CN109676613B - Error-controllable arc transition type smooth track generation method for four-axis industrial robot - Google Patents

Error-controllable arc transition type smooth track generation method for four-axis industrial robot Download PDF

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CN109676613B
CN109676613B CN201910150505.9A CN201910150505A CN109676613B CN 109676613 B CN109676613 B CN 109676613B CN 201910150505 A CN201910150505 A CN 201910150505A CN 109676613 B CN109676613 B CN 109676613B
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error
smooth
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CN109676613A (en
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何姗姗
颜昌亚
李振瀚
邓炎超
黄昆涛
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Wuhan Institute of Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

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Abstract

An error-controllable arc transition type smooth track generation method for a four-axis industrial robot comprises the following steps: step 1, track preprocessing of a four-axis industrial robot: traversing all track points, dividing the linear track into a track section needing smooth control and a track section not needing smooth control according to the position distance and the included angle, and preprocessing the four-axis posture to ensure that a minor arc track is formed between the two track points; step 2, four-axis track is smooth: traversing the sections of the track needing to be smoothed, which are generated in the step 1, and generating the arc transition type smooth track for each section of the track needing to be smoothed by adopting a geometric iteration method according to the track point error threshold, the position point chord height error threshold and the continuity requirement. The arc transition type smooth track generation method has G1 continuity with synchronous position and posture, meets track point errors and position chord height errors between track points, can be directly used under the condition that the existing instruction format of the four-axis industrial robot is not changed, and improves the operation efficiency and quality of the four-axis industrial robot.

Description

Error-controllable arc transition type smooth track generation method for four-axis industrial robot
Technical Field
The invention belongs to the field of track optimization of industrial robots, and particularly relates to an error-controllable arc transition type smooth track generation method for a four-axis industrial robot.
Background
Four-axis industrial robot, also known as a plane joint (SCARA) robot, has three rotation axes and a translation axis, is widely used in operations such as transport, assembly and gluing.
The existing motion instruction of the four-axis industrial robot comprises a linear instruction, an arc instruction and an axis joint motion instruction. Most complex trajectories consist of a linear motion command and a small number of circular motion commands. In order to accurately reach the track point, the speed of the robot must be reduced to zero in the operation process of the robot, the speed of the robot is accelerated from zero when the next instruction is executed, the operation efficiency is reduced due to frequent acceleration and deceleration of the robot in the whole operation process, and meanwhile, the operation quality is reduced due to the vibration of the robot.
The smooth transition instruction can enable the TCP point of the robot to smoothly and quickly approach the target point, but the TCP point cannot pass through the target point. Although the smooth instruction improves the continuity of the track, the precision is lost, and the operation quality is affected. In other transition instructions, an interpolation curve is used to ensure that the TCP point of the robot is smooth and passes through the track point (such as a spline instruction of KUKA), but the error between the two track points cannot be controlled, so that the operation quality is influenced.
The accepted patent application 201710097192.6 proposes a smooth motion trajectory generation method for an industrial robot capable of controlling both position point error and chord height error, but the transition curve only provides cubic B-spline and quartic B-spline, and the posture of the double track is not necessarily continuous at the connecting point of the straight line segment and the sample bar segment.
The accepted patent application document 201811468150.X provides an error-controllable three-dimensional track point track smoothing method, and the method can generate continuous and conformal three-dimensional tracks and smooth tracks meeting precision. However, when the four-axis industrial robot is popularized, the smooth posture of the track point needs to be considered, and the synchronous smoothness of the position and the posture needs to be considered, so that the four-axis smooth track with synchronous and smooth posture is obtained. In the track (position and attitude, short for pose) expression of the existing four-axis industrial robot, smooth track expression which simultaneously meets high continuity (pose synchronous continuity) and high precision (meets track point errors and position chord height errors between track points) is not provided.
Disclosure of Invention
The invention aims to solve the technical problem that aiming at the defects existing in the track expression of the existing four-axis industrial robot, the invention provides the error-controllable arc transition type smooth track generation method of the four-axis industrial robot, the generation method is simple in calculation, has the G1 continuity (synchronous and smooth position and posture) with synchronous pose, can meet the track point error and the position chord height error between the track points, and the generated track can be directly used without changing the existing instruction format of the four-axis industrial robot.
The technical scheme adopted by the invention for solving the technical problems is as follows:
an error-controllable arc transition type smooth track generation method for a four-axis industrial robot comprises the following steps:
step 1, track preprocessing of a four-axis industrial robot: dividing the linear track into a track section needing to be smoothed and a track section not needing to be smoothed according to the position distance and the included angle;
step 2, smoothing the track section needing to be smoothed: traversing the sections of the track needing to be smoothed, which are generated in the step 1, and generating the arc transition type smooth track for each section of the track needing to be smoothed by adopting a geometric iteration method according to the track point error threshold, the position point chord height error threshold and the continuity requirement.
According to the scheme, the step 1 specifically comprises the following steps:
step 1.1, calculating a segmentation index set according to a segmentation threshold, inputting track segments above two continuous track points, and outputting a segmentation index; set the trajectory points of the four-axis industrial robot as input
Figure BDA0001981387100000021
The number N of the track points is more than or equal to 2, wherein each track point Pi(xi，yi，zi，θi) Is a position (x)i，yi，zi) And attitude, i.e. angle of rotation about the Z axisiComposed four-dimensional vector, position segmentation condition by position distance threshold deltadAnd angle of position threshold deltaaAs an index;
traversing the track point index i as 1,2,. N-1, and respectively judging whether the index i meets the position segmentation condition:
firstly, calculating a track segment Pi-1PiAnd PiPi+1The distance d between the two sectionsi-1And diIf d isi-1Or diLess than a position distance threshold deltadIf so, the index i is considered to meet the position segmentation condition; otherwise, calculating the track segment Pi-1PiAnd PiPi+1At a position angle ofiIf sin aiLess than a position angle threshold deltaaIf so, the index i is considered to meet the position segmentation condition; otherwise, the index i is considered not to meet the position segmentation condition; adding indexes i meeting the position segmentation condition into a segmentation index set, and adding indexes 0 and N from beginning to end into the beginning and the end of the segmentation index set by default;
step 1.2, segmenting according to the segmented index set, and dividing a whole track segment into a plurality of track segments according to the segmented index set, wherein the track segment with the track point number larger than 2 in the track segments is marked as a track segment needing to be smoothed and is used for smoothing the track in the next step; otherwise, marking as a section without a smooth track, and outputting the section to the smooth track according to the linear track;
step 1.3, posture preprocessing, traversing track point indexes i which are 1,2,.. N according to the principle that minor arcs between two tracks are prior, and if two adjacent track points P arei-1，PiIs included angle distance oi-1Greater than 180 DEG, oi-1＝|θi-θi-1If P is modifiediAttitude angle of the fourth dimension of (1): if thetai> 0, modified to Pi(xi，yi，zi，θi-360 °); otherwise, modifying to Pi(xi，yi，zi，θi+360°)。
According to the scheme, the method for generating the arc transition type smooth track for each section of the track needing to be smoothed in the step 2 specifically comprises the following steps:
step 2.1, setting initial iteration parameters, and setting four-dimensional track points of the current linear track segment as
Figure BDA0001981387100000022
Marking as an original track point, and setting a position distance error threshold value which needs to be met by the smooth track as epsilonmaxPosition chord height error threshold is recorded as deltamaxError threshold o of included angle of attitude pointmaxThreshold value k of iteration numbermaxSetting the current iteration number as k equal to 0, and recording the iteration track point as
Figure BDA0001981387100000023
Step 2.2, traversing the index i as 1, 2.. N-1, and respectively generating iterative track points according to position point chord height error constraint, G1 continuity constraint and shape-preserving constraint
Figure BDA0001981387100000031
Transition circular arc track of
Figure BDA0001981387100000032
First separately calculate
Figure BDA0001981387100000033
And
Figure BDA0001981387100000034
is located a distance di-1And diAnd angle of position point betaiAnd calculate di-1And diSmaller value d ofmmin＝min(di-1，di)；
And then calculating according to the position point chord height error constraint and the conformal constraint
Figure BDA0001981387100000035
Transition ratio r of front and rear position pointsi-1And riFirstly, calculating the transition length of the front and rear position points as follows:
Figure BDA0001981387100000036
two transition ratios are then calculated:
Figure BDA0001981387100000037
wherein alpha is more than 0 and less than 1, which is a shape-preserving parameter and represents the distance proportion of a linear track section between two transition arcs in the whole track section;
finally, the starting point of the four-dimensional arc is calculated according to the G1 continuous condition and the two transition ratios
Figure BDA0001981387100000038
Terminal point
Figure BDA0001981387100000039
And the center of the circle
Figure BDA00019813871000000310
Figure BDA00019813871000000311
Step 2.3, traversing the index i as 1,2
Figure BDA00019813871000000312
With the original track point QiTrack point error of (2):
firstly, a four-dimensional arc is constructed according to the starting point, the end point and the circle center in the step 2.2
Figure BDA00019813871000000313
The first three dimensions are circular arc curves of European space, and the fourth dimension is a curve generated by the rotation angle along with the synchronous change of the circular arcs;
Figure BDA00019813871000000314
wherein t is ∈ [0,1 ]]，αi＝180°-βi，
Figure BDA00019813871000000315
Then calculating the parameter midpoint of the four-dimensional arc track as the maximum error point of the track point:
Figure BDA00019813871000000316
finally calculating a transition arc
Figure BDA00019813871000000317
With the original track point QiThe track point error comprises a position distance error and an attitude point included angle errorDifference, respectively calculating
Figure BDA00019813871000000318
The three-dimensional position distance and the attitude included angle distance are used as a position distance error and an attitude point included angle error;
step 2.4, calculating the maximum position distance errors of all transition arcs and original track points
Figure BDA00019813871000000319
Error of included angle with maximum attitude point
Figure BDA00019813871000000320
If it is
Figure BDA00019813871000000321
Less than a position distance error threshold epsilonmaxAnd is and
Figure BDA00019813871000000322
less than the error threshold o of the included angle of the attitude pointmaxOr the current iteration number k is larger than the iteration number threshold kmaxTerminating iteration and outputting arc smooth track
Figure BDA00019813871000000323
Turning to step 2.6; otherwise, turning to step 2.5;
step 2.5, traversing i & lt1 & gtiMaximum point of sum track point error
Figure BDA00019813871000000324
Calculating an offset vector
Figure BDA0001981387100000041
And updating iteration track points:
Figure BDA0001981387100000042
changing k to k +1, and turning to step 2.2;
step 2.6, the smooth tracks output in the step 2.4 are sorted and output, and the smooth tracks are formed by N four-dimensional linear tracksThe track and (N-1) four-dimensional arc smooth tracks are combined, and sequentially comprise: linear trajectory
Figure BDA0001981387100000043
Smooth circular arc track
Figure BDA0001981387100000044
Linear trajectory
Figure BDA0001981387100000045
Smooth circular arc track
Figure BDA0001981387100000046
Smooth circular arc track
Figure BDA0001981387100000047
Linear trajectory
Figure BDA0001981387100000048
Through the technical scheme, compared with the prior art, the invention has the advantages that:
1. the arc transition type smooth track of the four-axis industrial robot with high continuity, shape retention and high precision can be generated, and the generated track can be directly executed under the condition that the existing track instruction structure of the robot is not changed;
2. compared with the existing linear track of the four-axis industrial robot, the arc transition type smooth track generated by the method has higher continuity, so that the working efficiency of the four-axis industrial robot is improved, and the machine abrasion is reduced;
3. compared with the existing smooth track of the four-axis industrial robot, the track generated by the invention has higher execution precision besides continuity, can reach the position and the posture of a preset track, ensures the requirement of the position precision among track points, and further improves the operation precision;
4. compared with the problem of discontinuous postures of the four-axis industrial robot track, the continuous track of the four-axis industrial robot generated by the method can simultaneously meet the continuity of the position and the posture, and the position and the posture are synchronous and continuous, so that sufficient conditions are provided for finally generating the execution track of the four-axis industrial robot with high precision and high efficiency.
Drawings
FIG. 1 is a flow chart of arc transition type smooth track generation of a four-axis industrial robot in the embodiment of the invention;
FIG. 2 is a flow chart of track preprocessing of a four-axis industrial robot according to an embodiment of the invention;
FIG. 3 is a schematic diagram illustrating attitude angle preprocessing according to an embodiment of the present invention;
FIG. 4 is a flow chart of track smoothing according to an embodiment of the present invention;
FIG. 5 is a schematic view of a circular arc transition according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The invention discloses an error-controllable arc transition type smooth track generation method for a four-axis industrial robot, which is mainly characterized in that a main flow chart is shown as an attached figure 1 and mainly comprises two main steps of preprocessing and track smoothing:
step 1, track preprocessing of a four-axis industrial robot: dividing the linear track into a track section needing to be smoothed and a track section not needing to be smoothed according to the position distance and the included angle; traversing all track points of the whole track, segmenting according to the position distance and the included angle of the track, and dividing the whole track into a plurality of track segment sets; and preprocessing the four-axis postures to ensure that a minor arc track is formed between two track points, wherein the flow chart of the preprocessing step is shown in the attached figure 2 and mainly comprises two main steps of calculating a segmentation index and segmenting according to a segmentation index set:
step 1.1, calculating a segmentation index set according to a segmentation threshold value, inputting track segments above two continuous track points, outputting the segmentation index, and setting the input track point set of the four-axis industrial robot as
Figure BDA0001981387100000051
(the number N of the track points is more than or equal to 2), wherein two track points Pi-1，PiThe position distance between the two is calculated by
Figure BDA0001981387100000052
Setting three adjacent track points Pi-1，Pi，Pi+1The position points of the former three-dimensional composition of (1) are respectively pi-1，pi，pi+1. If d isi-1diNot equal to 0, the included angle of the three position points is
Figure BDA0001981387100000053
Position segmentation condition by position distance threshold deltadAnd adjacent position angle threshold deltaaAs an index, according to the fact that the position segmentation is consistent with a three-dimensional point segmentation method of an accepted patent application (201811468150.X), namely, a track point index i is traversed to be 1,2,.. N-1, and whether the index i meets the position segmentation condition is judged respectively; if d isi-1＜δdOr di＜δdIf the index i meets the position segmentation condition, otherwise, if sin ai＜δaIf so, the index i is considered to meet the position segmentation condition; adding an index i meeting the position segmentation condition into a segmentation index set, and adding a head index 0 and a tail index N-1 into the head and the tail of the segmentation index set by default;
step 1.2, segmenting according to the segmented index set, wherein the segmented index set obtained in the step 1.1 is set to be {0, 1, 14,16} in the embodiment, the track segment is segmented
Figure BDA0001981387100000054
Is divided into three sections: { P0～P1}，{P1～P14}，{P14～P16Recording track sections with track points more than 2 in the track sections as track sections needing to be smoothed, wherein the track sections are used for smoothing the track in the next step, such as the second section and the third section in the example; otherwise, the first segment is output to the smoothed trajectory according to the linear trajectory。
Step 1.3, preprocessing the posture according to the principle that the minor arc between the two tracks is prior, traversing track point indexes i which are 1,2i-1，PiThe method for calculating the distance between the included angles of the postures is oi-1＝|θi-θi-1If o |, ifi-1Greater than 180 deg., modify PiAttitude angle of the fourth dimension of (1): if thetai> 0, modified to Pi(xi，yi，zi，θi-360 °); otherwise, modifying to Pi(xi，yi，zi，θi+360 deg. as shown in FIG. 3, θi-1＝-170°，θiAt 30 deg. and from thetai-1To thetaiLinear interpolation, will go through the major arc, then will be θiModified to-330 deg., i.e. PiPoint modified to (x)i，yi，zi，-330°)。
Step 2, traversing the track sections needing to be smoothed, which are generated in the step 1, and generating an arc transition type smooth track for each section by adopting a geometric iteration method according to a track point error threshold, a position point chord height error threshold and continuity requirements, wherein the whole track smoothing flow is shown in fig. 4, and the arc transition type smooth track generation method is introduced into one section of track sections needing to be smoothed as follows:
step 2.1, setting initial iteration parameters, and setting four-dimensional track points of the current linear track segment as
Figure BDA0001981387100000055
Marking as an original track point, and setting a position distance error threshold value which needs to be met by the smooth track as epsilonmaxPosition chord height error threshold is recorded as deltamaxError threshold o of included angle of attitude pointmaxThreshold value k of iteration numbermaxThree error thresholds are set according to practical application requirements, and the requirement is epsilonmax≤δmax，kmaxIs not less than 1, when kmaxWhen the current iteration number is equal to 0, the iteration track point is recorded as 1, the obtained arc transition type smooth track is consistent with the existing arc smooth track, and the current iteration number is set as k
Figure BDA0001981387100000061
Step 2.2, traversing the index i as 1, 2.. N-1, and respectively generating iterative track points according to position point chord height error constraint, G1 continuity constraint and shape-preserving constraint
Figure BDA0001981387100000062
Transition circular arc track of
Figure BDA0001981387100000063
As shown in FIG. 5, the method of step 1.1 is first used to calculate the location distance di-1And diAnd angle of position point betaiAnd calculate di-1And diSmaller value d ofmmin＝min(di-1，di) (ii) a And then calculating according to the position point chord height error constraint and the conformal constraint
Figure BDA0001981387100000064
Transition length of front and rear position points:
Figure BDA0001981387100000065
further calculate two transition ratios ri-1And ri：
Figure BDA0001981387100000066
And finally, calculating the starting point, the end point and the circle center of the four-dimensional arc according to the transition proportion to ensure the synchronous continuity of the position and the posture.
Wherein, alpha is more than 0 and less than 1, which is a shape-preserving parameter and represents the distance proportion of the linear track section between two transition circular arcs in the whole track section, and if alpha is 0.2, the linear part between two circular arcs represents 20% of the distance of the whole linear part.
Finally, the starting point of the four-dimensional arc is calculated according to the G1 continuous condition and the two transition ratios
Figure BDA0001981387100000067
Terminal point
Figure BDA0001981387100000068
And the center of the circle
Figure BDA0001981387100000069
Figure BDA00019813871000000610
The addition, subtraction and multiplication operation rules of the four-dimensional vectors are consistent with the operation rules of the three-dimensional vectors, and a four-dimensional arc can be constructed according to the starting point, the end point and the circle center, wherein the former three dimensions form an arc curve of a Euclidean space, and the fourth dimension is a curve generated by the synchronous change of the attitude along with the arc.
Step 2.3, traversing the index i as 1,2
Figure BDA00019813871000000611
With the original track point QiTrack point error of (2):
firstly, a four-dimensional arc is constructed according to the starting point, the end point and the circle center in the step 2.2
Figure BDA00019813871000000612
Figure BDA00019813871000000613
Wherein t is ∈ [0,1 ]]，αi＝180°-βi，
Figure BDA00019813871000000614
Then calculating the parameter midpoint of the four-dimensional arc track as the maximum error point of the track point:
Figure BDA00019813871000000615
the reason why the midpoint of the parameter node is used as the maximum error point of the track point is that the three-dimensional position track is a three-dimensional space circular arc and the position of the midpoint is symmetricalThe point setting error is maximum;
finally calculating a transition arc
Figure BDA00019813871000000616
With the original track point QiRespectively calculating the track point errors including position distance errors and attitude point included angle errors
Figure BDA00019813871000000617
The three-dimensional position distance and the attitude included angle distance are used as a position distance error and an attitude point included angle error;
step 2.4, calculating the maximum position distance errors of all transition arcs and original track points
Figure BDA0001981387100000071
Error of included angle with maximum attitude point
Figure BDA0001981387100000072
Traverse index i ═ 1, 2.. N-1, compute
Figure BDA0001981387100000073
Three-dimensional position distance of
Figure BDA0001981387100000074
Included angle with the posture
Figure BDA0001981387100000075
Maximum distance of position
Figure BDA0001981387100000076
Is composed of
Figure BDA0001981387100000077
Maximum value of (d); maximum attitude angle distance
Figure BDA0001981387100000078
Is composed of
Figure BDA0001981387100000079
Maximum ofA value;
if it is
Figure BDA00019813871000000710
Less than a position distance error threshold epsilonmaxAnd is and
Figure BDA00019813871000000711
less than the error threshold o of the included angle of the attitude pointmaxOr the current iteration number k is larger than the iteration number threshold kmaxTerminating iteration and outputting arc smooth track
Figure BDA00019813871000000712
Turning to step 2.6; otherwise, turning to step 2.5;
step 2.5, traversing i & lt1 & gtiMaximum point of sum track point error
Figure BDA00019813871000000713
Calculating an offset vector
Figure BDA00019813871000000714
And updating iteration track points:
Figure BDA00019813871000000715
changing k to k +1, and turning to step 2.2;
step 2.6, the smooth tracks output in step 2.4 are sorted and output, the tracks after being smooth are formed by combining N four-dimensional linear tracks and (N-1) four-dimensional smooth tracks, and the steps are sequentially as follows: linear trajectory
Figure BDA00019813871000000716
Smooth circular arc track
Figure BDA00019813871000000717
Linear trajectory
Figure BDA00019813871000000718
Smooth circular arc track
Figure BDA00019813871000000719
Smooth circular arc track
Figure BDA00019813871000000720
Linear trajectory
Figure BDA00019813871000000721
The obtained smooth track can ensure the synchronous continuity of the position and the posture, namely the position p of any track point on the smooth track is ensuredi(xi，yi，zi) G1 is continuous and guarantees an arbitrary point qi＝pi+RZ(θi) G1 continuity of v, where v is an arbitrary three-dimensional vector, RZ(θi) Is a matrix of rotations about the z-axis.

Claims (2)

1. An error-controllable arc transition type smooth track generation method for a four-axis industrial robot is characterized by comprising the following steps:
step 1, track preprocessing of a four-axis industrial robot: dividing the linear track into a track section needing to be smoothed and a track section not needing to be smoothed according to the position distance and the included angle;
step 2, smoothing the track section needing to be smoothed: traversing the sections of the track needing to be smoothed, which are generated in the step 1, and generating the arc transition type smoothing track for each section of the track needing to be smoothed by adopting a geometric iteration method according to a track point error threshold, a position point chord height error threshold and continuity requirements, wherein the method specifically comprises the following steps:
step 2.1, setting initial iteration parameters, and setting four-dimensional track points of the current linear track segment as
Figure FDA0003155750330000011
The number N of the track points is more than or equal to 2, the track points are marked as original track points, and the position distance error threshold value required to be met by the smooth track is set as epsilonmaxPosition chord height error threshold is recorded as deltamaxError threshold o of included angle of attitude pointmaxThreshold value k of iteration numbermaxSetting the current iteration number as k equal to 0, and recording the iteration track point as
Figure FDA0003155750330000012
Step 2.2, traversing the index i to be 1,2, … N-1, and respectively generating iterative track points according to the position point chord height error constraint, the G1 continuity constraint and the shape-preserving constraint
Figure FDA0003155750330000013
Transition circular arc track of
Figure FDA0003155750330000014
First separately calculate
Figure FDA0003155750330000015
And
Figure FDA0003155750330000016
is located a distance di-1And diAnd angle of position point betaiAnd calculate di-1And diSmaller value d ofmin＝min(di-1,di)；
And then calculating according to the position point chord height error constraint and the conformal constraint
Figure FDA0003155750330000017
Transition ratio r of front and rear position pointsi-1And riFirstly, calculating the transition length of the front and rear position points as follows:
Figure FDA0003155750330000018
two transition ratios are then calculated:
Figure FDA0003155750330000019
wherein 0<α<1 is a shape-preserving parameter which represents the distance proportion of a linear track section between two transition arcs in the whole track section;
finally, the start of the four-dimensional arc is calculated according to the G1 continuous condition and two transition proportionsDot
Figure FDA00031557503300000110
Terminal point
Figure FDA00031557503300000111
And the center of the circle
Figure FDA00031557503300000112
Figure FDA00031557503300000113
Step 2.3, calculating the ith transition arc by traversing the index i equal to 1,2, … N-1
Figure FDA00031557503300000114
With the original track point QiTrack point error of (2):
firstly, a four-dimensional arc is constructed according to the starting point, the end point and the circle center in the step 2.2
Figure FDA00031557503300000115
The first three dimensions are circular arc curves of European space, and the fourth dimension is a curve generated by the rotation angle along with the synchronous change of the circular arcs;
Figure FDA0003155750330000021
wherein t is ∈ [0,1 ]],αi＝180°-βi,
Figure FDA0003155750330000022
Then calculating the parameter midpoint of the four-dimensional arc track as the maximum error point of the track point:
Figure FDA0003155750330000023
finally calculating a transition arc
Figure FDA0003155750330000024
With the original track point QiRespectively calculating the track point errors including position distance errors and attitude point included angle errors
Figure FDA0003155750330000025
The three-dimensional position distance and the attitude included angle distance are used as a position distance error and an attitude point included angle error;
step 2.4, calculating the maximum position distance errors of all transition arcs and original track points
Figure FDA0003155750330000026
Error of included angle with maximum attitude point
Figure FDA0003155750330000027
If it is
Figure FDA0003155750330000028
Less than a position distance error threshold epsilonmaxAnd is and
Figure FDA0003155750330000029
less than the error threshold o of the included angle of the attitude pointmaxOr the current iteration number k is larger than the iteration number threshold kmaxTerminating iteration and outputting arc smooth track
Figure FDA00031557503300000210
Turning to step 2.6; otherwise, turning to step 2.5;
step 2.5, traversing i to 1, … N-1 according to the original track point QiMaximum point of sum track point error
Figure FDA00031557503300000211
Calculating an offset vector
Figure FDA00031557503300000212
And updating iteration track points:
Figure FDA00031557503300000213
changing k to k +1, and turning to step 2.2;
step 2.6, the smooth tracks output in step 2.4 are sorted and output, and the tracks after being smooth are formed by combining N four-dimensional linear tracks and (N-1) four-dimensional circular arc smooth tracks, which are sequentially: linear trajectory
Figure FDA00031557503300000214
Smooth circular arc track
Figure FDA00031557503300000215
Linear trajectory
Figure FDA00031557503300000216
Smooth circular arc track
Figure FDA00031557503300000217
Smooth circular arc track
Figure FDA00031557503300000218
Linear trajectory
Figure FDA00031557503300000219
2. The error-controllable arc transition type smooth trajectory generation method for the four-axis industrial robot according to claim 1, wherein the step 1 specifically comprises the following steps:
step 1.1, calculating a segmentation index set according to a segmentation threshold, inputting track segments above two continuous track points, and outputting a segmentation index; set the trajectory points of the four-axis industrial robot as input
Figure FDA00031557503300000220
Wherein each track point Pi(xi,yi,zi,θi) Is a position (x)i,yi,zi) And attitude, i.e. angle of rotation about the Z axisiComposed four-dimensional vector, position segmentation condition by position distance threshold deltadAnd angle of position threshold deltaaAs an index;
and traversing the track point index i as 1,2 and … N-1, and respectively judging whether the index i meets the position segmentation condition:
firstly, calculating a track segment Pi-1PiAnd PiPi+1The distance d between the two sectionsi-1And diIf d isi-1Or diLess than a position distance threshold deltadIf so, the index i is considered to meet the position segmentation condition; otherwise, calculating the track segment Pi-1PiAnd PiPi+1At a position angle ofiIf sin aiLess than a position angle threshold deltaaIf so, the index i is considered to meet the position segmentation condition; otherwise, the index i is considered not to meet the position segmentation condition; adding indexes i meeting the position segmentation condition into a segmentation index set, and adding indexes 0 and N from beginning to end into the beginning and the end of the segmentation index set by default;
step 1.2, segmenting according to the segmented index set, and dividing a whole track segment into a plurality of track segments according to the segmented index set, wherein the track segment with the track point number larger than 2 in the track segments is marked as a track segment needing to be smoothed and is used for smoothing the track in the next step; otherwise, marking as a section without a smooth track, and outputting the section to the smooth track according to the linear track;
step 1.3, posture preprocessing, traversing track point index i to be 1,2, … N according to the principle that minor arcs between two tracks are prior, and if two adjacent track points P arei-1,PiIs included angle distance oi-1Greater than 180 DEG, oi-1＝|θi-θi-1If P is modifiediAttitude angle of the fourth dimension of (1): if thetai>0, then is modified to Pi(xi,yi,zi,θi-360 °); otherwise, modifying to Pi(xi,yi,zi,θi+360°)。
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