R

[

k
1
C

,

k
2
C

,

k
3
C

⁢

…
⁢

k
N
C

]

=

[

k
1
W

,

k
2
W

,

k
3
W

⁢

…
⁢

k
N
W

]

;

(

1
-
4

)

- let matrices M₁=[k₁^C, k₂^C, k₃^C. . . k_N^C] and M₂=[k₁^W, k₂^W, k₃^W. . . k_N^W], then:

RM
1

=

M
2

;

(

1
-
5

)

A parameter λ₁is a value representing a magnitude relationship between ks. For the four-degree-of-freedom robot, since the execution component of the robot only rotates around the z-axis, all data satisfy k₁^C=λ₁k₂^C=λ₂k₃^C. . . =λ_N-1k_N^Cand k₁^W=λ₁k₂^W=λ₂k₃^W. . . λ_N-1k_N^W, that is, for different collection positions, rotation transformations between the world coordinate system and the camera coordinate system only differ in the rotation angles, but have the same rotation axis direction, so that a rank of the matrices M₁and M₂is 1, and a correct rotation matrix R cannot be obtained by solving (1-5).

As can be seen from the above, although the automated calibration for the six-degree-of-freedom robot can be achieved by means of existing hand-eye calibration, for the the four-degree-of-freedom robot (the movement of the end of the execution component of the robot only includes spatial translation and rotation around the z axis), the hand-eye calibration result cannot be solved by means of the existing hand-eye calibration.

Although the hand-eye calibration of the rotation matrix of the four-degree-of-freedom robot cannot be solved by using the equation (1-1) or (1-3), it can be found by observing the equation (1-2) that if it is required that, an initial position posture of the end of the robot is kept unchanged in the whole data collection process, then R^W=I, hence the equation (1-2) can be simplified to:

RT
C

=

T
W

;

(

1
-
6

)

- where T^Wis a first translation matrix corresponding to the world coordinate system, and T^Cis a second translation matrix corresponding to the camera coordinate system.

Multiple sets (at least three sets) of T^Wand T^Care acquired. For example:

T
C

=

[

1
⁢
7
⁢

1
.
3

⁢
6

-
1

⁢
2
⁢
2
⁢

7
.
7

⁢
5

2
⁢
3
⁢
2
⁢

4
.
8

⁢
4

]

,

T
W

=

[

-
1

⁢
9
⁢

3
.
2

⁢
3

-
1

⁢
2
⁢
7
⁢

9
.
2

⁢
4

-
2

⁢
2
⁢
9
⁢

5
.
1

⁢
7

]

.

Three-dimensional affine transformation processing is performed on the basis of the multiple sets of T^Wand T^C, so as to determine the rotation matrix R. For example, the rotation matrix R may be determined by using an estimatedAffine3D function of OpenCV and performing three-dimensional affine transformation processing on the basis of the multiple sets of T^Wand T^C, which can ensure orthogonality of the rotation matrix R. For example, the determined rotation matrix R is:

[

-
0.9989

0.0451

0.0143

0.0447

0.9987

-
0.0262

-
0.0155

-
0.0255

-
0.9996

]

.

In addition, the equation (1-6) can be subjected to matrix transposition and transformed to (T^C)^tR^t=(T^W)^t, which can be solved by using singular value decomposition.

As can be seen from the above, the calculation process of the calibration can be divided into two steps: a first step of fixing an end posture of the execution component of the robot, only controlling the execution component of the robot to perform translation movement, collecting a sufficient quantity of data (N≥4), and calculating the rotation matrix R between the camera coordinate system and a robot coordinate system (the world coordinate system) by using the equation (1-6); and a second step of collecting other data for the calculation of the translation matrix T of the two coordinate systems.

The translation matrix can be determined on the basis of the first coordinate information, the deviation vector, the rotation matrix, and the third coordinate information in various manners. FIG. 4 is a schematic flow diagram of determining a translation matrix in some embodiments of a coordinate system calibration method for a robot according to the present disclosure, as shown in FIG. 4:

step 401, calculating a translation deviation on the basis of the rotation matrix and the third coordinate information. For example, a product of the rotation matrix and the third coordinate information is calculated as the translation deviation.

Step 402, according to the first coordinate information, the deviation vector, and the translation deviation, determining the translation matrix. For example, coordinate information of the calibration board is determined on the basis of a sum of the first coordinate information and the deviation vector, and according to a difference between the coordinate information of the calibration board and the translation deviation, the translation matrix is determined.

In some embodiments, after collecting the data for calibrating the rotation matrix R, two sets of data are collected: collecting one set of data at a certain position (or using a last set of data collected for calibrating the rotation matrix R), then controlling the end of the execution component of the robot not to perform translation movement but to perform 180° (or 90° and the like) rotation, and collecting one set of data after the rotation is finished. Using the two sets of data, the translation matrix T is calculated in combination with point cloud data provided by the camera.

Because the four-degree-of-freedom robot only rotates around the z axis, and there is a 180° rotation relationship between the two sets of data, for two sets of spatial positions of a same reference point on the calibration board, their center point is necessarily on an axis (rotation axis) parallel to the z axis of the world coordinate system of the robot, coordinates regarding the same reference point are extracted from point clouds of the two sets of data, and a coordinate (in the camera coordinate system) of their center point is calculated; and by using the coordinate of the end of the execution component of the robot in the world coordinate system of the robot when the data is collected and the distance dz of the end of the execution component of the robot to the calibration board, the translation matrix T between the camera coordinate system and the world coordinate system is solved.

Assuming that the coordinate of the center point of the positions collected for the same reference point twice on the calibration board is X, and the coordinate of the end of the execution component of the robot when the data is collected is Y, T is calculated by the following relation:

T
=

Y
+

Δ
⁢
Y

-
RX

;

(

1
-
7

)

Δ
⁢
Y

=

[

0

0

δ
⁢
z

]

,

where namely ΔY is the deviation vector, δz is the perpendicular distance between the calibration board and the corresponding coordinate plane of the world coordinate system; R is the rotation matrix calculated in the first step, RX is the translation deviation, X is the third coordinate information, Y is the first coordinate information, namely Y is the last set of first coordinate information which is collected for calibrating the rotation matrix R, and Y+ΔY is the coordinate information of the calibration board, namely Y+ΔY is the coordinate information of the center point of the positions collected for the same reference point twice on the calibration board in the world coordinate system.

For example, when the four-degree-of-freedom robot is calibrated, a customized calibration board is mounted. A posture of the end of the execution component of the robot is fixed, the execution component of the robot is controlled to perform translation movement, and by using random translation movement, N (N≥4) sets of data are automatically collected, denoted as data A; one set of data is collected at a certain position, the translation movement of the end of the execution component of the robot is fixed, the execution component is controlled to perform rotatory movement and 180° rotation, and a second set of data is collected, denoted as data B.