NL2031103A

NL2031103A - Global optimal feed speed planning method for cradle-type machine tools

Info

Publication number: NL2031103A
Application number: NL2031103A
Authority: NL
Inventors: Gao Xiaoshan; Shen Liyong; Yuan Chunming; Ma Hongyu
Original assignee: Univ Cas
Priority date: 2021-05-20
Filing date: 2022-02-28
Publication date: 2022-07-07
Also published as: CN113189939B; CN113189939A; NL2031103B1

Abstract

An embodiment of the description discloses a global optimal feed speed planning method for a cradle—type machine tool, the method comprising: the method constrains the velocity, acceleration and jerk in the machine coordinate system and the chord error in a 5 workpiece coordinate system. Then the optimal feed speed planning problem. is converted, into an optimal control problem, and the optimal solution is illustrated as “bang—bang” control. Finally, an iterative control vector parameter method is used to solve the optimal solution of the optimal control problem, and the solution 10 satisfies the “bang—bang” control. The present invention makes it possible to fully utilize the performance of the machine tool by fully considering the constraints of the whole machining process of the cradle—type machine, including the motion constraints and the geometric error constraints, and completing the whole cutting 15 task in the shortest time.

Description

GLOBAL OPTIMAL FEED SPEED PLANNING METHOD FOR CRADLE-TYPE MACHINE TOOLS

TECHNICAL FIELD The present application relates to the technical field of nu- merical control machine tools, and in particular to a global opti- mal feed speed planning method for cradle-type machine tools.

BACKGROUND ART Numerical control machining has been widely used in manufac- turing industry, and its machining quality and efficiency are the key factors affecting the competitiveness of numerical control ma- chining, so how to shorten the machining time under the con- straints of machine performance and workpiece accuracy is crucial. In the five-axis numerical control machining, because the machine tool has two more axes of rotation than the three-axis machine tool, the nonlinear transformation between the workpiece coordi- nate system and the machine tool coordinate system must be consid- ered in the speed planning. This transformation makes the optimal feed speed planning problem more difficult, so the optimal feed speed planning problem for five-axis machine tool is still chal- lenging.

The traditional five-axis fold line path is discontinuous at the corners, which leads to feed speed fluctuations, poor pro- cessing quality, etc. To overcome these drawbacks, some CNC com- mercial software uses parametric curves (e.g., spline curves, cir- cular arc curves) to represent smooth tool paths. Therefore, it is necessary to study the optimal time feed speed planning for five- axis parametric tool path. An object of Minimun Time Feedrate Planning (MTFP) is to drive to cut along a predetermined path with constraints determined by machine performance and geometric accu- racy {i.e.,;, motion constraints and geometric error constraints for each axis) and completing the entire cutting task in the shortest time. These constraints are closely related to the smooth motion, processing stability and quality of the axes. In the prior art,

some can strictly abide by kinematic constraints and geometric ac- curacy constraints, but the iteration number of the solution meth- od is closely related to the complexity of the tool path, so that the robustness is poor and the calculation efficiency is low. Some discretize the tool path into sampling points, so that only the sampling points are restrained, resulting in the motion curve of each axis exceeding the preset limit. Therefore, it is a technical problem to be solved how to ful- ly consider the constraints of the whole machining process of the cradle-type machine tool, including motion constraints and geomet- ric error constraints, and complete the whole cutting task in the shortest time, so as to fully utilize the performance of the ma- chine tool.

SUMMARY An embodiment of the description provides a global optimal feed speed planning method for a cradle-type machine tool, making it possible to fully utilize the performance of the machine tool by fully considering the constraints of the whole machining pro- cess of the cradle-type machine, including the motion constraints and the geometric error constraints, and completing the whole cut- ting task in the shortest time, which is a technical problem to be solved. In order to solve the above technical problem, the embodi- ments of the present description are realized as follows: an embodiment of the description provides a global optimal feed speed planning method for a cradle-type machine tool, com- prising: step S1, inputting a continuous six-dimensional parameter curve R(#) 5 [P(8).0(£)] of C in a workpiece coordinate sys- tem WCS; wherein B represents a normalized parameter and B e [0.1] ; P is a tool point path curve of the machine tool PPP]. 0 is a knife axis direction of the corresponding knife site, 0 = [0,,0,.0,1 step S2, mapping the continuous six-dimensional parameter R( 2 curve 8) of C in WCS to the machine tool coordinate system MCS to obtain a parameter tool path O(B) = [X(B).Y(B).Z(B)A(B)L(B)], ne . ; wherein X represents an X axis in the machine tool coordinate system, Y represents a Y axis in the machine tool coordinate system, Z represents a Z axis in the machine tool coordinate system, A represents a first rotation- al coordinate axis in the machine tool coordinate system, and C represents a second rotational coordinate axis in the machine tool coordinate system; step $3, dividing the parameter interval [0, 1] into N sub- intervals; step S4, acquiring an initial control vector H 2nd an ini- : : : V tial objective function value ©; step S5, presetting a convergence coefficient 7; : : V =V, step S6, making a first iteration parameter 1 6 and a sec- ond iteration parameter u=u, c=7 +1, and step 87, for each sub-interval k of the N sub-intervals, carrying out the following operations: calculating to obtain a maximum value (Brim of a speed Vv Co Vv J mink of each axis [, a minimum value (Bom) of a speed V of each a ack axis l a maximum value (Butt) of an acceleration @ of each Co a nk axis [, a minimum value (Barwin) of an acceleration 4 of each axis [ a maximum value JB) of a jerk J of each axis l, a minimum value (Bme) of a jerk J of each axis / and a maximum d < value (Bins) of a chord error d in the k** sub-interval accord- ing to the speed, acceleration, jerk curve and chord error curve of each axis / in the k* sub-interval;

= XY,Z, AC k=12,..,N wherein axis ’ ’ . step S8, on the basis of step S7, using an iterative method to solve a nonlinear programming problem represented by the fol- lowing formula (21) to obtain an optimal control vector u,

N 22 max’ = 29: i tzt SL Vp Sv (B )S vn, “Vig Vy Borat} < Vis Vig Vy (Bint) SV: “ay <a,(f.)<ay,, “Ay <a, (Bans ) S Apt Sd, (Brine ) < Cp» Fin = Ji (B, ) = Jigs Ji = Ji VAE Jo" Îe = JB) < Ji (21) keEl=X,Y,Z AC, d(B.)<6.d( Bums) <5. kB, 22 wr By = 0, Bx =0 OV | wherein represents an objective function value and d()<6 ‚ represents an chord error constraint. Preferably, the six-dimensional parametric curve in the step Sl is a NURBS curve or a B-spline curve. Preferably, step S2 specifically comprises: 2 2 0;(8)+0;(B) A(B )= arctan| +" 0,(B) . 0 C'(B)= arctan (8) 0,(8) (1) X(B)] [cos(C{8) -sin(C'(B)) 0 P(8) Y(8) |=|cos(A(B))sin(C(B)) cos(A(B))cos(C(B}) -sin(A(B)) || PAB) 20) | [sin(a(f)stnC(£) sn(a(@eosC(8) eostaB) P(A) Preferably, an initial value of the convergence coefficient in the step S6 is as follows:

V 0 Y= 10° Preferably, the specific contents of step S8 include: step S81, solving the non-linear programming problem repre- sented by the formula (21) to obtain a new objective function val-

V, i ue 'Y9 and a new control variable #, and judging whether

TV I VIS . U _ I | < — step S82, if "i 0 7, an optimal control vector # is ob- tained, and the iteration is stopped; V=V|> v 5 if V 0 a a new objective function value Ys and a new control variable # are obtained, proceeding to step S583, and the iteration is carried out again; = . . y a step S83, let V, Vo, calculating a maximum value (Borma) CL V nk of a speed V of each axis [, a minimum value (Borman) of a speed da ax V of each axis a maximum value (Bums) of an acceleration 4 CL a i. of each axis l a minimum value (Barwin) of an acceleration ¢ of each axis l, a maximum value JB mast) of a jerk J of each axis / ‚ a minimum value ! Birman of a jerk J of each axis l and a max- d imum value (Bamast ) of a chord error d, re-solving the nonlinear programming problem represented by the formula (21) to obtain a new control variable UH and a new ob- V, jective function value %; in step S84, proceeding to step S82. At least one embodiment provided in the description can achieve the following beneficial effects: the present invention makes it possible to fully utilize the performance of the machine tool by fully considering the constraints of the whole machining process of the cradle-type machine, including the motion con- straints and the geometric error constraints, and completing the whole cutting task in the shortest time.

DETAILED DESCRIPTION OF THE EMBODIMENTS To further clarify the objects, aspects, and advantages of one or more embodiments of the present disclosure, one or more em- bodiments of the present disclosure will be set forth in part in the description which follows. It is to be understood that the de- scribed embodiments are only a partial embodiment of the descrip- tion and not all embodiments. Based on the embodiments in the pre- sent description, all other embodiments obtained by a person of ordinary skill in the art without inventive effort fall within the scope of protection of one or more embodiments of the present de- scription.

The technical concept of the present invention will be brief- ly described as follows: embodiments of the present disclosure provide a complete and novel five-axis parametric path optimal feed speed planning method. The method restrains speed, accelera- tion and jerk in a Machine Tool Coordinate System (MCS), and re- strains a Chord Error in a Workpiece Coordinate System (WCS). Then the optimal feed speed planning problem is converted into an opti- mal control problem, and the optimal solution is illustrated as “bang-bang” control. Finally, the Iteration Control Vector Para- metrization Method (ICVP) is used to solve the optimal solution of the optimal control problem, which satisfies the “bang-bang” con- trol.

The technical solution of the present invention is described in detail below. Problem establishment and description The feed speed planning problem is converted into an optimal control problem, while speed, acceleration and jerk are restrained in MCS and chord error is restrained in WCS. Since the input of this method is a six-dimensional parametric curve in the WCS, a conversion formula between WCS and MCS is given first. Secondly, the mathematical expressions of motion constraint and chord error constraint are given. Finally, the optimal feed speed planning problem is obtained and converted into a time-optimal control problem.

Conversion between WCS and MCS It is mainly considered the rotary table (there are two rota- ry axes on the table, which are also called cradle-type) machine tools. The two rotary axes are respectively axis A and axis C. The axis A rotates around the axis X of the MCS, while the axis C ro- tates around the axis Z of the MCS. The machine tool coordinate system (MCS) is composed of two rotating axes plus three linear axes (X, Y, Z axes). The MCS does not change as the table changes. The workpiece coordinate system (WCS) is a coordinate system used to describe the actual shape of the workpiece, which is rigidly fixed on the worktable of the machine tool and moves with the movement of the worktable. The reason why the cradle-type machine tool is selected is that most of the numerical control machining centers on the market now use this type, and its main advantage is that the load on the spindle is less, which can effectively reduce tool vibration and improve machining accuracy.

The input to the method is a continuous six-dimensional para- metric curve of C in the WCS: R(8)=[P(8).0(8)], 8 € [0,1]. such as NURBS or B-EK. Where P=|P,P,P]| is a knife site path curve and 0=[0.0,.0,] describes the knife axis direction (in direction vectors) corresponding to the knife site. During machining, the machine tool drives the motion machining of each axis in the MCS, so the motion constraint needs to be considered in the MCS. The final workpiece quality effect is present in the WCS, so the geo- metric error, i.e., chord error, needs to be restrained in the WCS. Therefore, it is also necessary to obtain a representation of the input parametric curve in the MCS to describe the motion con- straints. The curve R(8) is mapped into MCS: 0=[X.Y.Z.A.C] this mapping transformation R—=>0Q is called an inverse kinematics transformation.

The inverse kinematic transformation is determined based on the configuration of the selected machine tool. For cradle-type machine tools, the specific transformation is as follows (assuming the origin of WCS and MCS coincide, and there is no singularity in the transformation process):

JO (B)+ 02 (8) 7 A(B) = arctan 2 0,(8) (1) O,(8 C(B) = | 29) 0, (8) X(p) cos(C(8)} -sin(C'(£)) 0 Pf) F(R) | = |cos(A(p))sin(C(B)) cos(A(f))cos(C(A)) -sin{A(f)) PAA (2) Z(B) sin(4(B)}sin(C(8)) sin{A(S))cos(C(B)) cos(A(S)) Pf) motion constraint and chord error constraint After transformation, the expression R(8) of the input curve in the WCS and the expression ols] in the MCS can be obtained.

This results in motion constraints in the MCS and chord error con- straints in the WCS.

According to the differential chain law, the velocity V, acceleration 4 and jerk J of the curve in the MCS can be expressed as follows: ® rot v()=Ó()=0'B'(), . 05 ys a(r)=0(t)= QB) +0"p(1), (NN AE ye oe ma wo JO=00=0F0)+30"0)F0)+0"5 (1) considering the limitation of the driving force and torque performance of the machine tool, and the purpose of reducing the vibration during machining and smoothing the generated trajectory, the speed, acceleration and jerk of the five axes need to be re- strained: VV, SVV, —a,<a<a,, 3) — Js = J < Jas Where V, = bv VVV | a, = la,.a,.a,.a,.a,] and ’ Js [Josse jo js] are a velocity, acceleration and jerk boundary constraint vectors respectively.

For each axis, the mo- tion constraints can be expressed in the form:

_ qt 92 “Vi <v‚(8) T MBB SV, qr 0 ” 32 Gs a, > I'(B)B + M(B)B S ds (4) . . == 32 ! ” as mm 22 . Ja JAE) = NB (Bw + 3(B)B +1"(B)B )< J, 7 wherein, [=X.Y,Z,4.C and b . Thus the motion con- i straint can be expressed as an inequality with respect to , and u.

For chord error constraints, because chord error describes the actual machining accuracy of the workpiece, it is the chord [PPP | error limit on the curve °° +’ +4 in the WCS.

In WCS, the curvature of the input knife point parametric curve path is: P(5)<P'(5) K(B)= — zj (5) PB) 2

1 AB) a radius of curvature is . From the chord error formula, the following can be obtained: ‚ (p\-8óp(B) 48’ 85p(B) 85 V eed (B ) - 2 ~ 2 == 22 T T x(B)T where 7 is an interpolation period and 0 is the upper bound V eg ( of the given chord error; 7 represents the feed speed which can be obtained by the following formula: i. 12 ?2 12 F ve =A P+ PPS (6) the chord error d(B) at the parameter value is defined as B to obtain the chord error constraint: 2 n 1? r (BP + PP) d(B)= <8 (7) 8 thus, the chord error constraint can be described as an ine-

B quality with respect to . Time optimal control problem For the Minimun Time Feedrate Planning (MTTP) problem, an ob- jective is to calculate a parametric rate A ) to minimize the machining time for cutting along the tool path R(8) and to satis- fy motion and geometry constraints in the process. So the MTFP problem can be expressed as: » gn oy E w pl 3 zn min Tee jo ledi jj odd st rm SV) ro SUL Foe AU di 8 vp Sv Sova, ag Nall} Sap, ja SHE Se == (8) Since the above-mentioned problem is time dependent, it is difficult to solve the problem directly. In order to facilitate the calculation, it is necessary to transform the problem into an equivalent equivalent time independent optimal control problem, and then solve it with nonlinear optimization.

A linear system is constructed by defining a state variable * "a T 1 "on 0 2 0 A X'(B)= B|=AX+Bu, A= 0 ol B51, (9) yb where B is a control variable, available from the linear system (2), the integral of U is B and the integral of 28 is

2 B=[ u(x), B = [2 B) « B . 6 ? 0 Then, both the motion constraint (4) and the chord error con- straint (7) can be expressed as inequalities with respect to the control variable U.

Finally, the initial control problem (8) can be converted in- to the following equivalent time-independent optimal control prob- lem: i v1 min T=[-df u p ’ st X'(B)= AX+Bu,X(0)=X,.,X(1)=X,, du)<s, —v, <v(u)<v,, (10) — da, < a,(u) < a. jus <jw)<j,.1=XY,Z,AC wherein X, =X, =[0.0]" since the system of this optimal con- trol problem is linear, its optimal solution is a “bang bang” con- trol, i.e., at least one constraint (speed, acceleration, jerk constraint or chord error constraint) reaches a limit at any time. Iterative control vector parameter method In this section, an Iteration Control Vector Parametrization (ICVP) is proposed to solve the optimal control problem (10) men- tioned in the previous section. The process is divided into three steps: firstly, the parameter interval [0,1] is divided into sev- eral sub-intervals according to the geometric characteristics of the cutter path; the problem (10) is converted into a solvable nonlinear programming problem and an initial solution thereof is calculated; the optimal solution is iteratively computed until convergence is reached if the motion constraints and chord error constraints are satisfied. Parameter interval segmentation The parameter interval [0, 1] is first densely divided into N, equal subintervals:

0=p<B << B < B, =1 (11) For each sub-interval, a constant value is used to approxi- mate the control variable #, and the approximated control variable u(B) Be[01] can be expressed as: ‚ up) =a.if pel] (2) the above-mentioned sub-intervals are trivial without consid- ering the geometric features of the path of the knife site. When a subinterval 16.5] contains a curvature maximum point, the motion performance of this curvature maximum point cannot be controlled by controlling the motion performance (velocity, acceleration and jerk) of the sampling point Bebe, because the velocity at this point is usually minimized and the motion curve is non-linear, it is difficult to describe this state by only a constant control

A quantity k Therefore all curvature maxima points of the knife point path in the WCS should be added to the above sampling points (11) so that the feed speed satisfies the motion constraints and

N chord error constraints for each axis. Assuming that there are 2 curvature maximum points in the path of knife point, these points (sampling point and curvature maximum point) are rearranged from N ( small to large to obtain sub-intervals: 0=8<B<.<B.,<B.=1 (13) wherein, N=N+N, For each new subinterval 16.5.1 a constant control quantity u, is given to describe the rate infor- mation. Thus, the required solution is this piecewise constant control function u(B).

For a regular curve, there are always a finite number of cur- vature maximum points, and these points satisfy the eduation K(B)=0 _ Y . However, the curvature expression (5) is generally high- ly nonlinear and difficult to solve directly. Therefore, they are calculated using the following approximate discrete method.

Discrete method For the discrete method, m points are uniformly sampled in each subinterval 15.5] of (11). Then a curvature maximum point P(Ê) | ( x(Ê)> (B) 1s selected from the m points. If It / , and x(6)> KB) ‚ 7, this point is selected as the curvature maximum point. All subintervals in (11) are then traversed to obtain all curvature maximum points. Intuitively, however, this method is too A — coarse and there may be an error between the calculat- | 2 ed curvature maximum point 1 (8) and the actual curvature maximum oT point 1 (B).

In general, the output of current industrial CNC interpola- tion software is a series of discrete interpolation points, which are separated by an interpolation period, so these interpolation points already contain the feed speed information, so as to dis- cretize each sub-interval in (11) to the interpolation level, the error € can be ignored. Now, a suitable discrete number m needs to

IV be determined. Suppose 7 is an interpolation period, fed js an upper bound of the feed speed of the machine tool, and S is a to- tal arc length of the path of the tool point. The technical solu- tion of the present embodiment uses half of feed to approximately 7 Ved calculate the arc length of two interpolation points: 2 . The discrete number m can thus be determined by the following formula: m=—25_ (14) V oee EN, wherein N, is the initial discrete number of the entire pa- rameter interval in (11).

Nonlinear optimization The optimal control problem is now converted into a non- linear programming problem by the piecewise constant control func- tion u(B) mentioned in the previous section.

According to the dif- ferential relation {9) obtained previously, the B curve is piece- 1 1 ee Boas B wise quadratic, so that a concrete expression of and can be given.

Let Ap =p.

B and = =1,2,N} With the linear system or Aj (9) and the control function u(B), a value of B and B at a point B can be obtained: se se k Bi =u, (B —- B) u = NS úAB.k Ez, (15) i=l . B . . 2 _ € A 2 bi 2] (0 (8- B) + BB + kt ~ 2 == u AB k=1 = k-1 (16) A 2 A 52 p— UAB +208, GAB + BL ke E\{1} i=1 the constraints in problem (10) can then all be expressed as | in | i=(a,..0,) inequalities on the control variables > NJ, Minimizing the machining time is equivalent to maximizing the feed speed at each point when the jerk constraint at the MCS is ignored.

Inspired by this conclusion, the initial optimal control problem is converted into the following form.

No Noo max V, — PR — 20, 0 Q k=l k=1 st. “Vi = v(B,)< Vig -d, = a,(B,)< As Js = J(B)s Jas ke Z,lI=X,Y,Z,A4,C, (17) dB )<d.keZ, 2 so B, == 0, PD, == 0, 12 12 2 — ozb, (8,)+P, (B.)+P, (B.).ke= wherein, ’ . The speed, ac- celeration and jerk of each axis at the point B are: ‘ | 2 v(8)=1 (B) D, » . ol ; oe nl; 5 a,(B)=! (È,)È, +/ (2). 2 (18) . [O2 {yr A ”" rs. " 22 i (B.)= B; ( (B.)i, +31 (B,)È, + [ (8,)B; ) wherein IEX Y ZAC.

The above nonlinear programming problem can be solved numeri- cally by conventional gradient-based optimization techniques such as Sequential Quadratic Programming (SQP), so that the initial so- lution U of problem (10) can be obtained.

Optimal solution calculation In the previous section, the optimal control problem is con- verted into a nonlinear programming problem, but this nonlinear programming only constrains the sampling points, not the entire parameter space. And the motion profile (speed, acceleration, jerk of each axis) is non-linear with respect to the parameters. Re- stricting only sampling points does not guarantee that the whole parameter space satisfies motion constraints. Therefore, it is al- so necessary to constrain the maximum and minimum points of each sub-interval motion curve.

The optimal solution will be obtained by iterative calcula- tion. The initial control amount ¥ can be obtained from the previ- ous section. For each subinterval [B.B] p and B is calculat- ed according to the differential relation (9) and U. Ali) = dl Bt BEP Sl (1%)

SHEMET HIN CCE SE SLE J ax, ZE 18a, fil. {20 Tl ff — Be 2B iS = Bed + FL REEL FE Pen A According to Equations (4) and (7), the velocity, accelera- tion, jerk curve and chord error curve of each axis in the inter- val [BB] can be calculated. The maximum and minimum values of oo AV: the axis velocities CV: Botan )v, Borman J, the maximum and minimum | fap a(n | values of the axis accelerations A / Bots) Bi) ‚+, the maximum and minimum values of the axis accelerations CUB it IB oane) , d and the maximum value of a chord error ( (B mee) of this sub- interval can be obtained, wherein kek and 1=X,Y,Z,A,C These parameters are added to the constraints to obtain a new nonlinear programming problem:

N Fo NC 2A max’ = 204; i #1 SL. “Vig = vi(8,) = Vig, “Vig s Vv; (Bans) S Vig, Vig S Vi (Bink) Vp, “yy = a, (8) S Ay, Og S d (Basi) = yg, Cp = a, (B, int) = fp, Js SJ (5. ) She Je ST (Bim) Spd Eh (8 mint) < Jg (21) keE1=XY.Z AC, MBE (B re) EEK SE, 7 . Br = 0, By = 0 a new control vector U can be obtained by solving this non- linear programming problem. A convergence coefficient V is set. If V : the absolute value of the objective function value obtained by solving the problem (17) subtracted from the objective function value V obtained by solving the problem (21) is less than 7 VVS , on: 1 ‚ then, the optimal solution is obtained.

If not, let VEV ng 7 1 1 , and will be taken into equations (19), (20), (4) and (7), resulting in new individual axis velocity (velocity, acceler- ation and jerk) curves and chord error curves.

The maximum and minimum values of the motion curves for each interval and the max- imum value of a chord error are then calculated and taken into the non-linear programming to re-solve the problem (21). A new objec- tive function value V, and a control variable u may be calculat- ed.

If VV, sy is satisfied, then the optimal solution is u.

If not, the above steps are repeated until VVS, satisfied.

The whole process is shown in Method 1 in Table 1. For the convergence coefficient 7, the objective function V, , value 9 corresponding to the solution of the problem (17) can be 7 calculated first.

Then, let ‚ the change of the objective function B corresponding to the solution obtained after each iter- ation, be less than 1 part per million of vs, consider the objec- tive function of the nonlinear programming problem (21) to be con- vergent.

However, for step 10 of method 1, since the motion curves and chord error curves of each interval are non-linear, it is dif- ficult to directly calculate the maximum and minimum values of these curves.

For ease of calculation, newton's iteration or the discrete method mentioned can be used to solve for these maxima and minima.

In the non-linear programming (21), the objective function can be rewritten as: V‚=>0;8, =(QU) (QU) k=l wherein

U [i A A A | |M 4, 4. U, and the matrix is equal to: Fy ARE {} Ce { ra 2 ‚ 5 > - is 3 3 dy (Ag? +2 3 Ah Ad yoo {A32 +3 3 AF AGE) er 0 des iss ’ at AFT +23 ARAS} oNlASi 2) Ai As) + enASL gam E is Therefore, the nonlinear programning problem is a quadratic 5 programming problem with nonlinear constraints. And the While loop in method 1 is convergent after many iterations.

Algorithm 1 ICVP Method the following is input: R{(fF)=|P ,O 10,1 parameter tool path (8) | (5) (8)} B [0.1] in WCS; . Vg dp Js.

Motion constraints: 5, and ‘5; and the upper bound of chord error: 8.

The following is output: control vector: u.

Q=[X.¥,Z,4C] | 1: the parametric tool path in the MCS is ob- tained by inverse kinematics transformation (1} (2).

2: the parameter space [0, 1] is divided into several sub- intervals according to the curvature information (see Section

3.1): OB SB hs Bl 3: each subinterval is assumed to have a constant control variable: u(f)=w if Be[ 8,8] 4: the SQP method is used to solve the nonlinear programming problem (17), and the initial control vector H and objective

5: planned convergence factor: 10 6: let Mh, Goy and U =u

7. while {>vy do 8: let VEV, 9: each axis motion (velocity, acceleration and jerk) curve {4) and chord error curve (7) are calculated from the control vector U.

10: the maximum value and minimum value of motion curve of each axis in each sub-interval, and the maximum value of chord error constraint are calculated.

11: the parameters corresponding to the above values are added to the constraints to form a new nonlinear programming problem (21).

12: the SQP method is used to solve the problem (21), and the objective function value V of the control vector u is ob- tained.

13 LeV Vl 14: return optimal control vector: u.

Table 1: pseudo code of the ICVP method in the technical so- lution of the embodiment of the present invention Experimental results: The optimal feed speed planning method is implemented in the environments of Matlab platform and RAM 166, Intel Windows 10. Engine impeller machining curves and S-specimen curves were used to verify the feasibility of the method. The results show that the feed speed calculated by the method satisfies the “bang-bang” con- trol. The motion constraints for the five axes at MCS are given in Table 1. The interpolation period 7 is set to 0.002 seconds. The upper bound of chord error © is set to 0.125 HM In the example, the velocity and acceleration of each axis are set to zero at both the beginning and end of the tool path.

Axis Speed {unit/s) Acceleration (unit/s?) Acceleration (unit/s®) X/Y/Z (mm) 250 500 3000 A/C {rad} 4 8 60 Table 2: five-axis motion constraint The present invention makes it possible to fully utilize the performance of the machine tool by fully considering the con- straints of the whole machining process of the cradle-type ma- chine, including the motion constraints and the geometric error constraints, and completing the whole cutting task in the shortest time.

The above-described embodiments are merely illustrative of the principles of the invention and its efficacy, and are not in- tended to limit the invention. Modifications and variations to the embodiments described above will occur to a person skilled in the art without departing from the spirit and scope of the invention.

Therefore, it is intended that the appended claims cover all such equivalent modifications or changes as fall within the true spirit and scope of the invention.

Claims

CONCLUSIONS

1. Global optimal feed rate planning method for a cradle-type machine tool, characterized in that it comprises: step S1, input of a continuous six-dimensional parameter curve R(8) = [P(8).0(8)] … C° B B ' B of C in a workpiece coordinate system WCS; where B represents a normalized parameter and Beo], p ; represents a tool point path curve of the working . : Pp = [PPP] O : . machine tool rma, stands for a knife axis direction of the | 0 =[0,.0,,0,] corresponding knife location, J tT step S2, mapping the continuous six dimensional parameter curve of “ in WCS to the machine tool coordinate system MCS to obtain of a parameter ready- O(B) = [X(8)-Y(8).Z(8).A(B).C(8)] ; shelf path ; where X represents an X axis in the machine tool coordinate system, Y represents a Y axis in the machine tool coordinate system, Z represents a Z axis in the machine tool coordinate system, A represents a first rotational coordinate axis in the machine tool coordinate system and C represents a second rotational coordinate axis in the machine tool coordinate system; step S3, dividing the parameter interval [0, 1] into N sub-intervals; step S4, obtaining an initial control vector U and an initial objective function value "0; step S5, presetting a convergence coefficient '>u} V=V step S6, making a first iteration parameter !9 and a second iteration parameter u=4u, c=7+1, and step 87, for each sub-interval k of the N sub-intervals, performing the following operations: performing a calculation to obtain a maxi-Vv eb ‚ Comum value (Bima) of a speed VY of each axis in a minimum V ng mum value (Bonini) of a speed V of each axis a maximum I . 7 mum value (Bums) of an acceleration d of each axis l, a d „7 min Ie . minimum value (Bains) of an acceleration 4 of each axis l, a maximum value Ji Bonne) of a shock J of each axis [ a minimum value Ji (Biman) of a shock J of each axis / and a maximum d value (Banas) of a chord error d in the k&° sub-interval according to the velocity, acceleration, shock curve and chord rd error curve of each axis / in the k° sub-interval; where ae [=X ZAC k= 12..N step S8, based on step S7, using an iterative method to solve a nonlinear programming problem represented by the following formula (21) to obtain an optimal control vector u,

N 202 maxV: = 20:5 u k=1 SL Vips v (B) SV, Vip SV, (Bomac ) VV SV (Bink) S Vips ty S a;(B;) Sdy, psd, (mar) Sd,- dp S ay (Bguini) S dp, Jip S Ji(B) S Jigs Jg S Ji Bani) < Jim" Jie S JB mint) © Ji (21) keE,l=X,Y,Z, AC, d(B ,)<8,d(B 1) SÖK EE, … Bs = 0, B =0 ‚V | d()<6 where represents an objective function value and represents a chord error limitation.

The global optimal feed rate planning method applied to five-axis numerical control according to claim 1, characterized in that the six-dimensional parametric curve in step S1 is a NURBS curve or a B-spline curve.

The global optimal feed rate planning method applied to five-axis numerical control according to claim 1, characterized in that the step S2 includes specifically: 2 0:(8)+0;(8) A(B) = arctan 0,(B) 0 C(B)= arctan ol) 0,(8) + (1) X(B)] [cos(C(8)) -sin(C(B)) 0 P(B) Y(B) =|cos(A(B))sin(C(8)) cos(A(B))cos(C(B)) - sin(A(B)) P.(B) Z(8 )| [sin(A(B)sin(C(8) sin(A(B)))eos(C(8) cos(A(B)) | P(8) (2).

The global optimal feed rate planning method applied to five-axis numerical control according to claim 1, characterized in that the initial values of the convergence coefficients in step S6 are as follows: Vy y == 10°.

The global optimal feed rate planning method applied to five-axis numerical control according to claim 1, characterized in that the specific contents of step S8 include: step S81, solving the nonlinear programming problem represented by the formula (21) for obtaining

AND ARS i step S82, if an optimal control vector # is obtained, and the iteration is stopped; Vi—V,|> if V 0 7, a new objective function value Vy and a new control variable u are obtained, proceeding to step S83, and the iteration is performed again; V,=V, | step 3583, suppose , calculate a maximum value 3 (Bs) | i.

{ Priva of a speed V of each axis l a minimum value ! Bimini of a speed V of each axis l, a maximum value a (Brim) | ! Par mass of an acceleration ad of each axis l, a minimum value 7 Batu of an acceleration d of each axis l, a maximum value JB rasa ); - ! B task of a shock J of each axis l a minimum value ! Biman of a shock J of any axis / and a maximum value d(B, (Bomar) of a chord error d by using the new control variable HU obtained by the solution; resolving the nonlinear programming problem which is represented by the formula (21) for obtaining a new control variable ¥# and a new objective function-

V value 39; in step S84, proceed to step S82.