CN116408793A - Industrial robot path fairing method and system with continuous curvature - Google Patents

Abstract

The invention belongs to the field of robot path planning, and discloses a method and a system for fairing paths of industrial robots with continuous curvatures, wherein the method comprises the following steps: reading a robot path and calculating curvatures at all position points on the path; dividing the path into curvature abrupt change areas and continuous areas; calculating the curvature, unit tangent vector and unit normal vector at two endpoints of the curvature abrupt change region; defining a quintic Bezier curve containing shape adjustment parameters; defining an objective function of the shape adjusting parameter according to boundary conditions, curvature change energy and length energy composite minimum principle at the end points; converting the shape adjustment parameter solving problem into a minimized objective function problem, respectively solving partial derivatives of the shape adjustment parameters through the objective function, solving the objective function to obtain the numerical value of the shape adjustment parameters, and substituting the numerical value into a five-time Bezier curve to obtain the fairing path of the curvature mutation region. The invention realizes the continuous smooth transition of the robot path curvature and improves the motion smoothness of the robot passing through the abrupt change region.

Description

Industrial robot path fairing method and system with continuous curvature

Technical Field

The invention belongs to the field of robot path planning, and particularly relates to an industrial robot path fairing method and system with continuous curvature.

Background

The motion path of the industrial robot generally comprises a circular arc segment path, a straight line segment path and other different types of paths. In order to improve the motion stability and speed of the robot and solve the problem of frequent acceleration and deceleration of the robot at corners, fairing treatment is needed between different types of path sections. The straight-line segment path is characterized in that the curvature is zero everywhere on the path, so when moving from one straight-line segment path to another straight-line segment path, the curvature before and after fairing at the corner is not changed, and the corresponding fairing method is quite large. However, for a circular arc segment path, the curvature may be any value greater than zero, so when moving from one circular arc segment path to another circular arc segment path or another straight line segment path, the characteristic of different curvatures at the two ends of the transition curve needs to be considered in the corner fairing process. Therefore, how to smooth according to different types of path sections at two sides of the transition section becomes a problem to be solved by the robot path fairing.

In the prior art, a person adopts a cubic B spline curve to transition a tiny straight line segment to carry out local fairing at a transition corner, so that C3 continuity at the corner is realized, but the method aims at an off-line command and does not involve a curve with curvature change; it has also been proposed to use a C3 continuous short segment transition fairing algorithm that uses a PH curve to ensure smooth transition at the junction of a straight segment and a PH curve, which also smoothes the path for the G01 command.

In the prior art, different curves such as NUBRS curve, five-time and seven-time PH curve, bezier curve and the like are adopted to smooth the path of the tiny straight line segment. For example, the transition between the arc segment path and the arc segment path, the arc segment path and the straight line segment path is adopted by adopting an assumed arc transition method, and although the fairing can be realized aiming at the arc segment paths with different diameters and different processing error requirements, the set assumed arc curvature is unequal to the curvature before and after transition, so that the motion is unstable at the motion connecting point.

Therefore, the gesture path after the fairing in the prior art realizes certain curvature continuity and motion stability, but the continuous fairing of the curvature is not considered for a complex arc path, and a plurality of different path fairing methods are lacked.

Disclosure of Invention

In order to overcome the defects in the prior art, the invention provides the industrial robot path fairing method and system with continuous curvature, which adopt the quintic Bezier curve containing the shape adjusting parameters, realize the continuous curvature when the arc section path and other types of paths are switched, and improve the motion stability of the industrial robot when passing through different paths.

The technical scheme adopted by the method is as follows: a method of fairing an industrial robot path with a continuous curvature, comprising the steps of:

reading a path of a given industrial robot, and calculating curvatures at all position points on the path;

dividing the path into a curvature abrupt change region and a curvature continuous region according to whether the curvature is abrupt;

for the curvature abrupt change region, calculating the curvature, unit tangent vector and unit normal vector at two endpoints of the region;

defining a quintic Bezier curve containing shape adjustment parameters for a fairing curvature mutation zone;

defining an objective function of a shape adjustment parameter in the quintic Bezier curve according to boundary conditions at the end points and according to a curvature change energy and length energy composite minimum principle;

converting the shape adjustment parameter solving problem into a minimized objective function problem, respectively solving partial derivatives of the shape adjustment parameters through the objective function, and solving the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjustment parameters;

substituting the numerical value of the shape adjusting parameter into the quintic Bezier curve to obtain the fairing path of the curvature mutation region.

The technical scheme adopted by the system of the invention is as follows: an industrial robot path fairing system with continuous curvature comprising the following modules:

the curvature calculation module is used for reading the path of the given industrial robot and calculating the curvatures at all the position points on the path;

the abrupt change area dividing module is used for dividing the path into a curvature abrupt change area and a curvature continuous area according to whether the curvature is abrupt or not;

the abrupt region processing module is used for calculating the curvature, the unit tangent vector and the unit normal vector at the two end points of the curvature abrupt region;

the Bezier curve definition module is used for defining a five-order Bezier curve fairing curvature mutation zone containing shape adjustment parameters; defining an objective function of a shape adjustment parameter in the quintic Bezier curve according to boundary conditions at the end points and according to a curvature change energy and length energy composite minimum principle;

the shape adjustment parameter solving module is used for converting the shape adjustment parameter solving problem into a minimized objective function problem, respectively solving partial derivatives of the shape adjustment parameters through the objective function, and solving the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjustment parameters; substituting the numerical value of the shape adjusting parameter into the quintic Bezier curve to obtain the fairing path of the curvature mutation region.

Compared with the prior art, the invention has the following technical effects:

the invention adopts the quintic Bezier curve containing the shape adjusting parameters, realizes the curvature continuity when the arc section path and other types of paths are switched, realizes the smooth transition of the robot path curvature continuity, and improves the motion stability of the industrial robot when passing through different paths.

Drawings

FIG. 1 is a flow chart of an industrial robot path fairing method in an embodiment of the invention.

FIG. 2 is a schematic illustration of a non-smooth path in example 1 of the present invention.

Fig. 3 is a schematic diagram of the path after fairing in example 1 of the present invention.

Fig. 4 is a graph showing the curvature change of a given path and a fairing curve in example 1 of the present invention.

FIG. 5 is a schematic illustration of a non-smooth path in example 2 of the present invention.

Fig. 6 is a schematic diagram of the path after fairing in example 2 of the present invention.

Fig. 7 shows the curvature change of a given path and a fairing curve in example 2 of the present invention.

Detailed Description

The technical scheme of the present invention will be described in further detail with reference to examples and drawings, but embodiments of the present invention are not limited thereto.

Example 1

The embodiment provides a fairing method with continuous curvature when an industrial robot arc path and an arc path transition, as shown in fig. 1, the method comprises the following main steps:

step 1, reading a path of a given industrial robot, and calculating curvatures at all position points on the path;

the path c (t) of the industrial robot comprises two arc segment paths, namely c₁ (t) and c₂ (t) the circle centers are respectively marked as C₁ And C₂ The radii are denoted as r respectively₁ And r₂ Each circular arc segment path comprises 300 position points P_i (x_i ,y_i ,z_i ) Where i=1, 2,3 …, N, x_i 、y_i 、z_i Coordinates of the ith position point on the x-axis, the y-axis and the z-axis are respectively, N is the number of position points included in the path, and N is 300 in the embodiment; the non-smooth pose path is shown in fig. 2.

Calculating the curvature k (t) at all the position points on the two circular arc path sections:

where c' (t) is the first derivative of path c (t), c "(t) is the second derivative of path c (t), t is the normalized parameter variable, t ε [0,1].

Step 2, dividing the path into a curvature abrupt change area and a curvature continuous area according to whether the curvature is abrupt;

the invention uses the contact point P of two arc segment paths_j The point is defined as the point of abrupt change of curvatureThe embodiment normalizes the parameter variable t E [0.9,1 ]]The included arc segment path is divided into curvature abrupt change areas.

Step 3, calculating the curvature, unit tangent vector and unit normal vector of the curvature mutation region at two end points of the region;

the location points of the two end points of the curvature mutation region are denoted as p₀ And p₁ Obtaining the curvature k of two endpoints of the curvature abrupt change region according to the curvature calculation formula in thestep 1₀ And k₁ The method comprises the steps of carrying out a first treatment on the surface of the Contact point P of two arc segment paths_j And two end points p of curvature mutation region₀ And p₁ Respectively calculating two arc segment paths c₁ (t) and c₂ (T) tangential vectors at both end points, and performing unitization processing on the tangential vectors to obtain a unitized tangential vector T₀ And T₁ The method comprises the steps of carrying out a first treatment on the surface of the The two unit tangential vectors are respectively rotated by 90 degrees anticlockwise to obtain two normal vectors, and the two normal vectors are respectively subjected to unitization treatment to obtain a unit normal vector N₀ And N₁ 。

That is, the unit tangent vector T in the present embodiment₀ And T₁ The vector unitization is obtained by two endpoints of the curvature mutation region and the curvature mutation point; the unit normal vector is denoted as N₀ And N₁ Is obtained by rotating the unit tangent vector by 90 degrees anticlockwise

Step 4, defining a quintic Bezier curve containing shape adjustment parameters for a fairing curvature mutation zone;

defining a quintic Bezier curve b (t) containing shape adjustment parameters as:

b(t)＝b₀ B₀ (t)+b₁ B₁ (t)+b₂ B₂ (t)+b₃ B₃ (t)+b₄ B₄ (t)+b₅ B₅ (t)

wherein b₀ 、b₁ 、b₂ 、b₃ 、b₄ And b₅ As control points, the expression of each control point is:

wherein alpha is₀ 、α₁ 、β₀ 、β₁ The shape adjusting parameter is larger than zero and is used for adjusting the tangential directions of the starting point and the end point of the spline curve and the positions of the control points;

B₀ (t)、B₁ (t)、B₂ (t)、B₃ (t)、B₄ (t) and B₅ (t) is a control point Bernstant polynomial of the formula:

step 5, defining an objective function of a shape adjustment parameter in the quintic Bezier curve according to boundary conditions at the end points and according to a curvature change energy and length energy composite minimum principle;

the objective function is defined as:

where b '"(t) is the third derivative of b (t) and b' (t) is the first derivative of b (t).

Step 6, converting the problem of solving the shape adjusting parameters into a minimized objective function problem, respectively solving partial derivatives of the shape adjusting parameters through the objective function, and solving the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjusting parameters;

the minimization of the objective function problem is: minf (alpha)₀ ,α₁ ,β₀ ,β₁ )；

The partial derivatives of the shape adjustment parameters are respectively calculated through the objective function, and the method comprises the following steps:

and->

Solving an objective function by adopting a quasi-Newton method, comprising the following steps of:

given an initial value

And a stop condition epsilon; wherein the superscript 0 denotes alpha atiteration 0₀ ,α₁ ,β₀ ,β₁ Is set to an initial value of (1);

setting a hessian matrix H_k Initial value H of₀ Is a unit matrix and calculates the gradient g of the objective function_k ：

Determining a search direction d_k ＝H_k g_k ；

Updating shape adjustment parameters

Wherein the superscript k denotes alpha at the kth iteration₀ ,α₁ ,β₀ ,β₁ Is a value of (2); the superscript k+1 denotes α at the k+1th iteration₀ ,α₁ ,β₀ ,β₁ Is a value of (2); lambda (lambda)_k Representing step size factor, by solving->

Lambda at the time of obtaining the minimum value;

judging

If so, outputting a shape adjustment parameter, otherwise, updating the gradient of the hessian matrix and the objective function:

and performs the next calculation of the shape adjustment parameters.

And 7, substituting the numerical value of the shape adjusting parameter into the quintic Bezier curve to obtain the fairing path of the curvature mutation region.

Fig. 2 shows a path before the two circular arc segment paths are straightened in this embodiment, and fig. 3 shows a curvature change condition after the two circular arc segment paths are straightened in this embodiment. It can be seen that after the path fairing is performed by adopting the method of the embodiment, the curvatures of the two transition path sections are equal to the curvatures of the two arc path sections respectively, as shown in fig. 4, so that the continuous change of the path curvature after the fairing is ensured, and therefore, the movement speed of the robot can not be reduced to zero, thereby improving the movement efficiency and ensuring the movement stability.

Example 2

The present embodiment provides a path fairing method with continuous curvature when the arc path and the straight path of the industrial robot are transited, and the main steps are basically the same as those ofembodiment 1. Fig. 5 shows a path before a circular arc section and a straight path section of the present embodiment, fig. 6 shows a path after a circular arc section and a straight path section of the present embodiment, fig. 7 shows a curvature change condition after a circular arc section and a straight path section of the present embodiment are straightened, a curvature of the path after the straightening continuously changes, a movement speed of the robot may not be reduced to zero, a movement efficiency of the robot is improved, and a movement stability of the robot is also ensured.

Example 3

The present embodiment, based on the same inventive concept asembodiment 1 andembodiment 2, provides an industrial robot path fairing system with continuous curvature, which includes the following modules:

the curvature calculation module is used for reading the path of the given industrial robot and calculating the curvatures at all the position points on the path;

the abrupt change area dividing module is used for dividing the path into a curvature abrupt change area and a curvature continuous area according to whether the curvature is abrupt or not;

the abrupt region processing module is used for calculating the curvature, the unit tangent vector and the unit normal vector at the two end points of the curvature abrupt region;

the Bezier curve definition module is used for defining a five-order Bezier curve fairing curvature mutation zone containing shape adjustment parameters; defining an objective function of a shape adjustment parameter in the quintic Bezier curve according to boundary conditions at the end points and according to a curvature change energy and length energy composite minimum principle;

the shape adjustment parameter solving module is used for converting the shape adjustment parameter solving problem into a minimized objective function problem, respectively solving partial derivatives of the shape adjustment parameters through the objective function, and solving the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjustment parameters; substituting the numerical value of the shape adjusting parameter into the quintic Bezier curve to obtain the fairing path of the curvature mutation region.

Wherein, the defined quintic Bezier curve b (t) containing the shape adjusting parameters is:

b(t)＝b₀ B₀ (t)+b₁ B₁ (t)+b₂ B₂ (t)+b₃ B₃ (t)+b₄ B₄ (t)+b₅ B₅ (t)

wherein b₀ 、b₁ 、b₂ 、b₃ 、b₄ And b₅ As control points, the expression of each control point is:

B₀ (t)、B₁ (t)、B₂ (t)、B₃ (t)、B₄ (t) and B₅ (t) is the control point, the Bernstan polynomial:

wherein p is₀ And p₁ The two end points of the curvature mutation zone are position points; alpha₀ 、α₁ 、β₀ And beta₁ The shape adjusting parameter is larger than zero and is used for adjusting the tangential directions of the starting point and the end point of the spline curve and the positions of the control points; k (k)₀ And k₁ Is the curvature of the two endpoints of the curvature abrupt region.

The shape adjustment parameter obtaining module is used for defining an objective function of the shape adjustment parameter in the quintic Bezier curve according to boundary conditions at the end points and the curvature change energy and length energy composite minimum principle, wherein the objective function is as follows:

wherein b '"(t) is the third derivative of b (t) and b' (t) is the first derivative of b (t);

the shape adjustment parameter solving module converts the shape adjustment parameter solving problem into a minimized objective function problem, respectively solves partial derivatives of the shape adjustment parameters through the objective function, solves the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjustment parameters, and comprises the following steps:

the minimization of the objective function problem is: minf (alpha)₀ ,α₁ ,β₀ ,β₁ )

The partial derivatives of the shape adjustment parameters are respectively calculated through the objective function, and the method comprises the following steps:

and->

Solving an objective function by adopting a quasi-Newton method, comprising the following steps of:

given an initial value

And a stop condition epsilon;

setting the initial value H of the Heisen matrix₀ Calculating the gradient of the objective function as an identity matrix

Determining a search direction d_k ＝H_k g_k Wherein H is_k Is a hessian matrix;

updating shape adjustment parameters

Wherein the superscript k denotes alpha at the kth iteration₀ ,α₁ ,β₀ ,β₁ The superscript k+1 denotes α at the k+1th iteration₀ ,α₁ ,β₀ ,β₁ Is a value of (2); lambda (lambda)_k Representing step size factor, by solving->

Lambda at the time of obtaining the minimum value;

judging

If so, outputting a shape adjustment parameter, otherwise, updating the gradient of the hessian matrix and the objective function:

and performs the next shape adjustment parameter calculation.

Please refer toembodiment 1 andembodiment 2 for the implementation process of each module in this embodiment, which is not repeated.

Although embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and not to be construed as limiting the invention, and that variations, modifications, alternatives, and variations may be made in the above embodiments by those skilled in the art without departing from the spirit and principles of the invention.

Claims (9)

1. A method for smoothing a path of an industrial robot having a continuous curvature, comprising the steps of:

reading a path of a given industrial robot, and calculating curvatures at all position points on the path;

dividing the path into a curvature abrupt change region and a curvature continuous region according to whether the curvature is abrupt;

for the curvature abrupt change region, calculating the curvature, unit tangent vector and unit normal vector at two endpoints of the region;

defining a quintic Bezier curve containing shape adjustment parameters for a fairing curvature mutation zone;

defining an objective function of a shape adjustment parameter in the quintic Bezier curve according to boundary conditions at the end points and according to a curvature change energy and length energy composite minimum principle;

converting the shape adjustment parameter solving problem into a minimized objective function problem, respectively solving partial derivatives of the shape adjustment parameters through the objective function, and solving the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjustment parameters;

substituting the numerical value of the shape adjusting parameter into the quintic Bezier curve to obtain the fairing path of the curvature mutation region.

2. The method according to claim 1, wherein the path of the industrial robot is c (t), the path comprising a position point P_i (x_i ,y_i ,z_i ) Where i=1, 2,3 …, N, x_i 、y_i 、z_i Coordinates of the ith position point on the x-axis, the y-axis and the z-axis respectively, and N is the number of position points included in the path;

curvature k (t) at all points on the path:

where c' (t) is the first derivative of path c (t), c "(t) is the second derivative of path c (t), and t is the normalized parameter variable.

3. The method of claim 1, wherein the computing curvature, unit tangent vector, and unit normal vector at two endpoints of the region for the region of abrupt curvature comprises:

the unit tangent vector is denoted as T₀ And T₁ The vector unitization is obtained by two endpoints of the curvature mutation region and the curvature mutation point; the unit normal vector is denoted as N₀ And N₁ Is obtained by rotating the unit tangent vector by 90 degrees anticlockwise.

4. The method according to claim 1, characterized in that the defined quintic bezier curve b (t) containing shape adjustment parameters is:

b(t)＝b₀ B₀ (t)+b₁ B₁ (t)+b₂ B₂ (t)+b₃ B₃ (t)+b₄ B₄ (t)+b₅ B₅ (t)

wherein b₀ 、b₁ 、b₂ 、b₃ 、b₄ And b₅ As control points, the expression of each control point is:

B₀ (t)、B₁ (t)、B₂ (t)、B₃ (t)、B₄ (t) and B₅ (t) is the control point, the Bernstan polynomial:

wherein p is₀ And p₁ The two end points of the curvature mutation zone are position points; alpha₀ 、α₁ 、β₀ And beta₁ Shape adjustment parameters greater than zero for adjusting spline curve start and end cutsLine direction and control point position; k (k)₀ And k₁ Is the curvature of the two endpoints of the curvature abrupt region.

5. The method of claim 4, wherein the objective function of the shape adjustment parameters in the five-order bezier curve defined according to the boundary conditions at the end points and according to the curvature change energy and length energy composite minima principle is:

where b '"(t) is the third derivative of b (t) and b' (t) is the first derivative of b (t).

6. The method of claim 5, wherein converting the shape adjustment parameter solving problem to a minimized objective function problem, respectively solving the shape adjustment parameters by the objective function as partial derivatives, and solving the objective function by quasi-newton method to obtain the values of the shape adjustment parameters, comprises:

the minimization of the objective function problem is: minf (alpha)₀ ,α₁ ,β₀ ,β₁ )

The partial derivatives of the shape adjustment parameters are respectively calculated through the objective function, and the method comprises the following steps:

and->

Solving an objective function by adopting a quasi-Newton method, comprising the following steps of:

given an initial value

And a stop condition epsilon; wherein the superscript 0 denotes alpha at iteration 0₀ ,α₁ ,β₀ ,β₁ Is set to an initial value of (1);

setting the initial value H of the Heisen matrix₀ Calculating the gradient of the objective function as an identity matrix

Determining a search direction d_k ＝H_k g_k Wherein H is_k Is a hessian matrix;

updating shape adjustment parameters

Wherein the superscript k denotes alpha at the kth iteration₀ ,α₁ ,β₀ ,β₁ Is a value of (2); the superscript k+1 denotes α at the k+1th iteration₀ ,α₁ ,β₀ ,β₁ Is a value of (2); lambda (lambda)_k Representing step size factor, by solving->

Lambda at the time of obtaining the minimum value;

judging

If so, outputting a shape adjustment parameter, otherwise, updating the gradient of the hessian matrix and the objective function:

and performs the next shape adjustment parameter calculation.

7. An industrial robot path fairing system with continuous curvature, comprising the following modules:

the curvature calculation module is used for reading the path of the given industrial robot and calculating the curvatures at all the position points on the path;

the abrupt change area dividing module is used for dividing the path into a curvature abrupt change area and a curvature continuous area according to whether the curvature is abrupt or not;

the abrupt region processing module is used for calculating the curvature, the unit tangent vector and the unit normal vector at the two end points of the curvature abrupt region;

the Bezier curve definition module is used for defining a five-order Bezier curve fairing curvature mutation zone containing shape adjustment parameters; defining an objective function of a shape adjustment parameter in the quintic Bezier curve according to boundary conditions at the end points and according to a curvature change energy and length energy composite minimum principle;

the shape adjustment parameter solving module is used for converting the shape adjustment parameter solving problem into a minimized objective function problem, respectively solving partial derivatives of the shape adjustment parameters through the objective function, and solving the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjustment parameters; substituting the numerical value of the shape adjusting parameter into the quintic Bezier curve to obtain the fairing path of the curvature mutation region.

8. The system of claim 7, wherein the defined quintic bezier curve b (t) containing shape adjustment parameters is:

b(t)＝b₀ B₀ (t)+b₁ B₁ (t)+b₂ B₂ (t)+b₃ B₃ (t)+b₄ B₄ (t)+b₅ B₅ (t)

wherein b₀ 、b₁ 、b₂ 、b₃ 、b₄ And b₅ As control points, the expression of each control point is:

B₀ (t)、B₁ (t)、B₂ (t)、B₃ (t)、B₄ (t) and B₅ (t) is the control point, the Bernstan polynomial:

wherein p is₀ And p₁ The two end points of the curvature mutation zone are position points; alpha₀ 、α₁ 、β₀ And beta₁ The shape adjusting parameter is larger than zero and is used for adjusting the tangential directions of the starting point and the end point of the spline curve and the positions of the control points; k (k)₀ And k₁ Is the curvature of the two endpoints of the curvature abrupt region.

9. The system of claim 8, wherein the objective function of the shape adjustment parameters in the five-order bezier curve defined according to the boundary conditions at the end points and according to the curvature change energy and length energy composite minima principle is:

wherein b '"(t) is the third derivative of b (t) and b' (t) is the first derivative of b (t);

converting the shape adjustment parameter solving problem into a minimized objective function problem, respectively solving partial derivatives of the shape adjustment parameters through the objective function, and solving the objective function by adopting a quasi-Newton method to obtain the numerical value of the shape adjustment parameters, wherein the method comprises the following steps:

the minimization of the objective function problem is: minf (alpha)₀ ,α₁ ,β₀ ,β₁ )

The partial derivatives of the shape adjustment parameters are respectively calculated through the objective function, and the method comprises the following steps:

and->

Solving an objective function by adopting a quasi-Newton method, comprising the following steps of:

given an initial value

And a stop condition epsilon; wherein the superscript 0 denotes alpha at iteration 0₀ ,α₁ ,β₀ ,β₁ Is set to an initial value of (1);

setting the initial value H of the Heisen matrix₀ Calculating the gradient of the objective function as an identity matrix

Determining a search direction d_k ＝H_k g_k Wherein H is_k Is a hessian matrix;

updating shape adjustment parameters

Wherein the superscript k denotes alpha at the kth iteration₀ ,α₁ ,β₀ ,β₁ Is a value of (2); the superscript k+1 denotes α at the k+1th iteration₀ ,α₁ ,β₀ ,β₁ Is a value of (2); lambda (lambda)_k Representing step size factor, by solving->

Lambda at the time of obtaining the minimum value;

judging

If so, outputting a shape adjustment parameter, otherwise, updating the gradient of the hessian matrix and the objective function:

and performs the next shape adjustment parameter calculation.

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