Abstract : Objective: This paper employs impedance control, applying a fuzzy PD self-tuning controller to impedance control, and using a fuzzy regulator to effectively adjust the impedance model coefficients, achieving force/position control of an industrial robot under uncertain environments. Methods: A position-based impedance control method is used. In the inner loop of position control, a fuzzy self-tuning PD controller is employed to ensure that the PD parameters are in optimal state at each stage of the system's dynamic process. In the outer loop of impedance control, a fuzzy regulator is used to adjust the impedance model coefficients. Results: The outer loop of impedance control using the fuzzy regulator provides good trajectory correction feedback to the system, and the inner loop of position control using the fuzzy self-tuning PD controller provides accurate control torque for the industrial robot, thus enabling the robot's force/position impedance control system to exhibit good force/position tracking performance. Conclusion: Using a dual-joint SCARA robot as a model, the effects of simple PD control and fuzzy PD control are compared through Matlab computer simulation. It can be seen that the impedance control system using the fuzzy controller has good robustness and force/position tracking performance.
1 Introduction
Since their inception, industrial robots have been replacing humans in high-intensity or hazardous tasks. With the increasing application of industrial robots and continuous technological advancements, the tasks they can perform can be divided into two categories: non-contact operations, such as handling and manipulating objects in free space, which can be accomplished simply by position control; and contact operations, such as polishing and grinding. For these tasks, simple position control is insufficient because the magnitude of the contact force is critical, and even a slight deviation in the robot's end effector can result in a large contact force, damaging both the robot and the target object. Therefore, contact force control functions must be added to improve the robot's effective operational accuracy.
Hongan proposed an impedance control method for robots in his literature. Robot impedance control indirectly controls the interaction forces between the robot and its environment. Its design philosophy is to establish a dynamic relationship between the robot's end effector force and its position. By controlling the robot's displacement, the end effector force is controlled, ensuring that the robot maintains the desired contact force in the constrained direction. Since the concept of impedance control was proposed, many different specific application methods have emerged. This literature review summarizes two basic methods of impedance control: force-based impedance control and position-based impedance control.
Since industrial robots are equipped with high-performance position controllers, position-based impedance control strategies have been widely adopted. This paper selects the widely used position-based impedance control as the control strategy. In the inner loop of position control, a fuzzy self-tuning PD controller is used to ensure that the PD parameters are in the optimal state at each stage of the system's dynamic process. In the outer loop of impedance control, a fuzzy regulator is used to adjust the impedance model coefficients, providing the system with good trajectory correction, thereby enabling the robot's force/position impedance control system to exhibit good force/position tracking performance.
2 Robot Dynamics Model
Dynamic model of the robot in joint space:
(1)
In the formula: τ is the joint driving force or torque vector; q is the angle vector of each joint of the robot; M(q) is the robot's inertia matrix; is the vector of centrifugal force and Coriolis force; is the robot's gravity vector.
The robot dynamics model shown in (1) has the following characteristics:
Property 1: The inertia matrix M(q) is symmetric positive definite and uniformly bounded for all q, i.e.
, where d is a positive constant. (2)
Characteristic 2: The Coriolis force matrix satisfies:
, is a positive constant. (3)
Characteristic 3: Oblique symmetry: For a suitably chosen Coriolis force matrix, we have:
(4)
The study of robot dynamics models will contribute to the analysis of robot systems and the design of controllers.
3. Location-based impedance control
Figure 1 is a structural diagram of a position-based impedance control system for an industrial robot .
As shown in Figure 1, the combination of position inner-loop control and impedance outer-loop control constitutes position-based impedance control. The function of the impedance outer loop is to obtain the position correction amount, which is calculated based on the interaction force between the robot and the operating environment and the set target impedance parameters. The position inner loop combines the position correction amount obtained from the outer loop, the reference position, and the actual position to enable the manipulator to accurately track the desired position, ultimately achieving the target dynamic characteristics of the robot. The accuracy of position control determines the quality of the overall system control. In position-based impedance control, the force/torque sensor collects and measures the contact force, then sends the detected force to the impedance model. The impedance model then generates a positive position correction vector, which satisfies the following equation:
(5)
Therefore, the impedance function can be expressed in the frequency domain as:
(6)
The ideal impedance model parameters M, B, and K are taken as diagonal matrices. Therefore, equation (6) can be regarded as a second-order low-pass filter for each element in the contact force. Adding E to the robot reference displacement, the robot displacement control command is obtained. When the robot end is not in contact with the environment, the external force is zero, and the corresponding correction is zero. From the equation, we get: When the robot end is in contact with the environment, assuming that there is no error in the position control, i.e., we have: In summary, by adopting the control structure shown in Figure 1, we can establish the ideal impedance relationship between the force and the displacement deviation represented by equation (6).
4 Design of an Impedance Control System Based on a Fuzzy PD Controller
For robot force/position control problems in uncertain environments, conventional impedance control is no longer adequate. This paper proposes a fuzzy impedance control method based on fuzzy PD control, the system structure of which is shown in Figure 2. In the outer impedance loop, the impedance model parameters are dynamically adjusted by a fuzzy regulator. The inputs to the fuzzy regulator are the position error and the error change, while the outputs are the impedance model adjustment coefficients. The coefficients of the inner loop PD controller, which controls the position of the robot joints, are adjusted by the joint position, velocity error, and fuzzy inference system. and represent the desired end effector position and the desired end effector force, respectively. and represent the forward and inverse kinematics solutions.
4.1 Design of PD Control Law
The designed independent PD control law is:
(7)
Where: is the position tracking error provided by the outer impedance loop to the inner position loop;
Kp and Kd are the proportional coefficient and differential coefficient, respectively;
τ is the given target torque.
Selecting Lyapunov functions:
(8)
Since the positive definiteness of is known, it is globally positive definite.
(9)
Using the oblique symmetry known from property 3,
,
so
(10)
From the above equation, we know that V is semi-negative definite and Kd is positive definite. When V=0, e=0, and thus e=0. According to LaSalle's theorem [10], it is the equilibrium point of global asymptotic stability of the controlled robot, that is, starting from any initial condition, we have .
4.2 Design of Fuzzy PD Controller
Fuzzy PD controllers optimize PD parameters at each stage to achieve satisfactory control results.
Based on the experience gained in designing existing control systems, the self-tuning rules for PD parameters can be derived as follows:
1. When the value is large, the value of should be relatively small to speed up the system's response.
2. When the system is at a moderate level, the value of should be relatively small in order to reduce overshoot caused by system response, and the value should be appropriately selected.
3. When the value is small, the value of should be relatively large to ensure that the system response achieves better stability. An appropriate value should be selected to prevent jitter from occurring near the equilibrium point after it has been reached.
Based on the characteristics of the robot's various states and the experience summarized above, the fuzzy rules that can be formulated are shown in Tables 1 and 2.
4.3 Design of Fuzzy Impedance Regulator
Once the target impedance coefficient is given, the robot cannot accurately complete the given task when the position and stiffness of the environment change, or when the controlled object is a time-varying system and is subject to external interference [8]. Through a large number of experiments, it has been found that fixing the target impedance coefficient does not achieve a good control effect and will produce a lot of overshoot. Therefore, it is considered that if the impedance coefficient changes with the environment, it can be adjusted to achieve an adaptive effect. By adopting steps such as fuzzy reasoning and fuzzy judgment, the control of the controlled object can be achieved. This algorithm has strong robustness and real-time performance.
In this design, the position error and the change in error are used as the inputs to the fuzzy controller. Since the target impedance coefficient is the main factor affecting the quantity, and the magnitude of affects the overshoot of the system and can play a role in suppression, the changes of and are used as the system outputs, forming a dual-input fuzzy inference system.
The fuzzy regulator has two inputs and two outputs. The linguistic values of both input and output variables are divided into seven fuzzy subsets. The input error universe, output variable universe, and the regularized output variable universe are given. The regulator's adjustment rules are shown in Tables 3 and 4. When the position error is large, the coefficients B and K are increased.
5 Simulation
The robot model used in this simulation is a two-jointed SCARA (Selective Compliance Assembly Robot Arm) robot system, and the coefficients of its dynamic model are:
,
in
The Jacobian matrix is:
in
The robotic arm's end effector is set to perform a circular motion with a radius of 1 (the time-displacement waveforms on the X and Y axes are sine and cosine waves, respectively), and the desired end effector force is a step waveform that jumps to 4N in 1.5s. The four membership functions of the two fuzzy inference systems are set to be trapezoidal on both sides and triangular in the middle.
Based on the impedance control block diagram shown in Figure 2, a simulation model was established using Matlab/Simulink. The simulation structure diagram is shown in Figure 3.
Figure 3 Simulation structure of the Matlab/Simulink fuzzy impedance control system
To compare the control effects, this paper compares the force/position tracking effects of simple PD impedance control and fuzzy PD impedance control. The time-displacement trajectories on the X and Y axes and the force tracking trajectory are shown in Figure 4-6.
Figure 4. Motion trajectory in the X-axis direction
Fig . 4 The trajectories in X-axis
Figure 5. Motion trajectory in the Y-axis direction
Fig . 4 The trajectories in Yaxis
In the simulation results shown in Figure 4-6, the three curves represent the desired end-effector force/position trajectory, the end-effector force/position trajectory using fuzzy impedance control, and the end-effector force/position trajectory using only a PD controller with impedance control, respectively. By comparing the tracking performance of the two actual end-effector trajectories with the desired trajectory, it can be seen that the tracking trajectory using only PD impedance control fluctuates significantly within the desired trajectory, while the tracking curve using fuzzy impedance control is smoother and the deviation gradually decreases. Regarding convergence time, the end-effector force/position trajectory using only PD impedance control converges to the desired trajectory and stabilizes within 6–7 seconds, while the fuzzy impedance control trajectory only takes 3–4 seconds.
Figure 6 Force trajectory at the end of the robotic arm
Figures 7 and 8 show the XY diagrams of the end effector position trajectory of the robotic arm under simple PD impedance control and fuzzy PD impedance control, respectively.
Figure 7. Robot arm end effector trajectory under PD impedance control
Figure 8. Robot arm end-effector trajectory under fuzzy PD impedance control
The comparative experiments above show that the robot control system using a fuzzy self-tuning PD controller and a fuzzy impedance coefficient adjuster has better force/position tracking performance. This is manifested in the fact that the force/position trajectory at the end of the robotic arm converges to the desired trajectory quickly, with small overshoot, short settling time, low jitter frequency, and smoother trajectory. As a result, the force/position control error at the end of the robotic arm is lower, and the control accuracy is higher.
6 Conclusions
This paper addresses the force/position impedance control problem of industrial robots by applying a fuzzy self-tuning PD controller and a fuzzy impedance coefficient regulator to impedance control, achieving force/position tracking control of the end effector of the industrial robot. The design and simulation results of the control system demonstrate that the fuzzy self-tuning PD controller and fuzzy impedance coefficient regulator designed in the force/position impedance control system are not only simple to design, have high tracking accuracy, fast response, and good tracking performance, but also allow the control system to adjust the PD controller coefficients and impedance model coefficients through fuzzy logic under uncertain external environments, further enhancing the adaptability and robustness of the control system.
