The main function of hydraulic support is to support the roof, provide working space for shearer and scraper conveyor, and successfully ensure coal mining in the working face [
25]. The posture of the hydraulic support includes its own posture and the posture between the two adjacent supports. In order to realize the accurate control of the hydraulic support group, the self-pose and relative-pose of the hydraulic support should be determined at first; the detection scheme of the self-pose and relative-pose of the hydraulic support is as shown in
Figure 1. In this scheme, the extension of the balance cylinder and the piston of the column can be measured by the displacement sensor; the posture of the support itself can be determined with the help of the self-pose solving model; and the relative-pose of the top beam of the two adjacent supports can be obtained by the relative-pose detection device and the relative-pose model and then the relative-pose of the two adjacent supports can be obtained by solving the self-pose model. Finally, the spatial position and posture information of the full working face of the hydraulic support group can be obtained, as shown in
Figure 1. Based on this, in order to realize the detection of the group pose information of the hydraulic support, firstly, according to the structural parameters of the hydraulic support and the relationship between the top beam position and the posture of the adjacent support, the self-pose solution model and the relative-pose solution model of the hydraulic support are constructed.
2.1. Self-Pose Solving Model of Hydraulic Support
When studying the position of the hydraulic support itself, the number of degrees of freedom of the hydraulic support is analyzed to determine the dimension of the control variable. The degree of freedom of hydraulic support can be calculated according to GK formula [
26] as shown in Equation (1).
where
F is the degrees of freedom of the system;
n is the number of elements of the system;
m is the number of motion pairs;
fi is the relative degrees of freedom of the
i-th motion pair;
v is the redundant constraints of the system;
q is the number of local degrees of freedom of the system.
It can be seen from Equation (1) that this type of hydraulic support has two degrees of freedom, and the balance cylinder and column cylinder are taken as the testing objects. The current position of the hydraulic support can be obtained by calculating the posture control schematic diagram of the hydraulic support, as shown in
Figure 2.
Based on this, the coordinates of each hinged point of the hydraulic support can be expressed in which point A, point B, point M, point P, point Q and point N are fixed points, and their coordinate values are as follows: the coordinate of point P is , the coordinate of point Q is , the coordinate of point N is , the coordinate of point A is , the coordinate of point B is , and the coordinate of point M is . Next, the coordinates of the active point will be solved.
Point C is the hinged point between the front rocker and the shield beam, and its coordinates are shown in Equation (2).
where
is the angle between front rocker and horizontal plane.
Point D is the hinged point between the rear rocker and the shield beam, and its coordinates are shown in Equation (3).
where
is the angle between the rear rocker and the negative direction of the x axis.
Point E is the hinged point between the top beam and the shield beam, and its coordinates are shown in Equation (4).
where
is the angle between the connecting rod CD and the positive direction of the x axis;
is the angle between the connecting rod DE and the connecting rod DC.
Point F is the hinge point between the balance cylinder and the shield beam, and its coordinates are shown in Equation (5).
where
is the angle between the connecting rod ED and the negative direction of the x axis;
is the angle between the connecting rod ED and the connecting rod EC;
is the angle between the connecting rod EC and the connecting line EF.
Point G is the hinge point between the balancing Jack and the jacking beam, and its coordinates are shown in Equation (6).
where
is the angle between the connecting rod EG and the connecting rod EF.
Point H is the fixed point between the connecting rod HF and the shield beam, and its coordinates are shown in Equation (7).
The coordinates of point I are shown in Equation (8).
where
is the angle between the connecting rod EI and the connecting rod EG.
The coordinates of point J are shown in Equation (9).
where
is the angle between the top beam and the positive direction of the x axis.
The coordinates of point K are shown in Equation (10).
The coordinates of point L are shown in Equation (11).
where
is the angle between the column cylinder ML and the positive direction of the x axis.
The coordinates of point R are shown in Equation (12).
Point
is the origin of the coordinate system
of the top beam relative position and posture detection device, and its coordinates are shown in Equation (13).
Suppose
. According to the knowledge of trigonometric functions and algebraic geometry, it can be expressed as Equation (14) to Equation (23).
Write the vector ring in the
Figure 2 as a phasor,
According to the Euler Formula (26),
Equations (24) and (25) are simplified, respectively, and Equations (27) and (28) can be obtained.
When the real part and the imaginary part are equal, respectively, in Equation (27), Equation (29) can be obtained.
When the real part and the imaginary part are equal, respectively, in Equation (28), Equation (30) can be obtained.
The above equations can be combined to obtain the system of equations shown in Equation (31)
where
,
,
,
are known position angles,
,
,
,
,
,
,
and
are displacement and position angles to be measured. Displacement sensors are used to measure the displacement of bracket posts and balance cylinders, so
and
can be known. Equation (31) is a set of nonlinear equations about (
,
,
,
,
,
). When Equation (31) is solved, the position of the hydraulic support itself can be obtained, and the relative-pose detection of two adjacent hydraulic supports can be realized.
2.2. Relative-Pose Solving Model of Hydraulic Support
When taking the top beam of hydraulic support as the research object, a hydraulic support is selected as the reference support, and the adjacent support is the hydraulic support to be tested. The fixed coordinate system is established on the top beam of the reference hydraulic support, and the top beam of the hydraulic support to be tested is defined as points P, A, B, respectively. If the coordinates of points P, A and B in the fixed coordinate system can be solved, the position and posture of the hydraulic support relative to the reference hydraulic support can be obtained. The principle diagram of the top beam position and posture detection of the hydraulic support is shown in
Figure 3.
The coordinates of points P, A, B can be obtained through calculating the module length and bearing of the vectors , , . Based on the existing technology, it is not convenient to solve the mode length and bearing of three vectors at the same time. In that way, three coordinate systems are established on the top beam of the benchmark hydraulic support, which are defined as coordinate, coordinate and coordinate, and each coordinate system solves a vector, respectively. After calculating the coordinates of points P, A and B in their respective coordinate systems, they can be transformed into the fixed coordinate system by coordinate transformation from which the coordinates of points P, A and B can be obtained.
As shown in
Figure 4, taking the solution of the point P coordinate in
as an example, the angle between the projection of the vector
in the plane
and the y axis is defined as the angle
. The angle between the projection of
in the plane
and the y axis as the angle
. The module length of the vector
is
. Then, the vector
coordinates are uniquely determined by them, and the coordinate solution formula of point P is as shown in Equation (32)
The coordinates of point P are defined as
, the coordinates of point A are
, and the coordinates of point B are
. When only the coordinate translation transformation is considered and the rotation transformation is not considered, the coordinate solution formulas of points P, A and B are as follows:
where Matrix D is shown in Equation (34)
In the formula, , and are respectively the deflection angles of points P, A and B around the x-axis, and , and are the deflection angles of points P, A and B around the y axis, respectively. From Equation (33), the coordinates of points P, A and B in the coordinate system can be obtained and then the relative positions of the top beam of the hydraulic support and the top beam of the reference hydraulic support can be obtained.