Method for solving worst stress attitude parameters of Stewart platform moving at low speed

Abstract

The invention discloses a method for solving worst stress attitude parameters of a Stewart platform moving at a low speed, and solves the problems that in the prior art, the stress of a connecting rod of the Stewart platform under a specific attitude is only analyzed, the maximum axial force of the connecting rod under all possible attitudes cannot be accurately solved, and when constraint conditions are relatively complex, the maximum axial force cannot be accurately solved. The problem that the worst stress posture is generally difficult to accurately find through a traditional method is solved; according to the method, the worst stress posture of the Stewart platform connecting rod under the complex constraint condition is efficiently and accurately predicted, the linear relation between the connecting rod axial force and the load weight is given, and the method has important guiding significance for strength design and driving device design of the Stewart platform and safe use of the Stewart platform.

Description

Technical Field

[0001] The invention relates to a method for calculating parameters of a parallel mechanism, and in particular to a method for solving the worst force posture parameters of a Stewart platform moving at a low speed. Background Art

[0002] The Stewart platform is a six-degree-of-freedom spatial parallel mechanism composed of a dynamic platform, a static platform, connecting rods and a hinge seat. It has the advantages of strong load-bearing capacity, good rigidity, no cumulative error and high precision. It has been increasingly widely used in the fields of robots, parallel machine tools, simulated driving, optical alignment, etc.

[0003] In the field of optical alignment, a low-speed Stewart platform is usually used to adjust the posture of the load. The axial force of the Stewart connecting rod under different postures often has significant differences. Therefore, finding the worst force posture of the Stewart platform and establishing the corresponding relationship between the load and the axial force of the connecting rod under this posture are of great significance for the design and use of the Stewart platform.

[0004] Although scholars have conducted a lot of research on the forces of the Stewart platform, they generally only analyze the forces of the connecting rods of the Stewart platform under specific postures, and cannot accurately solve the maximum axial force of the connecting rods under all possible postures. Some scholars also use the connecting rod length as a constraint condition for the posture of the Stewart platform, and evaluate the design of the oil source pressure and hydraulic cylinder of the Stewart platform by listing the forces of the connecting rods under various worse postures. However, this method requires relying on experience to determine the worse posture of the Stewart platform in advance. When the constraints are more complex, such as constraining the motion range of the dynamic platform, it is generally difficult to accurately find the worst force posture through traditional methods. Summary of the invention

[0005] The purpose of the present invention is to solve the problem that in the prior art, only the force of the connecting rod of the Stewart platform under a specific posture is analyzed, and the maximum axial force of the connecting rod under all possible postures cannot be accurately solved, and when the constraint conditions are relatively complex, it is generally difficult to accurately find the worst force posture through traditional methods; and to provide a method for solving the worst force posture parameters of a Stewart platform moving at a low speed.

[0006] In order to achieve the above object, the present invention adopts the following technical solution:

[0007] A method for solving the worst force posture parameters of a Stewart platform moving at a low speed is special in that it includes the following steps:

[0008] Step 1: According to the kinematic model of the Stewart platform, a relationship model between the posture parameters of the Stewart platform and the length of the connecting rod is established;

[0009] Step 2: Analyze the force of the Stewart platform according to the relationship model between the posture parameters of the Stewart platform and the length of the connecting rod in step 1, and solve the axial force of the connecting rod under fixed posture;

[0010] Step 3: According to the axial force of the connecting rod under the fixed posture in step 2, the objective function is constructed with the maximum axial force of the connecting rod under all fixed postures as the target;

[0011] Step 4: Establish constraints for the Stewart platform posture;

[0012] Step 5: Using the particle swarm algorithm, when the constraint conditions of the Stewart platform posture in step 4 are satisfied, the maximum value of the objective function constructed in step 3 is obtained, and the worst force posture parameters of the Stewart platform are determined according to the maximum value of the objective function.

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

[0014] Step 1.1, define the Stewart platform's moving coordinate system O′X′Y′Z′ as a coordinate system fixed to the moving platform, the centroid of the moving platform as the origin of the moving coordinate system, and the static coordinate system OXYZ as a coordinate system fixed to the static platform. According to the Stewart platform attitude parameters (x, y, z, α, β, γ), x, y, z are the coordinates of the origin of the moving coordinate system in the X, Y, and Z axes of the static coordinate system, α, β, γ are the components of the attitude angle of the moving platform relative to the static platform in the OYZ, OXZ, and OXY planes, respectively. It can be seen that the matrix equation for the Stewart platform's moving coordinate to static coordinate conversion is:

[0015]

[0016] In the formula, c is the abbreviation of cos, and s is the abbreviation of sin;

[0017] Step 1.2, take each hinge seat as a hinge support, and the homogeneous coordinate matrix of each hinge support of the moving platform in the Stewart platform in the moving coordinate system is:

[0018]

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

[0020] Step 1.3, from equation (1) and equation (2), the homogeneous coordinate matrix of each hinge 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 of the origin of the moving coordinate system in the Stewart platform along the Z axis in the static coordinate system as H. 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] Where, θ is the angle between two adjacent hinge points on the static platform and the center of the circle, and R is the radius of the static platform;

[0024] According to equations (1)-(3), the length L of the kth connecting rod is established k The relationship model between (k=1,2,…6) and posture parameters:

[0025]

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

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

[0028] Step 2.1, define the load mass center O2 on the moving platform with a horizontal coordinate of q1 and a vertical coordinate of q2 in the moving coordinate system. The initial height difference between the moving platform and the static platform is h. The homogeneous coordinate matrix of the load mass center O2 on the moving platform in the moving coordinate system is:

[0029] E′=[q1 q2 h 1] T (5)

[0030] Then the homogeneous coordinate matrix of the load mass center O2 in the static coordinate system is:

[0031]

[0032] Step 2.2, define n1, n2, n3 as unit vectors along the X-axis, Y-axis, and Z-axis respectively, F k is the force exerted by the kth connecting rod on the moving platform, and f is F k The direction vector, l k is the direction vector of the space line where the kth connecting rod (k=1,2,…6) is located, m is the mass of the load, g is the gravity acceleration of the load, and a is the inertial acceleration of the load, then:

[0033]

[0034]

[0035] For a Stewart platform moving at a low speed, the load inertia force is negligible, and the dynamic platform is subjected to force balance in the X-axis direction, and we get:

[0036]

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

[0038]

[0039] In the above formula, if the load on the kth connecting rod is a tensile load, then F k With l k In the opposite direction, f·l k =-1; if the load on the kth connecting rod is a compressive load, then F k With l k The same direction, f·l k =1;

[0040] Step 2.3: Use the same method as step 2.2, and balance the forces on the moving platform in the Y and Z directions to obtain:

[0041]

[0042]

[0043] Step 2.4: The component of the force exerted by the kth connecting rod on the moving platform in the XOY plane is:

[0044]

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

[0046]

[0047] The direction vector of the normal line of the projection line of the connecting rod 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′ in the XOY plane to the load center of mass O2 is:

[0050]

[0051] According to the moment balance of the moving platform around the Z axis, we can get:

[0052]

[0053] Step 2.5: Using the same method as step 2.4, according to the moment balance of the moving platform around the X-axis and Y-axis, we can obtain:

[0054]

[0055]

[0056] Equations (10)-(12) and (17)-(19) together constitute the mechanical equilibrium equation of the moving platform. Solving them together can obtain the force F exerted by the kth connecting rod 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 connecting rod in all postures, and |N1|~|N6| represents the axial force of each connecting rod in a certain posture.

[0060] Furthermore, in step 4, the constraint conditions include the value range of the worst force posture parameter of the Stewart platform and the constraint on the connecting rod length through a penalty function.

[0061] Furthermore, the constraint on the connecting rod length by the penalty function is specifically:

[0062]

[0063] In the formula, |N max | represents the nominal maximum axial force of the six connecting rods in the current posture, |N′ max | represents the true maximum axial force of the six connecting rods in the current posture, L k represents the length of the kth connecting rod, L max Indicates the maximum allowable length of the connecting rod, L minIndicates the minimum allowable length of the connecting rod.

[0064] Furthermore, the specific process of step 5 is as follows:

[0065] Step 5.1, set the maximum evolutionary generation T′, construct a particle swarm including multiple sub-populations, each sub-population includes multiple particles, the position of each particle corresponds to a set of Stewart platform posture parameters, and all meet the constraints in step 4, and initialize the parameters and particles in the particle swarm;

[0066] Step 5.2, evolve the sub-population based on the adaptive inertia weight, and solve the objective function to obtain the objective function value, and retain the larger objective function value and the corresponding optimal solution by comparing it with the objective function value before evolution;

[0067] Step 5.3: Randomly sort the order of each sub-population by a random function, and replace the worst individual of the previous sub-population with the best individual of the next adjacent sub-population in the new order to complete the interactive evolution between sub-populations;

[0068] Step 5.4: Repeat steps 5.2 and 5.3 until the maximum evolutionary generation T is reached. The larger objective function value output is the maximum axial force of the connecting rod under all allowable postures. The optimal solution is the position information of the particle. The worst force posture parameters of the Stewart platform are obtained based on the particle position information.

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

[0070]

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

[0072] Beneficial effects of the present invention:

[0073] The invention discloses a method for solving the worst force posture parameters of a Stewart platform moving at a low speed, which can efficiently and accurately predict the worst force posture of a connecting rod of the Stewart platform under complex constraint conditions, and give a linear relationship between the connecting rod axial force and the load weight, which has important guiding significance for the strength design, drive device design and safe use of the Stewart platform. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 11 is a schematic structural diagram of a Stewart platform in an embodiment of a method for solving the worst force posture parameters of a Stewart platform moving at a low speed of the present invention;

[0075] Figure 2 1 is a schematic diagram of a Stewart platform in a coordinate system in an embodiment of a method for solving the worst force posture parameters of a Stewart platform moving at a low speed of the present invention;

[0076] Figure 3 It is an evolution curve diagram of the maximum axial force of the connecting rod in an embodiment of a method for solving the worst force posture parameters of a Stewart platform moving at a low speed of the present invention;

[0077] In the figure, 1. load, 2. moving platform, 3. hinge seat, 4. connecting rod, 5. static platform. DETAILED DESCRIPTION

[0078] The technical solution of the present invention will be clearly and completely described below in conjunction with the accompanying drawings and embodiments. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work 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, it includes a load 1, a moving platform 2, a hinge seat 3, a connecting rod 4, and a static platform 5. Among them, there are six connecting rods 4. The size parameters of the Stewart platform are R = 350mm, θ = 15°, R′ = 250mm, θ′ = 60°, H = 560mm, the load mass m = 100kg, and the coordinates of the load center of mass in the moving coordinate system in the initial position are (0, 0, 80mm).

[0080] The present embodiment provides a method for solving the worst force posture parameters of a Stewart platform moving at a low speed, comprising the following steps:

[0081] Step 1: According to the kinematic model of the Stewart platform, establish the relationship model between the posture parameters of the Stewart platform and the length of the connecting rod; the specific process is as follows:

[0082] Define the moving coordinate system of the Stewart platform O 'X'Y'Z' is the coordinate system fixed to the moving platform. The centroid of the moving platform is the origin of the moving coordinate system. OXYZis the coordinate system fixed to the static platform. According to the Stewart platform attitude parameters (x, y, z, α, β, γ), x, y, z are the coordinates of the origin of the dynamic coordinate system in the X, Y, and Z axes of the static coordinate system, respectively. α, β, γ are the components of the attitude angle of the dynamic platform relative to the static platform in the OYZ, OXZ, and OXY planes, respectively. It can be seen that the matrix equation for the transformation of the Stewart platform dynamic coordinates to the static coordinates is:

[0083]

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

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

[0086]

[0087] Where θ′ is the angle formed by the line connecting two adjacent hinge points on the moving platform and the moving coordinate origin, and R′ represents the radius of the moving platform.

[0088] From equations (1) and (2), the homogeneous coordinate matrix of each hinge point of the moving platform in the Stewart platform in the static coordinate system is obtained as 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, in order to simplify the calculation, the homogeneous coordinate matrix of each hinge point of the static platform in the static coordinate system is:

[0090]

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

[0092] According to equations (1)-(3), the length L of the kth connecting rod is established k The relationship model between (k=1,2,…6) and posture parameters:

[0093]

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

[0095] Step 2: Analyze the force on the Stewart platform and solve the axial force of the connecting rod in a fixed posture; the specific process is:

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

[0097] E′=[q1 q2 h 1] T (5)

[0098] Then the homogeneous coordinate matrix of the load mass center O2 in the static coordinate system is:

[0099]

[0100] Define n1, n2, n3 as unit vectors along the XYZ axes, F k is the force exerted by the kth connecting rod on the moving platform, and f is F k The direction vector, l k is the direction vector of the space line where the kth connecting rod (k=1,2,…6) is located, m is the mass of the load, g is the gravity acceleration of the load, and a is the inertial acceleration of the load, then:

[0101]

[0102]

[0103] For a Stewart platform moving at a low speed, the load inertia force is negligible, and the dynamic platform is subjected to force balance in the X direction, and we get:

[0104]

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

[0106]

[0107] In the above formula, if the load on the kth connecting rod is a tensile load, then F k With l k In the opposite direction, f·l k =-1; if the load on the kth connecting rod is a compressive load, then F k With l k The same direction, f·l k =1;

[0108] Similarly, the dynamic platform is subjected to force balance in the Y direction and the Z direction, and we get:

[0109]

[0110]

[0111] The component of the force exerted by the kth connecting rod on the moving platform in the XOY plane is:

[0112]

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

[0114]

[0115] The direction vector of the normal line of the projection line of the connecting rod 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′ in the XOY plane to the load center of mass O2 is:

[0118]

[0119] According to the moment balance of the moving platform around the Z axis, we can get:

[0120]

[0121] Similarly, according to the moment balance of the moving platform around the X-axis and Y-axis, we can get:

[0122]

[0123]

[0124] Equations (10)-(12) and (17)-(19) together constitute the mechanical equilibrium equation of the moving platform. Solving them together can obtain the force F exerted by the kth connecting rod on the moving platform: k , let the axial force of the connecting rod be N k , then N k =-F k .

[0125] Step 3: Take the maximum axial force of the connecting rod under all postures as the target and construct the objective function; the objective function expression is:

[0126] max|N|=max{|N1||N2|…|N6|} (20)

[0127] In the formula, max|N| represents the maximum axial force of the connecting rod in all postures, and |N1|~|N6| represents the axial force of each connecting rod in a certain posture.

[0128] Step 4: Establish the constraints of the Stewart platform posture; the constraints include the value range of the worst force posture parameters of the Stewart platform and the constraint on the connecting rod length through the penalty function.

[0129] In this embodiment, the value range of the worst force posture parameters of the Stewart platform is: |x|≤250mm, |y|≤250mm, |z|≤250mm,

[0130] The penalty function is used to constrain the connecting rod length:

[0131]

[0132] In the formula, |N max | represents the nominal maximum axial force of the six connecting rods in the current posture, |N′ max | represents the true maximum axial force of the six connecting rods in the current posture, L k represents the length of the kth connecting rod, L max Indicates the maximum allowable length of the connecting rod, L min Indicates the minimum allowable length of the connecting rod.

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

[0134] Step 5: The sampling particle swarm algorithm is used to solve the maximum value of the objective function that satisfies the constraints of the Stewart platform posture, and the worst force posture parameters of the Stewart platform are determined based on the maximum value of the objective function. The specific process is:

[0135] Step 5.1, set the maximum evolutionary generation T', construct a particle swarm including multiple sub-populations, each sub-population includes multiple particles, the position of each particle corresponds to a set of Stewart platform posture parameters, and all meet the constraints in step 4, and initialize the parameters and particles in the particle swarm.

[0136] In this embodiment, the number of sub-populations I = 5, the number of particles contained in each sub-population J = 20, the maximum evolutionary number of the population T = 300, and the initial inertia weight w max =0.9, minimum inertia weight w min=0.4, self-learning factor c1=2, social learning factor c2=2, the criterion for falling into the local optimal solution is △T=10 and When the change of the optimal solution of a certain exploration population in △T consecutive generations is less than △N, it is considered that the subpopulation may be trapped in the local optimal solution. The significance of the △N value is that in the early stage of evolution, when t is small, the search results are far from the true optimal solution. In order to avoid the exploration population wasting too much time on the local optimal solution, a larger △N value is given to jump out of the local optimal solution as soon as possible; as the evolution proceeds, each particle generally gets closer to the global optimal solution, and the value of △N decreases with the evolutionary generation t, so as to reduce the speed and achieve a more refined search.

[0137] Particle initialization: Within the range allowed by the constraints, 100 particles are randomly generated. The position of each particle satisfies the constraints in step 4. The initial change speed of the translation posture is a random number between -25 and 25, and the initial change speed of the angle posture is a random number between -0.05 and 0.05.

[0138] Step 5.2, evolve the sub-population based on the adaptive inertia weight, and solve the objective function to obtain the objective function value, and retain the larger objective function value and the corresponding optimal solution by comparing it with the objective function value before evolution;

[0139] First, a single evolutionary population is divided into multiple sub-populations, each with the same initial inertia weight. After the first generation of evolution is completed, the optimal solutions of each sub-population in the current stage are compared. The population with the largest optimal solution is used as the development population, and the remaining populations are used as the exploration population. The inertia weight of the development population changes according to formula (22):

[0140]

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

[0142] In this embodiment, after the first generation of evolution is completed, the optimal solutions of each sub-population in the current stage are compared. The population with the largest optimal solution is used as the development population, and the remaining populations are used as the exploration population. The inertia weight of the development population changes according to formula (22), and the inertia weight decays rapidly in the early stage of evolution. The exploration population has good global optimization ability, and its inertia weight first changes according to formula (22). When an exploration population falls into a local optimal solution, it is provided with a larger inertia weight to jump out of the local optimal solution. The specific method is to reset the inertia weight of the sub-population to w max, and its evolutionary generation t is reset to zero, and then its inertia weight is decayed again according to formula (22). After each generation of evolution, the actual evolutionary generation of the particle swarm is t=t+1, and the optimal solutions of each sub-population are compared, and the development population and exploration population are re-divided.

[0143] According to the results of each generation of evolution, the population is divided into a development population and an exploration population. The development population has a smaller inertia weight, which is conducive to local fine search; the exploration population has a larger inertia weight, which is conducive to global search.

[0144] Step 5.3: Randomly sort the order of each sub-population by a random function, and replace the worst individual of the previous sub-population with the best individual of the next adjacent sub-population in the new order to complete the interactive evolution between sub-populations;

[0145] Step 5.4: Repeat steps 5.2 and 5.3 until the maximum evolutionary generation T is reached. The larger objective function value output is the maximum axial force of the connecting rod under all allowable postures. The optimal solution is the position information of the particle. The worst force posture parameters of the Stewart platform are obtained based on the particle position information.

[0146] This embodiment repeats four sets of operations to obtain the evolution curve of the maximum axial force of the connecting rod as follows: Figure 3 As shown in Table 1, after different evolution paths, each group of operations converges around 1.756 kN, so the maximum axial force of the connecting rod under the current constraint is 1.756 kN. The posture parameters of the Stewart platform 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 postures of the first group and the fourth group are almost the same. The average of the two groups of parameters gives posture I, which is (0.107, 0.220, -0.524, -226.6, -188.5, -69.9); the postures of the second group and the third group are almost the same. The average of the two groups of parameters gives posture II, which is (-0.105, 0.220, 0.524, -226.4, 188.4, -69.7). Posture I and posture II are two postures symmetrical about the XOZ plane. Both postures are the worst force postures of the Stewart platform under the current constraints. When solving the problem using the method of the present invention, one of the postures can be randomly searched each time.

[0150] The above description is only a specific embodiment of the present invention, and a comparison of the effects of the specific embodiments and the related comparative examples, but the protection scope of the present invention is not limited thereto. Any changes or substitutions within the technical scope disclosed by the present invention should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A method for solving the worst force posture parameters of a low-speed Stewart platform, characterized in that: The following steps are involved: Step 1: According to the kinematic model of the Stewart platform, a relationship model between the posture parameters of the Stewart platform and the length of the connecting rod is established; Step 2: Analyze the force of the Stewart platform according to the relationship model between the posture parameters of the Stewart platform and the length of the connecting rod in step 1, and solve the axial force of the connecting rod under fixed posture; Step 3: According to the axial force of the connecting rod under the fixed posture in step 2, the objective function is constructed with the maximum axial force of the connecting rod under all fixed postures as the target; Step 4: Establish constraints for the Stewart platform posture; Step 5: Using the particle swarm algorithm, when the constraint conditions of the Stewart platform posture in step 4 are satisfied, the maximum value of the objective function constructed in step 3 is obtained, and the worst force posture parameters of the Stewart platform are determined according to the maximum value of the objective function.

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

3. The method for solving the worst force posture parameters of a Stewart platform moving at a low speed according to claim 2, characterized in that: The specific process of step 2 is: Step 2.1, define the load mass center O2 on the moving platform with a horizontal coordinate of q1 and a vertical coordinate of q2 in the moving coordinate system. The initial height difference between the moving platform and the static platform is h. The homogeneous coordinate matrix of the load mass center O2 on the moving platform in the moving coordinate system is: E′=[q1 q2 h 1] T (5) Then the homogeneous coordinate matrix of the load mass center O2 in the static coordinate system is: Step 2.2, define n1, n2, n3 as unit vectors along the X-axis, Y-axis, and Z-axis respectively, F k is the force exerted by the kth connecting rod on the moving platform, and f is F k The direction vector, l k is the direction vector of the space line where the kth connecting rod (k=1,2,…6) is located, m is the mass of the load, g is the gravity acceleration of the load, and a is the inertial acceleration of the load, then: For a Stewart platform moving at a low speed, the load inertia force is negligible, and the dynamic platform is subjected to force balance in the X-axis direction, and we get: Substituting equation (7) and equation (8) into equation (9), we get: In the above formula, if the load on the kth connecting rod is a tensile load, then F k With l k In the opposite direction, f·l k = -1; If the load on the kth connecting rod is a compressive load, then F k With l k The same direction, f·l k =1; Step 2.3: Use the same method as step 2.2, and balance the forces on the moving platform in the Y and Z directions to obtain: Step 2.4: The component of the force exerted by the kth connecting rod on the moving platform in the XOY plane is: The coordinates of the upper and lower hinge points of the kth link projected in the XOY plane are (B 1k ,B 2k ,0) and (C 1k ,C 2k ,0), the coordinates of the projection of the origin O′ of the moving platform in the XOY plane are (x, y); according to the distance formula from a point to a straight line, the distance from the projection of point O′ in the XOY plane to the projection line of the kth link is: The direction vector of the normal line of the projection line of the connecting rod 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′ in the XOY plane to the load center of mass O2 is: According to the moment balance of the moving platform around the Z axis, we can get: Step 2.5: Using the same method as step 2.4, according to the moment balance of the moving platform around the X-axis and Y-axis, we can obtain: Equations (10)-(12) and (17)-(19) together constitute the mechanical equilibrium equation of the moving platform. Solving them together can obtain the force F exerted by the kth connecting rod on the moving platform: k , let the axial force of the connecting rod be N k , then N k =-F k , which is the axial force of the connecting rod in a fixed posture.

4. The method for solving the worst force posture parameters of a low-speed Stewart platform 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 connecting rod in all postures, and |N1|~|N6| represents the axial force of each connecting rod in a certain posture.

5. The method for solving the worst force posture parameters of a low-speed Stewart platform according to claim 1, characterized in that: In step 4, the constraint conditions include the value range of the worst force posture parameter of the Stewart platform and the constraint on the connecting rod length through a penalty function.

6. A method for solving the worst force posture parameters of a low-speed Stewart platform according to claim 5, characterized in that: The constraint on the connecting rod length by the penalty function is specifically: In the formula, |N max | represents the nominal maximum axial force of the six connecting rods in the current posture, |N′ max | represents the true maximum axial force of the six connecting rods in the current posture, L k represents the length of the kth connecting rod, L max Indicates the maximum allowable length of the connecting rod, L min Indicates the minimum allowable length of the connecting rod.

7. The method for solving the worst force posture parameters of a low-speed Stewart platform according to claim 5, characterized in that: The specific process of step 5 is as follows: Step 5.1, set the maximum evolutionary generation T′, construct a particle swarm including multiple sub-populations, each sub-population includes multiple particles, the position of each particle corresponds to a set of Stewart platform posture parameters, and all meet the constraints in step 4, and initialize the parameters and particles in the particle swarm; Step 5.2, evolve the sub-population based on the adaptive inertia weight, and solve the objective function to obtain the objective function value, and retain the larger objective function value and the corresponding optimal solution by comparing it with the objective function value before evolution; Step 5.3: Randomly sort the order of each sub-population by a random function, and replace the worst individual of the previous sub-population with the best individual of the next adjacent sub-population in the new order to complete the interactive evolution between sub-populations; Step 5.4: Repeat steps 5.2 and 5.3 until the maximum evolutionary generation T is reached. The larger objective function value output is the maximum axial force of the connecting rod under all allowable postures. The optimal solution is the position information of the particle. The worst force posture parameters of the Stewart platform are obtained based on the particle position information.

8. A method for solving the worst force posture parameters of a low-speed Stewart platform according to claim 7, characterized in that: In step 5.2, the relationship between inertia weight and evolutionary generations is: In the formula, w t represents the inertia weight of the tth generation, w max represents the maximum inertia weight initially defined, w min represents the minimum inertia weight initially defined, t represents the evolutionary generation, and T′ represents the maximum evolutionary generation.

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