Solution method for worst force posture parameters of low-speed moving stewart platform

Abstract

The application discloses a method for solving the worst force attitude parameter of a low-speed moving Stewart platform, and solves the problem that in the prior art, only the force of a connecting rod of the Stewart platform in a specific attitude is analyzed, the maximum axial force of the connecting rod in all possible attitudes cannot be accurately solved, and when the constraint condition is relatively complex, the worst force attitude is generally difficult to accurately find out by using a traditional method; the method is high in efficiency and accuracy in predicting the worst force attitude of the connecting rod of the Stewart platform under a complex constraint condition, and a linear relationship between the axial force of the connecting rod and the load weight is given, which has important guiding significance for the strength design of the Stewart platform, the design of a driving device and the safe use of the Stewart platform.

Description

Technical Field

[0001] This invention relates to a method for calculating parameters of parallel mechanisms, specifically a method for solving the worst-case force posture parameters of a Stewart platform moving at low speed. Background Technology

[0002] The Stewart platform is a six-degree-of-freedom spatial parallel mechanism consisting of a moving platform, a stationary platform, connecting rods, and hinges. It has advantages such as high load-bearing capacity, good rigidity, no cumulative error, and high precision. It is now being used more and more widely in fields such as robotics, parallel machine tools, driving simulators, and optical alignment.

[0003] In the field of optical alignment, the Stewart platform, which moves at low speed, is often used to adjust the attitude of the load. However, the axial force of the Stewart link often varies significantly under different attitudes. Therefore, it is of great significance to find the worst stress attitude of the Stewart platform and establish the correspondence between the load and the axial force of the link under this attitude for the design and use of the Stewart platform.

[0004] While scholars have conducted extensive research on the forces acting on the Stewart platform, these studies generally only analyze the forces acting on the connecting rods under specific postures, failing to accurately determine the maximum axial force of the connecting rods under all possible postures. Some scholars have used the connecting rod length as a constraint on the Stewart platform's posture, evaluating the design of the oil source pressure and hydraulic cylinders by listing various worst-case postures. However, this method relies on experience to pre-determine the worst-case postures of the Stewart platform. When the constraints are complex, such as constraining the range of motion of the moving platform, traditional methods generally struggle to accurately identify the worst-case posture. Summary of the Invention

[0005] The purpose of this invention is to address the problem that existing technologies only analyze the forces acting on the links of a Stewart platform under specific postures, and cannot accurately solve for the maximum axial force of the links under all possible postures and complex constraints, making it difficult to accurately find the worst-case stress posture using traditional methods; and to provide a method for solving the worst-case stress posture parameters of a Stewart platform moving at low speeds.

[0006] To achieve the above objectives, the present invention adopts the following technical solution:

[0007] A method for solving the worst-case force attitude parameters of a Stewart platform moving at low speeds, characterized by the following steps:

[0008] Step 1: Based on the kinematic model of the Stewart platform, establish a model relating the attitude parameters of the Stewart platform to the link length;

[0009] Step 2: Analyze the forces acting on the Stewart platform based on the relationship model between the Stewart platform attitude parameters and the link length in Step 1, and solve for the axial force of the link under a fixed attitude.

[0010] Step 3: Based on the axial force of the link under fixed attitude in Step 2, construct the objective function with the maximum axial force of the link under all fixed attitudes as the objective.

[0011] Step 4: Establish constraints for the Stewart platform's pose;

[0012] Step 5: Using the particle swarm optimization algorithm, when solving the constraint conditions that satisfy the Stewart platform attitude in Step 4, the maximum value of the objective function constructed in Step 3 is obtained, and the worst-case force attitude parameters of the Stewart platform are determined based on the maximum value of the objective function.

[0013] Furthermore, the specific process of step 1 is as follows:

[0014] Step 1.1: Define the moving coordinate system O′X′Y′Z′ of the Stewart platform as a coordinate system fixed to the moving platform, with the centroid of the moving platform as the origin of the moving coordinate system. Define the static coordinate system OXYZ as a coordinate system fixed to the static platform. Based on the Stewart platform attitude parameters (x, y, z, α, β, γ), where x, y, and z are the coordinates of the origin of the moving coordinate system on the X, Y, and Z axes of the static coordinate system, and α, β, and γ are the components of the attitude angles of the moving platform relative to the static platform in the OYZ, OXZ, and OXY planes, respectively, the matrix equation for the transformation from moving to static coordinates of the Stewart platform is:

[0015]

[0016] In the formula, c is the abbreviation for cos and s is the abbreviation for sin.

[0017] Step 1.2: Taking each hinge as a hinge fulcrum, the homogeneous coordinate matrix of each hinge fulcrum of the moving platform in the moving coordinate system of the Stewart platform is:

[0018]

[0019] In the formula, θ′ is the angle formed by the lines connecting two adjacent hinge points on the moving platform to the origin of the moving coordinate system, and R′ represents the radius of the moving platform;

[0020] Step 1.3: From equations (1) and (2), the homogeneous coordinate matrix of each hinge support point of the moving platform in the Stewart platform in the static coordinate system is B = AB′;

[0021] Step 1.4: Define the initial height H of the origin of the moving coordinate system in the Stewart platform along the Z-axis in the static coordinate system. To simplify the calculation, let the homogeneous coordinate matrix of each hinge point of the static platform in the static coordinate system be:

[0022]

[0023] In the formula, θ is the angle between two adjacent hinge points on the static platform and the center of the circle, and R represents the radius of the static platform;

[0024] Based on equations (1)-(3), the length L of the k-th link is established. k The relationship model between (k=1,2,…6) and attitude parameters:

[0025]

[0026] In the formula, B 1k B 2k B 3k These represent the values ​​in the k-th column of the 1st, 2nd, and 3rd rows of matrix B, respectively; C 1k C 2k C 3k These represent the values ​​in the k-th column of the 1st, 2nd, and 3rd rows of matrix C, respectively.

[0027] Furthermore, the specific process of step 2 is as follows:

[0028] Step 2.1: Define the load centroid O2 on the moving platform as having an abscissa of q1 and a ordinate of q2 in the moving coordinate system. The initial height difference between the moving and stationary platforms is h. The homogeneous coordinate matrix of the load centroid O2 on the moving platform in the moving coordinate system is:

[0029] E′=[q1 q2 h 1] T (5)

[0030] The homogeneous coordinate matrix of the load centroid O2 in the static coordinate system is:

[0031]

[0032] Step 2.2: Define n1, n2, and n3 as unit vectors along the X-axis, Y-axis, and Z-axis, respectively, and F k Let f be the force exerted by the k-th link on the moving platform. k Direction vector, l k Let be the direction vector of the spatial line containing the k-th link (k = 1, 2, ..., 6), m be the mass of the load, g be the gravitational acceleration of the load, and a be the inertial acceleration of the load. Then we have:

[0033]

[0034]

[0035] For a Stewart platform moving at low speed, the load inertial force is negligible. Therefore, from the force balance of the moving platform in the X-axis direction, we get:

[0036]

[0037] Substituting equations (7) and (8) into equation (9), we get:

[0038]

[0039] In the above formula, if the load on the k-th link is a tensile load, then F k With l k Opposite directions, f·l k =-1; If the load on the k-th link is a compressive load, then F k With l k Same direction, f·l k =1;

[0040] Step 2.3: Using the same method as in Step 2.2, by balancing the forces on the moving platform in the Y and Z directions respectively, we obtain:

[0041]

[0042]

[0043] Step 2.4, the component of the force exerted by the k-th link on the moving platform in the XOY plane is:

[0044]

[0045] The coordinates of the upper and lower hinge points of the k-th link projected onto the XOY plane are (B... 1k B 2k ,0) and (C 1k C 2k The coordinates of the projection of the origin O′ of the moving platform onto the XOY plane are (x, y). According to the formula for the distance from a point to a line, the distance from the projection of point O′ onto the projection line of the kth link in the XOY plane is:

[0046]

[0047] The direction vector of the normal to the projection line of link k in the XOY plane is:

[0048] n kxy =(C 2k -B 2k )n1+(B 1k -C1k )n2 (15)

[0049] The radial vector of the projection of point O′ onto the load centroid O2 in the XOY plane is:

[0050]

[0051] Based on the torque balance of the moving platform about the Z-axis, we can obtain:

[0052]

[0053] Step 2.5: Using the same method as in Step 2.4, based on the torque balance of the moving platform around the X and Y axes, we can obtain:

[0054]

[0055]

[0056] Equations (10)-(12) and (17)-(19) together form the mechanical equilibrium equations of the moving platform. Solving these equations simultaneously yields the force F exerted by the k-th link on the moving platform. k Let the axial force of the connecting rod be N. k Then N k =-F k .

[0057] Furthermore, in step 3, the objective function expression is:

[0058] max |N|=max{|N1| |N2|…|N6|} (20)

[0059] In the formula, max|N| represents the maximum axial force of the link under all attitudes, and |N1|~|N6| represent the axial force of each link under a certain attitude.

[0060] Furthermore, in step 4, the constraints include the range of values ​​for the worst-case force attitude parameters of the Stewart platform and the constraint on the link length through a penalty function.

[0061] Furthermore, the constraint on the link length using the penalty function specifically refers to:

[0062]

[0063] In the formula, |N max | represents the nominal maximum axial force of the six links under the current attitude, | N′ max | represents the true maximum axial force of the six links under the current attitude, L k L represents the length of the k-th link. max L represents the maximum allowable length of the link. minThis indicates the minimum allowable length of the link.

[0064] Furthermore, step 5 specifically involves the following process:

[0065] Step 5.1: Set the maximum number of generations T′, construct a particle swarm including multiple subpopulations, each subpopulation includes multiple particles, and the position of each particle corresponds to a set of Stewart platform attitude parameters, all of which satisfy the constraints in step 4. Initialize the parameters and particles in the particle swarm.

[0066] Step 5.2: Evolve the subpopulation based on adaptive inertia weights and solve the objective function to obtain the objective function value. By comparing it with the objective function value before evolution, retain the larger objective function value and the corresponding optimal solution.

[0067] Step 5.3: Randomly sort the order of each subpopulation using a random function, and replace the worst individual of the previous subpopulation with the best individual of the next adjacent subpopulation according to the new order, thus completing the interactive evolution between subpopulations.

[0068] Step 5.4: Repeat steps 5.2 and 5.3 until the maximum evolution generation T is reached. The larger output objective function value is the maximum axial force of the link under all allowed attitudes. The optimal solution is the particle position information. Based on the particle position information, the worst-case force attitude parameters of the Stewart platform are obtained.

[0069] Furthermore, in step 5.2, the relationship between inertia weight and evolutionary generation is as follows:

[0070]

[0071] In the formula, w t w represents the inertial weight of generation t. max w represents the initially defined maximum inertia weight. min T represents the initial minimum inertia weight, t represents the number of generations, and T′ represents the maximum number of generations.

[0072] The beneficial effects of this invention are:

[0073] This invention provides a method for solving the worst-case stress posture parameters of a Stewart platform moving at low speeds. It efficiently and accurately predicts the worst-case stress posture of the Stewart platform links under complex constraints and provides a linear relationship between the link axial force and the load weight. This method has important guiding significance for the strength design, drive device design, and safe use of the Stewart platform. Attached Figure Description

[0074] Figure 1This is a schematic diagram of the Stewart platform in an embodiment of the method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to the present invention.

[0075] Figure 2 This is a schematic diagram of the Stewart platform in the coordinate system in an embodiment of the method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to the present invention.

[0076] Figure 3 This is an evolution curve of the maximum axial force of the connecting rod in an embodiment of the method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to the present invention.

[0077] In the diagram, 1. Load, 2. Moving platform, 3. Hinge, 4. Linkage rod, 5. Static platform. Detailed Implementation

[0078] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and 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.

[0079] In this embodiment, the Stewart platform structure is as follows: Figure 1 As shown, the system includes a load 1, a moving platform 2, a hinge 3, connecting rods 4, and a stationary platform 5. There are six connecting rods 4. The dimensional parameters of the Stewart platform are R = 350 mm, θ = 15°, R′ = 250 mm, θ′ = 60°, H = 560 mm, the load mass m = 100 kg, and the initial coordinates of the load's center of mass in the moving coordinate system are (0, 0, 80 mm).

[0080] This embodiment presents a method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed, comprising the following steps:

[0081] Step 1: Based on the Stewart platform kinematic model, establish a model relating the Stewart platform attitude parameters to the link lengths; the specific process is as follows:

[0082] Define the moving coordinate system of the Stewart platform O Let 'X', 'Y', and 'Z' be a coordinate system fixed to the moving platform, with the centroid of the moving platform being the origin of the moving coordinate system, and the static coordinate system being... OXYZFor a coordinate system fixed to the static platform, based on the Stewart platform attitude parameters (x, y, z, α, β, γ), where x, y, and z are the coordinates of the origin of the moving coordinate system on the X, Y, and Z axes of the static coordinate system, and α, β, and γ are the components of the attitude angles of the moving platform relative to the static platform in the OYZ, OXZ, and OXY planes, respectively, the matrix equation for the transformation from moving to static coordinates of the Stewart platform is:

[0083]

[0084] In the formula, c is the abbreviation for cos and s is the abbreviation for sin.

[0085] Taking each hinge as a hinge support point, the homogeneous coordinate matrix of each hinge support point of the moving platform in the Stewart platform in the moving coordinate system is:

[0086]

[0087] In the formula, θ′ is the angle formed by the lines connecting two adjacent hinge points on the moving platform to the origin of the moving coordinate system, and R′ represents the radius of the moving platform.

[0088] From equations (1) and (2), the homogeneous coordinate matrix of each hinge support point of the moving platform in the Stewart platform in the static coordinate system is B = AB′.

[0089] Considering that the initial height of the origin of the moving coordinate system in the Stewart platform along the Z-axis in the static coordinate system is H, to simplify the calculation, let the homogeneous coordinate matrix of each hinge point of the static platform in the static coordinate system be:

[0090]

[0091] In the formula, θ is the angle between two adjacent hinge points on the static platform and the center of the circle, and R represents the radius of the static platform;

[0092] Based on equations (1)-(3), the length L of the k-th link is established. k The relationship model between (k=1,2,…6) and attitude parameters:

[0093]

[0094] In the formula, B 1k B 2k B 3k These represent the values ​​in the k-th column of the 1st, 2nd, and 3rd rows of matrix B, respectively; C 1k C 2k C 3k These represent the values ​​in the k-th column of the 1st, 2nd, and 3rd rows of matrix C, respectively.

[0095] Step 2: Analyze the forces acting on the Stewart platform and solve for the axial force of the connecting rod under a fixed attitude; the specific process is as follows:

[0096] The homogeneous coordinate matrix of the load centroid O2 located on the moving platform in the moving coordinate system is:

[0097] E′=[q1 q2 h 1] T (5)

[0098] The homogeneous coordinate matrix of the load centroid O2 in the static coordinate system is:

[0099]

[0100] Define n1, n2, and n3 as unit vectors along the X, Y, and Z axes, respectively, and F k Let f be the force exerted by the k-th link on the moving platform. k Direction vector, l k Let be the direction vector of the spatial line containing the k-th link (k = 1, 2, ..., 6), m be the mass of the load, g be the gravitational acceleration of the load, and a be the inertial acceleration of the load. Then we have:

[0101]

[0102]

[0103] For a Stewart platform moving at low speed, the load inertial force is negligible. Therefore, from the force balance of the moving platform in the X direction, we get:

[0104]

[0105] Substituting equations (7) and (8) into equation (9), we get:

[0106]

[0107] In the above formula, if the load on the k-th link is a tensile load, then F k With l k Opposite directions, f·l k =-1; If the load on the k-th link is a compressive load, then F k With l k Same direction, f·l k =1;

[0108] Similarly, from the fact that the moving platform is in force equilibrium in the Y and Z directions respectively, we get:

[0109]

[0110]

[0111] The component of the force exerted by the k-th link on the moving platform in the XOY plane is:

[0112]

[0113] The coordinates of the upper and lower hinge points of the k-th link projected onto the XOY plane are (B... 1k B 2k ,0) and (C 1k C 2k The coordinates of the projection of the origin O′ of the moving platform onto the XOY plane are (x, y). According to the formula for the distance from a point to a line, the distance from the projection of point O′ onto the projection line of the kth link in the XOY plane is:

[0114]

[0115] The direction vector of the normal to the projection line of link k in the XOY plane is:

[0116] n kxy =(C 2k -B 2k )n1+(B 1k -C 1k )n2(15)

[0117] The radial vector of the projection of point O′ onto the load centroid O2 in the XOY plane is:

[0118]

[0119] Based on the torque balance of the moving platform about the Z-axis, we can obtain:

[0120]

[0121] Similarly, based on the torque balance of the moving platform around the X and Y axes, we can obtain:

[0122]

[0123]

[0124] Equations (10)-(12) and (17)-(19) together form the mechanical equilibrium equations of the moving platform. Solving these equations simultaneously yields the force F exerted by the k-th link on the moving platform. k Let the axial force of the connecting rod be N. k Then N k =-F k .

[0125] Step 3: Construct an objective function with the maximum axial force of the link under all attitudes as the objective; the expression of the objective function is:

[0126] max|N|=max{|N1||N2|…|N6|} (20)

[0127] In the formula, max|N| represents the maximum axial force of the link under all attitudes, and |N1|~|N6| represent the axial force of each link under a certain attitude.

[0128] Step 4: Establish the constraints on the attitude of the Stewart platform; the constraints include the range of values ​​for the worst-case stress attitude parameters of the Stewart platform and the constraint on the link length through the penalty function.

[0129] In this embodiment, the worst-case stress attitude parameters of the Stewart platform are in the following ranges: |x|≤250mm, |y|≤250mm, |z|≤250mm.

[0130] The constraint on the link length using the penalty function is specifically as follows:

[0131]

[0132] In the formula, |N max | represents the nominal maximum axial force of the six links under the current attitude, | N′ max | represents the true maximum axial force of the six links under the current attitude, L k L represents the length of the k-th link. max L represents the maximum allowable length of the link. min This indicates the minimum allowable length of the link.

[0133] In this embodiment, L max =680mm, L min =550mm.

[0134] Step 5: Use the sampling particle swarm optimization algorithm to find the maximum value of the objective function that satisfies the attitude constraints of the Stewart platform, and determine the worst-case force attitude parameters of the Stewart platform based on the maximum value of the objective function. The specific process is as follows:

[0135] Step 5.1: Set the maximum number of generations T′, construct a particle swarm including multiple subpopulations, each subpopulation includes multiple particles, and the position of each particle corresponds to a set of Stewart platform attitude parameters, all of which satisfy the constraints in step 4. Initialize the parameters and particles in the particle swarm.

[0136] In this embodiment, the number of subpopulations is I = 5, the number of particles in each subpopulation is J = 20, the maximum number of generations is T = 300, and the initial inertia weight is w. max =0.9, minimum inertia weight w min=0.4, self-learning factor c1=2, social learning factor c2=2, the criterion for getting trapped in a local optimum is ΔT=10 and When the change in the optimal solution of a certain exploratory population over consecutive generations ΔT is less than ΔN, it is considered that the subpopulation may be trapped in a local optimum. The significance of the value of ΔN is that in the early stages of evolution, when t is small, the search results are far from the true optimal solution. To avoid the exploratory population wasting too much time on local optima, a larger value of ΔN is assigned to help it escape local optima as quickly as possible. As evolution progresses, each particle generally gets closer to the global optimum, and the value of ΔN decreases with the number of generations t to slow down the search and achieve a more refined search.

[0137] Particle initialization: Within the constraints, 100 particles are randomly generated. The position of each particle satisfies the constraints in step four. The initial rate of change of translational attitude is a random number between -25 and 25, and the initial rate of change of angular attitude is a random number between -0.05 and 0.05.

[0138] Step 5.2: Evolve the subpopulation based on adaptive inertia weights and solve the objective function to obtain the objective function value. By comparing it with the objective function value before evolution, retain the larger objective function value and the corresponding optimal solution.

[0139] First, the single evolutionary population is divided into multiple subpopulations, each with the same initial inertia weight. After the first generation of evolution is completed, the optimal solutions of each subpopulation at the current stage are compared. The subpopulation with the largest optimal solution is designated as the development subpopulation, and the remaining subpopulations are designated as the exploration subpopulation. The inertia weight of the development subpopulation is changed according to equation (22):

[0140]

[0141] In the formula, w t w represents the inertial weight of generation t. max w represents the initially defined maximum inertia weight. min T represents the initial minimum inertia weight, t represents the number of generations, and T′ represents the maximum number of generations.

[0142] In this embodiment, after the first generation of evolution is completed, the optimal solutions of each subpopulation at the current stage are compared. The subpopulation with the largest optimal solution is designated as the development subpopulation, and the remaining subpopulations are designated as the exploration subpopulations. The inertia weight of the development subpopulation changes according to equation (22), and the inertia weight decays rapidly in the early stage of evolution. The exploration subpopulation has good global optimization ability. Its inertia weight is first changed according to equation (22). When an exploration subpopulation gets stuck in a local optimum, it is given a larger inertia weight to escape the local optimum. Specifically, the inertia weight of the subpopulation is reset to w. maxThe evolutionary generation t is reset to zero, and its inertial weight is then decayed again according to equation (22). After each generation of evolution, the actual evolutionary generation t = t + 1 of the particle swarm is used to compare the optimal solutions of each sub-swarm and to re-divide the development and exploration populations.

[0143] Based on the results of each generation of evolution, the population is divided into an exploration population and an exploratory population. The exploration population has a smaller inertia weight, which is beneficial for local fine-grained search; the exploratory population has a larger inertia weight, which is beneficial for global search.

[0144] Step 5.3: Randomly sort the order of each subpopulation using a random function, and replace the worst individual of the previous subpopulation with the best individual of the next adjacent subpopulation according to the new order, thus completing the interactive evolution between subpopulations.

[0145] Step 5.4: Repeat steps 5.2 and 5.3 until the maximum evolution generation T is reached. The larger output objective function value is the maximum axial force of the link under all allowed attitudes. The optimal solution is the particle position information. Based on the particle position information, the worst-case force attitude parameters of the Stewart platform are obtained.

[0146] This embodiment repeated four sets of calculations to obtain the evolution curve of the maximum axial force of the connecting rod, as shown below. Figure 3 As shown, after different evolutionary paths, all sets of calculations converge around 1.756 kN, so the maximum axial force of the link under the current constraint is 1.756 kN. The attitude parameters of the Stewart platform at this time are shown in Table 1.

[0147] Table 1

[0148] α / rad β / rad γ / rad x / mm y / mm z / mm Group 1 0.110 0.220 -0.524 -227.7 -187.7 -70.1 Group 2 -0.092 0.222 0.524 -222.2 191.3 -68.6 Group 3 -0.119 0.219 0.524 -230.7 185.6 -70.8 Group 4 0.103 0.221 -0.524 -225.6 -189.4 -69.6

[0149] Comparison shows that the attitudes of groups 1 and 4 are almost identical. Averaging the parameters of the two groups yields attitude I as (0.107, 0.220, -0.524, -226.6, -188.5, -69.9); the attitudes of groups 2 and 3 are almost identical. Averaging the parameters of the two groups yields attitude II as (-0.105, 0.220, 0.524, -226.4, 188.4, -69.7). Attitudes I and II are two attitudes symmetric about the XOZ plane, and both attitudes represent the worst-case stress attitudes of the Stewart platform under the current constraints. When solving using the method of this invention, one of these attitudes can be randomly searched each time.

[0150] The above description is merely a specific embodiment of the present invention and a comparison of the effects of the specific embodiments with relevant comparative examples. However, the scope of protection of the present invention is not limited thereto. Any changes or substitutions within the technical scope disclosed in the present invention should be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed, characterized in that, Includes the following steps: Step 1: Based on the kinematic model of the Stewart platform, establish a model relating the attitude parameters of the Stewart platform to the link length; Step 2: Analyze the forces acting on the Stewart platform based on the relationship model between the Stewart platform attitude parameters and the link length in Step 1, and solve for the axial force of the link under a fixed attitude. Step 3: Based on the axial force of the link under fixed attitude in Step 2, construct the objective function with the maximum axial force of the link under all fixed attitudes as the objective. Step 4: Establish constraints for the Stewart platform's pose; Step 5: Using the particle swarm optimization algorithm, when solving the constraint conditions that satisfy the Stewart platform attitude in Step 4, the maximum value of the objective function constructed in Step 3 is obtained, and the worst-case force attitude parameters of the Stewart platform are determined based on the maximum value of the objective function.

2. The method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to claim 1, characterized in that, Step 1 is as follows: Step 1.1: Define the moving coordinate system of the Stewart platform. O′X′Y′Z′ Let OXYZ be a coordinate system fixed to the moving platform, with the centroid of the moving platform as the origin of the moving coordinate system. The static coordinate system is fixed to the static platform. Based on the Stewart platform attitude parameters (x, y, z, α, β, γ), where x, y, and z are the coordinates of the origin of the moving coordinate system on the X, Y, and Z axes of the static coordinate system, and α, β, and γ are the components of the attitude angles of the moving platform relative to the static platform in the OYZ, OXZ, and OXY planes, respectively, the matrix equation for the transformation from moving to static coordinates of the Stewart platform is: In the formula, c is the abbreviation for cos and s is the abbreviation for sin. Step 1.2: Taking each hinge as a hinge fulcrum, the homogeneous coordinate matrix of each hinge fulcrum of the moving platform in the moving coordinate system of the Stewart platform is: In the formula, θ′ is the angle formed by the lines connecting two adjacent hinge points on the moving platform to the origin of the moving coordinate system, and R′ represents the radius of the moving platform; Step 1.3: From equations (1) and (2), the homogeneous coordinate matrix of each hinge support point of the moving platform in the Stewart platform in the static coordinate system is: B = AB′; Step 1.4: Define the initial height H of the origin of the moving coordinate system in the Stewart platform along the Z-axis in the static coordinate system. To simplify the calculation, let the homogeneous coordinate matrix of each hinge point of the static platform in the static coordinate system be: In the formula, θ is the angle between two adjacent hinge points on the static platform and the center of the circle, and R represents the radius of the static platform; Step 1.5: Based on equations (1)-(3), establish the length L of the k-th link. k The relationship model between (k=1,2,…6) and attitude parameters: In the formula, B 1k B 2k B 3k These represent the values ​​in the k-th column of the 1st, 2nd, and 3rd rows of matrix B, respectively; C 1k C 2k C 3k These represent the values ​​in the k-th column of the 1st, 2nd, and 3rd rows of matrix C, respectively.

3. The method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to claim 2, characterized in that, Step 2 is as follows: Step 2.1: Define the load centroid O2 on the moving platform as having an abscissa of q1 and a ordinate of q2 in the moving coordinate system. The initial height difference between the moving and stationary platforms is h. The homogeneous coordinate matrix of the load centroid O2 on the moving platform in the moving coordinate system is: E′=[q1 q2 h 1] T (5) The homogeneous coordinate matrix of the load centroid O2 in the static coordinate system is: Step 2.2: Define n1, n2, and n3 as unit vectors along the X-axis, Y-axis, and Z-axis, respectively, and F k Let f be the force exerted by the k-th link on the moving platform. k Direction vector, l k Let be the direction vector of the spatial line containing the k-th link (k = 1, 2, ..., 6), m be the mass of the load, g be the gravitational acceleration of the load, and a be the inertial acceleration of the load. Then we have: For a Stewart platform moving at low speed, the load inertial force is negligible. Therefore, from the force balance of the moving platform in the X-axis direction, we get: Substituting equations (7) and (8) into equation (9), we get: In the above formula, if the load on the k-th link is a tensile load, then F k With l k Opposite directions, f·l k =-1; If the load on the k-th link is a compressive load, then F k With l k Same direction, f·l k =1; Step 2.3: Using the same method as in Step 2.2, by balancing the forces on the moving platform in the Y and Z directions respectively, we obtain: Step 2.4, the component of the force exerted by the k-th link on the moving platform in the XOY plane is: The coordinates of the upper and lower hinge points of the k-th link projected onto the XOY plane are (B... 1k B 2k ,0) and (C 1k C 2k The coordinates of the projection of the origin O′ of the moving platform onto the XOY plane are (x, y). According to the formula for the distance from a point to a line, the distance from the projection of point O′ onto the projection line of the kth link in the XOY plane is: The direction vector of the normal to the projection line of link k in the XOY plane is: n kxy =(C 2k -B 2k )n1+(B 1k -C 1k )n2 (15) The radial vector of the projection of point O′ onto the load centroid O2 in the XOY plane is: Based on the torque balance of the moving platform about the Z-axis, we can obtain: Step 2.5: Using the same method as in Step 2.4, based on the torque balance of the moving platform around the X and Y axes, we can obtain: Equations (10)-(12) and (17)-(19) together form the mechanical equilibrium equations of the moving platform. Solving these equations simultaneously yields the force F exerted by the k-th link on the moving platform. k Let the axial force of the connecting rod be N. k Then N k =-F k This refers to the axial force of the connecting rod in a fixed posture.

4. The method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to claim 1, characterized in that, In step 3, the objective function expression is: max |N|=max{|N1| |N2|…|N6|} (20) In the formula, max|N| represents the maximum axial force of the link under all attitudes, and |N1|~|N6| represent the axial force of each link under a certain attitude.

5. The method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to claim 1, characterized in that, In step 4, the constraints include the range of values ​​for the worst-case force attitude parameters of the Stewart platform and the constraint on the link length through a penalty function.

6. The method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to claim 5, characterized in that, The constraint on the link length using the penalty function specifically refers to: In the formula, |N max | represents the nominal maximum axial force of the six links under the current attitude, | N′ max | represents the true maximum axial force of the six links under the current attitude, L k L represents the length of the k-th link. max L represents the maximum allowable length of the link. min This indicates the minimum allowable length of the link.

7. The method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to claim 5, characterized in that, Step 5 is as follows: Step 5.1: Set the maximum number of generations T′, construct a particle swarm including multiple subpopulations, each subpopulation includes multiple particles, and the position of each particle corresponds to a set of Stewart platform attitude parameters, all of which satisfy the constraints in step 4. Initialize the parameters and particles in the particle swarm. Step 5.2: Evolve the subpopulation based on adaptive inertia weights and solve the objective function to obtain the objective function value. By comparing it with the objective function value before evolution, retain the larger objective function value and the corresponding optimal solution. Step 5.3: Randomly sort the order of each subpopulation using a random function, and replace the worst individual of the previous subpopulation with the best individual of the next adjacent subpopulation according to the new order, thus completing the interactive evolution between subpopulations. Step 5.4: Repeat steps 5.2 and 5.3 until the maximum evolution generation T is reached. The larger output objective function value is the maximum axial force of the link under all allowed attitudes. The optimal solution is the particle position information. Based on the particle position information, the worst-case force attitude parameters of the Stewart platform are obtained.

8. The method for solving the worst-case force attitude parameters of a Stewart platform moving at low speed according to claim 7, characterized in that, In step 5.2, the relationship between inertia weight and evolutionary generation is as follows: In the formula, w t w represents the inertial weight of generation t. max w represents the initially defined maximum inertia weight. min T represents the initial minimum inertia weight, t represents the number of generations, and T′ represents the maximum number of generations.

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