Asymmetric velocity curve dimensionality reduction planning method based on high-order polynomial

Abstract

The invention belongs to the technical field of high-stability motion speed planning in semiconductor packaging, and discloses an asymmetric speed curve dimensionality reduction planning method based on a high-order polynomial. The method comprises the following steps: firstly, constructing a movement speed curve with an endpoint derivative of zero based on a two-point cubic Hermite polynomial; secondly, proposing an equivalent jerk thought, and accurately calculating the running time of each section in a high-order velocity curve through a dimension reduction mode according to boundary velocity constraint; and finally, establishing a dynamic simplified model of the system, and analyzing the vibration response of the high-order motion curve through Matlab simulation. According to the method, the motion stability can be effectively improved, the operation efficiency can be guaranteed, and the method has important significance on high-speed and high-precision surface mounting of IC packaging in engineering practice.

Description

Technical Field

[0001] The invention belongs to the technical field of high-speed and stable motion planning in semiconductor packaging, and relates to a dimensionality reduction planning method for an asymmetric speed curve based on a high-order polynomial. Background Art

[0002] The accuracy of semiconductor back-end packaging reflects the core competitiveness of the integrated circuit industry to a certain extent. With the increasing miniaturization and integration of semiconductor products, higher requirements are placed on the accuracy of semiconductor packaging. Therefore, in the field of semiconductor packaging, there are extremely high requirements for the stability of packaging motion. In order to improve the motion stability during packaging, the packaging process usually has a smooth start-stop motion. The S-type speed curve has become the main choice for the built-in motion planning method of the driver due to its low computational complexity and good operating stability. It is widely used in the motion speed planning of medium and high-speed placement machines. However, high-precision chip placement has higher requirements for operating stability. The acceleration discontinuity of the conventional S-type curve will aggravate the periodic excitation in high-speed motion, induce residual vibration at the end of the pickup arm, and reduce the chip placement accuracy. Therefore, designing an excellent acceleration profile is the guarantee for improving motion stability and patch accuracy. Increasing the order of the motion curve can improve motion stability. Therefore, high-order acceleration profiles have been widely studied. However, high-order motion curves increase the amount of calculation, which is not conducive to online real-time planning and has high requirements on processor performance. Therefore, studying a high-order point motion speed real-time planning method is of great significance to improving the accuracy of semiconductor packaging and improving the amount of motion planning calculation.

[0003] In the paper “Frequency optimal feed motion planning in computer numerical controlled machine tools for vibration avoidance” by Sencer et al., the ninth-order acceleration trajectory proposed in this paper has good smoothness in the start-stop stage. However, the calculation amount of the ultra-high-order motion trajectory is larger than that of the conventional curve and the start-stop is too gentle, so it is not suitable for short-stroke and high-speed working conditions such as high-speed patch. In the paper “Smoothand time-optimal S-curve trajectory planning for automated robots and machines” by Fang et al., an S-curve trajectory planning method that takes into account both efficiency and smoothness is proposed, which is suitable for high-speed precision motion and reduces the vibration at the end of the high-speed motion cantilever structure to ensure stability. However, the essence is planning based on the S-curve. Summary of the invention

[0004] In view of the defects of the prior art, the present invention invents a dimensionality reduction planning method for asymmetric speed curves based on high-order polynomials. The method first constructs a high-order motion speed curve equation based on the Hermite interpolation polynomial endpoint derivative smoothing; secondly, the academic idea of ​​equivalent acceleration is proposed, and the boundary speed is used as a constraint to accurately calculate the running time and required displacement of each section of the high-order speed curve; then, a speed planning method is constructed based on the high-order motion curve; finally, a simplified dynamic model of the motion system is constructed, and the vibration response is simulated and analyzed. The proposed method is suitable for the high-stability motion process of chip patches, and plays an important role in improving the stability of patch motion and improving patch accuracy, and provides theoretical and technical support for high-speed and high-precision motion control in semiconductor packaging.

[0005] The technical solution of the present invention:

[0006] A dimensionality reduction planning method for asymmetric velocity curves based on high-order polynomials is proposed. First, a motion velocity curve with zero endpoint derivative is constructed based on two-point cubic Hermite polynomials. Secondly, the idea of ​​equivalent jerk is proposed. According to the boundary velocity constraint, the running time of each segment in the high-order velocity curve is accurately calculated by dimensionality reduction. Finally, a simplified dynamic model of the system is built, and the vibration response of the high-order motion curve is analyzed by Matlab simulation. The specific steps of the method are as follows:

[0007] Step 1: Construct the motion equation based on the Hermite interpolation polynomial;

[0008] Chip patching is a typical high-speed, high-acceleration motion. A smooth motion curve is the key to ensuring the stability of high-speed patching. Therefore, designing an excellent acceleration profile is the guarantee for improving motion stability and improving patch accuracy. The endpoint positions of the Hermite interpolation polynomial have smooth connectivity, which provides a basis for the design of high-order motion trajectories. In order to reduce the residual vibration at the end of the patch actuator (pickup arm), an acceleration curve profile with a smooth transition profile is designed based on a two-point cubic Hermite interpolation polynomial. The acceleration motion profile consists of multiple piecewise polynomial curves, including the acceleration stage [0, t3], the constant speed stage [t3, t4], and the deceleration stage [t4, t7].

[0009] The curve passing through the starting point (x1, y1) and the end point (x2, y2) satisfies the boundary function value and derivative constraints f(x1) = y1, f(x2) = y2, f'(x1) = g1 and f'(x2) = g2. Therefore, the curve can be expressed by two-point cubic Hermite interpolation as:

[0010] H3(x)=y1α1(x)+y2α2(x)+g1β1(x)+g2β2(x) (1)

[0011] In the formula, αi (x) and β i (x) needs to satisfy the corresponding function value and derivative conditions,

[0012] Taking the acceleration region as an example, when 0 < t ≤ t1, the acceleration-time curve satisfies the constraint condition:

[0013]

[0014] In the formula, H3′(t) represents the first derivative with respect to time;

[0015] Therefore, based on the two-point cubic Hermite interpolation polynomial, the acceleration-time trajectory can be expressed as:

[0016]

[0017] In the formula, a a,max is the maximum acceleration in the acceleration phase, t is the running time, and t1 is the time of the acceleration increasing phase.

[0018] Taking the derivative of Equation (3) can calculate the relationship between jerk and time, that is:

[0019]

[0020] In the formula, the jerk-time relationship can be expressed as a parabola with zero boundaries. Setting the derivative to zero gives the maximum value of jerk as

[0021] So far, a motion equation with continuous acceleration is constructed based on the Hermite polynomial.

[0022] Step 2: Construction of the high-order polynomial motion parameter and running time expression;

[0023] When the time t satisfies the condition 0 < t ≤ t1, the relationships between jerk-time, acceleration-time, velocity-time, and displacement-time can be expressed as:

[0024]

[0025] In the formula, v0 represents the initial velocity. According to the above formula, the specific values of jerk J(t), acceleration a(t), velocity v(t), and displacement x(t) in the stage of 0 < t ≤ t1 can be calculated. At t = t1, J(t1), a(t1), v(t1), and x(t1) can be solved. When t i-1 < t ≤ t i (i = 2, …, 7), J(t i )、a(t i )、v(t i)、x(t i ) can be solved for subsequent calculation and expression.

[0026] When the time t satisfies t1 < t ≤ t2, the acceleration remains at its maximum value. At this time, the velocity is still in the increasing stage. Therefore, the motion profile can be expressed as:

[0027]

[0028] When the time satisfies t2 < t ≤ t3, the jerk is opposite to that in the [0, t1] stage and is still in the acceleration stage. The acceleration gradually decreases to 0. The specific calculation is as follows:

[0029]

[0030] Since the running distance during patching is short, the existence of the uniform velocity section [t3, t4] needs to be determined according to the displacement constraint. The motion trajectory in this stage can be calculated as:

[0031]

[0032] In the deceleration section, the calculation method is similar to that in the acceleration section. At t4 < t ≤ t5, the acceleration during deceleration increases, and the corresponding motion equation can be calculated as:

[0033]

[0034] In the formula, a d,max represents the maximum deceleration in the deceleration section.

[0035] In the time period t5 < t ≤ t6, the acceleration during deceleration remains at its maximum value, and the motion equation can be expressed as:

[0036]

[0037] In the time period t6 < t ≤ t7, the acceleration in the deceleration stage gradually decreases to 0. The specific motion relationship is calculated as follows:

[0038]

[0039] Based on the above analysis, the relationship between the motion parameters and the running time is constructed using the Hermite interpolation polynomial to form a motion curve with continuous acceleration, which is defined as the super S curve. When a a,max = a d,max , the motion profile is symmetric. And when a a,max = γa d,max (γ ≠ 1), the motion profile is in a general asymmetric form. Usually, γ > 1 makes the motion smoother.

[0040] Step 3: Use the equivalent jerk to perform dimensionality reduction and solve for the running time;

[0041] Precise time series Δt of the super S-curve motion profile i (i = 1, …, 7) can be calculated by decoupling the maximum velocity, acceleration, and higher-order acceleration. However, the displacement-time curve constructed by Hermite interpolation is a fifth-degree polynomial, and the running time solution calculation is computationally intensive, making it difficult to ensure real-time calculation, which exacerbates the high-speed calculation burden of the hardware. By the method of equivalent jerk, its computational complexity is reduced from fifth-order to third-order, and based on the S-curve running time solution method in the aforementioned patent, two typical running time solution strategies of equal motion parameter constraint and equal running time constraint are introduced.

[0042] In the solution of the equal motion parameter constraint, under the premise of the equal maximum jerk constraint, the running time of the super S-curve is increased compared with the conventional S-curve. Under the equal running time constraint, the maximum jerk of the super S-curve running is increased compared with the maximum jerk of the S-curve. According to Equation (4), when t = 0.5t1 * the maximum jerk is obtained by differentiation as

[0043] To utilize the third-order S-curve to solve the running time of the super S-curve, the equivalent jerk is defined as:

[0044]

[0045] In the formula, J eq represents the equivalent acceleration, which can be calculated according to the equal motion parameter constraint and the equal running time constraint respectively.

[0046] (1) Equivalent jerk based on equal motion parameter constraint

[0047] The super S-curve motion velocity curve can obtain a smoother trajectory by sacrificing a small amount of running time. The maximum jerk of the super S-curve under the equal motion parameter constraint is equal to that of the conventional S-curve. Based on this, the following relationship can be obtained:

[0048]

[0049] In the formula, t1 is the running time of the acceleration phase where the acceleration increases on the S-curve; t1 * represents the corresponding time on the super S-curve. From the above formula, the following relationship can be obtained:

[0050]

[0051] When 0 < t ≤ t1, the equivalent jerk can be calculated by Equation (11), and its expression is:

[0052]

[0053] In the formula, the equivalent jerk Jeq =2J max / 3.

[0054] (2) Equivalent jerk based on equal running time constraints

[0055] Considering that the jerk of the linear motor is constrained within a safe range, in order to avoid the high-order motion trajectory reducing the motion efficiency, the equivalent jerk of the super S curve needs to be equal to the jerk of the conventional S curve, thus the following relationship can be established:

[0056]

[0057] In the formula, the equivalent acceleration J eq =J max .

[0058] The equivalent jerk of the super S curve is equal to the maximum jerk of the conventional S curve, and the running time of the acceleration / deceleration section will remain unchanged. According to equations (12)-(13), the maximum jerk can be calculated as:

[0059]

[0060] Therefore, the super S-curve motion profile was established based on the Hermite interpolation polynomial, and the running time solution model based on equal running time and equal motion parameters was established using the equivalent acceleration method to meet the time dimension reduction calculation of different constraints, greatly simplifying the high-order motion solution process. eq , maximum acceleration a a,max and speed v max , the running time series t of the high-order motion curve i (i=1,…,7) can be reduced to a smaller order using the exact solution method for the running time.

[0061] Step 4: Simulate and analyze the vibration response based on the simplified single-degree-of-freedom system vibration model

[0062] According to the transfer function relationship of the single-axis dynamic model of the linear motor, the control model of the motion system is established, and the transient vibration response of the foundation structure caused by the high-speed actuator driven by different acceleration curves is simulated and analyzed to verify that the super S curve proposed by this method has better motion stability. The dynamic vibration simulation analysis of the foundation structure is carried out using MATLAB / Simulink. Under the constraints of motion parameters, the conventional S curve and the super S curve proposed by this method are used as the motion input of the high-speed actuator, and the vibration response of the foundation structure in forward and reverse motion is compared and analyzed in combination with the reciprocating motion of the high-speed actuator. Furthermore, based on the equal running time constraint, the super S curve with equal time constraint is extended to an asymmetric super S curve, and the residual vibration generated by the asymmetric super S curve and the symmetric super S curve in the acceleration / deceleration area is compared. It is verified that the residual vibration of the asymmetric super S curve after reaching the terminal position in the deceleration stage is the smallest, and it is easier to obtain higher terminal positioning accuracy and shorter position adjustment time.

[0063] The beneficial effect of the present invention is a dimensionality reduction planning method for asymmetric speed curves based on high-order polynomials. The method effectively solves the problem of insufficient point motion stability during high-precision patching, avoids the phenomenon of poor high-speed operation stability and low stable operation efficiency, and is of great significance to improving operation stability. In order to improve the operation stability, the method first constructs a high-order motion speed curve equation based on the smoothing of the endpoint derivatives of the two-point cubic Hermite interpolation polynomial to ensure ultra-smooth transition in the start-stop phase; secondly, in order to improve the calculation speed, a new method of running time dimensionality reduction analysis based on equivalent acceleration is proposed with the boundary speed as a constraint to accurately calculate the running time and required displacement of each section of the high-order speed curve; finally, a simplified dynamic model of the motion system is constructed, and the vibration response of the high-order motion curve is analyzed by Matlab simulation. The proposed method is suitable for the high-stability motion process of chip patching, plays an important role in improving the stability of patch motion and improving patch accuracy, and provides theoretical and technical support for high-precision motion control in semiconductor packaging. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 This is the overall flow chart of the motion speed dimension reduction planning method based on high-order polynomials.

[0065] Figure 2a ) is the high-order speed curve running speed reaching the set value under the current parameters, where the abscissa represents time in seconds, and the legend of each graph is given in the figure;

[0066] Figure 2b ) is the high-order speed curve running speed that does not reach the set value under the current parameters, where the horizontal axis represents time in seconds, and the legend of each graph is given in the figure.

[0067] Figure 3a) is the relationship between acceleration and time of the equal-parameter super S curve, equal-time super S curve and conventional S curve, where the horizontal axis represents time in seconds and the vertical axis represents acceleration in m / s 2 ;

[0068] Figure 3b ) is the relationship between acceleration and time of the equal-parameter super S curve, the equal-time super S curve and the conventional S curve, where the horizontal axis represents time in seconds and the vertical axis represents acceleration in m / s 2 .

[0069] Figure 4 It is the vibration response under the excitation of equal-parameter super S curve, equal-time super S curve and conventional S curve, where the abscissa represents time in seconds and the ordinate represents vibration response in mm.

[0070] Figure 5 It is the vibration response under the excitation of asymmetric super S curve, symmetric super S curve and conventional S curve, where the horizontal axis represents time in seconds and the vertical axis represents vibration response in mm. DETAILED DESCRIPTION

[0071] The specific implementation of the present invention is described in detail below in combination with the technical solution and the accompanying drawings

[0072] During the semiconductor chip packaging process, the pick-up arm is driven by a linear motor to place the chip from the pick-up position to the patch position. The patch accuracy is an important indicator of the process. Compared with conventional point-to-point motion, chip patching has the characteristics of high running speed and short stroke. During frequent start-stop processes, it is necessary to strictly control the stable output of the running speed. Therefore, reasonable planning of the running speed is of great significance to ensure the motion stability of the chip patch and improve the patch accuracy. In view of the fact that conventional motion speed planning algorithms are difficult to meet the needs of high-precision patching, a motion speed dimensionality reduction planning method based on high-order polynomials is invented. The process of the method is shown in the attached figure. Figure 1 shown.

[0073] The present invention aims at the speed planning problem in the chip mounting process, and describes the specific implementation process of the present invention in detail through MATLAB simulation speed planning curve.

[0074] First, based on the two-point cubic Hermite polynomial, the acceleration-time relationship with zero initial and final position derivatives is constructed, and the result is By taking its derivative, first integration and second integration, we can get the correlation between jerk-time, velocity-time and displacement-time respectively.

[0075] Secondly, based on the above process, the expression relationship between jerk-time, acceleration-time, velocity-time, and displacement-time in the entire operation cycle [0, t7] can be promoted; based on this, using the idea of ​​equivalent jerk, the analysis of the running time in the high-order motion curve constructed by the Heimite polynomial is reduced to the calculation of the running time of the conventional S-curve, thereby planning the parameters of the high-order motion velocity curve under each scanning cycle.

[0076] Finally, the constructed motion system dynamics model is used to analyze the vibration responses of the high-order speed curve and conventional speed curve proposed by this method, and the effectiveness of the method in improving operational stability is verified through simulation results.

[0077] To verify the correctness of the algorithm, take the patch displacement of 100mm as an example, and the given acceleration is 5000m / s 3 , acceleration is 50m / s 2 When the maximum running speed is set to 1.6m / s, the maximum running speed can be reached under the current motion parameters, and the running time is 110ms; when the maximum running speed is set to 2m / s, the current motion parameters cannot reach the running speed of 2m / s, the maximum speed is 1.89m / s, and the running time at this time is 105ms. The relationship between motion parameters and running time is shown in Figure 2.

[0078] To verify the robustness of the algorithm, change the acceleration during motion to 100m / s 2 , the acceleration is 10000m / s 3 The acceleration-time curve of the high-order motion curve obtained by the patent method and the acceleration-time curve of the conventional method are shown in Figure 3. The blue line in the figure represents the conventional S curve as a comparison curve. The red and green lines represent the proposed super S curves of equal motion parameters and equal motion time, respectively.

[0079] The vibration response of the basic structure of the motion system under different motion trajectory excitation is different. The results are as follows: Figure 4 As shown in the figure, different acceleration curve inputs have different effects on the vibration of the foundation structure, and the maximum vibration amplitude during operation is 140μm. The vibration after the movement is terminated represents the magnitude of the residual vibration. The residual vibration after reaching the terminal position is related to the motion curve profile. The length of the adjustment time can reflect the magnitude of the residual vibration. The longer the adjustment time, the greater the residual vibration. Figure 4The results shown in the figure show that under the constraint of positioning error, the adjustment time of the super S curve is significantly shorter than that of the flexible S curve. Moreover, the residual vibration energy envelope range of the equal time / equal parameter super S curve proposed by this method is smaller after the motion is terminated, indicating that the motion curve has smaller vibration energy and shorter adjustment time. The super S curve is extended to an asymmetric super S curve, and the residual vibration generated by the asymmetric super S curve and the symmetric super S curve in the acceleration / deceleration area is analyzed. The results are shown in Figure 5 Simulation and data analysis show that compared with the conventional S-type motion curve, the proposed super S-curve can effectively reduce the residual vibration amplitude and shorten the adjustment time.

[0080] After comprehensive analysis of the simulation data, it can be seen that the judgment result is consistent with the simulation result. This shows that the motion speed dimension reduction planning method based on high-order polynomials of the present invention can effectively improve motion stability and ensure operation efficiency, provide technical support for high-speed and high-precision placement of IC packaging in engineering practice, and play an important guiding role in achieving low-cost and high-precision chip packaging.

Claims

1. A method for dimension reduction planning of asymmetric speed curve based on high-order polynomials, characterized in that: The steps are as follows: Step 1: Construct the motion equation based on the Hermite interpolation polynomial; High-speed chip placement is a high-speed and high-acceleration motion. A smooth motion curve is the key to ensuring the stability of high-speed chip placement. Therefore, the motion stability is improved and the chip placement accuracy is increased by designing the acceleration profile. To reduce the residual vibration at the end of the chip placement actuator, i.e., the pick-up arm, based on the two-point cubic Hermite interpolation polynomial, an acceleration curve profile with a smooth transition contour is designed. The acceleration curve profile consists of multiple piecewise polynomial curves, including an acceleration phase [0, t3], a constant velocity phase [t3, t4], and a deceleration phase [t4, t7]; The curve passing through the starting point (x1, y1) and the ending point (x2, y2) satisfies the boundary function value and derivative constraints f(x1) = y1, f(x2) = y2, f’(x1) = g1, and f’(x2) = g2. This curve is represented by the two-point cubic Hermite interpolation as: H3(x) = y1α1(x) + y2α2(x) + g1β1(x) + g2β2(x) (1) In the formula, α i (x) and β i (x) must satisfy the corresponding function value and derivative conditions, For the acceleration region, when the running time is 0 < t ≤ t1, the acceleration-time curve satisfies the constraint conditions as: In the formula, H3′(t) represents the first derivative with respect to time; Therefore, based on the two-point cubic Hermite interpolation polynomial, the acceleration time trajectory is expressed as: In the formula, a a,max is the maximum acceleration in the acceleration phase, t is the running time, and t1 is the running time in the acceleration increase phase; Taking the derivative of Equation (3) calculates the relationship between the jerk and time, that is: In the formula, the jerk-time relationship is expressed as a parabola with a zero boundary. By setting the derivative to zero, the maximum value of the jerk is So far, a motion equation with continuous acceleration is constructed based on the Hermite interpolation polynomial; Step 2: Construct the high-order polynomial motion parameter and running time expression; When the time t satisfies the condition 0 < t ≤ t1, the relationships between the jerk-time, acceleration-time, velocity-time, and displacement-time are expressed as: where \(v_0\) represents the initial velocity, and the specific values of the jerk \(J(t)\), acceleration \(a(t)\), velocity \(v(t)\), and displacement \(x(t)\) in the stage of \(0 \lt t \leq t_1\) are calculated according to the above formula; at \(t = t_1\), \(J(t_1)\), \(a(t_1)\), \(v(t_1)\), and \(x(t_1)\) are solved; when \(t\) i-1 \lt t \leq t i , \(i = 2, \ldots, 7\), \(J(t i \)), \(a(t i \)), \(v(t i \)), \(x(t i \)) can all be solved for subsequent calculation and expression; When the time t satisfies t1 < t ≤ t2, the acceleration remains at the maximum value. At this time, the velocity is still in the increasing stage. Therefore, the motion profile is expressed as: When the time satisfies t2 < t ≤ t3, the jerk is opposite to the [0, t1] stage and is still in the acceleration stage, and the acceleration gradually decreases to 0. The specific calculation is as follows: Since the running distance during chip placement is short, the existence of the constant velocity interval [t3, t4] needs to be determined according to the displacement constraint. The motion trajectory in this stage is calculated as: In the deceleration section, the calculation method is similar to that in the acceleration region; Specifically: at t4 < t ≤ t5, the acceleration during deceleration increases, and the corresponding motion equation is calculated as: In the formula, a d,max Indicates the maximum deceleration in the deceleration section; In the time period t5 < t ≤ t6, the acceleration during deceleration remains at the maximum value, and the motion equation is expressed as: In the time period t6 < t ≤ t7, the acceleration in the deceleration stage gradually decreases to 0. The specific motion relationship is calculated as follows: According to the above analysis, the relationship between motion parameters and running time is constructed based on the Hermite interpolation polynomial to form a motion curve with continuous acceleration, which is defined as a super S curve; when a a,max =a d,max When , the motion profile is symmetrical; a a,max =γa d,max ,γ≠1, the motion profile is asymmetric, and γ>1 makes the motion smoother; Step 3: Use the equivalent jerk to reduce the dimension and solve the running time; The precise time series Δt of the super S-curve motion profile i , i=1,…,7 is calculated by the maximum velocity, acceleration and acceleration high-order decoupling; the displacement-time curve constructed by Hermite interpolation is a fifth-order polynomial, and the running time solution calculation is large. Therefore, the equivalent acceleration method is used to reduce its computational complexity from fifth order to third order, and based on the super S-curve running time solution method, two running time solution strategies are introduced: equal motion parameter constraints and equal running time constraints; In solving the constraints of equal motion parameters, under the premise of equal maximum jerk constraints, the running time of the super S curve is longer than that of the S curve; under the constraint of equal running time, the maximum jerk of the super S curve is higher than that of the S curve; according to formula (4), when t = 0.5t1 * When , the maximum jerk is obtained by derivation: To use the third-order S-curve to solve the running time of the super S-curve, the equivalent jerk is defined as: In the formula, J eq represents the equivalent acceleration, which is calculated according to the equal motion parameter constraints and the equal running time constraints respectively; (1) The equivalent jerk based on the equal motion parameter constraint The super S-shaped motion speed curve can obtain a smoother trajectory by sacrificing a small amount of running time. Under the equal motion parameter constraint, the maximum jerk of the super S-curve is equal to that of the S-curve. Based on this, the following relationship is obtained: Where t1 is the running time of the acceleration phase of the S curve with increasing acceleration; t1 * It represents the time corresponding to the super S curve. The following relationship is obtained from the above formula: When \(0 < t\leq t_1\), the equivalent jerk is calculated by Equation (11), and its expression is: In the formula, the equivalent acceleration J eq =2J max / 3; (2) Equivalent jerk based on equal running time constraint Considering that the jerk constraint of the linear motor is within the safe range, in order to avoid the reduction of motion efficiency caused by high-order motion trajectories, the equivalent jerk of the super S-curve needs to be equal to the jerk of the S-curve. Therefore, the following relationship is established: In the formula, the equivalent acceleration J eq =J max ; The equivalent jerk of the super S-curve is equal to the maximum jerk of the S-curve, and the running time of the acceleration / deceleration section will remain unchanged; according to Equations (12)-(13), the maximum jerk is calculated as: Therefore, the super S-curve motion profile was established based on the Hermite interpolation polynomial, and the running time solution models based on equal running time and equal motion parameters were established using the equivalent acceleration method; thus, for a given equivalent acceleration J in the patch process eq , maximum acceleration a a,max and speed v max , the running time series t of the high-order motion curve i ,i=1,…,7 Reduce the order calculation according to the exact solution method of running time; Step 4: Simulate and analyze the vibration response based on the simplified single degree of freedom system vibration model Based on the transfer function relationship of the single-axis dynamic model of the linear motor, the control model of the motion system is established, and the transient vibration response of the basic structure excited by the high-speed actuator under different acceleration curves is simulated and analyzed to verify that the super S-curve has better motion stability; Use MATLAB / Simulink to perform dynamic vibration simulation analysis of the basic structure; under the constraint of motion parameters, use the S-curve and the super S-curve as the motion inputs of the high-speed actuator, and compare and analyze the vibration responses of the basic structure during forward and reverse motions in combination with the working conditions of the reciprocating motion of the high-speed actuator; furthermore, based on the equal running time constraint, the equal-time-constrained super S-curve is extended to an asymmetric super S-curve, and the residual vibrations generated by the asymmetric super S-curve and the symmetric super S-curve in the acceleration / deceleration region are compared, so as to verify that the residual vibration of the asymmetric super S-curve is the smallest after reaching the terminal position in the deceleration stage, and it is easier to obtain higher end positioning accuracy and shorter position adjustment time.

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