Local transition smoothing method with C3 continuity and error control strategy

Abstract

The invention discloses a local transition smoothing method with C3 continuity and an error control strategy, and mainly aims to solve the problems of insufficient path continuity and difficult tool direction error control in five-axis machining. According to the method, two specific quintic B-spline curves are constructed, and local transfer fairing is performed on a linear path, so that the continuity of the path is improved, one curve is used for performing local fairing on a position path, and the other curve realizes local fairing of a direction path on a unit spherical surface. And meanwhile, a control model of the direction path error is established from the perspective of geometry, and an analytical expression of the direction path error is deduced by utilizing the cosine law and the Western theorem, so that accurate prediction and control of the error are realized, and parameter synchronization between the position path and the direction path of the cutter is realized.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of Computer Numerical Control (CNC) machines, in particular to the field of algorithms for improving the continuity and error control of small line segments in five-axis linear machining paths. BACKGROUND

[0002] With the rapid development of modern industry in China, complex curved surface parts are widely used in aerospace, automobile manufacturing, shipbuilding industry and medical equipment, etc. important fields, its high efficiency and high precision machining has become one of the key issues of the development of China's manufacturing industry. Compared with three-axis CNC machine tools, five-axis CNC machine tools significantly improve the flexibility and diversity of tool posture adjustment by adding two rotary axes, reduce the collision between the tool and the workpiece, have higher machining precision and machining efficiency, so it plays a more important role in complex curved surface part machining. However, the machining of complex curved surfaces is usually approximated by a large number of discrete small line segments to the target curve. However, although such discrete line segments are continuous in position, they have discontinuity in high-order derivatives such as speed and acceleration, which easily leads to frequent start and stop of the machine tool, and mechanical vibration caused by acceleration and deceleration process, thereby reducing the machining precision and efficiency. In order to avoid equipment resonance and other phenomena, smooth optimization of tool path becomes a key research direction to achieve efficient and high-quality machining.

[0003] At present, the smoothing method of tool path mainly includes two categories: global smoothing and local transition smoothing. The global smoothing method approximates the whole section of the linear tool path by a parametric curve, although it can improve the continuity of the path, but the error control is difficult and the calculation cost is high. In contrast, the local transition smoothing method replaces the original line segment connection with a parametric curve at the path inflection point, which can realize high-order continuity in a local range and has good controllability, so it is widely used in three-axis machining path. However, extending this method to five-axis machining still faces significant challenges, namely how to effectively control the error of the tool axis direction path.

[0004] Existing local transition smoothing research mainly focuses on the improvement of the continuity of position path and direction path and the parameter synchronization of the two, while the control of the error of the direction path is relatively less concerned. Although the research on path continuity has been relatively in-depth, it is difficult to achieve C 3The research on continuity is still rare, and the approximate estimation method is usually used for the control of the orientation path error, and there is a lack of accurate error modeling and analytical expression method. However, the tool orientation has a significant influence on the surface quality and tool interference control in five-axis machining, and the accumulation and mutation of the orientation path error may lead to the increase of the profile deviation, the decline of the machining stability, and even the abnormal operation of the machine tool. Therefore, in order to solve the problems of the insufficient path continuity and the difficult control of the tool orientation error in five-axis machining, a five-axis local transition smoothing method with high-order continuity and a strategy for accurate analytical control of the error are proposed, which has important theoretical significance and application value for improving the machining stability of the five-axis numerical control machine tool and the surface quality of the complex surface. SUMMARY

[0005] The present application aims at the problems of the insufficient path continuity and the difficult control of the tool orientation error in five-axis machining, and proposes a five-axis local transition smoothing method with C 3 continuity and its error control method. The method ensures the high-order continuity of the path by constructing two specific quintic B-spline curves for local smoothing processing of the tool position path and the tool orientation path. Meanwhile, the control model of the orientation path error is established from the geometric point of view, the orientation path error is accurately and analytically controlled, and the parameter synchronization between the tool position path and the orientation path is realized.

[0006] The technical scheme adopted by the present application to solve the above technical problems is as follows:

[0007] 1) Construct a local transition smoothing model, and ensure the high-order continuity of the path by constructing two specific quintic B-spline curves for local transition smoothing processing of the tool position path and the tool orientation path.

[0008] 2) Normalize the local transition smoothing orientation path to ensure the physical feasibility of the tool posture.

[0009] 3) Analyze the generation mechanism of the orientation path error, establish the control model of the orientation path error from the geometric point of view, and accurately and analytically control the orientation path error by using the cosine theorem and the Wedderburn theorem. DRAWINGS

[0010] Figure 1 is a local transition smoothing model diagram of the position path.

[0011] Figure 2 is a local transition smoothing model diagram of the orientation path.

[0012] Figure 3 is a local transition smoothing model diagram of the normalized orientation path.

[0013] Figure 4is a direction path error generation mechanism diagram.

[0014] Figure 5 is a direction path error control model diagram.

[0015] Figure 6 is a technical flowchart of five-axis machining direction path error control. DETAILED DESCRIPTION

[0016] A position path local transition smoothing model is constructed as shown in Figure 1 Two adjacent linear paths P i-1 P i and P i P i+1 are defined by points P i-1 , P i , P i+1 . Wherein, the black dotted line is the original path, and the red solid line is the inserted quintic B-spline curve, and β is the included angle of the two linear paths. A quintic B-spline curve is introduced at the corner to perform local fairing. The expression of the quintic B-spline used is as follows:

[0017]

[0018] Wherein, p i is a control point, u∈[0,1] represents a spline parameter of the B-spline curve, N i,5 (u) is a B-spline basis function defined on the node vector U==[0,0,0,0,0,0,0.5,1,1,1,1,1,1], which can be recursively calculated by equation (2),

[0019]

[0020] To ensure that the connecting points of the micro-line segments satisfy C 3 continuity, the conditions that the second and third derivatives of the B p -spline curve with respect to the tool path length s are both zero at the connecting points need to be satisfied:

[0021]

[0022] The conditions that need to be satisfied at the connecting points of the inserted curve and the remaining straight line segment, i.e., the transition curves u=0 and u=1, are as follows:

[0023]

[0024] When u=0, the position vector p is differentiated with respect to the spline parameter u to obtain the second and third derivatives at the entrance p0 of the inserted curve, and the following expressions are obtained:

[0025]

[0026] The relationship between the control points can be solved to satisfy the following equation.

[0027]

[0028] wherein m1=P i P i-1 / ||P i P i-1 ||, m2=P i P i+1 / ||P i P i+1 ||, l1=‖p3p2‖, l2=‖p3p4‖:

[0029] To ensure that the inserted B-spline is symmetrical, let l1=l2=l, in which case the length between each control point satisfies:

[0030]

[0031] The approximation error on the position path is defined as the geometric deviation between the transition B-spline curve and the original trajectory P i-1 P i P i+1 Since the inserted B-spline curve is symmetrical, in this case, the maximum approximation error ε occurs at the midpoint B(0.5) of the inserted B-spline curve, and can be analytically calculated. To ensure that the error satisfies the user's preset position error tolerance limit ε * p , the length of the transition section must be reasonably constrained. Specifically, the transition length should satisfy the following condition to ensure that the approximation error does not exceed the allowable range:

[0032]

[0033] Since each linear tool path segment is used to smooth the two corner points corresponding to its start and end points respectively, when inserting the transition curve, it should be ensured that the smooth curve does not exceed the space range provided by the original path segment. Therefore, the maximum allowable length of the straight line segments p0p3 and p6p3 is half the length of the tool path segments P i P i-1 and P i P i+1 , respectively, so the condition that the transition length l should satisfy is:

[0034]

[0035] For example, Figure 2As shown, the smoothing of the transition path in the workpiece coordinate system is similar to that of the smoothing of the position path; both are processed by inserting a B-spline five times. The red solid line represents the inserted B-spline. o (u), where the black dashed line represents the original directional path, and α is the angle between the two linear directional paths. To ensure the directional path reaches C at the corner... 3 For continuity, spline control points that satisfy the continuity requirement of higher-order derivatives need to be constructed. Similarly, as with positional paths, the directional path control points that satisfy this continuity condition are as follows:

[0036]

[0037] Where t1=θ i θ i-1 / ||θ i θ i-1 ||,t2=θ i θ i+1 / ||θ i θ i+1 If r1 = ||o3o2|| and r2 = ||o3o4||, then the lengths of the control points satisfy:

[0038]

[0039] Since the maximum allowed length of o0o3 and o6o3 is the original path θ i θ i-1 and θ i θ i+1 To ensure the geometric feasibility of the inserted curve, the transition length r of the direction path must at least satisfy r0 = min(||θ). i θ i-1 || / 5,||θ i θ i+1 || / 5).

[0040] Since the tool length remains constant during machining, the magnitude of the direction vector must always be a unit length to ensure the physical feasibility of the tool orientation. Normalizing the direction path yields the following: Figure 3 As shown, the normalized smooth path O(u) can be calculated by the following formula:

[0041]

[0042] During toolpath transition smoothing, transition errors occur when smoothing the directional path, and these errors are converted into directional path errors. For example... Figure 4where P and θ are the original linear path, p and o are the fitted position path and orientation path, and the orientation path error is the angle between the fitted orientation path o and the original orientation path θ, denoted as ε o , which is used to evaluate the deviation degree of the tool posture orientation.

[0043] To realize the analytical control of the orientation path error, the maximum transition error between the inserted B-spline curve and the original orientation path needs to be converted into the angular error. As shown in Figure 5 , the maximum transition error e and the maximum orientation path error ε o are positively correlated during the fairing process of the orientation path. Therefore, the effect of controlling ε o can be achieved by controlling e.

[0044] Since the inserted orientation path B-spline curve does not have strict symmetry in general cases, an asymmetry parameter can be introduced to model the transition interval when calculating the maximum transition error e of the orientation path. Specifically, let r1 = r and r2 = kr1, where r is the transition length, and k is the symmetry coefficient, k = r2 / r1, which is used to describe the degree of asymmetry of the lengths on both sides of the transition segment. The maximum transition error e can be calculated by the following formula:

[0045] e = || θ i -E|| (13)

[0046] where E is the B-spline midpoint, which can be calculated by the following formula:

[0047]

[0048] By combining equations (10), (13), and (14), we can get

[0049]

[0050] According to the cosine theorem, the constraint relationship between the error ε o and the preset error upper limit ε * o can be expressed by equation (16).

[0051]

[0052] where O is the origin of the spherical coordinate, Oθ i is the unit vector, then according to equations (12) and (16), we can get OE as

[0053]

[0054] According to equations (13), (14), (16), and (17), we can get the relationship between r and ε*o as follows:

[0055]

[0056] Let x1=3 / 8(t1+kt2)·θ i , x2=9 / 64(1+k 2 +2kcosa), according to formula (18), after substituting x1 and x2, the inequality can be converted to:

[0057]

[0058] Discriminant is only when (t1+kt2) and Oθ i parallel x2-x1 2 is 0, at this time the equality holds, the rest of the case discriminant is greater than 0, therefore, r must exist solution:

[0059]

[0060] Classification of the solution of the inequality equation, so as to find the solution of r in the case of inserting B spline non-intersection. Let a=(x1 2 -x2cos 2 ε * o ), b=2(1-cos 2 ε * o )x1, c=1-cos 2 ε * o At this time can be divided into the following cases:

[0061] (1) parabola open upward, that is, a>0, at this time, the inequality has two real roots r - and r + , according to the Wedderburn theorem, since c≥0, the two real roots are of the same sign. When x1>0, b>0, the sum of the two roots-b / a<0, at this time the inequality equation has two negative roots, and in combination with the actual situation, r>0, at this time the inequality is always true; When x1<0, at this time the inequality equation has two positive roots, the solution set is r≤r - or r≥r + , in consideration of the transition length, r needs to take the minimum value among r - , r + , r0, and because r - must be less than r + in the two positive roots, therefore, only need to judge the size of r0 and r - , that is, r=min(r0, r - ), in which r - can be obtained by the root formula:

[0062]

[0063] (2) When parabola opens downward, a < 0, and since c > 0, there are two real roots with different signs. According to the root-finding formula, the positive root is:

[0064]

[0065] (3) When a = 0, we have:

[0066]

[0067] If x1> 0, the inequality is always true when r > 0; otherwise, x1< 0, and in this case, 0 < r < -1 / 2x1.

[0068] In summary, combined with the transition length requirement, r can be calculated according to equation (24):

[0069]

[0070] After obtaining the transition length r, to ensure that the change in the displacement of the direction path relative to the position path has good continuity at the connection, the first, second, and third order differences of the direction vector with respect to the arc length of the position path should be continuous at the connection point of the straight line and the spline, i.e.:

[0071]

[0072] where B o is the B-spline curve inserted at the corner of the direction path. Equation (25) can be rewritten as:

[0073]

[0074] where the parameter differential of the spline curve related to the position path is:

[0075]

[0076] Combined with equations (7), (25), and (27), the sufficient condition for the synchronization of the direction path and the position path is:

[0077]

[0078] According to the quintic B-spline formula, we have:

[0079]

[0080] According to equations (11), (28), and (29), the sufficient condition for the synchronization of the position path and the direction path is:

[0081]

[0082] Since r1 = r, r2 = kr1, the condition that the symmetry factor k satisfies for the position path and the direction path to be synchronized can be obtained according to equation (30):

[0083]

Claims

1. A type of C 3 The continuous local transition smoothing method and error control strategy are characterized by constructing two specific quintic B-spline curves to locally smooth the linear path and deriving an analytical expression for the direction path error from a geometric perspective using the cosine theorem and Vieta's formulas, thereby achieving accurate error prediction and control. This is done to satisfy C... 3 Continuity: The lengths of the fifth-order B-spline control points for the constructed position and direction paths satisfy the following relationship. (in, r1 = r, r2 = kr1, k is the symmetry coefficient (k = r2 / r1), and then the transition length is calculated based on the geometric relationship. This limits the error and the length of the inserted curve. To ensure the physical feasibility of the tool orientation, the magnitude of the direction vector must always remain at a unit length. After normalizing the direction path, to achieve analytical control of the direction path error, the maximum transition error between the inserted B-spline curve and the original direction path needs to be converted into an angular error. During the smoothing process of the direction path, the maximum transition error e and the maximum direction path error ε... o There is a positive correlation between the two, and e can be represented by the directional path transition length r, i.e. Therefore, ε can be controlled by controlling r. o The effect is achieved by deriving the transition length r and the preset upper limit of error ε based on the law of cosines. * o Constraints between (where x1 = 3 / 8(t1 + kt2)·θ) i x2=9 / 64(1+k) 2 +2kcosa), t1=θ i θ i-1 / ||θ i θ i-1 ||,t2=θ i θ i+1 / ||θ i θ i+1 (where α is the angle between two linear directional paths). According to Vieta's formulas, the transition length of the directional path can be derived. (where r0=min(||θ) i θ i-1 || / 5,||θ i θ i+1 || / 5), This transition length can be used to limit the directional path error and the length of the inserted curve, and then the conditions satisfied by parameter synchronization can be considered. Further, the sufficient condition for the synchronization of position path and direction path is derived as follows:

Images