RV reducer comprehensive transmission error modeling method based on RLS-PSO hybrid algorithm

Abstract

The invention discloses an RV speed reducer comprehensive transmission error modeling method based on an RLS-PSO hybrid algorithm. The modeling process comprises the following steps that an RV speed reducer is decomposed into a plurality of equivalent mechanisms, and a geometric error model is established; establishing a stress deformation model of the equivalent mechanism; and constructing a data set of the input torque and the output rotation angle. Performing signal processing on real-time input torque data by using a least square method (RLS), and removing noise; performing feature optimization on the signal processed by the RLS algorithm by using a particle swarm optimization (PSO); and an optimal input variable is obtained. The input variable is substituted into the stress deformation model to obtain a stress deformation transmission error curve rule reflecting an actual input torque fluctuation rule, and a speed reducer complete machine transmission error curve rule is obtained by combining geometric transmission errors, so that a theoretical basis is provided for RV speed reducer error analysis and key part error analysis; and a technical support is provided for design and analysis of transmission precision of the RV speed reducer.

Description

Technical Field

[0001] The present application relates to the fields of computer technology and mechanical mechanisms, and in particular to a comprehensive transmission error modeling method for an RV reducer based on an RLS-PSO hybrid algorithm. Background Art

[0002] RV reducers have the characteristics of high motion precision. Transmission performance is an important indicator to measure the transmission characteristics of RV reducers. The main sources of transmission errors of RV reducers are manufacturing errors, assembly errors and stress deformation. Due to the complex structure of RV reducers and the large number of key transmission parts, it is very difficult to analyze the error sources of RV reducers.

[0003] At present, many scholars have done a lot of research on the modeling and analysis of RV reducer transmission errors. Yang (A modelling approach for kinematic equivalent mechanism and rotational transmission error of RV reducer) proposed an error analysis model for RV reducer with an over-constrained structure. The model describes the relationship between the original error of the mechanism and the transmission accuracy. Zihui (Effect of Multifactor Interaction on the Accuracy of RV Reducers) established a response approximation model to analyze the influence of the initial errors of different geometric components on the overall transmission error of the system. Lin (Kinematic error analysis and tolerance allocation of cycloidal gear reducers) proposed a method for cycloidal transmission motion error analysis and tolerance allocation. Regarding elastic contact analysis, Huang and Tsai (A study on loaded tooth contact analysis of a cycloid planetary gear reducer considering friction and bearing roller stiffness) used the LTCA method to establish the relationship between the deformation and displacement of the cycloid transmission. Chen (Vibration characteristics analysis of the new pin-cycloid speed reducer) also applied the LTCA method to analyze the contact deformation on the meshing line. Li (Analysis of a cycloid speed reducer considering tooth profile modification and clearance-fit output mechanism) proposed an analytical model for load distribution analysis of cycloid reducers in the presence of profile modification, eccentricity and assembly errors. Wang (Transmission Performance Analysis of RV Reducers Influenced by Profile Modification and Load) proposed a multi-tooth contact model and transmission error model for RV reducers.

[0004] In recent years, the transmission error modeling of RV reducers is mainly based on the analysis of a single key factor. At present, the error source is not comprehensively considered, and the fluctuation of the curve of the overall transmission error cannot be well matched with the actual fluctuation curve. The above modeling method does not comprehensively consider the error source of the RV reducer and cannot accurately reflect the actual transmission error change law.

[0005] The current mainstream research on dynamic transmission error of RV reducers is mainly based on theoretical models. Although the calculation results of the reducer transmission error model are in good agreement with the experimental data within the error range, they cannot reflect the real-time characteristics of the actual transmission process, resulting in significant differences from the actual test results. Summary of the invention

[0006] In order to make up for the shortcomings of the existing mathematical model, the present invention proposes a comprehensive transmission error modeling method for RV reducer based on RLS-PSO hybrid algorithm. On the basis of comprehensively considering the influence of error sources on the transmission performance of RV reducer, the hybrid algorithm of partial least squares (Recursive Least Squares, RLS) and particle swarm optimization (PSO) is used to optimize the input torque of the RV reducer force deformation model, solve the instability of the model input caused by external factors and extract high-quality features in the input torque, so as to improve the accuracy of the RV reducer transmission error model. The main process of this modeling method is as shown in the attached figure. Figure 1 As shown, it can be divided into the following three steps: constructing the geometric transmission error model of the substructure, constructing the force and deformation model of the substructure, and inputting the RLS-PSO hybrid filtering algorithm of the torque.

[0007] (1) Construction of geometric transmission error model of substructure

[0008] The research object of this paper is RV reducer. In order to effectively analyze the influence of the error sources of key components on the transmission performance of RV reducer, firstly, according to the structural characteristics of RV reducer, it is decomposed into involute transmission mechanism of sun gear and planetary gear, parallelogram mechanism of cycloid gear and bearing, cycloid pin gear transmission mechanism of cycloid gear and pin gear, and output disc mechanism. The manufacturing error and assembly error of key components are simulated, the transmission direction vector of substructure is determined, the geometric error is projected to the transmission direction of the mechanism, and the relationship between the error of components and the transmission performance of reducer is obtained.

[0009] (2) Construction of the stress-deformation model of the substructure

[0010] For the stress-deformation model of the substructure, the deformation between the components of the substructure is obtained by combining Newton's third law. The deformation is converted into a rotation angle and projected onto the transmission direction. The transmission error caused by the deformation is converted into the output angle of the reducer through the reduction ratio of the reducer. Finally, the relationship between the stress-deformation of key components and the transmission performance of the reducer can be obtained.

[0011] (3) RLS-PSO adaptive hybrid algorithm

[0012] The present invention proposes a method of combining the partial least squares (RLS) adaptive filtering algorithm and the particle swarm algorithm (PSO) to extract features from the input quantity (input torque) of the force-deformation model. First, the RLS adaptive filtering algorithm is used to extract features and trends from the original signal (input torque data information) to filter out interference noise. Secondly, the PSO algorithm is used to extract features from the optimized signal again to further optimize the noise suppression effect of the signal, thereby ensuring the variation law of the actual input torque and removing the noise in the original signal.

[0013] The present invention provides a comprehensive transmission error modeling method for an RV reducer based on an RLS-PSO hybrid algorithm. By decomposing the reducer into multiple substructures, the analysis method focuses on the error sources and deformation of key components. The transmission errors caused by the manufacturing errors, assembly errors and force deformation of the RV reducer are modeled one by one, and a transmission error model of the whole machine is established. Based on the input quantity required for modeling, the data of the RLS-PSO hybrid algorithm of the input quantity is optimized, and the feature extraction of the experimental input torque data is performed, which not only retains the fluctuation law of the actual input torque but also removes the invalid data points (noise) of the input torque, providing technical support for the transmission accuracy design and key analysis of the RV reducer. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 It is a flow chart of the RV reducer transmission error modeling method provided by the present invention;

[0015] Figure 2 is a geometric transmission error diagram of the present invention;

[0016] Figure 3 is a force deformation transmission error diagram of the present invention;

[0017] Figure 4 The execution process of the RLS-PSO hybrid algorithm;

[0018] Figure 5 This is a diagram showing the effect of the RLS-PSO hybrid algorithm of the present invention on the data processing of the input torque;

[0019] Figure 6 It is a mathematical model solving flow chart of the present invention;

[0020] Figure 7 It is a theoretical curve diagram of transmission error of the reducer of the present invention. DETAILED DESCRIPTION

[0021] The following is a complete description of the implementation process of the present invention in conjunction with the accompanying drawings of the examples of the present invention;

[0022] The purpose of the present invention is to design a comprehensive transmission error modeling method for RV reducer based on RLS-PSO hybrid algorithm, overcome the shortcomings of the previous transmission error mathematical model, improve the trend of the theoretical transmission error curve to be more consistent with the actual transmission error curve, and focus on analyzing the influence of the errors of key components on the transmission performance of the reducer;

[0023] like Figure 1 As shown, the establishment process of the geometric transmission error model of the RV reducer;

[0024] Construct a transmission error calculation model for involute gear transmission mechanism;

[0025]

[0026] In the formula, e s E is the assembly eccentricity error of the sun gear; s is the error offset between the theoretical geometric center of the sun gear and the eccentric center; θ1 is the rotation angle of the sun gear; θ2 is the rotation angle of the planetary gear; β b1 is the angle between the offset of the sun gear and the x-axis of the generalized coordinate system; β b2 is the angle between the offset of the planetary gear and the x-axis of the generalized coordinate system; β n is the angle between the meshing line between the sun gear and the planetary gear and the x-axis; n1 is the actual transmission direction vector of the sun gear and the planetary gear.

[0027] Construct a transmission error calculation model for parallelogram mechanisms;

[0028]

[0029] In the formula, s c Represents the local coordinate system x c o c y c The vector of the geometric center of the eccentric circle of the middle bearing hole; e R is the bearing hole radius size deviation; θ' c Indicates the size offset of the bearing hole and the local coordinate system x c o c y c x c The angle between the axes; n2 represents the transmission direction vector of the parallelogram mechanism (i.e., the tangent direction of the geometric center distribution circle of the cycloid wheel bearing hole); Moc Represents the coordinate system S c To the generalized coordinate system S o The coordinate transformation matrix; e is the crank arm; θ represents the rotation angle of the cycloid wheel; θ g Indicates the input angle of the cycloid wheel.

[0030] Construct the transmission error calculation model of cycloid pin gear;

[0031] The pin tooth profile r in the generalized coordinate system f It is expressed as:

[0032]

[0033] Where K1 = eZ p / R p represents the short amplitude coefficient; r represents the radius of the needle tooth; R p Indicates the radius of the needle tooth distribution circle;

[0034] Theoretical tooth profile vector r of cycloid gear c :

[0035]

[0036] Where M co Denotes the generalized coordinate system S o Transform to the local coordinate system S c The coordinate transformation matrix of Z c Indicates the number of cycloid gear teeth; r c (θ g ) represents the crank arm input angle θ g The theoretical tooth profile vector of the cycloid gear with θ as the independent variable.

[0037] In the generalized coordinate system, the position vector r of the theoretical contact point between the cycloid wheel and the i-th pin tooth is mi for:

[0038]

[0039] Where i represents the y coordinate system from the generalized coordinate system o The i-th pin tooth counted in the counterclockwise direction of the axis; Z p Indicates the number of needle teeth; r mi (θ g ) represents the crank arm input angle θ g is the theoretical contact point position vector of the i-th pin tooth as the independent variable.

[0040] Continuous contact conditions between the cycloid wheel and the pin tooth contact surface:

[0041]

[0042] in:

[0043]

[0044] Where n mi (θ g ) is the crank arm input angle θ g is the normal vector of the contact point between the ith pin tooth and the cycloid wheel of the independent variable (the transmission direction vector between the cycloid wheel and the pin tooth on the pin tooth profile); n c (θ g ) is the crank arm input angle θ g is the normal vector of the cycloid gear surface of the independent variable (the transmission direction vector of the cycloid wheel and the pin teeth on the cycloid wheel tooth profile).

[0045] The manufacturing error and assembly error of the simulated pinion teeth lead to the transmission error of the reducer. By comparing r and R in equation (4), p Add offsets δr and δR respectively p Combined with the contact judgment condition of formula (7), the input angle θ of the theoretical cycloid wheel without adding variables is calculated: g And the angle θ' after adding the variable g The calculation formula of the transmission error of the cycloid pin-tooth transmission mechanism Δe3 is:

[0046]

[0047] In the formula, θ6 represents the theoretical output angle of the reducer output end;

[0048] Analyze the output disk transmission process separately and build the output disk transmission error calculation model;

[0049]

[0050] △e4=|(e pq +e pa )n3|(10)

[0051] In the formula, δP a and δP q They represent the eccentricity of the output disc and the eccentricity of the crankshaft hole on the planet carrier. p Respectively represent the offset δP a and the offset δP q The angle between the x-axis and the generalized coordinate system; α p is the relative position of the center of the planetary gear in the generalized coordinate system. Indicates the relative position of the geometric center of the cycloid gear. (j represents the jth cycloid gear, j = 1, 2); n3 represents the transmission direction vector of the output disc;

[0052] like Figure 1Flow chart of the process, construct the stress-deformation model;

[0053] Analyze the meshing force between the sun gear and the planetary gear separately, build the force-deformation model of gear meshing, and calculate the deformation Δα1;

[0054]

[0055] In the formula, k sp is the meshing stiffness of the spur gear; s N is the deformation displacement offset during meshing; r s Pitch radius of the sun gear; F pi is the force on the i-th planetary gear from the sun gear; T1 represents the input torque at the input end of the reducer; L represents the number of planetary gears;

[0056] The force between the crank bearing and the cycloid wheel is analyzed separately, the force-deformation model of the contact surface is constructed, and the deformation Δα2 is calculated;

[0057]

[0058] d ij represents the contact distance between the i-th pin tooth and the j-th cycloid gear tooth;

[0059]

[0060] In the formula, (x fc ,y fc ) represents the coordinates of the intersection of the contact force and the crank arm; (x c ',y c ') represents the coordinates of the maximum compression point C on the tooth profile of the cycloid wheel when the cycloid wheel and the i-th pin tooth are compressed and deformed; n x ' and n y ' represents the projection vector of the normal vector of point C on the x-axis and y-axis; F i represents the contact force between the i-th pin tooth and the cycloid wheel; α ij The contact force F i The angle between the crank arm line and the crank arm line; Δθ i is the torsion angle caused by the elastic deformation of the i-th pin tooth and the cycloid wheel; l i The contact force F i The force arm; δ i Indicates the elastic deformation; K cp represents the meshing stiffness between the cycloid wheel and the pin teeth; T2 is the torque of the pin teeth on the cycloid wheel; x pai and x pai The geometric center coordinates of the i-th needle tooth; x mi and mi represents the coordinates of the theoretical contact point between the cycloid wheel and the i-th pin tooth (obtained from formula (6));

[0061] The force between the cycloid wheel and the pin teeth is analyzed separately, the force-deformation model of the cylindrical contact surface is constructed, and the deformation Δα3 is calculated;

[0062]

[0063] In the formula, the force exerted by the crank bearing on the cycloid gear is decomposed into three directions: the x-axis direction, the y-axis direction, and the direction of the line connecting the crankshaft geometric center in the cycloid gear geometry. The decomposed forces in the three directions are expressed as F z1 、F z2 and F z3 ; r b F is the pitch radius of the cycloid wheel; bx and F by It represents the decomposition of the resultant force of the pin teeth on the cycloid wheel on the x-axis and y-axis of the generalized coordinate system; n p represents the number of crankshafts; F represents the force vector of the cycloid wheel on the crankshaft bearing; F r and F t It means that the force |F| exerted by the cycloid wheel on the crank bearing is decomposed into the local coordinate system S fixed to the bearing. b x b Axis and y b Axis direction; Δδ b It indicates the displacement of the crank bearing caused by the force of the cycloid wheel; Ks indicates the meshing stiffness between the crank bearing and the cycloid wheel.

[0064] The stress condition of the crankshaft transmission process is analyzed separately, and the deformation Δα4 is calculated:

[0065]

[0066] In the formula, F t1 and F r1 Indicates the force exerted by the planetary gear on the crankshaft. F t2 and F r2 Indicates the tangential force and radial force of the input disk of the planet carrier on the crankshaft. t5 and F r5 Indicates the tangential force and radial force of the output disc of the planetary carrier on the crankshaft; F r' and F t' represents the radial force and tangential force of the cycloid wheel on the crank bearing (same as F in formula (16) r and F t equal in magnitude and opposite in direction); L a Indicates the length of each section of the crankshaft (a=1,2,3,4).

[0067]

[0068] Where ω1 and ω2 represent the flexibility of the two bearing sections of the crankshaft respectively; ω3 represents the flexibility of the output end of the crankshaft; I b represents the section inertia moment of the crankshaft at section b (b = 1, 2, 3, 4); G represents the shear elastic modulus; E represents the elastic modulus; ψ q Represents the sum of the torsion angles of each section of the crankshaft; T 3i represents the torsion borne by the i-th section of the crankshaft.

[0069] Depend on Figure 4 From the execution flow of the RLS-PSO hybrid algorithm, we can see that the RLS-PSO hybrid algorithm is constructed:

[0070] Design a v-order filter, the RLS filter output is:

[0071]

[0072] in

[0073] Formula (19) can also be expressed as:

[0074]

[0075] Where M represents the amount of data of the original signal to be processed; T M (l) represents the vector of the original input signal consisting of M elements (the present invention represents the input quantity of force deformation, i.e., the input torque data vector); T(M-1) represents the value of the input torque at the M-1th moment; h M (v) represents the coefficient vector of the filter; h(M-1,v) represents the coefficient of the filter output function y(v) at the M-1th moment; v represents the order of the corresponding filter, and can also represent the vth equation of the filter linear equation system; where h t M (v) indicates h M (v) is the transpose of the matrix. The upper left corner t represents the transpose operation of the matrix. In order to linearize the original signal and ensure the uniqueness of the optimized signal, v=M. M is determined according to the amount of data collected by the test bench.

[0076] According to the recursive least squares method principle: the vector of the actual input torque data collected by the present invention is T M (l), l=0,1,…,n; at this time, the expected output is expressed as It can also represent the optimized signal data required by the present invention; l represents the input torque value at the lth moment in the input torque vector; then the error between the original signal and the expected signal is expressed as:

[0077]

[0078] Weighted sum ΔM of the squared error magnitudes:

[0079]

[0080] where w represents the weighting factor for adapting the filter coefficients to time-varying signal data, where v - l represents the weight value for weighting the data point at the l-th moment, which is achieved by using the exponent v - l for past data;

[0081] Coefficient vector h of the RLS filter M (v) is updated as follows:

[0082]

[0083] where T t M (l) represents the transpose of T M (l); T * M (l) represents the conjugate transpose of T M (l); R M (v) represents the autocorrelation matrix of the original signal, where R -1 M (v) is the inverse matrix of R M (v); h M (v - 1) represents the coefficient vector of the (v - 1)-th equation; P M (v - 1) is the inverse autocorrelation matrix composed of v - 1 original signal data, providing adaptive data for optimizing the v-th data and updating the coefficient vector; h M (v) updates the coefficients through the coefficient vector of the (v - 1)-th equation and the signal error e M (l, v); u M (v) is the updated data of the weighting factor; K M (v) is the Kalman gain vector.

[0084] Initialize the filter coefficient vector as h M (0) = [H, 0, 0, …, 0] T , (0 < H ≤ 1). Taking the condition that ΔM reaches the minimum value, write the RLS algorithm using MATLAB and calculate the optimized data of the RLS filtering algorithm for the input torque at each moment;

[0085] Update the coefficient vector by combining equations (21) and (22), and repeat equations (20) and (21) to optimize the data at the next moment. Finally, store the optimized data at each moment as the RLS-optimized data vector D.

[0086] The fitness function of the PSO algorithm is constructed, and the mean square error (MSE) is used as the fitness value. The formula is as follows:

[0087]

[0088] Where D i Represents the i-th input torque of the input torque data vector (also referred to as the input torque vector after RLS processing). D' i The input torque data of the expected output; R represents the number of particles;

[0089] Update the i-th particle; the update algorithm is as follows:

[0090]

[0091] Where r1 and r2 represent random functions generated from [0,1] during the MATLAB compilation process; θ outi Indicates the output angle of the reducer corresponding to the i-th input torque value after update (the maximum value is 360 degrees, if θ outi >360 degrees, then θ outi The value is 360°, which is the speed of updating and optimizing the search for the optimal data of the current i-th particle); θ ex_outi Indicates the output angle of the reducer corresponding to the previous torque value of the i-th input torque value; c1 and c2 are learning factors, usually with a value of 2; W is the inertia factor, whose value is greater than 0; pb exi represents the individual optimal value before the i-th input torque data; gb exi represents the optimal value of the group before the i-th input torque data (i.e., the optimal value from the best point of the population); D i represents the updated position of the particle in the particle swarm algorithm (also known as the updated data of the i-th input torque); D exi The particle position representing the previous position of the i-th input torque data (also referred to as the previous torque data of the i-th input torque);

[0092] Through multiple experiments on particle swarm optimization of input torque data, the number of particles R is set in the range of 20-100; the target value range of mean square error MSE is 0 <MSE<10 -5 ; Perform iterative calculations in the corresponding number of particles, optimize a series of individual optimal values ​​and select the extreme value from the individual optimal values ​​as the global optimal value; store the global optimal value as an element of the input torque vector optimized by the RLS-PSO hybrid algorithm, and finally constitute the input torque vector of the RLS-PSO hybrid algorithm;

[0093] The input torque data mentioned above is obtained according to the RV reducer transmission error test bench; sensors are installed at the input and output ends of the RV reducer to collect the input torque T at the input end and the output angle θ at the output end. out And the input angle θ at the input end in ; The test bench calculates the transmission error ΔE of the reducer as follows:

[0094]

[0095] In the formula, I is the transmission ratio of the reducer;

[0096] The present invention uses the output angle of the output end collected by the test as the independent variable and the input torque of the input end as the dependent variable, and substitutes the original input torque data into the RLS-PSO hybrid algorithm for optimization (the optimization result is shown in Figure 5 As shown in Figure 2), the output angle and the optimized input torque are substituted into the geometric error model and the force deformation model; the transmission error value (TE) of the whole machine is calculated as follows:

[0097]

[0098] In the formula, r b Indicates the distance between the geometric center of the bearing hole and the geometric center of the cycloid wheel; r d Indicates the distance between the geometric center of the bearing hole and the geometric center of the output disc; i b It represents the transmission ratio from the output end to the pinion when the involute gear is fixed, i b =-(1-I) / I; Δe1 represents the magnitude of the involute error offset; Δe2 represents the magnitude of the error offset of the parallelogram mechanism; Δe3 represents the magnitude of the error offset of the cycloid pinion gear transmission mechanism; Δe4 represents the magnitude of the error offset of the output disk; δe represents the total error offset of the geometric error at the output end of the reducer; Δα1 represents the force deformation of the involute gear transmission mechanism; Δα2 represents the force deformation of the cycloid wheel and the pinion; Δα3 represents the force deformation of the contact surface between the crank bearing and the cycloid wheel; Δα4 represents the total deformation of the crankshaft.

[0099] Finally, according to Figure 6 The MATLAB program execution flow is used to calculate the geometric transmission error ( Figure 2 ) and stress deformation error ( Figure 3 ), and finally the comprehensive transmission error of the whole machine is obtained ( Figure 7 ).

Claims

1. A comprehensive transmission error modeling method for RV reducer based on RLS-PSO hybrid algorithm, characterized in that: The following steps are involved: ① Establish a mathematical model for calculating the geometric transmission error of the RV reducer, which specifically includes the following steps: Step 1.1, according to the structural characteristics of the RV reducer, decompose it into four substructures: the involute transmission mechanism, the parallelogram mechanism of the crank bearing and the cycloid wheel, the transmission mechanism of the cycloid wheel and the pin gear, and the output disk of the planet carrier; Step 1.2, the geometric transmission error calculation formula of the involute transmission mechanism Δe1: In the formula, e s is the assembly eccentricity error of the sun gear; p E is the assembly eccentricity error of the planetary gear; s is the error offset between the theoretical geometric center of the sun gear and the actual assembled geometric center; θ1 is the rotation angle of the sun gear; θ2 is the rotation angle of the planetary gear; β b1 is the angle between the offset of the sun gear and the x-axis of the generalized coordinate system; β b2 is the angle between the offset of the planetary gear and the x-axis of the generalized coordinate system; β n E is the angle between the meshing line between the sun gear and the planetary gear and the x-axis; p is the error offset between the theoretical geometric center of the planetary gear and the actual geometric center; n1 is the actual transmission direction vector of the sun gear and the planetary gear; Δe1 represents the error vector in the transmission direction between the sun gear and the planetary gear; Step 1.3, the geometric transmission error calculation formula of the parallelogram transmission mechanism Δe2: In the formula, s c Represents the local coordinate system x c o c y c The vector of the geometric center of the eccentric circle of the middle bearing hole; e R is the manufacturing error of the bearing hole radius; θ' c Indicates the size offset of the bearing hole and the local coordinate system x c o c y c x c The angle between the axes; n2 represents the transmission direction vector of the parallelogram mechanism, which is the tangent direction of the geometric center distribution circle of the cycloid wheel bearing hole; M oc Represents the local coordinate system S c To the generalized coordinate system S o The coordinate transformation matrix; e is the crank arm; θ represents the rotation angle of the cycloid wheel; θ g It represents the input angle of the cycloid wheel, i.e. the rotation angle of the crank arm; Step 1.4, the geometric error calculation formula Δe3 of the transmission mechanism of the cycloid wheel and the pin gear: The tooth profile of the pinion tooth in the generalized coordinate system f It is expressed as: Where K1 = eZ p / R p represents the short amplitude coefficient; r represents the radius of the needle tooth; R p Indicates the radius of the needle tooth distribution circle; Theoretical tooth profile vector r of cycloid gear c : Where M co Denotes the generalized coordinate system S o Transform to the local coordinate system S c The coordinate transformation matrix of Z c Indicates the number of cycloid gear teeth; r c (θ g ) represents the crank arm input angle θ g The theoretical tooth profile vector of the cycloid wheel is the independent variable; In the generalized coordinate system, the position vector r of the theoretical contact point between the cycloid wheel and the i-th pin tooth is mi for: Where i represents the y coordinate system from the generalized coordinate system o The i-th pin tooth counted in the counterclockwise direction of the axis; Z p Indicates the number of needle teeth; r mi (θ g ) represents the crank arm input angle θ g is the theoretical contact point position vector of the i-th pin tooth of the independent variable; Conditions for continuous contact between the cycloid wheel and the needle: in: Where n mi (θ g ) is the crank arm input angle θ g is the normal vector of the contact point between the ith pin tooth and the cycloid wheel of the independent variable (the transmission direction vector between the cycloid wheel and the pin tooth on the pin tooth profile); n c (θ g ) is the crank arm input angle θ g is the normal vector of the cycloid gear surface of the independent variable (the transmission direction vector of the cycloid wheel and the pin teeth on the cycloid wheel tooth profile); In order to simulate the transmission error of the reducer caused by the manufacturing error and assembly error of the pin teeth, r and R in equation (4) are p Add offsets δr and δR respectively p ;δr represents the manufacturing error of the needle tooth radius;δR p represents the manufacturing error of the radius of the needle tooth distribution circle; combined with the contact judgment condition of formula (7), the input angle θ of the theoretical cycloid wheel without adding the offset is calculated g And the angle θ' after adding the variable g ; The calculation formula of the transmission error of the cycloid pin-tooth transmission mechanism Δe3: In the formula, θ6 represents the theoretical output angle of the reducer output end; Step 1.5, the geometric error calculation formula of the planet carrier output disk Δe4: Δe4=|(and pq +e pa )n3|(10) In the formula, δP a Indicates the eccentricity error of the output disk; δP q The eccentricity error of the crankshaft hole on the planet carrier; β3 represents the offset δP a The angle with the x-axis of the generalized coordinate system s0; β p Indicates the offset δP q The angle between the x-axis and the generalized coordinate system; α p is the angle between the line connecting the center of the planetary gear and the origin of the generalized coordinate system and the x-axis; Indicates the relative position of the geometric center of the cycloid gear; (j represents the jth cycloid gear, j = 1, 2); n3 represents the transmission direction vector of the output disc (i.e. the tangent direction of the crankshaft to the output disc), θ2 is the rotation angle of the planetary gear; e pq Indicates the offset error of the crank hole; e pa Indicates the offset error of the geometric center of the output disk; ② Establish the stress deformation calculation model of the key components of the reducer. The specific steps are as follows: Step 2.1, the force deformation calculation formula of the involute transmission mechanism Δα1: In the formula, k sp is the meshing stiffness of the spur gear; s N is the deformation during meshing; r s Pitch radius of the sun gear; F pi is the force on the i-th planetary gear from the sun gear; T1 represents the input torque at the input end of the reducer; L represents the number of planetary gears; Step 2.2, establish the contact deformation calculation formula Δα2 between the cycloid wheel and the pin gear: d ij represents the distance between the contact surface of the i-th pin tooth and the j-th cycloid gear tooth; In the formula, (x fc ,y fc ) represents the coordinates of the intersection of the contact force and the crank arm; (x c ',y c ') represents the coordinates of the maximum compression point on the cycloid wheel tooth profile when the cycloid wheel and the i-th pin tooth are compressed and deformed; n x ' and n y ' represents the projection of the normal vector of the maximum compression point of the cycloid wheel and the needle tooth on the x-axis and y-axis; F i represents the contact force between the i-th pin tooth and the cycloid wheel; α ij The contact force F i The angle between the crank arm line and the crank arm line; Δθ i is the torsion angle caused by the elastic deformation of the i-th pin tooth and the cycloid wheel; l i The contact force F i The force arm; δ i Indicates the elastic deformation; K cp represents the meshing stiffness between the cycloid wheel and the pinion gear; T2 is the torque of the pinion gear on the cycloid wheel; (x pai ,x pai ) represents the geometric center coordinates of the i-th needle tooth; x mi and mi The coordinates of the theoretical contact point between the cycloid wheel and the i-th pin tooth can be obtained by equation (6); Step 2.3, the calculation formula of the force deformation of the crank bearing and the cycloid wheel Δα3: In the formula, the force exerted by the crank bearing on the cycloid gear is decomposed into three directions: the x-axis direction, the y-axis direction, and the direction of the line connecting the crankshaft geometric center in the cycloid gear geometry. The decomposed forces in the three directions are expressed as F z1 、F z2 and F z3 ; r b F is the pitch radius of the cycloid wheel; bx and F by It represents the decomposition of the resultant force of the pin teeth on the cycloid wheel on the x-axis and y-axis of the generalized coordinate system; n p Indicates the number of crankshafts; F indicates the force vector of the cycloid wheel on the crankshaft bearing; F r and F t It means that the force vector |F| of the cycloid wheel on the crank bearing is decomposed into the local coordinate system S fixed to the bearing. b x b Axis and y b Axis direction; Δδ b It indicates the displacement of the crank bearing caused by the force of the cycloid wheel; Ks indicates the meshing stiffness between the crank bearing and the cycloid wheel; Step 2.4, the crankshaft stress deformation calculation formula Δα4: In the formula, F t1 and F r1 Indicates the force exerted by the planetary gear on the crankshaft; F t2 and F r2 Indicates the tangential force and radial force of the input disc of the planet carrier on the crankshaft; F t5 and F r5 Indicates the tangential force and radial force of the output disc of the planetary carrier on the crankshaft; F r' and F t' represents the radial force and tangential force of the cycloid wheel on the crank bearing (same as F in formula (16) r and F t equal in magnitude and opposite in direction); L a Indicates the length of each section of the crankshaft (a=1,2,3,4); Where ω1 and ω2 represent the flexibility of the two bearing sections of the crankshaft respectively; ω3 represents the flexibility of the output end of the crankshaft; I b represents the section inertia moment of the crankshaft in section b, b = 1, 2, 3, 4; G represents the shear elastic modulus; E represents the elastic modulus; ψ q Represents the sum of the torsion angles of each section of the crankshaft; T 3i represents the torsion borne by the i-th section of the crankshaft; ③ Establish the RLS-PSO hybrid optimization processing algorithm of the input torque of the force-deformation model. The specific steps are as follows: Step 3.1, design a v-order filter, the RLS filter output is: in Formula (19) can be expressed as: Where M represents the amount of data of the original signal to be processed; T M (l) indicates that the original input signal is a vector consisting of M elements, which represents the input force and deformation, i.e., the input torque data vector; T(M-1) indicates the input torque value at the M-1th moment; h M (v) represents the coefficient vector of the filter; h(M-1,v) represents the coefficient of the filter output function y(v) at the M-1th moment; v represents the order of the corresponding filter, or represents the vth equation of the filter linear equation system; where h t M (v) indicates h M (v) is the transpose of the matrix. The upper left corner t represents the transpose operation of the matrix. In order to linearize the original signal and ensure the uniqueness of the optimized signal, v = M. M is determined according to the amount of data collected by the test bench. Step 3.2, according to the principle of recursive least squares method: The vector of the actual input torque data collected is T M (l), l=1,2,…,M; at this time, the expected output is expressed as Or represents the required optimized signal data; l represents the input torque value at the lth moment in the input torque vector; then the error between the original signal and the expected signal is expressed as: The weighted sum of squared error margins ΔM: Wherein, w represents the weighting factor, which is used for the coefficient of the filter to adapt to the time-varying signal data, and vl represents the weight of the data point at the lth moment, which is achieved by assigning the index vl to the past data; The coefficient vector h of the RLS filter M The update process of (v) is: Where, T t M (l) indicates T M The transpose of (l); T * M (l) indicates T M The conjugate transpose of (l); R M (v) represents the original signal autocorrelation matrix, where R -1 M (v) is R M The inverse matrix of (v); h M (v-1) represents the coefficient vector of the v-1th equation; P M (v-1) is an inverse autocorrelation matrix composed of v-1 original signal data, which provides adaptive data for optimizing the vth data and updating the coefficient vector; h M (v) Through the coefficient vector of the v-1th equation and the signal error e M (l,v) updates the coefficient; u M (v) is the updated data of the weighting factor; K M (v) is the Kalman gain vector; Step 3.3: Initialize the filter coefficient vector as h M (0)=[H, 0, 0, …, 0] T , (0 < H ≤ 1); Taking the condition that ΔM reaches the minimum value, write the RLS algorithm using MATLAB and calculate the optimized data of the RLS filtering algorithm for the input torque at each moment; Step 3.4: Update the coefficient vector by combining equations (20) and (22), and repeat step 3.3 to optimize the data at the next moment, and finally store the optimized data at each moment as the RLS optimized data vector D; Step 3.5: Construct the fitness function of the PSO algorithm and use the mean square error (MSE) as the fitness value. The formula is as follows: Where D i The i-th input torque representing the input torque data vector may also be referred to as the input torque vector after RLS processing; D' i The input torque data of the expected output; R represents the number of particles; Update the i-th particle; the update algorithm is as follows: Where r1 and r2 represent random functions generated from [0,1] during the MATLAB compilation process; θ outi It indicates the output angle of the reducer corresponding to the i-th input torque value after update, and the maximum value is 360 degrees. If θ outi >360 degrees, then θ outi The value is 360°, which is the speed of updating and optimizing the search for the optimal data of the current i-th particle; θ ex_outi Indicates the output angle of the reducer corresponding to the previous torque value of the i-th input torque value; c1 and c2 are learning factors, usually with a value of 2; W is the inertia factor, whose value is greater than 0; pb exi represents the individual optimal value before the i-th input torque data; gb exi It represents the optimal value of the group before the i-th input torque data, that is, the optimal value from the best point of the population; D i Indicates the updated position of the particle in the particle swarm algorithm, which can also be called the updated data of the i-th input torque; D exi The particle position representing the previous position of the i-th input torque data may also be referred to as the previous torque data of the i-th input torque; Step 3.6: Test the particle swarm optimization of the input torque data multiple times; set the number of particles R in the range of 20-100; set the target value range of the mean square error MSE to 0 <MSE<10 -5 ; Perform iterative calculations in the corresponding number of particles, optimize a series of individual optimal values ​​and select the extreme value from the individual optimal values ​​as the global optimal value; store the global optimal value as an element of the input torque vector optimized by the RLS-PSO hybrid algorithm, and finally constitute the input torque vector of the RLS-PSO hybrid algorithm; Step 3.7, the input torque data described above is obtained according to the RV reducer transmission error test bench; sensors are installed at the input and output ends of the RV reducer to collect the input torque T at the input end and the output angle θ at the output end. out And the input angle θ at the input end in ; The test bench calculates the transmission error ΔE of the reducer as follows: In the formula, I is the transmission ratio of the reducer; The output angle of the output end collected in the test is used as the independent variable, and the input torque of the input end is used as the dependent variable. The original input torque data is substituted into the RLS-PSO hybrid algorithm for optimization, and the output angle and the optimized input torque are substituted into the geometric error model and the force deformation model; the calculation method of the transmission error value TE of the whole machine is: In the formula, r b Indicates the distance between the geometric center of the bearing hole and the geometric center of the cycloid wheel; r d Indicates the distance between the geometric center of the bearing hole and the geometric center of the output disc; i b It represents the transmission ratio from the output end to the pinion when the involute gear is fixed, i b =-(1-I) / I; as described in claims ① and ②, Δe1 represents the size of the involute error offset; Δe2 represents the size of the error offset of the parallelogram mechanism; Δe3 represents the size of the error offset of the cycloid pinion gear transmission mechanism; Δe4 represents the size of the error offset of the output disk; δe represents the total error offset of the geometric error at the output end of the reducer; Δα1 is the force deformation of the involute gear transmission mechanism; Δα2 is the force deformation of the cycloid wheel and the pinion; Δα3 represents the force deformation of the contact surface of the crank bearing and the cycloid wheel; Δα4 represents the total deformation of the crankshaft.

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