WO2024124794A1 - 一种五轴联动同步刀具路径插补方法和*** - Google Patents

一种五轴联动同步刀具路径插补方法和*** Download PDF

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WO2024124794A1
WO2024124794A1 PCT/CN2023/093939 CN2023093939W WO2024124794A1 WO 2024124794 A1 WO2024124794 A1 WO 2024124794A1 CN 2023093939 W CN2023093939 W CN 2023093939W WO 2024124794 A1 WO2024124794 A1 WO 2024124794A1
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Prior art keywords
tool
curve
interpolation
point
tool tip
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PCT/CN2023/093939
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English (en)
French (fr)
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高健
张桂鑫
罗于恒
张揽宇
邓海祥
陈云
陈新
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广东工业大学
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Priority to US18/583,904 priority Critical patent/US11994838B1/en
Publication of WO2024124794A1 publication Critical patent/WO2024124794A1/zh

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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/41—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by interpolation, e.g. the computation of intermediate points between programmed end points to define the path to be followed and the rate of travel along that path
    • G05B19/4103—Digital interpolation
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/34—Director, elements to supervisory
    • G05B2219/34169—Coarse interpolator, path calculator delivers position, speed, acceleration blocks

Definitions

  • the present invention relates to the technical field of numerical control machine tool control, and in particular to a five-axis linkage synchronous tool path interpolation method and system.
  • Five-axis CNC machine tools can flexibly control the tool position through the changes in the tool tip position coordinates and tool axis vector, thereby improving processing speed, accuracy and surface finish, and are widely used in the fields of power energy and aerospace.
  • parameter interpolation has higher accuracy and efficiency, and it is easier to obtain a smooth feed rate curve. Therefore, curve and surface interpolation processing technology in high-end CNC systems has gradually replaced traditional linear or circular interpolation processing technology.
  • NURBS Non Uniform Rational B-Spline curve interpolation method
  • non-uniform rational B-spline curve interpolation method is widely used in the industry due to its flexibility in fitting various different paths.
  • the NURBS curves used in the five-axis NURBS curve interpolation calculation are divided into tool tip NURBS curve C(u) and tool axis NURBS curve C(v).
  • the prior art also provides a method of using interval synchronization, setting the relationship between vi+1 and ui +1 , and reducing the deviation caused by the same parameter method.
  • the interval synchronization method is extremely dependent on the shape of the tool tip NURBS curve C(u) and the tool axis NURBS curve C(v). When the shapes of the two curves are inconsistent, the actual direction of the tool will still deviate from the expected direction.
  • the embodiment of the present invention provides a five-axis linkage synchronous tool path interpolation method and system, which is used to solve the technical problem that the existing five-axis NURBS curve interpolation method is prone to deviation between the actual direction of the tool and the expected direction, thereby causing poor morphology and contour of the processed parts.
  • a first aspect of the present invention provides a five-axis linkage synchronous tool path interpolation method, comprising:
  • the tool tip path curve and the tool axis path curve are fitted based on the quadratic NURBS curve to obtain the node vector of the tool tip path curve and the node vector of the tool axis path curve respectively;
  • the formula is used to calculate the i+1th interpolation point of the tool tip path curve;
  • the preset formula is:
  • ⁇ 2 is the proportional coefficient
  • V k is the left endpoint of the tool tip node vector interval
  • V k+1 is the right endpoint of the tool tip node vector interval
  • Q is the intersection of the first-order derivative of the i+1th interpolation point of the tool axis path curve and the first-order derivative curve of the tool axis path curve
  • C′(V k ) is the first-order derivative of the interpolation point corresponding to the left endpoint of the tool tip node vector interval
  • C′(V k+1 ) is the first-order derivative of the interpolation point corresponding to the right endpoint of the tool tip node vector interval
  • the i+1th interpolation point of the tool axis path curve linked with the i+1th interpolation point of the tool nose path curve is calculated according to the proportional coefficient, and the tool axis path curve is interpolated.
  • the formula for calculating the first-order derivative of the i+1th interpolation point of the tool tip path curve and the first-order derivative of the interpolation points corresponding to the left and right endpoints of the tool tip node vector interval is:
  • C′(u) is the first-order derivative curve of the tool tip path curve C(u)
  • N j,p (u) is the p-order B-spline basis function defined by the node vector of the tool tip path curve
  • N′ j,p (u) is the first-order derivative of N j,p (u)
  • P j is the j-th control point
  • w j is the j-th weight coefficient
  • n is the number of control points.
  • the i+1th interpolation point of the tool axis path curve linked to the i+1th interpolation point of the tool tip path curve is calculated according to the proportional coefficient.
  • vi +1 is the i+1th interpolation point of the tool axis path curve.
  • constraints of the CNC machine tool include sampling point velocity constraints, centripetal acceleration constraints, centripetal jerk constraints, chord error constraints, and contour error constraints;
  • f p,max is the maximum sampling speed
  • f t,max is the maximum feed speed determined by the performance of the CNC machine tool
  • v ⁇ ,max is the maximum speed of the axis
  • v ⁇ ,max is the maximum speed of the motor
  • centripetal acceleration constraint is:
  • f n,a is the centripetal acceleration
  • a n,max is the maximum centripetal acceleration
  • ⁇ is the curvature of the path curve
  • centripetal jerk constraint is:
  • f n,j is the centripetal acceleration
  • j n,max is the maximum centripetal jerk
  • chord error constraint is:
  • f chrd is the maximum speed constrained by the chord error
  • T s is the interpolation period
  • ⁇ max is the maximum chord error
  • the contour error constraint is:
  • fcntr is the maximum speed constrained by the contour error
  • ⁇ max is the maximum contour error
  • ⁇ n is the natural frequency of the CNC machine tool system
  • ⁇ is the damping of the CNC machine tool system.
  • a feed speed curve is planned, including:
  • f( ui ) is the maximum feed speed satisfying the constraint condition at the i-th sampling point
  • fip ,max is the maximum sampling speed satisfying the sampling point speed constraint at the i-th sampling point
  • fin ,a is the centripetal acceleration satisfying the centripetal acceleration constraint at the i-th sampling point
  • fin ,j is the centripetal jerk satisfying the centripetal jerk constraint at the i-th sampling point
  • fichrd is the maximum speed satisfying the chord error constraint at the i-th sampling point
  • ficntr is the maximum speed satisfying the contour error constraint at the i-th sampling point
  • a polynomial fitting method is used to obtain the feed rate curve that meets the constraint conditions on the entire curve path.
  • the second aspect of the present invention also provides a five-axis linkage synchronous tool path interpolation system, comprising:
  • a curve fitting module is used to fit the tool tip path curve and the tool axis path curve based on the quadratic NURBS curve according to the tool path information, and obtain the node vector of the tool tip path curve and the node vector of the tool axis path curve respectively;
  • the speed planning module is used to plan the feed speed curve according to the relationship between the constraints of the CNC machine tool and the feed speed;
  • a tool tip node vector interval judgment module is used to judge the tool tip node vector interval into which the i+1th interpolation point of the tool tip path curve falls;
  • the tool tip interpolation point first-order derivative calculation module is used to calculate the first-order derivative of the i+1th interpolation point of the tool tip path curve and the first-order derivative of the interpolation points corresponding to the left and right endpoints of the tool tip node vector interval;
  • An angular progress calculation module is used to project three first-order derivatives onto the XOY plane, and move the starting point to the same point, and respectively calculate the first angle between the first-order derivative of the i+1th interpolation point of the tool tip path curve on the XOY plane and the first-order derivative of the left end point of the tool tip node vector interval, and the second angle between the first-order derivative of the left end point of the tool tip node vector interval and the first-order derivative of the right end point of the tool tip node vector interval, and divide the first angle by the second angle to obtain the angular progress;
  • the first-order derivative calculation module of the first tool axis interpolation point is used to calculate the first-order derivatives of the tool axis node vector interval corresponding to the tool tip node vector interval and the interpolation points corresponding to the left and right end points of the tool tip node vector interval on the tool axis path curve;
  • the first-order derivative calculation module of the second tool axis interpolation point is used to determine the first-order derivative of the i+1th interpolation point of the tool axis path curve according to the angular progress;
  • the proportional coefficient calculation module is used to calculate the intersection of the first-order derivative vector of the i+1th interpolation point of the tool axis path curve and the first-order derivative curve of the tool axis path curve, and calculate the proportional coefficient according to the preset formula.
  • the preset formula is:
  • V k is the left end point of the tool tip node vector interval
  • V k+1 is the right end point of the tool tip node vector interval.
  • Q is the intersection of the first-order derivative of the i+1th interpolation point of the tool axis path curve and the first-order derivative curve of the tool axis path curve
  • C′(V k ) is the first-order derivative of the interpolation point corresponding to the left end point of the tool tip node vector interval
  • C′(V k+1 ) is the first-order derivative of the interpolation point corresponding to the right end point of the tool tip node vector interval;
  • the tool axis curve interpolation point calculation module is used to calculate the i+1th interpolation point of the tool axis path curve linked with the i+1th interpolation point of the tool tip path curve according to the proportional coefficient, and interpolate the tool axis path curve.
  • the formula for calculating the first-order derivative of the i+1th interpolation point of the tool tip path curve and the first-order derivative of the interpolation points corresponding to the left and right endpoints of the tool tip node vector interval is:
  • C′(u) is the first-order derivative curve of the tool tip path curve C(u)
  • N j,p (u) is the p-order B-spline basis function defined by the node vector of the tool tip path curve
  • N′ j,p (u) is the first-order derivative of N j,p (u)
  • P j is the j-th control point
  • w j is the j-th weight coefficient
  • n is the number of control points.
  • vi +1 is the i+1th interpolation point of the tool axis path curve.
  • constraints of the CNC machine tool include sampling point velocity constraints, centripetal acceleration constraints, centripetal jerk constraints, chord error constraints, and contour error constraints;
  • f p,max is the maximum sampling speed
  • f t,max is the maximum feed speed determined by the performance of the CNC machine tool
  • v ⁇ ,max is the maximum speed of the axis
  • v ⁇ ,max is the maximum speed of the motor
  • centripetal acceleration constraint is:
  • f n,a is the centripetal acceleration
  • a n,max is the maximum centripetal acceleration
  • ⁇ is the curvature of the path curve
  • centripetal jerk constraint is:
  • f n,j is the centripetal acceleration
  • j n,max is the maximum centripetal jerk
  • chord error constraint is:
  • f chrd is the maximum speed constrained by the chord error
  • T s is the interpolation period
  • ⁇ max is the maximum chord error
  • the contour error constraint is:
  • fcntr is the maximum speed constrained by the contour error
  • ⁇ max is the maximum contour error
  • ⁇ n is the natural frequency of the CNC machine tool system
  • ⁇ is the damping of the CNC machine tool system.
  • the speed planning module is specifically used for:
  • f( ui ) is the maximum feed speed satisfying the constraint condition at the i-th sampling point
  • fip ,max is the maximum sampling speed satisfying the sampling point speed constraint at the i-th sampling point
  • fin ,a is the centripetal acceleration satisfying the centripetal acceleration constraint at the i-th sampling point
  • fin ,j is the centripetal jerk satisfying the centripetal jerk constraint at the i-th sampling point
  • fichrd is the maximum speed satisfying the chord error constraint at the i-th sampling point
  • ficntr is the maximum speed satisfying the contour error constraint at the i-th sampling point
  • a polynomial fitting method is used to obtain the feed rate curve that meets the constraint conditions on the entire curve path.
  • the five-axis linkage synchronous tool path interpolation method provided by the present invention first plans a feed speed curve, calculates the interpolation points of the tool tip path curve according to the second-order Taylor expansion formula, and then obtains the first-order derivative of the corresponding interpolation point according to the tool tip node vector interval in which the interpolation point of the tool tip path curve falls, thereby calculating the angular progress, determining the first-order derivative of the interpolation point of the tool axis path curve according to the angular progress, and obtaining the proportional coefficient relationship of the first-order derivative curve of the tool axis path curve.
  • a more accurate interpolation point of the tool axis path curve is calculated from the parameters of the tool tip path curve, and there is no need to calculate the interpolation point of the tool axis path curve according to interval synchronization, and it is not affected by
  • the shape consistency of the tool tip path curve and the tool axis path curve is affected, thereby avoiding the deviation between the actual direction of the tool and the expected direction, and solving the technical problem that the existing five-axis NURBS curve interpolation method is prone to deviation between the actual direction of the tool and the expected direction, resulting in poor morphology and contour of the processed parts.
  • the five-axis linkage synchronous tool path interpolation system provided by the present invention is used to execute the five-axis linkage synchronous tool path interpolation method provided by the present invention. Its principle and technical effects are the same as those of the five-axis linkage synchronous tool path interpolation method provided by the present invention, and will not be repeated here.
  • FIG1 is a schematic flow chart of a five-axis linkage synchronous tool path interpolation method provided in an embodiment of the present invention
  • FIG2 is a schematic diagram of an angular progress calculation projection of a five-axis linkage synchronous tool path interpolation method provided in an embodiment of the present invention
  • FIG. 3 is a schematic diagram of a projection of a tool axis path curve and a corresponding first-order derivative curve provided in an embodiment of the present invention
  • FIG4 is a trajectory diagram after interpolation by the five-axis linkage synchronous tool path interpolation method provided in an embodiment of the present invention.
  • FIG5 is a schematic diagram showing a comparison of effects of an interpolation method using the prior art and an interpolation method provided by an embodiment of the present invention
  • FIG. 6 is a schematic structural diagram of a five-axis linkage synchronous tool path system provided in an embodiment of the present invention.
  • the present invention provides an embodiment of a five-axis linkage synchronous tool path interpolation method, comprising:
  • Step 101 According to the tool path information, a tool tip path curve and a tool axis path curve are fitted based on a quadratic NURBS curve to obtain a node vector of the tool tip path curve and a node vector of the tool axis path curve.
  • the control end of the CNC machine tool will first read the tool path information for processing the part: ( xn , yn , zn ), ( x2 , y2 , z2 ), ..., ( x1 , y1 , z1 ).
  • u and v are the parameters of the tool tip path curve and the tool axis path curve, respectively, where 0 ⁇ u ⁇ 1, 0 ⁇ v ⁇ 1.
  • the node vectors [0,0,0,U 1 ,U 2 ,U 3 ,...,U n ,1,1,1] of the tool tip path curve and [0,0,0,V 1 ,V 2 ,V 3 ,...,V n ,1,1,1] of the tool axis path curve can be obtained respectively.
  • Step 102 Plan a feed speed curve according to the relationship between the constraints of the CNC machine tool and the feed speed.
  • a feed speed curve f(u) that meets the constraints is planned, specifically, including sampling point speed constraints, centripetal acceleration constraints, centripetal jerk constraints, chord error constraints, and contour error constraints.
  • f p,max is the maximum sampling speed
  • f t,max is the maximum feed speed determined by the performance of the CNC machine tool
  • v ⁇ ,max is the maximum speed of the axis
  • v ⁇ ,max is the maximum speed of the motor
  • centripetal acceleration constraint is:
  • f n,a is the centripetal acceleration
  • a n,max is the maximum centripetal acceleration
  • ⁇ is the curvature of the path curve
  • centripetal jerk constraint is:
  • f n,j is the centripetal acceleration
  • j n,max is the maximum centripetal jerk
  • chord error constraint is:
  • f chrd is the maximum speed constrained by the chord error
  • T s is the interpolation period
  • ⁇ max is the maximum chord error
  • the contour error constraint is:
  • fcntr is the maximum speed constrained by the contour error
  • ⁇ max is the maximum contour error
  • ⁇ n is the natural frequency of the CNC machine tool system
  • ⁇ is the damping of the CNC machine tool system.
  • f( ui ) is the maximum feed speed satisfying the constraint condition at the i-th sampling point
  • fip ,max is the maximum sampling speed satisfying the sampling point speed constraint at the i-th sampling point
  • fin ,a is the centripetal acceleration satisfying the centripetal acceleration constraint at the i-th sampling point
  • fin ,j is the centripetal jerk satisfying the centripetal jerk constraint at the i-th sampling point
  • fichrd is the centripetal acceleration satisfying the centripetal jerk constraint at the i-th sampling point.
  • f i cntr is the maximum speed of the contour error constraint at the i-th sampling point that satisfies the contour error constraint.
  • a polynomial fitting method is used to obtain the feed rate curve f(u) that meets the constraint conditions on the entire curve path.
  • the i+1th interpolation point of the tool tip path curve is calculated according to the second-order Taylor expansion formula:
  • Ts is the interpolation period
  • ui +1 is the i+1th interpolation point of the tool tip path curve
  • ui is the i-th interpolation point of the tool tip path curve.
  • Step 104 determining the tool tip node vector interval into which the (i+1)th interpolation point of the tool tip path curve falls.
  • Step 105 calculating the first-order derivative of the (i+1)th interpolation point of the tool tip path curve and the first-order derivative of the interpolation points corresponding to the left and right end points of the tool tip node vector interval.
  • the first-order derivative of the i+1th interpolation point u i+ 1 of the tool tip path curve and the first-order derivatives of the interpolation points corresponding to the left and right endpoints of the tool tip node vector interval are calculated respectively according to the following formulas:
  • C′(u) is the first-order derivative curve of the tool tip path curve C(u)
  • N j,p (u) is the p-order B-spline basis function defined by the node vector of the tool tip path curve
  • N′ j,p (u) is the first-order derivative of N j,p (u)
  • P j is the j-th control point
  • w j is the j-th weight coefficient
  • n is the number of control points.
  • Step 106 project the three first-order derivatives onto the XOY plane, and move the starting point to the same point, calculate the first angle between the first-order derivative of the i+1th interpolation point of the tool tip path curve on the XOY plane and the first-order derivative of the left end point of the tool tip node vector interval, and the second angle between the first-order derivative of the left end point of the tool tip node vector interval and the first-order derivative of the right end point of the tool tip node vector interval, divide the first angle by the second angle to obtain the angular progress.
  • the first angle is divided by the second angle to obtain the angular deviation ⁇ 1 , namely as shown in picture 2.
  • Step 107 calculating the first-order derivatives of the tool axis node vector interval corresponding to the tool tip node vector interval and the interpolation points corresponding to the left and right end points of the tool tip node vector interval on the tool axis path curve.
  • the tool axis node vector interval [ Vk , Vk +1 ] corresponding to the tool tip node vector interval [Uk, Uk+1] on the tool axis path curve C(v) and the first-order derivatives C′( Vk ) and C′( Vk +1 ) of the interpolation points corresponding to the left and right endpoints of the tool tip node vector interval are calculated.
  • Step 108 Determine the first-order derivative of the i+1th interpolation point of the tool axis path curve according to the angular progress.
  • Step 109 calculate the intersection of the first-order derivative vector of the i+1th interpolation point of the tool axis path curve and the first-order derivative curve of the tool axis path curve, and calculate the proportional coefficient according to a preset formula.
  • the first-order derivative curve C′(v) of the tool axis path curve C(v) can be calculated, and the control points for controlling the multi-deformation can be obtained: [C′(0),C′(v 1 ),C′(v 2 ),...,C′(v n ),C′(1)].
  • V k is the left endpoint of the tool tip node vector interval
  • V k+1 is the right endpoint of the tool tip node vector interval
  • Q is the intersection of the first-order derivative of the i+1th interpolation point of the tool axis path curve and the first-order derivative curve of the tool axis path curve
  • C′(V k ) is the first-order derivative of the interpolation point corresponding to the left endpoint of the tool tip node vector interval
  • C′(V k+1 ) is the first-order derivative of the tool tip node vector interval.
  • Step 110 calculating the i+1th interpolation point of the tool axis path curve linked with the i+1th interpolation point of the tool tip path curve according to the proportional coefficient, and interpolating the tool axis path curve.
  • vi +1 is the i+1th interpolation point of the tool axis path curve.
  • step 103 After the calculation of the i+1th interpolation point of the tool axis path curve is completed, return to step 103 to start calculating the i+2th interpolation point of the tool axis path curve, and so on, all the interpolation points of the tool axis path curve can be calculated to complete the tool path interpolation.
  • FIG4 is a trajectory diagram after interpolation by the five-axis linkage synchronous tool path interpolation method provided in the embodiment of the present invention (the left side is a plane view, and the right side is a spatial view).
  • FIG5 is a schematic diagram comparing the effects of the interpolation method of the prior art and the interpolation method provided in the embodiment of the present invention, wherein the left side shows the error of the C axis (one of the five axes of rotation) after interpolation by the interpolation method of the prior art, with a maximum error of 0.06 rad (i.e., 3.5°), and the right side shows the error of the C axis after interpolation by the interpolation method provided in the embodiment of the present invention, with a maximum error of 1.5 ⁇ 10 -4 rad (i.e., 0.0086°).
  • the interpolation method provided in the embodiment of the present invention significantly improves the linkage interpolation accuracy.
  • the synchronization process of the existing interval synchronization method depends on the shape of the tool tip curve and the tool axis curve.
  • the shapes of the two curves are the same, the values of each node in the node vector and the length of the interval are also the same.
  • the shapes of the two curves are different, the values of each node in the node vector and the length of the interval are different. If the interval synchronization method is still used for calculation, there will be errors, and the greater the difference in shape, the greater the error.
  • the five-axis linkage synchronous tool path interpolation method does not calculate vi +1 from u i+1 according to interval synchronization, but utilizes the characteristics of spline curves: the strict proportional relationship between the points, control points and parameters on the first-order derivative curves of all spline curves, that is, the relationship between the proportional coefficient ⁇ 2 of the points and the control points and the parameters is fixed and will not change with the change of the curve shape. Therefore, for curves of different shapes, a more accurate vi +1 can be calculated from ui +1 using the fixed relationship of the first-order derivative curve, thereby avoiding the limitation affected by the shape consistency of the tool tip path curve and the tool axis path curve.
  • the five-axis linkage synchronous tool path interpolation method provided by the present invention first plans a feed speed curve, calculates the interpolation points of the tool tip path curve according to the second-order Taylor expansion formula, and then obtains the first-order derivative of the corresponding interpolation point according to the tool tip node vector interval in which the interpolation point of the tool tip path curve falls, thereby calculating the angular progress, determining the first-order derivative of the interpolation point of the tool axis path curve according to the angular progress, and obtaining the proportional coefficient relationship of the first-order derivative curve of the tool axis path curve.
  • the strict proportional coefficient relationship between the points, control points and parameters on the first-order derivative curve allows more accurate tool axis path curve interpolation points to be calculated from the parameters of the tool tip path curve. There is no need to calculate the tool axis path curve interpolation points according to interval synchronization, and it is not affected by the shape consistency of the tool tip path curve and the tool axis path curve, thereby avoiding the deviation between the actual direction of the tool and the expected direction. This solves the technical problem that the existing five-axis NURBS curve interpolation method is prone to deviation between the actual direction of the tool and the expected direction, resulting in poor morphology and contour of the processed parts.
  • FIG. 6 an embodiment of a five-axis linkage synchronous tool path interpolation system is provided in the present invention, comprising:
  • a curve fitting module is used to fit the tool tip path curve and the tool axis path curve based on the quadratic NURBS curve according to the tool path information, and obtain the node vector of the tool tip path curve and the node vector of the tool axis path curve respectively;
  • the speed planning module is used to plan the feed speed curve according to the relationship between the constraints of the CNC machine tool and the feed speed;
  • a tool tip node vector interval judgment module is used to judge the tool tip node vector interval into which the i+1th interpolation point of the tool tip path curve falls;
  • the tool tip interpolation point first-order derivative calculation module is used to calculate the first-order derivative of the i+1th interpolation point of the tool tip path curve and the first-order derivative of the interpolation points corresponding to the left and right endpoints of the tool tip node vector interval;
  • An angular progress calculation module is used to project three first-order derivatives onto the XOY plane, and move the starting point to the same point, respectively calculate the first angle between the first-order derivative of the i+1th interpolation point of the tool tip path curve on the XOY plane and the first-order derivative of the left end point of the tool tip node vector interval, and the second angle between the first-order derivative of the left end point of the tool tip node vector interval and the first-order derivative of the right end point of the tool tip node vector interval, and divide the first angle by the second angle to obtain the angular progress;
  • the first-order derivative calculation module of the first tool axis interpolation point is used to calculate the first-order derivatives of the tool axis node vector interval corresponding to the tool tip node vector interval and the interpolation points corresponding to the left and right end points of the tool tip node vector interval on the tool axis path curve;
  • the first-order derivative calculation module of the second tool axis interpolation point is used to determine the first-order derivative of the i+1th interpolation point of the tool axis path curve according to the angular progress;
  • the proportional coefficient calculation module is used to calculate the intersection of the first-order derivative vector of the i+1th interpolation point of the tool axis path curve and the first-order derivative curve of the tool axis path curve, and calculate the proportional coefficient according to the preset formula.
  • the preset formula is:
  • V k is the left endpoint of the tool tip node vector interval
  • V k+1 is the right endpoint of the tool tip node vector interval
  • Q is the intersection of the first-order derivative of the i+1th interpolation point of the tool axis path curve and the first-order derivative curve of the tool axis path curve
  • C′(V k ) is the first-order derivative of the interpolation point corresponding to the left end point of the tool tip node vector interval
  • C′(V k+1 ) is the first-order derivative of the interpolation point corresponding to the right end point of the tool tip node vector interval
  • the tool axis curve interpolation point calculation module is used to calculate the i+1th interpolation point of the tool axis path curve linked with the i+1th interpolation point of the tool tip path curve according to the proportional coefficient, and interpolate the tool axis path curve.
  • the formula for calculating the first-order derivative of the i+1th interpolation point of the tool tip path curve and the first-order derivative of the interpolation points corresponding to the left and right endpoints of the tool tip node vector interval is:
  • C′(u) is the first-order derivative curve of the tool tip path curve C(u)
  • N j,p (u) is the p-order B-spline basis function defined by the node vector of the tool tip path curve
  • N′ j,p (u) is the first-order derivative of N j,p (u)
  • P j is the j-th control point
  • w j is the j-th weight coefficient
  • n is the number of control points.
  • vi +1 is the i+1th interpolation point of the tool axis path curve.
  • the constraints of CNC machine tools include sampling point velocity constraint, centripetal acceleration constraint, centripetal jerk constraint, chord error constraint and contour error constraint;
  • f p,max is the maximum sampling speed
  • f t,max is the maximum feed speed determined by the performance of the CNC machine tool
  • v ⁇ ,max is the maximum speed of the axis
  • v ⁇ ,max is the maximum speed of the motor
  • centripetal acceleration constraint is:
  • f n,a is the centripetal acceleration
  • a n,max is the maximum centripetal acceleration
  • ⁇ is the curvature of the path curve
  • centripetal jerk constraint is:
  • f n,j is the centripetal acceleration
  • j n,max is the maximum centripetal jerk
  • chord error constraint is:
  • f chrd is the maximum speed constrained by the chord error
  • T s is the interpolation period
  • ⁇ max is the maximum chord error
  • the contour error constraint is:
  • fcntr is the maximum speed constrained by the contour error
  • ⁇ max is the maximum contour error
  • ⁇ n is the natural frequency of the CNC machine tool system
  • ⁇ is the damping of the CNC machine tool system.
  • the speed planning module is specifically used for:
  • f( ui ) is the maximum feed speed satisfying the constraint condition at the i-th sampling point
  • fip ,max is the maximum sampling speed satisfying the sampling point speed constraint at the i-th sampling point
  • fin ,a is the centripetal acceleration satisfying the centripetal acceleration constraint at the i-th sampling point
  • fin ,j is the centripetal jerk satisfying the centripetal jerk constraint at the i-th sampling point
  • fichrd is the maximum speed satisfying the chord error constraint at the i-th sampling point
  • ficntr is the maximum speed satisfying the contour error constraint at the i-th sampling point
  • a polynomial fitting method is used to obtain the feed rate curve that meets the constraint conditions on the entire curve path.
  • the five-axis linkage synchronous tool path interpolation system provided in the embodiment of the present invention is used to execute the five-axis linkage synchronous tool path interpolation method in the aforementioned embodiment. Its working principle is the same as the five-axis linkage synchronous tool path interpolation method in the aforementioned embodiment, and the same technical effect can be achieved, which will not be repeated here.

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Abstract

一种五轴联动同步刀具路径插补方法和***，利用样条曲线的一阶导曲线上的点、控制点和参数之间严格的比例系数关系，从刀尖路径曲线的参数计算出更准确的刀轴路径曲线插补点，不需要按照区间同步来计算刀轴路径曲线插补点，不受刀尖路径曲线与刀轴路径曲线的形状一致性影响，从而避免了刀具实际方向与期望方向出现偏差，解决了现有的五轴NURBS曲线插补方法容易出现刀具实际方向与期望方向存在偏差，从而造成被加工零件形貌轮廓差的技术问题。

Description

一种五轴联动同步刀具路径插补方法和*** 技术领域
本发明涉及数控机床控制技术领域，尤其涉及一种五轴联动同步刀具路径插补方法和***。
背景技术
五轴数控机床能够通过刀尖点位置坐标和刀轴矢量的变化达到灵活控制刀具位姿的目的，从而提高加工速度、精度和表面光洁度，在动力能源和航天航空等领域得到广泛应用。与传统的线性插补相比，参数插补精度和效率更高，更容易获得光滑的进给速度曲线，因此，高端数控***中的曲线和曲面插补加工技术逐渐取代传统的直线或圆弧插补加工技术。
NURBS(Non Uniform Rational B-Spline)曲线插补法，通常也称非均匀有理B样条曲线插补法，由于其拟合各种不同路径的灵活性而被业内广泛应用。五轴NURBS曲线插补计算用到的NURBS曲线分为刀尖NURBS曲线C(u)和刀轴NURBS曲线C(v)。为了能够使刀尖曲线与刀轴曲线联动从而形成刀具矢量，现有技术采用相同参数的方法，令刀轴曲线的参数与刀尖曲线的参数相同，即vi+1＝ui+1，这会造成刀具实际方向与期望方向存在偏差，影响加工精度。为此，现有技术中还提供了利用区间同步的方法，设置了vi+1和ui+1的关系，减少了采用相同参数法带来的偏差。但是区间同步法极度依赖于刀尖NURBS曲线C(u)和刀轴NURBS曲线C(v)的形状，当两条曲线形状不一致时，仍然会造成刀具实际方向与期望方向存在偏差。
发明内容
本发明实施例提供了一种五轴联动同步刀具路径插补方法和***，用于解决现有的五轴NURBS曲线插补方法容易出现刀具实际方向与期望方向存在偏差，从而造成被加工零件形貌轮廓差的技术问题。
有鉴于此，本发明第一方面提供了一种五轴联动同步刀具路径插补方法，包括：
根据刀具路径信息，基于二次NURBS曲线分别拟合刀尖路径曲线和刀轴路径曲线，分别得到刀尖路径曲线的节点矢量和刀轴路径曲线的节点矢量；
根据数控机床的约束条件和进给速度的关系，规划进给速度曲线；
根据进给速度曲线对刀尖路径曲线从第i＝0个插补点开始插补，依据二阶泰勒展开公 式计算刀尖路径曲线第i+1个插补点；
判断刀尖路径曲线第i+1个插补点落入的刀尖节点矢量区间；
计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
将三条一阶导矢投影到XOY平面，且起点移到同一个点，分别计算XOY平面上刀尖路径曲线第i+1个插补点的一阶导矢与刀尖节点矢量区间左端点的一阶导矢的第一夹角，以及刀尖节点矢量区间左端点的一阶导矢与刀尖节点矢量区间右端点的一阶导矢的第二夹角，用第一夹角除以第二夹角，得到角进度；
计算刀轴路径曲线上与刀尖节点矢量区间相对应的刀轴节点矢量区间和刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
根据角进度确定刀轴路径曲线第i+1个插补点的一阶导矢；
计算刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，根据预置公式计算比例系数，预置公式为：
其中，η2为比例系数，Vk为刀尖节点矢量区间左端点，Vk+1为刀尖节点矢量区间右端点，Q为刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，C′(Vk)为刀尖节点矢量区间左端点对应的插补点的一阶导矢，C′(Vk+1)为刀尖节点矢量区间右端点对应的插补点的一阶导矢；
根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点，对刀轴路径曲线进行插补。
可选地，计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢的公式为：
其中，C′(u)为刀尖路径曲线C(u)的一阶导曲线，Nj,p(u)为由刀尖路径曲线节点矢量定义的p次B样条基函数，N′j,p(u)为的Nj,p(u)一阶导，Pj为第j个控制点，wj为第j个权重系数，n为控制点数量。
可选地，根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1 个插补点的计算公式为：
vi+1＝Vk+η2(Vk+1-Vk)
其中，vi+1为刀轴路径曲线第i+1个插补点。
可选地，数控机床的约束条件包括采样点速度约束、向心加速度约束、向心加加速度约束、弦误差约束和轮廓误差约束；
采样点速度约束为：
fp,max＝min{ft,max,K1 -(v∧,max),K1 -(K2 -(v∨,max))}
其中，fp,max为最大采样速度，ft,max为数控机床性能决定的最大进给速度，v∧,max为轴的最大速度，v∨,max为电机的最大速度，为刀尖到轴的运动学逆变换，为轴到电机的运动学逆变换；
向心加速度约束为：
其中，fn,a为向心加速度，an,max为最大向心加速度，ρ为路径曲线的曲率；
向心加加速度约束为：
其中，fn,j为向心加加速度，jn,max为最大向心加加速度；
弦误差约束为：
其中，fchrd为弦误差约束的最大速度，Ts为插补周期，δmax为最大弦误差；
轮廓误差约束为：
其中，fcntr为轮廓误差约束的最大速度，εmax为最大轮廓误差，ωn为数控机床***的固有频率，ξ为数控机床***的阻尼。
可选地，根据数控机床的约束条件和进给速度的关系，规划进给速度曲线，包括：
根据数控机床的约束条件和进给速度的关系，计算每个采样点上满足约束条件的最大进给速度，其中，每个采样点上满足约束条件的最大进给速度为：
f(ui)＝min{fi p,max,fi n,a,fi n,j,fi chrd,fi cntr}
其中，f(ui)为第i个采样点上满足约束条件的最大进给速度，fi p,max为第i个采样点上满足采样点速度约束的最大采样速度，fi n,a为第i个采样点上满足向心加速度约束的向心加速度，fi n,j为第i个采样点上满足向心加加速度约束的向心加加速度，fi chrd为第i个采样点上满足弦误差约束的弦误差约束的最大速度，fi cntr为第i个采样点上满足轮廓误差约束的轮廓误差约束的最大速度；
根据全部采样点上满足约束条件的最大进给速度，采用多项式拟合的方式得到整条曲线路径上满足约束条件的进给速度曲线。
本发明第二方面还提供了一种五轴联动同步刀具路径插补***，包括：
曲线拟合模块，用于根据刀具路径信息，基于二次NURBS曲线分别拟合刀尖路径曲线和刀轴路径曲线，分别得到刀尖路径曲线的节点矢量和刀轴路径曲线的节点矢量；
速度规划模块，用于根据数控机床的约束条件和进给速度的关系，规划进给速度曲线；
刀尖曲线插补点计算模块，用于根据进给速度曲线对刀尖路径曲线从第i＝0个插补点开始插补，依据二阶泰勒展开公式计算刀尖路径曲线第i+1个插补点；
刀尖节点矢量区间判断模块，用于判断刀尖路径曲线第i+1个插补点落入的刀尖节点矢量区间；
刀尖插补点一阶导矢计算模块，用于计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
角进度计算模块，用于将三条一阶导矢投影到XOY平面，且起点移到同一个点，分别计算XOY平面上刀尖路径曲线第i+1个插补点的一阶导矢与刀尖节点矢量区间左端点的一阶导矢的第一夹角，以及刀尖节点矢量区间左端点的一阶导矢与刀尖节点矢量区间右端点的一阶导矢的第二夹角，用第一夹角除以第二夹角，得到角进度；
第一刀轴插补点一阶导矢计算模块，用于计算刀轴路径曲线上与刀尖节点矢量区间相对应的刀轴节点矢量区间和刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
第二刀轴插补点一阶导矢计算模块，用于根据角进度确定刀轴路径曲线第i+1个插补点的一阶导矢；
比例系数计算模块，用于计算刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，根据预置公式计算比例系数，预置公式为：
其中，η2为比例系数，Vk为刀尖节点矢量区间左端点，Vk+1为刀尖节点矢量区间右端 点，Q为刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，C′(Vk)为刀尖节点矢量区间左端点对应的插补点的一阶导矢，C′(Vk+1)为刀尖节点矢量区间右端点对应的插补点的一阶导矢；
刀轴曲线插补点计算模块，用于根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点，对刀轴路径曲线进行插补。
可选地，计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢的公式为：
其中，C′(u)为刀尖路径曲线C(u)的一阶导曲线，Nj,p(u)为由刀尖路径曲线节点矢量定义的p次B样条基函数，N′j,p(u)为的Nj,p(u)一阶导，Pj为第j个控制点，wj为第j个权重系数，n为控制点数量。
可选地，根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点的计算公式为：
vi+1＝Vk+η2(Vk+1-Vk)
其中，vi+1为刀轴路径曲线第i+1个插补点。
可选地，数控机床的约束条件包括采样点速度约束、向心加速度约束、向心加加速度约束、弦误差约束和轮廓误差约束；
采样点速度约束为：
fp,max＝min{ft,max,K1 -(v∧,max),K1 -(K2 -(v∨,max))}
其中，fp,max为最大采样速度，ft,max为数控机床性能决定的最大进给速度，v∧,max为轴的最大速度，v∨,max为电机的最大速度，为刀尖到轴的运动学逆变换，为轴到电机的运动学逆变换；
向心加速度约束为：
其中，fn,a为向心加速度，an,max为最大向心加速度，ρ为路径曲线的曲率；
向心加加速度约束为：
其中，fn,j为向心加加速度，jn,max为最大向心加加速度；
弦误差约束为：
其中，fchrd为弦误差约束的最大速度，Ts为插补周期，δmax为最大弦误差；
轮廓误差约束为：
其中，fcntr为轮廓误差约束的最大速度，εmax为最大轮廓误差，ωn为数控机床***的固有频率，ξ为数控机床***的阻尼。
可选地，速度规划模块具体用于：
根据数控机床的约束条件和进给速度的关系，计算每个采样点上满足约束条件的最大进给速度，其中，每个采样点上满足约束条件的最大进给速度为：
f(ui)＝min{fi p,max,fi n,a,fi n,j,fi chrd,fi cntr}
其中，f(ui)为第i个采样点上满足约束条件的最大进给速度，fi p,max为第i个采样点上满足采样点速度约束的最大采样速度，fi n,a为第i个采样点上满足向心加速度约束的向心加速度，fi n,j为第i个采样点上满足向心加加速度约束的向心加加速度，fi chrd为第i个采样点上满足弦误差约束的弦误差约束的最大速度，fi cntr为第i个采样点上满足轮廓误差约束的轮廓误差约束的最大速度；
根据全部采样点上满足约束条件的最大进给速度，采用多项式拟合的方式得到整条曲线路径上满足约束条件的进给速度曲线。
从以上技术方案可以看出，本发明提供的本发明提供的五轴联动同步刀具路径插补方法和***具有以下优点：
本发明提供的五轴联动同步刀具路径插补方法，先规划出进给速度曲线，依据二阶泰勒展开公式计算刀尖路径曲线的插补点，再依据刀尖路径曲线的插补点所落入的刀尖节点矢量区间求取对应的插补点一阶导矢，从而计算得到角进度，依据角进度确定刀轴路径曲线插补点的一阶导矢，得到刀轴路径曲线的一阶导曲线的比例系数关系，利用样条曲线的一阶导曲线上的点、控制点和参数之间严格的比例系数关系，从刀尖路径曲线的参数计算出更准确的刀轴路径曲线插补点，不需要按照区间同步来计算刀轴路径曲线插补点，不受 刀尖路径曲线与刀轴路径曲线的形状一致性影响，从而避免了刀具实际方向与期望方向出现偏差，解决了现有的五轴NURBS曲线插补方法容易出现刀具实际方向与期望方向存在偏差，从而造成被加工零件形貌轮廓差的技术问题。
本发明提供的五轴联动同步刀具路径插补***，用于执行本发明提供的五轴联动同步刀具路径插补方法，其原理和所取得的技术效果与本发明提供的五轴联动同步刀具路径插补方法相同，在此不再赘述。
附图说明
图1为本发明实施例中提供的一种五轴联动同步刀具路径插补方法的流程示意图；
图2为本发明实施例中提供的一种五轴联动同步刀具路径插补方法的角进度计算投影示意图；
图3为本发明实施例中提供的刀轴路径曲线和对应的一阶导曲线投影示意图；
图4为本发明实施例中提供的五轴联动同步刀具路径插补方法进行插补后的轨迹图；
图5为采用现有技术的插补方法和本发明实施例提供的插补方法的效果对比示意图；
图6为本发明实施例中提供的一种五轴联动同步刀具路径***的结构示意图。
具体实施方式
为了使本技术领域的人员更好地理解本发明方案，下面将结合本发明实施例中的附图，对本发明实施例中的技术方案进行清楚、完整地描述，显然，所描述的实施例仅是本发明一部分实施例，而不是全部的实施例。基于本发明中的实施例，本领域普通技术人员在没有做出创造性劳动前提下所获得的所有其他实施例，都属于本发明保护的范围。
为了便于理解，请参阅图1，本发明中提供了一种五轴联动同步刀具路径插补方法的实施例，包括：
步骤101、根据刀具路径信息，基于二次NURBS曲线分别拟合刀尖路径曲线和刀轴路径曲线，分别得到刀尖路径曲线的节点矢量和刀轴路径曲线的节点矢量。
需要说明的是，当有零件需要被加工时，数控机床的控制端会先读取加工该零件的刀具路径信息：(xn,yn,zn)，(x2,y2,z2)，...，(x1,y1,z1)。依据刀具路径信息，用二次NURBS曲线分别拟合刀尖路径曲线C(u)＝(x(u),y(u),z(u))和刀轴路径曲线C(v)＝(x(v),y(v),z(v))，u和v分别为刀尖路径曲线和刀轴路径曲线的参数，其中，0≤u≤1，0≤v≤1，分别可得到刀尖路径曲线的节点矢量[0,0,0,U1,U2,U3,...,Un,1,1,1]和刀轴路径曲线的节点矢量[0,0,0,V1,V2,V3,...,Vn,1,1,1]。
步骤102、根据数控机床的约束条件和进给速度的关系，规划进给速度曲线。
需要说明的是，根据数控机床的约束条件和进给速度的关系，规划出满足约束条件的进给速度曲线f(u)。具体地，包括采样点速度约束、向心加速度约束、向心加加速度约束、弦误差约束和轮廓误差约束。
采样点速度约束为：
fp,max＝min{ft,max,K1 -(v∧,max),K1 -(K2 -(v∨,max))}
其中，fp,max为最大采样速度，ft,max为数控机床性能决定的最大进给速度，v∧,max为轴的最大速度，v∨,max为电机的最大速度，为刀尖到轴的运动学逆变换，为轴到电机的运动学逆变换；
向心加速度约束为：
其中，fn,a为向心加速度，an,max为最大向心加速度，ρ为路径曲线的曲率；
向心加加速度约束为：
其中，fn,j为向心加加速度，jn,max为最大向心加加速度；
弦误差约束为：
其中，fchrd为弦误差约束的最大速度，Ts为插补周期，δmax为最大弦误差；
轮廓误差约束为：
其中，fcntr为轮廓误差约束的最大速度，εmax为最大轮廓误差，ωn为数控机床***的固有频率，ξ为数控机床***的阻尼。
因此，每个采样点上满足约束条件的最大进给速度为：
f(ui)＝min{fi p,max,fi n,a,fi n,j,fi chrd,fi cntr}
其中，f(ui)为第i个采样点上满足约束条件的最大进给速度，fi p,max为第i个采样点上满足采样点速度约束的最大采样速度，fi n,a为第i个采样点上满足向心加速度约束的向心加速度，fi n,j为第i个采样点上满足向心加加速度约束的向心加加速度，fi chrd为第i个 采样点上满足弦误差约束的弦误差约束的最大速度，fi cntr为第i个采样点上满足轮廓误差约束的轮廓误差约束的最大速度。
根据全部采样点上满足约束条件的最大进给速度，采用多项式拟合的方式得到整条曲线路径上满足约束条件的进给速度曲线f(u)。
步骤103、根据进给速度曲线对刀尖路径曲线从第i＝0个插补点开始插补，依据二阶泰勒展开公式计算刀尖路径曲线第i+1个插补点。
需要说明的是，在规划出进给速度曲线f(u)之后，将进给速度曲线f(u)的有效信息传给数控机床的控制端，并从第0个插补点开始插补，即u0＝0。刀尖路径曲线的第i+1个插补点依据二阶泰勒展开公式计算：
其中，Ts为插补周期，ui+1为刀尖路径曲线的第i+1个插补点，ui为刀尖路径曲线的第i个插补点。
步骤104、判断刀尖路径曲线第i+1个插补点落入的刀尖节点矢量区间。
需要说明的是，计算出刀尖路径曲线的第i+1个插补点ui+1之后，判断ui+1属于刀尖路径曲线的节点矢量[0,0,0,U1,U2,U3,...,Un,1,1,1]中的哪个节点矢量区间，假设ui+1∈[Uk,Uk+1]。
步骤105、计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢。
需要说明的是，根据以下公式分别计算刀尖路径曲线第i+1个插补点ui+1的一阶导矢、刀尖节点矢量区间左右端点(即Uk和Uk+1)对应的插补点的一阶导矢：
其中，C′(u)为刀尖路径曲线C(u)的一阶导曲线，Nj,p(u)为由刀尖路径曲线节点矢量定义的p次B样条基函数，N′j,p(u)为的Nj,p(u)一阶导，Pj为第j个控制点，wj为第j个权重系数，n为控制点数量。
因而可得到三条一阶导矢：C′(ui+1),C′(Uk),C′(Uk+1)。
步骤106、将三条一阶导矢投影到XOY平面，且起点移到同一个点，分别计算XOY平面上刀尖路径曲线第i+1个插补点的一阶导矢与刀尖节点矢量区间左端点的一阶导矢的第一夹角，以及刀尖节点矢量区间左端点的一阶导矢与刀尖节点矢量区间右端点的一阶导矢的第二夹角，用第一夹角除以第二夹角，得到角进度。
需要说明的是，将计算出来的三条一阶导矢，即C′(ui+1),C′(Uk),C′(Uk+1)，投影到XOY平面，且起点移到同一个点，分别计算XOY平面上刀尖路径曲线第i+1个插补点的一阶导矢C′(ui+1)与刀尖节点矢量区间左端点的一阶导矢C′(Uk)的第一夹角θ1，以及刀尖节点矢量区间左端点的一阶导矢C′(Uk)与刀尖节点矢量区间右端点的一阶导矢C′(Uk+1)的第二夹角θ2，用第一夹角除以第二夹角，得到角进度η1，即如图2所示。
步骤107、计算刀轴路径曲线上与刀尖节点矢量区间相对应的刀轴节点矢量区间和刀尖节点矢量区间左右端点对应的插补点的一阶导矢。
需要说明的是，计算刀轴路径曲线C(v)上与刀尖节点矢量区间[Uk,Uk+1]相对应的刀轴节点矢量区间[Vk,Vk+1]和刀尖节点矢量区间左右端点对应的插补点的一阶导矢C′(Vk)、C′(Vk+1)。
步骤108、根据角进度确定刀轴路径曲线第i+1个插补点的一阶导矢。
需要说明的是，在得到C′(Vk)和C′(Vk+1)之后，依据角进度可以确定刀轴路径曲线第i+1个插补点的一阶导矢C′(vi+1)。
步骤109、计算刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，根据预置公式计算比例系数。
需要说明的是，根据步骤105中的C′(u)公式，可以计算出刀轴路径曲线C(v)的一阶导曲线C′(v)，可获得其控制多变形的控制点：
[C′(0),C′(v1),C′(v2),...,C′(vn),C′(1)]。
刀轴路径曲线第i+1个插补点的一阶导矢C′(vi+1)与一阶导曲线C′(v)相交于一点Q，如图3所示，Q位于直线C′(vk)C′(vk+1)上，因而，计算比例系数η2的公式为：
其中，η2为比例系数，Vk为刀尖节点矢量区间左端点，Vk+1为刀尖节点矢量区间右端点，Q为刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，C′(Vk)为刀尖节点矢量区间左端点对应的插补点的一阶导矢，C′(Vk+1)为刀尖节点矢量区 间右端点对应的插补点的一阶导矢。
步骤110、根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点，对刀轴路径曲线进行插补。
需要说明的是，依据比例系数η2计算与ui+1联动的刀轴路径曲线插补参数vi+1，计算公式为：
vi+1＝Vk+η2(Vk+1-Vk)
其中，vi+1为刀轴路径曲线第i+1个插补点。
当刀轴路径曲线第i+1个插补点计算结束后，返回步骤103开始计算刀轴路径曲线第i+2个插补点，依次类推，可计算出刀轴路径曲线的所有插补点，完成刀具路径插补。
图4为本发明实施例中提供的五轴联动同步刀具路径插补方法进行插补后的轨迹图(左边为平面视图，右边为空间视图)。图5为采用现有技术的插补方法和本发明实施例提供的插补方法的效果对比示意图，其中，左边为现有技术的插补方法插补后C轴(五轴中的一个旋转轴)存在的误差，最大误差为0.06rad(即3.5°)，右边为本发明实施例提供的插补方法插补后C轴存在的误差，最大误差为1.5×10-4rad(即0.0086°)，与现有技术相比，本发明实施例提供的插补方法明显提高了联动插补精度。
现有的区间同步法的同步过程依赖刀尖曲线和刀轴曲线的形状，当两个曲线形状相同时，节点矢量中的各个节点的值和区间的长度也是相同的，则按ui+1在[Uk,Uk+1]区间中的进度计算属于区间[Vk,Vk+1]且有同样进度的vi+1是可行的。当两个曲线形状不同时，节点矢量中的各个节点的值和区间的长度不同，仍按区间同步法计算，则会存在误差，且形状相差越大，误差越大。本发明实施例中提供的五轴联动同步刀具路径插补方法不按照区间同步从ui+1计算vi+1，而是利用了样条曲线的特性：所有样条曲线的一阶导曲线上的点、控制点和参数三者之间的严格比例关系，即点和控制点的比例系数η2和参数间的关系是固定的，不会随着曲线形状的改变而改变。因此，对于不同形状的曲线，利用其一阶导曲线这个固定不变的关系，可以从ui+1计算出更准确的vi+1，从而避免了受刀尖路径曲线和刀轴路径曲线的形状一致性影响的限制。
本发明提供的五轴联动同步刀具路径插补方法，先规划出进给速度曲线，依据二阶泰勒展开公式计算刀尖路径曲线的插补点，再依据刀尖路径曲线的插补点所落入的刀尖节点矢量区间求取对应的插补点一阶导矢，从而计算得到角进度，依据角进度确定刀轴路径曲线插补点的一阶导矢，得到刀轴路径曲线的一阶导曲线的比例系数关系，利用样条曲线的 一阶导曲线上的点、控制点和参数之间严格的比例系数关系，从刀尖路径曲线的参数计算出更准确的刀轴路径曲线插补点，不需要按照区间同步来计算刀轴路径曲线插补点，不受刀尖路径曲线与刀轴路径曲线的形状一致性影响，从而避免了刀具实际方向与期望方向出现偏差，解决了现有的五轴NURBS曲线插补方法容易出现刀具实际方向与期望方向存在偏差，从而造成被加工零件形貌轮廓差的技术问题。
为了便于理解，请参阅图6，本发明中提供了一种五轴联动同步刀具路径插补***的实施例，包括：
曲线拟合模块，用于根据刀具路径信息，基于二次NURBS曲线分别拟合刀尖路径曲线和刀轴路径曲线，分别得到刀尖路径曲线的节点矢量和刀轴路径曲线的节点矢量；
速度规划模块，用于根据数控机床的约束条件和进给速度的关系，规划进给速度曲线；
刀尖曲线插补点计算模块，用于根据进给速度曲线对刀尖路径曲线从第i＝0个插补点开始插补，依据二阶泰勒展开公式计算刀尖路径曲线第i+1个插补点；
刀尖节点矢量区间判断模块，用于判断刀尖路径曲线第i+1个插补点落入的刀尖节点矢量区间；
刀尖插补点一阶导矢计算模块，用于计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
角进度计算模块，用于将三条一阶导矢投影到XOY平面，且起点移到同一个点，分别计算XOY平面上刀尖路径曲线第i+1个插补点的一阶导矢与刀尖节点矢量区间左端点的一阶导矢的第一夹角，以及刀尖节点矢量区间左端点的一阶导矢与刀尖节点矢量区间右端点的一阶导矢的第二夹角，用第一夹角除以第二夹角，得到角进度；
第一刀轴插补点一阶导矢计算模块，用于计算刀轴路径曲线上与刀尖节点矢量区间相对应的刀轴节点矢量区间和刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
第二刀轴插补点一阶导矢计算模块，用于根据角进度确定刀轴路径曲线第i+1个插补点的一阶导矢；
比例系数计算模块，用于计算刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，根据预置公式计算比例系数，预置公式为：
其中，η2为比例系数，Vk为刀尖节点矢量区间左端点，Vk+1为刀尖节点矢量区间右端点，Q为刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点， C′(Vk)为刀尖节点矢量区间左端点对应的插补点的一阶导矢，C′(Vk+1)为刀尖节点矢量区间右端点对应的插补点的一阶导矢；
刀轴曲线插补点计算模块，用于根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点，对刀轴路径曲线进行插补。
计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢的公式为：
其中，C′(u)为刀尖路径曲线C(u)的一阶导曲线，Nj,p(u)为由刀尖路径曲线节点矢量定义的p次B样条基函数，N′j,p(u)为的Nj,p(u)一阶导，Pj为第j个控制点，wj为第j个权重系数，n为控制点数量。
根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点的计算公式为：
vi+1＝Vk+η2(Vk+1-Vk)
其中，vi+1为刀轴路径曲线第i+1个插补点。
数控机床的约束条件包括采样点速度约束、向心加速度约束、向心加加速度约束、弦误差约束和轮廓误差约束；
采样点速度约束为：
fp,max＝min{ft,max,K1 -(v∧,max),K1 -(K2 -(v∨,max))}
其中，fp,max为最大采样速度，ft,max为数控机床性能决定的最大进给速度，v∧,max为轴的最大速度，v∨,max为电机的最大速度，为刀尖到轴的运动学逆变换，为轴到电机的运动学逆变换；
向心加速度约束为：
其中，fn,a为向心加速度，an,max为最大向心加速度，ρ为路径曲线的曲率；
向心加加速度约束为：
其中，fn,j为向心加加速度，jn,max为最大向心加加速度；
弦误差约束为：
其中，fchrd为弦误差约束的最大速度，Ts为插补周期，δmax为最大弦误差；
轮廓误差约束为：
其中，fcntr为轮廓误差约束的最大速度，εmax为最大轮廓误差，ωn为数控机床***的固有频率，ξ为数控机床***的阻尼。
速度规划模块具体用于：
根据数控机床的约束条件和进给速度的关系，计算每个采样点上满足约束条件的最大进给速度，其中，每个采样点上满足约束条件的最大进给速度为：
f(ui)＝min{fi p,max,fi n,a,fi n,j,fi chrd,fi cntr}
其中，f(ui)为第i个采样点上满足约束条件的最大进给速度，fi p,max为第i个采样点上满足采样点速度约束的最大采样速度，fi n,a为第i个采样点上满足向心加速度约束的向心加速度，fi n,j为第i个采样点上满足向心加加速度约束的向心加加速度，fi chrd为第i个采样点上满足弦误差约束的弦误差约束的最大速度，fi cntr为第i个采样点上满足轮廓误差约束的轮廓误差约束的最大速度；
根据全部采样点上满足约束条件的最大进给速度，采用多项式拟合的方式得到整条曲线路径上满足约束条件的进给速度曲线。
本发明实施例中提供的五轴联动同步刀具路径插补***，用于执行前述实施例中的五轴联动同步刀具路径插补方法，其工作原理与前述实施例中的五轴联动同步刀具路径插补方法相同，可取得相同的技术效果，在此不再进行赘述。
以上所述，以上实施例仅用以说明本发明的技术方案，而非对其限制；尽管参照前述实施例对本发明进行了详细的说明，本领域的普通技术人员应当理解：其依然可以对前述各实施例所记载的技术方案进行修改，或者对其中部分技术特征进行等同替换；而这些修改或者替换，并不使相应技术方案的本质脱离本发明各实施例技术方案的精神和范围。

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  1. 一种五轴联动同步刀具路径插补方法，其特征在于，包括：
    根据刀具路径信息，基于二次NURBS曲线分别拟合刀尖路径曲线和刀轴路径曲线，分别得到刀尖路径曲线的节点矢量和刀轴路径曲线的节点矢量；
    根据数控机床的约束条件和进给速度的关系，规划进给速度曲线；
    根据进给速度曲线对刀尖路径曲线从第i＝0个插补点开始插补，依据二阶泰勒展开公式计算刀尖路径曲线第i+1个插补点；
    判断刀尖路径曲线第i+1个插补点落入的刀尖节点矢量区间；
    计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
    将三条一阶导矢投影到XOY平面，且起点移到同一个点，分别计算XOY平面上刀尖路径曲线第i+1个插补点的一阶导矢与刀尖节点矢量区间左端点的一阶导矢的第一夹角，以及刀尖节点矢量区间左端点的一阶导矢与刀尖节点矢量区间右端点的一阶导矢的第二夹角，用第一夹角除以第二夹角，得到角进度；
    计算刀轴路径曲线上与刀尖节点矢量区间相对应的刀轴节点矢量区间和刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
    根据角进度确定刀轴路径曲线第i+1个插补点的一阶导矢；
    计算刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，根据预置公式计算比例系数，预置公式为：
    其中，η2为比例系数，Vk为刀尖节点矢量区间左端点，Vk+1为刀尖节点矢量区间右端点，Q为刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，C′(Vk)为刀尖节点矢量区间左端点对应的插补点的一阶导矢，C′(Vk+1)为刀尖节点矢量区间右端点对应的插补点的一阶导矢；
    根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点，对刀轴路径曲线进行插补。
  2. 根据权利要求1所述的五轴联动同步刀具路径插补方法，其特征在于，计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢的公式为：
    其中，C′(u)为刀尖路径曲线C(u)的一阶导曲线，Nj,p(u)为由刀尖路径曲线节点矢量定义的p次B样条基函数，N′j,p(u)为的Nj,p(u)一阶导，Pj为第j个控制点，wj为第j个权重系数，n为控制点数量。
  3. 根据权利要求1所述的五轴联动同步刀具路径插补方法，其特征在于，根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点的计算公式为：
    vi+1＝Vk+η2(Vk+1-Vk)
    其中，vi+1为刀轴路径曲线第i+1个插补点。
  4. 根据权利要求1所述的五轴联动同步刀具路径插补方法，其特征在于，数控机床的约束条件包括采样点速度约束、向心加速度约束、向心加加速度约束、弦误差约束和轮廓误差约束；
    采样点速度约束为：
    fp,max＝min{ft,max,K1 -(v∧,max),K1 -(K2 -(v∨,max))}
    其中，fp,max为最大采样速度，ft,max为数控机床性能决定的最大进给速度，v∧,max为轴的最大速度，v∨,max为电机的最大速度，为刀尖到轴的运动学逆变换，为轴到电机的运动学逆变换；
    向心加速度约束为：
    其中，fn,a为向心加速度，an,max为最大向心加速度，ρ为路径曲线的曲率；
    向心加加速度约束为：
    其中，fn,j为向心加加速度，jn,max为最大向心加加速度；
    弦误差约束为：
    其中，fchrd为弦误差约束的最大速度，Ts为插补周期，δmax为最大弦误差；
    轮廓误差约束为：
    其中，fcntr为轮廓误差约束的最大速度，εmax为最大轮廓误差，ωn为数控机床***的固有频率，ξ为数控机床***的阻尼。
  5. 根据权利要求4所述的五轴联动同步刀具路径插补方法，其特征在于，根据数控机床的约束条件和进给速度的关系，规划进给速度曲线，包括：
    根据数控机床的约束条件和进给速度的关系，计算每个采样点上满足约束条件的最大进给速度，其中，每个采样点上满足约束条件的最大进给速度为：
    f(ui)＝min{fi p,max,fi n,a,fi n,j,fi chrd,fi cntr}
    其中，f(ui)为第i个采样点上满足约束条件的最大进给速度，fi p,max为第i个采样点上满足采样点速度约束的最大采样速度，fi n,a为第i个采样点上满足向心加速度约束的向心加速度，fi n,j为第i个采样点上满足向心加加速度约束的向心加加速度，fi chrd为第i个采样点上满足弦误差约束的弦误差约束的最大速度，fi cntr为第i个采样点上满足轮廓误差约束的轮廓误差约束的最大速度；
    根据全部采样点上满足约束条件的最大进给速度，采用多项式拟合的方式得到整条曲线路径上满足约束条件的进给速度曲线。
  6. 一种五轴联动同步刀具路径插补***，其特征在于，包括：
    曲线拟合模块，用于根据刀具路径信息，基于二次NURBS曲线分别拟合刀尖路径曲线和刀轴路径曲线，分别得到刀尖路径曲线的节点矢量和刀轴路径曲线的节点矢量；
    速度规划模块，用于根据数控机床的约束条件和进给速度的关系，规划进给速度曲线；
    刀尖曲线插补点计算模块，用于根据进给速度曲线对刀尖路径曲线从第i＝0个插补点开始插补，依据二阶泰勒展开公式计算刀尖路径曲线第i+1个插补点；
    刀尖节点矢量区间判断模块，用于判断刀尖路径曲线第i+1个插补点落入的刀尖节点矢量区间；
    刀尖插补点一阶导矢计算模块，用于计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
    角进度计算模块，用于将三条一阶导矢投影到XOY平面，且起点移到同一个点，分别计算XOY平面上刀尖路径曲线第i+1个插补点的一阶导矢与刀尖节点矢量区间左端点的一阶导矢的第一夹角，以及刀尖节点矢量区间左端点的一阶导矢与刀尖节点矢量区间右端点的一阶导矢的第二夹角，用第一夹角除以第二夹角，得到角进度；
    第一刀轴插补点一阶导矢计算模块，用于计算刀轴路径曲线上与刀尖节点矢量区间相对应 的刀轴节点矢量区间和刀尖节点矢量区间左右端点对应的插补点的一阶导矢；
    第二刀轴插补点一阶导矢计算模块，用于根据角进度确定刀轴路径曲线第i+1个插补点的一阶导矢；
    比例系数计算模块，用于计算刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，根据预置公式计算比例系数，预置公式为：
    其中，η2为比例系数，Vk为刀尖节点矢量区间左端点，Vk+1为刀尖节点矢量区间右端点，Q为刀轴路径曲线第i+1个插补点的一阶导矢与刀轴路径曲线的一阶导曲线的交点，C′(Vk)为刀尖节点矢量区间左端点对应的插补点的一阶导矢，C′(Vk+1)为刀尖节点矢量区间右端点对应的插补点的一阶导矢；
    刀轴曲线插补点计算模块，用于根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点，对刀轴路径曲线进行插补。
  7. 根据权利要求6所述的五轴联动同步刀具路径插补***，其特征在于，计算刀尖路径曲线第i+1个插补点的一阶导矢、刀尖节点矢量区间左右端点对应的插补点的一阶导矢的公式为：
    其中，C′(u)为刀尖路径曲线C(u)的一阶导曲线，Nj,p(u)为由刀尖路径曲线节点矢量定义的p次B样条基函数，N′j,p(u)为的Nj,p(u)一阶导，Pj为第j个控制点，wj为第j个权重系数，n为控制点数量。
  8. 根据权利要求6所述的五轴联动同步刀具路径插补***，其特征在于，根据比例系数计算与刀尖路径曲线第i+1个插补点联动的刀轴路径曲线第i+1个插补点的计算公式为：
    vi+1＝Vk+η2(Vk+1-Vk)
    其中，vi+1为刀轴路径曲线第i+1个插补点。
  9. 根据权利要求6所述的五轴联动同步刀具路径插补***，其特征在于，数控机床的约束条件包括采样点速度约束、向心加速度约束、向心加加速度约束、弦误差约束和轮廓误差约束；
    采样点速度约束为：
    fp,max＝min{ft,max,K1 -(v∧,max),K1 -(K2 -(v∨,max))}
    其中，fp,max为最大采样速度，ft,max为数控机床性能决定的最大进给速度，v∧,max为轴的最 大速度，v∨,max为电机的最大速度，为刀尖到轴的运动学逆变换，为轴到电机的运动学逆变换；
    向心加速度约束为：
    其中，fn,a为向心加速度，an,max为最大向心加速度，ρ为路径曲线的曲率；
    向心加加速度约束为：
    其中，fn,j为向心加加速度，jn,max为最大向心加加速度；
    弦误差约束为：
    其中，fchrd为弦误差约束的最大速度，Ts为插补周期，δmax为最大弦误差；
    轮廓误差约束为：
    其中，fcntr为轮廓误差约束的最大速度，εmax为最大轮廓误差，ωn为数控机床***的固有频率，ξ为数控机床***的阻尼。
  10. 根据权利要求9所述的五轴联动同步刀具路径插补***，其特征在于，速度规划模块具体用于：
    根据数控机床的约束条件和进给速度的关系，计算每个采样点上满足约束条件的最大进给速度，其中，每个采样点上满足约束条件的最大进给速度为：
    f(ui)＝min{fi p,max,fi n,a,fi n,j,fi chrd,fi cntr}
    其中，f(ui)为第i个采样点上满足约束条件的最大进给速度，fi p,max为第i个采样点上满足采样点速度约束的最大采样速度，fi n,a为第i个采样点上满足向心加速度约束的向心加速度，fi n,j为第i个采样点上满足向心加加速度约束的向心加加速度，fi chrd为第i个采样点上满足弦误差约束的弦误差约束的最大速度，fi cntr为第i个采样点上满足轮廓误差约束的轮廓误差约束的最大速度；
    根据全部采样点上满足约束条件的最大进给速度，采用多项式拟合的方式得到整条曲线路径上满足约束条件的进给速度曲线。
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