CN1304178C - Method for testing collision between joint of robot with multiple mechanical arm - Google Patents

Method for testing collision between joint of robot with multiple mechanical arm Download PDF

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CN1304178C
CN1304178C CNB2004100232359A CN200410023235A CN1304178C CN 1304178 C CN1304178 C CN 1304178C CN B2004100232359 A CNB2004100232359 A CN B2004100232359A CN 200410023235 A CN200410023235 A CN 200410023235A CN 1304178 C CN1304178 C CN 1304178C
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joint
robot
arm
arm robot
line segment
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CN1623741A (en
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熊勇刚
凌四立
杨小娟
周友行
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熊勇刚
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Abstract

The present invention relates to a collision detection method between joints of a multi-mechanical arm robot, which uses a computer program control method to detect the collision between the joints of the multi-mechanical arm robot. The analysis on the multi-arm robot discovers that end-point spatial coordinates of each joint on the mechanical arm can be easily obtained as long as the sensors arranged on both ends of each joint of the multi-arm robot are utilized in the working process of the multi-arm robot; the space position of each joint on the mechanical arm of the multi-arm robot is transferred to a computer system by an electrical signal via the sensors in the mode of data signals of the spatial coordinates, and then a program controlled method is utilized to distinguish the mechanical interfere relation between two cylindrical objects with limitation of length according to the shortest distance between two line segments in arbitrary space. The collision detection between the joints of the multi-arm robot can be carried out by using the discrete detection algorithm.

Description

The interarticular collision checking method of a kind of multimachine tool arm robot
Technical field
The control method that to the invention belongs to a kind of motion with arm be the sequence controlled machine hand of characteristics is meant that especially each joint of a kind of multimachine tool arm robot system mechanics arm prevents the collision routine control detection method of mechanical interference.
Technical background
During for multimachine tool arm robot system works, each joint of mechanical arm all is in dynamic, the uncertain unstructuredness environment, collision phenomenon may often take place, and how judging and avoiding the collision in robot arm joint is the prerequisite that realizes multi-arm robot's automatic job.Be applied to mainly contain two kinds of systems at present, two kinds of levels, i.e. one-of-a-kind system and multi-computer system, complete machine level and joint level aspect the robot research at collision checking method.The complete machine level is more common in the mechanical interference research of multi-computer system, and mature computational methods are difficult to directly apply to multi-computer system on the level of one-of-a-kind system joint.And some multimachine tool arm system because it is huge relatively to constitute each articulation structure of mechanical arm, in working space, still can not be simplified it and handle as an independent integral body, how to handle this situation, still a sizable difficult problem.
Summary of the invention
The objective of the invention is to deficiency, propose a kind of detection method that can easyly detect each joint of mechanical arm collision of multimachine tool arm system robot at each joint of mechanical arm collision checking method of existing multimachine tool arm robot system.
Problem for convenience of explanation, the present invention introduces more relevant notions earlier.
According to the correlation between each mechanical arm of multimachine tool arm system, set up the simplified model in each each joint of mechanical arm, it mainly is the collision detection of considering between each joint of mechanical arm level that the multi-arm robot is closed that the internode collision detection changes into.At first must analyze the structure in each joint of mechanical arm, and carry out the simplification of appropriateness, set up the simplified model in each joint.Generally speaking, each joint of mechanical arm can be considered by columned rod member to be formed, therefore each joint of mechanical arm can be reduced to the hemisphere that two end portions has radius R, the intermediate portion is reduced to the cylindrical rod member of radius R, and the model after the simplification is shown in Figure 1.Can find out that from the simplification modeling of robot problem is differentiated in the interference between joint of robot, can be divided between the motion rod member collision detection problem between motion rod member and the static rod member.From the geometrical property of simplified model, its collision checking method is exactly how to carry out interarticular collision detection.According to simplified model, the beeline that the interarticular collision detection of multi-arm robot can be reduced between a plurality of two cylindric objects is interfered the differentiation problem.When space two cylindric objects were reduced to two line segments, the collision detection between two cylindric objects just was converted into the space interference problem of one two space line segment interrelationship.
From simplify modeling, can find out that the beeline that the interarticular collision detection of multi-arm robot can be reduced between a plurality of two cylindric objects is found the solution.When space two cylindric objects were reduced to two line segments, the collision detection between two cylindric objects just was converted into a space analysis geometrical issues of asking two space line segment interrelationships.Because the limited length system of space line segment, when how to judge the correlation between the two space line segments or finding out beeline between the two space line segments, only need that regular thing is summarized, is simplified to the various situations that may occur and just can find out and prevent the way of interfering.
Definition: make that AB is the two-end-point in any joint of simplified model; CD is the two-end-point in another joint; If the aerial gap of two movable joints is a Δ, when Δ ≠ 0, line segment AB, CD antarafacial, otherwise coplane.Show the center line of two movable joint simplified models of robot for two space line segment tables for appointing, the coordinate figure of four end points of line segment is known.From the basic theories of analytic geometry as can be known,
If l 1Be the distance of A point to line segment CD, l 2Be the distance of B point to line segment CD, l 1' be the distance of C point to line segment AB, l 2' for the D point to line segment AB distance; l ACFor the A point to C point distance, l ADFor the A point to D point distance, l BCFor the B point to C point distance, l BDFor the B point to D point distance, l ABBe the length of line segment AB, l CDLength for line segment CD.
According to the end points coordinate of the two line segments l that is easy to get 1, l 2, l 1', l 2', l AC, l AD, l BC, l BD, l AB, l CD, all more than or equal to zero, we are designated as it for they:
min(l 1，l 2，l 1′，l 2′，l AC，l AD，l BC，l BD，l AB?，l CD)≥0 (1)
If min (l 1, l 21 E can be found in)＞0 on line segment AB, make l AE/ l BE=l 1/ l 2Set up, order point E to the distance of line segment CD be l 3In like manner if min (l 1', l 2')＞0, also can on line segment CD, find 1 E ', allow l CE'/l DE'=l 1'/l 2' set up, order point E ' to the distance of line segment AB be l 3'; If min (l 1, l 2)=0 then makes l 3=0; In like manner, if min (l 1', l 2')=0, then make l 3'=0.Equally, l 3And l 3' also be easy to get.
Have only two kinds of situations in the locus by analyzing two rod members, a kind of is coplane, and a kind of is antarafacial.Therefore judge that line segment AB, the locus situation of CD should divide coplane and two kinds of situations of antarafacial to consider.
1, coplane situation
Can learn that by analyzing the relation of coplane two line segment mutual alignments can be summarized as following four kinds of situations:
1), intersect and intersection point on two line segments;
2), intersect but intersection point on the extended line of line segment;
3), conllinear;
4), parallel.
In conjunction with the correlation theory in the analytic geometry, we can obtain following inference:
Inference 1
If min (l 1, l 2, l 1', l 2', l AC, l AD, l BC, l BD, l AB, l CD, l 3, l 3')=0, line segment AB then, CD coplane.
Inference 2
If min (l 3, l 3')=max (l 3, l 3')=0, max (l 1, l 2, l 1', l 2')＞0, line segment AB then, CD intersect (promptly intersecting) mutually, and intersection point is necessarily at line segment AB, on the CD.
Inference 3
If min (l 1, l 2, l 1', l 2', l AC, l AD, l BC, l BD, l AB, l CD, l 3, l 3') ≠ 0, Δ=0, line segment AB, CD coplane and do not have intersection point, and have line segment AB not intersect with the straight line of crossing line segment CD, line segment CD does not also intersect with the straight line of crossing line segment AB.
If at coplane two line segment AB, among the CD, order
l 1 2 + l CD 2 < ( max ( l AC , l AD ) ) 2 - - - ( 2 )
If following formula is set up, we put A no homolographic projection point on line segment CD as can be known, and its subpoint is on the extended line of line segment CD, and some A is to the beeline l=min (l of line segment CD AC, l AD); If following formula is false, some A has the homolographic projection point at line segment CD, and some A is to the beeline l=l of line segment CD 1
l 2 2 + l CD 2 < ( max ( l BC , l BD ) ) 2 - - - ( 3 ) l 1 2 + l AB 2 < ( max ( l AC , l BC ) ) 2 - - - ( 4 )
l 2 2 + l AB 2 < ( max ( l AD , l BD ) ) 2 - - - ( 5 )
Inference 4
If min (l 1, l 2)=max (l 1, l 2)=0, that is, and l 1=0 and l 2=0, line segment AB then, CD conllinear: simultaneously min (l must be arranged 1', l 2')=max (l 1', l 2')=0, vice versa.Have following conclusion to set up simultaneously:
1), if min (l AC, l AD, l BC, l BD)=0, line segment AB, CD have end points to overlap, then they will beeline l=0;
2), min (l AC, l AD, l BC, l BD) ≠ 0, max (l AC, l AD, l BC, l BD)-min (l AC, l AD, l BC, l BD)＞(l AB+ l CD) set up then line segment AB, the beeline l=min (l between CD AC, l AD, l BC, l BD):
3), min (l AC, l AD, l BC, l BD) ≠ 0, max (l AC, l AD, l BC, l BD)-min (l AC, l AD, l BC, l BD)≤(l AB+ l CD) set up line segment AB then, the beeline l=0 between CD.
Inference 5
If max (l 1, l 2, l 1', l 2') ≠ min (l 1, l 2, l 1', l 2')=0, l is promptly arranged 1, l 2Be not 0 entirely, l 1', l 2' be not 0 entirely, then exist:
1), if max (l 1', l 2') ≠ min (l 1', l 2')=0 and max (l 1, l 2) ≠ min (l 1, l 2)=0, line segment AB, CD must intersect at certain end points, the beeline l=0 between them;
2), if max (min (l 1', l 2'), min (l 1, l 2)) ≠ min (min (l 1', l 2'), min (l 1, l 2))=0, i.e. min (l 1, l 2) and min (l 1', l 2') have only one to be 0.If min is (l 1, l 2)=0, line segment AB then, the beeline l=min (l between CD 1', l 2'); If min is (l 1', l 2')=0, line segment AB then, the beeline l=min (l between CD 1, l 2).
Inference 6
If min (l 1, l 2, l 1', l 2') ≠ 0, and min (l 3, l 3')=0, then cross line segment AB, two straight lines of CD necessarily intersect; Exist simultaneously:
1), if max (l 3, l 3')=min (l 3, l 3')=0, i.e. l 3=0 and l 3'=0, from inference 2 line segment AB as can be known, CD intersects at E point, the beeline l=0 between two line segments;
2), if max (l 3, l 3') ≠ min (l 3, l 3')=0, i.e. l 3=0, l 3'=0 does not set up simultaneously, if l 3=0, line segment AB, the beeline l=min (l between CD 1', l 2'); If l 3'=0, line segment AB then, the beeline l=min (l between CD 1, l 2).
Inference 7
If min is (l 3, l 3') ≠ 0 (min (l is arranged promptly 1, l 2, l 1', l 2', l 3, l 3') ≠ 0), and Δ=0, by inference 3 line segment AB as can be known, only there is intersection point in CD on the extended line of the extended line of line segment AB and line segment CD, perhaps line segment AB, CD is parallel; Also can be as drawing a conclusion:
1), if formula (2), (3) (4), (5) four formulas are all set up, line segment AB then, the last mutual no projection line segment of CD, the beeline l=min (l between them AC, l AD, l BC, l BD);
2), if formula (2), (3) (4), (5) four formulas all are false, line segment AB then, CD must be parallel and rectangularity distribute the beeline l=l between them 1
3), if formula (2), (3) (4), (5) any three formulas are set up, appoint to establish formula (2) and be false, line segment AB then, the beeline between CD is l=l 1, the rest may be inferred for other situations:
4), if formula (2), (3) (4), (5) have only any two formulas to set up, and appoint the formula (2) of establishing, formula (4) is set up, line segment AB then, the beeline between CD is l=min (l 1, l 1'), the rest may be inferred for other situations.
5), if formula (2), (3), (4), (5) have only any same form to set up, appoint to establish formula (2) and set up, line segment AB then, the beeline between CD is l=min (l 1', l 2, l 2').
2, antarafacial situation
Δ ≠ 0, space two line segment AB as can be known, must there be min (l in the CD antarafacial 3, l 3') ≠ 0.
Cross line segment AB arbitrarily, two spaces, two different surface beelines of CD can be configured to positive triangular pyramid, and have only three kinds of situations.Like this, antarafacial line segment AB, the line segment AB on the also available positive triangular pyramid of CD and with line segment C 1D 1The line segment CD that direction vector is identical represents that promptly line segment CD is crossing line segment C 1D 1Straight line on, particular location is indefinite.Then l 1 = l A C 1 , l 2 = l BD 1 , l 0 = l C 1 A 2 = l OO 1 .
By the fundamental theorem of analytic geometry as can be known, according to line segment two-end-point coordinate figure, easily know line segment AB, the angle μ between CD.
Describe as can be known according to the front: l AA 1 = l AB &times; sin &mu; ; - - - ( 6 )
With line segment AB, CD exchanges, and establishes: l CC 1 = l CD &times; sin &mu; ; - - - ( 7 )
Appoint and establish l 1=max (l 1, l 2), order: k = l 1 2 - l 0 2 - l AA 1 2 ; - - - ( 8 )
As can be known, if k＜0, line segment AB, the intersection point of the common vertical line of CD drop between line segment AB: if k=0, line segment AB, the intersection point of the common vertical line of CD drop on some end points of line segment AB; If k＞0, line segment AB, the intersection point of the common vertical line of CD drops on outside the line segment AB.
In like manner can get: k &prime; = l 1 &prime; 2 - l 0 2 - l CC 1 2 ; - - - ( 9 )
For k ' above-mentioned conclusion is arranged equally.
Therefore, we can draw inference 8:
Inference 8
1, if k≤0, k '≤0, antarafacial two line segment AB have the intersection point of common vertical line, the beeline l=l between them on the CD 0
2, as if k≤0, there is the intersection point of common vertical line k '＞0 on the line segment AB, and does not have the intersection point of common vertical line on the line segment CD.Simultaneously, if formula (4), set up (5), then the beeline l=min (l between them AC, l AD, l BC, l BD); If formula (4) has only the same form to set up in (5), appoint and establish formula (5) establishment, then l=l 2'; Same k '≤0, also there is similar conclusion k＞0 o'clock;
3, if k＞0, k '＞0 does not have the intersection point of common vertical line on the line segment AB, and does not have the intersection point of common vertical line on the line segment CD, and at this moment, 1,3,4,5 identical in beeline that line segment AB, CD see and the inference 7 promptly has:
If formula (2), (3), (4), set up (5), then the beeline l=min (l between them AC, l AD, l BC, l BD);
If any three formulas are set up, appoint and establish formula (2) and be false, then l=l 1, the rest may be inferred for other;
If any two formulas are set up, appoint and establish formula (2), be false (3), then l=min (l 1, l 2), the rest may be inferred for other;
If formula (2), (3), (4), same form establishment arbitrarily in (5) is appointed and is established formula (2) establishment, then l=min (l 1', l 2, l 2'), the rest may be inferred for other;
So far, we use inference 1 ..., inference 8 can obtain the beeline between oriented two line segments in any space.In view of the above, when we close the internode collision detection the research multi-arm robot, can design the beeline that program utilization computer obtains between two line segments of any space and differentiate two mechanical interference relations that the cylindric shape object of length restriction is arranged, and then detect the multi-arm robot and close internode and whether collide, abbreviate beeline as and find the solution collision checking method.This program detecting method comprises following subprogram:
Subprogram 1: find the solution l 1, l 2, l 1', l 2', l AC, l AD, l BC, l BD, l AB, l CD, l 3, l 3' Δ;
Make k k=min (l 1, l 2, l 1', l 2', l AC, l AD, l BC, l BD, l AB, l CD, l 3, l 3');
Subprogram 2: if k k=0, then find the solution beeline l between two line segments of space by inference 4,5 and inference 6;
Subprogram 3: if k k≠ 0, beeline l between two line segments of space is found the solution according to inference 7 in Δ=0 item;
Subprogram 4: if beeline l between two line segments of space is found the solution according to inference 7 in Δ ≠ 0 item.
According to above programmed algorithm to trace routine such as flow chart such as Fig. 2 show.
At T=T i-t adopted when dispersing detection algorithm, as T=t+ Δ t in the time iThe time, obtain space line segment end points after the coordinate figure under the cartesian space rectangular coordinate system, the step that beeline is found the solution the collision detection algorithm main program is as follows:
1, at T=t+ Δ t iThe time, runtime subroutine 1;
2, judge formula k kWhether=0 set up, if set up, runtime subroutine 2 then changes next step over to if be false;
3, continue judgement formula Δ=0 and whether set up, if set up, then runtime subroutine 3, otherwise runtime subroutine 4;
4, obtain the distance of space line segment after, judge whether the space rod-like articles collides, if workaround is then taked in collision; Otherwise continue;
5, judge T=t iWhether set up, if be false, T then I-1=T i+ Δ t I-1, continue to carry out said procedure, if set up EP (end of program).
Thereby can ask concrete position of intersecting point and the beeline obtained rapidly between the two space line segments judges between two rod-like articles whether bump by this algorithm.
Adopt said procedure can detect the state of each joint of robot.And available MATLAB Programming with Pascal Language is carried out simulation analysis to the program detection algorithm:
If n organizes two its radiuses of cylindric object and is respectively R i, r i(i=1 ..., n), at T=t n+ Δ t iConstantly, the center line end points coordinate between two cylindric objects is produced by a random function, with above-mentioned algorithm this n is organized two cylindric objects and carries out collision detection, differentiates a pair of rod member and whether collides consuming time need of two algorithms mutually about 0.003 second.
According to implementer's case that purpose of the present invention proposed be: adopt a kind of computer program control method to detect the collision in multimachine tool arm robot joint, by the multi-arm robot is analyzed, find as long as install sensor at each two ends, joint of multi-arm robot, in multi-arm robot's course of work, utilize these sensors, can easily obtain the end points space coordinates in each joint on the mechanical arm, each joint location in space on multi-arm robot's mechanical arm is delivered to computer system by sensor with the signal of telecommunication with the data-signal of space coordinates, differentiate two mechanical interference relations that the cylindric shape object of length restriction is arranged according to the beeline between two line segments of any space with programme controlled method again, adopt discrete detection algorithm just can carry out the interarticular collision detection of multi-arm robot, abbreviate beeline as and find the solution the collision detection program detecting method.This computer system comprises at least:
1) signals collecting and writing module are used for providing data to be tested to computer system;
2) be used to the data storage of the information that stores;
3) be used to judge the determination module that whether each joint bumps on the mechanical arm;
4) data of analyzing according to determination module are controlled each joint on the robot arm, in case control module that bumps and executing agency.
The characteristics of this detection method are:
1) this detection system is to utilize that the locus signal that sensor sent of each joint two-end-point carries out discriminant analysis on the robot arm;
2) determination module of detection system is that each joint of robot arm is reduced to line segment, according to each joint space beeline on the robot arm, utilizes discrete detection algorithm to carry out the multi-arm robot and closes that the internode collision detection judges;
3) after judgement, determination module is delivered to control module with being about to analyze data, and assigns fill order by control module, each joint space position on the robot arm is revised, in case each joint bumps on the robot arm.
Other characteristics of this detection method are: the signal that sensor is gathered is the coordinate figure under a kind of cartesian space rectangular coordinate system.
The advantage of state that adopts this program to detect each shoulder joint of multi-arm robot is:
1. this program detection algorithm is the two-end-point coordinate that utilizes joint of robot, derives and analysis by some simple geometric, differentiates two interarticular mutual collision status with length restriction.
2. this program detection algorithm is particularly suitable for computer realization, is easy to realize the modularization of programming, and algorithm complexity is lower.Collision detection on the joint level of multi-robot system had great using value.
3. this program detection algorithm also is a kind of collision detection algorithm based on discrete time, and the accuracy of its differentiation effect all instrument relies in Δ t iSize and robot in the movement velocity in joint, if can predict the movement velocity in each joint, then choose suitable Δ t i, can eliminate the erroneous judgement that may occur in the discrete logarithm.
Accompanying drawing and explanation
Fig. 1 is a principle of the invention structural representation;
Fig. 2 is a trace routine block diagram of the present invention.
The specific embodiment
Embodiment one
Adopt a kind of computer program control method to detect the collision in multimachine tool arm robot joint, by the multi-arm robot is analyzed, find as long as install sensor at each two ends, joint of multi-arm robot, in multi-arm robot's course of work, utilize these sensors, can easily obtain the end points space coordinates in each joint on the mechanical arm, each joint location in space on multi-arm robot's mechanical arm is delivered to computer system by sensor with the signal of telecommunication with the data-signal of space coordinates, differentiate two mechanical interference relations that the cylindric shape object of length restriction is arranged according to the beeline between two line segments of any space with programme controlled method again, adopt discrete detection algorithm just can carry out the interarticular collision detection of multi-arm robot, abbreviate beeline as and find the solution the collision detection program detecting method.This computer detection system comprises at least:
5) signals collecting and writing module are used for providing data to be tested to computer system;
6) be used to the data storage of the information that stores;
7) be used to judge the determination module that whether each joint bumps on the mechanical arm;
8) data of analyzing according to determination module are controlled each joint on the robot arm, in case control module that bumps and executing agency.
The characteristics of this detection method are:
4) this computer detection system is to utilize that the locus signal that sensor sent of each joint two-end-point carries out discriminant analysis on the robot arm;
5) determination module of computer detection system is that each joint of robot arm is reduced to line segment, according to each joint space beeline on the robot arm, utilizes discrete detection algorithm to carry out the multi-arm robot and closes that the internode collision detection judges;
6) after the data that determination module is imported sensor are judged, determination module is delivered to control module with being about to analyze data, and assign fill order by control module, each joint space position on the robot arm is revised, in case each joint bumps on the robot arm.
Other characteristics of this detection method are: the signal that sensor is gathered is the coordinate figure under a kind of cartesian space rectangular coordinate system.
Its control program comprises following subprogram:
Subprogram 1: find the solution l 1, l 2, l 1', l 2', l AC, l AD, l BC, l BD, l AB, l CD, l 3, l 3', Δ;
Make k k=min (l 1, l 2, l 1', l 2', l AC, l AD, l BC, l BD, l AB, l CD, l 3, l 3');
Subprogram 2: if k k=0, then find the solution beeline l between two line segments of space by inference 4,5 and inference 6;
Subprogram 3: if k k≠ 0, beeline l between two line segments of space is found the solution according to inference 7 in Δ=0 item;
Subprogram 4: if beeline l between two line segments of space is found the solution according to inference 7 in Δ ≠ 0 item.
According to above programmed algorithm to trace routine such as flow chart such as Fig. 2 show.
Adopt said procedure can detect the state of each joint of robot.And available MATLAB Programming with Pascal Language is carried out simulation analysis to the program detection algorithm:
If n organizes two its radiuses of cylindric object and is respectively R i, r i(i=1 ..., n), at T=t n+ Δ t iConstantly, the center line end points coordinate between two cylindric objects is produced by a random function, with above-mentioned algorithm this n is organized two cylindric objects and carries out collision detection, differentiates a pair of rod member and whether collides consuming time need of two algorithms mutually about 0.003 second.

Claims (4)

1, the interarticular collision checking method of a kind of multimachine tool arm robot, adopt a kind of computer program control method to detect the collision in multimachine tool arm robot joint, it is characterized in that: each two ends, joint the multi-arm robot are equipped with sensor, in multi-arm robot's course of work, utilize these sensors, can easily obtain the end points space coordinates in each joint on the mechanical arm, each joint location in space on multi-arm robot's mechanical arm is delivered to computer system by sensor with the signal of telecommunication with the data-signal of space coordinates, differentiate two mechanical interference relations that the cylindric object of length restriction is arranged according to the beeline between two line segments of any space with programme controlled method again, adopt discrete detection algorithm just can carry out the interarticular collision detection of multi-arm robot, this computer detection system comprises at least:
A, signals collecting and writing module are used for providing data to be tested to computer system;
B, be used to the data storage of the information that stores;
C, be used to judge the determination module that whether each joint bumps on the mechanical arm;
D, the data of analyzing according to determination module are controlled each joint on the robot arm, in case control module that bumps and executing agency.
2, the interarticular collision checking method of a kind of multimachine tool arm robot as claimed in claim 1, it is characterized in that: the determination module of computer detection system is that each joint of robot arm is reduced to line segment, according to each joint space beeline on the robot arm, utilize discrete detection algorithm to carry out the multi-arm robot and close that the internode collision detection judges.
3, the interarticular collision checking method of a kind of multimachine tool arm robot as claimed in claim 1, it is characterized in that: after the data that determination module is imported sensor are judged, determination module is delivered to control module with being about to analyze data, and assign fill order by control module, each joint space position on the robot arm is revised, in case each joint bumps on the robot arm.
4, the interarticular collision checking method of a kind of multimachine tool arm robot as claimed in claim 1, it is characterized in that: the signal that sensor is gathered is the coordinate figure under a kind of cartesian space rectangular coordinate system.
CNB2004100232359A 2004-05-24 2004-05-24 Method for testing collision between joint of robot with multiple mechanical arm Expired - Fee Related CN1304178C (en)

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