Abstract: This paper studies the force/position control problem of a robotic arm using an impedance control algorithm. To address the issues of contact force control between the end effector and the environment, and joint trajectory control being affected by inaccuracies in robotic arm dynamics, this paper establishes a robotic arm simulation model using the Simulink/SimMechanics module. This reduces the complexity of the robotic arm dynamics modeling process and mitigates modeling inaccuracies caused by various factors. Simultaneously, an impedance control algorithm is applied to solve the force/position control problem. Finally, a two-link robotic arm simulation model is constructed using Simulink/SimMechanics, and the designed impedance control method is used to conduct a simulation study on the force/position impedance control of the robotic arm. Simulation results show that the constructed robotic arm simulation model can realistically reproduce the dynamic characteristics of the robotic arm, and the designed impedance control method can effectively achieve force/position control of the robotic arm.
Keywords : Simulink/SimMechanics; impedance control; robotic arm; simulation study
1. Introduction
With the rapid development of machining and manufacturing, industrial robots are being used more and more widely in various fields, and the tasks they perform are becoming increasingly complex. Simple robotic arm position control can no longer meet the demands of industrial production. For example, in processes such as polishing, grinding, and deburring, it is necessary to control both the position of the robotic arm's end effector and the contact force generated when the end effector comes into contact with the working environment. Therefore, researching robotic arm force/position control, based on traditional robotic arm position control, is of great significance.
There are two basic methods for general robotic arm force/position control: the first is robotic arm impedance control; the second is robotic arm force/position hybrid control. Hogan proposed the basic theory of impedance control in the literature [1]. The literature points out that the impedance control algorithm maintains a mechanical impedance relationship between the position deviation of the robotic arm and the contact force between the end effector and the environment, so that the robotic arm exhibits compliance. Seul improved on the basis of Hogan's research [2]. Seul integrated the position control algorithm into the impedance control algorithm and used the torque compensation information of the previous sampling point to offset the uncertainty of the dynamic equation. Chan and Yao et al. introduced the sliding mode control method into the impedance control algorithm [3]. The sliding mode contains an ideal impedance relationship, but the implementation of this control method is based on the premise of accurate knowledge of the external environment.
Impedance control is an indirect algorithm for solving the force/position control of robotic arms. It indirectly controls the contact force between the end effector and the environment by controlling the end effector's position offset. This paper, based on research into robotic arm position control, applies the impedance control algorithm to study and analyze the force/position control problem of robotic arms, accomplishing tasks that cannot be achieved by single-function position control. An impedance model is established based on the relationship between the contact force between the robotic arm's end effector and the environment and the end effector's position offset. An impedance control law is designed, the stability of the impedance control system is analyzed, and a mechanical system simulation model of the robotic arm is built using the Simulink/SimMechnics module in Matlab. This avoids the complexity and inaccuracies in the mathematical modeling of robotic arm dynamics, allowing the constructed model to more effectively reproduce the characteristics of the robotic arm. Finally, the developed control algorithm is used to simulate the force/position control problem of the robotic arm. Simulation results show that the developed impedance control scheme for robotic arms is effective and feasible.
2. Impedance Controller Design and Analysis
2.1 Dynamics Model of the Robotic Arm
A typical n-joint robotic arm can be represented by a second-order nonlinear dynamic equation as follows [4]:
2.2 Impedance Controller Design
Impedance control simulates the interaction between a robotic arm and its environment as an impedance model, which can be simplified to a damped spring model. The impedance model of a robotic arm differs from its dynamic model; it reflects the relationship between the positional deviation of the robotic arm's end effector and the force applied at the end effector. Its task is to control the contact force between the robotic arm's end effector and the external environment while simultaneously achieving trajectory tracking, thus realizing a strategy of simultaneous force/position control of the robotic arm.
According to reference [5], the target impedance equation of the robotic arm is:
(2-3)
Where: Md, Bd, and Kd are the target inertia, damping, and stiffness matrices of the robotic arm, respectively; is the difference between the desired position and the actual position of the robotic arm; and is the end-effector contact force.
Figure 2-1 shows the schematic diagram of the impedance control principle of the robotic arm:
As shown in the impedance control principle diagram, the impedance control algorithm converts the forces acting on the robotic arm's end effector and the environment into positional deviations through an impedance controller. These positional deviations are then calculated using forward and inverse kinematics transformations to convert them into joint angles of the robotic arm, which are then fed into the robotic arm's position controller. This indirectly achieves end-effector force control of the robotic arm through position control.
Based on Figure 2-1, the control input torque of the robotic arm can be designed as follows:
3. Simulation Analysis and Research of Two-Link Robotic Arm
3.1 Two-link robotic arm model structure
The industrial robotic arm studied in this paper is shown in Figure 3-1. This robotic arm is a two-link rotary joint type industrial robotic arm, which consists of a series of rigid links connected by flexible rotary joints. The two-link robotic arm mainly consists of a rotatable base, an upper arm, a lower arm, and a robotic hand. For convenience, the role of the robotic hand is ignored in this paper. In Figure 3-1, d1 and d2 represent the base part of the robotic arm (this part is ignored in the simulation of this paper), l1 and l2 represent the lengths of the two links of the robotic arm, respectively, and q1 and q2 represent the joint angles of the two links of the robotic arm, respectively.
Figure 3-1 Robotic arm model
3.2 Simulink/SimMechanics Model Establishment of Two-Link Robotic Arm
Due to the nonlinearity, complexity, and strong coupling of robotic arm systems, their dynamic equations are often difficult to obtain precisely. Therefore, this paper uses the Simulink/SimMechanics module in Matlab to build a simulation model of the robotic arm, avoiding the adverse effects of inaccurate dynamic models in simulation research. SimMechanics is a cross-disciplinary research and analysis environment for controllers and object systems based on Simulink. SimMechanics provides intuitive and effective modeling and analysis tools for multibody dynamic mechanical systems and their control systems, with all work completed within the Simulink environment. SimMechanics provides forward and inverse kinematics simulation analysis, forward and inverse dynamics simulation analysis, linearization analysis, and equilibrium point analysis for mechanical systems, among other things.
After selecting all the necessary modules from the SimMechanics toolbox and placing them into the Simulink working environment, first set the environment parameters; second, set the input and output of the drive module and the initial position of the two links; finally, set the sensors for the quantities that need to be measured in the simulation. The established two-link robotic arm model is shown in Figure 3-2.
Figure 3-2 Two-link robotic arm model
3.3 Simulation of the Impedance Control System of the Robotic Arm
Based on the analysis and research on impedance control in Section 2, the designed impedance control system for the robotic arm is shown in Figure 3-3.
Simulation results show that the angle changes of the two joints of the robotic arm follow the sinusoidal law very well, and the position tracking error also decreases with increasing simulation time. Simulation experiments demonstrate that this control method can achieve position tracking of the robotic arm.
The simulation results of the contact force control at the end of the robotic arm are shown in Figure 3-6.
From the perspective of the joint position of the robotic arm, as shown in Figures 3-4 and 3-5, the actual trajectory of the robotic arm end effector at the two joint angles q1 and q2 can track the expected trajectory of the robotic arm end effector quite well, and can achieve the goal of the actual trajectory of the robotic arm end effector tracking the expected trajectory in a sinusoidal pattern.
Analyzing the contact force at the end effector of the robotic arm, as shown in Figure 3-6, the simulation results demonstrate that the contact force at the end effector eventually reaches a constant value, and the final force control curve tracks the desired rectangular wave change. Simulation experiments prove that this control method can achieve control of both the end effector position and the contact force.
4. Conclusion
This paper first establishes a dynamic mathematical model of the robotic arm, and based on this, a dynamic model of the Simulink/SimMechanics robotic arm is built. The kinematics and dynamics of the robotic arm model are analyzed for verification. Second, an impedance control algorithm for the robotic arm is proposed, and the stability of the algorithm is discussed. Then, an impedance controller is established to solve the force/position control problem of the robotic arm. Finally, the control algorithm is combined with the constructed robotic arm simulation model to conduct control simulation of the robotic arm system. Simulation results show that the joint angle changes of the robotic arm can track the desired sinusoidal curve changes, and the end contact force can eventually reach the desired constant value.
The impedance control algorithm is characterized by indirectly controlling the contact force between the robotic arm's end effector and the environment through robotic arm position control. An impedance model is established using the contact force at the robotic arm's end effector and the change in its position, with the position change indirectly reflecting the change in the force at the end effector. This control method is simple, convenient, and effective, and can be used to improve existing robotic arm position control algorithms.
Acknowledgments
The author acknowledges partial funding from the sub-project "Intelligent Handling and Processing Robots" of the Liaoning Provincial Major Science and Technology Innovation Project (Project No.: 201302001).
References
[1] N. Hogan, Impedance Control an Approach to Manipulation: Part I-Theory, Part II-Implementation, Part III-Application [J]. Journal of Dynamic Systems Measurement and Control, 1985:1-24.
[2] Jung Seul, Hsia TC Neural network impedance force control of robot manipulator[J]. Industrial Electronics, IEEE Transactions on. 1998, 45(3): 451-461.
[3] SP Chan, B. Yao, WB Gao, and M. Cheng, Robust impedance control of robot manipulators [J], International Journal of Robotics and Automation, Vol.6, No.4, pp.220-227, December 1991
[4] Zhou Fang, Zhu Qidan, Jiang Mai, Wang Tong, (2009), Adaptive wavelet sliding mode position/force hybrid control for constrained robotic arms [J]. Journal of Huazhong University of Science and Technology (Natural Science Edition), 37(11):9-12
[5] Lei Wang, Yongping Hao, Fei Wang, Hongyi Liu, Experimental study of force control based on intelligent prediction algorithm in open architecture robot system [C], 2007 IEEE International Conference on Robotics and Biomimetics (ROBIO), 15-18 Dec. 2007, 1675 - 1681
[6] Lei Wang, Yongping Hao, Fei Wang, Hongyi Liu, Experimental study of force control based on intelligent prediction algorithm in open architecture robot system [C], 2007 IEEE International Conference on Robotics and Biomimetics (ROBIO), 15-18 Dec. 2007, 1675 - 1681
