Optimal robust input shaper, design method, control system structure and applications

Abstract

The present application belongs to the technical field of multi-axis motion control, and discloses an optimal robust input shaper, a design method, a control system structure and an application. The method comprises the following steps: establishing an input shaper robustness evaluation index under the condition that the inherent frequency and the damping ratio parameters change simultaneously; designing an improved pulse limit constraint, selecting the number of pulse correction coefficients to be added; establishing an objective function optimization model containing the to-be-solved correction coefficient group; calculating the maximum area of the pulse correction coefficient by using a particle swarm algorithm, searching for the pulse correction coefficient group; updating the amplitude of the input shaper, and obtaining the corresponding optimal robust input shaper. The present application improves the robustness of the input shaper without affecting the production efficiency of the multi-axis motion equipment. The robustness evaluation index about the simultaneous change of the inherent frequency and the damping ratio double parameters is established, which is beneficial to the quantitative analysis of the robustness of the input shaper and the design of the robust input shaper.

Description

Technical Field

[0001] This invention belongs to the field of multi-axis motion control technology, and particularly relates to an optimal robust input shaper, its design method, control system structure, and application. Background Technology

[0002] Multi-axis motion platforms are the foundation for achieving automated industrial production. However, because most multi-axis motion platforms contain flexible components such as couplings, reducers, and drive shafts in their mechanical transmission devices, residual vibrations can easily occur on the load side of the mechanical equipment after rapid positioning movements. This not only affects the surface quality of the processed products but also reduces the service life of parts and production efficiency.

[0003] To suppress residual vibration, many technologies exist, among which input shaping technology is widely used in industrial production due to its advantages such as simple structure, strong scalability, and good vibration reduction effect. Filters designed using input shaping technology are called input shapers. Commonly used input shapers in multi-axis motion equipment include ZV, ZVD, and EI shapers. More robust input shapers include ZVDD, ZVDDD, and multi-peak EI (two-peak, three-peak, etc.). However, existing technologies have drawbacks or limitations:

[0004] (1) The current traditional input shapers such as ZV, ZVD, and EI mainly consider the constraint of the system's natural frequency in their robustness design. However, in actual industrial applications, the load change of the multi-axis motion platform will cause the system's natural frequency and damping ratio (represented by "two parameters") to change simultaneously. The simultaneous change of the two will make the residual vibration suppression effect of the traditional input shaper worse. Therefore, the traditional input shaper cannot well meet the robustness of the two parameter changes, and has not made optimal design for the robustness of the two parameter changes.

[0005] (2) In order to improve the robustness of ZV, ZVD, and EI shapers to changes in inherent frequency parameters, traditional design methods have also designed shapers such as ZVDD, ZVDDD, and multi-peak EI. Their biggest drawback is that while improving robustness, they also increase the lag time of the input shaper, which increases the total motion time and greatly reduces production efficiency. Summary of the Invention

[0006] To overcome the problems existing in related technologies, the present invention discloses an optimal robust input shaper, design method, control system structure and application, specifically relating to an optimal robust input shaper and design method for residual vibration suppression of multi-axis motion platforms.

[0007] The technical solution is as follows: a design method for an optimal robust input shaper, comprising the following steps:

[0008] S1, identify the vibration mode parameters of the mechanical transmission device of the multi-axis motion platform, and obtain the modeling natural frequency and modeling damping ratio;

[0009] S2 calculates the amplitude and timing of each pulse in the traditional input shaper;

[0010] S3, obtain the robustness evaluation index of the traditional input shaper under the condition that the modeling natural frequency and modeling damping ratio parameters change simultaneously, as shown in the two-parameter robustness evaluation index formula of the input shaper.

[0011] S4. Design an improved pulse constraint using the optimized pulse amplitude constraint formula, and select the number of pulse correction coefficients to add.

[0012] S5, Establish the robust optimization model of the input shaper, as shown in the formula, to optimize the objective function containing the set of correction coefficients to be determined;

[0013] S6. Calculate the area of ​​the region that maximizes the pulse correction coefficient using the particle swarm optimization algorithm, and search for the pulse correction coefficient set that satisfies the objective function of the robust optimization model formula of the input shaper to maximize the area of ​​the region.

[0014] S7. Substitute the pulse correction coefficient set into the formula for optimizing pulse amplitude constraint to update the amplitude of the input shaper and obtain the corresponding optimal robust input shaper.

[0015] In step S3, the formula for the two-parameter robustness evaluation index of the input shaper is:

[0016]

[0017] In the formula, V tol D represents the maximum permissible percentage of residual vibration in industrial applications. tol Not greater than the residual vibration percentage V tol The region where the natural frequency and damping ratio of the system vary is S = D. tol The area of ​​the irregular region, dω is a infinitesimal element with respect to ω, dζ is a infinitesimal element with respect to ζ, ω is the actual natural frequency of the system, ζ is the actual damping ratio of the system, and V(ω,ζ) is the percentage of residual vibration.

[0018] In step S4, the formula for optimizing the pulse amplitude constraint is:

[0019] A ORIS,i =A i +T i (a),i=1,2,3...n formula (13)

[0020] In the formula, A ORIS,i For the amplitude of the i-th pulse of the optimal robust input shaper, T i(a) is a function of any number of pulse correction coefficients, where n is the number of input shaper pulses; the amplitude of the optimal robust input shaper consists of the amplitude of the conventional input shaper and a series of pulse correction coefficients a, where a = [a1, a2…a2]. r ], r is the number of pulse correction coefficients added, A i The pulse amplitude of the traditional input shaper.

[0021] In step S4, the number of pulse correction coefficients to be added includes:

[0022] Let the initial value of the number r of pulse correction coefficients be n-2, where n≥2. Jump to steps S5-S6 for subsequent optimization design. The first obtained maximum area of ​​the region is S1=max{S(a)}. Then, increase the number of coefficients r+1 and jump to steps S5-S6 for subsequent optimization design. The second obtained maximum area of ​​the region is S2. Calculate the growth rate of the second maximum area of ​​the region relative to the previous maximum area. The expression is:

[0023]

[0024] In the formula, μ1 is the growth rate of the maximum area of ​​the region in the second time relative to the maximum area in the previous time, S1 is the maximum area of ​​the region obtained in the first time, and S2 is the maximum area of ​​the region obtained in the second time.

[0025] μ is obtained by iterative calculation m The termination condition is μ m ≤μ s μ s Set this value according to different operating conditions, not exceeding 5%, the default value is 1%, μ m μ is the growth rate after the m-th cycle. s To meet the growth rate requirements under this operating condition.

[0026] In step S5, the robustness optimization model of the input shaper, as shown in the formula, includes the objective function optimization model containing the set of correction coefficients to be determined, which includes:

[0027] Substituting the formula for optimizing the pulse amplitude constraint into the formula for the two-parameter robustness evaluation index of the input shaper, we obtain the following formula for the robustness evaluation function of the traditional input shaper with respect to the two-parameter variation under the constraint of varying pulse amplitude:

[0028]

[0029] In the formula, S(a) is the objective function optimization model containing the set of correction coefficients to be determined, and D... tol Not greater than the residual vibration percentage V tolThe region of variation of the system's natural frequency and damping ratio, dω is a differential element with respect to ω, dζ is a differential element with respect to ζ, ω is the actual natural frequency of the system, ζ is the actual damping ratio of the system, (ω,ζ) are all the variation points within the region of variation of the system's natural frequency and damping ratio, V(ω,ζ,a) is the percentage of residual vibration after adding pulse amplitude constraints; Based on formula (15), establish the robustness optimization model formula (16) of the input shaper:

[0030]

[0031] stD tol ={(w,ζ)|V(w,ζ,a)≤V tol}

[0032] ω min ≤ω≤ω max ,0≤ζ≤ζ max

[0033] -0.5 i,min <a<a i,max <0.5

[0034] In the formula, i = 1, 2, ..., r, a i,min The upper boundary value of the correction coefficient a is given by a. i,max To determine the lower boundary value of the correction coefficient 'a', the sum of the amplitudes of all pulses must equal 1, and the coefficient 'a' must be... i The upper boundary is less than 0.5; while the amplitude of a single pulse is greater than 0, and the coefficient a i The lower boundary is greater than -0.5; ω max As an upper bound for allowing variations in the system's natural frequency, ω min ζ is the lower bound that allows for variations in the system's natural frequency. max The upper bound for the allowable variation in the system damping ratio is max S(a), which represents the maximum robustness evaluation index of the pulse amplitude correction coefficient group to be determined.

[0035] In step S6, the particle swarm optimization algorithm includes:

[0036] The velocity update formula (17) and position update formula (18) of the d-th dimension of the i-th particle are respectively:

[0037]

[0038]

[0039] 1≤d≤D

[0040] In the formula, ​λ is the d-th dimension component of the velocity vector of the i-th particle after k+1 iterations. λ is a weighting coefficient that is not less than 0 and is used to adjust the search range of the solution space. c1 and c2 are acceleration constants used to adjust the maximum learning step size. r1 and r2 are random functions with values ​​in the range [0,1] to increase the randomness of the search. The optimal position experienced by an individual particle. This is the best position experienced globally. Let d be the d-th component of the velocity vector of the i-th particle after k iterations; It is the d-th dimension component of the position vector of the i-th particle after k iterations; The d-th dimension component of the position vector of the i-th particle after k+1 iterations; d is the number of components contained in the particle; D is the total number of components contained in the particle. Let d be the d-th dimension component of the position vector of the i-th particle after k+1 iterations;

[0041] λ must satisfy:

[0042]

[0043] In the formula, λ max λ represents the maximum value of the weighting coefficients. min k is the minimum value of the weighting coefficient. max λ is the maximum number of iterations. k+1 These are the weight coefficients after k+1 iterations;

[0044] In formula (16), the amplitude correction coefficient a for the i-th pulse to be determined is... i The position x of the i-th particle in the search space i The optimal global position gbest, satisfying the objective function formula (16), was found using the particle swarm optimization algorithm. i , where a is the required pulse correction coefficient set. i .

[0045] Another object of the present invention is to provide an optimal robust input shaper, which is designed using the design method of the optimal robust input shaper.

[0046] Furthermore, the optimal robust input shaper includes: the optimal robust input shaper ORIS-ZVD and the optimal robust input shaper ORIS-EI.

[0047] Furthermore, the multi-axis motion device control system structure inputs command x r (t) represents any industrial production motion trajectory. The motor control is assumed to be ideal. The dynamic relationship between the base and mass block m in the equivalent flexible system of the mechanical transmission device, after Laplace transform, is expressed by the transfer function G. 12 express:

[0048]

[0049] In the formula, x e (t) represents the position of the mass block relative to the base, y r (t) is derived from x r (t) The reference signal after being shaped by the optimal robust input shaper is used to suppress the transfer function G. 12 The residual vibration of a typical flexible second-order system after the motion ends, x b (t) represents the absolute position of the base, ω n For the undamped modeling natural frequency, ζ n To model the damping ratio, both the undamped modeling natural frequency and the modeling damping ratio are obtained by identifying the vibration modal parameters, X. e (s) is x e Laplace transform of (t), X b (s) is x b The Laplace transform of (t), where s is the Laplace transform parameter.

[0050] Another objective of this invention is to provide an application of the design method of the optimal robust input shaper in a PLC / motion control card. The design method of the optimal robust input shaper is encapsulated in the lower-level software of the PLC / motion control card, and the DSP in the hardware is used for calculation to control the operation of the mechanical transmission device of the multi-axis motion platform.

[0051] Combining all the above technical solutions, the advantages and positive effects of this invention are as follows: This invention proposes an optimal robust input shaper design method, which maximizes the robustness of traditional input shapers to simultaneous changes in both the system's natural frequency and damping ratio, ensuring residual vibration suppression under larger parameter variations (residual vibration percentage controlled below 5%). The optimal robust input shaper design method provided by this invention improves the robustness of the input shaper without increasing the lag time of traditional input shapers, and does not affect the production efficiency of multi-axis motion equipment. This invention establishes a robustness evaluation index for simultaneous changes in both natural frequency and damping ratio, which is beneficial for quantitative analysis of the robustness of input shapers and the design of robust input shapers. Attached Figure Description

[0052] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this disclosure and, together with the description, serve to explain the principles of this disclosure;

[0053] Figure 1 This is a flowchart of the design method for the optimal robust input shaper for residual vibration suppression provided in this embodiment of the invention;

[0054] Figure 2 This is a traditional ZVD method for ω / ω provided in the embodiments of the present invention. n Simulation diagram of the insensitive region of ζ;

[0055] Figure 3 The ORIS-ZVD method of the present invention, provided in this embodiment, is for ω / ω n Simulation diagram of the insensitive region of ζ;

[0056] Figure 4 This is a traditional EI for ω / ω provided in the embodiments of the present invention. n Simulation diagram of the insensitive region of ζ;

[0057] Figure 5 The ORIS-EI of the present invention, provided in the embodiments of the present invention, is for ω / ω n Simulation diagram of the insensitive region of ζ;

[0058] Figure 6 This is a schematic diagram of the structure of a multi-axis motion device control system with an optimal robust input shaper provided in an embodiment of the present invention. Detailed Implementation

[0059] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of the present invention. However, the present invention can be practiced in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0060] To facilitate a better understanding of the technical innovations of this invention, a brief analysis of the existing technology is presented below.

[0061] (1) Input Shaping Technology Principle: For a typical low-damping, low-stiffness second-order system, when the input is a unit pulse signal, the system responds with a sine curve whose amplitude decays. If the input is a two-pulse signal, the first pulse signal is A1δ(t-t1), which acts at time t1, resulting in residual vibration. By setting the amplitude and time of the two pulses, the vibration trajectory obtained by the second pulse A2δ(t-t2) at time t2 is made to be commensurate in magnitude and opposite in orientation to the vibration trajectory obtained by the first pulse at the same time. Then, from time t2 onwards, the sum of the pulse response amplitudes of A1δ(t-t1) and A2δ(t-t2) equals zero, thus completely eliminating the system's vibration. The process of removing residual vibration from the system using two-pulse signals is described in Appendix. Figure 1 If these two pulses A i δ(tt i(i = 1, 2), and convolve with the input instruction u(t-τ) to obtain the shaped signal A. i u(tt i -τ) is used as a new input signal to drive the system, thus eliminating residual vibrations. Input shaping is a process of convolving the original system input command with a specified pulse time-delay signal to obtain a new shaped input signal. The design of the input shaper involves adjusting the amplitude A. i and time position t i The design.

[0062] (2) Input Shaper Design: The input shaper can be represented in the complex frequency domain as follows:

[0063]

[0064] The evaluation index for the residual vibration suppression level of the input shaper is the residual vibration percentage V(ω,ζ), which is defined as the ratio of the total response (numerator) of the second-order system with the input shaper added to the unit impulse response (denominator) of the second-order system without the input shaper added.

[0065]

[0066] in,

[0067]

[0068]

[0069] In the formula, ω is the actual natural frequency of the system, ζ is the actual damping ratio of the system, and A i It is the amplitude of the i-th pulse of the input shaper, that is, the pulse amplitude of the traditional input shaper, t i It is the i-th time position.

[0070] The input shaper is designed around a residual vibration percentage function, but with different constraints.

[0071] 1) ZV Input Shaper (Zero Vibration Input Shaper)

[0072] When the model of the controlled system can be accurately obtained, it is only necessary to set the percentage of residual vibration to V(ω). n ,ζ n =0, and with the pulse amplitude limit and time optimal limit conditions, the system of constraint equations is as shown in formula (5). The input shaper designed in this way is called the ZV (Zero Vibration) input shaper, which contains two pulses, and the amplitude and time lag are as shown in formula (6).

[0073]

[0074]

[0075] in,

[0076]

[0077] In the formula, ω n This is the identification result of the system's natural frequency parameters, also known as the modeling natural frequency of the nominal model; ζ n This is the identification result of the system damping ratio parameter, also known as the modeling damping ratio of the nominal model; both are design values ​​input to the shaper, generally used for simplified design, with a default value of ζ. n =0.

[0078] 2) ZVD Input Shaper (Zero Vibration Derivative Input Shaper):

[0079] When there is a certain error in the model establishment of the controlled system, that is, when the damping coefficient and natural oscillation frequency of the second-order system are not accurately obtained, the vibration suppression effect of the ZV input shaper will deteriorate. In order to enhance the robustness of the vibration suppression algorithm, a constraint condition needs to be added to make the derivative of the residual oscillation percentage with respect to the natural frequency zero. The constraint condition equations are shown in Equation (7). This can ensure the vibration suppression effect when there is an error between the system's natural frequency identification result and the true value of the natural frequency. The input shaper designed in this way is called the ZVD (Zero Vibration and Derivative) input shaper, which contains three pulses. The amplitude and time lag are shown in Equation (8). The ZVDD, ZVDDD and other input shapers are based on ZVD and set second-order and third-order derivative constraints on V(ω,ζ) to calculate the amplitude and time position. They have more pulses. Each additional pulse brings a time delay of half a vibration cycle, thus improving the robustness to the natural frequency while increasing the lag time.

[0080]

[0081]

[0082] 3) EI Input Shaper (Ultra-insensitive Input Shaper):

[0083] In practical engineering applications, it is not required that the zero point of the input shaper exactly cancels the poles of the controlled system; it is only necessary to configure the zero point near the poles that need to be canceled. tol The maximum permissible percentage of vibration for the system is given, and the permissible range of the system's natural frequencies is ω1-ω2. The boundary conditions are defined by the single-peak EI input shaper design as shown in formula (10):

[0084]

[0085] The EI (Extra-Insensitive) input shaper contains three pulses. When the damping coefficient is zero, the amplitude and time lag are as shown in formula (11). In the design of a single-peak EI input shaper, the curve shape with a peak in the middle part is called a single-peak EI shaper. Based on this, there are also multi-peak EI shapers, which also have more pulses, thus increasing the lag time.

[0086]

[0087] (3) Robustness analysis of the input shaper:

[0088] Various input shapers designed using constraint equations exhibit a degree of insensitivity to system parameter modeling errors, and the normalized system natural frequency ω / ω n The percentage of residual vibration of the input shaper for different input shapers is a function of the percentage of residual vibration of the input shaper ZV, ZVD, and EI when the system's natural frequency changes. The corresponding parameter insensitivity (robustness) can be achieved by using a value less than the allowable vibration level V. tol The effective range of the input shapers is evaluated based on their width. As shown in the figure, when the allowable residual vibration percentage is 5%, the effective ranges of the ZV, ZVD, and EI input shapers are 0.064, 0.287, and 0.399, respectively (design parameters of the input shapers:). ζ n=0), gradually increasing, and the EI input shaper has the largest effective range and the best robustness; however, theoretical analysis shows that compared to the ZV input shaper, the ZVD and EI shapers increase the time delay by half a vibration cycle. Compared to the ZV input shaper, the ZVD and EI shapers significantly improve the robustness of the input shaper in residual vibration suppression against the system's natural frequency at the cost of a smaller lag time (acceptable for industrial production). Therefore, they are widely used in multi-axis motion equipment such as industrial multi-axis robotic arms, cranes, dispensing machines, and spacecraft. Their characteristic is that during normal operation, the natural frequency parameters of the mechanical transmission device change continuously within a certain range, which affects the effectiveness of other open-loop vibration suppression methods such as the ZV input shaper. However, the use of ZVD and EI shapers can ensure the residual vibration suppression effect within a certain range of natural frequency variation, that is, ensure that the maximum amplitude of residual vibration is within the allowable range. Specifically, in the field of industrial multi-axis robotic arms, input shapers can be used to suppress residual vibrations of the output shaft caused by flexible mechanical transmission devices such as RV reducers / harmonic reducers at the joints, thereby reducing the positioning error at the end of the robotic arm. More robust input shapers, such as ZVD and EI shapers, can ensure residual vibration suppression when the natural frequency parameters change due to changes in the configuration of the robotic arm or the end effector load. In cranes, more robust input shapers are widely used because cranes encounter two problems when lifting different weights with ropes: rope sway and the continuous changes in rope length and the mass of the load. Both of these problems can be solved by using more robust input shapers such as ZVD and EI shapers. Dispensing machines are often used in the dispensing and coating processes of chip manufacturing, thus requiring precise control and suppression of residual vibration. Furthermore, the quality of the adhesive in the dispensing valve changes continuously during multiple dispensing processes, necessitating the use of robust input shapers. Input shapers can also suppress torsional vibrations in spacecraft, as well as residual vibrations under changes in configuration and external loads.

[0089] Traditionally designed input shapers exhibit a degree of insensitivity to the system's natural frequency and robustness to changes in the system's damping ratio. Further considering the impact of simultaneous variations in both the system's natural frequency and damping ratio on the input shaper's vibration suppression capability, this can be reflected by three-dimensional sensitivity surface plots of the system frequency and damping ratio. The sensitivity variations of different input shapers under these two parameter changes include the three-dimensional sensitivity surfaces of ZVD and EI input shapers and the corresponding insensitive regions. The values ​​on the contour lines represent the percentage of residual vibration, and the insensitive regions include those with residual vibration percentages below 5%. From the sensitivity variations of different input shapers under these two parameter changes, it is evident that when the system's natural frequency and damping ratio change simultaneously, there is still a region that meets the system's allowable residual vibration level, indicating that ZVD and EI input shapers possess a certain degree of robustness to changes in both parameters.

[0090] Example 1: The design principle of the optimal robust input shaper for residual vibration suppression provided in this embodiment of the invention includes:

[0091] (1) Establish the two-parameter robustness evaluation index of the input shaper (the robustness evaluation index of the input shaper with respect to the natural frequency and damping ratio parameters (hereinafter referred to as "two parameters")) as follows:

[0092]

[0093] In the formula, V tol D represents the maximum permissible percentage of residual vibration in industrial applications. tol Not greater than the residual vibration percentage V tol The region where the natural frequency and damping ratio of the system vary is S = D. tol The area of ​​the irregular region, dω is a infinitesimal element with respect to ω, dζ is a infinitesimal element with respect to ζ, ω is the actual natural frequency of the system, ζ is the actual damping ratio of the system, and V(ω,ζ) is the percentage of residual vibration, i.e., formula (2).

[0094] Traditional input shapers calculate the amplitude and timing of each pulse based on constraints. The optimal robust input shaper, however, is designed by modifying only the pulse amplitude constraints, based on the traditional input shaper. Its amplitude is determined by the amplitude of the traditional input shaper and a series of pulse correction coefficients (a = [a1, a2…a2]). r The pulse amplitude constraint condition is composed of (where r represents the number of pulse correction coefficients added), and the formula for optimizing the pulse amplitude constraint is expressed as:

[0095] A ORIS,i =A o +T i (a),i=1,2,3...n formula (13)

[0096] In the formula, AORIS,i For the amplitude of the i-th pulse of the optimal robust input shaper, T i (a) is a function of any number of pulse correction coefficients, where n is the number of input shaper pulses; the amplitude of the optimal robust input shaper consists of the amplitude of the conventional input shaper and a series of pulse correction coefficients a, where a = [a1, a2…a2]. r ], r is the number of pulse correction coefficients added, A i The pulse amplitude of the traditional input shaper.

[0097] The design of the optimal robust input shaper also needs to satisfy the pulse amplitude constraint.

[0098]

[0099] In this context, having too many pulse correction coefficients (r) increases the computational burden and is detrimental to industrial applications, while having too few will result in insufficient pulse correction and hinder optimal design. The choice of r is also closely related to n. In principle, the minimum r should be selected while ensuring sufficient optimal robust design. That is, further increasing r will only slightly change the optimal value, and the r at this point is the minimum value that meets the requirements.

[0100] Substituting the optimized pulse amplitude constraint formula (13) into the two-parameter robustness evaluation index formula (12) of the input shaper, we obtain the robustness evaluation function formula (15) of the traditional input shaper with respect to the two-parameter variation under the variable pulse amplitude constraint:

[0101]

[0102] In the formula, S(a) is the objective function optimization model containing the set of correction coefficients to be determined, and D... tol Not greater than the residual vibration percentage V tol The region of variation of the system's natural frequency and damping ratio, dω is a differential element with respect to ω, dζ is a differential element with respect to ζ, ω is the actual natural frequency of the system, ζ is the actual damping ratio of the system, (ω,ζ) are all the variation points within the region of variation of the system's natural frequency and damping ratio, V(ω,ζ,a) is the percentage of residual vibration after adding pulse amplitude constraints; Based on formula (15), establish the robustness optimization model formula (16) of the input shaper:

[0103]

[0104] stD tol ={(w,ζ)|V(w,ζ,a)≤V tol}

[0105] ω min ≤ω≤ω max ,0≤ζ≤ζ max

[0106] -0.5 i,min <a<a i,max <0.5

[0107] In the formula, i = 1, 2, ..., r, a i,min The upper boundary value of the correction coefficient a is given by a. i,max To determine the lower boundary value of the correction coefficient 'a', the sum of the amplitudes of all pulses must equal 1, and the coefficient 'a' must be... i The upper boundary is less than 0.5; while the amplitude of a single pulse is greater than 0, and the coefficient a i The lower boundary is greater than -0.5; ω max As an upper bound for allowing variations in the system's natural frequency, ω min ζ is the lower bound that allows for variations in the system's natural frequency. max The upper bound for the allowable variation in the system damping ratio is max S(a), which represents the maximum robustness evaluation index of the pulse amplitude correction coefficient group to be determined.

[0108] (2) Solving for the optimal robust input shaper design parameters:

[0109] As can be seen from step (1) above, the key to designing the optimal robust input shaper lies in solving a series of pulse correction coefficients that satisfy equation (16). This is achieved by continuously searching for the maximum robustness evaluation index S, i.e., the maximum D. tol The area of ​​the region corresponds to the set of pulse correction coefficients. Since the objective function of the robustness optimization model formula (16) of the input shaper is a nonlinear equation, the analytical solution of the pulse correction coefficients cannot be obtained, but it can be calculated by some nonlinear numerical solution methods, such as particle swarm optimization (PSO).

[0110] In the embodiments of the present invention, it can be understood that the combination of the above technical solutions reflects the completeness of the proposed optimal robust input shaper design method.

[0111] Specifically, such as Figure 1 As shown, the design method for the optimal robust input shaper for residual vibration suppression provided in this embodiment of the invention includes:

[0112] S1, Identify the vibration modal parameters of the mechanical transmission device of the multi-axis motion platform and obtain the modeling natural frequency ω. n And let the modeling damping ratio ζ n =0;

[0113] S2, calculate the amplitude and time position of each pulse of the traditional input shaper according to step S1; calculate the amplitude and time position according to formula (6), formula (9), and formula (11).

[0114] ​S2, obtain the robustness evaluation index of the traditional input shaper under the simultaneous change of the modeling natural frequency and modeling damping ratio parameters as shown in formula (12);

[0115] S4, design improved pulse constraint, as shown in formula (13), where the number of pulse correction coefficients r is selected (selection principle: the selection of r is also closely related to n. In principle, the minimum r is selected under the premise of ensuring the optimal robust design is sufficient. That is, further increase of r will only slightly change the optimal value. At this time, r is the minimum value that meets the requirements.

[0116] The specific operation for selecting r is as follows: Let the initial value of the number of pulse correction coefficients r be n-2, where n≥2. Jump to step S5-S6 for subsequent optimization design. The first maximum area obtained is S1=maz{S(a)}. Then, increase the number r by 1 and jump to step S5-S6 for subsequent optimization design. The second maximum area obtained is S2. Calculate the growth rate of the second maximum area relative to the previous maximum area. The expression is:

[0117]

[0118] In the formula, μ1 is the growth rate of the maximum area of ​​the region in the second time relative to the maximum area in the previous time, S1 is the maximum area of ​​the region obtained in the first time, and S2 is the maximum area of ​​the region obtained in the second time.

[0119] μ is obtained by iterative calculation m The termination condition is μ m ≤μ s μ s Set this value according to different operating conditions, not exceeding 5%, the default value is 1%, μ m μ is the growth rate after the m-th cycle. s To meet the growth rate requirements under this operating condition.

[0120] S5, establish the optimization model of the objective function S(a) containing the set of correction coefficients to be determined, as shown in formula (16);

[0121] S6. Using particle swarm optimization or other nonlinear equation numerical solution methods, calculate the maximum region area S when the pulse correction coefficient is r. Based on the optimal r determined in step S4, search for the pulse correction coefficient set that satisfies the objective function of formula (16) to maximize the region area.

[0122] In this embodiment of the invention, the particle swarm optimization (PSO) algorithm for solving formula (16) using numerical methods includes:

[0123] The velocity update formula and position update formula for the d-th dimension of the i-th particle (1≤d≤D) are as follows:

[0124]

[0125]

[0126] 1≤d≤D

[0127] In the formula, Let λ be the d-th dimension component of the velocity vector of the i-th particle after k+1 iterations. λ is a weighting coefficient that is not less than 0 and is used to adjust the search range of the solution space. It has a large value at the beginning of the iteration and decreases linearly as the number of iterations increases to achieve fast convergence of the algorithm. c1 and c2 are acceleration constants used to adjust the maximum learning step size. r1 and r2 are random functions with values ​​in the range [0,1] to increase the randomness of the search. The optimal position experienced by an individual particle. This is the best position experienced globally. Let d be the d-th component of the velocity vector of the i-th particle after k iterations; It is the d-th dimension component of the position vector of the i-th particle after k iterations; The d-th dimension component of the position vector of the i-th particle after k+1 iterations; d is the number of components contained in the particle; D is the total number of components contained in the particle. Let d be the d-th dimension component of the position vector of the i-th particle after k+1 iterations;

[0128] To satisfy this requirement, λ must satisfy:

[0129]

[0130] In the formula, λ max λ represents the maximum value of the weighting coefficients. min k is the minimum value of the weighting coefficient. max λ is the maximum number of iterations. k+1 These are the weight coefficients after k+1 iterations;

[0131] In formula (16), the amplitude correction coefficient a for the i-th pulse to be determined is... i The position x of the i-th particle in the search space i The optimal global position gbest, satisfying the objective function formula (16), was found using the particle swarm optimization algorithm. i , where a is the required pulse correction coefficient set. i .

[0132] S7. Based on step S6, substitute the pulse correction coefficient group into formula (13) to update the amplitude of the input shaper, and the corresponding optimal robust input shaper can be obtained.

[0133] It is understandable that there are two parameters that determine the input shaper: pulse amplitude and time position. The proposed scheme only changes the pulse amplitude; therefore, by substituting the correction coefficient set into the formula to update the pulse amplitude, the corresponding optimal robust input shaper can be determined.

[0134] As can be seen from the above embodiments, the present invention improves the accuracy of rapid positioning motion of multi-axis motion equipment such as dispensing machines, high-precision machine tools, and cranes under variable load conditions, thereby improving production efficiency and product quality.

[0135] This invention fills the gap in the evaluation of the robustness of open-loop vibration suppression strategies—input shapers—to changes in both the system's natural frequency and damping ratio, as well as the design of optimal robustness.

[0136] This invention overcomes technical bias. Previous studies on input shaping technology have mostly focused on single parameters such as natural frequency or damping ratio, without addressing the optimal robust input shaper design under the combined change of two system parameters caused by load variations, or defining and quantitatively analyzing the robustness evaluation index of input shapers under dual parameter changes.

[0137] Example 2, based on the design method of the optimal robust input shaper for residual vibration suppression provided in Example 1 of the present invention, can be regarded as an algorithm in industrial applications, written on a host computer or encapsulated in a software library for direct calling. Furthermore, the addition of the PSO algorithm improves the design method of the optimal robust input shaper, ultimately resulting in the design of the optimal robust input shaper.

[0138] Example 3, based on the design method of the optimal robust input shaper for residual vibration suppression provided in Example 1 of the present invention, is an optimization algorithm, not limited to the form of the input shaper (effective for ZVD, ZVDD, ZVDDD…, EI, multi-peak EI…). Here, we take the ZVD and EI input shapers commonly used in industrial motion equipment as examples. The following positive effects can be achieved:

[0139] This invention maximizes the robustness of traditional input shapers to simultaneous variations in both the system's natural frequency and damping ratio, ensuring effective suppression of residual vibration (residual vibration percentage controlled below 5%) even under larger parameter variations. Under the same initial conditions, with a residual vibration percentage of ≤5%, the robustness evaluation index values ​​for ZVD and ORIS-ZVD are 2956 and 5539, respectively. ORIS-ZVD represents an 87.38% improvement over ZVD. The robustness evaluation index values ​​for EI and ORIS-EI are 2106 and 5569, respectively. ORIS-EI represents a 164.43% improvement over EI.

[0140] The optimal robust input shaper design method proposed in this invention improves the robustness of the input shaper without increasing the lag time of the traditional input shaper, and does not affect the production efficiency of the multi-axis motion equipment. See the case simulation in Part 5. Under the same initial conditions, the motion lag time brought by the input shaper / optimal robust input shaper in formulas (23)-(26) is 0.0355s.

[0141] The design method proposed in this invention establishes a robustness evaluation index for the simultaneous variation of two parameters, natural frequency and damping ratio, which is beneficial for the quantitative analysis of the robustness of the input shaper and the design of a robust input shaper.

[0142] This invention maximizes the robustness of traditional input shapers to simultaneous variations in both the system's natural frequency and damping ratio, ensuring effective suppression of residual vibration (residual vibration percentage controlled below 5%) even under larger parameter variations. Under the same initial conditions, with a residual vibration percentage of ≤5%, the robustness evaluation index values ​​for ZVD and ORIS-ZVD are 2956 and 5539, respectively. ORIS-ZVD represents an 87.38% improvement over ZVD. The robustness evaluation index values ​​for EI and ORIS-EI are 2106 and 5569, respectively. ORIS-EI represents a 164.43% improvement over EI.

[0143] The optimal robust input shaper design method proposed in this invention improves the robustness of the input shaper without increasing the lag time of the traditional input shaper, and does not affect the production efficiency of the multi-axis motion equipment. Under the same initial conditions, the motion lag time brought by the input shaper / optimal robust input shaper in formulas (23)-(26) is 0.0355s.

[0144] The design method proposed in this invention establishes a robustness evaluation index for the simultaneous variation of two parameters, natural frequency and damping ratio, which is beneficial for the quantitative analysis of the robustness of the input shaper and the design of a robust input shaper.

[0145] Example 4, based on the design method of the optimal robust input shaper for residual vibration suppression provided in the embodiment of the present invention provided in Example 1 above, improves the robustness of the traditional input shaper to the simultaneous variation of the system's natural frequency and damping ratio. Therefore, it is suitable for the suppression of residual vibration of multi-axis motion equipment when the natural frequency and damping ratio of the mechanical transmission device vary within a wider range during normal operation. In this range, the traditional input shaper cannot reach the level of residual vibration percentage allowed in industrial production, while the optimal robust input shaper can meet the requirements due to its further improved robustness. In practical applications, for example, when a crane drops a heavy object, it suppresses the swaying caused by changes in rope length and object mass over a wider range, increasing its applicability and eliminating the need to redesign the vibration suppression method parameters; in a dispensing machine, during multiple dispensing operations, it suppresses the vibration caused by changes in the weight of the adhesive in the dispensing valve over a wider range, allowing for a further increase in the valve's capacity and the number of dispensing operations after filling the valve once, thus indirectly increasing production efficiency; in industrial multi-axis robotic arms, it suppresses residual vibrations under greater weight variations at the end effector under different working conditions, improving positioning accuracy, such as in six-axis robotic arms used for welding, and multi-axis robotic arms used for handling or palletizing, etc.

[0146] In the above embodiments, the descriptions of each embodiment have different focuses. For parts that are not described in detail or recorded in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0147] To further demonstrate the positive effects of the above embodiments, the present invention conducts the following experiments based on the above technical solutions.

[0148] Numerical simulation case study of optimal robust input shaper design method:

[0149] To verify the effectiveness of the design method of this invention, the Optimal Robust Input Shaper (ORIS) method was applied to the commonly used industrial ZVD and EI input shapers, and ORIS-ZVD and ORIS-EI were designed. The design process is as follows:

[0150] 1) Initialize variables. The calculation parameters of the particle swarm optimization algorithm PSO are initialized as shown in Table 1. Based on formula (16), the position vector x is set during the iteration process. i The motion range of the (corresponding to the pulse correction coefficient group to be determined) is [-0.5, 0.5], and the variation range of the position vector vi during the iteration process is set to [-0.05, 0.25]. The design parameters of the ZVD and EI input shapers are initialized according to the identification results of the vibration modal parameters of the multi-axis motion platform mechanical transmission device as follows:

[0151] ω n =176.815 rad / s, ζ n =0,V tol =0.05 formula (20)

[0152] The inequality constraint parameters in the initialization optimization model formula (16) are as follows:

[0153] ζ max =0.21,ω min =0.7ω n ,ω max =1.3ω n Formula (21)

[0154] The initialization of the optimization model formula (16) involves the step size h of the change in the natural frequency ω and damping ratio ζ during the numerical solution of the objective function. ω and h ζ as follows:

[0155] h ω =0.00625,h ζ =0.0025 formula (22)

[0156] Table 1. Computational parameters of the Particle Swarm Optimization (PSO) algorithm

[0157] # Parameter Value 1 <![CDATA[λ min ]]> 0.2 2 <![CDATA[λ max ]]> 0.9 3 <![CDATA[c1]]> 1.3 4 <![CDATA[c2]]> 1.7 5 D 2 6 <![CDATA[k max ]]> 200 7 N 100

[0158] 2) The pulse amplitude and time position of the ZVD and EI input shapers are obtained according to formulas (20), (9), and (11) respectively as follows:

[0159] ZVD input shaper pulse amplitude and timing position:

[0160]

[0161] EI input shaper pulse amplitude and timing position:

[0162]

[0163] 3) Based on steps S4-S6, the optimal r value of the three-pulse input shaper is determined to be 2. The improved pulse amplitude constraints of ZVD and EI are set using formula (13) as follows:

[0164] A ORIS,1 =A1+a1a ORIS,2 =A2+a2A ORIS,3 =A3-a1-a2 formula (25)

[0165] 4) The pulse correction coefficient set corresponding to the ZVD and EI input shapers is obtained using the PSO algorithm, as follows:

[0166] ZVD input shaper pulse correction coefficient group:

[0167] Formula (26): a1 = 0.079, a2 = -0.026

[0168] EI input shaper pulse correction coefficient group:

[0169] Formula (27): a1 = 0.067, a2 = -0.001

[0170] 5) Substitute the above pulse correction coefficients into formula (25), and combine them with formula (23) and formula (24) to obtain the pulse amplitudes of the ZVD and EI input shapers after improvement, and then obtain ORIS-ZVD and ORIS-EI.

[0171] from Figures 2-5 As can be seen, compared with traditional IS, ORIS significantly increases the range of regions insensitive to natural frequencies and damping ratios (the area below 5% increases), thus improving overall robustness. The size of this range, S... num It can be obtained by discrete calculation of (14). Specifically, with step sizes of ω and ζ of 0.00625 and 0.0025 respectively, the S values ​​of ZVD and ORIS-ZVD are calculated. num The values ​​were 2956 and 5539 respectively. ORIS-ZVD increased by 87.38% compared to ZVD. The S values ​​of EI and ORIS-EI... num The numbers are 2106 and 5569 respectively. ORIS-EI represents a 164.43% increase compared to EI.

[0172] Application Example 1:

[0173] like Figure 6 The structure of a multi-axis motion device control system with an optimal robust input shaper is shown below. The input command is x. r (t) represents any industrial production motion trajectory. Assuming the motor control reaches an ideal state, the dynamic relationship between the base and the mass block m in the equivalent flexible system of the mechanical transmission device can be obtained by the Laplace transform of the transfer function G. 12 express.

[0174]

[0175] In the formula, x e (t) represents the position of the mass block relative to the base, y r (t) is derived from x r (t) The reference signal after being shaped by the optimal robust input shaper is used to suppress the transfer function G. 12 The residual vibration of a typical flexible second-order system after the motion ends, x b (t) represents the absolute position of the base, ω n For the undamped modeling natural frequency, ζ n To model the damping ratio, both the undamped modeling natural frequency and the modeling damping ratio are obtained by identifying the vibration modal parameters, X.e (s) is x e Laplace transform of (t), X b (s) is x b The Laplace transform of (t), where s is the Laplace transform parameter.

[0176] Application Example 2:

[0177] The optimal robust input shaper design method proposed in this invention can be encapsulated in lower-level software such as PLCs / motion control cards, and the calculations can be performed using the DSP in the hardware. Only the permissions to enable extended functions need to be granted. The circuit model remains unchanged.

[0178] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by those skilled in the art within the scope of the technology disclosed in the present invention and within the spirit and principles of the present invention should be covered within the scope of protection of the present invention.

Claims

1. A design method for an optimal robust input shaper, characterized in that, The method includes the following steps: S1, identify the vibration mode parameters of the mechanical transmission device of the multi-axis motion platform, and obtain the modeling natural frequency and modeling damping ratio; S2 calculates the amplitude and timing of each pulse in the traditional input shaper; S3, obtain the robustness evaluation index of the traditional input shaper under the condition that the modeling natural frequency and modeling damping ratio parameters change simultaneously, as shown in the two-parameter robustness evaluation index formula of the input shaper. S4. Design an improved pulse constraint using the optimized pulse amplitude constraint formula, and select the number of pulse correction coefficients to add. S5, Establish the robust optimization model of the input shaper, as shown in the formula, to optimize the objective function containing the set of correction coefficients to be determined; S6. Calculate the area of ​​the region that maximizes the pulse correction coefficient using the particle swarm optimization algorithm, and search for the pulse correction coefficient set that satisfies the objective function of the robust optimization model formula of the input shaper to maximize the area of ​​the region. S7, substitute the pulse correction coefficient set into the formula for optimizing pulse amplitude constraint to update the amplitude of the input shaper and obtain the corresponding optimal robust input shaper; In step S4, the number of pulse correction coefficients to be added includes: Let the number of pulse correction coefficients be added. The initial value is ,in Skip to steps S5-S6 for subsequent optimization design, and the maximum area of ​​the region obtained in the first step is: Then the number Skip to steps S5-S6 for subsequent optimization design. The maximum area of ​​the region obtained the second time is... The growth rate of the maximum area of ​​the region in a subsequent iteration relative to the maximum area in a previous iteration is expressed as: In the formula, This represents the growth rate of the region's maximum area in the subsequent iteration relative to the previous maximum area. This represents the maximum area of ​​the region obtained the first time. This represents the maximum area of ​​the region obtained the second time. Obtained by loop calculation The termination condition is , Set this value according to different working conditions, but not exceeding 5%. The default value is 1%. This represents the growth rate after the m-th cycle. To meet the growth rate requirements under this operating condition; In step S5, the robustness optimization model of the input shaper, as shown in the formula, includes the objective function optimization model containing the set of correction coefficients to be determined, which includes: Substituting the formula for optimizing the pulse amplitude constraint into the formula for the two-parameter robustness evaluation index of the input shaper, we obtain the following formula for the robustness evaluation function of the traditional input shaper with respect to the two-parameter variation under the constraint of varying pulse amplitude: Official (15) In the formula, For the objective function optimization model containing the set of correction coefficients to be determined, Not greater than the percentage of residual vibration The region where the system's natural frequency and damping ratio vary. For about The infinitesimal element, for The infinitesimal element, This is the system's actual inherent frequency. The actual damping ratio of the system. For all points of variation within the range of the system's natural frequency and damping ratio, This represents the percentage of residual vibration after applying pulse amplitude constraints. Based on formula (15), establish the robustness optimization model formula (16) for the input shaper: Official (16) In the formula, , Correction coefficient The upper boundary value, Correction coefficient The lower boundary value, the sum of the amplitudes of all pulses equals 1, coefficient The upper boundary is less than 0.5; while the amplitude of a single pulse is greater than 0, and the coefficient... The lower boundary is greater than -0.5; This is the upper bound for allowing variations in the system's inherent frequency. This is a lower bound that allows for variations in the system's natural frequency. This is the upper bound for allowing variations in the system's damping ratio. This represents the maximum robustness evaluation index of the pulse amplitude correction coefficient group to be determined.

2. The design method of the optimal robust input shaper according to claim 1, characterized in that, In step S3, the formula for the two-parameter robustness evaluation index of the input shaper is: Official (12) In the formula, This represents the maximum percentage of residual vibration permissible in industrial applications. Not greater than the percentage of residual vibration The region where the system's natural frequency and damping ratio vary. for The area of ​​the irregular region, For about The infinitesimal element, for The infinitesimal element, This is the system's actual inherent frequency. The actual damping ratio of the system. This represents the percentage of residual vibration.

3. The design method of the optimal robust input shaper according to claim 1, characterized in that, In step S4, the formula for optimizing the pulse amplitude constraint is: Formula (13) In the formula, For the optimal robust input shaper, the first The amplitude of each pulse, Let be a function of any number of pulse correction coefficients. It is the number of input shaper pulses; the amplitude of the optimal robust input shaper is determined by the amplitude of the conventional input shaper and a series of pulse correction coefficients. composition, , The number of pulse correction coefficients to include. The pulse amplitude of the traditional input shaper.

4. The design method of the optimal robust input shaper according to claim 1, characterized in that, In step S6, the particle swarm optimization algorithm includes: No. The first particle The velocity update formula (17) and the position update formula (18) are as follows: Official (17) Official (18) In the formula, for After the nth iteration The first particle velocity vector dimensional components, These are weighting coefficients that are not less than 0, used to adjust the search range of the solution space; These are all acceleration constants, used to adjust the maximum learning step size; All are random functions, and their range of values ​​is This increases the randomness of the search; The optimal position experienced by an individual particle. The best position experienced throughout the entire process. for After the nth iteration The first particle velocity vector Dimensional components; yes After the nth iteration The position vector of the nth particle Dimensional components; After the nth iteration The position vector of the nth particle Dimensional components; The components contained in a particle; D is the total number of components contained in a particle. for After the nth iteration The position vector of the nth particle Dimensional components; Must meet: In the formula, The maximum value of the weighting coefficient. This represents the minimum value of the weighting coefficient. The maximum number of iterations, for Weight coefficients after the next iteration; In formula (16), the unknown number is the first... Pulse amplitude correction coefficient Corresponding to the The position of each particle in the search space The overall optimal position satisfying the objective function formula (16) is found by using the particle swarm optimization algorithm. , is the required pulse correction coefficient set .

5. An optimal robust input shaper, characterized in that, It is designed using the design method of the optimal robust input shaper described in any one of claims 1-4.

6. The optimal robust input shaper according to claim 5, characterized in that, The optimal robust input shaper includes: the optimal robust input shaper ORIS-ZVD and the optimal robust input shaper ORIS-EI.

7. A control system structure for a multi-axis motion device with the optimal robust input shaper as described in claim 5, characterized in that, The multi-axis motion device control system structure input command For arbitrary industrial production motion trajectories, with motor control in an ideal state, the mechanical transmission device is equivalent to a flexible system with a base and mass block. The dynamic relationship between them, after being transformed by the Laplace transform, is expressed by the transfer function. express: In the formula, This represents the position of the mass block relative to the base. It is by The reference signal, shaped by the optimal robust input shaper, is used to suppress the transfer function. The residual vibrations of a typical flexible second-order system after the motion ends. The absolute position of the base. The modeled natural frequency for undamped circuits. To model the damping ratio, both the undamped modeling natural frequency and the modeling damping ratio are obtained by identifying the vibration modal parameters. for Laplace transform, for Laplace transform, For Laplace variation parameters.

8. The application of a design method for an optimal robust input shaper as described in any one of claims 1-4 in a PLC / motion control card, characterized in that, The design method of the optimal robust input shaper is encapsulated in the lower-level software of the PLC / motion control card, and the DSP in the hardware is used for calculation to control the operation of the mechanical transmission device of the multi-axis motion platform.

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