1. Introduction
Mining electric shovels (MES), also known as electro-mechanical excavators and electric rope shovels, are widely used in open-pit mining operations because of their compact structure, high working capacity, large digging range and adaptability to the environment [
1]. The working devices of MES in the excavation process are mainly the crowd and hoist mechanisms. With the widespread use of MES, more requirements are put forward for the working device of the MES. Therefore, it is necessary to analyze the configuration synthesis of the working device of the MES and provide a method for the structural innovation of the shovel, and thereby to optimize the parameters and improve work efficiency.
The configuration synthesis of mechanisms is an effective tool for mechanism design and structural innovation. The purpose of configuration synthesis is to find the relationship between the motion subsets and the position of the motion subsets in space from the expected degrees of freedom or the expected functional properties, and to design a mechanism type that satisfies the conditions. Herve [
2], a French scholar, first introduced the Lie group theory in the field of configuration synthesis and proposed a method of motion synthesis of displacement subgroups based on group theory. Li et al. [
3] and Meng et al. [
4] analyzed and synthesized parallel mechanisms based on the Lie group of displacements. Gogu [
5,
6,
7] analyzed a configuration synthesis method for parallel mechanisms based on the linear transformation idea. Yang et al. [
8] proposed a configuration synthesis method based on single open chain (SOC) composition principle. Based on the constraint screw theory, Kong and Gosselin [
9,
10] proposed a virtual chain configuration synthesis method to solve the configuration synthesis problem of parallel mechanisms. Han et al. [
11] synthesized four new configurations of annular deployable antennas by using the screw-theory constraint synthesis method. Compared with other configuration synthesis methods, the screw-theory-based constraint synthesis method has the advantages of clear physical meaning, simple mathematical expression and algebraic operations.
Kinematics analysis of the mechanism is the basis for studying the performance of the mechanism, thereby improving it, and designing new mechanisms. Zhao et al. [
12] analyzed the inverse kinematics of a hyper-redundant bionic trunk-like robot by a closed-loop vector equation. Wang et al. [
13] proposed a new 5-DOF parallel mechanism with 5PUS-UPU. The position equations of the mechanism are analyzed by the closed-loop vector method, and the mapping relationships of position, velocity and acceleration are obtained. Based on the closed-loop vector method, Chen et al. [
14] constructed a kinematic model of an over-constrained parallel mechanism and derived the velocity Jacobian matrix of the mechanism. The main analytical methods for the workspace are the boundary search [
15,
16,
17] and the numerical method [
18,
19,
20]. Among them, the numerical method can solve the problem by programming, which has the advantages of high efficiency and accuracy. Therefore, this paper uses the numerical method to analyze the workspace of the MES.
At present, many experts and scholars have carried out a substantial amount of research on the configuration synthesis of mechanisms, but for the time being, there is no report on the configuration synthesis of the MES. In this paper, based on the screw theory, we propose a configuration synthesis method for MES working devices and optimize the mining electric shovel working device with a larger excavation range, and then take the working device as an example for dimensional optimization and simulation analysis. The first part of the paper introduces the background of the study. The second part analyses the configuration synthesis method of the MES working devices. The third part solves the position equation of the working device. In combination with the constraints of the mechanism, the numerical method is used to solve its workspace. The fourth part analyses the influence of rod parameters on the working space and optimizes the working device dimensions based on genetic algorithms. In the fifth part, two kinds of scaled-down model test bench of MES were built. Simulation analysis and experimental verification are carried out on the excavation results of the two MES, the working devices. Finally, a summary of the full paper is presented.
2. Mining Electric Shovel Working Device Configuration Synthesis Based on Screw Theory
2.1. Overview of the Screw Theory
According to the screw theory [
21,
22], any motion can be represented by a screw. A unit screw in space can be denoted as a vector, and its expression is as follows:
where
means the screw;
is the direction vector along the screw axis;
denotes the dual part of the screw;
r is the position vector of any point on the screw axis; and
h is the pitch of the screw.
When the pitch of the screw h = 0, the screw can be used to represent the kinematic screw of the revolute pair (R) in space. When the pitch of the screw h → ∞, the screw can be used to represent the kinematic screw of the prismatic pair (P) in space.
The prismatic (P) and revolute (R) pair of the mechanism are single-degree-of-freedom kinematic parts, while the cylindric pair(C), universal joint (U), spherical pair (S) and other compound parts can be regarded as combinations of single-degree-of-freedom kinematic parts.
For any two screws
and
in space, the reciprocal product equation is
where
—the reciprocal product operator.
If the solution of the equation is 0, the screw and are said to be reciprocal. The reciprocal screw theory shows that if screw , , ,…, represents the kinematic screw of a branch chain of the mechanism, then screw represents the constraint screw of that branch chain.
The configuration synthesis method based on screw theory is as follows: Firstly, the motion screw system of the mechanism is listed according to the expected degrees of freedom of the mechanism. Then, based on the reciprocal screw theory, the constraint screw system of the mechanism is found, and the kinematic screw and constraint screw of the branched chain are calculated. Finally, the branch chain structure is constructed, and the branch chain structure is optimally configured to obtain the new configuration of the mechanism.
2.2. Analysis of Mining Electric Shovels: A Working Device
The working devices of the MES are mainly the crowd and hoist mechanisms, of which the main components are boom, saddle, push shaft, bucket rod, pushing gear, bucket, sheave pulley and lifting wire rope, as shown in
Figure 1.
With the boom as the frame, and the bucket rod and bucket as the moving platform, the structure of the rack and pinion in the pushing mechanism is simplified to one prismatic (P) and one revolute (R) pair, and the structure of the lifting wire rope and sheave pulley in the lifting mechanism is simplified to one prismatic (P) and two revolute (R) pairs. The brief diagram is shown in
Figure 2.
In
Figure 2, the motion screw system OXYZ is established with the point A as the coordinate origin, where the X-axis is perpendicular to the paper surface and faces outward, and the Y-axis and Z-axis represent the horizontal and vertical directions, respectively. In the working device of the MES, subchain I is the RP branch chain structure, and subchain II is the RPR branch chain structure. Based on the screw theory, the kinematic screw of each branch chain structure in the working device is analyzed, and then the constrained screw of each branch chain can be found according to the reciprocal product equation, as shown in
Table 1. In
Table 1,
ai and
bi (i = 1, 2, 3, 4) represent the coordinates of a point on the axis of the screw.
A combined analysis of the constrained screw system of subchain I and subchain II shows that the moving platform of the working device will be subject to four constraint screws:
Therefore, there are three common constraints for the moving platform of the working device, namely
,
and
. The degrees of freedom of the working device can be calculated according to the GK formula [
23]:
Combined with the constraint screw of the moving platform, it can be seen that the moving platform is subject to four independent constraints. Therefore, to obtain a new configuration of the MES requires one rotational degree of freedom and one translational degree of freedom.
2.3. Configuration Synthesis of MES, in a Working Device
As the moving platform of the working device of the MES is subject to four independent constraints, there are two ways of synthesizing its mechanism type. Integrated solution I: a new configuration of the electric shovel is completed using one branch chain to provide all the constraints and two six-degree-of-freedom branch chain structures to provide the drive. Integrated solution II: consisting of two branched chains, the two active branched chains apply drive and restraint together.
2.3.1. Synthesis Scheme I
New configurations of MES are synthesized by a three-bar strut structure. In order to simplify the analysis, the structure of the passive branch is considered as the RP structure, and a new configuration is synthesized by using a 6-degree-of-freedom branch structure with two connecting rods and three kinematic pairs. The active branches of 6 degrees of freedom are shown in
Table 2.
According to the screening conditions of the branch chain, some unreasonable branch structure types are removed:
(1) It is necessary to ensure that the mechanism can maintain the advantages of high stiffness and low inertia.
(2) The kinematic pair connected to the moving platform needs to be connected by the spherical pair as much as possible, so that the force of the connecting rod and the moving platform is simple.
The RUS, PUS, UPS, URS, USR, USP, RSS, PSS, CUS and SCS active branched chain structures that satisfy the conditions are obtained through optimization screening. Two active branched chains were selected from these branched chain structures, and the RP passive branched chain structure was added to form a variety of new mining shovel configurations. The MES working devices of RUS-RSS-RP, SCS-PSS-RP, URS-RUS-RP and SCS-PSS-RP synthesized by this method are shown in
Figure 3.
2.3.2. Synthesis Scheme II
We use two branch chain structures with few degrees of freedom to synthesize the new configuration of the MES working device. Considering the specialty of the constraint screw
, and to simplify the complexity of the mining shovel configuration synthesis, branch chain 1 is selected as either an RP or SP structure, The structure of branch chain 2 provides constraints such as
,
and
that branch chain 1 missed. The branch-chain-structure type of the configuration scheme II of the MES working device is obtained, as shown in
Table 3.
The MES working devices of RP-RPS, RP-RRR, SP-PRR and SP-RRR synthesized by this method are shown in
Figure 4.
In summary, considering the similarity of the mechanism and the reliability of the bar mechanism, taking the RP-RRR MES working devices as an example, the kinematics and workspace performance of the mechanisms are analyzed and studied. The RP-RRR MES working devices use a linkage mechanism to lift the bucket, replacing the lifting mechanism, consisting of lifting wire rope and sheave pulley in the MES working device, as shown in
Figure 5.
6. Conclusions
(1) For the first time, the screw theory is applied to analyze the problem of configuration synthesis of MES device. Based on the constrained synthesis method of screw theory, the working device of the MES is analyzed. By synthesizing and optimizing the branched chain structure, 100 kinds of MES working device synthesis scheme I, and 14 kinds of MES working device synthesis scheme II were obtained.
(2) The position equation of the RP-RRR-type MES working device is solved by the closed-loop vector equations, and the correctness of the kinematics equation is verified by simulation analysis. The numerical method is used to calculate the point set in the space that satisfies the constraint conditions, which is used to describe the workspace of the MES working device.
(3) The influence of bar parameters on the working space is analyzed. The genetic algorithm is used to optimize the parameters, and the working space of the optimized working device is increased by 13.4789%. The scale model test bench of MES and RP-RRR-type MES is built, and the excavation test is carried out. By comparing the experimental results with the simulation results, the feasibility of the RP-RRR MES working device is verified, and the effectiveness of the comprehensive method is proved.
This research can provide new ideas and methods for the mechanism design and innovation of excavators, and also provide a theoretical basis for the intelligent and unmanned development of the mechanism, which has certain feasibility and applicability. In subsequent research, the energy consumption of RP-RRR-type MES working devices can be further analyzed. On this basis, combined with the motion/force transmissibility, the high-quality workspace of the MES mechanism can be further analyzed and optimized to better meet the actual use requirements.