1. Introduction
Truck-mounted cranes can be applied to missile weapon equipment transportation, semi-automatic missile loading systems, hazardous environment rescue, and fruit tree and agricultural product spraying and harvesting. When applied to missile weapon equipment, missiles require transportation from production to combat training and battlefield use, including ship transport, vehicle transport, and personnel transport. Equipment reliability is closely related to the quality of its transportation performance, and the transportation environment affects missile performance. In semi-automatic missile loading systems, the function is to load missiles ready for launch onto the launch pad. The loading device mainly consists of a machine that loads or unloads missiles from the launch pad. To replace manual loading and ensure missiles automatically enter their precise positions on the launch pad, the missiles to be loaded are first transported to the corresponding positions on the launch pad, and then, through a simple rotation, the missiles are loaded individually or simultaneously onto the corresponding launch pads. In the field of rescue robots, facing increasingly complex and dangerous rescue strategies and frequent man-made and natural disasters worldwide, robots assisting or even replacing humans in rescue work has become an important issue and means of rescue efforts. The application of robots in agriculture can greatly reduce labor intensity, increase labor productivity, and solve the problem of labor shortage. Modern agricultural robots are machines that integrate multiple technologies such as detection technology, sensing technology, artificial intelligence technology, image recognition technology, and communication technology.
The kinematic model of a truck-mounted crane's robotic arm is fundamental to structural analysis. It facilitates the analysis of the mechanism's velocity and acceleration, and provides a convenient basis for studying dynamic-related problems. Common methods include the DH parameter representation and the homogeneous transformation matrix method. The advantage of this method is its ease of use in calculating the Jacobian matrix and force analysis. However, it has a fundamental technical problem: while it can fully describe robotic arms that move only along the x and z axes, it becomes inapplicable when considering motion along the y-axis. Furthermore, the mechanical mechanisms of automatic loading systems for truck-mounted cranes are complex, often involving multiple joints such as rotation, slewing, and extension. This poses significant challenges to setting joint coordinates using the DH parameter method. Therefore, employing the homogeneous transformation rotation matrix method for kinematic modeling of hydraulic truck-mounted crane robotic arms used in outdoor operations offers a simpler and more intuitive approach.
The end effector of a truck-mounted crane's robotic arm performs trajectory planning in a Cartesian coordinate system, while the drive device controls the robotic arm's movement by controlling joint variables. Traditional methods for solving the inverse kinematics of a robotic arm include the method of separation of variables, geometric methods, and iterative methods. For robotic arms with complex structures and dimensions, the control calculations are complex due to the high coupling between attitude and position, increasing the difficulty of inverse kinematics and making variable separation and planar geometric decomposition impossible, thus requiring the use of numerical algorithms. When joint variables change linearly, the trajectory of the robotic arm's end effector is not a straight line, and trajectory planning can be achieved in joint space. In this case, interpolation calculations are required for the joint variables. Commonly used interpolation algorithms include polynomial function interpolation, spline function interpolation, and linear interpolation.
2. Establishment of the DH method mathematical model
This paper first establishes a coordinate system using the Denavit-Hartenberg method (DH method), and then derives the motion equations of the robotic arm based on the coordinate system established by the DH method. The DH method is a matrix method for establishing relative pose. Proposed by Denavit and Hartenberg in 1995 and named after them, the DH method uses homogeneous transformations to describe the spatial geometric relationships of each link relative to a fixed reference frame. The spatial relationship between two adjacent links is described by a 4x4 homogeneous transformation matrix, thus allowing the derivation of the coordinate transformation matrix of the end effector's coordinate system relative to the base coordinate system, and establishing the motion equations of the manipulator.
To solve the forward kinematics of a robotic arm, the angles of each joint must be known. Then, the desired pose of the end effector must be calculated based on the angles of each joint. The standard method for studying robotic arm kinematics is to use the DH method for analysis and modeling. The basic idea is: first, establish a reference coordinate system at each joint of the robotic arm; second, determine the transformation matrix between any two adjacent coordinate systems and multiply the resulting matrices sequentially; finally, obtain the total transformation matrix from the base to the end effector.
Figure 1. Coordinate system of a six-DOF modular robotic arm
A robotic arm can be viewed as a series of links connected by joints. Based on the structural characteristics of this robotic arm, the DH method is used to establish a base coordinate system and coordinate systems for each joint. A six-degree-of-freedom robotic arm is a spatial mechanism with six joints. Only by establishing coordinate systems at each joint can the attitude and position of the robotic arm's end effector in space be described. Figure 1 shows the coordinate system of a six-degree-of-freedom modular robotic arm.
Substitute the DH parameters into the following formula:
3. Trajectory Planning
Figure 2. Cartesian spatial trajectory planning curve
4. MATLAB simulation
MATLAB not only complements the general functions of software, such as processing and plotting 2D curves and 3D surfaces, but also demonstrates outstanding processing capabilities in areas not found in other software, such as color processing, 4D data processing, and lighting manipulation. MATLAB also features data visualization capabilities, a feature inherent from its development, which facilitates user interaction by expressing matrices and vectors graphically. Advanced plotting, including 2D and 3D image processing, visualization, expression, and animation, can be used for engineering drawing and scientific computing. MATLAB has also designed corresponding functions for users at different levels who require specific graphical interfaces.
Kinematic simulation of a robotic arm is performed. To make the robotic arm's movements more three-dimensional and intuitive, the simulation results can be represented graphically. This includes whether the robotic arm can meet the spatial operation requirements, whether the movements are smooth, and whether there are any interference phenomena. Compared with simulation using data curves or the data itself, graphical simulation can analyze a large amount of important information, while simulation using data curves or the data itself cannot reveal the robot's motion laws.
Figure 3. Initial pose of the robotic arm in forward kinematics modeling.
Figure 4. End-effector pose of the eighth robotic arm
The vertical grasping trajectory was planned using interpolation. This paper simulates the spatial curve trajectory, velocity, and acceleration. Figure 5 shows the spatial curve trajectory. Figure 6 shows the joint velocity curve.
Figure 5 Spatial Curve Trajectory
Figure 6 Joint velocity curves
5. Conclusion
This paper uses the DH method to establish a coordinate system for forward kinematics analysis of a truck-mounted crane, solving for the DH parameters of the robotic arm. Based on the coordinate position of the end effector of the truck-mounted crane robotic arm and the forward kinematic solution, this paper also uses interpolation to plan the trajectory of the truck-mounted crane robotic arm. The kinematics is divided into seven stages using the Cartesian space trajectory planning method, and the uniform velocity stage and the variable acceleration stage are solved separately. Finally, this paper simulates the forward and inverse kinematics, setting the length of the rod and the assumed joint degrees to obtain eight sets of solutions. Comparing these eight sets of solutions with the original data, it is found that the two sets of data are similar, which fully proves that the solution of the inverse kinematics is correct.
For more information, please visit the Industrial Robots channel.
