Method for collaborative optimization of feed speed and printing direction for multi-axis additive

Abstract

A collaborative optimization method for the feed speed and printing direction of multi-axis additive is provided, which performs layer slicing and path rules on a printing model; a mathematical model of the optimization problem is established by constructing design variables, constraint conditions and optimization objective functions; by considering the indentation depth constraint of the nozzle relative to the printing surface when the normal vector is offset, the maximum offset angle of the normal vector of each printing point in the printing surface is calculated; based on the chain rule of differentiation, the gradient is calculated according to the final mathematical model, and the gradient-based optimization algorithm is used to collaboratively optimize the normal vector offset angle, the normal vector rotation angle and the feed speed of the nozzle relative to the workpiece. By comprehensively considering the normal vector offset and the feed speed matching, the two are collaboratively optimized, the optimization algorithm based on the gradient is used for optimization calculation, and according to the final optimization result, the printing direction offset and the feed speed matching during multi-axis machining are completed. The printing efficiency, printing quality and the stability of the printer operation are effectively improved.

Description

Technical Field

[0001] This invention relates to the field of 3D printing technology, specifically to a method for the coordinated optimization of feed rate and printing direction in multi-axis additive manufacturing. Background Technology

[0002] Compared to traditional three-axis additive manufacturing, multi-axis additive manufacturing can form parts with complex curved surfaces while effectively reducing support structures and avoiding stair-step effects. In three-axis additive manufacturing, the three axes generally move linearly in the spatial coordinate system, exhibiting similar behavior in terms of velocity, acceleration, and jerk. However, in multi-axis additive manufacturing, taking "3+2" five-axis additive manufacturing as an example, the different normal vector directions of each printing point necessitate the rotation of the turntable in conjunction with the movement of the printhead to achieve curved surface printing. This results in different motion patterns for the rotation and translation axes, leading to significant differences in velocity, acceleration, and jerk. Furthermore, when the normal vector direction of the printing point approaches the Z-axis, the inverse kinematic transformation can cause singularity problems, thus affecting print quality and efficiency.

[0003] Therefore, in multi-axis simultaneous printing, the CNC system needs to smoothly handle the speed, acceleration, and jerk of each axis to ensure that it does not violate the physical limitations of the drivers and motors. Simultaneously, to prevent printer jitter or vibration and ensure print quality and efficiency, the printhead feed speed must be carefully planned and the singularity problem addressed. Many scholars both domestically and internationally have proposed solutions to these problems. For example, Chen Zhengying et al.'s patent "AC Angle Optimization Method for Linear Cutting Path Planning in Five-Axis 3D Printing" (publication number CN113119450A) provides an AC angle optimization method based on normal vector offset, primarily addressing the singularity problem, as large differences in rotation angles between adjacent points on the machining path can lead to overcutting of the workpiece. Wu Yijie et al.'s patent "Linear Path CNC Machining Feed Speed ​​Control Method Based on Equivalent Acceleration" (publication number CN113759830A) provides a linear path CNC machining feed speed control method based on equivalent acceleration, primarily addressing feed speed matching considering acceleration constraints. In summary, most existing technologies address single problems, and the reliability and stability of these methods need improvement. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a collaborative optimization method for feed rate and printing direction in multi-axis additive manufacturing. This method addresses the singularity problem and machine tool motion constraint problem easily caused by the inverse kinematic transformation of multi-axis motion during multi-axis surface machining. By comprehensively considering normal vector offset and feed rate matching, the two are collaboratively optimized using a gradient-based optimization algorithm. Based on the final optimization result, the printing direction offset and feed rate matching during multi-axis machining are achieved. Thus, while considering the constraints of printer motion characteristics, this method effectively improves printing efficiency, printing quality, and printer operational stability, providing technical support for efficient multi-axis machining of complex curved surfaces.

[0005] To achieve the above technical objectives, the adopted technical solution is: a collaborative optimization method for feed rate and printing direction in multi-axis additive manufacturing, including the following steps:

[0006] Step 1: Perform layer slicing and path rules on the printed model;

[0007] Step 2: Construct the design variables, constraints, and objective function to establish a mathematical model of the optimization problem, including the following steps:

[0008] (21) The design variables for the optimization problem are: Where n is the number of print points in the print path, and the normal vector deviation angle is... Rotation angle of normal vector θ = (θ1, θ2…θ) n ) T The feed velocity of the nozzle relative to the workpiece is v = (v1, v2, ..., v...). n-1 ) T ;

[0009] (22) Based on the servo capability of the motor and the range of values ​​for the design variables, the constraints for the optimization problem are as follows: in, V represents the upper limit of the normal vector offset angle; threshold The set axial velocity threshold is greater than 0; a threshold The set axial acceleration threshold greater than 0; v imax The upper limit of the set feed rate value greater than 0; V vector This is a vector composed of the axial velocities of all printing points during the printing process; a vector V is a vector composed of the axial accelerations of all printing points during the printing process; vector || P Represents vector V vector The P-norm; ||avector || P Represents vector a vector The P-norm;

[0010] (23) Construct the optimization objective function, which includes the smoothness of the feed rate, the sum of the smoothness of the axial acceleration of each axis, and the theoretical total printing time, as shown in formula (3): f(x) = α·s(v re )+β·s(a)+γ·t (3) Where α, β, and γ are weighting coefficients; s(v re ) represents the smoothness of the feed rate, s(a) represents the sum of the smoothness of the axial acceleration of each axis, and t represents the theoretical total printing time;

[0011] (24) Formulas (1), (2), and (5) together form the final optimized mathematical model, as shown in formula (4):

[0012] Step 3: By considering the indentation depth constraint of the printhead relative to the printing surface when the normal vector is offset, calculate the maximum offset angle of the normal vector of each printing point in the printing surface, and use it as an optimization variable. Upper limit of values

[0013] Step 4: Based on the chain rule, and according to the final optimization mathematical model in step (24), calculate the objective function f(x) and the constraint condition ||V. vector || P 、||a vector || P The gradient of the design variable x is calculated using the following formula: in, Represents vector V vector The k-th element in Represents vector a vector The k-th element in the array, where m is the number of axes in the printer;

[0014] Step 5: Based on the gradient calculated in Step 4, use a gradient-based optimization algorithm to adjust the deviation angle of the normal vector. The normal vector rotation angle θ and the nozzle feed speed v relative to the workpiece are optimized together.

[0015] The smoothness s(v) of the feed rate of this invention re The calculation method for ) is as follows Where n is the number of print points in the print path, v i This represents the feed rate of the i-th printing interval.

[0016] The method for calculating the sum of the axial acceleration smoothness s(a) of each axis in this invention is as follows: Where n is the number of print points in the print path; m is the number of axes of the printer; and a ij This represents the i-th acceleration value on the j-th axis.

[0017] The method for calculating the total theoretical printing time t of this invention is as follows: Where, Δl i Let be the arc length of the i-th printing interval in the printing path. If the printing points are selected densely during path planning, it is approximately equal to the distance between two adjacent printing points; n is the number of printing points in the printing path; v i This represents the feed rate of the i-th printing interval.

[0018] The upper limit of the normal vector offset angle in this invention The calculation is performed using formula (11), which is as follows, and is divided into two cases: convex surface and concave surface. Where d is the diameter of the printhead; h is the interlayer height; r k δ is the maximum radius of curvature at that point on the printed surface; δ is the maximum indentation depth, and 0≤δ≤h; l is calculated using formula (12).

[0019] In summary, compared with the prior art, the collaborative optimization method for feed rate and printing direction for multi-axis additive manufacturing provided by the present invention has the following advantages: (1) The technical solution conceived in this invention, under the premise of considering the constraints of the printer's motion characteristics, optimizes the normal vector offset and speed matching in a coordinated manner, which effectively improves printing efficiency, printing quality and printer operation stability, and provides technical support for efficient multi-axis machining of complex curved surfaces. (2) Users can independently change the axial velocity and acceleration constraint values ​​according to the motion characteristics of different printers, which improves the flexibility of this technical solution; (3) Users can design the weight coefficients of the objective function according to the specific circumstances and optimize it for different optimization objectives; (4) Users can design the maximum indentation depth according to specific circumstances, thereby controlling the range of the normal vector deviation angle. (5) Users can independently design the transformation relationship between design variables and objective functions, thereby applying this method to a variety of processing technologies, including but not limited to five-axis FDM printing, five-axis continuous fiber printing, five-axis DED printing, and robotic arm additive manufacturing, with a wide range of applications. Attached Figure Description

[0020] Figure 1 This is a flowchart of the technical solution disclosed in this invention.

[0021] Figure 2 This is the surface model diagram selected in the embodiment of the present invention.

[0022] Figure 3 This is a path planning diagram of the surface model in an embodiment of the present invention.

[0023] Figure 4 This is a schematic diagram of the normal vector offset.

[0024] Figure 5 It is the geometric model of the maximum offset angle of the normal vector.

[0025] Figure 6 This is a comparison chart of the normal vectors before and after optimization.

[0026] Figure 7 It shows the optimized feed rate curve and a magnified view of some areas.

[0027] Figure 8 It is a comparison chart of axial velocity, acceleration, jerk and theoretical printing time before and after optimization. Detailed Implementation

[0028] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0029] This embodiment uses a randomly generated complex surface, and its mesh model diagram is as follows: Figure 2 As shown, a five-axis FDM 3D printer was selected for printing. The five-axis 3D printer includes an XYZ three-axis linkage nozzle based on a Cartesian structure and an AC axis turntable.

[0030] like Figure 1 As shown, the collaborative optimization method for feed rate and printing direction for multi-axis additive manufacturing disclosed in this invention mainly includes the following steps:

[0031] Step 1: Slice and plan the path for the printed model. This can be done using existing 3D printing research techniques. The final path planning result is as follows: Figure 3 As shown.

[0032] Step Two: Based on the optimization objective and printer motion characteristics constraints, construct the optimization objective function, constraints, and design variables, ultimately building a mathematical model for the optimization problem. Step Two includes the following sub-steps:

[0033] (21) The design variables of the optimization problem in this invention include: the deviation angle of the normal vector. Rotation angle of normal vector θ = (θ1, θ2…θ) n ) T The feed velocity of the nozzle relative to the workpiece is v = (v1, v2, ..., v...). n-1 ) T , and θ Figure 4 As shown. The design variables are as shown in formula (1): Where n is the number of print points in the print path.

[0034] (22) The constraints of the optimization problem of this invention include: the axial velocity and acceleration of each axis during printing do not exceed the servo capability of the motor; and the range of values ​​for the design variables. As shown in formula (2): in, V represents the upper limit of the normal vector offset angle; threshold For the axial velocity threshold, a threshold This is the axial acceleration threshold. The specific threshold can be set according to the actual situation of the motor, but it cannot exceed the upper limit set by the motor. A value greater than 0 is acceptable. imax The upper limit of the feed rate can be set manually, as long as the value is greater than 0; V vector This is a vector composed of the axial velocities of all printing points during the printing process; a vector V is a vector composed of the axial accelerations of all printing points during the printing process; vector || P Represents vector V vector The P-norm is calculated using formula (3); ||a vector || P Represents vector a vector The P-norm is calculated using formula (4); when P takes a large value, the P-norm of the vector can approximate the maximum value in the vector. Where m is the number of axes of the printer, for example, when using a five-axis printer to print a sample, m = 5; n is the number of print points in the print path; Represents vector V vector The k-th element in; Represents vector a vector The k-th element in.

[0035] In this embodiment, V is set threshold =200, a threshold =500, v imax =10, select a five-axis FDM 3D printer to print the sample.

[0036] (23) Construct the optimization objective function, which includes the smoothness of the feed rate, the smoothness of the acceleration of each axis, and the theoretical total printing time, as shown in formula (5): f(x) = α·s(v re )+β·s(a)+γ·t (5) Where α, β, and γ are weighting coefficients, which can be defined by the user; s(v re ) represents the smoothness of the feed rate, s(a) represents the sum of the smoothness of the axial acceleration of each axis, calculated using formulas (6) and (7); t represents the theoretical total printing time, calculated using formula (8). In formula (6), v i v represents the feed rate of the i-th printing interval; i+1 This represents the feed rate of the (i+1)th printing interval; v i-1 This represents the feed rate of the (i-1)th printing interval. In formula (7), m is the number of axes of the printer. For example, when using a five-axis printer to print samples, m = 5; a ij This represents the i-th acceleration value on the j-th axis. In formula (8), Δl i Let be the arc length of the i-th printing interval in the printing path. If the printing points are selected densely during path planning, it is approximately equal to the distance between two adjacent printing points. n is the number of printing points in the printing path.

[0037] In this embodiment, α = 0.4, β = 1e-6, and γ = 0.6.

[0038] (24) Formulas (1), (2), and (5) together form the final optimization model, as shown in formula (9).

[0039] Step 3: A geometric model was established, such as Figure 5 As shown, by considering the indentation depth constraint of the printhead relative to the printing surface when the normal vector is offset, the maximum offset angle of the normal vector of each printing point in the complex printing surface is calculated as an optimization variable. Upper limit of values Taking a printed point as an example, the upper limit of the value of its normal vector offset angle. The calculation is performed using formula (10), which is as follows, and is divided into two cases: convex surface and concave surface. Where d is the diameter of the printhead; h is the interlayer height; r k δ is the maximum radius of curvature at that point on the printed surface; δ is the maximum indentation depth, and 0≤δ≤h. The value can be set according to the specific situation to control the range of the deviation angle of the normal vector at that printed point; l is calculated by formula (11), which is as follows:

[0040] In this embodiment, d = 0.8, h = 0.5, and δ = 0.05.

[0041] Step 4: Based on the chain rule, and according to the mathematical model of the optimization problem shown in formula (9), calculate the objective function f(x) and the constraint condition ||V vector || P 、||a vector || P The gradient of the design variable x is calculated using the following formula: in, Represents vector V vector The k-th element in Represents vector a vector The k-th element in the array, where m is the number of axes in the printer, x i It can be any one of the following: normal vector deviation angle, normal vector rotation angle, or feed rate.

[0042] Step 5: Based on the gradient calculated in Step 4, use a gradient-based optimization algorithm to adjust the deviation angle of the normal vector. The normal vector rotation angle θ and the nozzle feed speed v relative to the workpiece are optimized in a coordinated manner. The specific optimization algorithm can be selected by the user, including but not limited to GCMMA, Fmincon, etc.

[0043] Step Six: Calculate the offset angle in the design variable x based on the optimization results from Step Five. The rotation angle θ is used to calculate the direction coordinates of each printing point after the normal vector offset, obtained through the spatial coordinate transformation formula. The surface model diagram after the normal vector offset is shown below. Figure 6 As shown in the figure. Blue represents the initial normal vector of each printed point, and red represents the optimized normal vector of each printed point.

[0044] Step 7: Based on the feed rate v in the design variable x after optimization calculation in Step 5, calculate the axial velocity of each axis in each printing interval using formula (15). The calculation formula is as follows, taking a "3+2" five-axis printer as an example: Where, ΔX i Let ΔY be the axial displacement length along the X-axis of the i-th printing interval in the printing path. i ΔZ i ΔA i ΔC i Similarly; V Xi V is the axial velocity of the X-axis when printing the i-th printing interval. Yi V Zi V Ai V Ci Similarly.

[0045] Optimized feed rate parameters are as follows Figure 7 As shown in the figure, the optimized feed rate is smoother, ensuring the stability of the feed rate.

[0046] Step 8: Based on the normal vector direction coordinates, axial velocities of each axis, and coordinates of each point in the printing path obtained in Steps 6 and 7, the user can convert them into the parameters required for actual printing according to the control system of the printer.

[0047] like Figure 8 As shown in the figure, the axial velocity, acceleration, and jerk are all significantly reduced before and after optimization. The maximum value of the axial velocity after optimization meets the constraint of 200 mm / s, and the maximum value of the acceleration meets the constraint of 500 mm / s. 2 Constraints were imposed, reducing the maximum jerk value to less than 10% of the pre-optimization value.

[0048] Matters not covered in this invention are common knowledge.

[0049] In summary, this invention discloses a method for co-optimizing feed rate and printing direction in multi-axis additive manufacturing.

[0050] This invention discloses a collaborative optimization method for feed rate and printing direction in multi-axis additive manufacturing. This method addresses the singularity problem and printer motion constraint issues that easily arise during multi-axis machining of complex curved surfaces due to the inverse kinematic transformations of multi-axis motion. By comprehensively considering normal vector offset and feed rate matching, the method collaboratively optimizes these two aspects. A gradient descent-based optimization method is used to calculate the global optimal solution. Based on the final optimization result, the printing direction offset and feed rate matching are achieved during multi-axis machining of curved surfaces. Therefore, while considering printer motion constraints, this method effectively improves printing efficiency, print quality, and printer operational stability, providing technical support for efficient multi-axis machining of complex curved surfaces.

[0051] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for co-optimizing feed rate and printing direction in multi-axis additive manufacturing, characterized by: Includes the following steps: Step 1: Perform layer slicing and path rules on the printed model; Step 2: Construct the design variables, constraints, and objective function to establish a mathematical model of the optimization problem, including the following steps: (21) Construct the set of design variables for the optimization problem as follows: Where n is the number of print points in the print path, and the normal vector deviation angle is... Rotation angle of normal vector θ = (θ1, θ2, ... θ) n ) T The feed velocity of the nozzle relative to the workpiece is v = (v1, v2, ..., v...). n-1 ) T (22) Based on the servo capability of the motor and the range of values ​​for the design variables, the constraints for the optimization problem are as follows: in, V represents the upper limit of the normal vector offset angle; threshold The set axial velocity threshold is greater than 0; a threshold The set axial acceleration threshold greater than 0; v imax The upper limit of the set feed rate value greater than 0; V vector This is a vector composed of the axial velocities of all printing points during the printing process; a vector V is a vector composed of the axial accelerations of all printing points during the printing process; vector || P Represents vector V vector The P-norm; ||a vector || P Represents vector a vector The P-norm; (23) Construct an optimization objective function, which includes the smoothness of the feed rate, the sum of the smoothness of the axial acceleration of each axis, and the theoretical total printing time. The objective function is: f(x) = a • s(v re ) + b • s(a) + g • t (3) wherein a, β, γ are weight coefficients; s(v re ) is the smoothness of the feed speed, s(a) is the sum of the axial acceleration smoothness of each axis, and t is the total theoretical printing time. (24) Formulas (1), (2), and (3) together form the final optimized mathematical model: Step 3: By considering the indentation depth constraint of the printhead relative to the printing surface when the normal vector is offset, calculate the maximum offset angle of the normal vector of each printing point in the printing surface, and use it as an optimization variable. Upper limit of values Step four, based on the chain rule of derivation, according to the final optimization mathematical model of step (24), the objective function f(x) and the constraint condition ||V vector || P ||a vector || P The gradient of the design variable x is calculated as follows: in, Represents vector V vector The k-th element in Represents vector a vector The k-th element in the array, where m is the number of axes in the printer; Step 5: Based on the gradient calculated in Step 4, use a gradient-based optimization algorithm to adjust the deviation angle of the normal vector. The normal vector rotation angle θ and the nozzle feed speed v relative to the workpiece are optimized together.

2. The method for collaborative optimization of feed rate and printing direction for multi-axis additive manufacturing as described in claim 1, characterized in that: The calculation method of the smoothness s(v re ) of the feed speed is Where n is the number of print points in the print path, v i This represents the feed rate of the i-th printing interval.

3. The method for collaborative optimization of feed rate and printing direction for multi-axis additive manufacturing as described in claim 1, characterized in that: The method for calculating the sum of the axial acceleration smoothness of each axis, s(a), is as follows: Where n is the number of print points in the print path; m is the number of axes of the printer; and a ij This represents the i-th acceleration value on the j-th axis.

4. The method for collaborative optimization of feed rate and printing direction for multi-axis additive manufacturing as described in claim 1, characterized in that: The method for calculating the theoretical total printing time t is as follows: Where, Δl i Let be the arc length of the i-th printing interval in the printing path. If the printing points are selected densely during path planning, it is approximately equal to the distance between two adjacent printing points; n is the number of printing points in the printing path; v i This represents the feed rate of the i-th printing interval.

5. The method for collaborative optimization of feed rate and printing direction for multi-axis additive manufacturing as described in claim 1, characterized in that: Upper limit of the normal vector offset angle The calculation is performed using formula (11), specifically for two cases: convex and concave surfaces. Where d is the diameter of the printhead; h is the interlayer height; r k δ is the maximum radius of curvature at that point on the printed surface; δ is the maximum indentation depth, and 0≤δ≤h; l is calculated using formula (12).

Images