the individual optimum, update the velocity and position of each particle; the update formula is as follows: Among them, v i k Let x be the velocity of the i-th particle. i k p represents the position of the particle. best,i For the individual optimal solution of the particle, g best For the global optimal solution, w is the inertia weight, c1 and c2 are acceleration constants, and r1 and r2 are random numbers; Step 3.4, Iteration stopping condition: When the global optimal solution of the particle swarm reaches the set accuracy requirement or the maximum number of iterations, the optimization process stops and the optimized structural parameter configuration is output.

[0009] Furthermore, when performing simulation analysis using the finite element analysis (FEA) software in step 4, the platform's motion accuracy is evaluated by calculating the difference between the platform's motion trajectory error and the designed trajectory, and its stability is evaluated by the vibration amplitude and frequency response of the platform under different working conditions.

[0010] Furthermore, the optimization strategy adjustment in step 5 includes adjusting parameters such as the number of particles, maximum number of iterations, inertia weight, and acceleration constant of the PSO algorithm to adjust the search range and convergence speed of the algorithm, so as to achieve higher optimization accuracy and shorter optimization time.

[0011] Furthermore, the optimized structural parameters include specific values ​​for branch length, platform layout, and actuator position. Specifically, the optimized range for branch length is [500mm, 1500mm], the optimized range for platform layout is [1000mm, 2000mm], and the optimized range for actuator position is [±30°], ensuring the platform's motion stability and accuracy under heavy loads. The structural parameter optimization method for the Stewart platform of the present invention has the following advantages: The Stewart platform structural parameter optimization method proposed in this invention overcomes the limitations of existing technologies that rely on experience to select parameters and have a single optimization objective. By establishing a comprehensive mathematical model and combining it with the particle swarm optimization (PSO) algorithm, key parameters such as branch length, platform layout, and actuator position are globally optimized, thus simultaneously considering workspace, rigidity, motion accuracy, and stability during the design phase. Compared with traditional experience-based or single-objective optimization methods, this invention effectively avoids local optima problems, achieves more reasonable parameter configurations, and ensures the feasibility and reliability of the optimization results through finite element and multibody dynamics simulations.

[0012] Compared to existing optimization methods employing genetic algorithms or simple numerical iterations, this invention offers significant advantages in global search capability and convergence speed. The PSO algorithm has a simple structure and few parameters, enabling it to obtain the global optimum in a short time, effectively improving optimization efficiency. Furthermore, the optimization process of this invention is flexible, allowing for adjustment of algorithm parameters according to actual needs, achieving a balance between accuracy and efficiency. This method is not only applicable to the Stewart platform but can also be extended to the optimization of other parallel mechanism structures, possessing broad engineering application value and promotional significance. Attached Figure Description

[0013] Figure 1 This is a schematic diagram of the Stewart platform structure of the present invention; Figure 2 This is a mathematical schematic diagram of the Stewart platform of the present invention; Figure 3 This is a flowchart of the PSO algorithm optimization process of the present invention; Figure 4 This is a comparison diagram of the Stewart platform's workspace before and after optimization according to the present invention; Figure 5 This is a schematic diagram of the finite element simulation results of the optimized platform of the present invention. Detailed Implementation

[0014] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0015] Those skilled in the art will understand that although some embodiments herein include certain features included in other embodiments but not others, combinations of features from different embodiments are intended to be within the scope of the invention and form different embodiments. For example, in the claims, any of the claimed embodiments can be used in any combination.

[0016] The following is a reference to the appendix. Figure 1 To be continued Figure 5 This invention describes a method for optimizing the structural parameters of the Stewart platform.

[0017] Reference Figure 1 , Figure 2 In this embodiment, the Stewart platform is first analyzed for its degrees of freedom. The Stewart platform consists of an upper platform, a lower platform, and six independent branches. Each branch is connected to the upper and lower platforms via universal joints or ball joints. Therefore, this mechanism is a typical 6-SPS (ball joint-pussor-ball joint) parallel mechanism. The degree of freedom analysis results show that the platform has six degrees of freedom, including translational degrees of freedom along the X, Y, and Z directions and rotational degrees of freedom about the X, Y, and Z axes, enabling movement in any spatial orientation. The lower platform's fixed coordinate system is {O}. m The coordinate system of the upper platform is {O}. p The coordinates of the lower hinge point of the branch are B. i The coordinates of the upper hinge point are P. i The pose of the platform is determined by the translation vector t=[x,y,z]. T The rotation matrix R represents the branch length L. i It can be calculated using the following formula: Among them, L i Let B be the length of the i-th branch. i and P i Let L be the coordinates of the connection points of the branches on the lower and upper platforms, respectively. This formula describes the process of solving for the branch lengths from a known pose. When the target pose (t, R) is known, the length L of each branch can be directly solved using this formula. i ,Right now: This process provides the basis for optimization and control, enabling the platform to move along the desired trajectory.

[0018] The dynamic model was established using the Lagrange method, yielding the platform's dynamic equations: Where L is the system Lagrangian function, and q is the generalized coordinate. Let Q represent the generalized velocity and Q represent the generalized force. Through the above modeling process, the dynamic differential equations of the platform can be obtained, allowing for the analysis of its force characteristics, dynamic response, and stability. This model not only provides a theoretical basis for subsequent structural parameter optimization but also allows for integration with simulation analysis (finite element analysis and multibody dynamics analysis) to verify the optimization effect.

[0019] Based on the established kinematic and dynamic model of the Stewart platform, a structural parameter optimization method based on particle swarm optimization (PSO) is further proposed. The optimization objectives, tailored to the application requirements of the Stewart platform, include: Maximizing workspace: Ensuring the platform can cover the largest possible pose range; Rigidity enhancement: Reduces the deformation of the platform under load; Motion accuracy: Reduce motion errors and improve trajectory tracking accuracy; Enhanced stability: Avoids resonance and structural instability.

[0020] Therefore, a comprehensive fitness function is designed: Among them, w1, w2, w3, and w4 are weighting coefficients used to balance the influence of different optimization objectives.

[0021] Reference Figure 3 The PSO optimization process includes the following steps: (1) Initialize the population: Set the particle swarm based on the branch length, driver position, and platform layout range, and the particle position vector x i Represents a set of structural parameters, velocity vector v i This indicates the magnitude of the parameter change.

[0022] (2) Calculate the fitness function: Substitute each particle into the Stewart platform model and calculate its fitness value F(x). i This is used to evaluate the performance of this parameter combination.

[0023] (3) Update individual optimal and global optimal: record the historical optimal solution pbest and the global optimal solution gbest for each particle in the particle swarm.

[0024] (4) Velocity and position updates: Where w is the inertia weight, c1 and c2 are learning factors, and r1 and r2 are random numbers.

[0025] (5) Termination condition judgment: If the global optimal solution meets the accuracy requirement or reaches the maximum number of iterations, then stop the optimization and output the optimal structure parameters; otherwise, return to step 2 to continue the iteration.

[0026] Initialize the particle swarm with 50 particles and a maximum of 200 iterations. Through continuous iteration, obtain the globally optimal structure parameter configuration.

[0027] The optimal structural parameter configuration output by the PSO algorithm was input into finite element analysis (FEA) and multibody dynamics simulation (MBD) software for verification. Motion accuracy verification: Compare the error distribution between the platform trajectory and the expected trajectory; Stability verification: Calculate the frequency response under different operating conditions to confirm that the system does not resonate; Rigid verification: Analyze branched deformation under heavy loads to ensure that the platform meets engineering requirements.

[0028] Reference Figure 4 This invention demonstrates a significant difference in the workspace before and after optimizing the structural parameters of the Stewart platform using the Particle Swarm Optimization (PSO) algorithm. Before optimization, the platform's workspace was unevenly distributed, with noticeable contraction in the boundary regions, restricting the platform's movement in certain directions. After optimization, the platform's effective workspace significantly expanded, with a more balanced overall distribution and a more regular hexagonal boundary shape. This indicates that the optimized design can significantly improve the platform's range of motion and flexibility, thereby meeting more complex operational requirements.

[0029] Reference Figure 5 This invention employs finite element structural simulation analysis on the optimized Stewart platform to verify its stability and reliability under stress conditions. Simulation results show that under typical loads, the platform exhibits uniform stress distribution with no stress concentration, maximum displacement is controlled within a reasonable range, and overall stiffness is guaranteed. Compared to the original structure, the optimized platform demonstrates better load-bearing capacity and deformation resistance under the same load conditions, further validating the feasibility and effectiveness of the optimization design method of this invention in practical applications.

[0030] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively describe all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A method for optimizing the structural parameters of a Stewart platform, characterized in that, Specifically, the following steps are included: Step 1: Establish a mathematical model of the Stewart platform, clarifying the platform's degrees of freedom, actuator positions, and constraints; by modeling the geometric parameters of each component of the platform, obtain the platform's dynamic equations. This model considers the platform's kinematic and dynamic characteristics and can effectively describe the platform's pose changes and force conditions. Step 2: Determine the optimization objectives; the optimization objectives include the platform's workspace, rigidity, motion accuracy, and stability; the platform's workspace is defined as the maximum range of motion that the platform can cover, rigidity refers to the platform's ability to deform under external forces, and stability considers whether the platform is prone to resonance or instability during motion. Step 3: Optimize structural parameters using the Particle Swarm Optimization (PSO) algorithm; considering the structural characteristics of the Stewart platform, the target parameters for optimization include: (1) Branch length: Optimize the length of the branches to maximize the platform’s workspace and ensure that the platform maintains sufficient rigidity under various loads; (2) Platform layout: Adjust the relative positions of the upper and lower platforms to improve motion accuracy and control accuracy; (3) Actuator position: Optimize the position and angle of the actuator to reduce energy consumption and motion error when the platform performs complex actions; The PSO algorithm simulates the process of a group of particles "flying" continuously in the search space. Guided by individual and global optimal solutions, it gradually adjusts platform parameters to find the optimal structural configuration. The algorithm has strong global search capabilities, avoids local optima problems, and ensures the reliability and global optimality of the optimization results. Step 4: Perform simulation analysis on the optimized structural parameters; use finite element analysis (FEA) and multibody dynamics (MBD) simulation software to verify the motion accuracy, stability and mechanical properties of the optimized Stewart platform under different working conditions, and ensure that the optimization results meet the design requirements; Step 5: Adjust the optimization strategy based on the simulation results; if the simulation results do not meet the requirements, further adjust the parameters of the PSO algorithm, such as the number of particles and the maximum number of iterations, and perform optimization iterations until the optimal parameter configuration is obtained.

2. The method for optimizing the structural parameters of a Stewart platform according to claim 1, characterized in that, Step 1 establishes the mathematical model of the Stewart platform, mainly including the degree-of-freedom analysis, kinematic modeling, and dynamic modeling of the Stewart platform. The degree-of-freedom analysis focuses on the fact that the Stewart platform has six degrees of freedom: three translational degrees of freedom (movement in the X, Y, and Z directions) and three rotational degrees of freedom (rotation about the X, Y, and Z axes). These degrees of freedom are connected to the base and upper platform via six branches of the platform. The kinematic modeling mainly includes forward kinematics and inverse kinematics. The forward kinematics model primarily calculates the position and attitude of the platform using the branch lengths, i.e., based on the known branch length L. i Calculate the position and attitude matrix of the platform (Transformation matrix between the base and the upper platform): It is a homogeneous transformation matrix describing the transformation from the base to the upper platform, representing the position and orientation of the platform, L. i It is the length of the i-th branch; this transformation matrix is ​​usually calculated through geometric relationships, taking into account the branch lengths of the platform and the geometric relationships between each connection point; The task of the inverse kinematics model is to solve for the length L of the branches based on the target position and orientation of the platform. i : The formula describes the transformation through a known matrix. The length of each branch is determined based on the target position and orientation. The dynamic modeling includes dynamic equations and torque balance equations. For the Stewart platform, the dynamic model can employ the Lagrangian method, a common method for describing the dynamics of multibody systems. This method derives the platform's Lagrangian equations based on the platform's generalized coordinates and generalized velocities. The platform's generalized coordinates are q, and its generalized velocity is... .

3. The method for optimizing the structural parameters of a Stewart platform according to claim 1, characterized in that, The optimization process of the Particle Swarm Optimization (PSO) algorithm in step 3 specifically includes the following steps: Step 3.1, Initialize the particle swarm: Set the initial position and velocity of the particle swarm according to the initial structural parameters of the Stewart platform. The particle position represents the structural parameters of the platform, and the particle velocity represents the magnitude of parameter change. During initialization, set an appropriate number of particles and search space to ensure the global search capability of the algorithm. Step 3.2, Fitness Function Calculation: Based on the optimization objectives of the platform's workspace, rigidity, motion accuracy, and stability, a fitness function is designed to evaluate the fitness of the particles. The fitness function comprehensively considers factors such as the platform's motion accuracy, rigidity, load capacity, and stability, and uses this as the basis for optimization. Step 3.3, Velocity and Position Update: Based on the current particle's fitness, the global optimum, and the individual optimum, update the velocity and position of each particle; the update formula is as follows: Among them, v i k Let x be the velocity of the i-th particle. i k p represents the position of the particle. best,i For the individual optimal solution of the particle, g best For the global optimal solution, w is the inertia weight, c1 and c2 are acceleration constants, and r1 and r2 are random numbers; Step 3.4, Iteration stopping condition: When the global optimal solution of the particle swarm reaches the set accuracy requirement or the maximum number of iterations, the optimization process stops and the optimized structural parameter configuration is output.

4. The method for optimizing the structural parameters of a Stewart platform according to claim 1, characterized in that, When performing simulation analysis using the finite element analysis (FEA) software in step 4, the platform's motion accuracy is evaluated by calculating the difference between the platform's motion trajectory error and the designed trajectory, and its stability is evaluated by the vibration amplitude and frequency response of the platform under different working conditions.

5. The method for optimizing the structural parameters of a Stewart platform according to claim 1, characterized in that, The optimization strategy adjustment in step 5 includes adjusting parameters such as the number of particles, maximum number of iterations, inertia weight, and acceleration constant of the PSO algorithm to adjust the search range and convergence speed of the algorithm, so as to achieve higher optimization accuracy and shorter optimization time.

6. The method for optimizing the structural parameters of a Stewart platform according to claim 1, characterized in that, The optimized structural parameters include specific values ​​for branch length, platform layout, and actuator position. The optimized range for branch length is [500mm, 1500mm], the optimized range for platform layout is [1000mm, 2000mm], and the optimized range for actuator position is [±30°], ensuring the stability and accuracy of the platform's motion under heavy loads.

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