JPH0377106A - Three-dimensional shift control method for industrial robot - Google Patents

Abstract

PURPOSE:To attain the three-dimensional shift with high accuracy for three key points by using a least square method to correct the coordinates obtained by displaying three corresponding points of a real work against the coordinates of the corresponding points obtained by transforming the former coordinates and obtaining a transformation matrix based on the corrected coordinates. CONSTITUTION:Three key points P1 - P3 are selected out of the reference points set on a model work. Then the points Q1 - Q3 are obtained by teaching three points of a real work in response to the points P1 - P3. The coordinate values of points Q1 - Q3 are corrected so as to obtain the least sum of square of distances between the points Q1 - Q3 and the position coordinates P1' - P3' obtain through the coordinate transformation of the three points of the real work corresponding to the points P1 - P3. Thus the points of the model work are shifted in terms of coordinates in response to the points of the real work respectively. Thus it is possible to attain a transformation method to secure the optimum three dimensional shift with application of the position coordinates causing an error.

Description

[Detailed Description of the Invention] [Industrial Application Field] The present invention is applicable to teaching work of an industrial robot, when there are points that have already been taught for a certain workpiece and the same workpiece is installed at different locations. The present invention relates to a three-dimensional shift control method used when automatically determining other points by simply teaching the positions of three representative points.

[Conventional technology]

Conventionally, such a three-dimensional shift has been proposed in Japanese Patent Application Laid-Open No. 57-182.
As described in Publication No. 205, for the model work used during teaching, a representative point P out of a large number of points is
1. P2. Set Ps. Then, during actual work, the point p of the actual workpiece corresponding to the representative point mentioned above is
IL, p,′, p,′ is given by teaching. Then, when finding the corresponding points P,' with respect to other points, for example, P, the four points P, which are the standards of the model work, are calculated.
1, P1, P1, P, and the corresponding points p,', p2', p,' of the actual workpiece,
In p4', the coordinates of point P4' are determined based on vectors P, 'P,', P, 'P,', on the premise that there is no relative positional shift.

[Problem to be solved by the invention]

However, in reality, there are errors in teaching and measuring the points P,', P2', P,', and the vector P l' P2
Since there are deviations in '+・P and 'P,', the points found using them also have deviations in their relative positions, and are distorted as a whole.

An object of the present invention is to provide a conversion method that performs an optimal three-dimensional shift when position coordinates with such errors are given.

[Means to solve the problem]

In order to achieve this objective, the three-dimensional shift control method for an industrial robot of the present invention has in advance the position coordinates of a plurality of reference points on a model work in a three-dimensional space, and calculates the relative positions of these points. In three-dimensional transformation that moves all points to locations corresponding to the actual workpiece without changing them, three representative points P1, P2.
Q1, Q1, Q3 obtained by selecting P, and teaching the three points on the actual workpiece that correspond to these points during work.
and the three representative points PI, P2. P) Position PAp obtained by coordinate transformation as a point on the actual workpiece corresponding to the point
,',P*'.

The teaching point Q1. Q2. It is characterized in that the coordinate values of Q3 are corrected and the coordinates of other points on the model work are shifted to corresponding points on the actual work.

[Effect]

In the present invention, a three-dimensional shift transformation matrix is provided based on the fact that the relative positions of reference points do not change even after transformation. As a condition at that time, let the points to which they are transferred by conversion be Q1, Q1, Q for the three representative points PI + P1, P from the reference point,
These points are the points p,', given by teaching or measurement.
This is a transformation that makes it as close as possible to p,', p,', and specifically, Q+P+' + (hP2'
+QsP3'...This is a three-dimensional shift such that (L) is minimized. This is shown in Figure 1.

In model #1, the position coordinates of all points are given in advance, and the three points are represented by PlI P2+ P3. =2 model is transferred by transformation,
Q+, Q2. Qs corresponds to the representative point. This Q1
, Q1, Q, and p given by teaching or measurement, ',
The relationship between p2', p, and ' is shown in Figure 2.

These points do not necessarily match due to errors. The present invention is a transformation that minimizes the sum of the squares of the distances of each point, and will be described in detail below.

First, set the coordinates of each point as follows.

Q+ (Xt, Y++ 21), Q2 (lh,
y2. zz) + Q3 (Xl, y3. Zlip, /
(X1, Y1, Z,), P2' (X2. Yt
, za), p, / (X3. Y3. za) where the coordinates of the points p,', p2', p3' are known.

Also, triangles P, P, P, and triangles Q, Q 2 Q,
are congruent, and since the lengths of the sides are known, the following conditional expression holds.

(X2 x,)2+(y2L+)2+(zz-zl)
”=f+(X3 X2>' + (ya V2
)2 + (23Z2)2= , i! z(Xl x3
)'+(yt 3'3)2+(21Z*)"= , f
! s Here, C, lx, and Rs are the lengths of the three sides of the triangle, respectively.

Minimizing equation (1) means minimizing the following evaluation function J.

J= (xt-XI)2+ (yt-L>2+ (z+
-Z, )2+(X2-X2)"+ (1)12-Y2)
'+ (Z, -22>2+(X3-X3)2+ (y3
-Y3)'+ (Z, -Z,)2 From this and the previous conditional expression, using Lagranche's undetermined multiplier method, L=J+λI +(X2 xt)2+(yz yt
>2+(z2z,)2-R1')+λ2((X3
1h)”(!I'3Y2>'+(Zz Z2)2-
j! 22) + Aa ((xt x3)”(yt
y3)2+(z+ 23)2-! 3') From this, aL7 ex+ =2(XI-XI)-2λ+ (×2
-XI)"2λ, (X, -X,) = OaL/a
x, =2(x2-Xi)-2λ2(X3-XI)+
2λ, (X2−X→=OaL/ aXy −2<
Xz-X >>-2λ 3(XI-X3)”2 λ 2
(X3-X2)・0 When these three equations are satisfied, the following equation stands out.

X I+ X 2 + X 3= X I+ X a
+ X 3 Similarly, yt + 'j 2 + y3= Y 1+ Y 2
+ Y 3Z+ 10 Z2+ Z3= Zl + Z2
+ Z3 rises. From this triangle Q IQ 2 Q
It can be seen that the positions (coordinates) of the centers of gravity of 3 and triangle Pl'P2'P3' match.

Therefore, as shown in Figure 3, the triangle P,'
Consider a coordinate system whose origin is the center of gravity of P2'P,'.

On the other hand, the center of gravity of the triangle Q 1Q x Q 3 is also fixed at this origin and can freely rotate around this point.

At this time, let R be the rotation matrix that acts on this triangle Q1Q2Q, and find R that minimizes equation (1).

First, let the position vectors of p l'+ p, ', p, and' from the origin be F1, 'F1, L, respectively. Mata, Q1, Q.

Let the initial position vector of Q before rotation be as follows.

q+ = (X+, y+, z+) 1Q! −(
X*, y3.2J J These positions can be given as long as triangle Q 1Q 2 Q 3 is congruent with triangles P and P2P, and the center of gravity is at the origin.There are countless ways to give these positions, but here they are generally expressed. I'll keep it. These points are qlR by rotation matrix R.

Move on to Q2R and Q3R. This is the evaluation function j of formula (1)
becomes as follows.

Ko , .

J-Σ(pt-q,R)・(p q+R) Here, the rotation matrix R is expressed as a row vector, and R=
y ]. However, the row vectors x, y, and z are unit orthogonal vectors and satisfy the following conditions.

x'x2y1y=z゛z=l X sand y=y-2-2-x=0 ......Pa...
(3) If Lagranche's undetermined multiplier method is used again for this conditional expression and the evaluation function of the above expression, the following relational expression is obtained.

However, a 3 → → 3 → → 3
→Σx, p1, b-ΣyIpi C=Σ2+Il) It is sufficient to find x, y, and z that satisfy this relational expression.

Therefore, the three points Q1, Q2 in equation (2). Place the first point of Q on the xy plane. That is, in equation (2), 2l-
22=z3-0, and from this, 7-6 gives 3=z=Q b · z=0. This shows that vector 2 is orthogonal to the plane formed by vectors a and b. On the other hand, vector a,
b is a vector l)1. p21 p3 - next conclusion 4
These are vectors on the triangles P, 'P2' and P3'. Therefore, vectors a and b are also on this triangle, and vector 2 intersects this triangle perpendicularly.

Therefore, 2 can be found as follows.

Since vector 2 is the unit vector in the Z-axis direction applied with rotation matrix R, the following cross product (Q2 ql)
If two components of X (Ql q+) are positive, 2 is a unit vector of (p2 = p+ ) x (p3-pt), and if it is negative, it is a unit vector with the sign reversed.

Next are the vectors X and y, which are 2 from equation (3)
and are orthogonal to each other. In other words, it lies on the plane formed by vectors a and b, that is, on the triangle p , l p 2 / p %.

First, let t be the unit vector of vector a, and create a unit vector, t, such that j = zxc. t and J are triangle P,'
These are orthogonal vectors on P2' and P3', and the vectors X and y obtained using these are expressed as follows.

x=cosθt+S+n0j y=-5in (it +cosθJ)
Substituting into the first equation of 4) and solving for COSθ and sinθ, two vectors with different signs will appear for good.

In this way, the rotation matrix R is obtained. After performing this rotation, all that is required is a parallel movement to shift the center of gravity.

To summarize the above, a three-dimensional shift is performed using a combination of transformation matrices as shown in FIG.

1. S: Move the center of gravity of triangle p, P2P3 parallel to the origin (see Figure 4(a)).

2. S2: Rotate around the origin so that the triangle is on the xy plane. The rotation matrix R is determined from the coordinates at that time (see FIG. 4(b)).

3. R: Rotate R around the origin (see Figure 4 (C)).

4, Ss: At the center of gravity of triangle p, / p2/ p,',
Translate the triangles so that their centers of gravity overlap (see Figure 4 (a)).

The transformation matrices S1, S2. R.

A three-dimensional shift is performed by acting on all points with S as a reference. Since this transformation consists of rotation and translation, the relative positions of all points are maintained, and the three representative points are placed as close as possible to the points given by teaching or measurement.

〔Example〕

FIG. 5 shows an embodiment of the present invention. Workpiece with the same shape #1
．． #2. There is #3, and their setting positions are different. Now, the teaching of only work #l is finished, and the representative point P l+ P2. It is assumed that point P 3 has also been taught. The teaching for work #2 only requires the positions of points p, ', p2', p, ', which are the representative points of #1, and by performing the three-dimensional shift of the present invention, the teaching points at the same locations as work #1 can be taught. is automatically rawhide. Also, for work #3, the representative point P +' * P 2
The same thing can be done only by displaying 'rPs'.

FIG. 6 shows an example of offline programming.

There is already a three-dimensional work model inside the computer. In order to reproduce the real environment inside a computer, the position of each point on the workpiece model is required.

Therefore, by inputting the coordinates of only three points representing the actual work into a computer and performing the three-dimensional shift of the present invention, the position of each point can be determined.

〔Effect of the invention〕

As mentioned above, in the present invention, three representative
Regarding the point, the coordinates obtained by teaching three corresponding points on the actual workpiece are corrected by the least squares method between these coordinates and the coordinates of the corresponding point obtained by coordinate transformation calculation, and based on the coordinates of this corrected point. I am trying to find the transformation matrix. This makes it possible to perform highly accurate three-dimensional shifting by correcting distortions caused by measurement errors and the like during three-point teaching on an actual workpiece.

Furthermore, since this transformation only involves rotation and translation, it is possible to perform a three-dimensional shift while maintaining the relative position of each point before transformation.

[Brief explanation of drawings]

Figure 1 is an explanatory diagram showing the three-dimensional shift and its representative points, Figure 2
The figure is an explanatory diagram showing the moved representative point and the point given by teaching or measurement, and Figure 3 is an explanatory diagram when calculating the rotation matrix R in a coordinate system with the center of gravity of two triangles coincident and with that as the origin. , FIG. 4 is an explanatory diagram of three-dimensional shift conversion according to the present invention, FIG. 5 is an explanatory diagram showing an embodiment using a welding robot, and FIG. 6 is an explanatory diagram showing an example of an offline program. Figure Figure Figure Figure PI'P2' Figure Figure Figure

Claims (1)

[Claims] 1. A location in which the position coordinates of a plurality of points serving as references on a model work are previously provided in a three-dimensional space, and all points correspond to the actual work without changing their relative positions. In the three-dimensional transformation to move the three representative points P_ from the reference point group on the model work.
Q_1, Q obtained by selecting 1, P_2, P_3 and teaching the three points on the actual workpiece that correspond to these points during work.
_2, Q_3 and the three representative points P_1, P_2, P
The teaching points Q_1, Q are set so that the sum of the squares of the distances between the position coordinates P_', P_2', P_3' obtained by coordinate transformation as points on the actual workpiece corresponding to the _3 points is minimized.
A three-dimensional shift control method for an industrial robot, characterized in that the coordinate values of _2 and Q_3 are corrected and the coordinates of other points on a model work are shifted to corresponding points on an actual work.