CN114002996B - C3 continuous five-axis path switching fairing method for hybrid robot - Google Patents
C3 continuous five-axis path switching fairing method for hybrid robot Download PDFInfo
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Abstract
The invention discloses a C3 continuous five-axis path switching fairing method of a hybrid robot, which comprises the following steps: the point position path of the tool nose is defined in a Cartesian coordinate system; the cutter shaft direction path is defined on the surface of the unit sphere; the tool point position path and the tool shaft direction path are switched and smoothed by adopting spline curve segments, and the path after smoothing consists of an inserted spline curve segment and a residual path segment; respectively deducing the continuous conditions of curvature differentiation of the two at the transfer point, and further establishing a relation function of the fairing length of the point position path of the tool nose, the fairing angle of the path in the direction of the tool shaft and the respective fairing errors; determining a fairing curve of a point position path of the tool nose and a path in the cutter shaft direction by combining constraint conditions synchronously introduced by parameters; and introducing a parameter synchronization curve to ensure that the third derivative of the five-axis path after geometric fairing with respect to time is continuous. The invention can realize the accurate prediction and control of the fairing error and improve the calculation efficiency of the fairing method.
Description
Technical Field
The invention relates to the field of robot numerical control processing, in particular to a C3 continuous five-axis path switching fairing method of a hybrid robot.
Background
At present, with the rapid development of the fields of aviation, aerospace, railway, shipping and the like, the processing demands on large-size complex parts are increasing. Manufacturing ultra-large machine tools with serial travel is a common solution for achieving large-size part machining. However, the above-described machining scheme is not an economical and efficient option when machining features are scattered over different areas on a large part, such as aircraft wing skin hole making. The five-degree-of-freedom hybrid robot is regarded as a plug and play module, and the in-situ processing of large-size parts can be realized by combining a long-stroke guide rail. The robot processing technology is a combination of numerical control processing technology and robot motion control technology. The machining path is usually measured in an operation space, and a high-order continuous path planning technology is an important guarantee for realizing the motion control of the robot. In order to solve the problem of frequent acceleration and deceleration of the robot caused by non-high-order continuous phenomenon of the processing path at the connecting point, the processing efficiency is improved, the precision is improved, the linear path of the hybrid robot is required to be subjected to switching fairing treatment, and the spline curve is inserted to replace the corner of the original path, so that the path after fairing reaches curvature differential continuity (G3 continuity).
The patent CN111230864 discloses a series-parallel robot tool path planning which divides an original path into a plurality of long straight line segments and a plurality of short straight line segments, respectively carries out switching fairing on break points, and carries out fitting fairing on segment straight line segment groups. Although the method focuses on path transfer fairing and adopts a direct fairing method to process the path of the cutter direction, a specific construction method of a fairing curve control point is not provided, and a fairing curve solving model under the restriction of a fairing error is not established.
The patent CN112506139 discloses a local corner fairing method of a short straight-line section track of a series-parallel robot, which provides a three-order continuous real-time corner fairing method based on an asymmetric PH curve. Although this method focuses on the hybrid robot linear path switching fairing, it does not involve the case that the point position path of the tool tip includes a circular arc segment. In addition, this method maps the arbor direction path to a linear path, and it is difficult to improve arbor movement performance in the vicinity of the connection point.
The series-parallel robot path switching fairing needs to solve two types of problems: (1) a fairing error control; (2) parameter synchronization of position and orientation paths. In terms of directional error control, existing methods can be divided into two categories. Firstly, a numerical iteration method is adopted to directly control the direction error, so that the calculated amount is greatly increased; and secondly, the direction deviation at the middle point of the spline curve is used as a predicted value to indirectly control the direction error, but under specific conditions, larger deviation occurs between the actual value and the predicted value, and the error control capability is deteriorated. Therefore, there is a need for a direction path fairing method that improves computational efficiency, and accurately predicts and controls the fairing error.
In terms of parameter synchronization, there are two solutions. Firstly, parameter synchronization is realized through the re-parameterization of the residual path, and the method can lead to larger cutter shaft angular acceleration and angle jump near a path transfer point due to neglecting the geometric characteristics of the position and direction path. Secondly, parameter synchronization is ensured by establishing a constraint relation between the position path fairing length and the direction path fairing angle, but the method needs to map the direction path into a series of intermediate paths consisting of straight line segments, and error control capability is reduced. Therefore, there is a need for a parameter synchronization method that is suitable for both directional path fairing of a unit ball surface and improving the degree of arbor motion fairing.
Considering the multi-branched coupling characteristic of the hybrid robot, in order to avoid inertial forced vibration of the robot, the motion track is generally required to meet the requirement of C3 continuity. In addition, in order to avoid singular phenomena caused by the indirect fairing method, a direct fairing method is adopted to treat the path of the cutter shaft in the surface of the unit sphere.
Disclosure of Invention
The invention provides a C3 continuous five-axis path switching fairing method of a hybrid robot for solving the technical problems in the prior art.
The invention adopts the technical proposal for solving the technical problems in the prior art that: a method for switching and smoothing a C3 continuous five-axis path of a hybrid robot comprises the following steps: setting the end point track of the unit vector of the cutter axis as a cutter shaft direction path; setting the point position track of the tool nose as the point position path of the tool nose; the point position path of the tool nose is defined in a Cartesian coordinate system; the cutter shaft direction path is defined on the surface of the unit sphere; the tool point position path and the cutter shaft direction path are switched and smoothed by adopting spline curve segments, the paths after smoothing both are composed of inserted spline curve segments and residual path segments, wherein the inserted spline curve segments comprise odd control points, and the maximum deviation between the middle control point of the inserted spline curve and the original path is assumed; respectively deriving the continuous filling conditions of curvature differentiation of the point position path of the tool nose and the cutter shaft direction path at the transfer point, and further respectively establishing the relation functions of the fairing length of the point position path of the tool nose, the fairing angle of the cutter shaft direction path and the respective fairing errors; based on the technological requirements, determining the fairing error limit value corresponding to the point position path of the tool nose and the path in the direction of the tool shaft, and determining the fairing length of the point position path of the tool nose and the fairing angle of the path in the direction of the tool shaft by combining constraint conditions synchronously introduced by parameters, so as to determine the fairing curves of the point position path of the tool nose and the path in the direction of the tool shaft.
Further, the method comprises the following steps: introducing a parameter synchronization curve, and reconstructing a point position path of the tool nose and a path analysis expression in the direction of the tool shaft; seven-order B-spline with eight control points is selected as a parameter synchronization curve, and a node vector is set as (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1) T The method comprises the steps of carrying out a first treatment on the surface of the Wherein, the first control point=0 and the eighth control point=1 of the parameter synchronization curve; obtaining an h-1 section of path C from the point position path of the tool nose after fairing h-1 (u), h-th segment Path C h (u), h+1th segment Path C h+1 (u); obtaining an h-1 segment path Q from the cutter shaft direction path after fairing h-1 (u), h segment Path Q h (u), h+1th segment Path Q h+1 (u); the continuous filling condition of C3 at the junction of the two paths is as follows:
C′ h (u)| u＝1 ＝C′ h+1 (u)| u＝0 ；C″ h (u)| u＝1 ＝C″ h+1 (u)| u＝0 ；C″′ h (u)| u＝1 ＝C″′ h+1 (u)| u＝0 ；
Q′ h (u)| u＝1 ＝Q′ h+1 (u)| u＝0 ；Q″ h (u)| u＝1 ＝Q″ h+1 (u)| u＝0 ；Q″′ h (u)| u＝1 ＝Q″′ h+1 (u)| u＝0 ；
wherein u represents a B spline curve parameter; h represents the path segment number of the point position path and the cutter shaft direction path after the smoothing; "'", "" and "'" respectively denote first, second and third derivatives of the curve with respect to the parameter u; determining values of a second control point and a seventh control point of the parameter synchronization curve according to continuous filling conditions of the angular speed of the cutter shaft; determining a third control point and a sixth control point value according to continuous charging conditions of the angular acceleration of the cutter shaft; and determining the values of the fourth control point and the fifth control point according to the continuous filling conditions of the angle of the cutter shaft.
Further, the method comprises the following steps: after finishing the smoothing of the point position path and the cutter shaft direction path, generating a continuous point speed curve of the cutter point by correcting the jump curve in the S acceleration and deceleration movement; and performing parameter interpolation according to the cutter point speed curve and combining with a pre-estimation-correction method to generate an interpolation point sequence required in the robot processing process.
Further, the method comprises the following steps: according to the interpolation point sequence, based on a hybrid robot kinematic model, determining joint variables of each driving joint of the robot at each moment; controlling the motion of the series-parallel robot according to the joint variable of each driving joint of the robot at each moment; based on the PMAC motion control card, the following errors of all joints in the motion process of the robot are collected in real time, and the fairing effectiveness of the point position path of the tool nose and the path in the direction of the tool shaft is evaluated.
Further, the interpolated spline curve segment is a cubic B-spline.
Further, the specific step of establishing a relationship function of the fairing length of the knife tip position path and the fairing error thereof comprises the following steps:
step A1, setting the (i-1) th knife point P i-1 The ith point P i I+1th tip point P i+1 Three sequentially connected cutter point points in the original cutter point position path; adopting a cubic B spline curve with seven control points as spline curve segments inserted in the process of switching the three nose points; the inserted spline curve segment expression is as follows:
Wherein C is i (u) represents at point P i Spline curves inserted at the positions; b (B) j,i Representing spline curve C i The control point of (u), wherein i represents the number of the original knife point, j=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters; n (N) j,5 (u) represents a fifth-order B-spline basis function;
step A2, establishing the following conditions for continuous curvature differentiation of the point position path of the tool nose at the connecting point:
wherein m is e,i Representing the point P i Pointing point P i+1 Wherein "e" is a flag bit; s represents the accumulated arc length of the point position path of the tool nose;
step A3, assume a path segmentThe length of (2) satisfies B 0,i B 1,i ||＝c 1 ||B 0,i B 3,i I, wherein c 1 E (0, 1) is a predetermined constant; assume Path section +.>The length of (2) satisfies B 1,i B 2,i ||＝c 2 ||B 0,i B 3,i I, wherein c 2 E (0, 1) is a predetermined constant; to avoid the curves crossing themselves, 0 < c should be ensured 1 +c 2 < 1; from the above assumption, a position spline curve C is established i (u) each control point and the cutting point P i Is a function of (a);
step A4, determining the lower limit value of the length of the residual path section according to monotonically increasing arc length of the residual path section along with curve parameters; based on the functional relation among the control points of the cutter point position path curve, the relation function of the fairing length of the cutter point position path and the fairing error is established as follows:
||B 0,i B 3,i ||＝||B 3,i B 6,i ||；
In the method, in the process of the invention,
ε pos,max a fairing error limit value representing a set point position path of the tool nose;
φ i representing a unit vector m s,i And m e,i Wherein m is s,i Representing the point P i Pointing point P i-1 Unit vector of m e,i Representing the point P i Pointing point P i+1 Is a unit vector of (2);
L i representing original path segmentsIs a length of (2);
L i+1 representing original path segmentsIs a length of (2);
representing the residual position path length +.>Wherein B is a lower limit value of 6,i-1 Indicated at point P i-1 The last control point of the spline curve inserted is located;
representing the residual position path length +.>Wherein B is a lower limit value of 0,i+1 Indicated at point P i+1 At the first control point of the interpolated spline.
Further, the specific step of establishing a relation function of the fairing angle of the cutter shaft direction path and the fairing error thereof comprises the following steps:
step B1, an original cutter shaft direction path consists of a series of concentric arcs on the surface of a unit sphere, and the sphere center of the unit sphere is set as W; setting the i-1 th tool axis unit vector end point O i-1 Ith tool axis unit vector end point O i I+1th tool axis unit vector end point O i+1 The unit vector end points of the cutter axis are sequentially connected with three paths in the original cutter shaft direction; a modulus constant based on a cutter axis unit vector is 1, and a quintic B spline curve comprising seven control points is used as a spline curve segment inserted when the path switching fairing in the cutter shaft direction; set Q i (u) is at point O i Spline curve of arbor direction path inserted at: q (Q) i The expression of (u) is as follows:
in phi, phi i (u) represents a continuous condition differentiated according to the path curvature in the arbor direction, at the point O i An initial spline curve constructed at the location; "|W phi i (u) || represents the point W to the curve Φ i (u) the distance of the point corresponding to the parameter u; d (D) k,i Representing spline curve Q i The control point of (u), wherein i represents the number of the original knife site, and k=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters;
step B2, establishing a continuous charging condition of curvature differentiation of the cutter shaft direction path at the connecting point:
wherein s is o Representing the accumulated arc length of the path in the cutter shaft direction;representing arc +.>Upper D 6,i A unit tangent vector at the position, wherein 'e' is a flag bit; d, d 6,i Representing the point D pointed at by the center W of the unit sphere 6,i Is a vector of (2);
step B3, establishing a curve Q according to the continuous charge condition of curvature differentiation and the derivative characteristic of the B-spline curve i Functional relationship of each control point of (u) with the original tool axis unit vector end point; assumption curve Q i (u) the control point satisfies the following relationship: line segmentAnd line segment->The length ratio of (2) h is 1:h, and h is more than or equal to 2 and less than or equal to 8; determining a fairing curve Q according to continuous charge conditions of path curvature differentiation in the cutter shaft direction i Control point D of (u) 3,i The method comprises the steps of carrying out a first treatment on the surface of the According to continuous charging conditions of path curvature in the cutter shaft direction and the central angle D of the switching section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining curve Q i Control point D of (u) 2,i And D 4,i The method comprises the steps of carrying out a first treatment on the surface of the According to tangential continuous filling conditions of cutter shaft direction paths and central angle D of the transfer section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining curve Q i Control point D of (u) 1,i And D 5,i ；
Let { R s,i The expression is in a circular arc segmentUpper point O i Frenet coordinate system built in place is provided with +.>Representation ofArc section->At point O i The unit tangent vector at the position is provided with +.>Representing arc segment +.>At point O i The principal normal unit vector at the position is provided with +.>Representing arc segment +.>At point O i A secondary normal unit vector at the location;
let { R e,i The expression is in a circular arc segmentUpper point O i Frenet coordinate system built in place is provided with +.>Representing arc segment +.>At point O i The unit tangent vector at the position is provided with +.>Representing arc segment +.>At point O i The principal normal unit vector at the position is provided with +.>Representing arc segment +.>At point O i A secondary normal unit vector at the location;
let d 3,i ＝o i The method comprises the steps of carrying out a first treatment on the surface of the Then there are:
in the method, in the process of the invention,
o i a cutter axis unit vector corresponding to the ith cutter position point is represented;
the representation represents a circular arc +.>Upper D 0,i A unit tangent vector at the position, wherein's' is a flag bit;
represents angle D 0,i WD 3,i Wherein "s" represents a flag bit;
represents angle D 3,i WD 6,i Wherein "e" represents a flag bit;
d 0,i Representing the direction from point W to point D 0,i Is a vector of (2);
d 1,i representing the direction from point W to point D 1,i Is a vector of (2);
d 2,i representing the direction from point W to point D 2,i Is a vector of (2);
d 3,i representing the direction from point W to point D 3,i Is a vector of (2);
d 4,i representing the direction from point W to point D 4,i Is a vector of (2);
d 5,i representing the direction from point W to point D 5,i Is a vector of (2);
d 6,i representing the direction from point W to point D 6,i Is a vector of (2);
step B4, according to curve Q i Deducing an analytical expression of the fairing error and the fairing angle of the path in the cutter shaft direction according to the functional relation between each control point of (u) and the unit vector end point of the original cutter axis; expressing the path fairing error of the cutter shaft direction asAnd theta n,i Wherein θ n,i Representing unit vector +.>And->Is included in the plane of the first part; obtaining the fairing error along with the fairing angle of the cutter shaft direction path through drawing a universal variation trend chart of the fairing error of the cutter shaft direction path>Monotonically increasing; based on the limiting conditions of the path fairing error and parameter synchronization in the cutter shaft direction, adopting a numerical method to solve the fairing angle +.>And->
Further, step B4 includes the following sub-steps:
step B4-1, consider the following limitationCondition determining a fairing angleUpper limit value of (2):
1) In order to avoid intersection of path curves of adjacent tool tip positions, the fairing angle should not exceed half of the original central angle;
2) Determining the central angle D of the residual path segment according to the monotonically increasing arc length of the residual path segment along with the curve parameter 0, i WD 3,i Sum +.D 3,i WD 6,i Lower limit value of (2);
then there are:
in the method, in the process of the invention,representing the fairing angle +.>Upper limit value of (2); delta i Representing arc segment +.>Is a central angle of (2); delta i+1 Representing arc segment +.>Is a central angle of (2);Represents the central angle D of the remained path section 6,i-1 WD 0,i Lower limit value of (2);Represents the central angle D of the remained path section 6,i WD 0,i+1 Lower limit value of (2);
step B4-2, according to the fairing angleCalculating the fairing error of the cutter shaft direction path; judging whether the path fairing error in the cutter shaft direction meets the following conditions:
(1-ε t )ε ori,max ≤ε ori,i ≤ε ori,max ；
wherein ε t Control precision epsilon for expressing path fairing error of cutter shaft direction ori,i Indicating that the path of the cutter shaft direction is at the point O i A fairing error at the location; epsilon ori,max A predefined cutter shaft direction path fairing error limit value;
if the path fairing error in the cutter shaft direction meets the condition, then the cutter shaft direction path fairing error is made to be
If the path fairing error in the cutter shaft direction does not meet the condition, adopting a dichotomy to solve the fairing angle; according to the dichotomy solving strategy, continuously updating the value of the fairing angle until the constraint condition is met;
and step B4-3, further completely determining a path fairing curve of the cutter shaft direction according to the calculated fairing angle.
Further, the method for improving the cutter shaft movement performance of the hybrid robot by adjusting the fairing length of the cutter point position path and the fairing angle of the cutter shaft direction path comprises the following steps:
Step D1, judging whether the fairing length of the point position path of the tool nose and the fairing angle of the path in the cutter shaft direction meet the following conditions according to the planned processing path of the hybrid robot:
if the condition is satisfied, then make
If the condition is not satisfied, make
Step D2, judging whether the fairing length of the point position path of the tool nose and the fairing angle of the path in the cutter shaft direction meet the following conditions according to the planned processing path:
if the condition is satisfied, then make
If the condition is not satisfied, make
Step D3, after finishing the adjustment of the fairing length of the point position path of the cutter point and the fairing angle of the path in the cutter axis direction, recalculating a fairing curve C of the point position path of the cutter according to the updated fairing length of the point position path of the cutter point and the updated fairing angle of the path in the cutter axis direction i (u) and arbor direction Path fairing curve Q i (u)；
And D4, forming a global C3 continuous series-parallel robot processing path after tool planning by the C3 continuous tool point position path and the C3 continuous cutter shaft direction path.
The invention has the advantages and positive effects that: the invention provides an analytical construction method for a straight line segment and circular arc segment switching fairing curve control point according to a constraint condition of curvature differential continuity.
The invention realizes direct fairing of the direction path on the surface of the unit sphere, does not need to map the direction path into a linear path, avoids singular phenomenon caused by an indirect method, ensures accurate prediction and control of the direction fairing error, improves the calculation efficiency of the fairing method, and further improves the motion performance of the hybrid robot.
According to the invention, geometric fairing path parameter synchronization is realized by parameterizing the residual path section, and the control point of the synchronization curve and the length or angle margin value of the residual path section can be determined without iterative calculation. In order to improve the motion performance of the cutter shaft, the invention provides a fairing length or angle adjusting method.
The smoothing method is not only suitable for five-axis linear paths, but also suitable for point position paths of tool points comprising circular arc sections. Therefore, the invention can further obtain a general five-axis path fairing method, and the related algorithm can be used as a pre-processing module independent of the numerical control system of the robot.
Drawings
Fig. 1: and the series-parallel robot processing path C3 is continuously switched to a fairing flow chart.
Fig. 2: the point position of the tool nose is a path switching fairing schematic diagram.
Fig. 3: a cutter shaft direction path switching fairing schematic diagram.
Fig. 4: a direction error change trend graph.
Fig. 5: and calculating a direction path fairing angle flow chart by adopting a dichotomy method.
Fig. 6: the S-shaped path C3 continuously switches the fairing results.
Fig. 7: based on the processing result of the S-shaped test piece.
In fig. 1:
c3 continuous represents the third derivative of the parameter curve with respect to time continuous;
g3 continuous means that the third derivative of the parametric curve with respect to the curve arc length is continuous, i.e. the curvature differential is continuous;
CAD representation ComputerAided Design, i.e., computer aided design;
CAM represents Computer Aided Manufacturing, computer aided manufacturing;
in fig. 2:
P i representing the point position of the tool nose corresponding to the ith tool position;
P i-1 representing the point position of the tool nose corresponding to the i-1 th tool position;
P i+1 representing the point position of the tool nose corresponding to the (i+1) th tool position;
C i (u) represents at point P i A position fairing curve inserted at the position;
B 6,i-1 representing a position fairing curve C i-1 A seventh control point of (u), wherein C i-1 (u) represents at point P i-1 A position fairing curve inserted at the position;
B 0,i representing a position fairing curve C i A first control point of (u);
B 3,i representing a position fairing curve C i A fourth control point of (u);
B 6,i representing a position fairing curve C i A seventh control point of (u);
B 0,i+1 representing a position fairing curve C i+1 A first control point of (u), wherein C i+1 (u) represents at point P i+1 A position fairing curve inserted at the position;
L i representing path segments of originIs a length of (2);
L i+1 representing path segments of originIs a length of (2);
l r,i representing a path segment of a residual locationWherein, the subscript "r" is a flag bit;
l r,i+1 representing a path segment of a residual locationWherein, the subscript "r" is a flag bit;
φ i representing vector m s,i And m e,i Wherein m is s,i Represented by P i Pointing to P i-1 The subscript "s" is the flag bit, m e,i Represented by P i Pointing to P i+1 The subscript "e" is a flag bit;
in fig. 3:
w represents the origin of the coordinate system of the workpiece;
O i representing a unit vector end point of the cutter axis corresponding to the ith cutter position point;
O i-1 representing a cutter axis unit vector end point corresponding to the i-1 cutter position point;
O i+1 representing a cutter axis unit vector end point corresponding to the (i+1) th cutter position point;
Q i (u) represents at point O i A direction fairing curve of the point insertion;
D 6,i-1 representing a directional fairing curve Q i-1 A seventh control point of (u), wherein Q i-1 (u) represents at point O i-1 A direction fairing curve of the point insertion;
D 0,i representing a directional fairing curve Q i A first control point of (u);
D 3,i representing a directional fairing curve Q i A fourth control point of (u);
D 6,i representing a directional fairing curve Q i A seventh control point of (u);
D 0,i+1 representing a directional fairing curve Q i+1 A first control point of (u), wherein Q i+1 (u) represents at point O i+1 A direction fairing curve of the point insertion;
{R s,i the expression is in a circular arc segmentUpper point O i The Frenet coordinate system built at the position, the subscript "s" is a marker bit, wherein ∈R is->Representing unit cut vector, & lt & gt>Representing principal normal vector, ++>Representing a secondary normal vector;
{R e,i the expression is in a circular arc segmentUpper point O i The Frenet coordinate system built at the position, the subscript "e" is a marker bit, wherein ∈F is used for the- >Representing unit cut vector, & lt & gt>Representing principal normal vector, ++>Representing a secondary normal vector;
θ n,i representing two secondary normal vectorsAnd->Wherein, the subscript "n" is a flag bit;
δ i representing a circular arc segmentIs a central angle of (2);
δ i+1 representing a circular arc segmentIs a central angle of (2);
representing arc +.>Upper D 0,i A unit tangent vector at the position, wherein, the subscript "s" is a flag bit;
representing arc +.>Upper D 6,i A unit tangent vector at the position, wherein, the subscript "e" is a flag bit;
in FIG. 4
ε ori,i Representation point O i A directional fairing error at the location;
in fig. 5:
represents an angle determined by considering the restriction conditions such as the self-disagreement of the directional fairing curve and the synchronization of parameters>Upper limit value of (2), wherein->Represents angle D 3,i WD 6,i I represents an index value of a knife site;
ε t indicating the accuracy of directional fairing error control, where the subscript "t" is a flag bit, e.g. epsilon t ＝0.1％；
b s Representing the use of intermediate variables in an iterative algorithm, wherein the subscript "s" is a flag bit;
b e representing the use of intermediate variables in the iterative algorithm, wherein the subscript "e" is a flag bit;
Detailed Description
For a further understanding of the invention, its features and advantages, reference is now made to the following examples, which are illustrated in the accompanying drawings in which:
referring to fig. 1 to 7, a method for transferring fairing of a C3 continuous five-axis path of a hybrid robot includes the following steps: setting the end point track of the unit vector of the cutter axis as a cutter shaft direction path; setting the point position track of the tool nose as the point position path of the tool nose; the point position path of the tool nose is defined in a Cartesian coordinate system; the cutter shaft direction path is defined on the surface of the unit sphere; the tool point position path and the cutter shaft direction path are switched and smoothed by adopting spline curve segments, the paths after smoothing both are composed of inserted spline curve segments and residual path segments, wherein the inserted spline curve segments comprise odd control points, and the maximum deviation between the middle control point of the inserted spline curve and the original path is assumed; respectively deriving the continuous filling conditions of curvature differentiation of the point position path of the tool nose and the cutter shaft direction path at the transfer point, and further respectively establishing the relation functions of the fairing length of the point position path of the tool nose, the fairing angle of the cutter shaft direction path and the respective fairing errors; based on the technological requirements, determining the fairing error limit value corresponding to the point position path of the tool nose and the path in the direction of the tool shaft, and determining the fairing length of the point position path of the tool nose and the fairing angle of the path in the direction of the tool shaft by combining constraint conditions synchronously introduced by parameters, so as to determine the fairing curves of the point position path of the tool nose and the path in the direction of the tool shaft.
The origin of the tool axis unit vector is the origin of the workpiece coordinate system.
Fairing length of knife point position path: the point position path of the tool nose is replaced by a spline curve to obtain the length corresponding to the part.
Fairing angle of arbor direction path: and replacing part of the corresponding central angle by the spline curve on the path in the cutter shaft direction.
The fairing error refers to: the maximum deviation between the spline and the original path, e.g., the position fairing error, refers to the hausdorff distance between the spline and the original path.
Preferably, the method may further comprise the steps of: introducing a parameter synchronization curve, and reconstructing a point position path of the tool nose and a path analysis expression in the direction of the tool shaft; seven-order B-spline with eight control points is selected as a parameter synchronization curve, and a node vector is set as (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1) T The method comprises the steps of carrying out a first treatment on the surface of the Wherein, the first control point=0 and the eighth control point=1 of the parameter synchronization curve; obtaining the h-1 section from the point position path of the tool nose after fairingPath C h-1 (u), h-th segment Path C h (u), h+1th segment Path C h+1 (u); obtaining an h-1 segment path Q from the cutter shaft direction path after fairing h-1 (u), h segment Path Q h (u), h+1th segment Path Q h+1 (u); the continuous filling condition of C3 at the junction of the two paths is as follows: c'. h (u)| u＝1 ＝C′ h+1 (u)| u＝0 ；C″ h (u)| u＝1 ＝C″ h+1 (u)| u＝0 ；C″′ h (u)| u＝1 ＝C″′ h+1 (u)| u＝0 ；
Q′ h (u)| u＝1 ＝Q′ h+1 (u)| u＝0 ；Q″ h (u)| u＝1 ＝Q″ h+1 (u)| u＝0 ；Q″′ h (u)| u＝1 ＝Q″′ h+1 (u)| u＝0 ；
Wherein u represents a B spline curve parameter; h represents the path segment number of the point position path and the cutter shaft direction path after the smoothing; "'", "" and "'" respectively denote first, second and third derivatives of the curve with respect to the parameter u; determining values of a second control point and a seventh control point of the parameter synchronization curve according to continuous filling conditions of the angular speed of the cutter shaft; determining a third control point and a sixth control point value according to continuous charging conditions of the angular acceleration of the cutter shaft; and determining the values of the fourth control point and the fifth control point according to the continuous filling conditions of the angle of the cutter shaft.
The parameter synchronization curve refers to: and the curve for synchronously adopting the point position path of the tool nose and the path parameters in the direction of the tool shaft is realized.
Preferably, the method may further comprise the steps of: after finishing the smoothing of the point position path and the cutter shaft direction path, generating a continuous point speed curve of the cutter point by correcting the jump curve in the S acceleration and deceleration movement; and performing parameter interpolation according to the cutter point speed curve and combining with a pre-estimation-correction method to generate an interpolation point sequence required in the robot processing process.
Preferably, the method may further comprise the steps of: according to the interpolation point sequence, based on a hybrid robot kinematic model, determining joint variables of each driving joint of the robot at each moment; controlling the motion of the series-parallel robot according to the joint variable of each driving joint of the robot at each moment; based on the PMAC motion control card, the following errors of all joints in the motion process of the robot are collected in real time, and the fairing effectiveness of the point position path of the tool nose and the path in the direction of the tool shaft is evaluated.
Preferably, the interpolated spline curve segment may be a cubic B-spline.
Preferably, the specific step of establishing a function of the relationship between the fairing length of the knife tip position path and its fairing error may comprise:
step A1, the i-1 th point P can be set i-1 The ith point P i I+1th tip point P i+1 Three sequentially connected cutter point points in the original cutter point position path; a cubic B spline curve with seven control points can be used as spline curve segments inserted when the three tool nose points are switched to smooth; the inserted spline curve segment expression is as follows:
wherein C is i (u) represents at point P i Spline curves inserted at the positions; b (B) j,i Representing spline curve C i The control point of (u), wherein i represents the number of the original knife point, j=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters; n (N) j,5 (u) represents a fifth-order B-spline basis function;
step A2, the following conditions can be established for continuous curvature differentiation of the point position path of the tool nose at the point position:
wherein m is e,i Representing the point P i Pointing point P i+1 Wherein "e" is a flag bit; s represents the accumulated arc length of the point position path of the tool nose;
step A3, the path segment can be assumedThe length of (2) satisfies B 0,i B 1,i ||＝c 1 ||B 0,i B 3,i I, wherein c 1 E (0, 1) is a predetermined constant; assume Path section +.>The length of (2) satisfies B 1,i B 2,i ||＝c 2 ||B 0,i B 3,i I, wherein c 2 E (0, 1) is a predetermined constant; to avoid the curves crossing themselves, 0 < c should be ensured 1 +c 2 < 1; from the above assumption, a position spline curve C is established i (u) each control point and the cutting point P i Is a function of (a);
step A4, determining the lower limit value of the length of the residual path section according to the monotonic increment of the arc length of the residual path section along with the curve parameter; based on the functional relation among the control points of the cutter point position path curve, the relation function of the fairing length of the cutter point position path and the fairing error is established as follows:
||B 0,i B 3,i ||＝||B 3,i B 6,i ||；
in the method, in the process of the invention,
ε pos,max a fairing error limit value representing a set point position path of the tool nose;
φ i representing a unit vector m s,i And m e,i Wherein m is s,i Representing the point P i Pointing point P i-1 Unit vector of m e,i Representing the point P i Pointing point P i+1 Is a unit vector of (2);
L i representing original path segmentsIs a length of (2);
L i+1 representing original path segmentsIs a length of (2);
representing the residual position path length +.>Wherein B is a lower limit value of 6,i-1 Indicated at point P i-1 The last control point of the spline curve inserted is located;
representing the residual position path length +.>Wherein B is a lower limit value of 0,i+1 Indicated at point P i+1 At the first control point of the interpolated spline.
Preferably, the specific step of establishing a relation function of the fairing angle of the arbor direction path and the fairing error thereof may comprise:
and B1, an original cutter shaft direction path consists of a series of concentric arcs on the surface of the unit sphere, and the cutter shaft direction path is a three-dimensional path formed by a unit vector end point of a cutter axis on the surface of the unit sphere in the processing process of the hybrid robot.
The sphere center of the unit sphere can be set as W; can set the i-1 th tool axis unit vector end point O i-1 Ith tool axis unit vector end point O i I+1th tool axis unit vector end point O i+1 The unit vector end points of the cutter axis are sequentially connected with three paths in the original cutter shaft direction; the modulus constant based on the unit vector of the cutter axis is 1, five including seven control points can be adoptedB spline curve is used as spline curve segment inserted when the path of cutter shaft direction is switched to smooth; set Q i (u) is at point O i Spline curve of arbor direction path inserted at: q (Q) i The expression of (u) may be as follows:
in phi, phi i (u) represents a continuous condition differentiated according to the path curvature in the arbor direction, at the point O i An initial spline curve constructed at the location; "|W phi i (u) || represents the point W to the curve Φ i (u) the distance of the point corresponding to the parameter u; d (D) k,i Representing spline curve Q i The control point of (u), wherein i represents the number of the original knife site, and k=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters;
step B2, the following conditions for continuous curvature differentiation at the connection points of the arbor direction path can be established:
wherein s is o Representing the accumulated arc length of the path in the cutter shaft direction;representing arc +.>Upper D 6,i A unit tangent vector at the position, wherein 'e' is a flag bit; d, d 6,i Representing the point D pointed at by the center W of the unit sphere 6,i Is a vector of (2);
step B3, establishing curve Q according to continuous curvature differentiation filling condition and derivative characteristic of B-spline curve i Functional relationship of each control point of (u) with the original tool axis unit vector end point; can assume curve Q i (u) the control point satisfies the following relationship: line segmentAnd line segment->The length ratio of (2) h is 1:h, and h is more than or equal to 2 and less than or equal to 8; the fairing curve Q can be determined according to continuous conditions of path curvature differentiation in the cutter shaft direction i Control point D of (u) 3,i The method comprises the steps of carrying out a first treatment on the surface of the Can be according to continuous requirement of cutter shaft direction path curvature and central angle D of transfer section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining curve Q i Control point D of (u) 2,i And D 4,i The method comprises the steps of carrying out a first treatment on the surface of the Can tangentially and continuously charge conditions according to the path of the cutter shaft direction and the central angle D of the switching section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining curve Q i Control point D of (u) 1,i And D 5,i ；
Can be provided with { R s,i The expression is in a circular arc segmentUpper point O i Frenet coordinate system built in place is provided with +.>Representing arc segment +.>At point O i The unit tangent vector at the position is provided with +.>Representing arc segment +.>At point O i The principal normal unit vector at the position is provided with +.>Representing arc segment +.>At point O i A secondary normal unit vector at the location;
can be provided with { R e,i The expression is in a circular arc segmentUpper point O i Frenet coordinate system built in place is provided with +.>Representing arc segment +.>At point O i The unit tangent vector at the position is provided with +.>Representing arc segment +.>At point O i The principal normal unit vector at the position is provided with +.>Representing arc segment +.>At point O i A secondary normal unit vector at the location;
can be provided with d 3,i ＝o i The method comprises the steps of carrying out a first treatment on the surface of the Then there are:
in the method, in the process of the invention,
o i a cutter axis unit vector corresponding to the ith cutter position point is represented;
the representation represents a circular arc +.>Upper D 0,i A unit tangent vector at the position, wherein's' is a flag bit;
represents angle D 0,i WD 3,i Wherein "s" represents a flag bit;
represents angle D 3,i WD 6,i Wherein "e" represents a flag bit;
d 0,i representing the direction from point W to point D 0,i Is a vector of (2);
d 1,i representing the direction from point W to point D 1,i Is a vector of (2);
d 2,i representing the direction from point W to point D 2,i Is a vector of (2);
d 3,i representing the direction from point W to point D 3,i Is a vector of (2);
d 4,i representing the direction from point W to point D 4,i Is a vector of (2);
d 5,i representing the direction from point W to point D 5,i Is a vector of (2);
d 6,i Representing the direction from point W to point D 6,i Is a vector of (2);
step B4, according to curve Q i Deducing an analytical expression of the fairing error and the fairing angle of the path in the cutter shaft direction according to the functional relation between each control point of (u) and the unit vector end point of the original cutter axis; expressing the path fairing error of the cutter shaft direction asAnd theta n,i Wherein θ n,i Representing unit vector +.>And->Is included in the plane of the first part; obtaining the fairing error along with the fairing angle of the cutter shaft direction path through drawing a universal variation trend chart of the fairing error of the cutter shaft direction path>Monotonically increasing; based on the limiting conditions of the path fairing error and parameter synchronization in the cutter shaft direction, adopting a numerical method to solve the fairing angle +.>And->
The Frenet coordinate system refers to: a coordinate system consisting of three mutually orthogonal unit vectors defined at a point on the spatial curve.
Preferably, step B4 may comprise the following sub-steps:
in step B4-1, the following limitation is considered to determine the fairing angleUpper limit value of (2):
1) In order to avoid intersection of path curves of adjacent tool tip positions, the fairing angle should not exceed half of the original central angle;
2) Determining the central angle D of the residual path segment according to the monotonically increasing arc length of the residual path segment along with the curve parameter 0, i WD 3,i
Sum +.D 3,i WD 6,i Lower limit value of (2);
Then there are:
in the method, in the process of the invention,representing the fairing angle +.>Upper limit value of (2); delta i Representing arc segment +.>Is a central angle of (2); delta i+1 Representing arc segment +.>Is a central angle of (2);Represents the central angle D of the remained path section 6,i-1 WD 0,i Lower limit value of (2);Represents the central angle D of the remained path section 6,i WD 0,i+1 Lower limit value of (2);
step B4-2, according to the fairing angleCalculating the fairing error of the cutter shaft direction path; judging whether the path fairing error in the cutter shaft direction meets the following conditions:
(1-ε t )ε ori,max ≤ε ori,i ≤ε ori,max ；
wherein ε t Control precision epsilon for expressing path fairing error of cutter shaft direction ori,i Indicating that the path of the cutter shaft direction is at the point O i A fairing error at the location; epsilon ori,max A predefined cutter shaft direction path fairing error limit value;
if the path fairing error in the cutter shaft direction meets the condition, the cutter shaft direction can be made to
If the path fairing error in the cutter shaft direction does not meet the condition, a dichotomy method can be adopted to solve the fairing angle; according to the dichotomy solving strategy, continuously updating the value of the fairing angle until the constraint condition is met;
and step B4-3, further completely determining a path fairing curve of the cutter shaft direction according to the calculated fairing angle.
Preferably, the motion performance of the cutter shaft of the hybrid robot can be improved by adjusting the fairing length of the position section and the fairing angle of the direction section, and the method for adjusting the fairing length of the position section and the fairing angle of the direction section can comprise the following steps:
Step D1, judging whether the fairing length of the point position path of the tool nose and the fairing angle of the path in the cutter shaft direction meet the following conditions according to the planned processing path of the hybrid robot:
if the condition is satisfied, the method can make
If the condition is not satisfied, the method can make
Step D2, judging whether the fairing length of the point position path of the tool nose and the fairing angle of the path in the cutter shaft direction meet the following conditions according to the planned processing path:
if the condition is satisfied, the method can make
If the condition is not satisfied, the method can make
Step D3, after finishing the adjustment of the fairing length of the point position path of the cutter point and the fairing angle of the path in the cutter axis direction, the fairing curve C of the point position path of the cutter can be recalculated according to the updated fairing length of the point position path of the cutter point and the fairing angle of the path in the cutter axis direction i (u) and arbor direction Path fairing curve Q i (u)；
And D4, forming a global C3 continuous series-parallel robot processing path after tool planning by the C3 continuous tool point position path and the C3 continuous cutter shaft direction path.
The workflow and principles of the present invention are further described in the following by a preferred embodiment of the present invention:
step 1, acquiring original pose information of a cutter from a cutter position file, wherein the original pose information comprises cutter point position information and cutter shaft direction information; determining an original processing path according to the original pose of the cutter; the position of the tool nose point is defined in a Cartesian coordinate system, two adjacent tool nose point position vectors are given, and a middle position vector is generated by adopting two modes of linear interpolation and circular interpolation respectively; the cutter shaft direction is defined on the surface of a unit sphere, unit vectors of the axes of two adjacent cutters are given, and a spherical linear interpolation method is adopted to generate a middle direction vector;
Step 2, generating an original linear cutter point position path in a linear interpolation mode, wherein the original linear cutter point position path only reaches position continuity or tangential continuity; the curvature differentiation continuity of the point position path of the cutter point is realized by inserting parameter curves such as a fifth-order B spline curve and the like at the corner of the point position path of the original cutter point, namely G3 continuity; the fairing error of the knife point position path is limited, so that the knife point position path after fairing meets the actual processing requirement; by constructing a functional relation among control points of the point position path fairing curve of the cutter point, the efficient solution of the point position path fairing curve of the cutter point is realized under the limiting conditions of the point position path fairing error of the cutter point, parameter synchronization and the like.
The functional relation among all control points of the cutter point position path curve is established by the continuous curvature filling condition and the derivative characteristic of the fifth-order B spline curve, and the method specifically comprises the following sub-steps:
step 2.1, setting the i-1 th knife point P i-1 The ith point P i I+1th tip point P i+1 Three sequentially connected cutter point points in the original cutter point position path; adopting a cubic B spline curve comprising seven control points as spline curve segments inserted in the process of switching the three nose points;
Wherein C is i (u) represents at point P i Spline curves inserted at the positions; b (B) j,i Representing spline curve C i The control point of (u), wherein i represents the number of the original knife point, j=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters; n (N) j,5 (u) represents a fifth-order B-spline basis function;
step 2.2, according to the i-1 th nose point P i-1 The ith point P i And the (i+1) th nose point P i+1 The filling condition of curvature differentiation continuity of the residual path segment and the spline curve segment at the connecting point is deduced. To ensure the continuity of the point position path of the tool nose, a control point B of the smoothing curve of the point position path of the tool nose 0,i Should be located in straight line segmentApplying; to ensure continuity of the point position path of the tool nose, a fairing curve control point B of the point position path of the tool nose 6,i Should be located in straight line segment +.>Further, the remaining path section of the tip position path +.>Can be controlled by control point B 6,i And control point B 0,i+1 Description is made in which B 0,i+1 Indicated at point P i+1 The point of the knife point is inserted into the control point of the path fairing curve. Further, the remaining path segment->Can be controlled by control point B 6,i-1 And control point B 0,i Description is made in which B 6,i-1 Indicated at point P i-1 The point of the knife point is inserted into the control point of the path fairing curve.
The following conditions are used for establishing continuous curvature differentiation of the point position path of the tool tip at the connecting point:
Wherein m is e,i Representing the point P i Pointing point P i+1 Is a unit direction vector of (a); s represents the curve cumulative arc length;
step 2.3, assume Path segmentThe length of (2) satisfies B 0,i B 1,i ||＝c 1 ||B 0,i B 3,i I, wherein c 1 E (0, 1) is a predetermined constant; assume Path section +.>The length of (2) satisfies B 1,i B 2,i ||＝c 2 ||B 0,i B 3,i I, wherein c 2 E (0, 1) is a predetermined constant; to avoid the curves crossing themselves, 0 < c should be ensured 1 +c 2 ＜1；
And establishing a functional relation among control points of the tool point position path fairing curve according to the continuous curvature filling conditions and the derivative characteristics of the fifth-order B-spline curve. Specifically, the fairing curve C can be determined based on the charge conditions for curvature differential continuity i Control Point B of (u) 3,i The method comprises the steps of carrying out a first treatment on the surface of the According to the continuous filling condition of curvature and the fairing length B 0,i B 3,i Sum of I I B 3,i B 6,i The fairing curve C can be determined i Control Point B of (u) 2,i And B 4,i The method comprises the steps of carrying out a first treatment on the surface of the According to tangential continuous filling conditions and fairing length B 0,i B 3,i Sum of I I B 3,i B 6,i The fairing curve C can be determined i Control Point B of (u) 1,i And B 5,i ；
Step 3, determining a fairing length B by considering a tool point position path fairing error and parameter synchronization limiting conditions based on a functional relation among control points of the tool point position path fairing curve 0,i B 3,i Sum of I I B 3,i B 6,i I. To ensure that the nose point position path fairing error is at the parameter u=0.5, it can be assumed that l B 0,i B 3,i ||＝||B 3,i B 6,i And deriving the tool tip point position path fairing error expression according to the Haussdorff distance. The tip position path fairing error can be expressed by the analysis of the fairing length. Further, the value of the fairing length can be determined from three aspects: (1) To avoid intersection of adjacent position fairing curves, the fairing length should not exceed half of the original path length; (2) The fairing error of the point position path of the tool nose is smaller than a predefined error limit value; (3) Remaining path segment Length B 6,i-1 B 0,i Sum of I I B 6,i B 0,i+1 The follow-up parameter synchronization requirement should be satisfied.
Based on the functional relation among the control points of the cutter point position path curve and based on the cutter point position path fairing error and parameter synchronization limiting conditions, the following fairing length is determined:
||B 0,i B 3,i ||＝||B 3,i B 6,i ||；
in the method, in the process of the invention,
ε pos,max a fairing error limit value representing a set point position path of the tool nose;
φ i representing a unit vector m s,i And m e,i Wherein m is s,i Representing the point P i Pointing point P i-1 Unit vector of m e,i Representing the point P i Pointing point P i+1 Is a unit vector of (2);
L i representing original path segmentsIs a length of (2);
L i+1 representing original path segmentsIs a length of (2);
representing the residual position path length +.>Wherein B is a lower limit value of 6,i-1 Indicated at point P i-1 The last control point of the spline curve inserted is located;
Representing the residual position path length +.>Wherein B is a lower limit value of 0,i+1 Indicated at point P i+1 A first control point of the spline curve inserted;
step 4, the original cutter shaft direction path consists of a series of concentric arcs on the surface of the unit sphere, and the original direction path generated by an arc interpolation mode only reaches position continuity or tangential continuity; the curvature differentiation continuity of the path in the cutter shaft direction is realized by carrying out switching smoothing on the path in the original cutter shaft direction on the surface of the unit ball; and analyzing the mathematical relationship between each control point of the constructed direction fairing curve and the unit vector end point of the original cutter axis by deducing the continuous filling condition of the curvature differentiation of the direction path.
Setting the sphere center of the unit sphere as W; setting the i-1 th tool axis unit vector end point O i-1 Ith tool axis unit vector end point O i I+1th tool axis unit vector end point O i+1 The unit vector end points of the cutter axis are sequentially connected with three paths in the original cutter shaft direction; analyzing and constructing the functional relation between each control point of the inserted spline curve of the cutter shaft direction path and the unit vector end point of the original cutter axis based on continuous requirement conditions of curvature differentiation of the cutter shaft direction path; the method specifically comprises the following sub-steps:
And 4.1, deducing the continuous filling condition of curvature differentiation of the directional path according to the i-1 th cutter axis unit vector end point, the i-1 th cutter axis unit vector end point and the i+1 th cutter axis unit vector end point. To ensure continuity of the cutter shaft direction path, a control point D of the cutter shaft direction path smoothing curve 0,i Should be located at the arc sectionApplying; to ensure continuity of the cutter shaft direction path, a control point D of the cutter shaft direction path smoothing curve 6,i Should be located in the arc section +.>Further, the path remaining path section in the arbor direction +.>Can be controlled by a control point D 6,i And control point D 0,i+1 Description is made in which D 0,i+1 Indicated at point O i+1 At the control point of the inserted fairing curve. Further, the path remaining path section in the arbor direction +.>Can be controlled by a control point D 6,i And control point D 0,i+1 Description is made in which D 0,i+1 Indicated at point O i+1 At the control point of the inserted fairing curve.
Based on arbor direction sheetThe modulus of the bit vector is 1, and a standardized quintic B spline curve Q comprising seven control points is adopted i (u) as a transitional fairing curve;
in phi, phi i (u) represents a continuous condition differentiated according to the path curvature in the arbor direction, at the point O i An initial spline curve constructed at the location; "|W phi i (u) || represents the point W to the curve Φ i (u) the distance of the point corresponding to the parameter u; d (D) k,i Representing spline curve Q i The control point of (u), wherein i represents the number of the original knife site, and k=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters;
step 4.2, establishing a continuous condition of curvature differentiation of the cutter shaft direction path at the connecting point:
wherein s is o Representing the accumulated arc length of the path in the cutter shaft direction;representing arc +.>Upper D 6,i A unit tangent vector at the position; d, d 6,i Represents the point D pointed by the center of the unit sphere 6,i Is a vector of (2);
step 4.2, establishing a curve Q according to the continuous charge condition of curvature differentiation and the derivative characteristic of the B-spline curve i Each control point of (u) as a function of the original tool axis unit vector end point; assumption curve Q i (u) the control point satisfies the following relationship: line segmentAnd line segment->The length ratio of (2) is 1:5; determining curve Q based on continuous curvature differential charge conditions i Control point D of (u) 3,i The method comprises the steps of carrying out a first treatment on the surface of the According to the continuous curvature filling condition and the central angle D of the transfer section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining curve Q i Control point D of (u) 2,i And D 4,i The method comprises the steps of carrying out a first treatment on the surface of the According to tangential continuous filling conditions and the central angle D of the transfer section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining a directional fairing curve Q i Control point D of (u) 1,i And D 5,i The method comprises the steps of carrying out a first treatment on the surface of the Let d 3,i ＝o i The method comprises the steps of carrying out a first treatment on the surface of the Then there are:
in the method, in the process of the invention,
o i representing a cutter shaft direction unit vector corresponding to the ith cutter position point;
The representation represents a circular arc +.>Upper D 0,i A unit tangent vector at the position, wherein's' is a flag bit;
represents angle D 0,i WD 3,i Wherein "s "represents a flag bit;
represents angle D 3,i WD 6,i Wherein "e" represents a flag bit;
d 0,i representing the direction from point W to point D 0,i Is a vector of (2);
d 1,i representing the direction from point W to point D 1,i Is a vector of (2);
d 2,i representing the direction from point W to point D 2,i Is a vector of (2);
d 3,i representing the direction from point W to point D 3,i Is a vector of (2);
d 4,i representing the direction from point W to point D 4,i Is a vector of (2);
d 5,i representing the direction from point W to point D 5,i Is a vector of (2);
d 6,i representing the direction from point W to point D 6,i Is a vector of (2);
{R s,i the expression is in a circular arc segmentUpper point O i A Frenet coordinate system built at the site, wherein +.>Representing a circular arc segmentAt point O i Unit tangent vector at>Representing arc segment +.>At point O i Principal normal vector at>Representing an arc of a circleSection->At point O i A secondary normal vector at;
{R e,i the expression is in a circular arc segmentUpper point O i A Frenet coordinate system built at the site, wherein +.>Representing a circular arc segmentAt point O i Unit tangent vector at>Representing arc segment +.>At point O i Principal normal vector at>Representing a circular arc segmentAt point O i A secondary normal vector at;
step 5, ensuring that the direction path after the fairing meets the processing requirement by limiting the fairing error of the direction path of the cutter shaft; according to the mathematical relationship between each control point of the cutter shaft direction path fairing curve and the direction vector end point, under the limiting conditions of fairing error, parameter synchronization and the like, the efficient solution of the direction fairing curve is realized;
Step 5.1, according to curve Q i Deriving an analytical expression of the path fairing error and the fairing angle of the cutter shaft direction according to the function of each control point and the unit vector end point of the original cutter axis; if the value of the parameter u corresponding to the maximum error of the path fairing in the cutter shaft direction is set to be u=0.5Further obtaining an analysis expression of the path fairing error of the cutter shaft direction; expressing the path fairing error of the cutter shaft direction as +.>And theta n,i Wherein θ n,i Representing unit vector +.>And->Is included in the plane of the first part; through drawing a general variation trend graph of the path fairing error in the cutter shaft direction, the fact that the path fairing error in the cutter shaft direction varies with the fairing angle is found out>Monotonically increasing; thereby realizing the solution of the fairing angle under the restriction of the fairing error;
step 5.2, rapidly solving the fairing angle by adopting a numerical method based on the limiting conditions of the path fairing error in the cutter shaft direction and parameter synchronizationAnd->
(1) Determining the angle of fairingAs the basis for the solution of the fairing angle under error constraints. To determine the upper value of the fairing angle, consider the following constraint: 1) In order to avoid intersection of path curves of adjacent tool tip positions, the fairing angle should not exceed half of the original central angle; 2) Central angle D of remained path section of cutter shaft direction path 0,i WD 3,i Sum +.D 3,i WD 6,i Should meet the backContinuous parameter synchronization requirements;
in the method, in the process of the invention,representing the fairing angle +.>Upper limit value of (2); delta i Representing arc segment +.>Is a central angle of (2); delta i+1 Representing arc segment +.>Is a central angle of (2);Represents the central angle D of the remaining direction section 6,i-1 WD 0,i Lower limit value of (2);Represents the central angle D of the remaining direction section 6,i WD 0,i+1 Lower limit value of (2);
(2) According to the upper limit value of the fairing angle, the fairing angle solving under the error limit is realized; order theCalculating the direction fairing error of the cutter shaft direction path; judging whether the path fairing error in the cutter shaft direction meets the following conditions:
(1-ε t )ε ori,max ≤ε ori,i ≤ε ori,max ；
wherein ε t Representing control accuracy of the path-fairing error in the arbor direction, e.g. taking epsilon t ＝0.1％；ε ori,i Indicating direction vector end point O i A directional fairing error at the location; epsilon ori,max Is a predefined directional fairing error limit.
If the direction fairing error meets the condition, then make
If the direction fairing error does not meet the condition, a dichotomy is adopted to solve the fairing angle. And continuously updating the value of the fairing length according to the dichotomy solving strategy until the constraint condition is met.
The directional fairing curve is further fully determined from the fairing angle.
And 6, further, in order to improve the cutter shaft movement performance of the hybrid robot, the fairing length of the cutter point position path and the fairing angle of the cutter shaft direction path are required to meet a specific mathematical relationship. To achieve a particular mathematical relationship, the position segment fairing length and the direction segment fairing angle need to be adjusted. The method specifically comprises the following steps:
(1) Judging whether the fairing length of the point position path of the tool nose and the fairing angle of the path in the cutter shaft direction meet the following conditions according to the planned processing path of the hybrid robot:
if the condition is satisfied, then make
If the condition is not satisfied, make
(2) Judging whether the fairing length of the cutter point position path and the fairing angle of the cutter shaft direction path meet the following conditions according to the planned processing path:
if the condition is satisfied, then make
If the condition is not satisfied, make
After finishing the adjustment of the fairing length of the point position path of the cutter point and the fairing angle of the path in the cutter axis direction, recalculating the fairing curve C of the point position path of the cutter according to the updated fairing length of the point position path of the cutter point and the updated fairing angle of the path in the cutter axis direction i (u) and arbor direction Path fairing curve Q i (u). And forming a global G3 continuous series-parallel robot processing path after tool planning by the G3 continuous tool point position path and the G3 continuous tool shaft direction path.
Step 7, for the inserted fairing curve, assuming that the point position path of the tool nose and the path in the cutter shaft direction share the same parameter; for the remaining path section, if the tool point position path and the tool axis direction path share the same parameter, the third derivative of the planned tool axis direction path with respect to time is discontinuous; in order to achieve continuous processing path C3 of the planned hybrid robot, namely continuous three-order derivative of the cutter shaft direction path with respect to time, a residual path section re-parameterization strategy is adopted. The residual path section re-parameterization strategy is to reconstruct the analytic expressions of the point position path residual path section of the cutter point and the path residual path section in the cutter shaft direction by introducing a parameter synchronization curve.
Seven-order B-spline with eight control points is selected as a parameter synchronization curve, and a node vector is set as (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1) T . Wherein, in order to ensure that the positions of the fairing curve section and the remained path section are continuous at the connecting point after parameter synchronization, the control points should satisfy the first control point=0 and the eighth control point=1.
And 7.1, alternately forming a path with the tool point position path and the tool axis direction path which are smooth by the inserted B-spline smooth curve and the residual path. Obtaining an h-1 section of path C from the point position path of the tool nose after fairing h-1 (u), h-th segment Path C h (u), h+1th segment Path C h+1 (u); cutter shaft after fairingThe h-1 segment of path Q is obtained from the direction path h-1 (u), h segment Path Q h (u), h+1th segment Path Q h+1 (u); according to the planned continuous requirement of the series-parallel robot processing path C3, the continuous charging condition of C3 at the connecting point can be further deduced as follows:
C′ h (u)| u＝1 ＝C′ h+1 (u)| u＝0 ，C″ h (u)| u＝1 ＝C″ h+1 (u)| u＝0 ，C″′ h (u)| u＝1 ＝C″′ h+1 (u)| u＝0
Q′ h (u)| u＝1 ＝Q′ h+1 (u)| u＝0 ，Q″ h (u)| u＝1 ＝Q″ h+1 (u)| u＝0 ，Q″′ h (u)| u＝1 ＝Q″′ h+1 (u)| u＝0
wherein u represents a curve parameter; h represents the path segment number in the point position path and the cutter shaft direction path after the smoothing; in order to realize the synchronization of the point position path of the tool nose and the path parameters in the direction of the tool shaft, the path sections are divided and numbered consistently; "'", "" and "'" denote the first, second and third derivatives, respectively, of the curve with respect to the parameter u.
And 7.2, determining each control point of the synchronous curve v (u) according to the continuous filling condition of C3 for the remaining path section of the point position path of the tool nose. Note that the solution process does not require the application of an iterative solution algorithm. According to continuous charging conditions of the angular speed of the cutter shaft, the value of the second control point and the value of the seventh control point can be determined; according to continuous charging conditions of the angular acceleration of the cutter shaft, the value of the third control point and the value of the sixth control point can be determined; according to the continuous condition of the tool axis angle, the fourth control point value and the fifth control point value can be determined, please refer to the following formula:
in the formula, { v 1 ,v 2 ,v 3 ,v 4 ,v 5 ,v 6 And the value from the second control point to the seventh control point of the parameter synchronization curve v (u). v 1 A second control point representing a parameter synchronization curve v (u); v 2 A third control point value representing a parameter synchronization curve v (u); and so on.
Further, to ensure that the arc length of the re-parametric curve monotonically increases with the curve parameter, the first derivative of the synchronization curve v (u) with respect to the curve parameter u should be constantly greater than 0. Thereby deducing the length of the remaining path section of the point position path of the tool noseThe lower limit of (2) is as follows:
in the method, in the process of the invention,indicating the length of the remaining path section of the point position path of the tool nose >The lower limit value of the index (R) is a marker bit; l (L) i+1 Representing the original path segment +.>Is a length of (2);
according to the length of the remaining path section of the point position path of the tool noseThe lower limit value of (2) and (3) are combined to further completely determine the point P i The point of the knife point is inserted into the position of the knife point to form a smooth curve.
And 7.3, determining each control point of the synchronization curve w (u) according to the continuous filling conditions of C3 for the path remaining path segments in the cutter shaft direction. Note that the solution process does not require the application of an iterative solution algorithm. According to continuous charging conditions of the angular speed of the cutter shaft, the value of the second control point and the value of the seventh control point can be determined; according to continuous charging conditions of the angular acceleration of the cutter shaft, the value of the third control point and the value of the sixth control point can be determined; according to the continuous condition of the tool axis angle, the fourth control point value and the fifth control point value can be determined, please refer to the following formula:
represents angle D 3,i WD 6,i ；
Represents angle D 0,i+1 WO i+1 Wherein D is 0,i+1 Indicated at point O i+1 A first control point of the inserted directional fairing curve;
represents angle D 6,i WD 0,i+1 ；
{w 1 ,w 2 ,w 3 ,w 4 ,w 5 ,w 6 And the value from the second control point to the seventh control point of the parameter synchronization curve w (u). w (w) 1 A second control point representing a parameter synchronization curve w (u) takes a value; w (w) 2 A third control point representing a parameter synchronization curve w (u) takes a value; and so on.
To ensure that the arc length of the re-parametric curve increases monotonically with the curve parameters, the first derivative of the synchronous curve with respect to the curve w (u) parameters should be constantly greater than 0. And deducing the central angle D of the path in the remaining direction 6,i WD 0,i+1 Lower limit value of (2);
in the method, in the process of the invention,central angle D of residual path section of path in cutter shaft direction 6,i WD 0,i+1 The lower limit value of the index (R) is a marker bit; delta i+1 Representing arc segment +.>Is a central angle of (2);
central angle D of path section remained according to cutter shaft direction path 6,i WD 0,i+1 Further, the lower limit value of (2) in combination with step 4 and step 5 can completely determine the point O i The path of the cutter shaft inserted in the position is smooth.
And 8, planning a continuous cutter point speed curve of the jump degree by correcting the jump degree curve in the S acceleration and deceleration movement rule according to the planned processing path of the C3 continuous series-parallel robot. And carrying out parameter interpolation according to the cutter point speed curve and combining with a pre-estimation-correction method to generate an interpolation point sequence required in the robot processing process.
Step 9, determining joint variables of each driving joint of the robot at each moment based on a hybrid robot kinematic model according to the interpolation point sequence; controlling the motion of the series-parallel robot according to the joint variable of each driving joint of the robot at each moment; and based on the PMAC motion control card, the following errors of all joints in the motion process of the robot are collected in real time, and the effectiveness of the fairing of the point position path of the tool nose and the path in the direction of the tool shaft is verified.
The above embodiments of the tip position path are only for illustrating the technical idea and features of the present invention, and are intended to enable those skilled in the art to understand the content of the present invention and implement it accordingly, and the scope of the present invention is not limited to the embodiments, i.e. equivalent changes or modifications made by the spirit of the present invention fall within the scope of the present invention.
Claims (9)
1. A method for switching and smoothing a C3 continuous five-axis path of a hybrid robot is characterized by comprising the following steps: setting the end point track of the unit vector of the cutter axis as a cutter shaft direction path; setting the point position track of the tool nose as the point position path of the tool nose; the point position path of the tool nose is defined in a Cartesian coordinate system; the cutter shaft direction path is defined on the surface of the unit sphere; the tool point position path and the cutter shaft direction path are switched and smoothed by adopting spline curve segments, the paths after smoothing both are composed of inserted spline curve segments and residual path segments, wherein the inserted spline curve segments comprise odd control points, and the maximum deviation between the middle control point of the inserted spline curve and the original path is assumed; respectively deriving the continuous filling conditions of curvature differentiation of the point position path of the tool nose and the cutter shaft direction path at the transfer point, and further respectively establishing the relation functions of the fairing length of the point position path of the tool nose, the fairing angle of the cutter shaft direction path and the respective fairing errors; based on the technological requirements, determining the fairing error limit value corresponding to the point position path of the tool nose and the path in the direction of the tool shaft, and determining the fairing length of the point position path of the tool nose and the fairing angle of the path in the direction of the tool shaft by combining constraint conditions synchronously introduced by parameters, so as to determine the fairing curves of the point position path of the tool nose and the path in the direction of the tool shaft.
2. The method for switching and smoothing a C3 continuous five-axis path of a hybrid robot according to claim 1, further comprising the steps of: introducing a parameter synchronization curve, and reconstructing a point position path of the tool nose and a path analysis expression in the direction of the tool shaft; seven-order B-spline with eight control points is selected as a parameter synchronization curve, and a node vector is set as (0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1) T The method comprises the steps of carrying out a first treatment on the surface of the Wherein, the first control point=0 and the eighth control point=1 of the parameter synchronization curve; obtaining an h-1 section of path C from the point position path of the tool nose after fairing h-1 (u), h-th segment Path C h (u), h+1th segment Path C h+1 (u); obtaining an h-1 segment path Q from the cutter shaft direction path after fairing h-1 (u), h segment Path Q h (u), h+1th segment Path Q h+1 (u); the continuous filling condition of C3 at the junction of the two paths is as follows:
C′ h (u)| u＝1 ＝C′ h+1 (u)| u＝0 ；C″ h (u)| u＝1 ＝C″ h+1 (u)| u＝0 ；C″′ h (u)| u＝1 ＝C″′ h+1 (u)| u＝0 ；
Q′ h (u)| u＝1 ＝Q′ h+1 (u)| u＝0 ；Q″ h (u)| u＝1 ＝Q″ h+1 (u)| u＝0 ；Q″′ h (u)| u＝1 ＝Q″′ h+1 (u)| u＝0 ；
wherein u represents a B spline curve parameter; h represents the path segment number of the point position path and the cutter shaft direction path after the smoothing; "'", "" and ""' "denote the first, second and third derivatives of the curve with respect to the parameter u, respectively; determining values of a second control point and a seventh control point of the parameter synchronization curve according to continuous filling conditions of the angular speed of the cutter shaft; determining a third control point and a sixth control point value according to continuous charging conditions of the angular acceleration of the cutter shaft; and determining the values of the fourth control point and the fifth control point according to the continuous filling conditions of the angle of the cutter shaft.
3. The method for switching and smoothing a C3 continuous five-axis path of a hybrid robot according to claim 1, further comprising the steps of: after finishing the smoothing of the point position path and the cutter shaft direction path, generating a continuous point speed curve of the cutter point by correcting the jump curve in the S acceleration and deceleration movement; and performing parameter interpolation according to the cutter point speed curve and combining with a pre-estimation-correction method to generate an interpolation point sequence required in the robot processing process.
4. The method for switching and smoothing a C3 continuous five-axis path of a hybrid robot according to claim 3, further comprising the steps of: according to the interpolation point sequence, based on a hybrid robot kinematic model, determining joint variables of each driving joint of the robot at each moment; controlling the motion of the series-parallel robot according to the joint variable of each driving joint of the robot at each moment; based on the PMAC motion control card, the following errors of all joints in the motion process of the robot are collected in real time, and the fairing effectiveness of the point position path of the tool nose and the path in the direction of the tool shaft is evaluated.
5. The method for series-parallel robot C3 continuous five-axis path switching fairing of claim 1, wherein the inserted spline curve segment is a cubic B-spline.
6. The method for switching and fairing a C3 continuous five-axis path of a hybrid robot according to claim 1, wherein the specific step of establishing a function of the relationship between the fairing length of the path at the point of the cutter and the fairing error thereof comprises:
step A1, setting the (i-1) th knife point P i-1 The ith point P i I+1th tip point P i+1 Three sequentially connected cutter point points in the original cutter point position path; adopting a cubic B spline curve with seven control points as spline curve segments inserted in the process of switching the three nose points; the inserted spline curve segment expression is as follows:
wherein C is i (u) represents at point P i Spline curves inserted at the positions; b (B) j,i Representing spline curve C i The control point of (u), wherein i represents the number of the original knife point, j=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters; n (N) j,5 (u) represents a fifth-order B-spline basis function;
step A2, establishing the following conditions for continuous curvature differentiation of the point position path of the tool nose at the connecting point:
wherein m is e,i Representing the point P i Pointing point P i+1 Wherein "e" is a flag bit; s represents the accumulated arc length of the point position path of the tool nose;
step A3, assume a path segmentThe length of (2) satisfies B 0,i B 1,i ||＝c 1 ||B 0,i B 3,i I, wherein c 1 E (0, 1) is a predetermined constant; assume Path section +.>The length of (2) satisfies B 1,i B 2,i ||＝c 2 ||B 0,i B 3,i I, wherein c 2 E (0, 1) is a predetermined constant; to avoid the curves crossing themselves, 0 < c should be ensured 1 +c 2 < 1; from the above assumption, a position spline curve C is established i (u) each control point and the cutting point P i Is a function of (a);
step A4, determining the lower limit value of the length of the residual path section according to monotonically increasing arc length of the residual path section along with curve parameters; based on the functional relation among the control points of the cutter point position path curve, the relation function of the fairing length of the cutter point position path and the fairing error is established as follows:
||B 0,i B 3,i ||＝||B 3,i B 6,i ||；
in the method, in the process of the invention,
ε pos,max a fairing error limit value representing a set point position path of the tool nose;
φ i representing a unit vector m s,i And m e,i Wherein m is s,i Representing the point P i Pointing point P i-1 Unit vector of m e,i Representing the point P i Pointing point P i+1 Is a unit vector of (2);
L i representing original path segmentsIs a length of (2);
L i+1 representing original path segmentsIs a length of (2);
representing the residual position path length +.>Wherein B is a lower limit value of 6,i-1 Indicated at point P i-1 The last control point of the spline curve inserted is located;
representing the residual position path length +. >Wherein B is a lower limit value of 0,i+1 Indicated at point P i+1 At the first control point of the interpolated spline.
7. The method for transferring and smoothing the continuous five-axis path of the hybrid robot C3 according to claim 6, wherein the specific step of establishing a relation function of the smoothing angle of the path in the arbor direction and the smoothing error thereof comprises the following steps:
step B1, an original cutter shaft direction path consists of a series of concentric arcs on the surface of a unit sphere, and the sphere center of the unit sphere is set as W; setting the i-1 th tool axis unit vector end point O i-1 Ith tool axis unit vector end point O i I+1th tool axis unit vector end point O i+1 The unit vector end points of the cutter axis are sequentially connected with three paths in the original cutter shaft direction; the modulus based on the unit vector of the cutter axis is constant at 1, and seven units are adoptedThe quintic B spline curve of the control point is used as a spline curve segment inserted when the path of the cutter shaft direction is switched to be smooth; set Q i (u) is at point O i Spline curve of arbor direction path inserted at: q (Q) i The expression of (u) is as follows:
in phi, phi i (u) represents a continuous condition differentiated according to the path curvature in the arbor direction, at the point O i An initial spline curve constructed at the location; "|W phi i (u) || represents the point W to the curve Φ i (u) the distance of the point corresponding to the parameter u; d (D) k,i Representing spline curve Q i The control point of (u), wherein i represents the number of the original knife site, and k=0, 1, …,6 represents the number of the control point; u E [0,1 ]]Representing spline curve parameters;
step B2, establishing a continuous charging condition of curvature differentiation of the cutter shaft direction path at the connecting point:
wherein s is o Representing the accumulated arc length of the path in the cutter shaft direction;representing arc +.>Upper D 6,i A unit tangent vector at the position, wherein 'e' is a flag bit; d, d 6,i Representing the point D pointed at by the center W of the unit sphere 6,i Is a vector of (2);
step B3, establishing a curve Q according to the continuous charge condition of curvature differentiation and the derivative characteristic of the B-spline curve i Functional relationship of each control point of (u) with the original tool axis unit vector end point; assumption curve Q i (u) the control point satisfies the following relationship: line segmentAnd line segment->The length ratio of (2) h is 1:h, and h is more than or equal to 2 and less than or equal to 8; determining a fairing curve Q according to continuous charge conditions of path curvature differentiation in the cutter shaft direction i Control point D of (u) 3,i The method comprises the steps of carrying out a first treatment on the surface of the According to continuous charging conditions of path curvature in the cutter shaft direction and the central angle D of the switching section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining curve Q i Control point D of (u) 2,i And D 4,i The method comprises the steps of carrying out a first treatment on the surface of the According to tangential continuous filling conditions of cutter shaft direction paths and central angle D of the transfer section 0,i WD 3,i Sum +.D 3,i WD 6,i Determining curve Q i Control point D of (u) 1,i And D 5,i ；
Let { R s,i The expression is in a circular arc segmentUpper point O i Frenet coordinate system built in place is provided with +.>Representing arc segment +.>At point O i The unit tangent vector at the position is provided with +.>Representing arc segment +.>At point O i The principal normal unit vector at the position is provided with +.>Representing arc segment +.>At point O i A secondary normal unit vector at the location;
let { R e,i The expression is in a circular arc segmentUpper point O i Frenet coordinate system built in place is provided with +.>Representing arc segment +.>At point O i The unit tangent vector at the position is provided with +.>Representing arc segment +.>At point O i The principal normal unit vector at the position is provided with +.>Representing arc segment +.>At point O i A secondary normal unit vector at the location;
let d 3,i ＝o i The method comprises the steps of carrying out a first treatment on the surface of the Then there are:
in the method, in the process of the invention,
o i a cutter axis unit vector corresponding to the ith cutter position point is represented;
the representation represents a circular arc +.>Upper D 0,i A unit tangent vector at the position, wherein's' is a flag bit;
represents angle D 0,i WD 3,i Wherein "s" represents a flag bit;
represents angle D 3,i WD 6,i Wherein "e" represents a flag bit;
d 0,i representing the direction from point W to point D 0,i Is a vector of (2);
d 1,i representing the direction from point W to point D 1,i Is a vector of (2);
d 2,i representing the direction from point W to point D 2,i Is a vector of (2);
d 3,i representing the direction from point W to point D 3,i Is a vector of (2);
d 4,i representing the direction from point W to point D 4,i Is a vector of (2);
d 5,i representing the direction from point W to point D 5,i Is a vector of (2);
d 6,i Representing the direction from point W to point D 6,i Vector of (2)An amount of;
step B4, according to curve Q i Deducing an analytical expression of the fairing error and the fairing angle of the path in the cutter shaft direction according to the functional relation between each control point of (u) and the unit vector end point of the original cutter axis; expressing the path fairing error of the cutter shaft direction asAnd theta n,i Wherein θ n,i Representing unit vector +.>And->Is included in the plane of the first part; obtaining the fairing error along with the fairing angle of the cutter shaft direction path through drawing a universal variation trend chart of the fairing error of the cutter shaft direction path>Monotonically increasing; based on the limiting conditions of the path fairing error and parameter synchronization in the cutter shaft direction, adopting a numerical method to solve the fairing angle +.>And->
8. The method for switching and smoothing the continuous five-axis path of the hybrid robot C3 according to claim 7, wherein the step B4 comprises the following sub-steps:
step B4-1, determining the fairing angle in consideration of the following limitation conditionsUpper limit value of (2):
1) In order to avoid intersection of path curves of adjacent tool tip positions, the fairing angle should not exceed half of the original central angle;
2) Determining the central angle D of the residual path segment according to the monotonically increasing arc length of the residual path segment along with the curve parameter 0,i WD 3,i Sum +.D 3,i WD 6,i Lower limit value of (2);
then there are:
in the method, in the process of the invention, Representing the fairing angle +.>Upper limit value of (2); delta i Representing arc segment +.>Is a central angle of (2); delta i+1 Representing arc segment +.>Is a central angle of (2);Represents the central angle D of the remained path section 6,i-1 WD 0,i Lower limit value of (2);Represents the central angle D of the remained path section 6,i WD 0,i+1 Lower limit value of (2);
step B4-2, according to the fairing angleCalculating the fairing error of the cutter shaft direction path; judging whether the path fairing error in the cutter shaft direction meets the requirementThe following conditions were:
(1-ε t )ε ori,max ≤ε ori,i ≤ε ori,max ；
wherein ε t Control precision epsilon for expressing path fairing error of cutter shaft direction ori,i Indicating that the path of the cutter shaft direction is at the point O i A fairing error at the location; epsilon ori,max A predefined cutter shaft direction path fairing error limit value;
if the path fairing error in the cutter shaft direction meets the condition, then the cutter shaft direction path fairing error is made to be
If the path fairing error in the cutter shaft direction does not meet the condition, adopting a dichotomy to solve the fairing angle; according to the dichotomy solving strategy, continuously updating the value of the fairing angle until the constraint condition is met;
and step B4-3, further completely determining a path fairing curve of the cutter shaft direction according to the calculated fairing angle.
9. The method for switching and smoothing the continuous five-axis path of the hybrid robot C3 according to claim 7, wherein the method for improving the cutter shaft movement performance of the hybrid robot by adjusting the smoothing length of the cutter point position path and the smoothing angle of the cutter shaft direction path comprises the following steps:
Step D1, judging whether the fairing length of the point position path of the tool nose and the fairing angle of the path in the cutter shaft direction meet the following conditions according to the planned processing path of the hybrid robot:
if the condition is satisfied, then make
If the condition is not satisfied, make
Step D2, judging whether the fairing length of the point position path of the tool nose and the fairing angle of the path in the cutter shaft direction meet the following conditions according to the planned processing path:
if the condition is satisfied, then make
If the condition is not satisfied, make
Step D3, after finishing the adjustment of the fairing length of the point position path of the cutter point and the fairing angle of the path in the cutter axis direction, recalculating a fairing curve C of the point position path of the cutter according to the updated fairing length of the point position path of the cutter point and the updated fairing angle of the path in the cutter axis direction i (u) and arbor direction Path fairing curve Q i (u)；
And D4, forming a global C3 continuous series-parallel robot processing path after tool planning by the C3 continuous tool point position path and the C3 continuous cutter shaft direction path.
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