Abstract: Objective This paper takes a two-joint SCARA robotic arm as the object and studies the force control problem under the condition that its joint angles are measurable but its angular velocity is not. Method Based on observer theory, this paper uses a robust sliding mode observer, which is an improvement of the Walcott-Zak observer, to observe the state variables of the system and uses the sliding mode control method to realize the force control of the industrial robot. Results The observation effect of the observer and the control effect of the controller are simulated in Matlab. The results show that the estimated value converges to the actual value very quickly, and the end force of the robotic arm tracks the expected force very well. Conclusion The improved robust sliding mode observer has a good effect on the estimation of joint angular velocity of the force control system of the robotic arm, with good robustness, and provides a relatively accurate angular velocity value for sliding mode force control. Under the action of the sliding mode force controller, good force/position tracking effect is obtained.
Keywords: industrial robot; sliding mode observer; force control; sliding mode control
1. Introduction
An industrial robot joint is a complex, multi-input, multi-output nonlinear system, susceptible to load variations and various unpredictable external disturbances and uncertainties [1,2]. Therefore, controlling industrial robots presents numerous challenges. Particularly in force control, we face issues such as control model accuracy, high control noise, and significant interference.
Sliding mode variable structure control is a variable structure control proposed by the Soviet scholar Emeyano in the 1950s. With its unique advantages, it provides a promising system control synthesis method for uncertain systems [3]. In the force/position control of industrial robots, the hybrid control method can achieve good force-position control decoupling, but its main drawback is that the accuracy requirement of the system model is high. In order to solve this problem, the literature [4,5] uses sliding mode control to directly compensate the system parameters and eliminate the uncertainty of the parameters. The literature [6,7] uses an adaptive control method based on the reference model and designs an adaptive adjustment rate to solve the blindness of parameter compensation. It can be seen from the above literature that the sliding mode control theory is effective in the control of industrial robots.
Our common goal is to make control systems more accurate and faster using simple and effective methods.
2. Problem Statement and Solution
In force control of robotic arms, the actual force needs to be detected as the quantity required for control. However, in reality, while the joint angles and positions of the robotic arm can be measured, the angular acceleration of the joints cannot be measured using sensors. How to overcome these uncertainties and achieve accurate force control using effective methods is the key research problem of this paper.
Walcott and Zak et al. proposed the Walcott-Zak observer using variable structure technology, which is robust to nonlinearity/uncertainty in the system [8]. However, in actual design, the design parameter matrix must meet the assumed conditions, and the design process is cumbersome. When the system dimension is high, it is difficult to design. The improved robust sliding mode observer can achieve good results, that is, it has strong robustness and the design process becomes simple.
The proposed solution is to use an improved Walcott-Zak observer to observe joint angular acceleration and to use a sliding mode force-position hybrid controller to achieve force control of the robotic arm.
3. Model Establishment
Taking the SCARA robot as the research object, as shown in Figure 1, its dynamic model is as follows:
(1)
Where represents the position vector of each joint, represents the angular velocity and angular acceleration of each joint, respectively, represents the torque vector acting on the joint, is the robot's symmetric and positive definite bounded inertia matrix, is the robot's centripetal force and Coriolis force matrix, and represents the Coulomb friction force. Furthermore, matrices M and H can be accurately calculated using the following formula:
(2)
It is a constant.
Figure 1 SCARA robot model
The dynamic model characteristics are as follows:
The robot dynamics equation shown in equation (1) has some characteristics that are beneficial for the study of the system model and the design and analysis of its observer. The characteristics are as follows [9]:
Property 1: The inertia matrix is symmetric positive definite and uniformly bounded for all x, i.e.
, is a positive integer. (4)
Property 2: The Coriolis force matrix satisfies:
, is a positive integer. (5)
Property 3: Oblique symmetry: For a suitably chosen Coriolis force matrix, we have:
(6)
4. Design of dynamic state equations and state observers
4.1 Dynamic Equations of State
The robot dynamics model derived from equation (1) can be transformed to obtain [10]:
(7)
in,
(8)
and
Let variables be , where: , , then we have
(9)
in,
The robot's dynamic model can then be written in the following form:
(10)
in,
4.2 Design of the State Observer
Equation (10) can be transformed into the following form
(11)
in,.
And it satisfies, where is a positive real number and is a known function.
For the system above, a robust sliding mode observer of the following form is proposed.
(12)
From equations (11) and (12) above, we obtain the deviation equation as follows:
(13)
In the formula, the sliding mode design is as follows:
(14)
In the formula, C is the output matrix of the system, and there exists a Lyapunov pair.
satisfy
, (15)
Assuming the deviation between the system and the observer is bounded and is a positive constant, the following sliding mode strategy is proposed:
(16)
And when p satisfies the condition, we design the parameter matrix F such that (see the proof below) is the Hurwitz matrix, which makes the system asymptotically stable, that is, its deviation system will asymptotically converge to the equilibrium point after reaching the sliding surface.
The following algorithm is used to select matrix F:
(1) Select the spectrum and calculate the corresponding matrix L;
(2) Represent matrix M using the symbols of each element of matrix F.
(3) Design F such that its matrix is a Hurwitz matrix.
Convergence proof: Select the Lyapunov function along the bias system (13), and
(17)
The design parameters and deviations are substituted into strategy (16) to obtain
(18)
Therefore, according to reference [11], the deviation system reaches the sliding mode plane within a linear time.
After reaching the sliding plane
(19)
The formula represents the segmentation of the sliding surface.
The formula (13) is divided into
(20)
(twenty one)
Define matrix
(twenty two)
We can obtain the result from equation (19).
(twenty three)
Substituting into (20), we get
(twenty four)
Therefore, during the design, F is ensured to be a Hurwitz matrix, which makes the system (24) asymptotically stable and ensures that the deviation system (13) tends to a steady state after reaching the sliding surface.
5. Slippage Force Controller Design
In terms of force controller design, a force-position hybrid control scheme is adopted, and its position controller and force controller are designed one by one.
Figure 2. Force-position hybrid control structure diagram
5.1 Design of the Position Control Controller
The position controller used in this paper is a simple sliding mode controller. The position control part realizes a trajectory tracking control. Through the controller, the actual position of the robot arm is tracked to the desired position in the position control direction.
Define the error matrix for position tracking in the control system:
(25)
Where represents the tracking error of the two joint angles and angular velocity, respectively, and and represent the estimated values of the two joint angles and angular velocities, respectively, and and represent the expected joint angles and angular velocities of the system.
The sliding mode function vector of the sliding mode controller is designed as follows:
(26)
The Lyapunov functions selected are as follows:
(27)
And from equation (25), we get:
(28)
According to the state equation, we can obtain:
(27)
, (28)
Therefore, the matrix represents the observation error of joints 1 and 2.
According to the approach law method, we have
(29)
And among them
(30)
In the formula;
,
;
For design parameters, and
The control law can be designed using the observer's observations and observation errors, provided that the control gain matrix is invertible.
(31)
The robust sliding mode observer, improved from the Walcott-Zak observer, obtains the angular velocity observations of the two joints. Then, using the sliding mode controller, the robotic arm achieves excellent position tracking in the position control direction.
5.2 Design of the Controller for the Force Control Section
To design a robot force controller, the dynamic equations need to be transformed as follows:
(32)
In the formula, is the force vector acting on the robot's end effector in the operating space, and J is the Jacobian matrix;
X represents the displacement of the robot's end effector in the operating space;
.
Take the desired end force as, and use
(33)
The ideal robot end-effector position is defined by the sliding mode function.
(34)
in
but
(35)
Therefore, the sliding force controller is
(36)
The position control torque obtained by the position controller is added to the force control torque obtained by the force controller to obtain the final control torque of the robotic arm.
,Right now:
(37)
6. Experimental parameter settings and simulation results
Based on the dynamic model of the dual-joint SCARA robot and equations (10) and (11), Matlab simulation was performed to verify that the designed robust sliding mode observer can successfully estimate the robot's angular velocity and the tracking effect of the force position controller.
Given the initial state of the system.
The initial state of the observer is...
And given the desired joint angle and joint acceleration.
For the system described above, the proposed design method for a robust sliding mode observer is as follows:
1) To satisfy the condition that it is a Hurwitz matrix, select the appropriate matrix;
2) Select, according to, to ensure that P is symmetric, and make it a Hurwitz matrix.
3) Based on the given information, find the value of , and ensure that it is symmetrical and positive definite.
4) Find and select.
The simulation results are shown in Figures 3-6.
Figure 3 Actual values vs. observed values
Figure 4 Actual values vs. observed values
Figure 5 Actual values vs. observed values
Figure 6 Actual values vs. observed values
Simulation results from the observer show that, in the first few seconds, the observer's estimated value deviates somewhat from the actual value of the system, but it quickly converges to the actual value, ensuring that the error during the observation transition gradually approaches zero. This guarantees excellent observation results.
Based on the observer simulation above, we perform force-position control simulation and parameter setting. The position controller parameters are as follows:
, is a zero matrix. Force controller parameters
.
Figure 7. Actual and expected values of joint 1
8 joints 2 actual values and expected values
Figure 9. Force tracking effect at the end effector of the robotic arm
Simulation results show that the observer-based robotic arm force control method has some initial errors, but achieves excellent force position tracking in a very short time.
7. Conclusion
This paper addresses the problem in robot force control where measured joint angles and angular velocities are needed as control parameters, but accurate measurement of joint angular velocities using sensors is often impossible in actual control systems. Taking a SCARA dual-joint robotic arm as an example, this paper designs a robust sliding mode observer based on an improved Walcott-Zak observer to observe the system's state variables by transforming the robot's dynamic model. Force control of the robot is then achieved through a sliding mode force-position hybrid controller. Simulation results show that the improved robust sliding mode observer has excellent observation performance. Furthermore, the designed force-position hybrid control exhibits good tracking control capability and strong robustness in actual force tracking control.
Acknowledgements! This work was supported by the Liaoning Provincial Science and Technology Innovation Major Project (201302001) and the National Natural Science Foundation of China (61100159).
