Traditionally, the construction of tunnel support is to manually move the manipulator through the remote control. During the manual construction process, there is a large rebound rate and a reduction in the compactness of the concrete. Skilled operators can obtain better shot blasting quality, but training of a skilled worker is costly. In this study, an automatic shotcrete manipulator is proposed to complete spraying and improve the spraying quality. The model of the manipulator is simplified into a planar two-link manipulator, solved the inverse kinematics of the shotcrete manipulator, and implemented the automatic trajectory planning process. By simplifying the complex mechanical arm structure model into a simple planar two-link manipulator model, the inverse kinematics of the shotcrete manipulator is solved, and automatic trajectory planning software is developed, which can better help construction and reduce injury to personnel’s health and can also help the construction party save materials and improve economic efficiency and construction efficiency.
Due to the large number of tunnel excavations and constructions in the construction of subways, high-speed rail, trains, and highways, the use of concrete wet spray machines is required to complete the tunnel support. The State Council of China had issued and implemented the “13th Five-Year Plan for the Development of Modern Comprehensive Transportation System” on February 3, 2017, see Table
Main indicators for the development of comprehensive transportation during the “13th Five-Year Plan.”
Operating mileage | 2015 (year) | 2020 (year) | Increase |
Railway operating mileage (10,000 km) | 12.1 | 15 | 2.9 |
High-speed railway operating mileage (10,000 km) | 1.9 | 3 | 1.1 |
Highway operating mileage (10,000 km) | 458 | 500 | 42 |
Expressway operating mileage (10,000 km) | 12.4 | 15 | 2.6 |
Operating mileage of urban rail transit (km) | 3300 | 6000 | 2700 |
Construction site.
Many algorithms have been proposed for the automatic control of industrial robotic arms, involving inverse kinematics, trajectory planning, motion planning, force control, and so on. Some general 6-degree-of-freedom inverse kinematics can solve the inverse solutions of some robotic arms that meet the Pieper principle [
Many researchers have studied the remote operation mode of the shotcrete manipulator, data acquisition during shotcrete, simulators, etc. These studies can help workers perform shotcrete [
The objective of this research is to model and develop an analytical solution of the shotcrete trolley of Henan Gengli Engineering Equipment Company. The manipulator has 7 degrees of freedom. By reducing the complex structure model of the robot arm to a simple planar two-link manipulator model, the inverse kinematics can be solved based on the analytical solution. Based on this, the motion trajectory of shotcrete was planned, and the control platform was developed. The structure of this paper is as follows. Section
The shotcrete manipulator depicted in Figure
Structural model of the shotcrete manipulator.
The Denavit and Hartenberg (D-H) modeling method was used to construct the coordinate system of the shotcrete manipulator as shown in Figure
Kinematic model of the shotcrete manipulator.
D-H parameters for the shotcrete manipulator.
Joint no. | Angle limit ( | ||||
1 | 0 | 0 | 0 | −180∼180 | |
2 | 90 | 0 | 0 | 0∼70 | |
3 | 0 | −20∼110 | |||
4 | −90 | 0 | 0 | 0∼4600 | |
5 | 0 | 0 | 0 | −180∼180 | |
6 | 90 | 0 | 0 | −120∼120 | |
7 | 0 | 0 | − |
The initial values are
In the inverse kinematics solution, the coordinate system
The vector
The 1 to 4 numbers of the joint of the manipulator (joint 1, joint 2, joint 3, and joint 4) are described as seen in Figure
Simplified manipulator model 1: (a) top view; (b) front view.
Separating variables
Let (2, 1) and (2, 2) elements on both sides of equation (
Equations (
By dividing equations (
When computing
The four joint variables determine the position of the tip nozzle and their functional relationship is:
Detailed description of their specific relationship is given through the following formulas:
As explained earlier, only three joint angles are required to determine the position of the end nozzle. There are four joint angles, and redundant degrees of freedom are introduced. The following is an analysis of the motion trajectory of the shotcrete manipulator to add constraints so that the equation can be solved. There are formulas for calculating joint angle 2 and joint angle 3 without adding constraints; however, they cannot really calculate the joint value:
Two-link model.
The moving area of the spray nozzle at the end of the shotcrete manipulator is divided into four pieces (see Figure
Changes in the joint angle during nozzle movement.
In the aforementioned simplified kinematics model, we ignored the offset value
Two-link model (
Then, we can compute the virtual desire position (
To improve the performance of the manual operation of the shotcrete manipulator, reasonable trajectory planning of the spraying robotic arm is required. The nozzle needs to maintain a distance of 1.5–2 m from the sprayed surface, and the nozzle needs to be vertical to the sprayed surface. Therefore, the end nozzle needs to control position and attitude parameters to represent its spatial pose. For the position parameters, it needs to control 3 parameters (
Roll (
Illustration tunnel geometry.
Tunnel models of various sizes were not considered in this research because environmental sensing such as Lidar is not incorporated. It is assumed that the geometric model of the tunnel is a semicircle, and the real blasted surface is not considered when trajectory planning is performed. The attitude of the nozzle is calculated by a mathematical model. In general, the trajectory of the nozzle is rectangular, and this geometry is used in the present study. By manually adjusting the shotcrete manipulator to specify the initial position and the distance between the tunnel trajectories, path length, and tunnel geometry, automatic trajectory planning program can generate a complete shot trajectory (see Figure
Desired shotcrete trajectory.
During the inverse kinematics solution, the movement of some joints is restricted, and artificially added constraints may cause joint values to jump during the movement (see Figure
The problem of movement of the nozzle: (a) nozzle work in mode 1 and (b) nozzle work in model 2.
Solutions to problems in movement of the nozzle: (a) change mode 1 to mode 2 and (b) change model 2 to mode 1 and move in a circular path.
The inverse kinematics solution calculates the current joint angle value based on the previous joint angle vector value and the current motion mode. As it can be seen from Table
Solutions of inverse kinematics in mode 1.
Solution no. | ||||||
1 | 0.080018 | 0.456693 | −0.227912 | 3.406000 | 0.757625 | 1.029767 |
2 | 0.079828 | 0.456685 | −0.229316 | 3.406000 | 0.758306 | 1.030003 |
3 | 0.079638 | 0.456667 | −0.230703 | 3.406000 | 0.758987 | 1.030239 |
4 | 0.079448 | 0.456642 | −0.232074 | 3.406000 | 0.759668 | 1.030475 |
5 | 0.079258 | 0.456607 | −0.233430 | 3.406000 | 0.760349 | 1.030712 |
6 | 0.079068 | 0.456566 | −0.234769 | 3.406000 | 0.761029 | 1.030949 |
7 | 0.078879 | 0.456516 | −0.236094 | 3.406000 | 0.761710 | 1.031188 |
8 | 0.078689 | 0.456458 | −0.237403 | 3.406000 | 0.762390 | 1.031426 |
Pose no. | ||||||
1 | 6.157767 | 0.947736 | 2.294252 | −0.773213 | −0.150869 | 2.833495 |
2 | 6.159044 | 0.946654 | 2.289542 | −0.773320 | −0.152263 | 2.833642 |
3 | 6.160322 | 0.945572 | 2.284829 | −0.773430 | −0.153649 | 2.833789 |
4 | 6.161599 | 0.944491 | 2.280118 | −0.773545 | −0.155028 | 2.833936 |
5 | 6.162878 | 0.943409 | 2.275403 | −0.773662 | −0.156400 | 2.834083 |
6 | 6.164155 | 0.942327 | 2.270694 | −0.773784 | −0.157764 | 2.834230 |
7 | 6.165432 | 0.941244 | 2.265984 | −0.773910 | −0.159120 | 2.834378 |
8 | 6.166711 | 0.940163 | 2.261270 | −0.774039 | −0.160470 | 2.834526 |
Solutions of inverse kinematics in mode 2.
Solution no. | ||||||
1 | 0.005281 | 0.046679 | −0.031888 | 3.440453 | 1.028433 | 1.200126 |
2 | 0.005119 | 0.046679 | −0.033265 | 3.441585 | 1.029013 | 1.200762 |
3 | 0.004957 | 0.046679 | −0.034638 | 3.442723 | 1.029592 | 1.201399 |
4 | 0.004794 | 0.046679 | −0.036011 | 3.443869 | 1.030171 | 1.202038 |
5 | 0.004632 | 0.046680 | −0.037382 | 3.445021 | 1.030749 | 1.202678 |
6 | 0.004470 | 0.046682 | −0.038755 | 3.446179 | 1.031327 | 1.203319 |
7 | 0.004308 | 0.046683 | −0.040128 | 3.447345 | 1.031905 | 1.203963 |
8 | 0.004146 | 0.046682 | −0.041498 | 3.448517 | 1.032482 | 1.204607 |
Pose no. | ||||||
1 | 6.700704 | 0.487896 | 0.291913 | −0.999891 | −0.300959 | 2.949828 |
2 | 6.701981 | 0.486814 | 0.287199 | −1.000311 | −0.301886 | 2.950126 |
3 | 6.703259 | 0.485732 | 0.282486 | −1.000733 | −0.302810 | 2.950425 |
4 | 6.704536 | 0.484650 | 0.277768 | −1.001155 | −0.303733 | 2.950724 |
5 | 6.705813 | 0.483568 | 0.273064 | −1.001579 | −0.304650 | 2.951024 |
6 | 6.707091 | 0.482486 | 0.268356 | −1.002004 | −0.305566 | 2.951324 |
7 | 6.708368 | 0.481404 | 0.263644 | −1.002430 | −0.306481 | 2.951625 |
8 | 6.709646 | 0.480322 | 0.258920 | −1.002857 | −0.307394 | 2.951926 |
Solutions of inverse kinematics between mode 1 and mode 2.
Solution no. | ||||||
1 | 0.094466 | 0.541054 | −0.283932 | 3.488624 | 0.706087 | 1.013583 |
2 | 0.094271 | 0.541054 | −0.285317 | 3.488475 | 0.706780 | 1.013781 |
3 | 0.094076 | 0.392774 | 0.000274 | 3.406000 | 0.707473 | 1.013979 |
4 | 0.093880 | 0.399768 | −0.014836 | 3.406000 | 0.708166 | 1.014177 |
5 | 0.011003 | 0.058919 | −0.007261 | 3.406000 | 1.007931 | 1.178817 |
6 | 0.010838 | 0.046679 | 0.015329 | 3.406000 | 1.008522 | 1.179401 |
7 | 0.010674 | 0.046678 | 0.013928 | 3.406900 | 1.009113 | 1.179987 |
8 | 0.010509 | 0.046681 | 0.012523 | 3.407808 | 1.009704 | 1.180574 |
Pose no. | ||||||
1 | 6.061955 | 1.028885 | 2.647606 | −0.736393 | −0.116284 | 2.818948 |
2 | 6.063232 | 1.027803 | 2.642895 | −0.736463 | −0.117689 | 2.819082 |
3 | 6.064507 | 1.026721 | 2.638190 | −0.792489 | 0.007869 | 2.822333 |
4 | 6.065786 | 1.025639 | 2.633476 | −0.789817 | 0.000319 | 2.822551 |
5 | 6.655990 | 0.525765 | 0.456821 | −0.983323 | −0.278522 | 2.938954 |
6 | 6.657269 | 0.524683 | 0.452097 | −0.986286 | −0.268043 | 2.939934 |
7 | 6.658546 | 0.523601 | 0.447382 | −0.986667 | −0.269050 | 2.940218 |
8 | 6.659823 | 0.522519 | 0.442673 | −0.987049 | −0.270055 | 2.940503 |
A linear motion path containing two motion modes and conversion between modes is selected to verify the proposed inverse kinematics algorithm. Tables
The end position is
Based on the correct inverse kinematics algorithm and trajectory planning algorithm, we use OpenGL and Qt to develop host computer control software for controlling the shotcrete manipulator (see Figure
Robot control platform software interface.
By setting reasonable set of parameters, control software generates a set of executable trajectories, and simulations in software show this process (see Figure
Motion simulation of the shotcrete manipulator.
A simple and effective inverse kinematics calculation method was proposed. Based on this inverse kinematics, trajectory planning of the shotcrete manipulator was realized. The two-link manipulator model is used to solve the inverse kinematics of the 6-degree-of-freedom shotcrete manipulator. The process is simple and efficient, does not require a large number of matrix operations, and can easily be obtained.
A combination of the motion trajectory shotcrete manipulator and its control principles is presented. Therefore, reasonable constraints enable the inverse kinematics of the robotic arm to be solved. In the simplified geometry model, it is described that, by giving some initial parameters, spraying trajectory can be automatically generated. Based on the inverse kinematics solution and automatic trajectory generation proposed, manipulator control software was developed using OpenGL and QT, which provides a friendly human-computer interaction interface.
In this study, a simple geometry tunnel model is used. In the future, the real tunnel model will be obtained by sensing the environmental information with Lidar. At the same time, the normal vector of the tunnel section after blasting can be calculated to give the attitude of the tip nozzle, which can be more perpendicular to the sprayed surface to a large extent, ensuring a smaller rebound rate and better compaction.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors thank Sha Hao and Song FeiFei for their help in writing the article and colleague Wang Hu for his help in the work. This research was supported by the Changsha Important Science and Technology Specific Projects (grant no. kq1703022).