CN109918807B - Local tool path smoothing method for optimizing curvature - Google Patents

Abstract

The invention relates to a local tool path fairing method for optimizing curvature, which comprises the steps of firstly establishing a quintic quasi-uniform B spline which is provided with 7 control vertexes and symmetrically distributed on the control vertexes to realize G3 continuity at the joint of a straight line and a curve, and optimizing the curvature of the curve by minimizing the strain energy and the jump energy of the curve to obtain a fairing path with the optimal curvature. The analytical solution of the fairing track of the local corner is realized by establishing a fitting function relation between the corner angle value and the control vertex distribution proportion, the analytical solution of the fairing track of the local corner is realized, the curvature peak value is greatly reduced, and the stability in the processing process is improved.

Description

Local tool path smoothing method for optimizing curvature

Technical Field

The invention relates to a local tool path smoothing method, in particular to a G3 continuous local tool path smoothing method with optimized curvature.

Background

Document 1 "s.tulssan, y.altitas, Local pitch smoothing for five-axis Machine Tools, International Journal of Machine Tools and manufacturing, 2015,96: 15-26" discloses a G3 continuous Local tool path smoothing method, which realizes G3 continuity by inserting five quasi-uniform B splines between corners formed by two adjacent straight line segments during smoothing of straight line path tracks formed by tool nose position points, and can realize precision control of smooth rear tracks. However, when solving the distribution of the control vertexes satisfying the G3 continuity, only one group of feasible solutions which are convenient to calculate is selected, and the curve curvature is not optimized.

Document 2 "X.Beudaert, S.Lavernhe, C.Tournier,5-axis local burner rounding of linear tool path discontinuities, International Journal of Machine Tools and manufacturing, 2013,73: 9-16" discloses a continuous local path smoothing method of G2. In the method, a straight line segment track formed by the position points of the cutter point is smoothened by adopting a cubic quasi-uniform B-spline, and a group of empirical values are directly selected when the distribution proportion of control vertex points is determined. Although G2 continuity of the smooth trajectory and precision control can be realized, the continuous acceleration in the machining process cannot be ensured, and curvature optimization is not carried out.

Typical features of the above documents are: the tool path smoothed by the literature method meets the precision requirement and a certain continuity requirement, but the curvature is not optimized, the optimal smoothing path cannot be achieved, the peak values of acceleration and acceleration in the machining process are large, and the machining efficiency is influenced.

Disclosure of Invention

Technical problem to be solved

In order to overcome the defects that the curvature is not optimized, the curvature peak value is large, the feeding speed is reduced, the machining efficiency is influenced, and the acceleration peak value are large in the machining process in the conventional G3 continuous local tool path smoothing method, the invention provides a G3 continuous local tool path smoothing method with the optimal curvature. The method comprises the steps of firstly establishing a quintic quasi-uniform B spline which is provided with 7 control vertexes and symmetrically distributed on the control vertexes to realize G3 continuity of a joint of a straight line and a curve, and optimizing the curvature of the curve by adopting a minimum energy method. In order to solve the control vertex by analysis, the invention establishes a fitting function of the corner angle value and the control vertex distribution ratio, and realizes the analysis and the solution of the local corner fairing track.

Technical scheme

A local tool path smoothing method for optimizing curvature is characterized by comprising the following steps:

step 1: in the form of two adjacent straight line segments p_k-1p_kAnd p_kp_k+1Five quasi-uniform B-splines with symmetrically distributed control vertices inserted on the local corners, wherein the point p_k-1、p_kIs a straight line segment p_k-1p_kEnd point of (1), point p_kAnd p_k+1Is a straight line segment p_kp_k+1The endpoint of (1); it controls the vertex P_k,iCalculated using the following formula, i ═ 0,1, 2.

P_k,3＝p_k

P_k,0＝p_k-(d_k,1+d_k,2+d_k,3)m_k

P_k,1＝p_k-(d_k,2+d_k,3)m_k

P_k,2＝p_k-d_k,3m_k

P_k,4＝p_k-d_k,3n_k

P_k,5＝p_k-(d_k,2+d_k,3)n_k

P_k,6＝p_k-(d_k,1+d_k,2+d_k,3)n_k

Wherein

In the formula (d)_k,1To control the vertex P_k,0And P_k,1Distance between d_k,2To control the vertex P_k,1And P_k,2Distance between d_k,₃To control the vertex P_k,2And P_k,3The distance between them;

step 2: the node vector of the B spline is U ═ 0000000.5111111;

and step 3: establishing an inserted B spline P according to the B spline control vertex solved in the step 1 and the node vector selected in the step 2_k(u)：

Wherein

Definition of

In the formula, n is the number of B control vertexes minus one, i represents the sequence number of the basis function, and p represents the number of the basis function is 5;

and 4, step 4: since the control vertexes of the inserted B-splines are symmetrically distributed, the maximum error occurs at the midpoint P of the spline_k(0.5), substituting the parameter u to 0.5 in step 3 may result in:

tolerance constraint of inserting B-spline as e_max＝‖P_k,3-P_k(0.5)‖≤ε_wIn which epsilon_wRepresenting a given tolerance value;

and 5: the control vertex P obtained in the step 1_k,iI is 0,1,2, 6, the tolerance constraint expression of step 4 is substituted into

In the formula

Is a straight line segment p_k-1p_kAnd p_kp_k+1The formed corner angle value;

step 6: the length of the straight line segment inserted into the B-spline is constrained to

In the formula I_k＝‖p_k-1p_k‖，l_k+1＝‖p_k p_k+1‖；

And 7: let x_k＝[d_k,1d_k,2d_k,3]Establishing an objective function of the curvature optimization by an energy method, and solving x_k＝[d_k,1d_{k, 2}d_k,3]；

minE_k(x_k)

s.t.

C(x_k)≤O

Wherein

In the formula beta_kIs a proportionality coefficient for balancing strain energy of curve

Energy of harmonious degree

And 8: calculating x according to steps 1-7_k＝[d_k,1d_k,2d_k,3]Let control vertex distribution ratio μ be d_k,1/d_k,3，ν＝d_k,2/d_k,3Establishing theta_kAnd a fitting function of μ, ν:

in the formula C_A,iAnd C_B,iIs a fitting coefficient, i ═ 1, 2.., 6;

and step 9: and (3) resolving the local smooth track:

1) calculating a corner angle value θ_k；

2) Determining control vertex distribution proportion coefficients mu and v according to the step 8;

3) computing

4) Calculating d_k,1＝μd_k,3、d_k,1＝μd_k,3；

5) Calculating a control vertex P according to step 1_k,iObtaining a locally smoothed trajectory, i ═ 0,1,2,. 6:

。

advantageous effects

The invention provides a local tool path fairing method for optimizing curvature, which comprises the steps of firstly establishing a quintic quasi-uniform B spline which is provided with 7 control vertexes and symmetrically distributed on the control vertexes to realize G3 continuity at the joint of a straight line and a curve, and optimizing the curvature of the curve by minimizing the strain energy and the jump energy of the curve to obtain a fairing path with the optimal curvature. The analytical solution of the fairing track of the local corner is realized by establishing a fitting function relation between the corner angle value and the control vertex distribution proportion, the analytical solution of the fairing track of the local corner is realized, the curvature peak value is greatly reduced, and the stability in the processing process is improved.

Drawings

FIG. 1 is a schematic diagram of a spline curve inserted in a local tool path smoothing process by using the present invention.

FIGS. 2 and 3 are graphs of the results of fitting functions between the corner angles and the control vertex distribution ratios established by the present invention.

FIGS. 4, 5, 6 and 7 are graphs showing the comparison of the distribution of the curvature of the track after smoothing of example 1 by the method of the present invention and the method of reference [1 ].

Fig. 8 and 9 are graphs showing the comparison of the acceleration during processing of the trajectory after smoothing of example 2 by the method of the present invention and the method of document [1 ].

FIGS. 10 and 11 are graphs showing a comparison of the jerk during processing of the trajectory after smoothing of example 2 by the method of the present invention and the method of reference [1 ].

Detailed Description

The invention will now be further described with reference to the following examples and drawings:

example 1 was used to demonstrate the advantage of the invention in terms of curvature of the smooth trail. Example 2 was used to demonstrate the advantages of the present invention during processing.

Example 1:

(1) the selected angles are theta respectively in the experiment_k＝15°、θ_k＝30°、θ_k＝45°、θ_k60 ° corner, taking l 10mm, the coordinates of the points making up the corner are [ p ]_k-1；p_k；p_k+1]＝[00；l 0；l+lcos(π-θ)lsin(π-θ)]Selecting a tolerance value epsilon_w＝0.1mm。

(2) Passing the corner angle value θ according to step (8)_kAnd determining the control vertex distribution proportion coefficients mu and v.

(3) Computing a control vertex P_k,2And P_k,3A distance d between_k,3

(4) Calculating d_k,1And d_k,2The value of (c): d_k,1＝μd_k,3、d_k,1＝μd_k,3

(5) Calculating the control peak P of the smooth trajectory_k,i(i＝0,1,2,...,6)，

P_k,3＝p_k

P_k,0＝p_k-(d_k,1+d_k,2+d_k,3)m_k

P_k,1＝p_k-(d_k,2+d_k,3)m_k

P_k,2＝p_k-d_k,3m_k

P_k,4＝p_k-d_k,3n_k

P_k,5＝p_k-(d_k,2+d_k,3)n_k

P_k,6＝p_k-(d_k,1+d_k,2+d_k,3)n_k

Wherein the content of the first and second substances,

will control the vertex P_k,iAnd substituting the expression of the B spline to obtain the track after the local fairing.

(6) The distribution of the curvature of the trajectory after fairing in the interval of the spline parameter u ═ 0,1 is calculated, and the specific results are shown in fig. 4, 5, 6 and 7. It can be seen that the method smoothed trace has a significantly reduced peak curvature compared to the typical G3 continuous local method smoothed trace.

Example 2:

(1) verification test selection of the starting Point [ p ]_k-1；p_k；p_k+1]＝[00；100；10+10cos(π-30°)10sin(π-30°)]And (4) forming a track for smoothing, wherein the unit length of a coordinate axis is 1 mm. Selected tolerance value epsilon_wThe feed rate was 3mm/s at 0.1 mm.

(2) Calculating a corner angle value

(3) Root of herbaceous plantPassing the corner angle value theta according to step (8)_kAnd determining the control vertex distribution proportion coefficients mu and v.

(4) Calculating the distance d_k,3

(5) Calculating d_k,1And d_k,2The value of (c): d_k,1＝μd_k,3、d_k,1＝μd_k,3。

(6) Calculating the control peak P of the smooth trajectory_k,i(i＝0,1,2,...,6)

P_k,3＝p_k

P_k,0＝p_k-(d_k,1+d_k,2+d_k,3)m_k

P_k,1＝p_k-(d_k,2+d_k,3)m_k

P_k,2＝p_k-d_k,3m_k

P_k,4＝p_k-d_k,3n_k

P_k,5＝p_k-(d_k,2+d_k,3)n_k

P_k,6＝p_k-(d_k,1+d_k,2+d_k,3)n_k

Wherein the content of the first and second substances,

and substituting the control vertex into an expression of a B spline to obtain a track subjected to local fairing.

(7) The smooth track is processed at a constant feed speed of 3mm/s, and the acceleration in the processing process is shown in the attached figures 8 and 9, so that the acceleration peak value in each axis motion of the smooth track is smaller under the same processing condition. The x-axis can be reduced by 9.03 percent, and the y-axis can be reduced by 13.37 percent.

(8) The jerk during the machining process is compared with that shown in fig. 10 and 11, and it can be seen that under the same machining conditions, the jerk peak value in each axis motion of the smooth track of the invention is smaller. The x-axis can be reduced by 10.79 percent, and the y-axis can be reduced by 12.17 percent.

Claims (1)

1. A local tool path smoothing method for optimizing curvature is characterized by comprising the following steps:

step 1: in the form of two adjacent straight line segments p_k-1p_kAnd p_kp_k+1Five quasi-uniform B-splines with symmetrically distributed control vertices inserted on the local corners, wherein the point p_k-1、p_kIs a straight line segment p_k-1p_kEnd point of (1), point p_kAnd p_k+1Is a straight line segment p_kp_k+1The endpoint of (1); it controls the vertex P_k,iCalculated using the following formula, i ═ 0,1, 2.

P_k,3＝p_k

P_k,0＝p_k-(d_k,1+d_k,2+d_k,3)m_k

P_k,1＝p_k-(d_k,2+d_k,3)m_k

P_k,2＝p_k-d_k,3m_k

P_k,4＝p_k-d_k,3n_k

P_k,5＝p_k-(d_k,2+d_k,3)n_k

P_k,6＝p_k-(d_k,1+d_k,2+d_k,3)n_k

Wherein

In the formula (d)_k,1To control the vertex P_k,0And P_k,1Distance between d_k,2To control the vertex P_k,1And P_k,2Distance between d_k,3To control the vertex P_k,2And P_k,3The distance between them;

step 2: the node vector of the B spline is U ═ 0000000.5111111;

and step 3: establishing an inserted B spline P according to the B spline control vertex solved in the step 1 and the node vector selected in the step 2_k(u)：

Wherein

Definition of

In the formula, n is the number of B control vertexes minus one, i represents the sequence number of the basis function, and p represents the number of the basis function is 5;

and 4, step 4: since the control vertexes of the inserted B-splines are symmetrically distributed, the maximum error occurs at the midpoint P of the spline_k(0.5), substituting the parameter u to 0.5 in step 3 may result in:

tolerance constraint of inserting B-spline as e_max＝‖P_k,3-P_k(0.5)‖≤ε_wIn which epsilon_wRepresenting a given tolerance value;

and 5: the control vertex P obtained in the step 1_k,iI is 0,1,2, 6, the tolerance constraint expression of step 4 is substituted into

In the formula

Is a straight line segment p_k-1p_kAnd p_kp_k+1The formed corner angle value;

step 6: the length of the straight line segment inserted into the B-spline is constrained to

In the formula I_k＝‖p_k-1p_k‖，l_k+1＝‖p_k p_k+1‖；

And 7: let x_k＝[d_k,1 d_k,2 d_k,3]Establishing an objective function of the curvature optimization by an energy method, and solving x_k＝[d_k,1 d_k,2d_k,3]；

minE_k(x_k)

s.t.

C(x_k)≤O

Wherein

In the formula beta_kIs a proportionality coefficient for balancing strain energy of curve

Energy of harmonious degree

And 8: calculating x according to steps 1-7_k＝[d_k,1 d_k,2 d_k,3]Let control vertex distribution ratio μ be d_k,1/d_k,3，ν＝d_k,2/d_k,3Establishing theta_kAnd a fitting function of μ, ν:

in the formula C_A,iAnd C_B,iIs a fitting coefficient, i ═ 1, 2.., 6;

and step 9: and (3) resolving the local smooth track:

1) calculating a corner angle value θ_k；

2) Determining control vertex distribution proportion coefficients mu and v according to the step 8;

3) computing

4) Calculating d_k,1＝μd_k,3、d_k,1＝μd_k,3；

5) Calculating a control vertex P according to step 1_k,iAfter obtaining local fairingI-0, 1,2, 6:

。

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