Rotation center calibration based on line laser rotating platform

The center of rotation can also be obtained by dividing the plane edge of the rotating platform in the left and right images, and then performing ellipse fitting on the obtained plane edge, and obtaining the center coordinates of the ellipse by the least squares fitting method. It is the center of rotation of the rotating platform [10].

As shown in Figure 2 above, Point F(x, y) is a point on the ellipse, and the focus of the ellipse is L₁(x₁, x₂) and L₂(x₂, x₂).

Figure 2.

(1) O L 1 + O L 2 = 2 c

Since the sum of the distances from any point on the ellipse to the two focal points is a constant 2a, there is an error between the sum of the actual distances, and the error is θ. As shown in the following formula (2): (2) F L 1 + F L 2 = 2 a + θ

The above formula (2) is specifically expanded as follows: (3) ( x − x 1 ) 2 + ( y − y 1 ) 2 + ( x − x 2 ) 2 + ( y − y 2 ) 2 = 2 a + θ

Using the above formula, the point cloud of the elliptical edge can be fitted by the least squares method, and the estimated values of the five parameters are obtained, and then linearized and solved iteratively.

The criteria for the iterative solution process are:

It is necessary to eliminate the edge point cloud data whose absolute value of the error θ is greater than or equal to 2.6β after each ellipse fitting. (β: standard error of unit weight after each iterative fitting)

Between two adjacent iterative fittings, the absolute value of the distance of any elliptical focus position must be less than a certain pixel value (typically 0.001 pixels). If it is greater than or equal to this pixel, the iteration will end immediately.

In the formula (3), the order: (4) l 1 = ( x − x 1 ′ ) 2 + ( y − y 1 ′ ) 2 (5) l 2 = ( x − x 2 ′ ) 2 + ( y − y 2 ′ ) 2

Let FL₁, FL₂ be partial to x₁, x₂, y₁, y₂. Then the above formula expands to: (6) x − x 1 ′ l 1 Δ x 1 + y − y 1 ′ l 1 Δ y 1 + x − x 2 ′ l 2 Δ x 2 + y − y 2 ′ l 2 Δ y 2 + 2 Δ a = l 1 + l 2 − 2 a 0 + θ

x 1 ′ , y 1 ′ , x 2 ′ , y 2 ′ represents the initial value of the coordinate of the focus, and the focus satisfies the relationship as follows: (7) x 1 = x 1 ′ + Δ x 1 , x 2 = x 2 ′ + Δ x 2 (8) y 1 = y 1 ′ + Δ y 1 , y 2 = y 2 ′ + Δ y 2 (9) a = a 0 + Δ a

The initial position obtained is used as the initial value, and the point cloud data of the ellipse edge is fitted by the least squares method step by step iteration [11], then the parameter (x₁, x₂, y₁, y₂, a) can be solved, and the elliptical center O(x₀, y₀) of the last fitting is: (10) x 0 = x 1 + x 2 2 , y 0 = y 1 + y 2 2

Since the image on the rotating platform has symmetry after being rotated by 180 degrees and is symmetrical about the center of rotation, a plurality of laser lines can be selected on the auxiliary pattern to perform calculation of the center of rotation to ensure accuracy. Finally, the parameters obtained can be optimized by methods such as BP neural network.

In order to accurately calibrate the center of rotation, the straight line equations of the two straight lines s₁s₂ and s 1 ′ s 2 ′ are fitted on the premise that the rotation angle of the rotating platform is known. Find the intersection O between the two lines. The intersection point O is the rotation center of the rotating platform. The point s₁ is located at the position of the point s 1 ′ after being rotated by 180 degrees, and point s₂ is rotated 180 degrees and is located at point s 2 ′ . s₁, O, s 1 ′ three points are collinear, and after rotation, are symmetric with respect to the rotation center O before rotation [12]. The principle of symmetry is shown in Figure 4.2 below.

Figure 3.

The calibration process of the symmetrical rotation platform is:

In the three-dimensional reconstruction of the rotating scanning mode, the obtained rotating point cloud is plane-fitted to fit the rotation center of the rotating platform [13]. The line laser emitter emits a laser beam that intersects the surface of the calibration block. As shown in the figure below, point A emits a line laser, and BC is the line of intersection between the laser and the calibration block. The plane formed between points A, B, and C is a light plane. The points in the light plane (Δ ABC) conform to the plane equation (11).

In online laser scanning, the acquired point clouds are all in the light plane, so all point clouds conform to the plane equation (11): (11) a x + b y + c z + d = 0

The n point cloud data obtained by the rotation scan are sequentially brought into the upper equation of the light plane. (12) { a x 1 + b y 1 + c z 1 + d = 0 a x 2 + b y 2 + c z 2 + d = 0 ⋮ a x n + b y n + c z n + d = 0

Figure 4.

There are 4 unknown n equations, and the values of four unknowns a, b, c, d can be solved by least squares method. The center of rotation of the rotating platform can be expressed as: (13) O = ( 1 n Σ i = 1 n x i , 1 n Σ i = 1 n y i , 1 n Σ i = 1 n z i )

In general, by gradually increasing the height of the rotating platform, a series of center points can be fitted in a straight line, so that the rotation axis can be calibrated. Then through the Rodrigo rotation formula [14], we can complete the coordinate transformation and so on.

The comparison between the calibration methods of the center of rotation is shown in Table 1 below.