A time-frequency domain active design method for time B-spline instructions of point-to-point linear motion

Abstract

The present invention relates to the technical field of servo system motion planning, and discloses a method for actively designing time-frequency domain of time B-spline instructions for point-to-point linear motion. The construction process of the motion instruction generation model includes: defining the motion instruction through a time B-spline curve to obtain an expression of the motion instruction; regarding the feed system as a dual-mass underdamped flexible system, analyzing the resonant frequency band of the system; constructing a wavelet packet coefficient matrix according to the resonant frequency band, and extracting the resonant frequency band energy of the motion instruction; utilizing the least squares method to minimize the energy of the motion instruction that can excite system resonance in the time-frequency domain, and introducing kinematic constraints to establish a motion instruction generation model. The present invention realizes minimization of the frequency band that is easy to excite resonance in the motion instruction in the frequency domain, and realizes the energy optimization of the motion instruction in the time domain, aiming to fundamentally reduce the resonance phenomenon that is easy to occur during high-speed motion.

Description

Technical Field

[0001] The present invention relates to the technical field of motion planning, and in particular to a method for actively designing time-frequency domains of time B-spline instructions for point-to-point linear motion. Background Art

[0002] In the field of precision manufacturing such as semiconductor processing, how to achieve a significant reduction in positioning time while strictly ensuring positioning accuracy, so as to promote productivity improvement and product optimization, has always been the key point of technological innovation. However, in the processing process of point-to-point motion such as bonding, speed and accuracy are a pair of contradictory relationships. Due to the lightly damped vibration characteristics of the pickup end on the bonding device, excessive speed will cause the system to produce a large resonance during the movement process, and it will take a long time to adjust to meet the accuracy requirements. Therefore, this application introduces the wavelet packet transform theory for the lightly damped cantilever beam high-speed feeding system at the pickup end of the bonding mechanical equipment, actively designs the time-frequency domain of the motion command, attenuates the frequency band energy that causes the system to resonate in the motion command, and prevents the system from resonating during the movement process of executing the motion command, fundamentally solving the problem that the system is prone to resonance during high-speed motion.

[0003] Reference 1 "Yano A, Vaughan J, Singhose W. Robust Negative Input Shapers for Vibration Suppression [J]. Journal of Dynamic Systems, Measurement, and Control, 2009, 131 (3): 031014-1-031014-9. " discloses an effective method for suppressing residual vibration based on input shaping, and the derivation of a given insensitive formula makes this method highly robust. However, this method introduces a time lag factor, which greatly increases the positioning time of the machine.

[0004] Reference 2 "Fang, Hu, Liu, et al. Smooth and time-optimal S-curve trajectory planning for automated robots and machines [J]. Mechanism and Machine Theory, 2019, 137127-153.." discloses a smooth and time-optimal S-curve trajectory planning method. This method introduces the Sigmoid function when establishing the piecewise S-shaped function, so that the generated trajectory is infinitely continuous and differentiable under given velocity, acceleration and excitation constraints. However, this method does not consider the dynamic characteristics of the flexible system, such as resonant frequency and damping, which are closely related to residual vibration.

[0005] Reference 3 "[1] Erkorkmaz K, Hosseinkhani Y. Control of ball screw drives based on disturbance response optimization [J]. CIRP Annals - Manufacturing Technology, 2013, 62 (1): 387-390. " discloses a method of proposing a linear parameter variation (LPV) control strategy based on directly optimizing the disturbance response of the load side to the cutting force. This method has good low-frequency anti-interference ability and also provides active vibration reduction, thereby reducing the tracking error near the mechanical resonance of the drive. However, this method must design a robust controller for actual implementation, and the controllability and observability of the mode are limited.

[0006] The typical characteristics of the above literature are: various methods can reduce residual vibration and improve the positioning accuracy of the system, but they cannot have high robustness, consider the dynamic characteristics of the system and low implementation difficulty at the same time. Algorithms without high robustness have low applicability and are not conducive to practical applications; not considering the dynamic characteristics of the system can easily cause resonance and reduce mechanical performance; introducing time lag factors will cause large contour errors. Summary of the invention

[0007] In order to solve the above technical problems, the present invention provides a method for actively designing time-frequency domains of time B-spline instructions for point-to-point linear motion.

[0008] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0009] A method for actively designing time-frequency domain of time B-spline instructions for point-to-point linear motion, the construction process of the motion instruction generation model adopted includes:

[0010] The motion command is defined by the time B-spline curve to obtain the expression of the motion command;

[0011] The lightly damped cantilever beam high-speed feeding system is regarded as a double-mass underdamped flexible system, and the resonant frequency band of the cantilever beam is analyzed.

[0012] According to the resonant frequency band, a wavelet packet coefficient matrix is ​​constructed to extract the energy of the resonant frequency band in the motion instruction;

[0013] The least squares method is used to minimize the energy of the resonant frequency band in the motion instruction, and kinematic constraints are introduced to establish a motion instruction generation model. The motion instruction can be designed according to the dynamic characteristics of the lightly damped cantilever beam high-speed feed system to reduce the vibration amplitude of the cantilever beam during movement.

[0014] Furthermore, the motion instruction is defined by the time B-spline curve to obtain an expression of the motion instruction, which specifically includes:

[0015] Based on the B-spline curve theory, the motion time of the motion instruction is used as the curve parameter to establish the node vector of the time B-spline curve. Through a set of control vertices and the linear combination of the basis functions of the B-spline curve, the analytical expression of the motion instruction is constructed. By setting the node vector and control vertices of the time B-spline curve, the planned motion instruction can be made high-order continuous.

[0016] Furthermore, based on the B-spline curve theory, the motion time of the motion instruction is used as a curve parameter to establish the node vector of the time B-spline curve, and the analytical expression of the motion instruction is constructed through a linear combination of a set of control vertices and the basis function of the B-spline curve, which specifically includes:

[0017] Based on the B-spline curve theory, the motion time of the motion instruction is used as the parameter of the curve to establish the node vector of the time B-spline curve. :

[0018] ;

[0019] in, Plan the total time for the movement command, is the predetermined time interval of the motion instruction, Represents the node vector elements, ;

[0020] Construct the motion command analytical expression based on the quintic time B-spline curve:

[0021] ;

[0022] in, is the motion instruction about the curve parameter t of the time B-spline curve; is the total time of movement command planning; is the i-th control vertex of the time B-spline curve; represents the basis function of the i-th B-spline curve.

[0023] Furthermore, it also includes: according to the motion instruction expression of the time B-spline curve, a motion instruction sequence for guiding the actuator of the light damping cantilever beam high-speed feed system to move from one point to another can be obtained :

[0024] ;

[0025] in, , , , represents the control vertex vector, Represents the matrix expression of the B-spline curve basis function, Indicates the length of the motion instruction sequence, represents the basis function of the pth B-spline curve, Indicates the number of control vertices.

[0026] Furthermore, the lightly damped cantilever beam high-speed feeding system is regarded as a dual-mass underdamped flexible system, and the resonant frequency band of the cantilever beam is analyzed, specifically including:

[0027] The actual lightly damped cantilever beam high-speed feed system is regarded as a dual-mass under-damped flexible system, that is, the linear motor and motion platform part in the lightly damped cantilever beam high-speed feed system are regarded as the base of the dual-mass under-damped flexible system, and the end of the cantilever beam is regarded as the second mass of the dual-mass under-damped flexible system. The dynamic equations of the base and the second mass are established to analyze the relationship between the resonance phenomenon of the lightly damped cantilever beam high-speed feed system and the stiffness and mass of the cantilever beam. By analyzing the undamped natural frequency, the resonant frequency band of the cantilever beam, that is, the frequency band where the cantilever beam is most likely to resonate, is determined.

[0028] Furthermore, the actual lightly damped cantilever beam high-speed feed system is regarded as a dual-mass under-damped flexible system, that is, the linear motor and the motion platform part in the lightly damped cantilever beam high-speed feed system are regarded as the base of the dual-mass under-damped flexible system, and the end of the cantilever beam is regarded as the second mass of the dual-mass under-damped flexible system. The dynamic equation of the base and the second mass is established to analyze the relationship between the resonance phenomenon of the lightly damped cantilever beam high-speed feed system and the stiffness and mass of the cantilever beam. By analyzing the undamped natural frequency, the resonant frequency band of the cantilever beam is determined, which specifically includes:

[0029] The actual lightly damped cantilever beam high-speed feeding system is regarded as a dual-mass underdamped flexible system, and the dynamic equations of the base composed of a linear motor and a motion platform and the end of the cantilever beam clamped on the motion platform are established:

[0030] ;

[0031] in, are the absolute displacement and absolute acceleration of the base; y, and They represent the displacement, velocity and acceleration of the relative vibration of the mass m at the end of the cantilever beam relative to the base; c represents the damping of the cantilever beam; k represents the stiffness of the system;

[0032] Then the transfer function between the base and the end mass m of the cantilever beam of the dual-mass underdamped flexible system is expressed as:

[0033] ;

[0034] in, is the undamped resonant frequency of the cantilever beam; is the damping ratio of the cantilever beam; transform the above formula from the complex domain s into the frequency domain , that is, , the frequency response function of the dual-mass underdamped flexible system is obtained :

[0035] ;

[0036] represents an imaginary unit;

[0037] The amplitude ratio of the output signal to the input signal of the lightly damped cantilever beam high-speed feed system is simulated by the frequency response function of the lightly damped cantilever beam high-speed feed system. Indicates that, then The frequency range with a larger value is the resonant frequency band of the system. , They are the lower and upper limits of the resonant frequency band of the system respectively; among them, the output signal of the lightly damped cantilever beam high-speed feeding system is the positioning information of the end of the cantilever beam, and the input signal of the lightly damped cantilever beam high-speed feeding system is the motion command.

[0038] Furthermore, constructing a wavelet packet coefficient matrix according to the resonant frequency band to extract the energy of the resonant frequency band in the motion instruction specifically includes:

[0039] The wavelet packet transform principle is used to determine the subspace of wavelet packet decomposition, so that the frequency band in the determined wavelet packet subspace can completely contain the resonant frequency band of the cantilever beam; the base wavelet for extracting the resonant frequency band energy of the motion command is selected, and the time-frequency domain of the motion command signal is decomposed based on the Mallat algorithm, and the wavelet packet coefficient matrix of the decomposed motion command to the determined wavelet packet subspace is pre-calculated, and combined with the expression of the motion command, the resonant frequency band energy in the motion command that can best excite the lightly damped cantilever beam high-speed feed system to produce resonance is extracted.

[0040] Furthermore, the wavelet packet transform principle is used to determine the subspace of the wavelet packet decomposition, so that the frequency band in the determined wavelet packet subspace can completely contain the resonant frequency band of the cantilever beam; the base wavelet for extracting the resonant frequency band energy of the motion command is selected, and the time-frequency domain of the motion command signal is decomposed based on the Mallat algorithm, and the wavelet packet coefficient matrix of the decomposed motion command to the determined wavelet packet subspace is pre-calculated, and the expression of the motion command is combined to extract the resonant frequency band energy that can best excite the lightly damped cantilever beam high-speed feeding system to generate resonance in the motion command, specifically including:

[0041] The motion instructions include acceleration motion instructions;

[0042] By using the wavelet packet transform principle, the number of layers j and the number of nodes v of the wavelet packet subspace containing the resonant frequency band after wavelet packet decomposition are determined; the db12 wavelet function is selected as the base wavelet, and the acceleration motion instruction a(t) is decomposed in the time-frequency domain to the jth layer, thereby obtaining 2j wavelet packet subspaces:

[0043] ;

[0044] in, represents the total space containing motion instructions, is the orthogonal operation of the wavelet packet subspace, and They represent the jth layer after decomposition by db12 wavelet packet. and Time-frequency information, Represents an integer;

[0045] The vth wavelet packet subspace The wavelet packet coefficient sequence in By acceleration motion command and wavelet packet function based on db12 wavelet function Convolution obtains:

[0046] ;

[0047] in, , is the motion instruction about the curve parameter t of the time B-spline curve, is the scaling function of the db12 wavelet function and wavelet function Determine the wavelet packet function sequence;

[0048] Using the Mallat algorithm, we can get The convolution process is discretized, and the vth wavelet packet subspace The wavelet packet coefficient sequence in By recursively passing through a low-pass filter and high pass filter With the wavelet packet coefficients of the previous layer Vector inner product representation;

[0049] The wavelet packet coefficient sequence of the e-th wavelet subspace in the j-th layer After low-pass filtering, it is expressed as:

[0050] ;

[0051] in, is the wavelet packet coefficient sequence of the e-th wavelet packet subspace of the j-1th layer, The low-pass filter matrix corresponding to the subspace of the j-1th layer decomposed from the wavelet packet subspace to the jth layer;

[0052] If the wavelet packet coefficient sequence of the vth wavelet subspace of the jth layer After convolution with a high-pass filter, we get:

[0053] ;

[0054] is the high-pass filter matrix corresponding to the subspace of the j-1th layer decomposed from the wavelet packet subspace; then according to the original length n of the motion instruction y(t), the low-pass filter coefficient matrix and high-pass coefficient matrix of each layer are pre-calculated ; Since the low-pass filter and high-pass filter matrices of each layer will not work at the same time during wavelet packet transformation, the number of layers and nodes of the wavelet packet subspace is determined, and the coefficient matrix required for each layer is determined for multiplication; then for the motion instruction sequence The wavelet packet coefficient sequence in the j-th layer v-th node number subspace Further expressed as:

[0055] ;

[0056] in, Indicates that the motion instruction sequence Decompose into a decomposition matrix corresponding to the j-th layer v-th wavelet packet subspace; the wavelet packet coefficient sequence contains the resonant frequency band energy in the motion instruction that can best excite the lightly damped cantilever beam high-speed feeding system to resonate.

[0057] Furthermore, the least square method is used to minimize the energy of the resonant frequency band in the motion instruction, and kinematic constraints are introduced to establish a motion instruction generation model, which can design motion instructions according to the dynamic characteristics of the lightly damped cantilever beam high-speed feed system to reduce the vibration amplitude of the cantilever beam during the movement process, specifically including:

[0058] Based on the resonant frequency band information of the motion instructions, the least squares method is used to establish the objective function of minimizing the energy of the resonant frequency band, so that the generated motion instructions contain the minimum energy that can excite the resonant frequency band of the system and reduce the vibration amplitude during the movement of the system; the boundary conditions of the trajectory and the limit constraints of the speed, acceleration and jump of each discrete point in the motion instructions are introduced to establish a motion instruction generation model to ensure that the generated motion instructions meet the kinematic and dynamic requirements of the system.

[0059] Furthermore, based on the resonant frequency band information of the motion command, the least square method is used to establish an objective function of minimizing the energy of the resonant frequency band, so that the generated motion command contains the minimum energy that can excite the resonant frequency band of the system and reduce the vibration amplitude during the system movement process; the boundary conditions of the trajectory and the limit constraints of the speed, acceleration and jump of each discrete point in the motion command are introduced to establish a motion command generation model, which specifically includes:

[0060] By using the least squares method, the resonance frequency band information in the motion command is minimized, that is, the wavelet packet coefficients in the wavelet packet subspace of the corresponding acceleration motion command are minimized, and the objective function of the active design trajectory generation model in the time-frequency domain is obtained as follows:

[0061] ;

[0062] is the vth wavelet packet subspace The wavelet packet coefficient sequence in represents the decomposition matrix of the motion instruction sequence into the corresponding j-th layer v-th wavelet packet subspace; represents the acceleration sequence of motion instructions;

[0063] Considering the kinematic constraints such as speed and acceleration in the actual point-to-point motion process and the continuity requirements at the beginning and end, the following constraints are established; among them, in order to avoid unnecessary jitter of the velocity curve, the control vertex vector of the velocity curve needs to be Upper bound constraint and lower bound constraints :

[0064] ;

[0065] p represents the number of control vertices; represents the final displacement of the lightly damped cantilever beam high-speed feed system motion;

[0066] In order to meet the system's kinematic requirements for the boundary conditions of the motion command, the initial state of the motion command is established and the final motion state The equality constraints are:

[0067] ;

[0068] They represent the basis functions of the pth B-spline curve and the corresponding first-order derivative, second-order derivative, and third-order derivative respectively; , and They represent the displacement, velocity, acceleration and jump of the lightly damped cantilever beam high-speed feed system at the starting point of the motion respectively; , and They represent the displacement, velocity, acceleration and jump of the lightly damped cantilever beam high-speed feeding system at the end point of the motion respectively;

[0069] In order to avoid exceeding the dynamic limits of the system during movement, the motion instructions should satisfy the preset speed, acceleration, and jump limit inequality constraints at each interpolation point:

[0070] ;

[0071] , and Respectively indicated by and The matrix form is composed of , , Indicates the preset speed, acceleration, and jump limits; Indicated by The matrix composed of Indicated by The matrix composed of

[0072] Establish a motion command generation model:

[0073] ;

[0074] : ;

[0075] Solve the motion command generation model and obtain the control vertex vector of the motion command , and substituting it into the analytical expression of motion instructions, we can obtain the motion instructions actively designed in the time-frequency domain according to the dynamic characteristics of the lightly damped cantilever beam high-speed feed system.

[0076] Compared with the prior art, the present invention has at least the following beneficial effects:

[0077] In order to reduce the resonance generated during the motion of the lightly damped cantilever beam high-speed feed system, the present invention proposes a method for actively designing the time-frequency domain of the time B-spline instruction of point-to-point linear motion. This method combines the time-frequency analysis capability of the wavelet packet transform with the high flexibility of the definition of the time B-spline curve. The motion instruction is actively designed according to the resonant frequency band of the lightly damped cantilever beam high-speed feed system. The obtained motion instruction has the minimum energy within the resonant frequency band, which fundamentally reduces the resonance phenomenon that is easy to occur during high-speed motion, and has the advantages of high robustness and high stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] Figure 1 It is a schematic diagram of the structure of a dual-mass underdamped flexible system;

[0079] Figure 2 It is a comparison diagram of the fast Fourier transform (FFT) results of the motion command solved by the method proposed by the present invention and the motion command solved by the S-shaped speed planning method;

[0080] Figure 3 It is a comparison diagram of the vibration response generated by the motion command solved by the method proposed in the present invention and the S-shaped motion command obtained by the S-shaped speed planning method. DETAILED DESCRIPTION

[0081] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution of the present invention will be clearly and completely described below in conjunction with the accompanying drawings. Obviously, the embodiments described below are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0082] As a specific implementation method of a certain working condition of the present invention, the present invention proposes a time-frequency domain active design method for point-to-point linear motion time B-spline instructions. Specifically, based on the B-spline curve theory, the motion time of the motion instruction is used as the parameter of the curve to establish the curve node vector :

[0083] ;

[0084] in, The total movement time planned for the movement instruction, is a predetermined motion time interval for the motion instruction. In a preferred embodiment, ; Represents the node vector Middle elements, .

[0085] In a preferred embodiment, the present invention constructs a position motion instruction expression based on a quintic time B-spline (t-Bspline) curve, which can be expressed as:

[0086] ;

[0087] in, is the motion instruction (displacement trajectory) about the curve parameter t of the time B-spline curve; is the i-th control vertex of the time B-spline curve, ; represents the basis function of the i-th quintic B-spline curve, which is defined as:

[0088] ;

[0089] Then the point-to-point motion instruction sequence It can be expressed as:

[0090] ;

[0091] in, , , , represents the control vertex vector, Represents the matrix expression of the B-spline curve basis function, Indicates the length of the motion instruction sequence, represents the basis function of the pth B-spline curve, Indicates the number of control vertices.

[0092] speed , acceleration , jump The first, second and third order derivatives of displacement with respect to time are obtained respectively.

[0093] Since the basis function Related to the time parameter, the speed , acceleration , jump Respectively expressed as:

[0094] ;

[0095] in, , and For Find the matrix form after the first, second and third order derivatives. The recursive formula for finding the k-th derivative is as follows:

[0096] ;

[0097] Secondly, the lightly damped cantilever beam high-speed feed system is considered as Figure 1 The double-mass underdamped flexible system shown in FIG. 1 is a double-mass underdamped flexible system. That is, the linear motor and the motion platform in the lightly damped cantilever beam high-speed feed system are regarded as the base of the double-mass underdamped flexible system, and the end of the cantilever beam is regarded as the second mass of the double-mass underdamped flexible system. The dynamic equations of the base and the second mass are established:

[0098] ;

[0099] Among them, x and is the displacement and acceleration of the motion of the base; y, and are the displacement, velocity and acceleration of the relative vibration of the cantilever beam end m relative to the base; c is the damping of the cantilever beam; k is the stiffness of the cantilever beam.

[0100] The transfer function of the above formula is expressed as:

[0101] ;

[0102] in, is the undamped resonant frequency of the cantilever beam, ; is the damping ratio of the two-mass underdamped flexible system, .

[0103] Convert the above formula from the complex domain (s domain) to the frequency domain ( domain), that is, , the frequency response function of the dual-mass underdamped flexible system is obtained It is expressed as:

[0104] .

[0105] According to the model of the frequency response function of the dual-mass underdamped flexible system, the resonant frequency band of the lightly damped cantilever beam high-speed feed system is analyzed. , using the wavelet packet transform principle, determine the wavelet packet decomposition contains The number of layers j and the number of nodes v in the corresponding wavelet packet subspace.

[0106] Since the acceleration motion command contains more frequency domain information, the db12 wavelet basis function is used to decompose the acceleration motion command a(t) in the time-frequency domain by using the wavelet packet transform principle, and the wavelet packet coefficient sequence in the jth layer vth wavelet packet subspace in the motion command is extracted. , acceleration motion command a(t) and wavelet packet function based on db12 wavelet Convolution is obtained, expressed as:

[0107] ;

[0108] in, , is the scaling function of the db12 wavelet and wavelet function Determine the wavelet packet function sequence.

[0109] The wavelet packet function sequence can be expressed as:

[0110] .

[0111] in, and It represents the low-pass and high-pass filters of db12. The specific values ​​can be obtained by referring to the data.

[0112] In order to achieve the real-time requirement of the calculation method, the above convolution process is discretized, based on the Mallat algorithm, The high-pass filter can be obtained recursively from the db12 wavelet basis function and low pass filter With the wavelet packet coefficients of the previous layer Vector inner product representation:

[0113] ;

[0114] in, is the wavelet packet coefficient in the e-th wavelet packet subspace of the j-1th layer, which is obtained by the inner product of the j-1th layer's wavelet packet subspace through the low-pass filter h(n) and the high-pass filter g(n).

[0115] Furthermore, the above formula can be expressed by vector as:

[0116] ;

[0117] in, is the wavelet packet coefficient sequence of the e-th wavelet packet subspace of the j-1th layer, whose length is l, They are the low-pass filter matrix and high-pass filter matrix corresponding to the subspace of the j-1th layer decomposed from the wavelet packet subspace to the jth layer. It should be noted that after the downsampling operation of the wavelet packet, The number of rows is The integer portion of the value.

[0118] Then, according to the original length n of the motion command y(t), the low-pass filter coefficient matrix of each layer can be pre-calculated according to the above formula: and the high-pass coefficient matrix . Since the low-pass filter and high-pass filter matrices of each layer will not work at the same time during wavelet packet transformation, the number of layers and nodes of the wavelet packet subspace can be selected to determine the coefficient matrix required for each layer for multiplication. Then for the wavelet packet coefficient sequence in the j-th layer and the v-th node number subspace of the motion instruction Y It can be further expressed as:

[0119] ;

[0120] in, It represents the decomposition of the motion instruction sequence Y into the wavelet packet coefficient matrix corresponding to the vth wavelet packet subspace of the jth layer.

[0121] Then you can use The matrix extracts the time-frequency information of the resonant frequency band of the motion command. Suppressing the components in the resonant frequency band of the motion command is to minimize the wavelet packet coefficients in the wavelet packet subspace corresponding to the resonant band. Since the acceleration motion command contains richer frequency information, the least squares method is used to minimize the time-frequency information of the resonant frequency band in the acceleration motion command, and the objective function of the trajectory generation model can be obtained as:

[0122] ;

[0123] Then, in order to avoid unnecessary jitter of the speed curve, the control vertices of the curve need to be The upper and lower bounds are constrained, expressed as:

[0124] ;

[0125] Keep the motion instructions meeting the set boundary conditions and establish equation constraints for the displacement, velocity, and acceleration at the start and end points of the motion instructions:

[0126] ;

[0127] In order to avoid system failure due to command overrun, the motion command established at each discrete interpolation point should satisfy the preset speed, acceleration and jump limit inequality constraints:

[0128] ;

[0129] Finally, the time-frequency domain active design motion command generation model can be expressed as:

[0130] ;

[0131] : ;

[0132] The model can be calculated using the interior point method to solve the control vertex of the motion command and obtain a motion trajectory that satisfies the kinematic and dynamic constraints and avoids the vibration amplitude of the exciting system in motion engineering. From the perspective of motion planning, the wavelet packet transform theory is introduced to solve the problem of strong vibration of the feed system during high-speed motion.

[0133] The dual-mass underdamped flexible system corresponding to the lightly damped cantilever beam high-speed feed system is as follows: Figure 1 As shown, in combination with the above implementation mode, the following example conditions are provided:

[0134] (1) The total displacement of this embodiment is 100 mm, the maximum speed limit is 100 mm / s, the maximum acceleration limit is 800 mm / s2, and the maximum jump limit is 10000 mm / s; the total planning time uses the S-shaped speed planning time as the time template.

[0135] (2) For comparison, this example uses the widely used S-shaped velocity curve for velocity planning to obtain the displacement, velocity, and acceleration instructions of the initial motion trajectory.

[0136] (3) Using the high-speed vibration suppression method for point motion based on wavelet packet transform theory proposed in this paper, the S-shaped velocity trajectory is re-optimized under the conditions of satisfying kinematic and dynamic constraints to obtain the optimized displacement, velocity and acceleration instructions. Figure 2 The FFT comparison of the motion instructions solved by the method proposed in the present invention and the motion instructions solved by the S-shaped velocity planning method is shown. It can be observed that the motion instructions solved by the method proposed in the present invention significantly reduce the amplitude of the resonant frequency band [17Hz, 23Hz].

[0137] (4) Apply the motion instructions solved by the method proposed in the present invention and the motion instructions solved by the S-shaped speed planning method to the lightly damped cantilever beam high-speed feeding system, and observe the vibration amplitude and total positioning time of the cantilever beam end during the movement process. Figure 3 As shown, it can be observed that the vibration caused by the motion command solved by the method proposed in the present invention is significantly reduced compared with the S-shaped motion command obtained by the S-shaped speed planning method.

[0138] Finally, it should be noted that the above-described embodiments are only specific implementations of the present invention, which are used to illustrate the technical solutions of the present invention, rather than to limit them. The protection scope of the present invention is not limited thereto. Although the present invention is described in detail with reference to the above-described embodiments, ordinary technicians in the field should understand that any technician familiar with the technical field can still modify the technical solutions recorded in the above-described embodiments within the technical scope disclosed by the present invention, or can easily think of changes, or make equivalent replacements for some of the technical features therein; and these modifications, changes or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims

1. A method for actively designing time-frequency domain of time B-spline instructions for point-to-point linear motion, characterized in that: The construction process of the adopted motion command generation model includes: The motion command is defined by the time B-spline curve to obtain the expression of the motion command; The lightly damped cantilever beam high-speed feeding system is regarded as a double-mass underdamped flexible system, and the resonant frequency band of the cantilever beam is analyzed. According to the resonant frequency band, a wavelet packet coefficient matrix is ​​constructed to extract the energy of the resonant frequency band in the motion instruction: the subspace of the wavelet packet decomposition is determined by using the wavelet packet transformation principle, so that the frequency band in the determined wavelet packet subspace can completely contain the resonant frequency band of the cantilever beam; a base wavelet for extracting the resonant frequency band energy of the motion instruction is selected, and the time-frequency domain of the motion instruction signal is decomposed based on the Mallat algorithm, and the wavelet packet coefficient matrix of the decomposed motion instruction to the determined wavelet packet subspace is pre-calculated, and combined with the expression of the motion instruction, the resonant frequency band energy that can best excite the lightly damped cantilever beam high-speed feeding system to generate resonance is extracted from the motion instruction; The least square method is used to minimize the energy of the resonant frequency band in the motion command, and kinematic constraints are introduced to establish a motion command generation model. The motion command can be designed according to the dynamic characteristics of the lightly damped cantilever beam high-speed feed system to reduce the vibration amplitude of the cantilever beam during movement: Based on the resonant frequency band information of the motion instructions, the least squares method is used to establish the objective function of minimizing the energy of the resonant frequency band, so that the generated motion instructions contain the minimum energy that can excite the resonant frequency band of the system and reduce the vibration amplitude during the movement of the system; the boundary conditions of the trajectory and the limit constraints of the speed, acceleration and jump of each discrete point in the motion instructions are introduced to establish a motion instruction generation model to ensure that the generated motion instructions meet the kinematic and dynamic requirements of the system.

2. The method for actively designing time-frequency domain of point-to-point linear motion time B-spline instructions according to claim 1, characterized in that: The motion command is defined by the time B-spline curve to obtain the expression of the motion command, which specifically includes: Based on the B-spline curve theory, the motion time of the motion instruction is used as the curve parameter to establish the node vector of the time B-spline curve. Through a set of control vertices and the linear combination of the basis functions of the B-spline curve, the analytical expression of the motion instruction is constructed. By setting the node vector and control vertices of the time B-spline curve, the planned motion instruction can be made high-order continuous.

3. The method for actively designing time-frequency domain of point-to-point linear motion time B-spline instructions according to claim 2, characterized in that: Based on the B-spline curve theory, the motion time of the motion instruction is used as a curve parameter to establish the node vector of the time B-spline curve. The analytical expression of the motion instruction is constructed through a linear combination of a set of control vertices and the basis function of the B-spline curve, which specifically includes: Based on the B-spline curve theory, the motion time of the motion instruction is used as the parameter of the curve to establish the node vector of the time B-spline curve : ； in, Plan the total time for the movement command, is the predetermined time interval of the motion instruction, Represents the node vector elements, ; Construct the motion instruction analytical expression based on the quintic time B-spline curve: ； in, is the motion instruction about the curve parameter t of the time B-spline curve; is the total time of movement command planning; is the i-th control vertex of the time B-spline curve; represents the basis function of the i-th B-spline curve.

4. The method for actively designing time-frequency domain of point-to-point linear motion time B-spline instructions according to claim 3, characterized in that: Also includes: According to the motion command expression of the time B-spline curve, the motion command sequence for guiding the actuator of the light-damped cantilever beam high-speed feed system to move from one point to another can be obtained. : ； in, , , , represents the control vertex vector, Represents the matrix expression of the B-spline curve basis function, Indicates the length of the motion instruction sequence, represents the basis function of the pth B-spline curve, Indicates the number of control vertices.

5. The method for actively designing time-frequency domain of point-to-point linear motion time B-spline instructions according to claim 1, characterized in that: The lightly damped cantilever beam high-speed feeding system is regarded as a dual-mass underdamped flexible system, and the resonant frequency band of the cantilever beam is analyzed, specifically including: The actual lightly damped cantilever beam high-speed feed system is regarded as a dual-mass under-damped flexible system, that is, the linear motor and motion platform part in the lightly damped cantilever beam high-speed feed system are regarded as the base of the dual-mass under-damped flexible system, and the end of the cantilever beam is regarded as the second mass of the dual-mass under-damped flexible system. The dynamic equations of the base and the second mass are established to analyze the relationship between the resonance phenomenon of the lightly damped cantilever beam high-speed feed system and the stiffness and mass of the cantilever beam. By analyzing the undamped natural frequency, the resonant frequency band of the cantilever beam, that is, the frequency band where the cantilever beam is most likely to resonate, is determined.

6. The method for actively designing time-frequency domain of point-to-point linear motion time B-spline instructions according to claim 5, characterized in that: The actual light-damped cantilever beam high-speed feeding system is regarded as a dual-mass under-damped flexible system, that is, the linear motor and the motion platform part in the light-damped cantilever beam high-speed feeding system are regarded as the base of the dual-mass under-damped flexible system, and the end of the cantilever beam is regarded as the second mass of the dual-mass under-damped flexible system. The dynamic equation of the base and the second mass is established to analyze the relationship between the resonance phenomenon of the light-damped cantilever beam high-speed feeding system and the stiffness and mass of the cantilever beam. By analyzing the undamped natural frequency, the resonant frequency band of the cantilever beam is determined, which specifically includes: The actual lightly damped cantilever beam high-speed feeding system is regarded as a dual-mass underdamped flexible system, and the dynamic equations of the base composed of a linear motor and a motion platform and the end of the cantilever beam clamped on the motion platform are established: ； in, is the acceleration of the base's motion; y, and They represent the displacement, velocity and acceleration of the relative vibration of the mass m at the end of the cantilever beam relative to the base; c represents the damping of the cantilever beam; k represents the stiffness of the cantilever beam; Then the transfer function between the base and the end mass m of the cantilever beam of the dual-mass underdamped flexible system is expressed as: ； in, is the undamped resonant frequency of the cantilever beam; is the damping ratio of the cantilever beam; transform the above formula from the complex domain s into the frequency domain , that is, , the frequency response function of the dual-mass underdamped flexible system is obtained : ； represents an imaginary unit; The amplitude ratio of the output signal to the input signal of the lightly damped cantilever beam high-speed feed system is calculated by using the frequency response function of the lightly damped cantilever beam high-speed feed system. Indicates that, then The frequency range with a larger value is the resonant frequency band of the system. , They are the lower and upper limits of the resonant frequency band of the system respectively; among them, the output signal of the lightly damped cantilever beam high-speed feeding system is the positioning information of the end of the cantilever beam, and the input signal of the lightly damped cantilever beam high-speed feeding system is the motion command.

7. The method for actively designing time-frequency domain of point-to-point linear motion time B-spline instructions according to claim 1, characterized in that: The wavelet packet transform principle is used to determine the subspace of the wavelet packet decomposition, so that the frequency band in the determined wavelet packet subspace can completely contain the resonant frequency band of the cantilever beam; the base wavelet for extracting the resonant frequency band energy of the motion command is selected, and the time-frequency domain of the motion command signal is decomposed based on the Mallat algorithm, and the wavelet packet coefficient matrix of the decomposed motion command to the determined wavelet packet subspace is pre-calculated, and the expression of the motion command is combined to extract the resonant frequency band energy that can best excite the lightly damped cantilever beam high-speed feeding system to generate resonance in the motion command, which specifically includes: The motion instructions include acceleration motion instructions; By using the wavelet packet transform principle, the number of layers j and the number of nodes v of the wavelet packet subspace containing the resonant frequency band after wavelet packet decomposition are determined; the db12 wavelet function is selected as the base wavelet, and the acceleration motion instruction a(t) is decomposed in the time-frequency domain to the jth layer, thereby obtaining 2j wavelet packet subspaces: ； in, represents the total space containing motion instructions, is the orthogonal operation of the wavelet packet subspace, and They represent the jth layer after decomposition by db12 wavelet packet. and Time-frequency information, Represents an integer; The vth wavelet packet subspace The wavelet packet coefficient sequence in By acceleration motion command and wavelet packet function based on db12 wavelet function Convolution obtains: ； in, , is the motion instruction about the curve parameter t of the time B-spline curve, is the scaling function of the db12 wavelet function and wavelet function Determine the wavelet packet function sequence; Using the Mallat algorithm, we can get The convolution process is discretized, and the vth wavelet packet subspace The wavelet packet coefficient sequence in By recursively passing through a low-pass filter and high pass filter With the wavelet packet coefficients of the previous layer Vector inner product representation; The wavelet packet coefficient sequence of the e-th wavelet subspace in the j-th layer After low-pass filtering, it is expressed as: ； in, is the wavelet packet coefficient sequence of the e-th wavelet packet subspace of the j-1th layer, The low-pass filter matrix corresponding to the subspace of the j-1th layer decomposed from the wavelet packet subspace to the jth layer; If the wavelet packet coefficient sequence of the vth wavelet subspace of the jth layer After convolution with a high-pass filter, we get: ； is the high-pass filter matrix corresponding to the subspace of the j-1th layer decomposed from the wavelet packet subspace; then according to the original length n of the motion instruction y(t), the low-pass filter coefficient matrix and high-pass coefficient matrix of each layer are pre-calculated ; Since the low-pass filter and high-pass filter matrices of each layer will not work at the same time during wavelet packet transformation, the number of layers and nodes of the wavelet packet subspace is determined, and the coefficient matrix required for each layer is determined for multiplication; then for the motion instruction sequence The wavelet packet coefficient sequence in the j-th layer v-th node number subspace Further expressed as: ； in, Indicates that the motion instruction sequence Decompose into a decomposition matrix corresponding to the j-th layer v-th wavelet packet subspace; the wavelet packet coefficient sequence contains the resonant frequency band energy in the motion instruction that can best excite the lightly damped cantilever beam high-speed feeding system to resonate.

8. The method for actively designing time-frequency domain of point-to-point linear motion time B-spline instructions according to claim 1, characterized in that: The objective function of minimizing the energy of the resonant frequency band is established based on the resonant frequency band information of the motion command by using the least square method, so that the generated motion command contains the minimum energy that can excite the resonant frequency band of the system and reduce the vibration amplitude during the motion of the system; the boundary conditions of the trajectory and the limit constraints of the speed, acceleration and jump of each discrete point in the motion command are introduced to establish a motion command generation model, which specifically includes: By using the least squares method, the resonance frequency band information in the motion command is minimized, that is, the wavelet packet coefficients in the wavelet packet subspace of the corresponding acceleration motion command are minimized, and the objective function of the active design trajectory generation model in the time-frequency domain is obtained as follows: ； is the vth wavelet packet subspace The wavelet packet coefficient sequence in represents the decomposition matrix of the motion instruction sequence into the corresponding j-th layer v-th wavelet packet subspace; represents the acceleration sequence of motion instructions; Considering the kinematic constraints of velocity and acceleration in the actual point-to-point motion process and the continuity requirements at the beginning and end, the following constraints are established; among them, in order to avoid unnecessary jitter of the velocity curve, the control vertex vector of the velocity curve needs to be Upper bound constraint and lower bound constraints : ； p represents the number of control vertices; represents the final displacement of the lightly damped cantilever beam high-speed feed system motion; In order to meet the system's kinematic requirements for the boundary conditions of the motion command, the initial state of the motion command is established and the final motion state The equality constraints are: ； They represent the basis functions of the pth B-spline curve and the corresponding first-order derivative, second-order derivative, and third-order derivative respectively; , and They represent the displacement, velocity, acceleration and jump of the lightly damped cantilever beam high-speed feed system at the starting point of the motion respectively; , and They represent the displacement, velocity, acceleration and jump of the lightly damped cantilever beam high-speed feed system at the end of the motion respectively; In order to avoid exceeding the dynamic limits of the system during movement, the motion instructions should satisfy the preset speed, acceleration, and jump limit inequality constraints at each interpolation point: ； , and Respectively indicated by and The matrix form is composed of , , Indicates the preset speed, acceleration, and jump limits; Indicated by The matrix composed of Indicated by The matrix composed of Establish a motion command generation model: ； ： ； Solve the motion command generation model and obtain the control vertex vector of the motion command , and substituting it into the analytical expression of motion instructions, we can obtain the motion instructions actively designed in the time-frequency domain according to the dynamic characteristics of the lightly damped cantilever beam high-speed feed system.

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